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Local Mass Conservation and Velocity Splitting in PV-Based Balanced Models. Part I: The Hyperbalance Equations

Ali R. MohebalhojehSchool of Mathematics and Statistics, University of St Andrews, St Andrews, United Kingdom, and Institute of Geophysics, University of Tehran, Tehran, Iran

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Michael E. McIntyreCentre for Atmospheric Science,* Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge, United Kingdom

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Abstract

This paper considers stratified and shallow water non-Hamiltonian potential-vorticity-based balanced models (PBMs). These are constructed using the exact (Rossby or Rossby–Ertel) potential vorticity (PV). The most accurate known PBMs are those studied by McIntyre and Norton and by Mohebalhojeh and Dritschel. It is proved that, despite their astonishing accuracy, these PBMs all fail to conserve mass locally. Specifically, they exhibit velocity splitting in the sense of having two velocity fields, v and vm, the first to advect PV and the second to advect mass. The difference vvm is nonzero in general, even if tiny. Unlike the different velocity splitting found in all Hamiltonian balanced models, the present splitting can be healed. The result is a previously unknown class of balanced models, here called “hyperbalance equations,” whose formal orders of accuracy can be made as high as those of any other PBM. The hyperbalance equations use a single velocity field v to advect mass as well as to advect and evaluate the exact PV.

* The Centre for Atmospheric Science is a joint initiative of the Department of Chemistry and the Department of Applied Mathematics and Theoretical Physics (more information available online at www.atm.damtp.cam.ac.uk/)

Corresponding author address: A. R. Mohebalhojeh, School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews KY16 9SS, United Kingdom. Email: arm@mcs.st-and.ac.uk

Abstract

This paper considers stratified and shallow water non-Hamiltonian potential-vorticity-based balanced models (PBMs). These are constructed using the exact (Rossby or Rossby–Ertel) potential vorticity (PV). The most accurate known PBMs are those studied by McIntyre and Norton and by Mohebalhojeh and Dritschel. It is proved that, despite their astonishing accuracy, these PBMs all fail to conserve mass locally. Specifically, they exhibit velocity splitting in the sense of having two velocity fields, v and vm, the first to advect PV and the second to advect mass. The difference vvm is nonzero in general, even if tiny. Unlike the different velocity splitting found in all Hamiltonian balanced models, the present splitting can be healed. The result is a previously unknown class of balanced models, here called “hyperbalance equations,” whose formal orders of accuracy can be made as high as those of any other PBM. The hyperbalance equations use a single velocity field v to advect mass as well as to advect and evaluate the exact PV.

* The Centre for Atmospheric Science is a joint initiative of the Department of Chemistry and the Department of Applied Mathematics and Theoretical Physics (more information available online at www.atm.damtp.cam.ac.uk/)

Corresponding author address: A. R. Mohebalhojeh, School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews KY16 9SS, United Kingdom. Email: arm@mcs.st-and.ac.uk

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