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[ACG36-P02] Stability of a Flow over Bottom Topography: A Linear Stability Analysis and a Numerical Simulation in a Two-Layer Quasi-Geostrophic Model
Keywords:Two-Layer Quasi-Geostrophic Flow, Bottom Topography, Pseudoenergy, Linear Stability Analysis, Barotropic and Baroclinic Instability
The stability of a two-layer quasi-geostrophic flow over bottom topography is examined by combining a method of obtaining a sufficient condition for stability, a linear stability analysis, and a numerical simulation. First, using a conserved quantity called pseudoenergy that is proportional to the square of the disturbance amplitude, a sufficient condition for stability is derived for the simplest steady background field in which the potential vorticity and the stream function are proportional to each other. The condition enables us to judge the stability of various background fields by explicitly taking into account the limitation imposed on the scale of the disturbance by the domain size and/or boundary conditions. In particular, the condition can predict a stable flow field with negative-definite pseudoenergy and thus is sometimes stricter than the well-known Charney–Stern–Pedlosky condition.
The stability condition is then applied to a specific case of a sinusoidal background flow and topography. The results show that the background flow is more stabilized when it flows with shallower water on the right in both the upper and lower layers. Furthermore, the stable range of the background flow velocity is found to broaden as the disturbance becomes limited to smaller horizontal scales. This theoretical prediction is confirmed by a linear stability analysis under various domain sizes and boundary conditions. The linear stability analysis further shows that the broadening of the stable range is mainly due to the suppression of barotropic instability that works effectively at large scales.
Finally, a series of numerical experiments are performed assuming a realistic parameter range over a low-rise ridge and slope. The results demonstrate that the stability condition is well applicable despite the existence of the nonlinear and viscous effects. Putting all the results together, we conclude that the stability condition obtained in this study is useful to identify a stable flow that can be achieved over bottom topography confined within a basin in the real ocean.
The stability condition is then applied to a specific case of a sinusoidal background flow and topography. The results show that the background flow is more stabilized when it flows with shallower water on the right in both the upper and lower layers. Furthermore, the stable range of the background flow velocity is found to broaden as the disturbance becomes limited to smaller horizontal scales. This theoretical prediction is confirmed by a linear stability analysis under various domain sizes and boundary conditions. The linear stability analysis further shows that the broadening of the stable range is mainly due to the suppression of barotropic instability that works effectively at large scales.
Finally, a series of numerical experiments are performed assuming a realistic parameter range over a low-rise ridge and slope. The results demonstrate that the stability condition is well applicable despite the existence of the nonlinear and viscous effects. Putting all the results together, we conclude that the stability condition obtained in this study is useful to identify a stable flow that can be achieved over bottom topography confined within a basin in the real ocean.