As the climate changes, the ability to predict changes in the frequency of tropical cyclogenesis is becoming of increasing interest. A unique approach is proposed that utilizes threshold values in potential intensity, wind shear, vorticity, and normalized saturation deficit. Prior statistical methods generally involve creating an index or equation based on averages of important meteorological parameters for a given region. The new method assumes that threshold values exist for each important parameter for which cyclogenesis is unlikely to develop. This technique is distinct from previous approaches that seek to determine how each of these parameters interdependently favors cyclogenesis.
To determine three of the individual threshold values (shear, potential intensity, and vorticity), an idealized climate is first established that represents the most advantageous but realistic (MABR) environment. An initial numerical simulation of tropical cyclone genesis in the MABR environment confirms that it is highly favorable for cyclogenesis. Subsequent numerical simulations vary each parameter individually until no tropical cyclone develops, thereby determining the three threshold values. The new method of point downscaling, whereby background meteorological features are represented by a single vertical profile, is used in the simulations to greatly simplify the approach. The remaining threshold parameter (normalized saturation deficit) is determined by analyzing the climatological record and choosing a value that is statistically observed to prevent cyclogenesis. Once each threshold value is determined, the fraction of time each is exceeded in the location of interest is computed from the reanalysis dataset. The product of each fraction for each of the relevant parameters then gives a statistical probability as to the likelihood of cyclogenesis. For predicting regional and monthly variations in frequency of genesis, this approach is shown to generally meet or exceed the predictive skills of earlier statistical attempts with some failure only during several off-season months. This method also provides a more intuitive rationale of the results.
a. Background and motivation
As the climate changes and the oceans become warmer, it is imperative that we understand the response hurricanes will have to this changing world. Questions such as will there be more hurricanes? and will they be stronger? become of interest not only to the scientific community but to the general population as well. As to the question regarding the number of hurricanes in a warmer climate, the research remains somewhat contradictory with some studies suggesting an increase in hurricane frequency (Ryan et al. 1992; Haarsma et al. 1993) while others suggest that the number will not change significantly (Walsh and Katzfey 2000; Tsutsui 2002; Walsh et al. 2004) or will even decrease (Sugi et al. 2002; Knutson et al. 2008, 2010). This study hopes to merge the benefits of two areas of research: pure statistical approaches and pure modeling approaches. By approaching the question as to the number of cyclogenesis events in a warming climate from a “threshold” perspective (to be explained further in section 2), the merging of numerical simulations with empirical data hopefully creates a more robust and physically meaningful methodology.
b. The genesis parameter approach
To predict the frequency of tropical cyclogenesis events there are basically two methods. The first involves the use of global climate models (GCMs) (Broccoli and Manabe 1990; Haarsma et al. 1993) whereby the models are used to count the number of cyclogenesis events that would occur in a warmer climate (Druyan et al. 1999; Camargo and Sobel 2004; Camargo et al. 2005). There are two distinct disadvantages to this approach: first, the simulations are costly and, second, such models do not accurately represent current tropical cyclone (TC) climatology. The second approach is to create an index (or equation) that contains previously acknowledged parameters that favor cyclogenesis and to use this index to make predictions. The first published index (Gray 1975) takes the form
where GP represents the “genesis parameter,” f the Coriolis parameter, ζ the low-level relative vorticity at 950 hPa, |δV/δp| the absolute wind shear between 950 and 200 hPa, EOCEAN the energy content of the oceans (later termed ocean heat content, see Shay et al. 2000), ∂θe/∂p the change in potential temperature versus height, and RH the average relative humidity between the 500-hPa and 700-hPa levels. Ryan et al. (1992) and Royer et al. (1998) applied this formula to a global warming environment and found a significant increase in cyclogenesis events. Royer et al. (1998) later revised this genesis parameter to match simulation outputs and also found a significant increase in the number of TCs, due primarily to “enhanced oceanic energy.”
where GP is once again the genesis potential, η is absolute vorticity at 850 hPa, H is relative humidity at the 600-hPa level, VPOT is the potential maximum wind speed (Emanuel 1987), and VSHEAR is the absolute difference in horizontal vector winds at the 200-hPa and 850-hPa levels. Although its predictive skills are not considered perfect and many improvements have been made (Emanuel 2010), this genesis parameter has been used in several studies to predict cyclogenesis events in various climates (Vecchi and Soden, 2007; Knutson et al. 2008).
Although each of the independent parameters of (1) may be conducive for cyclogenesis, it is widely known that a preexisting disturbance is required. Disturbances in the tropics are mostly created from several major sources including easterly jets, such as the African easterly jet or Caribbean low-level jet (Molinari et al. 1997; Thorncroft and Hodges 2001), the intertropical convergence zone (Frank and Clark 1979; Ferreira and Schubert 1997; Frank and Roundy 2006), frontal zones (Keyser and Shapiro 1986; Moore and Peltier 1987; Davis and Bosart 2001), and to a lesser extent, Kelvin waves (Frank and Roundy 2006). Although the yearly global disturbance count is in the hundreds, from those only approximately 80–90 TCs form (Gray 1979; Anthes 1982; Gray 1985; Frank 1987; McBride and Zehr 1981).
Although local conditions likely modulate the disturbance behavior (once the disturbance has formed), the disturbance frequency is not likely changed significantly by these same local conditions. This leads to an important question: If disturbance frequency is not highly correlated to these synoptic features and a cyclogenesis event is an on–off event, why would a doubling of, for example, VPOT (1) give an eightfold increase in cyclogenesis events? A possible explanation follows.
2. The GFI method
The genesis parameter index (1) equates a probabilistic outcome based on four environmental parameters. This equation proposes that as each parameter becomes more favorable more cyclogenesis events will occur (usually with some nonlinear behavior). The success of the GP method in predicting the frequency of TCs may, however, come from the fact that, as each parameter becomes more favorable, this creates a more frequent occurrence of exceeding a threshold value necessary for cyclogenesis. Central to this argument is the assumption that the seeds for cyclogenesis (preexisting disturbances) remain somewhat fixed in frequency for a given period of time and location. Therefore each cyclogenesis event is dependent only on each of the parameters (1) being met or exceeded. The problem becomes one of determining how often each threshold value is met within a certain region (assuming disturbances exist) and not on the actual value of each parameter. The threshold method is newly defined by the genesis frequency index (GFI).
The GFI uses threshold values believed to prevent cyclogenesis for each parameter, regardless of the remaining independent parameters. When each threshold value is found, a frequency is determined by empirically counting the number of times each threshold value is met divided by the total time in a historical dataset. The product of the separate frequencies then forms a total likelihood of cyclogenesis. The formula takes the form
where fshear represents the frequency of finding vertical wind shear below the derived threshold value, fVpot represents the frequency of finding maximum potential intensity above the derived threshold value, fvorticity represents the frequency of finding absolute vorticity above the derived threshold value, and fhumidity represents the frequency of finding humidity above the derived threshold value.
The range of GFI is therefore from 0 to 1, where 0 represents zero likelihood of cyclogenesis and 1 reflects that each of the four threshold values is met 100% of the time. An important consideration must be made for each extreme. If the GFI at some location equals 0, this implies that any number (or all) of the threshold values were never met. Finding a GFI 0 in a particular region of the ocean and also finding the cyclogenesis of a TC within that same region (and time) represents a failure of the GFI method. This “failure” interestingly provides a unique and useful opportunity. Most research in cyclogenesis focuses on the interdependency of meteorological parameters required to form a TC; for example, it is often believed that a higher sea surface temperature (SST) would allow development of a disturbance to overcome a slightly greater wind shear. The GFI method runs counter to this approach by determining the value of each parameter that would be needed to prevent cyclogenesis regardless of the remaining parameters. Upon finding a cyclogenesis event where GFI = 0, the question may now be asked, what unique conditions allowed this highly unlikely TC to form? At the other extreme, a GFI equal to 1 with no cyclogenesis events is not necessarily a failure of the method since other parameters may exist that must be added to (2) that have not been discovered or there may be a lack of precursor disturbances within that region that would act as a seed for development.
The greatest challenge of this method is the estimation of each threshold value. This problem is made simpler by first determining the “most advantageous but realistic” (MABR) value for each parameter where a threshold value is searched. When the MABR values are found, it then becomes a problem of individually varying each single parameter until it is found that a TC cannot develop. This defines each threshold value. We use the MABR here because, although there exist many theoretically “advantageous” conditions (e.g., zero wind shear), the environment is highly unlikely to have a vertical profile of winds without some change in wind speed. The term “realistic” must become an important consideration when searching for the most advantageous condition.
To describe the GFI method and the research involved, section 3 will describe the sources of data for determining threshold values and the methods used in calculating each threshold parameter. Section 4 will discuss the initial setup, while section 5 will describe in detail how each threshold value is found. Section 6 will show the results of the GFI method and discuss some of the findings. Section 7 will discuss and compare the GP method to the GFI method, and section 8 will provide conclusions.
3. Data and methods
a. Data used in analysis
Vertical profiles of temperature, relative humidity, and wind data along with sea level pressure (SLP) needed for the calculation of (2) for the Atlantic, eastern and western Pacific, northern and southern Indian Ocean, and South Pacific are taken from the 6-hourly National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996). SST values used in VPOT calculations of (2) are taken from the monthly NCEP–NCAR reanalysis. The SST used to determine the MABR value in the calculation of VPOT is taken from the Tropical Rainfall Measuring Mission Microwave Imager/Advanced Microwave Scanning Radiometer for Earth Observing System (TMI/AMSR-E) dataset (Wentz 1998; Wentz et al. 2000, 2001). Data for tropical cyclogenesis locations (for all basins, 1969–2008) is taken from the National Climatic Data Centers International Best Track Archive for Climate Stewardship (IBTrACS version 2, release 1; Knapp et al. 2010).
b. Model, domain, and initialization
The model used to simulate the dependence of cyclogenesis for various values of VPOT, shear, and environmental vorticity is version 3.1.1 of the Weather Research and Forecast model (WRF) (Skamarock et al. 2008). The WRF model domain consists of an outer grid with 240 × 240 points and 18-km horizontal resolution. Embedded within this domain are two nested, vortex-following grids with 120 × 120 points at 6-km resolution and 210 × 210 points at 2-km resolution. The model uses 40 vertical levels between the surface and approximately 20-km altitude, equally spaced in the WRF hydrostatic pressure coordinates. For parameterizations, the WRF single moment 6-class microphysics scheme (Hong and Lim 2006) and the Yonsei University (YSU) planetary boundary layer scheme (Noh et al. 2003; Hong et al. 2006) are used. Surface fluxes in the YSU PBL scheme are modified following Dudhia et al. (2008) to be more consistent with recent results regarding surface fluxes of heat and moisture in high wind speeds. Cumulus parameterization is not used, even on the 18-km grid, to prevent spurious cyclogenesis in the outer domain.
The modeling framework is a variation on that used in previous studies by Nolan and Rappin (2008) and Rappin et al. (2010). The goal using this framework is to create a large domain with a completely homogenous environment represented by a single profile of wind, temperature, and humidity. In the real atmosphere, large-scale wind shear must be balanced by temperature gradients according to the thermal wind equation (Holton 2004). To avoid this temperature gradient, Nolan and Rappin (2008) added an artificial force term to the momentum equations that exactly balances the Coriolis force, fU(p), generated by the initial wind profile. As discussed in Rappin et al. (2010, see appendix) this will maintain the desired wind profile and its associated wind shear across the domain, while allowing the flow and temperature fields to deviate locally around the developing cyclone. Doubly periodic boundary conditions on the outer domain allow the large-scale flow to remain as homogenous as possible. Since this modeling framework allows simulations of TC genesis in a large-scale environment described by a single sounding, it has come to be called “point-downscaling” [for an evaluation of this method see Nolan (2011)]. While idealized wind profiles U(p) were used in the previous studies, here we use wind profiles derived from monthly mean observations in the main development region (MDR, see Fig. 1). These will be described further below.
For each simulation, the model domain is initialized with the constant values of SST, Coriolis parameter, and the desired profiles of temperature, humidity, and zonal wind. As representative of a weak, midlevel vortex that precedes cyclogenesis, an axisymmetric, balanced vortex is introduced into the domain. In the radial direction, the tangential wind profile is that of a modified Rankine vortex with a radius of maximum winds of 80 km and a decay parameter a = 0.4. In the vertical, the winds are maximized at z = 3720-m height with a Gaussian decay above and below this level such that the peak surface wind is exactly half the peak midlevel wind. The pressure and temperature anomalies that hold this vortex in hydrostatic and gradient wind balance are computed using the iterative technique described in Nolan et al. (2001). Our procedure for choosing the strength (peak tangential wind) of the initial vortex is described in section 5a.
Certainly, the homogenous environment provided by this method neglects many other factors that can influence cyclogenesis, such as horizontal wind shear (and vorticity), baroclinicity, and large-scale divergence. However, the advantage is that it limits the factors that may be determining whether genesis occurs and is, thus, ideal for determining threshold values of the environmental parameters of interest.
4. Initial setup
a. Most advantageous but realistic values
The first of the four threshold parameters shown in (2) is wind shear. Vertical wind shear is well known to be an important parameter for cyclogenesis (Gray 1968; Frank and Ritchie 2001) and is generally considered a negative influence except during possible interaction with upper-level troughs (Molinari and Vollaro 1989; Molinari 1998; Bosart et al. 2000; Zehr 2003). To determine the MABR value for vertical wind shear, a definition of vertical wind shear must be made. An almost universally used definition of vertical wind shear is the absolute vector difference in horizontal winds at the 200-hPa and 850-hPa levels. Figure 2a shows a principal component analysis (PCA) of zonal winds in the MDR from 1969 to 2008 for the month of September, showing the first four empirical orthogonal functions (EOFs). The first EOF, providing ~41% of the variance (Fig. 2b), supports the standard practice of using the 200-hPa and 850-hPa levels as a good proxy for vertical wind shear (although one could argue the highest variance actually exists at the 200-hPa and boundary layer levels). For the purpose of this research, the customary 200-hPa and 850-hPa levels will be used since the difference is insignificant. The meridional winds were not included in the PCA as they are much weaker than the zonal winds.
Figure 2c shows the mean winds (dashed line) for the MDR during the same time period. The vertical wind shear is approximately 7 m s−1 and, although this is a realistic value, it is far from being advantageous. The dark thick line represents the mean zonal winds minus the product of the square root of the first eigenvalue (representing variance) and the first eigenvector (or EOF). This creates a new profile with a 1.4 m s−1 wind shear, a much more favorable value. This most advantageous profile is also realistic as it is simply a subtraction of the first EOF from the mean.
The second of the four threshold parameters shown in (2) is maximum potential intensity (VPOT). Along with SST, a vertical profile of water vapor and temperature is needed to compute VPOT. To find the MABR value of VPOT, each of these parameters must be analyzed. Figure 3 shows a histogram of SST taken from TMI/AMSR-E microwave satellites within the MDR for September 1999–2008. The narrow time frame was chosen due to the fact that TMI/AMSR-E represents the highest-resolution dataset available for SST; unfortunately the microwave satellites were not available before 1999. For subsequent GFI analysis, however, the high resolution of the microwave satellites will not be essential and the reanalysis data (coarser resolution but longer time frame) is used for SST. Figure 3 clearly shows that the MABR SST is close to 30°C. The second parameter that contributes to VPOT is the vertical profile of temperature. Figure 4 shows the mean vertical profile of temperature (solid line) within the MDR for September 1969–2008 along with two standard deviations above and below the mean (dashed lines). As expected in the tropics, the temperature deviations from the mean profile are minimal (Holton 2004). Although lower atmospheric temperatures are more beneficial to TC development because of the instability created by the larger temperature imbalance between the boundary layer and upper atmosphere, the mean profile was chosen as the MABR profile in determining VPOT as the variations were considered negligible. The third and final component of VPOT is water vapor. Figure 5a shows the variance of the vertical profile of water vapor by plotting the first four EOFs within the MDR for the same time frame as for temperature. Figure 5b shows the contribution from each of the nine EOFs; 42% of the variance is provided by the first EOF, similar in importance to the first EOF from zonal winds (Fig. 2b). The variability from the first EOF reflects the fact that any change in boundary layer water vapor is highly magnified at upper levels. To calculate the MABR profile of water vapor, the first EOF of relative humidity is added to the mean profile (Fig. 5c, dark line). Figure 5d is the resulting water vapor profile after the first EOF is added. The remaining EOFs were not used because the variation from water vapor in the boundary layer varied counter to the water vapor aloft and therefore created an offsetting effect. Using a SST of 30°C, the temperature profile of Fig. 4, and the humidity profile of Fig. 5d, the VPOT calculation using the methods of Emanuel (1988, 1999) is determined to be 82.8 m s−1. This is a very favorable environment for cyclogenesis—one that could realistically be expected to occur in nature.
The third parameter shown in (2) is absolute vorticity. Unfortunately, point downscaling cannot create horizontal variations in winds. Without the contribution of relative vorticity, the only contribution that can affect absolute vorticity is planetary vorticity through a change in latitude. Since the MDR was chosen as the analysis region for the previous profiles, a latitude dividing the meridional extent of the MDR at 15° was chosen as the MABR latitude, with a Coriolis value of 3.8 × 10−5 s−1. Although a higher initial latitude for the embedded vortex increases the planetary vorticity, previous studies have already suggested that the likelihood of cyclogenesis is not enhanced as the environmental vorticity increases above some minimum (or threshold) value (Nolan et al. 2007a; Tippett et al. 2011).
With the model environment set to the MABR values (latitude 15°, SST 30°C, vertical profile of zonal winds shown in Fig. 2c, temperature profile shown in Fig. 4, and water vapor profile shown in Fig. 5d) the next objective in the GFI calculation is to vary each parameter to a value where cyclogenesis is prevented.
b. Cyclogenesis definition
To define cyclogenesis the Atlantic Hurricane Database (IBTrACS will be used. The National Hurricane Center (NHC) is tasked with identifying when a disturbance becomes organized to a system where there is closed circulation in all quadrants in an earth-relative reference frame. The initial central pressure entry in the IBTrACS dataset therefore reflects the instance where cyclogenesis has occurred. Analysis of surface pressure fields from reanalysis of the Atlantic MDR reveals that the average pressure of a disturbance (generally an African easterly wave) is approximately 1012.0 hPa. Analysis of 40 years (1969–2008) of the IBTrACS dataset showed that the mean value for the initial pressure when NHC classified the TC was 1010.0 hPa, a 2-hPa drop in pressure from the disturbance stage. Cyclogenesis will therefore be defined as the time in the numerical simulation when the central pressure below the vortex falls below 1010.0 hPa and either continues to fall or remains below this pressure. The domain size of the simulation allowed for approximately six days of simulation time before the moving nests encountered the outer domain boundary and therefore all simulations were limited in time to six days. This is a reasonable time frame, as a disturbance that has favorable conditions to form rarely takes longer than six days to develop.
5. Threshold values
a. Initial conditions
With the exception of the embedded vortex, the point-downscaling method creates a single SLP across the entire domain. Therefore, the strength of the pressure anomaly created by the initial vortex is an important consideration. If the vortex is too weak, only the most advantageous meteorological conditions would create a TC and, if the vortex is too strong, the TC has already exceeded the definition for cyclogenesis before the numerical simulation has begun. Since the vortex is intended to be a representation of a tropical wave seen in or near the tropics, the strength of the vortex must closely represent those typical features. To simplify the approach, an axisymmetric vortex will be used. As discussed in section 3b and in keeping with the initial studies of Raymond et al. (1998) and subsequent work by Pytharoulis and Thorncroft (1999), Nolan (2007), Nolan et al. (2007a,b), and Nolan and Rappin (2008), the symmetric vortex will have peak winds at z = 3720 m with a radius of maximum winds (RMW) of 80 km. The pressure and temperature anomalies of the embedded vortex are established by gradient wind balance and therefore the central pressure of the vortex is dependent on the radial pressure gradient with the higher outer pressure set at the mean SLP within the MDR. By adjusting the midlevel peak winds, and therefore the vortex central pressure, the vortex can be set to represent a typical tropical wave or disturbance.
The strength of the initial vortex is determined in the following manner: using the MDR as a reference for tropical wave characteristics, every TC classified by the NHC and documented in the IBTrACS dataset (Knapp et al. 2010) was analyzed using NCEP–NCAR reanalysis the day before TC genesis to find the lowest pressure seen within the wave. As stated in section 4b, from 40 years (1969–2008) of data, the mean SLP of a tropical wave that subsequently became a TC was 1012.0 hPa. The mean SLP of the entire MDR within the same period was 1013.5 hPa. It was found that 12 m s−1 tangential winds at the z = 3720 m, decaying to 6 m s−1 at the surface, created a 1012.0-hPa central pressure relative to the 1013.5-hPa outer pressure. All simulations for this research, therefore, use these vortex characteristics.
The first threshold parameter to be determined is wind shear. The first EOF shown in Fig. 2a represents ~41% of the variance in zonal winds within the MDR and can therefore be used as a reliable profile in adjusting wind shear. The variance suggests that any increase in boundary layer wind speed is accompanied by a larger increase in wind speeds aloft but in the opposite direction. By amplifying the first EOF and adding it to the mean profile, wind shear in the numerical simulation can be adjusted to find the value for which cyclogenesis cannot occur. The value for wind shear in the subsequent numerical simulations will be the absolute difference between the zonal winds at the 200-hPa and 850-hPa levels. Figure 6a depicts the various wind shear profiles used in the various WRF simulations and Fig. 6b shows the results of each simulation. The solid black line represents the ideal conditions listed in section 4a. A vortex embedded in 8 m s−1 of wind shear clearly develops as the central pressure drops below 1010.0 hPa and continues to fall to a value of 1002.0 hPa. Interestingly, between 1.25 and 2 days, the higher wind shear of 8 m s−1 developed slightly faster than the MABR profile of 1.4 m s−1. A wind shear of 10 m s−1 can also be classified as favorable for TC development as the central pressure drops below 1010.0 hPa and remains constant at 1007.0 hPa. The final wind shear of 12 m s−1 can no longer be considered as favorable for TC development. Although the central pressure fell below 1010.0 hPa, it did not remain below this value and eventually increased back toward the 1010.0-hPa value. We therefore chose 11 m s−1 for the threshold shear value.
The second threshold value to be determined is VPOT. Since temperature and humidity are highly interrelated, we change VPOT in these simulations by leaving the temperature and humidity profiles fixed while simply varying the SST. Figure 7 shows the results of simulations using various SST values and their corresponding values of VPOT with the solid black line again representing the MABR conditions. In these simulations, there is a clear threshold value. A weak midlevel vortex over SST = 28° and 27°C (VPOT = 64.1 and 53.3 m s−1) clearly developed into a TC where the same vortex over SST = 26°C (VPOT = 41.3 m s−1) did not. A reasonable threshold value for VPOT is therefore chosen to be 47 m s−1.
The third threshold value to be determined is vorticity. Although the point-downscaling method can simulate planetary vorticity by setting the constant Coriolis parameter to represent the latitude-dependent f plane, the inability of the point-downscaling technique to provide horizontal variations prevents the calculation of relative vorticity and therefore excludes absolute vorticity (η = f + ς). However, as a validation of the threshold claim for vorticity, simulations using only planetary vorticity were performed by simply varying f (equivalent to varying latitude). Figure 8 shows results for various equivalent latitudes with the solid black line again representing the ideal conditions. As with the threshold value for SST, there is a clear distinction between the simulations that developed into a TC and the single case that did not. A vortex at 7° latitude did not develop. A reasonable threshold value for a latitude of 8° could therefore be chosen.
Although it appears that planetary vorticity behaves as a threshold value to cyclogenesis, it was found that by slightly increasing the strength of the initial vortex, storms could form as low as 2° latitude (results not shown). Although the strength of the initial vortex had a large impact on the threshold value for planetary vorticity, threshold values for shear and VPOT were not significantly affected by such changes, validating to some extent the methods described in sections 5b and 5c. To compensate for the effect of initial vorticity strength on cyclogenesis in a changing vorticity environment, the idea of absolute vorticity must therefore be reconsidered. Tippett et al. (2011) recently used a Poisson regression to show that absolute vorticity values up to 4.0 × 10−5 s−1 had a large effect on cyclogenesis while larger values did not further increase the likelihood. Neglecting relative vorticity, the value of 4.0 × 10−5 s−1 corresponds to a latitude of 15°, in agreement with our choice of this latitude for the MABR, as any vorticity beyond that does not enhance the likelihood of cyclogenesis.
To further investigate the importance of absolute vorticity on cyclogenesis, Fig. 9a shows a histogram of absolute vorticity measured at 850 hPa for all cyclogenesis events from 1969 to 2008 in all basins. Figure 9a somewhat reinforces the claim by Tippett et al. (2011) that values of absolute vorticity above 4.0 × 10−5 s−1 do not significantly improve the likelihood for cyclogenesis, as the peak frequency shown by Fig. 9a is only slightly higher at 5.0 × 10−5 s−1. As an aside, the decrease in events beyond the peak is not due to unknown physics relating to excess vorticity but relates to the fact that large absolute vorticity values correspond to high latitudes where low ocean temperatures offset any possible vorticity advantages.
To now consider a threshold value for absolute vorticity, Fig. 9a shows that the lowest absolute vorticity for storm development is approximately 0.25 × 10−5 s−1. Although this value could be chosen as the new threshold value for vorticity, the frequency at which absolute vorticity is above this value would remain near 100% throughout most of the domain as values less than this are rarely observed (as shown by a random sampling of absolute vorticity in regions where cyclogenesis is likely to occur, shown in Fig. 9b,1). The previous analysis using only planetary vorticity would be just as useful, as any latitude above 8° would also provide a frequency of 100% throughout most of the domain. However, one possible advantage in using absolute vorticity is to resolve the previously discussed problem of cyclogenesis caused by a stronger than normal initial vortex in regions of weak planetary vorticity. Considering a threshold value for cyclogenesis that remained near 100% at higher latitudes but would be selective in regions near the equator would indicate cyclogenesis events when absolute vorticity was significant even if the initial vortex was weak. Figure 8 showed a threshold value of planetary vorticity for 8° latitude. Converted to planetary vorticity (2Ω sinθ), this equals 2.0 × 10−5 s−1 (shown as the vertical line in Fig. 9). Although this threshold value was found using only planetary vorticity, if an assumption is made that a threshold of absolute vorticity (which would include relative vorticity) can be represented by the threshold planetary vorticity, this value can be used in further calculations. Therefore, when calculating fvorticity (the fraction exceeding 2.0 × 10−5 s−1), the procedure will determine the fraction of time that the absolute vorticity exceeds 2.0 × 10−5 s−1, not just when latitude exceeds 8° latitude.
Although numerical simulations are an intuitive and relativity easy way to calculate threshold values, there is nothing in the equation for GFI that mandates each variable be derived through this method. Empirical data may also play an important role in the calculations, as described shortly.
The final threshold parameter to be determined is humidity. Although relative humidity (usually measured at 600 hPa) is a commonly used parameter when it relates to developing a genesis index, recent research suggests that normalized saturation deficit (NSD) is a more robust measurement (Rappin et al. 2010; Emanuel 2010), defined as
where is the saturation mixing ratio at midlevels (600 hPa), qm is the mixing ratio at midlevels, qb is the mixing ratio in the boundary layer (1000 hPa), and is the saturation mixing ratio just above the ocean surface, computed using the SST for temperature. Equation 3 indicates how close midlevels are to saturation (as with relative humidity) but contains an additional normalizing parameter that relates to the moisture disequilibrium between the atmospheric boundary layer and the ocean surface. This disequilibrium is important for cyclogenesis since a small difference means less enthalpy flux from the ocean, a key component in providing moisture to a nascent disturbance by means of wind-enhanced surface moisture fluxes. Note that decreasing values of NSD indicate a more favorable environment for genesis, as this corresponds to either decreasing midlevel humidity (decreasing the numerator) or increasing air–sea disequilibrium (increasing the denominator).
Unfortunately, numerical simulations proved to be problematic in providing a threshold value for NSD. Numerous simulations were performed using exactly the methods described above, where the midlevel humidity was decreased by subtracting the first EOF, with increasing amplitudes, from the mean profile. Doing so increased the time it took the disturbance to achieve genesis but, regardless of how low the midlevel humidity was reduced (even to zero), the disturbance would eventually develop into a storm. As it turns out, with virtually no wind shear, convection in the core of a disturbance will eventually saturate the core, leading to genesis, as previously noted by Nolan (2007). To address this problem, a possible modification of our approach would be to establish an arbitrary time threshold for genesis in the simulations. However, a time threshold could not be found that provided consistent results for all parameters.
For these reasons, to determine a threshold value for NSD we turned to an empirical approach. Figure 10a is a histogram of NSD for all cyclogenesis events from 1969 to 2008 in all basins. Interestingly, although a lower NSD should be conducive for development, the plot shows no cyclogenesis events for values of NSD below 0.6. As of yet, no explanation exists as to why a lower limit exists. One possibility is that tropical environments during times for which hurricanes form never have values below 0.6. However, Fig. 10b 2 shows a histogram from a random sampling of NSD for tropical oceans during hurricane season and therefore excludes that possibility. As of this writing, no reason is apparent but an explanation is not vital to the present research.
Although no explanation exists for the lower limit, the upper limit of NSD is consistent with prior results. Subjectively, Fig. 10a indicates that the threshold value for NSD may be as high as 1.7. However, this value did not generate good results with the GFI. A rational alternative is to use the mean value plus two standard deviations; this value is equal to 1.05 and is shown as the vertical line in Fig. 10.
a. Contribution of each parameter
With four threshold values now established (shear at 11 m s−1, VPOT at 47 m s−1, absolute vorticity at 2.0 × 10−5 s−1, and the NSD of 1.05), the final calculations involve finding the frequency of events for each.
For the fshear component of (2) we use the NCEP–NCAR reanalysis data at each location within the 40-yr time frame to find the fraction (or frequency) of occurrences that the wind shear from 850 to 200 hPa is below the threshold value 11 m s−1. Although the original threshold calculations involved only zonal winds, the shear calculations from the NCEP–NCAR reanalysis data were not limited to zonal winds but included both zonal and meridional winds. Figure 11a reflects GFISHEAR for the month of July 1969–2008 in the Atlantic basin using only fshear in (2). The black dots with white circles represent cyclogenesis events as recorded by IBTrACS for the same time frame. High values of GFISHEAR exist within the Atlantic basin with the exception of the southwest–northeast tongue beginning near 20°N, 60°W. The low GFISHEAR reflects the fact that shear values are frequently above 11 m s−1 within this region, likely due to the frequently encroaching frontal boundaries that approach the mid-Atlantic in the summertime. In addition, low GFISHEAR values exist between South America and the islands of Jamaica and Hispañola. As will be shown later, the area of low genesis activity in the central Caribbean, which we refer to as the “dead zone,” is consistently present in the observational data.3 A recent study (Shieh and Colucci 2010) found that a Caribbean low-level jet exists within this region and suggested that the higher zonal motion of tropical waves in this area prohibits TC genesis. However, Fig. 11a and the previously mentioned EOF analysis suggests that the low-level jet is likely accompanied by a magnified reversal in the winds aloft creating higher than normal wind shear. Other regions where GFISHEAR is low exist poleward of 40°N and in a separate region near 5°N, 10°W. Although Fig. 11a shows that low GFISHEAR correlates well to regions of low cyclogenesis activity, the corollary is not true. For example, the GFI in Fig. 11a has very high values south of 10° latitude and in the region northeast of 30°N, 50°W; however no cyclogenesis events occur.
For the fVPOT component of (2), the NCEP–NCAR reanalysis was again used to find the fraction of occurrences when VPOT is above 47 m s−1. Although the original VPOT derivation involved only the independent SST component (since temperature and humidity profiles were held constant), Nolan et al. (2007b) argue that regardless of how the independent parameters lead to a value of 47 m s−1, it likely has the same effect on cyclogenesis. Figure 11b shows GFIVPOT for the same time frame as before, but if only fvpot is used in (2). Low GFIVPOT is shown to exist over the oceans where SSTs are low, such as in the northern Atlantic or off the northwest coast of Africa. As with GFISHEAR, low values of GFIVPOT correlate well with regions of low cyclogenesis activity. Again, the corollary is not true as many regions of the Atlantic show a large GFIVPOT but no cyclogenesis events.
For the fvorticity component of (2), two separate calculations are made for comparison. First, the planetary vorticity alone is used as a threshold using 8° latitude; therefore fvorticity = 1.0 at or above 8° latitude and is equal to zero below. Figure 11c shows GFIVORTICITY using this criterion. As a comparison, the NCEP–NCAR reanalysis data is used at each location for the 40-yr time frame to find the fraction of occurrences when absolute vorticity measured at 850 hPa is above the threshold value 2.0 × 10−5 s−1. Figure 11d shows GFIVORTICITY using absolute vorticity as the threshold. As stated in section 5d, small differences exist between the use of planetary vorticity and absolute vorticity except near low latitudes where a larger fraction of above-average absolute vorticity should support the development of weak vortices.
For the final component of (2), fhumidity, the NCEP–NCAR reanalysis data is used at each location for the 40-yr time frame to find the fraction of occurrences when the NSD is below the empirically derived threshold value of 1.05 (see section 5e). Figure 11e shows GFIHUMIDITY when only fhumidity is used. The mapping of GFIHUMIDITY to cyclogenesis events shows a high likelihood in most regions throughout the Atlantic basin except for the northern and northeastern regions where the high NSD are primarily caused by the lower sea surface temperatures. Owing to the large dependence on SST, Fig. 11e is similar to Fig. 11a except for the dead zone region north of South America where high values of GFIHUMIDITY suggest that many storms could occur where none exist.
Alone, fshear, fVpot, fvorticity, or fhumidity cannot accurately reflect cyclogenesis events. However, the product of the four as shown in Fig. 11f reveals the strength of the GFI method. The GFI equation maps high GFI values to regions where cyclogenesis events are common such as within the MDR, the Gulf of Mexico, and the midlatitude regions off the eastern coast of the United States, while low GFI values map to regions where cyclogenesis events are scarce such as the dead zone and the midlatitude tongue north of the MDR. Each frequency contribution can also identify the reasons for the cyclogenesis distribution. For example, the likely reason for the dead zone is excessive shear, as no other contribution to GFI exists to explain the low GFI values. In addition, the low cyclogenesis events south of 8° latitude are primarily due to the low absolute vorticity as shown by the fact that the remaining three frequencies support cyclogenesis (with the exception of a small region of low GFI from shear just west of Africa). The remaining region of low cyclogenesis events occurs within the tongue extending northeast from 20°N, 60°W and is likely a combined factor of low VPOT to the east and high shear farther to the west.
b. Results for all oceans and months
Figure 11 is a persuasive plot for the GFI method, but how does it perform in the other basins and for other time frames? Figures 12–23 show results for the Atlantic, the eastern Pacific, the western Pacific, the northern and southern Indian, and South Pacific ocean basins. The mapping of GFI to cyclogenesis events is remarkable considering the simplicity of the approach. There are some noticeable discrepancies, however. Assuming that the GFI method has incorporated all important parameters, a GFI value of 0 indicates that cyclogenesis events should not occur. However, there are quite a few noticeable events for which this fails. For example, several storms developed in the Atlantic for the months of April, May, November, and December; in the northern Indian Ocean for the months of June and August; and in the South Pacific for the months of September, October, and November. GFI values were close to 0 in all of these regions.
For significant GFI values with no cyclogenesis events, failures occurred in the Atlantic for the month of May; in the eastern Pacific in April, June, July, and October; in the northern Indian Ocean from January through April; in the southern Indian Ocean for June (only two events in a large GFI section), and in the South Pacific from July through August.
Unfortunately, GFI values of 0 where cyclogenesis events occur reflect a failure in the method. However, additional research on the individual events may contribute important information as to the reasons why a TC would develop given that every meteorological parameter signaled an impediment. For example, many tropical cyclones formed in the northern Indian Ocean in June (Fig. 18f) although GFI values were close to 0. Further investigation revealed that GFI was almost 0 solely due to the shear threshold of (less than) 11 m s−1 not being satisfied. Although the current research advocates a threshold paradigm, the authors are well aware that interdependency between the separate parameters likely exists. The advantage of the threshold method is that one can focus future research on the select failures, for example, what essential characteristic exists in the northern Indian Ocean that would allow storms to develop with such high shear?
For the opposite failure (a relatively large GFI with no cyclogenesis events) several comments need to be made. If the threshold method is valid, two likely scenarios exist to explain the discrepancy. First, as was discussed in the introduction, storms need a preexisting disturbance for development. There is nothing in the GFI equation (with the possible exception of absolute vorticity) that recognizes and accounts for the possible disturbance count in the region of interest. It is highly likely that a region can be conducive to TC development but lacks the essential disturbance. Second, although Eq. (2) likely incorporates the most important parameters for cyclogenesis, it may be lacking in other variables for which important values exist. For example, ocean heat content (Shay et al. 2000) and/or diurnal heating (Chen and Houze 1997), to name a few, may prove to be essential for developing storms and may need to be included in future GFI calculations.
To further explore the issue that the GFI method neglects the seeding mechanism (disturbances), an interesting detail emerges. When calculating GFI, the solution details the fraction of times the meteorological conditions are conducive for cyclogenesis. A logical result can therefore be summarized in the following equation:
where the total number of tropical cyclones (TCCOUNT) is dependent on the fraction of time the atmosphere allows development multiplied by the total number of disturbances within that region. Obviously no cyclones would develop if the atmosphere was unfavorable (regardless of the number of disturbances) and furthermore no cyclones could develop if there were no disturbances (regardless of the atmosphere’s favorability). Rearranging Eq. (4) , the number of disturbances within a basin could therefore be estimated. Figure 24 shows the implied number of disturbances during hurricane season (1 June–30 November) for 1996–2008 within the Atlantic basin using the rearranged Eq. (4). Care must be exercised that, when GFI approaches zero, the Ndisturbances would approach infinity since GFI is now in the denominator. This only means that, as GFI approaches zero, it is impossible to estimate the number of disturbances. Neglecting this issue, Fig. 24 details several regions where disturbance counts are high and are well documented (as discussed in the introduction), namely, off the west coast of Africa, likely caused by the African easterly jet (Molinari et al. 1997); within the MDR, likely caused by the baroclinic breakdown of the ITCZ (Frank and Clark 1979; Ferreira and Schubert 1997; Frank and Roundy 2006); and off the eastern coast of the United States and in the Gulf of Mexico, likely caused by frontal zones (Keyser and Shapiro 1986; Moore and Peltier 1987). An inherent advantage of the GFI method is it is dimensionally consistent and also provides an intuitive framework for additional information. However, to determine if the GFI method is more advantageous than its predecessors, a detailed comparison must be made.
To derive a “score” for a given index an objective method must be created that judges each index fairly. Unfortunately a simple correlation will not work since seasons can produce zero storms and it is difficult to judge a correlation between many nonevents and an index that consistently scores near zero.4 It was found that the simplest and most straightforward scoring method involves dividing each basin into 10° × 10° boxes in regions where cyclogenesis events are counted. While the GFI naturally falls between 0 and 1, genesis-parameter-type indices must also be normalized, here by the maximum value found in all basins and for all times. The score is then defined as
where n is the total number of 10° × 10° boxes in the region of interest, indexi is the normalized index, and TCcounti is the normalized tropical cyclone count (both within each box subscripted with i). The tropical storm count for all basins was taken from the IBTrACS database of the NOAA National Climatic Data Center (see section 3a). Since the last component of Eq. (5) involves a root-mean-square value and since all parameters are normalized, the maximum theoretical score is therefore 1.0 while the lowest is 0.0. Presently we will evaluate GFI, the Emanuel and Nolan (2004) GP [Eq. (1)], as well as an updated index presented by Emanuel (2010):
where η, VPOT, and VSHEAR are identical to those in (1) but include different exponents and biases. The humidity of (1) is replaced by the normalized saturation deficit described in section 5e. Although the original equation in Emanuel (2010) uses a normalized moist entropy, as temperatures do not change much in regions of cyclogenesis (Fig. 4), this parameter is nearly equal to NSD.
In calculating the individual parameters of Eqs. (1) and (6), the mean values using NCEP–NCAR reanalysis data for the years 1969–2008 for absolute vorticity, midlevel NSD, VPOT, and vertical wind shear were used to calculate the indices for each 10° × 10° box. The results are shown in Fig. 25 with the thick black line showing the performance of the GFI [Eq. (2)], the dashed line the performance of GP [Eq. (1)], and the dashed–dotted line the performance of the revised GP [Eq. (6)]. Added to the figure is a reference score calculated by setting the index in each box to zero (thin black line). Results from Fig. 25a show that the GFI score meets or exceeds all other methods for the Atlantic basin. This is not surprising considering the threshold values were derived using climatological data within the Atlantic MDR. If comparisons are made in the eastern Pacific (Fig. 25b), the GFI method continues to perform equally well, validating the use of the Atlantic MDR to compute MABR values for other basins. In the western Pacific (Fig. 25c) and the northern Indian Ocean (Fig. 25d), the GFI method outperforms the other indices during hurricane season but underperforms the other indices during some of the off-season months when the GFI method estimates a wider area of cyclogenesis events where few are documented (Figs. 16b,c; 18a–c). Results are similar in the Southern Hemisphere where seasons are reversed and the GFI method outperforms others during hurricane season but underperforms in several of the off-season months (June–September), again forecasting cyclogenesis events where few occur. Even considering the underperformance of the GFI method in some of the off-season months, the overall skill is encouraging.
However, to further understand the effectiveness of the GP method and explore why the GFI method is likely a more robust technique, Figs. 26a,b show probability distribution functions (PDFs) of values of vertical wind shear for two regions depicted in Fig. 1 for the month of July (white boxes). Figure 26a shows the PDF corresponding to the region centered at 15.0°N, 75.0°W, while Fig. 26b shows the PDF centered at latitude 12.5°N, 45.0°W. In both regions, values of VPOT and environmental vorticity are similar. If we were to assume that shear, VPOT, and absolute vorticity are the most important independent parameters, then the number of cyclogenesis events would be strongly dependent on vertical wind shear alone. The PDF in Fig. 26a shows that the data distribution is highly Gaussian with a mean shear of 17.0 m s−1. The GP method (1) with an inverse square function for shear will portend low probability for cyclogenesis events. The GFI method also predicts low probability because the 11 m s−1 threshold value is met less than 10% of the time (Fig. 11a).
If, however, the data is highly skewed as illustrated in Fig. 26b, the advantage of the GFI method becomes more apparent. While the peak of the distribution is near 7 m s−1, the skewness of the data gives a higher mean of 7.9 m s−1, thereby reducing the GP forecast in a region where there are, in fact, numerous cyclogenesis events. The GFI method, using the threshold technique, shows that the 11 m s−1 threshold is met approximately 85% of the time (Fig. 11a) and thereby gives a much higher prediction of events. In summary, the GP method generally works well if the distribution of data is approximately Gaussian but may fail if the distribution of data becomes skewed or noncontinuous. The GFI method will likely work well in all instances.
Forecasting the future frequency of cyclogenesis remains a difficult task. Although it is well known which meteorological parameters are important in the development of a TC, their exact interdependencies remain elusive. Prior methods using an empirically fit equation that included important meteorological parameters have shown some skill. However, the approach neglected an important distinction, namely, if TCs are born from atmospheric disturbances and the frequency and distribution of these disturbances are not highly correlated to the background meteorological conditions, why would the number of TCs be altered by increasing SST, for example? Cyclogenesis is a single event, either an occurrence or not. The GFI method considers this distinction by determining threshold values of each important meteorological parameter. To determine each threshold value, an idealized but realistic profile of meteorological conditions is first identified. These profiles are then used as background states for idealized simulations of TC genesis. Subsequent simulations adjust the individual parameters until cyclogenesis is prevented, thus providing four separate threshold values: vertical wind shear 11 m s−1, VPOT 47 m s−1, absolute vorticity 2.0 × 10−5 s−1, and a normalized saturation deficit of 1.05. At each grid point in the NCEP–NCAR reanalysis the fraction of occurrences for each of the threshold values (fraction of shear below 11 m s−1, fraction of VPOT above 47 m s−1, fraction of absolute vorticity above 2.0 × 10−5 s−1, and the normalized saturation deficit below 1.05) is calculated. The product of these four fractions therefore provides a quantitative index (GFI) of the likelihood of cyclogenesis for a particular time frame. With few exceptions, the GFI mapping performs remarkably well in representing regions of high cyclogenesis events with high values of GFI and regions of few cyclogenesis events with low values of GFI.
b. Future improvements
An important consideration in the use of the GFI method is the use of the VPOT threshold value. Three independent parameters are included in VPOT: SST, the vertical profiles of temperature, and water vapor. In this research, the latter two were considered fixed while only SST was used to adjust VPOT. Although there is considerable evidence to suggest that VPOT can be used as a proxy for cyclogenesis (Emanuel and Nolan 2004; Nolan et al. 2007b), it is not certain whether a specific threshold value can be considered an accurate threshold in other regions. For example, if SST is held constant while the remaining two independent parameters of water vapor and temperature are adjusted to the same VPOT threshold value, will the potential reflect the same transition from a case when cyclogenesis can happen to a case when it will not? Future research needs to resolve this issue.
The impetus for this research began with asking the question, will there be more hurricanes in a warmer climate? The Intergovernmental Panel on Climate Change (IPCC) (Alley et al. 2007) is tasked with assessing how climate conditions are likely to change as anthropogenic warming from carbon dioxide emissions increases. The most recent (fourth) assessment includes thousands of hours of numerical simulations designed to represent how the meteorological conditions will be affected in a warmer climate. With the strength of the GFI method and the results from these and future IPCC simulations, a more reliable prediction of future cyclogenesis frequency can hopefully be made.
The authors thank Dr. Eric Rappin for his continuous support and Dr. Kerry Emanuel for insightful discussions that led to the ideas of this research. The authors would also like to gratefully acknowledge support from the National Science Foundation under Grant ATM-0851021. Computing resources were provided by the Center for Computational Science at the University of Miami, and NCEP data was provided by the NOAA/OAR/ESRL PSD (Boulder, Colorado) from their Web site (http://www.esrl.noaa.gov/psd/). TMI data are produced by Remote Sensing Systems and sponsored by the NASA Earth Science MEaSUREs DISCOVER Project (data available online at www.remss.com). AMSR-E data are also produced by Remote Sensing Systems and sponsored by the NASA Earth Science MEaSUREs DISCOVER Project as well as the AMSR-E Science Team (data available online at http://www.remss.com).
The National Hurricane Center occasionally refers to this region as the “Hurricane Graveyard” (i.e., NHC, “advisory of TD Joyce,” 5 p.m. EDT October 1, 2000).
Correlation coefficients use the variance of both variables in the denominator. With either variance equal to zero, the solution becomes undefined.