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The Exner equation is solid-volume mass conservation: a downstream increase in sediment transport flux removes material and lowers the bed. The factor ${\varphi }_b$ converts bed-height change into solid-volume change. The saturation length equation models downstream adjustment of actual sediment transport to the equilibrium sediment flux. Grain acceleration and entrainment/deposition need a finite distance, so $q$ lags $q_{\mathrm{sat}}$. The last relation is an empirical transport law above the sediment entrainment threshold, with $\chi >0$ and normally $\gamma >0$; below threshold it must be interpreted with a positive part rather than raising negative excess stress to an arbitrary power. It is a constitutive closure, not a consequence of mass conservation.

Take the physical parameters $L_{\mathrm{sat}}>0$, ${\varphi }_b>0$, $\chi >0$ and $\gamma >0$. Linearize about a horizontal, uniformly transporting bed with $\Delta ={\tau }_0-{\tau }_{\mathrm{th},0}>0$ and $q_0={\varphi }_b\chi {\Delta }^{\gamma }$. Set $\lambda =\sigma -ikc$ and use the real parts ofThe locally planar approximation uses the small instantaneous bed slope to compute the inclined-bed sediment threshold. Since $\alpha ={\eta }_x+O({\eta }_x^3)$,A sinusoidal disturbance has both uphill and downhill slopes: the printed $0<\alpha \ll 1$ describes the uphill derivation, and its first-order continuation applies to both signs. The zero-slope reference is the one implicit in the printed decomposition with constant term ${\tau }_{\mathrm{th},0}$; a finite mean inclined bed would require a shifted base threshold.

Define $C=\chi \gamma {\Delta }^{\gamma -1}$. The linearized transport equations areThe linearization requires $|{\eta }_x|\ll 1$ and $|\hat{\tau }-{\hat{\tau }}_{\mathrm{th}}|\ll \Delta$.

A fluid-dynamical shear-response closure is missing from the printed question. The three sediment equations and local slope correction do not determine $\hat{\tau }$ from $\hat{\eta }$. Write the general bed shear response as $\hat{\tau }=\mathcal{T}(k)\hat{\eta }$. A frequently used scale-invariant closure for $k>0$ isHere $A$ is the component in phase with bed height and $B$ represents an upstream phase lead in bed shear stress. They require an independent flow model and can depend on $k$. Introducing them explicitly makes the linear stability analysis complete conditional on a specified flow response; it does not turn them into data supplied by the question. An example of this hydrodynamic closure is given in Fourrière, Claudin and Andreotti's bedform-instability analysis.

Eliminate the transport amplitudes in the linear stability analysis. For a general bed shear response $\mathcal{T}(k)$, the dispersion relation isThis is the most specific result available without adding the missing hydrodynamic closure. With $\mathcal{T}(k)={\tau }_{0}k(A+iB)$, setMultiplication by $1-ix$ gives the growth rate and migration velocityThe packed-bed volume fraction cancels. For a real physical closure the negative wavenumber response is the complex conjugate; the displayed $k>0$ formulas cover the independent normal modes.

If $A>0$, $B>0$ are treated as constants, the upstream shear stress phase lead promotes growth, while the inclined-bed sediment threshold reduces it and the finite saturation length delays sediment transport. There is an unstable band precisely when $b>0$: $0<k<b/(a{L}_{\mathrm{sat}})$. Within this band $c>0$, so the growing bedforms migrate downstream. The fastest normal mode satisfiesThis last maximization assumes constant $A,B$; if the hydrodynamic coefficients vary with wavenumber, their derivatives also enter the selection condition. When $b\le 0$ and $a>0$, all nonzero normal modes decay.

Just above the sediment entrainment threshold,Thus the common regime $B<1/{\mu }_s$ is stable near onset: the slope penalty overwhelms the hydrodynamic phase lead. This conclusion is conditional on that inequality, not universal. For constant $B>0$, instability begins only when ${\tau }_0>{\tau }_{\mathrm{th},0}/({\mu }_{s}B)$ as well as ${\tau }_0>{\tau }_{\mathrm{th},0}$. The transport sensitivity $C=\chi \gamma {\Delta }^{\gamma -1}$ vanishes at onset for $\gamma >1$, is finite for $\gamma =1$, and is singular for $0<\gamma <1$. Exactly at threshold, the positive-part transport law needs separate treatment; the above linearization with perturbations small compared with $\Delta$ is unavailable.

Far above the sediment entrainment threshold, $b\simeq {\tau }_{0}B$. If $A,B>0$, the unstable cutoff tends to $B/(A{L}_{\mathrm{sat}})$, and the dominant wavelength is set by the saturation length. Both growth rate and migration velocity scale with $C{\tau }_0\sim \chi \gamma {\tau }_0^{\gamma }$, apart from their length factors. Their numerical values and any detailed dependence on wavenumber still require a flow closure.

For example, prescribing uniform shear stress independently of the bed gives $\mathcal{T}=0$ andThis equally admissible closure illustrates why instability cannot be asserted from the printed sediment equations alone.

The plots use constant $a>0$ and three illustrative ratios $b/a$. They show how the sign of the shear phase lead minus the slope correction determines stability; they are not numerical predictions for unspecified flow conditions.

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 $a={\tau }_{0}A$, $b={\tau }_{0}B-{\tau }_{\mathrm{th},0}/{\mu }_s$, $C=\chi \gamma ({\tau }_0-{\tau }_{\mathrm{th},0}{)}^{\gamma -1}$, and positive wavenumber $k$, the growth rate is $\sigma =C{k}^2(b-ak{L}_{\mathrm{sat}})/(1+k^2{L}_{\mathrm{sat}}^2)$. Thus $b>0$ permits long-wave growth when $a>0$, while a finite saturation length stabilizes shorter waves.