Flow over finite isolated topography

Abstract

One and two layer models are used to study flow over axisymmetric isolated topography. Inviscid or nearly inviscid flow in which non-linear effects have order one importance is considered, and both the effects of β and finite topography are included. A one-layer quasi-geostrophic model is used to find the shape of Taylor columns on both the f-plane and the β-plane in the inviscid limit of the frictional problem. In this limit, the boundary of the Taylor column is a streamline, and the velocity in both directions vanishes on the boundary. The fluid within the Taylor column is stagnant, corresponding to the solution that Ingersoll (1969) found for flow over a right circular cylinder on the f-plane. In this case, the Taylor column is circular. An iterative boundary integral technique is used to find the solutions for flow over a cone on the f-plane. In this case the Taylor column has a tear drop shape. Solutions are also found for flow on the β-plane over a cylinder, and the Taylor column is approximately elliptical for westward flow with the major axis in the x direction, while it is slightly elongated in the y direction for eastward flow. The stagnation point of the Taylor column is located on the edge of the topography for all the solutions found. It was not possible to find solutions for smooth topographic shapes. Steady solutions for flow over a right circular cylinder of finite height are studied when the quasi-geostrophic approximation no longer applies. The solution consists of two parts, one which is similar to the quasi-geostrophic solution and is driven by the potential vorticity anomaly over the topography and the other which is similar to the solution of potential flow around an cylinder and is driven by the matching conditions on the edge of the topography. When the effect of β is large, the transport over the topography is enhanced as the streamlines follow lines of constant background potential vorticity. For eastward flow, the Rossby wave drag can be much larger than predicted from quasi-geostrophic theory. A two-layer model over finite topography using the quasi-geostrophic approximation is developed. The topography is a right circular cylinder which goes all of the way through the lower layer and an order Rossby number amount into the upper layer, so that the quasi-geostrophic approximation can be applied consistently. This geometry allows description of flow in which an isopycnal intersects the topography. The model is valid for a different regime than existing models of steady flow over finite topography in a continuously stratified fluid in which the bottom boundary is an isopycnal surface. The solutions contain the two components that are found in the the barotropic model of flow over finite topography. The model breaks down when the interface goes above the topography which occurs more easily when the stratification is weak. Closed streamlines occur more readily over the topography when the stratification is weak, whereas in traditional quasi-geostrophic theory…