Abstract
A Boussinesq fluid inside a stably thermally stratified square container whose walls are inclined ${45}^{\circ }$ with respect to gravity, with two opposite walls kept at constant temperatures and the other two insulated is nearly isothermal in the regions above and below the horizontal diagonal. The flow is concentrated in the wall boundary layers and a shear layer centred about the horizontal diagonal. The equilibrium is maintained by the balance between dissipation in the shear and boundary layers, the heat fluxes at the constant temperature walls, and the induced flow resulting from the no-flux condition at the inclined insulated walls. The dynamical response of the fluid to vertical oscillations of the container is studied over a range of forcing frequencies. For a small forcing amplitude and below a viscosity-dependent cutoff forcing frequency, this response exhibits a modal cellular structure localized about the shear layer. With increasing forcing amplitude, the response experiences instabilities, studied here numerically at a forcing frequency above the cutoff frequency, that are similar to those encountered in the Faraday wave problem, such as parametric subharmonic instability, triadic resonance and resonant collapse.
Highlights
Due to their ubiquitous presence in a wide range of geophysical flow phenomena, there is considerable interest in the instabilities of sheared density interfaces (Thorpe 1987; Fernando 1991; Ivey, Winters & Koseff 2008; Caulfield 2020, 2021)
Much insight has been gained from studying three idealized settings (Sutherland 2010): when the density interface and the velocity shear layer coincide, Kelvin–Helmholtz instability leads to the interface rolling up into billows; when they do not coincide, Holmboe instability ensues with waves travelling in opposite directions either side of the density interface; and when there are two density interfaces subjected to shear, Taylor instability results in billows of a more complicated nature
Lopez are problematic in that the initial density and shear profiles are not equilibrium states. This leads to compromises: the characteristic time scales for the evolution of the initial profiles and of their instabilities need to be carefully tuned in experiments, while linear stability analysis usually is based on the assumption that the so-called basic state is quasisteady, evolving on a slower diffusive time scale (Thorpe 1971; Parker, Caulfield & Kerswell 2020)
Summary
Due to their ubiquitous presence in a wide range of geophysical flow phenomena, there is considerable interest in the instabilities of sheared density interfaces (Thorpe 1987; Fernando 1991; Ivey, Winters & Koseff 2008; Caulfield 2020, 2021). The Θ response at RN = 105 and ω = 0.78 (shown at maximal phase) includes signatures of both the ω = 0.76 and ω = 0.80 peak responses at RN = 105.5, with a strong horizontal response emanating from the horizontal corners, together with bicorne-shaped cells in the centre of the cavity (see supplementary movie 2 for animations over one forcing period). Such splitting of peaks in response diagrams is expected as viscous effects are reduced and the separation between different modal responses increases
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
