Abstract
A detailed analysis of the energy transfer system between ExB turbulence and zonal flows is given. Zonal flows, driven by the ExB Reynolds stress of the turbulence, are coupled to pressure disturbances with sinusoidal poloidal structure in toroidal geometry through the geodesic curvature. These pressure ‘sidebands’ are nonlinearly coupled not only back to the turbulence, but also to the global Alfvén oscillation whose rest state is the Pfirsch–Schlüter current in balance with the pressure gradient. The result is a statistical equilibration between turbulence, zonal flows and sidebands, and additionally the various poloidally asymmetric parallel dynamical subsystems. Computations in three-dimensional flux surface geometry show this geodesic transfer effect to be the principal mechanism which limits the growth of zonal flows in tokamak edge turbulence in its usual parameter regime, by means of both control tests and statistical analysis. As the transition to the magnetohydrodynamic (MHD) ballooning regime is reached, the Maxwell stress takes over as the main drive, forcing the Reynolds stress to become a sink.
Highlights
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT modulational instability [15], that is, a direct transfer to the flow, seeded by the flow itself, rather than a local cascade
Having performed presence and absence tests on both geodesic coupling and interchange forcing effects, we find that the main qualitative difference in drift Alfven turbulence between slab and toroidal geometry is not interchange forcing, which produces linear instabilities, but geodesic coupling, whose role is to statistically saturate the zonal flows, act as geodesic transfer
We find that even for nominal parameters well below the MHD beta limit the turbulence is robustly electromagnetic enough to knock the Pfirsch– Schlüter currents out of equilibrium so that their involvement in the energetics is comparable to the zonal flows and geodesic acoustic dynamics
Summary
This study uses the same model as in [9], the isothermal version of the drift Alfven model described in detail in [28]. The geometry is described in terms of a field aligned flux tube geometry under the shifted metric treatment, with coordinates {x, y, s} representing the radial, perpendicular drift and parallel directions, noting that s is a projection of the parallel direction onto the poloidal angle [37]. Where sis the standard parameter representing the magnetic shear. This follows from the fact that the coordinate system is defined separately at each location in s, such that perpendicular operators are always evaluated with an orthogonal metric. Τi = Ti/Te is the background ion/electron temperature ratio, sgives the magnetic shear and the curvature operator is scaled with ωB, which can be set independently (slab geometry is ωB = 0).
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