Island Circulation Integrals 

Island Circulation Integrals
One of the
most important ingredients of the dynamics of flow in a sea strait is the
alongstrait momentum balance. In it, an
alongstrait 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 D_{f} 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 nonormal 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 righthand 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.

