In the last lecture, we derived an expression, (37), for the maximum wind speed in mature tropical cyclones, which depends, among other things, on the surface pressure, , at the radius of maximum winds. To obtain the wind speed and pressure separately, we need another relationship between wind and pressure to complement (37). Here, we need to make a more detailed analysis of the dynamics of tropical cyclones. We begin with the assumption, well verified by observations, that the basic wind and pressure fields in tropical cyclones are in a state of hydrostatic and gradient wind balance, as expressed by (1) and (2). Cross-differentiating to eliminate gives

We next use Maxwell's relation, (19), to express the specific volume, , in terms of entropy, , in (40):

Now we have expressed the right-hand side of the thermal wind equation in terms
of the conserved variable . We can also express the left-hand side of
(41) in terms of another conserved variable: the *angular momentum per
unit mass*, :

It is easy to show that is conserved for axisymmetric displacements in the absence of friction. If we replace by in (41) we get

We next assume that surfaces of constant and of constant are congruent. In the eyewall, this will be so because both and are conserved following the upflowing air. Elsewhere, it turns out that the congruency of the two surfaces constitutes a generalized condition for neutral stability with respect to convection in the presence of centrifugal as well as gravitational forces. This critical condition appears to be well satisfied in ``full physics" numerical simulations of tropical cyclones (Rotunno and Emanuel, 1987) [8]. This condition allows us to write (43) as

This may be regarded as an equation for the slope of and surfaces. Since the and surfaces are congruent, the two quantities are just functions of one another. Thus we may integrate (44) up along surfaces, from their base in the boundary layer to the point at which they intersect the tropopause, to get

where and are the radii of particular or surfaces where they intersect, respectively, the surface and the tropopause. In a well-developed tropical cyclone, these surfaces flare out to very large radii at the tropopause, so we can neglect the second term on the right of (45) in comparison to the first term. Thus, in the boundary layer, we may write (45) as

We cannot quite integrate (46) because of the term in it. But it turns out that we can express (46) in terms of perfect differentials of some other quantities. First, we substitute (42) for in (46):

Now we again substitute (42) for in the first term in (47) and also make use of (2) expressing gradient wind balance, to get

Finally, we make use of the ideal gas law, (3), for and once again assume the constancy of both and to get

This is the sought-for second relationship between pressure, velocity and specific humidity. Integrating it from the environment of the storm inward to the radius of maximum winds, and using (9) to express the entropy, in terms of other variables gives

where is the ambient surface pressure, and are the
surface pressure and specific humidity at the radius of maximum winds, and
is the ambient boundary layer specific humidity. In writing (50) we have
neglected the Coriolis term, which is small unless the storm extends to very
large radii, and another small term proportional to . Now remember that we have assumed that the *relative* humidity
is constant between the environment and the radius of maximum winds. Using this
together with (34), under the approximation that , gives

Given the quantities , , and , (51) and (37) can be solved for both and . Eliminating between the two yields a transcendental equation:

where and

By plotting the two curves and , it is easy to see that (52) has two solutions, one solution, or no solutions, depending on the values of and . Once (52) is solved, the result can be substituted into (37) to find the maximum wind speed.

Before showing solutions of (52), there is one other quantity that is useful to derive, because it has become routine to measure it from reconnaissance aircraft: the central pressure. (Recall that the solution of (52) is for the surface pressure at the radius of maximum winds.) To explore the pressure distribution inside the radius of maximum winds, it would be necessary to discuss the fascinating dynamics of the eye, which we deem beyond the scope of this short course. Instead, we take a short cut by noting that, in most cases, the observed wind distribution inside the radius of maximum winds is not far from that of solid body rotation, in which the azimuthal wind speed increases linearly with radius. If we assume that this is so, we can simply integrate the gradient wind balance relation, (2), inward from the radius of maximum winds to the storm center. In doing so, we neglect the Coriolis term in (2), which is small compared to the centrifugal term in the eye. Thus we have

Integrating this inward from to gives

where is the central pressure. Note that (54) is independent of the radius of maximum winds. Given the solution for , and assuming that , (54) can be solved for . Thus equations (37), (52) and (54) can be solved for the pressure at the radius of maximum winds, the maximum winds speed, and the central pressure. We note in passing that, once this is done, we can integrate (49) coupled with (2) to find the boundary layer pressure and wind outside the radius of maximum winds, while the assumption of solid body rotation and (53) gives the pressure and wind inside the radius of maximum winds. Then given these, (44) can be integrated upward along angular momentum surfaces to give the wind and entropy distributions between the boundary layer and the top of the storm. This was done by Emanuel (1986) [3].

As mentioned before, (52) has no solutions, one solution or two solutions, depending on the values of abd . Also, because of the way it has been defined, is greater than or equal to 1. It is possible to show that, when two solutions exist, the smaller of the two is the only stable, physically realizable solution. We present solutions to this set of equations, showing the smaller root where it exists, in Figure 5.1 and Figure 5.2. The maximum wind speed and central surface pressure are shown as functions of and for given values of and . (The air temperature, , has been assumed to equal the sea surface temperature, .) Note that the maximum intensity of the storm increases with the sea surface temperature and decreases with the outflow temperature. In tropical cyclones, typical values of and are, respectively, around 300 K and 200 K (27 C and -73 C). Also note the existence of a ``hypercane" regime in which there are no solutions to (52). In this regime, the energy cycle of the tropical cyclone is unstable. As the intensity of the storm increases, the surface pressure falls, and this increases the saturation enthalpy and entropy of the sea surface, which depends on pressure. This creates the potential for greater heat input into the system. In the normal tropical cyclone regime, this positive feedback is countered by the negative feedback owing to dissipation, which increases as the cube of the surface wind speed. But in the hypercane regime, the positive feedback is overwhelming, and the system runs away. Emanuel et al. (1995) [5] postulated that hypercanes might form where bolide impacts with ocean or large undersea volcanic eruptions locally heat sea water to values that are supercritical for hypercanes, and used a ``full-physics" numerical model to simulate hypercanes. One property of such storms is that they penetrate well up into the stratosphere, where they can deposit large quantities of water. (The stratosphere is normally very dry.) Chemical reactions would then rapidly destroy the ozone layer. We postulated that this might serve as a link between past massive bolide impacts and mass species extinctions, such as occurred at the end of the Cretaceous. Hypercanes also produce supersonic wind speeds, violating some routine assumptions that were made in the numerical model, so more comprehensive simulations ought to be made using a code capable of simulating supersonic flows.

Routine atmospheric data together with sea surface temperature records can be used to calculate the maximum wind speeds and minimum central pressures possible in tropical cyclones in the present climate. Now go to the potential intensity climatology site and look at the various distributions. It is helpful to bear in mind that the minimum central pressure ever measured in a northwest Pacific typhoon was 870 mb, while the lowest pressure in the North Atlantic was 888 mb.

Today's potential intensity calculations are shown at this website. Here the atmospheric temperature profiles are updated daily while the ocean temperatures are updated weekly.

How do the intensities of actual tropical cyclones compare with potential intensities? There are reliable records of tropical cyclones in the North Atlantic going back to about 1958, when reconnaissance aircraft first began to directly measure wind speeds. In the western North Pacific, there was routine aircraft reconnaissance from the 1960's through 1987. tropical cyclone positions and maximum wind speeds are recorded every 6 hours. We can estimate the potential maximum wind speed from its monthly mean value, which was calculated using about 20 years of atmospheric and sea surface temperature data. We take each 6 hours intensity report and divide it by the climatological potential intensity for that position and that month to create a normalized intensity, . We then create a cumulative distribution function, or CDF, by summing the total number of observations of normalized intensity exceeding for each value of . We do this only for those storms for which the potential maximum wind speed exceeds to avoid counting those storms that have moved over land or cold water and are in the process of rapid decay. The results are shown for the North Atlantic and western North Pacific in Figure 5.3 and Figure 5.4, respectively. Also shown is a curve that seems to fit both data sets quite well; it has the functional form

where is a constant that is related to the total frequency of events of all normalized intensities.

We have essentially no understanding of why the data fit a curve like that given by (55). Note that no storms have intensities that exceed the potential intensity by any significant amount, but the vast majority of storms fall well short of their potential. Some of this is owing to the fact that some of the storms in the record are in the process of intensifying, but even if we confine the data to storms in nearly steady states, a curve not much different from (55) results. On the other hand, virtually all numerical simulations of tropical cyclones using axisymmetric models manage to spin up storms right to their potential intensity. Why are storms in nature less intense than the model storms? This is a point we shall return to in a future lecture.

Mon Apr 13 10:50:48 EDT 1998