2.1 Overview

 

Our model comprises a quasi-geostrophic atmosphere overlying a quasi-geostrophic ocean, characterized by their respective potential vorticities (QGPV) and streamfunction distributions and governed by prognostic QGPV equations on a beta-plane.

The atmosphere, imagined to be bounded above by a lid and below by the ocean, is governed by the equation:

$$\frac{D}{D{t}}q_{a}=f_{o}\frac{\partial}{\partial z}\left( \f... ...right) -\epsilon \nabla ^{2}\psi _{as}\qquad \mbox{ATMOSPHERE}$$ (1)

Here $\frac{D}{Dt}$ is the Lagrangian derivative and q_a is the quasi-geostrophic potential vorticity:

$$q_{a}=\nabla ^{2}\psi _{a}+\beta y+f_{o}^{2}\frac{\partial}{\... ...frac{1}{N_{a}^{2}}\frac{\partial}{\partial z}\psi _{a}\right)$$

expressed in terms of the atmospheric streamfunction $\psi _{a}$. f_o is a reference value of the Coriolis parameter f, the meridional gradient of <tex2html_comment_mark> f is $\beta$, N_a² $=\frac{1}{\theta _{a0}}\frac{\partial }{% \partial z}\overline{\theta _{a}}$ is the atmospheric Brunt-Väisälä buoyancy frequency, $\theta _{a}$ the atmospheric potential temperature with $\theta _{a0}$ a typical value, and Q_a is the diabatic heating rate of the atmosphere defined by:

$$\frac{D}{D{t}}\theta _{a}=Q_{a}$$ (2)

In (1), $\epsilon \nabla ^{2}\psi _{as}$ represents frictional sinks of vorticity associated with Ekman layers at the surface with $% \epsilon ^{-1}$ a frictional spin-down time.

We suppose that a radiative-convective equilibrium temperature, $\theta _a^{*}$, controls the thermal forcing of the atmosphere thus:

$$Q_{a}=-\gamma_a (\theta _{a}-\theta _{a}^{*})$$ (3)

Here $\gamma_a ^{-1}$ is a time-scale set by the radiative-convective process; $\theta _a^{*}$ is a radiative-convective temperature profile to which $% \theta _{a}$ relaxes, which is assumed to be a function of sea-surface temperature thus:

$$\theta _{a}^{*}=\theta _{a}^{*}(\mbox{SST})$$ (4)

The form, (3) and (4), makes sense as a simple and physically plausible representation of convective heating of the troposphere, permitting the heating field to be a function of the state of both the atmosphere and the ocean. That heating will initiate a dynamical response of the atmosphere and change the winds that blow over the ocean.

The equations governing the ocean are:

$$\frac{D}{D{t}}q_{o}=f_{o}\frac{\partial}{\partial z}\left( \f... ...a \times \frac{\partial}{\partial z}\tau \qquad % \mbox{OCEAN}$$ (5)

where q_o is the oceanic QGPV:

$$q_{o}=\nabla ^{2}\psi _{o}+\beta y+f_{o}^{2}\frac{\partial}{\... ...frac{1}{N_{o}^{2}}\frac{\partial}{\partial z}\psi _{o}\right)$$

$\psi _{o}$ is an oceanic streamfunction, N_o² is an oceanic Brunt-Väisälä frequency, Q_o is the diabatic heating of the interior of the ocean and $\tau$ is the mechanical stress supplied by the surface wind. The stress at the ocean's surface is a function of the velocity of the wind at the surface:

$$\tau _{s}=\tau _{s}(\psi _{s})$$ (6)

The evolution of the oceanic mixed-layer temperature, which we assume is synonymous with sea-surface temperature, is

$$\left( \frac{\partial}{\partial t}+v\cdot \nabla \right) \mbox{SST} = Q_{o} \qquad\mbox{SEA SURFACE TEMPERATURE}$$ (7)

Here the horizontal velocity in the mixed layer is v, the sum of an Ekman and geostrophic components $(v=v_{\mbox{\scriptsize ek}}+v_{g})$, and Q_o is the diabatic heating of the mixed layer induced by air-sea interaction and entrainment fluxes through the mixed-layer base. There is no vertical advection in (7) because the mixed layer is assumed to be vertically homogeneous.

Note that:

1.

(1) and (2) are the starting point of analytical studies of atmospheric planetary waves dating back to Charney and Eliassen (1949) and Smagorinski (1953).

2.

If v=w=0, then (7) reduces to a `slab ocean', which responds on timescales of several months (primarily via surface heat exchange and entrainment), depending on the depth of the `slab' - see, eg. Hasselman (1977); Frankignoul and Hasselman (1977). On decadal time-scales, however, advective processes may be important and SST changes may be dominated by gyre dynamics and subduction processes (v and w): see Hall and Manabe (1997).

3.

If the wind-curl is assumed to be a stochastic process and Q_o=0 in (5), then it reduces to the ocean model analysed by Frankignoul et al.(1996) in their study of the response of the ocean to stochastic atmospheric forcing.

Clearly, (1) through (7) are highly simplified representations of the respective fluids and their interaction. But the philosophy of our approach is to build our intuition about the coupled problem in stages, by first fitting together simple pieces, and then increasing the complexity of the component parts and their coupling. Heating of the atmosphere depends, through (3) and (4 ), on the state of the ocean which, in turn, depends on its forcing from the atmosphere via (6). We shall now go on to study whether the above system supports coupled modes. Their existence will depend on the form assumed for (3), (4), (6) and ( 7) i.e. on the nature of the boundary layers of the two fluids and the manner in which they are assumed to interact with one-another and the `free' atmosphere/ocean above/below. To make analytical progress our representations will, of necessity, be simple, but they are motivated by sound physical principles.