ADJUSTMENT OF STRATIFIED FLOW OVER A SLOPING BOTTOM

The evolution of a steady stratified along-isobath current flowing cyc lonically (shallower water on the right looking downstream) over a slo ping frictional bottom is examined using an idealized model. The flow is assumed to consist of an inviscid vertically uniform geostrophic in terior above a bottom boundary layer in which density is vertically we ll mixed. Within the bottom boundary layer, vertical shear in the hori zontal velocities is assumed to result only from horizontal density gr adients. Density advection is included in the model, but momentum adve ction is not. The downstream evolution of the current is described by two coupled nonlinear partial differential equations for surface press ure and boundary layer thickness, each of which is first order in the along-isobath coordinate and can be easily integrated numerically. An initially narrow along-isobath current over a uniformly sloping bottom spreads and slows rapidly owing to the effects of bottom friction, mu ch like the unstratified case. However, as the bottom boundary layer g rows, the resulting horizontal density gradients reduce the bottom vel ocity, which in turn, decreases both the transport in the bottom bound ary layer and the spreading of the current. An equilibrium is reached downstream in which the bottom velocity vanishes everywhere and the cu rrent stops spreading. This equilibrium flow persists indefinitely des pite the presence of a frictional bottom. The width of the equilibrium current scales as W similar to (f/N alpha)(F-0/f)(1/2), where f is is the Coriolis parameter, N the buoyancy frequency, alpha the bottom sl ope, and F-0 the inflow volume flux per unit depth. The thickness of t he bottom boundary layer scales as alpha W, while the along-isobath ve locity scales as (N alpha/f)(F(0)f)(1/2). Surprisingly, the downstream equilibrium flow is independent of the magnitude of bottom friction. Good approximations for the equilibrium scales are obtained analytical ly by imposing conservation of mass and buoyancy transports. Generaliz ations to variable bottom slope, nonuniform stratification, and coasta l currents are also presented.