Nonmodal growth and optimal perturbations in magnetohydrodynamic shear flows

Abstract

In astrophysical shear flows, the Kelvin-Helmholtz (KH) instability is generally suppressed by magnetic tension provided a sufficiently strong streamwise magnetic field. This is often used to infer upper (or lower) bounds on field strengths in systems where shear-driven fluctuations are (or are not) observed, on the basis that fluctuations cannot grow in the absence of linear instability. On the contrary, by calculating the maximum growth that small-amplitude perturbations can achieve in finite time for such a system, we show that perturbations can grow in energy by orders of magnitude even when the flow is sub-Alfvénic, suggesting that shear-driven turbulence is possible even in the presence of strong magnetic fields, and challenging inferences from the observed presence or absence of shear-driven fluctuations. We further show that magnetic fields introduce additional nonmodal growth mechanisms relative to the hydrodynamic case, and that 2D simulations miss key aspects of these growth mechanisms.

Figures (2)

• Figure 1: Linear optimal perturbations, i.e., perturbations attaining the largest growth over a finite time, for the "small box" domain. (Left) Details of the LOPs for several values of $\mathrm{M_A}$: (a) gains, (b) streamwise wavenumbers, and (c) spanwise wavenumbers. The points in panel (a) correspond to the cases shown in the middle column. (Middle) Spatial structure of $u$, $w$, $b_x$, and $b_z$ of the LOPs in the $xz$-plane. Panels (d)--(g) correspond to LOPs for a short target time $t_0=2$, while (h)--(k) correspond to an intermediate target time $t_0=20$. $\mathrm{M_A}=0.5$ for both. (Right) Evolution of (l) kinetic and (n) magnetic energy components, and corresponding production terms (m), (o), for the LOPs highlighted in the middle column.
• Figure 2: Linear optimal perturbations for the "large box" domain. (a) LOP gains for different $\mathrm{M_A}$ and a range of target times $t_0$; in the inset, large box calculations (solid) are compared against small box (dashed)---note all large-box curves overlap over this range. (b) Spanwise wavenumbers $k_{y, \mathrm{opt}}$ for the LOPs. (c) Fraction of energy in different components for $\mathrm{M_A}=0.25$ LOPs for different $t_0$. Initially, energy is mostly in $v$ (orange), $w$ (dark orange), $b_y$ (purple), and $b_z$ (dark purple). (d) As for (c), but for the evolved state. The energy is now mostly in $u$ (light orange) and $b_x$ (light purple). (e) Evolution of kinetic (orange), magnetic (purple), and total (black) energy for the $t_0=1000$ LOP (vertical black lines in (c-e), blue diamond in (a)). (f-h) As in panels (c-e), but for $\mathrm{M_A}=1$ and $t_0=7500$.