Linear stability of katabatic Prandtl slope flows with ambient wind forcing

Abstract

We investigate the stability of katabatic slope flows over an infinitely wide and uniformly cooled planar surface subject to an additional forcing due to a uniform downslope wind field aloft. We adopt an extension of Prandtl's original model for slope flows (Lykosov & Gutman 1972) to derive the base flow, which constitutes an interesting basic state in stability analysis because it cannot be reduced to a single universal form independent of external parameters. We apply a linear modal analysis to this basic state to demonstrate that for a fixed Prandtl number and slope angle, two independent dimensionless parameters are sufficient to describe the flow stability. One of these parameters is the stratification perturbation number that we have introduced in Xiao & Senocak (2019). The second parameter, which we will henceforth designate the wind forcing number, is hitherto uncharted and can be interpreted as the ratio of the kinetic energy of the ambient wind aloft to the damping due to viscosity and stabilizing effect of the background stratification. For a fixed Prandtl number, stationary transverse and travelling longitudinal modes of instabilities can emerge, depending on the value of the slope angle and the aforementioned dimensionless numbers. The influence of ambient wind forcing on the base flow's stability is complicated as the ambient wind can be both stabilizing as well as destabilizing for a certain range of the parameters. Our results constitute a strong counter-evidence against the current practice of relying solely on the gradient Richardson number to describe the dynamic stability of stratified atmospheric slope flows.

Investigation of oscillations in katabatic Prandtl slope flows

“…Numerical simulations are performed by solving the incompressible form of the buoyancy‐driven Navier–Stokes equations under the Boussinesq approximation. Specifically, we use a second‐order accurate finite‐difference solver Jacobsen and Senocak (2013) that was also adopted in the works of Xiao and Senocak (2019, 2020a, 2020b); Senocak and Xiao (2020). We apply the same grid resolution and boundary conditions as used by Xiao and Senocak to investigate the instabilities of the primary modes further for multiple slope flow configurations.…”

“…Local linear stability analysis has been used to study the propagation of infinitesimal disturbances to Prandtl's laminar solution for slope flows. Such studies reported in the literature have revealed the existance of two distinct types of primary instabilities for katabatic as well as anabatic Prandtl slope flows, with Xiao and Senocak (2020a) and without ambient winds aloft Senocak (2019, 2020b); Senocak and Xiao (2020). These modes of instability are denoted as transverse and longitudinal.…”

“…Additional insight based on Prandtl's slope flow solution has been obtained through linear stability analysis applied to the simple laminar solution to assess the growth of instabilities in the flow from infinitesimal disturbances. This technique has been applied to slope flows under katabatic conditions with both quiescent conditions aloft Xiao and Senocak (2019) and ambient winds Xiao and Senocak (2020a), and anabatic conditions Xiao and Senocak (2020b). These contributions have shown the existence of organized vortical structures in the unstable laminar regime and provide instability maps that predict the emergence of three types of instability, namely stationary longitudinal rolls aligned along the slope direction, called the transverse mode; traveling-wave rolls aligned along the cross-slope direction that move along the slope direction, called the longitudinal mode; and a higher-order mixed-mode regime, where both longitudinal and traveling-wave rolls are interwoven and coexist in the flow.…”

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

“…In recent studies, we have investigated the linear stability of the Prandtl model for katabatic (Xiao & Senocak 2019) and anabatic (Xiao & Senocak 2020 b ) slope flows, as well as the extended version of the Prandtl model with ambient winds (Xiao & Senocak 2020 a ), and discovered the existence of a stationary roll instability at small slope angles, and a travelling wave instability at steeper slopes. These studies have also helped to establish the importance of the dimensionless stratification perturbation parameter in controlling the dynamics of stably stratified slope flows.…”