Self-similarity of strongly stratified inviscid flows

It is well-known that strongly stratified flows are organized into a layered pancake structure in which motions are mostly horizontal but highly variable in the vertical direction. However, what determines the vertical scale of the motion remains an open question. In this paper, we propose a scaling law for this vertical scale L_u when no vertical lengthscales are imposed by initial or boundary conditions and when the fluid is strongly stratified, i.e., when the horizontal Froude number is small: F_h=U/NL_h ≪ 1, where U is the magnitude of the horizontal velocity, N the Brunt-Vïsälä frequency and L_h the horizontal lengthscale. Specifically, we show that the vertical scale of the motion is L_v=U/N by demonstrating that the inviscid governing equations in the limit F_h→O, without any a priori assumption on the magnitude of L_u, are self-similar with respect to the variable zN/U, where z is the vertical coordinate. This self-similarity fully accounts for the layer characteristics observed in recent studies reporting spontaneous layering from an initially vertically uniform flow. For such a fine vertical scale, vertical gradients are large, O(1/F_hL_h). Therefore, even if the magnitude of the vertical velocity is small and scales like F_hU, the leading order governing equations of these strongly stratified flows are not two-dimensional in contradiction with a previous conjecture. The self-similarity further suggests that the vertical spectrum of horizontal kinetic energy of pancake turbulence should be of the form E(k_z)∝N²k_z^-3, giving an alternative explanation for the observed vertical spectra in the atmosphere and oceans.