The Kelvin-Helmholtz (KH) instability of a shear layer with an initially uniform magnetic field in the direction of flow is studied in the framework of 2D incompressible magnetohydrodynamics with finite resistivity and viscosity using direct numerical simulations. The shear layer evolves freely, with no external forcing, and thus broadens in time as turbulent stresses transport momentum across it. As with hydrodynamic KH, the instability here features a conjugate stable mode for every unstable mode in the absence of dissipation. Stable modes are shown to transport momentum up its gradient, shrinking the layer width whenever they exceed unstable modes in amplitude. In simulations with weak magnetic fields, the linear instability is minimally affected by the field, but enhanced small-scale fluctuations relative to the hydrodynamic case are observed. These enhanced fluctuations coincide with increased energy dissipation and faster layer broadening, with these features more pronounced in simulations with stronger fields. These trends result from the magnetic field reducing the effects of stable modes relative to the transfer of energy to small scales. As field strength increases, stable modes become less excited, thus transporting less momentum against its gradient. Furthermore, the energy that would otherwise transfer back to the driving shear because of the stable modes is instead allowed to cascade to small scales, where it is lost to dissipation. Approximations of the turbulent state in terms of a reduced set of modes are explored. While the Reynolds stress is well-described using just two modes per wavenumber at large scales, the Maxwell stress is not.
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The authors thank K. Burns, D. Lecoanet, C. Sovinec, and J. Parker for insightful discussions, and N. Hurst for insightful discussions and a thorough reading of the manuscript. The authors also thank the Dedalus developers and the Dedalus user group for assistance with many aspects of installing and running Dedalus. Partial support for this work was provided by the National Science Foundation under Award No. PHY-1707236 and by the U.S. Department of Energy, Office of Science, Fusion Energy Sciences under Award No. DE-FG02-04ER54742. Computing resources were provided by the National Science Foundation through XSEDE computing resources, Allocation Nos. TG-PHY130027 and TG-PHY180047. This research was performed using the computer resources and assistance of the UW-Madison Center for High Throughput Computing (CHTC) in the Department of Computer Sciences. The CHTC is supported by UW-Madison, the Advanced Computing Initiative, the Wisconsin Alumni Research Foundation, the Wisconsin Institutes for Discovery, and the National Science Foundation, and is an active member of the Open Science Grid, which is supported by the National Science Foundation and the U.S. Department of Energy’s Office of Science.