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Plasma Physics Laboratory Princeton University MP lljMMWMUJWWIcLgt V ••&> HI i,*—. ^3^ .Jr <ST~ LINEAR OSCILLATIONS IN GENERAL MAGNETICALLY CONFINED PLASMAS BY Liu-Chen and Shih-Tung Tsai April 1982 PLASMA PHYSICS LABORATORY PRINCETON UNIVERSITY PRINCETON, NEW JERSEY mTWBUTIWl OF TD«S B0CU*a|T IS IffliimiTta o.g. oeuMR or BUSY, oa-juat2-7c-a»-3073. PPPL--1889 DE82 012760 Linear Oscillations in General Magnetically Confined Plasmas by Liu Chen Plasma Physics Laboratory, Princeton University Princeton, New Jersey 08544, U.S.A. and Shin-Tung Tsai Institute of Physics Chinese Academy of Sciences Beijing, The People's Republic of China Abstract A systematic formalism for investigating linear electromagnetic perturbations in general magnetic field configurations is developed. The formalism employs the small adiabaticity parameter p/L and is valid for o arbitrary frequencies. Here, p and L0 are, respectively, the Larmor radius and equilibrium scale length. Effects associated with plasma and magnetic field inhomogeneities as well as finite Larmor radii are contained. The specific case of axisymmetric tokamaks is then considered to illustrate the potential application. mmm * THIS oscumni is mmtm 2 I. INTRODUCTION It is recognized that linear wave propertied, such as stability, propagation, and absorption, in realistic magnetically confined plasmas can often be qualitatively different from those predicted by either simple extensions of infinite-medium theories or simplified model calculations. This is because plasma and magnetic field inhomogeneities can give rise to additional important effects such as wave trapping, mode conversion, free energy sources, stabilization via wave convection, as well as new channels of wave-particle resonance. It is therefore desirable to develop a systematic description for the wave dynamics, and this Is the principle motivation of the present research. This systematic approach in dealing with waves in general magnetic geometries was pioneered, independently, by Rutherford and Frieman (1968) as well as Taylor and Hastie (1968) for electrostatic waves with frequencies much lower than the cyclotron frequencies. Their approach, now known as the low- frequency linear gyrokinetic formalism, employs the small adiabaticity parameter \ = p/l, and consists of a transformation to the guiding-center phase space and then an averaging over the gyrophase angle. Here, p is the Larmor radius and LQ is the scale length of the equilibrium plasma and magnetic field. It also needs to be noted that finite Larmor-radius effects, which are crucial for a kinetic description of the wave dynamics, are retained within their formalism via the guiding-center transformation. Using the same principle, this low-frequency gyrokinetic formalism has been extended to Include the full electromagnetic perturbations (Antonsen and Lane, 1980; Catto, Tang and Baldwin, 1981) as well as to the nonlinear regime (Frieman and Chen, 1981). 3 In a recent paper (Chen and Tsai, 1981, hereafter referred to as Paper I), we have noted that averaging over the gyro phase angle la equivalent to taking the zeroth component of the Fourier series in the gyrophase angle. Thus, by keeping all the components in the Fourier series, we have been able to extend the linear^ gyrokinetic formalism to arbitrary frequencies. While the basic principles underlying our formalism were developed in Paper I, only electrostatic perturbations were considered there. In the present work, *e further extend our work to allow the full electromagnetic perturbations (Taai and Chen, 1981), and thereby provide a more complete description of linear wave dynamics In general plasma equilibria. In Sec. II, we present the theoretical formulation and derivation of the corresponding gyrokinetic equation. The theoretical results are further explored using the WKB eikonal ansaty. in Sec. III. We then consider, in Sec. IV, the specific geometry of axisymmetric tokamaks. Section V contains final conclusions and discussion. II. THEORETICAL ANALYSES In Paper I as well as in the low-frequency formalism (Catto, 1978; Catto, Tang and Baldwin, 1981), it is more convenient to perform the present analysis in the guiding-center phase space (X, V) which is related to the particle phase space (x, v) via the following guiding-center transformation X =• x + v x e /Q , (1) V - Ce, |i, o) , (2) where u - v./2B, e • v*/2 + qfc /m, 4 and B are, respectively, the equilibrium electric potential and magnetic field, 0" qB/nc Is the cyclotron frequency, e„ - B/B, a is the gyrophase angle, and vi * v. (ei cos a + e-> sin °0 i (3) with e,, e_, and e. being the local orthogonal unit vectors. —1 ~2 ~ll The unperturbed Vlasov propagator in (x, v) phase space L = a/at + vV + (v x 0).V + (q/m)E «V , (4) V ~ "K ~~^v ~C~V then becomes, in (X, V) phase space, a/8t + v e + + V » ~»^ r(iB1 iB2) + (q/m) E »[(v /B)(a/3u) + (e /v )(a/Ba)l + v % - C a/Bcc , (5) where \n1 - v x V (e„/QVvv , (6) ~Bl ~ ~xv~ll • ~X v ' + 2*2 * C?«»0(8/6M) (^alO/aa) , (7) V^H = -(u^B + v( V^-vJ/B , (8) + v x 9 V" (%* s.2)*s,i (V ite* *rfei i|) • < > 5 We first consider the solution for the equilibrium distribution function, F„, where the subscript g denotes functions of the guiding-center variables (X, V). Noting \ = p/L « 1 and assuming the following ordering |pVxFg|/|Fg| ~ |vE/vt| ~0(\) , (10) we have (Rutherford and Frieman, 1968; Taylor and Hastie, 1968) F - F + F + ... , (11) g go gl v ' where F = F (e, u, X, 1 ; fio) go go^ ' ~1; v L*> that is, e, »7„ F - 0, -II ~X go Fgl - (P/B)(&Fgo/au) , (13) ~ a + da /c v P - -(XI'XD / < ' >[ ||fel% e,*^)- v^ Vt«e(/2]} , (14) and x *d " *l [(^2)?, XaV + v) a,%a,]/0 . (16) Here, Fgl is the a-dependent part of Fgl and, for the purpose of Che present work, knowing Fg0 and f' la sufficient. Next, we consider the perturbed distribution function, 6F , which obeys g the linearized Vlasov equation L 6F - -(q/m)[6a «V + (6a ne./al'LJF , (17) where 6a » 6E + v x 6P /c . (18) Adopting 6* and 6A as field variables such that 6B » 7 x M and 6E - -(V « + a&A/cat) sad letting F » 6F + 6U (19) g ag g ' with 6F .1 M ft(»- ' ' ) I » + "* ~* • V„lF , (20) ag m l g 5e v c •'g B Op. cQ -«J go ' ' Eq. (17) becomes, after soae algebra, L R (q/M)(R + R + R + R g*8 " " 8 " " l 2 3 «) • <"> where x Rx - [(»6*g/8t) a/Be- (V^ •|/0)'5c]'go . (22) 7 6<1> = 6S - v&A /c , (23) g 8 ~ ~« R = 2 (Wgo/Ba^ta/at + v, e,.^ )a*g , (24) R, - L[6$ fv, e. + v_J-VJ - 6* v«7 Jin BJ ] 3 g *• I -II ~D ~Xv g ~1 ~* - (&?)•?» + 0(6$g?)^ (C/de+ a/B3n)](oFgo/Ba|i) , (25) R, = -(l/c){v fvV e )«oA + 6ft, (vV el-v+ (q/m) e «E 6A„ 4 l Iv x ~l "g Bg ~ ~x ~l ~ ~B ~o Kg " v«6A«g x% ta B " (VMScfe'^g) - fe%)%] + Q(P v, 6A„g); (a/Bau)} (a F§O/B a»i) , (26) vv!+(vHi%xe~i » (27) and (a)' = aa/aoc. a In the following analyses, our approach deviates from the conventional low-frequency formalism, where the Q6/ oct term in L„ is taken to be 0(1) and the rest of the terms to be 0(X). We instead adopt the following formal ordering for the perturbed quantities: |3/3t| ~v |e .7 I ~v |e x v | ~|Q| -0(1) . (28) L 6G then becomes, correspondingly, 0(1) 0(1) 0(1) 0(X) L 6G - l{a/St + v. e -V - Q&/3a + vf V x , + X „ ) g g I ~l ~X ~ ~B1 ~B2' 0(X) + (q/m)E .[(v,/B)(o/d(i) + (e /v,)(d/aa)] 0(\) , where we have Indicated the formal ordering of each term. Fourier decomposing the perturbed quantities In the gyrophase angle a; e.g., 6G - I <5G > exp(-iJla) , (30) 6 £=•-» 8 •* where 2 w <6G > - (1/2-ri) / da 6G (X, \i, e, a) expUAa) g X „ g ~ o and assuming exp(-lut) time dependence, Eq. (21) then becomes, for all .!., <L 6G >c « <L >.<6G >.+ I L „ „,<6G >., - - <R >. (31) g g I g X g I x^x gX,X' g A' g X where (v + V* " l ~l ^^-X - i(» - •« + **a) , (32) and - <W a>„ - vie. »(e «V e,)-e»(V x ,)/2] . (33) a e 9 (i) , thus, corresponds to the gyrophase-averaged correction to the cyclotron 2 frequency. Furthermore, In Eq. (31), we have Ignored 0(\ ) terms In L„ and the L , operator corresponds to cyclotron-harmonic coupling due gX, X to 0(X) terms which are sinusoidal in a. Equation (31) can be formally solved as 1 <5G > - -<L„>_1 (<R„>„ + [ !.„,„<«;„>„) . (34) ^r-^'g 0 (%>^ ln^,r<^>V) • Expanding <6G > - <6G > + <fiG > + ... in ascending power of X, Eq. (34) 8 x g xo g xi. can be further reduced order by order. Here, since the perpendicular (to B) wavelengths are formally of 0(p), the <R > , are taken to be of the same "* g X S order; i.e., <R > ~ <R > , Near resonance with the oth cyclotron harmonic (including m - 0), we have, for m' + m, |<L > ,1 ~ 0(1) » I <L > I ~ g m • 8 ^ |L ,| ~0(X). Equation (34) then yields <aG > - -<L >~l <R > , (35) g mo g m g m ' <&3 > . = -<L > 1 I L , <6G > , , (36) g ml g m mfym gm.m' g m'o and <&G > , - -<L >~* (<R > , + L , <6G > ) . (37) g m'o g m' l g m' gm',m g mo; v ' Here we note that |<6G > , | ~ I <6G > I ~0(X) |<6G > I, and hence the 1 g m'o ' g ml1 ' ' g mo1 ' perturbed distribution function 6G is dominated by <6G > given by g g mo ' Eq.
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