Accelerating dense-water flow down a slope

Where water is denser on a shallow shelf than in the adjacent deep ocean, it tends to flow down the slope from shelf to ocean. The flow can be in a steady bottom boundary layer for moderate combinations of up-slope density gradient -ρx∞ and bottom slope (angle θ to horizontal): b ≡ |ρx∞| g sinθ / (f**2 ρ0) < 1. Here g is acceleration due to gravity, ρ0 is a mean density and f is twice the component of earth’s rotation normal to the sloping bottom. For stronger combinations of horizontal density gradient and bottom slope, the flow accelerates. Analysis of an idealised initial-value problem shows that when b ≥ 1 there is a bottom boundary layer with down-slope flow, intensifying exponentially at a rate fb**2 (1+b)**-1/2 /2, and slower-growing flow higher up. For stronger stratification b > 2**1/2, i.e. relatively weak Coriolis constraint, the idealised problem posed here may not be the most apposite but suggests that the whole water column accelerates, at a rate [ρ0**-1 |ρx∞| g sinθ]**1/2 if f is negligible.