The impact of magnetic fields on momentum transport and saturation of shear-flow instability by stable modes

Abstract

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.

Mechanism for Sequestering Magnetic Energy at Large Scales in Shear-Flow Turbulence

“…The process that counteracts the small scale generation of magnetic energy, which is quantified in this Letter, can be traced to the nonlinear excitation of large-scale stable linear eigenmodes [12]. The dispersion relation that is associated with the shear-flow instability present in this study has a stable (damped) root which acts to return energy and momentum to the mean profile, and which is excited to a high level by the nonlinearity [13,14]. The nonlinear excitation of stable modes and their effect on turbulence levels and transport has been studied previously, particularly in the context of fusion-relevant gyroradius-scale turbulence [12,[15][16][17][18][19][20][21][22][23].…”

“…The nonlinear excitation of stable modes and their effect on turbulence levels and transport has been studied previously, particularly in the context of fusion-relevant gyroradius-scale turbulence [12,[15][16][17][18][19][20][21][22][23]. Recent studies have also shown that stable modes are excited in macro-scopic shear flow driven turbulence and transiently affect momentum transport [13,14]. However, critical features of counteracting straining motion of the stable modes, essential for large-scale sequestering of magnetic energy, have only come to light in the study described here.…”

“…However, we show here that the reversible process by which stable modes return energy and momentum to the mean flow, in opposition to extraction by the instability, is quite effective at blunting the energy cascade to small scales. Stable modes have been connected to the shear-layer contraction (i.e., build up of mean flow) [14], but only as a transient process. Here we demonstrate that for a driven flow profile, the return of momentum to the mean profile occurs continuously at a rate that nearly matches the rate at which unstable modes attempt to flatten the flow profile.…”

“…(1a), to ensure an initial equilibrium state. We add small-amplitude perturbations to the flow and magnetic field [14,25] and perform the time-integration of the above set of equations using the pseudospectral code Dedalus [28] for long times (> 1000; the instability's e-folding growth time is ∼ 6). Apart from the simulation of Pm = 10 that uses spectral resolution of 4096×4096 for the box size of (L x , L z ) = (6π, 8π), all other simulations are performed at 2048 × 2048 spectral modes/resolution for the box size of (L x , L z ) = (10π, 20π), while convergence checks utilize resolutions as high as 8192 × 8192, with no substantial impact on dissipation rates and spectral energy densities.…”

“…We also perform a corresponding eigenvalue calculation of the non-dissipative equations, derived from Eqs. (1a) & (1b), by linearizing about the initial flow and magnetic field profiles [14]. A priori, one does not know whether such an eigenmode basis is useful to understand turbulent features.…”