The general formulation for the open ocean slab model is taken from Hansen et al. , although we have modified it to allow for a fractional sea ice coverage. The governing equation for ocean mixed layer temperature is:
The geographic structure of ocean mixed layer depth is specified from Levitus . 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:
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:
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
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:
The adjusted () is then used to update all ocean points due to deep ocean heat exchange and transport as:
The quantity is nonzero only if the temperature of the slab ocean falls below the freezing point:
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: