The general formulation for the open ocean slab model is taken from Hansen et al. [67], although we have modified it to allow for a fractional sea ice coverage. The governing equation for ocean mixed layer temperature is:

where is the ocean mixed layer temperature, is the density of ocean water, is the heat capacity of ocean water, is the annual mean ocean mixed layer depth (m), is the fraction of the ocean covered by sea ice, is the net atmosphere to ocean heat flux (Wm), is the internal ocean mixed layer heat flux (Wm), simulating deep water heat exchange and ocean transport, is the heat exchanged with the sea ice (Wm) (including solar radiation transmitted through the ice, see Eq. 6.39) and is the heat gained when sea ice grows over open water (Wm). and are constants (see Table 5.1 for values of the constants), and the nomenclature is such that all right-hand-side fluxes are positive down.

The geographic structure of ocean mixed layer depth is specified from Levitus [105]. Monthly mean mixed layer depths are determined using this dataset's standard measure of salinity ( is the density of sea water for a specified salinity, temperature, and atmospheric pressure) where the equality (surface) = .125 is satisfied on a grid. These data are then averaged to the standard CAM 3.0 grid (all data falling within a CAM 3.0 grid box are equally weighted), horizontally smoothed 10 times using a 1-2-1 smoother, and capped at 200m (to prevent excessively long adjustment times in coupled atmosphere ocean experiments). The resulting mixed layer depths in the tropics are generally shallow (10m-30m) while at high latitudes in both hemispheres there are large seasonal variations (from 10m up to the 200m maximum). The annually-averaged geographically-varying mixed layer depth, which is used for purposes related to energy conservation, is produced by averaging the monthly mean values.

The geographic distribution of the internal heat source is generally specified on a monthly basis using a control CAM 3.0 integration as described below. During a SOM numerical integration is linearly interpolated between monthly values (taken as mid month) to the appropriate model time step. The energy fluxes associated with ice formation and ice melt ( and respectively) are explicitly predicted.

The net atmosphere-to-ocean heat flux in the absence of sea ice, , is defined as:

where is the net solar flux absorbed by the ocean mixed layer, is the net longwave energy flux of the ocean surface to the atmosphere, is the sensible heat flux from the ocean to the atmosphere, and is the latent heat flux from the ocean to the atmosphere. The surface temperature used in evaluating these fluxes is .

The evolution of the mixed-layer temperature field, , is
evaluated using an explicit forward time step. At iteration *n*
the required information to advance the forecast include
, and , where
is time invariant and is linearly interpolated in time
between prescribed mid-monthly values. It is assumed that the exchange
between the ocean mixed layer and the atmosphere occurs faster than
deep adjustments. Hence, the first adjustment to is evaluated
as:

where is the model time step. We note that is computed from the fraction of the total CAM 3.0 grid box that is not covered by land, since only ocean and sea ice covered portion of the grid cell are considered for the SOM configuration:

where is the fraction of ice in the CAM 3.0 grid cell and is the fraction of land in the CAM 3.0 grid box.

The flux is then adjusted since it is possible (using monthly specified values of ) to introduce a non-physical cooling of the mixed layer when its temperature is at the freezing point. Therefore, if and , then

where , and is the ocean freezing temperature of -1.8C (where is expressed in units of C). This adjustment smoothly reduces the loss of heat from the mixed layer (if any) to zero as its temperature approaches the specified freezing point of sea water.

To ensure that the predicted SOM sea ice distribution compares favorably with the control simulation, and is bounded against unchecked growth or loss for atmospheric conditions significantly different from present day, an additional adjustment to under sea ice is applied:

(5.6) |

where

(5.7) |

is the local ice thickness, and is a tuning constant which may have different values for the Northern and Southern hemispheres. The coefficient ensures this adjustment only occurs under sea ice covered ocean. The function is empirical, and is designed to ensure that the hemispheric adjustments asymptote properly for very small and very large values of ice thickness. For present-day climate simulations the values of which yield good control sea ice distributions are +15 and -10 for the Northern and Southern hemispheres respectively.

The adjusted () is then used to update all ocean points due to deep ocean heat exchange and transport as:

(5.8) |

where is the energy flux associated with any ice melt and shortwave radiation transmitted through the sea ice from the previous time step.

The quantity is nonzero only if the temperature of the slab ocean falls below the freezing point:

(5.9) |

If is nonzero, new ice forms over the ice-free portion of the grid cell and is returned to the freezing temperature:

(5.10) |

A renormalization is necessary to ensure energy is conserved when is adjusted as described above. We distinguish warm ocean as those points for which C. An adjustment for warm ocean points is computed after all modifications to are completed. Let be the original unadjusted , and let be the global (area weighted) mean. The final (total) applied to warm ocean points is:

(5.11) |

where is the global area over all ocean, and the corresponding area over warm ocean. Taking the global mean of the bracketed quantity (which is zero over non-warm oceans) results in a multiplicative factor . Thus, , satisfying global energy conservation of for every time step. In practice, the bracket term adjustment is applied to warm ocean points after the Q redistribution is completed.