Extensions to CAM

7. Slab Ocean Model

The Slab Ocean Model (SOM) configuration enables a simple but tightly coupled ocean modeling component combined with a thermodynamic sea ice component based on the CCSM3 sea ice model. This configuration of the atmospheric model allows for a fully-interactive treatment of surface exchange processes in the CAM5.0. The ocean prognostic variable is the mixed layer temperature T_{o}, while the thermodynamic sea ice model treats snow depth, surface temperature, ice thickness, ice fractional coverage, and internal energy at four layers for a single thickness category. The ocean mixed layer contains an internal heat source Q (also called a Q flux), whose values are generally specified by a CAM control run, representing seasonal deep water exchange and horizontal ocean heat transport. For example, using prescribed sea surface temperatures and sea ice distributions, the net surface energy flux over the ocean surface can be evaluated to yield the heat source Q. Additional exchange of heat occurs between the ocean mixed layer and the sea ice model during ice formation and ice melt. To ensure the CAM5.0 SOM sea ice simulation compares well to the observed ice distribution, and to moderate sea ice changes in climate change experiments, the Q flux term is adjusted under the ice in a globally conserving manner.

7.1. Open Ocean Component

The general formulation for the open ocean slab model is taken from Hansen et al. (1984), although we have modified it to allow for a fractional sea ice coverage. The governing equation for ocean mixed layer temperature T_o is:

(1)\rho_o C_o h_o \frac{\partial T_{o}}{\partial t} = (1-A) F + Q + A F_{oi} + (1-A) F_{frz}

where T_o is the ocean mixed layer temperature, \rho_o is the density of ocean water, C_o is the heat capacity of ocean water, h_o is the annual mean ocean mixed layer depth (m), A is the fraction of the ocean covered by sea ice, F is the net atmosphere to ocean heat flux (Wm:math:^{-2}), Q is the internal ocean mixed layer heat flux (Wm:math:^{-2}), simulating deep water heat exchange and ocean transport, F_{oi} is the heat exchanged with the sea ice (Wm:math:^{-2}) (including solar radiation transmitted through the ice) and F_{frz} is the heat gained when sea ice grows over open water (Wm:math:^{-2}). \rho_o and C_o are constants (see Table [table:somconst] for values of the constants), and the nomenclature is such that all right-hand-side fluxes are positive down.

Temperatures
T_f = -1.8 \ \mathrm{^{\circ}C}
Ocean
\rho_{o} = 1.026 \times 10^{3} \ \mathrm{kg \: m^{-3}}
C_o = 3.93 \times 10^{3} \ \mathrm{J \: kg^{-1} \: K^{-1}}
Ice
L_i = 3.014 \times 10^{8} \ \mathrm{J \: m^{-3}}

Table: [table:somconst]Constants for the Slab Ocean Model

The geographic structure of ocean mixed layer depth h_o is specified from Levitus (1982). Monthly mean mixed layer depths are determined using this dataset’s standard measure of salinity \sigma_t = (\rho_S - 1) \cdot 10^3 (\rho_S is the density of sea water for a specified salinity, temperature, and atmospheric pressure) where the equality \sigma_t(h_o)-\sigma_t(surface) = .125 is satisfied on a 1^\circ \times 1^\circ grid. These data are then averaged to the standard CAM5.0 grid (all data falling within a CAM5.0 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 Q is generally specified on a monthly basis using a control CAM5.0 integration as described below. During a SOM numerical integration Q 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 (F_{frz} and F_{mlt} respectively) are explicitly predicted.

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

(2)F = FS - FL - SH - LH

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

The evolution of the mixed-layer temperature field, T_o, is evaluated using an explicit forward time step. At iteration n the required information to advance the forecast include T_o^n, h_o, F^n, Q^n, and A^n, where h_o is time invariant and Q^n 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 T_o is evaluated as:

(3)T_o^{(n+1)'} = T_o^n + \frac{(1-A^{n}) F^{n}} {\rho_{o} C_o h_{o}} \Delta t

where \Delta t is the model time step. We note that A^n is computed from the fraction of the total CAM5.0 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:

(4)A^n = \frac{{icefrac}^n} {(1 - landfrac)}

where icefrac is the fraction of ice in the CAM5.0 grid cell and landfrac is the fraction of land in the CAM5.0 grid box.

The Q^n flux is then adjusted since it is possible (using monthly specified values of Q) to introduce a non-physical cooling of the mixed layer when its temperature is at the freezing point. Therefore, if Q^n > 0 and T_o^{(n+1)'} < 0^\circ C, then

(5)Q^{n'} = Q^{n} f_T

where f_T = {(T_f - T_o^{(n+1)'})}/{T_f}, and T_f is the ocean freezing temperature of -1.8:math:^circC (where T_o is expressed in units of ^\circC). 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 Q under sea ice is applied:

Q^{n''} = Q^{n'} + [ A^n f(h_i) q_{hem} ]

where

\begin{aligned} f(h_i) &= h_i / (1 + h_i) \;\; q_{hem} < 0 \nonumber\\ f(h_i) &= 1 / (1 + h_i) \;\; q_{hem} > 0\end{aligned}

h_i is the local ice thickness, and q_{hem} is a tuning constant which may have different values for the Northern and Southern hemispheres. The coefficient A^n ensures this adjustment only occurs under sea ice covered ocean. The function f(h_i) 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 q_{hem} which yield good control sea ice distributions are +15W/m^2 and -10:math:W/m^2 for the Northern and Southern hemispheres respectively.

The adjusted Q^{n} (Q^{n''}) is then used to update all ocean points due to deep ocean heat exchange and transport as:

T_o^{(n+1)''} = T_o^{(n+1)'} - \frac{Q^{n''} + A^n F_{oi}^n} {(\rho_o C_o h_o )} \Delta t

where F_{oi}^{n} is the energy flux associated with any ice melt and shortwave radiation transmitted through the sea ice from the previous time step.

The quantity F_{frz}^{n} is nonzero only if the temperature of the slab ocean falls below the freezing point:

F_{frz}^{n+1} = (\rho_o C_o h_o) max(T_f - T_o^{(n+1)''},0)/ \Delta t

If F_{frz}^{n+1} is nonzero, new ice forms over the ice-free portion of the grid cell and T_o^{n+1} is returned to the freezing temperature:

T_o^{(n+1)''} = max(T_o^{(n+1)''},T_f)

A renormalization is necessary to ensure energy is conserved when Q is adjusted as described above. We distinguish warm ocean as those points for which T_o > 0^\circC. An adjustment for warm ocean points is computed after all modifications to Q are completed. Let Q_o be the original unadjusted Q, and let <Q_o> be the global (area weighted) mean. The final (total) Q applied to warm ocean points is:

Q''' = Q'' + [ (<Q_o>-<Q''>) (A_o/A_w) ]

where A_o is the global area over all ocean, and A_w 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 (A_w/A_o). Thus, <Q'''> = <Q_o>, satisfying global energy conservation of Q for every time step. In practice, the bracket term adjustment is applied to warm ocean points after the Q redistribution is completed.

7.2. Thermodynamic Sea Ice Model

After the slab ocean component computes the atmosphere-ocean heat fluxes and updates T_o and F_{frz}, the thermodynamic sea ice model takes the latter two variables as input and computes the atmosphere-ice and ocean-ice heat fluxes and advances the state of the sea ice, including snow depth, surface temperature, ice thickness, ice fractional coverage, and internal energy profile in the ice. The physics of the sea ice component model in CAM5.0 are discussed in detail in the next chapter.

7.3. Evaluation of the Ocean Q Flux

The ocean Q flux is generally evaluated using a CAM5.0 control simulation driven by prescribed sea surface temperature and sea ice distributions. Let

F_{net} = FS - FL - LH - SH

over ocean (regardless of whether the ocean surface is open or ice covered), for each of 12 ensemble mean months (n=1,…,12). The Q flux distribution for each month n is then evaluated: (note that here we use the CAM5.0 sign convention on the Q flux).

Q = Q_{ocean} - Q_{ice} - F_{net}

where:

Q_{ocean} = (\rho_o C_o h_o/\texttt{daysmonth}(m)) {\{(1-A(m+1)) T_o(m+1) - (1-A(m-1)) T_o(m-1)\}}

Q_{ice} = L_i {\{A(m+1)h_i(m+1) - A(m-1)h_i(m-1)\}}/\texttt{daysmonth}(m)

where \texttt{daysmonth} is the number of days in each month, L_i is the latent heat of fusion for ice, and h_i is the regionally specified ice thickness. We then define an annual average using the monthly mean data:

\overline{Q} = \sum_{m=1,12} \texttt{daysmonth}(m) Q(m)/365

By definition

\overline{Q_{ocean}} = 0

\overline{Q_{ice}} = 0

so that

\overline{Q} = -\overline{F_{net}}

Since F_{net} is the monthly mean flux into the ocean directly from the control, Q must be constrained to ensure that the actual Q applied in the SOM configuration has the same annual mean as -\overline{F_{net}}. Otherwise, the application of the Q flux would introduce a source or sink of heat with respect to the control.

The actual Q applied in the SOM configuration is based on linear interpolation between monthly means, taken as midpoints. Since the months have different lengths, in general the annual mean of the Q flux applied to the SOM will not equal -\overline{F_{net}}. Thus, we must define another annual mean, based on the time interpolated Q, to ensure that the SOM applied Q has the identical annual mean as the fluxes F_{net} from the control run.