Except for the convective parameterization and a few minor
changes, the hurricane
model used here is identical to that of Emanuel (1989, hereafter
E89)
. We here summarize the properties of the model and the changes that
have been made to E89; the actual model equations are presented in the
Appendix.
The model is axisymmetric and phrased in angular momentum
coordinates, using the potential radius 
, defined such that
Here 
is the Coriolis parameter (assumed constant), 
the
(physical) radius from storm center and 
is the azimuthal
velocity. The right side of (15) is the total angular momentum per
unit mass.
The model consists of 2 parts: a subcloud layer and the rest of the
troposphere (see Figure 1).
The latter is assumed always to be in hydrostatic and
gradient wind balance, and to be neutral to slantwise moist
convection, a condition approximated by constant 
along angular
momentum () surfaces. The primary dynamic variables of the model
are
All the entropy variables are replaced by a new quantity,
, defined
where 
and 
are the surface and tropopause absolute
temperatures (assumed constant), 
is the entropy, and 
is
the entropy of the ambient subcloud layer. Thus the subcloud-layer
entropy variable used in the model is 
, the troposphere
entropy variable is 
, and the saturation entropy of the
troposphere is 
:
Clearly, 
in the ambient environment (which is
assumed to be convectively neutral) and 
.
All the entropy variables, , are scaled in the model by the
ambient value of at saturation at sea surface temperature:
where 
is the ambient saturation entropy of the ocean
surface. (
does vary with radius because of the dependence
of on pressure.) The quantity 
has the units of
velocity squared, with typical values around 
. E89 showed that a characterisitic maximum surface wind speed in
tropical cyclones is 
, of order 
.
All the dependent and independent variables are made dimensionless according to the scaling in Table 1, which also shows typical numerical values of the scaling parameters, and the definitions and values of the nondimensional model parameters are given in Table 2.
The fundamental change from E89 consists of replacing the buoyancy closure for convection used there with the subcloud layer equilibrium scheme described here in section 1. That is, we use, with some modification, the equations (7), (11), (12) and (13) phrased in potential radius coordinates and scaled according to Table 1. In addition, the scaled form of the subcloud layer entropy equation, (1), is actually solved, but subject to the constraint (14), which takes the form
One important modification of the new closure is necessary in applying it to the hurricane problem. This consists of taking into account radial entropy advection in the subcloud layer. Accordingly, the dimensionless version of (1) contains a radial advection term, and consequently so does the equilibrium mass flux, whose dimensionless form is given in the Appendix by (31). In other words, the equilibrium downdraft mass flux into the subcloud layer is assumed to balance surface fluxes and radial entropy advection by the Ekman flow.
There are two further critical aspects of the new closure that must be
addressed. First, E89 showed that the quintessential physical process
leading to tropical cyclone genesis in the model is the cessation of
convective downdrafts owing to saturation of the middle and lower
troposphere on the mesoscale. Thus, unlike in the linear analyses of
tropical intraseasonal oscillations (Emanuel, 1993), we must here allow for
a variable bulk precipitation efficiency, 
. Qualitatively, one
would expect to vary with the relative humidity of the lower
and middle troposphere. Accordingly, we choose the function
where 
is the initial entropy deficit of the middle troposphere.
Thus is zero in the environment and approaches unity as
. At first, this may appear to be an extreme
variation of , but it should be noted that we are also applying
a Newtonian cooling in place of real radiative cooling. As this cooling
vanishes in the environment of the storm, so too must the convective
heating for a balance to be maintained.
The second critical process is the moistening of the lower and middle troposphere by convection, since this is the essential process that saturates the troposphere in the incipient cyclone core and allows downdraft-free convection to develop there. The effect of convection on free atmosphere entropy can be broken into two parts: entropy advection by subsiding air in the cloud environment, and detrainment of high entropy, cloudy air. As we shall see, the development of tropical cyclones in the model is sensitive to the distribution and magnitude of moistening of the lower troposphere by convection. In the present model, the detrainment of entropy into the lower troposphere is formulated as
where 
and 
are parameters that govern the rate
of detrainment of high entropy, cloudy air into the lower troposphere. Both
and are less than unity, and it is assumed that
that portion of the entropy detrainment that does not occur in the lower
tropopshere takes place in the upper troposphere so as to satisfy integral
entropy conservation.
When 
, it is assumed that a greater fraction of detrainment
occurs in the lower troposphere, while when is close to unity,
most of the detrainment happens in the upper troposphere. Note that the
entropy of the upper troposphere is not a model variable.
Aside from the representation of convection, the model differs from that of E89 in the following respects:
, and 
used in
E89 are not needed here.
Most of these differences are minor in comparison to the implementation of the new representation of convection, and have the effect of simplifying the model.
