 Solution

For $A=2\cot \alpha /3+8\mathrm{Re}/15>0$, maximize the growth rate $\sigma (k)=\epsilon (A{k}^2-S{k}^4)$ over real wavenumbers. Its derivative is $2\epsilon k(A-2S{k}^2)$, soThe second derivative at either nonzero maximizer is $-4\epsilon A<0$. If $A\le 0$, there is no positive growth rate in this model.

To compare growth with transport, restore dimensions. With $\epsilon =h_0/L$ and $S=2\gamma /(3\rho g{L}^2\sin \alpha )$, the maximal dimensional growth rate isThe corresponding dimensional wavenumber is $k^{\ast }/L=\sqrt{3\rho g\sin \alpha A/(4\gamma )}$, independent of the arbitrary reference length $L$. Let $d$ be the physical slope length. A disturbance convected at the leading kinematic wave speed $2U_0$ has residence time $d/(2U_0)$ and accumulates the growth exponentThus $d\ll \ell$ implies $N\ll 1$, and the amplitude multiplier $e^N$ remains close to one while the disturbance passes down the slope. This is the requested convective amplification length criterion. With the exact coefficients used here, one e-fold of convected growth occurs at $d=2\ell /3$; the stated much-less-than criterion is unaffected by this order-one factor. It is a transport-versus-growth estimate within the long-wave approximation, rather than a boundary-value stability theorem for every possible inlet and outlet condition.