Momentum and heat balances in steady coastal currents

Abstract

The momentum balance for coastal currents forced by a longshore wind stress and a longshore density gradient is examined for coastlines of various orientations. We isolate the coastal circulation by requiring that the longshore wind stress and density gradient become uniform in the adjacent deep ocean in which the longshore bottom pressure gradient is assumed to be zero, and derive a vorticity equation for the longshore current for a prescribed cross-shelf variation of wind stress and density structure. Solutions of this equation are obtained for a uniform longshore wind stress along a coastline with an exponential shelf in two situations: (A) For a uniform cross-shelf density on the inner shelf and a zero longshore bottom pressure gradient on the outer shelf, the division between the two regions occurring approximately at the depth of the main thermocline, and (B) for a thermohaline structure which is independent of the cross-shelf co-ordinate such that the coastal current is barotropic. It is shown that on a zonal coast the coastal transports due to the wind stress forcing are of O(σ τ Wl ϱ 0β ) , and due to the density gradient forcing of O(− σ ϱ 0 gh T 2/ β∂δ 0/ ∂l) where τ Wl and − ϱ 0 ∂δ 0/ ∂l are respectively, the longshore wind stress and surface density gradient, ϱ 0 is a reference density for seawater, g is the acceleration of gravity, h T is the scale depth for the main thermocline, β= df dy where f=2Ω sinϕ in which Ω is the angular speed of rotation of Earth, and ϕ is the latitude, and σ= W T W is a shelf parameter in which W T is the exponential shelf scale depth and W is an eddy scale. On eastern boundaries the transports are less [especially in (A)] and are identically zero on western boundaries on which the transport is determined by external (non-local) processes. The longshore velocity structure on the western boundary depends strongly on σ and incorporates a system of currents and counter-currents. The heat balance for the coastal circulation is investigated by considering the solution of the heat conservation equation for the surface temperature with the cross-shelf boundary conditions of zero heat flux through the coast and into deep ocean. It is found that the sea surface temperature anomaly is zero for wind stress forcing and positive for density gradient forcing (at constant salinity). Mixed forcing however may lead to upwelling which is characterized by a negative sea surface anomaly. Upwelling occurs on non-western boundaries on which there is a longshore wind stress of sufficient intensity or on western boundaries where there is an external transport of sufficient magnitude to generate a net transport up the longshore surface density gradient. This occurs on eastern coasts where there is poleward advection of warm water and sufficiently strong equatorward meridional winds, and on western coasts in a polar gyre where there is a strong equatorward external transport and a poleward temperature gradient. The Ekman circulation maintains the upwelling anomalies.