Island Circulation Integrals

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(Larry Pratt)

Island Circulation Integrals            

One of the most important ingredients of the dynamics of flow in a sea strait is the along-strait momentum balance. In it, an along-strait pressure gradient is usually balanced by bottom drag or, in the presence of hydraulic effects, by fluid acceleration and form drag. Archipelagos lack such directionality and it may be more meaningful to express the momentum balance in terms of a circulation integral. The simplest forms arise when the integration is carried out around the islands. As a simple example, consider a homogeneous ocean governed by the shallow water momentum equation:

where is the surface wind stress and f may vary with y.  Assuming a linear bottom drag law,


where Df is a drag coefficient.  Now consider a closed contour Γ that circles the coast of the island. Integration of the product of the depth H and (1) about this contour and use of the no-normal flow condition leads to

(Note that a similar result is obtained by integration about any contour of constant depth if it is known that there is no flow across the contour.)

The balance expressed in (2) is particularly simple because pressure gradient terms and nonlinear advection have been eliminated. A further simplification can be made by time averaging over a sufficiently long period, which eliminates the term on the right-hand side. Thus over long time periods the input from the wind must balance dissipation due to bottom drag. If the wind stress and u•t is known, perhaps from high frequency radar, the drag coefficient can be estimated.  If only the wind stress is known, the predominant direction of flow around the island can be estimated. Performing the calculation for many islands in the archipelago can allow one to build up a picture of how flow circulates through the archipelago.


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Last updated October 19, 2007
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