Tokamak edge turbulence and the approach to adiabatic response

Turbulent fluctuations just inside the last closed flux surface of a diverted tokamak are strongly influenced by the combination of parallel electron motion and magnetic geometry. Simple considerations of isothermal drift wave physics lead to a natural decomposition into adiabatic (a) and nonadiabatic (b) state variables, defined as linear combinations of the electron density and electro-static potential. In the limit of weak parallel impedance, the nonadiabatic variable is small even for strongly nonlinear turbulence. Even when small, the nonadiabatic variable remains important since it controls the gradient drive and the dissipation. In the limit of weak parallel impedance, the adiabatic, nonadiabatic decomposition may be systematically exploited in a 2D model to obtain an approximate closure for the nonadiabatic variable in terms of the adiabatic one, reducing the n, ϕ, j∥ equations to a single dynamical equation plus a constitutive relation. A linear solution of the full three-field model, spanning most of the parameter and wave number space, reveals that the linear behavior at drift-scale frequencies is always weakly-nonadiabatic for realistic edge turbulence parameters and k⊥rho s, less than a number not much smaller than 1. Even when the parallel impedance is not weak, the quadratic invariants of the adiabatic: nonadiabatic formulation reveal the special role of the polarization nonlinearity in exciting nonadiabatic fluctuations, a spectral division between regions of polarization and curvature drive, and a steady-state constraint on the spectra of a and b due to their significantly different dissipation mechanisms. For a diverted tokamak, the distortion of flute-like perturbations near the poloidal field null has a strong effect on the parallel dynamics of fluctuations over the cut ire radial extent of the near-separatrix layer. Using field-line following co-ordinates in an approximate X-point magnetic geometry, one may calculate an approximate parallel form for the fluctuations. The parallel extent of fluctuations in the adiabatic variable is limited by energy considerations. For weaker parallel impedance, the nonadiabatic variable b is limited by a sharp evanescence, following either from enhanced X-point curvature drift effects or from a cross-field diffusive current. Use of the approximate X-point parallel envelope in that limit yields a 2D, one-field reduction of the full 3D, three-field dynamics that closely resembles the resistive limit of a model that uses a single effective k∥. Using either the heuristic or X-point model, the systematic correlations of n and ϕ due to parallel electron motion---including effects of the Alfven wave and (for the diverted tokamak) the X-point ---have been incorporated into an approximate one-field model.