In the Prandtl model for anabatic slope flows, a uniform positive buoyancy flux at the surface drives an upslope flow against a stable background stratification. In the present study, we conduct linear stability analysis of the anabatic slope flow under this model and contrast it against the katabatic case as presented in Xiao & Senocak ( J. Fluid Mech. , vol. 865, 2019, R2). We show that the buoyancy component normal to the sloped surface is responsible for the emergence of stationary longitudinal rolls, whereas a generalised Kelvin–Helmholtz (KH) type of mechanism consisting of shear instability modulated by buoyancy results in a streamwise-travelling mode. In the anabatic case, for slope angles larger than $9^{\circ }$ to the horizontal, the travelling KH mode is dominant whereas, at lower inclination angles, the formation of the stationary vortex instability is favoured. The same dynamics holds qualitatively for the katabatic case, but the mode transition appears at slope angles of approximately $62^{\circ }$ . For a fixed slope angle and Prandtl number, we demonstrate through asymptotic analysis of linear growth rates that it is possible to devise a classification scheme that demarcates the stability of Prandtl slope flows into distinct regimes based on the dimensionless stratification perturbation number. We verify the existence of the instability modes with the help of direct numerical simulations, and observe close agreements between simulation results and predictions of linear analysis. For slope angle values in the vicinity of the junction point in the instability map, both longitudinal rolls and travelling waves coexist simultaneously and form complex flow structures.
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Speaker-wire vortices in stratified anabatic Prandtl slope flows and their secondary instabilities
Stationary longitudinal vortical rolls emerge in katabatic and anabatic Prandtl slope flows at shallow slopes as a result of an instability when the imposed surface buoyancy flux relative to the background stratification is sufficiently large. Here, we identify the self-pairing of these longitudinal rolls as a unique flow structure. The topology of the counter-rotating vortex pair bears a striking resemblance to speaker-wires and their interaction with each other is a precursor to further destabilization and breakdown of the flow field into smaller structures. On its own, a speaker-wire vortex retains its unique topology without any vortex reconnection or breakup. For a fixed slope angle $\alpha =3^{\circ }$ and at a constant Prandtl number, we analyse the saturated state of speaker-wire vortices and perform a bi-global linear stability analysis based on their stationary state. We establish the existence of both fundamental and subharmonic secondary instabilities depending on the circulation and transverse wavelength of the base state of speaker-wire vortices. The dominance of subharmonic modes relative to the fundamental mode helps to explain the relative stability of a single vortex pair compared to the vortex dynamics in the presence of two or an even number of pairs. These instability modes are essential for the bending and merging of multiple speaker-wire vortices, which break up and lead to more dynamically unstable states, eventually paving the way for transition towards turbulence. This process is demonstrated via three-dimensional flow simulations with which we are able to track the nonlinear temporal evolution of these instabilities.
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- Award ID(s):
- 1936445
- NSF-PAR ID:
- 10355588
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 944
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2022.508
