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Annu. Rev. Fluid Mech. 1991.23 : 179 96 Copyright 1991 by Annual Reviews Inc. All rights reserved THE THEORY OF HURRICANES
Kerry A. Emanuel Center for Meteorology and Physical Oceanography, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 KEYWORDS: tropical cyclones, convection,moist convection,finiteamplitude instability INTRODUCTION The hurricane remains one of the outstanding enigmas of fluid dynamics. This is so, in part, because the phenomenon comparatively difficult to is observe and because no laboratory analogue has been discovered. To this it must be added that hurricanes have received surprisingly little attention from the theoretically inclined fluid dynamicist, perhaps owing to an understandable tendency to avoid problems that involve complex thermodynamics and lack laboratory analogues. Yet hurricanes involve a rich spectrum of fluiddynamical processes, including rotating, stratified flow dynamics, boundarylayers, convection, and airsea interaction; as such, they provide a wealth of interesting and consequential research problems. This article reviews recent developments in the theory of hurricanes and delineates the important remaining scientific challenges. THE MATURE ENGINE HURRICANE: A NATURAL CARNOT About80 rotating circulations known generically as tropical cyclones form over the tropical oceans each year. Of these, roughly 60% reach an intensity (maximum winds in excess of 32 ms 1) that qualifies them as hurricanes, a term applied only in the Atlantic and eastern Pacific. (Similar storms in other parts of the world go by different names.) Anexcellent review of the climatology and observed characteristics of these storms is provided by Anthes (1982). Here we use the term hurricane in place of the generic term tropical cyclone. 179 0066M189/91/01150179502.00 Annual Reviews www.annualreviews.org/aronline 180 EMANUEL The mature hurricane may be idealized as an axisymmetric vortex in hydrostatic and rotational balance. The cyclonic azimuthal flow reaches its maximum intensity near the surface and decreases slowly upward, becominganticyclonic near the top of the storm, roughly 15 kmabove the surface. This flow configuration corresponds to a warmcore structure with maximum temperature perturbations on isobaric surfaces well in excess of 10C, highly concentrated at high levels near the center of the vortex. The radius at which the azimuthal winds peak ranges from 10 to 100 km near the surface and generally increases with height. Inside the radius of maximum winds the core is nearly in solidbody rotation, while outside the core the winds fall off gradually with radius, obeying approximately an r 1/2 law. Nolowlevel circulation can be detected outside a finite radius ranging from 100 to 1000 km. While the geometric size of hurricanes ranges over an order of magnitude, their intensity, as measured by maximum wind speeds or central pressure deficit, bears no perceptible relation to their size (Merrill 1984). While axisymmetry a good approxiis mation for the cyclonic flow, the upper anticyclone is usually highly asymmetric, with the bulk of the flow confined to one or two anticyclonically curving jets. The transverse circulation of a mature hurricane is thermally direct (except in the eye) and consists of radial inflow within a frictional boundary layer roughly 12 km deep, ascent mostly within a narrow, outwardsloping eyewall 5 to 100 kmfrom the center, and radial outflow in a thin layer at the storm's top. The eyewall looks like a coliseum of convective clouds surrounding an eye that is often nearly free of clouds. The transverse circulation in the eye is mechanically maintained and thermally indirect, with warmair slowly subsiding near the center. Clouds and precipitation outside the eyewall are usually organized in one or more cyclonically curved spiral bands of order 10 kmin Width, extending to a height of from 3 to 15 km. The axisymmetric structure of the mature hurricane is summarizedin Figure 1. Kleinschmidt(1951) first recognized that the energy source of hurricanes resides in the thermodynamicdisequilibrium between the tropical atmosphere and oceans. This is reflected not in an actual temperature difference between air and sea, which in the tropics is usually less than IC, but rather in the undersaturation of nearsurface air. The evaporation of water transfers heat from the ocean, whose effective heat capacity is enormous in comparison with the overlying atmosphere. To bring the troposphere into thermodynamicequilibrium with the ocean would require the transfer of roughly 108 J m 2 of energy from the ocean. The rate of transfer of heat from the ocean to the atmosphere is a function of the surface wind speed. If the ocean were a flat surface, the Annual Reviews www.annualreviews.org/aronline
THEORY OF HURRICANES 181 (I
~/ /',;7'", / ro r Figure 1 The hurricane Carnot cycle. Air begins spiraling in toward the s~orm center at point a, acquiring entropy from the ocean surface at fixed temperature T~. It then ascends adiab~tiCally from point c, flowing out near the storm top to some large radius, denoted symbolically by point o. The excess emropyis lost by export or by electromagnetic to space betwee~o and o' at a muchlower temperature To. The cycle is closed by integrating along an absolute vortex line between o' and a. The curves co and o'a also represent surfaces of constant absolute angular momentum about the sto~'s axis. transfer wouldincrease linearly with windspeed, but the increasing roughness of the sea surface leads to a somewhatgreater dependence on wind. The actual rate of heat transfer is a subject of muchcontroversy and research. The dependence of the transfer rate on wind is the principal feedback mechanism that allows hurricanes to develop. In its essence, the hurricane may be thought of as a windinduced surface heat exchange instability, in whichincreasing surface windslead to increased heat transfer from the sea, which leads to intensification of the storm winds, and so on. The energy cycle of the mature hurricane has been idealized by the author (Emanuel 1986) as a Carnot engine that converts heat energy extracted from the ocean to mechanical energy. In the steady state, this mechanicalenergy generation balances frictional dissipation, most of which occurs at the airsea interface. The idealized Carnot cycle is illustrated in Figure 1. Carnot's theoremmaybe easily derived from Bernoulli's equation and the first law of thermodynamics. former states that along The streamlines or absolute vortex lines in a steady system, d 7ivi +d(gz)+~zdp+F'dl = O, (1) Annual Reviews www.annualreviews.org/aronline 182 EMANUEL where V is the vector velocity, 9 the acceleration of gravity, z the height above the surface, ~ the specific volume, p the pressure, F the frictional force per unit mass, and di an incremental distance along a streamline or absolute vortex line. An energy equation may be derived by substituting from the first law of thermodynamics, which in a moist system may be written Tds = CpdT+d(Lvq)~ dp, (2) where s is the specific total entropy content of air (including the water vapor), o i s t he heat c apacity of a ir a t c onstant pressure, Lv i s t he l atent heat of vaporization, and q is the mass of water vapor per unit mass of air. The above neglects the heat capacity of water substance and the effect of water on the density of an airwater vapor mixture. [An exact treatment is provided in Emanuel(1988).] Eliminating ~ dp between (1) and (2) gives d IVl2+gz+CpT+Lvq Tds+F.di=O. (3) This can be integrated around a closed circuit (Figure 1), the first three branches of which are streamlines. The fourth branch is an absolute vortex line. Then ~Tds  ~F. dl, (4) whichillustrates that in the steady state, heating balances friction. Most of the heat input to a hurricane is from the sea surface. As air flows radially inward along the surface, its temperature is observed to be held nearly constant by a combination of turbulent fluxes and radiative transfer from the ocean. Thus, in the first branch, one has ~ T ds = TsAs, (5) where As( ScSa) is the difference betweenthe entropies of air near the storm center and in the ambient environment. The first law of thermodynamicscan be used to derive the entropy s (neglecting the heat capacity of water substance and other small terms): s = CplnTRlnp~ Lvq T ' (6) where R is the gas constant for air. Then Annual Reviews www.annualreviews.org/aronline
THEORY OF HURRICANES 183 (7) TsAs = RTs Inpa + Lv(qc qa), wherethe subscripts c and a refer to quantities evaluated along the surface at the storm center and at the starting point, respectively. The saturation of air near the storm center limits the entropy increase to
Lv , p~ TsAsmax RT~ln + ~,(q qa), = Pc . (8) whereq* is the saturation mixingratio at the stormcenter and is a function of pc and Ts. From an approximate integration of the ClausiusClapeyron equation, we have, to an excellent approximation, 3.802mbar [ 17.67Ts ] qc*  Pc exp L24~.5 ~ TsJ' (9) in whichTs is expressed in degrees Celsius. In the secondleg of the Carnot cycle, air ascends within deep convective clouds in the eyewall of the storm and then flows out to large radius. It is important to note that when water vapor is properly included in the description of the thermodynamic state of the system, this leg is very nearly reversible and adiabatic, so that ds0. Someresearchers define a dry entropy [without the last term in (6)] and are forced to deal with very large sources of dry entropy in the ascent region, wherethere is large conversion of latent to sensible heat. The problem with such an approach is that the diabatic source is purely a function of the flow itself and cannot be properly regarded as external. Attempts to regard the condensation heat source as external lead to the oftrepeated statement that hurricanes are driven by condensation of water vapor, a view rather analogous to that of an engineer who proclaims that elevators are driven upward by the downward acceleration of counterweights. Such a view, though energetically correct, is conceptually awkward;it is far more natural to consider the elevator and its counterweight as a single system driven by a motor. Here we adopt a similar strategy by dealing with the most conserved thermodynamic variable available, the total specific entropy. The altitude that the outflow asymptotically approaches is determined by the requirement that it be neutrally buoyant with respect to the environment. That is, the temperature in the outflow blends smoothly into the ambient temperature profile without shocks. The ability of the ambient atmosphere to control the interior structure of the vortex through this requirement is consistent with the fact that hurricanes are suberitieal vor Annual Reviews www.annualreviews.org/aronline 184 EMANUEL tices (i.e. internal gravityinertia wavesmaypropagate inward against the outflow). In the third leg of the Carnot cycle, air descends slowly in the lower stratosphere, retaining a nearly constant temperature To while losing heat by electromagnetic radiation to space. In this leg, then, o' Tds =  ToAs, (10) f with As given by (7). Real hurricanes are open systems that continually exchange mass with their environments. Nonetheless, the Carnot cycle can be closed by integrating the Bernoulli equation along a fourth branch that is an absolute vortex line of the system, which is also a surface of constant absolute angular molnentumabout the storm center. Since we are no longer following air parcels, the relationship implied by the first law of thermodynamics is not strictly valid. It has been shown by Emanuel (1988), however,that there is little thermodynamic contribution from this last leg, owingto the convective neutrality of the ambient atmosphere. Thus adding (7) and (10) gives an expression for the frictional work from eT, As = ~F dl, where e is the thermodynamic efficiency of the Carnot cycle, given by (11) TsTo Ts
For typical atmospheric conditions in the tropics, e ~ 1/3. Most of the frictional energy loss in the cycle occurs in the surface boundarylayer and at large radius in the outflow, where the air's original angular momentum ultimately be restored. This latter loss is idealized must as occurring at infinite radius. It maybe estimated from conservation of absolute angular momentum about the storm center: (M) 1 2 M = rV+ 2fi r , (12) where V is the azimuthal velocity, and .f is twice the local vertical component of the Earth's angular velocity. From(12), it follows that V2 = _ 1 22 (13) ~fr  ~ fM+ ~f r . The loss of kinetic energy in the third branch of the cycle is then Annual Reviews www.annualreviews.org/aronline
THEORY OF HURRICANES 185 F'di ~ !i~m~AV =  (MoM.) = ~f ra, (14) assumingthat V is zero at the beginning of the cycle. The frictional loss in the boundarylayer maybe related to the radial pressure drop by integrating the Bernoulli equation (1) inward along the first branch: F'dl =  pa, =  RTdlnp ~dp = RTsln P~ (15) where we have madeuse of the ideal gas law, e = RT/p. Thus, the frictional loss is directly proportional to the logarithm of the pressure deficit. Substituting (15) and (14) into (11) then
1 2 2 eT~As = RT~lnpa + ~f ra, Po (16) with an upper bound, for As provided by (8). The last term in (16) reflects the energy put into the upper anticyclone; this always detracts from the intensity of the surface cyclone, as reflected in the surface pressure deficit. The stipulation that in (P,/Pc) must be positive leads to a restriction on the magnitudeof ra. Given r,, p,, qa, Ts, and To, a lower bound on Pc is obtained by using (8) [with (9)] in (16). Unlessr~ is unusuallylarge, it has little effect on estimate. Figure 2 shows this lower bound(neglecting ra) as a function Ts and To, using a standard mean surface pressure p, and assuming an ambient nearsurface relative humidity of 75%.One curiosity of this calculation is that there is no solution for sufficiently large T~ or small To. In this regime (the hypercane regime) the Carnot cycle becomes unstable owing to a very large heat input from isothermal expansion: the more intense the storm, the lower the central pressure, giving greater isothermal heat input, whichintensifies the storm, and so on. An estimate of the minimumcentral pressure po from (16) using Septemberclimatological conditions is shownin Figure 3, together with locations and central pressures of the most intense hurricanes on record. Clearly, a few hurricanes reach the predicted upper bound on intensity, but the vast majority (not shownin Figure 3) do not. Twoenigmas emerge from Figure 3: Given that the energy potential for hurricanes is large over muchof the tropical oceans, why are hurricanes so rare? And even when a hurricane occurs, why do so few reach the theoretical upper bound on intensity? Annual Reviews www.annualreviews.org/aronline 186 EMANUEL RH = 75%
40 38 3~" 34. 32" 30' 2.8" ~'6' 24 975 ~ Hy percones ~0 0 15 =30 60 75 90 105 120 155 folOC)
Figure 2 The minimum sustainable central pressure (in millibars) as a function of seasurface temperature (T~) and meanoutflow temperature (~o), assuming an ambient surface pressure of 1015 mbar and an ambient nearsurface relative humidity (RH) of 75%. HURRICANE GENESIS AS AN EXAMPLE FINITEAMPLITUDE INSTABILITY OF Forecasters have knownfor some time that hurricanes never arise spontaneously, even if the environmental conditions are considered favorable. Rather, they always emerge from preexisting circulations of presumably independent dynamical origin. Yet some of the early theories attempted to treat tropical cyclogenesis using linear theory. The Failure of Linear Stability Theory Early investigations focused exclusively on the dynamics of moist convection (see the review by Yanai 1964). These studies suffered the result that disturbances of the smallest horizontal scale should develop most rapidly and so cannot explain the scale of hurricanes. Charney&Eliassen (1964) recognized this defect and proposed a theory called Conditional Annual Reviews www.annualreviews.org/aronline
THEORY OF HURRICANES tOOE 120 140 160 t80 t60 140 t20 00 80 187
60W 40 20 EQ 2O 4O 2,,,, tOOE t20 t40 ,~,, 160 ~, 180 , 160 , , , , , t00 , 80 , , ,~ 140 t20 60W Fi#ure 3 Minimum sustainable central pressure of tropical cyclones (in millibars) under September climatological conditions. The central pressures of someof the most intense tropical cycloneson record are shown italicized numbers crosses. by and Instability of the Second Kind (CISK) that sought to explain the scale hurricanes by requiring the convection to occur in proportion to the upwardmotion induced by the frictional boundarylayer of a vortex. Here, as in other similar works, dry thermodynamics was used and the heating was externalized as much as possiblein this case by making it proportional to the frictionally induced vertical velocity and by distributing it in the vertical by an arbitrary function. While not explicitly related to the concept of convective instability, CISKimplicitly requires a reservoir of convective energy to operate. By choosing the right ad hoc parameters, a largescale amplifying disturbance can be obtained. While always controversial, CISKhas remained popular amonga subset of meteorologists, in part because of the mathematical simplicity of the representation of cumulus convection suggested by Charney & Eliassen (1964), but perhaps also because of the enormousappeal of the notion the latter authors that "cumulus clouds and the large scale circulation cooperate, rather than compete." CISKentirely disregards the importance of enhanced heat fluxes from the sea, emphasized by earlier researchers such as Kleinschmidt (1951) and Riehl (1954), and can be shown to energetically on stored potential energy in the tropical atmosphere. This Annual Reviews www.annualreviews.org/aronline 188 EMANUEL stored energy wouldbe reflected in a thermodynamic stratification of the atmosphere that allows boundarylayer air to becomepositively buoyant whendisplaced upwarda sufficient distance; this is called a state of conditional instability. Recent work by Betts (1982) and Xu &Emanuel (1989), however, demonstrates that mean thermodynamic profiles over tropical oceans are almost precisely constant, as is true of most other forms of highRayleighnumber convection, so that there is very little variation of free atmospheric temperature in the absence of boundarylayer entropy variations. Stored energy occurs occasionally over midlatitude continents as a result of particular arrangements of topographical features. Thus CISKpredicts, in direct contradiction to observation, that nascent hurricanes should be common over midlatitude continents during the warmer months but absent over the oceans. The weight of evidence suggests that CISKbe rejected as a useful hypothesis and that attempts be undertaken to find a finiteamplitude instability based on the thermodynamicinteraction of the atmosphere and ocean. Results of Fully Nonlinear Integrations There have been a number of successful numerical simulations of hurricanes, dating back to the work of Ooyama(1969). Such modeling has advanced to the point where cumulus clouds themselves can be explicitly (albeit crudely) resolved, at least in axisymmetric models. Virtually all numerical simulations contain representations of the seaair heat flux necessary to maintain hurricanes, but nearly all begin with decidedly unstable thermal stratification. This creates an ambiguity in interpreting the initial spinup of the cyclone, since unstable convection artificially constrained to two dimensions will lcad to upscale energy transfer, as shownoriginally by Fjortoft (1953). This no doubt occurs in the axisymmetric numerical simulation by Yamasaki(1977), whose integration begins with highly unstable stratification but whosemodelallows no surface heat fluxes. The result is a thunderstormscale cyclone that reaches maturity in 40 hr, considerably less time than is generally observed in nature. At the other extremc, Rotunno & Emanuel (1987) integrate an axisymmetric, nonhydrostatic model that resolves cumulus convection, starting from an initial state that is neutral to cumulusconvection. The evolution of their simulated cyclone is portrayed in Figure 4, which shows three experiments that differ only in the amplitnde and geometryof the initial vortex. It is evident that amplification occurs only whenthe initial vortex is sufficiently intense and concentrated. In this simulation, the amplification may be considered to result from a finiteamplitude instability, in accord with nature; the resting atmosphere is metastable. The mature model cyclone Annual Reviews www.annualreviews.org/aronline
THEORY OF HURRICANES 5C 189 20 I'~ ,o i .,~b<......, ...,,...,.  " ~"..J \~ ",,,',,
I ' 2'0 4'0 6'o d0' 4o' t4o' t~0 ' ' ' do' 40'
TIME (hours) Figure 4 Maximum azimuthal velocity as a function of time (in hours) for three numerical simulations of hurricanes from Rotunno & Emanuel (1987). The solid curve denotes the control simulation. The dashed curve shows the results of an experiment identical to the control but starting with an amplitude of 2 m st. The dashdot curve denotes a third experimentin which the initial vortex has twice the radial dimensions of the control. has a structure and amplitude that are quite compatible with observed hurricanes, and the amplitude conformsnearly exactly with the theoretical prediction based on the Carnot cycle, as discussed previously. Assuming that the physical reasons for the finiteamplitude nature of the instability in the numerical model and in nature are similar, we mayinquire about the latter by using the fields producedby the model. As is often the case, however, the output of complex numerical models is as inscrutable as nature herself. A Minimal Nonlinear Model
One approach to understanding complex phenomenais to reduce them to their barest essence in idealized mathematical models. In doing so, one hopes to retain the fundamental physics while discarding embellishments, taking care not to throw awaythe baby with the bath water. (A worse fate yet is to end up with the wrong baby.) A minimal hurricane model has been constructed by the author (Emanuel 1989). (The model may be run on personal computers and is well documented.It is available on request.) The flow in the modelis taken to be axisymmetric and in hydrostatic and gradient wind balance, except in a frictional boundary layer. The equations of the model are transformed into a coordinate systemin whichone of the coordinates, R, is proportional to the absolute angular momentum unit mass about the storm center: per Annual Reviews www.annualreviews.org/aronline 190 EMANUEL (17) 1 u 1 2 ~fR = rV+ ~fr , where f is considered constant. Since R is conserved in the absence of friction, this formulation eliminates radial advections, except in the frictional boundary layer. Moreover, the theory of slantwise convection (e.g. Emanuel1983) shows that the relevant convectively neutral state one in which air parcels are neutrally buoyant along angularmomentum surfaces, rather than along vertical surfaces. The structure of the minimal model is summarized in Figure 5. The dependent variables are the physical radius r, predicted at the top of the model and at the top of the boundary layer; the mass streamfunction ~p, predicted at the middle level and diagnosed at the top of the boundary layer; temperature, represented by a saturation entropy s*, predicted at the middle level and in the lower layer; and actual entropy s, predicted in the boundarylayer and in the lower layer. (The saturation entropy is the entropy the air would have if it were saturated at the same temperature and pressure. It is a state variable.) The essence of the finiteamplitude nature of the hurricane instability appears to rely on the existence of a spectrum of different types of wet convection, represented in the model by only two categories: shallow convection, which penetrates only to the lower layer; and deep convection, which extends through the depth of the model atmosphere. These two forms differ in that the former does not precipitate while the latter does. Within deep, precipitating clouds, the latent heat acquired from the sea is mostly converted into sensible heat, while shallow clouds produce no net ~ V'o,rb Figure 5 Vertical structure of the simple balanced model. The dependent variables are physical radii r t and rb, saturation entropies s* and Sin*, and actual entropies sb and Sin. Deep clouds occur whenever Sb > S*, while shallow clouds are present whenSb > Sm*. The shallow clouds do not precipitate and thus produce no net heating. Annual Reviews www.annualreviews.org/aronline
THEORY OF HURRICANES 191 heating, since all the condensed water is ultimately reevaporated. But shallow clouds do stabilize the atmosphere by exchanging the highentropy, boundarylayer air with lowentropy air from the lower troposphere. The entropy minimum the lower troposphere above the boundin ary layer, which results from the large subsaturation there, is a normal feature of the tropical atmosphere and is crucial for understanding the finiteamplitude nature of tropical cyclogenesis. In the minimal model, shallow clouds occur wheneverthere is local convective instability in the lower atmosphere(i.e. whens > Sin*, where Sm* the saturation entropy of is the lower layer). Deepclouds occur whens > s*. The fundamental dimensional parameters governing the behavior of the model are the Coriolis parameter f, the depth H of the troposphere, the dimensionless exchange coefficient C D that appears in the bulk aerodynamic formulae by which heat and momentum fluxes from the ocean are calculated, and a measure of the seaair thermodynamic disequilibrium: ~ =(T~Tt)(sZSa), (18) where T is the ocean temperature, Tt the ambient temperature of the s tropopause, s~' the saturation entropy of the ocean surface, and sa the entropy of the normal tropical atmosphere near sea level. The quantity s~sa represents the thermodynamic disequilibrium of the atmosphereocean system. When the governing equations are suitably normalized, very few nondimensional parameters are needed to describe the system. The most important of these are those describing the initial vortex and the magnitude of the entropy minimum the lower troposphere. The scaling in factors for time, length, and velocity are listed in Table 1. Note that all of the scales dependon ks, and that these scales are quite different from those of baroclinic cyclones. Table 1 Hurricane scales Quantity Length Time Azimuthal velocity Radial velocity Vertical velocity a Scale Z~/~f ~ ~/~ C~ ~H)~7 Z~/2
~CD~sfCD~ls/2 Typical value 1000 km 16 hr 60 ms ~ 10 ms ~ 6 cms ~ ~H ~ "Scale parameters:H ~ depthof convectinglayer, f ~ twice the local vertical component Earth's angular velocity, Co=diof mensionless exchange coefficient for surface fluxes, ;(~ =thermodynamic disequilibriumparameter,definedby Equation (18). Annual Reviews www.annualreviews.org/aronline 192 EMANUEL Figure 6 shows the evolution of the maximum (dimensionless) wind with (dimensionless) time for four different experiments. The control run begins with a warmcorevortex of amplitude 0.22, which decays with time initially but ultimately amplifies to a quasisteady intense cyclone. The second experiment is identical to the first except that the initial amplitude is 0.05. This vortex never amplifies, which demonstrates the finiteamplitude nature of instability in the simple model. The third and fourth experiments are identical to the first two except that shallow clouds are omitted. These simulations exhibit linear instability in the sense that small perturbations can amplify. Clearly, the finiteamplitude nature of cyclogenesis in the simple modeldepends on the existence of weakly precipitating convection. Why? The Nature of Moist Convective Adjustment Consider two extremely different convective processes. In the first, we allow only nonprecipitating convection. Supposewe cool the middle troposphere by, say, adiabatic lifting of the (dry) middletroposphere air. This destabilizes the columnto convection, but since the convection does not precipitate, its ensembleaverage effects do not include heating and the adiabatic cooling is thus unopposed. But the convection does drive the column back toward neutrality by importing lowentropy air into the boundary layer from the middle troposphere (whose entropy is usually substantially less than that of the boundarylayer). Hence,nonprecipitatin9 convection forces the boundarylayer entropy to follow changes in freeatmosphere temperature. In the second process, we suppose that all of the water condensed in clouds falls out as rain. Now,when the middle troposphere is cooled, at least someof the cooling is opposed by the net heat released in precipitating clouds. Precipitatin 9 convection drives the atmosphere toward neutrality, in part, by forciny the freeatmosphere
1.0 .8 Vmox .6 .4 Exp.F2 .,~ Exp. Figure 6 Evolution of maximumdimensionless velocity with dimensionless time for four experiments with the simplified model. The control run is shown by the solid line. ExpcrimcntE is identical to the control except that it begins with a smaller amplitude. Experiments F~ and Fz are identical to the first two experimentsexcept that shallow clouds are omitted. Time 6 8 I0 t2 Annual Reviews www.annualreviews.org/aronline THEORYOF HURRICANES193 temperature to follow changes in the boundarylayer entropy. Generalizing from these two examples, we assert that only a fraction of forced cooling of the middle troposphere will be opposed by convective heating in a conditionally neutral mean atmosphere. This fraction is approximately equal to the precipitation efficiency, which is the fraction of condensed water that falls out of the system. From this argument we mayunderstand the essence of the finiteamplitude nature of tropical cyclogenesis. Whena weak vortex is placed in contact with the sea surface, frictional inflow forces upwardmotion in the free atmosphere near the vortex core. (Other processes mayforce upward motion in real cyclones.) Since the initial convection will no doubt have a precipitation eff~ciency less than unity, not all of the adiabatic cooling associated with the ascent of dry (lowentropy), middletropospheric air will be opposed by convective heating, and the central core will cool. Moreover, the boundarylayer entropy near the core will decrease by the action of downdrafts so as to keep the columnneutral to moist convection. Thus, we have no more than the classical spindown of a balanced, strati2 fied vortex on a rigid surface, but with the static stability N replaced by an effective stability approximately equal to N2(1ep), where ep is the precipitation efficiency. Opposing this tendency is the anomalous flux of (mostly latent) heat from the ocean into the boundary layer, associated with the anomalous surface winds. Initially, this flux can only partially compensatefor the reduction in boundarylayer entropy due to the import of lowentropy air from the middle troposphere by the convective downdrafts. The surface fluxes vary with approximately the first powerof the surface wind speed, while the drag (and thus the frictionally induced vertical motion) increases more nearly as the square of the wind speed, so that, if anything, the net cooling increases with initial vortex amplitude. But after some time has elapsed, the entropy of the middle troposphere increases owing to the upward flux of high, boundarylayer entropy by lowprecipitationefficiency convection. Whenthe entropy of the middle troposphere becomes large enough, the import of lowentropy air into the boundary layer by downdrafts can no longer compensatefor enhancedsurface heat fluxes, and the boundarylayer entropy increases together with the freeatmospheric temperature. The vortex amplifies. This is clearly a nonlinear effect. Examination of the thermodynamic fields of both the simple model and the complete model supports this heuristic view. The obvious observational test of this idea is to find out whether the saturation of a deep columnof tropospheric air is a necessary and sufficient condition for the development of tropical cyclones. Annual Reviews www.annualreviews.org/aronline 194 EMANUEL SUMMARY AND REMAINING PROBLEMS
The mature hurricane is a Carnot engine driven by the thermodynamic disequilibrium between the tropical oceans and atmosphere. Air spiraling radially inward in the boundarylayer is brought closer to thermodynamic equilibrium with the ocean by large windinducedheat fluxes; it then rises nearly (moist) adiabatically to great altitudes, where the excess heat exported or lost by electromagnetic radiation to space. The mechanical energy available from this cycle is a thermodynamic efficiency e, multiplied by the surface temperature and by the difference between the saturation entropy of the ocean surface (at the central pressure of the hurricane) and the undisturbed boundarylayer entropy. The efficiency is where T is the seasurface temperature, and T is the mean temperature S o at which heat is exported by or lost from the storm's highlevel outflow. Its typical magnitudeis 1/3. While the energy source for mature hurricanes has been recognized since at least the time of Kleinschmidt (1951), controversy remains about the energetics and dynamics of hurricane genesis. Observational evidence and forecasting experience favor the idea that hurricanes result from a finiteamplitude instability: Weak disturbances are often observed to decay even under favorable environmental conditions, and hurricanes are, after all, rare despite the nearly ubiquitous presence of an energy reservoir (see Figure 3). In the early 1960s the theory knownas Conditional Instability of the Second Kind (CISK) was proposed to explain hurricane genesis. In this theory, the ocean surface serves as a sink for momentum but not as a source of heat; the spinup relies on convective energy stored in the atmosphere. This theory is suspect for a variety of reasons. First, careful analyses of the maritime tropical atmosphereshowlittle stored convective energy, and the few locations (such as central North America in spring) that exhibit large amounts of stored convective energy are not knownto produce incipient cyclones. Also, CISKis a fundamentally linear instability; taken at face value, it predicts that incipient cyclones should be ubiquitous features of all convectively unstable atmospheres. The author (Emanuel 1986) has proposed an alternative theory that regards tropical cyclogenesis as resulting from finiteamplitude WindInduced Surface Heat Exchange(WISHE) instability of the tropical atmosphere, relying on a positive feedback between surface heat fluxes and Annual Reviews www.annualreviews.org/aronline
THEORY OF HURRICANES 195 surface wind. WISHE modesare regarded as occurring in ambient atmospheres that are neutral to adiabatic displacements of boundarylayer air and hence have no stored convective energy. The reevaporation of condensed water in the lowentropyair of the middle troposphereappears to be the reason for the finiteamplitude nature of the instability: The resulting downdrafts import lowentropyair into the boundary layer at a rate that exceedsthe enthalpyflux fromthe oceansurface. Intensification occurs whenthe entropy of the middletroposphericair has been raised enoughto substantially weaken lowentropyflux into the boundary the layer by downdrafts. Whilethese ideas are consistent with complex numerical simulations,they haveyet to be systematicallytested in real tropical cyclones. Evenif the reasons for the finiteamplitude nature of tropical cyclogenesis are correctly identified, the problem genesis wouldremainas of one of explainingthe initiating disturbance. Observations indicate that a variety of circulations of dynamicallyindependentorigin mayinitiate tropical cyclones. These include easterly waves, which are wavydisturbances in the easttowest flow of air in the tropics, especially over and west of subSaharanAfrica and in the central Pacific; baroclinic developments the subtropics; and continental mesoscalethunderstorm in complexes that occasionallydrift out over openwater. Acomplete finiteamplitude theory of hurricanes could presumablyspecify the required characteristics of the initiating disturbances. In addition to the problemof genesis, several aspects of hurricane behavior remain poorly understood. Giventhat "ignition" has occurred and a hurricane develops, very few reach the upper boundson intensity implied by Figure 3. Onthe other hand, many numericalsimulations(e.g. those of Rotunno Emanuel & 1987) consistently intensify storms right to the upper bound. Why nature different? Onepossibility is that most is hurricanes are limited by the cold water that they invariably mix upward frombeneath the oceanic seasonal thermocline. Observed seasurface temperature changes as large as 5C; only about 2.5Cof coolingis needed are to reverse the airsea thermodynamic disequilibrium. Preliminarystudies showthat the induced cooling mayindeed limit the intensity of many hurricanes, though a comprehensivesimulation with a coupled oceanatmospheremodelremains to be performed. Thespiral bandsof convectivecloudsthat give hurricanestheir characteristic appearance satellite photographs not well understood. in are Extant theories include Ekmanlayer instability and inertiagravity wavespropagating outwardfrom the eyewall. These theories have not been rigorously tested against observations. Some strong hurricanes exhibit concentric eyewalls that undergoa characteristic evolution in whichthe Annual Reviews www.annualreviews.org/aronline
196 EMANUEL eyewalls contract inward, the inner eyewall dissipates, and a new eyewall forms at a larger radius (Willoughby et al 1982). There is no wellaccepted theory of this phenomenon. Finally, the issue of hurricane steering remains the focus of lively research. Most theories pertain to the drift of barotropic vortices on/~planes (on which the vertical component of the Earth's rotation rate varies linearly with latitude). These theories predict that hurricanes should drift westward and poleward with respect to the mean wind. They do not account for the nonuniformity of the background potentialvorticity gradient in whichhurricanes are embedded, nor do they recognize hurricanes as strongly baroclinic vortices with anticyclonesin the uppertroposphere. It seemslikely that accountingfor such effects will radically alter our understanding of hurricane motion. Hurricanespresent a large number fascinating and unresolvedprobof lemsthat havereceivedsurprisinglylittle attention fromtheorists. Assuch, they remaina fertile and importantsubject of research in fluid dynamics.
Literature Cited Anthes,R. A. 1982.Tropical cyclones:their evolution, structure andeffects. Meteorol. Mono,qr. No. 41. Boston: Am.Meteorol. Soc. 298pp. Betts, A. K. 1982.Saturationpoint analysis of moistconvective overturning.J. A tmos. Sci. 39:14841505 Charney,J. G., Eliassen, A. 1964. Onthe growth of the hurricane depression. J. Atmos.Sei. 21:6875 Emanuel, A. 1983. The Lagrangian K. parcel dynamics moist symmetric of instability. J. Atmos.Sci. 40:236876 Emanuel, A. 1986. Anairsea interaction K. theory for tropical cyclones. Part I. J. Atmos. Sci. 43:585q504 Emanuel,K. A. 1988. The maximum intensity of hurricanes.J. Atrnos.Sci. 45:114355 Emanuel,K. A. 1989. The finiteamplitude nature of tropical cyclogenesis.,L Atmos. Sci. 46:343156 Fj~rtoft, R. 1953. Onthe changes of the spectral distribution of kinetic energyfor twodimensional, nondivergent flow. Tellus 5:22530 Kleinschmidt,E. Jr. 1951. Gundlagen einer Theorie des tropischen Zyklonen.Arch. Meteorol., Geophys.Bioklimatol., Ser. A 4:5372 Merrill, R. T. 1984. A comparison large of and small tropical cyclones. Mon.Weather Rev. 112:140818 Ooyama, 1969. Numericalsimulation of K. the life cycleof tropicalcyclones. A trnos. J. Sci. 26:340 Riehl, H. 1954. Tropical Meteorology.New York: McGrawHill. pp. 392 Rotunno, R., Emanuel, K. A. 1987. An airsea interaction theory for tropical cyclones.Part II. J. Atmos.Sei. 44: 54261 Willoughby, E., Clos, J. A., Shoreibah, H. M.G. 1982. Concentriceye walls, secondary wind maximaand the evolution of the hurricane vortex. J. Atmos.Sci. 39: 395~411 Xu,K., Emanuel, A. 1989.Is the tropical K. atmosphere conditionally unstable? Mon. WeatherRev. 117:147179 Yamasaki,M. 1977. A preliminary experimentof the tropical cyclonewithout parameterizing the effects of cumulusconvection. J. Meteorol.Soc. Jpn. 55:1131 Yanai, M. 1964. Formationof tropical cyclones. Rev. Geophys.2:367414 ...
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This note was uploaded on 01/08/2012 for the course MPO 551 taught by Professor Zhang,c during the Summer '08 term at University of Miami.
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