Analysis of Quasistatic MHD Perturbations and Stray Magnetic Fields in a Tokamak Plasma P

TOKAMAKS

Analysis of Quasistatic MHD Perturbations and Stray Magnetic Fields in a Tokamak Plasma P. V. Savrukhin Russian Research Centre Kurchatov Institute, pl. Kurchatova, Moscow, 123182 Russia Received October 23, 2000; in ﬁnal form, April 4, 2001

Abstract—Mechanisms for the development of quasistatic MHD perturbations in a viscous rotating tokamak plasma are considered. The inﬂuence of stray magnetic ﬁelds on the stability of MHD modes in the plasma of the TFTR tokamak is analyzed. © 2001 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION involve the three-dimensional modeling of magnetic Recently, the problem of quasistatic MHD perturba- fluxes (including the fluxes induced in the tokamak tions in large tokamaks has attracted considerable inter- conductors) at given positions of the controlling coils. est [1, 2]. An analysis shows that these perturbations Unfortunately, such modeling does not provide the can arise either due to the rotation of magnetic islands required accuracy in determining the stray fields or due to the destabilization of static (locked) MHD because of the complicated spatial structure of the modes. The stopping of rotation is usually accompa- induced fluxes and the mechanical deformation of the nied by a rapid increase in the perturbation amplitude tokamak construction during the experiment. Another and, sometimes, by discharge disruption (see [3]). One method is the direct measurement of helical perturba- of the factors favorable for the development of quasi- tions of the magnetic field with the help of magnetic static MHD perturbations is the breaking of the symme- probes and saddle loops. However, under the experi- try of the tokamak equilibrium magnetic configuration mental conditions, magnetic probes are situated outside (the generation of stray magnetic fields). The threshold the plasma (outside the vacuum chamber of the toka- amplitude of the stray magnetic field at which quasi- mak), which hampers the determination of the local static MHD perturbations develop decreases substan- field structure inside the plasma column. Magnetic tially as the tokamak size increases, the plasma rotation fields inside the vacuum chamber can be measured if slows down, and the plasma pressure increases (see [1, the tokamak is equipped with special movable probes 3]). For these reasons, this kind of instability may be (e.g., in experiments in the DIII-D tokamak [1]). How- dangerous for future tokamak reactors with a slowly ever, such measurements cannot be carried out during rotating plasma under burning conditions [3]. The pri- the discharge. The accuracy of the measurements of mary cause for the appearance of stray magnetic fields, stray magnetic fields can be substantially improved if which is associated with the imperfect fabrication of the tokamak is equipped with additional coils for gen- the tokamak magnetic system and the nonsymmetric erating helical magnetic fields with given amplitudes configuration of conductors, can be minimized by and spatial orientations [2]. In this case, the stray fields improving the accuracy of the device assembly. How- can be deduced by analyzing the thresholds for the ever, there are a number of factors that are fundamen- destabilization of quasistatic MHD perturbations at dif- tally unavoidable. First of all, there is the asymmetry of ferent amplitudes and phases of the external helical the tokamak mechanical elements (such as the divertor, magnetic fields. diagnostic ports, and neutral beam injectors), the inho- Unfortunately, experiments with helical magnetic mogeneity of internal plasma perturbations, and the fields produced by additional coils present serious local character of the interaction of the plasma with the problems to large tokamaks because of the high equip- chamber wall during major disruptions (the excitation ment cost and the rigorous schedule of the device oper- of halo-currents and the injection of impurities [4]). ation. In such a situation, the stray fields can be The diversity of sources exciting inhomogeneous mag- deduced by analyzing the dynamics of internal MHD netic fields makes it difficult to predict the onset of perturbations [5]. In this paper, we describe a procedure instabilities and hampers the development of stabiliza- of determining the dominant harmonics of external tion systems for future experiments. helical perturbations based on the numerical modeling At present, there are several methods for identifying of tearing modes in a viscous rotating plasma. By ana- the structure of stray fields. A direct method is based on lyzing internal perturbations in the TFTR plasma as an the calculations of magnetic fields excited by currents example, we identify the m = 2, n = 1 harmonic (where in the tokamak magnetic system. These calculations m and n are the transverse and longitudinal wavenum-

728 SAVRUKHIN bers, respectively), which plays a key role in the initi- one-fluid MHD theory [6], which describes the electro- ation of the discharge disruption. In order to improve magnetic effects in a plasma with finite viscosity and the accuracy of the analysis, the parameters of the inertia. The geometry of the model is shown schemati- numerical model are determined by comparing the cally in Fig. 1. results of calculations with the results of previous External magnetic field perturbations and MHD experiments in tokamaks equipped with external coils perturbations (modes) are represented in the form of (JET, DIII-D) at prescribed helical perturbations of the helical harmonics B = B exp( jmχ ) and B = magnetic field [5]. em e e rm χ χ ϑ ϕ ω χ Brexp( jm m). Here, m m = m – n + ∫ mdt , m e = ϑ ϕ ω χ χ 2. NUMERICAL MODEL OF MHD m – n + ∫ edt (where m and e are the phases and PERTURBATIONS CONTROLLED ω ω m and e are the angular frequencies of the modes and BY EXTERNAL MAGNETIC FIELDS external fields, respectively), and θ and φ are the trans- The evolution of MHD perturbations under the verse (poloidal) and longitudinal (toroidal) coordinates. ω action of external helical magnetic fields is analyzed The amplitude Br and the instantaneous frequency m = χ using a phenomenological model of tearing modes in a d m/dt are described by the equations viscous rotating plasma [5]. The model is based on the B dB /dt= c ∆' B r r 1 frb r (1) ()ω τ 2 ()ω 2 τ 2 ()χ χ 2 1 Ð c2 Br m w /1+ m w + c3 Be cos e Ð m , ω ()ω ω Brd m/dt= c4 p Ð m (2) 2ω τ ()ω 2 τ 2 ()χ χ Ð c5 Br m w/1+ m w + c6 Be sin e Ð m ; ∆ where 'frb is the stability parameter of the tearing τ mode in a plasma with a free boundary, w is the time ω constant of the conducting tokamak chamber, p is the instantaneous rotation frequency of the bulk plasma surrounding the magnetic island, and ci (i = 1–6) are numerical factors calculated using the measured plasma parameters. MHD perturbations were modeled and stray magnetic fields were then identified using the PLASCON 3 program [5] in the MATLAB programming environ- ment [7]. The block diagram of the program is shown ω (‡) in Fig. 2. The TEARING MODE block models the growth and rotation of MHD perturbations by numerically solving Eqs. (1) and (2) for given values of B and ω e p0 χ ∆ e. The stability parameter of the tearing mode 'frb is ω p specified using the current density profile calculated with the TRANSP code [4]. The transmission charac- (b) teristics of the set of magnetic probes (see below) are ω modeled by the MAGNETIC block. The calculated val- m ω Ω ues of Br, m (denoted by BR and m), and the signals r from magnetic detectors (BR-LMD and BP-MM) are rs rp rw compared with the measured values. The amplitude and phase of external fields are determined by fitting the Fig. 1. Schematic illustration of the tearing mode model and calculated values to the experimental results. the profiles of the angular plasma rotation velocity (a) in a ω quasi-stable configuration and (b) upon the onset of MHD The instantaneous angular frequency m and the perturbations. Magnetic islands ( ) are located in the vis- 1 perturbation amplitude Br depend on the mode phase cous plasma (2) confined in a chamber with conducting ω ω with respect to the external magnetic field [see Eqs. (1), walls (3). Here, m and p are the instantaneous angular χ χ frequencies of the mode and the plasma, respectively; r , r , (2)]. Depending on the phase shift ( e – m), the mode s p periodically accelerates and decelerates during the rota- and rw are the minor radii of the resonant magnetic surface, the bulk plasma, and the conducting wall, respectively; and tion period. At large amplitudes of the external field, ω p0 is the plasma rotation frequency in the absence of tear- such nonuniform rotation produces characteristic saw- ing modes. tooth signals from magnetic detectors, which allows us

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 ANALYSIS OF QUASISTATIC MHD PERTURBATIONS 729 to identify the phase of the stray field under given Experimental data experimental conditions. Magnetic detectors 3. EXPERIMENTAL RESULTS Plasma parameters BLMD, Bp AND COMPARISON WITH CALCULATIONS ω R0, rp, rw, p, The diagnostic complex of the TFTR tokamak [8] ∆' , B , T , frb τt e makes it possible to identify external and internal MHD ne, w perturbations and, concurrently, to measure the plasma MHD modes m/n, , , ω parameters used in numerical calculations (see table rs wi Br m and Fig. 3). The spatial structure of internal MHD modes (wavenumbers m and n), the angular frequency, Model and the magnetic island width are measured with the χ help of two microwave polychromators situated in the TEARING MODE Be, e cross sections separated by an angle of 126° in the toroidal direction. The amplitudes and phases of the qua- Stray sistatic and rotating perturbations of the magnetic field magnetic outside the plasma column were measured with the MAGNETIC fields help of saddle loops and sets of magnetic probes. The plasma rotation velocity was determined from spectro- Ω 6+ BR, m scopic measurements (charge exchange of C ions). BR-LMD Internal MHD perturbations and stray magnetic BP-MM Comparison fields were analyzed under conditions typical of the with experiment experiments in the TFTR tokamak described in [9]: the plasma current is Ip = 2.0 MA, the magnetic field is Bt = 5.0 T, the minor and major plasma radii are rp = 0.87 m Fig. 2. Block diagram of the PLASCON model used to identify quasistatic MHD modes and stray magnetic fields. and R0 = 2.51 m, and the neutral beam injection (NBI) The amplitude and phase of the external stray fields (Be and power is PNB ~ 24 MW. Figure 4 shows the time evolu- χ e) are determined by fitting the calculated amplitude and tion of the plasma parameters in those experiments. The frequency of MHD perturbations (BR and Ωm) and the sig- m = 2, n = 1 perturbations appear after an internal dis- nals from magnetic detectors (BR-LMD and BP-MM) to ≈ ω ruption at t 3.35 s. The subsequent increase in the the measured values Br, m, BLMD, and Bp. mode amplitude is accompanied by a decrease in the rotation frequency up to the full stopping (locking) of MHD perturbations at t ≈ 4.1–4.2 s. Spectroscopic mea- t ≈ 4.12 s). After several short pulses in the time interval surements of the velocity of C6+ ions show that the t ≈ 4.18–4.24 s, the given injector is switched on again mode grows simultaneously with the slowing-down of and the rotational moment TNBI reaches its initial value the bulk plasma rotation over the entire cross section of at t ≈ 4.27 s (Fig. 4). the plasma column. A rapid slowing-down of the mode immediately before the rotation stops is explained by An analysis of electron-cyclotron (EC) emission the fact that one of the neutral injectors is switched off with the help of polychromators shows that the growth at t ≈ 4.12 s. Switching-off is only accompanied by a of the m = 2, n = 1 perturbations is accompanied by the weak (about 10%) decrease in the total NBI power, appearance of flattened regions in the radial profile of which, however, results in the complete disappearance the electron temperature, which is associated with the of the rotational moment TNBI, determining the angular formation of magnetic islands. An analysis of the EC plasma motion under these conditions (see Fig. 5, emission profile allows us to determine the width (wi)

Diagnostic systems for the identification of internal and external MHD perturbations Plasma parameters Diagnostics Notation in Fig. 3 ω Helical structure of internal modes, the width m, n, wi, m Microwave polychromators GPC-1, GPC-2 and angular rotation velocity of magnetic islands

Amplitude of quasistatic MHD perturbations Br Detectors of locked modes LMD-D, LMD-F (saddle loops) LMD-N, LMD-P ω Helical structure, the amplitude and rotation m, n, Bp, ext Magnetic probes Mirnov coils frequency of external MHD perturbations ω Plasma rotation p(r) Spectroscopy of the charge- exchange C6+ ion lines

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N Figure 6 shows the results from the modeling of W E LMD-D tearing modes in the TFTR plasma. At the given exper- GPC-1 imental plasma parameters, the modeling allows us to describe the increase in MHD perturbations and the LMD-P LMD-F slowing-down and full stopping of the m = 2, n = 1 mode rotation. The modeling also allows us to repro- Mirnov GPC-2 duce sawtooth signals from magnetic probes, which probes reflect the nonuniform rotation of magnetic perturbations during the rotation period, depending on the phase LMD-N of the stray fields. By comparing the calculated and experimental signals under these conditions, we can determine the spatial orientation (phase) of the m = 2 Fig. 3. Arrangement of the TFTR diagnostic systems for studying quasistatic MHD perturbations and stray magnetic harmonic of the external magnetic field. An analysis fields (see [9]). The amplitude of external MHD perturba- shows that the experimentally observed signals corre- tions is measured with locked-mode detectors (LMD-D, spond to the permanent orientation of the external field, LMD-P, LMD-N, and LMD-P saddle loops) and magnetic which does not change when the mode amplitude pickup coils (Mirnov probes). The amplitude and frequency grows nor when MHD perturbations slow down. This of internal plasma perturbations is analyzed with two mul- tichannel polychromators (GPC-1 and GPC-2). The spiral indicates that the stray fields in the experiments under line shows the location of the m = 2, n = 1 mode, and the study are governed, first of all, by quasistatic inhomo- arrows W, N, and E show the spatial orientation of the toka- geneities of the tokamak magnetic system, rather than mak magnetic system. by possible inhomogeneous interaction between the plasma and the limiter (see [9]). At a specified (using experimental data) time evolu- and spatial location of magnetic islands. By analyzing tion of the amplitude and frequency of MHD perturba- the width of a magnetic island together with the signals tions, the calculated time during which the rotation of from external magnetic probes, we can reconstruct the the mode (t = tlock) stops is determined by the amplitude time evolution of the MHD perturbation amplitude (Br) of external stray fields. An analysis shows that the best on the resonant magnetic surface r = rs. agreement between the calculated and experimental , T 4 –4 LMD

10 2 B 0 4 , p

B 2 arb. units arb.

0 –3 ,

e 5 m n

19 3

10 5 , C m/s 0 4 V

10 –5 ,

NB 20 MW P 0 3 , e T keV 2 1 3 4 5 Time, s Fig. 4. Time evolution of the plasma parameters in the TFTR experiments [9] with the injection of neutral beams (t = 2.65Ð4.75 s) and a lithium pellet (t = 3.15 s). MHD perturbations with m = 2 and n = 1 arise after internal disruption (t = 3.35 s). The subsequent growth of the perturbation amplitude is accompanied by the slowing-down and full stopping of the mode at t = 4.1Ð4.2 s. Here, BLMD and Bp are the amplitudes of the radial and poloidal perturbations of the magnetic field, ne is the electron density, VC is the 6+ angular rotation velocity of C ions, PNB is the total power of neutral beams, and Te is the electron temperature near the q = 2 (R = 3.1 m) surface.

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 ANALYSIS OF QUASISTATIC MHD PERTURBATIONS 731

3 í í –4 2 –4 20 , 10 1 BR, 10 LMD B 0 0 3.5 4.0 4.5 1 0.1

kHz ∆ arb. units arb. 0 , TNBI , p m B –0.1 Ω 0 3.5 4.0 4.5 t, s 3 R = 3.1 m arb. units arb.

, 2 0.4 kHz ECE 0 , T m 0 Ω

–2 0.5 arb. units arb.

, –4 0 NBI arb. units arb. BR-LMD, T 4.1 4.2 –0.5 Time, s 1 Fig. 5. Time evolution of MHD perturbations in the TFTR plasma: BLMD, and Bp are the signals from a locked-mode BP-MM, detector and magnetic probes, respectively; T is the units arb. –1 ECE 4.20 4.21 4.22 4.23 t, s microwave emission intensity from the plasma near the q = 2 (R = 3.1 m) surface; and TNBI is the moment of momentum of the injected neutrals. Fig. 6. Results of the numerical modeling of tearing modes in the TFTR plasma. As in the experiment (Fig. 5), after the destabilization (at t ~ 3.5 s) and subsequent growth of magnetic perturbations, the rotation of tearing modes slows values of tlock is achieved at the q = 2 resonance surface down and they stop (t ~ 4.22 s). Here, BR and Ωm are the for the amplitude of the m = 2, n = 1 harmonic of exter- calculated amplitude and rotation frequency of the tearing ≈ × –4 modes, BR-LMD and BP-MM are the calculated signals nal fields equal to Be 7.5 10 T. ∆ from magnetic detectors, and TNBI is the change in the moment of momentum of neutrals injected into the plasma. Circles and squares show the amplitude of radial magnetic 4. DISCUSSION OF EXPERIMENTAL RESULTS field perturbations at the q = 2 surface and the rotation frequency of the m = 2, n = 1 mode (both measured with the An analysis of internal plasma perturbations help of microwave polychromators), respectively. observed in the TFTR experiments and the numerical modeling of tearing modes allow us to clarify the mechanisms for the growth of quasistatic MHD perturba- However, an analysis of signals from the magnetic tions in a viscous rotating plasma in the presence of probes and polychromators showed that, in the TFTR external stray magnetic fields. The developed proce- experiments under consideration, no substantial pertur- dure of comparing the calculated time evolution of tear- bations with the wavenumber m = 3 develop. This ing modes and the signals from the magnetic detectors allows us to ignore the m = 3 harmonic when analyzing with the values measured in the experiment allows us to stray fields in the TFTR. Experiments in the DIII-D, determine the spatial orientation and amplitude of the dominant m = 2 harmonic of external stray fields under however, demonstrate the significant role of the m = 3 conditions typical for experiments in large tokamaks. harmonic in certain discharge modes [1]. This difference can be attributed to the elongation of the cross sec- The numerical model used in this study was primarily developed for the modeling of the simultaneous tion of the plasma column or to the enhanced toroidal development of several MHD harmonics (see [5, 10]). effects when the DIII-D operates in modes with a low Calculations show, in particular, that the coupled m = 2 aspect ratio. To determine the stray fields under these and m = 3 modes stop rotating at a lower amplitude of conditions, it is necessary to model the simultaneous stray fields in comparison with the single m = 2 mode. time evolution of the m = 2 and m = 3 modes.

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The procedure described in this paper allows us to REFERENCES determine the stray magnetic fields under actual tokamak experimental conditions. Unfortunately, the 1. R. S. LaHaye, R. Fitzpatrick, T. C. Hender, et al., Phys. Fluids B 4, 2098 (1992). results of such an analysis depend on the accuracy of the measurements of the plasma discharge parameters 2. A. Santagiustina, S. AliArshad, D. J. Campbell, et al., in (the electron temperature and density, impurity compo- Proceedings of the 22nd European Conference on Con- sition, plasma rotation velocity, and current density). trolled Fusion and Plasma Physics, Bournemouth, 1995; For a higher reliability of the determination of stray ECA, 19C (4), 461 (1995). fields and for the possible compensation of MHD per- 3. ITER Expert Group on Disruption, Plasma Control, and turbations in future experiments, the tokamak should be MHD, ITER Physics Basis Editors, Nucl. Fusion 39, equipped with additional external coils. 2251 (1999). 4. H. Takahashi, E. Fredrickson, K. McGuire, and W. Mor- ris, Rev. Sci. Instrum. 66, 816 (1995). 5. CONCLUSION 5. P. V. Savrukhin, Fiz. Plazmy 26, 675 (2000) [Plasma An analysis of internal and external MHD perturba- Phys. Rep. 26, 633 (2000)]. tions in the TFTR plasma and numerical modeling make it possible to identify the parameters of the dom- 6. R. Fitzpatrick, Nucl. Fusion 33, 1049 (1993). inant m = 2, n = 1 mode and determine the spatial struc- 7. MATLAB User’s Guide / The MathWorks Inc. (Englew- ture of the stray magnetic fields. In order to more reli- wod Cliffs, Prentice Hall, 1992). ably identify the stray fields and to stabilize MHD per- 8. G. Taylor, E. Fredrickson, and A. Janos, Rev. Sci. turbations, the tokamak should be equipped with Instrum. 66, 830 (1995). additional external coils. 9. H. Takahashi, E. Fredrickson, K. McGuire, and A. Ram- sey, Bull. Am. Phys. Soc. 41, 1S27 (1996). ACKNOWLEDGMENTS 10. P. Savrukhin, D. J. Campbell, M. DeBenedetti, et al., I am grateful to I.B. Semenov, K.A. Razumova, and Numerical Simulation of Feedback Control of Coupled V.S. Strelkov for fruitful discussions. I am also grateful Tearing Modes at JET, JET Joint Undertaking Report to the reviewer for valuable remarks and bringing to my JET-R(95)-06 (1995). attention various mechanisms for the disruption insta- 11. F. Karger, H. Wobig, S. Corti, et al., Plasma Phys. Con- bility and the possible stabilizing role of helical mag- trolled Nucl. Fusion Res. 1, 207 (1975). netic fields in tokamaks under certain operating condi- 12. A. P. Popryadukhin, Prikl. Fiz., Nos. 3Ð4, 90 (1995). tions [11, 12]. This study was supported by the Russian Foundation for Basic Research, project nos. 00-02- 16275 and 01-02-16768. Translated by N. F. Larionova

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PLASMA TURBULENCE

Generation of Large-Scale Structures by Strong Drift Turbulence V. P. Lakhin Nuclear Fusion Institute, Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received January 21, 2001

Abstract—The previously existing quasilinear theory of the generation of a large-scale radial electric field by small-scale drift turbulence in a plasma is generalized for the case of strong turbulence which is usually observed in experiments. The geostrophic equation (i.e., the reduced CharneyÐHasegawaÐMima equation) is used to construct a systematic theory in the two-scale direct interaction approximation. It is shown that, as in the quasilinear case, drift turbulence results in a turbulent viscosity effect and leads to the renormalization of the Poisson bracket in the CharneyÐHasegawaÐMima equation. It is found that, for strong drift turbulence, the viscosity coefficient is represented as a sum of two parts, which are comparable in magnitude. As in quasilinear theory, the first part is determined by the second-order correlation functions of the turbulent field and is always negative. The second part is proportional to the third-order correlation functions, and the sign of its contribution to the turbulent viscosity coefficient depends strongly on the turbulence spectrum. The turbulent viscosity coefficient is calculated numerically for the Kolmogorov spectra, which characterize the inertial interval of the drift turbulence. © 2001 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION the nonlinear interaction of self-consistent turbulent pulsations. In [3], it was found that small-scale drift tur- The so-called CharneyÐHasegawaÐMima (CHM) bulence results in the negative turbulent viscosity in the model [1, 2] is a simple model for describing both drift equations for the large-scale field. This effect is related turbulence in a magnetized plasma and the turbulence to the term describing the adiabatic response of the of Rossby waves in the atmospheres of rotating planets. electrons in the HasegawaÐMima equation. This result Recently, this model has been used to investigate the agrees well with the conclusion drawn by Montgomery effect of small-scale turbulent pulsations on larger- and Hatori [10]: in the two-dimensional incompressible scale perturbations [3, 4]. The study by Gruzinov et al. hydrodynamics based on the Euler equation (i.e., in the [3] was motivated by experiments on LÐH transitions in absence of the above-mentioned term), small-scale tur- tokamaks [5, 6]. The authors supposed that a turbulent bulence does not give rise to turbulent viscosity. As was mechanism may be responsible for the generation of mentioned in [3], the instability driven by turbulent vis- radial electric fields and poloidal sheared flows, which cosity with a negative coefficient causes a spontaneous are both observed in the experiments cited. In order to growth of large-scale perturbations and can serve as a describe this phenomenon, they constructed a simple mechanism for the generation of the radial electric analytic theory based on the reduced CHM equation— fields observed in experiments on LÐH transitions. the geostrophic equation. This equation neglects the term that is proportional to the plasma density gradient A primary motivation for the study by Chechkin and accounts for linear waves in the CHM equation. As et al. [4] is its application to the physics of the atmo- was noted in [3], neglecting this term is justified only sphere and ocean. The analysis carried out in that paper for a strongly nonlinear turbulent regime dominated by was based on the CHM equation supplemented with the the nonlinear interaction of turbulent pulsations (rather terms describing molecular viscosity and the external than by linear waves). Experimental investigations of force. The role of the latter was merely to maintain tur- drift turbulence in tokamaks show that nonlinear fre- bulence at a steady level. The authors assumed that the quency shifts exceed the linear frequencies of the drift amplitude of the small-scale turbulent field is small waves [7], which provides evidence in support of such enough so that the Reynolds number for small-scale a simplification. On the other hand, in [3], the equation perturbations is low and the terms proportional to the describing the evolution of large-scale perturbations squared amplitudes of the small-scale perturbations in due to their interaction with small-scale turbulent pul- the equation for the turbulent field can be neglected. sations was derived in the quasilinear approximation, Thus, as in [3], the analysis of [4] also used the quasi- which is employed to construct standard turbulent linear approximation and revealed the effect of negative dynamo theories [8, 9]. However, in my opinion, the turbulent viscosity on large-scale perturbations: the quasilinear approximation can only be used when the term that reflects this effect originates from the com- turbulence is governed by the driving force rather than pressibility of fluid and is responsible for the same term

734 LAKHIN in the CHM equation as the adiabatic term in the equation function and the averaged Green’s function in the tions for drift waves in a magnetized plasma. In [4], it inertial interval of turbulence in order to derive analytic was also shown that small-scale turbulence leads to the expressions for the turbulent coefficients in the evolu- renormalization of the coefficient in front of the Pois- tionary equation for the large-scale field. Details of the son bracket, which, as is well known, describes nonlin- corresponding calculations are given in Appendix B. In ear effects in the CHM equation. this section, we also describe the results of the numeri- The purpose of this paper is to generalize the analy- cal computation of the turbulent viscosity coefficients sis of [3, 4] for the case of strong turbulence, which is with the help of the Mathematica-3 software package. more adequate for the experimental observations of Finally, in Section 5, we discuss the results obtained. drift turbulence in tokamaks [7]. Since this case cannot be described in the quasilinear approximation, we apply the so-called two-scale direct-interaction approx- 2. EVOLUTIONARY EQUATION imation. The corresponding formalism was developed FOR THE LARGE-SCALE FIELD by Yoshizawa [11] when constructing the theory of hydrodynamic turbulence with sheared flows. As 2.1. Basic Equations implied by its name, the formalism is based on a com- We start with the geostrophic equation in dimen- bination of the standard technique of multiscale expan- sionless variables, sions (in the case at hand, a two-scale expansion) and Kraichnan’s direct-interaction approximation [12], ∂ 2 2 which is used to calculate statistically averaged quanti------()λφ Ð — φ Ð0.[]—φ × —— φ z = (1) ties. In Section 2, we present a detailed formulation of ∂t the problem. We derive an equation for large-scale motions from the geostrophic equation. In accordance Here, —2 ≡ ∂2/∂x2 + ∂2/∂y2; the time is in units of the with the theory of multiscale expansions, we introduce inverse ion gyrofrequency; the electrostatic potential φ fast and slow variables. Then, we represent the electro- is in units of Te/e; and the spatial variables are normal- static potential as the sum of the mean (large-scale) and ized to the ion Larmor radius in terms of the electron random (small-scale) components and solve the equa- ρ ω 2 1/2 temperature, 0 = (Te/mi Bi ) . The Poisson bracket tion for the random (turbulent) component by applying × the method of expansion in the small parameter—the [f g]z denotes the z-component of the vector product. ratio of the spatial scale of the turbulent motion to that In the above dimensionless variables, the coefficient λ of the mean electrostatic field. Using the direct-interac- equals unity, λ = 1. We keep this coefficient merely in tion approximation, we calculate the mean values of the order to illustrate how different terms appear in expres- turbulence-related quantities to arrive at an evolution- sions describing the effect of turbulence on large-scale ary equation for the averaged electrostatic potential plasma motions (recall that our primary goal here is to expressed in terms of a second-order correlation func- calculate these terms). In Eq. (1), the term with the tion and a statistically averaged response function (a coefficient λ describes the adiabatic perturbation of the Green’s function) of the turbulent field. Details of the electron density. Formally, for λ = 0, Eq. (1) is a two- calculations are given in Appendix A. In Section 3, we dimensional Euler equation for the two-dimensional obtain expressions for the second-order correlation motions of an incompressible fluid. By its very nature, function and the averaged Green’s function of the the geostrophic equation is a simplified version of the small-scale drift turbulence. We assume that, in the HasegawaÐMima equation, in which, according to the absence of the mean field, drift turbulence is homoge- arguments given in the Introduction, the term that stems neous and isotropic. We also assume that the drift tur- from the nonuniformity of the equilibrium plasma den- bulence is in a highly nonlinear regime and is described sity and describes the drift waves is neglected. by the second-order correlation function and the averaged Green’s function characterizing the inertial inter- We represent the electrostatic potential as the sum of val of the turbulence; i.e., they are independent of both the averaged (mean) and random (turbulent) compo- the dissipation and the source of turbulence and are nents, completely determined by the nonlinear interaction of turbulent pulsations. To derive the correlation function φφφ==+ ˜ , φ 〈〉φ , (2) and Green’s function, we turn to the method based on the analysis of the invariant properties of the geo- where the angular brackets 〈...〉 stand for ensemble strophic equation with respect to the scale transforma- averaging. We average Eq. (1) in order to obtain the fol- tions. We consider two limiting cases in which the char- lowing evolutionary equation for the mean field: acteristic spatial scale of the turbulent motion is either much larger than or much smaller than the natural spa- ∂ 2 2 -----()λφ Ð — φ Ð []—φ × —∇ φ z tial scale for the geostrophic equation—the ion Larmor ∂t (3) radius in terms of the electron temperature. In Section ˜ 2 ˜ 4, we use the expressions for the second-order correla- Ð 〈〉[]—φ × —— φ z = 0.

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GENERATION OF LARGE-SCALE STRUCTURES BY STRONG DRIFT TURBULENCE 735

If we subtract Eq. (3) from Eq. (1), we find that the tur- 2 2 bulent field is described by the equation 1 k2 Ð k1 ˜ ˜ Ð --- []k × k ------φk ()φt k ()δt ()kkÐ Ð k 2 ∫ ∫ 1 2 z λ 2 1 2 1 2 ∂ 2 2 2 + k ˜ ˜ ˜ ˜ k1 k2 -----()λφ Ð — φ Ð []—φ × —∇ φ z Ð []—φ × —— φ z ∂t (4) 2 2 2 ()∂ []φ˜ × ∇ φ˜ 〈〉[]φ˜ × φ˜ ik ˜ 2i k—s ˜ Ð — — z +0.— —— z = = ------φk()t []k × — φ + ------φk()t 2 s z 2 ∂ λ + k λ + k t (10) 2.2. Introduction of Two Scales ()2 2 ik2 Ð k1 φ˜ ()[]× φ˜ ()δ() Ð ∫ ∫ ------k1 t k1 —s z k2 t kkÐ 1 Ð k2 Assuming that the spatial and temporal scales of the λ + k2 mean and turbulent fields are essentially different, we k1 k2 introduce two spatiotemporal scales, 4 xX,,()= ⑀x , tT()= ⑀ t . (5) 2i []× φ˜ ()()φ˜ () Ð ∫ ∫ ------k1 k2 z k1 t k2—s k2 t λ + k2 In other words, the mean field depends only on the k k slowly varying variables, while the turbulent field 1 2 depends on both the rapidly and slowly varying vari- ×δ()…kkÐ Ð k + , ables because of its interaction with the mean field: 1 2 φφ==()X, T , φ˜ φ˜ ()x,,t; X T . (6) where dots designate the terms of the second order and higher in the expansion in powers of the small parame- Transforming the Poisson bracket in an appropriate ter ⑀. These terms can be neglected, because they are manner, it is expedient to rewrite Eq. (3) as unimportant when calculating the effect of turbulent 2 2 viscosity. Here and below, we do not indicate the ∂ 2 ∂ ∂ ∂φ˜ ∂φ˜ ------()λφ Ð — φ + ------Ð ------∂ s 2 2 ∂ ∂ explicit dependence of the Fourier coefficients of the T ∂X ∂Y xˆ yˆ (7) turbulent field on the slowly varying variables. ∂2 ∂φ˜ 2 ∂φ˜ 2   []φ × 2φ Ð ∂------∂ - -----∂ - Ð -----∂ - Ð0,—s —s—s z = X Y xˆ yˆ 2.3. Expansions of the Turbulent Field ˆ ≡ ∂ ∂ ∂ ∂ ≡ ∂ ∂ 2 ≡ ∂2 ∂ 2 in the Small Parameter where — / x + / X, —s / X, and —s / X + ∂2/∂Y2. In terms of the variables introduced, Eq. (4) for Let us expand the Fourier coefficients of the electro- the turbulent field becomes static potential φ˜ in powers of the small parameter ⑀: ∂ ∂ 2 2 ----- + ------()λφ˜ Ð —ˆ φ˜ Ð []— φ × —ˆ ∇ˆ φ˜ () () () ∂ ∂ s z ˜ ˜ 0 ˜ 1 2 ˜ 2 t T φk = φk ()t ++⑀φk ()t; X, T ⑀ φk ()t; X, T +…. (11) 2 []ˆ φ˜ × 2φ []ˆ φ˜ × ˆ ˆ φ˜ (8) Ð — —s—s z Ð — —— Here, the first term is independent of the slowly varying 2 variables and corresponds to a uniform isotropic turbu- 〈〉[]ˆ φ˜ × ˆ ˆ φ˜ + — —— z = 0. lent field, which is assumed to be generated and main- tained at a steady level by small-scale drift instabilities. For convenience of further analysis, we represent The remaining terms describe the interaction between a the electrostatic potential of the turbulent field as the Fourier integral over the rapidly varying spatial vari- small-scale turbulent field and a large-scale electro- able: static field with the averaged potential φ . φ˜ (),, φ˜ (), ikx x t; X T = ∫dk k t; X T e We substitute Eq. (11) into Eq. (10) and collect terms of different orders in ⑀. As a result, we arrive at (9) the following zero-order equation for the Fourier coef- ≡φ˜ (), ikx ∫ k t; X T e . ficients of the electrostatic potential of the homoge- k neous isotropic turbulent field: Note that the Fourier coefficients of the electrostatic ∂ ˜ ()0 potential of the turbulent field depend on the slowly -----φk varying spatial variable X. According to Eq. (8), the ∂t Fourier coefficients satisfy the equation 2 2 (12) 1 k Ð k ()0 ()0 ∂ Ð --- []k × k ------2 1-φ˜ ()t φ˜ ()t δ()kkÐ Ð k = 0. φ˜ () ∫ ∫ 1 2 z 2 k1 k2 1 2 ∂----- k t 2 λ + k t k1 k2

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To first order, we obtain the homogeneous isotropic field, and the Green’s function: ∂ ()1 ˜ t -----φk 2 ∂ () () t φ˜ 1 ik []× φ ˆ , φ˜ 0 k ()t = ------2 k —s z ∫ dt'Gk()tt' k ().t' (16) 2 2 λ () () + k ∞ []× k2 Ð k1 φ˜ 0 φ˜ 1 δ Ð Ð ∫ ∫ k1 k2 z------k1 ()t k2 ()t ()kkÐ 1 Ð k2 λ + k2 The solution to the second-order equation (14) can be k1 k2 (13) found in an analogous manner: it is expressed in terms 2 () ()0 ˜ 0 ik φ˜ []× φ of φ , φ , the Green’s function, and the first-order = ------k k —s z. λ + k2 solution: φ˜ ()2 () 2i ()φ⋅ ˜ ()1 () In the second-order approximation, i.e., in the lowest k t = ------2 k —s k t order approximation in which the effect of turbulence λ + k on large-scale plasma motions can be described, the 2 t equation for the turbulent field has the form () ik []× φ ˆ ()φ, ˜ 1 () + ------2 k —s z ∫ dt'Gk tt' k t' () () λ + k ∂ 2 2i()k ⋅ — 1 Ð∞ ----- φ˜ Ð ------s φ˜ ∂ k 2 k t λ + k t 2 2 1 []× k2 Ð k1 δ() + --- ∫ ∫ ∫ dt' k1 k2 z------2- kkÐ 1 Ð k2 2 2 2 λ + k k Ð k ()0 ()2 k k Ð∞ []× ------2 1-φ˜ ()φ˜ ()δ() 1 2 Ð ∫ ∫ k1 k2 z 2 k1 t k2 t kkÐ 1 Ð k2 λ + k × ˆ ()φ, ˜ ()1 ()φ˜ ()1 () k1 k2 Gk tt' k1 t' k2 t' (17)

2 t ik ˜ ()1 i = ------φk []k × — φ Ð dt'------δ()kkÐ Ð k Gˆ k()tt, ' λ 2 s z ∫ ∫ ∫ ()λλ 2 ()2 1 2 + k ∞ + k + k2 k1 k2 Ð 2 2 (14) () () () () 1 k2 Ð k1 1 1 ˜ 0 2 ˜ 1 []× φ˜ φ˜ δ ×φk ()t' {2()λ + k []k × k ()φk ⋅ — k ()t' + --- ∫ ∫ k1 k2 z------k1 ()t k2 ()t ()kkÐ 1 Ð k2 1 1 1 2 z 2 s 2 2 λ + k2 k1 k2 ()λ 2 ()2 2 []× φ˜ ()1 ()} + + k2 k2 Ð k1 k1 —s z k2 t' . 2 2 ()()0 ()1 Having derived (with the desired accuracy) the correc- ik2 Ð k1 φ˜ []× φ˜ δ Ð ∫ ∫ ------k1 ()t k1 —s z k2 ()t ()kkÐ 1 Ð k2 tions to the small-scale turbulent field that come from λ + k2 k1 k2 the interaction with the large-scale field, we can calculate the statistical averages in Eq. (7) and thus analyze the effect of the small-scale drift turbulence on the aver- () () 2i []× φ˜ 0 φ˜ 1 δ aged field. Ð ∫ ∫------k1 k2 z k1 ()t ()k2—s k2 ()t ().kkÐ 1 Ð k2 λ + k2 k1 k2 2.4. Calculation of the Statistical Mean Values The function describing the response of turbulence to a We start by rewriting the averaged quantities in Eq. (7) perturbation (or, equivalently, the Green’s function) Gˆ k in terms of the Fourier transformed turbulent field. Tak- is introduced in a standard way as the solution to the ing into account Eqs. (9) and (11) and keeping the terms equation of the corresponding orders of smallness, we obtain

∂ ˜ ˜ ()⋅  ∂φ∂φ i k1 + k2 x ()0 ()0 -----Gˆ k()tt, ' ------ ( 〈〉φ˜ φ˜ - ∂ = ∫ ∫ e Ðk1xk2y k1 ()t k2 ()t t ∂xˆ ∂yˆ  k k 2 2 (15) 1 2 k Ð k ()0 2 1 ˜ ()1 ()1 Ð []k × k ------φk ()t Gˆ k ()tt, ' δ()kkÐ Ð k 〈〉)φ˜ φ˜ () ∫ ∫ 1 2 z 2 1 2 1 2 + k ()t k ()t Ð k1xk2y + k1yk2x λ + k 1 2 (18) k1 k2 ()1 ()0 ()2 ()0 ×φ()〈〉φ˜ ()t φ˜ ()t + 〈〉˜ ()t φ˜ ()t = δ()ttÐ ' . k1 k2 k1 k2

∂ ∂ ()1 ()0  Then, the solution to the ﬁrst-order equation (13) is ------〈〉φ˜ φ˜  () + ik2x + k2y k1 ()t k2 ()t , ˜ 0 ∂Y ∂X  expressed in terms of φ , the Fourier amplitude φk of

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The last statistically averaged quantity that is required ∂φ˜ 2 ∂φ˜ 2 ()⋅    i k1 + k2 x ()to calculate the effect of turbulence on the large-scale ------Ð ------= ∫ ∫ e  k1yk2y Ð k1xk2x --- ∂xˆ ∂yˆ  ﬁeld in the desired order of smallness can be found by k1 k2 substituting solution (16) into solution (17) and taking () () () () into account relationship (21): ×φ( 〈〉˜ 0 φ˜ 0 〈〉φ˜ 1 φ˜ 0 k1 ()t k2 ()t + 2 k1 ()t k2 ()t 2 () () () () (19) 〈〉φφ˜ 2 φ˜ 0 〈〉)˜ 1 φ˜ 1 〈〉φ˜ ()2 φ˜ ()0 2k1 ()⋅ []× φ +2 k1 ()t k2 ()t + k1 ()t k2 ()t k ()t k ()t = Ð------k1 —s k1 —s z 1 2 ()λ 2 2 + k1 ∂ ∂ ()1 ()0  +2ik ------Ð k ------〈〉φ˜ ()t φ˜ ()t . t 2x∂X 2y∂Y k1 k2  × (), ()δ, () ∫ dt'Gk1; tt' Ik1; t' t k1 + k2 We use the second-order correlation function and the Ð∞ averaged Green’s function as the fundamental statistical characteristics of the turbulent ﬁeld in the absence 4 t t' k1 []× φ 2 (), of a large-scale perturbation. Under the assumption that Ð ------k1 —s z ∫ dt' ∫ dt''Gk1; tt' ()λ 2 2 turbulence is homogeneous and isotropic, these func- + k1 Ð∞ Ð∞ tions depend only on the magnitude k of the wave vec- × (), ()δ, () tor and are described by the expressions Gk1; t' t'' Ik; t'' t k1 + k2 ()0 ()0 〈〉φ˜ ()φ˜ ()≡ ()δ, () t τ τ k1 t k2 t' Ik1; tt' k1 + k2 , (20) 1 τ []× + --- ∫ ∫ ∫ d ∫ dt' ∫ dt'' q1 q2 z 〈〉Gˆ k()tt, ' ≡ Gk(); tt, ' . 2 q q Ð∞ Ð∞ Ð∞ It is assumed that the higher order correlation functions 1 2 can be expressed in terms of functions (20) by using q 2q 2()q 2 Ð q 2 Kraichnan’s direct-interaction approximation [12]. In × ------1 2 2 1 -δ()k Ð q Ð q ()λλ 2 ()λ2 ()2 1 1 2 particular, from solution (16), we obtain + k1 + q1 + q2 (23) () () 〈〉φ˜ 1 ()φ˜ 0 () × []q × — φ []q × — φ Gk(); t, τ Gq(); τ, t' k1 t k2 t 1 s z 2 s z 1 1

2 t ˜ ()0 ˜ ()0 ˜ ()0 ik ()0 ()0 × Gq()φ; τ, t'' 〈〉q ()t' φq ()t'' φk ()t ------1 []× φ 〈〉ˆ ()φ, ˜ ()φ˜ () 2 1 2 2 = 2 k1 —s z ∫ dt' Gk1 tt' k1 t' k2 t λ + k t t' 1 Ð∞ 2 (21) q2 2 t + dt' dt''------δ()k Ð q Ð q ∫ ∫ ∫ ∫ 2 2 2 1 1 2 ik1 ()0 ()0 ()λλ () []× φ 〈〉φˆ , 〈〉˜ φ˜ ∞ ∞ + k1 + q2 = ------k1 —s z ∫ dt' Gk1()tt' k1 ()t' k2 ()t q1 q2 Ð Ð λ + k 2 1 Ð∞ × (), ()φ, 〈〉˜ ()0 φ˜ ()0 φ˜ ()0 Gk1; tt' Gq2; t' t'' q ()t' q ()t'' k ()t 2 t 1 2 2 ik1 = ------[]k × φ dt'Gk(); tt, ' Ik(); t', t δ()k + k 2 2 1 —s z ∫ 1 1 1 2 × { ()λ []× ()⋅ []× ∇ φ λ + k 2 + q1 q1 q2 z q2 —s q2 s z 1 Ð∞ and, analogously, ()λ2 2 ()2 []× ∇ []× ∇ φ } + q2 Ð q1 + q2 q1 s z q2 s z . 〈〉φ˜ ()1 ()φ˜ ()1 () k1 t k2 t We see that relationship (23) contains a third-order correlation function. In the direct-interaction approxima- k 2k 2 tion, this third-order correlation function can be = Ð------1 2 []k × — φ []k × — φ ()λλ 2 ()2 1 s z 2 s z expressed in terms of the second-order correlation + k1 + k2 functions and the averaged Green’s function by using t t the well-known procedure that was described in detail × 〈〉ˆ , ˆ , φ˜ ()0 φ˜ ()0 in, e.g., [12, 13]: ∫ dt' ∫ dt'' Gk1()tt' Gk2()tt'' k1 ()t' k2 ()t'' (22) () () () ∞ ∞ ˜ 0 ˜ 0 ˜ 0 Ð Ð 〈〉φk ()t φq ()t' φq ()t'' 2 1 2 (24) 4 t t = []q × q Ᏻ()δk ,,q q ; tt ,,' t'' ()k ++q q , k1 []× φ 2 , 1 2 z 2 1 2 2 1 2 = ------k1 —s z ∫ dt' ∫ dt''Gk()1; tt' ()λ 2 2 + k1 Ð∞ Ð∞ where × , × ()δ, () Ᏻ(),, ,, Gk()1; tt'' Ik1; t' t'' k1 + k2 . k2 q1 q2; tt' t''

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q2 × ()⋅ []× φ k1 —s k1 —s z,

k 1 []× 2 ()δ, () = q1 q2 z Fq1 q2 k1 Ð q1 Ð q2 q 1 ∫ ∫ – q q q 2 1 2 D × ()⋅ []× ∇ φ q2 —s q2 s z k 1 = – 4∆()k ,,q q (28) k1 q 1 (), 1 1 2 – = ∫∫dq1dq2q1q2Fq1 q2 ------4 - q q 2 k 1 + D 1 q 2 = k 2 2 2 1 × []∆ (),, ()⋅ []× φ k1 q2 Ð 8 k1 q1 q2 k1 —s k1 —s z. 0 k q 1 1 The region of integration, D, is shown in Fig. 1: it k q q Fig. 1. Region of integration in the space of the absolute val- restricts q1 and q2 so that the vectors 1, 1, and 2 form ues of the wave vectors. a triangle whose area is equal to ∆(),, k1 q1 q2

2 2 t 1 2 2 2 2 2 2 4 4 4 1/2 (29) = ---()2q q ++2q k 2k q Ð k Ð q Ð q . q2 Ð q1 τ (), τ (), τ (), τ 1 2 1 1 1 2 1 1 2 = ------∫ d Gk2; t Iq1; t' Iq2; t'' 4 λ + k 2 2 Ð∞ Also, in formula (26), we introduced the notation (25) t' k 2 Ð q 2 Λ(),, 4 ()2 2 2 2 2 τ (), τ (), τ (), τ k1 q1 q2 = k1 Ð q1 Ð q2 . (30) + ------∫ d Gq1; t' Ik2; t Iq2; t'' λ + q 2 1 Ð∞ After integration over the angles, Eq. (23) becomes ()2 ()0 2 2 t'' ˜ ˜ 〈〉φk ()φt k ()t q1 Ð k2 τ (), τ (), τ (), τ 1 2 + ------2 ∫ d Gq2; t'' Ik2; t Iq1; t' . λ 2 t + q2 Ð∞ 2k1 ()⋅ []× φ (), = Ð------k1 —s k1 —s z ∫ dt'Gk1; tt' In expression (23), integration can be carried out over ()λ 2 2 + k1 Ð∞ the angles in the spaces of the vectors q1 and q2 using the following formulas (the details of their derivation 4 × (), k1 []× φ 2 are given in Appendix B): Ik1; t' t Ð ------k1 —s z ()λ 2 2 + k1 2 []× ()δ, () t t' ∫ ∫ q1 q2 z Fq1 q2 k1 Ð q1 Ð q2 × (), (), (), q1 q2 ∫ dt' ∫ dt''Gk1; tt' Gk1; t' t'' Ik1; t'' t × []× φ []× φ Ð∞ Ð∞ q1 —s z q2 —s z τ τ t 3 3 2 2 1 q q ()q Ð q ∆()k ,,q q Ð --- dq dq dτ dt' dt''------1 2 2 1 - (), 1 1 2 ∫∫ 1 2 ∫ ∫ ∫ 4 2 2 = dq1dq2q1q2Fq1 q2 ------(26) 2 k ()λλ + k ()+ q ∫∫ 4 D Ð∞ Ð∞ Ð∞ 1 1 1 k1 D ∆(),, k1 q1 q2 2 × ------Ᏻ(),, ,, (), τ ×Λ{ ,, []× φ 2 k2 q1 q2; tt' t'' Gk1; t ()k1 q1 q2 k1 —s z λ + q2 (31) ∆2 ,, ⋅ φ 2 } × ()τ, ()Λτ, { (),, []× φ 2 Ð16 ()k1 q1 q2 ()k1 —s , Gq1; t' Gq2; t'' k1 q1 q2 k1 —s z 2 Ð16∆2()k ,,q q ()k ⋅ — φ } []× ()δ, () 1 1 2 1 s ∫ ∫ q1 q2 zFq1 q2 k1 Ð q1 Ð q2 t t' 3∆() q1 q2 4q1q2 k1q1q2 + ∫∫dq1dq2 ∫ dt' ∫ dt''------× []× []× φ 4()λλ 2 ()2 2 q1 —s z q2 —s z D Ð∞ Ð∞ k1 + k1 + q2 × Ᏻ(),, ,, (), (), ∆(),, (27) k2 q1 q2; tt' t'' Gk1; tt' Gq2; t' t'' (), 4 k1 q1 q2 = ∫∫dq1dq2q1q2Fq1 q2 ------k 4 ×λ{()2 ()2 2 2 ()λ 2 [ 2 2 D 1 + q2 q2 Ð q1 + 2 + q1 k1 q2

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3 3 2 2 2 πkk k ()k Ð k ∆()kk,,k Ð8∆ (k ,,q q )]}()k ⋅ — []k × — φ δ()k + k . × ------1 2 2 1 ------1 2- 1 1 2 1 s 1 s z 1 2 ()λλ 2 ()2 λ 2 2 + k + k1 + k2

t τ τ 2.5. Mean-Field Equation ×τ (), τ ()τ, ∫ d ∫ dt' ∫ dt''Gk; t Gk1; t' We substitute Eqs. (18) and (19) and relationships Ð∞ Ð∞ Ð∞ (21), (22), and (31) into Eq. (7) and use the formulas × ()τ, Ᏻ(),, ,, ∞ Gk2; t'' kk1 k2; tt' t'' . () πδ 3 () ∫dkkik jFk = ij∫k Fkdk, Expression (34) implies that, if the third-order correla- 0 tions (the second term on the right-hand side) are negligible, then the coefficient ν is determined by the first () T ∫dkkik jklkmFk (32) term and is nonzero only when λ ≠ 0, i.e., when the adi- ∞ abatic response of the electrons is important. The third- π order correlation functions can be neglected only in the = ---()δ δ ++δ δ δ δ k5Fk()dk 4 ij lm il jm im jl ∫ case of drift turbulence, which is dominated by the 0 source of turbulence rather than by the nonlinear interaction of turbulent pulsations. As was mentioned in order to carry out integration over the angles in the above, this case can be described in the quasilinear k vector space 1. As a result, we arrive at the final form approximation. It is not surprising that the conclusion of the evolutionary equation for the large-scale field: that the electron adiabatic term is responsible for turbu- ∂ lent viscosity agrees with the results obtained in [3, 4]. ()λφ 2φ ∂------Ð —s Since the Green’s function G and the spectral density I T (33) of turbulent pulsations are both positive, the second- ()β []φ × 2φ ν 4φ , order correlation functions always make a negative Ð1Ð T —s —s—s z +0T —s = ν contribution to the turbulent viscosity coefficient T. where On the other hand, even in the limiting case described by the two-dimensional Euler equation (λ = 0), taking ∞ t πλ 5 into account third-order correlations in a strongly tur- ν k (), (), T = Ð∫dk------∫ dt'Gk; tt' Ik; t' t bulent regime leads to a nonzero turbulent viscosity ()λ 2 2 0 + k Ð∞ coefficient, whose sign is very sensitive to both the turbulence spectrum and the Green’s function of the turbu- ∞ t t' π 3∆(),, lent field. Below, we will be interested in turbulent pul- 2 kk1k2 kk1 k2 Ð ∫dkk∫∫d 1dk2 ∫ dt' ∫ dt''------sations whose energy is concentrated in the wavenum- ()λλ 2 ()2 2 (34) ∞ ∞ + k + k2 ber range corresponding to the inertial interval of 0 D Ð Ð turbulence. In the next section, we will attempt to find × Ᏻ(),, ,, (), (), kk1 k2; tt' t'' Gk; tt' Gk2; t' t'' adequate expressions for the functions characterizing drift turbulence in this case. ×λ{}()2 ()2 2 2 ()λ 2 []2 2 ∆2 ,, + k2 k2 Ð k1 + + k1 k k2 Ð 8 ()kk1 k2 is the turbulent viscosity coefficient and 3. TURBULENCE SPECTRA IN THE INERTIAL INTERVAL ∞ t π 9 β k In the inertial interval of turbulence, the turbulent T = ∫dk------2 ∫ dt' 2()λ + k2 spectra are universal in the sense that they are governed 0 Ð∞ by the nonlinear interaction of turbulent pulsations and t are insensitive to dissipation and the sources of turbu- × (), (), (), lence. The shape of the turbulent spectrum in the inertial ∫ dt''Gk; tt' Gk; tt'' Ik; t' t'' interval can be determined using the method based on Ð∞ the scaling symmetries of the geostrophic equation (1) describing the drift turbulence. In the general case (λ Ӎ t' —2), this equation admits only one scaling symmetry, Ð2∫ dt''Gk(); tt, ' Gk(); t', t'' Ik(); t'', t which is characterized by the infinitesimal operator ∂ ∂ φ∂ ∂φ Ð∞ X1 = t / t – / . This operator implies that Eq. (1) is ∞ invariant under the following one-parameter scale transformation with the parameter α: []Λ(),, ∆2(),, Ð ∫dkk∫∫d 1dk2 kk1 k2 + 16 kk1 k2 (35) α φ αÐ1φ 0 D t' ===t, x' x, ' . (36)

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In the two limiting cases under analysis, the geo- so that the second-order correlation function (41) trans- strophic equation admits additional scaling symme- forms to tries. Specifically, when the adiabatic electron response 〈〉φ()φx , t ()x , t is negligible (λ Ӷ —2, which corresponds to the two- 1 1 2 2 dimensional Euler equation), the additional scaling 3abÐ a b c symmetry is characterized by the infinitesimal operator = ∑ C ,,α ()ε' ()τ' ()r' ∂ ∂ φ∂ ∂φ abc (43) X2 = x / x + 2 / , and the two-parameter group of abc,, scale transformations is defined as ≡α3abÐ 〈〉φ ()φ, (), ' x1' t1' ' x2' t2' . t' ==αt, x' βx, φ' =αÐ1β2φ. (37) On the other hand, the scale transformation (36) implies that In the opposite limit (λ ӷ —2), the geostrophic equation x∂ ∂x 2 admits the additional scaling symmetry X3 = / + 〈〉αφ()φx , t ()x , t = 〈〉φ'()φx' , t' '()x' , t' . (44) 4φ∂/∂φ and is thus invariant under the two-parameter 1 1 2 2 1 1 2 2 scale transformation We equate the powers of α in relationships (43) and (44) to obtain a = 2/3 + b/3. Inserting this equality into t' ==αt, x' βx, φ' =αÐ1β4φ. (38) formula (41), we see that the second-order correlation function of turbulence is an arbitrary function of the The geostrophic equation possesses two quadratic two parameters: integral invariants: the energy integral 〈〉εφ()φ, (), 2/3 ˆ ()ε1/3τ, x1 t1 x2 t2 = F r . (45)

1 2 2 Ww≡ ∫ dx = ---∫[]λφ + ()—φ dx (39) Taking into account the relationship 2 ⋅ ()φ, 〈〉()φ, (), Ðikr Ik; t1 t2 = ∫ x1 t1 x1 + r t2 e dr, (46) and the enstrophy integral we can see that the spectral function of the drift turbulence should have the general form 1 2 2 2 Uu≡ dx = --- []λ()—φ + ()— φ dx. (40) ∫ 2∫ ()ε, 2/3 ()ε1/3τ, Ik; t1 t2 = F1 k , (47)

If there is a source of turbulence that injects energy and where F1 is an arbitrary function. The Green’s function enstrophy, then, as a result of the nonlinear interaction of turbulence can be found in an analogous way. It of turbulent pulsations (see, e.g., [14]), the energy cas- should have the form cades toward large scales (an inverse cascade), while (), ()ε1/3τ, the enstrophy cascades toward small scales (a direct Gk; t1 t2 = R1 k . (48) cascade). The correlators and spectra of turbulence that For a constant density η of the enstrophy flux along the correspond to a constant energy flux or to a constant spectrum, we can also apply the above procedure. As a enstrophy flux can be found from the scaling properties result, we find that the turbulence is described by the of the geostrophic equation by applying the P theorem, spectral function and Green’s function as follows: which was proved as early as 1914 by Buckingham [15]. Thus, for a constant rate ε of the energy density ()η, 2/3 ()η1/3τ, Ik; t1 t2 = F2 k , transfer along the spectrum, the second-order two-time (49) (), ()η1/3τ, two-point correlation function of a steady uniform iso- Gk; t1 t2 = R2 k . tropic turbulence in the inertial interval can depend τ | | | | ε In the two limiting cases under analysis, additional only on three quantities: = t1 – t2 , r = x1 – x2 , and . Consequently, in the most general form, the second- scaling symmetries imply that the spectral function and order correlation function can be written as Green’s function are both arbitrary functions of only one parameter. In the limit λ Ӷ —2, which is described by the Euler equation, we arrive at the following spectra 〈〉φ()φx , t ()x , t = ∑ C εaτbrc, (41) corresponding to the constant energy and enstrophy 1 1 2 2 abc fluxes: abc,, ()ε, 2/3 Ð14/3 ()ε1/3 2/3τ Ik; t1 t2 = k F1 k , where Cabc are arbitrary tensor coefficients. Taking into (50) account the relationship ε ∝ dw/dt, we find that the Gk(); t , t = R ()ε1/3k2/3τ , scale transformation (36) converts the quantity ε 1 2 1 according to the law ()η, 2/3 Ð6 ()η1/3τ Ik; t1 t2 = k F2 , (51) Ð3 (), ()η1/3τ ε' = α ε, (42) Gk; t1 t2 = R2 .

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The Kolmogorov energy spectra that correspond to the λ Ӷ 2 0 for kc spectral functions (50) and (51) have the form (see, e.g., β =  (60) [14])  λ ӷ 2 2 for kc . Ek()∼∼ ε2/3kÐ5/3, Ek() η2/3kÐ3. (52) Analogously, in the opposite limit (λ ӷ —2), we obtain 4. TURBULENT COEFFICIENTS IN THE CASE OF STRONG DRIFT TURBULENCE ()ε, 2/3 Ð14/3 ()ε1/3 8/3τ ν Ik; t1 t2 = k F1 k , Here, we consider the turbulent coefficients T and (53) β (), ()ε1/3 8/3τ T in the particular case of strong turbulence such that Gk; t1 t2 = R1 k , the spectral function and Green’s function are independent of both the source of turbulence and the dissipation ()η, 2/3 Ð6 ()η1/3 2τ Ik; t1 t2 = k F2 k , mechanism (the inertial interval). The calculation of the (54) turbulent coefficients is straightforward but rather (), ()η1/3 2τ Gk; t1 t2 = R2 k . lengthy. Substituting relationships (56) into formula (34) yields the following expression for the turbulent The Kolmogorov energy spectra that correspond to the viscosity coefficient (see Appendix B for details): spectral functions (53) and (54) have the form ∞ 2/3 Ð11/3 2/3 Ð5 πλ 5σ() Ek()∼∼ ε k , Ek() η k . (55) ν k k T = Ð∫ dk------2 - 2()λ + k2 ω()k In the next section, we follow [11, 14] and set the arbi- kc trary functions in formulas (50)Ð(54) to be exponential, ∞ so that πkk k 3∆()kk,,k Ð dkkd dk ------1 2 1 2 ()σ, ()() ω() () ∫ ∫∫ 1 2 2 2 2 Ik; t1 t2 = k exp Ð k t1 Ð t2 HkÐ kc , ()λλ () k D + k + k2 (), c Gk; t1 t2 (56) σ()σk ()σk ()k 2 ()ω()()()() × 1 2 {()λ 2 ()2 2 = exp Ð k t1 Ð t2 Ht1 Ð t2 HkÐ kc , ------2 + k2 k2 Ð k1 ω()Ωk (61) where H(t) is the Heaviside step function and k is the c ()λ 2 []}2 2 ∆2(),, length of the wave vector of the largest scale vortices in +2 + k1 k k2 Ð 8 kk1 k2 the inertial interval of turbulence. The one-moment σ 2 2 2 2 (t1 – t2 = 0) spectral function (k) of turbulence and the k Ð k 1 k Ð k 1 Ω × ------1 ------+ ------2------1 + ------nonlinear frequency ω(k) are determined by the corre- 2 σ() 2 σ()ω()  λ k2 λ k1 2 k2 sponding equations presented above and have the form + k2 + k1

 2 2 2/3 Ð14/3 for the spectrum with a constant k Ð k 1 Ω  σ ε k + ------2 1------1 + ------Hk()Ð k Hk()Ð k , 1 2 σ()ω() 1 c 2 c  energy flux (57) λ k k2  σ()k =  + k σ η2/3 Ð6 for the spectrum with a constant Ω ≡ ω ω ω  2 k where (k) + (k1) + (k2). On the other hand, by  enstrophy flux, inserting relationships (56) into formula (35), we can see that the ﬁrst two terms in the resulting expression  cancel each other by virtue of the equality 1/3 α for the spectrum with a constant  f ε k t t  1 energy flux (58) ω()k =  ∫ dt' ∫ dt''Gk(); tt, ' Gk(); tt, '' Ik(); t', t''  1/3 β for the spectrum with a constant η Ð∞ Ð∞  f 2 k  enstrophy flux, t t' = 2∫ dt' ∫ dt''Gk(); tt, ' Gk(); t', t'' Ik(); t'', t (62) where the exponents α and β differ between the two limiting cases of drift turbulence (in which the charac- Ð∞ Ð∞ σ()k teristic spatial scales are much larger than or much = ------. smaller than the ion Larmor radius in terms of the elec- 2ω2()k tron temperature): Hence, in our model, in which the spectral functions λ Ӷ 2 2/3 for kc and Green’s functions depend exponentially on t1 – t2, α =  (59) the turbulent coefﬁcient β , which corresponds to the  λ ӷ 2 T 8/3 for kc , turbulence-related renormalization of the Poisson

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 742 LAKHIN bracket, is nonzero only at the expense of the third- and, for the spectra with a constant enstrophy flux, they order correlation function: are equal to ∞ λ Ӷ 2 4 for kc β = dkkd dk a ==5, b  c =β. (66) T ∫ ∫∫ 1 2  λ ӷ 2 6 for kc , kc D As for the turbulent viscosity coefficient, it depends on πkk3k 3()∆k 2 Ð k 2 ()σkk,,k ()σk ()σk ()k × ------1 2 1 2 1 2 1 2 both the quantity kc and the energy (enstrophy) flux 2()λλ + k2 ()λ+ k 2 ()ω+ k 2 ()Ωk 3 density. Let us determine this coefficient in the two lim- 1 2 iting cases in question. ×Λ[](),, ∆2(),, kk1 k2 + 16 kk1 k2 λ Ӷ 2  2 2 (63) 4.1. The Limit kc k2 Ð k1 1 Ω1 1 × ------1 + ------+ ------2 σ() ω() ω() In this limit, the terms λ can be neglected in all com-  λ + k k 4 k1 k2 binations of the form λ + k2 in the integrands, so that the 2 2 turbulent viscosity coefficients for the spectra with a Ω2 k Ð k 1 Ω + ------+ ------2------1 + ------constant energy and constant enstrophy flux density are 4ω()k ω()k λ 2 σ()k 4ω()k described by relatively simple expressions. 1 2 + k1 1 2 4.1.1. Spectrum with a constant energy flux. We 2 2 substitute the spectrum determined by relationships k Ð k 1 Ω  + ------1 ------1 + ------Hk()Ð k Hk()Ð k . (57)Ð(59) into formula (61) to obtain the following λ 2 σ()k 4ω()k 1 c 2 c + k2 2 1  expression for the turbulent viscosity coefficient: ν ε1/3 Ð4/3()κ λ 2 κ λ Ӷ 2 λ ӷ 2 β T = kc 1 /kc + 2 , (67) In both limits kc and kc , the coefficient T for a constant energy (enstrophy) flux along the spectrum where ε is independent of kc and the energy flux density (the κ = Ð3πσ /20 f , (68) enstrophy flux density η): 1 1 1 κ ∞ and the coefficient 2 is represented in integral form, 2 2 2 σ π∆(),, () ∞ β i xyz y Ð z 2 T = ------∫dxyz∫∫d d ------πσ ∆(),, 4 a + 2 a a 3 κ 1 xyz 1 f i ˜ x y z Ωˆ 2 = ------∫dxyz∫∫d d ------1 D 3 ()19/3 11/3 2/3 2/3 2/3 2 f 1 xz y ()x ++y z 1 D˜ × []2y2z2 ++x2 y2 x2z2 Ð x4 Ð z4 × {}y6 ++z6 x4 y2 Ð 2x2 y4 Ð y4z2 Ð x2z4 2 (69)  2 2 b c Ωˆ c c Ωˆ × ()() ()  2 2 10/3 1 2 2 8/3 2/3 2/3 2/3  z Ð y x yz ++---- y + z ------(64) × ()x Ð y z + ---()z Ð x y ()x ++y 3z  4 4  2

2 2 bc+ c Ωˆ 2 2 bc+ c Ωˆ   + ()x Ð z y z + ---- + ()y Ð x z y + ----  -+ ()y2 Ð z2 x8/3()x2/3 ++y2/3 2z2/3 Hy()Ð 1 Hz()Ð 1 . 4 4  

× Hy()Ð 1 Hz()Ð 1 . Expression (67) implies that the contribution of the second-order correlations to the turbulent viscosity coeffi- Ωˆ c c c ˜ 2 Here, = x + y + z ; the region of integration, D , is cient is formally as small as λ/k ; however, with the analogous to D and restricts y and z in such a way that c possible case κ Ӷ κ in mind, we retain this contribu- x, y, and z form a triangle; and the subscript i = (1, 2) 2 1 refers to the spectra with a constant energy flux and tion in the expression for the viscosity coefficient. κ constant enstrophy flux, respectively. For the spectra Since the coefficient 2 is impossible to determine with a constant energy flux, the exponents a, b, and c analytically because of the very complicated integrand, are equal to it was calculated by the Monte Carlo method with the help of the Mathematica-3 software package. By trun- λ Ӷ 2 8/3 for kc cating the turbulence spectrum at the upper limit (e.g., a ==13/3, b  by taking into account collisional dissipation), the  λ ӷ 2 (65) 14/3 for kc , numerical scheme was made convergent, and it was κ πσ 2 3 c = α, found that 2 = –0.17 1 / f 1 for kmax/kc = 10 and

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κ πσ 2 3 πσ 2 3 κ πσ 2 3 2 = –0.23 1 / f 1 for kmax/kc = 20. We thus see that the 2.73 2 / f 2 for kmax/kc = 10 and 2 = 11.86 1 / f 1 for third-order correlations of the turbulent field, like the kmax/kc = 20. Consequently, in the case at hand, the second-order correlations, make a negative contribu- third-order correlations of the turbulent field make a tion to the turbulent viscosity coefficient. With an infi- positive contribution to the turbulent viscosity coeffi- nite integration domain and different numbers of itera- cient, thereby canceling the negative contribution of the tions, the integral was always found to be finite and second-order correlations. That is why, in order to negative. On the other hand, in the standard iteration determine the sign of the turbulent viscosity coefficient, method, the step-by-step process of numerical integra- it is necessary to know the exact spectrum of the small- tion converges very slowly because of the nontrivial scale drift turbulence. However, because of the weight- integrand, which makes it impossible to complete the λ 2 iteration procedure and thus necessitates the use of ing factor /kc , the contribution of the second-order another numerical scheme. Since the details of the tur- correlations is small; therefore, it is highly probable σ that the total turbulent viscosity coefficient will be pos- bulence spectrum (i.e., the coefficients 1 and f1) are still unknown, the most important point here is to deter- itive and the spontaneous perturbations of the averaged mine the sign of the integral, so that we restrict our- field will be damped. selves to the results presented above. Hence, in the case at hand, the turbulent viscosity 4.2. The Limit λ ӷ k 2 coefficient is negative, so a spontaneous amplification c of the large-scale perturbations should take place. As In this limit, the integrands can be simplified by the amplitude of these perturbations increases due to neglecting the terms k2 in all combinations of the form the instability, nonlinear effects come into play (i.e., the λ + k2. The turbulent viscosity coefficients for the nonlinear term in the evolutionary equation for the spectra with a constant energy and constant enstrophy large-scale field becomes important). Presumably, it is flux density can be calculated in the same way as in these effects that cause the instability to saturate at a Section 4.1. certain level and the system to relax to a certain steady state. To answer this question, it is necessary to numer- 4.2.1. Spectrum with a constant energy flux. ically solve Eq. (33) for the large-scale field. Using relationships (57)Ð(59) and formula (61), we obtain the following expression for the turbulent vis- 4.1.2. Spectrum with a constant enstrophy flux. cosity coefficient: For the turbulence spectrum with a constant enstrophy flux, we obtain ν ε1/3 Ð4/3()γ γ T = kc 1 + 2 , (73) ν η1/3 Ð2()κ λ 2 κ T = kc 3 /kc + 4 , (70) where where γ πσ λ 1 = Ð3 1/8 f 1 , (74) κ πσ 3 = Ð 2/8 f 2, (71) γ and the coefficient 2 is represented in integral form, and the coefficient κ has the form 4 ∞ 2 ∞ πσ ∆(),, 2 γ 1 xyz 1 πσ ∆(),, 2 = ------∫dxyz∫∫d d ------κ 2 xyz{ 6 6 4 2 λ3 3 19/3 11/3 13/3 8/3 8/3 8/3 2 4 = ------∫dxyz∫∫d d ------y ++z x y f 1 x y z ()x ++y z 3 ()7 5 1 D˜ 9 f 2 xz y 1 D˜   × {}4 4 4 2 2 2 2 ()2 2 22/3 2 4 4 2 2 4} ()2 2 4 5()2 2 4 2y ++2z x Ð42x y Ð y z  x Ð y z --- Ð2x y Ð y z Ð x z  x Ð y z + --- z Ð x y (72)   2 (75) 1 2 2 14/3 8/3 8/3 8/3 2 2 14/3  + ---()z Ð x y ()x ++y 3z + ()y Ð z x ---+4()y2 Ð z2 x4 Hy()Ð 1 Hz()Ð 1 . 2   -× ()x8/3 ++y8/3 2z8/3 Hy()Ð 1 Hz()Ð 1 . Numerical integration shows that, for infinite integra-  κ κ ∞ tion limits, the quantity 4 diverges, 4 , so that the integral should be truncated at the upper limit. As an Numerical integration over an unbounded domain γ upper limit in the wavenumber, it is natural to choose showed that the coefficient 2 is infinite, so that a phys- the reciprocal of the scale length of the collisional vis- ically reasonable result can only be obtained by truncat- cosityÐrelated dissipation, which is neglected in our ing the turbulence spectrum at a large value of k. As an analysis. With the integral truncated in such a way, upper limit in the wavenumber, it is natural to choose κ numerical calculations yielded the expressions 2 = the reciprocal of the ion Larmor radius, i.e., to set

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2 λ kmax = , in which case numerical calculations give the lent viscosity with a negative coefficient. In turn, the γ πσ 2 λ3 3 instability driven by this turbulent viscosity causes the following expressions: 2 = 9.10 1 / f 1 for spontaneous growth of large-scale perturbations. κ πσ 2 3 kmax /kc = 10 and 2 = 40.54 1 / f 1 for kmax/kc = 20. Consequently, the third-order correlations of the turbu- 5. DISCUSSION OF THE RESULTS lent field make a positive contribution to the turbulent viscosity coefficient, thereby canceling the negative A systematic theory of the generation of large-scale contribution of the second-order correlations. Hence, in perturbations by the small-scale drift turbulence in a order to determine the sign of the turbulent viscosity plasma has been constructed by applying a geostrophic coefficient, it is necessary to know the exact spectrum equation (a simplified version of the HasegawaÐMima of the drift turbulence. model) and the two-scale direct interaction approximation technique. Thus, the previously existing theory, 4.2.2. Spectrum with a constant enstrophy flux. which is based on the quasilinear approximation and, We insert relationships (57), (58), and (60) into formula strictly speaking, is valid when the Reynolds number is (61) to obtain low (i.e., when the collisional dissipation dominates over the nonlinear interaction of turbulent pulsations) ν η1/3 Ð2()γ γ T = kc 3 + 4 . (76) or when the turbulence spectrum is determined by the source of turbulence, has been extended to the case of γ Here, the coefficient 3 is determined by the second- strong drift turbulence, whose spectrum is governed by order correlations, the nonlinear interaction of turbulent pulsations and is insensitive to dissipation and the source of turbulence. γ πσ λ 3 = Ð 2/2 f 2, (77) It has been shown that, as in quasilinear theory, accounting for small-scale drift turbulence gives rise to γ and the contribution 4 of the third-order correlations is a turbulent viscosity effect and leads to renormalization represented in integral form, of the nonlinear term with the Poisson bracket in the ∞ evolutionary equation for the large-scale (mean) field. πσ2 ∆(),, However, for the regime of strong drift turbulence γ 2 xyz 1 4 = ------∫dxyz∫∫d d ------under consideration, the turbulent viscosity coefficient λ3 3 7 5 5 2 2 2 2 f 2 x y z ()x ++y z 1 D˜ and renormalization coefficient are both represented as a sum of two parts, which are comparable in magnitude.  As in the quasilinear approximation, the first part is × {}2y4 ++2z4 x4 Ð42x2 y2 Ð y2z2 ()x2 Ð y2 z8 - determined by the second-order correlations and is  nonzero merely because the geostrophic equation (78) incorporates the adiabatic electron response. The con- 1 2 2 6 2 2 2 2 2 6 + ---()z Ð x y ()x ++y 3z + ()y Ð z x tribution of the second-order correlations to the turbu- 2 lent viscosity coefficient is negative regardless of the turbulence spectrum. In the limiting case corresponding  to the two-dimensional incompressible hydrodynamics -× ()x2 ++y2 2z2 Hy()Ð 1 Hz()Ð 1 . described by the Euler equation, the second-order cor-  relations do not contribute to the turbulent viscosity coefficient. The sign of the second part, which is asso- As in the case of the spectrum with a constant energy ciated with the third-order correlations of the turbulent λ Ӷ 2 field, depends on the spectral properties of turbulence. flux for kc , the integral in expression (78) converges even when the integration domain is unbounded. Explicit expressions for the turbulent viscosity coef- With different numbers of iterations, the integral was ficient and renormalization coefficient have been always found to be negative. On the other hand, derived for the model Kolmogorov spectra of drift tur- because of the nontrivial integrand, attempts to make bulence, which are thought to be characteristic of the the iteration process of numerical integration conver- inertial interval of turbulence. For the model used in the gent were unsuccessful. For this reason, we restrict our- analysis, the renormalization of the Poisson bracket is selves to the results obtained for a bounded region of completely governed by the third-order correlations, because the contribution of the second-order correla- integration, specifically, γ = –0.47πσ 2 / f 3λ3 for 4 2 2 tions is identically zero. For λ Ӷ k 2 (the case of turbu- γ πσ 2 3λ3 c kmax/kc = 10 and 4 = –0.86 2 / f 2 for kmax /kc = 20. lence described by the two-dimensional Euler equa- We see that the third-order correlations of the turbulent tion) and for the Kolmogorov spectrum with a constant field, like the second-order correlations, make a nega- energy flux, the contribution of the third-order correla- tive contribution to the turbulent viscosity coefficient. tions to the turbulent viscosity coefficient is negative. Hence, we can conclude that, in the case at hand, small- Accordingly, the turbulent viscosity coefficient is also scale drift turbulence on the whole gives rise to turbu- negative, which indicates the growth of large-scale per-

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 GENERATION OF LARGE-SCALE STRUCTURES BY STRONG DRIFT TURBULENCE 745 turbations. For the Kolmogorov spectrum with a con- y stant enstrophy flux, the contribution of the third-order correlations is positive, so that the determination of the sign of the turbulent viscosity coefficient requires a q2 q1 knowledge of the exact turbulence spectrum. Moreover, θ λ 2 2 since the weighting factor /kc of the contribution of θ the second-order correlations is small, it can be 1 k γ 1 expected that the resulting turbulent viscosity coefficient will be positive. x

The results obtained for λ ӷ k 2 differ radically from Fig. 2. Geometry of the fundamental triad of the wave vec- c tors. those obtained in the opposite limit. For the spectrum with a constant energy flux, the contribution of the third-order correlations to the turbulent viscosity coef- ∞ ∞ 2π 2π ficient is positive, so that the sign of this coefficient can- = q dq q dq dθ dθ k 2q 4 sin2 θ Fq(), q not be determined without knowing the details of the ∫ 1 1∫ 2 2 ∫ 1 ∫ 2 1 2 2 1 2 turbulence spectrum. On the other hand, for the spec- 0 0 0 0 (A.1) trum with a constant enstrophy flux, the contribution of ∂2φ ×θ()cos2 ()Ð γ Ð sin2 ()θ Ð γ ------the third-order correlations to the turbulent viscosity 2 2 ∂X∂Y coefficient is negative, so that the coefficient itself is also negative, which indicates the instability of large- ∂2φ ∂2φ scale perturbations. + sin()θθ Ð γ cos()Ð γ ------Ð ------2 2 ∂ 2 ∂ 2 The results obtained in this study may be of interest X Y in solving the problems of LÐH transitions in tokamaks ×δ()δk Ð q cosθ Ð q cosθ ()q sinθ Ð q sinθ . and the generation of large-scale vortices in the ocean 1 1 1 2 2 1 1 2 2 and in the atmospheres of rotating planets. Since, in the presence of the second δ function, the integral over the angles is nonzero only when the functions θ θ sin 1 and sin 2 have the same sign, the integral (A.1) ACKNOWLEDGMENTS can be represented as I am grateful to Prof. T. Schep for fruitful discus- 2π 2π θ θ ()… sions during my stay in the Netherlands and to Prof. ∫ d 1 ∫ d 2 V.S. Mikhailenko for bringing the papers by Yoshizawa 0 0 (A.2) to my attention. This study was carried out in the frame- π π 2π 2π work of the RussianÐDutch research cooperation under θ θ ()… θ θ ()… the auspices of the Netherlands Organization for Scien- = ∫d 1∫d 2 + ∫ d 1 ∫ d 2 . tific Research (project no. NWO.047.009.007). The 0 0 π π study was also supported by the Council of the Federal θ* π θ θ* π θ Program “Government Support of the Leading Scien- After the replacements 1 = 2 – 1 and 2 = 2 – 2 tific Schools,” project no. 00-15-96526. in the second integral in (A.2), we obtain ∞ ∞ 2 5 (), APPENDIX A Jq= ∫d 1∫dq2k1 q1q2 Fq1 q2 0 0 Derivation of Formulas (26)Ð(28) π π We illustrate the method for deriving formulas (26)Ð ×θ θ 2 θ  []()θ γ ∫d 1∫d 2 sin 2 cos 2 2 Ð (28) by calculating as an example the integral in expression (28). In the geometry shown in Fig. 2, this expres- 0 0 sion can be rewritten in the form ∂2φ 1 +2cos[])()θ + γ ------+ ---(sin[]2()θ Ð γ 2 ∂X∂Y 2 2 J = []q × q 2Fq()δ, q ()k Ð q Ð q ∫ ∫ 1 2 z 1 2 1 1 2 ∂2φ ∂2φ Ð2sin[]()θ + γ ) ------Ð ------q1 q2 2  ∂X2 ∂Y 2 (A.3) × ()⋅ []× φ ×δ()δθ θ ()θ θ q2 —s q2 —s z k1 Ð q1 cos 1 Ð q2 cos 2 q1 sin 1 Ð q2 sin 2

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∞ ∞ π π t' 2 5 ()θ, θ 2 θ θ τω[ ()()τ ω()()τ = ∫dq1∫dq2k1 q1q2 Fq1 q2 ∫d 1∫d 2 sin 2 cos2 2 + ∫d exp Ð k 2ttÐ ' Ð Ð k1 t' Ð 0 0 0 0 t''

t ∂2φ ∂2φ ∂2φ × 22cosγ ------Ð sin 2γ ------Ð ------ω()()]τωτ [ ()()τ ∂ ∂ 2 2 Ð k2 t' + Ð 2t'' + ∫d exp Ð k 2ttÐ ' Ð X Y ∂X ∂Y t' (B.2) ω()τ()ω()()]τ ×δ()δθ θ ()θ θ Ð k1 Ð t' Ð k2 t' + Ð 2t'' k1 Ð q1 cos 1 Ð q2 cos 2 q1 sin 1 Ð q2 sin 2 . θ In (A.3), we carry out integration over the angles 1 and 1 []ω ()ω ω θ γ = ---- exp Ð ()2k ()ttÐ ' Ð t'' Ð ()k1 + ()k2 ()t' Ð t'' 2 and take into account the relationships cos = k1x/k1 Ω γ and sin = k1y/k1. As a result, we arrive at expression (28). Formulas (26) and (27) can be derived in a similar 1 { [ ω()() way. + ω------() ω()ω()exp Ð2 k ttÐ ' k + k1 Ð k2 ω()()]ω[ ()() APPENDIX B Ð2 k2 t' Ð t'' Ð exp Ð k 2ttÐ ' Ð t'' Calculation of the Integrals over Time ν β 1 in the Expressions for T and T + ()ω()k Ð ω()k ()t' Ð t'' ]} + ------1 2 ω()k Ð ω()k Ð ω()k As an example, we turn to the second term in 1 2 ν expression (34) for T. In the triple integral over time, ×ω{ []()ω ω we calculate the part containing the ﬁrst of the three exp Ð ()k + ()k1 ()ttÐ ' Ð ()k2 ()tt+ '2Ð t'' terms in formula (25) for Ᏻ: []}ω()()ω()() Ð2exp Ð2k ttÐ ' Ð k2 t' Ð t'' . t t' t ≡ τ (), (), Inserting integral (B.2) into expression (B.1) and inte- Ft∫ d ' ∫ dt'' ∫ d Gk; tt' Gk2; t' t'' grating over t' and t'', we ﬁnd Ð∞ Ð∞ Ð∞ σ()σ() × (), τ (), τ (), τ k1 k2 1 1 Gk2; t Ik1; t' Ik2; t'' F = ------ω()Ω Ω---- +.------ω()- (B.3) 2 k k2 t t' t σ σ τ = ()k1 ()k2 ∫ dt'∫ dt''∫ d Ht()Ð t' Analogously, we can obtain (B.1) Ð∞ Ð∞ Ð∞ t t' t' × τ Ht()' Ð t'' Ht()Ð τ (), (), ∫ dt' ∫ dt'' ∫ d Gk; tt' Gk2; t' t'' ×ω[ ()()τ ω() τ ∞ ∞ ∞ exp Ð k 2ttÐ ' Ð Ð k1 t' Ð Ð Ð Ð ω()()]τ × (), τ (), τ (), τ Ð k2 t' Ð t'' + t'' Ð . Gk1; t Ik; t Ik2; t'' (B.4) The interval of integration over time can be divided into σ()σ() k k2 1 1 the following three subintervals: = ------ω()Ω Ω---- +,------ω() 2 k 2 k2 t ∫ dτHt()Ð t' Ht()' Ð t'' Ht()Ð τ t t' t'' τ (), (), Ð∞ ∫ dt' ∫ dt'' ∫ d Gk; tt' Gk2; t' t'' Ð∞ Ð∞ Ð∞ (B.5) ×ωexp[Ð ()k ()2ttÐ ' Ð τ Ð ω()k t' Ð τ 1 σ()σk ()k × Gk(); t, τ Ik(); t, τ Ik(); t', τ = ------1 . ω()()]τ 2 1 2 Ð k2 t' Ð t'' + t'' Ð 2ω()Ωk

t'' Integration over time in formula (35) gives = ∫ dτωexp[Ð ()k ()2ttÐ ' Ð τ t τ τ Ð∞ ∫ dτ ∫ dt' ∫ dt''Gk(); t, τ ()ω()ω()()]τ Ð k1 + k2 t' Ð Ð∞ Ð∞ Ð∞

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 GENERATION OF LARGE-SCALE STRUCTURES BY STRONG DRIFT TURBULENCE 747

t Substituting relationships (B.6)Ð(B.8) into formula ×τ(), τ ()τ, ()τ, (35), we arrive at expression (63). ∫ d 'Gk; t Gk1; t' Gk2; t'' (B.6) Ð∞ × (), τ (), τ (), τ REFERENCES Gk; t ' Ik1; t' ' Ik2; t'' ' 1. J. G. Charney, J. Mar. Res. 14, 477 (1955). σ σ ()k1 ()k2 1 1 1 1 1 = ------++------+ ------,2. A. Hasegawa and K. Mima, Phys. Fluids 21, 87 (1978). Ωω 2 ω ω Ωω ω ()k Ω 4 ()k1 ()k2 4 ()k1 ()k2 3. A. V. Gruzinov, P. H. Diamond, and V. B. Lebedev, Phys. Plasmas 1, 3148 (1994). τ τ t 4. A. V. Chechkin, M. I. Kopp, V. V. Yanovsky, and ∫ dτ ∫ dt' ∫ dt''Gk(); t, τ A. V. Tur, Zh. Éksp. Teor. Fiz. 113, 646 (1998) [JETP 86, 357 (1998)]. Ð∞ Ð∞ Ð∞ 5. F. Wagner, G. Becker, K. Behringer, et al., Phys. Rev. t' Lett. 49, 1408 (1990). ×τ(), τ ()τ, ()τ, ∫ d 'Gk; t Gk1; t' Gk2; t'' 6. R. J. Groebner, Phys. Fluids B 5, 2343 (1993). (B.7) Ð∞ 7. P. C. Liewer, Nucl. Fusion 25, 543 (1985). × Gk(); t', τ' Ik(); t, τ' Ik(); t'', τ' 8. S. I. Vainstein, Ya. B. Zel’dovich, and A. A. Ruzmaikin, 1 2 Turbulent Dynamo in Astrophysics (Nauka, Moscow, σ()σ() 1980), p. 208. k k2 1 1 = ------+,------9. H. K. Moffat, Magnetic Field Generation in Electrically 2 Ω ω() Ω ω()k 4 k2 Conducting Fluids (Cambridge Univ. Press, Cambridge, 1978; Mir, Moscow, 1980). t τ τ 10. D. Montgomery and T. Hatori, Plasma Phys. Controlled ∫ dτ ∫ dt' ∫ dt''Gk(); t, τ Fusion 26, 717 (1984). Ð∞ Ð∞ Ð∞ 11. A. Yoshizawa, J. Phys. Soc. Jpn. 46, 669 (1979).

t'' 12. R. H. Kraichnan, J. Fluid Mech. 5, 497 (1959). ×τ(), τ ()τ, ()τ, 13. R. H. Kraichnan, Phys. Fluids 6, 1603 (1963). ∫ d 'Gk; t Gk1; t' Gk2; t'' (B.8) 14. J. A. Krommes, in Basic Plasma Physics, Ed. by Ð∞ A. A. Galeev and R. N. Sudan (Énergoatomizdat, Mos- × (), τ (), τ (), τ cow, 1984; North-Holland, Amsterdam, 1984), Vol. 2. Gk2; t'' ' Ik; t ' Ik1; t' ' 15. E. Buckingham, Phys. Rev. 4, 345 (1914). σ()σ() k k1 1 1 = ------+.------2 Ω ω() Ω ω()k 4 k1 Translated by I. A. Kalabalyk

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001

Plasma Physics Reports, Vol. 27, No. 9, 2001, pp. 748Ð753. Translated from Fizika Plazmy, Vol. 27, No. 9, 2001, pp. 793Ð799. Original Russian Text Copyright © 2001 by Basmanov, Karpov, Model’, Nazarenko, Savchenko, Subbotin, Fedotkin, Shalata.

PLASMA DYNAMICS

High-Current Plasma Channel Produced with a Narrow Gas Column V. F. Basmanov, G. V. Karpov, B. I. Model’, S. T. Nazarenko, V. A. Savchenko, A. N. Subbotin, A. S. Fedotkin, and F. G. Shalata All-Russia Research Institute of Experimental Physics, Russian Federal Nuclear Center, Sarov, Nizhni Novgorod oblast, 607188 Russia Received January 25, 2001; in ﬁnal form, April 1, 2001

Abstract—A new method for creating high-current plasma channels is developed. The method uses a narrow gas column formed by the leading particles of a nonsteady gas jet outﬂowing into a vacuum. An electric discharge device with a system for the formation of a narrow gas column is experimentally studied. The parameters of emission from the plasma channel are measured. © 2001 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION sionless heating requires high electron current velocity (at least higher than the ion acoustic speed). However, The development of the methods for generating cur- in a self-contracted channel with a high electron density rent pulses with an amplitude of 10 MA and a rise time n , the electron current velocity may remain lower than of ~100 ns [1] motivated considerable interest in the e problem of producing self-contracted plasma channels the ion acoustic speed. Thus, in a channel whose radius (such as high-temperature Z-pinches) [2]. Among the is comparable to the skin depth, the electron current methods for solving this problem, we mention the fol- velocity will always be less than the ion acoustic speed if the electron linear density Ne exceeds a certain criti- lowing: (i) the use of multiwire metal arrays [3Ð5], 2 –2 (ii) pulsed gas puffing [6–8], and (iii) the electric explo- cal value, Ne > N0 = 2mic e . In particular, in a 1-mm- sion of a frozen deuterium fiber [9] or deuterium-con- diameter deuterium channel with the electron density 19 –3 × 16 –1 × 16 –1 taining fiber [10]. The latter method is of special impor- ne = 10 cm (Ne = 8 10 cm , N0 = 2.6 10 cm ), tance in thermonuclear fusion research because it the electron current velocity is approximately two times enables one to obtain high energy densities in a stable lower than the ion acoustic speed. In this case, colli- channel with an anomalously long lifetime [9]. Proba- sional heating is the only mechanism for ion heating; its bly, this channel would be even more stable if there rate can be estimated by the time of heat exchange were no instabilities leading to the formation of waists, between the plasma components, τ = 17AnT 3/2 –1 [13]. which, according to calculations [11], start to develop e well before the evaporation of the frozen deuterium is In a deuterium channel (A = 2) at a temperature of Te = completed. The studies of the electric explosion of a 107 K, we have τ > 100 ns. Hence, if the current rise hydrocarbon fiber with a preformed neck [12] provide time is on the order of 100 ns, we can expect that Te will indirect evidence in favor of this assumption. Hence, it grow faster than the ion temperature, which results in is of interest to change the initial conditions so as to the higher stability of the channel. The necessary con- achieve higher stability of the channel. An appropriate dition for this is a rapid (in the limiting case, synchro- method for solving this problem is to use a strongly nous with the magnetic pressure at the channel bound- ӷ nonisothermal plasma (Te Ti) produced from the ini- ary) increase in the electron thermal pressure at the tial gas channel under conditions such that the ion heat- channel axis. Otherwise, the radial charge separation ing is slower than the electron Joule heating. field inevitably arises, in which the ions can be accelerated toward the axis up to high energies. The only way Let us assume that, instead of a frozen deuterium to exclude or diminish the radial plasma implosion is to fiber, we have a narrow uniform gas column. In a dis- decrease the initial diameter of the gas column. The charge, the released Joule energy is spent mainly on question as to which parameters of the gas column are electron heating. Due to the high thermal conductivity, achievable still remains open because nobody has yet the electron temperature rapidly equalizes throughout dealt with the problem of completely excluding radial the channel cross section. Hence, if the current plasma implosion in gas puffs. increases rapidly and the initial diameter of the gas column is small enough, then the magnetic pressure can In this study, a new method for creating a narrow gas only be balanced by the electron thermal pressure. The column is developed. An electric discharge device in ions remain cold and immobile until they are heated via which the plasma channel is produced using such a col- either collisional or collisionless mechanisms. Colli- umn is experimentally studied.

HIGH-CURRENT PLASMA CHANNEL PRODUCED WITH A NARROW GAS COLUMN 749

2. FORMATION OF A NARROW GAS COLUMN Photographing 8 7 When a gas that was initially at rest expands through a long channel into a vacuum, the longitudinal velocity of the leading portion of the gas flow is several times higher than the speed of sound in an unperturbed gas Ä and the gas temperature here is nearly zero [14]. There- fore, after escaping from the channel into free space, the leading gas particles have high longitudinal and low transverse velocities, thus forming a gas jet with an extremely low divergence. If the channel is sufficiently narrow, the jet has the shape of a column whose diam- l l eter is much less than its length. The schematic of a gas- 0 1 Pumping out dynamic gas puff experiment based on the above con- Ä sideration is shown in Fig. 1. The gas is separated from the vacuum by an aluminum foil 15 µm thick. The foil is broken by an electric discharge in a time ∆t ~ 1 µs, 1 l0 = 150 mm which is much shorter than the time ts it takes for the 2 l1 = 35 mm leading particles to pass along the channel length l0: ∆ Ӷ 3 t ts. By virtue of this inequality, the gas flow is nonsteady, which distinguishes our device from gas puff systems used in high-current plasma facilities [6Ð8]. 5 The maximum initial gas pressure is less than 1 atm, 6 which is determined by the foil breaking strength. The 4 instant at which the gas appears at the channel exit, t = ts is indicated by the electric breakdown of an auxiliary spark gap. The time ts depends on the gas species, initial pressure, channel length, etc.; under our conditions, it is µ Fig. 1. Schematic of the narrow gas column formation: on the order of 100 s. (1) gas volume (V ≈ 10 cm3), (2) 15-µm-thick Al foil, (3) 0.5-mm-thick Ta disk 10 mm in diameter, (4) stainless In experiments, we encountered the problem of the steel grid, (5) needle electrode, (6) insulator, (7) capacitor visualization of the gas jet. Attempts to visualize the (C = 0.8 µF, U = 30 kV), and (8) controlled discharge gap. gas jet using laser techniques failed because of the low gas density and the small radius of the jet. Visualization with the help of a short low-current discharge along the flow did not provide the required image contrast. It was found that nitrogen, atmospheric air, and, particularly, oxygen jets are self-luminous; this property was employed for the jet visualization. The glow starts at t = ts and lasts for several tens of microseconds. The high-speed photography reveals no appreciable radial shift in the boundary of the glow region. The shape of the glowing gas column is adequately displayed by the time-integrated photographs. Figure 2 shows the time- integrated photograph obtained with atmospheric air puffing. It can be seen that the column diameter is 35 mm nearly constant and coincides with the channel diameter, which is equal to 3 mm. The glow is the most intense and contrasted at the channel outlet. Down- stream from the outlet, the glow intensity decreases and the radial boundary of the column becomes smeared. Fig. 2. Photograph of the glowing atmospheric air column. Perhaps, this is related to the axial inhomogeneity of the gas density in the column, which, in turn, is due to the highly nonuniform distribution of the gas density in the channel. It was found theoretically [14] that, in a varies strongly along the channel. Thus, for a diatomic nonsteady gas flow escaping into a vacuum through a 5 gas, we have n ~ cs . long channel, the speed of sound is equal to cs = 0 at the flow front and increases linearly along the channel as it The detailed diagnostics of the column are still dif- approaches the unperturbed gas. The gas density n also ficult to perform. Nevertheless, the results obtained

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001

750 BASMANOV et al. enabled us to carry out plasma experiments with a nar- spheric air, neon, argon, or xenon. The initial gas pres- row gas column and to plan subsequent steps in sure was less than 1 atm. improving the system for gas puffing. These future X-ray emission was measured using spectrometers improvements include (i) a decrease in the axial non- with silicon semiconductor detectors equipped with uniformity of the column by increasing the ratio of the various absorbing filters. The signals from seven detec- channel length l0 to the column length l1; (ii) the use of tors placed inside the discharge chamber near its axis at much higher initial pressures and, consequently, gas a distance of ~60 cm from the anode were recorded densities in the column; and (iii) a decrease in the diam- simultaneously. X radiation fell onto the detectors eters of the channel and gas column to less than 1 mm. through a central 8-mm-diameter hole in the cathode. We alternatively used spectrometers designed for measuring either soft (up to 10 keV) or hard (higher than 3. EXPERIMENTAL SETUP 10 keV) X radiation. In each of these spectrometers, one of the channels was the same to ensure the match- The plasma experiments were carried out in an ing of the measurements in the soft and hard X-ray experimental stand (Fig. 3) with a capacitive energy spectral ranges. Soft X radiation was measured with storage of 24 kJ at a charge voltage of 100 kV. The stor- SPPD-11 silicon detectors designed at the Research age bank consists of twelve IK-100-04 capacitors con- Institute of Pulsed Technologies (Moscow). A charac- nected in parallel; each of them is triggered by its own teristic feature of these detectors is the small thickness gas-filled discharge gap. The capacitors and gaps are of both the entrance window (the total thickness of the placed in a metal tank filled with transformer oil. The electrical contact and dead layer is less than 1 µm) and discharge chamber is mounted on the tank lid and is the sensitive region (~50 µm). Hard X radiation was connected to the energy storage bank via a low-induc- measured with silicon detectors based on sensitive ele- tance cable line. In the chamber, there are two elec- ments designed at the Research Institute of Pulsed trodes: a high-voltage anode and a grounded cathode. Technologies. The spectral sensitivity of the detectors The distance between the electrodes along the chamber was calculated using the Monte Carlo method with γ axis is l1 = 35 mm. The discharge chamber is preevacu- allowance for the transport of radiation, electrons, and ated to 10–4 torr. The gas is injected into the chamber positrons [15]. To attenuate the recorded radiation, each through the anode along a central channel with a diam- detector was equipped with its own aperture diaphragm with diameters of 2 to 9 mm. The time resolution of the eter of 3 mm and length of l0 = 15 cm. The foil separat- ing the working gas from the vacuum chamber is bro- different detectors varied from 1.5 to 20 ns. An assem- ken by the electric discharge of two IK-100-04 capaci- bly of the thermoluminescent dosimeters made of 0.2- tors connected in parallel and charged to 30 kV. The and 1-mm-thick IS-7 glass plates [16] were used to measure the integral dose and to estimate the effective delay time tc of triggering the energy storage bank with respect to triggering the controlled discharge gap in the energy of X-ray photons. A stack of five to ten glass plates was placed in a hollow Al cylinder with a 5-mm- foil-break circuit (Fig. 1) is regulated with an accuracy µ thick wall and was oriented so as to ensure the nearly no worse than 0.1 s. The working gas was alterna- normal incidence of radiation. From the source side, the tively deuterium, helium, nitrogen, oxygen, atmo- glass plates were covered with a mylar film either with or without aluminum deposition. The opposite end of the cylinder was closed with a 5-mm-thick Al cap. After 7 6 345 being irradiated, the glass plates were tested with a standard IKS-A dosimeter. The location of the X-ray source was determined from X-ray photos obtained using several pinhole cameras with different absorbing filters. The signals from several semiconductor detectors that simultaneously recorded X-ray emission from different regions of the interelectrode gap were also 2 used for this purpose. Both the hard X radiation with the photon energy above 100 keV and the neutron flux were recorded with an SSDI-8 detector [17] consisting of a plastic scintillator block with a photomultiplier. The time resolution of the detector was ~5 ns. To distinguish between the neutron and X-ray pulses, several 1 = 12 detectors were set at different distances from the dis- N charge chamber. The integral neutron yield was determined by the activation of an Ag foil placed in a paraf- Fig. 3. Schematic of the experimental facility: (1) capacitive fin moderator. In all discharges, the plasma channel energy storage, (2) feed cable line, (3) anode, (4) cathode, current was recorded with a time resolution no worse (5) insulator, (6) pulsed valve, and (7) current detector. than 10 ns.

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4. EXPERIMENTAL RESULTS 70 kA (‡) The electric breakdown of the gas column leads to the formation of a plasma current channel with a current rise time of several hundred nanoseconds. Due to the plasma processes in the channel, its resistance sharply increases. As a result, in the current oscillo- gram, the increase in the current is finished with a singularity in the shape of a kink. The current singularity, 110 kA (b) which is observed in all of the gases, is accompanied by an X-ray burst. For light gases, the burst duration is ~50 ns; for deuterium, neutron emission with an integral yield of 108 neutrons per pulse is also recorded. Figure 4 shows the waveforms of the current and the signals from the SSDI-8 detector for helium at an initial 0.5 µs pressure of 1 atm. The maximum current in the channel depends on the initial gas pressure and the delay time tc of the triggering of the energy storage bank. As tc and the gas pressure increase, the time during which the current grows also increases; as a result, the current Fig. 4. Waveforms of the current and the signals from an SSDI-8 detector for different delay times tc of triggering the attains a higher value before the singularity occurs. It capacitive storage: (a) 90 and (b) 95 µs; the working gas is µ can be seen in Fig. 4 that the increase in tc by 5 s helium at an initial pressure of 1 atm. increases the discharge current by a factor larger than 1.5. The further increase in tc provides a severalfold increase in the current by the instant of the kink; how- surements and pinhole images of the X-ray source ever, in this case, the reproducibility of the radiation obtained in a helium plasma channel with the use of a parameters substantially decreases. lead cap. In photo II (Fig. 6b), a metal grid welded into the side window of the discharge chamber casts a Semiconductor detector measurements show that shadow on the image of the lead cap emitting X-rays. the shortest X-ray pulses are generated when light gases are used. A comparison of the detector signals The efficiency of the electric discharge X-ray source obtained in the different spectral ranges during one dis- was estimated based on the dose measured by ther- charge reveals the following feature characteristic of all moluminescent dosimeters. The maximum dose recorded behind a 20-µm-thick mylar film with Al dep- the gases. Namely, the X-ray pulses show a double- µ humped shape which is most pronounced in the hard osition (~1 m) at a distance of 5.5 cm from the source × 3 X-ray spectral range. The signals from the detectors was 3 10 rad. The dose of radiation passed through recording X-ray photons with a lower energy also dem- the central hole in the cathode along the chamber axis µ onstrate the double-humped shape, but it is less pro- and measured behind a 20- m-thick Al foil at a dis- nounced. As the photon energy decreases, the signal tance of 55 cm from the anode edge was 20 rad. The maxima merge into one maximum. Such behavior was spectral distribution of the radiant energy was esti- observed throughout the entire spectral range that was mated based on the measurements of both the distribu- studied with semiconductor detectors, i.e., in the 1- to tion of the absorbed dose in the stack of IS-7 glasses 100-keV spectral range. and the signals from semiconductor detectors. Nearly one-half of the energy falls in the photon energy range Figure 5 shows the waveforms of double-humped of up to 10 keV, the spectral intensity being maximum X-ray pulses recorded in air and helium plasma chan- at 3 keV. Almost all of the remaining energy falls in the nels. We chose the most representative waveforms with range 10Ð60 keV. sufficiently long time intervals between the maxima in order to reliably resolve them in time. The semiconduc- When operating with light gases, a short (~20 ns) tor detectors were also used to locate the X-ray source; pulse of extremely hard X-ray radiation is observed. for this purpose, several identical detectors recorded the Since this pulse can be recorded (with the SSDI-8 emission from different regions of the plasma channel. detector) at a distance of up to 20 m from the source It is found that the main fraction of radiation with a behind a metal shield with a thickness of more than photon energy of ~10 keV and higher is generated near 5 mm, it should contain megaelectronvolt photons. the anode edge. When a cap was put on the anode, the Finally, it should be noted that, in the plasma chan- X-ray source was located at the cap edge, which was in nel, a rather intense axial plasma flow is produced, contact with the plasma. This is also confirmed by which extends beyond the interelectrode gap through X-ray photographs of the radiating region. In experi- the central hole in the cathode. The presence of the ments, we used thin lead caps with a wall thickness of plasma flow is indicated by the destruction of the films ~0.05 mm, which were evaporated during a single and foils placed on the axis behind the discharge gap. pulse. Figure 6 shows the schematic of pinhole mea- The flow divergence, which was determined from the

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 752 BASMANOV et al.

(a)

~3 keV (b)

~5 keV ~10 keV

~8 keV ~20 keV

~13 keV ~60 keV

t, 100 ns t, 100 ns

Fig. 5. Double-humped X-ray pulses recorded in different photon energy ranges in (a) an air plasma channel and (b) a helium plasma channel.

(‡)

3 2

5 mm

1 (b)

III5 mm 5 mm

Fig. 6. (a) Schematic of pinhole measurements: (1) pinhole camera I, (2) pinhole camera II, and (3) lead cap. (b) Pinhole images of the X-ray source.

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 HIGH-CURRENT PLASMA CHANNEL PRODUCED WITH A NARROW GAS COLUMN 753

flow imprints, was about one degree for deuterium or ACKNOWLEDGMENTS helium. The maximum flow velocity, which was esti- We are grateful to V.T. Punin for helpful remarks. mated by the instant of the flow appearance at a distance of ~50 cm behind the cathode flange, was equal to ~108 cm/s. REFERENCES 1. J. J. Ramírez, IEEE Trans. Plasma Sci. 25, 155 (1997). 5. CONCLUSION 2. V. A. Burtsev, V. A. Gribkov, and T. I. Filippova, Itogi Nauki Tekh., Ser. Fiz. Plazmy 2, 80 (1981). A new method for creating a plasma current channel 3. W. Clark, M. Gersten, J. Katzenstein, et al., J. Appl. has been developed. For this purpose, a 35-mm-long Phys. 53, 4099 (1982). 3-mm-diameter gas column formed by the leading par- 4. C. Deeney, T. J. Nash, R. B. Spielman, et al., Phys. Rev. ticles of a nonsteady gas jet outflowing into a vacuum E 56, 5945 (1997). is used. For all the gases under study (deuterium, 5. T. W. L. Sanford, T. J. Nash, R. C. Mock, et al., Phys. helium, nitrogen, oxygen, atmospheric air, neon, argon, Plasmas 4, 2188 (1997). and xenon), the waveforms of the current in the channel 6. J. Shiloh, A. Fisher, and N. Rostoker, Phys. Rev. Lett. 40, resemble those in Z-pinches [2]. The increase in the 515 (1978). current is finished with a singularity in the shape of a kink. The stable operation of the plasma channel lasts 7. W. Clark, R. Richardson, J. Brannon, et al., J. Appl. for several hundred nanoseconds until the singularity Phys. 53, 5552 (1982). occurs; this time is longer than the duration of a stable 8. A. V. Batyunin, A. N. Bulatov, V. D. Vikharev, et al., Fiz. plasma channel in experiments with a frozen deuterium Plazmy 16, 1027 (1990) [Sov. J. Plasma Phys. 16, 597 (1990)]. fiber [9]. The current singularity is accompanied by an X-ray burst; for deuterium, neutron emission with an 9. J. D. Sethian, A. E. Robson, K. A. Gerber, and A. W. De Silva, Phys. Rev. Lett. 59, 892 (1987). integral yield of 108 neutrons per pulse is also recorded. The instant of the singularity can be controlled by either 10. W. Kies, G. Decker, M. Mälzig, et al., J. Appl. Phys. 70, changing the initial pressure of the working gas or vary- 7261 (1991). ing the delay time of applying the high-voltage pulse to 11. I. R. Lindemuth, Phys. Rev. Lett. 65, 179 (1990). the gas column. The highest yield of X radiation, which 12. L. E. Aranchuk, S. A. Dan’ko, A. V. Kopchikov, et al., lasts for ~50 ns, is attained with light gases. A signifi- Fiz. Plazmy 23, 215 (1997) [Plasma Phys. Rep. 23, 194 cant fraction of radiation falls into the 10Ð60 keV pho- (1997)]. ton energy range. The X-ray source is located near the 13. L. A. Artsimovich, Controlled Thermonuclear Reac- anode edge. The maximum dose measured at a distance tions, Ed. by A. Kolb and R. S. Pease (Fizmatgiz, Mos- of 5.5 cm from the anode is 3 krad. At the instant of the cow, 1961; Gordon and Breach, New York, 1964). current singularity, the axial plasma flow directed away 14. K. P. Stanyukovich, Nonsteady Motions of Continuous from the anode is generated in the channel; its velocity Media (Gostekhteorizdat, Moscow, 1955), p. 154. attains 108 cm/s. 15. E. N. Donskoœ, Vopr. At. Nauki Tekh., Ser. Mat. Model. Fiz. Protsessov, No. 1, 3 (1993). In order to further develop the proposed method for 16. I. A. Bochvar, T. I. Gimadova, I. B. Keirim-Markus, creating the current plasma channel and perform exper- et al., Methods of Individual-Control-Glass Dosimetry iments with currents as high as 1 MA, a system for gas (Atomizdat, Moscow, 1977), p. 65. puffing with a 2–3 orders of magnitude higher initial 17. A. I. Veretennikov and K. N. Danilenko, Diagnostics of gas pressure must be designed. These experiments will Single-Pulse Radiation (AT, Moscow, 1999), p. 45. make it possible to draw the final conclusion about the influence of the initial discharge conditions on the stability of the plasma channel. Translated by N. N. Ustinovskiœ

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001

PLASMA DYNAMICS

Calculation of One-Dimensional Plasma Flows with Allowance for the Kinetics of Ion Collisions P. D. Gasparyan and N. V. Ivanov All-Russia Research Institute of Experimental Physics, Russian Federal Nuclear Center, Sarov, Nizhni Novgorod oblast, 607190 Russia Received November 13, 2000

Abstract—The inﬂuence of kinetic effects on the generation of line X radiation during the spherical implosion of a laser corona plasma with two ion species is studied under the conditions prevailing in experiments with thin-wall spherical targets in the Iskra-5 laser facility of the All-Russia Research Institute of Experimental Phys- ics (Sarov, Russia). Kinetic processes occurring in a multicharged plasma are investigated using a specially devised code for solving one-dimensional Landau equations for a nondegenerate multicomponent plasma by the Monte Carlo method (the KIN-MC code). The code was developed using the quasineutral plasma approximation under the assumption that the electron distribution function is locally equilibrium. The model equations are presented, the scheme of numerical solution is described, and the calculated results are discussed. © 2001 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION cesses in the region where the plasma flows interact should be studied using numerical methods based on The development of the theory of nonequilibrium kinetic plasma models [7Ð10]. processes in high-temperature multicharged plasmas is motivated by diverse practical and theoretical applica- Note that the plasma-related kinetic problems are tions. Such plasmas are created by the interaction of too complicated to be investigated numerically in full ultrashort laser pulses with solid bodies and other formulation, i.e., to simultaneously solve nonlinear objects in experiments aimed at investigating various Boltzmann equations for all subsystems of particles in targets used in laser fusion, inertial confinement fusion the plasma (photons, electrons, and ions) even in a one- (ICF), and laboratory X-ray lasers [1Ð3]. dimensional approximation. For this reason, we will apply an approach based on the solution of simplified In particular, in experiments on the laser irradiation physical problems. of targets, it is of considerable interest to study the Here, we study the violation of the gas-dynamic interaction of counterstreaming flows of a laser corona approximation in the model of a nondegenerate plasma plasma, because, in the interaction region, the tempera- with constant ion charge. The ion plasma component is ture and density of the plasma and its lifetime can described by the kinetic equation with the Landau col- increase substantially. This makes it possible to signifi- lision integral. The electron plasma component is cantly extend laboratory experiments to such issues of assumed to be equilibrium and to obey a locally Max- high-temperature plasma physics as (i) the emissivity wellian velocity distribution function. We use the of a multicharged plasma, (ii) the capability of a plasma quasineutral plasma approximation and neglect elec- to decelerate fast ion beams, and (iii) reaction rates in a tron inertia. An analogous model of the nonequilibrium plasma. plasma dynamics was applied in [7, 8] when studying The interaction of counterstreaming flows of a laser the role of kinetic effects in the dynamics of laser fusion corona plasma can be studied in experiments with var- targets. In our approach, the kinetic equations reduce to ious types of one-dimensional and two-dimensional a set of the modified nonlinear Landau kinetic equa- laser targets irradiated by laser light in different ways tions for ion plasma components and the electron (see, e.g., [4Ð11] and other related papers). Estimates energy balance equation. In the combined method show that, in the region where the flows of a laser developed for solving this set of equations numerically, corona plasma interact with each other, it is easy to the kinetic equations are solved by the Monte Carlo achieve conditions under which the mean free paths of (MC) method and the energy balance equation is solved multicharged plasma ions are comparable to or even in finite differences. The combined method was imple- larger than the characteristic dimensions of the flows. mented as the one-dimensional KIN-MC code. Under these conditions, the gas-dynamic approxima- When solving the nonlinear Boltzmann equations in tion, which is usually used to calculate the plasma finite differences, the plasma flows are especially diffi- parameters and to analyze them theoretically, may turn cult to calculate for ions with short mean free paths. out to be inaccurate; consequently, the physical pro- The method developed here is based on the MC method

CALCULATION OF ONE-DIMENSIONAL PLASMA FLOWS 755

mim j ij for solving kinetic equations and requires approxi- where mij = ------is the reduced mass, Akl = mately the same amounts of computational resources mi + m j for ions with short and long mean free paths. The pro- 4πe4Z 2Z 2Λ posed approach can also be applied to solving more ------i j ij (g2δ – g g ) is the diffusion tensor, g = 2 3 kl k l complicated problems, including those in which mijg Λ account should be taken of, e.g., the nonequilibrium v – c is the relative velocity, and ij are Coulomb loga- nature of plasma electrons, the kinetics of fusion reac- rithms for ionÐion collisions. tion, the non-Maxwellian character of ion distribution Using the approximation of a small electron-to-ion functions, and ion diffusion in a multicomponent mass ratio m m and assuming that the electrons obey plasma. For problems in kinetic formulation, the results e/ i of calculations of the dynamics of a simple plasma can a Maxwellian distribution function with a shifted argu- be used to analyze the accuracy of some other approxi- ment, we can write the ionÐelectron collision integral mations applied in the numerical modeling of the as [14] plasma processes, e.g., the multistream gas-dynamic ∂ ν T e ∆ mi () plasma approximation [7, 12]. Jie = ie ------f i +;------∂------vuÐ e f i (3) me me v The physical formulation of the dynamic plasma problems under consideration and the model equations where ue = (ue, 0, 0) is the mean electron velocity vec- 4 2 are presented in Section 2. The numerical method and 42π e Z L n m 3/2 the potentialities of the KIN-MC code are described in tor, ν = ------i ei e ------e is the ionÐelectron ie 3 2 T the Appendix, in which we also illustrate the results mi e Λ from test simulations of plasma flows in planar geome- collision frequency, and ie are Coulomb logarithms try (the problem of the interaction of unloading waves). for ionÐelectron collisions. Note that expression (3) can In Section 3, we discuss numerical results obtained easily be generalized to arbitrary electron distribution with the KIN-MC code when simulating the implosion functions, in particular, the bi-Maxwellian distribution of a laser corona plasma in experiments with thin-wall functions [16] used to describe fast electrons in the two- spherical targets with an internal input of the laser component diffusion approximation or even more gen- energy [11, 13]. We analyze how the collisional kinetics eral nonequilibrium electron distribution functions of the ions manifests itself in the spherical implosion of used in nonlocal extensions of plasma gas-dynamic the- a laser corona. We numerically investigate the influence ories [17, 18]. In such generalizations, the coefficients of kinetic effects on the resonant X-ray emission of the in front of the diffusion term and the term with a shifted Heα-line of Fe ions (the Fe Heα-line) in a nonequilib- argument in the ionÐelectron collision integral in rium high-temperature multicharged plasma under the expression (3) should be changed. conditions prevailing in experiments with thin-wall We assume that the electron distribution function is spherical targets in the Iskra-5 laser facility of the All- locally equilibrium. This allows us to apply the kinetic Russia Research Institute of Experimental Physics description solely to the ion plasma component. We (Sarov, Russia). also assume that the electrons obey a Maxwellian function described by the number density ne(t, x), mean 2. PROBLEM FORMULATION velocity ue(t, x), and temperature Te(t, x), which is AND MODEL EQUATIONS determined from the electron energy balance equation ∂ ∂ () () We consider an ideal two-component plasma in ----- neT e + ------ueneT e which the ions of different species have a constant ∂t ∂x charge. In planar geometry, the LandauÐVlasov kinetic 2 (4) ∂u ∂ ∂T m equations for the ion distribution function f (t, x, v) in 2 e λ e i ν () i = Ð ---neT e------∂ - ++∂------e------∂ 2∑ ------ieni T i Ð T e +Q, 3 x x x me the absence of a magnetic field have the form [14, 15] i = 1 2 where λ is the electron thermal diffusivity, ν is the ∂ f ∂ f eZ E ∂ f e ie i v i i i ionÐelectron collision frequency, and the term Q ------∂ ++1------∂ ------∂------= ∑ Jij + Jie; (1) t x mi v 1 j = 1 describes an additional electron energy source. Below, we restrict ourselves to considering a cur- where E(t, x) is the electric field and the integrals Jii and rentless plasma, which corresponds to quasineutral Jie describe ionÐion and ionÐelectron Coulomb colli- plasma motions. The conditions for the two-component sions, respectively. The Landau collision integrals plasma to be quasineutral and to carry no current have accounting for ionÐion collisions are represented as the form

2 ij ()∂ () ()∂ () ne = Z1n1 + Z2n2, (5) mij ∂ Akl f j c f i v f i v f j c Jij = ------∂ -∫------∂ - Ð ------∂ - dc, (2) mi v k 2 mi v l m j ul neue = Z1n1u1 + Z2n2u2; (6)

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 756 GASPARYAN, IVANOV where Zi, ni, and ui are the charges, densities, and mean where the collision integrals Jij are given by relation- velocities of the ions. T m ∂ ships (2) and J = ν -----e- ∆f + ------i ------(v – u )f . To describe the plasma flows completely, we are left ie ie m i m ∂v i i with the problem of determining the electric field e e E(t, x). We consider the following electron momentum The electron thermal diffusivity in Eqs. (4) is calcu- transport equation with allowance for the ion plasma lated from the interpolation formula [19, 20] component: aT5/2 ∂ ∂ ∂ λ = ------e . ue ue 1 Pe () e ()ΛZ + 3.44 me------∂ - + ue------∂ - = Ð ------∂ ++eE me Re1 + Re2 ; (7) ie t x ne x The Coulomb logarithms are calculated from the where Pe = neTe is the electron pressure and Rei = expressions ν ei (ui – ue) is the deceleration force exerted on the electrons by the ions. The electronÐion collision frequency a T TT ν Λ = max 1, ln------i ------e , ei can be reduced to the form ii 2 () Z ne TZT+ e π 4 2 3/2 ν 42e Zi Leinime ei = ------2 -------, (8) 3 T e T me Λ , e ie = max 1 lnaeT e ------(); in which we took into account the smallness of the elec- ne TZT+ e tron mass. We force me in Eq. (7) to zero and retain the where a, ai, and ae are constants. terms proportional to me to arrive at the following The method for the numerical solution of the set of expression for the electric field: kinetic equations (4) and (12) and the results of com- ∂ parative test numerical calculations are presented in the 1 Pe () E = ------∂ Ð me Re1 + Re2 . (9) Appendix. ene x Equations (1) and (4) and expression (9) provide a 3. SIMULATIONS OF THE SPHERICAL complete mathematical description of the plasma flows IMPLOSION OF A LASER CORONA PLASMA under consideration. Note that, in Eqs. (1), the collision integrals (3) are written in a frame of reference moving In order to investigate nonequilibrium physical pro- with velocity ue. However, for the numerical solution of cesses that occur during the implosion of a multi- the problem, it is convenient to transform them to a charged plasma, Gasparyan et al. [11] considered a thin-wall spherical target (TST), shown schematically frame moving with the mean ion velocity ui. In the lat- eZ ∂ f m ∂ in Fig. 1. ter frame, the expression ------i E------i – ν ------i ------(v – u )f ∂v ie ∂ e i A two-layer TST is a substrate plastic thin-wall mi 1 me v spherical shell whose inner surface is coated with a thin becomes layer of the material to be investigated. Laser light is ∂ f m ∂ fed into the target at an angle to the normal to the target F ------i Ð ν ------i ------()vuÐ f , surface through several entrance holes that are, on aver- i∂v ie ∂ i i 1 me v age, distributed uniformly over the surface. Inside the where target, laser light is absorbed by the laser corona plasma. The symmetry of the laser energy absorption ∂ eZ1 Pe ν () inside the target is ensured by such factors as the mini- Fi = Ð ------∂ + 1u u2 Ð u1 , m1ne x mum area of the holes, the focusing of laser light as it (10) ∂ passes through the holes, and multiple internal reflec- Z2 Pe ν () tions of the light until it is completely absorbed. The F2 = Ð ------∂ Ð 2u u2 Ð u1 , m2ne x thickness of the inner coating was chosen in such a way that the coating material completely evaporates during 4 2 2 42πe Z Z L n + L n m m 3/2 laser light absorption. The material of the spherical ν = ------1 2------e2 1 e1 2 ------e ------e n . (11) iu 2 n m T i shell is chosen to be transparent to X radiation gener- 3me e i e ated within a TST. When the laser pulse is short, the tar- Consequently, in the moving frame, the kinetic get goes through two stages: first; the formation of the equations for the ion distribution functions have the internal laser corona during laser light absorption and, form second, the implosion of a laser corona plasma toward the target center followed by the formation of a shock 2 ∂ f ∂ f ∂ f wave reflected from the center. Gas-dynamic simula- i v i i ------∂ ++1------∂ Fi∂------= ∑ Jij + Jie, (12) tions show that it is the second stage in which the t x v 1 j = 1 plasma temperature is maximum (several times the ini-

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 CALCULATION OF ONE-DIMENSIONAL PLASMA FLOWS 757 tial temperature of the laser corona) and the density of the hot plasma is highest (higher than the critical density of the initial corona by a factor of about 10 to 100). Hard X radiation is generated throughout both of the stages. Since the plasma inside the TST is relatively transparent to X radiation, the overall picture of the physical processes occurring during plasma implosion can be reconstructed from the measurements of the spatial and temporal evolutions of the X-ray spectra, thereby providing an interesting diagnostic tool for studying radiational and collisional kinetics of the plasma. The parameters of a TST and X-ray spectra were calculated using the CC-9 code [21] based on a nonequilibrium radiative gas-dynamic model in which the radiativeÐcollisional kinetics of the ion level popula- tions was described either by a chemical approach [21] Fig. 1. Schematic of a TST with internal input of the laser or by using the mean ion approximation [22, 23]. For energy. the conditions of experiments carried out in the Iskra-5 laser facility with Fe-coated TSTs [6], ideal spherically X-ray emission of the Fe Heα-line, we simulated the symmetric simulations showed that the maximum implosion of a laser corona plasma in a TST using the plasma density at the target center amounted to 0.1Ð KIN-MC code. 0.2 g/cm3, the plasma temperature being about 3 keV. Since ion kinetic effects play an insignificant role in The related experiments with Fe-coated TSTs in the the formation of a laser corona in the first stage of the Iskra-5 device were carried out in 1997 [13]. The TSTs TST dynamics (see the Appendix), we run the KIN-MC were made of substrate spherical CH shells with the ∆ µ ρ∆ code only to simulate the second stage. In the experi- radius RCH = 1 mm and thickness ëç = 5 m ( ëç = ments with TSTs in Iskra-5, the first stage comes to an 5 × 10–4 g/cm2). The thickness of Fe coatings on the end when the laser pulse terminates, i.e., by the time inner surfaces of the TSTs was about ∆ ~ 0.26 µm Fe t1 ~ 0.8 ns. By this time, about 4Ð5 kJ of the laser ρ∆ × –4 2 ( Fe = 2 10 g/cm ). The plasma inside the TSTs was energy is absorbed within a TST, the Fe coating of the created by light pulses with a total energy of about 7 kJ target is heated to a temperature of about 1Ð1.5 keV, τ and duration 0.5 ~ 0.3–0.4 ns from an iodine laser oper- and the tail of the unloading wave of a laser corona µ ating at the fundamental frequency (λ = 1.315 µm). The plasma approaches the radius Rl ~ 200–300 m. The total area of six holes for launching laser light into a profiles of the plasma velocity, plasma density, and TST was about 7% of the total area of the target surface. electron and ion temperatures obtained by the time t1 ~ The results from measurements of the integral spectra 0.8 ns through the gas-dynamic calculations with the of X-ray photons in the energy range ν > 6 keV and of CC-9 code served as the initial conditions for kinetic the spatial region where X radiation was generated in simulations with the KIN-MC code. the experiments were found to differ strongly from the We used the KIN-MC code to simulate the dynam- related numerical results obtained with the CC-9 code. ics of only the Fe coating of a TST, assuming that the As an example, the calculated diameter of the region of ν plastic spherical shell is immobile. The initial distribu- resonant X-ray emission of the Fe Heα-line ( ~ tion of the ions was modeled by a locally Maxwellian µ 6.8 keV) was about Dcalc ~ 300 m, while the measured distribution. The ion composition of the plasma was µ diameter was substantially larger, Dexp ~ 450–500 m. assumed to be constant, the ion charge number being One possible reason for such a large discrepancy 24, which corresponds to the results obtained with the between the theoretical and experimental results is that CC-9 code when calculating the degree of ionization of the gas-dynamic approximation may fail to describe the Fe atoms in the tail of the unloading wave of a laser spherical implosion of the plasma in the TST [13]. In corona plasma in a TST by the time t1 ~ 0.8 ns and in the fact, estimates show that, by virtue of the strong veloc- region where the counterstreaming ion flows interact. σ 4 –4 ity dependence of the ion mean free paths ( ii ~ Z /v ) The boundary conditions imposed at the left and right and the low density of a laser corona plasma, the ion boundaries of the computation region for the kinetic mean free paths in the interaction of the counterstream- equation assumed ideal ion reflection. The boundary ing ion flows in the central region are comparable with conditions imposed on the electron thermal diffusivity the initial dimensions of the target, in which case the assumed zero electron heat fluxes at the boundaries of gas-dynamic description of the implosion stage in a the Fe coating. Calculations with the CC-9 code TST may be inexact. In order to estimate the accuracy showed that, in the course of plasma implosion in a of the gas-dynamic approximation and to analyze how TST, all these boundary conditions are approximately the kinetics of ionÐion collisions affects the resonant satisfied over the time interval 0.8 < t < 3 ns.

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As the integral characteristic for gas-dynamic and We carried out the following three series of simula- kinetic simulations of plasma implosion in a Fe-coated tions with the KIN-MC code: in the first series, we used TST, we chose the emissivity of the Fe Heα-line in a the collisionless approximation, in which the ionÐion plasma, because this line makes the main contribution collision integral was artificially set to zero (the f ver- to hard (for ν > 6 keV) X radiation. The choice of this sion); in the second series, we modeled ionÐion colli- line was dictated by the ion composition of the com- sions in full measure (the c version); and, in the third pressed plasma and the transparency of the target wall series, we used the gas-dynamic approximation with a to Fe Heα-line radiation [13]. In the KIN-MC code, cal- Maxwellian ion distribution function (the m version). culations are carried out on a mixed EulerianÐ Numerical results from these three versions are Lagrangian spatial grid consisting of NX = 80 mesh illustrated in Figs. 2Ð6. points in space. The run time of the code was chosen to In Figs. 2Ð4, one can clearly see that the stage of be about 2 ns, which is close to the time at which the reflection of the shock wave from the center of a TST is pulse of resonant Fe Heα-line radiation comes to an well described in all calculation versions. Among the end. During this time, the shock wave reflected from three versions, the collisionless f version gives the max- the center travels a distance approximately equal to imum density (about 1 g/cm3) to which the Fe plasma one-half of the initial target radius and the thermal and is compressed, while the gas-dynamic m version gives gas-dynamic perturbations that propagate from the the highest ion temperature. The c version yields inner surface of the target approach the shock front. The parameter values intermediate between those obtained computation time over which the laser corona plasma in the f and m versions. In Fig. 3, we can see that, in the arrives at the target center is about 1 ns, which coin- stage of maximum compression, the kinetics of Fe ions cides with the beginning of the pulse of the resonant Fe in the central region is essentially collisionless. This Heα-line radiation. The resonant X radiation is most conclusion is evidenced by the fact that the wide lead- intense at a time of about tmax ~ 1.5–1.7 ns. ing edge of the ion heat wave extends to a radius of In the KIN-MC code, the effects of radiation from about 200 µm and significantly overtakes the jump in the plasma are described by the model of internal radi- density in the reflected shock wave. In the f version, the ative losses. The power of the internal radiative losses leading edge of the ion heat wave is double-humped, was calculated by the CC-9 code [21Ð23] in the corona which corresponds to the flows of high-energy ions that ρ × are scattered and unscattered near the target center. An approximation from the formula Q( , Te) = const ρ2F (ρ, T ), in which the function F was interpolated analysis of the ion density in phase space shows that, in 1 e 1 the f and c versions, the ion distribution function within using tabulated values at the mesh points. The internal the leading edge of the ion heat wave is far from being radiative losses were incorporated into the KIN-MC locally Maxwellian. At the time at which the reflected code as a negative source term in the electron energy shock wave forms, the ion distribution function over the balance equation. The specific power of the resonant Fe radial velocities is nonmonotonic: the beam component Heα-line radiation was interpolated in a similar way: of the function contains about 10Ð20% of the ions in the ρ × ρ2 ρ Jα( , Te) = const F2( , Te). The integral power and region R ≤ 130–200 µm. Such a significant beam com- the energy of the resonant Fe Heα-line radiation in the ponent of the distribution function facilitates the onset stage of implosion of a laser corona plasma were calcu- of various ion instabilities in the plasma and may hinder lated from the electron density and temperature profiles the implosion of a laser corona plasma even when the with the help of the above expression for the specific initial spherical plasma flow is ideally symmetric. power of the resonant X radiation: Although simulations with and without allowance R for collisions show distinctly different dynamics of the 2 2 7 plasma implosion, this difference has only a minor Jα()Rt, = 4πρ()rt, F ()ρ, T r dr × 10 []/ , ∫ 2 e J ns effect on the calculated electron temperature and the 0 calculated integral energy yield in the Fe Heα-line (Figs. 5, 6) and, for the conditions under consideration, JTα()Rt, results in discrepancies no higher than ~5%. The rea- t R sons for such a slight effect lie in the strong influence of π ρ2(), ()ρ, 2 × 7 [] the shape of the ion distribution function on the electron = 4 ∫ drtt∫ F2 T e r dr 10 J . temperature, strong coupling between the electron and 0.8 0 ion plasma densities, high electron thermal conducti- The distribution Jα(R, t) makes it possible to esti- vity, and the functional dependence of the intensity of mate the radius of the region where the resonant Fe the resonant Fe Heα-line emission in the corona approx- Heα-line radiation is generated during the implosion of imation. a laser corona plasma and to analyze the relative influ- Let us give a clearer insight into one of these factors. ence of the kinetic effects of the delocalization of mass The plasma differs radically from a system of individ- and heat transfers on the shape of the pulse of the reso- ual particles in that the electron and ion plasma compo- nant Fe Heα-line radiation and the radiation energy. nents are strongly coupled to one another. In our model,

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 CALCULATION OF ONE-DIMENSIONAL PLASMA FLOWS 759

ρ 3 (r, ti), g/cm 0.20

0.18

0.16

0.14 1 0.12

0.10 2 0.08

0.06 3 0.04

0.02

0 0.008 0.016 0.024 0.032 0.040 r, cm ρ Fig. 2. Density proﬁles (r, ti) computed at the times ti = (1) 1.1, (2) 1.6, and (3) 2 ns using the KIN-MC code. The solid curves are from the collisional c version, the dashed curves are from the collisionless f version, and the dotted curves are from the gas-dynamic m version.

Ti(r, ti), keV 120 1 108

36 2 24 3 12

0 0.008 0.016 0.024 0.032 0.040 r, cm

Fig. 3. Ion temperature proﬁles Ti(r, ti) computed at the times ti = (1)1.1, (2) 1.6, and (3) 2 ns using the KIN-MC code. The solid curves are from the collisional c version, the dashed curves are from the collisionless f version, and the dotted curves are from the gas-dynamic m version.

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Te(r, ti), keV 4.0

3.6 1

3.2

2.8

2.4 2

2.0

1.6

1.2

0.8 3

0.4

0 0.02 0.04 0.06 0.08 0.10 r, cm

Fig. 4. Electron temperature proﬁles Te(r, ti) computed at the times ti = (1) 1.1, (2) 1.6, and (3) 2 ns using the KIN-MC code. The solid curves are from the collisional c version, the dashed curves are from the collisionless f version, and the dotted curves are from the gas-dynamic m version.

Jα(r, ti), J/ns 60 3

42 2

6 1

0 0.008 0.016 0.024 0.032 0.040 r, cm

Fig. 5. Proﬁles of the integral (over space) power Jα(r, ti) of the resonant X radiation emitted from the sphere of radius R, computed at the times ti = (1) 1.1, (2) 1.6, and (3) 2 ns using the KIN-MC code. The solid curves are from the collisional c version, the dashed curves are from the collisionless f version, and the dotted curves are from the gas-dynamic m version. At the time t1 = 1.1 ns, the power Jα is nearly zero (the corresponding curves are close to the abscissa) because of the small mass of the hot plasma.

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Jα, J/ns implosion, the kinetic effects have no potential to 100 explain the experimental data. Presumably, the lower hard X-ray yield and the larger generation region in experiments with TSTs are associated with two-dimen- 80 sional and three-dimensional energy losses during the implosion of a laser corona plasma. These losses may 60 come from the unstable character of plasma implosion as well as from non-one-dimensional implosion condi- 40 tions in real targets [13].

20 4. CONCLUSION 0 0.8 1.0 1.2 1.4 1.6 1.8 2.0 (i) We have formulated the equations describing the t, ns ion kinetics in a simple multicomponent currentless ideal quasineutral plasma under the assumption than the electron distribution function is in local equilib- Fig. 6. Shape of the pulse of the integral energy flux Jα(t) of the resonant Fe Heα-line radiation from a TST, computed rium. The ionÐion and ionÐelectron collisions are using the KIN-MC code. The solid curve is from the colli- described by the Landau collision integrals. We have sional c version, the dashed curve is from the collisionless f developed the combined model and designed the version, and the dotted curve is from the gas-dynamic m KIN-MC code for numerically solving the kinetic prob- version. The dash-and-dotted curve is from the ff version lems of a simple plasma. In the combined model, the (or “fully free” version), in which the ionÐion and electronÐ ion collisions are both switched off. kinetic equations are solved by the Monte Carlo method and the energy balance equations are solved in finite differences. this coupling is accounted for by the quasineutrality condition. Because of this coupling, even in the absence (ii) We have presented the results from kinetic sim- of ionÐion and ionÐelectron collisions, the plasma elec- ulations of the spherically symmetric implosion of a trons are significantly heated adiabatically (provided laser corona plasma under the conditions of experi- that the radiative cooling is neglected) during the ments that were carried out in the Iskra-5 facility with implosion of a laser corona. In simulations in which all thin-wall spherical targets with an internal input of collisions are artificially “switched off,” the energy laser energy [11, 13]. We have investigated the characteristic features of plasma implosion with allowance for yield in the Fe Heα-line is lower by no more than a fac- ionÐion collisions. For a Fe-coated target with a diam- tor of about 2.5 (Fig. 6). The relatively weak sensitivity eter of about 1 mm in which about 4Ð5 kJ of the input of the electron temperature to the shape of the ion distribution function is governed to a large extent by the energy of the pulse from an iodine laser is absorbed, we quasineutrality condition, which is responsible for the have calculated the effect of the nonequilibrium charac- strong coupling between the electron and ion plasma ter of the ion velocity distribution function on the gen- components. Strong kinetic effects can also be reduced eration of the resonant Fe Heα-line radiation during by the spherical geometry of the target: during the plasma implosion. The radiative cooling of the plasma spherical implosion of the plasma flow, the plasma den- was included in the model of internal radiative losses sity increases substantially, thereby increasing the rate from the corona plasma. Our simulations show that the of ionÐion collisions. effect of the kinetics of ionÐion collisions on the generation of the resonant Fe Heα-line radiation is relatively Before carrying out simulations with the KIN-MC weak. Such a weak effect is explained by the strong code, we expected that, during the implosion of a laser integral influence of the shape of the ion distribution corona plasma, the delocalization of the energy loss of function on the electron temperature, strong coupling high-energy ions in the low-density tail of the unload- between the electron and ion plasma densities, the ing wave of a laser corona in a large plasma volume quasineutrality condition, and the functional depen- should significantly decrease the total energy yield in dence of the intensity of the resonant Fe Heα-line emis- the resonant Fe Heα-line and increase the diameter of sion in the corona approximation. Kinetic effects can the generation region. However, the above results of also be reduced by the spherical geometry of the target: kinetic and gas-dynamic simulations do not precisely during spherical implosion, the plasma density in the confirm this expectation in full measure (Fig. 6). In converging flow increases substantially, thereby Fig. 5, we can also see that the calculated diameter of increasing the ionÐion collision frequency. In the one- the region where the resonant radiation is generated is dimensional approximation, kinetic effects fail to µ about Dcalc ~ 300 m, which differs markedly from the explain the experimental data that were obtained in [13] µ experimentally measured diameter Dexp ~ 450–500 m. on X radiation spectra and on the diameter of the region Hence, in the case of an ideal spherically symmetric where the resonant Fe Heα-line radiation is generated.

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(iii) Nevertheless, taking into account the kinetic (ii) At the second substep, ionÐion collisions are taken effects of ionÐion collisions may be important when into account by the MC method: interpreting the detailed data from experiments with ∂ laser targets. First of all, this is true for the processes f 1 ------= J11 + J12, that are directly associated with the formation of the ion ∂t distribution function, such as the kinetics of fusion ∂ f 2 reactions and the formation of resonance lines in the ------= J22 + J21. radiation spectra. Our one-dimensional computations ∂t with the KIN-MC code show that the processes that (iii) At the third substep, the effect of the electron pres- have an indirect impact on the ion plasma component sure on the ions and on the electron temperature is cal- and are governed by, e.g., the electron temperature need culated from the equations not be so precisely incorporated into the detailed calcu- ∂ ∂ ∂ lation of the ion distribution functions. The radiative f 1 Z1 Pe f 1 ------∂ Ð0,------∂ ∂------= characteristics of these processes in a high-temperature t m1ne x v 1 multicharged plasma can be calculated with a reasonable accuracy in the gas-dynamic approximation using ∂ f Z ∂P ∂ f ------2 Ð0,------2 ------e ------2 = the methods of nonequilibrium radiative gas dynamics. ∂t m n ∂x ∂v Hence, we can expect that using the gas-dynamic 2 e 1 approximation is sufﬁcient to calculate the inﬂuence of ∂n T 2 ∂u ------e -e +0,---n T ------e = two-dimensional and three-dimensional effects on the ∂t 3 e e ∂x plasma implosion and the generation of X radiation in TST targets. Pe = neT e. (iv) At the fourth substep, the change in the ion velocity ACKNOWLEDGMENTS due to the frictional forces associated with plasma electrons is calculated from the equations We are grateful to V.V. Vatulin, B.A. Voinov, ∂ f ∂ f V.G. Rogachev, and A.N. Starostin for discussing this ------1 +0,ν ()u Ð u ------1 = ∂ 1u 2 1 ∂v study. This study was supported in part by the Interna- t 1 tional Science and Technology Center, project ∂ f ∂ f no. 1137. 2 ν ()2 ------∂ Ð0,2u u2 Ð u1 ∂------= t v 1

π 4 2 2 3/2 APPENDIX ν 42e Z1 Z2 Le2n1 + Le1n2 meme iu = ------2 ------------ni. 3m ne mi T e 1. Numerical Model and the KIN-MC Code e (v) At the fifth substep, ion–electron collision fre- The set of Eqs. (4) and (12) is solved numerically by quency and the related change in the electron tempera- a combination of particle-in-cell and difference meth- ture are calculated from the equations ods. The ion distribution function is modeled by an ∂ ∂ ensemble of particles whose evolution is traced by the f 1 ν T e ∆ m1 () MC method. The electron temperature is found by a ------∂ = 1e ------f 1 +,------∂------vuÐ 1 f 1 t me me v finite-difference approximation to the electron energy balance equation (4) on a grid whose spacing is appro- ∂ f T m ∂ ------2 = ν -----e-∆ f +,------2 ------()vuÐ f priately adjusted at each time step. In turn, each time ∂t 2e m 2 m ∂v 2 2 step is divided into the following six substeps. e e ∂ (i) At the first substep, the collisional ion motion and neT e m1 ν ()m2 ν () ------∂ -2= ------1en1 T 1 Ð T e + 2------2en2 T 2 Ð T e , the convective transport of the electron temperature are t me me calculated from the equations 4 2 42πe Z L n m 3/2 ν = ------i ei e ------e , ∂ f ∂ f ie 2  ------1 +0,v ------1 = 3 m T e ∂t 1 ∂x i (),, ui = ui 00. ∂ ∂ f 2 f 2 ------+0,v 1------= (vi) At the sixth substep, the electron thermal diffusiv- ∂t ∂x ity is calculated from the equations ∂ ∂ ∂ ∂ ∂T aT5/2 neT e neueT e () λ e λ e ------+0.------= ∂----- neT e ==∂------e------∂ , e ()Λ------. ∂t ∂x t x x Z + 3.44 ie

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Now, we briefly describe how each of the substeps which implies conservation of the total energy at this is implemented numerically. substep. This equation and the set of moment equations At the first substep, the electron heat transport is cal- are solved numerically by the Godunov method [26], in culated by a special procedure. First, each ion is which the required new electron velocities and pres- assigned an additional variable equal to the electron sures at the boundaries of the calculation cells are deter- temperature in the cell where the ion occurs. After the mined as follows. First, the following equations for ue ions are displaced, a new grid is constructed so as to sat- and Pe are constructed: isfy the condition that the number of ions in a cell be ∂u ∂P ∂P ∂u constant. Then, the new values of the density, mean e Z e Z e 2 e ------∂ - +0,------∂ - ==------∂ +0,nec ------∂ - velocity, and temperature of the ions and the electron t m ne x m t x temperature are calculated. In order to verify that the where convective transport of the electron temperature is modeled correctly, it is sufficient to multiply the first Z Z Z n Z Z n ---- = -----1------1 -1 + -----2------2 -2, two equations of this substep by Z1Te and the Z2Te, m m n m n respectively, and to integrate the resulting equations 1 e 2 e over ion velocities. Numerical experiments show that 2 2 Z P 2T Z c ==------e ------e ---- . adjusting the grid in such a manner appreciably reduces 3 m n 3 m statistical errors in calculating the ensemble-averaged e quantities. Then, the characteristic set of equations for these equa- The second substep is most difficult to implement tions, numerically. At this substep, Coulomb collisions are ∂u ∂ P ∂u ∂ P modeled by a specially developed version of the MC e 6 Z e e 6 Z e ------∂ - + ------∂ - +c------∂ - + ------∂ - = 0, method [24]. Specifically, the Landau operator is t ne m t x ne m x approximated by the Boltzmann collision integral, and, ∂ ∂ ∂ ∂ then, the methods of the rarefied gas dynamics [25] are ue 6 Z Pe ue 6 Z Pe ------∂ - Ð ------∂ - Ð c------∂ - Ð ------∂ - = 0, applied. The approximation error is controlled by t ne m t x ne m x appropriately choosing the dependence of the differential collision frequency on the time step. Fairly exact is used to solve the problem of the decay of the discon- results can be obtained only when the time between tinuity at the boundaries of the calculation cells. The model collisions is shorter than the time of relaxation to new values of the electron velocities and pressures are a locally Maxwellian distribution. Since this condition used to calculate the new ion velocities and the total is difficult to satisfy for cells in which the ion mean free energy and, then, to find the new electron temperature. paths are short, the rate of model collisions is chosen to The accuracy of the Godunov method is increased by satisfy the condition for a particle to experience only linearly interpolating the values of the quantities at the one collision event per time step. This condition guar- boundaries of the calculation cells from their values at antees relaxation to a locally Maxwellian distribution in the centers of the cells. a cell on each time step, so that the method used in the In order to implement the fourth substep numeri- second substep automatically starts to approximate the cally, it is also convenient to switch to the equivalent set corresponding gas-dynamic equations of motion of a of moment equations: quasineutral two-component plasma. ∂n ∂n At the third substep, the kinetic equations for the ------0,1 ==------2 0, ∂t ∂t distribution functions fi are replaced by the equivalent equations for the moments of the distribution functions: ∂n u ------1 1 = + ν ()u Ð u n , ∂n ∂n ∂t 1u 2 1 1 ------0,1 ==------2 0, ∂t ∂t ∂ n2u2 ν () ∂ ∂ ------∂ = Ð 2u u2 Ð u1 n2. n1u1 Z1n1 Pe t ------∂ +0,------∂ = t m1ne x These moment equations imply that the total mass velocity of the ions is conserved: ∂n u Z n ∂P ------2 2 +0.------2 2 ------e = ρ ρ ρ ρ ∂ ∂ 1u1 + 2u2 ==u u0, t m2ne x ρρ= + ρ . These moment equations and the equations for neTe 1 2 yield the equation These relationships give 2 2 ∂ ρ u ρ u 3 ∂u P ρ Ðνt ------1 1- ++------2 2- ---n T +0,------e -e ==ρ m n , u = u Ð -----1()u Ð u e , ∂t2 2 2 e e ∂x i i i 1 0 ρ 20 10

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ρ the nonlinear problems that are solved using a compar- 2()Ðνt u2 = u0 + ----ρ- u20 Ð u10 e , atively small number of particles. The KIN-MC code provides the possibility of a diffusive smoothing of sta- νν ν = 1u + 2u. tistical errors; moreover, the parameters of the smoothing procedure can be changed during a run of the code. When implementing the fifth substep numerically, ν the collision rate ie is calculated from the initial (at this substep) value of the electron temperature, while the 2. Comparative Calculations of Planar Flows electron temperature in front of the operator ∆f is i The accuracy of the calculations of the Landau col- assigned its final (at this substep) value. This way guar- lision integral by Monte Carlo methods was checked by antees that the energy is conserved. The final (at this Ivanov and Kochubeœ [24] for the problems of the ther- substep) electron temperature is determined as follows. malization of the nonequilibrium ion distribution func- The moment equations that are constructed for the ion tions in a homogeneous multicomponent plasma. The temperatures in each cell yield the following set of relevant Monte Carlo results were compared with the three ordinary differential equations for the ion and results obtained with the LAND code, which is based electron temperatures Ti and Te: on a completely conservative difference scheme for ∂ solving the Landau equations. Ivanov and Kochubeœ n1T 1 ν () ------∂ = 1εn1 T e Ð T 1 , [24] found that there is a good agreement between the t ion distribution function and its moments computed by ∂ different methods. They concluded that their numerical n2T 2 ν () ------∂ = 2εn2 T e Ð T 2 , scheme constructed for modeling Coulomb collisions t can also be used to calculate the spatial flows of a mul- ∂ ticomponent plasma. neT e ------= ν εn ()νT Ð T + εn ()T Ð T , ∂t 1 1 1 e 2 2 2 e The scheme developed in [24] was tested in many studies on the gas-dynamic flows of an ideal gas. 4 2 3/2 mi mi 42πe Zi Leineme Numerical modeling of the gas-dynamic problems that where ν ε = 2------ν = 2 ------. i m ie m 3 2 T admit analytic solutions (such as the problems of the e e mi e planar and spherical reflections of shock waves and The electron temperature obtained by solving this unloading waves) showed that this scheme gives cor- set of equations analytically is used for Monte Carlo rect numerical results, provided that the problems are simulations of ion–electron collisions by a Green’s solved numerically on spatial grids with relatively large function approach [27]. spacings [28]. Moreover, the results of simulations in At the sixth substep, the electron heat conduction which the ion distribution function was artificially equation is approximated by an implicit difference forced to be Maxwellian (see above) were found to scheme, in which case the electron thermal diffusivity coincide with the results obtained by modeling colli- is calculated from the temperature found at the fifth sions in the gas-dynamic approximation. This is substep. The resulting finite difference equations are because the methods implemented in the corresponding solved by the sweep method. numerical codes are, in fact, statistically analogous to the Godunov method [26]. The above model for solving the kinetic equations was implemented as the KIN-MC code. The code pro- Let us discuss the results obtained by simulating the vides the possibility of switching off the ionÐion colli- interaction of plane unloading waves in a simple sions; this version of simulations was referred to as a plasma by the KIN-MC code and, in the two-tempera- collisionless f version. Recall that, in the regions where ture gas-dynamic approximation, by the CC-9 code the ion mean free paths are short, our scheme with [21]. The calculations were carried out for the follow- model collisions results in the relaxation to a Max- ing initial parameter values: the thickness of the plasma wellian ion velocity distribution; consequently, in such corona is 40 µm (0 < x < 0.004 cm), the plasma density regions, the scheme reduces to the solution of the cor- is ρ = 0.1 g/cm3, the electron and ion temperatures are responding gas-dynamic equations of motion of a Te = Ti = 1.5 keV, the plasma flow velocity is u = 0, the quasineutral two-component plasma. With this circum- atomic weight of the Fe ions is A0 = 56, and their ion stance in mind, we arranged the KIN-MC code so as to charge number is Z0 = 26. The boundary conditions at be able to perform calculations with the ion distribution the left and right boundaries, x = 0 and x = 0.03 cm, functions that are artificially forced to be Maxwellian in assume ideal particle reflections. The radiative losses all of the calculation cells at each time step and, accord- from the plasma are neglected. These parameters are ingly, to solve the problems in the gas-dynamic approx- characteristic of the problems associated with laser tar- imation. This version of calculations was referred to as gets in which laser light is converted into X radiation a “Maxwellian” m version. (see Section 3). The simulations were carried out with- It is well known that statistical errors always occur out (version a) and with (version b) allowance for the in Monte Carlo simulations. This is primarily true of energy exchange between electrons and ions. The pro-

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Ti(r, ti), keV 200

175 1

150 2

125 3

100 4 5 75 6

0 0.010 0.014 0.018 0.022 0.026 0.030 r, cm

Fig. 7. Spatial proﬁles of the mean ion temperature Ti(r, ti) computed using the KIN-MC code (solid curves) and the CC-9 code (dotted curves) at the times ti = (1) 0.4, (2) 0.6, (3), 0.8, (4) 1.0, (5) 1.2, and (6) 1.4 ns without allowance for the energy exchange between the electron and ion subsystems.

ρ 3 (r, ti), g/cm 0.040

0.035 1 2 0.030 6 0.025 5 0.020

0.015 4

0.010 3 2 0.005 1

0 0.015 0.018 0.021 0.024 0.027 0.030 r, cm ρ Fig. 8. Spatial proﬁles of the plasma density (r, ti) computed using the KIN-MC code (solid curves) and the CC-9 code (dotted curves) at the times ti = (1) 0.3, (2) 0.4, (3) 0.5, (4) 0.6, (5) 0.7, and (6) 0.8 ns with allowance for the energy exchange between the electron and ion subsystems.

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Ti(r, ti), Tp(r, ti), keV 120

110 1 100 1 90 2 80 2 70 60 50 2 3 40 30 20 3 1 10 0 0.010 0.014 0.018 0.022 0.026 0.030 r, cm

Fig. 9. Spatial proﬁles of the mean ion temperature Ti(r, ti) (solid curves) and the transverse ion temperature Tp(r, ti) (dotted curves) at the times ti = (1) 0.2, (2) 0.3, and (3) 0.4 ns computed without allowance for the energy exchange between the electron and ion subsystems.

ρ 3 (r, ti), g/cm 0.030 0.028 1 0.026 2 0.024 0.022 3 0.020 0.018 4 0.016 0.014 0.012 0.010 2 0.008 0.006 1 0.004 0.002

0 0.006 0.012 0.018 0.024 0.030 r, cm ρ Fig. 10. Spatial proﬁles of the plasma density (r, ti) computed using the KIN-MC code (solid curves) and the CC-9 code (dotted curves) at the times ti = (1) 0.4, (2) 0.6, (3) 1.0, and (4) 1.5 ns with allowance for the energy exchange between the electron and ion subsystems.

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files of the gas-dynamic quantities calculated in version most profound difference between simulations with the a are shown in Figs. 7 and 8. CC-9 code and the KIN-MC code lies in the maximum In a multicharged (quasineutral) plasma, the ion values of the ion temperature Ti in the interaction velocity in the tail of the unloading wave is proportional between the plasma flows as well as in the width of the region where the ion temperature T is high. The ion to ~ ()Z + 1 /A (at the expense of the electron pres- i temperature T computed from simulations with the sure). In simulations, this velocity was found to be i about 0.15 cm/ns, which corresponds to the ion kinetic standard transport coefficients using the CC-9 code is energy E ~ 0.6 MeV/ion. Estimates of the mean free higher by a factor of about 1.5 to 2 and the width of the i shock front is accordingly smaller. In order to adjust the path of the ions with such energies with respect to Cou- results from gas-dynamic simulations to those from lomb collisions show that it is comparable with the kinetic simulations, we varied the ion viscosity and ion optical thickness of the expanding plasma, so that the thermal diffusivity in simulations with the CC-9 code. kinetic effects can be expected to come into play after The above results from calculations of version b were the plasma is reflected from the right boundary (i.e., in obtained for the ion viscosity and ion thermal diffusiv- the interaction of waves). Figures 7 and 8 show the pro- ity that were increased by factors of 5 and 15, respec- files of the ion temperatures and plasma density calcu- tively, in order to describe the high penetrability of an lated at the times close to the time at which the plasma anisotropic flow of fast ions in the tail of an unloading is reflected from the right boundary and the first shock wave when the waves collide. Such a “fitting” provides wave forms. The thermalization time for the ion distri- a qualitative modeling of the effects revealed in kinetic bution function in the interaction region was deter- simulations. mined from the relaxation time of the longitudinal and transverse ion temperatures and was found to be about 0.4Ð0.6 ns. As compared to the gas-dynamic simula- REFERENCES tions done with the help of the CC-9 code, which uses the standard transport coefficients, the reflected shock 1. J. D. Lindel, Phys. Plasmas 2, 3933 (1995). wave forms somewhat later and at a larger distance 2. R. C. Elton, X-Ray Lasers (Academic, Boston, 1990; from the boundary. Since, in the calculations of version a, Mir, Moscow, 1994). the ionÐelectron collisions are neglected, the ion tem- 3. R. Bock, Europhys. News 23 (5), 83 (1992). perature remains high throughout the computation 4. D. Shvarts, B. Yaakobi, P. Audebert, et al., Proc. SPIE time. It was found that, during the formation of a shock 831, 283 (1988). wave, about 50% of the kinetic energy acquired by the 5. T. Bochly, P. Audebert, D. Shvarts, et al., Proc. SPIE plasma during acceleration in the unloading wave is 831, 306 (1987). converted into the internal ion energy. 6. S. A. Bel’kov, A. V. Bessarab, A. V. Veselov, et al., The numerical results obtained in the calculations of Zh. Éksp. Teor. Fiz. 97, 834 (1990) [Sov. Phys. JETP 70, version b are illustrated in Figs. 9 and 10. 467 (1990)]. A comparison of Figs. 9 and 10 with Figs. 7 and 8 7. O. Larroche, Phys. Fluids B 5, 2816 (1993). shows that, in versions a and b, the shock waves form 8. F. Vidal, J. P. Matte, M. Casanova, and O. Larroche, in an essentially analogous fashion, thereby providing Phys. Fluids B 5, 3182 (1993). evidence that ionÐion collisions play a governing role in 9. C. Chenais-Popovics, O. Rancu, P. Renaudin, et al., the problem under investigation. The profiles of the J. Quant. Spectrosc. Radiat. Transf. 54 (1/2), 105 (1995). mean and transverse ion temperatures in Fig. 9 illustrate 10. M. E. Jones, D. Winske, S. R. Goldman, et al., Phys. the thermalization of the ion distribution function in the Plasmas 3, 1096 (1996). interaction between unloading waves. In version b, the characteristic spatial scale on which the plasma is 11. P. D. Gasparyan, V. M. Gerasimov, Yu. K. Kochubey, et al., in Proceedings of the 5th International Collo- nonequilibrium (or, in other words, the front width of µ quium on Atomic Spectra and Oscillator Strengths for the shock wave) was found to be about 100 m, the Astrophysical and Laboratory Plasmas, Meudon, 1995, time during which the plasma is essentially nonequilib- Ed. by W.-U. Tchang-Brillet, J.-F. Wyart, and C. J. Zeip- rium (i.e., the longitudinal and transverse ion tempera- pen (Publications de l’Observatoire de Paris, Meudon, tures are different) being about 0.2 ns. The develop- 1996), p. 198. ment of gas-dynamic processes in later stages is illus- 12. G. V. Dolgaleva and V. A. Zhmaœlo, Vopr. At. Nauki trated by Fig. 10. Tekh., Ser. MPMF, No. 2, 75 (1986). In the problem under consideration, an important 13. S. A. Bel’kov, A. V. Bessarab, B. A. Voinov, et al., in Pro- role is also played by the energy exchange between ions ceedings of the 5th International Zababakhin Scientific and electrons. As compared to version a, the electron Talks, Snezhinsk, 1998. temperature in the region of interaction between 14. F. Hinton, in Basic Plasma Physics, Ed. by A. A. Galeev unloading waves behind the front of the shock wave is and R. N. Sudan (Énergoatomizdat, Moscow, 1983; higher by a factor of about three. The relaxation time of North-Holland, Amsterdam, 1983), Vol. 1. Ti and Te is about 0.4 ns, which agrees with the results 15. V. M. Zhdanov, Transport Phenomena in Multicompo- of gas-dynamic simulations with the CC-9 code. The nent Plasma (Énergoatomizdat, Moscow, 1983).

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16. V. P. Bashurin and É. M. Antonenko, in Proceedings of 22. S. A. Bel’kov, P. D. Gasparian, G. V. Dolgolyova, and the Conference on Numerical Methods of Gas Dynam- Yu. K. Kochubey, J. Quant. Spectrosc. Radiat. Transf. ics, Abrau-Dyurso, 1998. 58, 471 (1997). 17. A. V. Brantov, V. Yu. Bychenkov, V. T. Tikhonchuk, and 23. S. A. Bel’kov, P. D. Gasparyan, Yu. K. Kochubeœ, and W. Rosmus, Zh. Éksp. Teor. Fiz. 110, 1301 (1996) [JETP E. I. Mitrofanov, Zh. Éksp. Teor. Fiz. 111, 496 (1997) 83, 716 (1996)]. [JETP 84, 272 (1997)]. 24. N. V. Ivanov and Yu. K. Kochubeœ, Vopr. At. Nauki Tekh., 18. A. V. Brantov, V. Yu. Bychenkov, V. T. Tikhonchuk, and Ser. Mat. Model. Fiz. Protses., No. 1, 3 (1998). W. Rosmus, Phys. Plasmas 5, 2742 (1998). 25. G. Berd, Molecular Gas Dynamics (Oxford Univ. Press, 19. L. Spitzer, Physics of Fully Ionized Gases (Interscience, Oxford, 1976; Mir, Moscow, 1981). New York, 1962). 26. S. K. Godunov, Mat. Sb. 47, 271 (1959). 20. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock 27. S. Chandrasekhar, Stochastic Problems in Physics and Waves and High-Temperature Hydrodynamic Phenom- Astronomy (AIP, New York, 1943; Inostrannaya Liter- ena (Nauka, Moscow, 1966; Academic, New York, atura, Moscow, 1947). 1966). 28. N. V. Ivanov and P. D. Gasparyan, Vopr. At. Nauki Tekh., Ser. Mat. Model. Fiz. Protses., No. 1, 17 (1997). 21. B. A. Voinov, P. D. Gasparyan, Yu. K. Kochubeœ, and V. I. Roslov, Vopr. At. Nauki Tekh., Ser. Mat. Model. Fiz. Protses., No. 2, 65 (1993). Translated by G. V. Shepekina

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PLASMA DYNAMICS

Dynamics of a Nonquasineutral Plasma in a Strong Magnetic Field A. V. Gordeev Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received September 28, 2000; in ﬁnal form, April 2, 2001

2 ӷ π 2 Abstract—The motion of a nonquasineutral plasma in a strong magnetic field such that B 4 nimic is analyzed. It is shown in simple examples that, when the plasma pressure and dissipation are neglected, the only dynamic process in a magnetized plasma is the evolution of the charge-separation electric field and the related magnetic field flux. The equations derived to describe this evolution are essentially the wave GradÐShafranov equations. The solution to these equations implies that, in a turbulent Z-pinch, a steady state can exist in which ӷ 3 the current at a supercritical level J mic /Ze is concentrated near the pinch axis. © 2001 MAIK “Nauka/Inter- periodica”.

1. Many problems in plasma theory are treated a plasma that is in equilibrium; the deviation from equi- under the quasineutrality condition librium is usually described with allowance for the inertia of the heavier particles—the ions. However, in a suf- ne = Zni, (1) ficiently strong magnetic field, the inertial motion of the ions can be neglected. If we also ignore thermal where ne and ni are the electron and ion densities and Z is the ion charge number. effects and dissipation, we can describe the ions by the hydrodynamic equation It is thought that this condition may fail to hold in the problems dealing with beams and diodes as well as dv Ze m ------i = ZeE + ------[]v × B . (2) in the problems of the interaction of high-frequency i dt c i electromagnetic waves with plasmas [1, 2]. In recent years, interest has grown in various plasma objects in For strong magnetic fields, we can estimate the deriva- strong magnetic fields, in which case the quasineutral- tive d/dt as the ratio of the speed of light c to the char- ity condition is also violated; these are, e.g., ion diodes acteristic scale length rB in order to see that ion inertia filled with magnetized electrons, electron vortices, and can be neglected under the condition laser plasmas [3Ð8]. Gott and Yurchenko published a 2 ӷ π 2 series of papers [9] in which they used an approach B 4 nimic . (3) based on the dimensionality analysis of tokamak plasmas and showed that, under the quasineutrality condi- For a hydrogen plasma density of n = 1012 cm–3, this tion, it is impossible to give a correct description of the condition corresponds to the magnetic field B ӷ 10 T. full variety of possible modes of tokamak operation; in Below, we will show that, in such magnetic fields, other words, the Debye radius should necessarily be the plasma in which the electrons and ions are both take into consideration. magnetized can exhibit evolution only if the quasineu- In investigating systems with a strong magnetic trality condition is violated. field, it is expedient to use, instead of the conventional 2. The basic set of equations used here consists of π 2 the drift equations for electrons and ions, Debye radius rD = T/4 e ne , the magnetic Debye π radius rB = B/4 ene, which is obtained from the Debye e[]× Ze[]× 0 ==Ð eE Ð -- ve B ,0 ZeE + ------vi B , (4) radius rD by replacing the temperature T with the quan- c c 2 π tity B /4 ne. The magnetic Debye radius rB introduced in such a way describes the screening of the electric and the electron and ion continuity equations, magnetic fields in a plasma under the condition B2 ӷ ∂n ∂n π ------e +0,∇ ⋅ ()n v ==------i +0,∇ ⋅ ()n v (5) 4 neT. An increase in the magnetic field causes the ∂t e e ∂t i i magnetization of the electrons and, then, of the ions, in which case the particle inertia can be neglected. There- Maxwell’s equations fore, the electromagnetic force acting on the particle is 4πe 1∂E equal to zero, so that the particles can be treated in the ∇ × B = ------()Zn v Ð n v + ------, (6) drift approximation. However, this approach is valid for c i i e e c ∂t

770 GORDEEV

1∂B ∂B E 1∂Eθ ∇ × E = Ð------, (7) ------z = -----r Ð ------. (15) c ∂t ∂r R c ∂t and Poisson’s equation Hence, under the above assumptions, the plasma ∇ ⋅ π () dynamics is completely described by Eqs. (12)Ð(15). E = 4 eZni Ð ne . (8) Note that, as was shown in [7], one of these equa- Although Poisson’s equation (8) is a consequence of tions, namely, Eq. (12), is equivalent to the induction Eqs. (5) and (6), we also include it in the basic set of equation. equations because of its importance for further analy- 3. Here, we derive the evolutionary equations for the sis. plasma in the problem as formulated. Expressing veloc- Equation (4) implies that the electron and ion velocity v from Eq. (4) in terms of Eθ and using Eqs. (13) ≡ r ities are the same, ve = vi v, so that we can introduce and (14), we can readily transform Eq. (12) into the the common total time derivative equation d ∂ ∂R ∂ ∂R ∂ ----- ≡ ----- + v ⋅ —. (9) ------()rE Ð0,------()rE = (16) dt ∂t ∂t ∂r r ∂r ∂t r First, we will consider a plasma system in which the which gives magnetic field is directed along the z-axis and the elec- rE = fR()≡ f , (17) tric field has two components: Er and Eθ. If the mag- r netic field in such a system is sufficiently strong, then where f(R) is an arbitrary function of R. Using formula the most important dynamic processes occur in the (17), we express R in terms of f and introduce the func- plane perpendicular to the magnetic field, in which tion F(f) such that case, in describing the two-dimensional plasma dynam- ∂F ics, we can neglect the dependence on the longitudinal RRf= ()≡ ------. (18) coordinate, ∂/∂z ≡ 0. ∂f Taking a curl of each of Eqs. (4) and using the cor- We substitute Eθ and B from Eqs. (13) and (14) into responding continuity equations, we obtain z Eq. (15) to obtain the following equation for Er: dI dI B B e i -----z -----z ------==------0,Ie =, Ii =. (10) ∂ 1 ∂ E 1 ∂ ∂E dt dt ne ni ----- R------()rE = -----r + ------R------r . (19) ∂ ∂ r 2 ∂ ∂ rr r R c t t We see that the Lagrangian invariants Ie and Ii are conserved along the trajectories of a particle moving with Using the identities velocity v. The initial conditions are assumed to be such ∂ ∂ ∂ ∂ that the quantities ZIe and Ii differ from one another, ()≡≡F ()F ≠ R----- rEr ------, R----- rEr ------, ZIe Ii; this assumption ensures that the quasineutrality ∂r ∂r ∂t ∂t condition fails to hold for all subsequent time. we easily arrive at the final form of Eq. (19): In the analysis to follow, it is convenient to introduce the quantity R, ∂ 1∂F 1 ∂2F f r------Ð ------= GF(), GF()≡ ------. (20) ∂rr ∂r 2 ∂ 2 ∂F/∂f 1 ≡ π Z 1 c t --- 4 e--- Ð ---- , (11) R Ii Ie This is a one-dimensional GradÐShafranov equation which has the dimensionality of length and obviously with an additional (second) term on the left-hand side. satisfies the equation With this term, the GradÐShafranov equation takes the form of an inhomogeneous wave equation. ∂ R ⋅ Combining Eqs. (13) and (18) yields F ≡ rAθ, where ------∂ +0.v —R = (12) t Aθ is the azimuthal component of the vector potential. Below, when deriving the main equations of the Of course, Eq. (20) can also be derived in a traditional approach presented here, we will restrict ourselves to way, by inserting the vector potential component Aθ cylindrically symmetric plasma states, ∂/∂θ ≡ 0, in into Maxwell’s equations and using Eq. (12) to estab- which case the basic equations describing the plasma lish a relationship between the quantities rEr and rAθ. dynamics take the form For a plasma with drifting electrons and ions, this relationship indicates the simultaneous wave dynamics of ∂ B 1 () z the magnetic flux, described by Aθ, and charge-separa- ---∂----- rEr = ----- , (13) r r R tion electric field Er. ∂ Interestingly, the approach used here implies that Eθ 1 Er ----- +0,------= (14) ӷ ω Ð1 R c ∂t rB/c pi ; in other words, the evolution of a non-

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 DYNAMICS OF A NONQUASINEUTRAL PLASMA IN A STRONG MAGNETIC FIELD 771 quasineutral plasma is much slower in comparison with Inserting this functional expression into Eqs. (25) the ion plasma oscillations. For the above parameter yields the relationship rE = Φ(A ), which puts Eq. (22) ӷ r z values, condition (3) yields rB 10 cm; consequently, in the final form the nonquasineutral plasma under consideration exhib- 2 its an evolution in the form of fairly large-scale oscilla- 1 ∂ ∂A 1 ∂ A 1 ------r------z Ð ------z = ---- Φ()A . (26) tions of the electric fields on a characteristic time scale r ∂r∂r 2 ∂ 2 2 z longer than 1 ns. In this case, fast plasma waves are sup- c t r pressed, because the magnetic field is frozen in the ions. This equation describes the dynamics of the mag- 4. Now, we derive an equation that is analogous to netic and electric fields in a plasma with a current flow- Eq. (20) and describes a plasma configuration with the ing along the z-axis. The characteristic currents whose magnetic field Bθ. This equation, which can be used for dynamics is described by Eq. (26) can be estimated investigating plasma currents far above the Alfvén ion from condition (3) in which B is replaced by the expres- ӷ ≡ 3 current, J JAi mic /Ze, will be derived using a dif- sion Bθ = 2J/rc, relating the magnetic field to the cur- ferent method, specifically, the method that was men- rent: tioned at the end of the previous section and in which all the quantities are regarded as functions of both the m c3 J ӷ J ≡ ------i . (27) coordinate r and time t. Ai Ze We turn to the z-component of Eq. (6): It is well known that, in turbulent regimes, plasma ∂ 1 ∂ 4πe 1 Ez turbulence tends to equalize some of the Lagrangian ------()rBθ = ------()Zn v Ð n v + ------, (21) r ∂r c i iz e ez c ∂t invariants [4, 10]. If the current flowing in a plasma gives rise to turbulence, then, as a result of such turbu- where ≡ ν Φ lent equalization, we have R* = const 1/ and (Az) = ∂ ∂ ν2 Az 1 Az Az, in which case Eq. (26) can be solved by separat- Bθ ==Ð------, E Ð------. ∂r z c ∂t ing the variables. Among the numerous possible solutions, we choose the solution With the electron and ion velocities deduced from Eq. (4) and with their z-components expressed as func- ∞ tions of the electric field, Eq. (21) becomes ∼ () () Az ∫ Nν rs sin cts ds, (28) 2 1 ∂ ∂A 1 ∂ A E 0 ------r------z Ð ------z = Ð------r . (22) ∂ ∂ 2 2 r r r c ∂t rR* where Nν(x) is the Neumann function. For an arbitrary value of ν, this solution is asymptotically (at ct ӷ r) ν In terms of the new Lagrangian invariants Ie* and Ii* , unsteady. However, in the important particular case = 1, the quantity R* on the left-hand side of Eq. (22) takes a the chosen solution Az has the form form similar to definition (11), ≡ < r ∼ ct > r Az 0, t -- , Az ------, t -- , (29) Bθ Bθ dI* * c 2 2 c e dIi rct()Ð r Ie* ==------, Ii* ------, ------==------0(23) rne rni dt dt which implies that, for ct ӷ r, the magnetic flux density and satisfies Eq. (12), near the plasma axis relaxes to a steady state such that A ~ 1/r. This particular case, in which we have R* = 1, 1 π Z 1 dR* z ------4==e----- Ð ----- , ------0. (24) corresponds to a plasma configuration whose character- R* * * dt Ii Ie istic scale length is on the order of the magnetic Debye radius rB. Let us estimate the magnetic Debye radius Note that the quantity R* is a dimensionless one, while π the quantity R has the dimensionality of length. rB ~ Bθ/(4 ene) for the current J = 100 MA and the 20 –3 The r-component of Eq. (6) and Eq. (8) give the fol- electron density ne = 10 cm . Using the expression that relates the magnetic field Bθ to the current, we can lowing two equations for Az and rEr: π × ∂ ∂ estimate rB as rB ~ J/2 enec , which gives rB ~ 0.5 Az ∂ Az ∂ ------+0,R*-----()rE ==------+0.R*----- ()rE (25) –2 ∂t ∂t r ∂r ∂r r 10 cm. It should be kept in mind that the drift approxima- Using Eq. (24), we can establish a relationship tion, in which expression (29) was obtained, becomes v between R* and rEr. We substitute r found from invalid near the axis and the solution has no singularity Eq. (4) into Eq. (24), express Ez and Bθ in terms of Az as r 0. The characteristic radius of this axial region and use Eq. (25) to ascertain that R satisfies an equa- 2 * is on the order of the ion Larmor radius r ~ m c /(eBθ) ~ tion that exactly coincides with Eq. (16). Consequently, Li i π 2 2 Ӷ we have R* = R*(rEr). rB4 nimic /Bθ rB.

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 772 GORDEEV

An analysis of the stability of the supercritical cur- ACKNOWLEDGMENTS rents flowing in a plasma yields the following equation δ This study was supported in part by the Russian for the perturbed radial electric field Er: Foundation for Basic Research (project no. 00-02- 16305) and INTAS (grant no. 97-0021). 1 d ρ d ------δE ρ ρ 2 ρ r d 1 + ρ d REFERENCES (30) Ω Ω2 2 1 δ 1. B. B. Kadomtsev, Collective Phenomena in a Plasma + ------+ ------Ð ----- Er = 0, (Nauka, Moscow, 1988). ()ρ2 2 ρ2 ρ2 1 + 1 + 2. Basic Plasma Physics, Ed. by A. A. Galeev and R. N. Sudan (Énergoatomizdat, Moscow, 1984; North-Holland, ρ ≡ Ω ≡ ω ω where kzr, /kzc, is the perturbation fre- Amsterdam, 1984). quency, and kz is the wave vector in the propagation 3. G. A. Askar’yan, S. V. Bulanov, F. Pegoraro, and direction of the current. A. M. Pukhov, Pis’ma Zh. Éksp. Teor. Fiz. 60, 240 (1994) [JETP Lett. 60, 251 (1994)]. In turn, an analysis of Eq. (30) shows that the inevi- 4. A. V. Gordeev, Fiz. Plazmy 23, 108 (1997) [Plasma table localization of the perturbing currents near the Phys. Rep. 23, 92 (1997)]. axis results in steady-state electric-field oscillations Ω 5. A. V. Gordeev and S. V. Levchenko, Fiz. Plazmy 25, 217 with = Ð1. This corresponds to nonzero perturbations (1999) [Plasma Phys. Rep. 25, 193 (1999)]. of the quantity R*. 6. A. V. Gordeev and S. V. Levchenko, Pis’ma Zh. Éksp. 5. The unsteady dynamics of a magnetized plasma Teor. Fiz. 67, 461 (1998) [JETP Lett. 67, 482 (1998)]. has been investigated under the assumption that the 7. A. V. Gordeev and T. V. Loseva, Pis’ma Zh. Éksp. Teor. plasma is inertialess. It is shown that, in a strong mag- Fiz. 70, 669 (1999) [JETP Lett. 70, 684 (1999)]. 2 ӷ π 2 netic field such that B 4 nimic , the only dynamic 8. T. Zh. Esirkepov, Y. Sentoku, K. Mima, et al., Pis’ma process in the plasma is the evolution of the charge-sep- Zh. Éksp. Teor. Fiz. 70, 80 (1999) [JETP Lett. 70, 82 aration electric field and the related magnetic field flux. (1999)]. 9. Yu. V. Gott and É. I. Yurchenko, Fiz. Plazmy 22, 16 The results obtained make it possible to qualitatively (1996) [Plasma Phys. Rep. 22, 13 (1996)]; Fiz. Plazmy explain the results from simulations of laser plasmas in 24, 315 (1998) [Plasma Phys. Rep. 24, 285 (1998)]; Fiz. which very intense laser radiation (~1022 W/cm2) gives Plazmy 25, 899 (1999) [Plasma Phys. Rep. 25, 827 rise to strong magnetic fields (B ~ 105 T) and the ions (1999)]. are accelerated to nearly the speed of light [11]. For 10. A. V. Gordeev, Preprint No. 5884/6 (Russian Research 21 –3 Centre Kurchatov Institute, Moscow, 1995). electron densities of about ne ~ 10 cm , the magnetic 11. S. V. Bulanov, N. M. Naumova, T. Zh. Esirkepov, et al., Debye radius in such plasmas is estimated to be rB ~ 1 µm, which corresponds to characteristic time scales Pis’ma Zh. Éksp. Teor. Fiz. 71, 593 (2000) [JETP Lett. 71, 407 (2000)]. of about τ ~ 10–14 s. An important role of the localization of the plasma current in the axial region was revealed in simulations carried by Bulanov et al. [11]. Translated by O. E. Khadin

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001

PLASMA OSCILLATIONS AND WAVES

Emission of Alfvén Waves from a Nonuniform MHD Waveguide N. G. Mazur, E. N. Fedorov, and V. A. Pilipenko Schmidt Joint Institute of Physics of the Earth, Russian Academy of Sciences, ul. Bol’shaya Gruzinskaya 10, Moscow, 123995 Russia Received February 15, 2001; in ﬁnal form, April 15, 2001

Abstract—The efficiency of the wave energy loss from a nonuniform MHD waveguide due to the conversion of the trapped magnetosonic waveguide modes into runaway Alfvén waves is estimated theoretically. It is shown that, if the waveguide parameters experience a jumplike change along the waveguide axis, the interaction between the waveguide modes and Alfvén waves occurs precisely at this “jump.” This effect is incorporated into the boundary conditions. A set of coupled integral equations with a singular kernel is derived in order to determine the transmission and reflection coefficients for the waveguide modes. The poles in the kernels of the integral operators correspond to the surface waves. When the jump in the waveguide parameters is small, analytic expressions for the frequency dependence of the transformation coefficients are obtained by using a model profile of the Alfvén velocity along the magnetic field. For the jump characterized by the small parameter value ε = 0.3, the wave-amplitude transformation coefficient can amount to 5Ð10%. Under the phase synchronization condition (when the phase velocities of the waveguide modes on both sides of the jump are the same), the wave- energy transformation coefficient is much higher: it increases from a fraction of one percent to tens of percent. The transformation of fast magnetosonic waves into Alfvén waves is resonant in character, which ensures the frequency and wavelength filtering of the emitted Alfvén perturbations. © 2001 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION plasmas. Such waveguides are able to absorb and accu- In research on controlled nuclear fusion in toka- mulate the energy of hydrodynamic perturbations. maks, auxiliary heating by RF fields at Alfvén frequen- Dense plasma sheets exist in the equatorial planes of cies [1Ð3] is successfully applied in addition to ohmic the magnetospheres of the giant planets, such as Jupiter heating. The maximum auxiliary heating power is and Saturn [11, 12]. Near the Earth, the plasma density achieved when the Alfvén mode is strongly absorbed is increased in the equatorial plane of the magneto- during the resonant excitation of a surface wave at a sphere [13] and in high-altitude cusps [14]. One of the steep gradient of the plasma parameters [4, 5]. The characteristic plasma configurations in which hydrody- highly nonuniform Alfvén velocity distribution and the namic waveguide modes can propagate is the plasma related surface waves also occur at the interfaces sheet in the Earth’s magnetotail [15, 16]. between the plasma media in astrophysical plasmas. The excitation and dissipation of surface waves may Information on the processes by which the wave play a role in the heating of the solar chromosphere [6]. energy is accumulated in the MHD waveguides near the In the Earth’s magnetosphere, surface waves at the Earth can be obtained by ground-based observations or plasmapause are observed as a sort of pulsations of the by satellite observations. This is possible because the geomagnetic field [7]. modes trapped in a waveguide are converted into The mathematical formalism for describing the con- Alfvén waves capable of propagating over large dis- version of fast magnetosonic (FMS) modes into Alfvén tances along the magnetic field lines without being waves is essentially identical to that for describing the scattered in space. In attempting to interpret the data conversion of electromagnetic waves into plasma oscil- from actual observations in terms of the conversion lations [8, 9]. The excited surface wave plays an impor- mechanism, it is necessary to estimate the wave energy tant role in the temporal evolution of the process of con- loss from an MHD waveguide due to the conversion of version of the electromagnetic wave into plasma oscil- the trapped magnetosonic waves into Alfvén waves. lations at a steep plasma density gradient [10]. Fedorov et al. [17] considered a waveguide where FMS All of the above wave processes in laboratory and waves are converted into Alfvén waves at smooth space plasmas were studied using one-dimensional plasma inhomogeneities to which the WentzelÐKram- models, which significantly simplify theoretical analy- ersÐBrillouin (WKB) approximation can be applied. sis. On the other hand, it is worth noting that the regions Here, we analyze the conversion of FMS waves into with an increased plasma density often form MHD Alfvén waves in a waveguide whose parameters change waveguides, which are frequently encountered in space along the axis in a stepwise manner.

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2. HYDROMAGNETIC WAVEGUIDE If the potentials obey the dependence ∝exp(–iωt), FOR FMS WAVES Eqs. (2) reduce to We consider an inhomogeneous plasma in a con- ϕ ()ψ2 LA ==0, —⊥ + LA 0, (3) stant magnetic field B0. We assume that, in a certain plasma layer, the plasma density is increased at the cen- ∂ 2 where LA = zz + kA is the Alfvén operator and kA = ter and decreased monotonically away from the center, ω/V is the Alfvén wavenumber. If the Alfvén velocity so that the Alfvén velocity V (z) at the center is mini- A A is independent of the transverse coordinates x and y, mum. Since there is a “well” in the Alfvén velocity dis- then Eqs. (3) for the potentials of FMS and Alfvén tribution, the layer serves as a waveguide for FMS per- waves are decoupled, so that the waves do not interact. turbations. We choose a Cartesian coordinate system (x, y, z) with the z- and x-axes directed along the magnetic Let us describe the structure of the modes that can propagate in the waveguide under consideration. To field B0 and the waveguide axis, respectively, and assume that the plasma is homogeneous along the simplify the model, we assume that kA(z) k∞ > 0 as 2 y-axis (Fig. 1). z ±∞. We introduce the function U(z) = k∞ – We neglect thermal and kinetic effects in order to 2 kA (z), which is negative and obeys the relationships describe the plasma perturbations by the ideal MHD ∞ ∞ equations. According to [17, 18], it is convenient to U(– ) = U( ) = 0. Since the plasma is homogeneous ψ ϕ in the y direction, different harmonics of the perturbed introduce the potentials and for FMS and Alfvén ∝ perturbations, respectively, in which case the electric potentials, exp(ikyy), can be treated separately. The ψ and magnetic fields, E and B, can be represented as E = equation for , the second one in Eqs. (3), takes the form EA + EM and B = BA + BM, where 2 2 ϕ []ψ× []ψ∂ Ð k ++k∞ ∂ Ð Uz() ()xz, = 0. EA ==Ð—⊥ , EM ez — , xx y zz (1) ∂ []∂× ϕ ∂ ∂ ψ ()2 ψ Since this equation is uniform in x, it has the solutions tBA ==ez — z , tBM —⊥ z Ð —⊥ ez. ψ(x, z) = exp(iκx)e(z), in which the profile e(z) along e ∂ e ∂ the z-axis is described by the following spectral prob- Here, we use the notation —⊥ = x x + y y. Substituting ν expressions (1) into Maxwell’s equations gives the lem with the parameter : equations for the potentials. []∂ () () ν2 () Ð zz + Uz eν z = eν z . (4) We start by considering a simple case in which the κ2 2 1/2 Alfvén velocity VA is uniform along the waveguide axis The wavenumber k⊥ = ( + ky ) is related to the spec- and depends only on z, V = V (z). For such a A A ν 2 2 ν2 κ2 waveguide, the potentials satisfy the equations [17] tral parameter by k⊥ = k∞ – , which yields = 2 2 2 k∞ – k – ν . ()ϕ∂ Ð2∂ y zz Ð V A tt = 0, (2) Equation (4) is a one-dimensional Schrödinger ()ψ2 ∂ Ð2∂ equation with the potential U(z), which has one mini- —⊥ + zz Ð V A tt = 0. mum, e.g., U(z = 0) = U0. If the function U(z) decreases sufficiently sharply at infinity, then the spectrum of problem (4) consists of a finite number of discrete ν 2 ν 2 eigenvalues n within the interval U0 < n < 0 and a ν2 SW continuum of eigenvalues in the positive z-axis (0, +∞). The properties of the continuous spectrum in the B spectral problem (4) that are required for further analysis are described in Appendix A. Here, we are interested in the eigenfunctions en (z) (n = 1, …, N) corresponding ν 2 to the eigenvalues n < 0, because these eigenfunctions WG describe the waveguide modes with the potentials ψ κ κ 2 2 2 ν 2 n(x, z) = exp(i nx)en(z) ( n = k∞ – ky – n ), which are trapped in the layer with a depressed Alfvén velocity. The eigenfunctions en(z) decrease exponentially ∝ SW with distance from the waveguide layer: en(z) −|ν | | | ∞ exp( n z ) for z . We assume that these eigen- Fig. 1. Geometry of a model MHD waveguide. The arrows functions satisfy the conventional normalization condi- ∞ show the propagation directions of the waveguide mode tion + e e dz = δ . (WG) and surface wave (SW). ∫Ð∞ m n mn

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 EMISSION OF ALFVÉN WAVES FROM A NONUNIFORM MHD WAVEGUIDE 775 | | µ 3. GENERAL THEORY OF THE EMISSION By x = +0 – By x = –0 = 0Iz serves to determine the field- OF SURFACE WAVES FROM AN MHD aligned surface current at the boundary. The remaining WAVEGUIDE WITH A SHARP PLASMA two conditions have the form INHOMOGENEITY E Ð0,E = In a longitudinally nonuniform waveguide such that y x = +0 y x = Ð0 V = V (x, z), the FMS waveguide modes propagating (7) A A Bz Ð0.Bz = along the waveguide axis can generate Alfvén waves. In x = +0 x = Ð0 this case, instead of the decoupled equations (2) for the In order to transform these relationships into the bound- potentials, we are faced with a far more complicated set ary conditions for the potential ψ, we turn to Eqs. (1), of coupled equations [17, 18]: which yield the following expressions for the compo- 2 2 nents Ey and Bz of the electromagnetic field of FMS —⊥()∂ + k —⊥ϕ = ()—⊥k ⋅ []ψe × — , zz A A z ∂ ψ ω 2 ψ waves: Ey = x and i Bz = —⊥ . We substitute these ()2 2 ψ ()2 ⋅ []ϕ× —⊥ —⊥ + kA —⊥ = Ð —⊥kA ez — . expressions into conditions (7) and take into account Eqs. (5) to obtain From these equations, we see that FMS waves (with the ∝ψ potential ) can generate Alfvén perturbations (with ∂ ψÐ ∂ ψ+ ∝ϕ x x = Ð0 = x x = +0, the potential ) if the Alfvén velocity VA changes (8) along the waveguide axis. Here, we are interested in the ()ψL Ð k 2 Ð = ()ψL Ð k 2 + . situation when the Alfvén velocity changes sharply Ð Ð x = Ð0 + + x = +0 over the wavelength of FMS waves, in which case the Now, we should specify the wave process that serves emitted Alfvén waves are similar in structure to the sur- as an energy source for an Alfvén wave generated at the face waves. Let the Alfvén velocity change in a step- boundary between the semi-infinite homogeneous Ð + wise manner from V A (z) to V A (z) at x = 0: plasma media. As was shown in Section 2, the equations for a medium that is homogeneous in the x direc- V Ð ()z for x < 0 tion have solutions describing the waveguide modes. (), A V A xz =  An FMS perturbation propagating along the waveguide  + () > V A z for x 0. is generally a superposition of all possible waveguide modes; for our purposes here, it is sufficient to consider In each of the half-spaces x < 0 and x > 0, the Alfvén only one of them, specifically, the mode that is incident velocity is uniform in x. Consequently, the electromag- on the boundary, e.g., from the left (Fig. 1). At the netic field on both sides of the boundary x = 0 can be boundary, this mode is partly reflected back into the described by simple equations (3). Since FMS waves original medium and partly transmitted into the second interact with Alfvén perturbations only at the boundary medium. Also, this mode may excite other waveguide between the half-spaces, the coupling between Eqs. (3) modes, which propagate away from the boundary in can be incorporated into the boundary conditions. both media. However, since we are interested in wave Hence, the potential ψ satisfies the conditions generation along the boundary between two homoge- ψψÐ ()ψ2 2 Ð < neous plasma media (in the direction of B0), we restrict = , —⊥ + kÐ Ð LÐ = 0 for x 0, (5) ourselves, for simplicity, to analyzing the situation in + 2 2 + which only one waveguide mode exists on both sides of ψψ= , ()ψ—⊥ + k Ð L = 0 for x > 0, + + the boundary. − ∂ 2 + 2 where L+− = – zz + U+− (z), U+− (z) = k+− – (kA ) , and In order to describe the wave process in question, we − − + −∞ + ∞ solve the boundary-value problem (5) with the bound- kA ( ) = kA (+ ) = k+− . ary conditions (8). According to the method for separat- The electromagnetic field components tangential to ing the Fourier variables, we expand the desired solu- the boundary between the half-spaces, namely, the tion in the spectra of the operators L+− : components Eτ and Bτ, are related by the conditions ψÐ()xz, = []exp()iκ Ð x + R exp()Ðiκ Ð x e Ð()z Eτ x = +0 Ð0,Eτ x = Ð0 = 1 1 1 1 (6) +∞ Bτ Ð Bτ = µ Ie× , x = +0 x = Ð0 0 x ()κ Ð Ð()ν + ∫ Rν exp Ði ν x eν z d , I where is the surface current flowing along the bound- ∞ ary. In what follows, we need only two of the four con- Ð (9) ψ+(), ()κ + +() ditions (6) imposed on the field components. In fact, xz = D1 exp i 1 x e1 z one of the conditions, specifically, the condition that the +∞ longitudinal component of the electric field be continu- ()κ + +()ν ous, holds automatically, because, in ideal magnetohy- + ∫ Dν exp i ν x eν z d . drodynamics, we have Ez = E|| = 0. The relationship Ð∞

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− − − κ + 2 2 2 ν + 2 κ + 2 2 2 ν2 Here, ()1 = k+− – ky – (1 ) , (ν ) = k+− – ky – ; found a solution to this equation, we can determine the remaining coefficients R , D , and Rν, because R1 and D1 are the reflection and transmission coeffi- 1 1 cients for the waveguide mode, respectively; and the Eqs. (11)Ð(13) relate them to Dν. analogous coefficients Rν and Dν refer to the contin- ()0 It is important to note that the quantities Dν , K , uum. The functions en(z) of the discontinuous spectrum D of the operator L were described in Section 2, and the ()0 Rν , and K contain the resonant denominator continuum is discussed in detail in Appendix A. R ψ We substitute expressions (9) for the potential into ∆ν() λ+κ Ð λ Ðκ + conditions (8) at the boundary x = 0 to obtain the rela- = ν ν + ν ν tionships ()2 ν2 2 2 ν2 ()2 ν2 2 2 ν2 = k+ Ð kÐ Ð ky Ð + kÐ Ð k+ Ð ky Ð, +∞ κ Ð()Ð() κÐ Ð ()µ 1 1 Ð R1 e1 z Ð ∫ µ Rµeµ z d which is the dispersion function of the surface wave ν ±ν generated at the boundary x = 0. The roots = s of the Ð∞ ∆ ν +∞ equation ( ) = 0 are the wavenumbers of this wave, in which case the values of |ν | lie between k and k . The κ + +() κ+ +()µ s – + = 1 D1e1 z + ∫ µ Dµeµ z d , representation of the wave field in the form of Eq. (12) Ð∞ was obtained by expanding the solution in the discrete ∞ (10) + and continuous spectra of the transverse operators L+− . λ Ð()Ð() λÐ Ð ()µ 1 1 + R1 e1 z + ∫ µ Rµeµ z d The surface perturbations are described in terms of the potentials ψ and ψ of the wave fields that are similar Ð∞ – + +∞ in structure to FMS perturbations in each half-space. However, since these potentials obey different condi- λ + +() λ+ +()µ = 1 D1e1 z + ∫ µ Dµeµ z d , tions at the boundary x = 0, it is impossible to introduce Ð∞ a unified potential ψ, which would be continuous over the entire space. This indicates that the potentials ψ +− 2 +− +− 2 – λ − ν 2 λ − ν2 ψ where 1 = k+ – (1 ) and ν = k+ – . To simplify and + describe the coupled FMS and Alfvén perturba- ψ ψ | 2 tions. The discontinuity ( + – –) x = 0 corresponds to the formulas, we use the notation k⊥ = λ with the cor- the field-aligned surface current I of a surface wave responding indices. z propagating along the jump in the Alfvén velocity, i.e., Taking the scalar product of relationships (10) with along the boundary x = 0. The field of the surface wave Ð Ð e and eν and carrying out simple but rather laborious will be explicitly calculated below by closing the inte- 1 gration contour in Eq. (12) in the plane of the complex manipulations, which are presented in Appendix B, we ν arrive at the final set of equations for the coefficients R , variable . As a result, the field will be represented as 1 the sum of the residues at the poles of the integrands D1, Rν, and Dν: ±ν (i.e., at the points s) and the integrals along contours +∞ along both sides of the cuts drawn from the branch ()0 points. The integrals along both sides of the cuts R = R + F ()ν Dνdν, (11) 1 1 ∫ R describe FMS waves propagating away from the Ð∞ boundary in both media, while the residues describe the +∞ surface waves that are localized in x and represent the ()0 coupled FMS and Alfvén perturbations. We emphasize D = D + F ()ν Dνdν, (12) 1 1 ∫ D that the surface waves experience no reflection and Ð∞ propagate away from the boundary in both directions ∞ along the z-axis. ()0 ()νµ, µ Rν = Rν + ∫ K R Dµd , (13) The kernels KD and KR possess an important prop- Ð∞ erty: each of them can be represented as a sum of the ∞ degenerate component (the product of the functions of ν and µ) and the nondegenerate component, which ()0 ()νµ, µ Dν = Dν + ∫ K D Dµd . (14) determines the singular operator with a Hilbert kernel: ∞ Ð ()νµ, ()νµ, 1 ()0 Φµ() CD The free terms and the kernels of these equations are K D = Ð ---Dν + ------2 2 , represented by formulas (B.2), (B.3), (B.8), and (B.9) 2 µ Ð ν in Appendix B. We direct attention to the general struc- (15) 1 ()0 C ()νµ, ture of the equations derived: Eq. (14) is an integral ()νµ, Φµ() R K R = Ð ---Rν + ------2 2-, equation of the second kind for the function Dν. Having 2 µ Ð ν

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kyL (a) (b) 2.5 + ω ky = k⊥( ) 2.0 + ω ky = k⊥( ) 1.5 – ω ky = k⊥( ) – ω 1.0 ky = k⊥( ) 0.5

0 0.5 1.0 1.5 0 0.5 1.0 1.5 ω L/VA

Fig. 2. Phase diagrams and synchronization conditions of the waveguide modes. where 2 2 ν 2 mode, e.g., the mode with k⊥ = k∞ – 1 . In the x direc- ω + + tion, the waveguide can transmit modes with ky < k⊥( ). λµ κµ + Ð Φµ()= ------+ ------()eµ, e , (16) Recall that the waveguide under investigation consists λ Ð κ Ð 1 1 1 of two homogeneous plasma media, each with its own dependence of k⊥ on ω. Depending on the relative ()0 ()0 shapes of the longitudinal profiles of V (z) on both and the quantities Dν , Rν , CD, and CR are described A by fairly involved explicit expressions (B.9) and (B.8) sides of the boundary between the media, the bound- Ð ω given in Appendix B. aries of the transmission bands, ky = k⊥ ( ) and ky = The efficiency for conversion of the waveguide + k⊥ (ω), either do not intersect (Fig. 2a) or intersect at a mode into a surface wave can be characterized by the ω quantities certain point ( ∗, k∗) (Fig. 2b). The latter case is of par- ω ω ∞ ticular interest because, at the frequency = ∗, the () () KS= []SW ()x, +∞ Ð S SW ()x, Ð∞ wavenumbers of the waveguide mode in both media ∫ z z coincide, which indicates synchronization of the ∞ Ð (17) entrance and exit channels of the waveguide. This situ- ∞ Ð1 ation can take place only under the following condition: × ()WG () the difference of the background Alfvén velocities far dx∫ Sx z dz + ±∞ Ð ±∞ Ð∞ from the waveguide, V A ( ) – V A ( ), and that at the + Ð and waveguide center, V A (0) – V A (0), should have oppo- ()SW ()WG Ð1 site signs (Fig. 3). If these differences have the same b± = max B± ()max B , (18) signs, then synchronization of the entrance and exit channels of the waveguide is impossible. which will be referred to as the wave-energy and wave- magnetic-field conversion coefficients, respectively. Here, S(WG) and B(WG) are, respectively, the averaged 4. PERTURBATION THEORY FOR A WEAKLY µ –1 × Poynting vector S = (2 0) Re[E B ] and the mag- NONUNIFORM WAVEGUIDE netic induction vector of the incident waveguide mode (SW) (SW) In a general formulation, the problem is very diffi- and S and B are the same vectors of the surface cult to solve mathematically. In order to give insight wave far from the waveguide axis. into the characteristic features of the wave process, we It is important to note that the very formulation of consider the limiting case of a small jump in the Alfvén the problem implies the existence of the waveguide velocity at the boundary between two homogeneous ψ(WG) Ð κ Ð plasma media: mode with the potential = e1 (z)exp(i 1 x ), which propagates in the medium to the left from the jump in + ω V A the Alfvén velocity. In other words, the parameters ------Ð 1 Ӷ 1. and k should belong to the transmission band of the Ð y V A entrance channel of the waveguide. In a uniform waveguide, the wavenumber is a function of frequency, In this case, we can derive explicit expressions for all of ω k⊥ = k⊥( ). Here, we speak of a particular waveguide the quantities of interest to us. We set U+(z) – U–(z) =

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V + Ð –A the relationship (eµ , e ) = O(ε), is nonuniform as 1.01 1 κ 0, because the function Φ(µ) contains the term proportional to ∝κ–1. 0.99 Hence, although the degenerate components of the ε2 0.97 kernels KD and KR are on the order of , they should be taken into account in obtaining uniform asymptotic 0.95 estimates, whereas the nondegenerate components can be neglected. The leading-order terms of the uniform 0.93 asymptotic estimates can be found from the following set of equations with nondegenerate kernels: 0.91 +∞ ()0 1 ()0 Dν = Dν Ð --- Dν Φµ()Dµdµ, 0.89 2 ∫ Ð∞ –3 –2 –1 0 1 2 3 ∞ (19) Z/L + ()0 1 ()0 Rν = Rν Ð --- Rν Φµ()Dµdµ. 2 ∫ Fig. 3. Model profile of the Alfvén velocity along the mag- ∞ netic field. Ð These equations have the solution ε 2 2 ε ε ()0 ()0 h(z) and k+ – kÐ = d, where is a small parameter. Dν ==ADν , Rν ARν , The function εh(z) can be regarded as a perturbation of ∞ + Ð1 (20) the operator L– across the boundary x = 0; i.e., we can 1 Φν() ()0 ν ε A = 1 + --- ∫ Dν d . set L+ = L– + h(z). The corrections to the eigenvalues 2 and eigenfunctions of the discrete spectra can be found Ð∞ using the standard Schrödinger perturbation theory ε [19]; the corresponding estimates for the continuum are By exploiting the smallness of , we can significantly ()0 presented in Appendix A. simplify expressions (B.8), (B.9), and (16) for Dν , ψ(SW) ()0 The potential (x, z) of the surface wave far from Rν , and Φ(ν). When analyzing the situation near the the waveguide axis can be calculated from the coeffi- boundary of the transmission band, we can also replace cients Dν and Rν, which should be found in advance by the quantities that remain finite as κ 0 by their val- solving Eqs. (14) and (13), and from the asymptotic ues at κ = 0. As a result, we obtain estimates (for z ±∞) of the integrals over the continuum in expansion (9). In this case, the perturbation ε 2 Ð ()0 ()0 C0k⊥ hν of the original waveguide mode is determined by the Dν ==ÐRν ------, ()νν2 2 ()2 ν 2 coefficients D1 and R1. 2 + a Ð s (21) It follows from further analysis that the efficiency Ð iεhν with which the waveguide mode is converted into a sur- Φν()= Ð------, face wave is the highest near the boundary of the trans- κν2 + a2 mission band of the entrance channel of the waveguide, κ Ð ω where we introduced the notation i.e., for 1 ( , ky) 0. Consequently, it is necessary to obtain uniform asymptotic estimates for Dν and Rν Ð ()Ð, Ð 2 ()ν Ð 2 > 2 ≡ λ Ð 2 2 that are valid for ε 0 up to the boundary of the hν = eν he1 , a = Ð0,1 k⊥ 1 = kÐ + a , transmission band. In order to simplify matters, we Ð1 κÐ2ε ()λ + λ Ð ε ε2 κ Ð κ C0 = 2() 1+ 1 + P1 1 Ð 1 = P1 + O(). denote 1 by .

()0 ()0 Using expressions (21), we find the coefficient A in For ε 0 at fixed κ, the free terms Dν and Rν solution (20): in Eqs. (14) and (13) are first-order quantities. The non- ∞ degenerate components of the kernels K and K , i.e., 2 2 + Ð 2 Ð1 D R iε C k⊥ hν dν the second terms on the right-hand sides of representa- A =  1 Ð ------0 - ∫ ------. (22) ε 4κ ()ν2 2 3/2[]ν2 ()2 tions (15), are also proportional to ; moreover, the cor- Ð∞ + a Ð kÐ + i0 responding proportionality coefficients are uniform in the parameter κ. On the other hand, the estimate Φ(µ) = Now, we estimate the integrals over the continuum O(ε), which follows from expression (16) at fixed κ and in expressions (9), restricting the analysis to the wave

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ν 2 2 ﬁeld far from the waveguide axis. For example, we con- Here, we take into account the fact that s kÐ as sider ε ν 2 0; recall that the value of s is intermediate ∞ between the values of k 2 and k 2 , and that k 2 – k 2 = ψ Ð (), ()κ Ð Ð()ν Ð + + Ð cont xz = ∫ Rν exp Ði ν x eν z d . (23) εd 0. In the limit z –∞, we arrive at analogous Ð∞ formulas:

∞ +− επ To be specific, we analyze the limit z +, in which ψ ()xz, = ±------AC ()γ Ð h Ð + δ Ð h Ð cont 0 kÐ kÐ kÐ kÐ we take into account expressions (A.5) and (A.6) to 2ikÐ (26) obtain × () exp Ð ky x Ð ikÐz . ∞ Using expressions (25) and (26) for the potentials of ψ Ð (), αÐ ()νκ Ð ν cont xz = ∫ ν Rν exp Ð i ν x + i z d the surface wave, we can calculate the conversion coef- Ð∞ ficients (17) and (18): ∞ (24) 2 2 2 k β Ð ()νκ Ð ν K = ε C A ------y -()q + q , + ∫ ν Rν exp Ð i ν x Ð i z d . 0 κ Ð 2 + Ð kÐ 1 k⊥ Ð∞ (27) k 2 In order to take the integrals on the right-hand side of 2 ε2 2 2 y b± = C0 A ------5 -q±, expression (24), we close the integration contours for k⊥ max wz() the first and second integrals through the upper and lower half-planes of the plane of the complex variable ν, where respectively. The integrands have poles at the points π2 2 ν ± ν α Ð Ð β Ð Ð = ( s + i0) (we displace the poles from the real axis q+ = ----- k hk + k hk , by infinitely small distances in order to take into 2 Ð Ð Ð Ð account an arbitrarily small dissipation and thus to π2 2 obtain a physically correct result). The calculation of q = ----- γ Ð h Ð + δ Ð h Ð , the integrals in expression (24) reduces to the calcula- Ð 2 kÐ kÐ kÐ kÐ tion of the residues at the indicated poles and integra- () Ð3()∂ Ð 2 Ð1()Ð 2 tion along contours along both sides of the cuts, which wz = k⊥ ze1 + k⊥ e1 . are necessary in order to deal with the single-valued α β γ δ branch of the integrand. For the first integral in expres- The coefficients ν, ν, ν, and ν are given by formulas sion (24), the cut is drawn from the branch point ν = (A.6) in Appendix A; the minus superscript indicates 2 2 that these coefficients are calculated from the “unper- iky Ð kÐ upward along the imaginary axis around the turbed” potential U–(z). Note that, for a symmetric lon- pole ν = ia, in which case, when considering the region gitudinal profile VA(z), we obviously have q+ = q–. adjacent to the boundary of the transmission band, we can restrict ourselves to the inequality ky > k–; i.e., we Ð 5. COEFFICIENTS FOR CONVERSION can assume that the branch point of the function κν lies OF THE WAVEGUIDE MODE INTO A SURFACE on the imaginary axis. For the second integral in ALFVÉN WAVE expression (24), the cut is drawn in a similar way— Let us consider in more detail the structure of expres- ν 2 2 sions (27) for the conversion coefficients K(ω, κ, ε) from the branch point = –iky Ð kÐ downward along and b(ω, κ, ε). The factors q± depend only on the fre- the imaginary axis. quency ω; consequently, when studying the behavior of For z +∞, the integrals along contours along K and b near the cutoff frequency, they can be regarded both sides of the cuts are negligibly small. Hence, we as being constant. The calculations carried out in Sec- are left with the problem of calculating the residues at tion 6 for a particular example show that each of the + functions q±(ω) vanishes in both limits ω 0 and the points ν = ±(ν + i0). The integral ψ (x, z) = s cont ω ∞ and has one or several maxima. ∞ + + Dν exp(iκν xe )ν (z)dν can be treated in a similar ∫Ð∞ When estimating different factors in expressions way. As a result, we obtain (27), we normalize them to the characteristic transverse scale length L of the waveguide layer. In order of mag- 3 +− επ nitude, the factor q±L is fairly small: it is about 0.1 or ψ ()xz, = ±------AC ()α Ð h Ð + β Ð h Ð cont 0 kÐ kÐ kÐ kÐ even smaller. Because of the smallness of the quantity 2ikÐ (25) ε2, the coefficient K, which contains the small denomi- × () exp Ð ky x + ikÐz . nator κL, is essentially nonzero only near the boundary

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 780 MAZUR et al. of the transmission band. As for the coefficient b, it 6. NUMERICAL ESTIMATES 3 FOR A MODEL WAVEGUIDE obeys the relationship b ∝ ε q± L , so that the region where it is not too small is wider. Let the longitudinal profile VA(z) in the region x < 0 of the incident waveguide mode be specified as Ð1 Ð2 3 –2 2 2 2 2 Ð1 Unlike the factor q±L , the factors ky kÐ k⊥ L and () []γ () V A z = V Ð 1 + /cosh z/L , 2 Ð5 Ð3 ky k⊥ L /maxw(z) in expressions (27) are easy to esti- where L is the characteristic thickness of the plasma mate analytically. They remain finite as κ k 0 ( y = layer. This shape of the profile VA(z) (see Fig. 2) is cho- 2 2 –1 sen in such a way that the potential U(z) in the k⊥ Ð κ . For moderate frequencies, ω V L , they ) ~ A Schrödinger equation (4) admits an explicit solution of are typically on the order of unity. Finally, we turn to the eigenvalue problem [19]. The parameter γ charac- ε2 2 | |2 the factor F = C0 A in expressions (27). This factor terizes the relative depth of the well of the profile VA(z): 2 γ2 γ2 depends in a fairly complicated fashion on the three 1 – min(VA/V–) = /(1 + ). In calculations, we nor- parameters ω, κ, and ε and plays an important role malize the frequency ω to the background Alfvén fre- (especially in combination with the factor κ in the denom- quency, thus introducing the dimensionless parameter Ω ω inator of K) in determining the behavior of K and b near = L/V–. After renormalization of the linear scale the boundary of the transmission band (as κ 0). length, ζ = z/L, the Schrödinger equation (4) becomes κ An analysis of the dependence of F on shows that the 2 2 Ð2 2 Ð κ 2 | κ |2 []∂ζζ ++γ Ω ()coshζ ()νL E()ν ()ζ = 0, functions C0( ) and A( ) increase monotonically L κ κ ∞ from zero at = 0 to unity for . The transition where the eigenfunctions are related to the original occurs near the boundary of the transmission band, in a Ð Ð ζ Ð ζ narrow region whose width depends on the value of the eigenfunctions eν by E()νL ( ) = eν (L ) for the contin- small parameter ε. The quantity C is a function of the Ð Ð 0 uum and by E (ζ) = Le (Lζ) for the discrete spec- dimensionless combination ξ = κ(εP )–1/2: C = 2ξ(ξ + 1 1 1 0 trum. In this waveguide model, the eigenvalue of the ξ2 + 1 )–1. We can see that, for κ 0, the function main state (the main waveguide mode) and the corre- Ð 2 ν 2 −τ2 κ2 κ sponding eigenfunction are represented as ()1 L = , C0 is proportional to and thus cancels the factor in κ Ð 1 2 2 the denominator in K( ), even without recourse to the and E (ζ) = N()coshζ –τ, where τ = --- (14+ γ Ω – 1) function |A|2. As a result, the conversion coefficient 1 2 K(κ) decreases to zero as κ 0, and, near the bound- and the normalizing coefficient N is defined according ary of the transmission band, it has a maximum on the ∞ 2 ζ to the condition ∫ ∞ E1 d = 1 and is expressed in order of ε3/2 P Ð1/2 L–1 at κ ~ εP . Ð 1 1 terms of the Γ function as N2 = π−1/2Γ(τ + 1/2)/Γ(τ). Note that, at the zero argument, the slope of the For numerical calculations, we normalize expres- κ function C0( ) increases without bound as P1 0. sions (27) to the linear scale length L in such a way that κ they represent the dependence of K and b on the dimen- For P1 = 0, the function C0( ) is identically unity. Con- ω sionless parameters Ω, γ, and k L. sequently, if the function P1( ) vanishes at a certain y value ω = ω∗ (in Section 3, this situation was called the Figure 4 shows the conversion coefficients K and b Ω synchronization of the entrance and exit channels of the calculated as functions of the frequency for different waveguide), then an increase in K(κ) as κ 0 is now values of the parameter kyL in the case when it is impos- | κ |2 ≡ sible to synchronize the wave parameters of the restricted by the factor A( ) . For C0 1, this factor is η κ ε2 entrance and exit channels of the waveguide. This cal- a function of the combination = L/ : culation was carried out for the model parameter values 2 2 2 2 Ð1 2 γ = 1, γ /γ = 1.383, and V /V = 0.940, which cor- α + iβ 2 η – + Ð Ð + A ==1 + ------, A ------. respond to ε = 0.3. The conversion coefficients are seen η ()ηα+ 2 + β2 to be very sensitive functions of the frequency. As ky ω ω α ω β ω decreases, the peaks in the profiles K( ) and b( ) Here, the meaning of the quantities ( ) and ( ) can become lower. As k increases, the peaks are displaced be understood by reference to expression (22). Conse- y ω ω κ to the right. The maximum values of the magnetic-field quently, for = ∗, the conversion coefficient K( ) conversion coefficient are about b(ω) Ӎ 2Ð4%, and the reaches its maximum at κL ~ ε2, the maximum value maximum values of the energy conversion coefficient being independent of ε!. As ω approaches ω∗, the amount to at most fractions of one percent. ε3/2 Ð1/2 –1 When synchronization of the wave parameters of above case with maxK ~ P1 L transforms grad- the entrance and exit channels of the waveguide is pos- 2 ω ε3 ually (for L P1( ) ~ ) to the case with maxK ~ 1. sible (Fig. 5), the conversion efficiency is significantly

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 EMISSION OF ALFVÉN WAVES FROM A NONUNIFORM MHD WAVEGUIDE 781

(a) K K (a) 0.004 0.25 0.365 1.0 0.20 0.003 0.75 0.15 0.36 0.002 0.5 0.10 0.33 0.001 0.05 0.30 0.37 0.25 0.40 0 0 b (b) b 0.33 0.10 0.30 0.36 0.05 (b) 0.365 1.0 0.08 0.04 0.75 0.5 0.06 0.40 0.03 0.25 0.04 0.02 0.02 0.37 0.01 0 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0 0.5 1.0 1.5 ω / ω L V– L/V– Fig. 4. Dependence of the (a) wave-energy and (b) wave- magnetic-field conversion coefficients on the dimensionless frequency in a model waveguide without phase synchroni- Fig. 5. Profiles of the (a) wave-energy and (b) wave-mag- zation. Numerals above the curves denote different values netic-field conversion coefficients in a model waveguide of the parameter kyL. with phase synchronization for kyL values close to k∗L. higher. The maximum values of K(ω) amount to several 7. SUMMARY AND CONCLUSIONS tens of percent; however, this is true of ky values in a We have shown that the plasma layer can serve as a ω narrow region near the value k∗ = k⊥( ∗). In this case, waveguide for FMS waves, which are insignificantly the coefficient b is also somewhat higher and the damped when propagating along the layer. The FMS decrease in the maxima of b(ω) as k decreases is far waves are converted into localized Alfvén waves at y small but sharp plasma inhomogeneities along the more gradual. waveguide axis. The conversion efficiency increases with the parameter k L and becomes maximum when The case ky < k∗ can be described as follows. As the y the parameter reaches values corresponding approxi- frequency decreases, the exit channel of the waveguide mately to the cutoff frequency of the waveguide mode, stops transmitting waves before the cutoff of the ω Ӎ kyVA. Note that, in the one-dimensional case, the entrance channel. The peak in the profile b(ω) is absorption of the energy of the incident wave at a steep ρ ӷ ρ approximately two times higher than that in the case plasma density gradient ( + –) is enhanced near the ky > k∗ and is far more pronounced. The peak in the pro- ω Ӎ + frequency s 2kzV A of the standing surface wave file K(ω) also becomes markedly higher. These effects [7]. On the whole, in a plasma that is inhomogeneous in are associated with the total reflection of the waveguide two directions and in which surface waves can escape mode; as a result, the amplitudes of the waves in the along the boundary between two different plasma conversion region double. The profiles in Fig. 5 were media, the efficiencies for conversion and absorption of obtained for the model parameter values γ = 1, the energy of the incident waveguide mode are lower – than those in a plasma that is inhomogeneous in one γ 2 γ 2 2 2 + /Ð = 0.438, and V Ð /V + = 1.600, which also corre- direction and where there are regimes in which the inci- spond to ε = 0.3. The conversion process acts as a kind dent wave can be absorbed essentially completely. of band-pass filter that permits only narrow-band The resonant character of the conversion of FMS Alfvén oscillations to reach the waveguide output. waves into surface Alfvén waves is responsible for fre-

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 782 MAZUR et al. quency and wavelength filtering of the emitted Alfvén Consequently, since the coefficients of this equation are perturbations. The phase synchronization condition can real, conditions (A.1) yield take place only for non-self-similar jumps in the profile () () VA(z). Phase synchronism takes place in the situation in uÐν z = uν z , (Ä.2) which the phase velocities of the waveguide mode with frequency ω∗ along the waveguide axis are the same on where the overbar denotes the complex conjugate. both sides of the jump in the Alfvén velocity. Under the The corresponding manipulations (which are omit- phase synchronization condition, the conversion effi- ted here for brevity) show that ciency is high: even in the case of a slight jump, it ∞ amounts to about ten percent. () () πδµ[]() ν δµ() ν ∫ uµ z uν z dz = 2 Ð + rµ + . Ð∞ ACKNOWLEDGMENTS We see that the functions uν(z) are not orthonormal. We are grateful to the participants of the seminar Consequently, in place of these functions, it is expedi- chaired by A.P. Kropotkin for fruitful discussions. This ent to use the functions eν(z) satisfying the conditions study was supported by the Russian Foundation for Basic Research, project no. 01-05-64710. ∞ () () δµ() ν ∫ eµ z eν z dz = Ð , (Ä.3‡) APPENDIX A Ð∞

Eigenfunctions of the Continuous Spectrum eν()z = e ν()z . (Ä.3b) of the Schrödinger Equation Ð We represent the electromagnetic fields in a Here, the latter condition is analogous to condition waveguide as a superposition of the fields of normal (A.2) and makes it possible to unambiguously deter- waves propagating along the waveguide axis. To do mine the functions eν(z). Each of the functions eν(z), this, we solve the equation for ψ in set (3) by the being a solution to Eq. (4), can be represented as a lin- method of expansion in the spectrum of the Alfvén ear combination of the functions uν(z) and u–ν(z) with the coefficients determined from conditions (A.3a) and ∂ 2 operator LA = zz + kA (z). In Section 2, we somewhat (A.3b): modified the problem at hand; specifically, we switched 2 1 []()1/2 ()Ð1/2 from the operator LA to the operator L = –LA + k∞ = eν = ------1 + dν uν Ð rν 1 + dν uÐν .(Ä.4) −∂ 2 πdν zz + U(z) in order to deal with the standard one- dimensional Schrödinger equation (4). In Section 2, we The coefficients in the asymptotic expressions also discussed the properties of the eigenfunctions of the discrete spectrum of the operator L in connection eν()z with the waveguide modes. Here, we analyze in detail the eigenfunctions of the continuum. αν exp()iνz + βν exp()Ðiνz for z +∞ (Ä.5) Ӎ  The continuous spectrum of Eq. (4) is doubly γ ν exp()iνz + δν exp()Ðiνz for z Ð∞ degenerate. Accordingly, we choose two linearly independent solutions to Eq. (4) in the following way. As ν2 are found by comparing expressions (A.1) and (A.4): runs the positive axis ν2 > 0, each of the two values of ν ν runs the entire real axis (except for the point = 0). α 1 ()1/2 ν ν = ------1 + dν , At a given value of , we single out the solution uν(z) to 2 π Eq. (4) by imposing the conditions β 1 ()Ð1()Ð1/2 ≈ ()ν ()∞ ν = Ð------rνdν dν 1 + dν , uν dν exp i z z + , π (Ä.1) 2 ≈ ()ν ()ν ()∞ (Ä.6) uν exp i z + rν exp Ði z z Ð . 1 Ð1 1/2 γ ν = ------dν ()dν ()1 + dν , If ν > 0, then the quantities rν and dν in conditions 2 π

(A.1) have the meaning of the reflection and transmis- 1 Ð1 Ð1/2 sion coefficients of the potential well U(z) for a plane δν = ------rν dν ()dν ()1 + dν . π wave. We can assume that the coefficients rν and dν for 2 a particular potential are known functions, because they Now, we consider two operators, L = –∂ u + U (z) can be obtained by integrating Eq. (4) numerically. + zz + ∂ + Ð Note that the functions uν(z) and u–ν(z) are solutions to and L– = – zzu + U–(z). We denote by eν (z) and eν (z) the differential equation (4) with the same parameter ν2. the solutions to the differential equation (4) with the

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 EMISSION OF ALFVÉN WAVES FROM A NONUNIFORM MHD WAVEGUIDE 783 potentials U+(z) and U–(z), respectively. We have to APPENDIX B determine the Fourier coefficients Derivation of the Set of Equations for the Coefficients ∞ R , D , Rν, and Dν ()+, Ð +()Ð() 1 1 eµ eν = ∫ eµ z eν z dz. Taking the scalar product of relationships (10) with Ð∞ Ð Ð Omitting the intermediate manipulations, we write out e1 and eν and using expression (A.7), we obtain the set the final expression of equations ()+, Ð ρ ()δµν () ν +∞ eµ eν ==ζ 1 Ð (Ä.7) κ Ð κ + κ Ð κ + µ ρ ()δµν ()γ ν 1 R1 + 1 p11 D1 = 1 Ð ∫ µ pµ Dµd , + ζ2 + + µν. Ð∞ γ µ2 ν2 –1 + Ð Here, µν = –hµν( – ) , hµν = ((U+ – U–),eµ eν ), +∞ + λ Ð λ + λ Ð λ + µ and the integrals resulting from the integration of (eµ , 1 R1 Ð 1 p11 D1 = Ð 1 + ∫ µ pµ Dµd , Ð Ð∞ eν ) over the spectral parameter µ and containing γµν are understood in terms of the Cauchy principal value. The κ Ð κ +[]ρ ()ν ρ ()ν coefficients in front of the δ functions depend on the ν Rν + ν ζ 1 Dν + ζ2 DÐν coefficients dν and rν in the asymptotic expressions +∞ (B.1) (A.1) in a fairly complicated fashion: κ + κ +γ µ = Ð 1 qν D1 Ð ∫ µ µνDµd , Ð +  Ð∞ ρ ()ν 1 dν dν ()+ Ð ζ 1 = ------1 + ------rν rν + Z 4 Z + Ð Ð + dν dν λν Rν Ð λν []ρζ ()ν Dν + ρζ ()ν D ν (Ä.8) 1 2 Ð Ð + dν dν ∞ + Ð + Ð rν rν 1 Ð ------, + + + Ð  λ ν λµ γ µν µ µ dν dν = 1 q D1 + ∫ D d , Ð∞ 2 ρ ()ν 1 [()δ+ Ð ( Ð Ð + ζ 2 = ------rν Ð rν + rν Ð rν rν Ð + where we introduced the notation 4 Zdν dν ()+, Ð ()+, Ð ()+, Ð Ð 2 Ð + Ð Ð 2 e1 e1 ===p11, eν e1 pν, e1 eν qν. + dν Ð dν dν )()1 + dν Ð rν δ (Ä.9) 2 2 Resolving the first two equations from set (B.1) in Ð()+ ()+ Ð + Ð + rν 1 + dν dν Ð dν + rν Ð rν R1 and D1 and the last two equations in Rν and the com- ρ ν ρ ν 2 2 bination ζ 1 ( )Dν + ζ 2 ( )D–ν, we transform Eqs. (B.1) Ðδ()]Ð + Ð Ð + Ð + rν rν rν Ð rν + dν dν Ð dν , to Ð + + Ð where Z = (1 + dν )(1 + dν ) and δ = dν – dν . ()∆λ +κ Ð λ Ðκ + Ð1 R1 = 1 1 Ð 1 1 11 For a small jump in the Alfvén velocity, the differ- +∞ + Ð + Ð + + + + Ð1 (B.2) ences rν – rν and dν – dν and the quantity δ are all on ()∆λ κ λ κ ν + ∫ ν 1 Ð 1 ν 11 pν Dνd , ε the order of . From dependence (A.9), we immediately Ð∞ ρ ν ε2 see that ζ 2 ( ) = O( ). Using dependence (A.8) and the Ð1 Ð Ð Ð1 2 2 λ κ ∆ Ð Ð D1 = 2 p11 1 1 11 relationship rν =1 – dν and taking the limit ε 0, +∞ we arrive at the asymptotic expression (B.3) Ð1∆ Ð1()λ +κ Ð λ Ðκ + ν Ð ∫ p11 11 ν 1 + 1 ν pν Dνd , Ð 2 dν Ð∞ ρ ()ν 1  ζ 1 ------Ð - 1 + ------Ð 2 41()+ dν  dν ()∆νλ +κ + λ +κ + ()Ð1 Rν = 1 ν Ð ν 1 qν D1 2 2 Ð Ð +∞ × []1 Ð1dν + ()+ dν = 1, (B.4) ()∆νλ +κ + λ +κ + ()Ð1γ µ + ∫ µ ν Ð ν µ µνDµd , which gives ρζ (ν) = 1 + O(ε). 1 Ð∞

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 784 MAZUR et al.

λ +κ Ð λ Ðκ + λ Ðκ Ð()λ +κ Ð λ Ðκ + ()ν 1 ν + ν 1 2 1 1 1 ν + ν 1 f ρ ()ν ρ ()ν Dν = Ð------ζ1 Dν ++ζ 2 DÐν ------qν D1 ∆ ∆ν()∆()ν ∆ν() p11 11 ρ +∞ (B.5) ∞ λ +κ Ð λ Ðκ + + ()λ +κ Ð λ Ðκ + ()λν ()+κ Ð λ Ðκ + µ ν + ν µ 1 ν + ν 1 f µ 1 + 1 µ pµ + ------γ µνDµdµ = 0, ∫ ∆ν() + ∫ ------∆ ∆ν()∆()ν (B.9) p11 11 ρ Ð∞ Ð∞

+ Ð Ð + + Ð Ð + + Ð Ð + ˜ ∆ λ κ λ κ ∆ ν λ κ λ κ ()λµ κν + λν κµ hµν where 11 = 1 1 + 1 1 and ( ) = ν ν + ν ν . µ + ------2 2 Dµd , Equation (B.5) contains both of the coefficients Dν and ∆ν()∆ρ()µν ()Ð ν D–ν. In order to eliminate the coefficient D–ν in (B.5) ˜ ρ ν ρ ν and to keep Dν, we replace ν by –ν. Taking into account where hµν = hµν ζ 1 (– ) – hµ, –ν ζ 2 ( ). Hence, we must, ± ± ± ± λ λ κ κ first, solve Eq. (B.9) to obtain the coefficient Dν, and, the relationships q–ν = qν , Ðν = ν , and Ðν = ν , we find then, from Eqs. (B.2), (B.3), and (B.8), we can determine the remaining coefficients R1, D1, and Rν. λ +κ Ð λ Ðκ + ρ ()ν ρ ()ν 1 ν + ν 1 ζ 1 Ð DÐ2ν ++ζ Ð Dν ------∆ν() qν D1 REFERENCES ∞ (B.6) 1. L. Chen and A. Hasegawa, Phys. Fluids 17, 1399 (1974). + λ +κ Ð λ Ðκ + µ ν + ν µ γ µ 2. J. A. Tataronis, J. Plasma Phys. 13, 87 (1975). + ∫ ------∆ν() µν, Ð Dµd = 0. 3. A. V. Timofeev, Usp. Fiz. Nauk 100 (2), 185 (1970) [Sov. Ð∞ Phys. Usp. 13, 51 (1970)]. 4. J. A. Tataronis and W. Grossmann, Nucl. Fusion 16, 667 ρ ν We subtract Eq. (B.6) multiplied by ζ 2 (– ) from (1976). Eq. (B.5) multiplied by ρζ (ν) to obtain the relationship 5. V. E. Golant and V. I. Fedorov, High-Frequency Methods 1 of Plasma Heating in Toroidal Fusion Facilities (Éner- goatomizdat, Moscow, 1986). λ +κ Ð λ Ðκ + ∆ ()ν 1 ν + ν 1 ()ν 6. J. Heyvaerts and E. R. Priest, Astron. Astrophys. 117, ρ Dν + ------∆ν() f D1 220 (1983). 7. L. Chen and A. Hasegawa, J. Geophys. Res. 79, 1033 ∞ (B.7) + λ +κ Ð λ Ðκ + (1974). µ ν + ν µ ()µν, µ + ∫ ------∆ν() g Dµd = 0, 8. K. N. Stepanov, Zh. Éksp. Teor. Fiz. 35 (6), 1002 (1965) [Sov. Phys. JETP 10, 773 (1965)]. Ð∞ 9. Yu. M. Aliev, S. Vukovic, O. M. Gradov, and A. Yu. Kure, where Pis’ma Zh. Éksp. Teor. Fiz. 25, 351 (1977) [JETP Lett. 25, 326 (1977)]. ∆ ()ν ρ ()ν ρ ()ν ρ ()ν ρ ()ν ρ = ζ 1 ζ1 Ð Ð ζ 2 ζ2 Ð , 10. T. A. Davydova, Fiz. Plazmy 7, 921 (1981) [Sov. J. Plasma Phys. 7, 507 (1981)]. ()ν ρ ()ν ρ ()ν 11. K. K. Khurana and M. G. Kivelson, J. Geophys. Res. 94, f = qν ζ1 Ð Ð qν ζ2 , 5255 (1989). 12. R. Cramm, K.-H. Glassmeier, M. Stellmacher, and ()γµν, ρ ()γν ρ ()ν g = µνζ1 Ð Ð Ðµ, ν ζ 2 . C. Othmer, J. Geophys. Res. 103, 11951 (1998). 13. A. V. Gul’el’mi, Magnetohydrodynamic Waves in Near- Finally, inserting Eq. (B.3) into Eqs. (B.4) and Earth Plasma (Nauka, Moscow, 1982). (B.7), we arrive at the formula 14. E. N. Fedorov, N. G. Mazur, V. A. Pilipenko, and S. Lep- idi, Geomagn. Aéron. 38 (2), 60 (1998). Ð Ð + + + + 15. P. M. Edwin, B. Roberts, and W. J. Hughes, Geophys. 2λ κ ()λ κν Ð λν κ qν 1 1 1 1 Res. Lett. 13, 373 (1986). Rν = ------∆ ∆ν() p11 11 16. W. Allan and A. N. Wright, J. Geophys. Res. 103, 2359 ∞ (1998). + ()λ +κ + λ +κ + ()λ +κ Ð λ Ðκ + 1 ν Ð ν 1 qν µ 1 + 1 µ pµ 17. E. Fedorov, N. Mazur, V. Pilipenko, and K. Yumoto, + ∫ Ð------∆ ∆ν() - (B.8) J. Geophys. Res. 103, 26595 (1998). p11 11 Ð∞ 18. E. N. Fedorov, N. G. Mazur, and V. A. Pilipenko, Fiz. + + + + Plazmy 21, 333 (1995) [Plasma Phys. Rep. 21, 311 ()λµ κν Ð λν κµ hµν (1995)]. Ð ------Dµdµ, ∆ν()µ()2 ν2 19. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Ð Non-Relativistic Theory (Nauka, Moscow, 1986; Perga- mon, New York, 1977). which relates Rν to Dν, and at the following integral equation of the second kind for the function Dν: Translated by I. A. Kalabalyk

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001

Plasma Physics Reports, Vol. 27, No. 9, 2001, pp. 785Ð793. Translated from Fizika Plazmy, Vol. 27, No. 9, 2001, pp. 831Ð840. Original Russian Text Copyright © 2001 by Popel, Gisko, Golub’, Losseva, Bingham.

DUSTY PLASMA

Influence of Electromagnetic Radiation on the Shock Structure Formation in Complex Plasmas S. I. Popel*, A. A. Gisko*, A. P. Golub’*, T. V. Losseva*, and R. Bingham** *Institute for Dynamics of Geospheres, Russian Academy of Sciences, Leninskiœ pr. 38-6, Moscow, 117979 Russia **Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 OQX, UK Received March 16, 2001; in ﬁnal form, April 2, 2001

Abstract—Electromagnetic radiation effects are calculated for the case of the solar radiation spectrum in the vicinity of the Earth. The inﬂuence of the photoelectric effect on the propagation of nonlinear waves in complex plasmas is studied when the dust grains acquire large positive charges. Exact solutions to nonlinear equations in the form of steady-state shocks that do not involve electronÐion collisions are found, and the conditions for their existence are obtained. In contrast to the classical collisionless shock waves, the dissipation due to the dust charging involves the interaction of the electrons and ions with the dust grains in the form of microscopic grain currents and the photoelectric current. The nonsteady problem of the evolution of a perturbation and its transformation into a nonlinear wave structure is considered. The evolution of an intense, initially nonmoving region with a constant increased ion density is investigated. It is shown that the evolution of a rather intense nonmoving region with a constant increased ion density can result in the formation of a shock wave. In addition to the compressional wave, a rarefaction region (dilatation wave) appears. The presence of a dilatation wave ﬁnally leads to the destruction of the shock structure. The possibility is discussed of the observation of shock waves related to dust charging in the presence of electromagnetic radiation in active rocket experiments, which involve the release of a gaseous substance in the Earth’s ionosphere in the form of a high-speed plasma jet at altitudes of 500Ð600 km. © 2001 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION the plasma electrons and ions with dust grains in the form of microscopic grain currents. The case in which At present, a large number of plasma investigations the shock waves related to dust grain charging are are devoted to multicomponent plasmas containing rather intense corresponds to the ion acoustic wave electrons, ions, charged microspheres (dust grains), and propagation. The main results concerning this new kind neutrals. Such plasmas are usually referred to as “com- of ion acoustic shocks are presented in [4Ð6]. Recently, plex plasmas.” Complex (dusty) plasma cannot usually the first results of laboratory experiments confirming survive in the absence of either external sources of elec- the effect of negatively charged dust on the formation trons and ions (e.g., due to ionization) or plasma parti- of ion acoustic shocks were obtained [7, 8]. cle fluxes from dust-free regions. The fluxes of electrons and ions are absorbed by dust grains, which The importance of shock waves in complex plasmas results in variations in their charges. Variations in the is associated with different astrophysical applications dust grain charge can also be caused by an external [5]. For example, according to modern concepts [9], the electromagnetic radiation via the photoelectric effect. formation of stars occurs mainly in interstellar dustÐ The strong dissipativity of the complex plasma due to molecular clouds after compressional shock waves dust grain charging [1] points to the exceptional role of have propagated through them, thus creating an initial dissipative structures (such as shock waves) in complex density condensation for further gravitational contrac- plasmas. tion. The presence of dust in interstellar clouds can sig- Shock waves often arise in nature because of the nificantly influence the sound velocity, not to mention balance between the wave breaking nonlinear force and the shock wave propagation. The investigation of shock the wave damping dissipative force. Collisional and waves related to the dissipation caused by dust grain collisionless shock waves can appear due to friction charging can also be important [5, 10] for the descrip- between the particles [2] and the waveÐparticle interaction of shocks in supernova explosions, particle accel- tion [3], respectively. In complex plasmas, an anoma- eration in shocks, the explanation of the effects in active lous dissipation due to dust charging results in the exist- geophysical and space experiments involving the ence of a new kind of shock wave related to this dissi- release of a gaseous substance in the Earth’s ionosphere pation. These waves are collisionless in the sense that and magnetosphere, etc. they do not involve electronÐion collisions. However, in In astrophysical applications, the effect of electro- contrast to classical collisionless shock waves, the dis- magnetic radiation (as well as the influence of the pho- sipation due to dust charging involves the interaction of toelectric effect on the dust grain charge) cannot often

786 POPEL et al. be ignored. This effect can lead to new qualitative According to the orbit-limited probe model [14, 15], results in comparison with the situation in which this for equilibrium electrons and ions, we have the follow- effect is neglected. In particular, the photoelectric effect ing expressions for the microscopic currents to the can result in the positive charge of dust grains, while in grain surface (cf. [11, 16Ð18]): the case where the dust grain charge is varied only due 1/2 to microscopic electron and ion currents, the dust grain ≈ π 2 8T e eqd charge is negative (see, e.g., [11]). Ie Ð a eπ------ne1 + ------, (1) me aTe This paper deals with the case where the photoelectric effect strongly influences the dust grain charge. The π 2 electromagnetic radiation effects are calculated for the Ii = ---a v Tieni solar radiation spectrum in the vicinity of the Earth. We 2 investigate solutions in the form of dust ion acoustic 2 2  v + v , ()q v v , ()q shock waves. In particular, we consider the nonsteady × 2expÐ------i min i d cosh ------i min i d - 2 2 problem of the evolution of a perturbation and its trans-  2v Ti v Ti formation into a nonlinear wave structure. In Section 2, (2) we describe the basic assumptions and equations and πv v 2 2eq present steady-state shock wave solutions. In Section 3, ------Tii d + --- 1 + ------2- Ð ------2- 2 v i  we consider the evolution of an intense, initially non- v Ti amiv Ti moving region with a constant, increased ion density using a numerical method analogous to that developed v ()v v ()v  × min, i qd + i min, i qd Ð i in [12]. In Section 4, we discuss the possibility of erf------Ð erf------,  observing the shock waves under consideration in 2v Ti 2v Ti active space rocket experiments that involve the release 1/2 of a gaseous substance in the Earth’s ionosphere. A where me is the electron mass, Tj and vTj = (Tj /mj) are summary of our findings and conclusions are given in the temperature and thermal velocity of particles of Section 5. species j (j = i, e), vi is the ion fluid velocity, vmin, i (qd) = 1/2 (2eqd/ami) , and erf(x) is the error function. Here, we 2. STEADY-STATE SHOCK WAVE SOLUTIONS take into account that, when qd is positive (in contrast to the situation with qd < 0), only the ions with velocities The average radius a of grains in a typical complex |v| > v (q ) can reach the grain surface, while there plasma is much smaller than the electron Debye length min, i d λ is no limitation on the absolute value of the electron D, the spatial scale of perturbations, and the distance velocity. We emphasize that this fact results in different between the plasma particles. Since the dust grains are (from the case of q < 0 considered, e.g., in [4]) expres- Ӷ d massive (miZd md, where mi and md are the ion and sions for the electron (I ) and ion (I ) currents. grain masses, respectively), they can be considered e i immobile and their density nd can be assumed to be The photoelectric current produced by the electrons constant on the ion acoustic time scale [13]. Further- emitted from the dust grain surface in the presence of more, in the absence of perturbations, the quasineutral- external electromagnetic radiation with spectrum Φ(ω), ity condition ni0 = ne0 + Zd0nd holds. Here, qd = –Zd e is where Φ = ∫Φ (ω)dω is the luminous flux, is given by the mean dust grain charge; –e is the electron charge; n i the formula and ne are the ion and electron densities, respectively; and the subscript 0 denotes unperturbed quantities. ∞ πβea2 Φω() We will assume that the grain charge varies due to ω I ph = ------ប ∫ ------ω d . (3) both the photoelectric electron current and the micro- ω 2 ប scopic electron and ion grain currents (originating from R Ð e Zd/a the potential difference between the plasma and the Here, β is the probability of the emission of an electron grain surface). The photoelectric electron current is due under the action of one photon on the dust particle sur- to the photoelectric effect, which results in the separa- ប បω face, is Planck’s constant, and R is the photoelec- tion (and removal) of the electrons from the dust grain tric work function. The limits of integration are deter- surface. When electromagnetic radiation is sufficiently mined by the fact that photons with frequencies ω > intense, the photoelectric effect leads to the positive ω 2 ប charge of dust grains. In this paper, we restrict our- R – (e Zd/a ) can only produce the photoelectric cur- selves to the case in which the dust grain charge is pos- rent. As was mentioned above, in this paper, electro- itive. This case differs qualitatively from the case in magnetic radiation effects are calculated for the solar which the photoelectric current is absent [4, 5, 12]; its radiation spectrum in the vicinity of the Earth. For sim- consideration allows us to distinguish the effects plicity, the spectrum is approximated by the black body Φ ω Φ ω3 បω caused by the influence of electromagnetic radiation on spectrum ( ) = 0 /[exp( /Ts) – 1] with an effec- Φ × –55 the shock wave propagation. tive temperature of Ts = 6000 K and 0 = 5.5 10 g s

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 INFLUENCE OF ELECTROMAGNETIC RADIATION 787 ξ (in this case, Φ = Φ (ω)dω = 1.4 × 106 erg/(cm2 s) is parameters depend only on the variable = x – Vt. From s ∫ the ion conservation equations, we obtain the solar constant). ()2 ϕ Ð1/2 The mean charge of immobile dust grains is gov- ni = Mni0 M Ð 2 , (6) erned by the charge conservation equation v = c ()MMÐ 2 Ð 2ϕ , (7) ∂ () () () i s tqd = Ie qd ++Ii qd I ph qd . (4) ϕ ϕ where we use the normalization e /Te , V/cs M, ξ λ ξ 1/2 The unperturbed mean charge qd0 satisfies the equation and / D . Here, cs = (Te/mi) is the ion acoustic Ie(qd0) + Ii(qd0) + Iph(qd0) = 0. The typical parameters of speed in the absence of dust. the Earth’s ionosphere at altitudes of 500Ð600 km (see, 3 –3 × 2 –3 From Poisson’s equation, we obtain e.g., [19]) are ne0 = 10 cm , ni0 = 8 10 cm , Te = 2 eV, and T = 0.5 eV. Then, for the dust parameters δ () i 2ϕϕ() z M 1 + Zd0d (see, e.g., [20]) a = 10–4 cm, λ ≡ 2πc/ω = 2 × 10–5 cm dξ = exp+  1 + ----- Zd0d Ð ------, (8) R R z0 2 ϕ (which is typical for most materials), and β = 0.1, the M Ð 2 ≈ × 3 unperturbed grain charge is equal to Zd0 –1.92 10 , 2 where d = nd0/ne0 and z0 = Zd0e /aTe. The normalized which confirms the above assumption qd = –Zde > 0. δ δ δ perturbed charge z = –e qd/aTe, where qd = qd – qd0 The electron density is assumed to obey a Boltz- is governed by the equation ϕ mann distribution ne = ne0exp(e /Te) with a constant δ ()ϕ ()ϕ ()ϕ ()ϕ dξ zj==je ++ji j ph , (9) electron temperature Te. We note that, in the vicinity of a dust grain, the Boltzmann distribution for the electron with density can fail to hold because of the attraction between the electrons and the dust grain. However, for A 8m T j = Ð------i -e exp()ϕ ()1 Ð z Ð δz , (10) Ti < Te (which is satisfied in the above example, in e ()π 0 M 1 + Zd0d meT i which Te = 2 eV and Ti = 0.5 eV), the electron density distribution can be considered to be Boltzmann at dis-  tances longer than or on the order of the ion Debye A 2 []()α2 γ 2 ()αγ ji = ------ --- exp Ð + cosh 2 length. In all subsequent considerations, the minimum 2  π characteristic parameter of the problem is the electron M Ð 2ϕ Debye length, which is larger than the ion Debye 1 γ 2 ()δ T e length. + ------12++2 z0 + z ----- (11) 22 T i In the situation under consideration, in which the photoelectric effect results in a positive grain charge erf()αγ+ erf()αγÐ  (cf. [11]), the ion current I is several orders of magni- × ------Ð ------, i γ γ  tude lower than Ie and Iph. Thus, on a time scale characteristic of ion acoustic waves, we can neglect the change in the number of ions and the loss of the ion 2 J ph w momentum due to dust charging and can use the ion j ph = ------∫ ------()dw. (12) M expT ew/T s Ð 1 continuity equation and the ion momentum equation. ww> ()δz Furthermore, we use Poisson’s equation for the electro- R λ 1/2 static potential Here, A = a[(1 + Zd0d)/4 D](Ti/Te) , Jph = πβe2a2 T 3/2m 1/2 Φ (λ /a)/ប4, γ = [M – (M2 – ∂ 2ϕ π () e i 0 D x = 4 ene + Zdnd Ð ni . (5) ϕ 1/2 1/2 α δ 1/2 1/2 δ 2 ) ](Te/2Ti) , = [–(z0 + z)] (Te/Ti) , wR( z) = δ បω In this paper, the problem is solved in one-dimen- wR0 – (z0 + z), and wR0 = R/Te. We also used rela- sional plane geometry, which is applicable when the tionships (6) and (7). Note that j is the normalized per- width of the shock front is far less than the characteris- turbed total current density. Clearly, all of the solutions tic transverse scale on which the parameters of the must satisfy the inequality ϕ ≤ M2/2. Equations (8) and problem vary. All of the parameters are assumed to (9) can be numerically integrated provided that the depend only on the time variable t and the spatial vari- existence conditions for nonlinear wave solutions are able x. known. The solutions to equations (8) and (9) cannot simultaneously satisfy the condition that ϕ 0 as ξ We consider a quasi-steady structure propagating tends to +∞ or –∞. This means that, in the problem as with velocity V in the x direction and assume that the formulated, there are no nonlinear solitary waves. For a Ӷ condition vTi < V vTe is satisfied, which is character- shock wave solution to exist, there must be two differ- istic of the ion acoustic wave propagation. Thus, all the ent asymptotic ϕ values at ξ ±∞ and the derivatives

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 788 POPEL et al. ϕ of the perturbed quantities must vanish there. In this the same as that of the function f2 ( ) in [4] (see Fig. 2 case, from (8) and (9) we obtain the relationship for the in [4]). perturbed grain charge The set of equations (13) and j = 0 has two different solutions ϕ = 0 and ϕ = ϕ only if the following condi- () A δ z0 M 1 + Zd0d ()ϕ tion is satisﬁed: zz= Ð 0 + ------Ð,exp (13) Zd0d 2 ϕ Ð1 M Ð 2 2 2 1 + Z d z G M > M ≡ 1 + ------d0 z G ------0 - + E , (14) 0 Z d 0 Z d and the current balance condition j = 0. The qualita- d0 d0 tive behavior of the solution to the equation j = 0 is where

()()()()Ð1 expz0T e/T i T e/T i 2 + z0T e/T i ++ 1 + Zd0d miT e/meT i F ph G = ------()()------, exp z0T e/T i 1 + z0T e/T i ()Ð1 () 1 + Zd0d miT e/meT i 1 Ð z0 E = ------()()-, exp z0T e/T i 1 + z0T e/T i π ()()2 {}[]() F ph = /8 J ph/A wR0 Ð z0 / expT e wR0 Ð z0 /T s Ð 1 .

Figure 1 presents the current density j [obtained two orders of magnitude less than that of solar radiation with allowance for relationship (13)] for M = 0.894 (we in the vicinity of the Earth show that the inequality ϕ Ӷ 2 emphasize that M is normalized to the ion acoustic A M /2 also remains valid in these cases. When this speed cs without dust) and the typical parameter values inequality is valid, we can neglect ji as compared to je 3 –3 × 2 –3 ϕ ne0 = 10 cm , ni0 = 8 10 cm , Te = 2 eV, Ti = and jph within the entire range of the potential corre- 0.5 eV, a = 10–4 cm, λ = 2 × 10–5 cm, β = 0.1, and Φ = sponding to the nonlinear wave solution (i.e., for R ϕ ϕ Φ × 6 2 0 < < A). In this case, inequality (14) is simpliﬁed to s = 1.4 10 erg/(cm s). For these parameters, we have z ≈ –1.41 and ϕ ≈ 0.024. We note that, in this ()2 () 0 A 2 2 HJ+ phwR 1 + Zd0d z0 ϕ Ӷ 2 M > M ≈ ------, (15) case, A M /2. The calculations carried out with 0 2 []() luminous ﬂuxes two orders of magnitude higher and J phwR z0 Ð HZd0d Ð 1 z0 Ð Zd0d where

j 8m T T w i e ------e R × –6 HA= π------expÐ. 1 1.5 10 meT i T s Inequality (14) is necessary for the existence of two ϕ ϕ ϕ 1.0 × 10–6 asymptotic solutions, namely, = 0 and = A (e.g., at ξ +∞ and –∞, respectively). The requirement that the set of equations (13) and 5.0 × 10–7 j = 0 have a solution with the asymptotic values ϕ = 0 ϕ ϕ ξ ±∞ and = A at results in the inequality 2 ≤ 2 ≡ 0 M M1 1 + Zd0d. (16) Condition (16) can be obtained in a way analogous × –7 to that in the case where the dust grain charge varies –5.0 10 only due to the microscopic electron and ion currents and the photoelectric effect is ignored [4]. When deriving this condition, it is taken into account that the ine- –1.0 × 10–6 ϕ ϕ 0 0.01 0.02 0.03 quality j < 0 remains valid for 0 < < A (see Fig. 1). ϕ ϕ ϕ ϕ We note that = 0 and = A are also exact solutions to Eqs. (8) and (9), because these differential Fig. 1. Current density j vs. ϕ for M = 0.894 and the param- equations are homogeneous. Furthermore, both solu- 3 –3 × 2 –3 eter values ne0 = 10 cm , ni0 = 8 10 cm , Te = 2 eV, Ti = tions correspond to E ≡ –∇ϕ = –dξϕ = 0. Now, we can –4 λ × –5 β Φ Φ 0.5 eV, a = 10 cm, R = 2 10 cm, = 0.1, and = s = find a solution bridging these two asymptotic solutions. 1.4 × 106 erg/(cm2 s). The solutions to the equation j = 0 are Figures 2a and 2b show the profiles of the potential ϕ(ξ) ϕ ϕ ≈ = 0 and A 0.024. and the electric field E = –dξϕ, respectively. A similar

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 INFLUENCE OF ELECTROMAGNETIC RADIATION 789 proﬁle for the normalized perturbed charge δz can be 0.04 deduced from Eq. (9) taking into account that j is (‡) always negative in the region of interest (see Fig. 2c). ϕ 0.02 Figure 2d shows the proﬁle of the ion density ni normalized to the unperturbed electron density ne0. 0 In Figs. 2a and 2b, oscillation regions that correspond to the separation of the electron and ion charges 6 × 10–6 (b) cannot be distinguished (see, e.g., [12]). However, this 4 × 10–6

effect actually takes place. Figure 3 presents a portion E × –6 of the profile of the electric field E on an enlarged scale. 2 10 The effect of charge separation increases when the 0 luminous flux decreases and/or M increases. 0.008 (c)

The solution presented in Figs. 2a and 2b corre- z sponds to the balance between the wave breaking non- δ 0.004 linear force and the wave damping dissipative force in a uniform complex plasma with Boltzmann electrons, 0 inertial ions, and immobile but variable-charge dust (d) grains. This solution can be treated as a steady-state 0 e 0.82 shock wave solution. The dissipativity in this wave is n / i related to variations in dust grain charges due to the n microscopic electron and ion grain currents and the 0.80 photoelectric electron current. The width of the shock ν ξ front can be expressed in terms of the charging rate q 20000 40000 ∆ξλ ν and the ion acoustic speed as D ~ cs/ q, where Fig. 2. Proﬁles of (a) the potential ϕ(ξ), (b) electric ﬁeld E = –dξϕ, (c) perturbation of the normalized charge δz, and (d) ∂()I ++I I ν ≡ Ð------e i ph- ion density ni normalized to the unperturbed electron den- q ∂ sity n in a shock wave structure for the parameters of qd q = ÐZ e e0 d d0 Fig. 1. 2 2 ω a z T z T ω a = ------pi exp ------0 -e 2 + ------0 -e + ------pe π T T π (17) 2 v Ti i i 2 v Te E 3.0 × 10–6 πβae2Φ T 2 ()w Ð z 2 + ------0 e ------R0 0 -. 4 []() ប expT e wR0 Ð z0 /T s Ð 1

3 –3 × 2 –3 For example, for ne0 = 10 cm , ni0 = 8 10 cm , –4 λ × –5 × –6 Te = 2 eV, Ti = 0.5 eV, a = 10 cm, R = 2 10 cm, 2.9 10 β Φ Φ × 6 2 = 0.1, and = s = 1.4 10 erg/(cm s), the width ∆ξλ 4λ ≈ × 5 of the shock front is D ~ 10 D 3 10 cm, while ν 5 cs/ q ~ 10 cm. We emphasize that, for the above plasma parameters, the charging rate in the presence of electro- ν ≈ –1 2.8 × 10–6 magnetic radiation ( q 20 s ) is much higher than the ν ≈ –1 charging rate in the absence of radiation ( q 0.3 s ). 36500 36550 36600 36650 36700 The shock speed M must simultaneously satisfy ine- ξ qualities (14) and (16). For the majority of complex Fig. 3. A portion of the proﬁle of the electric ﬁeld E on an plasmas in which ne0 ~ ni0, this speed is close to the ion ξ 1/2 enlarged scale. The range of corresponds to the wave front acoustic speed (Te/mi) . The range of M in which region. shock solutions exist is fairly narrow. For the above parameters, we have M ≈ 0.879 and M ≈ 0.8944. 0 1 dissipative effects due to dust charging to dominate In the above considerations, we neglected the effect over those due to Landau damping is of Landau damping on the shock wave formation. This ()λ2 λ 2 3 ӷ ()ν()ω is valid when dissipation due to dust charging is stron- a / D D ne0 me/mi q/ pi , (18) ger than that due to Landau damping. On time scales ω π 2 1/2 characteristic of ion acoustic wave propagation in the where pi = (4 ni0e /mi) is the ion plasma frequency. 3 –3 × 2 –3 presence of electromagnetic radiation, the condition for For ne0 = 10 cm , ni0 = 8 10 cm , Te = 2 eV, Ti =

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 790 POPEL et al.

–4 λ × –5 β Φ 0.5 eV, a = 10 cm, R = 2 10 cm, = 0.1, and = The LCPFCT transport algorithm consists of the Φ × 6 2 following four sequential steps: s = 1.4 10 erg/(cm s), inequality (18) is easily satisfied (the left-hand side is three orders of magnitude (i) the computation of the transported and diffused larger than the right-hand side). values and the choice of the diffusion coefficients to The shock waves discussed here are collisionless in satisfy monotonicity, the sense that they do not involve electronÐion colli- (ii) the computation of the raw antidiffusive fluxes, sions (the source of dissipation for ordinary collisional (iii) the correction or limitation of these fluxes to ion acoustic shock waves). However, in contrast to clas- assure monotonicity, and sical collisionless shock waves [3], in which dissipation (iv) the performance of the antidiffusive correction. is only related to the turbulent waveÐparticle interaction, dissipation due to dust charging involves the inter- To solve Eq. (4) for dust grain charging, we use the action of electrons and ions with dust grains in the form fourth-order RungeÐKutta method [22]. Poisson’s of microscopic grain currents and the photoelectric cur- equation is solved numerically using the sweep method rent. A similar situation takes place in the absence of [21]. electromagnetic radiation [4]. This is because, for the The total set of equations is solved using the follow- ν majority of complex plasmas, the charging rate q is ing sequence of operations (at each time step): much higher than the ionÐion collision frequency and (i) the integration of conservation equations, the electron collision frequency. We emphasize that (ii) the integration of the equation for dust grain dust shocks have unique conditions of existence. Thus, charging, and they can be useful in studying astrophysical complex plasmas. (iii) the integration of Poisson’s equation. These three stages are related to each other by an iteration process that is verified by the charge density 3. EVOLUTION OF AN INITIALLY NONMOVING convergence. REGION WITH A CONSTANT INCREASED ION DENSITY Now, we consider the situation in which the evolution of an initial perturbation can result in the formation The above investigation concerned the problem of of a shock wave structure that is close to the exact the existence of steady-state shock wave solutions. steady-state shock wave solution. For simplicity, we However, in order to understand whether shock wave consider a nonmoving region with a constant increased structures are significant nonlinear wave structures in ion density as an initial perturbation. The amplitude of ϕ dusty plasmas in the presence of electromagnetic radi- the initial perturbation 0 should be larger than that of ϕ ation, it is necessary to answer the question of whether the steady-state shock A. Otherwise, a quasi-steady the evolution of a perturbation leads to the formation of shock does not form; the amplitude of the evolving per- shocks in a dusty plasma with a variable grain charge. turbation progressively decreases; and, finally, the per- Furthermore, the solution of the problem of the evolu- turbation disappears. tion of a perturbation and the possibility of its transfor- The results of calculations describing the evolution mation into a shock wave is important from the stand- of an initially nonmoving region with a constant point of the description of real phenomena like super- increased ion density that corresponds to the initial nova explosions, as well as laboratory experiments and amplitude ϕ = 0.048 for the plasma parameters n active space and geophysical experiments. 0 e0 = 3 –3 × 2 –3 10 cm , ni0 = 8 10 cm , Te = 2 eV, Ti = 0.5 eV, a = In order to investigate the problem of the evolution –4 λ × –5 β Φ Φ × of a perturbation and its transformation into a nonlinear 10 cm, R = 2 10 cm, = 0.1, and = s = 1.4 wave structure in a dusty plasma with a variable grain 106 erg/(cm2 s) are presented in Figs. 4 and 5. It is charge in the presence of electromagnetic radiation, we assumed that the initial charge of dust grains is equal to used a computational method analogous to that devel- its equilibrium value in the absence of wave perturba- ≈ λ oped in [12]. tions (z0 –1.41). We use the normalization x/ D x λ To solve the ion continuity equation and the ion for the spatial variable and tcs/ D t for time. The momentum equation, we use the LCPFCT modification evolution of the perturbation results in the formation of a quasi-steady structure propagating at t > 100 with a of the flux-corrected transport (FCT) algorithm with ≈ fourth-order phase accuracy, second-order time accu- constant speed corresponding to the Mach number M 0.894. Figure 4 shows the profiles of the potential ϕ(x), racy, and minimum residual diffusion [21]. The FCT δ algorithm is monotonic, conservative, and positivity- the normalized charge perturbation z, the ion speed vi preserving. This means that the algorithm is accurate normalized to cs, and the ion density ni normalized to and resolves steep gradients (including scales on the the unperturbed electron density ne0 at the instants t = order of the grid size). When a convected quantity (such 100, 10000 and 20000. The initial profiles (at t = 0) of ϕ as the ion density) is initially positive, it remains posi- the potential and the normalized ion density ni/ne0 are tive and no new maxima or minima are introduced due shown by the light curves on the left of the correspond- to numerical errors during convection. ing panels. The light curves on the right of the panels

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 INFLUENCE OF ELECTROMAGNETIC RADIATION 791

t = 0 (‡) Ion density, arb. units 0.04 10 km 100 10000 20000 å = 0.894 0.02 ϕ

0 20 km 0.012 (b) 0.008 z δ 0.004 30 km 0 (c) 0.03 40 km s c / i v 0 50 km (d) 0.84 0 e n / i n 0.80 0 100000 200000 300000 0 2 4 6 Time, s x Fig. 5. Evolution of the initial perturbation of the ion den- Fig. 4. Profiles of (a) the potential ϕ(x), (b) normalized δ sity on the real spatial and time scales. The parameters are charge perturbation z, (c) ion speed vi normalized to cs, the same as in Fig. 4. and (d) ion density ni normalized to the unperturbed electron density ne0 at the instants t = 100, 10000 and 20000 for ϕ 3 –3 the following parameters: 0 = 0.048, ne0 = 10 cm , ni0 = scales. In this figure, the regions corresponding to the × 2 –3 –4 λ × 8 10 cm , Te = 2 eV, Ti = 0.5 eV, a = 10 cm, R = 2 compression and dilatation waves are clearly seen. –5 β Φ Φ × 6 2 10 cm, = 0.1, and = s = 1.4 10 erg/(cm s). The We have also studied the influence of the initial dust initial charge of dust grains is equal to the equilibrium grain grain charge on the evolution of an initially nonmoving ≈ − charge in the absence of wave perturbations (z0 1.41). region with a constant increased ion density. Analo- The initial profiles (at t = 0) of the potential ϕ and the nor- gously to [12], the evolution of an initial perturbation malized ion density ni/ne0 are shown by the light curves on for different initial grain charges is almost the same at the left of the corresponding panels. The light curves on t > 100. This is related to the fact that the characteristic the right of the panels (a)Ð(d) show the profiles corresponding to the exact steady-state shock wave solution charging time of dust grains is far less than the time with M = 0.894. during which a structure similar to the exact steady- state shock wave solution is formed. show the corresponding profiles of the exact steady- state shock wave solution with M = 0.894. 4. ACTIVE EXPERIMENTS AND THE POSSIBILITY OF SHOCK WAVE In Fig. 4, we can see that the evolution of an intense, OBSERVATION initially nonmoving region with a constant increased Let us discuss the possibility of the observation of ion density results in the formation of a shock wave shock waves related to dust charging in the presence of similar to the exact steady-state shock wave solution electromagnetic radiation. As was shown above, the with the Mach number M = 0.894. The difference width of the front of a shock wave related to dust charg- between these two solutions is the presence in the ing for ionospheric plasma parameters at altitudes of former of a rarefaction region (dilatation wave) in addi- 500Ð600 km and dust grain sizes on the order of tion to a compression region. In the course of the shock −4 wave evolution, the distance between the rarefaction 10 cm can attain several kilometers. Thus, it is of and compression regions decreases. Finally, the pres- interest to investigate the possibility of both the appear- ence of the dilatation wave leads to the destruction of ance of charged dust grains in active rocket experi- the shock structure. ments that involve the release of a gaseous substance in the Earth’s ionosphere and the formation of shock Figure 5 illustrates the spatial evolution of the initial waves related to dust charging in such experiments. The perturbation of the ion density at real spatial and time idea of shock wave observations was forwarded in con-

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 792 POPEL et al. nection with active space experiments carried out by (a) For velocities U lower than 10 km/s, the effect of the Active Magnetospheric Particle Tracer Explorers dust charging is significant and the characteristic dis- (AMPTE). One of the main purposes of the AMPTE tance L over which the grain acquires a significant experiment (see, e.g., [23]) was to study collisionless charge does not exceed the width of the shock front shock waves with very wide fronts. We pay main atten- (which, in our case, is on the order of 1 km). For exam- tion to experiments conducted in the daytime (when ple, for a molecular nitrogen jet propagating with a ≈ × 3 electromagnetic radiation makes an important contribu- velocity of U = 0.5 km/s, we have Zd0 –2.98 10 and tion). We assume that the experiments are carried out at L ~ 103 cm; for an iron jet and U = 9.2 km/s, we obtain altitudes of 500Ð600 km and the scheme of experiments Z ≈ –1.19 × 105 and L ~ 103 cm. is analogous to that of the Fluxus-1 and Fluxus-2 exper- d0 iments, which were carried out at an altitude of 140 km (b) The increase in the velocity U results in the [24, 25]. In those experiments, the source of charged weakening of the effect of dust charging and an particles was the generator of high-speed plasma jets. increase in the distance L. For an iron jet velocity of ≈ × 2 × The shock wave front is associated with the fore part of U = 15.5 km/s, we obtain Zd0 Ð6.55 10 and L ~ 3 the jet, i.e., the boundary between the jet plasma and the 105 cm (which is on the order of the shock front width). ≈ 7 ionospheric plasma. The possibility of observing shocks For U = 21.4 km/s, we have Zd0 –12 and L ~ 10 cm. related to the dissipation caused by dust charging in the For the above parameters, the ion acoustic speed, absence of electromagnetic radiation (when experi- which determines the speed of the shock front, is cs ~ ments are carried out at night) was discussed in [10]. 10 km/s. Thus, the optimum velocities U for manifest- In experiments conducted at altitudes of 500Ð600 km, ing the dust charging effect and observing the related aerosols (dust grains) appear as a result of condensation shock waves do not exceed 10 km/s. The distances L [26]. The period of the formation of the centers of con- corresponding to these velocities are reasonable from densation is very short, and all the drops have approxi- the standpoint of active experiments. By analogy to the mately the same size a. The size a was estimated for AMPTE experiment [23], we can expect that the active two situations [10]. In the first case, calculations were experiments will make it possible to study the structure performed by N.A. Artem’eva for an air jet. The char- of the shock wave front and the physical processes acteristic expansion velocity of molecular nitrogen is occurring at the front. Furthermore, the active experi- U = 0.3Ð0.5 km/s. Condensation starts when the jet ments described here and in [10] can help to model dif- passes a distance on the order of 10 cm. The degree of ferent physical phenomena occurring in nature, e.g., condensation is equal approximately to 0.72, and the during a large meteoroid impact with the Moon’s sur- grain size is a = 1.5 × 10–4 cm. In the second case, the face [27]. The evolution of the impact plume can lead gaseous substance was iron. The estimates were to the formation of a shock wave structure associated obtained with the use of data from [26]. In this case, the with the appearance of charged grains produced due to degree of condensation is x ≈ 0.44. The important result the condensation of both the plume substance and the here is that the size a decreases significantly as the jet vapor thrown from the crater and the surrounding velocity increases (a = 6 × 10–3 cm for U = 9.2 km/s; regolith layer. a = 3.3 × 10–5 cm for U = 15.5 km/s; a = 6 × 10–7 cm –8 for U = 21.4 km/s; and a = 10 cm for U = 27.2 km/s). 5. SUMMARY We see that, for the U values exceeding 25 km/s, the size a is on the order of the characteristic size of a mol- We have studied the influence of the photoelectric ecule. This means that, at sufficiently high jet veloci- effect on the propagation of nonlinear waves in com- ties, condensation does not lead to the formation of plex plasmas. The photoelectric effect results in the grains. appearance of electrons emitted from the surface of dust grains, which, in turn, leads to the generation of the The charge acquired by dust grains can be estimated photoelectric electron current in addition to the usually as an unperturbed dust grain charge from the balance considered microscopic electron and ion currents. We condition for the microscopic electron and ion grain have calculated the electromagnetic radiation effects currents and the photoelectric electron current. For ion- for the solar radiation spectrum in the vicinity of the ospheric parameters at altitudes of 500Ð600 km (ne0 = Earth. In complex plasmas, an important role is played 3 –3 × 2 –3 10 cm , ni0 = 8 10 cm , Te = 2 eV, Ti = 0.5 eV, and by nonlinear shock wave structures. The physical rea- Φ Φ × 6 2 λ ≡ = s = 1.4 10 erg/(cm s)), the parameters R son for their existence is an anomalous dissipation orig- π ω × –5 β inating from the dust charging. In contrast to the case in 2 c/ R = 2 10 cm and = 0.1, and the above values of the jet velocity U, we found the values of the charge which radiation is absent, the dust grains acquire large ν positive charges. We have found exact solutions to non- number Zd0, the charging rate q, and the characteristic distance L ~ U/ν over which the dust grain acquires the linear equations in the form of steady-state shocks and q have obtained the conditions for their existence. These charges q Z e. d ~ – d0 shock waves are collisionless in the sense that they do The main results of this investigation are the follow- not involve electronÐion collisions. However, in con- ing: trast to classical collisionless shock waves, in which

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 INFLUENCE OF ELECTROMAGNETIC RADIATION 793 dissipation is only due to the turbulent waveÐparticle and M. Y. Yu, in Plasma Physics, Ed. by P. Martin and interaction, the dissipation due to dust charging J. Puerta (Kluwer, Dordrecht, 1998), p. 107. involves the interaction of electrons and ions with dust 6. S. I. Popel, A. P. Golub’, T. V. Losseva, and R. Bingham, grains in the form of the microscopic grain current and Pis’ma Zh. Éksp. Teor. Fiz. 73, 258 (2001) [JETP Lett. the photoelectric current. In the case under consider- 73, 223 (2001)]. ation, ion acoustic shocks can exist within a rather nar- 7. Y. Nakamura, H. Bailung, and P. K. Shukla, Phys. Rev. row range of the shock wave velocities determined by Lett. 83, 1602 (1999). inequalities (14) and (16). For the majority of complex 8. Q.-Z. Luo, N. D’Angelo, and R. L. Merlino, Phys. Plas- plasmas, the shock wave velocity is close to the ion mas 6, 3455 (1999). 1/2 9. S. A. Kaplan and S. B. Pikel’ner, The Interstellar acoustic speed (Te/mi) . We have considered the nonsteady problem of the evolution of a perturbation and Medium (Nauka, Moscow, 1963; Harvard Univ. Press, its transformation into a nonlinear wave structure. We Cambridge, 1970). have investigated the evolution of an intense, initially 10. S. I. Popel and V. N. Tsytovich, Astrophys. Space Sci. nonmoving region with a constant increased ion den- 264, 219 (1999). sity. The evolution of such a region can result in the for- 11. O. Havnes, U. de Angelis, R. Bingham, et al., J. Atmos. mation of a shock wave solution that is similar to the Terr. Phys. 52, 637 (1990). exact steady-state solution. The difference between 12. S. I. Popel, A. P. Golub’, T. V. Losseva, et al., Fiz. Plazmy 27, 483 (2001) [Plasma Phys. Rep. 27, 455 these two solutions is that, in the former, there is a rar- (2001)]. efaction region (dilatation wave) in addition to a com- 13. S. I. Popel and M. Y. Yu, Phys. Rev. E 50, 3060 (1994); pression region. Finally, the presence of the dilatation Contrib. Plasma Phys. 35, 103 (1995). wave leads to the destruction of the shock structure. We 14. F. F. Chen, Plasma Diagnostic Techniques, Ed. by have discussed the possibility of observing shock R. H. Huddlestone and S. L. Leonard (Academic, New waves related to dust charging in the presence of elec- York, 1965), Chap. 4. tromagnetic radiation in active rocket experiments that 15. M. S. Barnes, J. H. Keller, J. C. Forster, et al., Phys. Rev. are based on the scheme of the Fluxus-1 and Fluxus-2 Lett. 68, 313 (1992). experiments [24, 25] and involve the release of a gas- 16. L. Spitzer, Jr., Physical Processes in the Interstellar eous substance in Earth’s ionosphere in the form of a Medium (Wiley, New York, 1978; Mir, Moscow, 1981). high-speed plasma jet at altitudes of 500Ð600 km. For 17. B. T. Draine and E. E. Salpeter, Astrophys. J. 312, 77 the shock waves related to dust charging to be (1979). observed, the jet velocity should not exceed 10 km/s. 18. O. Havnes, T. W. Hartquist, and W. Pilipp, in Physical Because of the unique conditions of the existence of Processes in Interstellar Clouds, Ed. by G. E. Morﬁll dust shock waves in the presence of electromagnetic and M. Scholer (D. Reidel, Dordrecht, 1987). radiation, they can be useful in studying astrophysical 19. D. Bilitza, International References Ionosphere 1990, complex plasmas. Science Data Center, NSSDC/WDC-A-R&S 90-20 (Greenbelt, Maryland, 1990). ACKNOWLEDGMENTS 20. E. C. Whipple, Rep. Prog. Phys. 44, 1206 (1981). 21. E. S. Oran and J. P. Boris, Numerical Simulation of This study was supported by INTAS (grant no. 97- Reactive Flow (Elsevier, New York, 1987). 2149) and INTASÐRFBR (grant no. IR-97-775). 22. G. E. Forsythe, M. A. Malcolm, and G. B. Moler, Com- puter Methods for Mathematical Computations (Pren- REFERENCES tice-Hall, Englewood Cliffs, 1977). 23. R. Bingham, V. D. Shapiro, V. N. Tsytovich, et al., Phys. 1. V. N. Tsytovich, Usp. Fiz. Nauk 167, 57 (1997) [Phys. Fluids B 3, 1728 (1991). Usp. 40, 53 (1997)]. 24. B. G. Gavrilov, A. I. Podgorny, I. M. Podgorny, et al., 2. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Geophys. Res. Lett. 26, 1549 (1999). Waves and High-Temperature Hydrodynamic Phenom- 25. R. E. Erlandson, P. K. Swaminathan, C.-I. Meng, et al., ena (Nauka, Moscow, 1966, 2nd ed.; Academic, New Geophys. Res. Lett. 26, 1553 (1999). York, 1966). 26. Yu. P. Raœzer, Zh. Éksp. Teor. Fiz. 37, 1741 (1959) [Sov. 3. R. Z. Sagdeev, in Reviews of Plasma Physics, Ed. by Phys. JETP 10, 1229 (1960)]. M. A. Leontovich (Atomizdat, Moscow, 1964; Consult- ants Bureau, New York, 1968), Vol. 4. 27. I. V. Nemtchinov, R. E. Spalding, V. V. Shuvalov, et al., in Proceedings of the 31st Lunar and Planetary Science 4. S. I. Popel, M. Y. Yu, and V. N. Tsytovich, Phys. Plasmas Conference, Houston, TX, 2000, p. 1334. 3, 4313 (1996). 5. S. I. Popel, V. N. Tsytovich, and M. Y. Yu, Astrophys. Space Sci. 256, 107 (1998); S. I. Popel, V. N. Tsytovich, Translated by the authors

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001

EDGE PLASMA

Structure of Fluctuations in the Floating Potential and Ion Saturation Current in the Edge Plasma of the L-2M Stellarator Yu. V. Kholnov Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia Received November 2, 2000; in ﬁnal form, January 18, 2001

Abstract—Results are presented from the measurements of the ion saturation current, the floating potential, and their fluctuations in the edge plasma of the L-2M stellarator. Distinguishing features in the distribution of the ion saturation current and the floating potential near the separatrix are revealed and examined. Based on the cross correlation measurements with probes positioned at different toroidal angles, it is concluded that fluctuations in the ion saturation current are related to fast vertical displacements of the plasma column, whereas fluctuations in the floating potential have the form of waves propagating in the radial direction. © 2001 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION In this paper, we present results from the studies of In order to better understand transport mechanisms the radial distributions of the floating potential and ion in tokamaks and stellarators, it is necessary to study the saturation current, fluctuations in these parameters, and edge plasma processes in these devices. Here, edge the cross correlation functions for probes positioned at plasma means a plasma region near the separatrix. Out- different toroidal and poloidal angles in the L-2M stel- side the separatrix, the magnetic field lines terminate at larator. the limiter, divertor plates, or vacuum chamber wall. Near the chamber wall, all of the edge plasma parame- 2. EXPERIMENTAL SETUP ters are strongly inhomogeneous. In the presence of a limiter, the direction of the poloidal rotation near the The main parameters of the L-2M stellarator are the chamber wall changes from the electron to ion diamag- following: the major radius is R = 100 cm, the mean netic drift direction [1]. The strong plasma inhomoge- plasma radius (the mean radius of the vacuum separa- neity is the main reason for the increased level of fluc- trix) is rs = 11.5 cm, and the toroidal magnetic field is tuations in the plasma parameters. BT = 1.2Ð1.4 T [6]. The edge plasma is usually investigated with Lang- The rotational transform of the L-2M magnetic con- muir probes. The validity of the probe measurements is figuration can be written in the form confirmed by reflectometry, electron-cyclotron emission ι ≈ ()2 ()4 ι measurements, and other diagnostics. At present, studies * 0.175++ 0.26 r/rs 0.27 r/rs , * = 1/q, of the edge plasma turbulence are in progress. Thus, a where r is the mean radius of a magnetic surface and q new line of investigation—the study of the statistical is the safety factor. The plasma was created and heated properties of turbulence in toroidal devices—has been with a gyrotron operating at a frequency of 75 GHz. A developed. Nevertheless, a number of problems that have microwave beam (180Ð250 kW) was launched at the been attacked for the last twenty years still remain unre- toroidal angle ϕ = 0°. The maximum electron tempe- solved. First of all, this concerns the theoretical descrip- rature was T ~ 1 keV, the average plasma density was tion of the edge plasma turbulence. e n ~ 1.5 × 1013 cm–3. In a number of studies on fluctuations in the floating e potential and ion saturation current in the edge plasma The parameters of the edge plasma in the L-2M stel- of toroidal devices (see, e.g., [2Ð4]), it was found that, larator were measured with several probe sets posi- ϕ Ӎ if the temperature fluctuations are ignored, then the tioned at different toroidal angles: in the top port at 347° and in the bottom ports at ϕ = 0°, ϕ Ӎ 231° and behavior of fluctuations in the plasma density and ϕ plasma potential cannot be adequately described by the = 90° (in Fig. 1, the corresponding probe sets are denoted as a, b, c, and d, respectively). Note that the φ˜ Boltzmann formula n˜ ~ ------. In [5], it was shown that microwave power was launched in the cross section in kTe which probe set c was installed; however, we did not only 45% of the density fluctuations correlated with the observe that this had any influence on the results of Boltzmann distribution. probe measurements.

STRUCTURE OF FLUCTUATIONS IN THE FLOATING POTENTIAL 795

Each probe set consisted of several individual ‡ probes; with these probes, we measured the poloidal and radial correlations. The probes had 2-mm-long molybdenum electrodes 0.5 mm in diameter. The probe wires were isolated with quartz pipes. The ion saturation current was measured using load resistances of 3 or 10 Ω. The recording system allowed us to analyze signals with frequencies up to 1 MHz. The rotational transform at the last magnetic surface was equal to ~0.7 × 2π. Thus, probes a and c separated by an interval equal to ~0.7 of the major circumference were located on nearly the same magnetic field line. The distance between probes b and c along the magnetic field line was longer than the major circumference by a factor of 1.5. The position of the last magnetic sur- b, c, d face r = 113 mm in the cross section of probes b was determined previously with magnetic measurements. Calculations show that, in the regions under study, the Fig. 1. Arrangement of the probes in the cross section of the outward shift of the separatrix due to the plasma pres- vacuum chamber of the L-2M stellarator. Probe a is posi- sure is relatively small (~5 mm). Note that, a similar tioned at the toroidal angle ϕ ≈ 347°, probe b is at ϕ ≈ 231°, probe c is at ϕ = 0°, and probe d is at ϕ = 90°. The last closed outward shift may be produced by an external vertical magnetic surface is schematically shown inside the vacuum Ӎ magnetic field of Bp 40 G. Unfortunately, the data chamber. from magnetic measurements for the other angles under study are lacking.

In [7], the parameters of the edge plasma were mea- V , V × 2 ~ sured with a probe positioned in the inner port in the f (a) Js 10 , A J/J 5 1.0 cross section ϕ Ӎ 90°. It was shown that the edge 3 0 plasma consists of three characteristic regions: a wall 0.5 region, a region with a high gradient of the electron –5 1 2 density, and the edge of the plasma column. The plasma –10 0 Ӎ (b) density in the wall region was on the order of ne 10 1.0 9 –3 10 cm , the electron temperature was Te ~ 2–4 eV, and 1 3 0 the floating potential Vf was close to zero. Near the separatrix, the plasma density increased by three orders of –10 2 0 magnitude. At the edge of the plasma column, the (c) plasma density increased only slightly, but the temper- 5 1.0 ature increased substantially. 3 0 1 2 0.5 –5 3. MEASUREMENTS OF THE ION SATURATION 0 CURRENT AND FLOATING POTENTIAL 80 90 100 110 120 r, mm Figure 2 shows typical distributions of the ion satu- ˜ ration current Js, fluctuations Js /Js, and the floating Fig. 2. Distributions of (1) the ion saturation current Js, potential Vf near the last magnetic surface in the cross ˜ (2) Js /Js fluctuations, and (3) floating potential Vf at the sections corresponding to probes a, b, and c. It can be plasma edge. The measurements were carried out (a) in the seen in these figures that the profiles measured in these top port by probe a, (b) in the bottom port by probe b, and cross sections are somewhat different, but their behav- (c) in the bottom port by probe c. The load in Js measure- ior is similar: the ion saturation current arises at a cer- ments is 10 Ω. tain distance from the chamber wall and increases as the probe is displaced into a plasma, whereas the current fluctuations increase toward the plasma edge, as can be seen to almost coincide with the region in which was also observed in other devices. The floating poten- the potential changes its sign. This was also observed in tial at the edge is positive (below 10 V) and varies only previous experiments in which the last closed magnetic slightly with radius; deeper in the plasma, it becomes surface was measured with probes in different cross negative and increases in magnitude. sections. The vertical arrow in Fig. 2b shows the position of The difference between the edge plasma profiles at the separatrix for probe b. The position of the separatrix different toroidal angles seems to be related, first of all,

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 796 KHOLNOV

Cii 0.4 (‡) 0.2 0 –0.2

0.4 (b) 0.2 0 –0.2

0.4 (c)

0.2

–0.2 0 0.2 τ, ms

Fig. 3. Cross correlation functions between fluctuations in the ion saturation current at probes b and c (solid line) and probes a and c (dashed line) under different experimental conditions: (a) the heating power is P = 200 kW and the vertical magnetic field is Bp = 40 G, (b) P = 200 kW and Bp = 0, and (c) P = 250 kW and Bp = 0. to an error in determining the coordinate with respect to several shots, which allowed us to enhance the contrast. the chamber wall. Second, the difference may be due to However, even this procedure did not reveal any corre- the presence of the resonant magnetic surface with lation between probes a and b, although the distance m/n = 2/1. The vacuum magnetic measurements between them along the magnetic field line was shorter. showed that this resonance takes place at a distance of ~2 cm from the last magnetic surface; however, in the Figure 4 shows the results of measurements in which presence of the plasma, the magnetic structure some- probe b was shifted radially with respect to probe c. what changes because of the plasma pressure. We can see that the correlation function varies slightly; i.e., the size of the region in which fluctuations are localized is at least ~1 cm. As can be seen in Figs. 3 and 4, 4. FLUCTUATIONS IN THE ION SATURATION the time delay in the cross correlation function is CURRENT absent, which indicates that the oscillation phase of the fluctuation signal does not change. The aim of the fluctuation measurements was to determine the structure of fluctuations in the plasma of The coefficient of correlation between the fluctua- the L-2M stellarator. To do this, we measured the cor- tions measured by the bottom probes b and c is higher relation between the probe signals. than that for probes a and b, although the distance Figure 3 shows the cross correlation functions mea- between the latter probes along the magnetic field line sured under different experimental conditions (for dif- is shorter. This indicates that long-wavelength fluctua- ferent heating powers with and without applying an tions along the magnetic field lines are absent. As is external vertical magnetic field, which leads to the shift seen from Figs. 3 and 4, the maximum value of the cor- of the plasma edge by several millimeters). The figure relation coefficient usually varies in the range 0.2Ð0.4 demonstrates a rather good correlation of fluctuations and, in some cases, attains 0.6. However, in some shots, in the ion saturation current measured with probes b the correlation between the bottom probes was signifi- and c, both located in the bottom ports. For probes a cantly lower, which indicated to the absence of a corre- (located in the top port) and c, the correlation level is lation between these probes. In this case, the relative ˜ substantially lower; nevertheless, the results of three fluctuation Js /Js increased from 10Ð15 to 30Ð40%. It different experiments show that, in this case, anticorre- may be supposed that, when the correlation between lation is observed. The correlation function Cii was cal- the signals from the probes separated by a large dis- culated by superimposing the functions measured in tance along the torus was high, the oscillation modes

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 STRUCTURE OF FLUCTUATIONS IN THE FLOATING POTENTIAL 797 with small k|| were excited. When no correlation was C ii (‡) observed, this mode, probably, was suppressed. 0.4

0.2 5. FLUCTUATIONS IN THE FLOATING POTENTIAL 0 The behavior of fluctuations in the floating potential –0.2 differs substantially from that in the ion saturation cur- –0.3 0 0.3 rent. We recall that no correlation was observed for the τ, ms ion saturation currents of probes a and b. At the same 0.4 (b) time, the cross correlation function between the floating potentials measured with these probes Cvv was rather 0.2 high, as is seen from Fig. 5. Figure 5a shows how the cross correlation function Cvv varies as the top probe shifts by ~3 cm. The results correspond to two positions 0 of probe b. We note that C varies with radius by a 100 110 120 vv r, mm sinusoidal law. A similar situation is observed in Fig. 5b, which shows a fragment of such sinusoidal τ Fig. 4. (a) Cross correlation function Cii( ) between fluctu- oscillations as the bottom probe c shifts relative to ations in the ion saturation current measured with probes b probe b, and in Fig. 5c, which represents another series and c positioned in bottom ports (rb = 110 mm, rc = 95 mm) of experiments in which the bottom probe d positioned and (b) Cii(0) vs. the radial position of probe b for rc = at ϕ = 90° shifts relative to probe b. It should be noted 95 mm. that the same distribution was observed independent of whether the measurements of the ion saturation current were carried out; i.e., this effect cannot be attributed to Cvv(0) plasma perturbation caused by the measurements of the 0.3 (‡) ion saturation current. 120 0 Variations in the cross correlation function Cvv with 100 110 130 radius can be interpreted as variations in the oscillation –0.3 phase, i.e., as the radial propagation of a wave. As fol- 0.3 (b) lows from Fig. 5a, the characteristic wavelength of the 0 wave propagating in the transverse direction is ~3 cm 90 100 110 120 and the characteristic period, which is determined by µ –0.3 the correlation time, is ~100Ð150 s. 0.3 (c) To understand the character of the radial wave prop- 0 agation (i.e., to determine whether the wave is standing 100 110 120 130 or running), we measured the correlation functions for –0.3 r, mm two probes of set b that were spaced by 4 mm in the radial direction. Figure 6 shows the correlation func- Fig. 5. Cross correlation function Cvv(0) between the float- tions calculated using the results of four series of exper- ing potential signals from (a) probes a and b vs. the position iments. In all four series, the delay times in the correla- of probe a for two positions of probe b, rb1 = 110 mm (left tion functions point to the radial propagation of fluctu- curve) and rb2 = 108 mm (right curve); (b) probes b and c ations. The sign of the delay time corresponds to the vs. the position of probe c for the fixed probe b (rb = outward propagation, and the velocity estimated from 110 mm); and (c) probes b and d vs. the position of probe d the delay time of the correlation function (3Ð5 µs) is for the fixed probe b (rb = 100 mm). ~105 cm/s. neutral gas flux from the wall increases, the point at 6. DISCUSSION OF THE RESULTS which Vf changes its sign shifts toward the plasma center. This allows us to suppose that the region in The measurements of the distributions of the aver- which the potential is positive and varies only slightly age values of the ion saturation current and the floating with radius is associated with ionization processes in potential showed that the last magnetic surface calcu- the plasma. Previous measurements of the electron lated with account for the plasma pressure nearly coin- temperature showed that the increase in Te correlates cides with the boundary of the region in which we with the decrease in V . observe the increase in the ion saturation current. The f floating potential in this region remains positive and There is another plausible explanation of the differ- varies slightly in magnitude. Deeper in the plasma, the ence between the observed Js and Vf distributions. The potential drops sharply. In [8], it was shown that, as the stable last closed magnetic surface is produced by the

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 798 KHOLNOV

related to the effects observed with the probes posi- Cvv (‡) 0.5 tioned at different toroidal angles. 0 (b) 7. CONCLUSION 1.0 0.5 (i) It is shown that, at any toroidal angle, the distri- 0 bution of the floating potential Vf in the edge plasma is (c) the same. Two radial regions may be distinguished: the 0.5 region in which the potential is positive and varies only 0 slightly and the region in which the potential is negative and its magnitude increases toward the plasma center. –0.5 (d) The plasma density becomes measurable in the region 0.5 where the potential is positive. 0 (ii) It is shown that, in the edge plasma of the L-2M –0.01 0 0.01 0.02 stellarator, the distributions of n˜ /n (which corresponds τ, ms φ˜ to Js fluctuations) and /kTe (which corresponds to Vf Fig. 6. Cross correlation functions Cvv(τ) between the float- fluctuations) are different. This indicates that fluctua- ing potential signals from two probes in set b that are spaced tions are not electrostatic drift fluctuations. The corre- by 4 mm in the radial direction for different positions of the lation between Js fluctuations at different radii evi- first probe: (a) r1 = 106 mm, (b) r2 = 104 mm, (c) r3 = dences the fast vertical displacements of the plasma 102 mm, and (d) r4 = 110 mm. column as a whole. external magnetic field, and the behavior of Vf outside ACKNOWLEDGMENTS this surface depends on the high plasma conductivity along the magnetic field lines. As for the electron den- I am grateful to K. A. Sarksyan and O. I. Fedyanin sity, the diffusion density distribution at these radii may for useful discussions. be related to the particle loss across (rather than along) magnetic field lines. REFERENCES Fluctuation measurements confirmed that the spatial distributions of fluctuations in the ion saturation current 1. C. Hidalgo, M. Pedrosa, B. van Milligen, et al., and fluctuations in the floating potential are different, as J. Plasma Fusion Res. 1, 96 (1998). was observed in many devices. The correlation between 2. S. J. Levinson, J. M. Beal, E. J. Powers, and R. D. Bengt- Js fluctuations measured by the bottom probes, the son, Nucl. Fusion 24, 527 (1984). weak dependence of this correlation on the radius, and 3. H. Zushi, T. Mzuuchi, O. Motojima, et al., Nucl. Fusion the anticorrelation between the signals from the top and 28, 433 (1988). bottom probes allow us to suggest that the effects observed can be attributed to the displacement of the 4. R. R. Domínguez and R. E. Waltz, Nucl. Fusion 27, 65 plasma column. The characteristic time of correlation (1987). between the signals from the top and bottom probes 5. V. Pericoli-Ridolfini, A. Pietropaolo, R. Cesario, and decreases during discharge (~10 ms) from 60Ð100 to F. Zonca, Nucl. Fusion 38, 1745 (1998). 15Ð30 µs and increases after the heating pulse ends. 6. V. V. Abrakov, D. K. Akulina, E. D. Andryukhina, et al., Hence, we may suppose that the fluctuations are fast Nucl. Fusion 37, 233 (1997). vertical displacements of the plasma column. 7. M. S. Berezhetskii, V. P. Budaev, and R. S. Ivanov, As was shown above, Vf fluctuations have the form J. Nucl. Mater. 162/164, 831 (1989). of waves propagating in the radial direction. The possi- 8. G. M. Batanov, L. V. Kolik, A. E. Petrov, et al., Pis’ma bility of the radial propagation of fluctuations is con- Zh. Éksp. Teor. Fiz. 68, 560 (1998) [JETP Lett. 68, 585 firmed by the results of [9], in which it was shown that (1998)]. the coupling between modes can provide a mechanism 9. X. Garbet, L. Laurent, A. Samain, and J. Chinardet, for such propagation. Unfortunately, we cannot cer- Nucl. Fusion 34, 963 (1994). tainly state whether these waves are running or standing, because the shifted correlation functions in Fig. 6 may correspond to other oscillation modes that are not Translated by N. F. Larionova

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001

LOW-TEMPERATURE PLASMA

Structures of the Plasma Channels in a Besselian Beam L. N. Pyatnitsky Associated Institute for High Temperatures, Russian Academy of Sciences, Izhorskaya ul. 13/19, Moscow, 127412 Russia Received February 28, 2001

Abstract—A mechanism for the formation of the structure of an optical discharge in Besselian laser beams is proposed on the basis of analyzing numerous experiments. The discharge structure is determined by the peri- odicity of the ﬁeld of a Besselian beam in the radial and longitudinal directions and also depends on the power and duration of the heating pulse. In the initial stage of the plasma channel formation, the conﬁguration of the channel inhomogeneities follows the discharge structure. If the spatial scale of the discharge structure is small, then the developing channel evolves into a homogeneous state. The time required for the structural inhomogeneities of the plasma channel to be smoothed out is estimated as a function of their scale length. © 2001 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION zeros of the Bessel function, J0(xm) = 0. The radius of γ ≈ The problems related to laser spark plasmas have the central region is equal to x0 = 2.40, which gives λ λ ≈ µ µ been widely discussed in the literature (see, e.g., [1, 2]). 0.4 /a0. Thus, for 1 m, 2a0 = 50 m, and 2R = Laser sparks are usually produced by a spherical lens, 4.5 cm, we have γ ≈ 1°. A comparison between these µ which focuses laser light into a small-diameter Gauss- two beams for 2a0 = 50 m shows that the cross-sec- ian beam. However, the distance L over which the tional area of a Gaussian beam increases tenfold over a intensity of such a beam can be assumed to be constant distance of L ~ 2 mm, while the cross section of a is restricted by the phenomenon of diffractive spread- Besselian beam remains constant over the entire focal ing. The divergence angle γ of the beam depends on the distance L ≈ 130 cm. A photograph of a laser spark in a λ ratio of the laser wavelength to the beam radius a0, Besselian beam with the above parameters is shown in γ λ ӷ ~ /a0, so that, over distances L a0, the beam radius Fig. 1 [5]. γ can be represented as a = L. The very high axial symmetry of Besselian beams The diffractive spreading, which is in principle makes it possible to overcome several challenging unavoidable, can, nonetheless, be compensated by problems in developing new plasma-based technolo- forming a laser beam with a conical wave front con- gies [5Ð9], such as super-high-speed commutation, the verging toward the beam symmetry axis at an angle γ. generation of short-wavelength laser pulses, the plasma A conical wave front is produced by a conical lens, or acceleration of charged particles, and inertial confine- an axicon (see, e.g., [3]). Over the focal distance L = ment fusion. In the latter case, the use of Besselian laser R/ tanγ behind an axicon with an aperture R, the beam beams should automatically provide a uniform irradia- radius a0 is independent of the wavelength and remains tion of laser targets (recall that, in spherical geometry, essentially constant. The radial distribution of the laser the degree of nonuniformity should be no higher than 1Ð2% in order for instabilities not to develop [10]). field is described by the Bessel function J0(x) [4], γ π λ where x = krsin , k = 2 / , and r is the distance from However, the matter is not so simple as it was origi- an arbitrary spatial point to the beam axis. nally thought to be. The photograph shown in Fig. 1 The field of such a Besselian beam has maximums was taken with a long exposure time (the camera shut- in the cylindrical regions whose boundaries xm are the ter was held open) in the intrinsic light from a plasma

0 50 cm 100 cm

Fig. 1. Laser spark produced by a Besselian beam with the parameters γ = 1°, λ = 1.06 µm, and R = 2.25 cm.

800 PYATNITSKY heated by a 200-J, 50-ns laser pulse. Further investiga- full width at half-maximum (FWHM) of the laser tions revealed several different regimes of breakdown pulse, and a2L ~ γ–3. of a gas by a Besselian beam (including the running- focus regime [6]) and the formation of structures with Figure 2 shows two instantaneous shadowgraphs of unusual configurations in the developing plasma chan- parts of the plasma channels produced by a laser pulse with the energy E = 70 J and duration τ = 40 ns in a nel. The formation and evolution of such structures γ indicate the onset and development of instabilities. In Besselian beam with the convergence angle = 7.5° order to gain insight into the nature of this phenome- (L = 17 cm). The shadowgraph of a channel in Fig. 2a non, the structures of plasma channels at different was recorded 10 ns after the breakdown of air at atmo- stages were investigated in a number of gases at pres- spheric pressure, and the shadowgraph in Fig. 2b shows sures from 0.05 to 10 atm (see, e.g., [7, 9]). It was estab- an analogous channel at the same pressure but in argon lished that the structures observed depend on the prop- 20 ns after the breakdown. It can be seen that the erties of Besselian beams [4]. The objective of this plasma channel that forms in air (Fig. 2a) is continuous study is to clarify the mechanism for the formation of and uniform; it also remains uniform throughout all structures with different configurations. subsequent stages of the discharge. In contrast, the channel in argon (Fig. 2b) has a beaded structure even 20 ns after the breakdown; to a greater or lesser extent, 2. STRUCTURES OF THE PLASMA CHANNELS the channel remains beaded throughout its lifetime. This structural difference can only be explained by The structures of plasma channels were investigated the fact that the intensity of the Besselian beam in argon using laser plasma diagnostics. The distribution of opti- is far above the threshold level, because, under the same cal inhomogeneities in a plasma channel was visualized conditions, the threshold for breakdown in argon is by shadowgraphy, schlieren photography, and interfer- lower in comparison with that in air. If this explanation ometry. The optical inhomogeneities were recorded by is true (i.e., if the structure of the forming plasma chan- image converter cameras and charge-coupled device nel is actually affected by the intensity of the Besselian (CCD) cameras. In the initial stage of the process, when beam), this characteristic feature should also persist in the elements of the structure were still small-scale and other gases. This important conclusion was verified in a the breakdown centers were opaque to probing radia- special series of experiments with a step profile of the tion, the structure of the plasma channels was deter- specific power along the beam. mined from the scattering of heating or probing radiation. These experiments were carried out with a profiled axicon capable of producing beams with a conical wave When comparing the plasma channels formed in front whose convergence angle γ varied in the range Besselian beams with different parameters, it should be 13° ≤ γ ≤ 20°. Over a focal distance of L ≈ 10 cm, the kept in mind that the longitudinal profile I(z) of the axicon focused approximately one-quarter of the laser beam intensity depends on the form of the axicon gen- energy onto a 1-cm-long interval, thereby providing the eratrix and the radial profile of the intensity of the inci- conditions under which the heating radiation intensities dent radiation. For a straight-line generatrix and a rect- at the neighboring sites of the focal region differed by angular (or Gaussian) profile, the longitudinal profile several times in the course of a discharge. Also, in these I(z) is peaked at z ≈ L (or, respectively, at z = L/2). An experiments, the laser pulse energy was increased to exact expression for the beam intensity will be ana- E = 150 J (which corresponded to τ = 40 ns). In Fig. 3, lyzed below. To compare the results obtained under dif- which shows a schlieren photograph of part of the ferent conditions, we can use the approximate formulas plasma channel in air at atmospheric pressure, the zone for the radiation intensity w ~ (E/τ)γ3 and the specific of elevated pulse intensity is indicated by two small energy ε ~ Eγ 3, where E is the laser pulse energy, τ is the strips, installed in advance under the Besselian beam.

(‡) (b)

2 mm 2 mm

Fig. 2. Plasma channels (a) 10 ns after the breakdown in air at a pressure of 1 atm and (b) 20 ns after the breakdown in argon at the same pressure for E = 70 J, τ = 40 ns, and γ = 7.5°.

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 STRUCTURES OF THE PLASMA CHANNELS IN A BESSELIAN BEAM 801

1 cm

Fig. 3. Plasma channel in a beam focused by a proﬁled axicon (13° ≤ γ ≤ 20°).

80 (‡) 100 120 140

(b) 80

100

120

0 50 100 150 200 250 Pixel number

τ γ Fig. 4. Interferograms of a channel produced in N2O at pressures of (a) 0.27 and (b) 0.67 atm for E = 0.6 J, = 100 ps, and = 18°.

The schlieren photograph was taken 180 ns after the zone where the intensity of the heating radiation beginning of the heating pulse. On the left side of the changes in a jumplike manner. In what follows, we will photograph, one can distinctly see a transparent homo- assume that the perturbation waves may also be driven geneous channel with a sharp boundary. Approaching in local zones of elevated radiation intensity. the indicated region, the increase in the specific input energy is accompanied by a gradual increase in the In a relatively late stage of the channel evolution plasma channel diameter. However, in the indicated (about 160 ns after the beginning of breakdown and region, where the pulse intensity is elevated, this ten- later), it is impossible to recognize the original seed dency is far more pronounced: the diameter of the chan- small-scale perturbations, which, presumably, had nel increases abruptly and the channel itself becomes enough time to damp. On the other hand, they may opaque to probing radiation, thereby indicating the influence the channel structure, in which case, however, change in the structure. We can thus conclude that the the heating pulse should be long enough so that, over channel structure may actually change when the laser the time of its action, the perturbation wave can travel a distance comparable with the characteristic scale length intensity becomes significantly higher than the thresh- τ old intensity. of the Besselian beam. Thus, for a pulse duration of = 40 ns and velocity v ~ 2 × 106 cm/s, this distance is In Fig. 3, we can also see a perturbation that about r ≈ 1 mm; this indicates that, for the primary per- emerges from the region of elevated pulse intensity and turbations to be observable in a beam of radius a = propagates to the left along the channel. Judging from 10 µm, the pulse should be made shorter by at most two the image of the projection of the perturbation onto the and a half orders of magnitude. plane of observation, the perturbation is spherical in shape and develops near the boundary of the indicated Plasma channels created by short laser pulses were region. The distance the perturbation passed over a time investigated in a device [11] operating at the laboratory of about 160 ns is estimated to be about 0.3 cm, which headed by Prof. H.M. Milchberg (University of Mary- indicates that the perturbation propagated at a mean land, USA) in joint experiments with the participation velocity of v ~ 2 × 106 cm/s. These results make it pos- of L.Ya. Margolin and the author of this paper. The sible to identify the perturbation as a wave driven in the channels shown in Fig. 4 were produced in nitrous

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 802 PYATNITSKY oxide, which was chosen as the working medium we again arrive at the conclusion that, for short heating because of the low threshold for optical breakdown and pulses, the channel structure should change when the thus provided the possibility of investigating the forma- beam intensity becomes higher than the threshold tion of the channels in a broad pressure range. The intensity. parameters of the Besselian beams were as follows: λ = Note that the pattern of the displaced fringes in 1.06 µm, E = 0.6 J, τ = 100 ps, and γ =18°. The diameter Fig. 4b provides evidence for numerous discrete pri- µ of the central part of such beams was 2a0 = 2.6 m, and mary breakdown centers. Therefore, we can assume the beam length was L = 1.5 cm. The state of the chan- that, on the whole, the channel is formed against the nels illuminated with 70-ps, 0.53-µm laser pulses was background of the perturbation waves driven by the pri- monitored by a MachÐZehnder interferometer. The mary breakdown centers and that the onset and the con- magnified interferogram images were recorded by a figuration of these centers play a dominant role in the 512-µm-long CCD camera. In Fig. 4, the numerals formation of the channel structure. denote the numbers of the pixels of the CCD camera, each of the pixels being 1.6 µm in size. 3. CONFIGURATIONS OF THE OBSERVED The interferograms shown in Figs. 4a and 4b were BREAKDOWN CENTERS taken at pressures of 0.27 and 0.67 atm, respectively. The contours of the plasma channels are seen against It was most expedient to display the configuration of the background of the interference fringes of equal the primary breakdown centers in the scattered laser inclination. The amount by which the fringes are dis- light. This made it possible to visualize, first of all, per- placed reflects the value of the plasma density. The turbations in which the electron density is the highest channels were created by identical Besselian beams and to automatically satisfy the synchronization condi- and were recorded 250 ps after the beginning of the tions. The representative images of the distribution of heating pulse. Nevertheless, the channel in Fig. 4a the primary breakdown centers in Besselian beams with appears to be completely uniform, while the interfero- different parameters during breakdown in air and argon gram of the channel in Fig. 4b provides evidence for are shown in Fig. 5, in which the wave front propagates highly developed perturbations. The threshold for from left to right. breakdown at a pressure of 0.67 atm (Fig. 4b) should be Figure 5a illustrates the breakdown in argon at a lower than that at a pressure of 0.27 atm (Fig. 4a). Thus, pressure of 0.2 atm in a Besselian beam with the param-

(‡) 10 cm (d)

(b) 1 mm (e)

Fig. 5. Primary breakdown centers in (a) argon at a pressure of 0.2 atm (E = 10 J, τ = 0.8 ns, γ = 1°), (b) air at a pressure of 1 atm (E = 17 J, τ = 0.8 ns, γ = 2.5°), (d) air at a pressure of 1 atm (E = 20 J, τ = 20 ns, γ = 5°), (e) air at a pressure of 1 atm (E = 70 J, τ = 40 ns, γ = 7.5°), and (f) argon at a pressure of 1 atm (E = 70 J, τ = 40 ns, γ = 7.5°). Frame (c) is a magniﬁed fragment of image (b).

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 STRUCTURES OF THE PLASMA CHANNELS IN A BESSELIAN BEAM 803 eters E = 10 J, τ = 0.8 ns, and γ = 1°. The images cover channel structure is herringbone-shaped. The spatial a beam portion of length 50 cm, the total length of the period of the extensions in the axial direction is about beam being about L ≈ 130 cm. Closer to the axicon (on l ≈ 0.12 mm, and the extensions themselves are sepa- the left side of the figure), the beam intensity is slightly rated from the discrete breakdown centers. above the threshold and is about one-quarter of the The structure of the plasma channels is most pecu- beam intensity averaged over the beam length. In this liar in optical discharges in argon. Figure 5f illustrates region, the breakdown centers occur at the beam axis an optical discharge driven in argon by a Besselian and form a periodic sequence of points with a spatial beam with the same parameters as in Fig. 5e (E = 70 J, ≈ separation of about l 7 mm. Farther away from the τ = 40 ns, γ = 7.5°). The most pronounced structural axicon, the beam intensity increases and the individual inhomogeneities of the channel are funnel-shaped or breakdown centers start to merge, tending to form a beaded, the spatial separation between them being larger continuous breakdown channel. than 1 mm. It is well known [1] that, in argon, the thresh- × 10 2 Figure 5b illustrates an optical discharge in air at old intensity for breakdown (Ith = 1.5 10 W/cm ) is × 10 2 atmospheric pressure in a beam with the parameters lower than that in air (Ith = 8 10 W/cm ) by a factor E = 17 J, τ = 0.8 ns, and γ = 2.5° (L ≈ 52 cm), the beam of 5.3. That is why, for the same beam parameters, the intensity being two orders of magnitude higher than excess of the beam intensity above the threshold in that for a discharge in argon. Judging from the image in argon plays a greater role and the structure of the Fig. 5b, we may speak of the structural blocks rather plasma channels in argon (Fig. 5f) differs radically than of the breakdown centers. The spatial separation from that in air (Fig. 5e). However, from Fig. 5f, we can between the blocks along the beam axis is about l ≈ see that, in argon, the breakdown centers also form a 1.1 mm. In turn, each block consists of smaller cells periodic structure whose period in the axial direction with a size of 0.02–0.05 mm. Figure 5c is a magnified (l ≈ 0.12 mm) coincides with the period of the channel fragment of the image shown in Fig. 5b. We can dis- structure in air (Fig. 5e). tinctly see the individual cells and the lines formed by According to the above experimental data, the struc- them. The angle β at which the lines are inclined to the γ ture of the plasma channels was always observed to be beam axis is significantly larger than . For this reason, periodic in the longitudinal direction. The spatial scale the appearance of the lines of cells cannot be explained of the structure is independent of the sort of gas, the gas as being due to the propagation of the breakdown front pressure, and the energy and duration of the heating upward laser radiation, as is usually done when investi- pulse and is determined exclusively by the convergence gating optical discharges at the focus of a spherical angle γ of the conical wave front of a Besselian beam. lens. Other types of structures, with larger and smaller char- The images shown in Figs. 5aÐ5c were obtained acteristic scale lengths, were also observed. It was with relatively short heating pulses in Besselian beams established that the properties of a larger scale structure with small convergence angles γ. Next, Figs. 5dÐ5f depend not only on the parameters of the Besselian illustrate the action of beams with modified parameters beam but also on the sort of gas. For a smaller scale and longer pulses. Thus, Fig. 5d shows the image of the structure, this dependence is far less obvious. However, structure of a plasma channel created in air at atmo- a detailed analysis of the experimental data shows that, spheric pressure by a Besselian beam with the parame- to a greater or lesser extent, the smaller scale structure ters E = 20 J, τ = 20 ns, and γ = 5° (L ≈ 26 cm). As com- always manifests itself in plasma channels produced by pared to the experiment illustrated in Fig. 5b, the beam Besselian beams. For this reason, we briefly address the intensity is lower by a factor of about three, but the spe- structure of the Besselian beam itself. cific energy of radiation is one order of magnitude higher. The structure of the breakdown discharge is again periodic in the axial direction, but the spatial 4. STRUCTURE OF THE BESSELIAN BEAM period is shorter (l ≈ 0.28 mm) and the structural blocks, The formation of a periodic sequence of breakdown which again occurred in the axial region, are now con- centers along the symmetry axis was observed when tinuous plasma formations. the first attempt was made to create a plasma channel in For a higher specific energy of radiation and longer the field of a Besselian beam [3]. Originally, the point laser pulses, the configuration of a breakdown dis- breakdown centers were explained as a result of mis- charge in air at atmospheric pressure becomes radically alignment. Only after the analysis of the streak images different. Figure 5e shows the image of an optical dis- of breakdown propagation [12], was the appearance of charge driven in a Besselian beam with the parameters the breakdown centers attributed to the laser field distri- E = 70 J, τ = 40 ns, and γ = 7.5° (L ≈ 17 cm). As com- bution formed behind the axicon in a nonlinear pared to Fig. 5d, the beam intensity in this discharge is medium. higher by a factor of 6 and the specific energy is higher The properties of the medium are determined by its by a factor of 12. The channel structure is characterized dielectric function by the appearance of extensions, which are inclined to εε ε ε ()2 the symmetry axis at an angle of β ӷ γ; as a result, the = 0 ++i '' NL E , (1)

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 804 PYATNITSKY

(‡) (b) E ------r3 E0

r ()0 1 1.5 E ------0 Ecr r0 1.0 0.812 0.749 r2 0.5 0.687 r 4 036912z/l 0.625

Fig. 6. Field structure of a Besselian beam with γ = 5°: (a) the measured radial field distribution and (b) the longitudinal field profiles calculated for a cubic nonlinearity of the gas. where the imaginary part ε'' describes the wave damp- gradually in the axial direction, ing and the nonlinear functional ε is determined by NL () π()λ 1/2 γ ()γ ()π the equation of state of the matter and describes the E0 z = 2 z/ sin Ein zsin exp Ði /4 . (5) characteristic features of the interaction. Consequently, behind the axicon, the field distribution By analogy with the focusing of a Gaussian beam can be described with good accuracy by the Bessel γ [13], we describe the field distribution behind the axi- function J0(krsin ): con in a medium with the nonlinear dielectric function () (1) by the following equation for the complex field E 0 ()rz, = E ()z J ()krsinγ , (6) amplitude E(r, t) = Re{eE(r, z)exp[–i(ωt – kz)]} [4]: 0 0 in which case the beam intensity can be represented as ∂ ∂ ω 2 E 1E []ε ε ()2 I ~ E 2 (z)(J 2 krsinγ). The photograph of the beam field 2ik------∂ ++---r------∂ ---- i '' + NL E E = 0,(2) 0 0 z r r c distribution in the experiment with a Besselian beam γ 2 ω 2ε ( = 5°) is shown in Fig. 6a, where the calculated values where k = ( /c) 0. We supplement this equation with of the zeros of the Bessel function rm (where m is the the following boundary condition for the focusing of number of the ring) are indicated at the scale on the laser radiation by an axicon with aperture R: right. (), () ()γ Erz= 0 = Ein r exp Ðikrsin . (3) The properties of a Besselian beam in a nonlinear interaction with the medium are determined by the non- ≤ 1/2 ε | |2 Here, Ein(r > R) = 0 and Ein(r R) = I (r), where I(r) linear component NL( E ) of the dielectric function of is the radial profile of the intensity of the incident beam. the medium. The particular form of the expression for ε depends on the degree of ionization, the electron In the axial (r < zsinγ, kr2 < z) zone of the focal NL temperature, the relationship between the electron region of the axicon (λ/sin2γ Ӷ z < L), the linear (ε'' = ε (0) mean free path and the scale on which the beam field NL = 0) solution E to Eq. (2) with the boundary con- varies, etc. In the strong field of a Besselian beam, the dition (3) has the form dielectric function of the gas with excited atoms can be considered to be a locally nonlinear power function ()0 (), (),,, E rz = Ea fkraz because of the multiphoton transitions from the meta- () ()()γ2 (4) stable state in gas atoms. According to [14], for the non- + E0 z exp Ðikz/2 sin linear effects to come into play when the beam intensity × []()γ ()γ γ 11 2 J0 krsin Ð irJ1 krsin /2zsin . at the axis is about 10 W/cm , it is sufficient that the fraction of the atoms in the excited metastable state be –3 Here, the first term Ea f, which accounts for the diffrac- 10 of the total number of gas atoms. tion at the edge of the axicon, is as small as (λ/z)1/2 com- During the avalanche ionization of the gas, the non- pared to the second term, which describes, to within a linear effects depend on the properties of the produced small term on the order of r/(2zsinγ), the beam field plasma. In a weakly ionized plasma, the electron tem- behind the axicon. The transverse (with respect to the perature changes as a result of the electron heating due beam axis) component of the wave vector of the beam to inverse-bremsstrahlung absorption and the energy field is equal to k⊥ = ksinγ (and, accordingly, δk|| = loss associated mainly with electronÐneutral collisions. 2γ λ 1/2ksin ), and the amplitude E0(z) of the field varies If, in this case, the electron mean free path e is small

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 STRUCTURES OF THE PLASMA CHANNELS IN A BESSELIAN BEAM 805 compared to the scale length LE on which the electro- odic structure whose period l = 0.28 mm coincides with λ δ 1/2 δ the period l obtained from the experimental data illus- magnetic field varies, e/LE < e (where e is the frac- trated in Fig. 5d. tion of the energy of an electron that is transferred to a neutral atom in a collision event), then the nonlinear For a particular nonlinear dielectric function, the part of the dielectric function ε depends on the extent to field of a Besselian beam may be either below or far which the electron density ne deviates from its initial above the critical field. The weaker the beam field, the δ smaller the modulation depth of the structure is. In con- quasisteady value n0, n = ne – n0, so that the nonlinear ε δ trast, as the beam field increases above the critical level, part NL = – n/nc of the dielectric function can be written as a large-scale structure with a period of l1 ~ 10l [4] forms simultaneously with the main structure with period (8). n 2 The configuration of a breakdown discharge in Fig. 5f ε = -----0 Ᏹ . (7) NL n corresponds precisely to this doubly periodic structure c ≈ ≈ with the periods l 0.13 mm and l1 1.4 mm (on aver- Ᏹ πδ 1/2 Here, = E/ET, where ET = (12 enÒT) is the plasma age), which are close to the calculated periods l ≈ field characteristic of thermal nonlinearity, nc = 0.124 mm and l1 = 1.24 mm. According to Fig. 5f and ω2 π 2 me /4 e , and n0 and T are the quasisteady electron theoretical predictions, the large-scale modulation δ 1/2 λ manifests itself not only in the axial focal region of the density and temperature. For e < e/LE < 1, the non- beam but also in the ring regions of the radial distribu- linear effects become nonlocal, in which case the tion of the beam field intensity. For this reason, the ε Ᏹ dependence of the nonlinear functional NL on can be diameter of the doubly periodic structure is larger than determined from the electron heat-conduction equa- that in the absence of the large-scale modulation; this tion. Expression (7) is also valid for a relatively hot conclusion is confirmed by a comparison of the break- λ |δ | plasma such that e > LE ( n/n0 < 1), in which case, down structures illustrated in Figs. 5e and 5f. however, the field amplitude Ᏹ should be normalized to π 1/2 the plasma field Es = (16 ncT) characteristic of the instability related to ponderomotive effects. 5. MECHANISM FOR THE FORMATION OF THE STRUCTURE OF AN OPTICAL For different power-law local nonlinearities (as well DISCHARGE as for a nonlocal response function of the medium), an analysis of the solution to Eq. (2) [4] shows that, in the We begin our analysis of the structure of an optical longitudinal profile E0(z) of the field amplitude, there is discharge by considering a plasma channel produced by a longitudinal structure with the characteristic scale a 0.6-J laser pulse with τ = 100 ps. The structure of this length plasma channel is shown in Fig. 4b. Because of the small dimensions of the primary perturbations and very π δ λ 2 γ l ==2 / k|| 2 /sin . (8) short times during which they developed, we failed to This structure forms when the radiation power in the record their positions and measure their parameters. central region and, accordingly, in each ring region of Hence, let us turn to the interferogram shown in Fig. 4b the radial distribution (6) corresponds to the value of and try to reconstruct the discharge structure by calcu- |Ᏹ| Ᏹ lations. that is close to the critical value cr for self-focusing. Thus, for the dielectric function with the nonlinear In the interferogram, the amount by which the part (7), this value lies in the parameter range fringes of equal inclination are displaced reflects the structure of the perturbations. The interferogram ()2 n 1 E 0 obtained with the CCD camera contains 56 fringes 0.2 Շ ------0 ------≤ 1, (9) n Ð n 2 γ 2 along the z-axis, so that, for a given radius r, we can c 0 sin Ecr construct the displacement of the interference fringes δ where E(0) is the value of the linear solution to Eq. (2) (N) as a function of N = z/h (the z-coordinate normal- at the axis and E = E . ized to the fringe spacing h). This dependence reflects cr T the distribution of the density perturbations along the For the experimental conditions of Fig. 5d, the channel. Figure 7 displays the dependence δ(N) calcu- numerical solution to Eq. (2) with the boundary condi- lated for different values of the ratio r/R, where 2R = tion (3) and the dielectric function with the nonlinear 41.0 µm is the mean diameter of the plasma channel part (7) is illustrated in Fig. 6b, which shows the longi- shown in Fig. 4b. In Fig. 7, the upper plot characterizes | | tudinal profiles of the dimensionless field E /E0 calcu- the displacement of the fringes far from the channel (at lated as a function of the dimensionless length z/l at the r/R = 5) and demonstrates that the fringes are in fact symmetry axis for four values of the parameter spaced periodically. The middle plots (from top to bot- | (0)| E /EÒr. We can see that, in the parameter range (9), the tom) show the displacement of the fringes at the chan- electric field in the plasma channel produced by a nel axis (r/R = 0) and at the relative radii r/R = 0.25, Besselian beam actually possesses a longitudinal peri- 0.50, and 0.75. The lower plot (at r/R = 1) characterizes

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 806 PYATNITSKY

Displacement of the fringes, pixel number down centers are dispersed among the beam intensity 5 r/R = 5.00 maxima (there are 24 points over a distance equal to 0 512 µm). For convenience, when comparing the calcu- –5 r/R = 0 lated results with the measured data, we normalize the 5 z coordinate and the amplitude of pulsations of the 0 plasma density to the fringe spacing h in the z direction –5 and investigate the channel structure over the same r/R = 0.25 5 interval {z1, z2} = {0, 55} that was observed experimen- 0 tally. We also take into account the fact that the dis- –5 r/R = 0.50 placement of the interference fringes corresponds to the 5 total phase increment along a chord of the cylindrical 0 channel. We specify the chord by the coordinates z = –5 (z2 – z1)/2 and r0 = R/2, in which case, along half of the 5 r/R = 0.75 ϕ 0 chord, the y coordinate changes from 0 to r0 tan , –5 where ϕ is the azimuthal angle. The summation along 5 r/R = 1.00 the entire chord should include all perturbations that are 0 driven at the points pz within the interval {z1, z2} on the –5 τ 0102030405060axis. Over a time interval t0 less than the duration of Distance along the channel axis, number of the fringe the heating pulse, the points pz may obey a random temporal distribution pt. We normalize the quantities pt, t , and τ to the time of observation of the channel, Fig. 7. Longitudinal profiles of the amplitude of pulsations 0 of the plasma density in the channel shown in Fig. 4b at dif- 250 ps, so the dimensionless duration of the pulse is ferent dimensionless radii r/R. equal to τ = 0.4. Let us write out the complete set of initial parameter the radial displacements of the upper and lower bound- values for our problem: aries of the channel. {},, τ This distribution of the density perturbations con- a ===0.063 0.145 0.227 , k 25, 0.4, {}, {}, firms the discrete nature of gas breakdown. Every t0 ==0.4, z1 z2 Ð561 , (10) breakdown center gives rise to a region in which the {}, {}, ()π () plasma temperature and density are both elevated and y1 y2 ==0 r0 tan /3 , fr 1, which starts expanding as a spherical ionization wave. m ===11, Q 1, q 0. In turn, every spherical wave is an expanding spherical layer with pulsating hydrodynamic parameters [15]. Along with the values specified above, this set includes The superposition of these waves gives rise to the the following parameters: the number k of perturbations observed pulsations of the plasma density. It seems rea- and the length of the interval of the channel under con- sonable to suppose that the breakdown centers form at sideration, {z1, z2} (with allowance for the processes at the points at which the beam intensity is maximum. the ends); the profiles f(r) (r ≤ a) of the parameters of According to Section 4, the symmetry properties of the primary perturbations; the number m of terms in the the problem enable us to describe the structure of a summation along the chord; the displacement Q of the Besselian beam by two coordinates, specifically, the entire uniform sequence of the points pz of primary per- longitudinal and radial coordinates, z and r. The longi- turbations (Q < h); and the range q of random devia- tudinal structure of the beam is characterized by a tions of the coordinates pz of the points in a partially sequence of points at which the beam intensity is max- destroyed periodic sequence of primary perturbations. imum and which are separated by the distance l = Since the time required for the wave front of a Besse- 2λ/sin2γ (or l ≈ 21 µm for the case of Fig. 4b). The radial lian beam to cross the interval {z1, z2} is equal to 1.6 ps structure of the beam is characterized by a sequence of (t = 0.01), we neglect the delay between the heating rings whose dimensionless (normalized to the channel µ pulse and the breakdown discharge driven by it and radius R = 20.5 m) radii (a = 0.063, 0.145, 0.227, ….) assume that the energy release in the breakdown is correspond to the zeros of the Bessel function. The instantaneous. question then arises: what are the rings at which the breakdown occurs? This question can be answered by Let us consider an initial (t = 0) breakdown center in reference to numerical modeling [16]. the form of a ball of radius ‡. Let the ball be centered at We assume that the breakdown centers appear either the origin of the coordinates. Inside the ball (r < a), the in the central focal region of the beam or at a structural perturbed density is described by an arbitrary function ring with the corresponding diameter 2a . We also f(r). Outside the ball (r > a), the unperturbed density is m ρ ρ assume that, in the longitudinal direction, the break- 0 = 1 and the excess density is = 0. According to

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 STRUCTURES OF THE PLASMA CHANNELS IN A BESSELIAN BEAM 807

δ/h, arb. units A[δ/h], arb. units 1.0 1.00 E E 1 0.5 0.75 1 0 0.50 –0.5 0.25 –1.0 0 1.0 1.00 23 2 3 0.5 0.8 0 0.4 –0.5 0.2

–1.0 0 1020304050 0 510152025 Length z/h, arb. units Frequency, N = ν + 1

Fig. 8. Longitudinal (at r0 = 0.5) structure of pulsations of the plasma density in a plasma channel according to the experimental data (curves E) and the calculational results for a = (1) 0.063, (2) 0.145, and (3) 0.227.

[15], for a perturbation propagating at the speed of case, the resonance phenomena begin to manifest them- sound c, we have selves. Agreement between the results obtained for the first ring (a = 0.145) and the experimental data is much rctÐ > a for ρ = 0, better. For this reason, the characteristics of the struc- ()rctÐ (11) ture of the plasma channels were refined and the initial rctÐ ≤ a for ρ ∼ fr()Ð ct------. r parameters were adjusted precisely for a = 0.145. The adjustable parameters were as follows: the number k of At r ӷ a, a perturbation of an arbitrary initial shape the initial perturbations, the distribution pz (including becomes a spherical layer of thickness 2‡, in which the the displacement Q of the entire sequence of the initial density pulsations are described by the function f(|r – perturbations and the range q of random deviations), ct|). Consequently, formula (11) determines the config- the time interval t0 during which the breakdown centers uration and the internal structure of a propagating per- originate, and the shape of the function f(r). turbation wave. The problem of the superposition of It was found that the structure of perturbations in a perturbation waves with the initial conditions (10) was plasma channel was described most exactly for the fol- solved by means of the Mathematica-4 software pack- lowing values of the adjustable parameters: k = 25, t0 = age. 0.2, Q = 0.17, q = 0, and f(r) = 1. It is these parameter Figure 8 illustrates the results calculated for the ini- values for which profiles 2 in Fig. 8 were calculated. tial perturbations with the radii a = 0.063, 0.145, and Hence, during the time interval t0 = 0.2 (about 50 ps), 0.227, which correspond to the first three roots of the the perturbations originate at each longitudinal maxi- Bessel function (i.e., to the axial region and the first and mum in the intensity of a Besselian beam and the spac- second rings); the related profiles are denoted by 1, 2, ing between the perturbations coincides with l = 21 µm and 3, respectively. The figure also shows the experi- to within several percent. For a heating pulse with the mentally obtained dependences of E corresponding to duration τ = 100 ps and energy E = 0.6 J, the zone of the perturbations illustrated in Fig. 7 for the relative perturbations extends to the second ring (of radius radius r/R = 0.5, which corresponds to the r0 value 2.9 µm) of the distribution of the Besselian beam inten- adopted for simulations. The experimental profiles of E sity. At rings with larger radii, the beam intensity is and the profiles computed for a = 0.145 are displayed insufficient to produce the breakdown centers. Note by the heavy curves. Shown on the left and on the right that the value t0 = 0.2 indicates a breakdown at a level of Fig. 8 are the longitudinal density profiles and their of 0.8 of the maximum beam intensity. spatial spectra, respectively. Having checked the assumption of the localization A comparison of numerical results with the experi- of the breakdown centers at the points of the maximum mental data shows that the profiles calculated for a = intensities of a Besselian beam, we turn to the analysis 0.063 and 0.227 contradict the measured dependence. of the structure of an optical discharge, which can be For a = 0.063, the scale of the structure is too small, affected by the breakdown propagation. A rich store of and, for a = 0.227, the scale is too large. In the latter data on particular breakdown processes was acquired in

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 808 PYATNITSKY

I 1.0

0.5

–10 z 1 –5 0 x 5 10 0

Fig. 9. Schematic structure of a Besselian beam. experiments aimed at investigating laser sparks in the peaks in the radial and longitudinal directions can be focal region of a spherical lens [1, 2]. According to estimated as δa ≈ λ/2sinγ and l = 2λ/sin2γ, respectively. these data, the threshold intensity for breakdown is proportional to the ionization potential and decreases with When the beam intensity is only slightly above the increasing pressure, diameter of the focal region, and threshold, the breakdown occurs exclusively in the laser pulse duration. The breakdown front can propa- focal region of the beam (x = 0). Under condition (9) (or 7 8 a similar condition), the intensity profile of the beam gate at a velocity of 10 –10 cm/s upward laser radiation becomes modulated with period l along the z-axis and as an ionization wave, a breakdown wave, or a light det- the beam evolves to the structure shown in Fig. 9. In onation wave. The optical discharge develops nonuni- contrast, a beam with an intensity below the threshold formly, because the velocity with which it expands in either remains unmodulated or becomes modulated the radial direction is close to the speed of sound in the very slightly (Fig. 6b). However, even under condition plasma produced and is thus appreciably lower than the (9), the sequence of breakdown centers occurs in the propagation velocity upward laser radiation. focal region of the smallest diameter (x0 = 2.4), and the Analogous phenomena can also take place in an perturbations are damped over short distances. Accord- optical discharge that is driven by a laser pulse focused ingly, after the neighboring breakdown centers merge by an axicon. In this case, the formation of the structure together, the plasma channel rapidly becomes uniform. of the discharge is, as before, affected by the energy and These two scenarios of the development of the process duration of the pulse. However, it is also necessary to are illustrated by Figs. 2a and 4a and by the left part of take into account the field structure of a Besselian the channel shown in Fig. 3. beam. To make the interpretation of the experimental As the laser energy increases, the beam becomes data more illustrative, we show in Fig. 9 a fragment of sufficiently intense for the breakdown centers to origi- the beam structure. In the figure, the beam radius is nate at the next inner cylindrical surfaces at which the characterized by the argument x = krsinγ of the Bessel beam intensity is maximum, so that the zone of the pri- function and the beam length z is specified as the num- mary breakdown successively extends to the rings of ber of the peak in the longitudinal profile of the beam larger radii (xm = 5.52, 8.65, …). As the diameter of the intensity. In order to make the effect of interest to us primary perturbation zone increases, the distance over more pronounced, the figure shows the varying compo- which the primary perturbation can affect the channel nent of the beam intensity I rather than the total inten- structure becomes longer (in proportion to am/a0). sity. The value x = 0 corresponds to the symmetry axis However, for a short laser pulse (τ = 0.1 ns), the plasma of the beam. The spacings between the neighboring produced by the breakdown does not have enough time

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 STRUCTURES OF THE PLASMA CHANNELS IN A BESSELIAN BEAM 809

ε × 3 to expand to the larger rings of the beam structure. As we = 91.2, e = 4.56 10 , re = 0.31 mm; the pulse comes to an end, the interaction of the heating ε × 4 radiation with the perturbations that it itself generates wf = 1250, f = 6.3 10 , rf = 1.1 mm. does not influence the discharge structure, so that the From the experimental data on wi, the Mathematica-4 plasma channel forms as a result of the spontaneous software package makes it possible to determine the propagation of perturbations. The channel structure number m of the ring at which the relative laser power shown in Fig. 4b was obtained by simulating precisely is sufficient for breakdown and the argument x = krsinγ this mechanism. of the Bessel function: The structure of the plasma channel is formed by this mechanism when the heating pulse is sufficiently xb = 25.7, mb = 8; xd = 10.0, md = 3; τ ≤ short ( 0.1 ns). Now, we consider the channel struc- xe = 59.0, me = 19; xf = 803, mf = 251. tures in the case of nanosecond laser pulses (Fig. 5). In Fig. 5a, which illustrates the experiment on laser pulses However, the same parameters can also be determined with τ = 0.8 ns in argon at a pressure of 0.2 atm, one can from the experimental data on the radii ri: distinctly see that pronounced breakdown centers occur xb = 25.8, mb = 8; xd = 51.7, md = 16; precisely at the points of the central maxima of the intensity of a Besselian beam. The spatial structure of xe = 238, me = 74; xf = 850, mf = 266. the forming plasma channel was resolved by recording For the same pulse duration, the power density in the optical inhomogeneities in the scattered laser light and channels in Figs. 5b and 5c is higher than that in the γ by using a beam with a small convergence angle, = 1° channel in Fig. 5a by a factor of 41. As a result, the ≈ (l 7 mm). Although, in this experiment, the pulse channels in Figs. 5b and 5c are more complicated in duration was almost one order of magnitude longer, the structure. Instead of individual breakdown centers, distance to the next row of the maximum intensities in there are numerous centers, which combine into the first ring of a Besselian beam was also found to groups. The distances between the centers of the groups δ ≈ increase by approximately the same amount ( a are equal to the period of the longitudinal beam struc- µ 30 m). ture, l = 1.1 mm (for the convergence angle γ = 2.5°). In the channel region that is closer to the axicon (on Note that both of the methods yield the same number of the left side of Fig. 5a), the beam intensity is about the ring of the radial structure of a Besselian beam: threshold intensity and its value actually corresponds to mb = 8. ≈ the pulse energy E 2.5 J. Let us use this value of the From Fig. 5c, we can see that the structural groups beam intensity when comparing the breakdown condi- of the breakdown centers lie within a cylindrical region tions in different experiments. Note that the threshold with a radius of about 0.1 mm and a length of about intensity for breakdown in argon at a pressure of 0.5 mm. In the middle of the cylindrical region, the 0.2 atm is lower than that in air at atmospheric pressure breakdown centers concentrate at the periphery (near by a factor of 2.6 [2]. For this reason, we adopt the fol- the cylindrical surface), while, near the edges, they con- lowing threshold parameter values for the beam in air: γ τ centrate in the axial region. This distribution of the E = 6.5 J, = 1°, and = 0.8 ns, assuming that, for these breakdown centers is similar to the distribution of the values, the breakdown centers in air at atmospheric beam field at a longitudinal maximum in the beam pressure originate in the central focal region of the intensity. We can thus conclude that the breakdown beam at the points at which the beam intensity is max- centers most likely originate at the points where the imum and which are separated by the distance l given longitudinal maxima in the beam intensity occur at the in formula (8). outermost ring of the radial beam structure at which the In order to estimate the conditions for the formation beam intensity is still above the threshold. It should be of a structured plasma channel, we determine the ratios noted that, in the experiment at hand, the laser pulse of the power and energy densities of radiation in the had a peculiar shape: for τ = 0.8, the rise time of the beams that produce the plasma channels shown in pulse front was 100 ps. Fig. 5 to their threshold values. To do this, we use the The period l = 0.28 mm of the longitudinal structure above relationships w ~ Eγ3/τ and ε ~ Eγ3. We also of the plasma channel shown in Fig. 5d coincides present the radii ri of the breakdown zone, which can be exactly with the period computed for a beam with the obtained from the data of Fig. 5 (the subscripts aÐf cor- convergence angle γ = 5°. However, the above two ε respond to the frames of this figure). Setting w‡ = ‡ = 1 methods give different numbers of the rings of the and taking into account the fact that the threshold inten- radial structure of a Besselian beam: md = 3 and 16. sity for breakdown in argon at a pressure of 0.2 atm is Clearly, this discrepancy is associated with the pulse lower than that at atmospheric pressure by a factor of duration, which is equal to τ = 20 ns and exceeds the ε ε 5.3, we obtain (wb = wc, b = c, rb = rc) duration of the beam in the previous experiment by a ε factor of more than 20. Accordingly, for a low specific wb = 40.6, b = 40.6, rb = 0.1 mm; radiation power, wd = 15.4, the pulse energy is higher ε ε wd = 15.4, d = 384, rd = 0.08 mm; by a factor of approximately 10; i.e., d = 384. Presum-

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 810 PYATNITSKY ably, for such a power, the primary breakdown centers threshold level. However, if the breakdown centers actually originate at the first three inner rings of the arise before the intensity of the heating pulse becomes radial beam structure (md = 3). However, a heating maximum and if the maximum intensity of the beam is pulse with a length of 6 m maintains plasma production sufficiently high, then the breakdown centers merge by breakdown even after the primary breakdown cen- into continuous plasma formations. In Fig. 5d, these ters have already originated. formations are seen as side extensions. In the experiment τ Under these conditions, the breakdown front in the illustrated in Fig. 5d, the pulse duration is = 20 ns, focal region of a spherical lens moves upward laser so that the beam intensity remains nearly maximum at least over a time interval of t τ = 2 ns. During this radiation at the velocity v 7 cm/s. The velocity of ~ 0.1 ~ 10 time, the region where the intensity threshold for break- the simultaneous thermal expansion of the plasma in down is lowered extends (at a velocity of u ≈ 2 × other directions is far lower, u ~ 106–5 × 106 cm/s. The 106 cm/s) for δr ~ 40 µm, which satisfies the main con- plasma is produced by the ionization wave. In the case δ δ dition for the process to occur, r > rm, because, in the of beam focusing by an axicon, the front of the primary δ µ breakdown can also propagate upward laser radiation. case at hand, we have rm = 6 m. However, unlike the case of spherical focusing, the The channel structure in Fig. 5e quantitatively intensity of a Besselian beam can also be maximum at (rather than qualitatively) differs from that in Fig. 5d. the points on the outside of the breakdown zone Indeed, the method for determining the number of the (Fig. 9). At each such point, the beam field has the ring of the radial structure of the Besselian beam from potential to maintain plasma production by breakdown. the measured radiation intensity gives me = 19 (xe = When passing through the point of maximum beam 59.0), while the method for determining the same num- intensity, the ionization wave (which reduces the ber from the measured radius of the breakdown zone ≈ threshold for breakdown) gives rise to a new break- yields me 74 (xe = 238). Of course, the channel struc- down center. We thus arrive at a certain combination of ture forms by the same mechanisms as in the previous the radiative (ionizational) mechanism and the break- case, and the larger number of rings on which the down-wave mechanism. breakdown centers originate is associated with the Let v be the propagation velocity of the breakdown higher power and energy densities of radiation. There- front upward laser radiation, u be the radial velocity of fore, the side extensions of the channel structure are the ionization wave, δr be the radial distance between longer than those in Fig. 5d. The higher specific power m γ the neighboring cylindrical surfaces of the radial struc- of radiation and the larger convergence angle lead to δ δ γ the larger inclination angle of the extensions (the ture of a Besselian beam, and rγ = rm/sin be the dis- β γ tance between the same surfaces but along the laser angle ). However, the large angle hinders the merg- “ray” (i.e., upward laser radiation, along a line directed ing of the breakdown centers, which are thus observed to merge together only in the axial region of the chan- at the angle γ). Since the angle γ is small and δr Ӷ δrγ, m nel structure. Using relationship (12), we can estimate the front of the spherical ionization wave passes the velocity ratio u v. For the channel structure shown through the neighboring ring earlier than the break- / in Fig. 5e, the inclination angle of the extensions is down front propagating upward laser radiation. Hence, β ≈ , which gives u v ≈ . the angle at which the breakdown wave propagates 36° ( / ) 0.56 deviates from γ. In addition, the beam intensity In the context of the mechanism under discussion, decreases along the laser ray because of the finite length we consider the configuration of an optical discharge of the axial structure maximum. As a result, the break- illustrated in Fig. 5f. In the relevant experiment, the ε down wave propagates at an angle β ӷ γ from point to specific power wb = 1250 and the specific energy b = point in the structure of a Besselian beam. 6.3 × 104 were both one order of magnitude higher than The longitudinal and radial components of the total those in Fig. 5e, the pulse duration being the same. velocity can be represented as v ≈ v cosγ and v ≈ Accordingly, the beam intensity is far above the thresh- z r old, so that the breakdown zone includes the ring num- v γ u. Then, the angle β is determined by the rela- sin + ber m ≈ (which corresponds to x = 803). On the tionship f 250 f other hand, measurements of the radius of the break- v sinγ + u down zone yield m ≈ 266 and x = 851. Hence, we may tanβ ≈ ------. (12) f f v cosγ suppose that the channel structure is formed by basi- cally the same mechanism. However, such a high For example, for v = 107 cm/s and u = 2 × 106 cm/s, we excess of the beam intensity above the threshold has have β = 16°, which agrees with the inclination angle important consequences. β = 13°–17° measured from the image of the channel in In Fig. 5f, the breakdown centers resemble large Fig. 5d. bright spots and occur at the peripheral rings of the For a long heating pulse, the ionization wave, which radial beam structure. Presumably, such factors as high lowers the intensity threshold for breakdown, passes electron plasma density in the spots and their large through the region between the primary breakdown dimensions are, to a great extent, responsible for the centers, in which the beam field can be below the screening of the central regions of the plasma channel

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 STRUCTURES OF THE PLASMA CHANNELS IN A BESSELIAN BEAM 811 from radiation. For this reason, it was possible to detect over short distances, so that, during a time interval of only fragments of the side extensions of the channel t ~ 5 × 10–11sin–2γ s, the breakdown centers merge structure; moreover, these fragments were observed to together, giving rise to a uniform plasma channel. terminate radially without reaching the axial region of When the diameter of the primary breakdown zone is the beam. From Fig. 5f, one can see that the beam inten- large (m > 0), the distance over which perturbations can sity at the center of the channel is sufficient to ensure affect the channel structure increases in proportion to breakdown only along discrete intervals in the axial am/a0, so that, as the pulse comes to an end, the plasma focal region of the axicon. The screening may also be channel forms as a result of spontaneous propagation of responsible for wide spacing between the structural the perturbations. groups of the breakdown centers. If the laser pulse is long, the breakdown front moves Let us estimate the velocity ratio u/v. By measuring upward laser radiation, as in the case of focusing by a the angle β from the inclination of the contours of the spherical lens. Simultaneously, the plasma is produced extensions, we find β ≈ 48°, which corresponds to the by a spherical ionization wave. However, unlike the ratio (u/v) ≈ 1, thereby providing evidence for the det- case of spherical focusing, there may be other maxima onation breakdown mechanism. Hence, we can con- of the intensity of a Besselian beam on the outside of clude that, when the beam intensity is far above the the breakdown zone. At each of these points, the beam threshold intensity, the structure of an optical discharge field has the potential to maintain plasma production by degenerates into a sequence of coaxial breakdowns and breakdown. When passing through a point of the maxi- the forming breakdown centers evolve according to the mum beam intensity on the outside of the breakdown detonation mechanism. zone, the ionization wave lowers the threshold for breakdown and gives rise to a new breakdown center. We can thus conclude that the breakdown front propa- 6. CONCLUSION gates in a jumplike manner, because of the peculiar A conical lens makes it possible to focus laser light combination of the radiative (ionizational) mechanism in such a way as to prevent the diffractive spreading of and the breakdown-wave mechanism. laser pulses. In this case, the radial profile of the laser The radial expansion velocity of the plasma is mark- field is described by the Bessel function and the longi- edly lower than the velocity of the breakdown wave. tudinal extension of the field is independent of the laser However, the spatial scale of the radial structure of a wavelength. Such Besselian beams can be used to cre- Besselian beam is smaller than the period of the longi- Ӷ ate an extended uniform plasma channel. However, tudinal beam structure, (rm + 1, k – rmk) (rm, k + 1 – rmk). such a channel is subject to instabilities resulting in var- As a result, the breakdown front propagates from the ious structural inhomogeneities. points r0k along trajectories whose directions are deter- The presence of structural inhomogeneities is asso- mined by the ratio of the expansion velocity of the ciated primarily with the field structure of the Besselian spherical ionization wave to the velocity of the break- beam, which is characterized by radially (m) and axi- down wave. The length of the trajectories depends on ally (k) ordered sets of the points of the field intensity the intensity and duration of the heating pulse. maximums. The radial ordering is determined by the For intensities far above the threshold level, the rings whose radii correspond to the zeros of the Bessel breakdown centers can originate at rings with large function. The longitudinal structure (zk) forms due to radii, in which case the breakdown front propagates as the nonlinear interaction of a Besselian beam with a a light detonation wave, producing large plasma gas. If this interaction is only slightly nonlinear, the domains. High plasma electron density in the domains beam gives rise to a uniform plasma channel. In a may be responsible for the screening of the central strongly nonlinear interaction, an optical discharge regions of the Besselian beam from radiation and for develops from breakdown centers that originate at the the onset of conical breakdown fronts. The radii am of longitudinal maxima zk in the beam intensity. the conical fronts depend on the beam intensity, and the The channel structure depends on the duration of the distances between the fronts are determined by the heating pulse and the excess of the beam intensity inclination angle of their generatrix. above the threshold intensity. The pulse is assumed to The configuration of inhomogeneities of the plasma be short if its duration is insufficient for the produced channel follows the structure of an optical breakdown plasma to reduce the threshold intensity for breakdown for a long time. The inhomogeneities may be smoothed at the neighboring structural elements of the beam. A out only as a result of interpenetration and intermixing. short pulse gives rise to breakdown centers at every The duration t of this smoothing depends on the scale structural inhomogeneity in the axial direction, in length δr of the inhomogeneities and can be easily esti- which case the diameter of the breakdown zone (the mated from the propagation velocity u of the perturba- number m of rings) depends on the pulse energy. The tions in the channel: t ~ δr/u. The larger the scale of the minimum diameter 2a0 is the diameter of the central channel structure, the longer the time interval is during region of a Besselian beam. In this case, the perturba- which the inhomogeneities are smoothed out and tion waves driven by the breakdown centers are damped damped. Thus, for the velocity u = 2 × 106 cm/s, the

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 812 PYATNITSKY structural inhomogeneities with the characteristic 7. L. N. Pyatnitsky and L. Ya. Polonsky, in Invited Papers dimensions δr ≤ 0.1 mm are smoothed out during the of the XIX International Conference on Phenomena in time interval t ≤ 5 ns, while, for δr ~ 1 mm, this time Ionized Gases, ICPIG, Belgrade, 1989, p. 342. interval increases at least up to t ~ 50 ns. 8. V. V. Korobkin and M. Yu. Romanovskiœ, Tr. Inst. Obshch. Fiz. Akad. Nauk 50, 3 (1995). 9. V. V. Korobkin, L. Ya. Polonsky, and L. N. Pyatnitsky, REFERENCES Tr. Inst. Obshch. Fiz. Akad. Nauk 41, 23 (1992). 10. S. Yu. Luk’yanov and N. G. Koval’skiœ, Hot Plasma and 1. Yu. P. Raœzer, Usp. Fiz. Nauk 132, 549 (1980) [Sov. Controlled Nuclear Fusion (Mosk. Inzh.-Fiz. Inst., Mos- Phys. Usp. 23, 789 (1980)]; Laser-Induced Discharge cow, 1999), p. 408. Phenomena (Nauka, Moscow, 1974; Consultants Bureau, 11. R. Clark and H. M. Milchberg, Phys. Rev. E 57, 3417 New York, 1977). (1998); J. Fan, T. R. Clark, and H. M. Milchberg, Appl. 2. G. V. Ostrovskaya and A. N. Zaœdel’, Usp. Fiz. Nauk Phys. Lett. 73, 3064 (1998). 111, 579 (1973) [Sov. Phys. Usp. 16, 834 (1973)]. 12. V. V. Korobkin, M. Yu. Marin, V. I. Pil’skiœ, et al., Kvan- tovaya Élektron. (Moscow) 12, 959 (1985); Preprint, 3. F. V. Bunkin, V. V. Korobkin, Yu. A. Kurinyœ, et al., IVTAN (Institute of High Temperatures, USSR Acad- Kvantovaya Élektron. (Moscow) 10, 443 (1983). emy of Sciences, Moscow, 1984). 4. N. E. Andreev, Yu. F. Aristov, L. Ya. Polonsky, and 13. V. N. Lugovoœ and A. M. Prokhorov, Usp. Fiz. Nauk 111, L. N. Pyatnitsky, Zh. Éksp. Teor. Fiz. 100, 1756 (1991) 203 (1973) [Sov. Phys. Usp. 16, 658 (1973)]. [Sov. Phys. JETP 73, 969 (1991)]. 14. N. E. Andreev, S. V. Kuznetsov, and L. N. Pyatnitsky, Fiz. Plazmy 17, 1123 (1991) [Sov. J. Plasma Phys. 17, 5. L. Ya. Polonsky and L. N. Pyatnitsky, Opt. Atmos. 1, 86 652 (1991)]. (1988); The Most Important Results of Research Engi- neering of Institute of High Temperatures, USSR Acad- 15. L. D. Landau and E. M. Lifshitz, Fluid Mechanics emy of Sciences (Nauka, Moscow, 1987), p. 93. (Nauka, Moscow, 1988; Pergamon, New York, 1987). 16. L. N. Pyatnitsky, Zh. Éksp. Teor. Fiz. 113 (1), 191 (1998) 6. S. S. Bychkov, M. Yu. Marin, and L. N. Pyatnitsky, [JETP 86, 107 (1998)]. Tr. Inst. Obshch. Fiz. Akad. Nauk 50, 166 (1995); S. Bychkov, M. Marin, and L. Pyatnitsky, Inst. Phys. Conf. Ser. 125, 439 (1992). Translated by G. V. Shepekina

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001

Plasma Physics Reports, Vol. 27, No. 9, 2001, pp. 813Ð818. Translated from Fizika Plazmy, Vol. 27, No. 9, 2001, pp. 859Ð864. Original Russian Text Copyright © 2001 by Bazhenov, Ryabtsev, Soloshenko, Terent’eva, Khomich, Tsiolko, Shchedrin.

LOW-TEMPERATURE PLASMA

Investigation of the Electron Energy Distribution Function in Hollow-Cathode Glow Discharges in Nitrogen and Oxygen V. Yu. Bazhenov, A. V. Ryabtsev, I. A. Soloshenko, A. G. Terent’eva, V. A. Khomich, V. V. Tsiolko, and A. I. Shchedrin Institute of Physics, National Academy of Sciences of Ukraine, pr. Nauki 144, Kiev, 03039 Ukraine Received February 15, 2001

Abstract—The mechanism for the formation of the inverse electron distribution function is proposed and real- ized experimentally in a nitrogen plasma of a hollow-cathode glow discharge. It is shown theoretically and experimentally that, for a broad range of the parameters of an N2 discharge, it is possible to form a significant dip in the profile of the electron distribution function in the energy range ε = 2Ð4 eV and, accordingly, to produce the inverse distribution with df(ε)/dε > 0. The formation of a dip is associated with both the vibrational excitation of N2 molecules and the characteristic features of a hollow-cathode glow discharge. In such a discharge, the applied voltage drops preferentially across a narrow cathode sheath. In the main discharge region, the electric field E is weak (E < 0.1 V/cm at a pressure of about p ~ 0.1 torr) and does not heat the discharge plasma. The gas is ionized and the ionization-produced electrons are heated by a beam of fast electrons (with an energy of about 400 eV) emitted from the cathode. A high-energy electron beam plays an important role in the formation of a dip in the profile of the electron distribution function in the energy range in which the cross section for the vibrational excitation of nitrogen molecules is maximum. A plasma with an inverted electron distribution function can be used to create a population inversion in which more impurity molecules and atoms will exist in electronically excited states. © 2001 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION only by energy transfer from vibrationally and electronically excited molecules to the plasma. As a result, the In recent years, numerous technological applica- EEDF becomes peaked at energies that correlate with tions of low-pressure gas discharges in nitrogen and the excitation energies of the vibronic (vibrationalÐ nitrogen-containing mixtures have stimulated active electronic) levels of molecules. experimental and theoretical investigations of the electron energy distribution function (EEDF) in order to Our paper is devoted to an experimental and theoret- gain a better insight into the physics of the plasmo- ical investigation of the EEDF in discharges that differ chemical processes occurring in various plasma devices radically from both those studied in [1Ð4] and those (see, e.g., [1Ð8]). studied in [5–8]. Specifically, we are interested in First of all, we should note that the shape of the steady hollow-cathode glow discharges in nitrogen and EEDF is highly sensitive to the type of discharge and its oxygen. In such discharges, which are widely used in parameters even in experiments with the same gas mix- various technological applications, the applied voltage tures. Thus, in [1Ð4], the EEDF in N2 and N2–O2 mix- drops preferentially across a narrow cathode sheath. tures was studied under conditions such that the electric According to our measurements, the electric field E in field was stronger than the breakdown field and was the the main discharge region was at most 0.1 V/cm (at a only factor responsible for the heating of plasma elec- pressure of about p ~ 0.1 torr), which is one order of trons. In those papers, the EEDF f(ε) was found to be magnitude weaker than that in [1Ð4]. In our experi- monotonic (df(ε)/dε < 0), but with different tempera- ments, the gas was ionized and the plasma electrons tures in different energy ranges. Such a shape of the were heated by a beam of fast electrons (with an energy EEDF is associated with the fact that, under those con- of about 400 eV) emitted from the cathode. We show ditions, the distribution of the plasma electrons rapidly that, in such discharges in nitrogen, the formation of a becomes Maxwellian under the action of the electric significant dip in the EEDF in the energy range 2Ð4 eV field. and, accordingly, the inversion of the EEDF (df(ε)/dε > 0) On the contrary, in [5Ð8], it was shown that, in a are attributed to the vibrational excitation of N2 mole- weak electric field in the post-discharge plasmas of cules. In contrast, in discharges in oxygen, the EEDF is repetitive discharges in N2 and Ar–N2 mixtures, an monotonic because, on the one hand, it is cut off at low important role in the formation of the EEDF is played threshold energies for the excitation of the lowest lying by superelastic collisions, because, during the period electronically excited metastable states of O2 molecules between successive discharges, the electrons are heated and, on the other hand, the cross sections for the excita-

814 BAZHENOV et al. tion of the vibrational levels of oxygen molecules are tive to the anode, the anode voltage, and the discharge small. current. The change in the probe current at each step was calculated automatically during the measurements with the help of a specially developed algorithm for 2. EXPERIMENTAL SETUP optimizing the signal-to-noise ratio over the entire AND MEASUREMENT RESULTS range of measured currents (a complete cycle of measurements of one IÐV characteristic included from 1500 The experiments were carried out with a hollow to 2000 steps). The measurements were carried out in cylindrical cathode 280 mm in diameter and 400 mm in the stroboscopic regime at a rate of 100 Hz synchro- length. A 230-mm-diameter anode was placed near one nously with the oscillations of the supply voltage, end of the cathode. The vacuum chamber was pre-evac- thereby making it possible to avoid the influence of uated by a forepump to a residual pressure of 5 × −3 these oscillations on the experimental results. The gat- 10 torr. Then, the chamber was filled with the working time of the measured signals was 1 µs. The time ing gas (nitrogen or oxygen) in the pressure range from delay (7 ms) of the measurement with respect to the 3 × 10–2 to 1 × 10–1 torr. The discharge current and beginning of the half-wave of the supply voltage was applied voltage were varied in the ranges 0.5Ð0.9 A and chosen so as to optimize the signal-to-noise ratio. After 400Ð600 V, respectively. The plasma density, electric performing the measurements over the prescribed fields, and EEDF were measured by a pair of single range of probe current variations, the measured data in Langmuir probes [9] made of tungsten wires 100 µm in the form of a dependence of the probe current on the diameter, the receiving sections being from 10 to probe potential for a given discharge current and given 15 mm in length. The probes were designed so as to discharge voltage were stored as a computer file. With provide measurements along and across the symmetry fixed experimental parameters, the IÐV characteristic axis of the cathode. In order for the measured currentÐ was measured up to 30 times. Thereafter, the data voltage (IÐV) characteristics of the probes to be stored in the corresponding files were averaged over the immune to probe-surface contamination, the probes measurement cycles. were cleaned after each measurement by heating them to a temperature of about 800° C with a dc source. The plasma potential was assumed to be equal to the probe potential at which the second derivative of the The EEDF was determined numerically by differen- probe current with respect to the probe voltage van- tiating the IÐV characteristics twice with (if necessary) ishes. The plasma density was computed from the elec- preinterpolation of the measured data. The measure- tron saturation current to the probe. ment accuracy was increased using the modified method described in [2], which is based on a specially Figure 1 shows the radial profiles of the plasma den- devised programmable diagnostic system controlled by sity for different nitrogen pressures. We can see that the a personal computer. At each step of a measurement shape of the profiles is highly sensitive to the working cycle, the controlling computer code specified the gas pressure. For p = 0.1 torr, the plasma density is min- probe current with an accuracy of 0.1 µA and provided imum at the cathode axis and increases gradually in the ≈ simultaneous measurements of the probe voltage rela- radial direction, reaching a maximum value at R 11 cm. At lower pressures (p = 0.03, 0.06 torr), the situation is radically different: the plasma density ne is maximum at the axis of the system and decreases 10 –3 ne, 10 cm monotonically with radius. This dependence is peculiar to hollow-cathode discharges. At a pressure of 0.1 torr, 3.5 3 essentially all of the energy of the fast primary elec- 3.0 trons emitted from the cathode is expended on the exci- 2 tation and ionization of the working gas within a dis- 2.5 tance of several centimeters from the cathode; consequently, the gas at the system axis is ionized mainly by 2.0 the electrons that diffuse from the region where the plasma has already been produced. As a result, the 1.5 1 plasma density is the highest not at the cathode axis but 1.0 in the region where the loss of fast electrons is maximum and, accordingly, the plasma production is most 0 2 4 6 8 10 12 14 intense. As the working gas pressure decreases, the R, cm maximum of the plasma density profile shifts toward smaller radii and, at p ≈ 0.05 torr, it becomes bell- Fig. 1. Radial profiles of the electron density for different shaped. nitrogen pressures: (1) p = 0.1 torr, Id = 0.63 A, and Ud = Recall that, in gas discharges, a very important role 470 V; (2) p = 0.06 torr, Id = 0.73 A, and Ud = 580 V; and is played by the electric field, which heats the plasma (3) p = 0.03 torr, Id = 0.77 A, and Ud = 615 V. electrons and thus can substantially influence the shape

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INVESTIGATION OF THE ELECTRON ENERGY DISTRIBUTION FUNCTION 815

–2 –2 Er, V/cm 10 V/cm for p = 0.06 torr, and Ez < 10 V/cm for p = 0.10 0.03 torr. Below, we will show that such longitudinal 2 and radial electric fields are too weak to significantly 0.08 affect the shape of the EEDF. 1 0.06 The EEDF was measured at different points inside 3 the vacuum chamber at different pressures of the work- 0.04 ing gas. Figures 3 and 4 show typical EEDFs measured at nitrogen pressures of 0.1 and 0.03 torr. We can see 0.02 that the EEDF is strongly non-Maxwellian: it has a pronounced dip in the energy range 2Ð4 eV. Moreover, at a 0 ...... lower pressure (Fig. 3), there are two dips in the EEDF in this energy range. 0 2 4 6 8 10 12 R, cm We also established that, at low pressures, the EEDF is essentially independent of the radius, while, at p = Fig. 2. Radial profiles of the radial electric field for different 0.1 torr, the EEDF experiences fairly strong variations nitrogen pressures: (1) p = 0.1 torr, Id = 0.63 A, and Ud = in the radial direction. Moreover, at the chamber axis 470 V; (2) p = 0.06 torr, Id = 0.73 A, and Ud = 580 V; and (i.e., in the region where the reduced electric field E/N (3) p = 0.03 torr, Id = 0.77 A, and Ud = 615 V. is minimum), the dip in the EEDF is the smallest. This radial behavior of the EEDF is not associated with the of the EEDF. Figure 2 displays the experimentally mea- presence of an electric field, because the action of the sured radial profiles of the radial electric field E for dif- electric field should result in the formation of an EEDF r that depends on the radius in the opposite manner—the ferent nitrogen pressures. The radial behavior of Er is dip in the EEDF would be smallest at the discharge seen to be analogous to that of the plasma density. At periphery, where the field is maximum. The observed low nitrogen pressures, the radial electric field is posi- reduction in the dip in the EEDF near the axis of a hol- tive and increases monotonically with the radius. For low cathode at high pressures (Fig. 4) may be attributed p = 0.1 torr, the component Er is negative in the axial to the decrease in the number of high-energy electrons region (where the plasma density is the lowest); at (which serve as the major energy source in the plasma) larger radii, it passes through zero and starts increasing and, accordingly, to an increasingly important role of monotonically. The maximum value of Er does not the processes that force the EEDF to evolve into a Max- exceed 0.1 V/cm. The longitudinal electric field Ez was wellian function. As was mentioned above (Fig. 1), it is found to be even weaker: the measurements showed precisely this pressure range in which the plasma den- ≈ × –2 ≈ × that Ez 1–2 10 V/cm for p = 0.1 torr, Ez 0.5–1 sity ne is minimum at the chamber axis.

100 100 10–1 –1 10 –3/2 –2

10 ), eV –3/2

–2 ε 10 ( 0 f

), eV –3

ε 10 (

–3 0

10 f 10–4 1 10–4 0 0 2 2 1 2 4 3 4 2 ε 3 6 , eV 4 6 ε , cm , eV 4 R 8 , cm 8 5 R 5 6 10 10 12 6

Fig. 3. EEDFs measured in nitrogen at a pressure of p = Fig. 4. EEDFs measured in nitrogen at a pressure of p = 0.03 torr at different radial positions R. 0.1 torr at different radial positions R.

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ε –3/2 f0( ), eV is the electron density; SeN is the integral of electronÐ 100 neutral inelastic collisions; See is the integral of elec- tronÐelectron collisions; and A(ε) is the ionization term, which includes the source of primary electrons. The 10–1 1 ε 2 expressions for the terms SeN, See, and A( ) were taken from [11]. –2 ε 10 The EEDF f0( ) was normalized to satisfy the condition 10–3 ∞ ε1/2 ()εε 1 2 3 4 5 6 7 8 ∫ f 0 d = 1. (2) ε , eV 0 The electron processes that were taken into account Fig. 5. EEDF measured in oxygen at a pressure of p = 0.06 torr at R = (1) 0 and (2) 9 cm. when solving the Boltzmann equation for discharges in nitrogen and oxygen are listed in Tables 1 and 2. The superelastic scattering of electrons by vibrationally The EEDF in an oxygen plasma is illustrated in excited molecules was neglected, because, for the dis- Fig. 5, which shows that it is qualitatively different charges under investigation, the specific input power from the EEDF in N2, although measurements were and, accordingly, the vibrational temperature Tv were carried out in discharges with the same parameters. To both much lower than those in the experiments of [4Ð8]. a good accuracy, the EEDF in O2 can be described by a The cross sections for elastic and inelastic scattering Maxwellian function, but with different temperatures in of electrons by N2 and O2 molecules were taken from the energy ranges ε = 0Ð2.5 eV and ε > 2.5 eV. [12]. The physical reasons for such a large discrepancy The calculations were carried out with the values of between the EEDFs in N2 and O2 will be discussed in the electric field and electron density that were mea- the next section. sured experimentally at different regions of the dis- ε charge chamber. The energy n of the beam of primary electrons was assumed to be on the order of the poten- ε ≈ 3. THEORY tial drop across the cathode sheath, n 400 eV. We calculated the EEDF by solving the Boltzmann Equation (1) was solved using the same methods as equation in the two-term approximation [10]: in [10]. Figure 6 shows the EEDFs obtained theoretically for 1/2 ∂() 2 ∂ ε ∂ 1 m ε1/2 ne f 0 1E f 0 a hollow-cathode discharge in nitrogen. The EEDF was ------------∂ Ð ------- ∂ε----- ------∂ε ne N 2e t 3 N QT calculated for the parameter range (ne, E) correspond- (1) ing to the intervals of variation of ne and E in the radial ∂ m 2∂ f Ð ----- 2-----Q ε f + T------0 = S ++S A()ε , direction in the discharge chamber. For all values of the ∂ε M T 0 ∂ε eN ee discharge parameters, there are two pronounced dips in ε ε the EEDF in the energy range = 2Ð4 eV. The dips are where f0( ) is the symmetric part of the EEDF; T is the associated with the sharp peaks in the cross section for × –12 gas temperature (in eV); e = 1.602 10 erg/eV; M is the vibrational excitation of N2 molecules in this energy the mass of a molecule; N is the gas density; QT is the range. The EEDF calculated without allowance for the transport cross section; m is the mass of an electron; ne vibrational excitation of N2 molecules is monotonic. Another factor that causes the dips in the EEDF to vanish is an artificial increase in E/N by an order of magni- Table 1 tude, up to the values that correspond to the experiments of [1Ð4] and are large enough for the EEDF to Threshold No. Reaction become Maxwellian. energy ε , eV sh Figure 7 shows the EEDFs calculated for an oxygen 1N2 + e N2(v) + e, v = 1…8 1.5 plasma. An important property in which oxygen differs + from nitrogen is the presence of the metastable excited 2 N + e N (A3 Σ ) + e 6.7 2 2 u states of O molecules, O (1∆ ) and O ()b1Σ + , with low 3N + e N (a1Π ) + e 8.55 2 2 g 2 g 2 2 g threshold energies (0.95 and 1.64 eV, respectively). The 3Π 4N2 + e N2(B g) + e 11.5 EEDF calculated with allowance for the excitation of 5 N + e N+ + e + e 15.6 these states is cut off at energies of about 1Ð2 eV; this 2 2 effect is especially pronounced for weak electric fields 6N2 + e N + N + e 9.76 (Fig. 7). As for the processes of the vibrational excita-

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–3/2 –3/2 f0, eV f0, eV 101 101 1 1 0 2 10 2 100 3 –1 10 4 10–1 –2 10 10–2 –3 10 10–3 –4 10 10–4 –5 10 10–5 –6 10 10–6 10–7 01234567891011121314 0123456789101112131415 ε, eV ε, eV Fig. 6. EEDFs in nitrogen calculated for the parameter val- Fig. 7. EEDFs in oxygen calculated for p = 0.06 torr, ne = 10 –3 10 Ð3 ues (1) p = 0.03 torr, ne = 10 cm , and E = 0.01 V/cm; 10 cm , and E = (1) 0.01 and (2) 0.1 V/cm. 10 Ð3 (2) p = 0.03 torr, ne = 10 cm , and E = 0.1 V/cm; (3) p = 10 Ð3 0.1 torr, ne = 10 cm , and E = 0.01 V/cm; and (4) p = × 10 Ð3 (Fig. 4), because, as was noted above, the radial energy 0.1 torr, ne = 2 10 cm , and E = 0.06 V/cm. distribution of the beam of fast electrons emitted by the cathode is highly nonuniform. In turn, the radial nonuniformity of the beam may be responsible for the non- tion of O molecules, they have essentially no influence 2 local character of the EEDF, thereby giving rise to the on the EEDF because the vibrational excitation cross loss of high-energy electrons at the axis of the dis- section for O2 molecules is much smaller than that for charge chamber. However, our calculations were car- N2 molecules: up to the threshold energies for the exci- ried out under the assumption that all of the parameters tation of electronic levels, the EEDF in an oxygen governing the EEDF are spatially uniform. plasma can be described with good accuracy by a Max- wellian function with one or two (depending on the To within experimental error, the EEDF measured in electric field magnitude) temperatures in different an oxygen plasma (Fig. 5) also agrees well with the energy ranges. EEDF calculated for the electric field E = 0.01 V/cm with allowance for the above scattering cross sections (Fig. 7). Note that, in discharges in oxygen, the electric 4. DISCUSSION OF THE RESULTS field was measured to be weaker than 0.02 V/cm. To a A comparison between the EEDFs measured exper- first approximation, the experimental EEDF can be imentally in a nitrogen plasma at low pressures (Fig. 3) described by two Maxwellian functions with different and the related EEDFs calculated theoretically (Fig. 6) shows that the experimental data agree well with theoretical predictions both qualitatively and quantitatively. Table 2 In the energy range 2Ð4 eV, the positions of the mea- Threshold No. Reaction sured and calculated dips in the EEDF coincide within energy ε , eV an accuracy of 10Ð20%. In the measured EEDF, the dip sh that is closer to the right peak coincides with the rele- 1O2 + e O2(v) + e, v = 1…10 1.95 1∆ vant calculated dip within the limits of experimental 2O2 + e O2( g) + e 0.98 error. 1 Σ+ It should be noted that our experimental technique 3 O2 + e O2(b g ) + e 1.64 does not provide correct measurements of the EEDF in 3 Σ+ the energy range below 1 eV. Presumably, this is the 4 O2 + e O2(A u ) + e 4.5 reason why, in the range ε < 1 eV, the measured EEDF 5O2 + e O2(*) + e 6.0 f(ε) decreases to a lesser extent than the calculated 6O + e O (**) + e 8.0 EEDF. However, a comparison between the calculated 2 2 7O + e O (***) + e 9.7 and measured extents to which the EEDF decreases rel- 2 2 + ative to its value at an energy of 1 eV also shows good 8 O2 + e O2 + e + e 12.2 agreement between theory and experiment. 9O2 + e O + O + e 6.0 At higher pressures, the agreement between the cal- Ð culated and measured EEDFs is somewhat worse 10 O2 + e O + O 3.6

PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 818 BAZHENOV et al. temperatures. Up to an energy of about 7 eV, the EEDF sponding processes in more complicated plasma-form- calculated theoretically for E = 0.1 V/cm is a one-tem- ing media. perature Maxwellian function; however, in our experiments with discharges in oxygen, such strong electric fields were not observed. REFERENCES 1. N. L. Aleksandrov, A. M. Konchakov, and É. E. Son, Fiz. Plazmy 4, 169 (1978) [Sov. J. Plasma Phys. 4, 98 5. CONCLUSION (1978)]. Hence, we have investigated the EEDF in hollow- 2. V. Guerra and J. Loureiro, J. Phys. D 28, 1903 (1995). cathode glow discharges in which the plasma is prefer- 3. V. Guerra and J. Loureiro, Plasma Sources Sci. Technol. entially heated by fast electrons that are emitted from 6, 373 (1997). the cathode and are accelerated by the space charge 4. C. M. Ferriera and J. Lourero, Plasma Sources Sci. Tech- electric field in the cathode sheath. We have revealed nol. 9, 528 (2000). the following features of the EEDF: 5. C. Gorse, M. Capitelli, and A. Ricard, J. Chem. Phys. 82, (i) In discharges in nitrogen, there is a significant dip 1900 (1985). in the EEDF in the energy range ε = 2Ð4 eV. The dip is 6. N. A. Gorbunov, N. B. Kolokolov, and A. A. Kudryav- associated with the vibrational excitation of N2 mole- tsev, Zh. Tekh. Fiz. 58, 1817 (1988) [Sov. Phys. Tech. cules. Phys. 33, 1104 (1988)]. (ii) In discharges in oxygen, the monotonic nature of 7. N. A. Gorbunov, N. B. Kolokolov, and A. A. Kudryav- the EEDF stems from the following two factors: first, tsev, Zh. Tekh. Fiz. 61 (6), 52 (1991) [Sov. Phys. Tech. the cross section for the vibrational excitation of oxy- Phys. 36, 616 (1991)]. gen molecules is smaller than that for nitrogen mole- 8. N. A. Oyatko, Yu. Z. Ionikh, N. B. Kolokov, et al., cules and, second, the threshold energies for the excita- J. Phys. D 33, 2010 (2000). tion of the lowest lying metastable states of oxygen 9. Yu. A. Ivanov, Yu. A. Lebedev, and L. S. Polak, Contact molecules are low. Diagnostics in Nonequilibrium Plasmochemistry (Nauka, The results of theoretical calculations agree well Moscow, 1981). with the experimental data. 10. A. I. Shchedrin, A. V. Ryabtsev, and D. Lo, J. Phys. B 29, 915 (1996). The measured and calculated EEDFs can be used to model the plasma kinetics and, accordingly, to deter- 11. J. P. Shkarofsky, T. W. Johnston, and M. P. Bachynski, The Particle Kinetics of Plasmas (Addison-Wesley, mine the densities of all plasma components, thereby Reading, 1966; Atomizdat, Moscow, 1969). providing a good way to optimize the technologies 12. I. A. Soloshenko, V. V. Tsiolko, V. A. Khomich, et al., based on hollow-cathode glow discharges. In addition, Fiz. Plazmy 26, 845 (2000) [Plasma Phys. Rep. 26, 792 since the electric fields in such discharges are low, the (2000)]. EEDF adequately reflects inelastic electron processes and thus may give both qualitative and quantitative information about the cross sections for the corre- Translated by I. A. Kalabalyk

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