= Triangular-channel shallow water equations
In a channel with width $2\lambda xz$ at height $z$, a well-mixed <particle-laden gravity current> of depth $h$ occupies area $\lambda xh^2$. Its <volume conservation>, momentum and <particle deposition flux> balances give
$$
h_t+uh_x+(h/2)u_x=-uh/(2x),\quad
u_t+uu_x+d\phi h_x+(dh/3)\phi_x=0,\quad
\phi_t+u\phi_x=-2W_s\phi/h,
$$
where the <reduced gravity> is $g'=d\phi$. The $h/3$ coefficient is the cross-section-weighted mean of $h-z$ in the <hydrostatic pressure> force. The <characteristic curves> have speeds $u$ and $u\pm\sqrt{g'h/2}$.
Back to article page
