8.1 The continuum tracer concentration budget

Before starting, it is useful to summarize the terms appearing in a local budget for the tracer concentration, as determined by the tracer equation

 

$$-\nabla \cdot ( {\bf u} \, T + {\bf F} ).$$ T_t = (8.1)

In this equation, T represents the amount of a substance (generically called a tracer) per unit volume; i.e., it is a concentration. Equation (8.1) is the conservation equation for this tracer concentration (e.g., Gill 1982, chapter 4 or Apel 1987, chapters 3,4). Within the Boussinesq approximation employed by MOM, $T = \rho_{o} \, s$, representing the mass of salt per volume of seawater, satisfies this equation. Note that salinity s, which is a prognostic variable carried in MOM, is dimensionless since it represents the grams of salt per kilogram of seawater (ppt). Additionally, with the same Boussinesq approximation, the amount of potential heat per volume $T = \rho_{o} \, c_{p} \, \theta$, satisfies this equation (in this context, heat is considered a ``substance''). Note that c_p, the specific heat at constant pressure, is held fixed within the Boussinesq approximation (e.g., Chandrasekhar 1961). Passive tracers, where T represents the amount of tracer per volume, also satisfy this equation.

The terms on the right hand side were discussed in Section 4.2.4. In particular, ${\bf F}=\left({\bf F}^h, F^{z} \right)$ are the horizontal and vertical tracer flux components distinct from the advective flux ${\bf u}T$ due to the resolved velocity field. Typically in the mixed layer, which includes the region between the free surface $z=\eta$ and the bottom of the upper ocean model box z=z₁, the fluxes take the form ${\bf F}_{h}(T) = -A_{h} \nabla T$, and $F^{z}(T) = -\kappa_{h} T_{z}$. More general closures can be considered. In the following, these fluxes are not specified explicitly and they are generally dependent on the space-time resolution used in the model. For purposes of terminology, they will be referred to as diffusive fluxes.