Hurricane model



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Hurricane model

Except for the convective parameterization and a few minor changes, the hurricane model used here is identical to that of Emanuel (1989, hereafter E89)gif. 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 gif 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:

1.
Momentum diffusion has been added to the top layer. This has the cosmetic effect of preventing real discontinuities from forming at the model top, but is observed to have little effect otherwise.

2.
The enthalpy surface exchange coefficient is allowed to differ from the surface drag coefficient.

3.
The magnitude of the radiative cooling, while still capped, is not turned off in regions of positive Ekman pumping as in E89.

4.
The other variables, , and used in E89 are not needed here.

5.
There is no downward advection of entropy from the upper to the lower troposphere.

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.



next up previous
Next: Results Up: THE BEHAVIOR OF A Previous: Introduction



Kerry Emanuel
Mon Jan 5 07:19:46 EST 1998