David G. Dritschel

Applied Mathematics Courses 2009-10

Applied Mathematics Courses 2010-11

Applied Mathematics Course Choices

Research Interests

I have sought to combine theoretical analysis and numerical computation in the study of fundamental aspects of atmospheric and oceanic fluid dynamics, in particular vortex dynamics. The atmosphere and oceans are hugely influenced by both the background planetary rotation as well as the density stratification. These effects, together with the shallow flow geometry (typical horizontal scales are 10 to 100 times larger than typical vertical scales), constrain the motion to be approximately layerwise two-dimensional. This means that vertical motion tends to be very weak compared to horizontal motion over much of the atmosphere and oceans, and stratification surfaces tend to be nearly flat. On these surfaces, a scalar quantity called the "potential vorticity" is often, to a good approximation, conserved following fluid "particles". That is, the potential vorticity (PV) is advected or transported by the nearly horizontal flow on these surfaces.

I have developed a series of lagrangian or partly-lagrangian numerical methods that allow one to accurately conserve potential vorticity, something which is not easy to do using commonly-used numerical methods (see the paper by Dritschel, Polvani & Mohebalhojeh in Monthly Weather Review 1999 below). These methods have permitted a careful investigation of a range of fundamental processes, including vortex filamentation, stripping, merging, splitting, collapsing, etc, mainly in two-dimensional (height-independent) flows but recently also in three-dimensional flows (see papers with Dr. Jean Reinaud and Will McKiver below). Further recent numerical developments have also permitted a careful assessment of the role of gravity waves in single-layer shallow-water flows (see papers with Dr Ali R. Mohebalhojeh below), and internal gravity waves in two- and three-dimensional Boussinesq (ocean-like) flows (see papers with Dr. Alvaro Viudez below).

These methods have also permitted the ultra-high resolution study of both two-dimensional and three-dimensional (gravity-wave free, balanced quasi-geostrophic) turbulence. These studies have helped to clarify the nature of freely-decaying turbulence, and in particular have helped to identify the most common vortex shapes and forms of interaction.

Current research is aimed at building more realistic atmospheric and oceanic models based in part on the numerical methods above, and in collaboration with the European Centre for Medium-range Weather Forecasting and the UK Meteorological Office. In addition, a number of process studies are underway. One study is trying to understand how potential vorticity gradients (and indirectly model resolution) may modify the effects of planetary wave forcing on the stratospheric polar vortex (collaboration with Prof Lorenzo M. Polvani & Dr Richard Scott (Columbia University) and Prof Darryn W. Waugh (Johns Hopkins University)). And in another study, we are trying to elucidate the role played by potential vorticity in the emission, propagation, scattering, reflection and refraction of internal gravity waves. While potential vorticity is widely regarded as a key dynamical field --- principally because in practice it largely determines other dynamical fields like velocity, pressure and temperature via approximate "balance relations" hidden in the equations of motion --- it is not widely used theoretically and seldomly numerically due to its nonlinear character. This is particularly the case for the Boussinesq (ocean-like) equations: the explicit use of potential vorticity forces one to solve a nonlinear diagnostic equation, in fact a Monge-Ampere equation, to recover other dynamical fields. However, this approach gives direct insight into the fundamental structure of the equations, and on the practical side it is simple and efficient to implement, as well as being much more accurate than conventional approaches which fail to differentiate potential vorticity advection from internal gravity wave propagation. Several papers on this subject are available via anonymous ftp.

Other areas of research interest include the use of ellipsoidal vortices to model vortex interactions in quasi-geostrophic flows (with Jean Reinaud and Will McKiver), simulation of quasi-geostrophic flows confined within cylindrical and generally irregular domains (with Charlie Macaskill, Sydney University), and the simulation of infinite Prandt'l number Stokes flows (with Eckart Meiburg, University of Santa Barbara).


A full CV (as a pdf file) can be obtained here.


David G. Dritschel
Department of Applied Mathematics
School of Mathematics and Statistics
University of St Andrews
North Haugh
St Andrews
KY16 9SS
Scotland
Group page: http://www-vortex.mcs.st-and.ac.uk/

School web page: http://www.mcs.st-and.ac.uk/

Email: dgd@mcs.st-and.ac.uk

©2002, David G. Dritschel

Last modified: Monday, 8 October 2012