= Solution
The <water> <mass conservation> equation inherited from the earlier model is $\epsilon h_t+(hu)_x=0$. It is essential: without it the printed displayed equations are not closed. Take the slow bed branch $\epsilon=0$, and perturb the uniform state by $h=1+\hat h e^{\lambda t+ikx}$, $u=1+Ue^{\lambda t+ikx}$, $c=1+Ce^{\lambda t+ikx}$, $s=Se^{\lambda t+ikx}$ and $\eta=1+\hat\eta e^{\lambda t+ikx}$. For $k\ne0$, <mass conservation> gives $\hat h=-U$ and $\hat\eta=S-U$.
Put $m=E^{*\prime}(1)+1$; the extra $1$ comes from linearizing $c/u$. The linearized <momentum> and suspended-sediment equations are
$$
[3\delta+ik(F^2-1)]U=-ikS,\qquad (1+ik)C=mU,\qquad \lambda S=-mU+C.
$$
Eliminating $U,C$ gives the <dispersion relation>
$$
\boxed{\lambda=-\frac{m k^2}{(1+ik)[3\delta+ik(F^2-1)]}.}
$$
Away from $F=1$, for fixed nonzero $k$ and negligible $\delta$,
$$
\operatorname{Re}\lambda=\frac{m k^2}{(F^2-1)(1+k^2)},\qquad \frac{\hat\eta}{S}=\frac{F^2}{F^2-1}.
$$
Thus growth requires $m/(F^2-1)>0$. For $F>1$, any $E^{*\prime}(1)>-1$ gives an unstable branch; for $F<1$ growth would require $E^{*\prime}(1)<-1$. A physically increasing erosion <function> therefore gives the supercritical instability in this simplified model, rather than proving growth of ordinary subcritical dunes. Retaining drag gives
$$
\operatorname{Re}\lambda=\frac{m k^2[k^2(F^2-1)-3\delta]}{|(1+ik)[3\delta+ik(F^2-1)]|^2},
$$
which shows its stabilizing correction and the need to reconsider the limit near critical flow.
The negative bed-to-surface ratio for <subcritical flow> means bed crests lie beneath surface troughs: the <dune> phase relation. Its positive sign for <supercritical flow> gives aligned crests: an <antidune>. In the unstable supercritical case the imaginary part gives upstream migration, since $-\operatorname{Im}\lambda/k=-m/[(F^2-1)(1+k^2)]$. The figure shows these two phase relations. They identify the bedforms but do not by themselves establish growth. The qualitative limitation of the suspended-sediment mechanism is also discussed in the official https://people.maths.ox.ac.uk/hewitt/publications/C5_11_notes.pdf[Mathematical Geoscience notes, pages 41–42]{ref}.
Back to article page
