General linear stability properties of monoclinal shallow waves

TL;DR

This paper investigates the linear stability of monoclinal shallow waves on inclined planes using the shallow-water equations with resistive drag, revealing how flow shear and wave profile influence instability and mode amplification.

Contribution

It derives simple criteria for wave instability onset, incorporates shear effects into stability analysis, and examines the amplification of disturbances near wave fronts.

Findings

Shear flow can significantly alter instability thresholds.

Corrections for shear profiles tend to stabilize waves.

Disturbance amplification near wave fronts can become unbounded.

Abstract

We analyze the linear stability of monoclinal traveling waves on a constant incline, which connect uniform flowing regions of differing depths. The classical shallow-water equations are employed, subject to a general resistive drag term. This approach incorporates many flow rheologies into a single setting and enables us to investigate the features that set different systems apart. We derive simple formulae for the onset of linear instability, the corresponding linear growth rates and related properties including the existence of monoclinal waves, development of shocks and whether instability is initially triggered up- or downstream of the wave front. Also included within our framework is the presence of shear in the flow velocity profile, which is often neglected in depth-averaged studies. We find that it can significantly modify the threshold for instability. Constant corrections to the governing equations to account for sheared profiles via a 'momentum shape factor' act to stabilize traveling waves. More general correction terms are found to have a nontrivial and potentially important quantitative effect on the properties explored. Finally, we have investigated the spatial properties of the dominant (fastest growing) linear modes. We derive equations for their amplitude and frequency and find that both features can become severely amplified near the front of the traveling wave. For flood waves that propagate into a dry downstream region, this amplification is unbounded in the limit of high disturbance frequency. We show that the rate of divergence is a function of the spatial dependence of the wave depth profile at the front, which may be determined straightforwardly from the drag law.