The Use of an Effective Froude Number in Non-Uniform Flow

We use a three-dimensional numerical model to investigate the utility of the inverse ``Froude number'', ε = Nh/U (where N is the buoyancy frequency, U is the cross mountain wind speed, and h is the mountain height), in determining the flow response, a-priori, when an upstream inversion below the mountain top is present.

Our simulations are initialized with an upstream inversion at three levels below the mountain top and a control upstream condition where no inversion is present. We test two ways to estimate the stability of the upstream conditions; an average, where N is averaged below the top of the inversion, and a bulk estimate, where we calculate the total change of the potential temperature below the top of the inversion and use this to estimate N. We use our estimated value of N to determine an ``effective'' inverse Froude number, ε_e. We span a range of ε_e and mountain shapes, as defined by the ratio of the cross-stream to along-stream length scales, β, to map situations with upstream inversions into the parameter space ε_e × β. In particular we investigate where the threshold of upstream blocking, wave breaking, and lee vortex formation maps into this parameter space.

Our results suggest it is better to use a bulk estimate of N rather than an average of N when constructing ε_e, but even then, there are substantial differences in the threshold condition for upstream blocking, wave breaking, and lee vortex formation between layered atmospheres and a uniformly stratified flow with ε = ε_e.