**Equilibrium current-drive tearing mode in the hydrodynamic regime **

F. Cozzani, S.M. Mahajan

**Abstract**

The effect of the parallel equilibrium current on the linear stability of the drift-tearing mode in the collisional regime is investigated analytically.

In the appropriate parameter regime, a new unstable mode, driven by equilibrium current, is found and its relevance to tokamak discharges is discussed.

**The fast alpha particle distribution function in an open field-line plasma with electrostatic confining potential **

H. Hamnen, J.D. Hanson

**Abstract**

An appropriate solution to the Fokker-Planck equation for the alpha-particle distribution function in an open field-line fusion plasma with electrostatic confining potential is presented. Particle and energy pitch-angle scattering losses are computed. The electrostatic confining potential is found to substantially reduce particle and energy losses.

**A renormalized theory of dissipative dispersive turbulent system **

Y.Z. Zhang, S.M. Mahajan

**Abstract**

A systematic perturbation theory to deal with stationary, homogeneous turbulence in a dispersive dissipative system is developed and is shown to be renormalizable. General properties of the renormalized equations are discussed, and for the specific case of Vlasov-Poisson turbulence, it is shown that the present theory reduces to the conventional weak turbulence theory. Kramers-Kronig kind of dispersion relations are derived for the nonlinear dielectric.

**Role of multiple helicity nonlinear interaction of tearing modes in dynamo and anomalous thermal transport in the reversed field pinch **

Z.G. An, P.H. Diamond, R.D. Hazeltine, J.N. Leboeuf, M.N. Rosenbluth, R.D. Sydora, T. Tajima, B.A. Carreras

**Abstract**

A theory of magnetic fluctuation dynamics, the dynamo mechanism, and anomalous thermal transport in the reversed field pinch is presented. Nonlinear generation of and coupling to m ≥ 2 m codes is advanced as an m-1 tearing modes sustain the magnetic configuration and stabilize themselves by lowering the safety factor on axis (g(0)) are elucidated. The nonlinear dynamics of resistive interchange modes are discussed. Stochastic magnetic field transport arguments are used to estimate anomalous thermal conductivity and confinement scalings.

**Enhancement of tokamak ion transport due to electron collisions **

A.A. Ware

**Abstract**

In tokamas where auxiliary ion heating leads to T_{i} > T_{e}, the neglect of electron collisions is no longer a good approximation in determining ion transport coefficients. The enhancement of the ion heat conduction due to electron collisions is determined for (a) the Pfirsch-Schluter regime and (b) the banana regime for the case where Z_{eff}^{i} is large. The enhancement of ion viscosity is particularly important; the contribution due to ion-electron-collisions is approximately equal to the ion-ion-collision term even for T_{i} = T_{e}.

**Onset of stochasticity and the diffusion approximation in drift waves **

W. Horton

**Abstract**

The Hamiltonian structure of the E*B drift equations is exploited to describe the onset of stochasticity for test particles in drift waves. In contrast to a longitudinal plasma wave, a drift wave acts over the entire single particle phase space. This feature precludes a simple use of the techniques presently available for predicting the onset of chaos in Hamiltonian systems. For two drift waves a generalized Chirikov (1979) overlap criterion is derived. The present work gives conditions on the drift wave spectrum for global stochasticity and the validity of the diffusion approximation.

**Algebraic decay in self-similar Markov chains **

J.D. Hanson, J.R. Cary, J.D. Meiss

**Abstract**

A continuous-time Markov chain is used to model motion in the neighborhood of a critical invariant circle for a Hamiltonian map. States in the infinite chain represent successive rational approximants to the frequency of the invariant circle. For the case of a noble frequency, the chain is self-similar and the nonlinear integral equation for the first passage time distribution is solved exactly. The asymptotic distribution is a power law times a function periodic in the logarithm of the time. For parameters relevant to the critical noble circle, the decay proceeds ast ^{–4.05}.

**Diffusion of particles in a slowly modulated wave **

J.R. Cary, D.F. Escande, J. Tennyson

**Abstract**

Particle motion in a slowly modulated wave as studied. Recent results for the change in the adiabatic invariant due to separatrix crossing are used to calculate the diffusion coefficient for the adiabatic invariant of a particle moving in an amplitude modulated wave. It is argued that the calculated scaling also holds for the case of a narrow spectrum wave field. The present scaling law due to adiabatic invariant jumps, D_{J} ≈ k_{O}δv_{φ}^{3}, differs substantially from the resonance broadening theory result D_{RB} ≈ W_{O}^{3/4}k_{O}^{1/2}, where δv_{φ} is the spectral width of the phase velocity, and W_{O} is the energy density of the wave field.

**Shear-Alfven dynamics of toroidally confined plasmas (Part A & Part B) **

R.D. Hazeltine, J.D. Meiss

**Abstract**

Part A: Recent developments in the stability theory of toroidally confined plasmas are reviewed, with the intention of providing a picture comprehensible to non-specialists. The review considers a class of low-frequency, electromagnetic disturbances that seem especially pertinent to modern high-temperature confinement experiments. It is shown that such disturbances are best unified and understood through consideration of a single, exact fluid moment: the shear-Alfven law. Appropriate versions of this law and its corresponding closure relations are derived - essentially from first principles - and applied in a variety of mostly, but not exclusively, linear contexts. Among the specific topics considered are: flux coordinates (including Hamada coordinates), the Newcomb solubility condition. Shafranov geometry, magnetic island evolution, reduced MHD and its generalizations, drift-kinetic electron response, classical tearing, twisting, and kink instabilities, pressure-modified tearing instability (δ-critical), collisionless and semi-collisional tearing modes, the ballooning representation in general geometry, ideal ballooning instability, Mercier criterion, near-axis expansions, the second stability region, and resistive and kinetic ballooning modes. The fundamental importance of toroidal topology and curvature is stressed.

Part B: This part covers radial boundary layer theory and the eikonal theory of pressure driven modes. (MOW)

**Ion acoustic turbulence and anomalous transport **

W. Horton

**Abstract**

Theoretical interpretation of ion acoustic turbulence is shown to require the use of renormalized turbulence theory for calculating the turbulent spectra and transport coefficients. The physics of solitons, double layers, and ion phase space holes have an impact on the one-dimensional problem.

**Enhancement of ion transport due to (A) electron collisions and (B) non-Maxwellian ion distributions **

A.A. Ware

**Abstract**

This paper considers the enhancement of ion transport, namely ion heat conduction (q_{ir}) and ion viscosity (P_{||r}), due firstly, to electorn collisions and secondly, to non-Maxwellian ion distributions (f_{io}).

**Drift and tearing modes in a sheared cylinder **

J.R. Cary, B. Newberger

**Abstract**

Drift and tearing modes in a sheared cylindrical collisionless plasma column are studied. A set of differential equations in the radial coordinate is derived with small gyroradius and low-β expansion. The finite-β effects include curvature drifts, gradient-B drifts, and the parallel magnetic field perturbation. Algebraic elimination reduces the resulting set of equations to a fourth-order system. Analysis shows that bad curvature does not drive the collisionless modes unstable.

**Institute for fusion studies progress report for the period 1 September 1983 to 31 August 1984**

IFS Staff

**The nonlinear stability of ion velocity profiles in beam injected mirrors **

V. Mirnov, J.D. Meiss, J. Tennyson

**Abstract**

The nonlinear stability of ion velocity distributions in simple beam-injected mirror machines is investigated using a one-dimensional kinetic equation. The steady states and their linear response functions are calculated, and the nonlinear behavior is studied both analytically and numerically. It is shown that when charge exchange effects are negligible, the oscillatory instabilities found in a previous work [1] are suppressed. When unstable steady states are present, the eventual equilibrium of the system is shown to depend not only on the initial density, but on the initial velocity distribution as well.

**Nonlinear Landau damping of purely perpendicular Bernstein modes **

S. Riyopoulos

**Abstract**

Analytical calculation of the trajectories of the trapped particles using an approximation Hamiltonian resounding. The integration alongside the orbits gives the rate of non linear cushioning in a similar way to the one used by O' Neil for electrostatic not magnetized method cushioning. The results can be spread to the general case of the electrostatic methods, of short length of wave, almost perpendicular, close to the harmonic cyclotron

**Stochasticity in classical Hamiltonian systems-universal aspects **

D.F. Escande

**Abstract**

This review presents universal aspects of stochasticity of simple A.5- or 2-degree-of-freedom Hamiltonian systems. Stochasticity the seemingly erratic wandering of orbits of non-integrable Hamiltonian systems over some part of phase space, accompanied by exponential ivergence of nearby orbits. It is a large-scale phenomenon that spreads over larger and larger regions of phase space by the successive breakups of jrriers called Kolmogorov-Arnold-Moser (KAM) tori when some perturbation to an integrable Hamiltonian is increased. The main emphasis of this review is on the breakup of KAM tori which is described by a renormalization group for Hamiltonians of the KAM type. This paper also reports :cent progress in describing chaotic transport which is the large scale manifestation of stochasticity, but this is not the last word to chaos. The central model of this paper is the Hamiltonian of one particle in two longitudinal waves, H?(v, x, t)= v2l2- M cosx- P cos k(x- t), which is a iradigm for simple Hamiltonian systems. Simple approximate renormalization schemes for KAM tori of Hp are derived, and the way to exactly normalize a general Hamiltonian of the KAM type is explained as well.

**Theory of resistivity gradient driven turbulence **

L. Garcia, P.H. Diamond, B.A. Carreras, J.D. Callen

**Abstract**

A theory of the nonlinear evolution and saturation of resistivity driven turbulence, which evolves from linear rippling instabilities, is presented. The nonlinear saturation mechanism is identified both analytically and numerically. Saturation occurs when the turbulent diffusion of the resistivity is large enough so that dissipation due to parallel electron thermal conduction balances the nonlinearly modified resistivity gradient driving term. The levels of potential, resistivity, and density fluctuations at saturation are calculated. A combination of computational modeling and analytic treatment is used in this investigation.

**Nonlinear global Alfven eigenmodes **

F. Brunel, J.N. Leboeuf, S.M. Mahajan

**Abstract**

Global or discrete Alfvén waves have been investigated with a full nonlinear, three-dimensional, ideal magnetohydrodynamic simulation code. When driven strongly with an outside antenna, those modes grow to a very high level, δB / BT≈10%. No destabilization of the plasma is observed. These modes appear to be much more efficient than the ones in the continuum in absorbing the pump energy.

**The effect of noise on time-dependent quantum stochasticity **

E. Ott, T.M. Antonsen, J.D. Hanson

**Abstract**

The dynamics of a time-dependent quantum system can be qualitatively different from those if its classical counterpart when the latter is chaotic. It is shown that small noise can strongly alter this situation.

**Effects of a radial electric field on tokamak edge turbulence **

P.W. Terry, P.H. Diamond

**Abstract**

Turbulence associated with sheared radial electric fields such as those arising in tokamak edge plasmas is investigated analytically. Two driving mechanisms are considered: in the region of maximum vorticity (maximum electric field shear), the electric field is the dominant driving mechanism. Away from the maximum, turbulence is driven by the density gradient. In the latter case, previous work is extended to include the effects of the electric field on the spatial scales of density correlation in the frequency-Doppler-shifted density-gradient-driven turbulence. For radial electric field driven turbulence, the effects of magnetic shear on linear instability and on fully developed turbulence are examined. In the case of weak magnetic shear, saturation occurs through an entropy cascade process which couples regions of driving and dissipation in wavenumber space. For stronger magnetic shear, such that the resistive layer is comparable to the radial electric field scale length, saturation occurs through nonlinear broadening of the mode structure, which pushes entropy into the region of dissipation. Estimates of mode widths, fluctuation levels and scalings are obtained for both mechanisms. Comparison is made with the results of fluctuation measurements in the TEXT tokamak.

**Kinetic theory of the electromagnetic drift modes driven by pressure gradients **

W. Horton, D.I. Choi

**Abstract**

Kinetic equations for the electromagnetic drift modes are derived and analyzed for the stability of tokamaks in the local approximation. In the dissipationless, hydrodynamic limit, the fifth-order polynomial dispersion relation previously studied is recovered. The kinetic velocity space integrals in the ion dynamics are shown to modify the five principal modes of oscillation and their polarizations. It is shown that in kinetic stability theory the critical plasma pressure defined in magnetohydrodynamic theory determines a transition from microinstability to macroinstability.

**Nonlinear equations for the drift instabilities in a high-beta plasma **

A.Y. Aydemir, H.L. Berk, V. Mirnov, O.P. Poguste, M.N. Rosenbluth

**Abstract**

A nonlinear system of equations is derived for drift waves in a high-beta plasma (β>>1). The magnetic field pressure is taken small compared to the particle pressure. Pressure balance is established by having a uniform particle pressure with the density and temperature gradients in opposite directions. The primary purpose of the magnetic field is to inhibit radial heat flux. This is the principle of such plasma fusion systems as the wall sustained multiple mirror, compressed liner, and magnetic-insulated inertial fusion, where the heat is contained over a relatively short radial scale length and a long axial scale length. The nonlinear equations for the mathematical model contain drift instabilities that give rise to radial heat and particle fluxes that can enhance the losses expected from classical collisional effects. The linear and nonlinear evolution of the model is studied here.

**Compact form for the relativistic ponderomotive Hamiltonian **

J.R. Cary, B. Newberger

**Abstract**

The infinite series of functions present in the ponderomotive Hamiltonian are explicitly summed using Newberger's sum rule. The result is a compact and easily evaluated expression for the ponderomotive Hamiltonian. Application of the K_{-χ} theorem yields the linear susceptibility of relativistic magnetized plasma in agreement with, but generalizing, the previous result of Weiss and Weitzner.