Vector cylindrical harmonics for low-dimensional convection models

Abstract

Approximate empirical models of thermal convection can allow us to identify the essential properties of the flow in simplified form, and to produce empirical estimates using only a few parameters. Such "low-dimensional" empirical models can be constructed systematically by writing numerical or experimental measurements as superpositions of a set of appropriate basis modes, a process known as Galerkin projection. For three-dimensional convection in a cylinder, those basis modes should be vector-valued, mutually orthogonal, and defined in cylindrical coordinates. Here we construct such a basis set and demonstrate that it has these desired properties and boundary conditions when the exact constraint of incompressibility is relaxed. We show its use for representing sample simulation data and point out its potential for low-dimensional convection models.