Model formulation

The above ideas are encapsulated in the following coupled system of equations of non-dimensional quantities - a rigorous derivation based on a coupled thermodynamic model representing four different boxes on either side of the zero windstess-curl line in the atmosphere (extending over the depth of the troposphere) and the ocean (comprising the portion of the dynamics affected by the air-sea interactions) may be found in Marshall [2000a]:

   

$$\frac{{\rm d} \Delta T}{{\rm d}t} \, = - \alpha \tau \, - \, \lambda \Delta T \, + \, Q_0$$ (4)

$$\tau \, = N \, - \, f \Delta T$$ (5)

$$Q_0 \, = m \Psi_m \, + \, g \Psi g\vert _w$$ (6)

Here, $\Delta T$ denotes the strength of the SST dipole (T_N-T_S) that straddles the Gulf Stream, $\tau$ the amplitude of the NAO windstress, assumed to be decomposed into a stochastic component N and an SST - induced component ( $f \Delta T$). The heating rate Q_o due to anomalous advection of heat composed of $\Psi_m$, the strength of the anomaly of meridional overturning, and $\Psi g\vert _w$, the intergyre gyre streamfunction evaluated just inside the western boundary current. The model parameters are:
 $\bullet$ $\alpha$: scaling of stochastic windstress N into a surface heat flux anomaly;
 $\bullet$ $\lambda$: damping of $\Delta T$ due to air - sea interactions (cf. [FMZ97]);
 $\bullet$ f: feedback of $\Delta T$ on NAO windstress pattern; it is assumed that a small positive feedback (f>0) exists between the NAO and $\Delta T$ (see section 4.4);
 $\bullet$ g, m: efficiency of heat transport by anomalous gyres and meridional overturning, respectively;
 $\bullet$ s: efficiency of thermal dipoles in driving meridional overturning, eqn. (3).