Second-order models

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 ${\langle u'u' \rangle}$, ${\langle u' w' \rangle}$, ${\langle w'b' \rangle}$, 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 $u'_i$, the momentum fluxes can be expressed as

$$\dot{ \rule{0mm}{4mm} \langle u'_i u'_j \rangle} - {\cal D}_{ij} = P_{ij} + G_{ij} + F_{ij} + \Phi_{ij} - \epsilon_{ij} \; , \quad$$ (46)

where ${\cal D}_{ij}$ is the sum of the viscous and turbulent transport terms and $\dot{ {\langle \cdots \rangle} }$ denots the material derivative of the ensemble average. The shear-production, $P_{ij}$, and the buoyancy production, $G_{ij}$, on the right hand side are defined as

$$P_{ij} = - {\langle u'_i u'_m \rangle} \dfrac{\partial {U_j}}{\pa... ...3} {\langle u'_j b' \rangle} + \delta_{j3} {\langle u'_i b' \rangle} \; , \quad$$ (47)

where $b'$ is the fluctuating part of the buoyancy, defined analogously to the mean buoyancy, $B$, in (33). The tensor of the dissipation rate is assumed to be isotropic, leading to $\epsilon_{ij} = 2/3 \epsilon \delta_{ij}$. $\Phi_{ij}$ denotes the pressure redistribution terms discussed below. The influence of the Coriolis-acceleration can be summarized in the tensor $F_{ij}$ which is, however, neglected in the current version of GOTM.

The contraction of (46) yields the equation for the turbulent kinetic energy, (150), with production terms defined by

$$P = \dfrac{1}{2} P_{ii} \; , \quad G = \dfrac{1}{2} G_{ii} \quad .$$ (48)

Similar to (46), the transport equation for the turbulent buoyancy flux is given by

$$\dot{ \rule{0mm}{4mm} \langle u'_i b' \rangle} - {\cal D}^b_{i} =... ...rtial {x_m}} + F^b_i + 2 \delta_{i3} k_b + \Phi^b_i - \epsilon^b_{i} \; , \quad$$ (49)

where ${\cal D}^b_{i}$ denotes the viscous and turbulent transport terms, see Sander (1998). For the dissipation, one has $\epsilon^b_{i}=0$, following from isotropy. The redistribution terms $\Phi^b_i$ are discussed below. As in (46), the Coriolis term $F^b_i$ is neglected in the current version of GOTM.

Note that $k_b$ is half the buoyancy variance and relates to the turbulent potential energy, $E_p$, according to

$$k_b = {\langle b'^2 \rangle} /2 = E_p N^2 \; , \quad$$ (50)

where the square of the buoyancy frequency, $N^2$, is defined in (38).

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

$$\Phi_{ij} = - c_1 \tau^{-1}_u k \; b_{ij} + c_2 k S_{ij} + c_3 k \Sigma_{ij} + c_4 k Z_{ij} + c_5 k N_{ij} + c_6 \Gamma_{ij} \; , \quad$$ (51)

usually extended by the last term to account for the effects of buoyancy, see Gibson and Launder (1976), Gibson and Launder (1978). The model (51) is expressed here in terms of the dimensionless tensor of the stress anisotropies,

$$b_{ij} = \dfrac{ {\langle u'_i u'_j \rangle} }{2k} - \dfrac{1}{3} \delta_{ij} \; , \quad$$ (52)

and two traceless and symmetric tensors,

$$\Sigma_{ij} = S_{im} b_{mj} + S_{jm} b_{mi} - \frac{2}{3} S_{mn} b_{mn} \delta_{ij} \; , \quad Z_{ij} = W_{im} b_{mj} + W_{jm} b_{mi} \; , \quad$$ (53)

which depend on the symmetric and the anti-symmetric parts of the velocity gradient,

$$S_{ij} = \frac{1}{2} \left( L_{ij} + L_{ji} \right) \; , \quad W_{ij} = \frac{1}{2} \left( L_{ij} - L_{ji} \right) \quad L_{ij} = \dfrac{\partial {U_i}}{\partial {x_j}} \quad .$$ (54)

Buoyancy enters via the symmetric and traceless tensor

$$\Gamma_{ij} = - \left( G_{ij} - \frac{2}{3} G \delta_{ij} \right) \; , \quad$$ (55)

with $G_{ij}$ as defined in (47). In view of the derivation of Explicit Algebraic Models (EASMs), the models implemented in GOTM neglect the term $N_{ij}$ on the right hand side of (51), which is non-linear in $b_{ij}$, see Speziale et al. (1991). $c_1$-$c_6$ are model constants. In geophysical applications, in contrast to engineering, virually all authors used $c^*_1=0$ in (51). In GOTM, the return-to-isotropy time scale $\tau_u$ is identified with the dynamic dissipation time scale

$$\tau = \dfrac{k}{\epsilon} \; , \quad$$ (56)

which is a reasonable model assumption in many applications (Canuto et al. (2001), Jin et al. (2003)).

For Explicit Algebraic Heat Flux Models, a quite general model for the pressure buoyancy-gradient correlation appearing in (49) can be written as

$$\begin{array}{rcl} \Phi^b_i &=& - c_{b 1} \tau^{-1}_b \; {\la... ...rtial {x_j}} - 2 c_{b 5} k_b \delta_{i3} \; , \quad \end{array}$$ (57)

where $\tau_b=\tau$ is adopted for the return-to-isotropy time scale.

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 $\tau$ and the buoyancy variance $k_b$ are known. Models for the latter two quantities and simplifying assumptions reducing the differential equations to algebraic expressions are discussed in the following subsection.