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We show how two-dimensional mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo-uniform grids. This family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal C-grid finite difference methods. There are a few major advantages: the mixed finite element methods do not require an orthogonal grid, and they allow a degree of flexibility that can be exploited to ensure an appropriate ratio between the velocity and pressure degrees of freedom so as to avoid spurious mode branches in the numerical dispersion relation. These methods preserve several properties of the C-grid method when applied to linear barotropic wave propagation, namely: a) energy conservation, b) mass conservation, c) no spurious pressure modes, and d) steady geostrophic modes on the $f$-plane. We explain how these properties are preserved, and describe two examples that can be used on pseudo-uniform grids: the recently-developed modified RT0-Q0 element pair on quadrilaterals and the BDFM1-pdg element pair on triangles. All of these mixed finite element methods have an exact 2:1 ratio of velocity degrees of freedom to pressure degrees of freedom. Finally we illustrate the properties with some numerical examples.
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This article takes the form of a tutorial on the use of a particular class of mixed finite element methods, which can be thought of as the finite element extension of the C-grid staggered finite difference method. The class is often referred to as compatible finite elements, mimetic finite elements, discrete differential forms or finite element exterior calculus. We provide an elementary introduction in the case of the one-dimensional wave equation, before summarising recent results in applications to the rotating shallow water equations on the sphere, before taking an outlook towards applications in three-dimensional compressible dynamical cores.
In this essay, I outline a personal vision of how I think Numerical Weather Prediction (NWP) should evolve in the years leading up to 2030 and hence what it should look like in 2030. By NWP I mean initial-value predictions from timescales of hours to seasons ahead. Here I want to focus on how NWP can better help save lives from increasingly extreme weather in those parts of the world where society is most vulnerable. Whilst we can rightly be proud of many parts of our NWP heritage, its evolution has been influenced by national or institutional politics as well as by underpinning scientific principles. Sometimes these conflict with each other. It is important to be able to separate these issues when discussing how best meteorological science can serve society in 2030; otherwise any disruptive change - no matter how compelling the scientific case for it - becomes impossibly difficult.
We aim to study a finite volume scheme to solve the two dimensional inviscid primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions to the system of equations. In that respect, a version of a projection method is introduced to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. The resulting scheme allows for a significant reduction of the errors near the topography when compared to more standard finite volume schemes. In the numerical simulations, we first present the associated good convergence results that are satisfied by the solutions simulated by our scheme when compared to particular analytic solutions. We then report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated. The numerical results show that such a forcing is responsible for recurrent large-scale patterns to emerge in the temperature and velocity fields.
The formation of precipitation in state-of-the-art weather and climate models is an important process. The understanding of its relationship with other variables can lead to endless benefits, particularly for the worlds monsoon regions dependent on rainfall as a support for livelihood. Various factors play a crucial role in the formation of rainfall, and those physical processes are leading to significant biases in the operational weather forecasts. We use the UNET architecture of a deep convolutional neural network with residual learning as a proof of concept to learn global data-driven models of precipitation. The models are trained on reanalysis datasets projected on the cubed-sphere projection to minimize errors due to spherical distortion. The results are compared with the operational dynamical model used by the India Meteorological Department. The theoretical deep learning-based model shows doubling of the grid point, as well as area averaged skill measured in Pearson correlation coefficients relative to operational system. This study is a proof-of-concept showing that residual learning-based UNET can unravel physical relationships to target precipitation, and those physical constraints can be used in the dynamical operational models towards improved precipitation forecasts. Our results pave the way for the development of online, hybrid models in the future.
We prove that the Scott-Vogelius finite elements are inf-sup stable on shape-regular meshes for piecewise quartic velocity fields and higher ($k ge 4$).
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