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

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 .