Speaker-wire vortices in stratified anabatic Prandtl slope flows and their secondary instabilities

Abstract

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.

Investigation of oscillations in katabatic Prandtl slope flows

“…Figure 6a shows the contour plot of the normalised $$$ u $$$‐velocity component field for this case, which exhibits continuous and straight longitudinal rolls that are parallel to the along‐slope direction. Such smooth rolls, which are initially stationary, start to meander when the $$$ {\Pi}_{\mathrm{s}} $$$ value is increased and without requiring any additional perturbation due to a secondary roll instability as described in Xiao and Senocak (2022). Slightly increasing the stratification perturbation parameter to $$$ {\Pi}_{\mathrm{s}}=13 $$$, Figure 6b reveals how the instability rolls, while remaining aligned with the along‐slope direction, are now curved and create branches after colliding with each other as a result of secondary instabilities of vortical rolls.…”

“…Note that no length or velocity scales are applied externally in the prototype slope flow scenario and hence no such scales appear in the dimensionless parameters governing this type of flow. Most recently, Xiao and Senocak (2022) showed that $$$ {\Pi}_{\mathrm{s}} $$$ plays a nonlinear role in flows over flat terrain stratified by the combined action of surface cooling and a constant ambient stratification as well.…”

“…The region beneath both critical curves represents the laminar Prandtl flow regime, where the flow is one-dimensional without developing any instabilities. The dashed line represents a transition region where the longitudinal rolls start meandering [Colour figure can be viewed at wileyonlinelibrary.com] is increased and without requiring any additional perturbation due to a secondary roll instability as described in Xiao and Senocak (2022). Slightly increasing the stratification perturbation parameter to Π s = 13, Figure 6b reveals how the instability rolls, while remaining aligned with the along-slope direction, are now curved and create branches after colliding with each other as a result of secondary instabilities of vortical rolls.…”