Stabilization of the magneto-modulational instability in relativistic plasmas

Volume 101A, number 3

PHYSICS LETTERS

19 March 1984

STABILIZATION OF THE MAGNETO-MODULATIONAL INSTABILrI'Y IN RELATIVISTIC PLASMAS A.B. MIKHAILOVSKII I. V. Kurchatov Institute o f Atomic Energy, Moscow, USSR

G.I. SURAMLISHVILI Institute o f Physics, Academy o f Sciences o f the Georgian SSR, Tbilisi, USSR

and V.R. KUDASHEV and E.G. TATARINOV Moscow Physical- Technical Institute, Moscow, USSR

Received 11 July 1983

The magneto-modulational instability of a plasma with a monochromatic Langmuir wave is shown to be stabilized at ultrarelativistic electron temperatures.

1. The magneto-modulational instability (MMI) of a plasma with a monochromatic Langmuir wave has arisen recently much discussions [ 1 - 5 ] . The MMI is of interest, in particular, because its development is accompanied by the generation of a quasistationary magnetic field. It was shown in ref. [3] that, with the electron temperature being sufficiently high, the MMI growth rate may be greater than that of the modified instability [5]. Therefore, the MMI may be of importance in high temperature plasmas. Thus, it is of interest to consider whether the MMI may occur in a relativistic plasma. The problem is discussed in this paper. 2. As it was mentioned in ref. [4], the generation of the magnetic field in a plasma with a monochromatic Langmuir wave is influenced both by the magneto-modulational effect and wave anisotropy. The dispersion equation for the MMI allowing for the anisotropy is of the form [4] : e0 +e 1 + e 2 - c2k2/6o = 0 .

(1)

Here, similar to ref. [4], k and 6o are the wave vector and the frequency of magnetostatic perturbations, e 0 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

is the low-frequency transverse dielectric permittivity of a plasma with no Langmuir wave (in terms used in refs. [1-3] e 0 describes the non-linear Landau damping), el, e2, are the parts of the dielectric permittivity associated with the anisotropy and the magneto-modu. lational effect. According to ref. [4], the anisotropic effect given rise to by the term e 1 is a stabilizing one. In ref. [4] the stabilization of the MMI was considered under the assumption of isotropic electron momentum distribution. It is known, however, [6], that the presence of a monochromatic wave leads to the anisotropy of the particle momentum distribution. Hence, the situation considered in ref. [4] concerns a plasma that had anisotropy of opposite sign before switching on the wave field. In this case the stabilization of the MMI manifests itself in the nonrelativistic approximation. Now, let us follow ref, [1 ] and suppose the equilibrium particle momentum distribution to be isotropic before switching on the wave field. Then, switching on the field adiabatically causes the equilibrium particle anisotropy. According to ref. [7], the wave anisotropy and the particle anisotropy are competing effects with respect to the magnetic field 137

Volume 101A, number 3

PHYSICS LETTERS

19 March 1984

generation. The quantity e I that describes the total anisotropic effect is of the form

3. Within the context of (1), (2), (5) and (8) we find the MMI to occur for K < K . where

e I = -(cop/co)2(G1/g 1 ) w .

K 2 = (cop/C) 2 [H(a)/gl]co,

(2)

Here w = ((Eh)2)/47rnT is the Langmuir wave dimensionless energy density, E h is the wave electric field, n is the plasma density, T is the electron temperature, co2 = 41re2n/m is the square of the nonrelativistic plasma frequency; e, m are electron charge and rest mass. The quantities G 1 , g l are functions of the relativistic parameter ct = m c 2 / T . They are defined by the relations _ G1

ot f 2K2 (a) 0 el2

KE(X) (x - a)2dx x2

(3)

~o K2(x )

gl =K2(a ) j

x 2 dx,

(4)

(10)

with

H ( a ) = G2 /otg2 - G 1 .

(11)

In the nonrelativistic limiting case tx ~ 1, G 1 = l/a, G 2 = 1 ,g2 = 3/a, and t h u s H = ~ > 0. This implies the stabilizing effect of the anisotropy in nonrelativistic plasmas to be negligible as compared to the magnetomodulational one. In the ultrarelativistic limiting case t~,~ 1,G 1 = G 2 = ~-a2,g 2 =~ol. Then H = - ~ a 2 <0. Therefore, eq. (10) does not hold in the ultrarelativistic plasma and the MMI does not occur. The MMI disappears at temperatures which may be found from the condition k2, = 0. According to (10), this requirement reduces to the equation

where K 2 is the McDonald function. According to ref. [7], we have for co ~ 0

O2/c~g 2 = O 1 •

e 2 = (cop/co)2(G2/Oglg2)w,

The solution of the latter is c~ = 0.12. Hence, the MMI can occur for temperatures T < 8.3 mc 2 = 4.2 MeV.

(5)

(12)

where

References ix2

f

K2(x)"

(6)

(X

3ix2 7 KI(X) g2 = 2K2(t~ ) --~ (3or2 - x 2) d x .

(7)

The expression for e 0 is of the standard form [8]

e 0 = i(co2/c Ik Ico)g,

(8)

where g = -~lr(1 + c0 exp(--~)/ctK2(a ) .

138

(9)

[ 1 ] S.A. Bel'kov and V.N. Tsitovich, Sov. Phys. JETP 49 (1979) 656. [2] M. Kono, M.M. Skorie and D. ter Haar, J. Plasmas Phys. 26 (1981) 123. [3] M.M. Skorie and Lj. Stokic, Phys. Lett. 92A (1982) 389. [4] V.R. Kudashev, A.B. Mikhailovskii and G.I. Summlishvili, Phys. Lett. 93A (1983) 409. [5] V.E. Zakharov, Soy. Phys. JETP 35 (1972) 908. [6] V.P. Silin, Parametrieheskoe vozdeistvie izluchenla bol'shoi moschnosti na plasmu (Nauka, Moscow, 1973). [7] A.B. Mikhailovskii, V.R. Kudashev and G.I. Suramlishvili, Zh. Eksp. Teor. Fiz. 84 (1983) 1712. [8] A.B. Mikhailovskii, Plasma Phys. 22 (1980) 133.