NIFS—89
JP9109092
Nonlinear Behavior of Multiple-Helicity Resistive Interchange Modes near Marginally Stable States
H. Sugama, N. Nakajima and M. Wakatani
(Received - May 1, 1991)
NIFS-89 May 1991 This report was prepared as a preprint of work performed as a collaboration research of the National Institute for Fusion Science (NIFS) of Japan. This document is intended for information only and for future publication in a journal after some rearrange ments of its contents. Inquiries about copyright and reproduction should be addressed to the Research Information Center, National Institute for Fusion Science, Nagoya 464-01, Japan. Nonlinear behavior of multiple-helicity resistive interchange modes near marginally stable states
Hideo Su^ama and Noriyoshi Nakajima
.\'ilir.}ial hisfiliiti for Fusion Scinx-i, !\'tif/oya Jtu.'t 01, Japan
Masahiro Wakatani Plasma 1'hysics Lalovato.rij, fxijotn l-nirt rsihj, I'ji ''7/. Jupau
ABSTRACT
Xnnlinear behavior of restive interchange modes near marginally stable states is theoreti cally Mudied under (he tnnllipledielirify condition. Redievvl fluid equations in I lie sheared slab configuration are u>ed in order to treat a local transport prolilem. With the use of the invarianee properly of local reduced fluid model equations under a transformation between the modes with different rational surfaces, weakly nonlinear theories for single- helicity modes by Hamaguchi [Phys. Phricis R 1. btlfi (198!))] and Nakajima [Phys. Flui KEYWORDS: resistive interchange mode, weakly nonlinear theory, multiple-helicity. anomalous transport, reduced MUD model - 1 - I. INTRODUCTION hi magnei jcally confined plasmas inhomogoneities of magnetic fields, /urrents, pressure. density and temperature cause a variety of instability to grow and have finite amplitudes for which linear theory is no longer valid. A lot of phenomena in plasmas Mich as saw loo I h oscillations, disruptions in tokamaks and self-reversal of magnetic fields in \U; V plasma.-, are considered as essentially nonlinear processes. Another important example of nonlinear phenomena is anomalous transport1 •which is observed in most of magnetically confine ment systems and is considered to be the enhancement of transport due to fluctuations or turbulence in plasmas. Generally such nonlinear problems are complex and especially theoretical quantitative treatment of si rong nonlinear or turbulent systems is still difficult so that complete understanding of anomalous transport as strongly nonlinear processes is not yet achieved. However weakly nonlinear theories.' which treat nonlinear interaction (>|' the modi's with small amplitudes, have been developed and applied successfully to some problems of fluid mechanics and plasma physics. Landau presented a weakly nonlinear theory based on a general model and described bifurcation of the system from one steady state to anol her using his model equation (ailed the Landau equation.•' Palm derived the Landau equation from the partial differential equations describing the system for a problem of Benard convection.1 Malkus, Veronis and many authors applied a weakly nonlinear theory to problems of hydrodynamic stability.-''' Also based on pla.sma fluid models, Ilamaguchi and Nakajima developed a weakly nonlinear theory for sjugle-helicity modes of plasma instabilites.''-0 They derived some parameter dependence ()f the amplitude of the steady nonlinear solution and the couvective transport theoretically. These weakly nonlinear theories allow us quantitative and partially analytical treatment of nonlinear behavior of the system near the marginally stable slate and give us some clues to the understanding of properties in the strongly nonlinear regime and anomalous transport. In this paper we will develop the weakly nonlinear theory for resistive interchange modes will) multiple helicity. A single-helicily condition gives a two-dimensional problem in which 2 nonlinear interaction only among the modes localized around J he sanio rational surface is Healed while we must consider also nonlinear interaction among the mode.-, with different ratiuMal surfaces in the innltiple-holicily case which is essentially throe-dimensional. When we ;i]«' concerned with plasma transport in the sheared magnetic fields, it i^ natural to include (he contributions from all linearly unstable modes lying Hi different radial positions rather than from only modes localized around a single surface and necessarily we must 'onsidor I he multiplo-helicity problem. Generally mulliple-helicity or three-dimensional problems are complex although it will be shown that (he weakly nonlinear theory for the singlo-helicily modes is easily generalized to the multiple-holicity case by noting the iuvariance property of the fluid model equations under a certain transformation of the mulliple-helicity modes. We will find that this property holds for many kinds of reduced fluid mode] equations based on the local sheared slab geometry which is often assumed Un local transport problems. The equations governing the amplitudes of the multiple- helicity resistive interchange modes will be derived. Due to the invariance property of the local fluid model equations, they have symmetric solutions which are also invariant, under the transformation of the multiple-helicity modes. The symmetric solutions consist of a virtually in finite sequence of mutiple-helicity modes with the same radial slrurt ure localized a rou in 1 t heir own rational surfaces, which are similar to the sir net ure of ballooning modes.'1 I sing these symmetric solutions is more relevant for local description oft he system (which is analogous to the eikonal representation in geometrical optics) than using bounded solutions which vanishes M some boundaries and are confined within the finite radial region. For (he symmetric solutions, the amplitude equations reduce to the Landau equation, for which we have a simple analytical expression of the solutions. The resistive interchange modes are considered to be linearly unstable in I he peripheral region of slellarator/heliotron plasmas which have a magnetic hill there and therefore they are candidates for the cause of edge turbulence and anomalous transport /' in~M Convective transport in the nonlinoarly saturated states of multiple-helicity resistive interchange modes near marginally stable states will be studied by using the weakly nonlinear theory and comparison between theoretical results and those obtained by direct numerical simulation 3 will he shown. We will fin A II. MODEL EQUATIONS Resistive interchange modes are described by the following reduced MHD model610 1214 in the electrostatic limit, which consists of the vorticity equation: and the pressure convection equation: -_vVi + -sxV0.v)P=-^ (2) where is the electrostatic potential, p the pressure fluctuation, B0 the component of the sialic magnetic field along the c-axis, p„, the average mass density, r the light velocity in the vacuum, i; the resistivity, v the kinematic viscosity, \ the pressure HifTusivity, P„ = r//'a/d:r (< 0) the volmiie-avcragcd pressure gradient and fl' = iKi/dx (> 0) the average curvature of I he magnetic field line. V]_ = d] + d^ denotes the two-dimensional Laplacian. The gradient along the the static sheared magnetic field line is given by „ d x d Merc B0. /.,. p,„, t], v. \, PQ and Q' arc assumed to be constant since we treat a local transport problem. The electrostatic approximation is used in Eqs.(l) and (2) since we consider the low beta plasma in the peripheral region. Choosing the units: 2 [t] = (-PuW/pra)-'/ [x] = [«/] = cLtffti-pnPiWyH/Ba W = i. M = M7M = ] = cLrfUpyi-PirW'/Bo we obtain model equations in non-dimensional variables from Eqs.(l) and (2) as follows dtVl4> + [4>,Vl d,p + [4>,p] = -d„ — 5 — The Prandil number is defined by l'r=l'l\- («) All ihe nonlinear terms appea* in ilie form of Poisson brackets: [f,g} = ((U)(d,g)-(<\g)(dyf). (n) - 6 — III. SYMMETRY PROPERTY OF LOCAL SLAB MODEL WITH CONSTANT MAGNETIC SHEAR Hero we discuss (ho symmetry property of the lora] rnodrl equations (5) and ((J). The electrostatic potential = 2-, L exp27Ti(mi//Lv + n.-/t:) oo oo = E E \ \expik(my + nAz) (10) "=-»»=-•» \ Pmn(z) / where Ls and />.. are the maximum scale lengths of the fluctuations in the ;/ and ; directions, respectively, and we defined k = 2r/L„ A = L„/LZ. (11) The wavenumbers of the Fourier modes in the y and z directions are given by ky = Inm/Ly = mk and kz = 2wn/Ly = ufcA, respectively, where m,n = 0, ±1, ±2, • • • are the mode numbers. Here k = 2ir/Ly denotes the minimum wavenumber of the fluctuations in the y direction. We consider the following functional transformation T: / *(*, y, *) \ _/ (7»(:r, y, .-) \ = / #x - A, y - A.', z) ' ipfx.y,-) J" { (Tp)(x,y,z) ) { p(x - A,y - A;, -) which is expressed in terms of the Fourier mode as (13) Pm,„(i) / ( (rP)m.n(i) ) { pm,m+„(* - A) For arbitrary functions f(x,y,z) and g{x,y,z), we have 3x77 = TdJ, d,Tf = TdJ (M) [Tf,Tg] = T[f,gl V,|T/ = 7'VN/ -7 Defining Vi 0 * = p J 10 1 ' -V{<>-dsp+-Kp,\;\lp-{4>.^vi>} F[*]: (15) -d„ |->)S.(r)) and (G) are rewritten as dtA$ = F[*] (Hi; We (ind from Kqs.(H) that T rnmtnut.es with A ami F AT = TA. FT = TF (IT Thus il follows llial Kq.(U) is invariant with the transformation T. i.e.. for an arbitrary solution * of K(|.( Hi), T4> = (/, Tp)1' is also (lie solution fl,A[T*] = F[T*]. !l«) We should note that. not. only FCJS.(5) and (G) but. also other reduced fluid mndc-1 equations based on the local sheared slab geometry with const ant magnetic shear are invariant under the transformation T. Next wo define another functional transformation P : (ID) P(z,I/>~) / \ p{x,-y..-z) which is expressed in terms of the Fourier mode as -tj>- -„(z) *-,..(*) m 0) PmA*) ) \ P-m.-A*) J where a superscript * denotes a complex cunjugate. Here we used the reality of the value of the functions <&.„(*) = ^-™.-»(i), ;C,,» = ;>-„,-„(.T). (21) Equations (")) and ((i) (or fjq.(!6)) arc invariant with the transformation P, i.e., Eq.( !(i) gives fl,A[P#] = F[P*]. (22) It should he remarked that this invariant under the transformation P results from the far I that Kf(s.(.r<) an P# = * (2.!) which is valid at any time / if it is at the initial time / = 0. Then the solution <1> satisfying Kq.(2-'l) is expanded into the Fourier series as ( 0{.r.!/.:) | _A| '?on(j--)sintnA.- \ ^, * / ^ ;.'(.r.!).:) J n=o y ;;0„(.r)cosfcuAc J m=t.=-» ^ pm„(x)cosk(my + nA;) J where '/>„,„(./•) nm\ pJIUI(?) are the real-valued functions which are transformed by T in the same manner as in Eq.(l-l). 9 IV. WEAKLY NONLINEAR THEORY OF MULTIPLE- HELICITY RESISTIVE INTERCHANGE MODES Nop' wo treat the resistive inlerehange modes near the marginal stable Ma to by I h<- weakly nonlinear theory. Kquations (~>) and (fi) have the trivial equilibrium solution .;> -- p — (). Solving the linearized equation gives I ho spectrum of the eigenvalue or the linear growth rate •> for the peritubal on which varies in the form exp V and vanishes as ,r — ±oc. When the Prandtl number /',. is fixed, a critical dilfusivity \,. exists such I hat all the Fourier modes in Kq.(21) are linearly stable for \ > \,., only one of the eigenvalues of the w = I modes becomes 7 = 0 for \ = \(. arid the system is linearly unstable for \ < \,.. Since the neighborhood of the marginally stable ::t.at.es \ = \,. is considered, the magnitudes a/td the temporal variations of (j>(x, //, z) ami ;;(.r, ,7,.;) are small and therefore we make the following perturbation expansion with the parameter A PI W'l / \, P2 d , d ,, d 7JT = A—+ A-— + Of <)T\ <)T2 X = W + Ax, +>?\2 + • (2.r,) Substituting Kq.(25) into Eqs.(5) and (6) yields in 0(\) = 0. (26) Pi I \ rV< -Xr^lPi Equation (26) is just the linear equation for tho marginally stable state. As stated above the solution of Eq.(26) consists of the linear combination of the m = 1 modes «, \ _ » / 4i(x + nAyuik(i, + r,A.:) \ ;i, ) .i=-» ^ p,{? + »A)cos/-(j/ + nA.:) J Mere we used the fact that since the linear equation (26) is also invariant with the transfor mation T, ive can produce the linear solutions from one set of eigenfunrt.ioihs (ijit(s)s\nkii. — 10 — Pi(x)cirsky)1 l>.v operating T successively / ij](j')sin ky \ I 4>i(x - nA)sin k[y — nA;) , T" = VU (" (n = 0,±l,±_>.-..). (28) I /'i(.r) cos ky I \ p\(x — 7iA)cosk(y — nA~) Thus the marginally stable state is degenerate with the m= 1 mode eigenfunrlions (28). Here we employ the boundary condition that surfaces j- = nA (n = 0, ±1, ±2, • • •). In Eq.(27) An is a real-valued function of the time: ,-l„ = /l„(r,,r3,--)- (29) When the linear operator L defined by Eq.(26) acts on the (m,n) Fourier mode in En,.(24), il is replaced with L,„„ given by __ / m'k*(x + %Af- + xSAdl - m2k2)2 mk \ L„„, = . (30) V mk -Xc(dl-mV)J T T For arbitrary function vectors u = (ui(x),U2(*)) and v = (vi(x),v2{x)) , the inner product is defined by dxlui^v^x) + u2(x)v2{x)} (31) We see from Eq.(30) that the linear operator Lm„ is self-adjoint with respect to the inner bracket (31), i.e., (u,Lm„i>> = Equation (26) is rewritten by using Eq.(30) as 2 2 2 2 J 4>i(x + nA)\( k (x + nA) + XcPr(dl-k ) k \ ( fa{x + nA) \ _ "\PI(I + «A) ) " \ k -Xc(dl ~ k') } { Pi(* + nA) J ~ (33) which gives the profiles of the eigenfunctions the mode amplitude An. For the mode number m > 2, the linear groth rates are negative and \imn is invertible so that Lmt:"i, — 0 gives the trivial solution u — 0. — 11 — Equations (5) and (6) yield in 0(X2 LI ^ | = [ ^'^ " X,PrV^1 + ^Vi We expand the solution of Eq.(34) into the Fourier series as in Eq.(24) ( \ Pi I "=0 \ Pon(i)cos^nA; / 171=1 n=-oo I p„„(x)cosk\my + nAz) I From Eqs.(34) and (35), wo have for the m = 1 modes 2 2 01n(l) ' (ST1/l„)(aj - * )<*,l> + nA) - x,Pr/!n(5; - k f^(x + nA) 2 pU*) -R An)p,(x + nA) + X,M% - k )p,(.r + nA) (:«;; The solvability condition for the m = 1 mode equation (36) is given by Pi(i + nA)/ / \ \Pi{x + nA)J \ pln{x + nA) J I (37) where Eqs.(32), (33) and (36) are used. Equation (37) reduces to 2 2 r dx {(dTl An){\dxMx + "A)| + k 2 2 2 2 2 +X,A„{P,\(dl - k )4>,(x + nA)| + \dlPl(x + nA)\ + k p (x + nA)} ] = 0. (38) Since an arbitrariness remains in the perturbation method given by Eq.(25), we can impose an additional constraint. If we assume that Xi — 0 (or dnAn = 0), we have Xi = dTIA„ = 0. (39) Then Eq.(36) has the same form as Eq.(33) so that the solution is given by (<#i„(x), pi„(x))T = const. • {4>\(x + nA),pi(x + nA))T. Including this into the 0(A) solution makes it possible to put <\n(x) 0. (40) Pin(z) — 12 — For the m = 0, n = 0 mode, <*oo(z) \ [ XcPrd*>t>oo(x) \ I 0 Poo(2)/ \ -Xc^Poofl) / \ |*Er=-»-4nai{^l(^+nA)Pl(^ + "^)} («) For the m = 0, n > 1 mode, , 1 [ 8,W,(i + nii)^i(i + iui)-Wi+n14)^.(i + ni4)} ) - »,-»,=- ^ dI{j>1(x + n,A)pi{x + n2A) + For the m = 2 mode 1 / 9^,(1+ mA)9j<^i(s + n2A) -fa (x + njA)9^aii(3; + n2A) -fc 2^ ^ni-^nj I - n,+n,=n I -9^.0,(1 + n!A)pi(x + n2A)+ (^i(x+ niA)9a:p1(i+ n2A) (43) For the m > 3 mode, we have Lm„ ( 9mA |=0. (44) Pm»(l) Thus the fluctuations in 0(\2) consist of the m = 0 and m = 2 modes which are obtained from the m = 1 modes in O(X) by Eqs.(4.41)-(4.43). We have the equations in 0(A3) from Eqs.(5) and (6) ar,V XlP V 1 + [ V 2] + V L [ ^ ] = { ^' " ' ^ *" ^ t^' ^'l 1 (45) V P3 / I -3r,Pl + X2Vipi - [fa, Pi] - [fa, pi] If the Fourier expansion is done for (03,P3)T as in Eq.(35), we obtain for the m = 1 modes &*) PS?(«) — 13- 2 ; (0r,.'U(tfi - k^M-J- + »A) - \2l\A„{<); - k rM* + "A) + [*,, Vi2],„ + [fa, V*i>,],„ -(^,.'l,,);»i(^ + HA) + \,.-l„(r); - A'-');',(.r + "A) - [0,,?,],,, - [fa,/.,],„ where (fa',, (J), />,„ (,r))7 denotes (lie m = 1, u = n Fourier mode of (rt>,3\ ;>''")'' and [0].Vi0.],„ = -W' E E •4„-„1/l,„+n2.4„J0,(T + (n-n,)A)a^/-;,(3- + ,1,A) P + 7 E E ^H.,u 2 2 -T^ E E -4„r_„/l„,_„J/l„2{249i1(.T + (7Jl-n)A)(^-4i: )//„1_2„!(3: + n,,A) Ti| =-00 tl3=-00 + [fa'.•>, Vifa]i00 O„O 2 J*' E E >l»+„,/ln1+„1v4„J^-* W1(a: + (R + 7Jl)AR^1(a: + n2A) 2 + 7^ E E -4„,-n-4n,-n!^!{2(S^-<- a,)^1(x + (n1-n)A)i/„1-2„!(a: + „2A) 2 2 +(6> - k )Mx + ("i - n)A)d,H„^2n,(x + n2A)} [*i.Pa]i» i oo oo m =0 nj=—oo 2 ~4* E E An+nxAni+niA„24>i(^ + (n + ni)A)dTGni(x + n2A) _ 2 4* £ E AHl^nA„l.niAni{2dx4>i[x + (n1-n)A)Ini.2n3[x^n2A) +(*,(* + («] - nJAja,/,,.^^! + n2A)} [02,/>l]ln 2 = 7* E E -4„-„,/l„l+„;>l„Jp,(x + (n-n1)A)3IF„,(.T + n2A) n | = 1 n j = — oc 1 .2, ~7* E E A„+„,Ani^An,pilx + (n + ,i,)A)dIF„l(x + ,uA) -W £ E A,n-„A„l.„,A„1{2,%ri(x + (v, -Ii)A)//,ll_2„3(.T + n,A) TM=-OO n3=—TO +PI(.T + (n, - n)A)r)t/f,„_.,,„(r + n2A)}. Here /•'„, (/„, //„ and /„ are given by F x L »( )\ _ ( ax{0i(.t + «A)9>i(T)- 0„(T) / ^ 9r{ , tf„(a') 1 [ C*I^I(I + ;!A)r9;01(i:)-^,(j + nA)(5>1(j-) Li, /„(T) ) \ -dx In the same manner as in Eq.(37), the solvability condition of Eq.(46) is given by // 0,(T + ,IA) ,RHSof Eq.(4G)) = 0 (47) Pi(x + nA) which reduces to dA Dojr^ + X.0..4,, + £ £>„-«,,„-„, A„.„t+„,A„,A„7 = 0. (48) OT-, Rewriting 2 2 \A„->A„, r2^A i, A X2 - (x ~ Xc) (49) dA Do-^- + (x-Xc)D,An+ £ Dn-„J.„-„,/t»-„l+„J/l»,/l„! =0 (50) where A, = |_™ ^O^^fo:)!2 + ^(s) + p2(z)] (51) A = /" dxlPAidl-WMxtf + fcpMf + kVdx)] (52) ti],n2 ^[-^(i + nAJd^.Vl^^ + fe.Vl^,],,.} -oo +Pl(i + nA)((ft,p,],„ + [^2,pi]i,}]. (53) Equation (50) determines the behavior of the amplitude A„ in the solution of the leading order (27) near the marginally stable state. V. COMPARISON BETWEEN THEORY AND NU MERICAL SIMULATIONS OF SYMMETRIC SOLUTION In this section the results of the weakly nonlinear theory in the previous section are compared with those of the numerical simulations of Eqs.(5) and (6). Here we consider the symmetric solutions which are invariant under the transformations T and P given in Section III T# = P* = *. (51) Then it follows that A„ = ,4 (n = 0, ±1, ±2, •• •) and Eq.(50) reduces to the following Landau equation: 3 D0dtA + (x - xc)£M + D3A = 0 (55) where 1 /°° °° Di = -k dx[-pl(x)J2{ -Pi(z) J2 {2dj +Pi(*)£{Pi(z -nA) - p,(x + nA)}dx ~Pi(x) £ {2d*Pi(x + nA) • fan(x) + pi(x + nA^xfa^x)} n=—oo oo nA + +M*) E {2dxUx + nA)(dl-(2k¥)4>2n(x) + 2 -0,(r) J2(d; - k ){Mx - nA) -<*,,(* + nA)} • dx$0„(x) n = l 3 2 -Mx) £ {2(d x-k dx)Mx + nA)-dx4>2n(x) n = —oo 2 +(d t - k?) -16 — The solution of the Landau equation is easily obtained and written as A2 A2 = - (57) where A0 = A(t = 0) and ° = (Xc-X)D1/D0 (58) < = (Xc-X)D,/D3. (59) Equations (51) and (52) show that D0 > 0 and 03 > 0. We find from Eqs.(57) and (58) that if \ > \t. then a < 0 and A ~ A0cxp(crt), i.e., the linear theory holds as I. — oo for sufficiently small .40. As will be. found in the detailed calculations, D3 is positive in the cases treated here. In the case of \ < \C we have a > 0 and A ~ yloexp( for sufficiently small .40. In this case, A — ,4„ as /. — oc for arbitrary magnitudes of A0. The volume-averaged convective flux (pvE), in whicli the contributions from the m = 1 modes are dominant, is 0{\2) and given by (P«i)« = ~^~X)^JJ~ J_ dxpiix^ix). (61) In the right-hand side of Eq.(61), the calculation of D3 is most complicated as seen in Eq.(56). It is found from the numerical calculations that the most dominant contribution to Di is from the terms relating to the pressure gradient of the (m = 0, n = 0) mode 'hPooi?)- From Eq.(41), we have 9IPm(x) = -£-l E 01(x + nA)p,(.T + nA)-Co| (62) where the integral constant C0 is determined by the condition that the volume average of ^iPoo(^) vanishes and is given by J co rA 1 i-oo C'o = — V\ / rfx^>i(z + nA)pi(a: + nA) = — / rfi^1(a)p](ir) (63) — 17 — The contribution from the (onus shown by Eq.(G2) to W( is represented by I), : 0) /;i = k p rfJ-{-;.,;.r)o,(Jo,)r;lM(^} "V '-^ U=-TO / rfjpi(j-)0i(-r) 51 ifilr + nA^lr + nAj-if, 2\, F 2 * d? ]T ji,(.r + i(A)^,(.T + 7!.A)-C j (64) — / 0 Equation (64) shows that D3 is always positive and contributes to the saturation of the fluctuations. We find that D3 +0 as A — +0. Thus for sufficiently small A, the saturation mechanism due to D3 is negligibly weak and if the other terms in D3 are less effective the saturation amplitude becomes so large that the vc-akly nonlinear theory itself is supposed to give inaccurate results. When A is larger than the scale lengths of the m = 1 mode structures <^>i(x) and Pi{x), Eq.(64) is approximated by 2 0> k 2 M = : f^dx {Pl(x)Mx)} ~ A'' {[^dxpMM*)}'} (65) *Xc liquation (65) shows that, also in this case, D3 decreases and the saturation ampliliude becomes larger as A decreases. Figure 1 shows the lime evolution of the conveclive flux (pi^) for \ = 1.4 x 10"', PT = 1, h = 1 and A = 5. In this case the critical diffusivit-y for the m = 1 modes is V = 4.771 x 10"'. Here the thoretical result (a solid curve) obtained by Eqs.(57) (60) and the numerical result (solid circles) obtained by the simulations of Eqs.(5) and (6) are given. Both results are in good agreement. The units of the time I and the convective flux {pvx) are obtained from Eq.(4). In the simulations of Eqs.(5) and (6), tfi(x, y, ~) and p(?, y, ~) are expanded into the Fourier modes as in Eq.(24) and we imposed the boundary conditions for ??i > 1 |J|1nl^,,„(I) = |J|inl,I, (x) = 0 (66) and the symmetry constraint (54) which gives 6n.n(*) = 6n.m+n(*-A) (67) Pm.nM - pm.m+ri(j-A) (68) We hurl from Kqs.(66) and (67) lhat the in = 0 mode structures fpon(x) and pa„[x) are the periodic functions of x with the period A 0O.„(T + A) = ^o.n(^), Po.n(a: + A) = /W)- (69) Figures 2 show the electrostatic potential and the tn(ai pressure in the saturated state of the mulliple-helicity resistive inlerchange mode obtained from the numerical simulation _1 of Fqs.(5) and (6). Here \ = 4.4 x 10 , PT = 1, k = 1 and A = 4. In the numerical simulation we have included the modes with 0 < in < 5. Figure 2(a) shows the contours of the electrostatic potential in I he x — y plane for .: = 0. [n this case the m = ] mode structures appear dominantly, the profiles of which are shown in Fig.2(1)). The m ~ 1 mode structures are similar to those of the linear eigenfunctions obtained from Eq.(-J3) except that the former is somewhat broader 1 ha.n the latter due to the higher order nonlinear interaction. Figures 2(c) and (d) show the contours of the total pressure and the profile of thai averaged in the y direction. We see the flattening of the pressure around the m = 1 tnode rational surfaces. Figure '.\ shows the dependence of the convective flux {pvx) in the saturated state on I he difTusivity \ obtained from the theoretical expression Eq.(fil) and the simulations of Hqs.(5) and ((i). Here PT = 1, k = 1 and A = 5. The theoretical results are in good agreement wit.li the simulation results. For small \ the convective flux obtained from the simulations is smaller than that obtained from the theory due to the higher order nonlinear corrections. figure 4 shows the dependence of the saturated convective flux {p'-'x) multiplied by A 1 1 — \kA2 [J^ -19 remain unchanged and therefore A ' represents the measure for the ratio of the width of the m = J mode structure to the interval between the neighboring in = 1 mode rational surfaces (or the extent to which the neighboring m = 1 modes overlap). Here t he results of the theory [Kq(6I)] and the simulations of Eqs.(5) and (G) are shown. Wr find from Kq.((il) that (/;r,.)A depends on A~' only through D3 in the denominator. The dependence of DA 1 -1 on A" is also shown. As A is increased, Dz is monotonically decreased and therefore 1 (;wr)A is monotonically increased. For small A" , D:i lias the form of the first degree polynomial function of A-1 as predicted from Eq.(6.r>). The simulation results are well described hv the theoretical predictions but for larger A"1 the former show the more rapid increase of {;iPj.)A than the latter. Thus we see that the higher order nonlinear corrections enhance the saturation amplitude for larger A-1 while they lower that for smaller \ as seen from f'ig.3. — 20 — VT. CONCLUSIONS AND DISCUSSION In I his paper we have generalized the weakly nonlinear theory for siglo-helicity modes of llamaguchi and Nakajima lo the case of multiplo-holicity modes in the local sheared slab configuration. Wo have found that 1 ho model equations are invariant under the trans formation T defined by Kq.(12) or (13) and that this invariance property holds generally for other local reduced fluid model equations in the slab geometry with constant magnetic shear. From this property it follows thai there exist the multiplo-helicity modes which correspond to the critical order parameter (in this paper we used ihe diffusivity as a or der parameter but other quantities such as the pressure gradient or the average magnetic curvature can be used in the same way) or in other words thai the critical order param eter is degenerate due to I he multiplo-helicity modes, which are produced by operating T successively, '['hen the weakly nonlinear theory for tiie ninltiple-helicity modes is quite naturally developed from the single-helicily case and we have obtained the governing equa tions (50) for the amplitudes {A„} of multiple-helicity resistive interchange modes near marginal stability. We have compared the theretical results with those obtained by the numerical simulation of the model equations for 1 ho cases of the symmetric solutions, in which all the amplitudes .•1„ of the dominant. {?n = 1) modes have the same value. We have soon that both results are in good agreement in the parameter region near the marginal stability. The symmetry conditions used hero arc suitable to the local transport problem since they are free from boundary layers with steep pressure gradients as is seen under fixed boundary conditions, which produce artificially distorted pressure profile inappropriate: to the local model, hi this condition the weakly nonlinear behavior of t ho multiple-helicity modes is described by the Landau equation (55) in the same way as in the single-helicity case. When the diffusivity \ becomes smaller than the critical value \r (which depends on the minimum wavenumbor /• and the Prandtl number /',-), the new stationary states with convective colls bifurcates from static equilibria. Then the convective transport or the effective pressure dilTusivity defined by \.\/, = {pr.r)/{ — rfPn[r)frff) i* proportional to (\, — \). It was also shown that -21- when the ratio of I he inlerval A between the iit = 1 modes to the radial width of the mode profile decreases, the saturation amplitude per single mode becomes larger than that in the single-helicity case and the convnetive transport of the miilti-helicity modes is more enhanced than that predicted by the linear superposition of the saturated convection of I lie single-helicity cases. Here only the symmetric solutions are examined but the iionlim-ar amplitude equations (50) may have other various types of solutions and it is interesting as a future task to investigate spatial patterns and nonl'icar stability of such solutions. In the parameter range of actutual experimental devices plasmas are supposed to lie in the strong nonlinear or turbulent states, which' are far from marginal stability. In order to estimate the turbulent difFusivily \(t(r(.i in such strong nonlinear states, the following heuristic argument is often used. Replacing (he nonlinear convection v • Vj. with the m ( 1C turbulent diffusion —\(«r^I ' niodel equations, the nonlinear growth rale 7,V7,(£V, \) is expressed in terms of the linea growth raio 7f (fcv, \) as 7JV/,(^, \) = 7J.(*JM \ + Xturb) (70) where Pr ~ 1 is assumed. H is considered that the stationary state is realized when : 7,v/,(kv.\) < 0 for all wavenumbers kN. For example if we assume that ~,v/.(/' y) — "'i.{hy) ~ Xturb^x then we have a well-known expression of the turbulent diffusivity \f„ri, = r (liJk']_),„n.T f° the stationary state. From Fq.(70) we have more general expression V.-rfc = [\,.(*»)]m„ - \ (Tl) wliCi'e Xc{kv) is defined by the diffusivity required to linearly stabilize the mode with the wavenumber ky i.e. 7/,(&v, \>) = 0. In order to obtain the turbulent diffusivity of the resistive interchange modes Oarreras et a].in used the one-point renormalization theory and applied essentially the same argument as above to the stationary turbulent slates. It is interesting that the effective diffusivity of the convection obtained in this paper by the weakly nonlinear theory has the form similar to Eq.(7l), i.e., \^L = h'(\r(k) — \) where \r{k = (A?9)„,,„) = [vr(A.'y)]Tnox holds and the coefficient A' depends on k, Pr and A. 'This analogy between the expressions of the nonlinear difTusivity is naturally understood by noting thai the Landau equation (55) derived by the weakly nonlinear theory includes the — 22 - nonlinearity in the way thai it is obtained from the linear equation by replacing \ with 2 , 1 i! (nr \ + \A (\A — [D.\/ 1)\)A ) <*"«! ' ' stationary solution is given by \ + \A - \,. which is tin1 same logic as leading to Kq.(7I). As an example let us estimate the nonlinear thermal difFusivity for experimental plasma of Ileliotron K. of which in the peripheral region I lie resitive interchange modes arc un stable and supposed to cause the anomalous tranport. As local plasma parameters in I he r It% _1 peripheral region of lleliotron K we use tif = n, = > x IO m . 1\ — !\ — -10f \'. H = 17 . 1 ] A, = O.Gm, dQ/dr = 4.5m" and Lp = \d\\\ I\ldr\~ = :\ x lO-'m. Then we have the col- lisioTial difbisivity \ = 6.1 x 10_3m-/s. We consider the local transport around the radial position of r = 0. (8??? close to which there are no mode rational surfaces with !-?w puloidal - modenunibers mp = 1,2, ••• and we use kv — 5.9 x lO'm ' as the minimum wavenumber in the poloidal direction which corresponds to the poloidal iiiodenumber ID,, — 10. Kor r that poioidal wavenumber the critical difFusivity is \r = 0..>lm~/.s, the radial interval be- tween the modes is given by A = 5 x 10~3m and the radial mode structure of the form tt> oc oxp[— k{ix ~ xs)I^x}7] (where i\, denotes the radial position of the mode rational 4 surface) with lT = G x 10~ m for which the significant effects of the neighboring mode interaction on the convective transport appear. From Kq.(Gl) we obtain ihe nonlinear _ pressure difFusivity \/v/, = (p"x)/( dAo/rir) = f\'{\r — \) = 1.0?»"/.s (A' = 2.0). Using Ihe relations |(pi;r) = -n\tkdT/dx and d\n P/dx = V/[V - \)d\nT/dx [V = £) we have 2 the nonlinear thermal diffusivity \th = ^-\NL — 3.8m /.s. This gives a reasonable value of the anomalous transport coefficient compared to the experimental values in Heliotron K]i 2 \rj:p = 1 ^ ]0m /.s although more accurate estimation of anomalous transport and fluc tuation spectra require the theory treating strong nonlinearity and high degree of freedom in the turbulent system. Since direct numerical simulations of three-dimensional station ary turbulence in experimental plasmas with such parameters as above require huge sizes of computer memory and lime, further development of nonlinear theory to complement numerical simulation is indispensable. - 23 - - ACKNOWLEDGEMENTS The authors thank Professor Okamnto for his encouragements of this work. The numer ical simulation was done on the computer al the National Institute for Fusion Science. -24 — REFERENCES 1) I'. (.'. I.iower, Nurl. Fusion 25, 5-11 (1985). J) I'. (!. Drazin and W. H. Hoid, liydrodynamic Stability (Cambridge li. P., Cambridge, 1 9X I) • I) I.. I). Landau and p. M. Lifshitz, Fluid Mechanics (Porgamon P., London, 1950) •I) I-:. Palm, J. Fluid Merh. 8, 18.'J (19(50). 5) W. V. R. Malkus and (5. Veronis, .1. Fluid Modi. 4, 225 (1958). (i) S. Ilamaguchi. Phys. Fluids B 1, 1-116 (1989). T) \. Xakajima. Pliys. Fluids B 2, 1170 (1990). 8) \. Xakajiuw and S. Hamaguchi, Phys. Fluids B 2 118-1 (1990). 9) Y. ('. Loo and .1. W. Van Dam. in Finite Beta Thcnrij Workshop, Varonna, 1977 (P. S. Dopartnionl of F.nergy. Washington, IX'. 1977), (!O.\F-77091fi7, p.93 10) B. A. Carroras. I,. Ciarria. and P. II. Diamond, Phys. Fluids 30, 1388 (1987). 11) (I. S. Loo and B. A. Carroras, Phys. Fluids B 1, 119 (1989). 12) II. Sugama and M. Wakatani, .1. Phys. Soc. Jpn. 57. 2010 (1988). 13) II. Zushi. T. Mizuikhi, O. Motojima, M. Wakatani, F. Sano, M. Sato, A. Iiyoshi, and K. I.'o, Nucl. Fusion 28, -133 (1988). II) II. H. Strauss. Plasma Phys. 22, 733 (1980). 15) H. Sugama and M. Wakatani.. J. Phys. Soc. Jpn. 58, 3859 (1989). — 25 10° .1 i—r 1 •!—1—1 1 -1—] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ICT1 - v '. {P x) J lo-2 1 II I — Theory : , 1 . 10-3 • Simulation - 10 -4 _ icr5 7 P t '• 6 io~ 1 1 1 1 1 1 1 1 t 1 • 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.1 1 • 0 100 200 300 -1 Fig.l Time evolution of the convective flux (pvx) for x = 4.4 x 10 , PT = 1, it = 1 and A = 5. Theoretical and numerical results are shown by a solid curve and solid circles, respectively. X Fig. 2 The electrostatic potential and the total pressure in the satu rated state of the multiple-helicity resistive interchange modes obtained from the numerical simulation. Here x — 4.4 x 10_1, P, = 1, k = 1 and A = 4. (a) The contours of the electrostatic potential in the x — y plane for z = 0. (b) The m = 1 mode structures of the electrostatic potential. (c) The contours of the total pressure. (d) The profile of the total pressure averaged in the y direction. ~i—i—f-1—i—n—|~i—i—i—i—|-1—i—r—T—|—i—i—i- 1.0 Theory {pvx) 0.5 Simulation I • ' • I I I I I I I I I I • I I I I • I I I ik I r I • • 0 0.1 0.2 0.3 0.4 0.5 0.6 x Fig.3 The saturated convective flux {pvx) as a function of x obtained from the theory and the simulations. Here PT = 1, k = 1 and A = 5. 2.0 -i—i—i—i—i—i—i—r—|—i—i—i—i—i—i i i—i | i i—i—r—i—i—r-i—r 5.0 Simulation, 4.0 3.0 2.0 1.0 i • • i i i t • i i i i i i i i i i i i i i i i i i i i i 0 0.1 0.2 0.3 A"1 Fig.4 The saturated convective flux {pvx) multiplied by A as a function of A-1 obtained from the theory and the simulations. The values of D3 in the Landau equation is also shown. Here PT = 1, k = I and x = 4.4 x 10_1. Recen Issues of NIFS Series NIFS-39 O. Kaneko, S. Kubo, K. Nishimura, T. Syoji, M. Hosokawa, K. Ida, H. Idei, H. Iguchi, K. Matsuoka, S. Morita, N. Noda, S. Okamura, T. Ozaki, A. Sagara, H. Sanuki, C. Takahashi, Y. Takeiri, Y. Takita, K. Tsuzuki, H. Yamada, T. Amano, A. Ando, M. Fujiwara, K. Hanatani, A. Karita, T. Kohmoto, A. Komori, K. Masai, T. Morisaki, O. Motojima, N. Nakajima, Y. Oka, M. Okamoto, S. Sobhanian and J. Todoroki, Confinement Characteristics of High Power Healed Plasma in CHS; Sep. 1990 NIFS-40 K. Toi, Y. Hamada, K. Kawahata, T, Watari, A. Ando, K. Ida, S. Morita, R. Kumazawa, Y. Oka, K. Masai, M. Sakamoto, K. Adati, H. Akiyama, S. Hidekuma, S. Hirokura, O. Kaneko, A. Karita, T. Kawamoto, Y. Kawasumi, M. Kojima, T. Kuroda, K. Narihara, Y. Ogawa, K. Ohkubo, S. Okajima, T. Ozaki, M. Sasao, K. Sato, K.N. Sato, T. Seki, F. Shimpo, H. Takahashi, S. Tanahashi, Y. Taniguchi and T. Tsuzuki, Study ofLimiter H- and IOC- Modes hy Control of Edge Magnetic Shear and Gas Puffing in the JIPP T-IIU Tokamak; Sep. 1990 NIFS-41 K. Ida, K. Itoh, S.-l. Itoh, S. Hidekuma and JIPP T-IIU & CHS Group, Comparison of Toroidal!Poloidal Rotation in CHS HeliotronlTorsatron and JIPP T-IIU Tokamak; Sep .1990 NIFS-42 T.Watah, R.Kumazawa.T.Seki, A.Ando,Y.Oka, O.Kaneko, K.Adati, R.Ando, T.Aoki, R.Akiyama, Y.Hamada, S.Hidekuma, S.Hirokura, E.Kako, A.Karita, K.Kawahata, T.Kawamoto, Y.Kawasumi, S.Kitagawa, Y.Kitoh, M.Kojima, T. Kuroda, K.Masai, S.Morita, K.Narihara, Y.Ogawa, K.Ohkubo, S.Okajima, T.Ozaki, M.Sakamoto, M.Sasao, K.Sato, K.N.Sato, F.Shinbo, H.Takahashi, S.Tanahashi, Y.Taniguchi, K.Toi, T.Tsuzuki, Y.Takase, K.Yoshioka, S.Kinoshita, M.Abe, H.Fukumoto, K.Takeuchi, T.Okazaki and M.Ohtuka, Application of Intermediate Frequency Range Fast Wave to JIPP T-IIU and HT-2 Plasma; Sep. 1990 NIFS-43 K.Yamazaki, N.Ohyabu, M.Okamoto, T.Amano, J.Todoroki, Y.Ogawa, N.Nakajima, H.Akao, M.Asao, J.Fujita, Y.Hamada, T.Hayashi, T.Kamimura, H.Kaneko, T.Kuroda, S.Morimoto, N.Noda, T.Obiki, H.Sanuki, T.Sato, T.Satow, M.Wakatani, T.Watanabe, J.Yamamoto, O.Motojima, M.Fujiwara, A.liyoshi and LHD Design Group, Physics Studies on Helical Confinement Configurations with 1=2 Continuous Coil Systems; Sep. 1990 NIFS-44 T.Hayashi. 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Karita, K.Kawahata, T.Kawamoto, Y.Kawasumi, S.Kitagawa, Y.Kitoh, M.Kojima, T.Kuroda, K.Masai, S.Morita, K.Narihara, Y.Ogawa, K.Ohkubo, S.Okajima, T.Ozaki, M.Sakamoto, M.Sasao, K.Sato, K.N.Sato, F.Shinbo, H.Takahashi, S.Tanahashi, Y.Taniguchi, K.Toi and T.Tsuzuki, Application of Intermediate Frequency Range Fast Wave to JIPP T-IIU Plasma: Sep.1990 NIFS-49 A.Kageyama, K.Watanabeand T.Sato, Global Simulation of the Magnetosphere with a Long Tail: The Formation and Ejection of Plasmoids; Sep.1990 NIFS-50 S.Koide, 3-Dimensional Simulation of Dynamo Effect of Reversed Field Pinch; Sep. 1990 NIFS-51 O.Motojima, K. Akaishi, M.Asao, K.Fujii, J.Fujita, T.Hino, Y.Hamada, H.Kaneko, S.Kitagawa, Y.Kubota, T.Kuroda, T.Milo, S.Morimoto, N.Noda, Y.Ogawa, I.Ohtake, N.Ohyabu, A.Sagara, T. Satow, K.Takahata, M.Takeo, S.Tanahashi, T.Tsuzuki, S.Yamada, J.Yamamoto, K.Yamazaki, N.Yanagi, H.Yonezu, M.Fujiwara, A.liyoshi and LHD Design Group, Engineering Design Study of Superconducting Large Helical Device; Sep. 1990 NIFS-52 T.Sato, R.Horiuchi, K. 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