8.4 Comments on the surface tracer fluxes

The diffusive or turbulent part of the surface tracer flux

$$Q_{wT}^{diff} = {\bf F}_{h}(T) \cdot \nabla_{h} \eta - F^{z}(T)$$ (8.10)

must be specified by a boundary condition. This flux enters the ocean after passing a sequence of boundary layers between atmosphere and ocean. The tracer transport through these layers is governed by a complex superposition of several processes, as molecular and turbulent diffusion, wave breaking, Langmuir circulation, radiation processes, chemical reactions and biological processes. Strong local gradients as in a thermal skin layer may be built up. Thus, the calculation of Q_wT^diff is a problem in nonequilibrium thermodynamics, turbulence theory, physical chemistry, and/or biophysics and is a rather complex problem by itself. For an introduction see the book by de Groot and Mazur (1962), or the articles by Forland et al. (1988) or Doney (1995).

The set of basic equations of MOM does not provide information on the turbulent tracer transport, and the complicated vertical structure of the air-sea boundary layer is not resolved. However, especially for long time integrations, the surface boundary conditions are cruical for the accuracy of MOM integrations. Therefore it is useful to make a few general statements here in hopes of exposing the basic issues.

First, recall that the dimensions of the flux Q_wT are tracer concentration times velocity (see the definition of Q_wT in equation (8.8)). As such, Q^diff_wT represents the total mass (or energy for the case of heat) of a tracer passing through the sea surface per unit area and per unit time. The amount of the tracer substance crossing the air-sea interface fulfills certain conservation laws. For example, if the tracer is a substance i, the mass of the tracer passing the air-sea interface must be conserved, i.e., the flux at the ocean and the air side of the interface is the same,

Q^diff_wi = Q^diff_ai (8.11)

If chemical reactions at the sea surface are possible, more general expressions can be found from the conservation of mass for the involved chemical elements. If the tracer is an energy, the reaction heats Q_R from phase transitions or chemical reactions must be included into the total energy balance at the air-sea interface,

Q^diff_wi = Q^diff_ai+ Q_R. (8.12)

At the ocean side of the boundary layer, the flux (equation (8.8))

 

$$Q_{wT} = T \, q_{w} + {\bf F}_{h}(T) \cdot \nabla_{h} \eta - F^{z}(T) \qquad z=\eta$$ (8.13)

appears at the top of the uppermost ocean box. As mentioned earlier, the first contribution, $T(\eta) \, q_{w}$, brings about a change in tracer concentration due to changes in ocean volume upon introducing fresh water. The air-sea interface acts on a tracer like a filter with a transparency depending on the difference of the chemical potentials of the tracers in the air and in water. For example, ionic tracers such as dissolved salt have a total air-sea flux Q_wT which is zero due to the large hydration energy. On the other hand, many weakly dissolved trace gases leave the ocean together with evaporating water. For this reason, the remaining terms $Q_{wT}^{diff} = {\bf F}_{h}(T) \cdot \nabla_{h} \eta - F^{z}(T)$ in equation (8.13) are not independent of the fresh water flux term $T(\eta) \, q_{w}$. Rather, they describe the turbulent diffusive tracer flux at the top of the surface box, and this flux is established by both the tracer concentration gradients across the air-sea interface, and tracer gradients within the ocean boundary layer coming from the chemical tracer kinetics in connection with the fresh water flux. The next issue concerns how approximations for the tracer fluxes can be found. The complexity of the boundary layer prevents a direct coupling of an ocean circulation model with a boundary layer model which resolves the genuine dynamics of the boundary layer. Alternately, the tracer fluxes are often calculated from empirical approximations. The difference of the bulk tracers in the atmosphere and the ocean, T_a -T_s, is taken as the thermodynamic force for the diffusive tracer flux. Then the tracer flux has the general form

$$Q_{wT}^{diff} \sim C_{T}u^{wind}\left(T_a - T_s \right)$$ (8.14)

The empirical tubulent kinetic coefficient C_T summarizes the complex dynamics within the boundary layer. It depends mainly on the tracer properties, on the wind velocity and on the stability of the boundary layer. In the following, the boundary conditions for the fresh water flux, heat flux, and salt flux are specified in more detail.