Moment Approximations and Model Cascades for Shallow Flow

TL;DR

This paper introduces a moment-based approach to shallow flow modeling that retains vertical flow details, improving accuracy while maintaining computational efficiency, especially under certain physical conditions.

Contribution

It develops a generic hierarchy of shallow flow moment systems of arbitrary order derived from a fully resolved vertical model.

Findings

Shallow moment approximations reduce model error compared to standard shallow flow models.

Higher-order moments improve accuracy in specific flow regimes.

Numerical results demonstrate the potential for increased predictive power with minimal computational cost.

Abstract

Shallow flow models are used for a large number of applications including weather forecasting, open channel hydraulics and simulation-based natural hazard assessment. In these applications the shallowness of the process motivates depth-averaging. While the shallow flow formulation is advantageous in terms of computational efficiency, it also comes at the price of losing vertical information such as the flow's velocity profile. This gives rise to a model error, which limits the shallow flow model's predictive power and is often not explicitly quantifiable. We propose the use of vertical moments to overcome this problem. The shallow moment approximation preserves information on the vertical flow structure while still making use of the simplifying framework of depth-averaging. In this article, we derive a generic shallow flow moment system of arbitrary order starting from a set of balance laws, which has been reduced by scaling arguments. The derivation is based on a fully vertically resolved reference model with the vertical coordinate mapped onto the unit interval. We specify the shallow flow moment hierarchy for kinematic and Newtonian flow conditions and present 1D numerical results for shallow moment systems up to third order. Finally, we assess their performance with respect to both the standard shallow flow equations as well as with respect to the vertically resolved reference model. Our results show that depending on the parameter regime, e.g. friction and slip, shallow moment approximations significantly reduce the model error in shallow flow regimes and have a lot of potential to increase the predictive power of shallow flow models, while keeping them computationally cost efficient.