Asymptotics of unit Burger number rotating and stratified flows for small aspect ratio

Abstract

Rotating and stably stratified Boussinesq flow is investigated for Burger number unity in domain aspect ratio (height/horizontal length) δ < 1 and δ = 1 . To achieve Burger number unity, the non-dimensional rotation and stratification frequencies (Rossby and Froude numbers, respectively) are both set equal to a second small parameter ϵ < 1 . Non-dimensionalization of potential vorticity distinguishes contributions proportional to ( ϵ δ ) − 1 , δ − 1 and O ( 1 ) . The ( ϵ δ ) − 1 terms are the linear terms associated with the pseudo-potential vorticity of the quasi-geostrophic limit. For fixed δ = 1 / 4 and a series of decreasing ϵ , numerical simulations are used to assess the importance of the δ − 1 contribution of potential vorticity to the potential enstrophy. The change in the energy spectral scalings is studied as ϵ is decreased. For intermediate values of ϵ , as the flow transitions to the ( δ ϵ ) − 1 regime in potential vorticity, both the wave and vortical components of the energy spectrum undergo changes in their scaling behavior. For sufficiently small ϵ , the ( δ ϵ ) − 1 contributions dominate the potential vorticity, and the vortical mode spectrum recovers k − 3 quasi-geostrophic scaling. However, the wave mode spectrum shows scaling that is very different from the well-known k − 1 scaling observed for the same asymptotics at δ = 1 . Visualization of the wave component of the horizontal velocity at δ = 1 / 4 reveals a tendency toward a layered structure while there is no evidence of layering in the δ = 1 case. The investigation makes progress toward quantifying the effects of aspect ratio δ on the ϵ → 0 asymptotics for the wave component of unit Burger number flows. At the lowest value of ϵ = 0.002 , it is shown that the horizontal kinetic energy spectral scalings are consistent with phenomenology that explains how linear potential vorticity constrains energy in the limit ϵ → 0 for fixed δ .