Lee waves from a sphere in a stratified flow

Abstract

Two asymptotic analyses of the generation of lee waves by horizontal flow at velocity U of a stratified fluid of buoyancy frequency N past a sphere of radius a are presented, for either weak or strong stratification, corresponding to either large or small internal Froude number F=U/(Na), respectively. For F⋙1, the fluid separates into two regions radially: an inner region of scale a with three-dimensional irrotational flow unaffected by the stratification, and an outer region of scale U/N with small-amplitude lee waves generated by the O(1) vertical motion in the inner region. For F⋘1, the fluid separates into five layers vertically: from the lower dividing streamsurface situated at a distance U/N above the bottom of the sphere to the upper dividing streamsurface situated at a distance U/N below the top, there is a middle layer with two-dimensional horizontal irrotational flow; from the upper dividing streamsurface to the top of the sphere, and from the lower dividing streamsurface to the bottom, there are top and bottom transition layers, respectively, with three-dimensional flow; above the top and below the bottom, there are upper and lower layers, respectively, with small-amplitude lee waves generated by the O(F) vertical motion in the transition layers.The waves are calculated where they have small amplitudes. The forcing is represented by a source of mass: for F⋙1, the surface distribution of singularities equivalent to the sphere in three-dimensional irrotational flow; for F⋘1, the horizontal distribution of singularities equivalent, in the upper (resp. lower) layer, to the flat cut-off obstacle made up of the top (resp. bottom) portion of the sphere protruding above (resp. below) the upper (resp. lower) dividing streamsurface. The analysis is validated by comparison of the theoretical wave drag with existing experimental determinations. For F⋙1, the drag coefficient decreases as (ln F+7/4-γ)/(4F4), with γ the Euler constant; for F⋘1, it increases as $(32\surd2)/(15\upi)F^{3/2}$. The waves have the crescent shape of the three-dimensional lee waves from a dipole, modulated by interferences associated with the finite size of the forcing. For strong stratification, the hydrostatic approximation is seen to produce correct leading-order drag, but incorrect waves.