jats:pWe 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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline1" />jats:tex-math$\mathrm{Ri}f$</jats:tex-math></jats:alternatives></jats:inline-formula>, gradient Richardson number jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline2" />jats:tex-math$\mathrm{Ri}g$</jats:tex-math></jats:alternatives></jats:inline-formula> and turbulent flux coefficient jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline3" />jats:tex-math$𝛤$</jats:tex-math></jats:alternatives></jats:inline-formula> are constant across a large portion of the depth. Changing the slope angle jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline4" />jats:tex-math$𝛼$</jats:tex-math></jats:alternatives></jats:inline-formula> modifies these mixing parameters, and the flow approaches a maximum Richardson number jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline5" />jats:tex-math$\mathrm{Ri}max\approx 0.15$</jats:tex-math></jats:alternatives></jats:inline-formula> as jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline6" />jats:tex-math$𝛼\to 0$</jats:tex-math></jats:alternatives></jats:inline-formula> at which the entrainment coefficient jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline7" />jats:tex-math$E\to 0$</jats:tex-math></jats:alternatives></jats:inline-formula>. The turbulent Prandtl number remains jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline8" />jats:tex-math$O(1)$</jats:tex-math></jats:alternatives></jats:inline-formula> for all slope angles, demonstrating that jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline9" />jats:tex-math$E\to 0$</jats:tex-math></jats:alternatives></jats:inline-formula> is not caused by a switch-off of the turbulent buoyancy flux as conjectured by Ellison (jats:italicJ. Fluid Mech.</jats:italic>, vol. 2, 1957, pp. 456–466). Instead, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline10" />jats:tex-math$E\to 0$</jats:tex-math></jats:alternatives></jats:inline-formula> occurs as the result of the turbulence intensity going to zero as jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline11" />jats:tex-math$𝛼\to 0$</jats:tex-math></jats:alternatives></jats:inline-formula>, due to the flow requiring larger and larger shear to maintain the same level of turbulence. We develop an approximate model valid for small jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline12" />jats:tex-math$𝛼$</jats:tex-math></jats:alternatives></jats:inline-formula> which is able to predict accurately jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline13" />jats:tex-math$\mathrm{Ri}f$</jats:tex-math></jats:alternatives></jats:inline-formula>, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline14" />jats:tex-math$\mathrm{Ri}g$</jats:tex-math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline15" />jats:tex-math$𝛤$</jats:tex-math></jats:alternatives></jats:inline-formula> as a function of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline16" />jats:tex-math$𝛼$</jats:tex-math></jats:alternatives></jats:inline-formula> and their maximum attainable values. The model predicts an entrainment law of the form jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019004300_inline17" />jats:tex-math$E=0.31(\mathrm{Ri}max-\mathrm{Ri})$</jats:tex-math></jats:alternatives></jats:inline-formula>, 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.</jats:p>