Hydraulically Drained Flows in Rotating Basins. Part I: Method*
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An asymptotic method for coupling circulations in basins to hydraulically controlled overflows is introduced. The method is applicable when the forcing, dissipation, and coupling with the overflow are weak, in which case the lowest order solution for the homogeneous or 1 1/2-layer model consists of the natural basin modes including gravity, inertia-gravity, potential vorticity, Helmholtz, and steady geostrophic modes. At the next order of approximation, the mode amplitudes are found to vary slowly with time as the result of forcing, dissipation, interior nonlinear mode interactions, and, most importantly, coupling with the overflow. Even when the latter are absent, the overflow dynamics generally introduce nonlinearity. Although the basin dynamics are assumed linear to lowest order, the overflow is intrinsically nonlinear. To couple the two systems, the overflow model must be adapted to serve as a nonlinear boundary condition on the basin flow. To do so, a rotating-channel model introduced by Whitehead et al. valid for relatively shallow sills is employed. Although not the central focus, corresponding formulations are derived for straits acting as geostrophic controls or which are dominated by bottom drag. The principle aim of Part I is to derive the evolution equations governing the coupling between basin and sill. Parts II and III of this work contain a number of examples intended to illustrate the general method and provide insight into physical phenomena associated with hydraulically drained, time-dependent how in deep basins such as those that occur in the Nordic seas.