On The Richardson Number Dependence of Nonlinear Critical-Layer Flow over Localized Topography

Abstract

The dynamics of stratified shear flow over localized topography in two spatial dimensions is investigated through analysis of a sequence of nonlinear numerical simulations. The existence of a critical level for all stationary waves is ensured by employing a hyperbolic tangent profile of horizontal wind such that the shear is localized at some elevation ?c and the flow is asymptotic to a speed ?U0 above and +U0 below this level. Previous studies of this problem have investigated the dependence of the quasi?steady-state solution upon the critical-level elevation ?c and on the strength of the topographic forcing. In the present study these analyses are extended to include an investigation of the dependence of the solution on the third and final governing parameter, namely the minimum gradient Richardson number R?m. Contrary to previous implicit assumption, we find that the temporal evolution of the flow towards the high-drag state can be strongly influenced by the value of R?m, to a degree that depends upon the strength of the topographic forcing. The sense of this parameter dependence is such that, as the value of R?m is decreased (with R?m > 0.25 always), the transition to the severe downslope windstorm state is progressively delayed and for moderate topographic forcing (characterized by an inverse Froude number of 0.5) may possibly be prevented. Furthermore, we find that when this dependence is taken into account, there appears no evidence that would support the occurrence of a ?resonance shift? (i.e., the notion that the center of the critical layer must be shifted to lower elevation as the inverse Froude number is decreased in order to obtain maximal drag for the steady-state solution), which has been suggested in several previous analyses of this problem.