Term 2
Exploring mathematical formulations for a next-generation compatible finite element dynamical core
Speaker : Daniel Witt
Date : Feb 13th 2024
Abstract :
Dynamical cores can be considered the numerical heart of a numerical weather model, and consequently the choices in their mathematical formulation can have significant ramifications on the accuracy of the model. One such formulation is the use of compatible finite elements to discretise the governing equations. Compatible finite element methods are attractive for modelling geophysical fluids because they can replicate many of the desirable properties of the Arakawa C-grid, such as good wave dispersion. Compatible finite elements also facilitate alternative grid structures which avoid the clustering of grid points at the poles without the associated downsides these grids have with a finite difference scheme. A preliminary investigation into the impacts of altering this formulation is presented, specifically the effects of increasing the finite element order on the model in both horizontal and vertical directions.
Stormy Weather Generation : Does conditioning on storm types improve the skill of precipitation weather generators?
Speaker : Paul (Pasha) Bell
Date : Feb 20th 2024
Abstract :
Our primary tool for projecting into future climate are General Circulation Models (GCMs). These do not have the resolution to resolve small scale processes like local weather. One solution to bridge the scale gap between GCMs and local weather is statistical downscaling: training a statistical model on large scale atmospheric variables and local weather data. In the work presented one such model, a typical precipitation weather generator trained on ERA5 data and precipitation weather stations in Devon, has been conditioned on a discrete set of storm types developed by Catto and Dowdy (2021). A comparison has been made between conditioning with and without storm types using the Continuous Ranked Probability Score (CRPS) and including storm types has produced an improvement in the CRPS of 7-9%.
Catto, Jennifer L. and Andrew Dowdy (June 2021). “Understanding compound hazards from a weather system perspective”. en. In: Weather and Climate Extremes 32, p. 100313. issn: 2212-0947. doi: 10.1016/j.wace.2021.100313.
Shallow Water Sloshing Problems: Conservation and Long Term Numerics using Geometry
Speaker : James Arthur
Date : Feb 27th 2024
Abstract :
Many numerical simulations provide insight into the solution of PDEs but seem to struggle for long term prediction. This is usually because there is a discrepancy in between the properties of the physical system and the numerical system, for example conservation of mass, energy or enstrophy. Arakawa and Lamb [1] presented a numerical scheme using a C-grid that conserved these quantities and hence was able to do long term prediction for the shallow water equations model. Their method was clunky and presented in a way that was non-intuitive. Salmon [2] then presented a way to do this using the well-known Hamiltonian Mechanics and Geometry. I will provide insight into these methods and present some numerical simulations and a fourth order spatial discretisation that preserves mass, energy and enstrophy using these techniques for a sloshing problem.
[1] - Akio Arakawa and Vivian R Lamb. “A potential enstrophy and energy conserving scheme for the shallow water equations”. In: Monthly Weather Review 109.1 (1981), pp. 18–36.
[2] - Rick Salmon. “Poisson-bracket approach to the construction of energy-and potential-enstrophy-conserving algorithms for the shallow-water equations”. In: Journal of the atmospheric sciences 61.16 (2004), pp. 2016–2036
Tracking mesoscale convective systems over South America in Met Office convection-permitting model simulations
Speaker : Harriet Gilmour
Date : March 5th 2024
Abstract :
South America is highly vulnerable to storms and extreme precipitation. Mesoscale Convective Systems (MCSs), a prevalent storm type in tropical and subtropical South America, are particularly damaging due to the organised, deep convection that fuels heavy precipitation over wide areas. Future warming will likely bring changes to MCS characteristics and precipitation extremes across the region. However, the coarse horizontal resolution of current global climate models fails to explicitly resolve convective processes, making any future changes uncertain. Here, cutting-edge convection-permitting regional climate model simulations over South America (SA-CPRCM), run by the UK Met Office, are used to simulate MCSs in both a present and future climate. Convective parameterisation is switched off in the simulations so that deep convection is explicitly resolved. These ten-year simulations are compared with satellite observations using a cloud tracking algorithm (tobac) to generate storm track datasets and assess differences in storm representation. We find that the SA-CPRCM simulations resolve, and successfully capture, the structure of MCSs over South America when compared to satellite observations. The SA-CPRCM simulations perform well at capturing the spatial frequency, seasonal cycle and total accumulated precipitation of MCSs. However, the simulations produce too many storms compared with observations, and they struggle to capture nighttime initiation. Due to a lack of high-resolution observational precipitation datasets, it is hard to compare precipitation metrics. We tentatively conclude that the CPM produces MCSs with heavier rainfall than observations, but that a smaller proportion of the MCS cloud shield is raining at a given time. Here, the first decade-long climate simulation study of MCSs over South America will be presented, as well as cloud tracking methods and results that assess the representation of MCSs in the SA-CPRCM.
Picard groups of affinoid spaces
Speaker : Mark Heavey
Date : March 19th 2024
Abstract :
Rigid analytic geometry is one framework out of many for studying non-archimedean geometry; an attempt to extend the analytic geometry of manifolds over the real and complex numbers to fields with a complete non-archimedean norm. Such fields (eg the p-adic numbers) occur naturally in number theory, and affinoid spaces are a fundamental building block of rigid analytic spaces, akin to affine schemes in scheme theory. The Picard group of a rigid space has major connections to the étale fundamental group of the space, with numerous different interpretations available. In this talk, I will explain how, given an affinoid space and an associated formal model, we can relate the Picard groups of the affinoid space, its formal model and the canonical reduction. I will relate some early structural results for these groups. Much of the talk will serve as an introduction to rigid analytic spaces, and a familiarity with algebraic geometry will be helpful but not essential. Talk may be of interest to anyone whose research intersects with geometry and analysis. Research joint with supervisor Prof Mohamed Saidi.
An Introduction to the Hopf-Galois Theory of Separable Extensions
Speaker : Andrew Darlington
Date : March 26th 2024
Abstract :
Hopf-Galois theory was developed in the late 1960’s in an effort to come up with a Galois theory for non-Galois field extensions. Given a field extension L/K, it is possible to associate to it a Hopf algebra H such that H acts on L in a similar way to the Galois group acting on the top field of its Galois extension; we say that H along with this action endows L/K with a Hopf-Galois structure (HGS). In fact, where there can only ever be at most one Galois group, it may be that many different Hopf algebras give HGSs on the same extension. This now means that we have several lenses with which to study and gain information about field extensions (this greatly expands Galois module theory, for example), but it also means that we have a classification problem on our hands... Although the notion of a Hopf-Galois extension can actually be extended to commutative rings, in 1987, it was discovered that finding HGSs on separable (but not necessarily normal) field extensions could be approached entirely group-theoretically, with a further result in 1996 giving the explicit connection between HGSs on separable extensions L/K and transitive subgroups of the holomorph of groups of order the degree of L/K. In this talk, we will follow this journey from Galois theory to some key results in the Hopf-Galois theory of separable extensions, with examples sprinkled throughout. I will end by discussing some connections and applications Hopf-Galois theory has with other, seemingly unrelated areas of mathematics. Knowledge of basic group and field theory is assumed for this talk. Familiarity with Galois theory and algebraic structure is helpful but not essential.