Water-at-Rest Equilibrium Stability Analysis of a first-moment Shallow Water Exner Moment Model with Sediment Entrainment and Deposition: Extended Technical Report
This work addresses the analytical stability of a sediment transport model for geophysical flows, but the results are incremental as they do not resolve the weak hyperbolicity issue.
The authors derive the SWEMED1 model and analyze its water-at-rest equilibrium stability, finding weak hyperbolicity but no numerical instability, and propose a fast-slow scaling leading to a new equilibrium manifold. They identify the transport closure as the remaining issue and suggest directions for improved closures.
We derive the first-moment Shallow Water Exner Moment model with sediment entrainment and deposition (SWEMED1) and show that the full source term has a fully-settled water-at-rest equilibrium manifold. We prove that the model is only weakly hyperbolic at this equilibrium, which prevents the use of Yong's structural stability framework. However, a linear spectral analysis and numerical results do not indicate instability. Based on numerical results, we introduce a fast-slow scaling of the source term, and for the fast limit, we derive a new suspended water-at-rest equilibrium manifold, which has a different structure but is still only weakly hyperbolic. Our results show that the remaining obstruction is linked to the transport closure of the SWEMED1, and we give a constructive direction for the derivation of new closures leading to models with more desirable analytical properties.