Introduction

There are different types and levels of closure models available in GOTM to compute the vertical turbulent fluxes. Simple models rely on the idea that theses fluxes can be computed as the product of a positive turbulent diffusivity and a mean flow gradient. Contributions to the fluxes that are not `down-gradient', are summarized in so-called counter-gradient terms. Using these assumptions, the fluxes of momentum and buoyancy can be expressed as

$${\langle u' w' \rangle} = - \nu_t \dfrac{\partial {u}}{\partial {... ...e} = - \nu^B_t \dfrac{\partial {B}}{\partial {z}} + \tilde{\Gamma}_B \; , \quad$$ (43)

where $\tilde{\Gamma}_{(U,V,B)}$ denote the counter-gradient fluxes. They can be important under very strong stratification and in the case of convection. Note, that the current version of GOTM identifies the diffusivities of heat and salt with $\nu^B_t$ (see section 3.10 and section 3.11).

Using an analogy to the kinetic theory of gases, the vertical turbulent diffusivities, $\nu_t$ and $\nu^B_t$, are often assumed to be the product of a typical velocity scale of turbulence, $q$, times a typical length scale, $l$, see Tennekes and Lumley (1972). The velocity scale $q$ can e.g. be identified with the average value of the turbulent fluctuations expressed by the turbulent kinetic energy, $k= q^2 /2$. Then, the diffusivities of momentum and heat can be written as

$$\nu_t = c_\mu k^\frac{1}{2} l \; , \quad \nu^B_t = c_\mu' k^\frac{1}{2} l \; , \quad$$ (44)

where the dimensionless quantities $c_\mu$ and ${c_\mu}'$ are usually referred to as the `stability functions'. Depending on the level of turbulent closure, these stability functions can be either constants, empirical functions, or functions of some non-dimensional flow parameters resulting from a higher-order turbulence model. The same applies to the counter-gradient fluxes $\tilde{\Gamma}_{(U,V,B)}$ defined in (43).

There are different possibilities in GOTM to compute the scales $q$ (or $k$) and $l$ appearing in (44). According to the level of complexity, they are ordered in GOTM in the following fashion.

1. Both, $k$ and $l$ are computed from algebraic relations. The algebraic equation for $k$ is based on a simplified form of the transport equation of the turbulent kinetic energy. The equation for the length-scale may result from different approaches. The most simple models assume an empirically motivated, prescribed vertical distribution of the length-scale. This level of closure corresponds to the `level 2' model of Mellor and Yamada (1982), but also to more recent approaches, see Cheng et al. (2002). Algebraic models are an over-simplification in numerous situations.

2. At the next level, $k$ is computed from the differential transport equation for the turbulent kinetic energy (`energy models'). As before, the length-scale is computed from an empirically or theoretically based relation. Models of this type are quite popular in geophysical modelling. A description is given in section 4.19.

3. In the so-called two-equation models, both, $k$ and $l$, are computed from differential transport equations. As before, $k$ follows from the transport equation of the turbulent kinetic energy. Now, however, also the length-scale is determined from a differential transport equation. This equation is usually not directly formulated for the length-scale, but for a related, length-scale determining variable. Presently, there are different possibilities for the length-scale determining variables implemented in GOTM, such as the rate of dissipation, $\epsilon$, or the product $kl$. They are discussed in section 4.7.9.

   The main advantage of the two-equation models is their greater generality. There are, for example, a number of fundamental flows which cannot be reproduced with an algebraically prescribed length-scale. Examples are the temporal decay of homogeneous turbulence, the behaviour of turbulence in stratified homogeneous shear flows, and the spatial decay of shear-free turbulence from a planar source. A discussion of these points is given in section 4.7.3 and section 4.7.4. Also see Umlauf et al. (2003) and Umlauf and Burchard (2003).

In addition to the hierarchy of turbulence models in terms of their methods used to compute the turbulent kinetic energy and the length-scale, GOTM also supports an ordering scheme according to the extent to which transport equations for the turbulent fluxes are solved.

1. At the lowest level of this scheme, it is postulated that $c_\mu=c_\mu^0$ and $c'_\mu=c'^0_\mu$ are constant. Because these models implicitly assume an isotropic tensor relation between the velocity gradient and the tensor of the Reynolds-stresses, they usually fail in situations of strong anisotropy, most notably in stably stratified, curved or shallow flows. In unstratified flows with balanced aspect ratios (which seldom occur in nature), however, they may compute reasonable results. Models of this type are referred to as the `standard' models in the following.
2. Some problems associated with standard versions of the models can be ameliorated by making $c_\mu$ and $c'_\mu$ empirical functions of one or several significant non-dimensional flow parameters. At this level, the simplest approach would be to formulate empirical relations suggested from observations in the field or in the laboratory. An example of such a relation is the model of Schumann and Gerz (1995) which has been implemented in GOTM (see section 4.29).
3. Another, more consistent, approach results from the solution of simplified forms of the transport equations for the Reynolds-stresses and the turbulent heat fluxes in addition to the transport equations for $k$ and the length-scale determining variable. Surprisingly, it turns out that under some assumptions, and after tedious algebra, the turbulent fluxes computed by these models can be expressed by (44). The important difference is, however, that the existence of vertical eddy diffusivities is not a postulate, but a consequence of the model. The stability functions $c_\mu$ and $c'_\mu$ can be shown to become functions of some non-dimensional numbers like

   $$\alpha_M = \frac{k^2}{\epsilon^2} M^2 \; , \quad \alpha_N = \frac{k^2}{\epsilon^2} N^2 \; , \quad \alpha_b = \frac{kk_b}{\epsilon^2} \; , \quad$$ (45)

   
   

   with the shear-frequency, $M$, and the buoyancy frequency, $N$, computed as described in section 3.5 and section 3.14, respectively. $k$ and $k_b$ are the turbulent kinetic energy and the buoyancy variance, respectively and $\epsilon$ denotes the rate of dissipation.

   The most well-known models of this type have been implemented into GOTM. An up-to-date account of their derivation can be found in Canuto et al. (2001). Their evaluation for the oceanic mixed layer has been extensively discussed by Burchard and Bolding (2001).

4. Even more complete models include further differential equations for the buoyancy variance and for some or all of the turbulent fluxes. These models cannot be reduced to the form (44). The derivation of models of the type discussed in the latter two points are reviewed in section 4.2

Evidently, this short introduction cannot serve as an introductory text on one-point turbulence modelling. It serves merely as a place to define the most important quantities and relations used in this manual. Readers not familiar with this subject will certainly feel the need for a more in-depth discussion. An excellent introduction to turbulence is still the book of Tennekes and Lumley (1972). A modern and detailed approach to one and two-equation models for unstratified flows is given in the book of Wilcox (1998), and the effects of stratification are discussed e.g. by Rodi (1987) and by Burchard (2002b).