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 they occur more readily when the stratification is strong. Near the topography, the interface is depressed to the right and raised to the left (looking downstream). A hierarchy of time-dependent models is used to examine the initial value problem of flow initiation over topography on the f-plane. A modified contour dynamics method is developed that extends the range of problems to which contour dynamics can be applied. The method allows boundary and matching conditions to be applied on a circular boundary. A one-layer quasi-geostrophic model is used to show that more fluid that originates over the topography remains there when the flow is turned on slowly than when it is turned on quickly. Flow over finite topography in a one-layer model shows a variety of different behaviors depending on the topographic height. When the topography has moderate height, two cyclonic eddies are created; when the topography fills up most of the water column, the fluid oscillates on and off the topography as it moves around the topography in a clockwise direction, and none of the fluid is shed downstream. Two quasi-geostrophic stratified models are considered, one in which the topography is small, and the other in which it is finite. In the small topography model, an eddy is shed which is cyclonic, warm-core, and bottom-trapped. In contrast, the shed eddy is cyclonic, cold-core, and surfaceintensified in the finite depth model using the geometry described above.