Forces and moments within layers of driven tearing modes with sheared rotation

Abstract

For driven low amplitude tearing modes in a plasma with sheared rotation, forces on tearing layers due to Maxwell and Reynolds stresses are calculated. First moments about the center of the tearing layer, also due to Maxwell and Reynolds stresses, are also calculated. The forces tend to cause the tearing mode to lock to the phase of the driving perturbation, and the moments determine the evolution of the rotation shear within the layer. These forces and moments are calculated for two constant-ψ regimes of tearing modes, namely, the viscoresistive (VR) regime and the resistive-inertial (RI) regime, and an ordering in terms of the constant-ψ small parameter ϵ∼δΔ is introduced, with the velocity shear ordered as ∼ϵ. Here, δ is the layer width and Δ the logarithmic jump in the derivative of the flux function across the layer. The forces and first moments are reported to the lowest nonvanishing order in ϵ. The Reynolds moment is analogous to the effect that can drive zonal flows in other contexts. The treatment of the tearing layers is by means of variational principles using Padé approximants (A. J. Cole and J. M. Finn, Phys. Plasmas 21, 032508 (2014)). The usual result for the Maxwell force without rotation shear is recovered for both regimes. That is, the correction due to velocity shear is small; also, the lowest order contribution to the Reynolds force is zero. In the VR regime, we find no first moments up to second order in the constant-ψ parameter. In the RI regime, we find Nm is zero to at least order ϵ3/2. In the RI regime, the Reynolds moment Nr is found to be of order ϵ3/2 and is proportional to minus the rotation shear in the layer; it thus tends to damp out any velocity shear across the layer.