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Nonlinear Behavior of Multiple-Helicity Resistive Interchange Modes Near Marginally Stable States

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Nonlinear Behavior of Multiple-Helicity Resistive Interchange Modes Near Marginally Stable States 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<ls H 2. 1170 (!!)!)())] are extended to the multiple-hclicity ca.se and applied to the resistive interchange modes. We derive the nonlinear amplitude equations of the multiple-helicity modes, from which the convertive transport in the saturated state is obtained, ft is shown how the convective transport is enhanced by nonlinear interaction between modes with dif­ ferent rational surfaces compared with the single-helicity case. We confirm that theoretical results are in good agreement with direct numerical simulation?. 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<i how I ransport is changed by the effects of nonlinear interaction of modes with different rational surfaces comparer! with the single-hehVify case. This paper is organized as follows. In Sec. 11 the fluid model equations describing the resistive interchange modes is explained. In Seel 11 we find general symmetry properties of reduced fluid equations based on the local sheared slat) configuration. In Sec.IV we develop the weakly nonlinear theory for the multiple-holicity resistive interchange modes and derive the equations governing the amplitudes of niult.iple~helicil.y modes. In Sec.V symmetric solutions of the amplitude equations are given and convective transport in the iHHilineary saturated states is obtained. The effects of multiple-hclicity on transport are investigated and theoretical results are compared with numerical simulation results. Kin ally conclusions and discussion are given in Sec.VI. 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 = <?rt-PSWL\m (i) W = c,(-^)fi'i?/Bo [?>] = cLrfUpyi-PirW'/Bo we obtain model equations in non-dimensional variables from Eqs.(l) and (2) as follows dtVl4> + [4>,Vl<t>] = -V\4>-dyP+\Pr^\it> (5) d,p + [4>,p] = -d„<t> + \V]_p (6) where v., = a + jd,. (7) — 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 <p and the pressure fluctuation p are expanded into the Fourier series with respect. U, y and z as = 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.
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