Understanding mixing efficiency in the oceans: do the nonlinearities of the equation of state for seawater matter?

• Main Author: R. Tailleux
• Format: Article
• Language: English
• Published: Copernicus Publications 2009-07-01
• Series: Ocean Science
• Online Access: http://www.ocean-sci.net/5/271/2009/os-5-271-2009.pdf

There exist two central measures of turbulent mixing in turbulent stratified fluids that are both caused by molecular diffusion: 1) the dissipation rate <i>D</i>(APE) of available potential energy APE; 2) the turbulent rate of change <i>W</i><sub><i>r</i>, turbulent</sub> of background gravitational potential energy GPE<sub><i>r</i></sub>. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPE<sub><i>r</i></sub>, owing to the well known exact equality <i>D</i>(APE)=<i>W</i><sub><i>r</i>, turbulent</sub> for a Boussinesq fluid with a linear equation of state. Recently, however, Tailleux (2009) pointed out that the above equality no longer holds for a thermally-stratified compressible, with the ratio ξ=<i>W</i><sub><i>r</i>, turbulent</sub>/<i>D</i>(APE) being generally lower than unity and sometimes even negative for water or seawater, and argued that <i>D</i>(APE) and <i>W</i><sub><i>r</i>, turbulent</sub> actually represent two distinct types of energy conversion, respectively the dissipation of APE into one particular subcomponent of internal energy called the "dead" internal energy IE<sub>0</sub>, and the conversion between GPE<sub><i>r</i></sub> and a different subcomponent of internal energy called "exergy" IE<sub>exergy</sub>. In this paper, the behaviour of the ratio ξ is examined for different stratifications having all the same buoyancy frequency <i>N</i> vertical profile, but different vertical profiles of the parameter Υ=α <i>P</i>/(ρ<i>C<sub>p</sub></i>), where α is the thermal expansion coefficient, <i>P</i> the hydrostatic pressure, ρ the density, and <i>C<sub>p</sub></i> the specific heat capacity at constant pressure, the equation of state being that for seawater for different particular constant values of salinity. It is found that ξ and <i>W</i><sub><i>r</i>, turbulent</sub> depend critically on the sign and magnitude of <i>d</i>Υ/<i>dz</i>, in contrast with <i>D</i>(APE), which appears largely unaffected by the latter. These results have important consequences for how the mixing efficiency should be defined and measured in practice, which are discussed.