= Solution
{c}
For $A=2\cot\alpha/3+8\mathrm{Re}/15>0$, maximize the <growth rate> $\sigma(k)=\varepsilon(Ak^2-Sk^4)$ over real <wavenumbers>. Its <derivative> is $2\varepsilon k(A-2Sk^2)$, so
$$
\boxed{|k^*|=\sqrt{\frac A{2S}},\qquad
\sigma^*=\frac{\varepsilon A^2}{4S}.}
$$
The second <derivative> at either nonzero maximizer is $-4\varepsilon A<0$. If $A\leq0$, there is no positive <growth rate> in this model.
To compare growth with transport, restore dimensions. With $\varepsilon=h_0/L$ and $S=2\gamma/(3\rho gL^2\sin\alpha)$, the maximal dimensional <growth rate> is
$$
\Omega^*=\frac{U_0}{L}\sigma^*
=\frac{3\rho gh_0\sin\alpha\,U_0A^2}{8\gamma}.
$$
The corresponding dimensional <wavenumber> is $k^*/L=\sqrt{3\rho g\sin\alpha\,A/(4\gamma)}$, 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 exponent
$$
N=\frac{\Omega^*d}{2U_0}
=\frac{3\rho gh_0\sin\alpha\,A^2d}{16\gamma}
=\frac32\frac d\ell,
\qquad
\boxed{\ell=\frac{8\gamma}{\rho gh_0\sin\alpha\,A^2}.}
$$
Thus \b[$d\ll\ell$ implies $N\ll1$], 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.
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