= Steady depositing gravity current
{title2=$q,K,g'(x)$}
A dilute, vertically mixed <particle-laden gravity current> with no <fluid entrainment>, bed erosion or carrier-fluid leakage obeys <volume conservation> $q=uh=\text{constant}$. With downward <settling velocity> magnitude $v_s$, an absorbing bed gives $q\,dg'/dx=-v_sg'$. The <variable-buoyancy shallow water equations> conserve the kinematic momentum flux including <hydrostatic pressure>,
$$
g'(x)=g'_0e^{-v_sx/q},\qquad K=\frac{q^2}{h}+\frac12g'h^2.
$$
These equations describe a steady interior supplied continuously from upstream, not the moving nose of a finite-volume release. The depth follows one of the <depth branches of a steady depositing gravity current>.
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