The structure of low-Froude-number lee waves over an isolated obstacle

Abstract

We present new insight into the classical problem of a uniform flow, linearly stratified in density, past an isolated three-dimensional obstacle. We demonstrate how, for a low-Froude-number obstacle, simple linear theory with a linearized boundary condition is capable of providing excellent quantitative agreement with laboratory measurements of the perturbation to the density field. It has long been known that such a flow may be divided into two regions, an essentially horizontal flow around the base of the obstacle and a wave-generating flow over the top of the obstacle, but until now the experimental diagnostics have not been available to test quantitatively the predicted features. We show that recognition of a small slope that develops across the obstacle in the surface separating these two regions is vital to rationalize experimental measurements with theoretical predictions. Utilizing the principle of stationary phase and causality arguments to modify the relationship between wavenumbers in the lee waves, linearized theory provides a detailed match in both the wave amplitude and structure to our experimental observations. Our results demonstrate that the structure of the lee waves is extremely sensitive to departures from horizontal flow, a detail that is likely to be important for a broad range of geophysical manifestations of these waves.

The internal wavefield generated by a towed sphere at low Froude number

“…Semi-empirical relationships for the Froude number dependence of lee wave drag have been derived by Greenslade (2000), Voisin (2007) and Dalziel et al (2011) for both the wave and wake contributions. For the low-Fr limit, Fr → 0, the corresponding drag coefficients are C wave D ∝ Fr 3/2 and C wake D ∝ Fr 1/2 .…”

“…Also relevant are the related studies of lee wave flows over topographic obstacles (i.e. objects mounted on a fixed surface) (Castro, Snyder & Baines 1990;Vosper et al 1999;Dupont et al 2001;Dalziel et al 2011). Although many of these studies are at low Re, they exhibit the same wavefield features as objects towed in an unbounded flow.…”

Observations on stratified flow over a bank at low Froude numbers

“…Past observational and modeling efforts in the ocean have often been focused on studying energetic flows with Froude numbers approaching 1 or larger, i.e., critical and supercritical flows over rough topography with energy being released through, for instance, a hydraulic jump [e.g., Wesson and Gregg , ; Farmer and Armi , ; Hogg et al ., ; Moum and Nash , ]. Much of our understanding of flow dynamics for very low Froude numbers (much less than 1) comes mostly from the atmospheric research of subcritical airflows over a mountain [e.g., Hunt and Snyder , ; Smolarkiewicz and Rotunno , ; Smith and Grønås , ; Vosper et al ., ; Dalziel et al ., ]. These airflows are usually characterized by blocking, splitting, and moving around on the upstream side of the obstacle.…”

Energetics of Seamount Wakes. Part II: Wave Fluxes

“…The classical view that the flow is confined to horizontal planes below the cap appears to be violated by the observed isopycnal displacement, particularly when the Rossby number is below O(0.1) (Brighton 1978;Dalziel et al 2011). The isopycnal deflections raise interesting questions about flow patterns around 3D topography in a rotating, stratified regime.…”

“…The height of the cap h c corresponds to the vertical length scale for which Fr 5 1. There is evidence that the plane of separation that delineates the bifurcation between flow over and flow going around is tilted slightly down toward the lee side (Brighton 1978;Dalziel et al 2011), but this is a higher-order effect with respect to the cap height. The flow over the cap is associated with the generation of internal lee waves (Voisin 2007).…”