Steady depositing gravity current ( $q, K, g^{'} (x)$ )

A dilute, vertically mixed particle-laden gravity current with no fluid entrainment, bed erosion or carrier-fluid leakage obeys volume conservation

q = u h = constant

. With downward settling velocity magnitude

v_{s}

, an absorbing bed gives

q d g^{'} / d x = - v_{s} g^{'}

. The variable-buoyancy shallow water equations conserve the kinematic momentum flux including hydrostatic pressure,

g^{'} (x) = g_{0}^{'} e^{- v_{s} x / q}, K = \frac{q ^{2}}{h} + \frac{1}{2} g^{'} h^{2} .

(1)

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.

Table of contents
- Bidisperse gravity-current deposition Steady depositing gravity current
- Depth branches of a steady depositing gravity current Steady depositing gravity current

Bidisperse gravity-current deposition ( $f_{1} (x)$ )

 0  0

In a bidisperse particle suspension transported by a steady depositing gravity current, species with concentrations

c_{i}

and downward settling velocity magnitudes

v_{i}

satisfy

c_{i} (x) = c_{i 0} e^{- v_{i} x / q}

. Their local particle deposition fluxes are

v_{i} c_{i}

. The local deposited particle-volume fraction belonging to species one is

f_{1} (x) = \frac{v _{1} c _{10} e ^{- v_{1} x / q}}{v _{1} c _{10} e ^{- v_{1} x / q} + v _{2} c _{20} e ^{- v_{2} x / q}} .

(1)

For

v_{1} > v_{2} > 0

and both species present,

f_{1}

decreases strictly downstream, since

f_{1}^{'} = (v_{2} - v_{1}) f_{1} (1 - f_{1}) / q

. If

g_{i}^{'} = g_{0 i} c_{i}

, use

c_{i} = g_{i}^{'} / g_{0 i}

rather than buoyancy ratios unless

g_{01} = g_{02}

. A number fraction additionally divides each species volume flux by its single-particle volume. Integrating deposition over

[0, x]

instead gives cumulative weights

q c_{i 0} (1 - e^{- v_{i} x / q})

Depth branches of a steady depositing gravity current ( $h_{-} (x), h_{+} (x)$ )

 0  0

For positive

q, K, g^{'}

in a steady depositing gravity current, the function

K (h) = q^{2} / h + g^{'} h^{2} /2

has a unique minimum at

h_{c} = (q^{2} / g^{'})^{1/3}

, where

Fr = 1

and

K_{c} = 3 q^{4/3} (g^{'})^{1/3} /2

. When

K > K_{c}

there are two positive depth roots:

h_{-} < h_{c}

is supercritical flow and

h_{+} > h_{c}

is subcritical flow. Differentiating the momentum invariant and using

g_{x}^{'} = - v_{s} g^{'} / q

gives

\frac{d h}{d x} = \frac{v _{s} h}{2 q ( 1 - Fr ^{2} )} .

(1)

The supercritical depth decreases towards

q^{2} / K

g^{'} ↓ 0

, whereas the subcritical depth increases like

2 K / g^{'}

. Because

K_{c}

decreases downstream, an initially strict branch never reaches a finite-distance critical point. A critical inlet instead gives a singular depth derivative, so a smooth critical continuation is not justified by this model.