= Solution
The <shallow-water approximation> assumes a horizontal length scale $\ell$ much greater than the current depth, $h/\ell\ll1$. 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 $|\rho-\rho_0|/\rho_0\ll1$: <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 gravity>
$$
\boxed{g'=g\frac{\rho-\rho_0}{\rho_0}>0.}
$$
Thus the pressure excess over the ambient hydrostatic value at height $z$ within the current is $\rho_0g'(h-z)$. 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>.
Back to article page
