Asymmetry in the spin-up of a stratified fluid in a sliced, square cylinder

Previously, we have reported theory and experiments on the spin-up of a linearly stratified fluid contained in a sealed square cylinder with a bottom sloped at angle α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} (Munro & Foster, Phys. Fluids, 26, 026603, 2014). The motion is characterized by the small Rossby number, ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon$$\end{document}, the fractional change in the rotation rate, the Ekman number, E, and the Burger number, S, the square of the ratio of the buoyancy frequency (or Brunt-Väisälä frequency) to the base rotation rate. This paper, supplementing what has gone before, is focused on the horizontal and vertical structure of the propagating vorticity waves (Rossby waves), with alternating anti-cyclonic and cyclonic cells, which move across the tank from east to west, thereby accomplishing at least the early stages of the spin-up. We find strong dependence of the modal structure of these Rossby waves, and their speed, on the Burger number. However, what is particularly noteworthy is the observed strong north-south asymmetry of the wave field. We find that the asymmetry is not consistent with typical Rossby-wave theories (Pedlosky and Greenspan, J. Fluid Mech., 27, p. 291, 1967), but is attributable to second-order effects in the asymptotic expansion of the streamfuntion in powers of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}.