We explore the dynamics of inclined temporal gravity currents using direct numerical simulation, and find that the current creates an environment in which the flux Richardson number
$\mathit{Ri}_{f}$
, gradient Richardson number
$\mathit{Ri}_{g}$
and turbulent flux coefficient
$\unicode[STIX]{x1D6E4}$
are constant across a large portion of the depth. Changing the slope angle
$\unicode[STIX]{x1D6FC}$
modifies these mixing parameters, and the flow approaches a maximum Richardson number
$\mathit{Ri}_{max}\approx 0.15$
as
$\unicode[STIX]{x1D6FC}\rightarrow 0$
at which the entrainment coefficient
$E\rightarrow 0$
. The turbulent Prandtl number remains
$O(1)$
for all slope angles, demonstrating that
$E\rightarrow 0$
is not caused by a switch-off of the turbulent buoyancy flux as conjectured by Ellison (J. Fluid Mech., vol. 2, 1957, pp. 456–466). Instead,
$E\rightarrow 0$
occurs as the result of the turbulence intensity going to zero as
$\unicode[STIX]{x1D6FC}\rightarrow 0$
, due to the flow requiring larger and larger shear to maintain the same level of turbulence. We develop an approximate model valid for small
$\unicode[STIX]{x1D6FC}$
which is able to predict accurately
$\mathit{Ri}_{f}$
,
$\mathit{Ri}_{g}$
and
$\unicode[STIX]{x1D6E4}$
as a function of
$\unicode[STIX]{x1D6FC}$
and their maximum attainable values. The model predicts an entrainment law of the form
$E=0.31(\mathit{Ri}_{max}-\mathit{Ri})$
, which is in good agreement with the simulation data. The simulations and model presented here contribute to a growing body of evidence that an approach to a marginally or critically stable, relatively weakly stratified equilibrium for stratified shear flows may well be a generic property of turbulent stratified flows.