The Magnetohydrodynamic Kelvin-Helmholtz Instability: A Two-dimensional Numerical Study

Abstract

We have carried out two-dimensional simulations of the nonlinear evolution of unstable sheared magnetohydrodynamic flows. These calculations extend the earlier work of Miura (1984) and consider periodic sections of flows containing aligned magnetic fields. Two equal density, compressible fluids are separated by a shear layer with a hyperbolic tangent velocity profile. We considered two cases: a strong magnetic field (Alfvén Mach number, M a = 2.5) and a weak field (M a = 5). Each flow rapidly evolves until it reaches a nearly steady condition, which is fundamentally different from the analogous gasdynamic state. Both MHD flows relax to a stable, laminar flow on timescales less than or of the order of 15 linear growth times, measured from saturation of the instability. That timescale is several orders of magnitude less than the nominal dissipation time for these simulated flows, so this condition represents an quasi-steady relaxed state analogous to the long-lived single vortex, known as "Kelvin's Cat's Eye", formed in two-dimensional nearly ideal gasdynamic simulations of a vortex sheet.The strong magnetic field case reaches saturation as magnetic tension in the displaced flow boundary becomes sufficient to stabilize it. That flow then relaxes in a straightforward way to the steady, laminar flow condition. The weak magnetic field case, on the other hand, begins development of the vortex expected for gasdynamics, but that vortex is destroyed by magnetic stresses that locally become strong. Magnetic topologies lead to reconnection and dynamical alignment between magnetic and velocity fields. Together these processes produce a sequence of intermittent vortices and subsequent relaxation to a nearly laminar flow condition in which the magnetic cross helicity is nearly maximized. Remaining irregularities show several interesting properties. A pair of magnetic flux tubes are formed that straddle the boundary between the oppositely moving fluids. Velocity and magnetic fluctuations within those features are closely aligned, representing Alfvén waves propagating locally downstream.The flux tubes surround a low density channel of hot gas that contains most of the -3excess entropy generated through the relaxation process.

Growth and saturation of the Kelvin–Helmholtz instability with parallel and antiparallel magnetic fields

“…In the uniform MHD cases, we used the slightly diffusive minmod limiter. Our method is then essentially identical with the one used by Frank et al (1996). For all hydrodynamic and reversed MHD cases, we used the sharper Woodward limiter (see Tóth and Odstrčil (1996)).…”

“…We confirm and extend this result for the uniform field over a wider range of Alfvén and sound Mach numbers. Frank et al (1996) found that numerical dissipation, mimicking viscous and resistive effects, eventually led to similar end states consisting of a stable laminar flow with dynamically aligned velocity and magnetic fields.…”

“…They referred to these stages as the linear stage, the dissipative transient stage with intermittent reconnection events, and the saturation stage where small scale turbulent motions decay to form aligned structures. Their final stage is qualitatively similar to the laminar flow end state of Frank et al (1996). We focus attention to the two stages preceding turbulence, where the amplification and the dynamic influence of the magnetic field grows and eventually halts the KH instability.…”

“…Recently, Frank et al (1996) carried out non-linear 2D MHD calculations for two cases. They took the field parallel to the shear flow and compared the KH evolution of a 'weak' field case with a 'strong' field case.…”

The stability of astrophysical jets

“…Additionally, the non-linear development of KHI can be significantly modified. Equipartition or weaker magnetic fields aligned with the velocity in a shear layer have been shown to be capable of stabilizing a shear profile through non-linear saturation, e.g., Frank et al (1996); Jones et al (1997); Keppens & Tóth (2000); Ryu et al (2000). The development of mixing can also be slowed by a helically twisted equipartition magnetic field configuration (Rosen et al 1999).…”