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Choose $x$ downslope along the rigid plane and $z$ normal to it into the fluid, with the wall at $z=0$ and the flat free surface at $z=h$. The gravitational body force per unit volume has components $(\rho g\sin \alpha ,-\rho g\cos \alpha )$. The wind's tangential traction on the top surface is $-S$ in the $x$ direction; the atmospheric normal traction is $-p_{\mathrm{atm}}$ in the $z$ direction. At the wall there is no-slip boundary condition, and the wall supplies the opposing normal force and a tangential traction whose sign depends on the wind strength.

The velocity is parallel to the plane, $(u(z),0)$. Its velocity profile is a concave quadratic, with $u(0)=0$ and negative slope at the free surface when $S>0$. The diagram uses moderate wind, for which the flow is still everywhere downslope. Stronger wind can reverse the upper portion, or the entire moving layer; the later cases quantify this.

The volume flux per unit transverse width isThusAssume a genuinely inclined plane, $0<\alpha <\pi /2$. With $Z=z/h$ and $U=\rho g{h}^2\sin \alpha /\mu$, the zero-flux velocity profile is $u/U=Z/3-Z^2/2$. It is positive for $0<Z<2/3$, zero at $Z=2/3$, and negative above that level. Its maximum is $U/18$ at $Z=1/3$, and its surface value is $-U/6$. Thus equal integrated downslope and upslope transports cancel, rather than the fluid being motionless.