On coupling resolved and unresolved physical processes in finite element discretisations of geophysical fluids

At the heart of modern numerical weather forecasting and climate modelling lie simulations of two geophysical fluids: the atmosphere and the ocean. These endeavours rely on numerically solving the equations that describe these fluids. A key challenge is that the fluids contain motions spanning a range of scales. As the small-scale processes (unresolved by the numerical model) affect the resolved motions, they need to be described in the model, which is known as parametrisation. One major class of methods for numerically solving such partial differential equations is the finite element method. This thesis focuses on the coupling of such parametrised processes to the resolved flow within finite element discretisations. Four sets of research are presented, falling under two main categories. The first is the development of a compatible finite element discretisation for use in numerical weather prediction models, so as to avoid the bottleneck in computational scalability associated with the convergence at the poles of latitude-longitude grids. We present a transport scheme for use with the lowest-order function spaces in such a compatible finite element method, which is motivated by the coupling of the resolved and unresolved processes within the model. This then facilitates the use of the lower-order spaces within Gusto, a toolkit for studying such compatible finite element discretisations. Then, we present a compatible finite element discretisation of the moist compressible Euler equations, parametrising the unresolved moist processes. This is a major step in the development of Gusto, extending it to describe its first unresolved processes. The second category with which this thesis is concerned is the stochastic variational framework presented by Holm [Variational principles for stochastic fluid dynamics, P. Roy. Soc. A-Math. Phy. 471 (2176), (2015)]. In this framework, the effect of the unresolved processes and their uncertainty is expressed through a stochastic component to the advecting velocity. This framework ensures the circulation theorem is preserved by the stochastic equations. We consider the application of this formulation to two simple geophysical fluid models. First, we discuss the statistical properties of an enstrophy-preserving finite element discretisation of the stochastic quasi-geostrophic equation. We find that the choice of discretisation and the properties that it preserves affects the statistics of the solution. The final research presented is a finite element discretisation of the stochastic Camassa-Holm equation, which is used to numerically investigate the formation of ‘peakons’ within this set-up, finding that they do still always form despite the noise’s presence.Open Acces

The design of conservative finite element discretisations for the vectorial modified KdV equation

We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting

A numerical study of variational discretizations of the Camassa-Holm equation

We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa-Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare the variational discretizations to existing methods over several numerical examples.Comment: 45 pages, 14 figure