Dr. Gordan R. Stuhne

Atmospheric Physics Group, Room MP609
Department of Physics, University of Toronto
60 St. George Street, Toronto, ON, Canada M5S 1A7

Phone: (416) 946-3019

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Structured (orthogonal tripolar) vs. unstructured (icosahedral quadtree) spherical grids.

World ocean bathymetry contoured on coastally conforming unstructured triangular grid.

Jet formation in decaying shallow water turbulence.

World ocean circulation in an unstructured grid model (projected into latitude-longitude coordinates).

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Research Interests

Background

My research interests relate to the computational modeling of complex hydrodynamic systems. At present, I am engaged in the development of numerical techniques for oceanic and climatological simulation, but I have also worked in biophysical and biomedical application areas. A key challenge in all of these fields is the formulation and implementation of continuum-mechanical models of systems whose detailed, small-scale behavior is complex and unpredictable. My specific contributions (with various collaborators) and ongoing work are given, in point form, below, followed by a list of publications. To try to explain the overall motivation and theme, I will begin with this general discussion, for which literature references may be found in the listed publications.

Hydrodynamic equations express the continuous forms of the conservation laws for momentum and a variety of other physical quantities (e.g., internal energy, or the salt content of the ocean). These laws are statistical in the sense that they apply to the bulk properties of materials whose microscale behavior is not taken into account. In Eulerian terms, each quantity is assumed to be advected by a local mean current as it is scattered by random (i.e., unknown) fluctuations, which induce diffusion. Allowing also for sources and sinks, the redistribution of a scalar quantity, C, in space is governed by the conservation equation,

in which the local advecting velocity, u, itself evolves (nonlinearly) through time on account of an analogous redistribution of vector momentum. The form of the diffusive flux, F, depends upon the probability distribution of deviations from the mean motion, with the assumption of an isotropic random walk leading, for example, to the standard Fickian counter-gradient diffusion law: i.e., .

General Areas of Interest

The application of hydrodynamical approximation to molecular systems yields the well-studied Navier-Stokes equations, which are known to provide a very good description of fluids in most situations. Outstanding research issues at this level mostly relate to mesh generation and geometric transformation techniques for representing complex spatial domains in computational models, as well as to the formulation of discretizations that offer optimal accuracy while preserving fundamental hydrodynamical conservation laws.

When one considers complex fluid systems like the ocean or atmosphere, naive numerical solution of the Navier-Stokes equations is totally impractical. Instead, hydrodynamical reasoning is applied to a succession of models, with the high-frequency or eddy motions occurring in one model being identified with the randomizing mechanism in a less complete, but more useful, simplified model. There are a variety of well-known chains of approximation: e.g., in geophysical fluid dynamics, compressible -> anelastic -> Boussinesq; non-hydrostatic -> hydrostatic -> shallow-water; ageostrophic -> geostrophic -> nondivergent barotropic, etc. These chains often run perpendicular to each other and link together into a network of related models. As one traverses this network, the basic conservation structure of the Navier-Stokes equations remains essentially unchanged, but ad hoc assumptions are typically made about the diffusive fluxes with the aim of mimicking physics that is unresolved with a given hydrodynamical approximation and computational grid. Finding convincing a priori mathematical justifications for such parameterizations is very difficult, and most research in this area boils down to the discretization of different hydrodynamical models and parameterization schemes, and to the comparison of simulation results with each other and with empirical data. In this context, it is of interest to understand the relationships amongst different models and parameterizations at the level of the computational algorithms, and to systematize comparisons amongst their various predictions.

Finally, there is an interesting class of situations in which hydrodynamic reasoning can be applied fairly rigorously, but with inaccurate or counter-intuitive results. For example, if one aims to model the concentration of biological organisms like marine plankton, it can be proven mathematically that there are important cases where the continuous hydrodynamic quantity is strictly diffused, even though the discrete population dynamics results in the formation of spatially localized clusters (an inherently non-diffusive phenomenon). The inaccuracy of the hydrodynamic approximation results because there are large non-conservative statistical fluctuations in the local concentration field. Any introduction of biological components into hydrodynamic models must hence be at the level of the carbon cycle and/or other dynamical processes involving precise conservation laws. In a related vein, the nonlinear self-advection entailed in the momentum conservation law complicates the simplistic association of "eddies" with random mixing and diffusion. This is exemplified by the way in which coherent large-scale vortices and banded zonal flows can sometimes be generated, rather than dissipated, by unordered small-scale turbulence - an important phenomenon that may account for the observed structures of the atmospheres of the gaseous giant planets, and for the complex bands in the Antarctic circumpolar ocean current. Asymptotic analyses of so-called Kolmogorov flows (which represent a very simple idealized case) capture aspects of such anti-diffusive dynamics, which appear when the derived large-scale diffusion coefficient becomes negative. Studying oddities of this kind helps build understanding of the assumptions (and potential pitfalls) inherent in any application of hydrodynamical approximation.

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