Water Lifting and Outflow Gain of Kinetic Energy in Tropical Cyclones

Anastassia M. Makarieva; Victor G. Gorshkov; Andrei V. Nefiodov; Alexander V. Chikunov; Douglas Sheil; Antonio Donato Nobre; Paulo Nobre; Günter Plunien; Ruben D. Molina

doi:10.1175/JAS-D-21-0172.1

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Fig. 1.

Air streamlines (solid and dotted black arrows) and the infinitely narrow thermodynamic cycle (thick pink lines) considered in the text (cf. Fig. 1b of Makarieva et al. 2020). The z and r axes correspond to the altitude above the sea level and the distance from hurricane center, respectively; r_b′ = r_B′, r_c = r_C′. Points B and B′ are infinitely close and chosen in the vicinity of maximum wind. The atmosphere is inviscid for z > z_b = z_B′.

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Fig. 2.

Key steps of deriving the relation between turbulent dissipation and heat input in the lower atmosphere. The integrals over a closed contour refer to B′bcC′B′ in Fig. 1. The heat input δQ_d is normalized per unit dry air mass.

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Fig. 3.

Main assumptions (green rounded boxes) and results (blue rectangular boxes) of this work with comments (dotted boxes).

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Editorial Type:: Article

Article Type:: Research Article

Water Lifting and Outflow Gain of Kinetic Energy in Tropical Cyclones

Anastassia M. Makarieva

Anastassia M. Makarieva ^aTheoretical Physics Division, Petersburg Nuclear Physics Institute, Saint Petersburg, Russia
^bInstitute for Advanced Study, Technical University of Munich, Garching, Germany

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Victor G. Gorshkov

Victor G. Gorshkov ^aTheoretical Physics Division, Petersburg Nuclear Physics Institute, Saint Petersburg, Russia

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Andrei V. Nefiodov

Andrei V. Nefiodov ^aTheoretical Physics Division, Petersburg Nuclear Physics Institute, Saint Petersburg, Russia

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Alexander V. Chikunov

Alexander V. Chikunov ^cPrinceton Institute of Life Sciences, Princeton, New Jersey

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Douglas Sheil

Douglas Sheil ^dForest Ecology and Forest Management Group, Wageningen University and Research, Wageningen, Netherlands
^eFaculty of Environmental Sciences and Natural Resource Management, Norwegian University of Life Sciences, Ås, Norway

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Antonio Donato Nobre

Antonio Donato Nobre ^fCentro de Ciência do Sistema Terrestre, Instituto Nacional de Pesquisas Espaciais, São José dos Campos, Brazil

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Paulo Nobre

Paulo Nobre ^gCenter for Weather Forecast and Climate Studies, Instituto Nacional de Pesquisas Espaciais, São José dos Campos, Brazil

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Günter Plunien

Günter Plunien ^hInstitut für Theoretische Physik, Technische Universität Dresden, Dresden, Germany

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Ruben D. Molina

Ruben D. Molina ⁱEscuela Ambiental, Facultad de Ingeniería, Universidad de Antioquia, Medellín, Colombia

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Online Publication:: 08 Aug 2023

Print Publication:: 01 Aug 2023

DOI:: https://doi.org/10.1175/JAS-D-21-0172.1

Page(s):: 1905–1921

Received:: 25 Jun 2021

Final Form:: 02 Dec 2021

Accepted:: 07 Dec 2021

Published Online:: 08 Aug 2023

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

While water lifting plays a recognized role in the global atmospheric power budget, estimates for this role in tropical cyclones vary from no effect to a major reduction in storm intensity. To better assess this impact, here we consider the work output of an infinitely narrow thermodynamic cycle with two streamlines connecting the top of the boundary layer in the vicinity of maximum wind (without assuming gradient-wind balance) to an arbitrary level in the inviscid free troposphere. The reduction of a storm’s maximum wind speed due to water lifting is found to decline with increasing efficiency of the cycle and is about 5% for maximum observed Carnot efficiencies. In the steady-state cycle, there is an extra heat input associated with the warming of precipitating water. The corresponding positive extra work is of an opposite sign and several times smaller than that due to water lifting. We also estimate the gain of kinetic energy in the outflow region. Contrary to previous assessments, this term is found to be large when the outflow radius is small (comparable to the radius of maximum wind). Using our framework, we show that Emanuel’s maximum potential intensity (E-PI) corresponds to a cycle where total work equals work performed at the top of the boundary layer (net work in the free troposphere is zero). This constrains a dependence between the outflow temperature and heat input at the point of maximum wind, but does not constrain the radial pressure gradient. We outline the implications of the established patterns for assessing real storms.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Gorshkov: Deceased.

Corresponding author: Anastassia M. Makarieva, ammakarieva@gmail.com

Abstract

Gorshkov: Deceased.

Corresponding author: Anastassia M. Makarieva, ammakarieva@gmail.com

Keywords: Hurricanes; Hurricanes/typhoons; Precipitation; Sea surface temperature; Tropical cyclones

1. Introduction

Reliable predictions of storm intensity are vital for improving human safety. These predictions require a robust account of the major physical factors that determine the maximum wind speed that can be developed by the storm. Tropical cyclones do not just generate kinetic energy; they also lift water that subsequently precipitates. This lifting can diminish the power available for winds. Nonetheless, available estimates of this impact are inconsistent (Makarieva et al. 2020; Emanuel and Rousseau-Rizzi 2020).

Table 1 summarizes the situation. In steady-state large-scale circulations, the water lifting power W_P (W m⁻²) is within 20%–50% of total wind power. By analogy to hydropower, this lifting power is estimated from the known precipitation rate P and precipitation pathlength H_P (the mean height from which the hydrometeors are falling) (Gorshkov 1982, 1995; Pauluis et al. 2000; Pauluis and Dias 2012; Makarieva et al. 2013, 2017c).

Table 1.

Relative estimates of the contribution of water lifting to atmospheric energetics, by different authors, in chronological order.

For tropical cyclones, Emanuel (1988) estimated that water lifting reduces the central pressure drop in intense storms by about 5% and 20% for pseudoadiabatic and reversible ascent, respectively, and concluded that “the importance of water loading in limiting the hurricane intensity in the reversible case” is “very substantial.” Without referring to this prior work, Emanuel and Rousseau-Rizzi (2020) recently agreed with Makarieva et al. (2020) that in real cyclones the reduction of the squared maximum velocity due to water lifting should not exceed 10%.

In contrast, Sabuwala et al. (2015)—quoted by Emanuel (2018) but neglected by Rousseau-Rizzi and Emanuel (2019) and by Emanuel and Rousseau-Rizzi (2020)—used satellite-derived precipitation data and Emanuel’s potential intensity framework to report an approximately 50% reduction in the squared maximum velocity due to water lifting for pseudoadiabatic ascent. Unlike Sabuwala et al. (2015), who did not quote Emanuel (1988), Wang and Lin (2020) used the approach of Emanuel (1988) to account for the total water mixing ratio q_t in the pseudoadiabatic model of Emanuel and Rotunno (2011) and found that this reduces air velocity at the radius of maximum wind in a hurricane with reversible adiabats by about 10% (or squared velocity by 20%). At the same time, Emanuel and Rousseau-Rizzi (2020) indicated that the impact of the water lifting on storm intensity depends on the integral of dq_t/dt over a closed contour. For a reversible cycle, which conserves the total water content, this integral is exactly zero (Table 1).

The preceding issues raise several questions. First, is the impact of water lifting on storm intensity large or small, and if it is small, why is this different from the power budget of larger-scale circulations? Second, what is the reason for the high observation-derived estimates of Sabuwala et al. (2015)? Third, why is the impact of the water lifting maximized in reversible compared with pseudoadiabatic hurricanes?

Assessing the influence of water lifting on a storm’s steady-state intensity requires a consideration of the storm’s thermodynamic cycle. The original derivation of a storm’s maximum velocity by Emanuel (1986) was based on a scaling relation between velocity and temperature along a surface of constant moist saturated entropy and angular momentum [Emanuel 1986, his Eq. (13); Emanuel and Rotunno 2011, their Eq. (11)]. The derivation assumed the free troposphere to be in gradient-wind balance. Makarieva et al. [2018, their Fig. 1 and Eqs. (p5) and (p6)] showed that this assumption can be relaxed in the assessment of storm-integrated energy fluxes. Kerry Emanuel suggested^¹ that Makarieva et al.’s (2018) approach could be used locally to describe an infinitely narrow cycle in the vicinity of maximum wind. Without referring to Makarieva et al. (2018), Rousseau-Rizzi and Emanuel (2019) applied this suggestion but, as noted by Montgomery and Smith (2020) and Makarieva et al. (2020), their derivations were based on an incorrect configuration of air streamlines.

Here we consider an infinitely narrow thermodynamic cycle in the vicinity of maximum wind that comprises two streamlines connecting the top of the boundary layer to some arbitrary level in the free troposphere (Fig. 1). Our analysis assumes the atmosphere to be inviscid above the boundary layer, but does not require the gradient-wind balance at the radius of maximum wind. We show that the expression for work of this cycle is equivalent to the scaling relation in Emanuel’s (1986) original derivation (section 2). The new, more general formulation of Emanuel’s maximum potential intensity (E-PI) framework is useful in the following three aspects.

Fig. 1.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-21-0172.1

First, by allowing an explicit evaluation of the water lifting term that we perform in section 3, it responds to the above three questions concerning the contribution of water lifting to storm’s energetics.

Second, it allows the estimation of the gain of kinetic energy and angular momentum in the outflow region of the storm. Regarding this term, it has long been held that it can be large only when the outflow radius is very large (Emanuel 1986, p. 602; Emanuel 2004, p. 190). Rousseau-Rizzi and Emanuel (2019) noted that the outflow term “will be small if the radius at which this occurs is not too large.” However, Makarieva et al. (2019), see also Makarieva et al. (2020), showed that, conversely, this term “is significant when the outflow radius … is close to the radius of maximum wind,” i.e., when the outflow radius is small. Omitting to quote Makarieva et al. (2019) or to discuss their own previous opposing view, Emanuel and Rousseau-Rizzi (2020) made an effort to rederive the result of Makarieva et al. (2019) about the (in)significance of the outflow term at (large) small outflow radii. Emanuel and Rousseau-Rizzi’s (2020) derivations were not conclusive, however, as they were based on their Eq. (6), where the dimensions of the right-hand and left-hand sides do not match. As Makarieva et al. (2020) argued, this is due to an incorrect transition from volume to surface power fluxes. Here a consistent derivation of the outflow term is presented (section 3).

Third, the new formulation demonstrates that E-PI at the point of maximum wind corresponds to a thermodynamic cycle with zero work in the free troposphere (section 4). This strong constraint, together with the recently revealed relation between the inner core and outflow parameters in E-PI, is essential for evaluating “superintensity” (hurricane wind speeds exceeding their E-PI limits) (Makarieva and Nefiodov 2021).

2. An infinitely narrow thermodynamic cycle

a. Combining dynamics and thermodynamics

We consider two closed air streamlines, ABCDA and A′B′C′D′A′ (Fig. 1), in an axisymmetric atmosphere. Our goal is to find the relation between turbulent dissipation and heat input at the top of the boundary layer. From this relation, the maximum wind speed in E-PI can be estimated.

We assume hydrostatic equilibrium and apply the Bernoulli equation to path b′Bb that belongs to streamline ABCDA:

- α \frac{\partial p}{\partial z} = g,

(1)

- α d p = d (\frac{V^{2}}{2}) + g d z - F \cdot d l,

(2)

where α ≡ 1/ρ, ρ = ρ_d + ρ_υ + ρ_l is the density of moist air (including dry air ρ_d, water vapor ρ_υ and condensed water ρ_l), p is air pressure, g is the acceleration of gravity, V is air velocity, F is the turbulent friction force per unit air mass, and dl = V dt.

The connection between the dynamics and thermodynamics will be found through the common term αdp. The logic of our derivations is schematized in Fig. 2.

Fig. 2.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-21-0172.1

Applying Eqs. (1) and (2) to b′Bb results in Eq. (3) (Fig. 2), which relates work of the friction force to the sum of the horizontal differences in pressure and kinetic energy per unit moist air mass (halved squared velocity). In Eq. (3), we have additionally assumed that V_b′ = V_B′. This implies two possibilities. One is that ∂V/∂z = 0, which holds by definition at the point of maximum wind and is otherwise a plausible assumption at the top of the boundary layer, where turbulent viscosity becomes negligible (Bryan and Rotunno 2009a, p. 3045). Another possibility is that points b′ and B′ (and, respectively, B and b) coincide, such that path b′b is horizontal. This second case with z_b = 0 was considered by Rousseau-Rizzi and Emanuel (2019, their Fig. 1), who assumed that |F| = 0 for z > 0 but |F| ≠ 0 at B′b. For a derivation of Eq. (3) from the equations of motion, see appendix A.

Our next step is to consider the inviscid atmosphere above z = z_b. Applying the Bernoulli Eq. (2) with |F| = 0 to streamlines bc and B′C′ and assuming hydrostatic equilibrium at path cC′ (which is not a streamline) yields Eq. (4). It relates the horizontal change of αdp at B′b to the sum of the integral of αdp over the closed contour B′bcC′B′ and the changes of kinetic energy at B′b and in the outflow region cC′ (Figs. 1 and 2). [When these changes are zero, the first equality of Eq. (8) follows.]

At this point, we invoke the relation between the specific volumes of moist and dry air, α_d = (1 + q_t)α, where α_d ≡ 1/ρ_d is the specific volume of dry air, and q_t ≡ (ρ_υ + ρ_l)/ρ_d is the total water mixing ratio. This relation allows us to link the integrals of αdp and α_ddp over the closed contour B′bcC′B′; see Eq. (5).

On the other hand, the integral of α_ddp over a closed contour represents work done per unit dry air mass in the corresponding thermodynamic cycle. This work is converted from the heat input with the cycle’s efficiency ε in Eq. (6). By summing Eqs. (3)–(6) we combine the dynamic and thermodynamic constraints to obtain a relation between turbulent dissipation and heat input in the lower atmosphere; see Eq. (7).

In the particular case of q_t = constant, one can divide the functions under the integral signs in Eq. (6) by a constant factor 1 + q_t. Then we obtain the second equality in Eq. (8).

In the derivation, the hydrostatic equilibrium approximation (1) was applied at b′Bb and cC′, but it was not used at bc and B′C′. The thermodynamic processes in the cycle were not specified, so Eq. (6) can be viewed as defining the value of ε. The choice of the outflow point c along the streamline in Eqs. (4) and (7) was arbitrary. The assumption that the cycle is infinitely narrow was used in Eqs. (3) and (7), but not in Eqs. (4)–(6). Equation (8) describes an infinitely narrow cycle B′bcC′B′ with ∂V²/∂z = 0 at cC′ and ∂V²/∂r = 0 at B′b. It is also valid for a special case of B′bcC′B′ being a closed streamline with V_b = V_B′ (Emanuel 1988).

b. Conventional E-PI estimate

We will now demonstrate the equivalence of the framework depicted in Fig. 2 to E-PI in two ways: in terms of turbulent dissipation and in terms of angular momentum.

The ratio of the surface fluxes of turbulent dissipation and ocean-to-atmosphere heat is proportional to squared velocity [e.g., Bister and Emanuel 1998, their Eqs. (15) and (16)]. An independent estimate of this ratio would yield a constraint on velocity. Such an estimate can be deduced from Eq. (7) with some assumptions.

Emanuel (1986) did not discriminate between α and α_d (and accordingly between δQ and δQ_d) and thus neglected the second term on the right-hand side of Eq. (7). The third term on the right-hand side of Eq. (7), which is the change of kinetic energy in the outflow, was also neglected. That was because Emanuel (1986) assumed gradient-wind balance and, hence, V = υ, where υ is tangential velocity, and then chose point c in the outflow where V = υ = 0 and, hence, ∂V²/∂z = 0. Finally, Emanuel (1986) considered the thermodynamic cycle to be reversible, such that its efficiency ε equals Carnot efficiency ε_C ≡ (T_b − T_c)/T_b.

Applying these assumptions—neglecting the last two terms and putting ε = ε_C and

δ Q_{d} = T_{b} d s^{*}

in Eq. (7)—we lift the integral signs in the limit of the infinitely narrow cycle b′ → B′ and divide both sides of the equation by dt, to obtain

- F \cdot V = ε_{C} \frac{δ Q_{d}}{d t} = ε_{C} T_{b} \frac{\partial s^{*}}{\partial r} u,

(9)

where V = dl/dt is total air velocity, u = dr/dt is radial velocity, and

s^{*}

is moist saturated entropy.

Multiplied by ρ, Eq. (9) relates local volume-specific rates (W m⁻³) of turbulent dissipation and heat input into a horizontally expanding air parcel. Assuming that the ratio of these volume-specific rates is the same as the ratio of the corresponding surface fluxes (W m⁻²) of turbulent dissipation D = ρC_DV³ and heat input

J = ρ C_{k} V (k_{s}^{*} - k)

- \frac{F \cdot V}{δ Q_{d} / d t} = \frac{D}{J},

(10)

yields the original E-PI formula for maximum velocity [e.g., Emanuel and Rotunno 2011, their Eq. (22)]:

V_{\max}^{2} = \frac{D}{J} \frac{C_{k}}{C_{D}} (k_{s}^{*} - k) = ε_{C} \frac{C_{k}}{C_{D}} (k_{s}^{*} - k) .

(11)

Here C_k ≃ C_D are surface exchange coefficients for enthalpy and momentum, respectively; $k_{s}^{*}$ (J kg⁻¹) is the saturated enthalpy of air at surface temperature, and k is the actual enthalpy of air.

The second way of demonstrating the equivalence between E-PI and the framework in Fig. 2 is to show that the following combination of Eqs. (4)–(6),

- \int_{B^{'}}^{b} α d p - \int_{B^{'}}^{b} d (\frac{V^{2}}{2}) = ε \int_{B^{'}}^{b} δ Q_{d} + \oint q_{t} α d p + \int_{c}^{C^{'}} d (\frac{V^{2}}{2}),

(12)

is, under E-PI assumptions, equivalent to Emanuel and Rotunno’s (2011) Eq. (11),

\frac{ν_{1} - ν_{2}}{T_{1} - T_{2}} = - \frac{d s^{*}}{d M}, (z \geq z_{b}),

(13)

where T₁, T₂ and ν₁ ≡ υ₁/r₁, ν₂ ≡ υ₂/r₂ are, respectively, air temperatures and angular velocities at arbitrary distances r₁ and r₂ from the storm center on a surface of constant angular momentum M and moist saturated entropy

s^{*} = s^{*} (M)

defined by the given value of

d s^{*} / d M

Using the definition of angular momentum

M \equiv υ r + \frac{f r^{2}}{2},

(14)

where the Coriolis parameter f ≡ 2Ω sinφ is assumed constant (φ is latitude, Ω is the angular velocity of Earth’s rotation), and assuming gradient-wind balance at point b,

α \frac{\partial p}{\partial r} = \frac{υ^{2}}{r} + f υ,

(15)

we can write our Eq. (12) (with the second term on the right-hand side ignored) as

\int_{b}^{B^{'}} \frac{υ}{r} \frac{\partial M}{\partial r} d r = - ε \int_{b}^{B^{'}} δ Q_{d} + \frac{1}{2} (υ_{C^{'}}^{2} - υ_{c}^{2}) .

(16)

Here we have assumed, as did Emanuel and Rousseau-Rizzi (2020), that the change of velocity V over path cC′ is dominated by the change in tangential velocity υ, $V_{C^{'}}^{2} - V_{c}^{2} = υ_{C^{'}}^{2} - υ_{c}^{2}$ . Since the atmosphere at cC′ is frictionless and hydrostatic, it is equivalent to assuming local gradient-wind balance [see Eq. (A10)]. With path cC′ hydrostatic and in gradient-wind balance, all our results are invariant with respect to its orientation (whether/how cC′ is tilted about the vertical axis).

To lift the integral signs and describe an infinitely narrow cycle, we need to evaluate the last term in Eq. (16) in the limit r_B′ → r_b. Since the atmosphere is frictionless above z_b and since the pressure field is axisymmetric, paths bc and B′C′ conserve angular momentum M. Using Eq. (14) for M in the equation M_B′ − M_b = M_C′ − M_c and dividing this equation by r_c(r_B′ − r_b) we obtain

\frac{1}{r_{c}} [υ_{B^{'}} + r_{b} \frac{υ_{B^{'}} - υ_{b}}{r_{B^{'}} - r_{b}} + \frac{f}{2} (r_{B^{'}} + r_{b})] = \frac{υ_{C^{'}} - υ_{c}}{r_{B^{'}} - r_{b}} .

(17)

Taking the limit r_B′→r_b and multiplying Eq. (17) by υ_c we find

υ_{c} \frac{\partial υ_{c}}{\partial r} = \frac{υ_{c}}{r_{c}} (υ + r \frac{\partial υ}{\partial r} + f r) = \frac{υ_{c}}{r_{c}} \frac{\partial M}{\partial r}, (z = z_{b}) .

(18)

We now assume that our infinitely narrow cycle has Carnot efficiency ε_C = (T_b − T_c)/T_b and that the heat input at B′b can be expressed in terms of the increment of moist saturated entropy

δ Q_{d} = T_{b} d s^{*}

. Lifting the integral signs in Eq. (16) with the use of Eq. (18), we obtain

\frac{υ_{b}}{r_{b}} \frac{\partial M}{\partial r} = - (T_{b} - T_{c}) \frac{\partial s^{*}}{\partial r} + \frac{υ_{c}}{r_{c}} \frac{\partial M}{\partial r}, (z = z_{b}) .

(19)

Assuming, finally, that $s^{*} = s^{*} (M)$ , such that $d s^{*} / d M = (\partial s^{*} / \partial r) / (\partial M / \partial r)$ , we obtain Eq. (13) from Eq. (19) with points 1 and 2 corresponding to points b and c, respectively.

With point c chosen such that υ_c = 0, Eqs. (13) and (19) are equivalent to Emanuel’s (1986) Eq. (13) [see also Emanuel and Rotunno 2011, their Eq. (12)]. Emanuel (1986) made several assumptions about the boundary layer (specifically, that the surfaces of constant

s^{*}

and M are vertical and that the horizontal turbulent diffusion fluxes are small) to justify that the radial gradients of

s^{*}

and M relate as their surface fluxes τ_s = J/T_s and τ_M = −Dr/V, respectively:

\frac{\partial s^{*} / \partial r}{\partial M / \partial r} = \frac{τ_{s}}{τ_{M}} .

(20)

Then assuming that V ≃ υ_b, r ≃ r_b and T_s ≃ T_b, where T_s is sea surface temperature, Eqs. (19) and (20) yield Eq. (11). Under Eq. (19), assumptions (10) and (20) are equivalent.

Equation (11) relates local fluxes and is intended to estimate storm’s maximum potential intensity. However, neither Eq. (9) nor Eq. (19), from which the maximum potential intensity (11) can be derived, require ∂V²/∂r = 0 (the condition of maximum wind). This peculiarity of E-PI was noted by Montgomery and Smith (2017). Equation (9) requires ∂V²/∂z = 0 at z = z_b, while Eq. (19) does not.

As we will discuss in section 4, ∂V²/∂r = 0 is an important constraint on E-PI. Here we note that, according to Eq. (8), in the E-PI framework work in the free troposphere (along the path bcC′B′) is zero. The total work of the cycle, i.e., heat input at B′b multiplied by efficiency ε_C, equals work on B′b. Since ε_C depends on the outflow temperature T_c, the specification of the thermodynamic process at B′b and the choice of T_c cannot be independent (Makarieva and Nefiodov 2021).

3. Estimating the water lifting and outflow terms

a. Reversible and pseudoadiabatic hurricanes

For the considered thermodynamic cycle B′bcC′B′ to have Carnot efficiency, it should be reversible. This requires that the air is saturated (relative humidity $H$ is equal to 100%) and the total moisture mixing ratio q_t is constant, everywhere in the cycle.^²

With q_t = const, dividing both sides of Eq. (6) by (1 +q_t) we obtain

- \oint α d p = ε \int_{B^{'}}^{b} δ Q,

(21)

where δQ ≡ δQ_d/(1 + q_t) is the heat input per unit moist air mass. Summing Eqs. (3), (4), and (21) yields

- \int_{b^{'}}^{b} F \cdot d l = ε \int_{B^{'}}^{b} δ Q + \int_{c}^{C^{'}} d (\frac{V^{2}}{2}) .

(22)

Comparison of Eqs. (7) and (22) shows that the latter lacks the water lifting term. Rousseau-Rizzi and Emanuel (2019) similarly found that the water lifting term is comprised in the integral of the material derivative dq_t/dt over a closed streamline, which is zero when q_t = const (Table 1). The physical meaning of this result is that all the water that is lifted in the ascending branch of the cycle taking the energy away, goes down in the descending branch and performs work, with the net effect being zero.

However, if the storm circulation is composed of streamlines representing reversible processes (saturated isotherms and adiabats conserving q_t), but q_t differs between streamlines, the thermodynamic cycle B′bcC′B′ will not be reversible due to the change of q_t on paths B′b and cC′ that connect different streamlines, ABCDA and A′B′C′D′A′. The efficiency of such a cycle will be lower than Carnot efficiency.

b. Extra heat input to warm precipitating water

A cycle with reversible adiabats bc and C′B′ (q_t = const,

H = 100 %

) and saturated isotherms B′b and cC′, along which q_t, respectively, increases and decreases, has the following relation between work and heat input (for derivations, see appendix B):

- \oint α_{d} d p ≃ ε_{C} T_{b} Δ s^{*} + \frac{ε_{C}}{2} c_{l} (T_{b} - T_{c}) Δ q_{t},

(23)

where ε_C ≡ (T_b − T_c)/T_b is Carnot efficiency, Δq_t ≥ 0 and

Δ s^{*} \geq 0

are, respectively, the changes of total water mixing ratio q_t and moist saturated entropy

s^{*}

from B′ to b, c_l = 4.2 kJ kg⁻¹ K⁻¹ is the specific heat capacity of liquid water. The first term on the right-hand side of Eq. (23) is due to a heat input into an expanding air parcel with evaporating water. If, as we assume below, all water at the warmer isotherm is added in the form of water vapor, then Δq_t should be replaced by saturated water vapor mixing ratio

Δ q^{*}

in Eq. (23).

While in finite differences Eq. (23) is valid only when B′b and cC′ are isotherms, in the limit of an infinitely narrow cycle B′bcC′B′, when B′b and cC′ degenerate each to a point, Eq. (23) becomes valid even if the temperature along B′b and cC′ is not constant (see appendix B). As one of our reviewers pointed out, this is due to the small change of temperature along these paths compared to the finite difference T_b − T_c (see also Carnot 1890, p. 59). This explains how Emanuel (1986) obtained a Carnot efficiency multiplier at the radial heat input $T_{b} \partial s^{*} / \partial r$ in his Eq. (13) without assuming horizontal isothermy at the top of the boundary layer.

The last term in Eq. (23), see also Eq. (B11), corresponds to term “(c)” in Eq. (19) of Emanuel (1988), who described it as “the increase of entropy due to addition of water mass” and “the contribution of water substance to the heat capacity.” Without referring to Emanuel (1988), Pauluis (2011) rederived this term in his Eq. (B2) and interpreted it as “additional work,”^³ accounting for which elevates the cycle’s efficiency above Carnot efficiency. Pauluis (2011) explained that this elevation does not violate the second law of thermodynamics because the cycle is open (moisture is added and removed), while Carnot efficiency limits the efficiencies of closed cycles only. On the other hand, according to Pauluis (2011), it is not accidental that the same cycle has Carnot efficiency when c_l = 0: it is because this open cycle is thermodynamically equivalent to a closed cycle where the moisture removed at the colder isotherm is kept within the heat engine and added back to the cycle at the warmer isotherm. This interpretation raises the question of why with c_l ≠ 0 such a cycle is not equivalent to a closed one.

This is resolved by recognizing an additional heat input to the cycle. Warming the water removed at the colder isotherm with temperature T_c and returned at the warmer isotherm with T_b requires extra heat $c_{l} (T_{b} - T_{c}) Δ q^{*}$ . As the moist air ascends and cools from T_b to T_c, the mean temperature at which the water loses heat is, in the linear approximation, T_w ≃ (T_b + T_c)/2. Accordingly, the maximum efficiency with which this heat can be converted to work is (T_b − T_w)/T_b ≃ ε_C/2.

Thus, such a cycle is equivalent to a closed cycle with total heat input

Δ Q_{d} = T_{b} Δ s^{*} + c_{l} (T_{b} - T_{c}) Δ q^{*}

(24)

and efficiency

ε = ε_{C} [1 - \frac{1}{2} \frac{c_{l} (T_{b} - T_{c}) Δ q^{*}}{T_{b} Δ s^{*} + c_{l} (T_{b} - T_{c}) Δ q^{*}}],

(25)

which is lower than Carnot efficiency (ε_C/2 ≤ ε ≤ ε_C) due to the irreversibility associated with warming the precipitating water. This inherent thermodynamic imperfection of steam cycles was recognized already by Carnot (1890, p. 58).

In a pseudoadiabatic hurricane, all condensed water is immediately removed (precipitates) from the air parcel: ρ_l = 0 and $q_{t} = q^{*}$ . The extra heat reduces to $c_{l} (T_{b} - T_{P}) Δ q^{*}$ , where T_P is the mean temperature at which condensation and precipitation occur (T_c ≤ T_P < T_b). The last term in Eq. (23) can be approximated by ε_P $c_{l} (T_{b} - T_{P}) Δ q^{*} / 2$ , where $ε_{P} \equiv (T_{b} + T_{P} - 2 T_{c}) / T_{b}$ ; see Eq. (B16). For T_P = T_c, ε_P = ε_C.

c. Water lifting

Taking into account that paths B′b and cC′ are, respectively, horizontal and vertical (Fig. 1) and using

α_{d} = (1 + q^{*}) α

for B′b, we can combine Eqs. (3), (6), (7), and (23) in the following form:

\int_{b}^{B^{'}} (α \frac{\partial p}{\partial r} + \frac{1}{2} \frac{\partial V^{2}}{\partial r}) d r = - ε_{C} \int_{b}^{B^{'}} T \frac{\partial s^{*}}{\partial r} d r - c_{l} (T_{b} - T_{P}) \frac{ε_{P}}{2} \int_{b}^{B^{'}} \frac{\partial q^{*}}{\partial r} d r + \oint α q_{t} d p + \int_{c}^{C^{'}} d (\frac{V^{2}}{2}) .

(26)

Here we have assumed that evaporation occurs from the ocean surface (there is no condensate along B′b:

q_{t} = q^{*}

). We calculate the water lifting term in appendix C and lift the integral signs in Eq. (26) for the infinitely narrow cycle B′bcC′B′, i.e., considering the limit r_B′→r_b and z_C′→z_c. This gives for z = z_b and r= r_b

(1 + q^{*}) (α \frac{\partial p}{\partial r} + \frac{1}{2} \frac{\partial V^{2}}{\partial r}) = - ε_{C} T_{b} \frac{\partial s^{*}}{\partial r} + [\frac{1}{2} (V_{P}^{2} - V^{2}) + g (z_{P} - z_{b}) - c_{l} (T_{b} - T_{P}) \frac{ε_{P}}{2}] \frac{\partial q^{*}}{\partial r} + (1 + q_{t c}) \frac{1}{2} \frac{\partial V_{c}^{2}}{\partial r} .

(27)

Here q_t_c and V_c refer to the point (r_c, z_c);

\partial V_{c}^{2} / \partial r

is the limit of

(V_{C^{'}}^{2} - V_{c}^{2}) / (r_{B^{'}} - r_{b})

at r_B′ → r_b and z_C′ → z_c, cf. Eqs. (17) and (18). The quantities

V_{P}^{2}

, z_P and T_P are evaluated on the unclosed contour bcC′B′ (denoted by

↷

V_{P}^{2} \equiv - \frac{1}{Δ q^{*}} \int_{↷} V^{2} d q_{t}, z_{P} \equiv - \frac{1}{Δ q^{*}} \int_{↷} z d q_{t}, T_{P} \equiv - \frac{1}{Δ q^{*}} \int_{↷} T d q_{t},

(28)

where

Δ q^{*} \equiv q_{b}^{*} - q_{B^{'}}^{*}

. In a cycle with reversible adiabats, i.e., with constant q_t along bc and C′B′ and

Δ q^{*} = q_{t c} - q_{t C^{'}}

, the integrals in Eq. (28) in the considered limit are straightforwardly evaluated: V_P = V_c, z_P = z_c, and T_P = T_c.

Equation (27) summarizes the energy budget of the infinitely narrow cycle B′bcC′B′ and thus provides a relation between local variables. The first and second terms in the square brackets represent kinetic and potential energy increments associated with phase transitions. Term $(V_{P}^{2} - V^{2}) / 2$ formally accounts for water vapor being added to the air mixture with kinetic energy V²/2 of the air at B′b, while disappearing via precipitation with kinetic energy $V_{P}^{2} / 2$ in the free troposphere. We will neglect this term,^⁴ since for typical z_P ∼ 10 km and V ∼ 60 m s⁻¹, V²/2 is only about 2% of gz_P.

The water lifting term g(z_P − z_b) accounts for the net energy expended to lift water. In a real cycle, where there is no precipitation in the descending branch C′B′ and q_t does not change, z_P equals the mean precipitation height H_P in the ascending branch bc. In an infinitely narrow cycle with reversible adiabats, q_t is also constant in the “descending branch,” and z_P = z_c = H_P has the same meaning.

However, in an infinitely narrow cycle with q_t varying along bc and C′B′, moisture disappears along bc. It then arises anew along C′B′ with its own nonzero gravitational energy. In this hypothetical cycle, moisture performs work as it descends along C′B′ and consumes energy when it is raised along bc. Thus, the net energy $g (z_{P} - z_{b}) Δ q^{*}$ is equal to the difference in the gravitational energy of “precipitation” between bc and C′B′ [cf. Emanuel 1988, his Eq. (C12).

This net energy is equal to the energy $g (H_{P} - z_{b}) Δ q^{*}$ spent to raise the additional moisture $Δ q^{*}$ from z_b to the mean precipitation height H_P along bc plus the energy $g Δ q^{*} \partial H_{P} / \partial q_{b}^{*} (q_{b}^{*} - q_{c}^{*})$ spent to raise nearly all moisture $q_{b}^{*} - q_{c}^{*} \sim q_{b}^{*}$ to an additional altitude $Δ q^{*} \partial H_{P} / \partial q_{b}^{*}$ equal to the difference between the mean precipitation heights along bc and B′C′. For pseudoadiabats, H_P is approximately proportional to $q_{b}^{*}$ , so that $q_{b}^{*} \partial H_{P} / \partial q_{b}^{*} \sim H_{P}$ . In this case, z_P − z_b turns out to be approximately twice the mean precipitation height, i.e., around 10 km for T_b ≃ 300 K, Eq. (C6).

The third term in the square brackets represents the warming of the precipitating water, Eq. (24). It is of the opposite sign to the water lifting term. For reversible adiabats, T_P = T_c; with T_b = 300 K, T_b − T_P ≃ 100 K and z_P − z_b ≃ 17 km (which corresponds to mean lapse rate Γ = (T_b − T_P)/(H_P − z_b) ≃ 6 K km⁻¹), the water warming term constitutes about 40% of the water lifting term. Accounting for water warming somewhat, but not fully, compensates the impact of water lifting, and the more so, the larger the difference T_b − T_P. For pseudoadiabats, T_P is equal to the mean temperature of precipitating water [Eq. (C6)], the difference T_b − T_P is relatively small, and the impact of water warming is also smaller. With T_b = 300 K, T_b − T_P = 25 K and z_P − z_b = 10 km (Table C1), it constitutes 30% of the water lifting term.

In units of latent heat of vaporization L_υ, the cumulative contribution K₁ of the water lifting and warming

K_{1} \equiv \frac{1}{L_{υ}} [g (z_{P} - z_{b}) - c_{l} (T_{b} - T_{P}) \frac{ε_{P}}{2}]

(29)

is of similar magnitude K₁ ∼ 0.03–0.04 for reversible and pseudoadiabatic storms with high ε_C ∼ 0.3 and T_b ≃ 300 K. For a given T_b, the pseudoadiabatic K₁ values are smaller due to their lower values of z_P (Table C1).

d. The outflow gain of kinetic energy

The last term on the right-hand side of Eq. (27) describes how the kinetic energy in the outflow,

V_{c}^{2} / 2

, changes depending on the radius r = r_b where the streamline crosses the level z = z_b (Fig. 1). Assuming that we are at a point near the radius of maximum wind, such that υ/r ≫ ∂υ/∂r, we can neglect ∂υ/∂r in Eq. (18) and use the relation between tangential velocity and pressure gradient

α \frac{\partial p}{\partial r} \equiv B (\frac{υ^{2}}{r} + f υ),

(30)

where

B > 0

defines the degree to which the airflow is radially unbalanced (

B < 1

for the supergradient flow when the outward-directed centrifugal force exceeds the inward-pulling pressure gradient), to obtain from Eq. (18)

\frac{1}{2} \frac{\partial V_{c}^{2}}{\partial r} ≃ \frac{1}{2} \frac{\partial υ_{c}^{2}}{\partial r} = \frac{υ_{c}}{r_{c}} \frac{r}{υ} (\frac{υ^{2}}{r} + f υ) = α \frac{\partial p}{\partial r} \frac{1}{B} \frac{υ_{c}}{r_{c}} \frac{r_{b}}{υ_{b}}, (r = r_{b}, z = z_{b}) .

(31)

On the other hand, using Eq. (14) and the fact that M_b = M_c we can express υ_c as

υ_{c} = \frac{1}{r_{c}} [υ_{b} r_{b} + \frac{f}{2} (r_{b}^{2} - r_{c}^{2})] .

(32)

[Evaluating the derivative of Eq. (32) with respect to r_b and taking into account that ∂r_c/∂r_b = 0 is another way to obtain Eq. (18)]. Using Eq. (32) and neglecting q_t_c ≪ 1we can write the last term in Eq. (27) as K₂α∂p/∂r, where

K_{2} \equiv \frac{1}{B} \frac{υ_{c}}{r_{c}} \frac{r_{b}}{υ_{b}} = \frac{1}{B} [\frac{r_{b}^{2}}{r_{c}^{2}} + \frac{f r_{b}}{2 υ_{b}} (\frac{r_{b}^{2}}{r_{c}^{2}} - 1)] .

(33)

For (r_b/r_c)² ≪ 1, this term is small and negative. In contrast, Emanuel (1986, p. 602) incorrectly concluded that the outflow term becomes significant if the outflow radius is very large.^⁵ In their reevaluation of this issue, Emanuel and Rousseau-Rizzi (2020) considered the material derivative of angular momentum dM/dt along the path cC′ connecting the two streamlines. Their derivation is not valid, since cC′ is not a streamline and the air does not move along that path. Defending their configuration of streamlines, Emanuel and Rousseau-Rizzi (2020) noted that “the properties of D′ and D, and of A′ and A are identical” but said nothing about the properties of C′ and C (see Fig. 1 of Rousseau-Rizzi and Emanuel 2019).

e. Estimates of maximum velocity

We will now consider the point where ∂V/∂r = 0 [see Eq. (8) in Fig. 2]. This corresponds to the point of maximum wind if point b is chosen at the top of the boundary layer as in the derivations of Emanuel (1986) and Emanuel and Rotunno (2011) or to the point of maximum surface wind if point b is chosen at the surface as in the derivations of Rousseau-Rizzi and Emanuel (2019).

By analogy with Eq. (9), from Eq. (3) with ∂V/∂r = 0 we obtain [see also Makarieva et al. 2020, their Eq. (14)]

- F \cdot V = - α \frac{\partial p}{\partial r} u,

(34)

where u = dr/dt. Equation (34) is fundamental: it derives from the Bernoulli equation and hydrostatic equilibrium (Fig. 2). It predicts that if at the point of maximum wind |F| → 0 (as is the case at the top of the boundary layer), then u → 0. This feature is observed in numerical models (e.g., Bryan and Rotunno 2009a, p. 3054). It is also valid for a horizontal streamline at z_b = 0 as considered by Rousseau-Rizzi and Emanuel (2019, their Fig. 1), whereby u ≠ 0 and |F| ≠ 0.

On the other hand, for ∂υ/∂r = 0 we have from Eqs. (14) and (30)

α \frac{\partial p}{\partial r} = B \frac{υ}{r} \frac{\partial M}{\partial r} .

(35)

Finally, using the definitions of K₁ (29) and K₂ (33) and neglecting

q^{*} ≪ 1

, we can write Eq. (27) for ∂V/∂r = 0 as follows:

- α \frac{\partial p}{\partial r} = \frac{ε_{C} - β K_{1}}{1 - K_{2}} T_{b} \frac{\partial s^{*}}{\partial r}, β \equiv \frac{L_{υ} \partial q^{*} / \partial r}{T_{b} \partial s^{*} / \partial r} .

(36)

Here 0 ≤ β ≤ 1 is the share of latent heat in total heat input into the air parcel along B′b.

Now using two distinct assumptions, (10) and (20), about how the volume and surface energy fluxes relate, we obtain two expressions for V_max that differ by a factor of

B

. From Eq. (36), Eq. (35), and Eq. (20) we obtain

V_{\max}^{2} = \frac{ε_{C} - β K_{1}}{B (1 - K_{2})} \frac{C_{k}}{C_{D}} (k_{s}^{*} - k) .

(37)

From Eq. (36) multiplied by u, Eq. (34), and Eq. (10) we obtain the same expression, but without $B$ . Since $B$ can be as small as 0.5 (e.g., Bryan and Rotunno 2009a, Fig. 8), this is a significant source of uncertainty in V_max associated with assumptions (10), (20) and their modifications as discussed elsewhere (Makarieva and Nefiodov 2021).

For our present purpose of estimating the role of the water lifting and the outflow, this does not matter, since the values of K₁ and $K_{2}$ are compared with ε_C and unity, respectively. With K₁ ≃ 0.04 in the reversible case, β ∼ 0.7 estimated from a typical Bowen ratio, and ε_C = 0.3, K₁ reduces $V_{\max}^{2}$ by βK₁/ε_C × 100% ≃ 10% and V_max by 5%. This result is comparable to Emanuel (1988), who found that the central pressure drop is reduced by water lifting by about 20% in the reversible case (Table 1). Emanuel (1988) did not evaluate local maximum velocity but considered a large-scale thermodynamic cycle with a horizontally isothermal top of the boundary layer. Horizontal isothermy is generally not compatible with the other E-PI assumptions and leads to an underestimate of α∂p/∂r (Makarieva and Nefiodov 2021). This could cause the overestimate of K₁ for the reversible case. The details of pseudoadiabatic calculations were not reported by Emanuel (1988); the numerical value (5% reduction for pressure drop, Table 1) is similar to ours (5% reduction for $V_{\max}^{2}$ for maximum ε_C, Table C1).

As for the outflow term K₂, since r_b < r_c, the factor in square brackets in Eq. (33) is confined between −fr_b/(2υ_b) < 0 and unity. For characteristic values of r_b = 30 km, υ_b = 60 m s⁻¹, φ = 15°, and f ≃ 3.77 × 10⁻⁵ s⁻¹ we have fr_b/(2υ_b) ≃ 10⁻². With $B ≃ 1$ , K₂ reduces $V_{\max}^{2}$ by about 1% for large r_c.

On the other hand, if the outflow radius is relatively small, K₂ is positive and elevates rather than lowers the maximum velocity estimate. This may happen for many storms with $r_{b} / r_{c} ≳ \sqrt{f r_{b} / (2 υ_{b})} \sim 10^{- 1}$ . Interpreting K₂ as “dissipation” to occur at an arbitrary point c in an otherwise frictionless troposphere, is incorrect (cf. Emanuel and Rousseau-Rizzi 2020). When K₂ > 0 the kinetic energy increases from c to C′. Smith et al. (2014) noted that a mechanism that would provide an increment of angular momentum, and hence a kinetic energy increment, along B′b does not appear to exist. But Makarieva et al. (2017a) indicated that extra angular momentum can arise in the upper atmosphere as a real steady-state hurricane is an open system that moves through the atmosphere and can import angular momentum as it imports air and water vapor.

Sabuwala et al. [2015, their Eq. (2), the “adiabatic case”] did not derive their formulations from the original assumptions of E-PI. However, they included what seemed a plausible water lifting term in the hurricane’s power budget, as follows:

ε_{C} (D + J) = D + g H_{P} P,

(38)

from which one obtains that

V_{\max}^{2} = \frac{D}{J} \frac{C_{k}}{C_{D}} (k_{s}^{*} - k) = \frac{1}{1 - ε_{C}} (ε_{C} - \frac{P L_{υ}}{J} \frac{g H_{P}}{L_{υ}}) \frac{C_{k}}{C_{D}} (k_{s}^{*} - k) .

(39)

Here P (kg m⁻² s⁻¹) is the local precipitation in the region of maximum winds.^⁶ Equation (39) is obtained from Sabuwala et al.’s (2015) Eq. (2) and their additional equation

{\dot{Q}}_{in}

−

{\dot{Q}}_{out}

= P. [The “diabatic case” of Sabuwala et al. (2015) addressed thermal dissipation of the potential energy of the falling droplets (see Igel and Igel 2018).]

Compared to (37), the water lifting term gH_P/L_υ ∼ K₁ in Eq. (39) is multiplied by a large factor PL_υ/J ≫ 1 reflecting the ratio of local precipitation to local heat input. For typical Bowen ratios in hurricanes B ≡ J_S/J_L ≃ 1/3 (e.g., Jaimes et al. 2015) we have J = (1 + B)J_L = (1 + B)EL_υ and PL_υ/J = (P/E)/(1 + B), where J_S, J_L, and E are the local fluxes of sensible heat, latent heat, and evaporation, respectively. Ratio P/E between local precipitation and evaporation in the region of maximum winds is variable but on average on the order of 10 (see Makarieva et al. 2017a, their Table 1 and Figs. 2 and 3). For H_P ∼ 5 km and ε_C ∼ 0.3, Sabuwala et al.’s (2015) correction to $V_{\max}^{2}$ is thus 10(gH_P/L_υ)/[(1 + B)ε_C] ≃ 0.5, which is 5 times larger than our βK₁/ε_C ≃ 0.1 (Table C1). The unjustified replacement of evaporation by precipitation caused Sabuwala et al.’s (2015) estimate to be too high.

4. The physical meaning of E-PI at the point of maximum wind

Equation (34) shows that, at the point of maximum wind, the local volume-specific rate (W m⁻³) of turbulent dissipation, −ρF ⋅ V, is equal to the local volume-specific rate of sensible heat input into an isothermally and horizontally expanding air parcel, −(∂p/∂r)u. Thus, if all this turbulent dissipative power transforms locally to heat, the external sensible heat input into the air parcel must be zero.

In the general, nonisothermal case, we can relate turbulent dissipation to latent heat input. Inspecting Eq. (27), we notice that the water lifting term is proportional to

\partial q^{*} / \partial r

, while the outflow term is proportional to ∂p/∂r, see Eq. (31). On the other hand, using the definition of

q^{*}

and Clausius–Clapeyron law, the radial gradient of moist saturated entropy in Eq. (27) can also be expressed in terms of

\partial q^{*} / \partial r

and ∂p/∂r, see Eqs. (B5) and (B18):

T \frac{\partial s^{*}}{\partial r} = L_{υ} \frac{\partial q^{*}}{\partial r} (1 + ϰ_{1}) - α_{d} \frac{\partial p}{\partial r} (1 - ϰ_{2}) .

(40)

Coefficients

ϰ_{1}

and

ϰ_{2}

describe the deviation from horizontal isothermy:

ϰ_{1} \equiv \frac{1}{μ γ_{d}^{*} ξ^{2} (1 + γ_{d}^{*})} = \frac{ϰ_{2}}{γ_{d}^{*} ξ} ≃ 0.3,

(41)

ϰ_{2} \equiv \frac{1}{μ ξ (1 + γ_{d}^{*})} ≃ \frac{1}{μ ξ} ≃ 0.2,

(42)

where ξ ≡ L/RT ≃ 18, μ ≡ R/(c_pM_d) ≃ 2/7 and

γ_{d}^{*} \equiv (M_{d} / M_{υ}) q^{*}

are defined in appendix B. The numerical values correspond to T = 300 K and

γ_{d}^{*} \equiv p_{υ}^{*} / p_{d} = 0.04

. For ∂T/∂r = 0, the terms proportional to

ϰ_{1}

and

ϰ_{2}

jointly vanish from Eq. (40), see also Eq. (B18). Therefore, the isothermal case can be formally obtained by putting

ϰ_{1} = 0

and

ϰ_{2} = 0

in Eq. (40) and related.

Using Eqs. (31) and (40) we write Eq. (27) with ∂V²/∂r = 0 and

q^{*} ≪ 1

neglected, as follows:

α \frac{\partial p}{\partial r} = [- ε_{C} (1 + ϰ_{1}) + K_{1}] L_{υ} \frac{\partial q^{*}}{\partial r} + [ε_{C} (1 - ϰ_{2}) + K_{2}] α \frac{\partial p}{\partial r} .

(43)

Multiplying Eq. (43) by −u and using Eq. (34), we obtain

- F \cdot V = \frac{ε_{C} (1 + ϰ_{1}) - K_{1}}{1 - ε_{C} (1 - ϰ_{2}) - K_{2}} L_{υ} \frac{\partial q^{*}}{\partial r} u .

(44)

[This equation together with Eq. (36) show that E-PI constrains the latent-to-total heat ratio β at B′b. For

K_{1} = K_{2} = ϰ_{1} = ϰ_{2} = 0

, we have β = 1 − ε_C, which, as can be verified from the exact relation (B19), is a good approximation of β for small K₁ ≪ ε_C.] Assuming that turbulent dissipation relates to latent heat input equally at bB′ and at the surface [i.e., replacing in Eq. (10) total heat

T d s^{*}

and J with latent heat

L_{υ} d q^{*}

and

J_{L} = ρ C_{k} V L_{υ} (q_{s}^{*} - q)

, respectively],

- \frac{F \cdot V}{L_{υ} (\partial q^{*} / \partial r) u} = \frac{D}{J_{L}},

(45)

from Eq. (44) we obtain another expression for

V_{\max}^{2}

V_{\max}^{2} = \frac{D}{J_{L}} \frac{C_{k}}{C_{D}} L_{υ} (q_{s}^{*} - q) = \frac{ε_{C} (1 + ϰ_{1}) - K_{1}}{1 - ε_{C} (1 - ϰ_{2}) - K_{2}} \frac{C_{k}}{C_{D}} L_{υ} (q_{s}^{*} - q) .

(46)

When surface sensible heat J_S = ρC_kVc_p(T_s − T) is negligibly small, such that J_L ≫ J_S and J = J_L + J_S ≃ J_L [as assumed by Emanuel (1986), who put T = T_s], Eq. (46) coincides with the “dissipative heating” formulation [Bister and Emanuel 1998, Eq. (21)]. Indeed, for $K_{1} = K_{2} = ϰ_{1} = ϰ_{2} = 0$ , Eq. (46), derived without assuming any dissipative heating, is then equivalent to Eq. (11) but with ε_C replaced in the latter by ε_C/(1 − ε_C). This formal similarity encountered in a numerical model could lead to the dissipative heating formulation.

While Eq. (44) relates turbulent dissipation to latent heat input alone, this does not mean that “only surface latent heat fluxes can power tropical cyclones,” which is how Emanuel and Rousseau-Rizzi (2020) apparently misunderstood the isothermal version of Eq. (44) [see Eq. (15) of Makarieva et al. 2020]. Emanuel and Rousseau-Rizzi (2020) interpreted this relation as a contradiction in the reasoning of Makarieva et al. (2020), since it can be concluded that with no latent heat input from the ocean there can be no storms, while dry hurricanes were shown to exist at least in numerical models (Mrowiec et al. 2011; Cronin and Chavas 2019). We note, however, that whatever follows from Eq. (44), be that a contradiction or not, is an inherent feature of E-PI. All the equations that we have so far considered can be derived from E-PI’s key equations, and vice versa, as we demonstrated in section 2.

To elucidate the meaning of these relationships, let us consider yet another, equivalent, formulation of Eq. (44). The radial gradient of moist saturated entropy can be also expressed in terms of ∂T/∂r and ∂p/∂r (see Makarieva and Nefiodov 2021):

T \frac{\partial s^{*}}{\partial r} = - α_{d} \frac{\partial p}{\partial r} (1 + \frac{L γ_{d}^{*}}{R T}) (1 - \frac{1}{Γ} \frac{\partial T / \partial r}{\partial p / \partial r}),

(47)

where Γ is the moist adiabatic lapse rate (K Pa⁻¹). If, for simplicity, we neglect the water lifting by putting K₁ = 0, i.e., ignoring the term in square brackets in Eq. (27), we notice, taking into account Eq. (47), that all the remaining terms in Eq. (27) are proportional to the common factor α∂p/∂r, which can be canceled. Again, putting for simplicity

q_{t} = q^{*} = 0

and

B = 1

, and using Eq. (33), we obtain from Eq. (27)

1 = ε_{C} (1 + \frac{L γ_{d}^{*}}{R T_{b}}) (1 - \frac{1}{Γ} \frac{\partial T / \partial r}{\partial p / \partial r}) + \frac{υ_{c}}{r_{c}} \frac{r_{b}}{υ_{b}},

(48)

which can be rewritten in a form similar to Eq. (13) using ε_C = (T_b − T_c)/T_b and ν = υ/r:

\frac{ν_{b} - ν_{c}}{T_{b} - T_{c}} = \frac{ν_{b}}{T_{b}} (1 + \frac{L γ_{d}^{*}}{R T_{b}}) (1 - \frac{1}{Γ} \frac{\partial T / \partial r}{\partial p / \partial r}) .

(49)

In comparison to Eq. (13), in this equation point b is not arbitrary but pertains to the point of maximum wind, while point c remains of an arbitrary choice.

Equations (48) and (49) show that when ν_c = 0 (no kinetic energy change in the outflow), and the air is horizontally isothermal, E-PI framework presumes $ε_{C} [1 + (L γ_{d}^{*} / R T_{b})] = 1$ (Makarieva and Nefiodov 2021). This relation between the water vapor mixing ratio and the outflow temperature has the following meaning.

As we already noted, in the E-PI framework, total work of the cycle is equal to the work on the path B′b, where heat input occurs [see Eq. (8) in Fig. 2]. This can only be the case when the adiabat bc is, at least somewhere, warmer than the adiabat B′C′. Then the pressure deficit at b as compared to B′ can be compensated by the pressure surplus in the free troposphere at bc as compared to B′C′. Without this pressure surplus aloft, the work along bcC′B′ will be negative rather than zero (for a more detailed discussion, see Makarieva et al. 2017d, Fig. 1). When the air is horizontally isothermal, the required difference in the temperatures of the two adiabats can only be ensured by a higher water vapor mixing ratio $q^{*}$ and, accordingly, a higher partial pressure ratio $γ_{d}^{*}$ , at bc. How much of this water vapor condenses along bc producing the required temperature surplus, is dictated by the outflow temperature T_c. The lower T_c (i.e., the higher ε_C), the lower $γ_{d}^{*}$ is required to ensure net zero work in the free troposphere.

When the cycle is dry, achieved by putting L = 0, then Eq. (49), and the E-PI framework, lack a nontrivial solution for the isothermal case ∂T/∂r = 0. Otherwise, total work of such a cycle would exceed that of a Carnot cycle, where total work is always lower than work on the warmer isotherm. A dry Carnot cycle where total work is equal to the work at the warmer isotherm—as it is in E-PI at the point of maximum wind—is impossible. A dry E-PI hurricane must have local air temperature increasing toward the center at the point of maximum wind.

5. Conclusions

We considered an infinitely narrow steady-state thermodynamic cycle with the higher temperature corresponding to the point of maximum wind, Figs. 1 and 2, and demonstrated its equivalence to the E-PI framework as presented by Emanuel (1986) and Emanuel and Rotunno (2011). This revealed constraints not obvious in the original E-PI framework and clarified its physical meaning. A summary of our results is given in Fig. 3. Since this analysis required many detailed derivations, we leave a comparably detailed discussion and development of the implications of these results to subsequent studies. Here we outline what we consider most essential.

Fig. 3.

Main assumptions (green rounded boxes) and results (blue rectangular boxes) of this work with comments (dotted boxes).

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-21-0172.1

The water lifting term g(z_P − z_b), with z_P given by Eq. (28), can be evaluated for any cycle with the known distribution of q_t independent of the value of ε (Fig. 3). The E-PI thermodynamic cycle has a higher efficiency than the steady-state atmospheric circulation and tropical convection (e.g., Goody 2003), which explains the less significant impact we estimated for E-PI storms (Table 1). Water lifting constitutes part of the total work of the cycle, which cannot be larger than ε times the heat input. Since latent heat is a major part of heat input, its product with efficiency approximates total work. Both water lifting and latent heat input depend on the amount of evaporated water. Hence the ratio of the water lifting to total work is roughly equal to the potential energy of precipitation gz_P divided by the product of latent heat and efficiency, K₁/ε ∼ gz_P/(εL_υ). With K₁ ∼ 10⁻² and ε ∼ 10⁻² the energy needed to lift water can exceed the total work of the cycle.

The infinitely narrow E-PI cycle is not a real steady-state cycle where evaporation equals precipitation, as the air does not descend along the adiabat C′B′ (Fig. 1). Here, instead, the water formally arises anew with its own nonzero gravitational energy. For this reason, the water lifting term in E-PI is proportional to local evaporation $(Δ q^{*})$ rather than precipitation $(q_{b}^{*} - q_{c}^{*})$ . In real storms, precipitation P at the point of maximum wind is significantly higher than evaporation E. Replacement of E with P caused Sabuwala et al.’s (2015) overestimate of the water lifting term (section 3e).

If the hurricane is composed of closed streamlines, each with a constant q_t (the reversible case), the total work performed on lifting the water in such a storm is zero (section 3a). However, since the E-PI cycle is not a cycle along which the air moves, the water lifting term here is higher in the reversible case than in the pseudoadiabatic case. This is due to the higher effective precipitation height in the former [z_c > aH_P, Eq. (C7)]. Accounting for the water warming, which requires information about the cycle’s thermodynamics (Fig. 3), reduces this difference. For T_b ≃ 300 K and the largest observed ε_C ≃ 0.3 (DeMaria and Kaplan 1994), the magnitudes of βK₁/ε_C ≃ 0.1 for reversible and pseudoadiabatic cases are similar. This corresponds to a 5% reduction of V_max [Eq. (37)]. This reduction is larger for smaller ε_C but smaller for lower T_b (Table C1). The developed analytical framework can be used to evaluate corresponding magnitudes for different scenarios in numerical models.

Our analysis clarifies that ε in E-PI is not the actual efficiency of the cycle but the ratio of total work to heat input at B′b (Figs. 2 and 3). With additional heat inputs elsewhere in the cycle, ε in Eq. (7) can be higher than Carnot efficiency [cf. Eq. (24)]. This helps understand the phenomenon of “superintensity.” When the adiabaticity is violated near the tropopause due to an extra heat input, E-PI’s Eqs. (27) and (37) can significantly underestimate α∂p/∂r and the squared maximum velocity (for details, see Makarieva and Nefiodov 2021). Since at the tropopause q_t is approximately zero, this extra heat input will not affect the q_t distribution and, hence, will make the relative water lifting impact even smaller (same absolute magnitude of K₁ related to greater total work).

Our derivations exposed the sensitivity of $V_{\max}^{2}$ to E-PI’s key assumption about how the gradients of respective variables at the point of maximum wind relate to their local surface fluxes. If Eq. (20) is applied, the extension of E-PI to unbalanced winds consists in dividing the conventional E-PI squared velocity by factor $B$ [Eq. (30)] that describes the deviation from gradient-wind balance [for how this relates to the analysis of Bryan and Rotunno (2009a), see Makarieva and Nefiodov (2021)]. If, on the other hand, Eq. (10) is applied, the gradient-wind imbalance does not affect the E-PI formulation, see Eq. (37).

The choice of the outflow point c as a point where υ_c = 0, i.e., putting K₂ = 0, is equivalent to postulating that, for a cycle including the point of maximum wind, net work in the free troposphere is zero [see Eq. (8) in Fig. 2]. Since generally in the free troposphere the outflowing air has to move against the inward-pulling horizontal pressure gradient, compensating for this negative work requires extra warming at bc compared to C′B′. This extra warming is provided either by a higher mixing ratio at b compared to B′, or by a higher temperature, or by both. This constraint takes the form of the dependence between the outflow temperature, the mixing ratio, and the ratio of the horizontal gradients of temperature and pressure, see Eqs. (48) and (49). It follows that attempts to retrieve total pressure fall from E-PI by assuming ∂T/∂r = 0 cannot yield correct results under the conventional assumption of K₂ = 0 (cf. Emanuel 1986, p. 588).

When the horizontal gradient of air temperature is moist adiabatic (which corresponds to zero heat input), Eq. (48) reduces to υ_b/r_b = υ_c/r_c. This constancy of angular velocity combined with angular momentum conservation along bc gives two solutions [see Eq. (33)]. One is trivial, r_b = r_c. Another one is ν_b = ν_c = −f/2; it describes an atmosphere at rest in the inertial frame of reference. In either case, there is no storm. However, we now know that, at least in models, it is possible to have a tropical cyclone with zero heat input from the ocean (Kieu et al. 2020), although further tapering with the conventional model parameters might be required to make such a cyclone more stable. This prompts reconsidering the relevance of the local approach for the determination of maximum potential intensity.

A nonlocal constraint on the work in the free troposphere resulting from E-PI can be applied to the integral cycle ABCDA. In this case, we cannot put ∂V²/∂r = 0 in Eq. (8). The work in the free troposphere is equal to the nonzero increment of the kinetic energy in the boundary layer. Having reached the eyewall, the air must then have sufficient energy to flow away from the hurricane. If not generated in the boundary layer, this energy could derive from a pressure gradient in the upper atmosphere: if at the expense of the hurricane’s extra warmth the air pressure in the column above the area of maximum wind is higher than in the ambient environment, this pressure gradient will accelerate the air outward. However, a significant pressure deficit at the surface precludes the formation of a significant pressure surplus aloft (e.g., Makarieva et al. 2017d, Fig. 1d).

Moreover, this pressure deficit is what accelerates air in the boundary layer. If the pressure gradient is sufficiently steep and the radial motion sufficiently rapid, air expansion will be accompanied by a drop of temperature. The process is closer to an adiabat than to an isotherm, as it was, for example, in Hurricane Isabel 2003 (Montgomery et al. 2006; Aberson et al. 2006; Makarieva and Nefiodov 2021). As the warm air creates a pressure surplus aloft facilitating the outflow, cold air creates a pressure deficit. This enhances the pressure gradient in the upper atmosphere against which the air must work to leave the hurricane. Consequently, the storm cannot deepen indefinitely. Eventually, the kinetic energy acquired in the boundary layer becomes insufficient for the rising and adiabatically cooling air to overcome the pressure gradient in the upper atmosphere, and the outflow must weaken. This condition could provide distinct constraints on storm intensity. Further research is needed to see whether such processes are relevant in real storms.

Acknowledgments.

We are grateful to three anonymous referees for their useful comments. Work of A. M. Makarieva is partially funded by the Federal Ministry of Education and Research (BMBF) and the Free State of Bavaria under the Excellence Strategy of the Federal Government and the Länder, as well as by the Technical University of Munich–Institute for Advanced Study.

Data availability statement.

There were no raw data utilized in this study.

APPENDIX A

Deriving Eq. (3) from the Equations of Motion

One of our reviewers stated that Eq. (3), stemming from the Bernoulli equation, could not be obtained assuming hydrostatic equilibrium, (1), alone, but additionally requires the gradient-wind balance. To show that it is not the case and to facilitate a comparison of our approach with the available studies (e.g., Rotunno and Bryan 2012), we follow the reviewer’s suggestion to derive the Bernoulli equation from the equations of motion. These equations in a reference frame rotating with angular velocity Ω can be cast into the vector form as follows (see, e.g., Lorenz 1967; Vallis 2006):

\frac{d V}{d t} = - α \nabla p - 2 [Ω \times V] - \nabla Φ + F .

(A1)

Here, V is the total velocity of air motion, relative to rotating Earth; α ≡ 1/ρ is the specific volume; p is air pressure. On the right-hand side of Eq. (A1), the first term is the pressure-gradient force acting on an air parcel of unit mass from the side of its surrounding air; the second term describes the Coriolis acceleration; F is frictional force per unit mass. The geopotential Φ is defined such that g = −∇Φ, where g is the effective gravity, which in addition to the acceleration due to gravity also takes into account the centrifugal acceleration.

The material derivative is given by

\frac{d}{d t} = \frac{\partial}{\partial t} + (V \cdot \nabla) .

(A2)

Using the relation (V ⋅ ∇)V = ∇V²/2 − V × [∇ × V], one can write Eq. (A1) for the case of a steady state (∂V/∂t = 0) as

- α \nabla p = \frac{1}{2} \nabla V^{2} + [ω \times V] - g - F,

(A3)

where ω ≡ [∇ × V] + 2Ω is the absolute vorticity.

Let us find the inner product of Eq. (A3) and vector dl. The equality

[ω \times V] \cdot d l = 0

(A4)

holds if dl is the length element along a streamline, i.e., by definition is tangent to the local velocity V = dl/dt. Indeed, Eq. (A4) follows from the relations

[ω \times V] \cdot V = ω \cdot [V \times V] = 0 .

(A5)

As a result, one obtains the Bernoulli equation

- α \nabla p \cdot d l = \frac{1}{2} \nabla V^{2} \cdot d l - g \cdot d l - F \cdot d l,

(A6)

which is valid along a streamline. The meaning of Eq. (A4) is that the force [ω × V] (per unit mass) caused by the vorticity (including the Coriolis force) does not perform work and, accordingly, it does not change the energy of moving air parcel.

If the air motion possesses axial symmetry, it is convenient to use the cylindrical coordinate system with the basis vectors e_r, e_θ, and e_z. The position vector of a point r is now characterized by three components, namely, by radial distance r, azimuth θ, and height z. The length element of a streamline and the gradient operator read

d l = d r e_{r} + r d θ e_{θ} + d z e_{z} = (u e_{r} + υ e_{θ} + w e_{z}) d t = V d t,

(A7)

\nabla \equiv \frac{\partial}{\partial l} = e_{r} \frac{\partial}{\partial r} + e_{θ} \frac{1}{r} \frac{\partial}{\partial θ} + e_{z} \frac{\partial}{\partial z} .

(A8)

The curl of velocity (relative vorticity) may be written in the form of a determinant

curl V \equiv [\nabla \times V] = \frac{1}{r} | \begin{matrix} e_{r} & r e_{θ} & e_{z} \\ \partial / \partial r & \partial / \partial θ & \partial / \partial z \\ u & r υ & w \end{matrix} | = (\frac{1}{r} \frac{\partial w}{\partial θ} - \frac{\partial υ}{\partial z}) e_{r} + (\frac{\partial u}{\partial z} - \frac{\partial w}{\partial r}) e_{θ} + \frac{1}{r} [\frac{\partial (r υ)}{\partial r} - \frac{\partial u}{\partial θ}] e_{z} .

(A9)

The geopotential is given by Φ = gz, so that g = −∇Φ = −ge_z. It is usual to assume that the contribution of the Coriolis force to the vertical (z) component of the equations of motion is small with respect to the contribution of the centrifugal force and can be neglected. Then ω ≃ [∇ × V] + fe_z, where f = 2Ω sinφ is the Coriolis parameter (Ω = 2π/T is the rotation rate of Earth, T = 24 h is the rotation period of Earth and φ is latitude).

For axisymmetric motion, p and V are independent of the angle θ. In this case, the cylindrical components of Eq. (A3) have the form

- α \frac{\partial p}{\partial r} = \frac{1}{2} \frac{\partial V^{2}}{\partial r} + w ω_{θ} - υ ω_{z} - F_{r},

(A10a)

0 = u ω_{z} - w ω_{r} - F_{θ},

(A10b)

- α \frac{\partial p}{\partial z} = \frac{1}{2} \frac{\partial V^{2}}{\partial z} - u ω_{θ} + υ ω_{r} + g - F_{z},

(A10c)

where ω_r = −∂υ/∂z, ω_θ = ∂u/∂z − ∂w/∂r and ω_z = f + r⁻¹∂(rυ)/∂r. To derive the Bernoulli equation, we consider the streamline defined by dl; see Eq. (A7). Multiplying the above three equations by dr, rdθ, and dz, respectively, and summing them one obtains

- α \frac{\partial p}{\partial l} \cdot d l = \frac{1}{2} \frac{\partial V^{2}}{\partial l} \cdot d l + g d z - F \cdot d l,

(A11)

where ∂p/∂θ = 0 and ∂V/∂θ = 0. The cancellation of all terms involving the absolute vorticity components is a consequence of the more general formula (A4).

Applying hydrostatic equilibrium (1) and the condition ∂V/∂z = 0 turns Eq. (A10c) into −uω_θ + υω_r − F_z = 0 and the Bernoulli equation, Eq. (A11), into

- α \frac{\partial p}{\partial r} d r = \frac{1}{2} \frac{\partial V^{2}}{\partial r} d r - F \cdot d l,

(A12)

which yields our Eq. (3).

APPENDIX B

Extra Work due to Warming Precipitating Water: Deriving Eq. (23)

Saturated moist entropy per unit dry air mass is defined as [see Pauluis 2011, his Eq. (A4)

s^{*} = (c_{p d} + q_{t} c_{l}) \ln \frac{T}{T_{0}} - \frac{R}{M_{d}} \ln \frac{p_{d}}{p_{0}} + q^{*} \frac{L_{υ}}{T} .

(B1)

Here, L_υ = L_υ₀ + (c_pυ − c_l)(T − T₀) is the latent heat of vaporization (J kg⁻¹); $q^{*} = ρ_{υ}^{*} / ρ_{d}$ , q_l = ρ_l/ρ_d, and $q_{t} = q^{*} + q_{l}$ are the mixing ratio for saturated water vapor, liquid water, and total water, respectively; ρ_d, $ρ_{υ}^{*}$ , and ρ_l are the density of dry air, saturated water vapor, and liquid water, respectively; c_pd and c_pυ are the specific heat capacities of dry air and water vapor at constant pressure; c_l is the specific heat capacity of liquid water; R = 8.3 J mol⁻¹ K⁻¹ is the universal gas constant; M_d is the molar mass of dry air; p_d is the partial pressure of dry air; T is the temperature; p₀ and T₀ are reference air pressure and temperature.

The ideal gas law for the partial pressure p_υ = p − p_d of water vapor is

p_{υ} = N_{υ} R T, N_{υ} = \frac{ρ_{υ}}{M_{υ}},

(B2)

where M_υ and ρ_υ are the molar mass and density of water vapor, respectively. Using Eq. (B2) with

p_{υ} = p_{υ}^{*}

in the definition of

q^{*}

q^{*} \equiv \frac{ρ_{υ}^{*}}{ρ_{d}} = \frac{M_{υ}}{M_{d}} \frac{p_{υ}^{*}}{p_{d}} \equiv \frac{M_{υ}}{M_{d}} γ_{d}^{*}, γ_{d}^{*} \equiv \frac{p_{υ}^{*}}{p_{d}},

(B3)

and applying the Clausius–Clapeyron law

\frac{d p_{υ}^{*}}{p_{υ}^{*}} = ξ \frac{d T}{T}, L \equiv L_{υ} M_{υ}, ξ \equiv \frac{L}{R T},

(B4)

we obtain from Eq. (B1)

T d s^{*} = d (L_{υ} q^{*}) - α_{d} d p + (c_{p d} + c_{l} q_{t}) d T + c_{l} T \ln \frac{T}{T_{0}} d q_{t}

(B5a)

= L_{υ} d q^{*} - α_{d} d p + c_{p} d T + c_{l} T \ln \frac{T}{T_{0}} d q_{t},

(B5b)

where

c_{p} \equiv c_{p d} + q^{*} c_{p υ} + q_{l} c_{l}

and α_d ≡ 1/ρ_d. Integrating Eq. (B5a) with T₀ = T_b yields

- \oint α_{d} d p = \oint T d s^{*} - c_{l} \oint q_{t} d T - c_{l} \oint T \ln \frac{T}{T_{b}} d q_{t} .

(B6)

Choosing T₀ = T_b ensures that $T d s^{*}$ at B′b equals heat input.

If bc and C′B′ are reversible adiabats (dq_t = 0), we have

d s^{*} = 0

and

s_{b}^{*} = s_{c}^{*}

s_{B^{'}}^{*} = s_{C^{'}}^{*}

. Taking into account that paths B′b and cC′ are isotherms with temperatures T_b and T_c, respectively, we can write (Pauluis, 2011)

\int_{c}^{C^{'}} T d s^{*} = T_{c} (s_{C^{'}}^{*} - s_{c}^{*}) = T_{c} (s_{B^{'}}^{*} - s_{b}^{*}) = - \frac{T_{c}}{T_{b}} \int_{B^{'}}^{b} T d s^{*} $ .

(B7)

Then the first term on the right-hand side of Eq. (B6) can be written as follows:

\oint T d s^{*} = \int_{B^{'}}^{b} T d s^{*} + \int_{c}^{C^{'}} T d s^{*} = (1 - \frac{T_{c}}{T_{b}}) \int_{B^{'}}^{b} T d s^{*} .

(B8)

With q_t_b = q_t_c and q_t_B′ = q_t_C′, the second and third terms on the right-hand side of Eq. (B6) are

\oint q_{t} d T = - \oint T d q_{t} = \int_{C^{'}}^{c} T d q_{t} - \int_{B^{'}}^{b} T d q_{t} = Δ q_{t} (T_{c} - T_{b}),

(B9)

\oint T \ln \frac{T}{T_{b}} d q_{t} = T_{c} \ln \frac{T_{c}}{T_{b}} \int_{c}^{C^{'}} d q_{t} = - Δ q_{t} T_{c} \ln \frac{T_{c}}{T_{b}},

(B10)

where Δq_t ≡ q_t_b − q_t_B′ = q_t_c − q_t_C′.

Using Eqs. (B8)–(B10), integral (B6) can be cast into the following form:

- \oint α_{d} d p = (1 - \frac{T_{c}}{T_{b}}) \int_{B^{'}}^{b} T d s^{*} + c_{l} Δ q_{t} (T_{b} - T_{c} + T_{c} \ln \frac{T_{c}}{T_{b}}) = ε_{C} \int_{B^{'}}^{b} T d s^{*} + c_{l} T_{b} Δ q_{t} [ε_{C} + (1 - ε_{C}) \ln (1 - ε_{C})] .

(B11)

Taking into account that for

ε_{C} = 1 - T_{c} / T_{b} ≲ 1

ε_{C} + (1 - ε_{C}) \ln (1 - ε_{C}) ≃ \frac{ε_{C}^{2}}{2},

(B12)

we obtain Eq. (23) from Eq. (B11). For the largest observed ε_C = 0.35, approximation (B12) is a 13% underestimate.

For an infinitely narrow cycle, with B′ → b, Eqs. (B8)–(B11) are valid even if B′b and cC′ are not isotherms, due to the smallness of temperature change on these paths as compared to the finite difference T_b − T_c.

If bc and C′B′ are pseudoadiabats, we have

q_{t} = q^{*}

and δQ = 0, i.e., the sum of the first three terms in Eq. (B5b) is zero:

L_{υ} d q^{*} - α_{d} d p + c_{p} d T = 0 $ .

(B13)

In view of Eq. (B13), integrating ds^* (B5b) with T₀ = T_b over the closed contour B′bcC′B′ yields

\int_{c}^{C^{'}} d s^{*} = - \frac{1}{T_{b}} \int_{B^{'}}^{b} T d s^{*} - c_{l} \int_{b}^{c} \ln \frac{T}{T_{b}} d q^{*} - c_{l} \int_{C^{'}}^{B^{'}} \ln \frac{T}{T_{b}} d q^{*} .

(B14)

Using Eqs. (B13), (B5b) and (B14), Eq. (B6) is evaluated as follows:

- \oint α_{d} d p = ε_{C} \int_{B^{'}}^{b} T d s^{*} + c_{l} T_{b} Δ q^{*} + c_{l} \int_{↷} (T - T_{c} \ln \frac{T}{T_{b}}) d q^{*} .

(B15)

Here

Δ q^{*} \equiv q_{b}^{*} - q_{B^{'}}^{*}

and symbol ↷ denotes integration over unclosed contour bcC′B′. We estimate the last integral by assuming that ln T/T_b is a slowly varying function with respect to T, so that it can be taken out of the integral at the mean temperature T = T_P defined in Eq. (28). Then the water warming term in the pseudoadiabatic case is, cf. Eq. (B11),

c_{l} Δ q^{*} (T_{b} - T_{P} + T_{c} \ln \frac{T_{P}}{T_{b}}) ≃ \frac{ε_{P}}{2} c_{l} (T_{b} - T_{P}) Δ q^{*},

(B16)

where ε_P ≡ (T_b + T_P − 2T_c)/T_b. We discuss the accuracy of this expression in the end of the next section.

To obtain Eq. (40), we use Eqs. (B3) and (B4) to exclude the temperature differential from Eq. (B5):

\frac{d q^{*}}{q^{*}} = \frac{d γ_{d}^{*}}{γ_{d}^{*}} = (1 + γ_{d}^{*}) (\frac{d p_{υ}^{*}}{p_{υ}^{*}} - \frac{d p}{p}) = (1 + γ_{d}^{*}) (ξ \frac{d T}{T} - \frac{d p}{p}),

(B17)

d T = (\frac{d q^{*}}{q^{*}} + \frac{d p}{p_{d}}) \frac{T}{ξ (1 + γ_{d}^{*})} = (\frac{d q^{*}}{q^{*}} + α_{d} \frac{M_{d}}{R T} d p) \frac{T}{ξ (1 + γ_{d}^{*})} .

(B18)

Putting Eq. (B18) into Eq. (B5b) with T₀ = T and grouping the

d q^{*}

and dp terms yields Eq. (40). Combining Eqs. (36), (34), and (44) we obtain for β in Eq. (36)

β = \frac{1 - ε_{C} (1 - ϰ_{2}) - K_{2}}{(1 - K_{2}) (1 + ϰ_{1}) - K_{1} (1 - ϰ_{2})} .

(B19)

APPENDIX C

Water Lifting: Deriving Eq. (27)

In Eq. (26), we represent the last but one term on the right-hand side as

\oint α q_{t} d p = \int_{B^{'}}^{b} α q^{*} \frac{\partial p}{\partial r} d r - \int_{b}^{c} q_{t} d (\frac{V^{2}}{2} + g z) - \int_{c}^{C^{'}} q_{t} g d z - \int_{C^{'}}^{B^{'}} q_{t} d (\frac{V^{2}}{2} + g z)

(C1a)

= \int_{B^{'}}^{b} α q^{*} \frac{\partial p}{\partial r} d r - q_{t} {\frac{V^{2}}{2} |}_{b}^{c} - q_{t} {\frac{V^{2}}{2} |}_{C^{'}}^{B^{'}} + \int_{b}^{c} \frac{V^{2}}{2} d q_{t} + \int_{C^{'}}^{B^{'}} \frac{V^{2}}{2} d q_{t} - \int_{↷} g q_{t} d z

(C1b)

= \int_{B^{'}}^{b} α q^{*} \frac{\partial p}{\partial r} d r + q^{*} {\frac{V^{2}}{2} |}_{B^{'}}^{b} + q_{t} {\frac{V^{2}}{2} |}_{c}^{C^{'}} + \int_{b}^{c} \frac{V^{2}}{2} d q_{t} + \int_{C^{'}}^{B^{'}} \frac{V^{2}}{2} d q_{t} - \int_{↷} g q_{t} d z

(C1c)

= \int_{B^{'}}^{b} q^{*} (α \frac{\partial p}{\partial r} + \frac{1}{2} \frac{\partial V^{2}}{\partial r}) d r + \int_{B^{'}}^{b} \frac{V^{2}}{2} \frac{\partial q^{*}}{\partial r} d r + q_{t} {\frac{V^{2}}{2} |}_{c}^{C^{'}}_{}^{} + \int_{b}^{c} \frac{V^{2}}{2} d q_{t} + \int_{C^{'}}^{B^{'}} \frac{V^{2}}{2} d q_{t} - \int_{↷} g q_{t} d z

(C1d)

= \int_{B^{'}}^{b} q^{*} (α \frac{\partial p}{\partial r} + \frac{1}{2} \frac{\partial V^{2}}{\partial r}) d r + \int_{B^{'}}^{b} \frac{V^{2}}{2} \frac{\partial q^{*}}{\partial r} d r + \int_{c}^{C^{'}} \frac{q_{t}}{2} \frac{\partial V^{2}}{\partial z} d z + \int_{↷} \frac{V^{2}}{2} d q_{t} - \int_{↷} g q_{t} d z .

(C1e)

Here it is assumed that q_l = 0 at the isotherm B′b. In Eq. (C1a), we have used the Bernoulli equation for the two streamlines, bc and B′C′, and the hydrostatic equilibrium Eq. (1) for the vertical path cC′. The unclosed contour bcC′B′ is denoted by symbol $↷$ . Note that z_B′ = z_b and r_c = r_C_′.

Writing the last integral in Eq. (C1e) as

{\int_{↷} q_{t} d z = q^{*} z |}_{b}^{B^{'}} - \int_{↷} z d q_{t} \equiv (z_{P} - z_{b}) Δ q^{*},

(C2)

we obtain

\oint α q_{t} d p = - \int_{b}^{B^{'}} q^{*} (α \frac{\partial p}{\partial r} + \frac{1}{2} \frac{\partial V^{2}}{\partial r}) d r - \int_{b}^{B^{'}} \frac{V^{2}}{2} \frac{\partial q^{*}}{\partial r} d r - [g (z_{P} - z_{b}) + \frac{V_{P}^{2}}{2}] Δ q^{*} + \int_{c}^{C^{'}} \frac{q_{t}}{2} \frac{\partial V^{2}}{\partial z} d z .

(C3)

Here $Δ q^{*} \equiv q_{b}^{*} - q_{B^{'}}^{*}$ and z_P and $V_{P}^{2}$ are defined in Eq. (28).

Putting Eq. (C3) into Eq. (26) yields

\int_{b}^{B^{'}} (1 + q^{*}) (α \frac{\partial p}{\partial r} + \frac{1}{2} \frac{\partial V^{2}}{\partial r}) d r = - ε_{C} \int_{b}^{B^{'}} T \frac{\partial s^{*}}{\partial r} d r - c_{l} (T_{b} - T_{P}) \frac{ε_{P}}{2} \int_{b}^{B^{'}} \frac{\partial q^{*}}{\partial r} d r - \int_{b}^{B^{'}} \frac{V^{2}}{2} \frac{\partial q^{*}}{\partial r} d r - [g (z_{P} - z_{b}) + \frac{V_{P}^{2}}{2}] Δ q^{*} + \int_{c}^{C^{'}} (1 + q_{t}) \frac{1}{2} \frac{\partial V^{2}}{\partial z} d z .

(C4)

In the limit r_B′ → r_b and z_C′ → z_c we can lift the integral signs in Eq. (C4) and divide both parts of the equation by dr to obtain Eq. (27). Note that by chain rule $(\partial V_{c} / \partial z) d z |_{r = r_{c}} = (\partial V_{c} / \partial r) d r |_{z = z_{b}}$ .

To evaluate

V_{P}^{2}

, T_P and z_P for the case

q_{t} = q^{*}

, we can express the corresponding definitions (28) as follows:

X_{P} Δ q^{*} \equiv - \int_{↷} X d q^{*} = {\bar{X}}_{bc} (q_{b}^{*} - q_{c}^{*}) + {\bar{X}}_{c C^{'}} (q_{c}^{*} - q_{C^{'}}^{*}) + {\bar{X}}_{C^{'} B^{'}} (q_{C^{'}}^{*} - q_{B^{'}}^{*}),

(C5a)

{\bar{X}}_{k j} \equiv - \frac{1}{q_{k}^{*} - q_{j}^{*}} \int_{k}^{j} X d q^{*},

(C5b)

where

Δ q^{*} \equiv q_{b}^{*} - q_{B^{'}}^{*}

, X = {V², T, z} and

{\bar{X}}_{kj}

is the average value of X over path kj.

In the limit r_B′ → r_b and z_C′ → z_c, when the two streamlines coincide, we have from Eq. (C5)

X_{P} = {\bar{X}}_{bc} + \frac{\partial {\bar{X}}_{bc}}{\partial q_{b}^{*}} (q_{b}^{*} - q_{c}^{*}) + (X_{c} - {\bar{X}}_{bc}) \frac{q_{c}^{*} - q_{C^{'}}^{*}}{q_{b}^{*} - q_{B^{'}}^{*}},

(C6)

where

X_{c} = {\bar{X}}_{c C^{'}}

. When

(q_{c}^{*} - q_{C^{'}}^{*}) / (q_{b}^{*} - q_{B^{'}}^{*}) \sim q_{c}^{*} / q_{b}^{*} ≪ 1

(which is always the case when z_c is sufficiently large), the last term on the right-hand part of Eq. (C6) can be neglected.

Due to the Clausius–Clapeyron law (B4), the relative change of temperature is about ξ⁻¹ ∼ 0.05 of the relative change of $p_{υ}^{*}$ and, hence, of $q^{*} : q_{b}^{*} \partial {\bar{T}}_{bc} / \partial q_{b}^{*} \sim {\bar{T}}_{bc} / ξ$ . Therefore, for X = T, the second term on the right-hand side of Eq. (C6) can be neglected, and $T_{P} = {\bar{T}}_{bc}$ , which is the mean temperature of precipitation along bc.

With the mean temperature of precipitation approximately independent of

q_{b}^{*}

, the mean precipitation height

H_{P} \equiv {\bar{z}}_{b c}

scales as

H_{P} \sim T_{P} / Γ \sim q_{b}^{*}

, because for sufficiently large

q^{*}

the moist adiabatic lapse rate Γ is approximately inversely proportional to

q^{*}

[e.g., Makarieva and Nefiodov 2021, their Eq. (A10). Therefore, for X = z the second term in Eq. (C6) is approximately equal to the first one, such that

z_{P} = a H_{P}

(C7)

with a ∼ 2. For more accurate estimates, we evaluated hydrostatic saturated reversible adiabatic and pseudoadiabatic profiles with surface pressure p_s = 950 hPa, z_b = 1 km and variable T_b and z_c (Table C1). For the same z_c, the values of T_c are different, since the reversible moist adiabatic lapse rate is smaller than the pseudoadiabatic one (e.g., Mapes 2001, Fig. 1a). The dependence of H_P on $q^{*}$ in Eq. (C6) was evaluated by varying p_s at constant surface temperature T_s. The same calculations show that the approximate Eq. (B16) for the pseudoadiabatic water warming deviates by less than 5% from the exact value [sum of the last two terms in Eq. (B15)].

Table C1.

Estimates of $H_{P} = {\bar{z}}_{bc}$ in (C5b), X_P = {T_P, z_P} in (C6), a in (C7), β in (B19), and K₁ in (29) for hydrostatic pseudoadiabatic (in boldface) and reversible adiabatic profiles with surface temperature T_s, surface pressure p_s = 950 hPa, and z_b = 1 km.

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K. Emanuel made this suggestion in his signed review of (subsequently rejected) submission of Makarieva et al. (2018) to J. Geophys. Res. Atmos.

In the literature one can sometimes find a loose definition of a reversible process that only assumes q_t = const but allows the relative humidity to vary [e.g., Bryan and Rotunno 2009b, Eq. (23)].

Equation (B3) of Pauluis (2011) should have T_in instead of T_out in the denominator of the right-hand part; otherwise, it contradicts Eq. (B2) from which supposedly derives.

Term $(V_{P}^{2} - V^{2}) / 2$ can be explicitly accounted for by specifying the interaction between condensate and air (i.e., introducing a specific term to the equations of motion and Bernoulli equation). If condensate is assumed to have the same horizontal velocity as air (see, e.g., Ooyama 2001; Makarieva et al. 2017b), then as the precipitating condensate leaves the air at the surface, it will have the same velocity as the newly evaporated water vapor. The net contribution of this term to the total power budget of the hurricane will be exact zero (see Makarieva et al. 2017c, Fig. 1).

This conclusion stemmed from Emanuel’s (1986) Eq. (18), where an outflow term proportional to a large squared radius first appeared. While deriving this equation for a cycle with finite B′b, Emanuel (1986, p. 588), on the one hand, used the conservation of angular momentum along streamlines bc and B′C′ and, on the other hand, assumed r_B′ (interpreted as “the radial extent of the storm near the sea level”) to be large enough for ∂p/∂r to vanish and, at the same time, small enough for r∂p/∂r|_b ≫ r∂p/∂r|_B′ (for details, see Makarieva et al. 2019, their appendix A). With r∂p/∂r ∼ ρυ², ignoring this term at B′ means that $V_{b}^{2} - V_{B^{'}}^{2} \sim V_{b}^{2}$ , while at the point of maximum wind this difference is zero.

Note the following differences in notations between Sabuwala et al. (2015) and the present work: T_s → T_b, ${\dot{Q}}_{in} \to J$ , ${\dot{Q}}_{d} \to D$ , P → W_P = gH_PP. Factor 1/(1 − ε_C) is due to dissipative heating.

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Water Lifting and Outflow Gain of Kinetic Energy in Tropical Cyclones

Abstract

Abstract

1. Introduction

2. An infinitely narrow thermodynamic cycle

a. Combining dynamics and thermodynamics

b. Conventional E-PI estimate

3. Estimating the water lifting and outflow terms

a. Reversible and pseudoadiabatic hurricanes

b. Extra heat input to warm precipitating water

c. Water lifting

d. The outflow gain of kinetic energy

e. Estimates of maximum velocity

4. The physical meaning of E-PI at the point of maximum wind

5. Conclusions

Acknowledgments.

Data availability statement.

APPENDIX A

Deriving Eq. (3) from the Equations of Motion

APPENDIX B

Extra Work due to Warming Precipitating Water: Deriving Eq. (23)

APPENDIX C

Water Lifting: Deriving Eq. (27)

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Water Lifting and Outflow Gain of Kinetic Energy in Tropical Cyclones

Abstract

Abstract

1. Introduction

2. An infinitely narrow thermodynamic cycle

a. Combining dynamics and thermodynamics

b. Conventional E-PI estimate

3. Estimating the water lifting and outflow terms

a. Reversible and pseudoadiabatic hurricanes

b. Extra heat input to warm precipitating water

c. Water lifting

d. The outflow gain of kinetic energy

e. Estimates of maximum velocity

4. The physical meaning of E-PI at the point of maximum wind

5. Conclusions

Acknowledgments.

Data availability statement.

APPENDIX A

Deriving Eq. (3) from the Equations of Motion

APPENDIX B

Extra Work due to Warming Precipitating Water: Deriving Eq. (23)

APPENDIX C

Water Lifting: Deriving Eq. (27)

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