Strato-rotational instability without resonance

Abstract

Strato-rotational instability (SRI) is normally interpreted as the resonant interactions between normal modes of the internal or Kelvin variety in three-dimensional settings in which the stratification and rotation are orthogonal to both the background flow and its shear. Using a combination of asymptotic analysis and numerical solution of the linear eigenvalue problem for plane Couette flow, it is shown that such resonant interactions can be destroyed by certain singular critical levels. These levels are not classical critical levels, where the phase speed $c$ of a normal mode matches the mean flow speed $U$, but are a different type of singularity where $(c-U)$ matches a characteristic gravity-wave speed $\pm N/k$, based on the buoyancy frequency $N$ and streamwise horizontal wavenumber $k$. Instead, it is shown that a variant of SRI can occur due to the coupling of a Kelvin or internal wave to such ‘baroclinic’ critical levels. Two characteristic situations are identified and explored, and the conservation law for pseudo-momentum is used to rationalize the physical mechanism of instability. The critical level coupling removes the requirement for resonance near specific wavenumbers $k$, resulting in an extensive continuous band of unstable modes.

Inertio–elastic instability of a vortex column

“…Finally, the eigenmode of the usual stratorotational instability does not become singular in the limit of zero growth rate (Yavneh et al 2001;Vanneste & Yavneh 2007). The present scenario thus more closely resembles the recently studied case where the stratorotational instability results from a resonant interaction between a Kelvin/inertia-gravity wave with a baroclinic critical level (Wang & Balmforth 2018).…”

Nonlinear dynamics of forced baroclinic critical layers II

“…The disturbances are assumed to decay for and satisfy periodic boundary conditions in and ; in practice, we assume the same periodicity as the forcing. As in Wang & Balmforth (2020), we assume that the flow is linearly stable: we take to eliminate the centrifugal instability (Emanuel 1994); the lack of any reflective boundaries removes the possibility of the strato-rotational instability (Vanneste & Yavneh 2007; Wang & Balmforth 2018).…”

“…In inviscid linear theory, waves propagating through stratified, horizontally directed and sheared flow encounter singularities at a novel type of critical level. These ‘baroclinic’ critical levels arise where the phase speed relative to the basic flow matches a characteristic gravity wavespeed (Olbers 1981; Basovich & Tsimring 1984; Badulin, Shrira & Tsimring 1985; Staquet & Huerre 2002; Boulanger, Meunier & Le Dizès 2008; Wang & Balmforth 2018, 2020). Much like the classical critical level, the singularity must be resolved by weak viscosity, unsteadiness or nonlinearity.…”

Nonlinear dynamics of forced baroclinic critical layers

“…Because our interest lies in the forcing of the baroclinic critical layers of an internal wave, we consider basic flows that are linearly stable to prevent unstable modes from dominating the dynamics. Centrifugal instabilities arise when 0 < f < 1 (Emanuel 1994), so we set f > 1 or f < 0 to eliminate them; strato-rotational instability is not present because it requires reflective boundaries (Yavneh, McWilliams & Molemaker 2001;Wang & Balmforth 2018) which are absent here. Initially, there is no disturbance, implying u = v = w = ρ = p = 0 at t = 0.…”

“…The current configuration implies that waves are generated at y = 0 and develop with baroclinic critical levels to either side (although simplifications are afforded by the symmetry described presently). Had we placed the wavemaker along a boundary at y = 0, only one critical level would have featured, but the wall may also make the basic flow linearly unstable (Wang & Balmforth 2018). Other idealizations include wavemakers that gradually switch on (Béland 1976), that generates disturbances with finite phase speed (displacing the baroclinic critical levels), or that with finite thickness (as for the vortices of Marcus et al).…”