8.4.2 Heat flux into the free surface model

For temperature, the heat balance in the upper box has to be considered. The heat flux enters the ocean through a boundary layer which has an atmospheric and an oceanic component. There are four major contributions to the heat flux,

• the insolation,
• the infrared radiation balance between ocean and atmosphere,
• the sensible heat flux, which is basically a turbulent diffusion of heat,
• the heat transfer in connection with a fresh water flux. This effect involves a direct energy flux in connection with the flux of matter and a latent heat due to the liquid-vapour phase transition at the sea surface.

The insolation and the infrared radiation are not discussed here. For simple parameterization see e.g. Smith and Dobson (1984) or Rosati and Miyakoda (1988).

The enthalpy flux Q_ae through the top of the atmosphere-ocean boundary layer is,

Q_ae = Q_ae^fresh + Q_ae^rad + Q_ae^sens. (8.19)

The radiative component Q_ae^rad includes insolation and the infrared radiation from the ocean and the atmosphere. The thermal radiation is emitted or absorbed in a thin skin layer at the sea surface and the approximation of a surface flux is justified. For the structure of the thermocline it may be important to resolve the vertial absorption profile of short wave radiation. To do this, the short wave radiation must be removed from the surface flux and the vertical divergence of the short wave radiation must be included in the source term. Q_ae^sens describes the turbulent diffusion of heat, Q_ae^fresh is the heat flux in connection with the heat capacity of the fresh water advected relative to the sea surface. Under the assumption that the heat flux in the boundary layer has no vertical divergence, the enthalpy flux from the bottom of the boundary layer into the ocean, Q_we, is

Q_we = Q_ae + Q_e^lat, (8.20)

where Q_e^lat is the latent heat from that amount of fresh water which undergoes a phase transition at the air-sea interface and can be calculated from the water vapour flux q_w^V,

Q_e^lat = L Q_w^V. (8.21)

L is the evaporation heat of fresh water. Q_e^lat is positive if the ocean gains heat by condensation and negative if heat is used for evaporation. It is a common approximation that the latent heat flux goes directly into the ocean and leaves the atmosphere temperature unaffected.

For a simple parameterization of the sensible heat flux the difference of the bulk virtual potential temperature of the atmosphere, $\theta_{va}$ and the ocean, $\theta_{vs}$, is assumed as the thermodynamic forcing function,

$$\rho_a c_{ap} C_T u^{wind}\left(\theta_{va} - \theta_{vs}\right).$$ Q_ae^sens = (8.22)

As for the fresh water flux the kinetic coefficient C_T u^wind describes the turbulent vertical diffusion of heat and can be parameterized in terms on the wind speed and the stability of the atmosphere. c_ap is the specific heat of air at constant pressure, $\rho_{a}$ the density of air. The sign convention is to count a heat flux directed into the ocean as positive.

The heat flux between atmosphere and air-sea boundary layer due to the heat capacity of the fresh water is

$$\rho_w c_p T_R Q_{w}^R +\rho_a c_{ap} \theta_a Q_{w}^V.$$ Q_ae^fresh = (8.23)

T_R is the temperature of the liquid fresh water flux, i.e. of rain, $\rho_w$the fresh water density, T_a the temperature of vapour, which should be the atmosphere temperature. Usually, T_R is not known and simpler approximations are necessary.

Finally, the boundary condition for the potential temperature $\theta$ is

$$Q_{w\theta}^{diff} \theta(\eta) Q_w + \nabla_{h} \eta \cdot {\bf F}_{h}(\theta) - F^{z}(\theta),$$ =

$$(c_p \rho)^{-1} Q_{we}$$ =

$$(c_p\rho)^{-1} \left(Q_{ae}^{rad} + Q_{ae}^{sens} + L Q_{w}^V + c_p T_r Q_{w}^R +c_{ap} \theta_a Q_{w}^V\right).$$ = (8.24)