Wave drag on a sphere in a strongly stratified flow
Résumé
Lee-wave drag, namely the wave drag on an obstacle fixed in a stratified flow, is an important feature of geophysical fluid dynamics, with application for example to subgrid-scale dissipation in meso-scale numerical simulation. We consider the weakly and strongly stratified limits, corresponding respectively to high and small internal Froude numbers $F$. The analysis is developed for the simplest obstacle, a sphere of radius $a$, and the simplest flow, of uniform velocity $U$ at infinity, in an unbounded uniformly stratified fluid of buoyancy frequency $N$, with $F = U/(Na)$. Greenslade (2000) has recently summarized the available experimental data (Mason 1977; Lofquist & Purtell 1984) and proposed two asymptotic models. When $F \gg 1$ the flow is essentially that, irrotational, of a homogeneous fluid, wave generation acting as a small perturbation; the obstacle may be represented by the same distribution of monopoles as for irrotational flow, with wave drag $C_D \sim (\ln F+7/4-\gamma)/(4F^4) \approx (\ln F+1.17)/(4F^4)$ for $F \gg 1$, where $\gamma$ is the Euler constant (Gorodtsov & Teodorovich 1982). When $F \ll 1$ the flow is essentially horizontal, wave generation being due to small regions, of height $aF$, with vertical motion, located at the top and bottom of the obstacle (Hunt et al. 1997); based on geometrical considerations and on fits to experimental data, Greenslade (2000) obtained $C_D \approx 0.73F^{3/2}$ (Mason fit) or $0.86F^{3/2}$ (Lofquist & Purtell fit) for $F \ll 1$. We argue that, in the strongly stratified régime, in spite of the inherent non-linearity of the flow, wave generation is indeed a linear process, the small regions at the top (respectively bottom) of the sphere acting as flat paraboloidal obstacles, of height $aF$ and much larger length $2a(2F)^{1/2}$, lying on horizontal surfaces -- the so-called dividing streamsurfaces -- below (respectively above) which the motion is purely horizontal. The representation of such obstacles is a horizontal distribution of monopoles of strength proportional to the local streamwise slope, with drag $C_D \sim (2^{11/2})/(15\pi F^{3/2}) \approx 0.96F^{3/2}$ for $F \ll 1$. The mathematical analysis involves the expression of the drag as a weighted double integral of the distribution's spectrum squared, and the asymptotic evaluation of this integral by Mellin-Barnes theory. The same representation can be used for calculating the waves themselves.
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