Thermal Instability in Plasma with Finite Larmor Radius

Transcription

1 Thermal Instability in Plasma with Finite Larmor Radius R. C. Sharma and Kirti Prakash Department of Mathematics, Himachal Pradesh University, Simla (India) (Z. Naturforsch. 30 a, [1975] ; reveiced July 12, 1974) The effects of the finite Larmor radius of the ions on the thermal instability of a plasma are investigated. When the instability sets in as stationary convection, the finite Larmor radius is found to have a stabilizing effect. The conditions for the nonexistence of overstability are investigated. The case with horizontal magnetic field is discussed. 1. Introduction The thermal instability of a fluid layer heated from below has been discussed by Chandrasekhar 1 under varying assumptions of hydromagnetics. The effects of the finiteness of the ion Larmor radius, which exhibit itself in the form of a magnetic viscosity in the fluid equations, have been studied by many a u t h o r s 2-5. Melchior and Popowich 6 have considered the finite Larmor radius effect on the Kelvin-Helmholtz instability of a fully ionized plasma while that on the Rayleigh-Taylor instability has been studied by Singh and Hans 7. Recently, the author 8 has studied the finite Larmor radius and Hall effects on the thermal instability of a rotating plasma. The effect of the finite Larmor radius on the thermal instability of a plasma in the presence of a vertical magnetic field has also been studied by the author 9. It seems of some interest to study the effect of the finite Larmor radius on the thermal instability of a plasma in the form of an infinite, horizontal layer of thickness d, acted on by a horizontal magnetic field H(H, 0, 0) and gravity force g(0, 0, -g). This layer is heated from below such that a steady adverse temperature gradient ß( =! dt/dz j) is maintained. The plasma is assumed to be incompressible and of finite electrical conductivity. 2. Perturbation Equations and the Characteristic Value Problem The linearized hydromagnetic perturbation equations appropriate to the problem are: 3* 3«= _ V<3P + 4 V'<? = 0, ( f V ) 7 V + r]\72h,, = *V 2 Ä= 0, (3) 6>, (4) where q(u, v, iv), h, SP, 0, Sq denote, respectively, the perturbations in velocity, magnetic field H, pressure P, temperature T and density Q. g, x, v, / / e and r] denote, respectively, the gravitational acceleration, the thermal diffusivity, the kinematic viscosity, the magnetic permeability and the resistivity. For the magnetic field along the z-axis the stress tensor P, taking into account the finite ion gyration radius 3, has the components Pxx Px2 = P J Pxy Pyx ~ 2 P*T. = 2 J'n 0 dv dx yy = P~Qv P o QVQ + dx 3w du 3* (5) aw Pyz Pzy Q v0 ( g ~ i 37 Pzz = P + Q >'o + 3w where p is the scalar part of pressure and qv0 = N Tj4> cog, where coh is ion gyration frequency, while N and T denote, respectively, the number density and temperature of the ions. Analyzing the disturbances into normal modes, we seek solutions whose dependence on x, y, and t is given by exp { i kxx -f i ky y + n t}, (6) where kx, ky are wave numbers along the x and y directions respectively and n is the frequency. ovv2<? + 7 (VxA) 3h = (H-S7)q xh+gdg, (1) (2) Reprint requests to Dr. R. C. Sharma, Department of Mathematics, Himachal Pradesh University, Simla (India).

2 462 R. C. Sharma and K. Prakash Thermal Instability in Plasma with Finite Larmor Radius Using (5) and (6) and taking do = oqo in Boussinesq approximation we get, on eliminating dp: d 2 n ^ k 2 )w v k 2 \ w + g k 2 a & ^ ^ i kx \ ~j~2 ~ I ^ IV d 2 1 ^y I j _2 ~ 4 h 2 - V J -- 2ikx - k 2 I - _2-2k 2 -ky 2 ) v +2 kx ky k 2 u cr = 0, where = ikxv i ky u denotes the z-component of the vorticity. a is the coefficient of thermal expansion. Taking the curl of (1), we get n-v T- - /r a z~ c=-' k ~ v " l dz 2 r > 4 p (8) where = i kx hy i ky hx denotes the z-component of the vector curl h. Similarly, taking the curl of (3), we have (9) Equations (4) and (3) can be written as [n-x(d 2 /dz 2 -k 2 )]0 = ßw, [n r] (d 2 /d z 2 k 2 ) ] hz = i kx H w, (10) (11) Letting a = kd, o = nd 2 jv, p1 = v/y. and p2 =, Equations (7) (11) become (.D 2 - a 2 ) (D 2 - a 2 - o) w - f ^ H a i kx ^ ( 2 _ a2)h + \ V j 4 71 Q V (ikx v0 d 2 \f k 2 A (12) (D 2 a 2 o) ; = Z) 2 a 2 + 3,, a 2 it; i A; 9 \ v ( 1 kx v0 \ / 9 o, o 2 / 4 p v c >\. H kx d~. m* 2 s,ikxhd 2 \ [D- -a- -p2o) S = - \ _ I C, (13) (14) (Z) 2 -a 2 - P lo) = - ßd 2 and i (Z) 2 a 2 p2 o) hz - // d- V iv, (15, 16) rhere Z) = (/ d~ " 3. The Case of Stationary Convection When the instability sets in as stationary convection, the marginal state will be characterized by o = 0. In this case Eqs. (12) (16) reduce to (D 2 -a 2 ) 2 w-t 9 -^-)a 2 (9+ ik */* Hd2 (Z)2_a2)Ä +(iwfi\( Z )2_ a2 + 3M a 2\ C = o, (17) \ V J 4 71 QV \ V j \ k- J (n" f I 1 v o \ (n* 9 q ^V 9 \ /, u e H d~ (18) (Z)-a-)C=( - J(D--a- + 3 «-j 1 _ (19) (Z) 2 -a 2 ) = - (IM^LW

3 463 R. C. Sharma and K. Prakash Thermal Instability in Plasma with Finite Larmor Radius (D 2 - a 2 ) 6> = - {^Pj w, (.D 2 -a 2 )hz= - ^ ' * ) w. (20,21) Eliminating 0, hz, from Eqs. (17) (21), we get [ (D 2 a 2 ) 2 + a 2 Q cos 2 [( 2 -a 2 ) 3 + R a 2 + a 2 Q cos 2 <Z> (D 2 -a 2 )]w a 2 vj cos 2 <P\ a v0 cos (D2_a2)2 (Z)2_A2 +3 a 2 cos 2 ) 2 iv, (22) where R = g a ß d*/v x is the Rayleigh number, Q = H 2 <i 2 /4 n o v rj is Chandrasekhar number and cos = kx/k. This tenth degree equation with suitable boundary conditions constitutes the required characteristic value problem. We consider the case of a fluid layer with two free surfaces when the adjoining medium is assumed to be electrically nonconducting, the case having some relevance to the ionospheric layer. The appropriate boundary conditions for this case are: w = D 2 w 0, <9 = 0, Z) = 0, = 0 at 2 = 0,1 and h is continuous. Making use of the above boundary conditions in (17) - (21), we obtain D 4 w = 0, D 2 0 = 0, t = D 2 C = 0 at z = 0andl. Differentiating (17) twice with respect to z and using the above boundary conditions through Eqs. (18)-(21), we get D 6 w = 0, D*6 = 0, DH = 0 at 2 = 0 and 1. This process can be continued and it can be shown that all the even derivatives of w must vanish for z = 0, 1. The proper solution of Eq. (22) characterizing the lowest mode is therefore In Fig. 1, we have plotted Rx as a function of x for Qx = 100 and for three values of M ( = 100, 200, 500) for the set = 0. Similarly Rx against x for the same values of Qx and M, but for the sets & = 45 = 90 have been plotted in Figs. 2 and 3 respectively S w = A sin iz (23) where A is a constant. Substituting the solution (23) in Eq. (22) and letting a 2 = i 2 X, RX = R/A 4, QX = Q/JI 2, we obtain (l+z) &(!+*) ( /. i i -4- r i cos ens- 2 &'t' + M cos2 < (1 -fa;) 2 (1 +a:-3xcos 2 $)' (l+x) 2 + Q1xcos 2 & (24) where M = v0 2 /v 2 is a nondimensional parameter. Equation (24) expresses the modified Rayleigh number Rx as a function of the dimensionless wave number x and the parameters and M Fig. 1. The variation of 7?t with x for a fixed ^t = 100 and for three values of M (= 100, 200, 500) for the set $ = 0. It is clear from Figs. 1 and 2 that, for a given value of x, Rx increases continuously with increase in the value of M, thus showing the stabilizing effect of the finite Larmor radius. For the case 0 = 90 in Fig. 3, Rx is independent of the effects of Qx and M, and a minimum exists at x = I.

4 464 R. C. Sharma and K. Prakash Thermal Instability in Plasma with Finite Larmor Radius 4. The Case of Overstability In this section we discuss the possibility as to whether instability may occur as overstability. Here also we consider the case of two free boundaries. Eliminating 0, hz, and from Eqs. (12) (16), we obtain [ (D 2 - a 2 - o) (D 2 -a 2 - p, o) + Q a 2 cos 2 <Z>] [.D 2 - a 2 -o) (D 2 - a 2 - Pl o) (D 2 - a 2 - p2 o) -f i? a 2 (D 2 a 2 p2 o) + () a 2 cos 2 0 (D 2 a 2 ) (D 2 a 2 Pl a) ] w (25) (D 2 -a 2 -Plo) (D 2 -a 2 -p,o) 2 (D 2 - a a 2 cos 2 &) 2 w, where o may be complex. get The appropriate solution of (25) characterizing the lowest mode is w = As'm7iz. (26) Substituting the solution (26) in (25) and letting a 2 = 7i 2 x, R1 = R/ji 4, QX = Q/7i 2 and iox=o\n 2, we D / 1 +sc \.,., (1+z) ( 1 + i o l P l ) Äi= (l+ar + iöj) (l+a; + iolpl) -f^cos-0 V T Z T T ^ T \ X ] 1T*T' OJ P2 + Mcos 2 0 (1 +x + io1pi) (1 + X + joxp2) (1 +X-3XCOS 2 0 ) J (l+x + ioj) (1 + x\- io1p2) + Qxx cos 2 0 Since for overstability, our interest is to determine the critical Rayleigh number for the onset of instability via a state of pure oscillations, it will suffice to find conditions for which (27) will admit of solutions with ox real. Assuming, then, that ox is real, and equating, separately, the real and the imaginary

5 465 R. C. Sharma and K. Prakash Thermal Instability in Plasma with Finite Larmor Radius parts of Eq. (27), we obtain n l l + x \ f n ^. M o, & cos 2 < (!+*) {(1 + x)*+p1p2ol 2 } L [{(l+x) 2 - P L p20! 2 } {(1+x) 2 -POO^+^XCOS 2^} + 0^ (1 +*) 2 (1 + Pa) (P1 + P2)] {(1 +X) 2 -p 2Ö 1 2 +^ 1ICOS 2 + o 1 2 (l+x) 2 (l+p 2) 2 (1 + (1 + Pl) + ( i + * ) 2 + p 2 W, L(l+x) [(1 + x) 2 (Pl - 1) +Po 2 (Pl - 1K 2 + (Pl -J-pJ&s cos 2 {(1 + x) 2 - p2 ox 2 + Qt x cos 2 <Z>} 2 + öl 2 (1 + x) 2 (1 + p2) 2 (29) lere L = M cos 2 < (1 + x - 3 x cos 2 <P) 2. Equation (29) can be written in the form A1o1* + Bl V + Ci Oi 2 + ^i = 0, (30) where (l+x)(l+pi)p2 4, = y u t x i u -t-pijp*» ß 1 = (l+x)(l+pl) {(1 +X) 2 (2 +p2 2 ) -2p2(?1XCOS 2 0}p2 2 + & COS 2 <5 (1 + X) 2 p2 2 (Pl - p2) + L (1 + X) p2 4 (Pl - 1), C l = ( 1 ^ ) C 1 + )( 1 +Pi) (1 +2p2 2 ) +2 p2(l + x) 2 Q1xcos 2 &(p2 1) + p2 2 Q1 2 x 2 cos 4 <5} + <2LCOS 2 <Z>(l+X) 2 ( P L- P 2) {(1+X) 2 (l+p2 2 ) -2p2Q1xcos 2 <P} + p2 2 (pi-l) (1+x) 2 +p2 2 {( l+x) 2 ( P l-l) +(p1+p2)q1x COS 2 < }, >1 = (1 + X) 3 (1 + PL) { (1 + X) 2 + X COS 2 0} 2 + (?1C0S 2^(1+X) 2 ( P l- P 2) {(l+z) 2 -T-^XCOS 2^} 2 + L(l+X) 3 {(l+x) 2 ( P l_l) + (p1 + p2) ^XCOS 2^}. Equation (30) is cubic in o^. Applying Hurwitz' criterion to Eq. (30) we find, as all the coefficients of this equation are positive when pt > p2 > 1 and Q 2 cos 4 0 < 16, that either all the three roots of this equation are real and negative or there is one negative real root and the remaining two roots are complex with negative real parts. Hence (30) will have all values of oj 2 such that R e(a1 2 ) <0 if P l> P 2> 1, Qx 2 cos 4 #<16, (32) 1 S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Chap. IV, Clarendon Press, Oxford J. D. Jukes, Phys. Fluids 7, 52 [1964]. 3 Ju. V. Vandakurov, Prik. Mat. Mech. 1, 28, 69 [1964]. 4 K. V. Roberts and J. B. Taylor, Phys. Rev. Letters 8, 197 [1962]. which means r/2 J2 v>ri>>i and ^ <4a 2 sec 2 <P. (33) For overstability aa has to be real and it has been shown that not a single value of at 2 is positive. Equation (33) are, therefore, the sufficient conditions for the nonexistence of overstability, the violation of which does not necessarily imply occurrence of overstability. The authors are grateful to Prof. P. L. Bhatnagar for his help during the preparation of this paper. 5 M. N. Rosenbluth, N. Krall and N. Rostoker, Nucl. Fus. Suppl. Pt. L, 143 [1962]. 6 H. Melchior and M. Popowich, Phys. Fluids 11, 581 [1968]. 7 S. Singh and H. K. Hans, Nucl. Fus. 6, 6 [1966]. 8 R. C. Sharma, Phys. Fluids 15, 1822 [1972]. 8 R. C. Sharma, Nuovo Cim. 20 B, 303 [1974].