Scalar conservation equations in a turbulent ocean

Divergence of the instantaneous velocity field arises from molecular diffusion as well as compressibility. By contrast, the divergence of the turbulent flux of density does not contribute to the mean velocity divergence, which, in a turbulent ocean, arises from compressibility and nonlinearities of the equation of state. These nonlinearities also lead to “densification on mixing” in the equation for the mean density, though the contribution from vertical (but not horizontal) mixing is balanced by a divergence of the vertical eddy fluxes in a density profile. The advective forms of the conservation equations for scalar variables (except in situ density) are found to be accurate in their normal forms; in particular, there are no terms from the nonlinear equation of state in the normal advective form of the conservation equations for potential temperature and salinity. However, the flux forms of the same conservation equations have a “production” term proportional to the divergence of the mean velocity vector, ▿·$\text{u}$. While this extra production term is not small, the traditional approach of putting $\text{▿·}\text{u}\text{= 0}$ in ocean models is a valid procedure for circumventing the issue. Finally, it is shown that the conservation equations for scalar variance are not seriously affected through the neglect of terms involving the velocity divergence.