Magnetohydrodynamic counter-rotating vortices and synergetic stabilizing effects of magnetic field and plasma flow

Abstract

A nonlinear two-dimensional steady state solution in the framework of hydrodynamics describing a row of periodic counter-rotating vortices is extended to the magnetohydrodynamic (MHD) equilibrium equation with incompressible flow of arbitrary direction. The extended solution covers a variety of equilibria because four surface quantities remain free. Similar to the case of the MHD cat-eyes equilibrium [Throumoulopoulos et al., J. Phys. A: Math. Theor. 42, 335501 (2009)] and unlike linear equilibria, the flow has a strong impact on isobaric surfaces by forming pressure islands located within the counter-rotating vortices even for values of β (defined as the ratio of the thermal pressure over the external axial magnetic-field pressure) on the order of 0.01. Also, the axial current density is appreciably modified by the flow. Furthermore, a magnetic-field-aligned flow of experimental fusion relevance, i.e for Alfvén Mach numbers of the order of 0.01, and the flow shear in combination with the variation of the magnetic field perpendicular to the magnetic surfaces have significant stabilizing effects potentially related to the equilibrium nonlinearity. The stable region is enhanced by an external axial magnetic field.

Two-dimensional nonlinear cylindrical equilibria with reversed magnetic shear and sheared flow

“…The maximum of E takes larger values as the flow amplitude increases but its position is insensitive to the flow. For parallel flows application of a condition for linear stability implies that the equilibrium nonlinearity has a stabilizing effect together with a weaker stabilizing impact of the sheared flow in agreement with past nonlinear equilibrium studies [17,18,19]. Also even a small up-down asymmetry influences stability.…”

On the Equilibrium and Stability of ITER Relevant Plasmas

“…• In certain cases stability is improved by the equilibrium nonlinearity in agreement with past equilibrium studies in plane geometry [15,16]. A pertinent example supporting this result in connection with the equilibrium of Fig.…”

International thermonuclear experimental reactor-like extended Solovév equilibria with parallel flow

“…In particular, by including in the solution fourteen free parameters we have derived ITER-like equilibria with prescribed values of the safety factor on axis and toroidal beta for both Mach-function profiles, peaked on and off axis. Application of a sufficient condition for linear stability showed that this condition is satisfied only in a small region irrespective of flow and flow shear unlike the cases of the nonlinear cat-eyes [12] and counter-rotating-vortices equilibria [13]. This result is in favor to the conjecture that the equilibrium nonlinearity may activate flow stabilization.…”

“…Application of (16) for R in and R out furnishes relations (12) and (13). Relation (14) for the upper point can be derived in a similar way.…”

“…Since plasma flow is common in astrophysical systems and play a role in the transition to improved confinement regimes of magnetic confinement devices there has been an increasing interest to self consistent equilibria with flow [1]- [13]; they can be employed as starting points for stability and transport studies. In the static regime the most known and widely employed axisymmetric magnetohydrodynamic (MHD) analytic equilibrium is associated with the Solovév solution of the Grad-Sfafranov (GS) equation [14].…”