Three-Dimensional Numerical Simulations of Strongly Stratified Flow past Conical Orography

Abstract

Results from a series of numerical simulations of three-dimensional stably stratified flows past conical orography with unit slope are presented and are compared directly with laboratory results from a stratified towing tank. The simulations are conducted with a finite-difference model (configured to simulate flows in the towing tank) based on the inviscid nonhydrostatic equations of motion in σ (normalized pressure) coordinates. A free-slip lower boundary condition is implemented. The flows studied have values of the Froude number, Fh = U/Nh, between 0.1 and 0.8, where U is the mean flow speed, N is the buoyancy frequency, and h is the mountain height. To excite unsteadiness in the simulations an asymmetric perturbation is applied to the initial potential temperature (?) field. The resulting variation of the temporally averaged drag coefficient with Fh is found to compare reasonably well with the laboratory measurements and there is a general trend for the drag coefficient to decrease as Fh increases. For many of the simulations the temporal evolution of the drag is highly unsteady: when Fh ? 0.3 the unsteadiness is quasiperiodic and is due to vortex shedding in the lee of the orography. The nondimensional vortex shedding frequency is similar to that measured in the laboratory. Simulations conducted without the initial ? perturbation do not exhibit vortex shedding and in this case the drag is significantly reduced. The vorticity generated in both the perturbed and unperturbed flows is shown to be largely perpendicular to the isentropic surfaces and hence potential vorticity anomalies are present. These anomalies appear early on in the simulations and are caused by internal dissipation within the vortices due to numerical viscosity. Simple tests in which the free-slip lower boundary condition is replaced with one containing a surface friction parametrization show that one of the main effects of friction is to suppress the vortex shedding. Further, the results indicate that in the laboratory an inviscid mechanism (in which vertical vorticity is generated via the tilting of baroclinically generated horizontal vorticity) dominates over the generation of vertical vorticity in the boundary layer. At Fh = 0.4 a local maximum occurs in the temporally averaged drag and this corresponds to the occurrence of wave breaking in the lee of the mountain. The wave-breaking process itself is highly unsteady and after continuous growth of the wave amplitude (and drag) convective instability eventually leads to a complete collapse of the overturning region and a significant fall in the drag. Further unsteadiness occurs at higher values of Fh when a rigid-lid upper-boundary condition is enforced: for 0.45 ? Fh ? 0.6 the evolution of the simulated drag is quasiperiodic and this is shown to be caused by the generation of an unsteady wave motion upstream. Comparison with existing linear theory indicates that this is due to the existence of a wave mode whose horizontal group velocity is small but has a nonzero frequency. The effects of blockage, due to the presence of the towing-tank side walls, are investigated by enforcing radiative boundary conditions at the spanwise lateral boundaries. As found in the towing-tank experiments, blockage effects are shown to significantly increase the drag coefficient and nondimensional shedding frequency at low Froude numbers (Fh ? 0.4) and also alter the Froude number at which wave breaking occurs.