= Sediment-bed linear instability
Combining the <Exner equation>, <sediment transport saturation>, <bed shear response>, and <inclined-bed sediment threshold> gives a competition between upstream forcing and downstream relaxation. With $a=\tau_0A$, $b=\tau_0B-\tau_{\mathrm{th},0}/\mu_s$, $C=\chi\gamma(\tau_0-\tau_{\mathrm{th},0})^{\gamma-1}$, and positive <wavenumber> $k$, the <growth rate> is $\sigma=Ck^2(b-akL_{\mathrm{sat}})/(1+k^2L_{\mathrm{sat}}^2)$. Thus $b>0$ permits long-wave growth when $a>0$, while a finite <saturation length> stabilizes shorter waves.
Back to article page
