**Effect of turbulent diffusion on collisionless tearing instabilities **

J.D. Meiss, R.D. Hazeltine, P.H. Diamond, S.M. Mahajan

**Abstract**

The direct interaction approximation is used to obtain the kinetic electron response in the presence of drift-wave turbulence. Primary effects on the tearing mode equations are a diffusive broadening of the conductivity and an anomalous perpendicular viscosity. The first has a negligible effect on collisionless current channel modes. The viscous term, however, produces significant stabilization of the m = 1 mode for levels of turbulence much smaller than those expected.

**Rigorously diffusive deterministic map **

J.R. Cary, J.D. Meiss

**Abstract**

The statistical properties of periodic impulse maps may be obtained from the characteristic functions. Series representations for the characteristic functions, the force correlations, and the momentum diffusion coefficient are presented. These results are applied to the sawtooth map for integer values of the perturbation parameter &epsilon, in which case the series may be summed explicitly. It is found that the diffusion coefficient has the quasilinear value for chemically bond ε + 2 chemically bond ≥ 2, it vanishes for ε = -2 and ε = -1, and it is infinite for ε = -3.

**Theory of ion compton scattering for drift wave turbulence in a sheared magnetic field **

P.H. Diamond, M.N. Rosenbluth

**Abstract**

An approximate analytic theory of ion Compton scattering for drift wave turbulence in a cylinder with shear is presented. A renormalized gyrokinetic equation is derived and solved iteratively to obtain the turbulent ion response and the nonlinear damping rate. The result is applied to the regime where stochastic electron motion destabilizes the universal mode. The Compton scattering process causes a transfer of mode spectrum energy to longer, more stable wavelengths. For k_{θ}ρ_{s} = 1, the ratio of the electron decorrelation rate ω_{c} to the mode frequency ω at saturation is substantially less than unity. Radical changes in the parameter scaling also occur. Therefore, it is argued that ion nonlinearity __must__ be considered in constructing an anomalous electron heat transport theory based on drift wave turbulence.

**Forward raman instability and electron acceleration **

T. Tajima, C. Joshi, J.M. Dawson, H. Baldis, N. Ebrahim

**Abstract**

It is demonstrated by experiments and supporting particle simulations that the forward Raman instability is capable of producing extremely high-energy electrons in an underdense plasma. The instability has a high saturation level for the electrostatic wave component. Its consequences and applications to the laser electron accelerator and the laser-fusion pellet preheat are discussed.

**Ballooning stability in toroidal devices **

S. Yoshikawa

**Abstract**

The marginal stability condition of ballooning instabilities for toroidal confinement devices is derived for low critical stability β (β ≤ 10%). The stability condition derived here should be applicable to EBT and multipoles as well as tokamaks and stellarators. For EBT and multipoles a more compact expression for the stability condition is possible and is given here in the appendix.

**Theory of the renormalized dielectric for electrostatic drift wave turbulence in tokamaks **

P.H. Diamond, M.N. Rosenbluth

**Abstract**

A theory of collisionless electrostatic drift wave turbulence in a circular cylinder with shear is presented. A renormalized electron drift kinetic equation which is consistent with conservation of energy is derived. For low levels of turbulence, a perturbative solution indicates that the turbulence has a stabilizing effect and that total mode spectrum energy decreases. For strongly turbulent regimes, results quantitatively but not qualitatively different from those of Hirshman and Molvig are found. In particular, additional stabilizing terms lead to a lower saturation amplitude, but the basic picture of turbulent destabilization competing against linear and nonlinear shear damping persists.

**Stochasticity and the random phase approximation for three electron drift waves **

P. Terry, W. Horton

**Abstract**

The interaction of three nonlinearly coupled drift waves is investigated for the occurrence of stochastization of the phases and the applicability of the random phase approximation. The drift wave nonlinearities include the E x B and polarization drift couplings for waves that are linearly unstable for appropriate values of the perpendicular wavenumber. The conservation properties and sample numerical solutions for the exact three wave interaction are given along with the conservation properties and solutions of the corresponding random phase approximation equations.

**Energy principle with global invariants **

A. Bhattacharjee, R.L. Dewar

**Abstract**

A variational principle is proposed for constructing equilibrium with minimum energy in a toroidal plasma. The total energy is minimized subject to global invariants which act as constraints during relaxation of the plasma. These global integrals of motion are preserved exactly for all ideal motions and approximately for a wide class of resistive motions. We assume, specifically, that relaxation of the plasma in dominated by a tearing mode of single helicity. Equilibria with realistic current density and pressure profiles may be constructed in this theory, which is also used here to study current penetration in Tokamaks. The second variation of the free energy functional is computed. It is shown that if the second variation of any equilibrium constructed in this theory is positive, the equilibrium satisfies the necessary and sufficient conditions for ideal stability.

**Renormalization of plasma turbulence in toroidal geometry **

R.D. Hazeltine

**Abstract**

A renormalized kinetic equation is derived for electromagnetic turbulence in an axisymmetric torus, which is formally very similar to the electrostatic, cylindrical version which it generalizes.

**Quasi-linear momentum transport **

S.M. Mahajan, R.D. Hazeltine, D. Hitchcock

**Abstract**

The equations of quasilinear momentum and energy transport are studied to show that a suprathermal level of low-frequency fluctuations can cause highly enhanced electron energy transport without effectively changing the plasma resistivity.

**Interchange instabilities caused by impurity ions in rotating plasmas **

S. Yoskikawa

**Dynamic magnetic x-points **

J.N. Leboeuf, T. Tajima, J.M. Dawson

**Abstract**

Two and one half dimensional magnetostatic and electromagnetic particle simulations of time varying magnetic x-points and the associated plasma response are reported. The stability and topology depends on the crossing angle of the field lines at the x-point, irrespective of the plasma. It is stipulated that the electrostatic field and finite Larmor radius effects are important in current penetration and shaping of the plasma flow. The snapping of the field lines, and dragging of the plasma into, and confinement of the plasma at, an o-point is observed. Magnetic island coalescence with explosive growth of the coalescence mode occurs and is accompanied by a large increase of kinetic energy and temperature as well as the formation of hot tails on the distribution functions.

**Lie transform perturbation theory for Hamiltonian systems **

J.R. Cary

**Abstract**

A review of the theory of Lie transform perturbation theory for Hamiltonian systems is presented. The operator theory of Dewar for continuous families of canonical transformations is discussed. It is then used to derive the perturbation method of Deprit. Two examples of the use of this method are provided. In addition, the more efficient perturbation method of Dragt and Finn is discussed.

**Variational structure of the Vlasov equation **

H.L. Berk, R.R. Dominguez, E.K. Maschke

**Abstract**

The derivation of the variational structure of the Vlasov-Maxwell integral equations for the case of a plasma equilibrium having two ignorable coordinates shows that the kernel of the Maxwell equations is a self-adjoint integral operator. This operator may also be represented as a differential equation of arbitrary order, which is useful when the differential operator is truncated to finite order, yielding a system of intrinsically self-adjoint differential equations. The formalism is illustrated through consideration of a plasma situated in a uniform external magnetic field with no external forces.

**A generalized kinetic energy principle **

J.W. Van Dam, M.N. Rosenbluth, Y.C. Lee

**Abstract**

Using three single-particle adiabatic invariants, we derive an energy principle which generalizes that for the usual guiding center plasma in order to describe the low-frequency stability of a plasma containing an energetic nonhydromagnetic component (such as the annular electrons in an Elmo Bumpy Torus device).

**Magnetic trapped particle modes **

M.N. Rosenbluth

**Abstract**

It is shown that very-low-frequency magnetic modes may be unstable for systems such as tandem mirrors which contain plasma trapped in regions of unfavorable curvature. Onset of the instability occurs when the diamagnetic plasma pressure is sufficient to reverse particle drift velocities.

**Necessary stability condition for field-reversed theta pinches **

J.R. Cary

**Abstract**

Toroidal systems of arbitrary cross section without toroidal magnetic field are analyzed via the double adiabatic fluid equations. Such systems are shown to be unstable if there exists one closed field line on which the average of κrB^{2} is positive, where κ is the curvature. A similar criterion is derived for linear systems and is applied to a noncircular z-pinch.

**Drift modes in axisymmetric tandem mirrors **

W. Horton

**Abstract**

The drift mode analysis of the tandem mirror is developed for a large aspect ratio, axisymmetric model of the equilibrium. The axial drift wave eigenmodes are shown to change character as the plasma pressure varies with respect to the inverse aspect ratio. In the high beta regime the drift modes are transformed into a finite frequency convective cell and the flute-like ion drift wave. Quasi-linear formulas are given for the anomalous radial losses.

**Electron Temperature limit for poloidal equilibrium and tokamak containment scaling **

A.A. Ware

**Abstract**

The experimentally observed scaling laws for the electron temperature (Τ_{eo}) and energy containment time (τ_{Ee}) with Ohmic heating are derived from the two conditions: (a) the predicted critical Τ_{eo} for loss of poloidal equilibrium and (b) the proportionality of the ion hear flux to the Poynting vector. Condition (b) is inferred from the large fraction of Ohmic power shown to be transferred to the ions.

**Numerical computation of relaxation rates for the test wave model **

J.D. Meiss

**Abstract**

Numerical integration of the dynamical equations for three wave interactions can, in principle, determine the validity of the approximations used in the derivation of relaxation and transport equations. To provide a simple dynamic model, we use the test wave system. This consists of a single mode interacting with a spectrum of ambient modes; the ambient modes, however, do not interact among themselves. These "direct" interactions are just those required for the derivation of relaxation rates by lowest order perturbation theories. The test wave model, however, has been shown to be an integrable Hamiltonian system and therefore it is necessary to consider an ensemble average to obtain statistical behavior. Numerical computations of the test wave action, averaged over initial conditions of the ambient waves, do indeed exhibit relaxation to a steady state. It is necessary to include non-resonant triads in the system: to obtain a converged value for the relaxation rate, ν, triads with resonance mismatch, δ, of order several times ν must be included. Validity of the relaxation rate computations hinges on the degree a continuous set of ambient modes is approximated by the discrete set of modes in the computation. We discuss the effects of discreteness for our computations, and attempt to encourage further work along this line.

**Statistical characterization of periodic area-preserving mappings **

J.R. Cary, J.D. Meiss, A. Bhattacharjee

**Abstract**

A method of statistically characterizing an area-preserving, doubly periodic mapping is presented. This method allows one to calculate the characteristic functions, and the joint probabilities of the mapping.

**Summary of US-Japan Workshop on non-axisymmetric confinement **

W. Horton, D.W. Ross

**Unified kinetic theory in toroidal systems **

D. Hitchcock, R.D. Hazeltine, S.M. Mahajan

**Abstract**

The kinetic theory of toroidal systems has been characterized by two approaches: neoclassical theory which ignores instabilities and quasilinear theory which ignores collisions. In this paper we construct a kinetic theory for toroidal systems which includes both effects. This yields a pair of evolution equations; one for the spectrum and one for the distribution function. In addition, this theory yields a toroidal generalization of the usual collision operator which is shown to have many similar properties - conservation laws, H theorem - to the usual collision operator.

**Neoclassical energy transfer between electrons and ions in a tokamak **

A.A. Ware

**Abstract**

Expressions are derived for the rates of energy transfer between electrons and ions associated with the neoclassical, electrostatic and Pfirsh-Schluter contributions to the diffusion and also the anomalous contribution caused by electrostatic drift-wave turbulence. The rates vary in sign as well as magnitude. As a result, even when the resultant diffusion is small, there can be a substantial energy transfer between electrons and ions comparable with the collisional transfer rate.

**Magnetic X-points, Island coalescence and intense plasma heating **

J.N. Leboeuf, T. Tajima, J.M. Dawson

**Abstract**

Release of a large amount of magnetic energy and intense plasma heating take place and an extremely hot ion tail is created, if and when tearing instability induced magnetic islands merge, according to fully self-consistent particle simulations. Although the tearing instability exponentiates the kinetic energy, it does not give rise to intense heating because of the small amount of magnetic energy available by reconnection. The stability and topology of the x-point depend on the crossing angle of the field lines at the x.

**Anomalous ion conduction from toroidal drift modes **

W. Horton

**Abstract**

The anomalous ion thermal transport produced by the toroidal drift instability is analyzed for finite amplitude single modes and the transition to turbulent flows.