# Analysis of models for large-scale geophysical flows

Lead Research Organisation:
Heriot-Watt University

Department Name: S of Mathematical and Computer Sciences

### Abstract

The rigorous analysis of the highly nonlinear equations that model atmospheric and oceanic flows is a very difficult task, generally beyond the power of current mathematical tools. This is certainly true of the full governing equations, the famous compressible Navier-Stokes equations (called Euler equations when viscosity is neglected), whose solution is too complicated to compute, even numerically. Indeed, in practice the modelling that informs applications, such as forecasting the weather, is based on averaged versions or simplified reductions of the governing equations. While such equations are used ubiquitously to model complex physical phenomena and to perform numerical approximations, both the solvability of the models and the validity of the approximations computed rests largely on heuristics rather than on rigorous mathematical ground.

This proposal concerns a particular system of equations, the semi-geostrophic system, that models the large-scale dynamics of inviscid geophysical flows.

The importance of this particular model rests on the fact that, as an asymptotic reduction, the system is expected to be a more accurate approximation to the full model than other reductions used in practice. In addition the validity of this reduction persists also when certain parameters, for example the earth rotation coefficient, are taken to be variable. For this reason, the model can approximate the large-scale dynamics of the flow more accurately than models whose solutions are assumed close to a uniform reference state. Mathematically, the semi-geostrophic system supports singular solutions, thus it can capture rigorously phenomena such as front formation. This is important in view of the fact that the physical derivation of this system was guided precisely by the need to model the formation of atmospheric fronts.

The mathematical interest in the semi-geostrophic model has been revived by the discovery that a specific change of variables, well known to practitioners, transforms it into a system that can be analysed rigorously by using modern techniques of variational analysis and optimal transport theory. Activity in these areas in the past twenty years has seen very important results and advances, depending on delicate and sophisticated mathematical tools. The overarching aim of this project is to adapt and translate these techniques and, using new recent insights, to obtain results on the existence and uniqueness of solutions of the semigeostrophic system in increasingly realistic cases. The research proposed also aims at proving the validity and asymptotic order of the system as a reduction of the Euler equations, thus putting on rigorous foundations the numerical and physical modelling based on these equations.

This proposal concerns a particular system of equations, the semi-geostrophic system, that models the large-scale dynamics of inviscid geophysical flows.

The importance of this particular model rests on the fact that, as an asymptotic reduction, the system is expected to be a more accurate approximation to the full model than other reductions used in practice. In addition the validity of this reduction persists also when certain parameters, for example the earth rotation coefficient, are taken to be variable. For this reason, the model can approximate the large-scale dynamics of the flow more accurately than models whose solutions are assumed close to a uniform reference state. Mathematically, the semi-geostrophic system supports singular solutions, thus it can capture rigorously phenomena such as front formation. This is important in view of the fact that the physical derivation of this system was guided precisely by the need to model the formation of atmospheric fronts.

The mathematical interest in the semi-geostrophic model has been revived by the discovery that a specific change of variables, well known to practitioners, transforms it into a system that can be analysed rigorously by using modern techniques of variational analysis and optimal transport theory. Activity in these areas in the past twenty years has seen very important results and advances, depending on delicate and sophisticated mathematical tools. The overarching aim of this project is to adapt and translate these techniques and, using new recent insights, to obtain results on the existence and uniqueness of solutions of the semigeostrophic system in increasingly realistic cases. The research proposed also aims at proving the validity and asymptotic order of the system as a reduction of the Euler equations, thus putting on rigorous foundations the numerical and physical modelling based on these equations.

### Planned Impact

The impact of this work will be realised, first and foremost, in an interdisciplinary translation of advanced and sophisticated mathematical techniques into tools and applications in applied physics and meteorology. This will be assisted by the PI's connections to both communities, and by the visibility and activities of the Maxwell Institute in Edinburgh, where we plan to organise several short events on the topic of this proposal.

The importance of a rigorous understanding of the models underpinning the operational models of weather forecasting centres such as the MET Office or ECMWF (a European weather forecasting centre based in the UK) cannot be overestimated. These centres compete internationally for the fastest and most accurate models, while having to reduce their computational costs (and the energy consumption of the computational devices, which is becoming increasingly prohibitive). While it is beyond the possibility of current mathematical methods to understand these models in the full generality used in practice, mathematical tools and ideas arising in the analysis of even simplified version of the models have provided and continue to provide new ways to increase efficiency of the numerical approximation, sometimes entirely unexpected. An example is the use of optimal transport techniques to "move meshes" to generate the optimal computational grid for a specific computation, a technique that has exciting potential.

The increased flow of information between the community of applied physicists and modellers, and the community of pure mathematical analysts has reaped some tangible and important rewards, that have put the UK at the frontline of the efforts to use and translate mathematical results into enhanced techniques and improved understanding of physical phenomena and their modelling and prediction. The visibility of the EPSRC-funded CDT "Mathematics of Planet Earth", with which the PI maintains strong links, and the amount of interests it has attracted internationally put the UK in an optimal position for preparing and supporting a new generation of interdisciplinary mathematical scientists at ease with mathematically advanced results but also aware of the specific results needed or useful in applications. The impact of this project is a contribution to this important effort.

The importance of a rigorous understanding of the models underpinning the operational models of weather forecasting centres such as the MET Office or ECMWF (a European weather forecasting centre based in the UK) cannot be overestimated. These centres compete internationally for the fastest and most accurate models, while having to reduce their computational costs (and the energy consumption of the computational devices, which is becoming increasingly prohibitive). While it is beyond the possibility of current mathematical methods to understand these models in the full generality used in practice, mathematical tools and ideas arising in the analysis of even simplified version of the models have provided and continue to provide new ways to increase efficiency of the numerical approximation, sometimes entirely unexpected. An example is the use of optimal transport techniques to "move meshes" to generate the optimal computational grid for a specific computation, a technique that has exciting potential.

The increased flow of information between the community of applied physicists and modellers, and the community of pure mathematical analysts has reaped some tangible and important rewards, that have put the UK at the frontline of the efforts to use and translate mathematical results into enhanced techniques and improved understanding of physical phenomena and their modelling and prediction. The visibility of the EPSRC-funded CDT "Mathematics of Planet Earth", with which the PI maintains strong links, and the amount of interests it has attracted internationally put the UK in an optimal position for preparing and supporting a new generation of interdisciplinary mathematical scientists at ease with mathematically advanced results but also aware of the specific results needed or useful in applications. The impact of this project is a contribution to this important effort.

### Publications

Cullen MJP
(2019)

*The Stability Principle and global weak solutions of the free surface semi-geostrophic equations in geostrophic coordinates.*in Proceedings. Mathematical, physical, and engineering sciences
Lisai S
(2019)

*Smooth solutions of the surface semi-geostrophic equations*in Calculus of Variations and Partial Differential Equations
Wilkinson M
(2019)

*On the Non-uniqueness of Physical Scattering for Hard Non-spherical Particles*in Archive for Rational Mechanics and AnalysisDescription | Two new techniques fo rapproachign the solution fo this PDE system have been studied. The first is based on a different formulation of the equations. The second is based on a discrete version of optimal transport - a way to minimise discrete energy |

Exploitation Route | Powerful new techniques for studying PDEs |

Sectors | Aerospace, Defence and Marine |