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A Universal Hurricane Frequency Function

Evidence is provided that the global distribution of tropical hurricanes is principally determined by a universal function H of a single variable z that in turn is expressible in terms of the local sea surface temperature and latitude. The data-driven model presented here carries stark implications for the large increased numbers of hurricanes which it predicts for a warmer world. Moreover, the rise in recent decades in the numbers of hurricanes in the Atlantic, but not the Pacific basin, is shown to have a simple explanation in terms of the specific form of H(z), which yields larger percentage increases when a fixed increase in sea surface temperature occurs at higher latitudes and lower temperatures.


Introduction
There are numerous factors that either promote or inhibit the formation of hurricanes or tropical storms, and it is well-known that two of them have special significance: the sea surface temperature SST (or simply T) and the latitude  at the time and place of storm formation. The importance of latitude is that it governs the strength of the Coriolis Effect, a principal factor in creating some initial vorticity, while the SST is the source of an updraft of air that can create a tropical low. Here, we assert more specifically that the probability density of a hurricane or tropical storm forming can be expressed in terms of a simple mathematical function of those two variables, ) , (    T  H . Alternatively, we can even write H in terms of a single combined variable | | sin ) ( , where T is the SST, T 0 is a threshold value of 25.5 0 C, and  is the latitude. The functional form of H(z) most consistent with the data is found to be a simple power law, H(z) = Cz n , where C = 0.00073, and n = 3.5+0.5, for most regions of the globe. This result appears to be independent of time, and location, with the exception of those regional departures.
The study reported here rests on four key assumptions.

a. Existence
There exists a probability function H which describes whether a hurricane (also referred to as a cyclone or typhoon) forms. Furthermore, we assume that H depends on numerous variables, of which only the SST and the latitude play a universal role, with the other ``secondary" variables, being restricted in time or space.

b. Secondary variables
The secondary variables influencing H do so in a way that simply multiplies it by a constant factor that is regionally and temporally limited, and can be explained in terms of previously studied regional or oscillatory phenomena that enhance the value of H by a specific factor over that spatial or temporal region.

c. Data-driven
The function H can be found using the data on recorded hurricanes, without requiring a fundamental understanding of the basic physics describing exactly how a hurricane forms. This latter subject is one of current research interest, and the clue offered by the specific functional form of H presented here may advance that search for understanding.
Since the function H is justified by appealing to data rather than fundamental theory, the result is a model rather than a theory.

d. Validation
Given a specific form of H derived from the data, we can test the model by crosschecking it for consistency using the data, and more importantly, by seeing whether future hurricane data agrees with the model. 5

Methods
This work has relied on the data sets maintained by NOAA for the Atlantic and Eastern Tropical storms can be thought of as heat engines which derive their energy from the warm surface temperature of the ocean --or more precisely the temperature difference between the warm ocean and the much lower temperature at high altitude in the 6 atmosphere, which can represent a differential of around 100 0 C. It has been observed that tropical storms do not arise unless the SST, T, exceeds some threshold value T 0 cited in various sources as being 26 or 27 0 C. The other key variable in storm formation besides SST is the latitude, which is important because the horizontal component of the Coriolis force varies as its sine. The Coriolis force produces some initial vorticity that helps the storm self-organize. Thus, tropical storms do not form very close to the equator, and they have opposite senses of rotation in the two hemispheres.
In searching for a function H, we make use of the above well-known aspects of tropical storm formation, and hence assume that the relevant variables entering the distribution H involve the quantities 0 T T  and  sin . Since it is much easier searching an unknown function of a single variable than one involving two variables, we divide the search process into two steps: first, finding a single combined variable z expressed in terms of 0 T T  and  sin and second finding an appropriate functional form of z that best fits the data. One plausible combination of the variables when because we want the resulting H(z) generated using the data themselves to satisfy these six requirements: a. Threshold H (z) must have the correct threshold behavior. This requirement is guaranteed by our choice of z for all a and b, as long as H(z) vanishes when z approaches zero.

b. Symmetry
H (z) must be a symmetrical function between the hemispheres even though the actual numbers of storms is not symmetric.

c. Monotonicity
H (z) must be a monotonic function of z since we expect H to increase continually as either 0 T T  or  increasealthough we cannot rule out a priori the possibility that it levels off at some saturation value for large z.

d. Uniformity
H (z) must have the same functional form in each basin across the globe.

e. Normalization
H(z) ideally should also have the same normalization in each basin, or if not, the variations should be explainable in terms of known regional phenomena.

f. Zeroes
H(z) must not predict storms occurring in any regions of the globe where none are observed, e.g., in the South Atlantic.

8
Let us now explain exactly how H(z) is deduced from the data themselves. The point of origin of each recorded storm in the data base places it in a specific latitude-longitude cell (taken here to be 4 x 4 degrees), and since the time of occurrence of that storm is known, we can use the SST-record to find the T in that cell when the storm formed. Given a particular trial definition relating z to T and  , we can then compute the z-value associated with each storm. The z-values are binned, and we simply count the number of storms N j in the jth bin between z j and z j+1 . We then use the SST-record to obtain T for a given month for each latitude-longitude cell to find the number of months during the 47year record from 1960-2007 that a given cell had a temperature T. Finally combining that T with the  values of each cell, we find the number of months M j that cells in a given geographic basin (such as the Atlantic) had a specific binned z-value, z j . The computed value of H for that z j simply equals N j /M j , which represents the observed number of storms per month in 4x4 degree latitude-longitude cells for a given z. Since these numbers are always less than one, it is more appropriate to think of them as the probability of having a storm per month. Once we have deduced from the data a set of H-values with associated uncertainties (discussed later), we can proceed to find the functional form of H based on a best fit to the data-derived values. Given the fairly stringent requirements on the form of H(z) discussed earlier, the search for an H(z) that fits the data is fairly easy, given any trial choice of a and b used to define the variable z.

9
For example, we have found that the simplest choice: is found to give poor results, while the choice: , gives good ones as we shall see in the next section. Based on chi square for the Asian Pacific basin the best fitting exponent is actually b=0.49 +0.01.) One interesting property of our is that it yields a common temperature threshold T 0 for all latitudes, which may seem surprising. Other possible choices such as would not have this feature, but they would introduce one more free parameter, and would imply that H(z) is no longer separable into functions of T and  . Were our choice not found to yield a good fit to the data, one would need to explore such more complex definitions of z.

Results
In order to test how well the data from various regions of the world agree with a power law in z, i.e., | | sin , we have relied heavily on the data for the Asian Pacific basin. These data constrain H(z) far more than Atlantic data, since (a) the Asian Pacific basin includes over five times as many storms as the Atlantic basin, (b) it includes storms over a larger range of latitudes (31 bins in z versus only 17 for Atlantic data), and (c) the Asian Pacific includes appreciable numbers of storms in both hemispheres, whereas fewer than 1% of Atlantic storms are in the Southern Hemisphere.
We show in fig. 1  Before discussing the quality of fit for data in the other hurricane basins, we digress to consider the size of the error bars used in fig.1, which directly affects the quality of the fits.

a. Size of Uncertainties
Although one often only displays error bars on the dependent variable only, here the independent variable z also has a significant uncertainty that must be taken into account.
, we see that because the data is binned in 4 by 4 degree bins and also in increments of 0.25 0 C in T, we need to use an uncertainty (due to binning) of . When these errors are propagated and added in quadrature to find the uncertainty in z using representative values of T and  for storms, we find that 15 . 0   z .

11
Finally, since the quantity H(z)/z n should be a constant if H(z) satisfies a z n power law, we may use standard error propagation methods to find the variance in H(z)/z n by combining the separate uncertainties in z and H according to: Note that it is this resultant uncertainty that is used to compute chi square under the requirement that H(z)/z n remains constant, and thus:

The Other Hurricane Basins
Let us now consider how well the data from other basins fits H(z), starting with the Eastern Pacific, principally the region off the Southern coast of Mexico --a region of very frequent storms. This data (the filled triangles in fig. 2) appears to be at odds with what was seen in the Western (Asian) Pacific basin in two important respects. First, although most of the data again lies on a z 3.5 power law (upper) curve, its normalization is 3.6 times higher than for the Western Pacific data. Second, three data points with the largest z-values lie on the lower curve having the original normalization shown for fig. 1.

As it happens the Easter Pacific is one area significantly affected by the Madden-Julian
Oscillation (MJO). The MJO is a large-scale quasi-periodic modulation of tropical winds that travels eastward from Asia to America. It has been previously shown that hurricane activity in the Eastern Pacific is around four times more likely during the phase of the 12 oscillation associated with ascending rather than descending wind flowssee Maloney (2000). Furthermore, the MJO oscillation leads to a significant modulation of the SST by around 0.5 0 C during the cycle, with the phase of the increased hurricane activity being associated with a reduction in SSTsee i.e., no factor of 3.6 multiplier. Thus, the data from all portions of the globe having significant numbers of hurricanes are consistent with this same functional form, and even the same normalization, provided one includes a factor of 3.6 enhancement in those regions subject to constant or quasi periodic influences that have been found previously to increase hurricane formation. Furthermore, the function H(z) does not predict any storms should occur in regions of the globe where none are found.

c. Temporal Trends
The results presented until now all deal with the spatial distribution of hurricanes, and the function H(z) was deduced with no direct reference to temporal variations --except through the correlation between local SST and time. The model can, however, be used to predict temporal variations based on the SST of each latitude-longitude bin at any given time. The procedure is to sum the fitted H(z) over all cells with a given z for that particular time. Therefore, the model (if correct) should agree with the observed temporal variations in numbers of storms over some extended time period. In what follows, we make these comparisons with the data three different ways. One comparison (figs. 3, and 4), shows the numbers of hurricanes predicted to occur each month of the year (averaged over the period 1960-2007) by the model with those actually observed in three particular basins: the North Asian Pacific, the Southern Hemisphere, and the combined Atlantic and Eastern Pacific basins. In most cases the agreement is fairly good, with the Southern Hemisphere probably being the worst. It should be noted that the horizontal error bars extend 50% beyond the half-month associated with the bin width.
The reason for increasing these error bars is that it has been found that the process of hurricane genesis from some initial disturbance has a tendency to cluster in time sometimes over a period of several weeks. Gray (1979). Since we want to associate SST with the very start of the process, not the observed first point on a hurricane track, we therefore add an extra + 1 week uncertainty to the horizontal error bars. observed percentage increases were identical, the data points should lie on a line through the origin with unit slope. Despite the large error bars, it is reassuring that the best fit straight line passing through the origin has a slope 1.25 + 0.15 that is only slightly farther from 1.0 than one standard deviation. In fact the chi square for a line with unit slope (shown dotted) is 15.5, giving an acceptable probability of 26%.

What constitutes a test of the model?
As previously noted, all three types of time trend comparisons discussed above did not factor into the original selection of H(z) which relied on fitting data that was averaged over time, temperature and latitude for any given z-value, and hence there was no a priori guarantee that good fits to the temporal trends could be assured, or even that the data would be describable using a single variable z. Nevertheless, the good temporal trend fits cannot really be said to validate the model, because for any given \phi, the time and temperature are strongly correlated variables. On the other hand, even if the model is viewed as merely a data-fitting exercise, the finding of a function of a single variable that fits the data reasonably well could have great utility, should the model continue to describe the distribution and numbers of hurricanes in the coming years, as global temperatures continue to rise. The true test of the model therefore will come if the numbers of hurricanes in various basins increase in the manner predicted by the model.
In particular, one should be able to see the upturn in the Asian Pacific region seen in the model but not yet in the data. Failure to see any such increase in perhaps three to five years of further rising SST's would be fatal for the model. It has been shown by others that the trend in Atlantic hurricanes correlates well with mean SST's going as far back as 1850 --see Mann (2007), but the new element here is that of a universal function of local SST (at the time of hurricane formation) that fits the data world-wide, with regional adjustments in normalization. For example, our model offers a simple explanation why with warmer SST temperatures in recent decades, we have witnessed a significant rise in the number of Atlantic hurricanes, but very little if any rise in Pacific hurricanes --an inconsistency which has led some researchers to a (probably false) conclusion that SST is not that important in determining how often hurricanes form. The model explains the apparent inconsistency, because in comparison to the Western Pacific, much of the region showing the highest percentage increases in storms in the Atlantic basin occurs in cells at higher latitudes, where SST's are cooler.
For a given increment in temperature T  , the function H(z)=Cz 3.5 produces a much larger percentage increase for lower temperatures (above T 0 than for higher ones that are further from the 25.5 0 C threshold.

Implications
This is not a paper about global warming, but the implications for a warmer world are stark. While it has previously been shown that destructive potential of hurricanes is likely to increase in a warmer worldsee: Emanuel (2005), earlier model predictions on the numbers of hurricanes have been highly inconsistent, with many models even suggesting their numbers might decrease, even as they become more destructivesee:. Oouchi (2006) and . If our results stand the test of time, they imply not only that the numbers of hurricanes will increase as SST rises, but they will do so in proportion to the 3.5 power of the temperature excess above 25.5 0 C for any given location. Thus, consider for example a location where the temperature in a given month is 27.5 0 C, or two degrees above the 25.5 0 C threshold. An average 2 0 C rise in the temperature from global warming during that month would increase the numbers of