Reduced complexity models

From everything to something

The primary tool used to study theoretical aspects of geophysical fluid dynamics, and indeed of many other physical systems, is ‘reduced modelling’. This is the process by which we simplify complex physical systems or sets of mathematical equations into a form more amenable to analysis. There is a balancing act here: on the one hand we want to study systems close to those found in nature, but on the other hand our mathematical tools of analysis (and limited computer power) only enable us to understand simplified systems, predominantly highly simplified ones. We can gain important knowledge by first studying these simple systems, regarding them as a stepping stone to more realistic systems.

In most fields, at the very start simplifications have already been made just to begin. In fluid dynamics, we rarely take into account relativistic or quantum effects, for example, and we accept Boltzmann’s statistical description of molecular dynamics in order to justify viewing a fluid as a continuum. Hence, many simplifications have already been made, but they are vital to making any progress at all in many circumstances.

Equations of fluid dynamics

In the study of geophysical vortex dynamics, often (though not always) we consider an incompressible fluid (an important exception is the atmosphere, where compressibility is essential, see e.g. Scott & Dritschel 2005, and the references therein). This simplifies the system considerably by removing any consideration of thermodynamics. The fundamental variables are then velocity (3 components), density and pressure. These variables satisfy 4 ‘prognostic’ equations (involving a single time derivative) and one ‘diagnostic’ or constraint equation, namely the expression of incompressibility: zero velocity divergence.

Filtered ‘balanced’ models

This is already a reduced set of equations, and in a sense a reduced model. Yet, even this set is challenging mathematically and computationally. One can reduce further by ‘filtering’ relatively high-frequency motions associated with inertia-gravity waves (propagating on the background ‘planetary’ vorticity arising from the rotating Earth and gradients of density arising from natural stratification). Such filtering may be justified for typical motions occurring at intermediate to large scales: such motions are, to leading order, in geostrophic and hydrostatic balance. This balance, and the resulting asymptotic reduction of the equations to the ‘quasi-geostrophic’ (QG) ones, depends on several non-dimensional parameters being small compared to unity, namely the Rossby number Ro, the Froude number Fr, the Coriolis to buoyancy frequency ratio f/N, and a measure of the variation of the Coriolis frequency with latitude. Typically, these parameters are found to be comparably small, enabling one to derive a much reduced set of equations. These QG equations now have just one prognostic equation, for the advection of potential vorticity (a materially conserved field in the absence of friction, forcing or diabatic heating), and a series of simple, linear diagnostic relations that provide the flow field (and if desired the density and pressure fields) entirely in terms of potential vorticity (PV).

The QG equations, due to their relative simplicity, are the workhorse for understanding many general aspects of geophysical fluid dynamics. They approximately model the energetically dominant part of atmospheric and oceanic motion. They are deficient in the sense that they do not contain inertia-gravity waves, but more importantly in the sense that they neglect ageostrophic motions (these distinguish cyclonic from anticyclonic motions in real geophysical flows). Despite these limitations, they remain immensely useful. And, moreover, we still don’t know everything about these equations! PV advection is fundamentally nonlinear, and this can result in layerwise-two-dimensional turbulence. Even strictly two-dimensional turbulence is complicated; less still is known about QG turbulence.

The ellipsoidal model

Further reduction is possible if one restricts attention to modelling the interaction between isolated vortices. By exploiting the fact that an ellipsoidal volume of uniform PV anomaly (w.r.t. a uniform background having constant f and N) is an exact, steadily rotating solution of the QG equations (Zhmur & Shchepetkin 1991; Meacham 1992) and that these ‘vortices’ retain their ellipsoidal form (though distorted) even in a linear background straining flow, it is possible to derive a reduced model for the interaction of two or more ellipsoids. The ‘ellipsoidal model’ (ELM) does just this by neglecting only parts of the interaction which would result in non-ellipsoidal deformations. I.e., the idea is to retain only the ellipsoidal shape and ‘filter’ the non-ellipsoidal perturbations (see Dritschel, Reinaud & McKiver, 2004 for full details).

The justification of this approximation is illustrated in the adjoining figure, comparing the close interaction of two vortices (they are nearly touching!) in the full QG equations (top) with that obtained using the ELM (bottom). Note, the ELM only has 8 degrees of freedom per vortex, while the former has essentially infinite degrees of freedom: the ELM is very much a reduced model and, evidently a powerful one. It has been used widely in our research to understand basic aspects of vortex dynamics, and in particular enables one to well estimate margins of stability, which when breached lead to vortex merger or more complex vortex interactions.

A vortex interaction in the QG model (top) and ELM (bottom)

More information and animations related to the ELM can be found here.

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