Interacting vorticity waves as an instability mechanism for magnetohydrodynamic shear instabilities

Abstract

The interacting vorticity wave formalism for shear flow instabilities is extended here to the magnetohydrodynamic (MHD) setting, to provide a mechanistic description for stabilising and destabilising shear instabilities by the presence of a background magnetic field. The interpretation relies on local vorticity anomalies inducing a non-local velocity field, resulting in action at a distance. It is shown here that the waves supported by the system are able to propagate vorticity via the Lorentz force, and waves may interact. The existence of instability then rests upon whether the choice of basic state allows for phase locking and constructive interference of the vorticity waves via mutual interaction. To substantiate this claim, we solve the instability problem of two representative basic states, one where a background magnetic field stabilises an unstable flow and the other where the field destabilises a stable flow, and perform relevant analyses to show how this mechanism operates in MHD.

Stratified shear flow instabilities in the non-Boussinesq regime

“…Unlike the previous formulations where transformation matrices were present 18,21 , the complication here is from the (p/ρ) 1,2 term in equation (25). This contributes a (∂/∂t + U ∂/∂x)u term, which means we can no longer write the problem in the form ∂ζ/∂t = Aζ in a simple way, and the transformation matrix acting on A becomes complicated.…”

“…This contributes a (∂/∂t + U ∂/∂x)u term, which means we can no longer write the problem in the form ∂ζ/∂t = Aζ in a simple way, and the transformation matrix acting on A becomes complicated. In principle, since everything is linear, an alternative approach that one could take is to work out how the individual terms in the governing equation (25) should look like, and the equations for ζ ± 1,2 should have on the right hand side the interaction terms written in terms of the appropriate contributions from the terms in (25). We may postulate for example that the equation for ζ + 1 say should only be affected by all variables not including ζ − 1 , i.e., the governing equations without the modal solution assumption should be of the form…”

“…Starting first with the phase difference, since we defined it using the displacement rather than vorticity 25 , it may be seen that −π < ∆ǫ < 0 is the unstable regime, and with −π/2 < ∆ǫ < 0, we are in the 'hindering' regime where the configuration is such that the counter-propagating waves hinder each other's propagation to achieve phase-locking, typical of fast waves (see also Figure 8 here). We see in panels (a, d, g, j) this occurs for waves at higher values of R and small k, which is consistent with the dispersion relation (27), where faster waves occurs for larger R and smaller k. The reverse is true when −π < ∆ǫ < −π/2, and we are in the 'helping' regime.…”

On the opposing roles of the Boussinesq and non‐Boussinesq baroclinic torques in surface gravity wave propagation

“…Furthermore, from Equation we obtain, , thus Equations –(16b) gives the familiar DW phase speed relation: corresponding to Hence, when the interface displacement and vorticity are in (anti)phase, the wave propagates to the (left) right (Figure ). Such behaviour is common with other types of interfacial vorticity waves such as capillary (Biancofiore et al ., ), Rossby (Hoskins et al ., ) and Alfven (Heifetz et al ., ) waves.…”

Normal form of synchronization and resonance between vorticity waves in shear flow instability

“…Similarly, a counter-propagating wave in region II is obtained when the wave's vorticity and displacement are in phase. Examples of such vorticity waves are Rossby [3], gravity [7], capillary [8] and Alfven [9] waves.…”