Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence

In order that it be correctly characterized, irreversible turbulent mixing in stratified fluids must distinguish between adiabatic 'stirring' and diabatic 'mixing'. Such a distinction has been formalized through the definition of a diapycnal diffusivity, K-rho (Winters & D'Asaro, J. Fluid Mech., vol. 317, 1996, pp. 179-193) and an appropriate mixing efficiency, E (Caulfield & Peltier, J. Fluid Mech., vol. 413, 2000, pp. 1-47). Equivalent attention has not been paid to the definitions of a corresponding momentum diffusivity Km and hence an appropriately defined turbulent Prandtl number Pr-t = K-m/K-rho. In this paper, the diascalar framework of Winters & D'Asaro (1996) is first reformulated to obtain an 'Osborn-like' formula in which the correct definition of irreversible mixing efficiency E is shown to replace the flux Richardson number which Osborn (J. Phys. Oceanogr., vol. 10, 1980, pp. 83-89) assumed to characterize this efficiency. We advocate the use of this revised representation for diapycnal diffusivity since the proposed reformulation effectively removes the simplifying assumptions on which the original Osborn formula was based. We similarly propose correspondingly reasonable definitions for K-m and Pr-t by eliminating the reversible component of the momentum production term. To explore implications of the reformulations for both diapycnal and momentum diffusivity we employ an extensive series of direct numerical simulations (DNS) to investigate the properties of the shear-induced density-stratified turbulence that is engendered through the breaking of a freely evolving Kelvin-Helmholtz wave. The DNS results based on the proposed reformulation of K-rho are compared with available estimations due to the mixing length model, as well as both the Osborn-Cox and the Osborn models. Estimates based upon the Osborn-Cox formulation are shown to provide the closest approximation to the diapycnal diffusivity delivered by the exact representation. Through compilation of the complete set of DNS results we explore the characteristic dependence of K-rho on the buoyancy Reynolds number Re-b as originally investigated by Shih et al. (J. Fluid Mech., vol. 525, 2005, pp. 193-214) in their idealized study of homogeneous stratified and sheared turbulence, and show that the validity of their results is only further reinforced through analysis of the turbulence produced in the more geophysically relevant Kelvin-Helmholtz wave life-cycle ansatz. In contrast to the results described by Shih et al. (2005) however, we show that, besides Re-b, a vertically averaged measure of the gradient Richardson number Ri(b) may equivalently characterize the turbulent mixing at high Re-b. Based on the dominant driving processes involved in irreversible mixing, we categorize the intermediate (i.e. Re-b = O (10(1)-10(2))) and high (i.e. Re-b > O (10(2))) range of Reb as 'buoyancy-dominated' and 'shear-dominated' mixing regimes, which together define a transition value of Ri(b) similar to 0.2. Mixing efficiency varies non-monotonically with both Re-b and Ri(b), with its maximum (on the order of 0.2-0.3) occurring in the 'buoyancy-dominated' regime. Unlike K-rho which is very sensitive to the correct choice of E (i.e. K-rho proportional to E/(1 - E)), we show that Km is almost insensitive to the choice of E (i.e. K-m proportional to 1/(1 - E)) so long as E is not close to unity, which implies K-m approximate to Ri(b)Re(b) for the entire range of Re-b. The turbulent Prandtl number is consequently shown to decrease monotonically with Re-b and may be (to first order) simply approximated by Re-b itself. Assuming Pr-t = 1, or Pr-t = 10 (as is common in large-scale numerical models of the ocean general circulation), is also suggested to be a questionable assumption.