Linear Dynamics of the Semi-geostrophic Equations in Eulerian Coordinates on R3

Stefania Lisai; Mark Wilkinson

doi:10.1007/s00021-021-00574-2

Linear Dynamics of the Semi-geostrophic Equations in Eulerian Coordinates on R³

Stefania Lisai^*, Mark Wilkinson

^*Corresponding author for this work

School of Mathematical & Computer Sciences

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Abstract

We consider a class of steady solutions of the semi-geostrophic equations on R³ and derive the linearised dynamics around those solutions. The linear PDE which governs perturbations around those steady states is a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in L²(R³, R³) introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on R³. We investigate stability of the steady solutions of the semi-geostrophic equations by looking at plane wave solutions of the associated linearised problem, and discuss differences in the case of the quasi-geostrophic equations.

Original language	English
Article number	54
Journal	Journal of Mathematical Fluid Mechanics
Volume	23
Issue number	3
Early online date	5 May 2021
DOIs	https://doi.org/10.1007/s00021-021-00574-2
Publication status	Published - Aug 2021

Keywords

Atmospheric/oceanic fluid dynamics
Linear stability
Quasi-geostrophic equations
Semi-geostrophic equations

ASJC Scopus subject areas

Mathematical Physics
Condensed Matter Physics
Computational Mathematics
Applied Mathematics

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Cite this

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title = "Linear Dynamics of the Semi-geostrophic Equations in Eulerian Coordinates on R3",

abstract = "We consider a class of steady solutions of the semi-geostrophic equations on R3 and derive the linearised dynamics around those solutions. The linear PDE which governs perturbations around those steady states is a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in L2(R3, R3) introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on R3. We investigate stability of the steady solutions of the semi-geostrophic equations by looking at plane wave solutions of the associated linearised problem, and discuss differences in the case of the quasi-geostrophic equations.",

keywords = "Atmospheric/oceanic fluid dynamics, Linear stability, Quasi-geostrophic equations, Semi-geostrophic equations",

author = "Stefania Lisai and Mark Wilkinson",

note = "Funding Information: S. Lisai was supported by The Maxwell Institute Graduate School in Analysis and its Applications (MIGSAA) funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. M. Wilkinson was supported by the EPSRC Standard Grant EP/P011543/1. The authors would like to thank Mike Cullen, Beatrice Pelloni and Jacques Vanneste for helpful discussions on this project. Publisher Copyright: {\textcopyright} 2021, The Author(s).",

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AU - Lisai, Stefania

AU - Wilkinson, Mark

N1 - Funding Information: S. Lisai was supported by The Maxwell Institute Graduate School in Analysis and its Applications (MIGSAA) funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. M. Wilkinson was supported by the EPSRC Standard Grant EP/P011543/1. The authors would like to thank Mike Cullen, Beatrice Pelloni and Jacques Vanneste for helpful discussions on this project. Publisher Copyright: © 2021, The Author(s).

PY - 2021/8

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N2 - We consider a class of steady solutions of the semi-geostrophic equations on R3 and derive the linearised dynamics around those solutions. The linear PDE which governs perturbations around those steady states is a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in L2(R3, R3) introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on R3. We investigate stability of the steady solutions of the semi-geostrophic equations by looking at plane wave solutions of the associated linearised problem, and discuss differences in the case of the quasi-geostrophic equations.

AB - We consider a class of steady solutions of the semi-geostrophic equations on R3 and derive the linearised dynamics around those solutions. The linear PDE which governs perturbations around those steady states is a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in L2(R3, R3) introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on R3. We investigate stability of the steady solutions of the semi-geostrophic equations by looking at plane wave solutions of the associated linearised problem, and discuss differences in the case of the quasi-geostrophic equations.

KW - Atmospheric/oceanic fluid dynamics

KW - Linear stability

KW - Quasi-geostrophic equations

KW - Semi-geostrophic equations

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