Flow Regimes and Transient Dynamics of Two-Dimensional Stratified Flow over an Isolated Mountain Ridge

Abstract

Four regimes are identified for two-dimensional, unstructured, nonrotating, continuously stratified, hydrostatic, uniform Boussinesq flow over an isolated mountain ridge: (I) flow with neither wave breaking aloft nor upstream blocking (F≥1.12, where F = U/ NH; U and N are upstream basic flow speed and Brunt-Väisälä frequency, respectively; and h is the mountain height), (II) flow with wave breaking aloft in the absence of upstream blocking (0.9 < F≤1. 12), (III) flow with both wave breaking and upstream blocking, but where wave breaking occurs first (0.6≤F≤0.9), and (IV) flow with both wave breaking and upstream blocking, but where blocking occurs first (0.3≤F≤0.6). In regime I, neither wave breaking nor upstream blocking occurs, but columnar disturbance does exist. The basic flow structure resembles either linear or weakly nonlinear mountain waves. It is found that the columnar disturbance is independent of the wave breaking aloft. In regime II, an internal jump forms at the downstream edge of the wave-breaking region, propagates downstream, and then becomes quasi-stationary. The region of wave breaking also extends downward toward the lee slope. After the internal jump travels farther downstream, a stationary mountain wave becomes established in the vicinity of the mountain above the dividing streamline, which is induced by wave breaking. A high-drag state is predicted in this flow regime. In addition, a vertically propagating hydrostatic gravity wave is generated by the propagating jump and travels with it. Along the lee slope, a strong downslope wind develops. Static and Kelvin-Helmholtz instabilities may occur locally in the region of wave breaking. The critical F for wave breaking is about 1.12, which agrees well with the value 1.18 found by Miles and Huppert. This study also found that the flow responses in this flow regime, as well as in the other regimes, are similar for constant F. In regime III, the downstream internal jump propagates downstream in the early stage, retrogresses in the direction against the basic flow once blocking occurs, and then becomes quasi-stationary. The retrogression of the downstream jump may be caused by the modification of the upstream boundary conditions. The layer depth of blocked fluid is independent of F and h/a, where a is the mountain half-width. In regime IV, the internal jump quickly becomes stationary over the lee slope once it forms. It is found that the presence of wave breaking aloft is not necessary for upstream blocking to occur. A vertically propagating gravity wave is generated by the upstream reversed flow and travels with it. The speed of the upstream reversed flow is proportional to h/a. The surface drag increases abruptly from regime I to II, while it decreases gradually from regime II (III) to III (IV). The surface drag is a function of h/a and is a minimum for h/a=0.05 for constant F. The average dividing streamline height generated by wave breaking is roughly 0.85?z (HdN/U=5.34), and the level at which overturning initially occurs is found to be about 4.4 in the high-drag state, where ?z and Hd, are the dimensional hydrostatic vertical wavelength and dividing streamline height, respectively. This indicates that the initial wave overturning occurs at the level of the largest gradient of streamline deflection. It is found that nonlinearity tends to accelerate the upslope flow, decelerate the flow near the mountain peak and top of the leeslope, and accelerate the flow near the internal jump.