Multi-scale Mathematics applied to Parameterisation of Convection

Computer models designed for numerical weather prediction and climate modelling necessarily resolve the Earth's atmosphere using a finite grid. Physical processes that occur on scales smaller than this grid, for example atmospheric convective circulations that have horizontal scales of a kilometre or less, cannot be resolved explicitly. Nevertheless, the cumulative effects of these convective circulations on the larger scale must be represented, using a technique called parameterisation. Formulating an accurate parameterisation of convection is key to modelling the transport of momentum, moisture and entropy in the Earth's atmosphere. As highlighted by a recent major NERC programme (`Understanding and representing atmospheric convection across scales') coupling between current parameterisation schemes and resolved (large-scale) dynamics is a high priority for improvement. Inaccurate coupling is evident in systematic biases in the speeds
of those atmospheric waves that support convection, leading to poor model representation of important large-scale processes such as the Madden-Julian Oscillation in the tropics.
Our scientific hypothesis is that a key source of coupling error is the result of assumptions made in the current method used to implement parameterisations in models. Currently, the momentum, mass and moisture fluxes associated with (hypothesised) small-scale convective plumes are simply added directly to the large-scale model equations. The assumption here is the large-scale flow evolves essentially as if the (unresolved) small-scale convective circulations can be directly averaged out. However, a simple mathematical treatment of a related problem reveals that the propagation speed of large-scale inertia-gravity waves, moving through a variable environment, in fact shows very strong sensitivity to the presence of small but finite regions of reduced stratification, such as occur in convective plumes. In other words, the simple averaging process assumed is not correct, and a more sophisticated averaging technique is required. The aim of the studentship is to explore new techniques for averaging across the convective plumes using a systematic
and mathematically rigorous approach: `multi-scale mathematics' (MSM). MSM has been fundamental to advances in diverse fields such as hydrology, crystallography, and the science of meta-materials, each of which involves problems requiring systematic averaging across small-scale structure. The student will work under the guidance of experts in geophysical fluid dynamics (Esler), MSM (Smyshlyaev) and state-of-the- art computational modelling of convection (Whitall, Met Office). The approach will be systematic, first developing the students' intuition by working on relatively simple mathematical problems, while training proceeds in the computational aspects. Next, a relatively simple numerical model, in which convection can be explicitly resolved, will be explored in detail. The aim will be to evaluate the performance of a traditional parameterisation, and compare it to the new approach based on MSM. Finally, the impact of switching to an MSM-based parameterisation on tropical wave speeds will be estimated, and the feasibility of using MSM to modify the implementation of the convective parameterisation in the Met Office Unified
Model will be evaluated. The studentship opportunity includes being trained in, and then creatively exploring, skills at the cutting edge of mathematics and climate science, as well as developing expertise in state-of-the-art atmospheric modelling. The research is potentially extremely high-impact, as it could result in gains potential forecast accuracy of very high economic value, in addition to the high societal benefit of improved climate forecasts. Wider benefits include knowledge exchange between the Met Office, climate modelling communities and mathematicians, leading to the introduction of MSM techniques to a wide class of related problems.