Multidimensional stability and transverse bifurcation of hydraulic shocks and roll waves in open channel flow

TL;DR

This paper investigates the multidimensional stability and bifurcation phenomena of hydraulic shocks and roll waves in open channel flow using analytical and numerical methods, revealing complex behaviors and stability boundaries.

Contribution

It introduces new analytical and numerical techniques to analyze stability and bifurcations of hydraulic shocks and roll waves, including development of efficient Evans solvers and explicit stability boundary computations.

Findings

Identification of a novel transverse bifurcation leading to periodic patterns.

Discovery of metastable behaviors driven by mixed wave interactions.

Observation of Floquet-type bifurcations causing wave pattern changes.

Abstract

We study by a combination of analytical and numerical methods multidimensional stability and transverse bifurcation of planar hydraulic shock and roll wave solutions of the inviscid Saint Venant equations for inclined shallow-water flow, both in the whole space and in a channel of finite width, obtaining complete stability diagrams across the full parameter range of existence. Technical advances include development of efficient multi-d Evans solvers, low- and high-frequency asymptotics, explicit/semi-explicit computation of stability boundaries, and rigorous treatment of channel flow with wall-type physical boundary. Notable behavioral phenomena are a novel essential transverse bifurcation of hydraulic shocks to invading planar periodic roll-wave or doubly-transverse periodic herringbone patterns, with associated metastable behavior driven by mixed roll- and herringbone-type waves initiating from localized perturbation of an unstable constant state; and Floquet-type transverse ``flapping'' bifurcation of roll wave patterns.