The Role of Electrostatic Instabilities in the Critical Ionization Velocity Mechanism

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TRITA-EPP-77-18

THE ROLE OF ELECTROSTATIC INSTABILITIES IN THE CRITICAL IONIZATION VELOCITY MECHANISM

M.A. Raadu

July 1977

Department of Plasma Physics Royal Institute of Technology S-100 44 STOCKHOLM 70, Sweden THE ROLE OF ELECTROSTATIC INSTABILITIES IN THE CRITICAL IONIZATION VELOCITY MECHANISM M.A. Raadu Department of Plasma Physics, Royal Institute of Technology, Stockholm, Sweden

Abstract The role of electrostatic instabilities in the critical ionization velocity mechanism is investigated. The analysis is based on the theory developed by Sherman which interprets Alfven's critical velocity in terms of a circular process. This process involves the acceleration of electrons by a two-stream instability modified by the presence of a magnetic field. A general expression for the energy and momentum of ions and electrons associated with an electrostatic mode is derived in terms of the plasma dielectric constant. This is used in the case of the modified two-stream instability to determine the distribution of energy between ions and electrons. An extrapolation from the linear phase then gives an estimate of the energy delivered to the electrons which is compared to that required to ionize the neutral gas. 1. Introduction

The critical ionization velocity mechanism involves the interaction of neutral gas with a plasma. In general ionization of a neutral gas is possible if there is a source which can deliver the required energy. In a situation where plasma and neutral gas move relative to one another, energy is in principle available from the relative motion and may be used to produce ionization. Alfvén (1954, 1960) proposed that there should be a critical velocity for ionization to occur. He considered the situation where a continuous stream of neutral gas collides with a plasma in a magnetic field. The mechanism envisaged is a rapid increase in ionization when the velocity of the neutral gas u with respect to the plasma is such that the atoms have a corresponding kinetic energy equal to the ionization energy, eV.. Thus if the atomic mass is m_, Ck

\ Vc - eV. (1)

The required energy is released by the braking of the relative motion between the neutral gas and the magnetized plasma. The original proposal for such a critical ionization velocity was made by Alfvén (1954) in connection with a theory of the origin of the solar system. There is now ample confirmation from a variety of laboratory experiments (see Danielsson, 1973, for an extensive review).

In a collision between a neutral atom and an ion, where momentum is conserved, only a fraction of the kinetic energy of the neutral atom given by the relative velocity can be released. Such collisions would imply a value of th»j critical velocity larger than originally proposed and also dependent on the mass of the ions involved. However the original proposal has been found to be in close agreement with experiment. This indicates that collective plasma processes may be involved. A particularly vivid demonstration of the critical velocity phenomenon is given by the plasma-gas impact experiment (Danielsson 1970, 1974; Danielsson and Brenning, 1975) in which a hydrogen plasma with super-critical velocity penetrates a small cloud o^ neutral gas, usually halium, in a region of transverse magnetic field. A strong interaction occurs leading to rapid ionization and a braking of the motion to a velocity close to the critical value- Danielsson (1970) argues from the inefficiency of direct binary encounters that an efficient collective interaction between the plasma and the neutral gas must operate. High energy electrons are found to be produced by the interaction and these can account for the rapid ionization of neutral gas. Thus the critical velocity mechanism is a particularly interesting ionization process in which the collective plasma properties of the ionized gas play a vital role.

From a theoretical point of view an efficient process for energizing electrons is needed. The initial energy is in the ion component of tha plasma. A similar problem is raised in the interpretation of experiments involving collisionless shock waves, .«'here a mechanism is required to preheat the electrons. In this connection Krall and Liewer (1971) consider the role of instabilities driven by the relative drift between ions and electrons across a magnetic field. They describe instabilities resulting from the coupling between a drift wave in a nonuniform plasma to either the ion plasma oscillation or a lower hybrid oscillation or alternatively a form of the two-stream instability. The growth rates are of the order of the ion plasma or lower hybrid frequency. Sherman (1969, 1972) argues that the critical ionization velocity mechanism is driven by a circular process involving the energization of electrons by the electrostatic two-stream instability modified by the presence of a magnetic field, such as described by Buneman (1962). There are, therefore, useful comparisons to be made between the critical velocity mechanism and processes in shock waves.

If, in the critical velocity mechanism, electrons are being energized by the modified two-stream instability, it is important to know hov; the energy is shared between the ions and electrons. The situation considered is one in which the ions are inefficient- in producing ion ization. To account for ~l the observed critical velocity the bulk of the energy should go via the electrons to ionize the neutral gas. In this paper an estimate of the maximum electron energy is made extrapolating from the distribution of energy between the ions and the electrons in the linear phase and using general arguments about the saturation process. This analysis may be applied in a further development of the approach initiated by Sherman (1969) to estimate the critical velocity and is an extended version of a previous report ( Raadu, 1975 ).

2. Sherman's Critical Ionization Velocity Mechanism

Sherman (1969, 1972) proposes a circular process for the critical velocity mechanism in connection with Danielsson*s (1970) experiment. In this experiment a beam of plasma is shot into a region of transverse magnetic field into which a cloud of neutral gas has been released. If ionization of the neutral gas occurs, newly formed ions with no initial drift velocity will be introduced within the plasma beam and thbre will be a relative drift between these ions and the electrons in the beam. The plasma beam enters the region of trahsverse magnetic field by setting up a polarization electric field and the plasma moves with the E/B drift velocity. The newly formed ions and electrons are accelerated by this electric field and acquire a common drift velocity with the plasma beam. The polarization electric field is reduced by separation of the newly formed charges and the plasma beam is decelerated. Due to the relatively large inertia of the plasma beam ions, they will have a relative motion with rospect to the electrons during the deceleration of the beam. The motions of the beam particles and newly formed ions and electrons produced by ionization are indicated in Figure 1. Sherman argues that the relative drift motions between ions and electrons produced by these mechanisms can drive a two-stream instability modified by the presence of a magnetic field which accelerates electrons parallel to the magnetic field. These energetic electrons can then ionize the neutral gas. In this way the interaction is maintained by a circular process.

The ionization energy is derived from the relative motion between ions and electrons accompany ing the introduction of newly formed charges within the plasma beam. An essential link in the process is the electron acceleration. From an analysis of the modified two-stream instability, Sherman argues that the unstab1e modes propagating nearly perpendicular to the magnetic field are the most significant and that their growth stops when the electron energy is of the order ^ n^u^r where nu and u, are the mass and relative velocity of the ions drifting with respect to the electrons. This energy must be greater than the ionization energy eV. for the circular process to work. If the newly formed ions are considered, then m. is equal to the mass of the neutral atoms, m , and sufficiently energetic electrons to ionize should be produced if the drift velocity u, exceeds the . critical velocity u defined by Equation 1. Since the newly formed ions have initially no mean velocity with respect to the neutral gas their drift velocity u, with respect to the plasma beam is equal to the relative velocity between the plasma and the neutral gas. Thus consideration of the saturation of instabilities driven by the drift motion of newly formed ions leads directly to a critical velocity for the plasma interaction with the neutral gas. This is the point of view adopted in the following analysis.

Sherman (1969) develops the analysis of the modified two- stream instability given by Buneinan (1962) to describe the growth of the instability and the final saturation. The appropriate length and time scales for the development are such that the electron motion perpendicular to the magnetic field is restricted, but the ions may be regarded as moving freely. Under these circumstances the electrons are accelerated primarily in the magnetic field direction. In the critical ionization velocity interaction newly formed electrons acquire the E/B drift velocity on the time scale of the electron gyroporiod. Moreover the adjustment of the electron drift velocity to changes in the polarization electric field associated with braking of the plasma motion takes place on a similar time scale. Thus the? electrons may be treated as having a common E/B drift velocity on longer time scales. The plasma ions and newly formed ions will initially have different mean velocities and velocity distributions, which develop under the combined action of the polarization ele ^tric field and the magnetic field. Here it will be assumed that the essential mechanism is the two stream interaction between newly formed ions having their initial velocity and the electrons moving with the drift velocity. The plasma beam ions are assumed to play a secondary role.

3. The Modified Two-Stream Instability

The modified two-stream instability arises in a situation in which ions stream relative to electrons across a mag- . netic field. The treatment follows that of Buneman (1962) for cold ions and electrons and applies to a region sufficiently small to be treated as uniform with a time scale long compared with the required growth times of the instability. A frame of reference is chosen where the electrons are stationary and the ions drift across the magnetic field with a velocity u,. It is implicitly assumed that the length scale is small compared with the ion gyro radius so that the ion drift velocity u. can be treated as a constant.

Buneman (1962) considers electrostatic modes with frequency a) and wave vector k. The momentum equation for the electron perturbations is then,

-iwv = ^- k -f o. A v (2) -w

where v is the electron velocity, <|> the electrostatic poten- tial and w the electron gyrofrequency vector (parallel to the magnetic field). From the continuity equation for the electrons,

-iume = -i(k.v)no (3)

where n is the perturbation to the original electron density n , it then follows that the electron charge perturbation is ~l O 7 u 2 ku- (k.., e . eo4 JB. ene . e4 JB- - -* e

where co is the electron plasma frequency. Similarly for the ions moving with velocity u, but neglecting the magnetic forces the perturbation to the ion charge is u . 2,2 eni "V» r ,2 (5) ik)

where <*> is the ion plasma frequency. Purely electrostatic modes are considered and hence the perturbations to the magnetic field have been assumed to be zero.

In the present treatment it is convenient to describe the X situation macroscopically in terms of a complex plasma dielectric constant e which implicitly incorporates the

effects of plasma charge and current densities, p_, and jD. Thus from Poissonfs equation

2 2 -c k 't> + !>„ + Ps, •-= -ek tf + Pr = 0 (6)

where P is an externally introduced charge density. From Equations 4 and 5 the plasma dielectric constant is given by, 2 2 r £(k, w) = i - —-£±-1w .2 cos i-w ^ 2 sin r-f-t»w 2 (7)

where 9 is the angle between the wave vector k and the magnetic field. The free modes of the plasma are given by setting the external charge density to zero in Equation 6, the Poisson equation, and the dispersion relation is given by setting to zero the plasma dielectric constant, c, in Equation 7. If general electromagnetic modes are considered the plasma properties must be represented by a dielectric tensor (c.f. Ichimaru, 1973). In the present case the plasma source terms in Maxwell's equations can be incorporated in the definition of the displacement vector D as follows, r ~i b

where

D = e (k,w)E (10)

If the external sources, indicated by the subscript E, are zero Equations 8-10 give the plasma density and current fluctuations in terms of the electric field E for the free modes.

The dispersion relation which follows from Equation 7 re- N duces to that for no magnetic field for modes parallel to the magnetic field (sin9 = 0). The term containing sin8 is due to electron motion perpendicular to the magnetic field and that containing cosö to the parallel electron motion. The parallel motion may be understood as follows. For electrostatic modes the electric field is parallel to the wave vector and the component parallel to the magnetic field contains a factor cos8 (see Figure 2 ). This results in acceleration parallel to the magnetic field. The component of this acceleration which is significant in deter- mining charge fluctuations is that parallel to the electric field and this introduces a further factor cos8. Thus the response for electron motion parallel to the magnetic field is equivalent to that for a particle which is forced to move parallel to the applied electric field with an increased mass mH given by

m = y~ (11) COS f)

where m is the electron mass. The associated plasma frequency would be 2 (nne \I/2 OJ * = l_iL = cosOw (12) pe U,jn*7 pe This concept of an effective electron mass, also used by McBrido et a! '.1972), thus indicates why the cos'O factor occurs ir. tho JiRpersion relation ( c.f. Raadu, 1975 ).

Unstable c?i.r:etiostati.c modes are expected from the dispersion relation which are analogous to those for the ordinary two-stream instability in the absence of a magnetic field. These modes can be expected to grow rapidly until the components of the electron velocity and the drift velocity u-, parallel to the wave vector are comparable. For low frequency modes ;:.;

1 / v \2 1 * 2 2 me() - mv which io the energy of 2.particle with effective mass m moving with velocity v parallel to the wave vector. For modes which propagate nearly perpendicularly m can be large and in the case where the effective electron plasma frequency J **" equals the ion plasma frequency tu . it is equal to m., the ion mass. If the drift velocity u, is also nearly perpendicular tc the magnetic field the final 2 electron energy should then be of the order m.u,/2. This property is an essential link in the circular process for the critical ion i nation velocity mechanism proposed by Sherman.

We now precede to consider further properties of the modified two-stream instabilities, which are relevant to electron energization and tho critical velocity mechanism. A basic requirement is that during the growth of the instabilities the electron?; move mainly parallel to the magnetic field. This is the case if the term arising fror* the neroendicular motion in Equation 7 is small compared to that from the parallel motion (the; coy'" term) . Hence assuming j a> j << »> we also require

(xi •>> wj tan " (14) 10

For the critical velocity mechanism to work this should hold for growing modes with m* = n^. The characteristic frequency is then u . and since 8 is then close to 90 ^ pi the requirement is, we >y wpe

If this condition is not fulfilled the maximum possible effective electron mass will be less than the ion mass and the critical velocity deduced from Sherman's circular process must be larger than u_ as defined in Equation 1. Assuming now that the condition (14) is met the dispersion relation which follows from Equation 7 may be written approximately as, "> 1 Ei—. - "Ef- - 0 (15) (W-kUa) to Here it is assumed that the drift velocity JJ-, is perpendicular to the magnetic field and that the wave vector Jc, considered is nearly parallel to u,. In this form the dispersion relation is identical to that for electrons with an effective mass m* in the absence of a magnetic field. The dynamics of an electron constrained to move par£.il-'l to the magnetic field can in general be regarded as equivalent to that of a particle of mass m* accelerated in the direction of the applied forces. This principle will tr used in the further developments of the theory presented here.

An essential requir tent from the point of view of the theory for critical ionizo'ion velocity is that most of the energy released by the modified two-stream instability should go into the electron motion. In general energy is also gained by the to s. In principle all unstable modes should be 'reated as acting simultaneously. However, as the growth rate is essentially determined by the two fre- quencies to . , u) *' it can b-j shown to decrease as u * de- 11

creases, that is as m increa: JS. Moreover the saturation cf a particular mode can be expected when the spread of electron velocity components in the direction of the k vector considered is greater than the component of the drift velocity u, in the same direction. This means that the total electron saturation velocity increases as the angle between k and the magnetic field is increased, or equivalently as m*" increases. Thus modes with small equivalent masses m* grow most rapidly, but saturate first with a lower spread in electron velocity parallel to the magnetic field. In the following analysis we consider the distribution of energy between the ions and electrons for different modes and show that the smaller the value of m11" the greater the fraction of energy which goes to the electrons. Now, if we consider the growth of an unstable mode with a particular value of m*, only those modes with smaller values of m* can be expected to have grown significantly and reached saturation. They will have delivered a greater fraction of energy to the electrons than the mode under consideration can deliver. Thus if instead we assume that all the energy is released by one particular mode this will lead to an underestimate of the actual energy delivered to the electrons from the combined action of all modes which have gr^wn significantly.

4. The energy distribution for electrostatic modes

From the dispersion relation (15) two-stream unstable modes can be found which grow exponentially in the linear phase. The directed and oscillatory motions of the ions and electrons will change as a result. The associated kinetic energies will also change. That part of the kinetic energy associated with the mean velocity is dependent on the frame of reference; whereas that associated with the oscillatory motion is absolutely defined and is equivalent to a spread in the velocities. The spread in the electron velocities can be regarded as an energizing of the electrons eventually leading to ionization. The relative mean velocity between the ions and electrons decreases. r 12 ~1 To find the directed and oscillatory parts of the kinetic energy as defined above, in the general case of an electrostatic instability, the approach of Stringer (1964) could be followed. His results, which are based on a second order perturbation analysis, involve the real and imaginary parts of the dispersion equation. However, as far as the energy is concerned more general results may be derived from the definition of the plasma dielectric constant e. These general results must be consistent with the particular case dealt with by Stringer, as can be verified by taking the appropriate form of e, since they are ultimately based on a similar kinetic analysis. It is of interest to note that as e is derived from a first order analysis the appli- cation of general principles of energy and momentum avoids all consideration of the form of the second order perturbations.

The plasma charge and current densities p., and j_ can be derived from the basic definitions of the displacement D and the plasma dielectric constant e (Equations 8-10). The work done on the plasma current must go into the kinetic energy T, and the force acting on the plasm i charge changes the momentum P. The rates of change of T anc P are given by

at PP 5

For a particular electrostatic mode with wave vector k and frequency cu it follows that,

= Re{-io)(e-e0)E}Re{E} (16)

|| = Re{ik(c-eo)E}Re{E} (17)

Let Y be the growth rate, the imaginary part of a. Since T and P are proportional to the square of the electric field 13

their growth rate is 2y. From Equations 16 and 17 , taking space averages and dividing through by 2y it then follows that,

(18)

where U is the total energy, < > denotes the space average and the wave number k is real. These expressions are quite general and may be used both for cold and warm plasma provided that the appropriate form for the dielectric constant is used. For free modes e is zero and thus the total energy and momentum are zero, as they should be from conservation principles for modes which either grow from or decay to vanishingly small values. However e is the sum of terms associated with the components of the plasma for each of which the energy and momentum can be found using Equations 18 and 19 .

In the two-stream situation we are interested in the kinetic energy associated with oscillatory motions and so that part of the kinetic energy associated with a mean change of velocity and momentcii must be discarded. Dealing term by term, the oscillatory kinetic energy of the component o is,

= ImUw-ku )E } ,E> 9 (20) Y 4 where o denotes quantities associated with the component o and UQ is the drift velocity.

In the particular case of a cold plasma two-stream instabi lity in the presence of a magnetic field considered here, using Equations 15 and 20 we find:

K e 2 l r 14 ~l

i T ^2 7 1 i 2 where the frequency u is complex. This determines the distribution of energy once the dispersion equation is solved for the modes of interest.

5. The two-stream modes

In the special case where m* and m. are equal Equation 15 has solutions of a relatively simple form:

W = 3kud * * (23)

where V

and for ku, <2/2 to . there are growing modes with imaginary A. From Equations 21 and 22 it follows that

z T = T4 = ^- P^- <^ e E > (25)

for these growing modes. This is the case where the effective electron plasma frequency u> equals the ion plasma frequency u) ^. Equation 25 indicates that the ions and electrons will have equal energies. Their velocity components parallel to the wave vectoi will also be equal and when they are around half the drift velocity, u^, saturation of the instability should set in. Following the chain of argument for the critical velocity mechanism this would lead to a critical ionization velocity twice that observed.

Therefore we must consider the modes with m* > m. which for a given saturation velocity give greater electron energy. However as the effective electron inertia, the mass rri*, increases we anticipate that the ions will gain proportiona- tely more energy and that the electron saturation velocity will be decreased. It is these competing effects which must be assessed in the more detailed analysis which follows. r 15 ~l The modes with nt1*" > m. can be treated by the approximation method used by Buneman (1958) and Sherman (1969). It is assumed that [w^ku^ and writing u)=[w le1* Equation (15) can then be approximated by-

u *2 21 OJ . 2 2tu . I to I x . . e(kr »,»!-_£« e" * - -EL-, EiLL e * (26)

The modes are given by setting the real and imaginary parts of the dielectric constant to zero. Discarding the stable mode and looking for the mode with maximum growth rate given by t|>=2ir/3 it follows that,

(27)

where the corresponding growth rate is /3|w|/2. Using Equa- tions 21 and 22 the energy in the components is then,

!—I (28) \m*/ 2 ° TI - 4 EOE2> (29)

As anticipated the ions have more energy than the electrons. To determine the electron energy the saturation of the instability must be considered in more detail.

6. The electron energy

As the ions and electrons interact through the two-stream instability their relative velocity is reduced. This is the source of the energy for the instability that ultimately goes into heating the plasma.

In the situation of the plasma neutral gas impact experiment saturation of the instability is due both to the spread in the velocities of the ions and electrons and a reduction of their mean relative velocity. 16

On the basis of the electron dynamics discussed above it is possible to treat all motions as being in the same direction if the electron effective mass m* is used. Applying conservation of momentum and supposing that the original relative velocity u, is reduced to us the energy dissipated per unit volume is

m. m .> _

where nQ is the electron density. Assuming that this energy ultimately goes to the ions and electrons when the instability saturates and that it is distributed approximately in the way indicated by the linear phase, the electron energy can be estimated usinci Equations 28 and 29 :

2 2 Te = i ami(ud -us ) (31) where the electron energy parameter a is given by,

insEL _ L_ (32)

This function, plotted in Figure 3, has a maximum for an effectieffectiv\ e mass ratio m./m* •- 0.2075 and the electron energy is then

2 2 Te = 0.4012 ^m.(ud -ug ) (33)

An upper estimate is given by assuming the finil relative velocity to be zero and this gives a critical velocity greatei than that observed by a factor 1.58. This is an improvement on the factor 2 for the case of equipartition and indicates the significance of the modes considered.

The analysis of modes with high effective electron mass m1* is based on the assumption that the electrons are constrained r 17 ~1

to move parallel to the magnetic field (the initial thermal motion is negligible compared to the direct motion). From Equation 14 the condition for this is:

The dependence on the mass ratio is very weak and for the mode giving maximum electron energy the condition is w >> 1.03 which is not significantly different from the first estimate, w >> w „. e pe 7. Conclusions

In deriving the particular results of relevance to the critical ionization velocity mechanism a general method of deriving the energy for electrostatic plasma modes has been developed. Given the complex plasma dielectric constant which determines the macroscopic properties of a plasma, it is possible to deduce the oscillatory energy in the ions and electrons and their change in momentum, without having to consider details of the particle kinetics. These results are expressed in Equations 8-10 and should be of general use in interpreting the physical nature and consequences of electrostatic instabilities.

Applying this method to the modified two-stream instability we have shown that the electrons can gain energy of the right order of magnitude required for the critical velocity. Al- though an increasing fraction of energy goes to the ions for modes with large effective electron mass m*, the effective loss of energy to the ions does not prevent substantial energization of the electrons. The condition that the electron motion is predominantly parallel to the magnetic field for the instabilities effective in heating the electrons, Equation 14, is marginally satisfied in the plasma-gas inpact experiment (Danielsson, 1970) for which the electron gyrofrequency is comparable to the plasma frequency. The expected growth rate which is of the order of the ion plasma frequency is also sufficiently large in comparison to the experimentally r 18

observed interaction rate. Polarization measurements on He-lines ( Danielsson , 1974 ; Danielsson and Lindberg , 1974 ) indicate an anisotropic electron velocity distribution with the main velocity ••omponent along the magnetic field. This gives experimental confirmation of the requirement that the electrons are accelerated along the magnetic field in the plasma-gas impact experiment and supports the theory of electron energization. Thus it appears that the modified two-stream instability may play an essential role in some critical velocity situations.

Acknowledqeroents The author is indebted to Professor C-G Fälthammar, Dr. L. Lindberg and Dr. J. Sherman for stimulating discussions.

This work has been supported by the Swedish Atomic Research Council. r 19 ~1

References

Alfvén, H.: 1954, On the Origin of the Solar System, Oxford University Press, Oxford

Alfvén, H.: 1960, Collision Between a Nonionized Gas and a Magnetized Plasma, Rev. Mod. Phys. 32, 710

Buneman, O.: 1958, Instability, Turbulence, and Conductivity in current-Carrying Plasma, Phys. Rev. Letters ^1, 8

Buneman, O.: 1962, Instability of Electrons Drifting Through Ions Across a Magnetic Field, Plasma Physics (J. Nucl. Energy Part C) £, 111

Danielsson, L.: 1970, On the Interaction between a Plasma and a Neutral Gas, Phys. Fluids 13, 2288

Danielsson, L.: 1973, Review of the Critical Velocity of Gas-Plasma Interaction I: Experimental Observations, Astrophys. and Space Sci. 24, 459 " "' '" ••»•'— « •••III •.....! ...... ^^^

Danielsson, L.: 1974, Experiment on the Interaction Between a Plasma and a Neutral Gas II, TRITA-EPP-74-03, Royal Institute of Technology, Stockholm

Danielsson, L. and Lindberg, L.: 1974, Polarization Measurement of Spectral Lines of Low Intensity and Short Duration, J. Phys. E : Scl. Instr. J,' 817

Danielsson, L. and Brenning, N.: 1975, Experiment on the Interaction Between a Plasma and a Neutral Gas II, Phys. Fluids 18, 661

Ichimaru, S.: 1973 Basic Principles of Plasma Physics, A Statistical Approach, W.A. Benjamin Inc., Reading, Massachusetts

Krall, N.A. and Hewer, P.C.: 1971, Low Frequency Instabi- lities in Magnetic Pulses, Phys. Rev. A 4^ 2094

McBride, J.B., Ott, E., Boris, J.P. and Orens, J.H.: 1972, Th€;ory and Simulation of Turbulent Heating by the Modified Two-Stream Instabi 1 i ty, Plvys^_Fl^iids 15, 20 ~1

Raadu, M. A.: 1975, Critical Ionization Velocity and Electrostatic Instabilities, TRITA-EPP-75^28, Royal Institute of Technology, Stockholm

Sheritian, J.C.: 1969, Some Theoretical Aspects of the Inter- action Between a Plasma Stream and a Neutral Gas in a Magnetic Field, TRITA-EPP-69-29, Royal Insti- tute of Technology, Stockholm

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Stringer, T.E.: 1964, Electrostatic Instabilities in Current Carrying and Counterstreaming Plasmas, Plasma Phys. (J. Nucl. Energy Part C) 6, 267 Figure Captions

Fig. 1. Plasma moves from the left with the E/B drift velocity u. The electric field £ produced by the polarization charges indicated separates the ions and electrons produced by ionization at the point £ . The new charges tend to cancel the polarization charges reducing the electric field by Ag. The electrons ( e~ ) acquire the new E/B drift velocity. The new helium ions ( Hef ) are only slowly accelerated and the plasma hydrogen ions ( H+ ) are slowly decelerated.

Fig. 2. The electric field component parallel to the magnetic field E cos P produces an acceleration v = -eE cos 0 / m which in turn has a component 2e # v cos 6 = -eEcos 9 / m = -eE/m parallel to the electric field E.

Fig. 3. The electron energy parameter a giving the electron 1 2 2 saturation energy, T = •=• a m. ( u, -u ) as a function of the effective mass ratio m./m . The curve is calculated assuming that this ratio is much less than unity. For comparison the dash3d line ( a = 0.25 ) indicates the value for equipartition ( m./n#- 1 ) . m. is the ion mass. 1 -i- r ~i

BO u

Fig. 1 r ~i

Fig. 2 ELECTRON ENERGY PARAMETER oc

2) tu ~l

TRITA-EPP-77-18 Royal Institute of Technology, Department of Plasma Physics, Stockholm, Sweden THE ROLE OF ELECTROSTATIC INSTABILITIES IN THE CRITICAL IONIZATION VELOCITY MECHANISM N.A. Raadu July 1977/ 24p. incl. ill., in English i

si' The role of electrostatic instabilities in the critical *• • ionization velocity mechanism is investigated. The analysis '«'* is based on the theory developed by Sherman which interprets Alfven's critical velocity in terms of a circular process. This process involves the acceleration of electrons by a i" two-stream instability modified by the presence of a magnetic field. A general expression for the energy and momentum of ions and electrons associated with an electrostatic mode is derived in terms of the plasma dielectric constant. This is used in the case of the modified two-stream instability to determine the distribution of energy between ions and electrons. An extrapolation from the linear phase then gives an estimate of the energy delivered to the electrons which Is compared to that required to ionize the neutral gas.

Key words: Critical ionization velocity, Plasma-neutral gas interaction, Electrostatic instability, Two-stream instability, Magnetic field/ Electrostatic mode energy, Electron acceleration.