The Dissipation of Fluctuating Tracer Variances

Abstract

The evolution of the covariance of two tracers involves a quantity called codissipation, proportional to the covariance of the gradients of the two tracers, and analogous to the dissipation of tracer variance. The evolution of the variance of a composite tracer?a linear combination of two simple, primary tracers?depends on the?composite dissipation,? a combination of the individual simple tracer dissipations and the codissipation. The composite dissipation can be negative (implying growth of the variance of the composite tracer) for structures in which the correlation of the simple tracer gradients are large enough (i.e., large codissipation). This situation occurs in the phenomena of double diffusion and salt fingering. A particular composite tracer called watermass variation, a measure of water-type scatter about the mean tracer versus tracer relationship, lacks production terms of the conventional form?tracer flux multiplying tracer gradient?in its variance evolution balance. Only codissipation can produce variance of watermass variation. The requirements that watermass variance production and dissipation be in equilibrium, and that no other composite tracer variance be tending to grow due to codissipation, lead to a particular relation among codissipation and the simple dissipations and between the simple dissipations themselves. The latter are proportional to one another, the proportionality factor being the square of the slope of the mean tracer versus tracer relation. The same results can be obtained by modifying Batchelor?s argument to give the equilibrium cospectrum of two tracer gradients at high wavenumbers in a well-developed field of isotropic turbulence. As a consequence of these arguments, the turbulent eddy tracer fluxes are also proportional, with the mean tracer?tracer slope as proportionality factor. Further, the ratio of turbulent diffusivities of two tracers is unity. The dissipation of buoyancy, a composite tracer constructed from temperature and salinity, is proportional at equilibrium to thermal dissipation multiplied by a factor that depends on the stability ratio. This previously established result is obtained here under less restrictive conditions.