Mixing rates in stably stratified flows with respect to the turbulent froude number and turbulent scales

Abstract

In this analysis the quantification of diapycnal diffusivity $K_\rho$ in stratified flows such as those found in the ocean and atmosphere is explored. There are two simplifications that are routinely made when estimating mixing rates in stably stratified flows. First, a constant value is commonly assumed for the (irreversible) mixing coefficient $\Gamma$. Second, dissipation rates of turbulent kinetic energy $\epsilon$ are inferred using either the Thorpe (or Ellison) length scales or from microstructure measurements using the isotropy assumption. Data from three independent direct numerical simulations of homogeneous stratified turbulence are used as a testbed to highlight impacts of these assumptions on estimates of $K_\rho$. A systematic analysis compares the inferred diffusivities to exact DNS diffusivities as a function of the turbulent Froude number $Fr_t$. Use of a constant $\Gamma$ results in an under-prediction of $K_\rho$ by up to a factor of 5 for strongly stratified conditions (low $Fr_t$) and an over-prediction of $K_\rho$ by up to two orders of magnitude in weakly stratified conditions (high $Fr_t$). The use of inferred dissipation rates $\epsilon$ based on the assumption of isotropy results in an over-prediction of $K_\rho$ by a factor of 2 for low $Fr_t$ (which is within the instrumentation error) and converges on the exact $K_\rho$ for $Fr_t \ge 1$. However, the use of kinematic length scales, such as the Thorpe or Ellison scales, to infer $\epsilon$ result in significant errors. The implications of these findings are applied in a simple demonstration to show how these tools can be used for improved estimates of mixing rates in stably stratified flows.