Ill posedness in shallow multi-phase debris flow models

TL;DR

This paper analyzes the stability of multi-phase debris flow models, revealing that many are ill posed and proposing conditions under which adding diffusive terms can regularize them.

Contribution

It provides a comprehensive analysis of ill posedness in multi-phase debris flow models and develops a general framework for detecting and regularizing such issues.

Findings

Many multi-phase models are ill posed over relevant parameters.

Small diffusive terms can eliminate ill posedness.

Conditions for regularization are derived but often not met by existing models.

Abstract

Depth-averaged systems of equations describing the motion of fluid-sediment mixtures have been widely adopted by scientists in pursuit of models that can predict the paths of dangerous overland flows of debris. As models have become increasingly sophisticated, many have been developed from a multi-phase perspective in which separate, but mutually coupled sets of equations govern the evolution of different components of the mixture. However, this creates the opportunity for the existence of pathological instabilities stemming from resonant interactions between the phases. With reference to the most popular approaches, analyses of two- and three-phase models are performed, which demonstrate that they are more often than not ill posed as initial value problems over physically relevant parameter regimes - an issue which renders them unsuitable for scientific applications. Additionally, a general framework for detecting ill posedness in models with any number of phases is developed. This is used to show that small diffusive terms in the equations for momentum transport, which are sometimes neglected, can reliably eliminate this issue. Conditions are derived for the regularisation of models in this way, but they are typically not met by multi-phase models that feature diffusive terms.