Julian Mak

Interacting vorticity waves as an instability mechanism for magnetohydrodynamic shear instabilities

The interacting vorticity wave formalism for shear flow instabilities is extended here to the magnetohydrodynamic (MHD) setting, to provide a mechanistic description for the stabilising and destabilising of 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; 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.

Linear stability analysis for plane-Poiseuille flow of an elastoviscoplastic fluid with internal microstructure for large Reynolds numbers

Abstract We study the linear stability of Plane Poiseuille flow of an elastoviscoplastic fluid using a revised version of the model proposed by Putz and Burghelea (A.M.V. Putz, T.I. Burghelea, Rheol. Acta 48 (2009) 673–689). The evolution of the microstructure upon a gradual increase of the external forcing is governed by a structural variable (the concentration of solid material elements) which decays smoothly from unity to zero as the stresses are gradually increased beyond the yield point. Stability results are in close conformity with the ones of a pseudo-plastic fluid. Destabilizing effects are related to the presence of an intermediate transition zone where elastic solid elements coexist with fluid elements. This region brings an elastic contribution which does modify the stability of the flow.

Stratified shear flow instabilities in the non-Boussinesq regime

Effects of the baroclinic torque on wave propagation normally neglected under the Boussinesq approximation is investigated here, with a special focus on the associated consequences for the mechanistic interpretation of shear instability arising from the interaction between a pair of vorticity-propagating waves. To illustrate and elucidate the physical effects that modify wave propagation, we consider three examples of increasing complexity: wave propagation supported by a uniform background flow; wave propagation supported on a piecewise-linear basic state possessing one jump; and an instability problem of a piecewise-linear basic state possessing two jumps, which supports the possibility of shear instability. We find that the non-Boussinesq effects introduces a preference for the direction of wave propagation that depends on the sign of the shear in the region where waves are supported. This in turn affects phase-locking of waves that is crucial for the mechanistic interpretation for shear instability, and is seen here to have an inherent tendency for stabilisation.

Shear flow instabilities in shallow-water magnetohydrodynamics

Within the framework of shallow-water magnetohydrodynamics, we investigate the linear instability of horizontal shear flows, influenced by an aligned magnetic field and stratification. Various classical instability results, such as H{\o}ilands growth rate bound and Howards semi-circle theorem, are extended to this shallow-water system for quite general profiles. Two specific piecewise-constant velocity profiles, the vortex sheet and the rectangular jet, are studied analytically and asymptotically; it is found that the magnetic field and stratification (as measured by the Froude number) are generally both stabilising, but weak instabilities can be found at arbitrarily large Froude number. Numerical solutions are computed for corresponding smooth velocity profiles, the hyperbolic-tangent shear layer and the Bickley jet, for a uniform background field. A generalisation of the long-wave asymptotic analysis of Drazin & Howard (1962) is employed in order to understand the instability characteristics for both profiles. For the shear layer, the mechanism underlying the primary instability is interpreted in terms of counter-propagating Rossby waves, thereby allowing an explication of the stabilising effects of the magnetic field and stratification.

Magnetohydrodynamic shear instabilities arising from interacting vorticity waves

The presence of a backgroundmagnetic field is known to have a stabilising as well as a destabilising effect for shear flow instabilities. To explain the reason for this, we extend the Counter-propagating Rossby Waves mechanism, well known in the geophysical fluid dynamics community, to the magnetohydrodynamicsetting. It is demonstrated here that wave displacement leads to a magnetic field configuration that results in an appropriate vorticity distribution, and, via the non-local velocity field generated by the local vorticity anomalies, action-at-a-distance results in a constructive interaction between two waves and leads to shear instability. The existence of shear instability then rests upon whether the chosen basic state supports such a configuration required for constructive interference.