Alternative expression for the maximum potential intensity of tropical cyclones
AAlternative expression for the maximum potential intensity oftropical cyclones
Anastassia M. Makarieva and Andrei V. Nefiodov Theoretical Physics Division, Petersburg Nuclear Physics Institute, Gatchina 188300, St. Petersburg, Russia
Correspondence to:
A. M. Makarieva ([email protected])
Abstract.
Emanuel’s Maximum Potential Intensity (E-PI) derives the maximum velocity of tropical cyclones from environ-mental parameters based on distinct sets of assumptions applied to the upper atmosphere, the boundary layer and the air-seainterface. At the top of the boundary layer at the radius of maximum wind E-PI relates the centrifugal acceleration (squaredmaximum velocity divided by radius) to the radial gradient of saturated moist entropy. The proportionality coefficient equalsthe difference between air temperatures in the outflow region and at the top of the boundary layer. We show that a different re-lationship between the same quantities derives straightforwardly from the definition of saturated moist entropy and the gradientwind balance. Here the proportionality coefficient depends on the local mixing ratio of water vapor and the degree of adiabatic-ity of the radial gradient of air temperature. The robust alternative relationship reveals that when, as originally assumed in theE-PI derivation, the air is horizontally isothermal, E-PI at the top of the boundary layer underestimates the squared maximumvelocity by approximately twofold. This provides an explanation to the “superintensity” phenomenon (maximum wind speedsexceeding E-PI). The discrepancy increases (diminishes) when the air temperature at the point of maximum wind declines(grows) towards the storm center. The established theoretical relationships are illustrated with the data for Hurricane Isabel2003 and their implications for assessing the maximum intensity of tropical storms are discussed.
Tropical storms threaten human lives and livelihoods. Numerical models can simulate a wide range of storm intensities underthe same environmental conditions (e.g., Tao et al., 2020). Thus it is desirable to have a reliable theoretical framework thatwould, from the first principles, confine model outputs to the reality domain (Emanuel, 2020). The theoretical formulation formaximum potential intensity of tropical cyclones by Emanuel (1986) (E-PI) has been long considered as a robust upper limiton storm intensity (see discussions by Garner (2015), Kieu and Moon (2016) and Kowaleski and Evans (2016)). At the sametime, the phenomenon of “superintensity”, when the observed or modelled storm velocities exceed E-PI, has been perceivedas an important research challenge (e.g., Persing and Montgomery, 2003; Montgomery et al., 2006; Bryan and Rotunno, 2009;Rousseau-Rizzi and Emanuel, 2019; Li et al., 2020). Since the strongest storms are the most dangerous ones, it is important tounderstand when and why the theoretical limits can be exceeded. The principal way of approaching the superintensity problemwas to reveal how the E-PI assumptions can be modified to yield greater intensities. For example, Montgomery et al. (2006)suggested that superintensity could result from an additional heat source provided by the storm eye (a source of energy not1 a r X i v : . [ phy s i c s . a o - ph ] J a n onsidered in E-PI). Bryan and Rotunno (2009) investigated how superintensity could result from the flow being supergradient(while E-PI assumed the gradient wind balance). For a recent overview of superintensity assessments in modelling studies seeRousseau-Rizzi and Emanuel (2019).Here we present a different approach. We show that, even in the case when all the E-PI assumptions hold, E-PI will system-atically underestimate storm intensitites provided the air is horizontally isothermal at the point of maximum wind. Horizontalisothermy of the top of the boundary layer was assumed by Emanuel (1986) in his original derivation of E-PI. As we willdiscuss, it is indeed a physically plausible assumption.That under horizontal isothermy E-PI underestimates storm intensity, follows straightforwardly from the definition of sat-urated moist entropy. At the point of maximum wind E-PI relates the radial gradients ∂s ∗ /∂r and ∂p/∂r of saturated moistentropy s ∗ and air pressure p via an external parameter (the outflow temperature T o ). However, s ∗ being a state variable, itsradial gradient is a local function of the radial gradients of air pressure p and temperature T . Thus, specifying a relationshipbetween ∂s ∗ /∂r and ∂p/∂r uniquely sets ∂T /∂r . Conversely, setting ∂T /∂r = 0 relates ∂s ∗ /∂r and ∂p/∂r in a specific waythat, under common atmospheric conditions, is shown here to be incompatible with E-PI.We derive the alternative relationship between the radial gradient of saturated moist entropy and maximum velocity inSection 2, illustrate the obtained relationships with the data for Hurricane Isabel 2003 in Section 3 and discuss their implicationsfor assessing the intensity limits of tropical storms in Section 4. E-PI has three blocks, with distinct set of assumptions applied to each block: the upper atmosphere including the top of theboundary layer, the interior of the boundary layer and the ocean-atmosphere interface. Here we focus on the first block.For the upper atmosphere, the key relationship of E-PI is between saturated moist entropy s ∗ and angular momentum M (for a compact derivation see Emanuel and Rotunno, 2011, Eq. 13): − ( T − T o ) ds ∗ dM = vr + f , z ≥ z b , (1)where M = vr + 12 f r , (2) z b is the height of the boundary layer, r is the distance from the storm center, T is the local air temperature, T o corresponds to r → ∞ on a streamline defined by a given value of ds ∗ /dM , v is the tangential velocity. The Coriolis parameter f ≡
2Ω sin ϕ is assumed constant ( ϕ is latitude, Ω is the angular velocity of Earth’s rotation). For the definition of saturated moist entropysee Eq. (A.2) in the Appendix.Relationship (1) derives from the assumptions of hydrostatic and gradient wind balance and from the statement that surfacesof constant s ∗ and M coincide (Emanuel, 1986; Emanuel and Rotunno, 2011). Using the definition of angular momentum (2),2q. (1) can be re-written as follows (see Emanuel, 1986, Eq. 13): ( T − T o ) ∂s ∗ ∂r = − Mr ∂M∂r . (3)At the point ( r m , z m ) of maximum wind, provided that f (cid:28) v m /r m and ∂v/∂r (cid:28) v m /r m , Eq. (3) becomes (see, e.g., Belland Montgomery, 2008, their Eq. 5 and our Eq. (A.1) in the Appendix): εT b ∂s ∗ ∂r = − v m r m , (4)where ε ≡ ( T b − T o ) /T b is the Carnot efficiency. E-PI assumes that the point of maximum wind is located at the top of theboundary layer, i.e. T = T b at ( r m , z m ) , where z m = z b (Emanuel, 1986; Emanuel and Rotunno, 2011).In gradient wind balance, α ∂p∂r = v r + f v, (5)where p is air pressure and α ≡ /ρ is inverse air density, Eq. (4), with f v neglected, becomes εT b ∂s ∗ ∂r = − α ∂p∂r . (6)We recapitulate that Eq. (6) is valid at the point of maximum wind under the same assumptions as does Eq. (1). Since saturated moist entropy s ∗ is a state variable, its radial gradient can be expressed in terms of the radial gradients of airpressure and temperature (see Eq. (A.9) in the Appendix):
11 + ζ T ∂s ∗ ∂r = − α d ∂p∂r (cid:18) − ∂T /∂r∂p/∂r (cid:19) , (7)where ζ ≡ Lγ ∗ d / ( RT ) , R = 8 . J mol − K − is the universal gas constant, γ ∗ d ≡ p ∗ v /p d , p ∗ v is the saturated water vapor pressure, p d is the partial pressure of dry air, L ≈ kJ mol − is the latent heat of vaporization, Γ (K Pa − ) is the moist adiabatic lapserate of air temperature (see its definition (A.10) in the Appendix), α d ≡ /ρ d is the inverse dry air density.Equation (7) does not contain any assumptions but follows directly from the definition of saturated moist entropy. Beinggenerally valid, Eq. (7) can be used to examine the validity of E-PI’s Eq. (6). For simplicity, taking into account that q ∗ ∼ − (cid:28) , we assume in Eq. (7) α d = (1 + q ∗ ) α ≈ α . Referring to observations and to Frank (1977), Emanuel (1986, p. 588) in his original derivation of E-PI assumed the top of theboundary layer to be isothermal. With ∂T /∂r = 0 , the expression in braces in the right-hand side of Eq. (7),
C ≡ − ∂T /∂r∂p/∂r , (8)3
88 293 298 303 308 313 T b H K L H a L P a P a - o C - o C - o C ¶ (cid:144) H + Ζ L
288 293 298 303 308 313 T H K L - - - - - - H b L iiiiii
12 Sep13 Sep14 Sep G d H K hPa - M G H K hPa - M Figure 1.
Parameters ε versus / (1 + ζ ) (a) and moist Γ and dry Γ d adiabatic temperature gradients (b) as dependent on temperature. The ε ≡ ( T b − T o ) /T b curves correspond to different outflow temperatures T o ; the / (1 + ζ ) curves correspond to p d values of and hPa,see Eq. (A.10); Γ and Γ d are calculated for p = 850 hPa. In (b), the circles indicate the mean temperature gradients (K hPa − ) observed inHurricane Isabel 2003 on September 13 between the eyewall and the outer core at the surface (i) and at the top of the boundary layer (ii)and between the eye and the eyewall at the top of the boundary layer (iii); the squares indicate the local temperature gradients (K hPa − )at the point of maximum wind calculated from Eq. (11) and the data of Table 1 for Hurricane Isabel 2003 on 12, 13 and 14 September. SeeSection 3.2 for calculation details. equals unity. In this case E-PI’s Eq. (6) can only be valid if ε = 11 + ζ . (9)In the terrestrial atmosphere such conditions appear to be rare (Fig. 1a). The maximum Carnot efficiency estimated from thetemperatures observed in the outflow and at the top of the boundary layer is ε = 0 . (DeMaria and Kaplan, 1994). Theminimum value of / (1 + ζ ) = 0 . is . -fold larger. It corresponds to the largest γ ∗ d ≈ . for T b = 303 K ( °C) and p d ≈ p = 800 hPa.The partial pressure p ∗ v of saturated water vapor and, hence, γ ∗ d depend exponentially on air temperature. The realistictemperatures at the top of the boundary layer are commonly significantly lower than K. Thus, the discrepancy betweenE-PI and Eq. (7) should be commonly significantly higher (Fig. 1a).In the general case, instead of the gradient wind balance (5), we can write α ∂p∂r ≡ B v r , (10)where B defines the degree to which the flow is radially unbalanced: B < for the supergradient flow when the outward-directed centrifugal force is larger than the inward-pulling pressure gradient. For example, in Hurricane Isabel 2003 on Septem-ber 13 the tangential wind was estimated at the top of the boundary layer as being supergradient (Bell and Montgomery,4008). This corresponds to B = 1 / (1 . ≈ . . For the supergradient wind in the numerical experiment of Bryan and Ro-tunno (2009, Fig. 8), B ≈ . . Note that at the point of maximum wind the vertical and radial velocities are usually muchsmaller than the tangential velocity, so v ≈ V , where V is total air velocity.Using the definitions of C (8) and B (10), Eq. (7) can be written as ζ ) T BC ∂s ∗ ∂r = − v r . (11)This equation has a general validity. Its comparison with Eq. (4) reveals that the flow being supergradient ( B < ) and airtemperature declining towards the storm center ( C < ) cause E-PI to underestimate v m more than it does in the radially bal-anced isothermal case ( B = 1 , C = 1 ). Conversely, for E-PI’s Eq. (4) to be consistent with observations for a radially balanced( B = 1 ) or supergradient ( B ≤ ) flow, the temperature at the point of maximum wind must increase towards the hurricanecenter ( C > ). When, as is the case in the stronger storms, the pressure gradient is sufficiently steep and the radial motion sufficiently rapid, theradial expansion of air is accompanied by a drop of temperature. In the well-studied Hurricane Isabel 2003 the surface air cooledby about 4 K while moving from the outer core (150-250 km) to the eyewall (40-50 km) (Montgomery et al., 2006, Fig. 4c).Over the same distance, the surface pressure fell by less than about 50 hPa (from less than 1013 hPa to 960 hPa) (Aberson et al.,2006, Fig. 4). (Air pressure at the outermost closed isobar ∼ km from the center was 1013 hPa, hence at 150-250 km fromthe center it should have been smaller.) With ∆ p ≈ − hPa and ∆ T ≈ − K, at T = 297 K and p ≈ hPa, the horizontaltemperature gradient at the surface ∆ T / ∆ p = 0 . K hPa − approaches the dry adiabatic gradient Γ d = µT /p = 0 . K hPa − (see Fig. 1b).At the top of the boundary layer the radial flow is weaker than it is on the surface, and the mean horizontal temperaturegradient is smaller. At the level of maximum wind z m = 1 km in Hurricane Isabel 2003 the temperature difference betweenthe eyewall and the outer core was ∆ T ≈ − K (Montgomery et al., 2006, Fig. 4c). Assuming that the pressure difference atthis level is about . of its value at the surface, ∆ p ≈ − . × hPa, we have ∆ T / ∆ p = 0 . K hPa − . The mean horizontaltemperature gradient between the outer core and the eyewall at the top of the boundary layer approaches the moist adiabaticgradient Γ = 0 . K hPa − for T = 293 K and p = 850 hPa (Fig. 1b).In the eye, the sunlight and the descending air motion work to elevate the air temperature above that at the eyewall andin the outer core. The air temperature in the eye rises towards the storm center and ∂T /∂r < . For Hurricane Isabel 2003,with pressure and temperature differences at z m = 1 km between the eye and the eyewall ∆ p ≈ − hPa (Aberson et al., 2006,Fig. 4) and ∆ T = 3 K (Montgomery et al., 2006, Fig. 4c), for p = 850 hPa and T = 293 K we have ∆ T / ∆ p = − . K hPa − ≈− . (Fig. 1b).That the horizontal temperature gradient changes its sign somewhere in the eyewall suggests that ∂T /∂r = 0 at the point ofmaximum wind is a plausible assumption. However, the magnitudes of horizontal temperature gradients on both sides of theeyewall are large enough to significantly impact the maximum velocity estimates (Fig. 1b).5or example, if at the point of maximum wind the horizontal temperature gradient were close to Γ (as it was on averagebetween the eyewall and the outer core in Hurricane Isabel 2003), then C → and BC (1 + ζ ) ε → . In this case E-PI wouldformally infinitely underestimate v m . Physically, this limit corresponds to the situation when the moist adiabat is locallyhorizontal, ∂s ∗ /∂r → , such that the dependence between saturated moist entropy and maximum velocity vanishes, see inEq. (11).If, on the other hand, at the point of maximum wind the horizontal temperature gradient were equal to − . (as it was onaverage between the eye and the eyewall in Hurricane Isabel 2003), then C = 3 . and E-PI’s Eq. (4) would overestimate v m by BC (1 + ζ ) ε = 1 . -fold for a balanced flow ( B = 1 ). With B = 0 . as discussed above, the overestimate reduces to . (about ).For a known B , the value of C can be derived from the observed values of variables entering Eq. (11). The data for HurricaneIsabel 2003 suggest that at the point of maximum wind the air temperature increases towards the center C > , but not enoughto bring E-PI in agreement with observations: on September 12 and 14, E-PI underestimates v m by about and ,respectively (Table 1).The closest agreement is observed on September 13, when C is the largest (Fig. 1b). Given that the flow at this date wassupergradient with B ≈ . (Bell and Montgomery, 2008), this agreement does not indicate that the storm is in thermal windbalance (cf. Montgomery et al., 2006, p. 1345). Rather, it suggests that the large value of C = 2 . nearly compensated theunderestimate that would have otherwise resulted from B < . The underestimate is greatest on September 12, when the localtemperature gradient is closest to zero (Fig. 1b) and C is close to unity (Table 1). Table 1.
Parameters of Eqs. (11) and E-PI’s Eq. (4) estimated from observations for Hurricane Isabel 2003.Date r m , km T o , °C θ e , K ∂θ e /∂r , K km − ˜ v m , m s − v m , m s − ε (1 + ζ ) ε ( v m / ˜ v m ) C
12 September −
65 360 − . .
29 0 .
43 0 .
46 1 .
13 September −
58 357 − . .
27 0 .
40 0 .
96 2 .
14 September −
56 357 − .
35 74 61 0 .
26 0 .
39 0 .
68 2 . Notes. Observed values of r m , T o , ˜ v m and ∂θ e /∂r are taken from, respectively, the first and the third columns of Table 2, and equivalentpotential temperature θ e from Fig. 5, of Bell and Montgomery (2008). At the top of the boundary layer, temperature T b = 293 K (20 °C)is assumed for all the three days based on Fig. 4c of Montgomery et al. (2006), ζ = Lγ ∗ d /RT b = 0 . for p d = 850 hPa and p ∗ v = 23 hPa.The values of v m are E-PI estimates of maximum velocity obtained from Eq. (4), where ∂s ∗ /∂r = ( c p /θ e ) ∂θ e /∂r , c p = 1 kJ kg − K − (see Montgomery et al., 2006, Eq. A2). Factor (1 + ζ ) ε < , see Eq. (9), indicates by how much E-PI’s Eqs. (6) and (4) would underes-timate the squared maximum velocity under isothermal conditions in gradient wind balance; C is estimated from Eq. (11) with B = 0 . , v = ˜ v m and r = r m . Conclusions
In intense tropical cyclones, the radial air inflow significantly impacts the radial temperature gradient, which may change itssign in the vicinity of maximum wind. Thus, assuming that at the point of maximum wind the top of the boundary layeris horizontally isothermal appears justified. Here we showed that in this case one of the key relationships of E-PI, Eq. (4),underestimates the maximum wind speed by about twofold under common atmospheric conditions. This conclusion followsrobustly from the definition of saturated moist entropy.For E-PI’s Eqs. (4) and (6) to conform to observations, the air temperature at the point of maximum wind must rise consid-erably towards the storm center. Such a pattern is possible if the storm is sufficiently intense to have a warm eye (where thehigher temperature is ensured by the descending air motion and the incoming solar radiation under clear sky conditions). Weconsidered this feature for Hurricane Isabel 2003 (Table 1). Our conclusion, that the warm eye can reduce the underestimateand drive the E-PI estimate up and closer to observations, runs somewhat counter to the suggestion of Bell and Montgomery(2008) that the extra heat from the eye may be the cause of E-PI underestimating the actual velocity. Without a warm eye (andhence without C > ), the E-PI’s relationship at the top of the boundary layer would have underestimated the storm intensityeven more significantly.In the weaker cyclones without a well-defined eye, the existence of a pronounced temperature surplus at the storm centermight be less likely. In such cyclones, E-PI at the top of the boundary layer should considerably underestimate maximumvelocity (Fig. 1). E-PI in the weaker cyclones providing, instead, an upper limit on their intensities, indicates that a certainovercompensation occurs in the assumptions pertaining to the remaining two E-PI blocks, the boundary layer interior and theair-sea interface (Table 2, second and third rows).In particular, Bryan and Rotunno (2009) showed that the E-PI key relationship for the boundary layer interior, that ds ∗ /dM is equal to the ratio τ s /τ M of the surface fluxes of entropy and momentum, overestimates the actual ds ∗ /dM ratio by up to . This overestimate of ds ∗ /dM by τ s /τ M can partially compensate the underestimate of maximum velocity by E-PI’sEqs. (1) and (4) (see also Table 2, first row).Another compensating overestimate results from E-PI’s assumptions concerning the disequlibrium ∆ k = k ∗ s − k = c p ∆ T + L v ∆ q at the air-sea interface at the radius of maximum wind (Table 2, second row). Since the local enthalpy difference ∆ k is unknown, E-PI limits it from above by assuming that the local difference in mixing ratios ∆ q is less than the water vapordeficit (1 − H a ) q ∗ sa in the ambient environment (Table 2, third row).However, as Emanuel (1986) and Emanuel and Rotunno (2011) pointed out, in reality ∆ k tends to decline from the outercore towards the storm center. Indeed, if the radial inflow is sufficiently slow, as it is the case in the weaker storms, the surfaceair can remain in approximate thermal equilibrium with the oceanic surface. In his original evaluations of E-PI Emanuel (1986,p. 591) assumed ∆ T = 0 . On the other hand, evaporation into the air parcels that are spiraling inward increases the relativehumidity and diminishes ∆ q . In the result, in the weaker cyclones the actual ∆ k at the radius of maximum wind can be muchlower than its ambient constraint (1 − H a ) q ∗ sa . This would overcompensate the underestimate of v m at the top of the boundary7 able 2. Three logical blocks of E-PI.Atmospheric region Assumptions Key relationship ReferencesUpper atmosphereand the top ofboundary layer( z ≥ z b ) The air is in hydrostatic and gradient wind balance; sur-faces of constant saturated moist entropy s ∗ and angularmomentum M coincide − ( T − T ) ds ∗ dM = v r − v r Emanuel and Rotunno(2011, Eq. 11)Boundary layernear the radius ofmaximum wind( < z ≤ z b ) Horizontal turbulent fluxes of s ∗ and M are negligiblecompared to vertical ones; surfaces of constant s ∗ and M are approximately vertical; turbulent fluxes of s ∗ and M vanish at z = z b ds ∗ dM = τ s τ M = C k C D k ∗ s − kT s rv Emanuel (1986,Eqs. 32, 33), Emanueland Rotunno (2011,Eqs. 17, 19, 20)Air-sea interfacenear the radius ofmaximum wind The upper limit for the air-sea disequilibrium is set by theambient relative humidity H a q ∗ s − q (cid:46) (1 − H a ) q ∗ sa (Emanuel, 1995,p. 3971), Emanuel(1989, Eq. 38)Final E-PI estimate For ε ≈ . , C k /C D ≈ , H a = 0 . and T s = 300 K, v m (cid:46) m s − v m (cid:46) ε C k C D L v (1 − H a ) q ∗ sa Emanuel (1989, Eq. 38and Table 1)Notes. In the first row, T , T and v , v are, respectively, air temperatures and tangential wind speeds at arbitrary points r and r on a stream-line defined by the given value of ds ∗ /dM . In E-PI, r is chosen at the top of the boundary layer ( z = z b ) near the point r = r m of maximumwind, r is chosen in the outflow in the upper atmosphere, such that either v = 0 or r → ∞ and v /r → , see Eq. (1). In the second row, τ s and τ M are the surface fluxes of, respectively, entropy and angular momentum, C k and C D are exchange coefficients for enthalpy and momen-tum, r is local radius, k ∗ s is saturated enthalpy at sea surface temperature T s , k ∗ s − k = c p ( T s − T ) + L v ( q ∗ s − q ) , where c p is the specific heatcapacity of air at constant pressure, L v is the latent heat of vaporization, q ∗ s is the saturated mixing ratio at T s , v , k , q and T are the tangentialwind, enthalpy, water vapor mixing ratio and air temperature at a reference height (usually about 10 m above the sea level). In the third row, H a and q ∗ sa are the relative humidity and saturated mixing ratio at the surface temperature in the ambient environment outside the storm core. layer by E-PI’s Eq. (4) and explain why in many cases the E-PI final expression (Table 2, forth row) goes above the observedmaximum velocities.In the stronger storms like hurricanes, however, the air streams so quickly towards the center that it cools significantlycompared to the isothermal oceanic surface it moves above. As discussed by Camp and Montgomery (2001) and Montgomeryet al. (2006), this cooling tends to offset the increase in relative humidity, such that the mixing ratio q does not considerablygrow, and ∆ q does not diminish significantly, towards the center. In this case the E-PI’s assumption, ∆ q ≈ (1 − H a ) q ∗ sa ,becomes valid. No overcompensation occurs in the third block of E-PI. As a result, in the strongest storms the underestimatestemming from the first block of E-PI becomes explicit.On the other hand, that the final E-PI expression (Table 2, forth row) produces a plausible if not 100% robust upper limit onmaximum intensity (despite deriving from assumptions that systematically underestimate and overestimate intensities), can beexplained by the fact that the quantitative parameters in the final expression for kinetic energy incidentally combine into thepartial pressure of water vapor (see Eqs. (A.12)–(A.14) in the Appendix). Partial pressure of water vapor has been suggested8o be the key parameter of hurricane dynamics (Makarieva et al., 2014). Its exponential temperature dependence would explainthat of E-PI.The alternative relationship between the radial gradient of saturated moist entropy and maximum velocity, Eq. (11), revealsthe profound influence of the radial temperature gradient on storm intensity. While until now the radial temperature gradientat the top of the boundary layer has not received much attention in the assessments of E-PI’s validity (e.g., Montgomery et al.,2006; Bryan and Rotunno, 2009; Emanuel and Rotunno, 2011), our results highlight the need to pay more attention to theradial change of air temperature in further studies. Appendix: Details of derivations
Using the definition of angular momentum M (2), we obtain Mr ∂M∂r = (cid:18) vr + f (cid:19) (cid:18) v + f r + r ∂v∂r (cid:19) = v r + f v + v ∂v∂r + f (cid:18) v + f r + r ∂v∂r (cid:19) = α ∂p∂r (cid:18) f r v (cid:19) . (A.1)The last equality assumes the condition ∂v/∂r = 0 (the point of maximum wind) and the gradient wind balance (5).Saturated moist entropy is defined as (see Pauluis, 2011, Eq. A4) s ∗ = ( c pd + q t c l ) ln TT − RM d ln p d p + q ∗ L v T . (A.2)Here, L v = L v + ( c pv − c l )( T − T ) is the latent heat of vaporization (J kg − ); q ∗ = ρ ∗ v /ρ d , q l = ρ l /ρ d , and q t = q ∗ + q l arethe mixing ratio for saturated water vapor, liquid water, and total water, respectively; ρ d , ρ ∗ v , and ρ l are the density of dry air,saturated water vapor and liquid water, respectively; c pd and c pv are the specific heat capacities of dry air and water vapor atconstant pressure; c l is the specific heat capacity of liquid water; R = 8 . J mol − K − is the universal gas constant; M d isthe molar mass of dry air; p d is the partial pressure of dry air; T is the temperature; p and T are reference air pressure andtemperature.From Eq. (A.2) we have T ds ∗ = ( c pd + q t c l ) dT − RTM d dp d p d + L v dq ∗ + q ∗ dL v − q ∗ L v dTT = (cid:18) c p − q ∗ L v T (cid:19) dT − RTM d dp d p d + L v dq ∗ , (A.3)where c p ≡ c pd + q ∗ c pv + q l c l and dL v = ( c pv − c l ) dT . Equation (A.3) assumes q t = const (reversible adiabat).The ideal gas law for the partial pressure p v of water vapor is p v = N v RT, N v = ρ v M v , (A.4)where M v and ρ v are the molar mass and density of water vapor. Using Eq. (A.4) with p v = p ∗ v in the definition of q ∗ q ∗ ≡ ρ ∗ v ρ d = M v M d p ∗ v p d ≡ M v M d γ ∗ d , γ ∗ d ≡ p ∗ v p d , (A.5)9nd applying the Clausius-Clapeyron law dp ∗ v p ∗ v = LRT dTT , L ≡ L v M v , (A.6)we obtain for the last term in Eq. (A.3) L v dq ∗ = L v M v M d (cid:18) dp ∗ v p d − p ∗ v p d dp d p d (cid:19) = L v M v M d (cid:18) p ∗ v p d dp ∗ v p ∗ v − p ∗ v p d dp d p d (cid:19) = L v q ∗ (cid:18) LRT dTT − dp d p d (cid:19) . (A.7)Using the Clausius-Clapeyron law (A.6), the ideal gas law p d = N d RT , where N d = ρ d /M d , and noting that p = p ∗ v + p d ,we obtain for the last but one term in Eq. (A.3) RTM d dp d p d = RTM d (cid:18) dpp d − dp ∗ v p d (cid:19) = dpM d N d − RT p ∗ v M d p d dp ∗ v p ∗ v = dpρ d − L v M v p ∗ v M d p d dTT = dpρ d − q ∗ L v dTT . (A.8)Putting Eqs. (A.7) and (A.8) into Eq. (A.3) yields T ds ∗ = (cid:18) c p + L v q ∗ T L (1 + γ ∗ d ) RT (cid:19) dT − (cid:18) Lγ ∗ d RT (cid:19) dpρ d = − (1 + ζ ) α d dp (cid:18) − dTdp (cid:19) . (A.9)Here Γ ≡ α d c p ζ µζ ( ξ + ζ ) , ξ ≡ LRT , ζ ≡ ξγ ∗ d , µ ≡ RC p = 27 , (A.10)where α d ≡ /ρ d is the volume per unit mass of dry air and C p = c p M d is the molar heat capacity of air at constant pressure.Approximating air molar mass by molar mass M d of dry air and c p by c pd , we can conveniently express Γ as Γ ≈ Tp µ (1 + ζ )1 + µζ ( ξ + ζ ) ≈ Tp µ (1 + ξγ ∗ d )1 + µξ γ ∗ d . (A.11)Using Eq. (A.5) and q ≡ H q ∗ , where H ≡ p v /p ∗ v is relative humidity, the final E-PI expression for v m (Table 2, forth row)can be re-written as follows: v m ∼ (cid:18) ε C k C D LRT s − H a H a (cid:19) p vsa ρ sa . (A.12)Here p vsa = H a p ∗ vsa is the actual partial pressure of water vapor in surface air in the ambient environment, p ∗ vsa is the partialpressure of saturated water vapor at sea surface temperature in the ambient environment, ρ sa is ambient air density at thesurface. Using typical tropical values T s = 300 K, H a = 0 . , ρ sa ≈ . kg m − , ε = 0 . (Table 1 of Emanuel, 1989) and C k /C D = 1 , we have p vs = 28 hPa and v m ≈ m s − in agreement with Table 1 of Emanuel (1989).The coefficient in parentheses in Eq. (A.12) for the same typical parameters is close to unity and depends only weakly on airtemperature: ε C k C D LRT s − H a H a ≈ . (A.13)This means that numerically the scaling of maximum velocity in E-PI practically coincides with the scaling ρ sa v m p vsa (A.14)10roposed within the concept of condensation-induced atmospheric dynamics (for a more detailed discussion see Makarievaet al., 2019, Section 5). Introducing the dissipative heating leads to additional factor / (1 − ε ) ∼ in Eq. (A.13) and the finalE-PI expression for v m (Table 2, forth row) (for a discussion, we refer to Makarieva et al., 2010; Bister et al., 2011; Kieu,2015; Bejan, 2019; Makarieva et al., 2020; Emanuel and Rousseau-Rizzi, 2020) . Emanuel and Rousseau-Rizzi (2020) equate quantities of different dimensions (see their Eq. (6) and related). 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