US-Japan Workshop on statistical physics and chaos in fusion plasma

W. Horton


Charged particle motion near a linear magnetic well

J.S. Kim, J.R. Cary


Charged particle motion near the null of a two-dimensional magnetic field is studied. Specifically, the magnetic field is given by the vector potential A=zψ0[(y/a)2 + (εx/a)2)], in which ψ0, a, and ε are constants with ε parametrizing the ellipticity of the flux surfaces. Conversation of canonical z-momentum pz reduce the number of nontrivial degrees of freedom to two. Scaling reduces the number of parameters in the system to two, ε and σ, the sign of pz. Analytical and numerical methods are used to study the nature of the orbits. The results are expressed conveniently in terms of ε and Q≡√2mE/pz. When ε is unity, the additional symmetry implies integrability. When ε is less than unity (the case ε > 1 is trivially related) three regimes are found: (1) |Q| >> 1 particle orbits are regular. (2) For ε3/2 ∼< |Q| ∼< 1 most particle orbits are stochastic. (3) For |Q| << ε 2/3 particle orbits are regular, with the third invariant being the magnetic moment.


Proceedings of US-Japan workship on hot electron physics

J.W. Van Dam


3-D nonlinear incompressible MHD calculations

A.Y. Aydemir, D.A. Barnes


n algorithm is developed for 3D nonlinear, resistive, incompressible magneto-hydrodynamic calculations in a clindrical geometry. The nonreduced primitive MHD equations are used. The state variables are expanded in Fourier series in the poloidal and axial coordinates, while a finite difference scheme is used in the radial direction. Applications to m = 1 tearing mode calculations in tokamaks and the self-reversal of a reversed field pinch are presented.


Kinetic theory of global N=1 instabilities in toroidal plasmas

S.I. Itoh, K. Itoh, T. Tuda, S. Tokuda


The kinetic theory of the global n= 1 instabilities of finite-pressure tokamakplasmas with circular cross-section is investigated in collisionless limit (m, n :poloidal and toroidal mode numbers). Wave-particle interactions and finitegyroradius effect are included. The radial-poloidal eigenmode equations aredirectly solved numerically. The 777=1 internal/tearing mode, 77=1 ballooningmode and m=2 tearing mode are identified with a fixed boundary condition.The m=I internal mode turns out to be the collisionless tearing mode in lowpressure regimes. The pressure driven ballooning mode is connected with theelectrostatic-like ballooning mode. The toroidal coupling further stabilizes them=2 tearing mode, which is destabilized by the parallel current and can bestabilized by coupling with drift branch. Analytical studies are made by using theenergy integral.


Kinetic Theory of RF Wave in a plasma in an inhomogeneous magnetic field

S.I. Itoh, K. Itoh


Kinetic effects on propagation and absorption of radio frequency (rf) waves in inhomogeneous and dispersive plasmas are studied, where effects of finite gyroradius and wave-particle interactions are included. The generalized linear propagator in the presence of the inhomogenetiy of magnetic field strength along the field line is calculated for obtaining the nonlocal conductivity tensor. Instead of a plasma dispersion function a new function is introduced. The influence of the inhomogeneity to the rf wave-energy deposition scheme is found to be appreciable in high temperature plasmas.


Drift waves in a stellarator

A. Bhattacharjee, J.E. Sedlak, P.L. Similon, M.N. Rosenbluth, D.W. Ross


The eigenmode structure of drift waves in a straight stellarator using the ballooning mode formalism is investigated. The electrons are assumed to be adiabatic and the ions constitute a cold, magnetized fluid. The effective potential has an overall parabolic envelope but is modulated strongly by helical ripples along it. Two classes of solutions were found: those that are strongly localized in local helical wells, and those that are weakly localized and have broad spatial extent. The weakly localized modes decay spatially due to the existence of Mathieu resonances between the periods of the eigenfunction and the effective potential.


Hot plasma decoupling condition for long wavelength modes

H.L. Berk, J.W. Van Dam, D.A. Spong


The stability of layer modes is analyzed for z-pinch and bumpy cylinder models. These modes are long wavelength across the layer and flute-like along the field line. The stability condition can be expressed in terms of the ratio of hot to core plasma density. It is shown that to achieve conditions close to the Nelson, Lee-Van Dam core beta limit, one needs a considerably smaller hot to core plasma density than is required to achieve stability at zero core beta.


Institute for Fusion Studies progress report for the period 1 September 1981 to 31 August 1982

IFS Staff


Local WKB dispersion relation for the Vlasov-Maxwell equations

H.L. Berk, R.R. Dominguez


A formalism for analyzing systems of integral equations, based on the Wentzel--Kramers--Brillouin (WKB) approximation, is applied to the Vlasov--Maxwell integral equations in an arbitrary-β, spatially inhomogenous plasma model. It is shown that when treating frequencies comparable with and larger than the cyclotron frequency, relevant new terms must be accounted for to treat waves that depend upon local spatial gradients. For a specific model, the response for very short wavelength and high frequency is shown to reduce to the straight-line orbit approximation when the WKB rules are correctly followed.


Finite Larmor radius stability theory of EBT plasmas

H.L. Berk, C.Z. Cheng, M.N. Rosenbluth, J.W. Van Dam


An eikonal ballooning mode formalism is developed to describe curvature driven modes of hot electron plasmas in bumpy tori. The formalism treats frequencies comparable to the ion cyclotron frequency, as well as arbitrary finite Larmor radius and field polarization, although the detailed analysis is restricted to E|| = 0. Moderate hot electron finite Larmor radius effects are found to lower the background beta core limit, whereas strong finite Larmor radius effects produce stabilization.


Implicit particle simulation of magnetic plasmas

D.C. Barnes, T. Kamimura, J.N. Leboeuf, T. Tajima


A second order accurate, direct method for the simulation of magnetized, multidimensional plasmas was developed. A time decantered particle push is combined with the direct method for implicit plasma simulation to include finite sized particle effects on an absolutely stable algorithm. A simple iteration is used to solve the field corrector equation. Details of the two dimensional, electrostatic, constant magnetic field, periodic case are given. Numerical results for ion acoustic fluctuations and for an unstable gravitational interchange confirm the accuracy and efficacy of the method applied to low frequency plasma phenomena.


Guiding center dispersion function

P.L. Similon, J.E. Sedlak, H.L. Berk, D. Stotler, W. Horton


Analytical properties of the linear Vlasov response function for guiding center particle motion in low frequency flute modes are investigated for a two temperature Maxwell-Boltzmann plasma. Algorithms are given for evaluating the family of analytic functions Gm,n(ω).


Curvature-driven instabilities in the elmo bumpy torus

H. Abe, H.L. Berk, C.Z. Cheng, M.N. Rosenbluth, J.W. Van Dam, D.A. Spong, N.A. Uckan, T.M. Antonsen, Y.C. Lee, K.T. Tsang, P.J. Catto, X.S. Lee, K.T. Nguyen, T. Kammash


Curvature-driven instabilities are analyzed for an EBT configuration which consists of plasma interacting with a hot electron ring whose drift frequencies are larger than the growth rates predicted from conventional magnetohydrodynamic (MHD) theory. Stability criteria are obtained for five possible modes: the conventional hot electron interchange, a high-frequency hot electron interchange (at frequencies greater than the ion-cyclotron frequency), a compressional instability, a background plasma interchange, and an interacting pressure-driven interchange. A wide parameter regime for stable operation is found, which, however, severely deteriorates for a band of intermediate mode numbers. Finite Larmor radius effects can eliminate this deterioration.


Nonlinear electron Landau damping of ion-acoustic solitons

J.D. Meiss, P.J. Morrison


Several authors have treated kinetic effects associated with the ion-acoustic soliton; e.g., Ott and Sudan investigated linear electron Landau damping and Karpman and Lotko have looked at damping due to ion reflection. Here an O`Neil-type frozen wave calculation that includes effects associated with electron orbits in a soliton is presented. This calculation differs from previous ones in that the usual three time scale argument is made: ωpe >>ωbe >>γL. The orbit effects included in this ordering become important at the modest amplitude eΦ/Te> or ≈(me/m i)2. Saturation at finite amplitude is predicted.


Large scale particle simulations in a virtual memory computer

P. Gray, J. Wagner, T. Tajima, R. Million


Virtual memory computers are capable of executing large-scale particle simulations even when the memory requirements exceed the computer core size. The required address space is automatically mapped onto slow disc memory by the operating system. When the simulation size is very large, frequent random accesses to slow memory occur during the charge accumulation and particle pushing processes. Accesses to slow memory significantly reduce the execution rate of the simulation. We demonstrate in this paper that with the proper choice of sorting algorithm, a nominal amount of sorting to keep physically adjacent particles near particles with neighboring array indices can reduce random access to slow memory, increase the efficiency of the I/O system, and hence, reduce the required computing time.


Topics in plasma instabilities: Trapped particle modes and MHD

M.N. Rosenbluth


A theory for trapped-particle modes in tandem mirrors is given.


Electromagnetic drift modes driven by ion pressure gradients in tokamaks

W. Horton, D.I. Choi, B.G. Hong


A hybrid of hydrodynamics and kinetics is used to study the effect of finite plasma pressure on the pressure-gradient driven toroidal drift modes. The linear drift modes of the system are given by a fifth-order polynomial describing the coupling of the electron-drift, the ion-acoustic, and the shear Alfvén oscillations. The characteristic frequencies, growth rates, and polarization of the electromagnetic modes are investigated as functions of the parameters of toroidicity, plasma gradients, and plasma pressure.


Solitary drift waves in the presence of magnetic shear

J.D. Meiss, W. Horton


he two-component fluid equations describing electron drift and ion acoustic waves in a nonuniform magnetized plasma are shown to possess nonlinear two-dimensional solitary wave solutions. In the presence of magnetic shear, radiative shear damping is exponentially small in Ls/Ln for solitary drift waves, in contrast to linear waves.


Fast growing trapped-particle modes in tandem mirrors

H.L. Berk, M.N. Rosenbluth, H.V. Wong, T.M. Antonsen, D.E. Baldwin


The variational structure of the plasma linear response function is used to demonstrate the relation of magnetohydrodynamic and trapped-particle instabilities. Though in most systems, where bending energy stabilizes ballooning modes, trapped-particle instabilities have a low growth rate, in tandem mirrors with thermal barriers the trapped-particle instability growth rate approaches that of MHD instabilities. In addition, the kinetic theory yields stabilizing effects due to the difference in electron and ion orbits, and destabilizing effects due to the variation of the E x B drifts along a field line.


Drift wave turbulence in a low-order K space

P. Terry, W. Horton


In the low-order isotropic k space introduced by Kells and Orszag for the two-dimensional Euler equation, the evolution of the fluctuations arising from the electron drift wave instability is studied. The two-dimensional drift wave model contains the E x B and polarization drift nonlinearities in the hydrodynamic ions and linear, dissipative electrons. The strength of the electron dissipation is shown to determine the spectral width and the level of the fluctuations.


Long time correlation of periodic area preserving maps

J.D. Meiss, J.R. Cary, C. Grebogi, J.D. Crawford, A.N. Kaufman


A simple analytical decay law for correlation functions of periodic, area-preserving maps is obtained. This law is compared with numerical experiments on the standard map. The agreement between experiment and theory is good when islands are absent, but poor when islands are present. When islands are present, the correlations have a long, slowly decaying tail.


Turbulent spectra from three drift-wave interactions

P. Terry, W. Horton


Hydrodynamic equations for the drift-wave instability containing the E X B convective nonlinearity are used to show that the three wave interactions lead to temporal chaos with broad-band frequency spectra in the saturated state.


Vacuum magnetic fields with dense flux surfaces

J.R. Cary


A procedure for eliminating resonances and stochasticity in nonaxisymmetric vacuum toroidal magnetic field is presented. The procedure is tested by the surface of section method. It is found that magnetic fields with increased rotational transform and decreased island structure can be obtained while retaining basically the same winding law.


Poisson brackets for fluids and plasmas

P.J. Morrison


Noncanonical yet Hamiltonian descriptions are presented of many of the non-dissipative field equations that govern fluids and plasmas. The dynamical variables are the usually encountered physical variables. These descriptions have the advantage that gauge conditions are absent, but at the expense of introducing peculiar Poisson brackets. Clebsch-like potential descriptions that reverse this situations are also introduced.


Proceedings of US-Japan Workshop on drift wave turbulence

W. Horton


Some minimum-energy toroidal equilibria

A. Bhattacharjee, J.C. Wiley


Equilibria with minimum energy are constructed from a variational principle in which the energy of a plasma is minimized subject to a recently proposed set of global invariants. The equilibrium equation is solved in axisymmetric, toroidal geometry. In order to compute toroidal equilibria the variational principle is exploited to obtain a reduced set of ordinary differential equations which we solve numerically. Tokamaklike and pinchlike solutions of minimum energy are found in toroidal geometry. Based on the ideal and resistive stability studies of the cylindrical limit of these solutions, it is argued that some of these equilibria should have robust stability to modes of low m and n number.


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