Diascalar flux and the rate of fluid mixing

Abstract

We define the rate at which a scalar θ mixes in a fluid flow in terms of the flux of θ across isoscalar surfaces. This flux θd is purely diffusive and is, in principle, exactly known at all times given the scalar field and the coefficient of molecular diffusivity. In general, the complex geometry of isoscalar surfaces would appear to make the calculation of this flux very difficult. In this paper, we derive an exact expression relating the instantaneous diascalar flux to the average squared scalar gradient on an isoscalar surface which does not require knowledge of the spatial structure of the surface itself. To obtain this result, a time-dependent reference state θ(t,z*.) is defined. When the scalar is taken to be density, this reference state is that of minimum potential energy. The rate of change of the reference state due to diffusion is shown to equal the divergence of the diffusive flux, i.e. (∂/∂z*)θd.This result provides a mathematical framework that exactly separates diffusive and advective scalar transport in incompressible fluid flows. The relationship between diffusive and advective transport is discussed in relation to the scalar variance equation and the Osborn–Cox model. Estimation of water mass transformation from oceanic microstructure profiles and determination of the time-dependent mixing rate in numerically simulated flows are discussed.