QUASI-LINEAR THEORY OF THE LOSS-CONE INSTABILITY

Transcription

1 IC/66/92 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS QUASI-LINEAR THEORY OF THE LOSS-CONE INSTABILITY A. A. GALEEV 1966 PIAZZA OBERDAN TRIESTE

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3 IC/66/92 International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHTSICS QUASI-LINEAR THEORY OF TEE LOSS-CONE INSTABILITY* A.A. GALEEV ** TRIESTE August 1966 * To be submitted to "Plasma Physics" ** Permanent address: Institute for Nuclear Physics, Novosibirsk, USSR

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5 ABSTRACT The loss-cone instability of plasma confined in a mirror-type trap is considered. Relaxation of the particle distribution in the trap with, a length larger than the mean free path "between "turbulent" collisions is described by conventional quasi linear theory. A quasilinear equation for the ion distribution is solved analytically for the case of a small loss-cone volume of particles in velocity space (it takes place, for instance, in the case of a trap with a large mirror ratio). -1-

6 QUA3I-LIH3AR TIE3QKY OF TUB LOS3-OOJ02 INSTABILITY 1. BASIC EQUATIONS Me consider here the loss-cone instability of a plasma confined in a mirror trap (Rosenbluth and Post, 1965) assuming ^"nat: a) the time of escape through the magnetic mirrors of a trap of finite length L is much greater than the time of the turbulent diffusion into the loss-cone Tp L» f^ VI where V is the ion velocity along the magnetic line of force. z b) the volume of the "loss cone" in velocity space is much smaller than the volume occupied by particles (it takes place, for instance, in the case of a magnetic trap with a large mirror ratio). The first assumption permits us to neglect the mode-mode coupling at the nonlinear stage of the instability and to describe the instability evolution within the framework of quasi-linear theory. The second assumption permits us to solve the quasi-linear equations for the wave amplitudes and the distribution functions of the particles. Besides, as Rosenbluth and Post, we assume that: c) unstable oscillations are electrostatic; d) the perturbation scale is sufficiently small for the plasma to be regarded as homogeneous. The electric field potential *p can then be expanded in a sum over the fields of the individual harmonic oscillations! (1) where K, and K^_ are the components of the wave vector respectively along and transverse to the unperturbed magnetic field ^""{.OjO^ H t, CO is the frequency, and ^nu) i s "tke amplitude of the oscillation; e) electrons are cold; f) the frequency and wave length of the oscillation are within the intervals (2) 2

7 where W M, Q_v_i are the gyrofrequencies of electrons and ions respectively; R.^ = V^' L /_O.n ^s "^e Larmor radius of ions with thermal velocities V^; 5 OH ^S "k^e electron Larmor radius. The development of the instability within the above interval of frequencies (2) is possible only in a dense plasma: *-H (3) where U)p,!. > are the plasma frequencies of the electrons and ions respectively. Under assumption (f) we neglect the influence of the magnetic field on the ion motion and use the drift approximation for the description of the electron motion. Then the kinetic equations for the ions and electrons respectively, can be written in the form (4) where M is the magnetic moment of the electron (i.e., M- The complete system of equations contains, besides Eqs. (4) and (5), the equation for the electric field potential x '2H (5) In the linear approximation we reduce Eqs. (4)-(6) to the dispersion equa tion K (7) In a very dense plasma U)p ^ ^0^ we must take into account in Eq.(5) the inertial drift of the electrons which corresponds to the additional term co J 2 -. /cq ^~ in Eq.(Y). We omit this term in the following caloula 3

8 "felons because it changes only the definition of the plasma frequencies in the following way: Ap In so far as K t «K x,we neglect the ion motion along magnetic field lines. Further,the unperturbed ion distribution does not depend on the azimuthal angle in the velocity space, and we can easily integrate over this angles Here (l/f\^/j i& the ion distribution with respect to the dimensionless velocities W = "^ /"it^ki» and it satisfies the normalization condition ( I y(w)aty = \_ ). The root of the integrand of F(y) corresponds to taking the integrals in (8), Q ' being in the upper half plane; so that for u = U + ^ G (9) Finally, we neglect the electron thermal motion according to the third assumption and rewrite the dispersion equation in the form For the plasma near the marginally stable state one may expand Eq over ^J*\ to * )( «U) r and find that the solution with J M GO "> 0-4-

9 corresponding to growing disturbances, appears only if The particle escape from "the loss cone through, the magnetic mirrors keeps this condition fulfilled because tf(o ) = ^ ( ) ~ Using 3q. (10) we can express the growth rate y K in terms of Then the equation for the wave amplitudes can be written in the usual form J Averaging the kinetic equation over the period of the rapid oscillations in theoo-ordinate space we derive the quasi-linear equation for the averaged distribution function i o j ) Under our first assumption L > ^x> ^i ( ^i> is til6 (12) time into the loss cone), no loss of the particles from the trap appears during the process of the ion-distribution relaxation. Then, due, to the small volume of the loss oone (our second assumption) the wave energy is small* Hence, "we can neglect mode-mode coupling and take the. i rapidly oscillating part of the distribution function i eleotric field potential ^ K to in " tlie linear approximation where 6) K is the frequency of the eigen-oscillation with wave vector and amplitude o > >K -5-

10 Within this approximation we reduce Eq.(l2) to the "quasi-linear equation" in the usual form, valid now even for strong instability Y K -^ U) K (13) We used here the axial symmetry of the distribution function in the.velocity space to average Sq. (12) over the azimuthal and longitudinal components of the ion velocity. We can do this even for the wave spectrum without axial symmetry in the j^ -space since the gyration of the particles in the strong magnetic field symmetrizes the velocity distribution. The quasi-linear kinetic Eq. (13) and the equation for the wave amplitudes (ll) are the basic system, which we have now to solve to find the time evolution of plasma with the loss cone for t ~ 0. This is actually a very complicated system. However, w can simplify Eq.(l3) if the main contribution in the integral ^ J /\rf&\]" '»\_ comes from that domain of the velocity space where \j^_» CO /^, If we take the initial distribution function in the form 1 shown qualitatively in Pig. 1, Fig. 1 w the condition» would mean that where U7 is the plateau size. Actually, any kind of initial distribution, even, say, of the type 2 very soon will have a form 1, sinoe the quasi-linear diffusion is very high for smaller ]$". -6-

11 1966) Thus, Eq.(l3 ) might be rewritten in the approximate form (Galeev, where 25 7>W fc Within the same approximation Eq. i 11 ) has a simple form fo:r "t 116 wave amplitude where P. Here the frequency of the oscillations changes in time due to the slow relaxation of the particle distribution. Of course we assume that the characteristic time of the relaxation L is much larger than the period of the osoillation Eq..(14) can be easily solved using Laplace transform. (Galeev, 1966*) The result is where Oi (^ ) is the initial ion distribution, is the Green funotion of Bq. (14)< -7-

12 2. SIMPLE MODEL OF THE ION DISTRIBUTION Let us consider the idealized ion distribution for which we can take the integrals of Eqs. (15) and (16) exaotly = A (J -A e where /j - > P/2/K ) I~ (it C s/ i)^jf 5 -^.7 " B ^^e norma li za '' ; i on constant j l"5 sk^o ( A = i corresponds to the case of empty loss cone: 5^,(0) - 0 ). Then the solution for the distribution function and the equation for wave amplitude can be written in the form = A e From the last expression we see that, if initially A > the growing oscillations cause the turbulent diffusion of ions into the loss cone. As we can see from expression (l8), for the particular choice of the initial ion distribution (17) the quasi-linear relaxation of the turbulent spectrum in time can be described by changing in time the parameters A(-t) t (&) and the effective "temperature" of the main body of the ion distribution only. This change can be represented, by using the formula -8-

13 := e (20) Now it is easy to describe the solution of Eq. (19) qualitatively. At some moment "t o we reach the marginally stable point A(-t ft )= \.(t ) and the oscillations stop growing. However, the ions continue to diffuse into the loss oone (see. Eq. (14)) and the oscillations become damping. It is obvious that the relaxation process stops only when the oscillation amplitude reaches the zero level, when > oc>, The final ion distribution has a margin of stability with respect to the considered perturbations (i.e., J'K <O for arbitrary wave vector K. ). The parameters of this distribution depend on!>*=, only and can be found from the energy conservation law* where CO 00 wdw - (21) =. an? 1 (18) and (20) we immediately obtain A * v< (22) * In the author's paper (Galeev, 1966) it was proposed to use for this aim the equation g (T>^) - 0 because after the first state of relaxation to the quasi-steady state (21) the particle loss through the mirrors supports the instability on a very low level. But in that case we must add at the r.h.s. of Eq. (14) the term describing this particle loss and change the expression for the growth rate (19). -9-

14 In the limit of large mirror ratio the parameter is small and the amplitudes of the oscillations remain small during the relaxation process (i.e., Jj << _ ). On the other hand, by reducing the quasi-linear Eq.(l3) to the form (14) we supposed that the width of the sink on the distribution function essentially increases due to the quasilinear diffusion. Hence, we can apply the result (22) only to the case of initially strong instability A» ^6 (23) Expanding (22) on small parameters Q, -^ao t 7 ^ ^lwe obtain (24) Here we can see directly that the sink on the ion distribution under condition (23) becomes *vider and less deep after relaxation, and wo can justify the approximation (14). The main part of the distribution function is only slightly disturbed by "quasi-linear diffusion". In order to write the equation for the waves in terms of one variable D only, we need to suppose, as usual in the quasi-linear theory, that the wave spectrum has a sharp maximum at the point K = J5-*» ^ C 5_* ^ "^ J( C ) ^ow we neglect the change of frequency for D (-t) > 2- a^d integrate Eq. (19) up to the quadrature with conditions Parameter }*& in the saturation regime can be easily found from Eq.(25) as well as from Eq.(2l) 6s-/^

15 3. DISCUSSION OF THE GEUERAL CASE Let us consider now the case of the arbitrary form of the initial ion distribution but within the approximation (b) of this paper. We represent the initial distribution function as a product of two funotions %M= Y«y(w) (i-% r M) (27) Here ftiva (W/ is. the main body of the ion distribution and changes essentially within the scale A VV f^j L» The second function O& ( W ) is not negative and differs from zero in the small region W <. << i_.of course 1 ^ b if(o) > % y(w/) > o Then in the final stage of the ion distribution relaxation we can expand Eq.(l6) over small parameters (29) The result is,ay oo Here we neglect the small change of the main body of the ion distribution and therefore we cannot describe rigorously the change of the particle energy. But we can use it to find the equation for the wave amplitudes. Substituting Eq,(3O) in Eq»(15) we obtain under assumptions (29) The solution of this equation is saturated at the value

16 3/-S This ansver can he fotind from the energy conservation law (21) "by taking into account in Eq. (30) the change of the main body of the ion distribution OT by independent expansion of the energy conservation law (21) in the integrated form: where. r ( $ ' $ > 'JJ'x is the degenerated hypergeometric. function. if, I 5 *5 ' _ D z 7^; "5^ IV In solving ovcc problem we introduced a lot of idealizations in order to get soluble equations. Of course, in the realistic situation, the development and nonlinear relaxation of the loss cone instability have a more complicate nature, due, for example, to the continuous loss of particles through the mirrors, which we did not take into account, and so on. Under these circumstances, the noniinear mode-mode coupling becomes very important, as was shown by tb.e author,

17 ACKNO WLEDOMEMT S The author is indebted to Professor B.Z.Sagdeev for helpful diaoussions and suggestions, to Dr. H.V. ong for reading the manuscript and for discussion. He thanks Professor Abdua Salam and the I.A.E.A. for hospitality at the International Centre for Theoretical Physios, Trieste* -13-

18 REFERENCES Galeev A.A., (1966) Plasma Physics and Controlled Nuclear Fusion Research, Culham vol. 1 p.393, IAEA, Vienna. Rosenbluth M JU f and Post R.F., (1965) Phys. Fluids B f M

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20 Available from the Office of the Scientific Information and Documentation Officer, International Centre for Theoretical Physics, Piazza Oberdan 6, TRIESTE, Italy 6*37