Geometric Methods in Geophysical Fluid Dynamics
A Research Network in Mathematics funded by EPSRC
|Publications||Simulation results with the HPM method||Links|
Geophysical Fluid Dynamics (GFD) is characterized by a level of complexity that makes straightforward numerical modelling nearly impossible. From a fundamental numerical modelling point of view, two distinct challenges can be identified. First, there is the presence of energetic flow on a range of spatial scales from millimetres to megametres, and corresponding temporal scales from seconds to millennia. It is well established that there are vigorous nonlinear energy cascades linking all these scales, even though the details of these cascades remain ill-understood and are often modelled in a crude way. Second, there are fundamentally different kinds of fluid motion on all these scales. For instance, there is the quasi-two-dimensional vortex dynamics that characterizes the large-scale flow along stratification surfaces. On the other hand, there are also inertia--gravity waves, which are oscillations of the stratification surfaces themselves. Combining the huge range of scales as well as the different kinds of fluid motion in a single numerical model is an extremely difficult goal: present-day numerical models are falling short both on quantitative performance and on qualitative aspects of the simulated flow. One aspect of nonlinear cascades that is generally accepted is that at intermediate scales they can be described to good approximation by the ideal fluid equations, even though one is dealing in principle with a forced--dissipative system driven by large-scale external forcing (i.e. solar insolation) and subject to various forms of small-scale dissipation. This makes available a large amount of ideal-fluid theory, such as Kelvin's circulation theorem, or the Hamiltonian structure of the ideal fluid equations and its geometric interpretations, but this theory is not used in present-day numerical models. Novel numerical modelling techniques are needed that explicitly preserve the geometric structure of the underlying equations in discrete form. One the other, the Hamiltonian framework can also be used to understand the energy exchange between fast gravity waves and slower vortical motion. Furthermore, there is quite an interest in developing statistical mechanics models based on a micro/macro-canonical ensemble theory. Again these ideas and concepts rely heavily on a geometric approach to fluid dynamics which provides the main focal point for the network.