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Tropical Cyclone Intensity and Sea Surface Temperature LARS R. SCHADE Berlin, Germany (Manuscript received 23 November 1998, in final form 22 October 1999) ABSTRACT It is hypothesized that the effect of the SST on the intensity of tropical cyclones can be separated into two distinct contributions: one from the large-scale SST field that is in equilibrium with the atmosphere, and another one from a local reduction of the SST under the eye of the cyclone due to the surface winds of the cyclone. A theory for the maximum possible reduction of the SST directly under the eye of a tropical cyclone is proposed and tested against model data. It implies a much higher sensitivity of tropical cyclone intensity to the SST under the eye of the storm than to the large-scale SST field. It therefore suggests a much higher importance of the SST feedback effect on the intensity of tropical cyclones than has previously been thought.

1. Introduction Heat fluxes from the ocean to the atmosphere are the energy source for tropical cyclones. These fluxes are promoted by the vigorous surface winds and the much reduced surface pressure in the cyclone. While the surface winds enhance the surface fluxes through increased turbulence close to the surface, the reduction of the surface pressure affects the surface fluxes by increasing the saturation mixing ratio and thus the thermodynamic disequilibrium at the sea surface. Even though this fundamental picture was established nearly half a century ago (e.g., Riehl 1950; Kleinschmidt 1951) the sensitivity of the intensity of tropical cyclones to the SST is still controversial. On the one hand, the dynamic theories for the maximum intensity of tropical cyclones imply a moderate sensitivity of the cyclone intensity to the SST. On the other hand, the static theories display a very large sensitivity of the cyclone intensity to the SST. The issue is further complicated by changes of the SST in response to the surface winds of the cyclone. This paper aims to bring together the dynamic and the static theories for the maximum intensity of tropical cyclones in an idealized quantitative theory for the role of the SST in tropical cyclones. The paper is structured as follows. We first summarize the dynamic and the static approach to the maximum intensity of tropical cyclones and highlight their differences. Then a theory for the maximum possible cooling

Corresponding author address: Dr. Lars R. Schade, Holsteinische Straße 23, D-12161 Berlin, Germany. E-mail: [email protected]

q 2000 American Meteorological Society

under the eye of a tropical cyclone is presented and tested against model data. Finally, we discuss some implications of this theory and conclude with an idealized picture of the effect of the SST on the intensity of tropical cyclones. 2. Theories for the maximum intensity of tropical cyclones Various theories for the intensity of tropical cyclones have been developed in the past 50 years. According to their basic approach they can be classified into dynamic theories and static theories. In the dynamic theories, individual parcels are traced along idealized trajectories through the entire circulation of the cyclone under the assumption of certain conservation principles and balance approximations. The mature cyclone intensity can then be calculated from the constraint of a balance between the generation of kinetic energy by the cyclone scale overturning and its destruction primarily due to friction. Thus dynamic theories basically describe tropical cyclones as heat engines and have been explicitly formulated as such. The early work of Kleinschmidt (1951) already contains many concepts fundamental to dynamic theories. The first closed formulation of a dynamic theory for the maximum intensity of tropical cyclones was presented by Emanuel (1988, hereafter Emanuel88) and shall be considered below. Static theories for the maximum intensity of tropical cyclones are based entirely on the thermodynamic structure of the eye and the eyewall and of the unperturbed environment. The thermodynamic structure in the eye can be determined from the environmental sounding and the SST under the eye with assumptions about the boundary layer structure and mixing. The cyclone in-

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TABLE 1. Theoretical and observed sensitivities of the MPI at the reference point SST a 5 298C, Ha 5 80%, and Tout 5 2748C.

SSTa Ha Tout SSTa with Tout 5 f (SSTa )

Emanuel88

Holland97

DK94

9.6 hPa 8C 21 4.9 hPa/% 22.6 hPa 8C 21 10–20 hPa 8C 21

33 hPa 8C 21 — — —

— — — 8–18 hPa 8C 21

tensity is then given as the hydrostatic depression of the surface pressure in the eye of the cyclone. The work of Miller (1958) is an early example of this class of theories. The recent formulation by Holland (1997, hereafter Holland97) shall be considered below.

range of relevant SSTs from 268 to 318C. In contrast, the cyclone intensity is very sensitive to H a . The third parameter, the outflow temperature, also significantly affects the cyclone intensity. b. Observations

a. The role of the SST in the dynamic theory of Emanuel88 In the theory of Emanuel88, tropical cyclones are embedded in a slantwise moist neutral atmosphere. Therefore the unperturbed atmospheric sounding is implicitly given by the specification of the SST and the relative humidity in the unperturbed environment. With the additional assumption that surface parcels become saturated in the eyewall the maximum potential intensity (MPI) depends on the SST, the relative humidity in the ambient boundary layer (H a ), and the entropy-weighted mean outflow temperature (Tout ) defined in Emanuel88. It is important to note that the SST is considered horizontally uniform and constant in time in the theory of Emanuel88. As the MPI is a nonlinear function of these three parameters the sensitivity of the MPI to these parameters is itself a function of the parameters. We chose a reference point of SSTa 5 298C, H a 5 80%, and Tout 5 2748C and calculated the sensitivities about this reference point (Table 1). The sensitivity to the SST of about 10 hPa 8C21 is rather small in light of the limited

FIG. 1. The relationship between Tout and SSTa . The solid line was calculated by DK94 to give a perfect match between their observations of the MPI and the theory of Emanuel88. The diamonds mark outflow temperatures calculated for observed hurricanes from climatological data.

While the three parameters SSTa , H a , and Tout are considered independent from one another in the theory of Emanuel88, the observational work of DeMaria and Kaplan (1994, hereafter DK94) suggests that the SST and the outflow temperature are actually strongly correlated. DK94 derived an empirical relationship between the MPI and the SST from a large set of observations of Atlantic hurricanes. They found an exponential increase of the MPI with increasing SST for SSTs from 158C up to about 288C. For even higher SSTs the MPI increases much more slowly than expected from the exponential behavior at lower SSTs. It is unclear whether this is primarily a sampling problem or a true effect. If it were a true effect the sensitivity of the MPI1 to the SST would decrease from 14 hPa 8C21 at 288C to 8 hPa 8C21 at 298C. In contrast, the exponential law yields a sensitivity of 18 hPa 8C21 at 298C. DK94 argue that the SST and Tout should not be considered independent since the climatological meridional gradient of the SST is paralleled by a gradient of the outflow temperature of the opposite sign. DK94 actually calculated which relationship between Tout and the SST would result in an exact match between their observations and the theoretical model of Emanuel88 (Fig. 1): For SSTs larger than 258C the outflow temperature has to decrease by about 3.58C for every degree Celsius increase of the SST. An ambient boundary layer relative humidity of 80% was assumed for this calculation. Can such a relationship between the SST and Tout in fact be seen in observational data of Atlantic hurricanes? To answer this question we calculated the entropyweighted mean outflow temperature for a set of observed hurricanes. As the theory of Emanuel88 applies to mature storms in a nearly steady state, we selected only hurricanes that met two criteria. First, they had to be over the open ocean and thus unaffected by land.

1 DK94 used maximum wind speed as the measure of hurricane intensity. As we use pressure depression in the storm center as the intensity measure, the results of DK94 were converted according to the empirical wind–pressure relationship for Atlantic hurricanes derived by Dvorak (1995).

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Second, their intensity as measured by the minimum central pressure had to be constant to within 4 hPa over a 30-h period. Seventy-eight post-1946 hurricanes met these two criteria. Besides the minimum central pressure of the storm, the ambient sounding needs to be known in order to calculate the outflow temperature on a tephigram. We used the climatologies by Robinson et al. (1979) and by Newell et al. (1972) to construct a climatological sounding for the date and location of each storm. These climatological soundings are considered to be an estimate of the ambient conditions the hurricanes were embedded in. They were used to calculate an entropy-weighted mean outflow temperature for each of the 78 hurricanes. The resulting outflow temperatures are plotted against the SST in Fig. 1 together with the relationship proposed by DK94. Our data nicely match the proposed functional relationship and thus support the suggestion of DK94 that Tout and the SST should not be treated as independent parameters. This holds particularly for SSTs below 288C, which are highly correlated with Tout with a regression coefficient of R 5 20.95 and a slope of the regression line of 23.8 6 0.28C 8C21 . Our data thus confirm the conclusions of DK94 that the observed correlation between the MPI and the SST can be explained with the theory of Emanuel88 as a combination of the dependence of the storm intensity on the SST and on the outflow temperature. The effective sensitivity of the storm intensity to the SST thus depends on the functional relationship between the SST and Tout . A lower limit on the sensitivity is found by keeping the outflow temperature constant. As suggested by the observational data, a reasonable upper limit on the sensitivity is found by lowering Tout by 48C for every degree Celsius increase of the SST. At the chosen reference point of SSTa 5 298C, H a 5 80%, and Tout 5 2748C, the effective sensitivity of the storm intensity to the SST is therefore between 10 and 20 hPa 8C21 . c. The role of the SST in the static theory of Holland97 In the theory of Holland97 the inner core of a tropical cyclone is idealized as a vertical moist neutral eyewall surrounding an eye with constant equivalent potential temperature. The SST and the relative humidity under the eyewall are treated as external parameters and determine the saturated equivalent potential temperature in the eyewall and the equivalent potential temperature in the eye. Holland97 assumes an idealized vertical profile of the relative humidity in the eye such that the temperature profile in the eye can be deduced from the equivalent potential temperature there. The MPI finally results from a hydrostatic integration of the virtual temperature anomaly in the eye. This integration is performed from the surface up to the level at which the temperature anomaly vanishes. The relative humidity under the eyewall and its profile in the eye can be con-

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sidered internal parameters of the theory of Holland97, such that the environmental sounding and the SST under the eye are the only parameters that must be specified externally. For the mean January sounding at Willis Island, Australia, Holland97 reports a sensitivity of the MPI to the SST of 33 hPa 8C21 . d. Closing the gap The static theory of Holland97 predicts a much higher sensitivity of the storm intensity to the SST than the dynamic theory of Emanuel88. In the remainder of this paper we shall try to resolve this apparent discrepancy between the two theories. The key to this resolution is the different physical meaning of variations of the SST in the theories of Emanuel88 and Holland97. The different sensitivities to the SST thus reflect different physical processes in the two theories. In the theory of Emanuel88 different SSTs correspond to different equilibrium states of the unperturbed tropical atmosphere. Therefore, it makes sense to compare the results of Emanuel88 with the climatological data of DK94. In the theory of Holland97, in contrast, the SST is treated as independent from the mean state of the atmosphere. Thus, the sensitivity to the SST reported in Holland97 should be interpreted as the sensitivity of tropical cyclones to deviations of the SST from the large-scale mean SST that the atmosphere has adjusted to. SST anomalies of small spatial extent, that is, of the order of an eye diameter, have generally little effect on the storm intensity because of the storm propagation that is typically many eye diameters per day. An important exception is the SST reduction induced by the storm. It is caused by the strong surface winds of the cyclone primarily via the vertical turbulent entrainment of cold water into the warm oceanic mixed layer (e.g., Price 1981; Sanford et al. 1987). As this SST anomaly moves along with the storm, it can have a pronounced effect on the storm intensity in spite of its spatial confinement to a narrow band of only a few eye diameters’ width. This so-called SST feedback effect has been investigated with numerical models by a number of research groups, for example, Chang and Anthes (1979), Bender et al. (1993), and most recently Schade and Emanuel (1999, hereafter SE99). The different authors arrived at qualitatively and quantitatively different results such that the importance of the SST feedback is still a very controversial topic. If the above interpretation of the physical meaning of the sensitivity to the SST in the theories of Emanuel88 and Holland97 is correct, then the following statement should be true: The sensitivity reported by Holland97 is identical to the sensitivity associated with the SST feedback effect in the framework of the theory of Emanuel88. As the original theory of Emanuel88 does not include the SST feedback effect, an extension to

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this theory is formulated in the next sections in order to test the above statement. 3. Isolating the effect of the negative SST feedback Let us consider an idealized setting in which a tropical cyclone can evolve to its maximum potential intensity over an ocean of uniform and constant SST. This is the intensity predicted by the theory of Emanuel88. It can be described by the surface pressure depression in the eye of the cyclone, pa 2 pMPI , where pa and pMPI are the unperturbed surface pressure in the ambient environment and the surface pressure in the eye of the cyclone, respectively. If we now allow the SST to respond to the wind forcing of the cyclone, we expect to see a reduction of the SST under and behind the cyclone, which in turn will lead to a weakening of the cyclone. This negative feedback effect was called SST feedback effect by SE99. In the idealized setting considered here the intensity reduction is entirely due to the SST reduction. Dimensional reasoning therefore yields that scaling parameters must exist such that the dimensionless weakening of the cyclone is a unique function of the dimensionless reduction of the SST. The appropriate scale for the weakening of the cyclone is the MPI, that is, pa 2 pMPI . For the SST reduction, the scaling is not as straightforward primarily because the SST reduction is a two-dimensional field. As suggested by both the dynamic and the static theories, the key element for the cyclone intensity is the moist entropy achieved in the boundary layer under the eyewall. This is also illustrated by the observation that tropical cyclones are typically only weakly affected by the vicinity of land as long as the eye of the storm remains over the open ocean. We therefore assume that the SST reduction under the eye of the cyclone can be considered the single relevant feature of the two-dimensional SST reduction. The appropriate scale for the reduction of the SST under the eye then is the theoretical upper limit on the SST reduction under the eye, that is, the SST reduction that corresponds to a complete disappearance of the cyclone. In this limit, even the saturation of surface parcels under the eyewall does not result in a radial gradient of moist entropy in the boundary layer. Put differently, the limit is characterized by the equality of the ambient boundary layer moist entropy and the saturation moist entropy at the reduced SST under the eye: cp ln(SST a ) 1

L y H a r*[SST a , pa ] SST a

ronment of the cyclone, H is the relative humidity, r*[T, p] the saturation mixing ratio, and DSSTmax the maximum possible reduction of the SST under the eye of the cyclone. The pressure term in the definition of the moist entropy cancels from both sides of the equation because the considered limit features no pressure depression. Equation (1) is an implicit equation for DSSTmax and can be solved iteratively. An approximate analytical solution is given by DSST max 5 (1 2 H a )

L y r*[SST a , pa ] cp

5

26

1

L r*[SST a , pa ] Ly 3 11 y 21 cp SST a R y SST a

As before, the index ‘‘a’’ refers to the ambient envi-

.

For relative humidities and temperatures of interest here, that is, 75% , H a and 278C , SSTa , 308C, this approximate solution (2) is accurate to within 64%. Equation (2) yields that the maximum possible SST reduction under the eye of a tropical cyclone is primarily a function of the relative humidity in the ambient environment of the cyclone. This can also be seen in Fig. 2, which shows both the iterative solution of Eq. (1) and the approximate solution given by (2). With the above definition of scales the expected nondimensional relationship between the intensity reduction and the SST reduction reads

[

(1)

21

(2)

]

pc 2 pMPI SST a 2 SST c 5F , pa 2 pMPI DSST max

5 cp ln(SST a 2 DSST max ) L r*[SST a 2 DSST max , pa ] 1 y . SST a 2 DSST max

FIG. 2. The maximum possible SST reduction under the eye of a tropical cyclone as a function of the ambient relative humidity. The solid black bar is the iterative solution of Eq. (1). The width of the bar results from varying the ambient SST between 268 and 318C. The two dotted lines correspond to the approximate solution (2).

(3)

where pc and SSTc are the actual surface pressure and SST in the eye of the cyclone, respectively. The lefthand side of Eq. (3) lies in the range [0; 1] and measures

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TABLE 2. Comparison of the modeled MPIs in the simulations of SE99 and the theoretical predictions of Emanuel88 (values in parenthesis). The MPIs are expressed as pressure depressions in hPa. SSTa 5 278C Ha Ha Ha Ha

5 5 5 5

78% 81% 84% 87%

69 59 48 37

(70) (60) (51) (42)

SSTa 5 288C 80 67 56 44

SSTa 5 298C

(79) (69) (59) (48)

92 78 65 51

(91) (79) (67) (55)

the amplitude of the SST feedback effect. Small values correspond to a weak feedback effect, whereas large values close to 1 correspond to a near disappearance of the cyclone due to an extremely strong feedback. Except for a factor of (21) this expression is actually identical to the definition of the SST feedback factor chosen by SE99. The function F needs to be determined from observations. A priori we only know that F must satisfy F[0] 5 0 and F[1] 5 1. Once the function F is known, the intensities of Emanuel88 can be transformed into intensities that account for the SST feedback effect. Also, the sensitivity of the cyclone intensity to a local SST reduction under its eye can be calculated: (4)

FIG. 3. The nondimensional intensity reduction as a function of the nondimensional SST reduction under the eye in the model data of SE99. The intensity reductions and the SST reductions were scaled by the modeled MPIs and by the theoretical values of DSSTmax , respectively. The solid line is the linear regression based on all cases with a nondimensional intensity reduction smaller than 0.5.

Ideally one would now take observations of pc and SSTc from real hurricanes to infer the function F. While the central pressure is routinely recorded in tropical cyclones, the SST in the eye of a tropical cyclone cannot easily be measured because of the extremely rough sea there. We therefore decided to use model data to deduce the function F, namely, the dataset described in SE99. This dataset was generated with a coupled atmosphere– ocean model. A three-dimensional ocean model was propagated at constant speed relative to an axisymmetric hurricane model, and the coupled system was integrated until a steady state was reached in the frame of reference of the atmospheric model. Next, 2318 integrations were performed to simulate various environmental conditions characterized by the translation speed and the MPI of the cyclone, and the temperature and the thickness of the oceanic mixed layer, among other parameters. Most important for the present purpose, the SST reduction under the eye of the hurricane can easily be extracted from the simulations. The modeled SST reduction under the eye is generally much smaller than the maximum SST reduction observed behind the storm. The SST reduction under the eye can be as small as a few tenths of a degree Celsius for fast moving storms over thick oceanic mixed layers. Conversely, very large SST reductions of up to 28C were simulated for slowly moving storms and thin oceanic mixed layers. Observational estimates of the SST reduction under the eye of a hurricane fall inside this range.

Let us now first compare the modeled MPIs with the predictions of Emanuel88. In the model Tout is a function of SSTa . Therefore the MPI is a function only of SSTa and H a . Three different values of SSTa and four different values of H a were used in the set of integrations of SE99 such that 12 different values of the MPI result. For each of these 12 cases the MPI was deduced from a control simulation in which the atmospheric model was integrated over an ocean of constant SST. In Table 2 the modeled MPIs are listed together with the theoretical predictions of Emanuel88. The modeled MPIs typically differ by no more than 3 hPa from the theoretically predicted MPIs. Only for the weakest storms can the difference become as large as 5 hPa or 14%. For reasons of consistency, the modeled MPIs will be used below to nondimensionalize the modeled cyclone intensities. In Fig. 3 the dimensionless intensity reduction is plotted against the dimensionless SST reduction under the eye for all 2318 integrations of SE99. The SST reductions were scaled by values of DSSTmax resulting from Eq. (1). Overall, the expectation of a unique functional relationship is supported by the model data. In particular for nondimensional intensity reductions smaller than 0.5 the relationship is nearly linear and takes the form F[x] 5 1.2x. As the theory requires that F[1] 5 1 this linear relationship cannot hold for large nondimensional intensity reductions close to 1. One could simply fit a nonlinear function to the data such that F[1] 5 1. Instead, we decided to infer values for DSSTmax from the model data and compare them to those predicted by our theory. For this purpose the entire dataset was split into

]( pa 2 pc ) p 2 pMPI ]F (x) 5 a . ]SST c DSST max ]x Deducing the function F

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TABLE 3. Comparison of the maximum possible SST reduction under the eye as deduced from the data of SE99 and as predicted by Eq. (1) (values in parenthesis). DSSTmax is given in 8C. SSTa 5 278C Ha Ha Ha Ha

5 5 5 5

78% 81% 84% 87%

2.67 2.22 1.79 1.43

(3.08) (2.64) (2.20) (1.77)

SSTa 5 288C 2.75 2.30 1.81 1.41

(3.14) (2.69) (2.25) (1.81)

SSTa 5 298C 2.80 2.37 1.93 1.46

(3.20) (2.74) (2.29) (1.84)

and Eq. (3) becomes pc 2 pMPI SST a 2 SST c 5 . pa 2 pMPI DSST max

FIG. 4. The nondimensional intensity reduction as a function of the dimensional SST reduction under the eye in the model data of SE99 for SSTa 5 288C and H a 5 81%. Intensity reductions were scaled by the modeled MPI. The solid line is the linear regression based on all cases with a nondimensional intensity reduction smaller than 0.5. The regression yields DSSTmax [288C, 81%] 5 2.308C.

12 subsets according to SSTa and H a . The simulations inside each subset then should be characterized by the same DSSTmax . For each of the subsets, we plotted the nondimensional intensity reduction against the dimensional SST reduction under the eye, as shown in Fig. 4, for SSTa 5 288C and H a 5 81%. The data clearly suggest a linear relationship. This feature can be exploited to deduce DSSTmax . For intensity reductions smaller than 0.5 a linear relationship is fit to the data. This linear fit is then extrapolated to a nondimensional intensity reduction of 1, which by definition corresponds to the maximum possible SST reduction. This procedure is illustrated in Fig. 4. In Table 3 the thus inferred values of DSSTmax are compared with the predictions of the theory, that is, the solution of Eq. (1). It turns out that the model-deduced values of DSSTmax are smaller than the theoretical solutions by 0.48C, independent of SSTa and H a . It is presently unclear to the author whether this discrepancy results from a deficiency of the model or of the theory. Besides this small offset, the theoretically predicted dependency of DSSTmax on SSTa and H a is nicely confirmed by the model data. If we now scale the SST reductions by the modeldeduced values of DSSTmax the nondimensional intensity reduction becomes directly proportional to the nondimensional SST reduction under the eye with a slope of 1. This means that the function F in Eq. (3) takes the simple form F[x] 5 x

(5)

(6)

Equation (6) is very strictly obeyed by the entire dataset of SE99. This can be seen in Fig. 5, which is a histogram of the relative differences between the nondimensional intensity reduction in the model data and the theoretical values according to Eq. (6). The mean difference is less than 3%. 4. Implications and conclusions We can now determine the sensitivity of the maximum intensity of a tropical cyclone to changes in the SST under its eye by use of Eqs. (4) and (5): ]( pa 2 pc ) p 2 pMPI 5 a . ]SST c DSST max

(7)

So far, we have either taken the MPI from the simulations of SE99 or calculated the MPI iteratively following the theory of Emanuel88. Both approaches agree reasonably well, as was shown in Table 2. For application in Eq. (7) it is more appropriate to use an ap-

FIG. 5. Histogram of the difference between the nondimensional intensity reduction in the data of SE99 and the theoretical prediction according to Eq. (6). Differences are expressed in percent of the modeled nondimensional intensity reduction.

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proximate explicit formulation for the MPI because the exact result is available only in an implicit form. A very close approximation to the exact solution of Emanuel88 reads pa 2 pMPI 5 e*(1 2 H a) a

le 1 H a kle 2 1 , 1 2 H a kle

(8)

with

k5

e*a , pa 2 e*a

l5

Ly , R y SST a

e5

SST a 2 Tout , Tout

(9) and

(10) (11)

where e*a is an abbreviation for e*[SSTa ], that is, the saturation vapor pressure at the ambient SST. For relative humidities, ambient SSTs, and outflow temperatures of interest here, that is, 75% , H a , 278C , SSTa , 308C, and 2808C , Tout , 2608C, the approximate MPI according to Eq. (8) differs from the exact solution of Emanuel88 by less than 5%. In this range of interest of the parameters, most of the variance of the MPI is explained by the factor e*a (1 2 H a ) on the right-hand side of Eq. (8). This means that the MPI is nearly proportional to (1 2 H a ) as already pointed out by Emanuel88. This proportionality translates into a very high sensitivity of the MPI to the ambient relative humidity and thus limits the practical utility of the theory of Emanuel88. The same is true of the theory for the maximum possible cooling under the eye of a tropical cyclone, which we derived in the previous section. Fortunately, the factor (1 2 H a ) in Eqs. (2) and (8) cancels out upon substitution into Eq. (7): ]( pa 2 pc ) ]SST c

1 1 kl(l 2 1)2(le 1 H kle 2 1) p 2 e* R cp

a

5

a

SST a

a

d

l (1 2 H a kle)

.

(12) The sensitivity of the storm intensity to local SST changes under its eye, ]( pa 2 pc )/]SSTc , is a function primarily of the ambient SST, to a lesser degree of the outflow temperature, and only weakly of the ambient relative humidity. For the range of interest of these parameters, that is, 75% , H a , 278C , SSTa , 308C, and 2808C , Tout , 2608C, the sensitivity ranges from 21 to 45 hPa 8C21 . Finally, we can now test the validity of the statement made at the end of section 2, namely, that the sensitivity reported by Holland97 is identical to the sensitivity associated with the SST feedback effect in the framework

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of the theory of Emanuel88. Holland97 used the Willis Island January sounding with a surface temperature of 28.48C. We estimate that the SST is 298C. Following DK94 the corresponding outflow temperature is 2748C. In January the relative humidity at Willis Island is between 65% and 85%. Using Eq. (12) these values translate into sensitivities between 31 and 34 hPa 8C21 , in close agreement with the value of 33 hPa 8C21 reported by Holland97. Our extension to the theory of Emanuel88 therefore quantitatively supports the hypothesis that the sensitivity of the MPI to the SST reported by Holland97 should be interpreted as the sensitivity of a tropical cyclone to local changes of the SST under its eye. An idealized picture of the role of the SST The effect of the SST on the intensity of tropical cyclones can now be summarized by rewriting Eq. (6):

5

pa 2 pc 5 ( pa 2 pMPI ) 1 2

6

SST a 2 SST c . DSST max

(13)

The maximum potential intensity and the maximum SST reduction under the eye can be calculated from Eqs. (8) and (2), respectively, if the ambient SST, the closely related outflow temperature, and the ambient relative humidity are known. The actual intensity of the storm deviates from the maximum potential intensity due to the SST feedback effect. As the SST under the eye is lowered with respect to the ambient SST, the storm intensity is reduced by the factor enclosed in the curly brackets on the right-hand side of Eq. (13). The different roles of the SST in the ambient environment (SSTa ) and under the eye of the storm (SSTc ) are illustrated in Fig. 6, which shows solutions of Eq. (13). The solid and the dashed lines correspond to ambient relative humidities of 75% and 85%, respectively. The outflow temperature was assumed to obey the functional dependence on SSTa suggested by DK94 and therefore does not enter the picture explicitly. In Fig. 6 the storm intensity is plotted against the SST under the eye of the storm. The heavy lines mark the maximum cyclone intensity for a given SSTc , which is realized when the ambient SST equals the SST under the eye, that is, when there is no negative SST feedback. It can easily be seen that the MPI depends strongly on the SST and on the ambient relative humidity. For a given H a the MPI increases with the SST at about 10–20 hPa 8C21 as seen by the slope of the heavy lines. The thin lines in Fig. 6 connect points of equal ambient SST. The slope of these lines therefore indicates the sensitivity of the cyclone intensity to a reduction of the SST under the eye. It ranges from 20 hPa 8C21 up to 45 hPa 8C21 and is thus significantly larger than the slope of the heavy lines.

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FIG. 6. Cyclone intensity as function of the SST under the eye. The solid and the dashed lines correspond to ambient relative humidities of 75% and 85%, respectively. The heavy lines mark the maximum possible intensity that is realized for SSTa 5 SSTc , i.e., without SST feedback. The thin lines connect points with the same ambient SST.

5. Summary It was proposed that the effect of the SST on the intensity of tropical cyclones can be idealized by two separate effects. On the one hand, the large-scale ambient SST field sets the stage for the tropical cyclone. The ambient atmosphere is in equilibrium with this ambient SST. Higher ambient SSTs provide the potential for stronger tropical cyclones. On the other hand, the intensity of a tropical cyclone is highly sensitive to a reduction of the SST under its eye. Such a reduction commonly occurs as a response of the ocean to the surface wind field of the cyclone. In the idealized picture, the two key aspects of the SST that influence the intensity of a tropical cyclone are the ambient SST and the SST reduction under the eye of the cyclone. An approximate equation for the intensity of a tropical cyclone as a function of its ambient environment and of the SST reduction under the eye of the storm was derived. It displays a much higher sensitivity of the storm intensity to the SST under the eye than to the ambient SST. Based on the relatively weak sensitivity of the MPI to the ambient SST, it was often argued (e.g., DK94) that the negative SST feedback effect can only account for a small fraction of the difference between the ob-

served and the maximum possible intensities of tropical cyclones. As shown here, the relevant sensitivity for this argument is the much stronger sensitivity of the cyclone intensity to changes of the SST under the eye of the cyclone. This result therefore suggests that the importance of the negative SST feedback effect has been underestimated previously and needs to be reassessed. REFERENCES Bender, M. A., I. Ginis, and Y. Kurihara, 1993: Numerical simulations of tropical cyclone–ocean interaction with a high resolution coupled model. J. Geophys. Res., 98, 23 245–23 263. Chang, S. W., and R. A. Anthes, 1979: The mutual response of the tropical cyclone and the ocean. J. Phys. Oceanogr., 9, 128–135. DeMaria, M., and J. Kaplan, 1994: Sea surface temperature and the maximum intensity of Atlantic tropical cyclones. J. Climate, 7, 1324–1334. Dvorak, V., 1995: Tropical cyclone intensity analysis using satellite data. NOAA Tech. Rep. NESDIS 11, 47 pp. Emanuel, K. A., 1988: The maximum intensity of tropical cyclones. J. Atmos. Sci., 45, 1143–1155. Holland, G. J., 1997: The maximum potential intensity of tropical cyclones. J. Atmos. Sci., 54, 2519–2541. Kleinschmidt, E., Jr., 1951: Grundlagen einer Theorie der tropischen Zyklonen. Arch. Meteor. Geophys. Bioklimatol., 4A, 53–72. Miller, B. I., 1958: On the maximum intensity of hurricanes. J. Meteor., 15, 184–195. Newell, R. E., J. W. Kidson, D. G. Vincent, and G. J. Boer, 1972:

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Mean Salinities of the Surface Layer. U.S. Naval Oceanographic Office, 104 pp. Sanford, T. B., P. G. Black, J. R. Haustein, J. W. Feeney, G. Z. Forristall, and J. F. Price, 1987: Ocean response to a hurricane. Part I: Observations. J. Phys. Oceanogr., 17, 2065–2083. Schade, L. R., and K. A. Emanuel, 1999: The ocean’s effect on the intensity of tropical cyclones: Results from a simple coupled atmosphere–ocean model. J. Atmos. Sci., 56, 642–651.