P. H. Diamond

Theory of ion-temperature-gradient-driven turbulence in tokamaks

An analytic theory of ion‐temperature‐gradient‐driven turbulence in tokamaks is presented. Energy‐conserving, renormalized spectrum equations are derived and solved in order to obtain the spectra of stationary ion‐temperature‐gradient‐driven turbulence. Corrections to mixing‐length estimates are calculated explicitly. The resulting anomalous ion thermal diffusivity χi=0.4[(π/2)ln(1+ηi)]2[(1+ηi)/τ]2 ρ2pcs/Ls is derived and is found to be consistent with experimentally deduced thermal diffusivities. The associated electron thermal and particle diffusivity, and particle and heat‐pinch velocities are also calculated. The effect of impurity gradients on saturated ion‐temperature‐gradient‐driven turbulence is discussed and a related explanation of density profile steepening during Z‐mode operation is proposed.

Theory of resistive pressure-gradient-driven turbulence

Saturated resistive pressure‐gradient‐driven turbulence is studied analytically and with numerical calculations. Fluid viscosity and thermal diffusivity are retained in the analysis and calculations. Such dissipation guarantees the existence of a stable, high‐m dissipation range, which serves as an energy sink. An accurate saturation criterion is proposed. The resulting predicted pressure diffusivity scales similarly to the mixing length estimate but is significantly larger in magnitude. The predictions of the analytic theory are in good quantitative agreement with the numerical results for fluctuation levels.

Momentum and thermal transport in neutral-beam-heated tokamaks

The relation between momentum and thermal transport in neutral‐beam‐heated tokamaks with subsonic toroidal rotation velocity has been investigated. A theory of diffusive momentum transport driven by ion‐temperature‐gradient‐driven turbulence (ηi turbulence) is presented. In addition, the level of ηi turbulence is enhanced by radially sheared toroidal rotation. The resulting ion shear viscosity is χφ=1.3{(1+ηi)/τ+[(Ln/2cs) (dV0/dr)]2}2(ρ2scs/ Ls). The associated ion thermal diffusivity, χi, is identical to χφ. Thus a scenario based on velocity‐shear‐enhanced ηi turbulence is consistent with the experimentally observed relationship between thermal and momentum confinement.

Theory of dissipative density‐gradient‐driven turbulence in the tokamak edge

We appreciate the interest of Krommes in our recent paper and welcome the opportunity to discuss his comments and other related issues. In our opinion, most of the objections hea has raised follow from a misunderstanding of the physics treated by clump and hole theory. In particular, throughout his critique Krommes attempts to extrapolate results and intuition of homogeneous Navier-Stokes turbulence (HN-ST) to the more complicated case of dissipative drift-wave turbulence (DD-WT). Since these two cases are so dissimilar with regard to their fundamental constituents, drive, characteristic scales and interaction mechanisms, extrapolations from one case to the other are unwarranted and misleading. Moreover, the hypotheses and results of clump and hole theories have fared well in several tests using laboratory and simulation data which is relevant to the theoretical models analyzed. 7 refs.

Theory of anomalous tearing mode growth and the major tokamak disruption

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: γ∼(∑k′u2009k′2θ‖φk′‖2)3/8 ×(Δ′)1/2. This analytical prediction is in good accord with computational results.

The effects of compressibility of the resistive ballooning mode

The linear stability of the resistive ballooning mode, as described by the resistive magnetohydrodynamic (MHD) model, is investigated both analytically and numerically. When the pressure evolution is approximated by fluid convection (reduced MHD model), an instability driven by geodesic curvature, with a growth rate γ∼η1/3 β2/3p, is found. For conditions relevant to the Impurity Study Experiment (ISX‐B), it is shown that for modest poloidal bets (βp≂1), high current, and relatively high temperatures, compressibility has a significant stabilizing influence, relative to the pressure convection model. However, at high βp (≳2), low current, and lower temperatures, compressibility has much less effect.

Thermally driven convective cells and tokamak edge turbulence

A unified theory for the dynamics of thermally driven convective cell turbulence is presented. The cells are excited by the combined effects of radiative cooling and resistivity gradient drive. The model also includes impurity dynamics. Parallel thermal and impurity flows enhanced by turbulent radial diffusion regulate and saturate overlapping cells, even in regimes dominated by thermal instability. Transport coefficients and fluctuation levels characteristic of the saturated turbulence are calculated. It is found that the impurity radiation increases transport coefficients for high density plasmas, while the parallel conduction damping, elevated by radial diffusion, in turn quenches the thermal instability. The enhancement due to radiative cooling provides a resolution to the dilemma of explaining the experimental observation that potential fluctuations exceed density fluctuations in the edge plasma (eΦ/Te >n/n0).

Nonlinear interaction of toroidicity‐induced drift modes

Drift modes in toroidal geometry are destabilized by trapped electron inverse dissipation and evolve to a nonlinearly saturated state. Using renormalized one‐point turbulence theory for the nonlinear gyrokinetic equation in the ballooning representation, it is shown that ion Compton scattering is an effective saturation mechanism. Ion Compton scattering transfers wave energy from short to long perpendicular wavelength, where it is absorbed by ion resonance with extended, linearly stable, long‐wavelength modes. The fluctuation spectrum and fluctuation levels are calculated using the condition of nonlinear saturation. Transport coefficients and energy confinement time scalings are determined for several regimes. Specifically, the predicted confinement time density scaling for an Ohmically heated discharge increases from n3/8 in the collisionless regime to n9/8 in the dissipative trapped electron regime.

Effects of a radial electric field on tokamak edge turbulence

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 enstrophy cascade process which couples regions of driving and dissipation in wavenumber space. For stronger magnetic shear, such that the width of the dissipation region resulting from parallel resistivity is comparable to the radial electr...

Resistive fluid turbulence in diverted tokamaks and the edge transport barrier in H‐mode plasmas

The thermal and particle diffusivities driven by resistive fluid turbulence in diverted tokamak edge plasmas are calculated. Diverted tokamak geometry is characterized by increased global shear near the separatrix and the tendency of field lines to linger near the x point. For resistive fluid turbulence, the dominant effect is increased global shear, which causes a reduction in the effective step size of the turbulent diffusion process and corresponding improvements in heat and particle confinement close to the separatrix. Stability of resistive kink modes resonant near separatrix is also ensured by the increased global shear. The relevance of these considerations to the L→H transition and to the edge transport barrier in H‐mode plasmas is discussed.