Real frequency tearing layers with parallel dynamics and the effect on error field locking and resistive wall modes

Abstract

Tearing modes with real frequencies in the plasma frame (i.e. in addition to the Doppler shift due to E×B rotation) are of potential importance because of their effect on the locking process. In particular, it has recently been shown [9] that the Maxwell torque on the plasma in the presence of an applied error field is modified significantly for tearing modes having real frequencies near marginal stability. In addition, it is known [10] that resistive wall tearing modes can be destabilized below their nowall limits by rotation, if the tearing modes have real frequencies near marginal stability. In this paper we first derive the tearing mode dispersion relation with pressure gradient, field line curvature and parallel dynamics in the resistive-inertial (RI) regime, neglecting the divergence of the E ×B drift and perpendicular resistivity. The results show that the usual Glasser effect, a toroidal effect which involves real frequencies, occurs in this simplified model, which ignores perpendicular resistivity and the divergence of the E × B drift. We also find, using a similar simple model, the surprising result that in the viscoresistive regime with pressure gradient, favorable curvature due to toroidal effects, and parallel dynamics, a similar Glasser-like effect is found. We show that in both regimes the existence of tearing modes with complex frequencies is related to nearby electrostatic resistive interchange modes with complex frequencies. We discuss the effect on locking to an error field and the significant lowering of the threshold for destabilization of resistive wall tearing modes, which can be much more pronounced than the weak effect observed for RI tearing modes without pressure-curvature drive in Ref. [10].

Nonlinear error field response in the presence of plasma rotation and real frequencies due to favorable curvature

“…The forms of the reconnected flux and Maxwell torque as functions of plasma rotation, emerging from the present linear simulations, are consistent with the quasilinear theory of tearing and error-field penetration in the presence of real frequencies that are driven by pressure gradient and magnetic curvature 4,7 . The above results show that these effects are present in a tearing regime other than the RI regime since these results involve ion inertia and viscosity, i.e.…”

“…4. It should be noted that the propagation frequencies are very slow here compared to Alfvén scales, reaching ω r τ A ≈ 10 −4 at marginal stability, but are comparable to the typical growth rates found in the RI and VR regimes 5,7 .…”

“…It has recently been shown in Ref. 7 that a similar behavior occurs in the a) Electronic mail: c akcay@tibbartech.com visco-resistive (VR) resistive MHD regime as well as in the RI regime, suggesting that the mode propagation and stabilization occur over a wider range of parameters than previously realized. This effect in either of these two resistive MHD regimes is much like the behavior of the two-fluid drift-tearing mode [8][9][10][11] , which introduces diamagnetic propagation ω r ∼ ω * and stabilization.…”

“…When β is raised past the second critical value, β c , corresponding to ∆ c , the (2, 1) TM is stabilized. The linear simulations contain both inertia and viscosity, so that the modes are either in the RI regime or the VR regime 7 or (most commonly) in the transition range between these regimes, verifying that the Glasser effect is indeed present across a wide range of parameters. Linear simulations for β just above the second critical limit with error field ψw and rotation Ω 0 show the characteristic form 4,7 of the reconnected flux ψt in response to the error field for tearing layers with real frequencies: a double-humped curve with peaks near Ω 0 = ±ω r /m, i.e.…”

Structure and overstability of resistive modes with runaway electrons

“…The real frequency of the mode has been studied as an effect of plasma pressure gradient( [31][32][33][34]), two-fluid effect [35][36][37][38], polarization drifts [39], energetic ion interaction [40,41], effects of favorable average curvature [42], and coupling with other modes [43]. It has been used to explain the Maxwell torque and plasma rotation as mode locking happens [44,45]. With multiple effects simultaneously present, a single frequency is generally found for the mode depending on the combined influences.…”