Dissipation estimates for stratified turbulence

Abstract

There is concern about the appropriateness of the isotropic formula for calculating the dissipation rate of turbulent kinetic energy in a stratified fluid, such as turbulence in the thermocline. An alternative approach, given that stratification acts directly on the vertical component of the motion, is the axisymmetric theory of Batchelor (1946). Under the assumption that 〈(∂ u 2 /∂ x 3 ) 2 〉 ≈ 〈(∂ u 2 /∂ x 1 ) 2 〉, which is consistent with our measurements, we derive an upper and lower bound for dissipation in axisymmetric turbulence. For a value of ε/ν N 2 > 20 the dissipation estimate is comparable to the isotropic formula (based on the gradient of a horizontal velocity component). For decreasing values of ε/ν N 2 the error increases but is limited to less than 35%. Interestingly enough, in an axisymmetric turbulence the correct dissipation rate may be larger or smaller than an appropriate isotropic estimation. The important result is that the isotropic formula based on the shear of a horizontal velocity component is a good estimate of the dissipation, given the uncertainty in the current oceanographic technique. Both vertical and horizontal sampling are appropriate to estimate the dissipation rate from the isotropic formula as long as one uses a horizontal component of the velocity.