\chapter{Slab Ocean Model}


The slab ocean model consists of a  prognostic equation at each ocean 
point for the oceanic mixed-layer temperature $T_{mix}$.
The prognostic equation for $T_{mix}$ is given by


\begin{equation}
\frac{dT_{mix}}{dt} = \frac{Q_A+Q_O}{\rho_w c_{p_w} h_{mix}}
\end{equation}

where $\rho_w$ (=1030kg/m$^3$) is the density and $c_{pw}$ (=4180J/kg/K) 
the heat capacity of ocean water. 
$h_{mix}$ is a prescribed ocean mixed layer depth (default = 50m). 
$Q_A$ denotes the net atmospheric heat flux into the ocean 
which consists of the net solar and long wave radiation 
and the sensible and latent heat fluxes. 

The ocean mixed layer heat flux ($Q_O$) represents the oceanic transport and 
the deep water exchange. Commonly $Q_O$ is prescribed from monthly mean data
which are obtained from climatologies of the uncoupled model by computing    

\begin{equation}
Q_{O} = <Q_{A}^{u}> - <\frac {dT_{mix}}{dt} \rho_w c_{p_w} h_{mix}>
\end{equation}

where $<Q_{A}^{u}>$ and $<dT_{mix}/{dt}>$ 
are the climatological (monthly) 
averages of the net
atmospheric heat flux and the mixed layer temperature tendency, 
respectively, both taken from the uncoupled (i.e.~prescribed SST) simulation.

In addition to a prescribed oceanic heat transport, horizontal and 
vertical diffusion can be switched on optionally. 
In the case of vertical diffusion 
a user defined number $n$ of layers with prescribed thicknesses $h_{mix}^{n}$ 
are coupled via diffusion

\begin{equation}
\frac{\partial T_{mix}}{\partial t} = \frac{\partial}{\partial z} \left( K_v
\frac{\partial T_{mix}}{\partial z} \right)
\end{equation}

with the (level depentend) diffusion coefficient $K_v$ (set to a default value
of 0.0001m$^2$/s for all levels). The equation is solved using a 
back-substitution method.

Horizontal diffusion of $T_{mix}$ is given by

\begin{equation}
\frac{\partial T_{mix}}{\partial t} = K_h {\nabla}^2 T_{mix} 
\end{equation}

for each level. The default value of $K_h$ is 1000m$^2$/s.



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