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.Jr 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 .!., 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 - (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 > - - Expanding <6G > - <6G > + B) wavelengths are formally of 0(p), the <&3 > . = - g ml g m mfym gm.m' g m'o and <&G > , - - 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. (35); i.e., 10 6G = <6G > exp(-imct) . (38) As we gradually move away from the mth cyclotron resonance, |<6G > , |/I<6G > I Increases and, furthermore, the cyclotron-harmonics ' g m'o1 ' g mo' coupling term, L . <6G > , In Eq. (37) becomes negligible. For instance, gm ,m g mo 1 /2 let us take ~ 1/2 |L , <6G > |/|<1 R > ,1 ~ |<6G > , 1I/|<6G > I v J * gm ,m g mo ' g m ' * g mo ' g mo' ~0(\ ) . Thus, away from the cyclotron resonances, we have | <*«>*>•-v? v* * (39) From Eqs. (35), (37), (38), and (39), it is then clear that in the zeroth- order approximation we may neglect effects due to the cyclotron-harmonic coupling. Equation (31) then reduces to We emphasize here that Eq. (41) is valid so long as the cyclotron- harmonics coupling term is small, i.e., more specifically, 'W « l°l • (42) 11 From Eq. (29), we find that the validity condition, Eq. (42), is satisfied if X « 1 and s fl (43) IritJ IrV - Noting that i Bl1 ' ~ ~B1 01 '-> Eq. (43) thus imposes an upper limit on Hie perpendicular (to B) wavenumber k ; that is, Eq. (41) is valid for |k p| « k p =L/p » 1 , (44) 1 Jin B where Lg specifically denotes the scale length of the equilibrium magnetic field to allow the possibility that the plasma inhomogenity scale length, Lp, oay be different from Lg. The case with k. ~k. has recently been treated by Lee, Myra, and Catto (1981). We remark that, for most cajes of practical interest, Eq. (44) is easily satisfied. Now, it is desirable to remove terms in <8. > involving the parallel g * (to B) propagator, v. e_ »V . Letting <&G >„ - -(q/m)<8<(; >„fdF /Bo|i) + <6H >„ , (45) g A g X. go g A Eq. (41) then becomes, after seme algebraic manipulations, 12 where (47) - ( V C - v, + <(X<-^Vr rP)> * LIX(~XV~ *%/"g ) v(~ ~&^ - v - + Q[<(fi6*g)jt>i(a/ae + d/B6[i) 48 <{hM(Am /c)'>0 (a/Bou)]}(dT /Ban) , < > and 6 . Is the Kronecker delta. In deriving Eqs. (46) to (48), we have noted that | also that the last two terms in Equation (46) along with the limits imposed by Eq. (44) and \ « 1 are the desired linear gyroklnetic equation valid for arbitrary frequencies. Equations (19), (30), (45), and (46) then determine the perturbed distribution function 6F and, hence, 6F. Knowir.g 6F, we can calculate Ihe linear density and current perturbations, which, when coupled with Maxwell's equations, 13 provide a complete description of the linear wave dynamics. In order to proceed beyond Eq. (46), we need to either specify the magnetic field geometry and/or make further assumptions. In the next section, we explore the results with the HKB eikont.1 assumption. III. RESULTS WITH THE EIKOHAL ANSATZ Since in many practical applications one is interested in the regime where the perpendicular wave length is much shorter than L„, it is suggestive to adopt the following WKB description for the perturbed quantities Si h 6F(x,v) = "5P"(x,v) exp(i / k'dxj - 6F(X,V) exp(i / k, -dX - ie(k)1 , ~ ~ ~ ~ ~1 ~1 ~ ~ ~1 ~1 ~1 (49) where 8(k ) = k «v x e /Q, Ik ,L I » 1, and 6F as well as k contain slow ~1 ~1 ~ ~l J. o ~1 spatial variations. Meanwhile, we also have Xi 6F (X,V) « "5F (X,V) exp(i / k »dX J , (50) g g ~1 ~1 and hence 6T - 6F exp [-ie(k )] . (51) Equation (46) then reduces to v/v* • ^( » wh - ^v* • i Qx * <» " kI«y0 - A(Q - »B) , (54) P,(F )- [w3/oe + JKJ d/B3u + (k xe„/Q)«V_]F , (55) Jr go' L v~l ~l ->XJ go - viJ^(Y)65||/kic , (56) J is the Bessel function, y - |k^ v ./Q|, J£ - dJ^z^dz, and higher order and ignorable here. Rirthermore, the Coulomb gauge, V »&A » 0, Is adopted here. From Eq. (53), <6H > may be solved by integrating along the magnetic field line and will be formally written as O^-lW^Xl^1^ - (57) The perturbed distribution function in (X,V) phase space, 6T, is then given by (58) and <67g>a - -(q/m)<6$g>A(8Fgo/Bdu) + Here in Eq. (58), we have neglected the fSCxe„/cQ)»V F tern [c.f. Eq. (20)] which can be shown to be generally of higher order. With 6F given by 15 Eq. (58), the corresponding Maxwell's equations can then be expressed as &P v &A oF n I*. o-i c ; B9nJ + I <«•>, J-CY)} . (60) 2 0 2 „, , v.6A. 9F 2 (») -v.— 8it -....*- B5jl C j « + { (k2_^)ffli.i^k Jq/MHp¥ij where k2 =k2 - (B-7 KB-1 e,«V ) , (63) 1 v^^jjk ~(~x-' j is the species, and sum over ±vy is understood in the velocity integration. Equations (57) and (59) to (62) thus provide a complete description of the linear wave properties within the framework of the elkonal ansatz. In passing we note also that, as in the electrostatic case discussed in Paper I, there are new channels for wave-particle resonances due to A additional velocity-dependent terms in IV. AXISYMMETRIC TOKAMAKS In this suction we consider the specific magnetic geometry of axisymmetric tokjimaks. Employing the usual <|> (poloidal flux), Z, (toroidal angle), and x (Poloidal angle-like) coordinates, we have B - V I x 7 H>+ K<|>, X)V I , (64) and F » F (([>, u, e) . (65) o o Motivated by experimental observations of fluctuations at tht ion cyclotron frequency and Its harmonics (Budny et al., 1981), we assume the perturbations have large poloidal and toroidal mode numbers [ [k p | ~ 0(1)] but long parallel wavelengths j|L e «V | ~0(1)] since modes with short parallel wavelengths tend to be stabilized by ion Landau damping. Therefore, we may adopt the ballooning-mode representation (Connor, Kastie and Taylor, 1979; Lee and Van Dam, 1979; Glasser, 1979; Dewar et al., 1979) to express the perturbations in the following form (Connor, Hastie and Taylor, 1980; Frieman et al., 1980; Pegoraro and Schep, 1981): fiH(x.v) - I exp(-imx/xj / <*T) exp(lmT)/x ) m x Sr( 2 _1 ls here, x *» the period In x, v - IJ/R , J - (? t|» * £E'V x) the Jacobian, R is the major radius, k(4) Is the global WKB radial wavenumber, 17 and 6TT contains the slow variations In i\> and n. Substituting Eq. (66) Into Eq. (46), we can readily carry out the integration along B (Frlemaa and Rutherford, 1968; Taylor and Haetie, 1968) to obtain h b i^> ^ " fir k»x %> - ""V. "SdT1"k)e^ • <68) and P'(F ) - (wd/os + JtQ3/Bd(i + nBd/B&4-)F . (69) X. O O Here we have taken v = v . We then find, for circulating particles with v > 0, and with v < 0, <6v*,c- - * m pi <6*pi - (6» - v^AOJ^y') - v^TOfij/kj^e , (72) and y' •• k (<|»,T)*)v /Q . As to trapped particles, we find 18 Vi-lrr^ JB + i<64-g>i2 BinKI^lcoBKip^])} , (73) *l. ««*g'>Jttco.[(IJl)^].inI 8ln[(ip^]) + J^dr,'^ «6ig->ju8ln[(IJt)^} COSKI^J'] +K6i^>J|28lnt where <&4-g>il - JA(y')6i - v^^YOfiB/k^ , (75) <&lpn - J/yOjvJ^/c , (76) and r\ and r| are the two turning points closest to r\ . With <6H > determined, we can use the results obtained In the g ^ preceding section, specifically, Eqs. (59) to (62) with k2 * k? , to derive tne corresponding wave equation, which in general takes the form of an i 19 integral equation. Here we remark that the velocity Integrations must be appropriately differentiated between the circulating and trapped particles. To further pursue the solutions of the integral wave equation, one has to either perform numerical calculations (e.g., Chen et al., 1981) or make simplifying approximations. However, this is beyond the scope intended for the present work. V. CONCLUSIONS AND DISCUSSIONS In this work, the systematic formalism developed in Paper I for investigating linear waves in general magnetically confined plasmas has been extended to include the full electromagnetic perturbations. A corresponding linear gyrokinetlc equation, valid for arbitrary frequencies, le derived In the guiding-center phase space and is shown to be applicable to a wide range of parameters. Effects due to plasma and magnetic field inhomogenities as. well as finite Larmor radii are retained. Furthermore, we have explored the results with the WKB eikonal ansatz, where the linear responses atul, hence, the wave dynamics can be readily found by solving the gyrokinetlc equal Ion v.,t integration along the magnetic field line. As a specific example we have considered In more detail the axisymmetric tokamaks and, employing eh-! ballooning-node representation, obtained the perturbed distribution function: for circulating as well as trapped particles. We remark that since the primary purpose of this work is to present the formalism, detailed investigations on the results derived here are therefore beyond the intended scope. It Is, however, clear that, with the existence of a systematic formalism, such investigations have become at least conceptually straightforward and, as In the case of low-frequency instability theories, accessible to analytical as well as numerical treatment. 20 Finally, we note that while colllslonal effects are neglected here to simplify the analyses, extensions to Include these effects are straightforward. ACKNOWLEDGMENT This work was supported by United States Department of Energy Contract No. DE-AC02-76-CHO3073. 21 REFERENCES Antonsen Jr., T. M., and Lane, B. (1980) Phys. Fluids 23, 1205. Budtiy, R., Motley, R., Hsuan, H., and Ho, D. (1981) Bull. Amer. Phys. Soc. 26, 864. Catto, P. J. (1978) Plasma Phys. J£, 919. Catto, P. J., Tang, W. M., and Baldwin, D. E. (1981) Plasma Phys. _23_, 639. 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(1979) in Proceedings of the Finite-Beta Theory Workshop, Varenna, Italy, edited by B. Coppi and W. Sadowski (National Technical Information Service, Springfield, VA) p. 93. 22 Pegoraro, F., and Schep, J. J. (1981) Phys. Fluids ljt> 478< Rutherford, P. H., and Frleman, E. A. (1968) Phys. Fluids 11, 569. Taylor, J. B., and aastle, R. J. (1968) Plasma Phys. JU), 479. Tsai, S. T., and Chen, L. (1981) Bull. Amer. Phys. Soc. "iA, 889. + 1 Jto <6