 Solution

The shallow-water approximation assumes a horizontal length scale $\ell$ much greater than the current depth, $h/\ell \ll 1$. Vertical acceleration is then small compared with gravity and pressure is given by the hydrostatic approximation. A depth-averaged model also assumes that the horizontal velocity and excess mass density can be represented by nearly uniform cross-sectional values. Here the ambient fluid is deep or sufficiently undisturbed that its motion and interface-pressure perturbations can be neglected.

The Boussinesq approximation assumes $|\rho -{\rho }_0|/{\rho }_0\ll 1$: mass density differences are retained in gravitational forces but a common reference mass density is used for inertia. A dense bottom current has the positive reduced gravityThus the pressure excess over the ambient hydrostatic value at height $z$ within the current is ${\rho }_0{g}^{\prime }(h-z)$. We neglect bed drag, molecular diffusion and horizontal turbulent stress in the leading balances, but retain the prescribed turbulent entrainment across the interface. Negligible bed drag does not imply that entrainment has no effect on the current's velocity or mass density.