Stellar mixing

III. The case of a passive tracer^⋆

¹ NASA, Goddard Institute for Space Studies, New York, NY 10025, USA
² Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA
e-mail: vmc13@columbia.edu

Received: 9 July 2010
Accepted: 25 August 2010

Abstract

In Papers I–II, we derived the expressions for the turbulent diffusivities of momentum, temperature T, and mean molecular weight μ. Since the scalar T-μ fields are active tracers (by influencing the density and thus the velocity field), whereas passive tracers such as ⁷Li are carried along by the flow without influencing it, it would be unjustified to use the diffusivities of the T-μ fields to represent the diffusivity of passive tracers. In this paper, we present the first derivation of a passive tracer diffusivity. Some key results are: a) In the general 3D case, the passive tracer diffusivity is a tensor given in algebraic form; b) the diffusivity tensor depends on shear, vorticity, T, and μ-gradients, thus including double diffusion and differential rotation; c) in the 1D version of the model, the passive tracer diffusivity is a scalar denoted by K_c; d) in doubly stable regimes, ∇_μ > 0,   ∇ − ∇_ad < 0, K_c is nearly the same as those of the T-μ fields; e) in semi-convection regimes, ∇_μ > 0,   ∇ − ∇_ad > 0, K_c is larger than that of the μ-field; f) in salt fingers regimes ∇_μ < 0,∇ − ∇_ad < 0, K_c is smaller than that of the μ-field; and finally, g) in the only case we know of a direct measurement of a passive tracer diffusivity, the oceanographic North Atlantic Tracer Release Experiment (NATRE), the model reproduces the data quite closely.