Moment Approximations and Model Cascades for Shallow Flow (Communications in Computational Physics, 2018)
Shallow flow models are used for a large number of applications includingweather forecasting, open channel hydraulics and simulation-based natural hazardassessment. In these applications the shallowness of the process motivates depth-averaging.While the shallow flow formulation is advantageous in terms of computationalefficiency, it also comes at the price of losing vertical information such as theflow's velocity profile. This gives rise to a model error, which limits the shallow flowmodel's predictive power and is often not explicitly quantifiable. We propose the useof vertical moments to overcome this problem. The shallow moment approximationpreserves information on the vertical flow structure while still making use of the simplifyingframework of depth-averaging. In this article, we derive a generic shallowflow moment system of arbitrary order starting from a set of balance laws, which hasbeen reduced by scaling arguments. The derivation is based on a fully vertically resolvedreference model with the vertical coordinate mapped onto the unit interval. Wespecify the shallow flow moment hierarchy for kinematic and Newtonian flow conditionsand present 1D numerical results for shallow moment systems up to third order.Finally, we assess their performance with respect to both the standard shallow flowequations as well as with respect to the vertically resolved reference model. Our resultsshow that depending on the parameter regime, e.g. friction and slip, shallowmoment approximations significantly reduce the model error in shallow flow regimesand have a lot of potential to increase the predictive power of shallow flow models,while keeping them computationally cost efficient.