**Energy principle with global invariants: applications **

A. Bhattacharjee, R.L. Dewar, A.H. Glasser, M.S. Chance, J.C. Wiley

**Abstract**

Minimum-energy equilibrium states of a pressureless cylindrical plasma are constructed subject to a recently proposed set of constraints appropriate to turbulent relaxation dominated by a tearing mode of single helicity. Stability to ideal and resistive modes is examined. It is found that the relaxed states of minimum energy are always stable to the dominant mode, as well as a class of other low m and n modes, but that there are typically high n residual instabilities in the case of the pitch branch, and a residual m=3, n=2 resistive instability is found for an otherwise stable narrow window in the tokamak branch. It is argued that the residual instabilities may be compatible with experimental observations.

**Curvature-driven instabilities in a hot electron plasma: radial analysis **

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

**Abstract**

A nonlocal analysis of curvature-driven instabilities for a hot-electron ring interacting with a warm background plasma has been made. Four different instability modes characteristic of hot-electron plasmas have been examined: the high-frequency hot-electron interchange (at frequencies larger than the ion-cyclotron frequency), the compressional Alfven instability, the interacting background pressure-driven interchange, and the conventional hot-electron interchange (at frequencies below the ion-cyclotron frequency). The decoupling condition between core and hot-electron plasmas has also been examined, and its influence on the background and hot-electron interchange stability boundaries has been studied. The assumed equilibrium plasma profiles and resulting radial mode structure differ somewhat from those used in previous local analytic estimates; however, when the analysis is calibrated to the appropriate effective radial wavelength of the nonlocal calculation, reasonable agreement is obtained. Comparison with recent experimental measurements indicates that certain of these modes may play a role in establishing operating boundaries for the ELMO Bumpy Torus-Scale (EBT-S) experiment. The calculations given here indicate the necessity of having core plasma outside the ring to prevent the destabilizing wave resonance of the precessional mode with a cold plasma.

**Fast magnetic reconnection processes **

F. Brunel, T. Tajima, J.M. Dawson

**Abstract**

It is found through theory and magnetohydrodynamic particle simulation that fast magnetic-field-line reconnection may consist of more than one stage. After the Sweet-Parker phase is established for an Alvén time, a faster "second phase" of reconnection takes over if the plasma is compressible: The reconnected flux varies as ψ=ψ_{0}tρ_{i} / ρ_{e}, where ρ_{e} and ρ_{i} refer to the plasma densities outside and inside of the current channel.

**Variational method for the three-dimensional inverse equilibrium problem in toroids **

A. Bhattacharjee, J.C. Wiley, R.L. Dewar

**Abstract**

A variational method is developed for three dimensional magnetostatic equilibria in toroids. We represent equilibria in cylindrical inverse variables R (v, θ, ζ), φ (v, θ, ζ), and Z (v, θ, ζ), where v is a radial flux surface label, theta, a poloidal angle, and zeta, a toroidal angle. We Fourier-expand in theta and zeta and derive, from the variational principle, a set of ordinary differential equations for the amplitudes in v. Truncation of the infinite Fourier series leads to a reduced set of equations which we solve numerically by collocation to obtain two and three dimensional toroidal equilibria.

**Integral equations formulation of electromagnetic mode equations **

S.M. Mahajan

**Abstract**

The coupled set of differential equations describing the electromagnetic perturbations in tokamak plasmas is reduced to a single simple integral equation with a symmetric kernel. Obvious analytical and computational advantages are discussed.

**Integrals of the test wave Hamiltonian: A special case **

J.D. Meiss

**Abstract**

The integrability of the test wave Hamiltonian has been analyzed and conditions for integrability have been determined.

**Fluctuation spectra of a drift wave soliton gas **

J.D. Meiss, W. Horton

**Abstract**

A theory of drift wave turbulence is presented based on a low-density gas of drift wave solitons. The Gibb`s ensemble for the ideal gas is used to calculate the dynamical scattering form factor S(k,ω). In contrast to renormalized turbulence theory, the spectrum has a broad frequency component with δω ∝ to the fluctuation level δn_{e}n_{0} at fixed k and peaks at a frequency ω>k_{y}v_{d}e.

**Analysis of chaotic, area-preserving maps **

J.R. Cary

**Impurity flow reversal in tokamaks without momentum input **

F. Hinton

**Abstract**

A new method for impurity flow reversal is suggested, which does not depend upon momentum input. The method uses neutral-beam injection or radio-frequency fields to drive the Pfirsch-Schluter ion current required for toroidal equilibrium. The necessary alteration of the momentum transfer between the hydrogen ions and impurity ions can cause the former to diffuse into the plasma and the impurities to diffuse out.

**Stabilization of axisymmetric mirror plasmas by energetic ion injection **

F. Hinton, M.N. Rosenbluth

**Abstract**

Plasmas in axisymmetric mirror devices can be made stable to MHD interchange modes by injecting energetic ions which contribute significantly to the pressure and spend a sufficiently large fraction of a bounce time in regions of favorable curvature. Pitch-angle scattering adversely affects the method by reducing this fraction. The ions must be sufficiently energetic that pitch-angle scattering is not detrimental for that part of a slowing-down time during which they contribute significantly to the pressure. We have solved the bounce-averaged Fokker-Planck equation, including drag and pitch-angle scattering, and calculated the energetic ion contribution to the stability integral. With specially tailored magnetic fields, the required injection energy and power drain are found to be reasonable.

**Relation of linear response to nonlinear motion **

J.R. Cary

**Abstract**

Ponderomotive force is the time-averaged force on a particle in an oscillating field. Ponderomotive force was encountered many decades ago in the form of radiation reaction. Use of the ponderomotive force concept is widespread in plasma theory. Ponderomotive force has been proposed as a mechanism of plasma confinement. It has been invoked as the dominant nonlinearity in wave coupling and filamentation of radiation. However, ponderomotive force calculations are typically specific to the problem at hand.

**The feasibility of a ring stabilized plasma reactor in the Lee-Van Dam Limit **

B. Saltzman

**Thesis**

**Confinement of high-beta plasma column **

F. Brunel, T. Tajima

**Abstract**

Hydromagnetic aspects of confinement of a high-beta linear plasma column, including the reverse field configurations, are investigated. The previous theoretically predicted diverging confinement time of a sharp-boundary plasma column as β > 1 is removed by properly taking into account the magnetic tension effect near the column throats. The obtained end loss rate is tested through simulation: the present simulation results in slab geometry are in good agreement with theory and are also close to other simulation and experimental values. The confinement time τ of a plasma with thickness a and length L at β = 1 is found to be τ = 3/4 (3/2π)^{1/2}(L/2c_{s})(L/a)^{1/2}. When the magnetic field inside the column is reversed, the plasma end loss much improves: the end loss induces the reconnection of the field lines near the throats, which produces closed field lines (islands). Those islands are found to be unstable against the tilting instability. The tilting-induced reconnection of the island field line and the mirror field line is a very fast process even for negligibly resistive plasmas and this helps to rapidly spill out the plasma confined in the islands. The significance of these processes for the spheromaks and reversed field (theta) pinches is discussed.

**Nonlinear collisionless tearing and reconnection **

T. Tajima

**Abstract**

A threshold phenomenon for collisionless reconnection of magnetic field-lines is found: when the plasma is compressible or with weak toroidal field, reconnection involves laminar but singular flow and is characterized by the collisionless Sweet-Parker process, followed by a faster second phase. When incompressible or with strong toroidal field, reconnection involves turbulent non-singular flow, characterized by saturated semi-turbulent-collisional tearing modes: island width w ∼ π/2 c/ω_{p} and private poloidal flux ψ ∝ t.

**Breakdown of onsager symmetry in neoclassical transport theory **

K. Molvig, L.M. Lidsky, K. Hizanidis, I.B. Bernstein

**Abstract**

Neoclassical transport theory is developed in a Lagrangian rather than the usual Eulerian formulation. We show that an underlying asymmetry exists in the neoclassical pinch and bootstrap effects and demonstrate the physical basis of the Onsager symmetry relationship in the pinch-bootstrap duality. It is suggested that low frequency turbulence can destroy the bootstrap current at levels too low to effect the Ware Pinch.

**Kinetic effects on the toroidal ion pressure gradient drift mode **

P. Terry, W. Anderson, W. Horton

**Abstract**

The question of the threshold of the ion pressure gradient drift mode in toroidal geometry is examined from the drift kinetic equation retaining the grad-B and curvature drift resonances of the ions with the mode. Analytic criteria for the onset of the instability are derived which exhibit the parametric dependence on toroidicity, pressure gradient and perpendicular wavenumber.

**Drift wave turbulence and anomalous transport **

W. Horton

**Abstract**

The properties of a magnetized plasma such as density and temperature are typically nonuniform in space. The presence of the spatial gradients across the magnetic field leads to the so-called diamagnetic drift currents of the particles in the direction mutually perpendicular to the gradient of the plasma and the direction of the magnetic field. In such nonuniform, magnetized plasmas there arise collective oscillations propagating along the diamagnetic currents called drift waves. Drift waves give rise to a motion of the charged particles across the magnetic field. The excursion of the particle across the magnetic field can be many times greater than its gyroradius. Through the particle dynamics in the collective modes of the system a net plasma transport occurs along the macroscopic plasma gradient. As reviewed in this chapter, the mechanisms for the transport are varied, depending on the regime of the magnetized plasma and the details of the drift modes. Quite generally, however, the effect of the plasma oscillations on the macroscopic scale is to produce a collective dissipation in the form of anomalous diffusion of particles and thermal energy along the gradients of he plasma density and temperature. The collective transport is independent of the binary particle collisions and may exceed the collisional transport by several orders of magnitude. For this reason, the transport is called anomalous, although it is the general mechanism of transport in high temperature plasmas and obeys the general thermodynamic laws of transport processes.

**Eigenmode stability analysis for a bumpy torus **

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

**Abstract**

The analysis of eigenmodes in a bumpy torus yielded several stability boundaries that indicate the existence of a parameter regime for generally stable operation consistent with current experiments. A relatively narrow band of parameters where instability persists was obtained and these parameters are discussed.

**Self-consistent spectrum of electrostatic drift wave fluctuations due to electron phase space correlations in a sheared magnetic field **

S. Hirshman, P.H. Diamond

**Abstract**

The spectrum of electrostatic universal mode fluctuations due to electron phase space correlations (clumps) in a sheared magnetic field is self-consistently calculated. The pair correlation equation for electrons in a sheared field is derived and renormalized, and an approximate solution for the correlation function is obtained. Using the density correlation function as a source term in Poisson`s equation, the renormalized dielectric operator is inverted to obtain the turbulent spectrum. Self-consistency is imposed by requiring that the diffusion coefficient derived from the calculated spectrum equal that used to obtain the spectrum originally.

**Toroidal drift-wave fluctuations driven by ion pressure gradients **

W. Horton, D. Brock

**Abstract**

Dynamic equations for the electrostatic potential and pressure fluctuations in the ballooning drift modes are shown to lead to a turbulent state described by the mixing length theory for saturation. The nonlinear states are compared with previously given formulae for the root-mean-square amplitudes of the fluctuations and the k, ω structure of the fluctuations. The parametric dependence of the root-mean-square fluctuation level, the rate of decay of the two-time correlation functions, and the anomalous thermal flux are reported.

**Anomalous transport from drift modes driven by temperature gradients and trapped electrons **

S.M. Mahajan

**Abstract**

Drift modes driven by trapped electrons in the presence of temperature gradients are used to construct a theory for anomalous electron transport in tokamak plasmas. Several crucial features of the observed electron transport are predicted by the theory.

**Turbulent response in a stochastic regime **

K. Molvig, J. Freidberg, R. Potok, S. Hirshman, J. Whitson, T. Tajima

**Abstract**

The theory for the non-linear, turbulent response in a system with intrinsic stochasticity is considered. It is argued that perturbative Eulerian theories, such as the Direct Interaction Approximation (DIA), are inherently unsuited to describe such a system. The exponentiation property that characterizes stochasticity appears in the Lagrangian picture and cannot even be defined in the Eulerian representation. An approximation for stochastic systems - the Normal Stochastic Approximation - is developed and states that the perturbed orbit functions (Lagrangian fluctuations) behave as normally distributed random variables. This is independent of the Eulerian statistics and, in fact, we treat the Eulerian fluctuations as fixed. A simple model problem (appropriate for the electron response in the drift wave) is subjected to a series of computer experiments. To within numerical noise the results are in agreement with the Normal Stochastic Approximation. The predictions of the DIA for this mode show substantial qualitative and quantitative departures from the observations.

**Ion cyclotron resonance heating **

T. Tajima

**Abstract**

Ion cyclotron resonance heating of plasmas in tokamak and EBT configurations has been studied using 1-2/2 and 2-1/2 dimensional fully self-consistent electromagnetic particle codes. We have tested two major antenna configurations; we have also compared heating efficiencies for one and two ion species plasmas. We model a tokamak plasma with a uniform poloidal field and 1/R toroidal field on a particular q surface. Ion cyclotron waves are excited on the low field side by antennas parallel either to the poloidal direction or to the toroidal direction with different phase velocities. In 2D, minority ion heating (v_{⊥}) and electron heating (v_{||} , v_{⊥}) are observed. The exponential electron heating seems due to the decay instability. The minority heating is consistent with mode conversion of fast Alfvén waves and heating by electrostatic ion cyclotron modes. Minority heating is stronger with a poloidal antenna. The strong electron heating is accompanied by toroidal current generation. In 1D, no thermal instability was observed and only strong minority heating resulted. For an EBT plasma we model it by a multiple mirror. We have tested heating efficiency with various minority concentrations, temperatures, mirror ratios, and phase velocities. In this geometry we have beach or inverse beach heating associated with the mode conversion layer perpendicular to the toroidal field. No appreciable electron heating is observed. Heating of ions is linear in time. For both tokamak and EBT is slight majority heating above the collisional rate is observed due to the second harmonic heating.

**Energy conservation and related constraints in drift wave turbulence **

D. Thayer, K. Molvig

**Abstract**

The problem of energy conservation for the renormalization of the drift wave instability in a sheared magnetic field is considered. It has been suggested previously that there is a connection between a certain constraint on the nonlinear term in the drift kinetic equation and energy conservation. Arguments are presented to dissolve this connection; and in turn, energy conservation is formulated in the physically meaningful statistically averaged sense. Finally, energy conservation is proven for the system of nonlinear equations, renormalized by the normal stochastic approximation, describing the drift wave instability.

**Computer simulation of synchrotron radiation in two dimensions **

L. Wang, J.M. Dawson, A. Lin, C. Menyk, T. Tajima

**Abstract**

Plasma effects on the emission of synchrotron radiation are investigated using a two dimensional electromagnetic relativistic simulation code. Results are compared to those for a vacuum; it is found that the emission lies between the vacuum emission for n_{c} and n_{c} + 1 where n_{c} is the critical harmonic for EM wave propagation (n_{c} = ω_{p} / Ω_{o}).