Since one-point second-order models are an essential part of GOTM, this section is devoted to a detailed discussion of the derivation and the properties of these models. Second-order models result from the full or approximate solution of the transport equations for the turbulent fluxes like , , , etc. Model equations for the turbulent momentum fluxes follow directly from the Navier-Stokes equations. The derivation of these equations for stratified and rotating fluids is discussed e.g. in Sander (1998).
Considering the one-point correlations for the velocity fluctuations , the momentum fluxes can be expressed as
The contraction of (46) yields the equation for the turbulent kinetic energy, (150), with production terms defined by
Similar to (46), the transport equation for the turbulent buoyancy flux is given by
Note that is half the buoyancy variance and relates to the turbulent potential energy, , according to
The crucial point in (46) is the model for the pressure-strain correlation. The most popular models in engineering trace back to suggestions by Launder et al. (1975) and Gibson and Launder (1976). With the modifications suggested of Speziale et al. (1991), this model can be written as
For Explicit Algebraic Heat Flux Models, a quite general model for the pressure buoyancy-gradient correlation appearing in (49) can be written as
The models (51) and (57) correspond to some recent models used in theoretical and engineering studies (So et al. (2003), Jin et al. (2003)), and generalize all explicit models so far adopted by the geophysical community (see Burchard (2002b), Burchard and Bolding (2001)). With all model assumptions inserted, (46) and (49) constitute a closed system of 9 coupled differential equations, provided the dissipation time scale and the buoyancy variance are known. Models for the latter two quantities and simplifying assumptions reducing the differential equations to algebraic expressions are discussed in the following subsection.
Karsten Bolding 2012-01-24