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theory to compute the atmospheric response. These models, however, neglect the
fact that the distribution of diabatic heating is highly dependent on the mean wind
and equivalent potential temperature distributions in the boundary layer. These
in turn depend on the surface pressure and the moisture distribution, which are
themselves dependent on the motion field. Thus, in a consistent model the diabatic
heating cannot be regarded as an externally specified quantity, but must be obtained
as part of the solution by, for example, using the cumulus parameterization scheme
illustrated in Section 11.3.
As indicated in (11.27) this scheme requires information on the vertical dis-
tribution of convective heating in order to solve for the temperature perturbation
throughout the troposphere. There is, however, evidence that the essential features
of stationary equatorial circulations can be partly explained on the basis of a model
that involves only the boundary layer. This is perhaps not surprising since the main-
tenance of a convective system depends on evaporation and moisture convergence
in the boundary layer. Over the tropical oceans the boundary layer can be approx-
imated as a mixed layer of about 2 km depth, which is capped by an inversion,
across which there is a density discontinuity δρ (see Fig. 5.2). The virtual temper-
ature in the mixed layer is strongly correlated with the sea surface temperature.
If we assume that the pressure field is uniform at the top of the mixed layer, the
surface pressure will be determined by hydrostatic mass adjustment within the
mixed layer. (The system is analogous to the two-layer model discussed in Sec-
tion 7.3.2.) The resulting pressure perturbation in the layer depends on the density
discontinuity at the top of the layer and the deviation, h, of the layer depth from
its mean depth H
b
. In addition, there will be a contribution from the perturbation
virtual potential temperature, θ
v
, within the mixed layer. Thus, the perturbation
geopotential in the mixed layer can be expressed as
=
g (δρ /ρ
0
) h
−
θ
v
(11.49)
where
(gH
b
/θ
0
) is a constant and ρ
0
and θ
0
are constant mixed layer reference
values of density and potential temperature, respectively.
According to (11.49), positive sea surface temperature anomalies and negative
boundary layer height anomalies will produce low surface pressures, and vice
versa. If the boundary layer depth does not vary too much, the surface pressure
gradient will thus tend to be proportional to the sea surface temperature gradient.
The dynamics of steady circulations in such a mixed layer can be approximated by a
set of linear equations analogous to the equatorial wave equations (11.29)-(11.31),
but with the time derivative terms replaced by linear damping terms.
Thus, in the momentum equations the surface eddy stress is taken to be pro-
portional to the mean velocity in the mixed layer. In the continuity equation the
perturbation in the mixed layer height is proportional to the mass convergence in
the layer, with a coefficient that is smaller in the presence of convection than in
its absence, due to ventilation of the boundary layer by convection. The x and y
≡
