## Publications

133 results found

Bock A, Cotter CJ, Kirby RC, 2024, Planar curve registration using Bayesian inversion, *Computers and Mathematics with Applications*, Vol: 159, Pages: 155-172, ISSN: 0898-1221

We study parameterisation-independent closed planar curve matching as a Bayesian inverse problem. The motion of the curve is modelled via a curve on the diffeomorphism group acting on the ambient space, leading to a large deformation diffeomorphic metric mapping (LDDMM) functional penalising the kinetic energy of the deformation. We solve Hamilton's equations for the curve matching problem using the Wu-Xu element (Wu and Xu (2019) [12]) which provides mesh-independent Lipschitz constants for the forward motion of the curve, and solve the inverse problem for the momentum using Bayesian inversion. Since this element is not affine-equivalent we provide a pullback theory which expedites the implementation and efficiency of the forward map. We adopt ensemble Kalman inversion (EKI) using a negative Sobolev norm mismatch penalty to measure the discrepancy between the target and the ensemble mean shape. We provide several numerical examples to validate the approach.

Benamou JD, Cotter CJ, Malamut H, 2024, Entropic optimal transport solutions of the semigeostrophic equations, *Journal of Computational Physics*, Vol: 500, ISSN: 0021-9991

The semigeostrophic equations are a frontogenesis model in atmospheric science. Existence of solutions both from the theoretical and numerical point of view is given under a change of variable involving the interpretation of the pressure gradient as an optimal transport map between the density of the fluid and its push forward. Thanks to recent advances in numerical optimal transportation, the computation of large scale discrete approximations can be envisioned. We study here the use of entropic optimal transport and its Sinkhorn Algorithm companion.

Cotter CJ, Shipton J, 2023, A compatible finite element discretisation for the nonhydrostatic vertical slice equations, *GEM: International Journal on Geomathematics*, Vol: 14, ISSN: 1869-2672

We present a compatible finite element discretisation for the vertical slice compressible Euler equations, at next-to-lowest order (i.e., the pressure space is bilinear discontinuous functions). The equations are numerically integrated in time using a fully implicit timestepping scheme which is solved using monolithic GMRES preconditioned by a linesmoother. The linesmoother only involves local operations and is thus suitable for domain decomposition in parallel. It allows for arbitrarily large timesteps but with iteration counts scaling linearly with Courant number in the limit of large Courant number. This solver approach is implemented using Firedrake, and the additive Schwarz preconditioner framework of PETSc. We demonstrate the robustness of the scheme using a standard set of testcases that may be compared with other approaches.

Cotter CJ, Holm DD, Pryer T, 2023, Singular solutions of the r-Camassa-Holm equation, *Nonlinearity*, Vol: 36, Pages: 6199-6223, ISSN: 0951-7715

This paper introduces the r-Camassa-Holm (r-CH) equation, which describes a geodesic flow on the manifold of diffeomorphisms acting on the real line induced by the W 1 , r metric. The conserved energy for the problem is given by the full W 1 , r norm. For r = 2, we recover the Camassa-Holm equation. We compute the Lie symmetries for r-CH and study various symmetry reductions. We introduce singular weak solutions of the r-CH equation for r ⩾ 2 and demonstrates their robustness in numerical simulations of their nonlinear interactions in both overtaking and head-on collisions. Several open questions are formulated about the unexplored properties of the r-CH weak singular solutions, including the question of whether they would emerge from smooth initial conditions.

Cecil T, Albert C, Anderson AJ,
et al., 2023, Fabrication Development for SPT-SLIM, a Superconducting Spectrometer for Line Intensity Mapping, *IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY*, Vol: 33, ISSN: 1051-8223

Cotter CJ, Kirby RC, Morris H, 2023, Weighted-norm preconditioners for a multilayer tide model, *SIAM Journal on Scientific Computing*, Vol: 45, Pages: A1789-A1811, ISSN: 1064-8275

We derive a linearized rotating shallow water system modeling tides, which can be discretized by mixed finite elements. Unlike previous models, this model allows for multiple layers stratified by density. Like the single-layer case [R. C. Kirby and T. Kernell, Comput. Math. Appl., 82 (2021), pp. 212–227], a weighted-norm preconditioner gives a (nearly) parameter-robust method for solving the resulting linear system at each time step, but the all-to-all coupling between the layers in the model poses a significant challenge to efficiency. Neglecting the inter-layer coupling gives a preconditioner that degrades rapidly as the number of layers increases. By a careful analysis of the matrix that couples the layers, we derive a robust method that requires solving a reformulated system that only involves coupling between adjacent layers. Numerical results obtained using Firedrake [F. Rathgeber et al., ACM Trans. Math. Software, 43 (2016), 24] confirm the theory.

Betteridge JDD, Cotter CJJ, Gibson THH,
et al., 2023, Hybridised multigrid preconditioners for a compatible finite-element dynamical core, *Quarterly Journal of the Royal Meteorological Society*, Vol: 149, Pages: 2454-2476, ISSN: 0035-9009

Compatible finite-element discretisations for the atmospheric equations of motion have recently attracted considerable interest. Semi-implicit timestepping methods require the repeated solution of a large saddle-point system of linear equations. Preconditioning this system is challenging, since the velocity mass matrix is nondiagonal, leading to a dense Schur complement. Hybridisable discretisations overcome this issue: weakly enforcing continuity of the velocity field with Lagrange multipliers leads to a sparse system of equations, which has a similar structure to the pressure Schur complement in traditional approaches. We describe how the hybridised sparse system can be preconditioned with a non-nested two-level preconditioner. To solve the coarse system, we use the multigrid pressure solver that is employed in the approximate Schur complement method previously proposed by the some of the authors. Our approach significantly reduces the number of solver iterations. The method shows excellent performance and scales to large numbers of cores in the Met Office next-generation climate and weather prediction model LFRic.

Yamazaki H, Cotter CJ, Wingate BA, 2023, Time-parallel integration and phase averaging for the nonlinear shallow-water equations on the sphere, *Quarterly Journal of the Royal Meteorological Society*, Vol: 149, Pages: 2504-2513, ISSN: 0035-9009

We describe a proof-of-concept development and application of a phase-averaging technique to the nonlinear rotating shallow-water equations on the sphere, discretised using compatible finite-element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables), so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: (1) we have developed a stable matrix exponential specific to finite elements and (2) we have developed a parallel finite averaging procedure. Following recent studies, we consider finite-width phase-averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm that there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi-implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work.

Ham DA, Kelly PHJ, Mitchell L, et al., 2023, Firedrake user manual, Firedrake User Manual

Cotter CJJ, 2023, Compatible finite element methods for geophysical fluid dynamics, *Acta Numerica*, Vol: 32, Pages: 291-393, ISSN: 0962-4929

This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

Bendall TM, Wood N, Thuburn J,
et al., 2023, A solution to the trilemma of the moist Charney–Phillips staggering, *Quarterly Journal of the Royal Meteorological Society*, Vol: 149, Pages: 262-276, ISSN: 0035-9009

The Charney–Phillips grid, used in many numerical models of the atmosphere, involves vertically staggering the nodes of the density variable with the nodes of the entropy-type variable. When moisture is included in such a model, it is either co-located with density so that moisture can be transported conservatively and consistently with dry mass, or with the entropy-type variable so that the coupling between moisture and temperature can be represented well. Both properties are desirable, yet at first it appears difficult to obtain both simultaneously. Here, we present a framework to resolve this problem, by co-locating the moisture mixing ratio with potential temperature but formulating its transport as that of a density on a vertically shifted mesh. Within this framework, particular choices of the operators involved provide the desired conservation and consistency properties of the moisture transport. The framework is described in the context of a finite-element approach. We also present an explicit Runge–Kutta time-stepping scheme that is appropriate for use within this framework. This approach is then illustrated through numerical tests, which demonstrate that it does indeed have the desired conservation and consistency properties.

Clare MCA, Wallwork JG, Kramer SC,
et al., 2022, Multi-scale hydro-morphodynamic modelling using mesh movement methods, *GEM: International Journal on Geomathematics*, Vol: 13, ISSN: 1869-2672

Hydro-morphodynamic modelling is an important tool that can be used in the protection of coastal zones. The models can be required to resolve spatial scales ranging from sub-metre to hundreds of kilometres and are computationally expensive. In this work, we apply mesh movement methods to a depth-averaged hydro-morphodynamic model for the first time, in order to tackle both these issues. Mesh movement methods are particularly well-suited to coastal problems as they allow the mesh to move in response to evolving flow and morphology structures. This new capability is demonstrated using test cases that exhibit complex evolving bathymetries and have moving wet-dry interfaces. In order to be able to simulate sediment transport in wet-dry domains, a new conservative discretisation approach has been developed as part of this work, as well as a sediment slide mechanism. For all test cases, we demonstrate how mesh movement methods can be used to reduce discretisation error and computational cost. We also show that the optimum parameter choices in the mesh movement monitor functions are fairly predictable based upon the physical characteristics of the test case, facilitating the use of mesh movement methods on further problems.

Egan CP, Bourne DP, Cotter CJ,
et al., 2022, A new implementation of the geometric method for solving the Eady slice equations, *Journal of Computational Physics*, Vol: 469, Pages: 1-30, ISSN: 0021-9991

We present a new implementation of the geometric method of Cullen & Purser (1984) for solving the semi-geostrophic Eady slice equations, which model large scale atmospheric flows and frontogenesis. The geometric method is a Lagrangian discretisation, where the PDE is approximated by a particle system. An important property of the discretisation is that it is energy conserving. We restate the geometric method in the language of semi-discrete optimal transport theory and exploit this to develop a fast implementation that combines the latest results from numerical optimal transport theory with a novel adaptive time-stepping scheme. Our results enable a controlled comparison between the Eady-Boussinesq vertical slice equations and their semi-geostrophic approximation. We provide further evidence that weak solutions of the Eady-Boussinesq vertical slice equations converge to weak solutions of the semi-geostrophic Eady slice equations as the Rossby number tends to zero.

Bauer W, Cotter C, Wingate B, 2022, Higher order phase averaging for highly oscillatory systems, *SIAM: Multiscale Modeling and Simulation*, Vol: 20, Pages: 936-956, ISSN: 1540-3459

We introduce a higher order phase averaging method for nonlinear oscillatory systems.Phase averaging is a technique to filter fast motions from the dynamics while still accounting fortheir effect on the slow dynamics. Phase averaging is useful for deriving reduced models that canbe solved numerically with more efficiency, since larger timesteps can be taken. Recently, Hautand Wingate [SIAM J. Sci. Comput., 36 (2014) pp. A693–A713] introduced the idea of computingfinite window numerical phase averages in parallel as the basis for a coarse propagator for a parallel-in-time algorithm. In this contribution, we provide a framework for higher order phase averagesthat aims to better approximate the unaveraged system while still filtering fast motions. While thebasic phase average assumes that the solution is independent of changes of phase, the higher ordermethod expands the phase dependency in a basis which the equations are projected onto. In this newframework, the original numerical phase averaging formulation arises as the lowest order version ofthis expansion in which the nonlinearity is projected onto the space of functions that are independentof the phase. Our new projection onto functions that arekth degree polynomials in the phase givesrise to higher order corrections to the phase averaging formulation. We illustrate the properties ofthis method on an ODE that describes the dynamics of a swinging spring due to Lynch (2002).Although idealized, this model shows an interesting analogy to geophysical flows as it exhibits a slowdynamics that arises through the resonance between fast oscillations. On this example, we showconvergence to the nonaveraged (exact) solution with increasing approximation order also for finiteaveraging windows. At zeroth order, our method coincides with a standard phase average, but athigher order it is more accurate in the sense that solutions of the phase averaged

Clare MCA, Leijnse TWB, McCall RT,
et al., 2022, Multilevel multifidelity Monte Carlo methods for assessing uncertainty in coastal flooding, *Natural Hazards and Earth System Sciences*, Vol: 22, Pages: 2491-2515, ISSN: 1561-8633

When choosing an appropriate hydrodynamic model, there is always a compromise between accuracy and computational cost, with high-fidelity models being more expensive than low-fidelity ones. However, when assessing uncertainty, we can use a multifidelity approach to take advantage of the accuracy of high-fidelity models and the computational efficiency of low-fidelity models. Here, we apply the multilevel multifidelity Monte Carlo method (MLMF) to quantify uncertainty by computing statistical estimators of key output variables with respect to uncertain input data, using the high-fidelity hydrodynamic model XBeach and the lower-fidelity coastal flooding model SFINCS (Super-Fast INundation of CoastS). The multilevel aspect opens up the further advantageous possibility of applying each of these models at multiple resolutions. This work represents the first application of MLMF in the coastal zone and one of its first applications in any field. For both idealised and real-world test cases, MLMF can significantly reduce computational cost for the same accuracy compared to both the standard Monte Carlo method and to a multilevel approach utilising only a single model (the multilevel Monte Carlo method). In particular, here we demonstrate using the case of Myrtle Beach, South Carolina, USA, that this improvement in computational efficiency allows for in-depth uncertainty analysis to be conducted in the case of real-world coastal environments – a task that would previously have been practically unfeasible. Moreover, for the first time, we show how an inverse transform sampling technique can be used to accurately estimate the cumulative distribution function (CDF) of variables from the MLMF outputs. MLMF-based estimates of the expectations and the CDFs of the variables of interest are of significant value to decision makers when assessing uncertainty in predictions.

Yamazaki H, Weller H, Cotter CJ,
et al., 2022, Conservation with moving meshes over orography, *Journal of Computational Physics*, Vol: 461, ISSN: 0021-9991

Adaptive meshes have the potential to improve the accuracy and efficiency of atmospheric modelling by increasing resolution where it is most needed. Mesh re-distribution, or r-adaptivity, adapts by moving the mesh without changing the connectivity. This avoids some of the challenges with h-adaptivity (adding and removing points): the solution does not need to be mapped between meshes, which can be expensive and introduces errors, and there are no load balancing problems on parallel computers. A long standing problem with both forms of adaptivity has been changes in volume of the domain as resolution changes at an uneven boundary. We propose a solution which achieves exact local conservation and maintains a uniform scalar field while the mesh changes volume as it moves over orography. This is achieved by introducing a volume correction parameter which tracks the cell volumes without using expensive conservative mapping.A finite volume solution of the advection equation over orography on moving meshes is described and results are presented demonstrating improved accuracy for cost using moving meshes. Exact local conservation and maintenance of uniform scalar fields is demonstrated and the correct mesh volume is preserved.We use optimal transport to generate meshes which are guaranteed not to tangle and are equidistributed with respect to a monitor function. This leads to a Monge-Ampère equation which is solved with a Newton solver. The superiority of the Newton solver over other techniques is demonstrated in the appendix. However the Newton solver is only efficient if it is applied to the left hand side of the Monge-Ampère equation with fixed point iterations for the right hand side.

Clare MCA, Piggott MD, Cotter CJ, 2022, Assessing erosion and flood risk in the coastal zone through the application of multilevel Monte Carlo methods, *Coastal Engineering*, Vol: 174, ISSN: 0378-3839

Coastal zones are vulnerable to both erosion and flood risk, which can be assessed using coupled hydro-morphodynamic models. However, the use of such models as decision support tools suffers from a high degreeof uncertainty, due to both incomplete knowledge and natural variability in the system. In this work, we showfor the first time how the multilevel Monte Carlo method (MLMC) can be applied in hydro-morphodynamiccoastal ocean modelling, here using the popular model XBeach, to quantify uncertainty by computing statisticsof key output variables given uncertain input parameters. MLMC accelerates the Monte Carlo approach throughthe use of a hierarchy of models with different levels of resolution. Several theoretical and real-world coastalzone case studies are considered here, for which output variables that are key to the assessment of flood anderosion risk, such as wave run-up height and total eroded volume, are estimated. We show that MLMC cansignificantly reduce computational cost, resulting in speed up factors of 40 or greater compared to a standardMonte Carlo approach, whilst keeping the same level of accuracy. Furthermore, a sophisticated ensemblegenerating technique is used to estimate the cumulative distribution of output variables from the MLMC output.This allows for the probability of a variable exceeding a certain value to be estimated, such as the probabilityof a wave run-up height exceeding the height of a seawall. This is a valuable capability that can be used toinform decision-making under uncertainty

Clare MCA, Kramer SC, Cotter CJ,
et al., 2022, Calibration, inversion and sensitivity analysis for hydro-morphodynamic models through the application of adjoint methods, *Computers and Geosciences*, Vol: 163, Pages: 1-13, ISSN: 0098-3004

The development of reliable, sophisticated hydro-morphodynamic models is essential for protecting the coastal environment against hazards such as flooding and erosion. There exists a high degree of uncertainty associated with the application of these models, in part due to incomplete knowledge of various physical, empirical and numerical closure related parameters in both the hydrodynamic and morphodynamic solvers. This uncertainty can be addressed through the application of adjoint methods. These have the notable advantage that the number and/or dimension of the uncertain parameters has almost no effect on the computational cost associated with calculating the model sensitivities. Here, we develop the first freely available and fully flexible adjoint hydro-morphodynamic model framework. This flexibility is achieved through using the pyadjoint library, which allows us to assess the uncertainty of any parameter with respect to any model functional, without further code implementation. The model is developed within the coastal ocean model Thetis constructed using the finite element code-generation library Firedrake. We present examples of how this framework can perform sensitivity analysis, inversion and calibration for a range of uncertain parameters based on the final bedlevel. These results are verified using so-called dual-twin experiments, where the ‘correct’ parameter value is used in the generation of synthetic model test data, but is unknown to the model in subsequent testing. Moreover, we show that inversion and calibration with experimental data using our framework produces physically sensible optimum parameters and that these parameters always lead to more accurate results. In particular, we demonstrate how our adjoint framework can be applied to a tsunami-like event to invert for the tsunami wave from sediment deposits.

Wimmer GA, Cotter CJ, Bauer W, 2022, Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction, *The SMAI journal of computational mathematics*, Vol: 7, Pages: 267-300

We present an energy conserving space discretisation based on a Poisson bracket that can be used to derive the dry compressible Euler as well as thermal shallow water equations. It is formulated using the compatible finite element method, and extends the incorporation of upwinding for the shallow water equations as described in Wimmer, Cotter, and Bauer (2020). While the former is restricted to DG upwinding, an energy conserving SUPG method for the (partially) continuous Galerkin thermal field space is newly introduced here. The energy conserving property is validated by coupling the Poisson bracket based spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved thermal field development with respect to stability when upwinding is included. An approximately energy conserving scheme that includes upwinding for all prognostic fields with a smaller computational cost is also presented. In a falling bubble test case used for the Euler equations, the latter scheme is shown to resolve small scale features at coarser resolutions than a corresponding scheme derived directly from the equations without the Poisson bracket framework.

Bock A, Cotter CJ, 2021, Learning landmark geodesics using the ensemble Kalman filter, *Foundations of Data Science*, Vol: 3, Pages: 701-727, ISSN: 2639-8001

We study the problem of diffeomorphometric geodesic landmark matching where the objective is to find a diffeomorphism that, via its group action, maps between two sets of landmarks. It is well-known that the motion of the landmarks, and thereby the diffeomorphism, can be encoded by an initial momentum leading to a formulation where the landmark matching problem can be solved as an optimisation problem over such momenta. The novelty of our work lies in the application of a derivative-free Bayesian inverse method for learning the optimal momentum encoding the diffeomorphic mapping between the template and the target. The method we apply is the ensemble Kalman filter, an extension of the Kalman filter to nonlinear operators. We describe an efficient implementation of the algorithm and show several numerical results for various target shapes.

Bendall TM, Cotter CJ, Holm DD, 2021, Perspectives on the formation of peakons in the stochastic Camassa-Holm equation, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 477, ISSN: 1364-5021

A famous feature of the Camassa–Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite-element discretization for it, which we use to explore the formation of peakons. Our simulations using this discretization reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

Bauer W, Behrens J, Cotter C, 2021, A structure-preserving approximation of the discrete split rotating shallow water equations, European Numerical Mathematics and Advanced Applications Conference 2019, Publisher: Springer Verlag, Pages: 103-113, ISSN: 1439-7358

We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework (Bauer, Behrens and Cotter, 2019), we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes' properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.

Clare MCA, Percival JR, Angeloudis A,
et al., 2021, Hydro-morphodynamics 2D modelling using a discontinuous Galerkin discretisation, *Computers and Geosciences*, Vol: 146, Pages: 1-13, ISSN: 0098-3004

The development of morphodynamic models to simulate sediment transport accurately is a challenging process that is becoming ever more important because of our increasing exploitation of the coastal zone, as well as sea-level rise and the potential increase in strength and frequency of storms due to a changing climate. Morphodynamic models are highly complex given the non-linear and coupled nature of the sediment transport problem. Here we implement a new depth-averaged coupled hydrodynamic and sediment transport model within the coastal ocean model Thetis, built using the code generating framework Firedrake which facilitates code flexibility and optimisation benefits. To the best of our knowledge, this represents the first full morphodynamic model including both bedload and suspended sediment transport which uses a discontinuous Galerkin based finite element discretisation. We implement new functionalities within Thetis extending its existing capacity to model scalar transport to modelling suspended sediment transport, incorporating within Thetis options to model bedload transport and bedlevel changes. We apply our model to problems with non-cohesive sediment and account for effects of gravity and helical flow by adding slope gradient terms and parametrising secondary currents. For validation purposes and in demonstrating model capability, we present results from test cases of a migrating trench and a meandering channel comparing against experimental data and the widely-used model Telemac-Mascaret.

Cotter CJ, Deasy J, Pryer T, 2020, The r-Hunter-Saxton equation, smooth and singular solutions and their approximation, *Nonlinearity*, Vol: 33, Pages: 7016-7039, ISSN: 0951-7715

In this work we introduce the r-Hunter–Saxton equation, a generalisation of the Hunter–Saxton equation arising as extremals of an action principle posed in Lr. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter–Saxton equation.

Bendall TM, Gibson TH, Shipton J,
et al., 2020, A compatible finite-element discretisation for the moist compressible Euler equations, *Quarterly Journal of the Royal Meteorological Society*, Vol: 146, Pages: 3187-3205, ISSN: 0035-9009

A promising development of the last decade in the numerical modelling of geophysical fluids has been the compatible finite‐element framework. Indeed, this will form the basis for the next‐generation dynamical core of the Met Office. For this framework to be useful for numerical weather prediction models, it must be able to handle descriptions of unresolved and diabatic processes. These processes offer a challenging test for any numerical discretisation, and have not yet been described within the compatible finite‐element framework. The main contribution of this article is to extend a discretisation using this new framework to include moist thermodynamics. Our results demonstrate that discretisations within the compatible finite‐element framework can be robust enough also to describe moist atmospheric processes.We describe our discretisation strategy, including treatment of moist processes, and present two configurations of the model using different sets of function spaces with different degrees of finite element. The performance of the model is demonstrated through several test cases. Two of these test cases are new cloudy‐atmosphere variants of existing test cases: inertia–gravity waves in a two‐dimensional vertical slice and a three‐dimensional rising thermal.

Kramer S, Wilson C, Davies R, et al., 2020, FluidityProject/fluidity: New test cases "Analytical solutions for mantle flow in cylindrical and spherical shells"

This release adds new test cases described in the GMD paper "Analytical solutions for mantle flow in cylindrical and spherical shells"

Cotter C, Crisan D, Holm DD, et al., 2020, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Publisher: arXiv

The stochastic variational approach for geophysical fluid dynamics wasintroduced by Holm (Proc Roy Soc A, 2015) as a framework for derivingstochastic parameterisations for unresolved scales. This paper applies thevariational stochastic parameterisation in a two-layer quasi-geostrophic modelfor a beta-plane channel flow configuration. We present a new method forestimating the stochastic forcing (used in the parameterisation) to approximateunresolved components using data from the high resolution deterministicsimulation, and describe a procedure for computing physically-consistentinitial conditions for the stochastic model. We also quantify uncertainty ofcoarse grid simulations relative to the fine grid ones in homogeneous (teamedwith small-scale vortices) and heterogeneous (featuring horizontally elongatedlarge-scale jets) flows, and analyse how the spread of stochastic solutionsdepends on different parameters of the model. The parameterisation is tested bycomparing it with the true eddy-resolving solution that has reached somestatistical equilibrium and the deterministic solution modelled on alow-resolution grid. The results show that the proposed parameterisationsignificantly depends on the resolution of the stochastic model and gives goodensemble performance for both homogeneous and heterogeneous flows, and theparameterisation lays solid foundations for data assimilation.

Cotter C, Crisan D, Holm D, et al., 2020, Data Assimilation for a Quasi-Geostrophic Model with Circulation-Preserving Stochastic Transport Noise, Publisher: SPRINGER

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Bauer W, Behrens J, Cotter CJ, 2020, A structure-preserving approximation of the discrete split rotating shallow water equations, EGU General Assembly 2020, Publisher: Copernicus GmbH, ISSN: 0090-8312

We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework [1,2], we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes' properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.

Gibson T, Mitchell L, Ham D,
et al., 2020, Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond, *Geoscientific Model Development*, Vol: 13, Pages: 735-761, ISSN: 1991-959X

Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have becomeincreasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point andelliptic systems arising from these discretizations continue to be an on-going challenge. One possible approach for addressingthis issue is to employ a method known as hybridization, where the discrete equations are transformed such that classic staticcondensation and local post-processing methods can be employed. However, it is challenging to implement hybridization as performant parallel code within complex models, whilst maintaining separation of concerns between applications scientistsand software experts. In this paper, we introduce a domain-specific abstraction within the Firedrake finite element library thatpermits the rapid execution of these hybridization techniques within a code-generating framework. The resulting frameworkcomposes naturally with Firedrake’s solver environment, allowing for the implementation of hybridization and static condensa-tion as runtime-configurable preconditioners via the Python interface to PETSc, petsc4py. We provide examples derived from second order elliptic problems and geophysical fluid dynamics. In addition, we demonstrate that hybridization shows greatpromise for improving the performance of solvers for mixed finite element discretizations of equations related to large-scalegeophysical flows.

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