AS1.2

Canopies represent sub-grid scale features in earth system models and interact as such with the large-scale processes resolved numerically. The canopy is implemented with a viscosity approach, resembling a roughness parameterization. However, the idea is that high viscosity is applied locally to an obstacle area whereas free spaces are assigned low viscosity. In a first step, we test this approach on a micro-scale, using an advection-diffusion equation to solve for tracer transport around obstacles. Available wind tunnel data are used for validation of a standard finite element implementation. In a second step, this approach is combined with a multi-scale finite element approach, such that a large-scale simulation can be coupled to the micro-scale representation of a canopy. Comparison of high-resolution standard finite element and low-resolution multi-scale finite element methods will allow for quantitative error analysis. This approach has the potential to lead to better parameterizations of subgrid-scale processes in large-scale simulations.

How to cite: Patel, H., Simon, K., and Behrens, J.: Towards Canopy parameterization for Multiscale Finite Element Method, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-3807, https://doi.org/10.5194/egusphere-egu22-3807, 2022.

This talk introduces WAVETRISK-OCEAN, an incompressible version of the atmosphere model WAVETRISK with a free surface. This new model is built on the same wavelet-based dynamically adaptive core as WAVETRISK, which itself uses DYNAMICO’s mimetic vector-invariant multilayer rotating shallow water formulation. Both codes use a Lagrangian vertical coordinate with conservative remapping. The ocean variant solves the incompressible multi-layer shallow water equations with inhomogeneous density layers. Time integration uses barotropic–baroclinic mode splitting via a semi-implicit free surface formulation, which is about 34-44 times faster than an unsplit explicit time-stepping. The barotropic and baroclinic estimates of the free surface are reconciled at each time step using layer dilation. No slip boundary conditions at coastlines are approximated using volume penalization. The vertical eddy viscosity and diffusivity coefficients are computed from a closure model based on turbulent kinetic energy. Results are presented for a standard set of ocean model test cases adapted to the sphere (seamount, upwelling and baroclinic turbulence). An innovative feature of WAVETRISK-OCEAN is that it could be coupled easily to the WAVETRISK atmosphere model, thus providing a first building block toward an integrated Earth-system model using a consistent modelling framework with dynamic mesh adaptivity and mimetic properties.

How to cite: Kevlahan, N. and Lemarié, F.: WAVETRISK-OCEAN: an adaptive dynamical core for ocean modelling, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-4949, https://doi.org/10.5194/egusphere-egu22-4949, 2022.

We introduce a stochastic representation of the rotating shallow water equations and a corresponding structure preserving discretization. The stochastic flow model follows from using a stochastic transport principle and a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved flow components. Similarly to the deterministic case, this stochastic model (denoted as modeling under location uncertainty (LU)) conserves the global energy of any realization. Consequently, it permits us to generate an ensemble of physically relevant random simulations with a good trade-off between the representation of the model error and the ensemble's spread. Applying a structure-preserving discretization of the deterministic part of the equations and standard finite difference/volume approximations of the stochastic terms, the resulting stochastic scheme preserves (spatially) the total energy. To address the enstrophy accumulation at the grid scale, we augment the scheme with a scale selective (energy preserving) dissipation of enstrophy, usually required to stabilize such stochastic numerical models. We compare this setup with one that applies standard biharmonic dissipation for stabilization and we study its performance for test cases of geophysical relevance. 

How to cite: Bauer, W., Brecht, R., Li, L., and Memin, E.: Towards structure preserving discretizations of stochastic rotating shallow water equations on the sphere, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-6267, https://doi.org/10.5194/egusphere-egu22-6267, 2022.

Successful operational weather forecasting with (semi-)implicit timestepping methods relies on obtaining an accurate solution to a very large system of equations in a timely manner. It is therefore crucial that the solver algorithm is fast and efficient, as this can account for up to a third of model runtime.

For models based on mixed finite element discretisations, the standard Schur-complement solver approach is not feasible since the Schur-complement system is dense and cannot be solved with iterative methods. To address this issue in its next-generation forecast model - codenamed LFRic - the Met Office is investigating a so called “hybridised” solver algorithm, which shows its full potential when combined with multigrid techniques.
We introduce both the hybridised discretisation and multigrid techniques on simplified problems, comparing and contrasting these with the current, non-hybridised multigrid solver algorithm used in the Met Office model. We will talk about how this is generalised to the full model and present results from this comparing several solver configurations.
Since our new hybridised multigrid solver reduces the number of global reduction operations, it is particularly promising when solving very large problems on a massively parallel computer. To explore this, we ran our code on large numbers of compute cores, and will present the results of those runs here.
The efficiency of our non-nested multigrid approach depends on the choice of the coarse level finite element space. To further improve the solver algorithm, we compare different coarse level spaces for a simplified setup in the Firedrake finite element code generation framework.

How to cite: Griffith, M., Mueller, E., and Melvin, T.: Accelerating climate- and weather-forecasts with faster multigrid solvers, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-10049, https://doi.org/10.5194/egusphere-egu22-10049, 2022.

The matrix model for the barotropic vorticity equation on the torus and the 2-sphere, introduced by Zeitlin, remains a reference discretization, since it provides N conserved quantities with N degrees of freedom. Modin and Vivani recently also demonstrated its relevance for the numerical study of geophysical fluid dynamics. The origins of the discretization and its connection to the Moyal bracket of quantum mechanics are, however, somewhat mysterious, hampering the prospect of generalizing the ansatz to the shallow water and primitive equations. We show how the matrix model can be understood in the framework of variational, structure preserving discretizations of fluids introduced by Pavlov and co-workers, which has recently been extended to the finite element setting by Natale and Cotter as well as Gay-Balmaz and Gawlik. Pavlov et al.’s approach is to discretely mirror the continuous theory, where the dynamics take place in the space of (divergence free) vector fields, i.e. the Lie algebra of the (volume preserving) diffeomorphism group, and the reduced Euler-Poincaré variational principle yields the dynamical equations. Specifically, one considers the representation of the group and its Lie algebra on a finite dimensional function space, i.e. through their action on scalar functions, yielding an appropriate matrix group and Lie algebra as discrete configuration space. Because of the finite dimensional setting, one has to deviate at this point from the continuous theory and introduce a non-holonomic constraint, which amounts to restricting the finite dimensional Lie algebra to elements that correspond to vector fields. The Euler-Poincaré-d’Alembert principle has consequently also to be used to obtain semi-discrete time evolution equations. A modification of this methodology is to insist on the Euler-Poincaré theory from the continuous side and modify how the Lie algebra is discretized so that it remains applicable. Specifically, one can start with the action of a symmetry group on the configuration space, e.g. SO(3) on the 2-sphere, and consider the associated infinitesimal action of the Lie algebra on functions, which corresponds to vector fields, as in the approach by Pavlov et al. When the action admits a momentum map, it can equivalently be written using the Poisson bracket and Hamiltonians linear in the Lie algebra. Building on this and requiring that a generalization of the action on functions beyond linear Hamiltonians should be consistent with the group action, one is led to the iterated action of the Poisson algebra, which is equivalent to the Moyal bracket Lie algebra for the symmetry group (through the universal enveloping algebra of the original Lie algebra). When one then fixes a finite-dimensional spectral basis to discretize functions, this corresponds to a sub-algebra of gl(n). Finally, using Euler-Poincaré theory, as in the continuous case, on this Lie sub-algebra, one obtains the matrix model by Zeitlin that retains N conserved quantities for N degrees of freedom. We hope that our rationalization of the derivation of the matrix model opens up the possibility to generalize it to other equations for geophysical fluid dynamics, and we discuss possible directions for the shallow water and primitive equations.

How to cite: Lessig, C. and da Silva, C. C.: The matrix model for the barotropic equation, connections to variational discretizations, and generalizations to the shallow water equations, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-11140, https://doi.org/10.5194/egusphere-egu22-11140, 2022.

Semi-Lagrangian advection schemes are accurate, efficient and retain accuracy and stability even for large Courant numbers, but are not conservative. Flux-form semi-Lagrangian schemes are conservative and used to achieve large Courant numbers. However, this is complicated and would be prohibitively expensive on grids that 
are not topologically rectangular. 

Strong winds or updrafts can lead to localised violations of Courant number restrictions which can cause a model with explicit Eulerian advection to crash. Schemes are needed that remain stable in the presence of large Courant numbers and general grids, while the accuracy in the presence of localised large Courant numbers may not be so crucial.

Implicit time stepping for advection is not popular in atmospheric science because of the cost of the global matrix solution and the phase errors for large Courant numbers. However, implicit advection is simple to implement (once appropriate matrix solvers are available) and is conservative on any grid structure and can exploit improvements in solver efficiency and parallelisation. This talk will describe an implicit version of the MPDATA advection scheme and show results of linear advection test cases. To optimise accuracy and efficiency, implicit time stepping is only used locally where needed. This makes the matrix inversion problem local rather than global. With implicit time stepping MPDATA retains positivity, smooth solutions and accuracy in space and time.

How to cite: Weller, H., Woodfield, J., Kuehnlein, C., and Smolarkiewicz, P.: Long Time Steps for Advection: MPDATA with implicit time stepping, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-11370, https://doi.org/10.5194/egusphere-egu22-11370, 2022.