Hurricane potential intensity from an energetics point of view

Transcription

1 Q. J. R. Meteorol. Soc. (2004), 130, pp doi: /qj Hurricane potential intensity from an energetics point of view By WEIXING SHEN, National Centers for Environmental Prediction, Camp Springs, USA (Received 8 April 2003; revised 15 March 2004) SUMMARY This article describes an approach to hurricane potential intensity with a model that considers total kinetic energy balance within a hurricane. The major kinetic energy source and sink are the kinetic energy converted from sea surface entropy flux and the surface dissipation, respectively, but the conversion efficiency is radiusdependent in this model. Also, the internal conversion due to convectively available potential energy (CAPE) as a source is considered. In contrast to previous potential intensity models, energy balance in the entire hurricane is used to get its steady state. With this model, the roles of environmental sea surface temperature (SST), CAPE and some features of the hurricane itself in determining hurricane potential intensity are investigated. The results indicate that existence of CAPE may lead to the steady state (or potential intensity) appreciably different from the otherwise surface entropy flux conversion and surface dissipation balanced state, although CAPE contribution to kinetic energy generation is usually much smaller than that of the surface flux. It is interesting that hurricane potential intensity seems less dependent on the underlying SST than was shown with previous theoretical models. The results also suggest that hurricane size may affect its potential intensity in that larger-sized hurricanes tend to have higher potential intensities. KEYWORDS: Inner core Kinetic energy Thermodynamic efficiency 1. BACKGROUND In an attempt to understand tropical cyclone intensity, maximum potential intensity (MPI) (Emanuel 1986; Holland 1997) was proposed, where the maximum achievable tropical cyclone intensity is controlled by the environmental thermodynamic state. These thermodynamic approaches view a tropical cyclone as an axisymmetric system and assume that the internal processes have no constraint on the maximum achievable intensity. However, Holland s (1997) approach to MPI relies heavily on the thermodynamic conditions within the hurricane eye and eyewall. The Emanuel (1986, 1988) MPI focused on the energy balance in a hurricane by following an air parcel, and this approach was refined by Emanuel (1991, 1995). Basically, it is assumed that all the dissipation occurs in the surface frictional layer and, for a steady-state hurricane, this dissipation is perfectly balanced by the mechanical energy generation due to sea-surface entropy input through a Carnot cycle. This implies a nearly neutral atmosphere so that the impact of atmospheric CAPE is ignored. Also, the parcel trajectory assumption that the inward-spiralling air is carried all the way into the centre of the storm seems not very realistic. Emanuel (1997) put forward the total energy balance in a hurricane as follows: ra ρεt s C h V s (S s S a )r dr = ρc d Vs 3 r dr, (1) r m r m where r m is the radius of maximum wind (RMW), r a is the outer radius, ρ is the air density, ε is the thermodynamic efficiency representing the efficiency for available potential energy conversion into mechanical energy, T s is the surface air temperature, V s is the surface wind, S s is the saturation entropy of the ocean surface and S a the air entropy. C h and C d are the surface entropy and momentum exchange coefficients respectively. In Emanuel (1997), however, only the local energy balance under the eyewall was actually used to attain the maximum surface wind for pedagogic purposes. Corresponding address: NCEP, Rm 207, Camp Springs, MD 20746, USA. weixing.shen@noaa.gov c Royal Meteorological Society, ra

2 2630 W. SHEN To avoid confusion, we point out that for all the relevant papers for the Emanuel MPI theory, both local eyewall energetics and thermal wind balance are followed to attain the potential intensity including both minimum surface pressure and maximum surface wind (e.g. Emanuel 1995; Bister and Emanuel 1998). In this study, an approach for potential intensity is proposed based on (1), with addition of CAPE release given its existence in the tropics. According to (1), surface dissipation and entropy flux (input) depend very differently on the surface wind (simply V 3 versus V ). Thus, a hurricane with a local balance under the eyewall should have the energy unbalanced outside the eyewall due to the rapid decrease of surface wind with increasing radius, unless the thermodynamic efficiency, ε, corresponding to the local entropy flux quickly decreases with increase of radius in such a way that the dissipation and generation are equally confined. This implies that the intensity of a hurricane may be sensitive to its size or the radial distribution of surface wind. It is notable that the Emanuel model (e.g. Emanuel 1995) takes into account the impact of hurricane size but its involvement is different from the present approach. In this paper, total energy balance will be used to estimate hurricane potential intensity with the radial wind profiles from Holland (1980). The change in potential energy, gz, is neglected because the energy balance in the entire system is considered. Since the hurricane size may affect the energy balance in (1) and the outer radius, r a, has not been well defined, this will be discussed. In section 3, we will find such an outer radius is not actually needed as the kinetic energy generation mainly occurs in a newly defined inner core of large-scale (relative to convection) upward motion. The size of such an inner core is internally determined by the surface wind profile. CAPE may be small in the real atmosphere but its involvement can appreciably push the storm steady state away from the energy balance determined by (1). Thus in this study we estimate the effect of such an internal energy conversion on hurricane intensity. We emphasize here that this study focuses on the dependence of hurricane intensity on the thermodynamic environmental conditions, such as SST and CAPE, as well as some features of the hurricane itself, rather than seeking an accurate solution to the hurricane potential intensity. Section 2 of this paper is a discussion of the empirical surface wind profiles used in this study and their implications. In section 3, a simple model based on hurricane energetics is proposed. The results of using this model on energy balance and potential intensity are presented in section 4. Section 5 gives the conclusions. 2. RADIAL PROFILE OF SURFACE WIND AND SOME IMPLICATIONS (a) Profiles of surface pressure and wind In Holland (1980), the surface pressure profile is approximated by a family of rectangular hyperbolae P = P c + (P e P c ) e (r m/r) B, (2) where B is a scaling parameter defining the shape of the profile, P c is the central pressure at surface and P e the environmental surface pressure. Using the gradient wind relationship, the wind profile becomes ( V g = Vc 2 + r2 f 2 ) 1 2 rf 4 2, (3) where { B ( V c = ρ (P rm ) } 1 B e P c ) e (r m /r) B 2 r

3 HURRICANE POTENTIAL INTENSITY 2631 is the cyclostrophic wind, an approximation for the inner area of the hurricane, and f the Coriolis parameter. The maximum wind (at r m where V c / r = 0) is approximately ( )1 B 2 V m = (Pe P c ) 1 2. (4) ρ e Similar relationships between V m and P e P c have been widely used for estimating the maximum winds in hurricanes, but in (4), the coefficient is B-dependent instead of being empirically determined. It is notable that, although such profiles may not fit with all the detailed features of hurricane wind, they do fit well with the overall pictures of various hurricane cases (Holland 1980). Importantly, such profiles allow us to discuss the possible roles or influences of some major features such as size in hurricanes. (b) Surface fluxes Considering a hurricane as a well-organized axisymmetric system, one can then calculate the entire surface entropy input and surface dissipation in the hurricane using the profiles from (3) and the surface thermodynamic conditions. Since the abovegradient wind profile by (3) applies to the free atmosphere above the surface boundary layer, a reduction factor, c r, is used to extend such a profile to the altitude of 10 m so that the conventional empirical surface exchange coefficients at this altitude can be used. The surface entropy flux into a hurricane is F = 2πρ ra 0 T s C h V s (S s S a )r dr, (5) where V s = c r V g is the 10 m wind and the radial variation in surface density is assumed to be negligible. For the current purpose, we divide F into F 1 for the inner area and F 2 for the outer area where r r m and the wind is nearly geostrophic. Using (2), the geostrophic wind in the outer area is V s = BrB m c r(p e P c ) ρr B+1. (6) f So, with r s as the separation radius, F 2 = C h ra 1 r s r B dr = C h C h 1 ( 1 (B 1) ( ) ln ra ln r s r B 1 s 1 ) ra B 1 if B 1 if B = 1, where C h = 2πT sc h Brm Bc r(s s S a )(P e P c )/f and the radial variations of (S s S a ), C h and T s in the outer area are considered small enough to ignore. Applying the area separation above to the surface kinetic energy dissipation we have where K diss = 2πρc 3 r ra { ( (K diss ) 2 = C d 1 1 (3B + 2) 0 C d Vg 3 r dr, (7) r 3B+2 s 1 )} ra 3B+2, C d = 2πC dc 3 r {BrB m (P e P c )} 3 /ρ 2 f 3.

4 2632 W. SHEN From the above equations for F 2 and (K diss ) 2, it is apparent that F 2 for r a with B 1, while (K diss ) 2 is always bounded. This is because the dissipation is proportional to Vs 3 and thus strongly confined to the inner area, while the surface entropy flux is proportional to V s and thus is not confined for large size hurricanes (B 1). From the Carnot cycle point of view (e.g. Emanuel 1988), the kinetic energy generation is simply the surface entropy flux multiplied by a thermodynamic efficiency. Such an efficiency is in reality related to the diabatic exchanges between the lifting air parcel and its ambient environment as well as the height the air parcel can reach. The diabatic exchange makes the lifted air parcel deviate from the wet adiabat and reach a lower altitude. In this case, one can still use the concept that the excessive energy gain from the surface is all lost at the outflow level, which should now be lower than the real outflow level due to the diabatic exchange during the lifting. This outflow level will be referred to later as the equivalent outflow level. For the kinetic energy generation to be bounded, the thermodynamic efficiency has to decrease quickly with increasing radius. The efficiency decrease is true because both the large-scale vertical motion (relative to the cumulus convection and its associated downdrafts) and the (circularly averaged) cumulus convection become less effective with regard to energy conversion with an increase of radius. In the following, the vertical motion associated with the secondary circulation will be referred to as the large-scale motion which is presumably resolvable by the wind profile. (c) Inner and outer cores We now use a steady barotropic solution as an approximation for the hurricane surface boundary layer. The barotropic approximation is based on the fact that the wind change in the vertical is small in the surface boundary layer. Therefore, the momentum equation for the tangential direction in a cylindrical coordinate can be written as Vn r = κ d (V g r) { f + 1 }, (8) d(v g r) r dr where Vn is the vertically averaged radial velocity in the surface boundary layer and κ d = A z /{D b h ln(h/z 0 )} (see Wang 1988), a surface drag measure equivalent to taking the momentum dissipation in the form of Rayleigh friction where A z is the turbulent viscosity, D b is the depth of boundary layer, h is the depth of surface layer, and z 0 the surface roughness. Meanwhile, the boundary layer continuity equation in this case is W t = d(v n r) D b, (9) r dr where W t is the vertical velocity at the top of the boundary layer. For the outer area in a hurricane, using (6), and also (8) with neglect of d(v g r)/r dr which is related to the nonlinearity and curvature effects, (9) becomes W t = κ db 2 rm B(P e P c ) ρf 2 r B+2 D b, (10) which implies large-scale subsidence at the top of the outer boundary layer. This is consistent with the prevailing large-scale subsidence in the outer area of a hurricane. In practice, this decrease is only used in the inner core in our model, since the outer core is very different (see the detailed discussion in section 3).

5 HURRICANE POTENTIAL INTENSITY 2633 With neglect of the curvature term, (8) is also applied as an approximation to find the radius where large-scale vertical motion reverses. One might think that the neglect of the curvature effect would appreciably affect the accuracy near the eyewall. But this radius for reverse, r 0, is generally much larger than r m, except for very confined hurricanes or hurricanes of very small size. We found that for hurricanes which are not highly confined (large B), direct use of (8) yields only a small difference in r 0 and a minor change in intensity at latitudes more than 15 from the equator. The small impact of the neglect on the intensity is not only because the two parts in d(v g r)/r dr partially cancel each other, but also because both the generation and the dissipation are highly confined to the eyewall, so that they are not sensitive to a small change in r 0. Therefore, using (9) with W t = 0 into (8) with neglect of d(v g r)/r dr, wehave d(v g r) dr = 0. (11) r0 Based on (11), we define the hurricane inner core as the inner area where upward motion prevails (r <r 0 ) and the outer core as the outer area of subsidence (r >r 0 ). These inner and outer cores are different from those which may have been widely used but loosely defined. It is worthy of note that (11) is attained without use of any assumed wind profiles. In the outer core (r >r 0 ), d(v g r)/dr >0, which also leads to infinite entropy flux (F 2 for r a ). This is consistent with the previous discussion based on the assumed wind profiles. 3. A MODEL BASED ON HURRICANE ENERGETICS We now propose a model based on the kinetic energy balance in a hurricane. We first divide a hurricane into two parts: an inner core with large-scale upward motion and an outer core with subsidence. Obviously, the upward motion is much faster than the subsidence because the outer core is much broader than the inner core. The inner and outer cores are denoted by the areas 1 and 2 in Fig. 1. In area 2, we assume that the air flowing from area 1 loses all its excessive energy at the outflow layer and moves down very slowly so that it is in thermodynamic equilibrium with its environment. So the environmental sounding also applies to this outer core. We also assume that the surface entropy flux (mainly by surface evaporation) into the surface boundary layer nearly balances the loss by dry air intrusion from the free atmosphere due to downward motion and the vertical diffusion near the top of the surface boundary layer. The vertical diffusion also acts to offset the drying by the subsidence in the free atmosphere. Thus, the surface boundary layer air temperature and its water vapour content remain the same as those of the environment (symbolically at r a ). Given the small surface pressure change in the outer core (from r a to r 0 ), the outer core CAPE has approximately the value of environmental CAPE. The nearly unchanged mixing ratio and temperature in the surface boundary layer in the outer core are evident in hurricane observations. We point out that the above division bears similarity to that from the simulations by Rotunno and Emanuel (1987) where the potential temperature in the surface boundary layer has an inward increase under the eyewall and its surrounding and remains nearly constant in the outer area (see Fig. 12 of Rotunno and Emanuel 1987). The feature of the inward increase the inner core as a necessity will be discussed later. Based on the above discussion, we consider that the convection, if any, in area 2 is very small and its contribution to the kinetic energy generation in a hurricane is negligible. However, there may be significant rainbands in the outer cores of some hurricanes, In the current estimate, the radial variation of κ d and D b is considered negligible.

6 2634 W. SHEN Figure 1. A schematic diagram of an axisymmetric hurricane. Regions 1 and 2, separated by r 0,representthe hurricane inner and outer cores. We assume the downward draughts between the convection cells in the inner core are insignificant, or the local vertical circulations in the inner core are nearly reversible processes, so that they do not contribute to kinetic energy changes and are thus not shown. and one can view these as part of the mechanism to maintain the environmental sounding in the outer area. This assumption is made in the current model, but it may not be very accurate in all cases. In area 1, the air entropy in the surface boundary layer increases towards the eyewall. Because of this inward increase, the entropy advection out from a given area of the surface boundary layer (inward and upward ) is larger than the entropy advection into the area for a steady state. This net loss is just balanced by the surface entropy flux into this area. It should be pointed out that the inward increase of the boundary layer entropy is a necessity for a hurricane. Without this increase, which implies the entropy gain from the sea surface is just lost from the top of the boundary layer, the boundary layer air under the eyewall can be considered roughly as the air converging from the environment through an adiabatic thermal expansion due to pressure decrease. In this case, such a pressure decrease can only come from atmospheric CAPE and its release. This is because without atmospheric CAPE in this case, a lifted air parcel near the eyewall would gain no buoyancy and thus no surface pressure drop (P e P c ). Given the nearly wet adiabat in the hurricane eyewall and hydrostatic balance, the larger the entropy increase, the larger the surface pressure drop. This is part of the Holland (1997) theory, except it used the air entropy and moisture near the surface. It is apparent that larger inward increase of boundary layer entropy results from larger surface entropy flux and less local (vertical) release by convection and upward motion. Note that vertical diffusion at the boundary layer top is usually insignificant in the inner core (i.e. Rotunno and Emanuel 1987). On the other hand, a larger entropy increase in turn leads to larger atmospheric instability and less surface thermodynamic disequilibrium and thus possible larger local release and less surface entropy flux. Therefore, in reality, the air moisture in the inner boundary layer, which is critical for the resulting air entropy, is not a choice. Nevertheless, the impact of the parameter choice under the eyewall on the entire surface entropy flux in the inner core is not as significant as its impact on the thermodynamic profile in the eyewall or the local entropy flux under the eyewall used in the Holland and Emanuel MPI models. So, the current model leads to less sensitivity to the choice of the air thermodynamic state under the eyewall. In addition, previous studies using the Holland and Emanuel MPI models followed the near-surface conditions instead of the whole boundary layer in deriving MPI. The vertical convection can be considered to be part of the upward advection at the price of neglecting any convection-associated downdraughts into the boundary layer; otherwise convection helps the net loss.

7 HURRICANE POTENTIAL INTENSITY 2635 The models also do not consider the details between the environment and the eyewall. With little change in surface air mixing ratio found, the surface pressure drop was emphasized for the surface air entropy increase and thus the hurricane intensity. We point out that such a surface air entropy increase should not be directly linked to the ocean flux role. This is because the surface entropy increase is basically due to the surface pressure drop with fixed surface air temperature not the moisture change. In fact, however, it is the moisture from the ocean that is most important in a hurricane. Thus, without considering the whole depth of the surface boundary layer where large vertical mixing occurs, it would be hard to understand the role of the ocean flux in a hurricane. In the current model, the boundary layer details are not used but the boundary layer entropy increase is implicitly assumed. Also, the entire surface entropy flux in the inner core is used to represent the role of ocean. According to Emanuel (1997), the kinetic energy generation due to the surface entropy flux in the inner core can be calculated from the following: r0 K sflx = 2πρc r εt s C h V g (S s S a )r dr. (12) r m In (12), the hurricane eye is ignored given its negligible influence. For deep convection or vertical advection such as that in the hurricane eyewall, the ascent is a nearly conserved process with ε 0 = T s T o, (13) T s where T o is the temperature at the outflow level or approximately the tropopause. As was previously concluded, the thermodynamic efficiency for the local entropy flux, ε, decreases as the radius increases. As a rough approximation, we decrease the efficiency (outside the eyewall) by multiplying ε 0 with a radius-dependent factor, (r m /r) Ɣ,whereƔ is a parameter that determines the confinement of the kinetic energy generation. Hence ( rm ) Ɣ ε = ε 0. (14) r The accurate picture, however, depends on the whole spectrum of the hurricane convection and secondary circulation as well as many other details. For the current approximation, Ɣ is treated as a variable dependent only on r m and will be further discussed in section 4. Although ε may also be dependent on other factors, such as hurricane size (or r 0 ), intensity and latitude, these kinds of dependence are assumed to be much weaker than those on r and r m, and are ignored. From (12), the overall thermodynamic efficiency can be written as r0 / r0 ε = εc h V s (S s S a )r dr C h V s (S s S a )r dr. (15) r m r m With inclusion of the CAPE (internal) contribution (see Appendix), the net kinetic energy change in a hurricane can be written as K t = K sflx + K CAPE K diss, (16) where K sflx is the kinetic energy generation due to surface entropy flux by (12), K CAPE = (ε/ε 0 )(K CAPE ) pot (by (A.7)) is the generation due to CAPE, and K diss is the sink due to surface dissipation by (7). In practice, only the integration from r m to r 0 is used for K diss in (7) due to the small contribution outside this range. Ignoring the

8 2636 W. SHEN dissipation outside r 0 basically increases the intensity but the increase is generally small. It is important that nearly identical results can be attained by choosing a slightly lower Ɣ in (14) with additional dissipation outside r 0. The determination of Ɣ can be based on an intensity observation and will be discussed later. Finally, we point out that in this study the hurricane size is treated as a factor independent of its thermodynamic environment because little is known about what determines the hurricane size. At least, no simple relationships between them have been indicated by observations. It is also possible that hurricane size and its evolution may be related to hurricane internal dynamics, its interaction with the environmental systems, and its initial state or formation. This issue is not pursued in this study. 4. HURRICANE ENERGETICS AND POTENTIAL INTENSITY (a) Hurricane size and CAPE The proposed model is now used to investigate the kinetic energy generation and dissipation for a given hurricane and its environment conditions. The following gives the parameter choices and justification for the investigation: We assume the relative humidity (RH) under the eyewall (at r m in Fig. 1) is always 90%. This is within the range of achievable values (Holland 1997) and agrees with numerical simulations (Shen et al. 2000). For the value of RH between r 0 and r m, linear interpolation is used. The averaged environmental SST is used for the surface air temperature from r a to r m. This may cause a slight difference in the surface entropy flux since both observations (e.g. Beckerle 1974; Cione and Black 1998) and numerical simulations (Shen et al. 2002) indicate that air temperature decreases towards the eyewall, falling between the SST and the temperature of a converging air parcel by an adiabatic thermal expansion. Although the environmental surface RH may vary in reality, the effect of such a variation is not discussed in this study. The value of 80% is used for all the experiments in this study. The surface entropy and momentum exchange coefficients are assumed to be the same: C h = C d = ( V s ) 10 3 for surface winds under 30 m s 1 (e.g. Garratt 1977; Wu 1980) and unchanged for winds over 30 m s 1. The relationship between the surface coefficients and surface wind under 30 m s 1 is quite consistent with the GFDL model calculations using a Monin Obukhov scheme (B. Thomas 2003, personal communication). The value of c r = V s /V g used in (7) and (12) reflects the vertical wind profile in the surface boundary layer. Recent hurricane observations (e.g. Black and Franklin 2000; Dodge et al. 2002) indicated that vertical mixing may lead to horizontal wind being barely changed in the vertical. An assumption that wind reduction is small in the vertical was already used in deriving (8). It is apparent from (7) and (12) that changing c r changes the energy balance. Simply put, decreasing c r reduces the energy dissipation more than creation and thus increases intensity. c r is normally below 1. A c r of 0.85 is used in this study. In determining the hurricane inner core size, r 0,weusedaκ d of in (8). It is equivalent to having the boundary layer velocity change direction by about 12 at 30 N and 18 at 20 NwithD b for the first 100 hpa. This implies that CAPE, whose contribution to kinetic energy generation is proportional to Vn in (A.7), is more important in lower latitudes. However, the effect of latitude is not much discussed in this paper, since most of the experiments in this study were performed at 30 N. Note that the value of is used only to estimate the inner core size so that it may not necessarily represent an accurate value of κ d everywhere.

9 HURRICANE POTENTIAL INTENSITY 2637 Figure 2. Ratio of kinetic energy generation due to surface entropy flux to its surface dissipation, as a function of minimum surface pressure and wind profile shape parameter, B. The thin dashed lines denote the radius (km) of the hurricane inner core. Environmental surface pressure, sea surface temperature and radius of maximum wind are taken as 1014 hpa, 300 K and 30 km, respectively. As was mentioned, Ɣ in (14) is assumed to be a function only of r m.inthis section, we start with r m =30 km and Ɣ = 1, which is equivalent to assuming a potential minimum surface pressure of 947 hpa in the case of SST = 300 K, r 0 = 120 km and no CAPE with all the above justified conditions/parameters. So, in this case, ε = ε 0 (r m /r). Ɣ will be further discussed later when change of r m is involved. Figure 2 shows the kinetic energy generation by the surface entropy flux versus dissipation as a function of central pressure and the parameter B. It is seen that B is primarily responsible for the inner core size. It should be kept in mind that a larger inner core size means larger hurricane size given the same hurricane eye size or r m.the contour of 1 indicates the steady state in the case of a neutral sounding. The details in the inner core for the case marked with a triangle are given in Fig. 3. Apparently, the introduction of the factor (r m /r) in ε largely reduces the role of the surface entropy flux in generating kinetic energy in a hurricane (from the thin line to the thick line). Even in this case, the kinetic energy generation due to surface entropy flux is still less confined to the eyewall than the energy dissipation by the surface friction (the dashed line).thisiswhy,infig.2,k sflx /K diss increases with increasing inner core size given the same hurricane intensity. Figure 2 also implies that energy balance in a hurricane and thus intensity and its change can be very sensitive to the surface boundary layer physics (K sflx /K diss is proportional to cr 2 according to (7) and (12)) and atmospheric details (see the ε involvement in (15)). To investigate the contribution of CAPE, the kinetic energy generation by surface entropy flux (K sflx ) and that by the CAPE release (K CAPE ) are computed for large ranges of SST and CAPE. The values of K CAPE /K sflx are given in Fig. 4. Given the CAPE value, its impact is more significant over colder oceans than over warmer oceans. Although the contribution of CAPE is generally small for CAPE below 1000 J kg 1,it is important to note that this contribution can change the steady state considerably from the otherwise K sflx and K diss balanced state. This will be discussed next.

10 2638 W. SHEN Figure 3. Radial profiles of the energy generation (thick solid) and dissipation (dashed) (units Js 1 km 1 ), corresponding to the case of the triangle in Fig. 2. The thin solid line gives the kinetic energy generation due to local entropy flux without radial decrease of thermodynamic efficiency. This is equivalent to a case in which all convection and upward motion occur in the eyewall. Figure 4. Ratio of kinetic energy generation due to CAPE to that due to surface entropy flux, as a function of CAPE and sea surface temperature (SST, K), when central pressure is 940 hpa and radius of maximum wind is 30 km. (b) Potential intensity The steady state or the potential intensity can be attained by setting K t in (16) to be zero. We reach this solution by iteration. The steps (for example with specified r m and B) are: (i) estimate the Holland surface wind and pressure profiles from (2) and (3) by using an initial guess of P c, and the given values of P e, f, r m and B; (ii) estimate the inner core size, r 0, by using (11) with the wind profile and the radial velocity Vn in the surface boundary layer by using (8) with κ, f and the wind profile;

11 HURRICANE POTENTIAL INTENSITY 2639 Figure 5. Maximum potential intensity (i.e. minimum surface pressure, hpa) as a function of sea surface temperature (K) and wind profile shape parameter B, which approximately denotes the inner core radius (km, shown as thin dashed lines). The radius of maximum wind is 30 km. (iii) estimate ε 0 from (13), ε from (14) and K sflx from (12) with r 0, the surface profiles and the given hurricane and its thermodynamic environmental conditions as well as the parameter choices discussed in the previous section; (iv) get K diss from (7) and K CAPE = (ε/ε 0 )(K CAPE ) pot by (A.7) with the results from steps (i), (ii) and (iii), and the given conditions about the hurricane and its thermodynamic environment; (v) compute K t by (16); (vi) repeat steps from (i) to (v) with a specified small increase (decrease) in P c if K t < (>) 0, until K t changes its sign. We first investigate the case without CAPE. Figure 5 shows the potential intensity as a function of SST and B (or the inner core size shown by the dashed lines). Intensity depends on the hurricane (inner core) size in that a larger hurricane has a higher potential intensity. This is basically because the kinetic energy generation corresponding to its underlying entropy flux is less confined than the surface dissipation (see Fig. 3 for an example) so that higher wind near the eyewall is required to balance the larger net generation for a larger hurricane. Note that less confined generation is not assumed but a reasonable radial profile of ε should lead to this. This is because otherwise the resulting potential intensity would be equal to or lower than that by using a local energy balance under the eyewall. Such a local energy balance with RH of 90% (or above) under the eyewall, which is possible in reality, would result in very low potential intensity even over warm oceans. The size effect is worth discussion. Weatherford and Gray (1988), using observational data in the North Atlantic basin, did not get any significant correlation between hurricane size and intensity, although many super- (intense) typhoons are of large size. Weatherford and Gray (1988), however, did not focus on mature hurricanes and the data used were from various stages in the hurricanes lives. It is known that during hurricane development, the wind increase under the eyewall is typically much quicker than outside, and the hurricane eye may become smaller during development. This may lead to the hurricane looking more confined at a developed stage. Also, a variety of environmental conditions were associated with the cases in the study by Weatherford

12 2640 W. SHEN and Gray (1988). For our results above, the size dependence really relates to mature hurricanes with all other conditions being the same. At the moment, what determines the hurricane size and its evolution is far from being understood. Nevertheless, the size of a mature hurricane is treated as an independent factor in this study although some of the sizes (given the potential intensity) may not be achievable in reality. Based on our argument on the confinement of kinetic energy creation, we suspect that the resulting qualitative dependence of potential intensity on hurricane size may be reliable. However, the quantitative magnitudes shown should have been affected by the uncertainties in parametrizing the radial profile of thermodynamic efficiency and may not be very accurate. We point out that, given the effect of parameter B on the relationship between the maximum wind and minimum surface pressure in (4), V m has only a slight increase with an increase in storm size (or a decrease of B) in the above cases in Fig. 5. In particular, V m decreases slightly with increasing storm size over the high SSTs in Fig. 5. This is due to the universal use of ε = ε 0 (r m /r) with fixed r m (of 30 km) for all intensities. Since little is known about the accurate form of ε, a more accurate solution for intensity with further modification in ε is not pursued for the current case. In the following discussion, the surface central pressure, P c, or surface pressure drop, P e P c, will be used to represent the hurricane intensity. Figure 5 also indicates that hurricane potential intensity is not sensitive to the SST as much as was indicated by previous theoretical models. For example, Emanuel (1988), by following a parcel trajectory, attained (ln P e ln P c ) (S s S a ), while in the current model, the dependence roughly follows (P e P c ) (S s S a ) 1/2. With the current model, even over the oceans of around 20 C (about 293 K), a pre-existing storm can maintain or intensify to a considerable magnitude. This may be used to explain why some hurricanes that moved to the cold oceans around 40 N or higher latitudes could maintain or re-attain a considerable intensity (e.g. Alex 2004). We speculate that the major reason for the majority of the hurricanes diminishing quickly when moving to middle and high latitudes is the adverse environment, such as strong vertical shears in the extratropical regions. Since the result above was attained using neutral soundings, we expect an intensity increase with the existence of CAPE in the environment. Figure 6 shows the resulting intensity with CAPE ranging from zero to 2000 J kg 1 over oceans of 300 K. The potential intensity increases with CAPE. The influence of CAPE is larger for smallersized hurricanes (larger B). The central pressure only decreases from 935 to 920 hpa for B = 0.8 with an inner core radius of about 320 km for CAPE of 1000 J kg 1, while it decreases from 955 to 920 hpa for B = 2.0 with an inner core radius of about 80 km. To illustrate the steady state shift due to CAPE, the total kinetic energy budget under different intensity conditions with CAPE of 1000 J kg 1 was computed and the results are shown in Fig. 7. This figure clearly shows how the steady state shifts from S1 to S2 and the maximum surface wind increases from 57 to 65m s 1 due to the impact of the CAPE. The slight increase of the CAPE-caused addition in generation at a higher intensity is mainly due to the increase of upward motion in the inner core or boundary layer convergence to the inner core. Figure 7 also shows a case at a lower latitude, where potential intensity is higher with the current model but the dependence on latitude is generally weak for not very low latitudes. The situation at very low latitudes is obviously beyond the current model. The net increases in maximum surface wind due to the CAPE are almost the same in these two cases with different latitude conditions, in spite of a larger addition of kinetic energy generation in the lower latitude case. This is because the dissipation curves have larger slopes at higher intensities.

13 HURRICANE POTENTIAL INTENSITY 2641 Figure 6. Maximum potential intensity (i.e. minimum surface pressure, hpa) as a function of CAPE and wind profile shape parameter, B, with sea surface temperature 300 K and radius of maximum wind 30 km. Figure 7. Total kinetic energy (10 12 Js 1 ) generation without CAPE (lower dashed) and with CAPE of 1000 Jkg 1 (upper dashed) and dissipation (thick solid) under different intensity conditions at 30 N/S, with profile shape parameter B = 1.5 and sea surface temperature 300 K. S1 and S2 are steady states (see text). The thin solid line shows the dissipation at 20 N/S, with + signs denoting steady states at these latitudes. Another indication of Fig. 7 is that CAPE existence may help the rate of hurricane growth and be critical for initial development because the ratio of the CAPE-induced generation addition to the generation due to surface entropy flux is higher at a lower intensity. In realistic cases, convection at a very weak stage may be loosely organized and the overall thermodynamic efficiency is thus smaller than we have used in this study. As an approximation, one can reduce the efficiency to mimic these situations, but details like these are not pursued in this study. We point out here that currently uncertainties exist regarding the possible values of atmospheric CAPE. Emanuel et al. (1994) argued that the tropical atmosphere is nearly convectively adjusted or neutral. The impact of CAPE on potential intensity is not included in the Emanuel MPI. The nearly neutral state argument is also supported by some observational studies (e.g. Betts 1982; Xu and Emanuel 1989). However, others

14 2642 W. SHEN Figure 8. Maximum potential intensity (i.e. minimum surface pressure, hpa, bold lines) as a function of radius of maximum wind (km) and wind profile shape parameter, B. Hurricane inner core radius (km) is shown as thin dashed lines. Sea surface temperature is 300 K, and no CAPE is used. (e.g. Williams and Renno 1993; Henderson-sellers et al. 1998) found that the value of atmospheric CAPE can be high particularly when freezing at very low temperatures is included. In actual calculations, the CAPE value can be affected by the path used (pseudoadiabatic or reversible ascent, or somewhere between) and where the air parcel starts. For the present model, the air parcel really means the whole boundary layer convergence into the inner core. For accurate calculation, it should be the averaged value for those with parcel starting levels from the surface to the boundary layer top, weighted by the radial velocity component. Such a CAPE should be somewhat smaller than the CAPE calculated by using a near-surface air parcel since CAPE usually decreases with an increase of starting level. So far, only a single fixed hurricane eye size, r m, of 30 km has been used. We now discuss the possible effect of a change in the size of the hurricane eye. Definitely, ε = ε 0 (r m /r) should not be used for hurricanes of differing eye sizes because it would result in much less effective overall convection for hurricanes with smaller eyes than with larger eyes. For simplicity, when a difference in r m is involved, Ɣ = 1 + η(r m r c ) is used in (14). To be consistent with the previous discussion, r c of 30 km is used. In theory, such a η can be determined by using a single observation of potential intensity with r m r c. Figure 8 shows the dependence of potential intensity on hurricane eye and inner core sizes with η = km 1, which is equivalent to assuming any single intensity in Fig. 8 (i.e. 938 hpa with r m = 20 km and r 0 = 120 km). This result indicates potential intensity increases with decreasing hurricane eye size. In general, with the present choice of η, the intensity dependence on r m is weak except for hurricanes of very large sizes and small eyes (r m ). Apparently, increasing η leads to more (or less) effective kinetic energy generation away from the eyewall for hurricanes of smaller (or larger) eyes and thus a larger intensity increase with decreasing hurricane eye size. To have a sense for what kind of constraint by using η = km 1 was placed on the hurricane convection with regard to the size change of hurricane eye, we assume: (i) the surface entropy from the ocean is totally lost to the free atmosphere by local convection,

15 HURRICANE POTENTIAL INTENSITY 2643 Figure 9. Radial distribution of equivalent convection depth for radius of maximum wind = 20, 30 and 40 km (solid, dashed and dotted, respectively). Sea surface temperature is 300 K, and no CAPE is used. The entropy flux under the eye is assumed to be transported to and released in the eyewall where convection is assumed to reach the tropopause. (ii) an air parcel has negligible exchanges with its environment during vertical displacement, and (iii) the atmospheric lapse rate does not vary with altitude. Under these conditions, the thermodynamic efficiency, ε, is proportional to the depth of convection or cloud height so that the radial distribution of ε reflects that of the cloud depth. Note that assumption (i) may overestimate the convection depth (outside of the eyewall) since part of the local surface flux is, by inward advection, released with a higher thermodynamic efficiency. But one can view this overestimation as roughly offset by the underestimation due to assumption (ii). We then have Fig. 9 showing the radial distribution of equivalent convection depth for cases of r m = 20, 30 and 40 km. The current parameter choice still seems to impose too strong suppression on the local convection for hurricanes with smaller eyes. We suspect η = km 1 may still be a conservative choice but further discussion is not made due to the complicated nature and our limited understanding of this issue. Since we attained the result in Fig. 5 with the same hurricane eye size, and stronger hurricanes generally have smaller eyes, we recalculated the intensity dependence on SST and B by incorporating the variation of r m into hurricanes. In this calculation, a simple linear relationship between the central pressure and r m is used in attaining the potential intensity with central pressure ranging from 1000 to 900 hpa corresponding to r m from 60 to 10 km. For the central pressure beyond this range, r m remains unchanged. The so-obtained potential intensity denoted by central pressure is shown in Fig. 10. It is quite similar to that in Fig. 5 except for a slight intensity increase (or decrease) for larger (or smaller) storm sizes at higher (or lower) SSTs. In both Figs. 5 and 10, the inner core size can approximately be denoted by B but their relationship is somewhat changed due to the change of hurricane eye size in Fig. 10. Note that the non-smooth change in the upper left corner is due to the specific pressure r m relationship used where r m is constant for the central pressure smaller than 900 hpa.

16 2644 W. SHEN Figure 10. As Fig. 5, but using an intensity-dependent radius of maximum wind (see text for details). 5. CONCLUSIONS An energy balance model for a hurricane is proposed and used to investigate the dependence of potential intensity on hurricane size, CAPE and underlying SST. The model distinguishes itself from the previous methods in that the entire fluxes under the hurricane inner core and CAPE are accounted for in the energy balance. In this model, the Holland (1980) radial profiles of surface wind, which allow storm size and shape changes, are used. A hurricane is conceptually divided into two areas: an inner core defined as the inner area of upward large-scale motion and a broad outer core of subsidence. In the outer core, we assume the sounding and the mixing ratio in the surface boundary layer are the same as those in the environment. Under these conditions, the kinetic energy generation due to convective activities in the outer core is neglected and the kinetic energy generation due to CAPE release in the inner core is estimated based on the environmental conditions and the boundary layer mass convergence from the outer core into the inner core. Since local convection away from the eyewall is less effective regarding energy conversion, a reduction of thermodynamic efficiency is used, which is also a theoretical need to bound the intensity for large size hurricanes. The efficiency reduction thus makes the kinetic energy generation due to surface entropy flux be confined to the eyewall but it is still less confined than the surface dissipation. This is because the hurricane potential intensity would otherwise be equal to or lower than that via a simple local balance under the eyewall which can lead to unreasonably low potential intensities. Hence, for a steady hurricane, the net kinetic energy generation away from the eyewall is balanced by the net sink around the eyewall. Since a larger hurricane has larger kinetic energy generation away from the eyewall due to its less confined surface wind and size, it demands larger dissipation around the eyewall for a balance and thus larger maximum surface wind or intensity. Therefore, the less confined kinetic energy generation (relative to the surface dissipation) is the key for understanding the size-dependence in this study: the larger the storm size, the higher the storm potential intensity. But the quantitative significance of such a size effect is beyond this paper due to the current uncertainties in treating the radial profile of the thermodynamic efficiency. The results of this model indicate that CAPE as another energy source may appreciably affect the maximum achievable intensity. Although CAPE of possibly realistic

17 HURRICANE POTENTIAL INTENSITY 2645 values only creates a small amount of kinetic energy compared to that by surface entropy flux, it may make the steady state considerably different from the otherwise surface entropy flux-related generation and surface dissipation balanced state. Under the current framework, the CAPE impact on hurricane intensity can be significant with a CAPE value of the order of 1000 J kg 1, particularly for hurricanes of small sizes. Although hurricane development and genesis is not a focus of the current study, the results imply that CAPE existence may be critical for energy creation to overcome the dissipation at early stages. With this model, the hurricane potential intensity is less sensitive to the SST than was suggested by previous MPI models. The current results indicate that once a hurricane is formed, it can reach a considerable magnitude even over cold oceans of SST near 20 C or below. This may be the reason why some individual storms were able to develop to or maintain a considerable magnitude over cold oceans at middle and high latitudes. We suspect that the general adverse environment in the extratropical regions may have helped to shape the statistical picture of hurricane confinement to the low latitudes. ACKNOWLEDGEMENTS The author wishes to express his thanks to the two anonymous reviewers for their insightful suggestions/comments that led to substantial improvement of the paper. The author also wishes to thank Dr Xiaofang Li of NESDIS and Dr Zavisa Janjic of NCEP for their helpful comments on the initial manuscript. APPENDIX Kinetic energy generation due to CAPE To estimate the CAPE contribution, we consider a hurricane as an axisymmetric eddy on a resting mean state so that w = v = 0. This implies that the eddy kinetic energy is the total kinetic energy in this case. As long as the entire hurricane system is considered, the energy conversion from available potential energy dominates the kinetic energy generation in the hurricane. This available potential energy in the hurricane can be due to both environmental CAPE and the underlying surface entropy flux. The kinetic energy generation can be approximately represented by the following (i.e. with Eq. (9) of Li et al. (2002), neglecting the kinetic energy change through the covariance between eddy vertical velocity and cloud mixing), K G = ν ρg T T w dν, (A.1) where ν denotes the integration (by volume) for the entire hurricane and T is the (virtual) temperature deviation of the lifted air parcel from the mean (T ) or approximately that of the large-scale environment, g is the gravity and w = dz/dt is the vertical velocity. Note that (A.1) is actually true for hurricane cases without vertical shear of mean flow, although resting mean state was assumed. Using P = ρrt and dp = ρg dz, (A.1) can be written as r0 { pb } K G = 2π ρrt w d(ln p) r dr, (A.2) r m p o where p b is the pressure in the surface boundary layer, p o is the pressure at the outflow level, and R the gas constant. r 0 is used instead of r a because the outer core has the

18 2646 W. SHEN environmental sounding. T can be written as (T T p ) + (T p T e ) where T is the temperature of an air parcel lifted from the surface boundary layer, T p is the air parcel temperature without the entropy addition due to the surface entropy flux (or that of an air parcel lifted from the surface boundary layer in the environment), and T e the environmental sounding temperature. For a neutral atmospheric environment, T p = T e and the temperature difference (T T e ) is due solely to the entropy increase in the surface boundary layer caused by the surface entropy flux. In this case, K G = K sflx by (12). In the presence of CAPE, (A.2) can be written as r0 { pb } K G = K sflx + 2π ρr(t p T e )w d(ln p) r dr. (A.3) r m p o In the case of ε<ε 0,p o can be viewed as the pressure at the equivalent outflow level (see its definition in section 2) so that T p follows the wet adiabat and K sflx in (A.3) fits in its original definition. We now assume the boundary layer entropy increase (towards eyewall) is barely affected by the CAPE existence given the storm intensity and size. Thus, K sflx can be calculated independently of CAPE except that the outflow level and thus the thermodynamic efficiency, ε, may be somewhat higher. We now use ε = ε + ε where ε refers to the efficiency in the case of no CAPE. To estimate the kinetic energy generation in the presence of CAPE, we first estimate its maximum potential with ε 0 everywhere. The potentials of the two terms in (A.3) can be written as r0 { pb } (K sflx ) pot = 2π ρr(t T p )w d(ln p) r dr r m p tro r0 = 2πρc r ε 0 T s C h V g (S s S a )r dr (A.4) r m and r0 { pb } (K CAPE ) pot = 2π ρr(t p T e )w d(ln p) r dr, (A.5) r m p tro where p tro is the pressure at the level where an adiabatically lifted air parcel loses buoyancy (usually near the tropopause). Since the horizontal convergence between the surface boundary layer and the top outflow layer (which are thin compared to the whole depth) is small, ρw can approximately be treated as a constant in (A.4). Hence, using (A.5) becomes where CAPE = R pb p tro (T p T e ) d(ln p), (K CAPE ) pot. = (CAPE)Fm (A.6) r0 F m = 2πρ b w t r dr is the total vertical mass flux at the top of the surface boundary layer. Using mass conservation, (A.6) can be written as the following: r m (K CAPE ) pot. = 2πρb r 0 D b V n r 0 (CAPE), (A.7)