**Generalized poisson brackets and nonlinear Lyapunov stability-application to reduced MHD **

R.D. Hazeltine, D.D. Holm, J.E. Marsden, P.J. Morrison

**Abstract**

A method is presented for obtaining Liapunov functionals (LF) and proving nonlinear stability. The method uses the generalized Poisson bracket (GPB) formulation of Hamiltonian dynamics. As an illustration, certain stationary solutions of ideal reduced MHD (RMHD) are shown to be nonlinearly stable. This includes Grad-Shafranov and Alfven solutions.

**Enhanced lower hybrid heating due to frequency modulation **

S. Riyopoulos

**Abstract**

The effect of frequency modulation during stochastic ion heating induced by lower hybrid waves is examined. The modulation occurs either in the ion cyclotron frequency due to the variation of the magnetic field in toroidal devices, or it can be externally imposed on the frequency of the lower hybrid waves. It has already been observed numerically [Phys. Fluids __27__, 184(1984)] that a small variation in the ion cyclotron frequency can induce velocity diffusion for wave amplitude well below the stochasticity threshold in a uniform magnetic field. Here a detailed study reveals that to the lowest order in the small parameters, the modulational effects can be incorporated in a two-dimensional Hamiltonian. It is found that a small amount of modulation, Δω / ω ∼< 1%, produces an order of magnitude reduction in the stochasticity threshold relative to the constant frequency case. The stochastic regime in velocity space also grows in size, resulting in a considerable increase of the number of heated particles in the case of devices with modest aspect ratio. Both ion cyclotron and wave frequency modulation lead to similar results. The modulation of the wave frequency offers the ability to control and optimize the modulation parameters and is proposed as a method to enhance RF heating.

**Validity of mixing length saturation theory and its implication for anomalous thermal transport in RFP **

Z.G. Zn, P.H. Diamond

**Abstract**

In the reversed-field pinch, resistive interchange modes produce magnetic perturbations which can tear flux surfaces, generate magnetic islands, and determine transport and a soft Β - limit. In the past, heuristic mixing length arguments have been used to estimate the saturated fluctuation level of resistive interchange modes and other instabilities. For the case of resistive interchange modes, the mixing length argument asserts that the instability saturates when the velocity turbulence level is sufficient to convect a pressure element through the distance of a mode width (λ) in one growth time γ^{-1}. This determines the fluctuation level to be |V_{rms}| ≅ <λγ>. The simple mixing length theory has yielded quantitatively accurate results for the fluctuation saturation level of resistive interchange modes in RFP ^{1,2}. However, certain ambiguities appear when heuristic mixing length arguments are applied to the problem of saturation of zero frequency (MHD-like) instabilities. For example, since the mode width Δ is a function of growth rate γ, the meaning of estimates such as δp/p ∼ Δ/L_{p} at saturation (γ=0) is rather unclear. In general, a more precise understanding of the spatial (mixing length) and temporal (correlation time) scales __at saturation__ is required. Here we describe a renormalized theory of the resistive interchange mode using the simplest possible model, which is incompressible reduced MHD. For electrostatic modes, using a set of ene rgy conserving renormalized equations for vorticity ∇^{2}_{⊥}∅, and pressure p we derive the fluctuation saturation level, which justifies the mixing length theory and recovers the results reported previously for transport and scaling laws. In particular, resistive interchange mode turbulence is a plausible explanation of the proportionality of temperature and current observed in current RFP experiments. For electromagnetic modes, balance of diffusion modified pressure driving with nonlinear J x B force allows one to determine the fluctuation level at saturation. Results thus obtained for δB/B, thermal transport and scaling laws agree reasonably well with numerical simulation and RFP experiments.

**Magnetic fluctuations, field reversal maintenance, and anomalous thermal transport in the reversed field pinch **

P.H. Diamond, Z.G. An, M.N. Rosenbluth, B.A. Carreras, R. Hender, H. Hicks, J. Holmes, V. Lynch

**Abstract**

A theory of magnetic fluctuations, field reversal maintenance (dynamo activity), and anomalous thermal transport in the Reversed Field Pinch is proposed. Nonlinear generation of and coupling to m greater than or equal to 2 modes is advanced as an m = 1 tearing mode saturation mechanism. The mechanism by which nonlinear m = 1 tearing modes sustain the toroidal magnetic field is elucidated. The predicted fluctuation levels and scalings are consistent with those required for maintaining the B_{z} configuration. Heat transport is estimated using stochastic magnetic field diffusion arguments.

**Nonlinear dynamics of magnetic islands with curvature and pressure **

M. Kotschenreuther, R.D. Hazeltine, P.J. Morrison

**Abstract**

Curvature and finite pressure are known to have a dramatic influence on linear magnetic tearing stability. An analytic theory of the nonlinear resistive growth of magnetic islands in tokamaks that includes the interchange driving term in presented here. A Grad-Shafranov equation to describe the magnetohydrodynamic (MHD) equilibrium of thin islands is derived. The resistive evolution of these islands is then obtained. Interchange effects are found to become progressively less important with increasing island width.

**Pressure induced islands in three-dimensional toroidal plasma **

J.R. Cary, M. Kotschenreuther

**Abstract**

he production of magnetic islands by plasma pressure in three-dimensional toroidal systems is analyzed. Far from the rational surfaces, a procedure based on linearization in the plasma pressure applies. This yields the solution in terms of delta-function currents at the surface. These currents are found by a nonlinear analysis valid in the vicinity of the island. The result is a set of coupled nonlinear equations determining the island widths. Scaling is found by using the approximation of nearly circular flux surfaces. The results indicate that for typical stellarators, which have a small ratio ιl_{0}m_{0} of field line rotational transform to coil rotational transform, the island size in the low-β limit depends dramatically on whether a magnetic well is present. In this case, if a magnetic well is present, islands are insignificant; in contrast, if a magnetic hill is present, island overlap occurs for arbitrarily low pressure.

**Relation of fluid drift to oscillation center drift **

J.R. Cary

**Abstract**

The fluid theory and oscillation center theory of the ponderomotive drift give apparently different results. It is shown that this difference is caused by a ponderomotive diamagnetic effect.

**Interaction of the precessional wave with free-boundary Alfven surface waves in tandem mirrors **

H.L. Berk, T.B. Kaiser

**Abstract**

A symmetric tandem mirror plugging a long central cell, with plugs stabilized by a hot-component plasma is considered. The system is taken to have a flat pressure profile with a steep edge gradient. The interaction of the precessional mode with Alfven waves generated in the central cell is then considered. This analysis is noneikonal and is valid when mδ/r<1 (m is the azimuthal mode number, r the plasma radius, and δ the radial gradient scale length) for long-wavelength radial modes. Without finite-Larmor-radius (FLR) effects the precessional mode is always destabilized by the excitation of the Alfven waves for m> or =2. For m = 1, it is possible to achieve stabilization with conducting walls. A discussion is given of how FLR affects stabilization of the m> or =2 long-wavelength modes and of FLR stabilization of modes described in the eikonal approximation.

**Stability of high energy particle plasmas to MHD-like modes **

H.L. Berk, H. Wong

**Abstract**

The stability of symmetric mirror systems, such as tandem mirrors and multiple mirrors (or equivalently bumpy tori of large aspect ratio) is investigated when stabilization is attempted with high energy particles. The analysis is derived from a zero Larmor radius variational form, and the stability criteria for eikonal and long wavelength layer modes are obtained. For eikonal modes it is shown that line-bending can stabilize the low 1-number modes and together with finite Larmor radius effects discussed elsewhere, complete stabilization is possible. For disc-shaped plasma pressure profiles it is shown that currents induced by conducting walls can stabilize l=1 layer mode, while the higher-l layer modes required finite Larmor radius effects for stabilization. For thin ring-like pressure profiles, wall stabilization of the l=1 mode cannot be achieved, although the line bending term reduces the core beta limit and the growth rate of low 1-number layer modes. The coupling of the precessional mode of a plasma ring to the surface Alfven wave in a multiple mirror plasma is also discussed.

**Stabilization of an axisymmetric tandem mirror cell by a hot plasma component **

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

**Abstract**

A hot plasma component formalism is used for a symmetric tandem mirror cell to relate the precessional mode, well known from Astron experiments, to more standard magnetohydrodynamic (MHD) modes. The l = 1 mode can be stabilized by conducting walls, while the higher l modes can be stabilized by finite Larmor radius. Such a configuration is free of remnant dissipative instabilities because all perturbations are positive energy excitations when stability is achieved.

**Nonlinear interaction of tearing modes-A: Comparison between the tokamak and the reversed field pinch configurations **

J. Holmes, B.A. Carreras, R. Hender, H. Hicks, V. Lynch, Z.G. An, P.H. Diamond

**Abstract**

The multiple helicity nonlinear interaction of resistive tearing modes is compared for the Tokamak and reversed field pinch configurations using the magnetohydrodynamic equations. Unlike the case of the Tokamak disruption, for which this interaction is destabilizing when islands overlap, the nonlinear coupling of the dominant helicities is shown to be a stabilizing influence in the reversed field pinch. The behavior of the coupled instabilities in the two configurations can be understood as a consequence of the stability properties of the nonlinearly driven modes. In the case of the Tokamak disruption, quasi-linear effects linearly destabilize the dominant driven mode, which then feeds energy to the driving mode. For the reversed field pinch, the driven modes remain stable, acting as more of a brake on the growth of the dominant instabilities than was observed in single helicity studies.

**Free-boundary stability of straight stellarators **

D.C. Barnes, J.R. Cary

**Abstract**

The sharp boundary model is used to investigate the stability of straight stellarators to free boundary, long wavelength modes. To correctly analyze the heliac configuration, previous theory is generalized to the case of arbitrary helical aspect ratio (ratio of plasma radius to periodicity lengths). A simple low β criterion involving the vacuum field and the normalized axial current is derived and used to investigate a large variety of configurations. The predictions of this low β theory are verified by numerical minimization of δW at arbitrary β. The heliac configuration is found to be remarkably stable, with a critical β of over 15% determined by the lack of equilibrium than the onset of instability. In addition, other previously studied systems are found to be stabilized by net axial plasma current.

**Noncanonical Hamiltonian field theory and reduced MHD **

J. Marsden, P.J. Morrison

**Abstract**

Aspects of noncanonical Hamiltonian field theory are reviewed. Many systems are Hamiltonian in the sense of possessing Poisson bracket structures, yet the equations are not in canonical form. A particular system of this type is considered, namely reduced magnetohydrodynamics (RMHD) which was derived for tokamak modelling. The notion of a Lie-Poisson bracket is reviewed; these are special Poisson bracket of a semi-direct product group. The process by which this bracket may be derived from a canonical Lagrangian description by reduction is described.

**Localization of waves in a fluctuating plasma **

D.F. Escande, B. Souillard

**Abstract**

We present the first application of localization theory to plasma physics: Density fluctuations induce exponential localization of longitudinal and transverse electron plasma waves, i.e., the eigenmodes have an amplitude decreasing exponentially for large distances without any dissipative mechanism in the plasma. This introduces a new mechanism for converting a convective instability into an absolute one. Localization should be observable in clear-cut experiments.

**Particle simulation of the resistive g-mode in a sheared magnetic field **

R. Sydora, J.N. Leboeuf, P.H. Diamond, Z.G. An, T. Tajima

**Abstract**

The linear and nonlinear evolution of the interchange instability of a resistive plasma (the resistive g mode) in a sheared magnetic field is studied using electrostatic particle simulation methods. Both the fast and slow interchange mode regimes are considered. In both cases the linear growth rates of the modes scale well with the theoretical values. The saturation of the instabilities is caused primarily by convective mixing of pressure over the width of the eigenmode. The saturation levels predicted by mixing length theory are in reasonable agreement with the simulation results.

**Renormalization theory of stationary homogeneous strong turbulence in a collisionless plasma **

Y.Z. Zhang

**Abstract**

A renormalization procedure for the perturbation expansion of the Vlasov-Poisson equation is presented to describe stationary homogeneous turbulence. By using the diagramatic scheme the theory is shown to be renormalizable to any order. The expressions for the renormalized propagator, the renormalized dielectric function, and the intrinsically incoherent source are given. The renormalization leads to a complete separation of the fluctuating distribution function f_{k} into two parts, the coherent part, which is proved to represent the dielectric effect of the medium, and the intrinsically incoherent part, which represents the effect of nonlinear source. The turbulent collisional operator in the transport equation is proved equal to ΓO, the frequency broadening when k = 0.

**Implicit algorithm for compressible three dimensional magnetohydrodynamic calculations **

A.Y Aydemir, D.C. Barnes

**Abstract**

An efficient algorithm for the solution of compressible magnetohydrodynamic equations in a three-dimensional geometry is presented. Compressional Alfven waves are treated implicitly, thus greatly increasing the stable time step of the calculation and making it possible to study magnetohydrodynamic (MHD) phenomenon on resistive time scales.

**Hamiltonian structure of the BBGKY hierarchy **

J. Marsden, P.J. Morrison, A. Weinstein

**Abstract**

The BBGKY hierarchy equations for the evolution of the i-point functions of a plasma with electrostatic interactions are shown to be Hamiltonian. The Poisson brackets are Lie-Poisson brackets on the dual of a Lie algebra. This algrebra is constructed from the algebra of n-point functions under Poisson bracket and the filtration obtained by considering subspaces of i-point functions, 1 ≤ i ≤ n.

**Quasilinear evolution of collisional tearing modes **

W. Horton, T. Tajima, R.M. Galvao

**Abstract**

A theoretical model is developed to describe the quasilinear evolution of tearing modes in configurations where many modes are unstable at a particular resonant surface. An expression for the positive definite anomalous resistivity is derived and used to study the self-consistent diffusion of the equilibrium magnetic field. It is shown that the anomalous diffusion ca be much faster than the collisional diffusion at a low level of magnetic fluctuations.

**Study of generalized toroidal cusp configurations **

J.N. Leboeuf, S. Ratliff, J.M. Dawson

**Abstract**

Hot electrons as well as hot ions are a desirable feature of any fusion device. We investigate by particle simulations (in slab geometry) and simple theory two variants of the Toroidal Cusp Experiment (TCX) with an eye to achieving hot electrons, while retaining its large direct ion heating feature. A toroidal field added to the TCX configuration, with cusp magnets arranged poloidally to partly cancel cusp trapping of the electrons insures electron flow across the cusps and electron heating. Helical winding of the cusp magnets around the torus with pitch theta greater than or equal to (m/M)^{1/2} allows trapped electron flow on average helical paths; however, ions can still flow across the cusps and the J . E heating of ions and electrons can be varied at will.

**Particle simulation of drift waves in a sheared magnetic field **

R. Sydora, J.N. Leboeuf, T. Tajima

**Abstract**

Electrostatic properties of density gradient drift waves (the universal mode) in a sheared magnetic field are studied using a two-and-one-half dimensional (2 1/2 -D) particle code. For the case of a single rational surface, the drift waves are found to be stable with an eigenmode structure that matches the linear theoretical prediction as long as the ion resonance layer is well within the system. This applies to both even and odd parity modes with respect to the rational surface. The dependence on various parameters such as the shear length is examined.

**Threshold of global stochasticity and universality in Hamiltonian systems **

D.F. Escande, M. Mohamed-Benkada

**Abstract**

The robustness of noble tori is interpreted as being due to a hierarchy of the fixed points of the renormalization group for KAM tori. The threshold of global stochasticity for a large class of area-preserving maps and 1.5 or 2 degree-of-freedom hamiltonian systems is estimated by a simple method which relies upon a new version of an approximate renormalization scheme for H(v, x, t) = 1/2v^{2} - M cos x - P cos k(x - t), that is consistent with that hierarchy.

**Nonlinear trapped electron response for drift wave turbulence **

K. Swartz, P.H. Diamond, S.M. Mahajan, R.D. Hazeltine

**Abstract**

The nonlinear trapped electron response to drift wave fluctuations is calculated using the coherent approximation to the DIA in action-angle variables appropriate for toroidal geometry. The bounce-averaged nonlinear response to low frequency electrostatic fluctuations is computed. Employing a spectrum of Pearlstein-Berk structure modes satisfying the symmetry of the ballooning representation, the nonlinear terms are evaluated explicitly. Nonlinear effects do not significantly modify the trapped electron response. Quasineutrality and the nonlinear ion response can then be used in a sheared slab to obtain the turbulence level at saturation level. The level of trapped electron diffusion is calculated.

**Theory of anomalous tearing mode growth and the major tokamak description **

P.H. Diamond, R.D. Hazeltine, Z.G. An, B.A. Carreras, H. Hicks

**Abstract**

An analytic theory of turbulence in reduced resistive magnetohydrodynamics is developed and applied to the major disruption in tokamaks. The renormalized equations for a long-wavelength tearing instability are derived. The theory predicts two principal nonlinear effects: an anomalous flux diffusivity due to turbulent fluid convection in Ohm`s law and a vorticity damping term due to turbulent magnetic stresses in the equation of motion. In the final phase of the disruption, when fine-scale fluid turbulence has been generated, detailed considerations show that anomalous diffusivity has the dominant effect at long wavelengths. For a low-m tearing mode, the solution of the renormalized equations during the turbulent phase yields a growth rate analogous to the classical case but increased by turbulent resistivity: &gamma ≈ (∑_{k'}k^{'2}_{θ}|φ_{k'}|^{2})^{3/8}x (Δ')^{1/2}. This analytical prediction is in good accord with computational results.