# American Institute of Mathematical Sciences

June  2019, 11(2): 139-151. doi: 10.3934/jgm.2019007

## Particle relabelling symmetries and Noether's theorem for vertical slice models

 1 Department of Mathematics, Imperial College London, London, SW7 2AZ, UK 2 Met Office, FitzRoy Road, Exeter, Devon, EX1 3PB, UK

* Corresponding author: C. J. Cotter

CJC is supported by NERC grant NE/K012533/1.
Michael John Priestley Cullen's contribution is Crown Copyright.

Received  December 2017 Revised  July 2018 Published  May 2019

Fund Project: This paper entilted "Particle relabelling symmetries and Noether's theorem for vertical slice models" is licensed under a Creative Commons Attribution 3.0 Unported License. See http://creativecommons.org/licenses/by/3.0/.

We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc Roy Soc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closed loops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can be derived directly from this circulation theorem. In this paper, we show that this property is due to these models having a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, not just those diffeomorphisms that preserve the potential temperature. This is developed using the methodology of Cotter and Holm (Foundations of Computational Mathematics, 2012).

Citation: Colin J. Cotter, Michael John Priestley Cullen. Particle relabelling symmetries and Noether's theorem for vertical slice models. Journal of Geometric Mechanics, 2019, 11 (2) : 139-151. doi: 10.3934/jgm.2019007
##### References:
 [1] C. J. Cotter and D. D. Holm, On Noether's theorem for the Euler–Poincaré equation on the diffeomorphism group with advected quantities, Foundations of Computational Mathematics, 13 (2013), 457-477.  doi: 10.1007/s10208-012-9126-8.  Google Scholar [2] C. J. Cotter and D. D. Holm, A variational formulation of vertical slice models, Proc. R. Soc. A, 469 (2013), 20120678, 17pp. doi: 10.1098/rspa.2012.0678.  Google Scholar [3] C. Cotter and D. Holm, Variational formulations of sound-proof models, Quarterly Journal of the Royal Meteorological Society, 140 (2014), 1966-1973.  doi: 10.1002/qj.2260.  Google Scholar [4] M. J. P. Cullen, A Mathematical Theory of Large-Scale Atmospheric Flow, Imperial College Press, 2006.   Google Scholar [5] M. J. P. Cullen, Modelling atmospheric flows, Acta Numerica, 16 (2007), 67-154.  doi: 10.1017/S0962492906290019.  Google Scholar [6] M. J. P. Cullen, A comparison of numerical solutions to the Eady frontogenesis problem, Q. J. R. Meteorol. Soc., 134 (2008), 2143-2155.  doi: 10.1002/qj.335.  Google Scholar [7] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler–Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar [8] M. Oliver, Variational asymptotics for rotating shallow water near geostrophy: A transformational approach, Journal of Fluid Mechanics, 551 (2006), 197-234.  doi: 10.1017/S0022112005008256.  Google Scholar [9] A. R. Visram, C. J. Cotter and M. J. P. Cullen, A framework for evaluating model error using asymptotic convergence in the eady model, Quarterly Journal of the Royal Meteorological Society, 140 (2014), 1629-1639.  doi: 10.1002/qj.2244.  Google Scholar

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##### References:
 [1] C. J. Cotter and D. D. Holm, On Noether's theorem for the Euler–Poincaré equation on the diffeomorphism group with advected quantities, Foundations of Computational Mathematics, 13 (2013), 457-477.  doi: 10.1007/s10208-012-9126-8.  Google Scholar [2] C. J. Cotter and D. D. Holm, A variational formulation of vertical slice models, Proc. R. Soc. A, 469 (2013), 20120678, 17pp. doi: 10.1098/rspa.2012.0678.  Google Scholar [3] C. Cotter and D. Holm, Variational formulations of sound-proof models, Quarterly Journal of the Royal Meteorological Society, 140 (2014), 1966-1973.  doi: 10.1002/qj.2260.  Google Scholar [4] M. J. P. Cullen, A Mathematical Theory of Large-Scale Atmospheric Flow, Imperial College Press, 2006.   Google Scholar [5] M. J. P. Cullen, Modelling atmospheric flows, Acta Numerica, 16 (2007), 67-154.  doi: 10.1017/S0962492906290019.  Google Scholar [6] M. J. P. Cullen, A comparison of numerical solutions to the Eady frontogenesis problem, Q. J. R. Meteorol. Soc., 134 (2008), 2143-2155.  doi: 10.1002/qj.335.  Google Scholar [7] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler–Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar [8] M. Oliver, Variational asymptotics for rotating shallow water near geostrophy: A transformational approach, Journal of Fluid Mechanics, 551 (2006), 197-234.  doi: 10.1017/S0022112005008256.  Google Scholar [9] A. R. Visram, C. J. Cotter and M. J. P. Cullen, A framework for evaluating model error using asymptotic convergence in the eady model, Quarterly Journal of the Royal Meteorological Society, 140 (2014), 1629-1639.  doi: 10.1002/qj.2244.  Google Scholar
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