Film hydrodynamics is crucial in water-driven morphological pattern formation. A
prominent example is given by icicle ripples and ice ripples, which are regular patterns developing
on freezing-melting inclined surfaces bounding open-channel flows. By a suitable
mathematical model based on conservation principles and the use of the cuspmap method, in
this paper we address the convective-absolute nature of these two kinds of instabilities. The
obtained results show that icicle ripples, which develop at inverted (overhang) conditions,
have subcentimetric wavelengths which are unstable when the Reynolds number of the
liquid flow (Re) is small and the supercooling is intensive. With the increase in Re, the
instability switches from absolute to convective. Ice ripples instead exhibit the opposite
dependance on Re and are highly affected by the surface slope. In addition, the evaluation
of the so-called absolute wave number, which is responsible for the asymptotic impulse
response, suggests a different interpretation of some recent experiments about ice ripples
