Critical-layer instability of shallow-water magnetohydrodynamic shear flows

Abstract

In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow, $c-U(y)$ , matches the Alfvén wave velocities $\pm B(y)/\sqrt {\mu \rho }$ , based on the local magnetic field $B(y)$ , the magnetic permeability $\mu$ , and the mass density of the fluid $\rho$ . It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at large wavenumbers and numerical solutions, and the mechanism of instability explained using the conservation of momentum. For the shallow-water magnetohydrodynamic system, the paper gives the general form of the local differential equation governing such coalescing critical layers for any generic field and flow profiles, and determines precisely how the magnetic field modifies the purely hydrodynamic stability criterion based on the potential vorticity gradient in the critical layer. The curvature of the magnetic field profile, or equivalently the electric current gradient $J' = - B''/\mu$ in the critical layer, is found to play a complementary role in the instability.

Two-dimensional ideal magnetohydrodynamic waves on a rotating sphere under a non-Malkus field: I. Continuous spectrum and its ray-theoretical interpretation

“…This situation is similar to that of equatorial Rossby waves in Taniguchi and Ishiwatari (2006), who studied eigenmodes in a linear shear flow on an equatorial β-plane. An alternative explanation may be their transformation into quasimodes (Spencer and Rasband, 1997;Schecter et al, 2000;Balmforth et al, 2001;Turner and Gilbert, 2007;Wang et al, 2022a), or the Landau damping in a broad sense. When embedded in a continuous spectrum due to the replacement of basic fields, a discrete real eigenvalue (on the principle Riemann sheet) may change into a complex pole on the next Riemann sheet (e.g.…”

“…Despite the fact that the eigenfrequencies of unstable modes are complex numbers, their eigenfunctions are affected by the positions of critical points, as demonstrated by, e.g. Gilman and Fox (1999a) and Wang et al (2022a). Moreover, a neutral mode sometimes interacts with continuous modes when the branch of the neutral one overlaps the continuous spectrum.…”

An analytical study of the MHD clamshell instability on a sphere

“…The asymptotic analysis clearly indicates that the critical layer plays a fundamental role in making the flow unstable. In our previous studies of instability induced by critical layers (Riedinger & Gilbert 2014; Wang & Balmforth 2018; Wang, Gilbert & Mason 2022), conservation of momentum provides a useful tool for understanding the mechanism of the instability. Indeed, it has been found that the critical layer provides a source of mean-flow momentum, which drives the exponential growth of the outer flow.…”

A systematic review of natural flood management modelling: Approaches, limitations, and potential solutions

“…Overall, lab‐based studies will be useful for validating fully 3D CFD models which, in turn, can be used to parameterise complex processes in 1D or 2D models, such as turbulent flows in ponds or around leaky barriers (Wang, 2017). Including these complex flow processes and other relevant factors, such as the effect of soil moisture content and proper representation of feature details, may help improve the simulation accuracy in large‐scale modelling.…”