Large quasineutral electron velocity oscillations in radial expansion of an ionizing plasma A R Karimov, M Y Yu and Lennart Stenflo Linköping University Post Print N.B.: When citing this work, cite the original article. Original Publication: A R Karimov, M Y Yu and Lennart Stenflo, Large quasineutral electron velocity oscillations in radial expansion of an ionizing plasma, 2012, Physics of Plasmas, (19), 9, 092118. http://dx.doi.org/10.1063/1.4754865 Copyright: American Institute of Physics (AIP) http://www.aip.org/ Postprint available at: Linköping University Electronic Press http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-85093 PHYSICS OF PLASMAS 19, 092118 (2012) Large quasineutral electron velocity oscillations in radial expansion of an ionizing plasma A. R. Karimov,1,a) M. Y. Yu,2,3,b) and L. Stenflo4 1Institute for High Temperatures, Russian Academy of Sciences, Izhorskaya 13/19, Moscow 127412, Russia 2Department of Physics, Institute for Fusion Theory and Simulation, Zhejiang University, 310027 Hangzhou, China 3Institut fur€ Theoretische Physik I, Ruhr-Universitat€ Bochum, D-44780 Bochum, Germany 4Department of Physics, Linkoping€ University, SE-58183 Linkoping,€ Sweden (Received 13 July 2012; accepted 12 September 2012; published online 28 September 2012) Radial expansion of an ionizing gas or plasma cylinder into vacuum is investigated. An exact model for the evolution of the density and velocity fields of the electrons, ions, and neutrals, including the effect of photo and electron-impact ionization on the flow characteristics is developed and solutions obtained. A quasineutral nonlinear electrostatic mode involving rapid oscillations in the electron velocity but not in the density can occur in the expanding plasma. The mode turns out to be almost unaffected by weak ionization. VC 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4754865] I. INTRODUCTION II. GOVERNING EQUATIONS Expansion of an ionizing gas into vacuum remains a We shall consider radially symmetric expansion15 of an topic of much interest because of its relevance in many areas unmagnetized cylindrical plasma consisting of neutral atoms, of research and applications, including laser-plasma interac- cold ions, and electrons, accounting for electron-impact ioni- tion, astrophysical jets and clouds, phenomena in ionospheric zation or photo ionization in the presence of an intense exter- and space plasmas, laboratory electrical discharges, etc.1–4 nal light source such as a sun or a laser beam. The domain of The process can be accompanied or even dominated by con- the expansion is infinite, i.e., there are no wall or other densation, chemical reactions, ionization, recombination, boundary effects. At t ¼ 0, the system is located in 0 < r < a etc., and can involve several time and space scales that are of and its density is given by similar, or differ by several orders of, magnitude. To investi- n ðrÞ¼N ; (1) gate such flows, perturbation methods based on a priori time- 0s 0s scale separation or fixing are inapplicable and one has to where the subscripts s ¼ e, i, and n denote electron, ion, and take into account all possible timescales of the system of neutral gas quantities, respectively. The constants a and N 5–13 0s interest. define the initial density profile of the plasma. For conven- Most existing works on the flow of reacting or ionizing ience, we assume N0e ¼ N0i ¼ N0n ¼ N0, although initially gases invoke linear, or nonlinear but small amplitude, pertur- nonneutral systems can also be included. Elastic collisions 7–13 bations around a steady state. However, for phenomena among the charged particles, such as that of electron-electron, initially far from equilibrium, especially in the interaction of electron-ion, etc, are usually ignorable in low-temperature intense laser with matter and in astrophysical or nuclear plasmas, which are usually weakly ionized and the plasma explosions, such perturbation approaches are not applicable density is low. since the field variations can be larger, or even much larger, We can write the governing fluid equations in the than the background fields. In this case, a fully nonlinear treat- dimensionless forms ment of the governing equations is called for. In the present paper, we examine how ionization influences the evolution of @tðrnsÞþ@rðrnsvsÞ¼rWs; (2) expanding plasma layers by allowing for large field variations. A nonperturbative approach based on the method of separa- @tvs þ vs@rvs ¼ lsE þ Wsvn; (3) tion of variables shall be used.14,15 In particular, we consider @rðrEÞ¼rðni À neÞ; (4) the effect of ionization16 on the flow dynamics and large- amplitude oscillations in a homogeneous expanding cylindri- where l 1; l m =m , and l 0. The radial coordi- 15,17 e ¼ i ¼ e i n ¼ cal plasma. It is found that in the expanding plasma, there À1 nate r and the time t are normalized by a and xpe can exist a quasineutral nonlinear electrostatic mode with fast 2 1=2 ¼ðme=4pN0e Þ , respectively, where e and me are the oscillations in the electron velocity that is not accompanied by charge and mass of the electrons. The densities ns, velocities density oscillations. Because of the quasineutral character of vs, and the electric field E have been normalized by N0; axpe, ionization, weak ionization does not significantly affect the and 4peN0a, respectively, and Ws is the normalized rate of expansion and the mode. ionization. Significant photoionization of a neutral gas occurs in the a)Email: [email protected]. presence of sufficiently intense radiation. Depending on the b)Email: [email protected]. properties of the radiation and the gas, it can appear in many 1070-664X/2012/19(9)/092118/5/$30.00 19, 092118-1 VC 2012 American Institute of Physics Downloaded 07 Nov 2012 to 130.236.83.30. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions 092118-2 Karimov, Yu, and Stenflo Phys. Plasmas 19, 092118 (2012) forms, including single and multiple photon ionization, and characteristic electron temperature, and k is Boltzmann’s avalanche ionization or optical breakdown, etc., as well com- constant. For the recombination, we have18 binations of them. For photoionization by an intense (say of 14 2 2 5 3 intensity I 10 W=cm ) radiation source, the normalized 4 p r0vene 2 WL ¼ ; (9) rate for the simplest photoionization mode can be written as 3 N0xpe IrphðxÞ 2 W ¼ W ¼W ¼ n ¼ n ; (5) where r0 ¼ 2e =3kTe is the Coulomb radius. The condition e i n hxx n ph n pe We WL then becomes where rph is the photoionization cross section of the neutral gas re Ui Ui 4p ne ne atom at the frequency x of the radiation. We have assumed þ 2 exp À ; (10) rc kTe kTe 3 nn nc axpe c vosc,wherevosc ¼ eE0=xme is the maximum os- cillation speed of an electron in the light-wave electric field where r ¼ pr2 and n ¼ 1=r3. Analogously, for photo ioni- 1=2 c 0 c 0 E0 ¼ð8pI=cÞ and c is the light speed, so that radiation zation, the corresponding condition is driven plasma dynamics can be neglected. If jvsjvosc,an external harmonic electric field, say E sinðxtÞ of the radia- re I 4p ne ne ext ; (11) tion, should be added to the self-consistent fields. As a result, rc hxneve 3 nn nc the plasma will also be driven by the imposed oscillating field as well as the resulting ponderomotive force and thus gain addi- so that Eqs. (7), (10), and (11) determine the conditions tional energy. However, this effect requires a separate treatment under which particle production by ionization is dominant. and is beyond the scope of the present work. They are easily satisfied for low-temperature plasmas. Note Electron-impact ionization of a gas occurs if the latter is that since photo ionization involves photon-atom interaction, of sufficiently (but not too) high temperature (say, >16 eV it is usually much faster than the collisional processes. for the common gases, but the ionization rate decreases for too high temperatures since the collision cross section III. EVOLUTION OF THE PLASMA becomes smaller, see Eq. (8) below.) For electron-impact ionization, the normalized ionization rates can be written as2 In order to solve Eqs. (2)–(4) without perturbation, we look for self-similar flow structures in the form k N W ¼ W ¼W ¼ i 0 n n ¼ n n ; (6) e i n n e coll n e vs ¼ rVsðtÞ; ns ¼ NsðtÞ; (12) xpe which corresponds to spatially uniform electron, ion, and where ki is the ionization constant. Thus, the inelastic ioniza- tion processes are characterized by the rates and . neutral densities and a given velocity profile that has a linear ph coll 14,15 We note that the photo ionization and electron-impact ioni- r dependence. We shall see that with this simple struc- zation nonlinearly affect the particle continuity and momen- ture, the space-time dependence of the collective plasma dy- tum equations in different manners. namics and the full nonlinearity of the given system can be The effect of elastic inter-particle collisions on the dy- included without approximation in obtaining the solution, namics of the plasma expansion and oscillations is negligible and the evolution of the system can still be complex and since the plasma component is rarefied and such collisions physically interesting. do not affect the ionization process.1,2 The condition for Substitution of Eq. (12) into Eq. (2) yields neglect of elastic collisions can be written as dtNs þ 2NsVs ¼ Ws; (13) sn vs À vn 1; s ¼ e; i; (7) where s ¼ e, i, and n, and W is given by Eq. (5) for photo xpe vs s ionization and Eq.