Stability of the Prandtl model for katabatic slope flows

Abstract

We investigate the stability of the Prandtl model for katabatic slope flows using both linear stability theory and direct numerical simulations. Starting from Prandtl’s analytical solution for uniformly cooled laminar slope flows, we use linear stability theory to identify the onset of instability and features of the most unstable modes. Our results show that the Prandtl model for parallel katabatic slope flows is prone to transverse and longitudinal modes of instability. The transverse mode of instability manifests itself as stationary vortical flow structures aligned in the along-slope direction, whereas the longitudinal mode of instability emerges as waves propagating in the base-flow direction. Beyond the stability limits, these two modes of instability coexist and form a complex flow structure crisscrossing the plane of flow. The emergence of a particular form of these instabilities depends strongly on three dimensionless parameters, which are the slope angle, the Prandtl number and a newly introduced stratification perturbation parameter, which is proportional to the relative importance of the disturbance to the background stratification due to the imposed surface buoyancy flux. We demonstrate that when this parameter is sufficiently large, then the stabilising effect of the background stratification can be overcome. For shallow slopes, the transverse mode of instability emerges despite meeting the Miles–Howard stability criterion of $Ri>0.25$. At steep slope angles, slope flow can remain linearly stable despite attaining Richardson numbers as low as $3\times 10^{-3}$.

Linear stability of katabatic Prandtl slope flows with ambient wind forcing

“…A plot of the critical Π s required for the onset of each instability mode at a specific slope angle α and wind forcing number Π w is shown in figure 4. The effect of the ambient wind on the transition slope angle α t at which the dominant instability mode switches from the transverse to longitudinal mode can be clearly observed: Due to the stabilizing effect of increasing ambient wind forcing on the longitudinal mode as discussed previously, for wind forcing number Π w sufficiently large, α t increases beyond the value of 62 • found by Xiao & Senocak (2019) in the absence of ambient wind Π w = 0 . The monotonic destabilizing effect of growing Π w on the transverse mode, i.e.…”

“…This means that the instability growth rate as a function of the wave vectors k x , k y attains its maximum on either the k x or the k y axis, with the other wave vector being zero. It turns out that the same also holds true for katabatic flows in the presence of ambient winds, hence the growth rate contours for disturbances in the wave vector space, looking qualitatively similar like those in Xiao & Senocak (2019), will not be shown here.…”

“…From the plots shown in figure 2, we observe that increasing the ambient wind tend to lower an instability's growth rate at the same Ri g number, i.e. the most unstable mode at fixed Ri g is found at u ∞ = 0. which is the original Prandtl model as analysed in Xiao & Senocak (2019). At the low slope angle of α = 4 • , it can be observed that for the wind forcing number Π w < 3, the base flow can be unstable despite possessing a larger Ri g number than critical value Ri g = 0.25.…”

“…The new dimensionless number in the above set is Π w . Π s was introduced in Xiao & Senocak (2019). We designate Π w the wind forcing number interpret it as the ratio of the kinetic energy in the ambient wind to the kinetic energy damping in the flow due to viscosity and stabilizing effect of stratification.…”

“…In order to visualise the flow field at these conditions, the Navier-Stokes equations (2.1)-(2.2) for katabatic slope flows are solved using a Cartesian mesh, three-dimensional, bouyancy-driven incompressible flow solver (Jacobsen & Senocak 2013). The settings for the direct numerical simulations are the same as adopted in Xiao & Senocak (2019), i.e. the simulation domain is chosen to be large enough to capture multiple vortex rolls along both cross-slope and along-slope directions, and the mesh resolution ensures that there are at least two points per length scale l 0 in each direction.…”