Distinct Modes Of Instability Research Articles

Thermally driven hydromagnetic convection in a rapidly rotating sphere

Buoyancy-induced convection is investigated in a rapidly rotating, self-gravitating fluid sphere internally heated by a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance ϖ * from the rotation axis. Atten­tion is restricted to relatively small magnetic field strengths (as measured by the parameter ∧ ) such that the dominant force balance remains geo-strophic. Convection is then confined to a thin cylindrical annulus, radius ϖ 0 *, about the rotation axis. A linear analysis is used to find the state of marginal stability together with the corresponding minimum critical value of the modified Rayleigh number, R c (a measure of the buoyancy force required to maintain convective motions). Two distinct modes of instability are found to operate: ‘Rossby’ and ‘magnetic’. When no magnetic field is applied ( ∧ = 0), and when the Prandtl number σ ≪ 1, the instability takes the form of a thermally driven Rossby wave propagating eastward but with group velocity westward. Modifications to this mode due to a non-zero magnetic field result in R c being a complicated function R c ( ∧, q, σ ), where q = k/η and σ = v/k with v , k and η denoting the viscous, thermal and magnetic diffusivities. When it has a significant effect [ σ ≤ O (1)], the magnetic field inhibits the Rossby mode and convection moves towards the axis where the field is weaker ( ϖ 0 *→ 0). The presence of the magnetic field also permits new modes of instability which are facilitated by increasing field strength, with R c ∝ ∧ ─1 . These magnetic modes propagate eastwards (westwards) when q < q m ( q > q m ) with q m ≈ 2.8. Of particular interest to the Earth is the limit q ≪ 1 where the marginally stable magnetic mode takes the form of an undamped buoyancy wave in an unstably stratified fluid. No diffusive processes are operating at leading order and the only effect opposing convection is the geostrophic constraint.

Some Characteristics of the Kelvin-Helmholtz and Resonant Overreflection Modes of Shear Flow Instability and of Their Interaction through Vortex Pairing

A stably stratified parallel shear flow near the ground is able to support two distinct modes of instability, these being the well known Kelvin-Helmholtz disturbance and a secondary wave whose growth in time derives from the process of critical level overreflection. Here we first consider the two modes of instability in circumstances in which they exist in isolation from one another and present some new results concerning their finite-amplitude characteristics. In particular, we comment on the likely mechanism of secondary instability which marks the transition to turbulence in a Kelvin-Helmholtz billow. Further, simulations of the finite-amplitude evolution of an isolated resonant mode shows that nonlinearity, of itself, is not able to compensate for a small linear growth rate in making the resonant mode a prominent feature of the flow. We suggest that such may be accomplished, however, by the interaction between the two modes through the process of vortex pairing since the first subharmonic of the fastest growing Kelvin-Helmholtz mode may be a resonant mode. We provide a simple, weakly nonlinear calculation which supports this notion.