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The validity of two-dimensional models of a rotating shallow fluid layer

Published online by Cambridge University Press:  12 August 2020

David G. Dritschel Open the ORCID record for David G. Dritschel [Opens in a new window]  and
Mohammad Reza Jalali Open the ORCID record for Mohammad Reza Jalali [Opens in a new window]
David G. Dritschel*
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St AndrewsKY16 9SS, UK
Mohammad Reza Jalali
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St AndrewsKY16 9SS, UK
*
†Email address for correspondence: david.dritschel@st-andrews.ac.uk
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Abstract

We investigate the validity and accuracy of two commonly used two-dimensional models of a rotating, incompressible, shallow fluid layer: the shallow-water (SW) model and the Serre/Green–Naghdi (GN) model. The models differ in just one respect: the SW model imposes the hydrostatic approximation while the GN model does not. Both models assume the horizontal velocity is independent of height throughout the layer. We compare these models with their parent three-dimensional (3-D) model, initialised with a height-independent horizontal velocity, and with a mean height small compared to the domain width. For small to moderate Rossby and Froude numbers, we verify that both models well approximate the vertically averaged 3-D flow. Overall, the GN model is found to be substantially more accurate than the SW model. However, this accuracy is only achieved using an explicit reformulation of the equations in which the vertically integrated non-hydrostatic pressure is found from a novel linear elliptic equation. This reformulation is shown to extend straightforwardly to multi-layer flows having uniform density layers.

JFM classification

Geophysical and Geological Flows: Shallow water flows Mathematical Foundations: General fluid mechanics Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A33
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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