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Let $X(t)$ be the advancing nose into previously unoccupied pores. In the advective limit the interior flux is $uh$, while the exterior current thickness and flux are zero. Applying the Rankine-Hugoniot condition to the storage $\varphi h$ givesThis is the leading-edge kinematic boundary condition. The outer depth immediately behind this nose is $h(X(t{)}^{-},t)=h_0$, whereas ahead it is zero. A first-order advection model allows this jump; imposing zero depth on its interior trace would contradict its characteristic solution.

For the full inclined porous gravity current, the nose is instead resolved by a continuous spreading layer with $h(X,t)=0$. Taking the flux divided by depth at that interface gives $\dot{X}=(u/\varphi )(1-\cot \theta h_x{|}_{X^{-}})$, when this one-sided derivative exists. Its leading outer value is $u/\varphi$ only when the nose correction is negligible on the scale of the approximation. The zero-depth condition belongs to that resolved layer, not to the discontinuous outer profile itself.

Choose $x$ upslope and $z$ normally downwards from the impermeable roof, so the buoyant layer occupies $0<z<h(x,t)$. Use the hydrostatic approximation and neglect the ambient return-flow pressure correction of relative order $h/H$. Pressure continuity at the interface gives the pressure inside the buoyant layer asThe along-slope Darcy velocity is thereforeDefine $u=kg\Delta \rho \sin \theta /\mu$. Integrating over depth gives the volume flux per unit width $F=uh-u\cot \theta h{h}_x$. Local fluid-volume conservation, including porosity, is $\varphi h_t+F_x=0$. Hence the inclined porous gravity current satisfiesThe first term is upslope advection driven by the mass density difference; the second is nonlinear hydrostatic spreading. The Darcy velocity $u$ and the corresponding pore velocity $u/\varphi$ must be distinguished.