[1,2]Anastassia M.Makarieva [1]Victor G.Gorshkov [1]Andrei V.Nefiodov
[3]Alexander V. Chikunov [4,5]DouglasSheil [6]Antonio DonatoNobre
[7]PauloNobre [8]GünterPlunien [9]Ruben D.Molina

1]Theoretical Physics Division, Petersburg Nuclear Physics Institute, Gatchina
188300, St. Petersburg, Russia 2]Institute for Advanced Study, Technical
University of Munich, Lichtenbergstrasse 2 a, D-85748 Garching, Germany
3]Princeton Institute of Life Sciences, Princeton, New Jersey 08540, USA
4]Forest Ecology and Forest Management Group, Wageningen University &
Research, PO Box 47, 6700 AA, Wageningen, The Netherlands 5]Faculty of
Environmental Sciences and Natural Resource Management, Norwegian University
of Life Sciences, Ås, Norway 6]Centro de Ciência do Sistema Terrestre INPE,
São José dos Campos, São Paulo 12227-010, Brazil 7]Center for Weather Forecast
and Climate Studies INPE, São José dos Campos, São Paulo 12227-010, Brazil
8]Institut für Theoretische Physik, Technische Universität Dresden, Dresden
01069, Germany 9]Escuela Ambiental, Facultad de Ingeniería, Universidad de
Antioquia, Medellín, Colombia

A. M. Makarieva<EMAIL_ADDRESS>
# Water lifting and outflow gain of kinetic energy in tropical cyclones

###### 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.

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 path length $H_{P}$ (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.

Author | Context | Value ($\%$) | Comment
---|---|---|---
Gorshkov (1982, p. 6), Gorshkov (1995, Table 2.1) |  Atmospheric circulation over land |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 20$ |  Precipitation over land $P=0.5$ t m-2 yr-1 (L’vovitch, 1979) falling from mid troposphere $H_{P}\sim 5$ km gives $S_{L}gPH_{P}=1.2\times 10^{14}$ W for total land area $S_{L}=1.5\times 10^{14}$ m2 or $\sim 20\%$ of total atmospheric power equal to the sum of $W_{P}=gPH_{P}=0.8$ W m-2 and kinetic energy dissipation $W_{K}=3.5$ W m-2. The latter value corresponds to $1\%$ contribution of total solar flux (Gustavson, 1979).
Emanuel (1988, Fig. 3a vs. Fig. 4a) |  Tropical cyclone, reversible ascent |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 20$ |  A storm that lifts all condensed water from the surface with relative humidity $80\%$ and temperature $T_{s}=30$℃ to the atmospheric layer with temperature $T_{o}\simeq-73$℃ is theoretically estimated to develop a 20% lower pressure drop ($\simeq 96$ hPa, from $1013$ hPa to $917$ hPa) than would a similar storm without lifting water ($\simeq 118$ hPa, from $1013$ hPa to $895$ hPa).
Emanuel (1988, p. 1149) |  Tropical cyclone, pseudoadiabatic ascent |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 5$ |  A storm where condensed water precipitates from the point of condensation is theoretically estimated to develop a 5% lower pressure drop than would a similar storm without lifting water.
Pauluis et al. (2000); Pauluis and Dias (2012) |  Tropical convection |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 50$ |  Satellite-derived $W_{P}=1.5$ W m-2 (Pauluis and Dias, 2012) and $W_{K}=1.4$ W m-2 (Pauluis et al., 2000) correspond to $W_{P}/(W_{P}+W_{K})\times 100\%\simeq 50\%.$
Sabuwala et al. (2015, Fig. 4a) |  Tropical cyclone, pseudoadiabatic ascent |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 50$ |  A hurricane with maximum velocity $\simeq 50$ m s-1 is theoretically estimated to develop a $50\%$ lower squared maximum velocity than would a similar storm without lifting water.
Makarieva et al. (2013, 2017c) |  Global atmospheric circulation |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 30$ |  $W_{K}=2.3$ W m-2 based on instantaneous MERRA re-analysis data and $W_{P}=1$ W m-2 based on observations and theoretical estimates correspond to $W_{P}/(W_{P}+W_{K})\times 100\%\simeq 30\%.$
Rousseau-Rizzi and Emanuel (2019) |  Tropical cyclone, reversible cycle |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 0$ |  Condensed water does not precipitate but travels with the air parcels both up and down, net water lifting power (the integral over $dq_{t}$ in Eq. (13) of Rousseau-Rizzi and Emanuel, 2019) is zero.
This work |  E-PI cyclone with $\varepsilon_{C}\simeq 0.3$ |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 10$ |  Cumulative account of water lifting and water warming, Eq. (29), reduces squared maximum velocity by about $10\%$ in both reversible and pseudoadiabatic storms.

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 exact zero (Table 1).

The preceding issues raise several questions. First, is the water lifting
impact 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, Eq. (13); Emanuel
and Rotunno 2011, Eq. (11)). The derivation assumed the free troposphere to be
in gradient-wind balance. Makarieva et al. (2018, their Fig. 1, Eqs. (p5) and
(p6)) showed that this assumption can be relaxed in the assessment of storm-
integrated energy fluxes. Kerry Emanuel suggested111K. Emanuel made this
suggestion in his signed review of (subsequently rejected) submission of
Makarieva et al. (2018) to the Journal of Geophysical Research: Atmospheres.
that Makarieva et al. (2018)’s 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
the original Emanuel (1986)’s derivation (Section 1). The new, more general
formulation of E-PI framework is useful in the following three aspects.

First, by allowing an explicit evaluation of the water lifting term that we
perform in Section 2, 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 re-derive the result of Makarieva et al. (2019) about the
(in)significance of the outflow term at (large) small outflow radii. Emanuel
and Rousseau-Rizzi (2020)’s derivations were not conclusive, however, as they
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 2).

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 3). 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).

##  1 An infinitely narrow thermodynamic cycle

###  1.a Combining dynamics and thermodynamics

We consider two closed air streamlines, $\rm ABCDA$ and $\rm
A^{\prime}B^{\prime}C^{\prime}D^{\prime}A^{\prime}$ (Fig. 1), in an
axisymmetric atmosphere. The goal of our derivations 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
$\rm b^{\prime}Bb$ that belongs to streamline $\rm ABCDA$:

$\displaystyle-\alpha\frac{\partial p}{\partial z}=g,$ (1)
$\displaystyle-\alpha dp=d\left(\frac{V^{2}}{2}\right)+gdz-\mathbf{F}\cdot
d\mathbf{l},$ (2)

where $\alpha\equiv 1/\rho$, $\rho=\rho_{d}+\rho_{v}+\rho_{l}$ is the density
of moist air (including dry air $\rho_{d}$, water vapor $\rho_{v}$ and
condensed water $\rho_{l}$), $p$ is air pressure, $g$ is the acceleration of
gravity, $V$ is air velocity, $\mathbf{F}$ is the turbulent friction force per
unit air mass, and $d\mathbf{l}=\mathbf{V}dt$.

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

Figure 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;
$r_{\rm b^{\prime}}=r_{\rm B^{\prime}}$, $r_{\rm c}=r_{\rm C^{\prime}}$.
Points $\rm B$ and $\rm B^{\prime}$ are infinitely close and chosen in the
vicinity of maximum wind. The atmosphere is inviscid for $z>z_{\rm b}=z_{\rm
B^{\prime}}$. Lower atmosphere: Bernoulli equation, hydrostatic equilibrium,
$V_{\rm b^{\prime}}=V_{\rm B^{\prime}}$ $\displaystyle-\int_{\rm
b^{\prime}}^{\rm b}{\mathbf{F}}\cdot d{\mathbf{l}}=-\int_{\rm B^{\prime}}^{\rm
b}\alpha dp-\int_{\rm B^{\prime}}^{\rm b}d\left(\frac{V^{2}}{2}\right)$ (3)
Upper atmosphere: Bernoulli equation, hydrostatic equilibrium at ${\rm
cC^{\prime}}$, ${\mathbf{F}}=0$ $\displaystyle-\int_{\rm B^{\prime}}^{\rm
b}\alpha dp=-\oint\alpha dp+\int_{\rm B^{\prime}}^{\rm
b}d\left(\frac{V^{2}}{2}\right)+\int_{\rm c}^{\rm
C^{\prime}}d\left(\frac{V^{2}}{2}\right)$ (4) Wet and dry mass units:
$(1+q_{t})\alpha\equiv\alpha_{d}$, $(1+q_{t})\delta Q\equiv\delta Q_{d}$
$\displaystyle-\oint\alpha dp\equiv-\oint\alpha_{d}dp+\oint q_{t}\alpha dp$
(5) Thermodynamics: Link between work and heat input
$\displaystyle-\oint\alpha_{d}dp=\varepsilon\int_{\rm B^{\prime}}^{\rm
b}\delta Q_{d}$ (6) Link between volume-specific frictional dissipation and
heat input $\displaystyle-\int_{\rm b^{\prime}}^{\rm b}{\mathbf{F}}\cdot
d{\mathbf{l}}=\varepsilon\int_{\rm B^{\prime}}^{\rm b}\delta Q_{d}+\oint
q_{t}\alpha dp+\int_{\rm c}^{\rm C^{\prime}}d\left(\frac{V^{2}}{2}\right)$ (7)
Reversible cycle with $q_{t}=\rm const$, no kinetic energy change at $\rm
B^{\prime}b$ (point of maximum wind) and in the outflow $\rm cC^{\prime}$
$\displaystyle-\int_{\rm B^{\prime}}^{\rm b}\alpha dp=-\oint\alpha
dp=\varepsilon\int_{\rm B^{\prime}}^{\rm b}\delta Q$ (8) Figure 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 $\rm
B^{\prime}bcC^{\prime}B^{\prime}$ in Fig. 1. The heat input $\delta Q_{d}$ is
normalized per unit dry air mass.

Applying Eqs. (1) and (2) to $\rm b^{\prime}Bb$ 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 $V_{\rm
b^{\prime}}=V_{\rm B^{\prime}}$. This implies two possibilities. One is that
$\partial V/\partial z=0$, 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, 2009b, p.
3045). Another possibility is that points $\rm b^{\prime}$ and $\rm
B^{\prime}$ (and, respectively, $\rm B$ and $\rm b$) coincide, such that path
$\rm b^{\prime}b$ is horizontal. This second case with $z_{\rm b}=0$ was
considered by Rousseau-Rizzi and Emanuel (2019, their Fig. 1), who assumed
that $|\mathbf{F}|=0$ for $z>0$ but $|\mathbf{F}|\neq 0$ at $\rm B^{\prime}b$.
For a derivation of Eq. (3) from the equations of motion, see Appendix A.

Our next step is to consider the inviscid atmosphere above $z=z_{\rm b}$.
Applying the Bernoulli equation (2) with $|\mathbf{F}|=0$ to streamlines $\rm
bc$ and $\rm B^{\prime}C^{\prime}$ and assuming hydrostatic equilibrium at
path $\rm cC^{\prime}$ (which is not a streamline) yields Eq. (4). It relates
the horizontal change of $\alpha dp$ at $\rm B^{\prime}b$ to the sum of the
integral of $\alpha dp$ over the closed contour ${\rm
B^{\prime}bcC^{\prime}B^{\prime}}$ and the changes of kinetic energy at $\rm
B^{\prime}b$ and in the outflow region $\rm cC^{\prime}$ (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, $\alpha_{d}=(1+q_{t})\alpha$, where $\alpha_{d}\equiv 1/\rho_{d}$
is the specific volume of dry air, and
$q_{t}\equiv(\rho_{v}+\rho_{l})/\rho_{d}$ is the total water mixing ratio.
This relation allows us to link the integrals of $\alpha dp$ and
$\alpha_{d}dp$ over the closed contour $\rm B^{\prime}bcC^{\prime}B^{\prime}$,
Eq. (5).

On the other hand, the integral of $\alpha_{d}dp$ 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
$\varepsilon$, 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, Eq. (7).

In the particular case of $q_{t}=\rm const$, 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 $\rm b^{\prime}Bb$ and $\rm cC^{\prime}$, but it was not used at $\rm bc$
and $\rm B^{\prime}C^{\prime}$. The thermodynamic processes in the cycle were
not specified, so Eq. (6) can be viewed as defining the value of
$\varepsilon$. The choice of the outflow point $\rm c$ 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 $\rm B^{\prime}bcC^{\prime}B^{\prime}$
with $\partial V^{2}/\partial z=0$ at $\rm cC^{\prime}$ and $\partial
V^{2}/\partial r=0$ at $\rm B^{\prime}b$. It is also valid for a special case
of $\rm B^{\prime}bcC^{\prime}B^{\prime}$ being a closed streamline with
$V_{\rm b}=V_{\rm B^{\prime}}$ (Emanuel, 1988).

###  1.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, 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 $\alpha$ and $\alpha_{d}$ (and
accordingly between $\delta Q$ and $\delta Q_{d}$) 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, $V=v$, where $v$ is tangential velocity, and then chose
point $\rm c$ in the outflow where $V=v=0$ and, hence, $\partial
V^{2}/\partial z=0$. Finally, Emanuel (1986) considered the thermodynamic
cycle to be reversible, such that its efficiency $\varepsilon$ equals Carnot
efficiency $\varepsilon_{C}\equiv(T_{\rm b}-T_{\rm c})/T_{\rm b}$.

Applying these assumptions –neglecting the last two terms and putting
$\varepsilon=\varepsilon_{C}$ and $\delta Q_{d}=T_{\rm b}ds^{*}$ in Eq. (7)–
we lift the integral signs in the limit of the infinitely narrow cycle $\rm
b^{\prime}\to B^{\prime}$ and divide both sides of the equation by $dt$, to
obtain:

$-\mathbf{F}\cdot\mathbf{V}=\varepsilon_{C}\frac{\delta
Q_{d}}{dt}=\varepsilon_{C}T_{\rm b}\frac{\partial s^{*}}{\partial r}u,$ (9)

where $\mathbf{V}=d\mathbf{l}/dt$ is total air velocity, $u=dr/dt$ is radial
velocity, and $s^{*}$ is moist saturated entropy.

Multiplied by $\rho$, 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 $D=\rho C_{D}V^{3}$ and heat input $J=\rho
C_{k}V(k_{s}^{*}-k)$,

$-\frac{\mathbf{F}\cdot\mathbf{V}}{\delta Q_{d}/dt}=\frac{D}{J},$ (10)

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

$V_{\rm
max}^{2}=\frac{D}{J}\frac{C_{k}}{C_{D}}(k_{s}^{*}-k)=\varepsilon_{C}\frac{C_{k}}{C_{D}}(k_{s}^{*}-k).$
(11)

Here $C_{k}\simeq C_{D}$ 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),

$-\int_{\rm B^{\prime}}^{\rm b}\alpha dp-\int_{\rm B^{\prime}}^{\rm
b}d\left(\frac{V^{2}}{2}\right)=\varepsilon\int_{\rm B^{\prime}}^{\rm b}\delta
Q_{d}+\oint q_{t}\alpha dp+\int_{\rm c}^{\rm
C^{\prime}}d\left(\frac{V^{2}}{2}\right),$ (12)

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

$\frac{\nu_{1}-\nu_{2}}{T_{1}-T_{2}}=-\frac{ds^{*}}{dM},\quad(z\geq z_{\rm
b}),$ (13)

where $T_{1}$, $T_{2}$ and $\nu_{1}\equiv v_{1}/r_{1}$, $\nu_{2}\equiv
v_{2}/r_{2}$ are, respectively, air temperatures and angular velocities at
arbitrary distances $r_{1}$ and $r_{2}$ 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 $ds^{*}/dM$.

Using the definition of angular momentum

$M\equiv vr+\frac{fr^{2}}{2},$ (14)

where the Coriolis parameter $f\equiv 2\Omega\sin\varphi$ is assumed constant
($\varphi$ is latitude, $\Omega$ is the angular velocity of Earth’s rotation),
and assuming gradient-wind balance at point $\rm b$,

$\alpha\frac{\partial p}{\partial r}=\frac{v^{2}}{r}+fv,$ (15)

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

$\int^{\rm B^{\prime}}_{\rm b}\frac{v}{r}\frac{\partial M}{\partial
r}dr=-\varepsilon\int^{\rm B^{\prime}}_{\rm b}\delta
Q_{d}+\frac{1}{2}(v^{2}_{\rm C^{\prime}}-v^{2}_{\rm c}).$ (16)

Here we have assumed, as did Emanuel and Rousseau-Rizzi (2020), that the
change of velocity $V$ over path $\rm cC^{\prime}$ is dominated by the change
in tangential velocity $v$, $V^{2}_{\rm C^{\prime}}-V^{2}_{\rm c}=v^{2}_{\rm
C^{\prime}}-v^{2}_{\rm c}$. Since the atmosphere at $\rm cC^{\prime}$ is
frictionless and hydrostatic, it is equivalent to assuming local gradient-wind
balance (see Eq. (A10)). With path $\rm cC^{\prime}$ hydrostatic and in
gradient-wind balance, all our results are invariant with respect to its
orientation (whether/how $\rm cC^{\prime}$ 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 $r_{\rm B^{\prime}}\to r_{\rm
b}$. Since the atmosphere is frictionless above $z_{\rm b}$ and since the
pressure field is axisymmetric, paths $\rm bc$ and $\rm B^{\prime}C^{\prime}$
conserve angular momentum $M$. Using Eq. (14) for $M$ in the equation $M_{\rm
B^{\prime}}-M_{\rm b}=M_{\rm C^{\prime}}-M_{\rm c}$ and dividing this equation
by $r_{\rm c}(r_{\rm B^{\prime}}-r_{\rm b})$ we obtain

$\frac{1}{r_{\rm c}}\left[v_{\rm B^{\prime}}+r_{\rm b}\frac{v_{\rm
B^{\prime}}-v_{\rm b}}{r_{\rm B^{\prime}}-r_{\rm b}}+\frac{f}{2}(r_{\rm
B^{\prime}}+r_{\rm b})\right]=\frac{v_{\rm C^{\prime}}-v_{\rm c}}{r_{\rm
B^{\prime}}-r_{\rm b}}.$ (17)

Taking the limit $r_{\rm B^{\prime}}\to r_{\rm b}$ and multiplying Eq. (17) by
$v_{\rm c}$ we find

$v_{\rm c}\frac{\partial v_{\rm c}}{\partial r}=\frac{v_{\rm c}}{r_{\rm
c}}\left(v+r\frac{\partial v}{\partial r}+fr\right)=\frac{v_{\rm c}}{r_{\rm
c}}\frac{\partial M}{\partial r},\quad(z=z_{\rm b}).$ (18)

We now assume that our infinitely narrow cycle has Carnot efficiency
$\varepsilon_{C}=(T_{\rm b}-T_{\rm c})/T_{\rm b}$ and that the heat input at
$\rm B^{\prime}b$ can be expressed in terms of the increment of moist
saturated entropy $\delta Q_{d}=T_{\rm b}ds^{*}$. Lifting the integral signs
in Eq. (16) with the use of Eq. (18), we obtain

$\frac{v_{\rm b}}{r_{\rm b}}\frac{\partial M}{\partial r}=-(T_{\rm b}-T_{\rm
c})\frac{\partial s^{*}}{\partial r}+\frac{v_{\rm c}}{r_{\rm c}}\frac{\partial
M}{\partial r},\quad(z=z_{\rm b}).$ (19)

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

With point $\rm c$ chosen such that $v_{\rm c}=0$, Eqs. (13) and (19) are
equivalent to Emanuel (1986)’s Eq. (13) (see also Emanuel and Rotunno, 2011,
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
$\tau_{s}=J/T_{s}$ and $\tau_{M}=-Dr/V$, respectively:

$\frac{\partial s^{*}/\partial r}{\partial M/\partial
r}=\frac{\tau_{s}}{\tau_{M}}.$ (20)

Then assuming that $V\simeq v_{\rm b}$, $r\simeq r_{\rm b}$, and $T_{s}\simeq
T_{\rm b}$, 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 $\partial
V^{2}/\partial r=0$ (the condition of maximum wind). This peculiarity of E-PI
was noted by Montgomery and Smith (2017). Equation (9) requires $\partial
V^{2}/\partial z=0$ at $z=z_{b}$, while Eq. (19) does not.

As we will discuss in Section 3, $\partial V^{2}/\partial r=0$ 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 ${\rm
bcC^{\prime}B^{\prime}}$) is zero. The total work of the cycle, i.e., heat
input at $\rm B^{\prime}b$ multiplied by efficiency $\varepsilon_{C}$, equals
the work on $\rm B^{\prime}b$. Since $\varepsilon_{C}$ depends on the outflow
temperature $T_{\rm c}$, the specification of the thermodynamic process at
$\rm B^{\prime}b$ and the choice of $T_{\rm c}$ cannot be independent
(Makarieva and Nefiodov, 2021).

##  2 Estimating the water lifting and outflow terms

###  2.a Reversible and pseudoadiabatic hurricanes

For the considered thermodynamic cycle $\rm B^{\prime}bcC^{\prime}B^{\prime}$
to have Carnot efficiency, it should be reversible. This requires that the air
is saturated (relative humidity $\mathcal{H}$ is equal to $100\%$) and the
total moisture mixing ratio $q_{t}$ is constant, everywhere in the cycle222In
the literature one can sometimes find a loose definition of a reversible
process that only assumes $q_{t}=\rm const$ but allows the relative humidity
to vary (e.g., Bryan and Rotunno, 2009a, Eq. (23))..

With $q_{t}=\rm const$, dividing both sides of Eq. (6) by $(1+q_{t})$ we
obtain

$-\oint\alpha dp=\varepsilon\int_{\rm B^{\prime}}^{\rm b}\delta Q,$ (21)

where $\delta Q\equiv\delta Q_{d}/(1+q_{t})$ is the heat input per unit moist
air mass. Summing Eqs. (3), (4) and (21) yields

$-\int_{\rm b^{\prime}}^{\rm b}{\mathbf{F}}\cdot
d{\mathbf{l}}=\varepsilon\int_{\rm B^{\prime}}^{\rm b}\delta Q+\int_{\rm
c}^{\rm C^{\prime}}d\left(\frac{V^{2}}{2}\right).$ (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 $q_{t}=\rm const$ (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 $\rm
B^{\prime}bcC^{\prime}B^{\prime}$ will not be reversible due to the change of
$q_{t}$ on paths $\rm B^{\prime}b$ and $\rm cC^{\prime}$ that connect
different streamlines, ${\rm ABCDA}$ and ${\rm
A^{\prime}B^{\prime}C^{\prime}D^{\prime}A^{\prime}}$. The efficiency of such a
cycle will be lower than Carnot efficiency.

###  2.b Extra heat input to warm precipitating water

A cycle with reversible adiabats $\rm bc$ and $\rm C^{\prime}B^{\prime}$
($q_{t}=\rm const$, $\mathcal{H}=100\%$) and saturated isotherms $\rm
B^{\prime}b$ and $\rm cC^{\prime}$, along which $q_{t}$, respectively,
increases and decreases, has the following relation between work and heat
input (for derivations, see Appendix B):

$-\oint\alpha_{d}dp\simeq\varepsilon_{C}T_{\rm b}\Delta
s^{*}+\frac{\varepsilon_{C}}{2}c_{l}(T_{\rm b}-T_{\rm c})\Delta q_{t},$ (23)

where $\varepsilon_{C}\equiv(T_{\rm b}-T_{\rm c})/T_{\rm b}$ is Carnot
efficiency, $\Delta q_{t}\geq 0$ and $\Delta s^{*}\geq 0$ are, respectively,
the changes of total water mixing ratio $q_{t}$ and moist saturated entropy
$s^{*}$ from $\rm B^{\prime}$ to $\rm b$, $c_{l}=4.2$ 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 $\Delta q_{t}$ should be replaced by
saturated water vapor mixing ratio $\Delta q^{*}$ in Eq. (23).

While in finite differences Eq. (23) is valid only when $\rm B^{\prime}b$ and
$\rm cC^{\prime}$ are isotherms, in the limit of an infinitely narrow cycle
$\rm B^{\prime}bcC^{\prime}B^{\prime}$, when $\rm B^{\prime}b$ and $\rm
cC^{\prime}$ degenerate each to a point, Eq. (23) becomes valid even if the
temperature along $\rm B^{\prime}b$ and $\rm cC^{\prime}$ 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
$T_{\rm b}-T_{\rm c}$ (see also Carnot, 1890, p. 59). This explains how
Emanuel (1986) obtained a Carnot efficiency multiplier at the radial heat
input $T_{\rm b}\partial s^{*}/\partial r$ in his Eq. (13) without assuming
horizontal isothermy at the top of the boundary layer.

The last term in Eq. (23), see also Eq. (Appendix B), 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) re-derived this term in his Eq. (B2) and interpreted it as “additional
work”333Equation (B3) of Pauluis (2011) should have $T_{\rm in}$ instead of
$T_{\rm out}$ in the denominator of the right-hand part, otherwise it
contradicts Eq. (B2) from which supposedly derives., 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. The question to this interpretation is why with
$c_{l}\neq 0$ 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 $T_{\rm c}$ and
returned at the warmer isotherm with $T_{\rm b}$ requires extra heat
$c_{l}(T_{\rm b}-T_{\rm c})\Delta q^{*}$. As the moist air ascends and cools
from $T_{\rm b}$ to $T_{\rm c}$, the mean temperature at which the water loses
heat is, in the linear approximation, $T_{\rm w}\simeq(T_{\rm b}+T_{\rm
c})/2$. Accordingly, the maximum efficiency with which this heat can be
converted to work is $(T_{\rm b}-T_{\rm w})/T_{\rm b}\simeq\varepsilon_{C}/2$.

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

$\Delta Q_{d}=T_{\rm b}\Delta s^{*}+c_{l}(T_{\rm b}-T_{\rm c})\Delta q^{*}$
(24)

and efficiency

$\varepsilon=\varepsilon_{C}\left[1-\frac{1}{2}\frac{c_{l}(T_{\rm b}-T_{\rm
c})\Delta q^{*}}{T_{\rm b}\Delta s^{*}+c_{l}(T_{\rm b}-T_{\rm c})\Delta
q^{*}}\right],$ (25)

that is lower than Carnot efficiency
($\varepsilon_{C}/2\leq\varepsilon\leq\varepsilon_{C}$) due to the
irreversibility associated with warming the precipitating water. This inherent
thermodynamic imperfection of steam cycles was recognized already by Sadi
Carnot (1890, p. 58).

In a pseudoadiabatic hurricane, all condensed water is immediately removed
(precipitates) from the air parcel: $\rho_{l}=0$ and $q_{t}=q^{*}$. The extra
heat reduces to $c_{l}(T_{\rm b}-T_{P})\Delta q^{*}$, where $T_{P}$ is the
mean temperature at which condensation and precipitation occur ($T_{\rm c}\leq
T_{P}<T_{\rm b}$). The last term in Eq. (23) can be roughly approximated by
$c_{l}(T_{\rm b}-T_{P})^{2}\Delta q^{*}/(2T_{\rm b})$.

###  2.c Water lifting

Taking into account that paths $\rm B^{\prime}b$ and $\rm cC^{\prime}$ are,
respectively, horizontal and vertical (Fig. 1) and using
$\alpha_{d}=(1+q^{*})\alpha$ for $\rm B^{\prime}b$, we can combine Eqs. (3),
(6), (7) and (23) in the following form:

$\int_{\rm b}^{\rm B^{\prime}}\left(\alpha\frac{\partial p}{\partial
r}+\frac{1}{2}\frac{\partial V^{2}}{\partial
r}\right)dr=-\varepsilon_{C}\int_{\rm b}^{\rm B^{\prime}}T\frac{\partial
s^{*}}{\partial r}dr-c_{l}T_{\rm b}\frac{\varepsilon^{2}_{P}}{2}\int_{\rm
b}^{\rm B^{\prime}}\frac{\partial q^{*}}{\partial r}dr+\oint\alpha
q_{t}dp+\int_{\rm c}^{\rm C^{\prime}}d\left(\frac{V^{2}}{2}\right),$ (26)

where $\varepsilon_{P}\equiv 1-T_{P}/T_{\rm b}$. Here we have assumed that
evaporation occurs from the ocean surface (there is no condensate along $\rm
B^{\prime}b$: $q_{t}=q^{*}$). We calculate the water lifting term in Appendix
C and lift the integral signs in Eq. (26) for the infinitely narrow cycle $\rm
B^{\prime}bcC^{\prime}B^{\prime}$, i.e., considering the limit $r_{\rm
B^{\prime}}\to r_{\rm b}$ and $z_{\rm C^{\prime}}\to z_{\rm c}$. This gives
for $z=z_{\rm b}$ and $r=r_{\rm b}$

$(1+q^{*})\left(\alpha\frac{\partial p}{\partial r}+\frac{1}{2}\frac{\partial
V^{2}}{\partial r}\right)=-\varepsilon_{C}T_{\rm b}\frac{\partial
s^{*}}{\partial
r}+\left[\frac{1}{2}\left(V_{P}^{2}-V^{2}\right)+g(z_{P}-z_{\rm
b})-c_{l}T_{\rm b}\frac{\varepsilon^{2}_{P}}{2}\right]\frac{\partial
q^{*}}{\partial r}+(1+q_{t\rm c})\frac{1}{2}\frac{\partial V_{\rm
c}^{2}}{\partial r}.$ (27)

Here $q_{t\rm c}$ and $V_{\rm c}$ refer to the point $(r_{\rm c},z_{\rm c})$;
$\partial V_{\rm c}^{2}/\partial r$ is the limit of $(V^{2}_{\rm
C^{\prime}}-V^{2}_{\rm c})/(r_{\rm B^{\prime}}-r_{\rm b})$ at $r_{\rm
B^{\prime}}\to r_{\rm b}$ and $z_{\rm C^{\prime}}\to z_{\rm c}$, cf. Eqs. (17)
and (18). The quantities $V^{2}_{P}$, $z_{P}$ and $T_{P}$ are evaluated on the
unclosed contour $\rm bcC^{\prime}B^{\prime}$ (denoted by $\curvearrowright$):

$V_{P}^{2}\equiv-\frac{1}{\Delta
q^{*}}\int_{\curvearrowright}V^{2}dq_{t},\quad z_{P}\equiv-\frac{1}{\Delta
q^{*}}\int_{\curvearrowright}zdq_{t},\quad T_{P}\equiv-\frac{1}{\Delta
q^{*}}\int_{\curvearrowright}Tdq_{t},$ (28)

where $\Delta q^{*}\equiv q^{*}_{\rm b}-q^{*}_{\rm B^{\prime}}$. In a cycle
with reversible adiabats, i.e., with constant $q_{t}$ along $\rm bc$ and $\rm
C^{\prime}B^{\prime}$ and $\Delta q^{*}=q_{t\rm c}-q_{t\rm C^{\prime}}$, the
integrals in Eq. (28) in the considered limit are straightforwardly evaluated:
$V_{P}=V_{\rm c}$, $z_{P}=z_{\rm c}$, and $T_{P}=T_{\rm c}$.

Equation (27) summarizes the energy budget of the infinitely narrow cycle $\rm
B^{\prime}bcC^{\prime}B^{\prime}$ 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 $V^{2}/2$ of the air at $\rm B^{\prime}b$, while
disappearing via precipitation with kinetic energy $V_{P}^{2}/2$ in the free
troposphere. We will neglect this term444Term $(V_{P}^{2}-V^{2})/2$ 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)., since for typical $z_{P}\sim 10$ km and
$V\sim 60$ m s-1, $V^{2}/2$ is only about 2% of $gz_{P}$.

The water lifting term $g(z_{P}-z_{\rm b})$ accounts for the net energy
expended to lift water. In a real cycle, where there is no precipitation in
the descending branch $\rm C^{\prime}B^{\prime}$ and $q_{t}$ does not change,
$z_{P}$ equals the mean precipitation height $H_{P}$ in the ascending branch
$\rm bc$. In an infinitely narrow cycle with reversible adiabats, $q_{t}$ is
also constant in the "descending branch", and $z_{P}=z_{\rm c}=H_{P}$ has the
same meaning.

However, in an infinitely narrow cycle with $q_{t}$ varying along $\rm bc$ and
$\rm C^{\prime}B^{\prime}$, moisture disappears along $\rm bc$. It then arises
anew along $\rm C^{\prime}B^{\prime}$ with its own non-zero gravitational
energy. In this hypothetical cycle, moisture performs work as it descends
along $\rm C^{\prime}B^{\prime}$ and consumes energy when it is raised along
$\rm bc$. Thus, the net energy $g(z_{P}-z_{\rm b})\Delta q^{*}$ is equal to
the difference in the gravitational energy of "precipitation" between $\rm bc$
and $\rm C^{\prime}B^{\prime}$ (cf. Emanuel, 1988, Eq. (C12)).

This net energy is equal to the energy $g(H_{P}-z_{\rm b})\Delta q^{*}$ spent
to raise the additional moisture $\Delta q^{*}$ from $z_{\rm b}$ to the mean
precipitation height $H_{P}$ along $\rm bc$ plus the energy $g\Delta
q^{*}\partial H_{P}/\partial q^{*}_{\rm b}(q^{*}_{\rm b}-q^{*}_{\rm c})$ spent
to raise nearly all moisture $q^{*}_{\rm b}-q^{*}_{\rm c}\sim q^{*}_{\rm b}$
to an additional altitude $\Delta q^{*}\partial H_{P}/\partial q^{*}_{\rm b}$
equal to the difference between the mean precipitation heights along $\rm bc$
and $\rm B^{\prime}C^{\prime}$. For pseudoadiabats, $H_{P}$ is approximately
proportional to $q^{*}_{\rm b}$, so that $q^{*}_{\rm b}\partial H_{P}/\partial
q^{*}_{\rm b}\sim H_{P}$. In this case, $z_{P}-z_{\rm b}$ turns out to be
approximately twice the mean precipitation height, i.e., around $10$ km for
$T_{\rm b}\simeq 300$ 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 $T_{P}=T_{\rm c}$; with $T_{\rm b}=300$ K,
$T_{\rm b}-T_{P}\simeq 100$ K and $z_{P}-z_{\rm b}\simeq 17$ km (which
corresponds to mean lapse rate $\Gamma=(T_{\rm b}-T_{P})/(H_{P}-z_{\rm
b})\simeq 6$ 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 $T_{\rm b}-T_{P}$. For pseudoadiabats, $T_{P}$ is equal to the mean
temperature of precipitating water (Eq. C6), the difference $T_{\rm b}-T_{P}$
is relatively small, and the impact of water warming is almost negligible.
With $T_{\rm b}=300$ K, $T_{\rm b}-T_{P}=25$ K and $z_{P}-z_{\rm b}=10$ km
(Table C1), it does not exceed $10\%$.

In units of latent heat of vaporization $L_{v}$, the cumulative contribution
$K_{1}$ of the water lifting and warming

$K_{1}\equiv\frac{1}{L_{v}}\left[g(z_{P}-z_{\rm b})-c_{l}T_{\rm
b}\frac{\varepsilon_{P}^{2}}{2}\right]$ (29)

is of similar magnitude $K_{1}=0.04$ for reversible and pseudoadiabatic storms
with high $\varepsilon_{C}\geq 0.3$ (Table C1).

###  2.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, $V_{\rm c}^{2}/2$, changes depending on the radius
$r=r_{\rm b}$ where the streamline crosses the level $z=z_{\rm b}$ (Fig. 1).
Assuming that we are at a point near the radius of maximum wind, such that
$v/r\gg\partial v/\partial r$, we can neglect $\partial v/\partial r$ in Eq.
(18) and use the relation between tangential velocity and pressure gradient

$\alpha\frac{\partial p}{\partial
r}\equiv\mathcal{B}\left(\frac{v^{2}}{r}+fv\right),$ (30)

where $\mathcal{B}>0$ defines the degree to which the air flow is radially
unbalanced ($\mathcal{B}<1$ for the supergradient flow when the outward-
directed centrifugal force exceeds the inward-pulling pressure gradient), to
obtain from Eq. (18):

$\frac{1}{2}\frac{\partial V_{\rm c}^{2}}{\partial
r}\simeq\frac{1}{2}\frac{\partial v_{\rm c}^{2}}{\partial r}=\frac{v_{\rm
c}}{r_{\rm c}}\frac{r}{v}\left(\frac{v^{2}}{r}+fv\right)=\alpha\frac{\partial
p}{\partial r}\frac{1}{\mathcal{B}}\frac{v_{\rm c}}{r_{\rm c}}\frac{r_{\rm
b}}{v_{\rm b}},\quad(r=r_{\rm b},z=z_{\rm b}).$ (31)

On the other hand, using Eqs. (14) and the fact that $M_{\rm b}=M_{\rm c}$ we
can express $v_{\rm c}$ as

$v_{\rm c}=\frac{1}{r_{\rm c}}\left[v_{\rm b}r_{\rm b}+\frac{f}{2}(r_{\rm
b}^{2}-r_{\rm c}^{2})\right].$ (32)

(Evaluating the derivative of Eq. (32) with respect to $r_{\rm b}$ and taking
into account that $\partial r_{\rm c}/\partial r_{\rm b}=0$ is another way to
obtain Eq. (18)). Using Eq. (32) and neglecting $q_{t\rm c}\ll 1$ we can write
the last term in Eq. (27) as $K_{2}\alpha\partial p/\partial r$, where

$K_{2}\equiv\frac{1}{\mathcal{B}}\frac{v_{\rm c}}{r_{\rm c}}\frac{r_{\rm
b}}{v_{\rm b}}=\frac{1}{\mathcal{B}}\left[\frac{r_{\rm b}^{2}}{r_{\rm
c}^{2}}+\frac{fr_{\rm b}}{2v_{\rm b}}\left(\frac{r_{\rm b}^{2}}{r_{\rm
c}^{2}}-1\right)\right].$ (33)

For $(r_{\rm b}/r_{\rm c})^{2}\ll 1$, 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 large555This conclusion
stemmed from Emanuel (1986)’s Eq. (18), where an outflow term proportional to
a large squared radius first appeared. While deriving this equation for a
cycle with finite $\rm B^{\prime}b$, Emanuel (1986, p. 588), on the one hand,
used the conservation of angular momentum along streamlines $\rm bc$ and $\rm
B^{\prime}C^{\prime}$ and, on the other hand, assumed $r_{\rm B^{\prime}}$
(interpreted as “the radial extent of the storm near the sea level”) to be
large enough for $\partial p/\partial r$ to vanish and, at the same time,
small enough for $r\partial p/\partial r|_{\rm b}\gg r\partial p/\partial
r|_{\rm B^{\prime}}$(for details, see Makarieva et al., 2019, Appendix A).
With $r\partial p/\partial r\sim\rho v^{2}$, ignoring this term at $\rm
B^{\prime}$ means that $V^{2}_{\rm b}-V^{2}_{\rm B^{\prime}}\sim V^{2}_{\rm
b}$, while at the point of maximum wind this difference is zero.. In their re-
evaluation of this issue, Emanuel and Rousseau-Rizzi (2020) considered the
material derivative of angular momentum $dM/dt$ along the path $\rm
cC^{\prime}$ connecting the two streamlines. Their derivation is not valid,
since $\rm cC^{\prime}$ 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 $\rm D^{\prime}$ and $\rm D$, and
of $\rm A^{\prime}$ and $\rm A$ are identical” but said nothing about the
properties of $\rm C^{\prime}$ and $\rm C$ (see Fig. 1 of Rousseau-Rizzi and
Emanuel, 2019).

###  2.e Estimates of maximum velocity

We will now consider the point where $\partial V/\partial r=0$ (see Eq. (8) in
Fig. 2). This corresponds to the point of maximum wind if point $\rm b$ 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 $\rm b$ is chosen at the surface as in the derivations of Rousseau-
Rizzi and Emanuel (2019).

By analogy with Eq. (9), from Eq. (3) with $\partial V/\partial r=0$ we obtain
(see also Makarieva et al., 2020, Eq. (14)):

$-\mathbf{F}\cdot\mathbf{V}=-\alpha\frac{\partial p}{\partial r}u,$ (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 $|\mathbf{F}|\to 0$ (as is the case at the top of the
boundary layer), then $u\to 0$. This feature is observed in numerical models
(e.g., Bryan and Rotunno, 2009b, p. 3054). It is also valid for a horizontal
streamline at $z_{\rm b}=0$ as considered by Rousseau-Rizzi and Emanuel (2019,
their Fig. 1), whereby $u\neq 0$ and $|\mathbf{F}|\neq 0$.

On the other hand, for $\partial v/\partial r=0$ we have from Eqs. (14) and
(30)

$\alpha\frac{\partial p}{\partial r}=\mathcal{B}\frac{v}{r}\frac{\partial
M}{\partial r}.$ (35)

Finally, using the definitions of $K_{1}$ (29) and $K_{2}$ (33) and neglecting
$q^{*}\ll 1$, we can write Eq. (27) for $\partial V/\partial r=0$ as follows:

$-\alpha\frac{\partial p}{\partial r}=\frac{\varepsilon_{C}-\beta
K_{1}}{1-K_{2}}T_{\rm b}\frac{\partial s^{*}}{\partial
r},\quad\beta\equiv\frac{L_{v}\partial q^{*}/\partial r}{T_{\rm b}\partial
s^{*}/\partial r}.$ (36)

Here $0\leq\beta\leq 1$ is the share of latent heat in total heat input into
the air parcel along $\rm B^{\prime}b$.

Now using two distinct assumptions, (10) and (20), about how the volume and
surface energy fluxes relate, we obtain two expressions for $V_{\rm max}$ that
differ by a factor of $\mathcal{B}$. From Eq. (36), Eq. (35) and Eq. (20) we
obtain

$V_{\rm max}^{2}=\frac{\varepsilon_{C}-\beta
K_{1}}{\mathcal{B}(1-K_{2})}\frac{C_{k}}{C_{D}}(k_{s}^{*}-k).$ (37)

From Eq. (36) multiplied by $u$, Eq. (34) and Eq. (10) we obtain the same
expression, but without $\mathcal{B}$. Since $\mathcal{B}$ can be as small as
$0.5$ (e.g., Bryan and Rotunno, 2009b, Fig. 8), this is a significant source
of uncertainty in $V_{\rm max}$ 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 $K_{1}$ and $K_{2}$ are
compared with $\varepsilon_{C}$ and unity, respectively. With $K_{1}\simeq
0.04$, $\beta\sim 0.7$ estimated from a typical Bowen ratio, and
$\varepsilon_{C}=0.3$, $K_{1}$ reduces $V_{\rm max}^{2}$ by $\beta
K_{1}/\varepsilon_{C}\times 100\%\simeq 10$% and $V_{\rm max}$ by $5\%$ both
for pseudoadiabatic and adiabatic cases at $T_{\rm b}=300$ K.

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
$\alpha\partial p/\partial r$ (Makarieva and Nefiodov, 2021). This could cause
the overestimate of $K_{1}$ for the reversible case. The details of
pseudoadiabatic calculations were not reported by Emanuel (1988); we can
hypothesize that neglecting the second term in Eq. (C6) for $z_{P}$ could
cause the underestimate of $K_{1}$ for the pseudoadiabatic case.

As for the outflow term $K_{2}$, since $r_{\rm b}<r_{\rm c}$, the factor in
square brackets in Eq. (33) is confined between $-fr_{\rm b}/(2v_{\rm b})<0$
and unity. For characteristic values of $r_{\rm b}=30$ km, $v_{\rm b}=60$ m
s-1, $\varphi=15^{\rm o}$ and $f\simeq 3.77\times 10^{-5}$ s-1 we have
$fr_{\rm b}/(2v_{\rm b})\simeq 10^{-2}$. With $\mathcal{B}\simeq 1$, $K_{2}$
reduces $V_{\rm max}^{2}$ by about $1\%$ for large $r_{\rm c}$.

On the other hand, if the outflow radius is relatively small, $K_{2}$ is
positive and elevates rather than lowers the maximum velocity estimate. This
may happen for many storms with $r_{\rm b}/r_{\rm c}\gtrsim\sqrt{fr_{\rm
b}/(2v_{\rm b})}\sim 10^{-1}$. Interpreting $K_{2}$ as “dissipation” to occur
at an arbitrary point $\rm c$ in an otherwise frictionless troposphere, is
incorrect (cf. Emanuel and Rousseau-Rizzi, 2020). When $K_{2}>0$ the kinetic
energy increases from $\rm c$ to $\rm C^{\prime}$. Smith et al. (2014) noted
that a mechanism that would provide an increment of angular momentum, and
hence a kinetic energy increment, along $\rm B^{\prime}b$ 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:

$\varepsilon_{C}(D+J)=D+gH_{P}P,$ (38)

from which one obtains that

$V_{\rm
max}^{2}=\frac{D}{J}\frac{C_{k}}{C_{D}}(k_{s}^{*}-k)=\frac{1}{1-\varepsilon_{C}}\left(\varepsilon_{C}-\frac{PL_{v}}{J}\frac{gH_{P}}{L_{v}}\right)\frac{C_{k}}{C_{D}}(k_{s}^{*}-k).$
(39)

Here $P$ (kg m-2 s-1) is the local precipitation in the region of maximum
winds666Note the following differences in notations between Sabuwala et al.
(2015) and the present work: $T_{s}\to T_{\rm b}$, $\dot{Q}_{in}\to J$,
$\dot{Q}_{d}\to D$, $P\to W_{P}=gH_{P}P$. Factor $1/(1-\varepsilon_{C})$ is
due to dissipative heating.. Equation (39) is obtained from Sabuwala et al.
(2015)’s 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 $gH_{P}/L_{v}\sim K_{1}$ in Eq. (39)
is multiplied by a large factor $PL_{v}/J\gg 1$ reflecting the ratio of local
precipitation to local heat input. For typical Bowen ratios in hurricanes
$B\equiv J_{S}/J_{L}\simeq 1/3$ (e.g., Jaimes et al., 2015) we have
$J=(1+B)J_{L}=(1+B)EL_{v}$ and $PL_{v}/J=(P/E)/(1+B)$, where $J_{S}$, $J_{L}$
and $E$ are the local fluxes of sensible heat, latent heat, and evaporation,
respectively. Ratio $P/E$ between local precipitaton and evaporation in the
region of maximum winds is variable but on average of the order of $10$ (see
Makarieva et al., 2017a, their Table 1 and Figs. 2 and 3). For $H_{P}\sim 5$
km and $\varepsilon_{C}\sim 0.3$, Sabuwala et al. (2015)’s correction to
$V_{\rm max}^{2}$ is thus $10(gH_{P}/L_{v})/[(1+B)\varepsilon_{C}]\simeq 0.5$,
which is five times larger than our $\beta K_{1}/\varepsilon_{C}\simeq 0.1$
(Table 1). The unjustified replacement of evaporation by precipitation caused
Sabuwala et al. (2015)’s estimate to be too high.

##  3 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,
$-\rho\mathbf{F}\cdot\mathbf{V}$, is equal to the local volume-specific rate
of sensible heat input into an isothermally and horizontally expanding air
parcel, $-(\partial p/\partial r)u$. 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, non-isothermal case, we can relate turbulent dissipation to
latent heat input. Inspecting our key equation (27), we notice that the water
lifting term in Eq. (27) is proportional to $\partial q^{*}/\partial r$, while
the outflow term is proportional to $\partial p/\partial r$, 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 $\partial p/\partial r$,
see Eqs. (B5) and (B17):

$T\frac{\partial s^{*}}{\partial r}=L_{v}\frac{\partial q^{*}}{\partial
r}\left(1+\varkappa_{1}\right)-\alpha_{d}\frac{\partial p}{\partial
r}\left(1-\varkappa_{2}\right).$ (40)

Coefficients $\varkappa_{1}$ and $\varkappa_{2}$ describe the deviation from
horizontal isothermy:

$\displaystyle\varkappa_{1}\equiv\frac{1}{\mu\gamma^{*}_{d}\xi^{2}(1+\gamma^{*}_{d})}=\frac{\varkappa_{2}}{\gamma^{*}_{d}\xi}\simeq
0.3,$ (41)
$\displaystyle\varkappa_{2}\equiv\frac{1}{\mu\xi(1+\gamma^{*}_{d})}\simeq\frac{1}{\mu\xi}\simeq
0.2,$ (42)

where $\xi\equiv L/RT\simeq 18$, $\mu\equiv R/(c_{p}M_{d})\simeq 2/7$ and
$\gamma^{*}_{d}\equiv(M_{d}/M_{v})q^{*}$ are defined in Appendix B. The
numerical values correspond to $T=300$ K and $\gamma^{*}_{d}\equiv
p_{v}^{*}/p_{d}=0.04$. For $\partial T/\partial r=0$, the terms proportional
to $\varkappa_{1}$ and $\varkappa_{2}$ jointly vanish from Eq. (40), see also
Eq. (B17). Therefore, the isothermal case can be formally obtained by putting
$\varkappa_{1}=0$ and $\varkappa_{2}=0$ in Eq. (40) and related.

Using Eqs. (31) and (40) we write Eq. (27) with $\partial V^{2}/\partial r=0$
and $q^{*}\ll 1$ neglected, as follows

$\alpha\frac{\partial p}{\partial
r}=\left[-\varepsilon_{C}\left(1+\varkappa_{1}\right)+K_{1}\right]L_{v}\frac{\partial
q^{*}}{\partial
r}+\left[\varepsilon_{C}\left(1-\varkappa_{2}\right)+K_{2}\right]\alpha\frac{\partial
p}{\partial r}.$ (43)

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

$-\mathbf{F}\cdot\mathbf{V}=\frac{\varepsilon_{C}(1+\varkappa_{1})-K_{1}}{1-\varepsilon_{C}(1-\varkappa_{2})-K_{2}}L_{v}\frac{\partial
q^{*}}{\partial r}u.$ (44)

(This equation together with Eq. (36) show that E-PI constrains the latent-to-
total heat ratio $\beta$ at $\rm B^{\prime}b$. For
$K_{1}=K_{2}=\varkappa_{1}=\varkappa_{2}=0$ we have $\beta=1-\varepsilon_{C}$,
which, as can be verified from the exact relation (B18), is a good
approximation of $\beta$ for small $K_{1}\ll\varepsilon_{C}$.) Assuming that
turbulent dissipation relates to latent heat input equally at $\rm
bB^{\prime}$ and at the surface (i.e., replacing in Eq. (10) total heat
$Tds^{*}$ and $J$ with latent heat $L_{v}dq^{*}$ and $J_{L}=\rho
C_{k}VL_{v}(q^{*}_{s}-q)$, respectively),

$-\frac{\mathbf{F}\cdot\mathbf{V}}{L_{v}(\partial q^{*}/\partial
r)u}=\frac{D}{J_{L}},$ (45)

from Eq. (44) we obtain another expression for $V^{2}_{\rm max}$:

$V_{\rm
max}^{2}=\frac{D}{J_{L}}\frac{C_{k}}{C_{D}}L_{v}(q_{s}^{*}-q)=\frac{\varepsilon_{C}(1+\varkappa_{1})-K_{1}}{1-\varepsilon_{C}(1-\varkappa_{2})-K_{2}}\frac{C_{k}}{C_{D}}L_{v}(q_{s}^{*}-q).$
(46)

When surface sensible heat $J_{S}=\rho C_{k}Vc_{p}(T_{s}-T)$ is negligibly
small, such that $J_{L}\gg J_{S}$ and $J=J_{L}+J_{S}\simeq J_{L}$ (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
$K_{1}=K_{2}=\varkappa_{1}=\varkappa_{2}=0$, Eq. (46), derived without
assuming any dissipative heating, is then equivalent to Eq. (11) but with
$\varepsilon_{C}$ replaced in the latter by
$\varepsilon_{C}/(1-\varepsilon_{C})$. 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 1.

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 $\partial T/\partial r$ and
$\partial p/\partial r$ (see Makarieva and Nefiodov, 2021):

$T\frac{\partial s^{*}}{\partial r}=-\alpha_{d}\frac{\partial p}{\partial
r}\left(1+\frac{L\gamma_{d}^{*}}{RT}\right)\left(1-\frac{1}{\Gamma}\frac{\partial
T/\partial r}{\partial p/\partial r}\right),$ (47)

where $\Gamma$ is the moist adiabatic lapse rate (K Pa-1). If, for simplicity,
we neglect the water lifting by putting $K_{1}=0$, 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
$\alpha\partial p/\partial r$, which can be canceled. Again, putting for
simplicity $q_{t}=q^{*}=0$ and $\mathcal{B}=1$, and using Eq. (33), we obtain
from Eq. (27):

$1=\varepsilon_{C}\left(1+\frac{L\gamma_{d}^{*}}{RT_{\rm
b}}\right)\left(1-\frac{1}{\Gamma}\frac{\partial T/\partial r}{\partial
p/\partial r}\right)+\frac{v_{\rm c}}{r_{\rm c}}\frac{r_{\rm b}}{v_{\rm b}},$
(48)

which can be re-written in a form similar to Eq. (13) using
$\varepsilon_{C}=(T_{\rm b}-T_{\rm c})/T_{\rm b}$ and $\nu=v/r$:

$\frac{\nu_{\rm b}-\nu_{\rm c}}{T_{\rm b}-T_{\rm c}}=\frac{\nu_{\rm b}}{T_{\rm
b}}\left(1+\frac{L\gamma_{d}^{*}}{RT_{\rm
b}}\right)\left(1-\frac{1}{\Gamma}\frac{\partial T/\partial r}{\partial
p/\partial r}\right).$ (49)

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

Equations (48) and (49) show that when $\nu_{\rm c}=0$ (no kinetic energy
change in the outflow), and the air is horizontally isothermal, E-PI framework
presumes $\varepsilon_{C}(1+L\gamma^{*}_{d}/RT_{\rm b})=1$ (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 $\rm B^{\prime}b$, where heat input occurs (see Eq.
(8) in Fig. 2). This can only be the case when the adiabat $\rm bc$ is, at
least somewhere, warmer than the adiabat $\rm B^{\prime}C^{\prime}$. Then the
pressure deficit at $\rm b$ as compared to $\rm B^{\prime}$ can be compensated
by the pressure surplus in the free troposphere at $\rm bc$ as compared to
$\rm B^{\prime}C^{\prime}$. Without this pressure surplus aloft, the work
along $\rm bcC^{\prime}B^{\prime}$ 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 $\gamma_{d}^{*}$, at $\rm
bc$. How much of this water vapor condenses along $\rm bc$ producing the
required temperature surplus, is dictated by the outflow temperature $T_{\rm
c}$. The lower $T_{\rm c}$ (i.e., the higher $\varepsilon_{C}$), the lower
$\gamma_{d}^{*}$ 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 non-trivial solution for the isothermal case $\partial
T/\partial r=0$. 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
towards the center at the point of maximum wind.

Point of maximum wind; frictionless troposphere above the point of maximum
wind; axial symmetry; saturated air; hydrostatic equilibrium in the outflow
$\rm cC^{\prime}$ No information about thermodynamics: $\varepsilon$ is
defined by Eq. (6) $\alpha\frac{\partial p}{\partial r}=$ $-\varepsilon T_{\rm
b}\frac{\partial s^{*}}{\partial r}$ $+g(z_{P}-z_{\rm b})\frac{\partial
q^{*}}{\partial r}$ $+\frac{1}{2}\frac{\partial V_{\rm c}^{2}}{\partial r}$
For $q_{t\rm b}=q_{t\rm c}$, $z_{P}=z_{\rm c}$ (Eq. 28); for $q_{t}=q^{*}$,
$z_{P}=\overline{z}_{\rm bc}$ (Eq. C6) $V^{2}_{\rm c}-V^{2}_{\rm C^{\prime}}$
$=v^{2}_{\rm c}-v^{2}_{\rm C^{\prime}}$ Streamlines $\rm bc$ and $\rm
B^{\prime}C^{\prime}$ are reversible adiabats or pseudoadiabats
$\alpha\frac{\partial p}{\partial r}=$ $-\varepsilon_{C}\,\,T_{\rm
b}\frac{\partial s^{*}}{\partial r}$ $+\,\left[-\frac{c_{l}T_{\rm
b}}{L_{v}}\frac{\varepsilon^{2}_{P}}{2}\,\right.$ $\left.+\frac{g(z_{P}-z_{\rm
b})}{L_{v}}\right]\frac{L_{v}\partial q^{*}}{\partial r}$
$+\,\,\,\frac{1}{\mathcal{B}}\frac{v_{\rm c}}{r_{\rm c}}\frac{r_{\rm
b}}{v_{\rm b}}\,\,\alpha\frac{\partial p}{\partial r}$ $K_{1}\simeq 0.04$ (Eq.
29)$K_{2}\simeq-0.01$ (Eq. 33) Assumption about how gradients at the point of
maximum wind relate to local surface fluxes: Eq. (10), (20), or (45) $V_{\rm
max}^{2}=\,\frac{1}{\mathcal{B}}\,\frac{\varepsilon_{C}-\beta
K_{1}}{1-K_{2}}\frac{C_{k}}{C_{D}}(k_{s}^{*}-k)$ Gradient-wind imbalance
$\mathcal{B}\sim 1$ (Eq. 30) $\beta\simeq 1-\varepsilon_{C}$ (latent-to-total
heat ratio) is constrained by Eqs. (36), (34), and (44) Eq. (27) Eq. (37)
OutflowTotal workWater liftingHeat input at $\rm bB^{\prime}$Water warming
Figure 3:  Main assumptions (green rounded boxes) and results (blue
rectangular boxes) of this work with comments (dotted boxes).

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.

The water lifting term $g(z_{P}-z_{\rm b})$, 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 $\varepsilon$ (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
$\varepsilon$ 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, $K_{1}/\varepsilon\sim gz_{P}/(\varepsilon L_{v})$. With
$K_{1}\sim 10^{-2}$ and $\varepsilon\sim 10^{-2}$ 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 $\rm C^{\prime}B^{\prime}$ (Fig. 1). Here, instead, the water formally
arises anew with its own non-zero gravitational energy. For this reason, the
water lifting term in E-PI is proportional to local evaporation ($\Delta
q^{*}$) rather than precipitation ($q^{*}_{\rm b}-q^{*}_{\rm c}$). 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.
(2015)’s overestimate of the water lifting term (Section 2.e).

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 2.a). 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 ($z_{\rm c}>aH_{P}$, Eq. (C7)).
Accounting for water warming, which requires information about the cycle’s
thermodynamics (Fig. 3), reduces this difference. For $T_{\rm b}\simeq 300$ K
and the largest observed $\varepsilon_{C}\simeq 0.3$ (DeMaria and Kaplan,
1994), the magnitudes of $K_{1}\simeq 0.04$ (Eq. 29) and $\beta
K_{1}/\varepsilon_{C}\simeq 0.1$ for reversible and pseudoadiabatic cases are
similar. This corresponds to a $5\%$ reduction of $V_{\rm max}$ (Eq. 37). This
reduction is larger for smaller $\varepsilon_{C}$ but smaller for lower
$T_{\rm b}$ (Table C1). The developed analytical framework can be used to
evaluate corresponding magnitudes for different scenarios in numerical models.

Our analysis clarifies that $\varepsilon$ in E-PI is not the actual efficiency
of the cycle but the ratio of total work to heat input at $\rm B^{\prime}b$
(Figs. 2 and 3). With additional heat inputs elsewhere in the cycle,
$\varepsilon$ 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 $\alpha\partial
p/\partial r$ 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 $K_{1}$ related to greater total work).

Our derivations exposed the sensitivity of $V^{2}_{\rm max}$ 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 $\mathcal{B}$ (30) that describes the
deviation from gradient-wind balance (for how this relates to the analysis of
Bryan and Rotunno (2009b), 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 $\rm c$ as a point where $v_{\rm c}=0$, i.e.,
putting $K_{2}=0$, 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 $\rm bc$ compared to $\rm
C^{\prime}B^{\prime}$. This extra warming is provided either by a higher
mixing ratio at $\rm b$ compared to $\rm B^{\prime}$, 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
$\partial T/\partial r=0$ cannot yield correct results under the conventional
assumption of $K_{2}=0$ (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 $v_{\rm b}/r_{\rm
b}=v_{\rm c}/r_{\rm c}$. This constancy of angular velocity combined with
angular momentum conservation along $\rm bc$ gives two solutions (see Eq. 33).
One is trivial, $r_{\rm b}=r_{\rm c}$. Another one is $\nu_{\rm b}=\nu_{\rm
c}=-f/2$; 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 non-local constraint on the work in the free troposphere resulting from E-PI
can be applied to the integral cycle $\rm ABCDA$. In this case, we cannot put
$\partial V^{2}/\partial r=0$ in Eq. (8). The work in the free troposphere is
equal to the non-zero 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. Our
response to them can be found in the Appendix. 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.

##  4 Appendices

###  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 $\bm{\Omega}$ can be cast
into the vector form as follows (see, e.g., Lorenz, 1967; Vallis, 2006)

$\frac{d\mathbf{V}}{dt}=-\alpha\nabla
p-2[\bm{\Omega}\times\mathbf{V}]-\nabla\Phi+\mathbf{F}.$ (A1)

Here, $\mathbf{V}$ is the total velocity of air motion, relative to rotating
Earth; $\alpha\equiv 1/\rho$ 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; $\mathbf{F}$ is frictional
force per unit mass. The geopotential $\Phi$ is defined such that
$\mathbf{g}=-\nabla\Phi$, where $\mathbf{g}$ 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

$\frac{d}{dt}=\frac{\partial}{\partial t}+(\mathbf{V}\cdot\nabla).$ (A2)

Using the relation $(\mathbf{V}\cdot\nabla)\mathbf{V}=\nabla
V^{2}/2-\mathbf{V}\times[\nabla\times\mathbf{V}]$, one can write Eq. (A1) for
the case of a steady state ($\partial\mathbf{V}/\partial t=0$) as

$-\alpha\nabla p=\frac{1}{2}\nabla
V^{2}+[\bm{\omega}\times\mathbf{V}]-\mathbf{g}-\mathbf{F},$ (A3)

where $\bm{\omega}\equiv[\nabla\times\mathbf{V}]+2\bm{\Omega}$ is the absolute
vorticity.

Let us find the inner product of Eq. (A3) and vector $d\mathbf{l}$. The
equality

$[\bm{\omega}\times\mathbf{V}]\cdot d\mathbf{l}=0$ (A4)

holds if $d\mathbf{l}$ is the length element along a streamline, i.e., by
definition is tangent to the local velocity $\mathbf{V}=d\mathbf{l}/dt$.
Indeed, Eq. (A4) follows from the relations

$[\bm{\omega}\times\mathbf{V}]\cdot\mathbf{V}=\bm{\omega}\cdot[\mathbf{V}\times\mathbf{V}]=0.$
(A5)

As a result, one obtains the Bernoulli equation

$-\alpha\nabla p\cdot d\mathbf{l}=\frac{1}{2}\nabla V^{2}\cdot
d\mathbf{l}-\mathbf{g}\cdot d\mathbf{l}-\mathbf{F}\cdot d\mathbf{l},$ (A6)

which is valid along a streamline. The meaning of Eq. (A4) is that the force
$[\bm{\omega}\times\mathbf{V}]$ (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 $\mathbf{e}_{r}$,
$\mathbf{e}_{\theta}$, and $\mathbf{e}_{z}$. The position vector of a point
$\mathbf{r}$ is now characterized by three components, namely, by radial
distance $r$, azimuth $\theta$, and height $z$. The length element of a
streamline and the gradient operator read

$\displaystyle d\mathbf{l}$
$\displaystyle=dr\mathbf{e}_{r}+rd\theta\mathbf{e}_{\theta}+dz\mathbf{e}_{z}=(u\mathbf{e}_{r}+v\mathbf{e}_{\theta}+w\mathbf{e}_{z})dt=\mathbf{V}dt,$
(A7) $\displaystyle\nabla$
$\displaystyle\equiv\frac{\partial}{\partial\mathbf{l}}=\mathbf{e}_{r}\frac{\partial}{\partial
r}+\mathbf{e}_{\theta}\frac{1}{r}\frac{\partial}{\partial\theta}+\mathbf{e}_{z}\frac{\partial}{\partial
z}.$ (A8)

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

$\mathrm{curl}\leavevmode\nobreak\
\mathbf{V}\equiv[\nabla\times\mathbf{V}]=\frac{1}{r}\begin{vmatrix}\mathbf{e}_{r}&r\mathbf{e}_{\theta}&\mathbf{e}_{z}\\\
\partial/\partial r&\partial/\partial\theta&\partial/\partial z\\\
u&rv&w\end{vmatrix}=\left(\frac{1}{r}\frac{\partial
w}{\partial\theta}-\frac{\partial v}{\partial
z}\right)\mathbf{e}_{r}+\left(\frac{\partial u}{\partial z}-\frac{\partial
w}{\partial
r}\right)\mathbf{e}_{\theta}+\frac{1}{r}\left(\frac{\partial(rv)}{\partial
r}-\frac{\partial u}{\partial\theta}\right)\mathbf{e}_{z}.$ (A9)

The geopotential is given by $\Phi=gz$, so that
$\mathbf{g}=-\nabla\Phi=-g\mathbf{e}_{z}$. 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
$\bm{\omega}\simeq[\nabla\times\mathbf{V}]+f\mathbf{e}_{z}$, where
$f=2\Omega\sin\varphi$ is the Coriolis parameter ($\Omega=2\pi/T$ is the
rotation rate of the Earth, $T=24$ h is the rotation period of the Earth and
$\varphi$ is latitude).

For axisymmetric motion, $p$ and $\mathbf{V}$ are independent of the angle
$\theta$. In this case, the cylindrical components of Eq. (A3) have the form

$\displaystyle-\alpha\frac{\partial p}{\partial r}$
$\displaystyle=\frac{1}{2}\frac{\partial V^{2}}{\partial
r}+w\omega_{\theta}-v\omega_{z}-F_{r},$ (A10a) $\displaystyle 0$
$\displaystyle=\hphantom{\frac{1}{2}\frac{\partial V^{2}}{\partial
r}}+u\omega_{z}-w\omega_{r}-F_{\theta},$ (A10b)
$\displaystyle-\alpha\frac{\partial p}{\partial z}$
$\displaystyle=\frac{1}{2}\frac{\partial V^{2}}{\partial
z}-u\omega_{\theta}+v\omega_{r}+g-F_{z},$ (A10c)

where $\omega_{r}=-\partial v/\partial z$, $\omega_{\theta}=\partial
u/\partial z-\partial w/\partial r$ and
$\omega_{z}=f+r^{-1}\partial(rv)/\partial r$. To derive the Bernoulli
equation, we consider the streamline defined by $d\mathbf{l}$, Eq. (A7).
Multiplying the above three equations by $dr$, $rd\theta$ and $dz$,
respectively, and summing them one obtains

$-\alpha\frac{\partial p}{\partial\mathbf{l}}\cdot
d\mathbf{l}=\frac{1}{2}\frac{\partial V^{2}}{\partial\mathbf{l}}\cdot
d\mathbf{l}+gdz-\mathbf{F}\cdot d\mathbf{l},$ (A11)

where $\partial p/\partial\theta=0$ and $\partial\mathbf{V}/\partial\theta=0$.
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 $\partial V/\partial
z=0$ turns Eq. (A10c) into $-u\omega_{\theta}+v\omega_{r}-F_{z}=0$ and the
Bernoulli equation (A11) into

$-\alpha\frac{\partial p}{\partial r}dr=\frac{1}{2}\frac{\partial
V^{2}}{\partial r}dr-\mathbf{F}\cdot d\mathbf{l},$ (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, Eq. (A4))

$s^{*}=(c_{pd}+q_{t}c_{l})\ln\frac{T}{T_{0}}-\frac{R}{M_{d}}\ln\frac{p_{d}}{p_{0}}+q^{*}\frac{L_{v}}{T}.$
(B1)

Here, $L_{v}=L_{v0}+(c_{pv}-c_{l})(T-T_{0})$ is the latent heat of
vaporization (J kg-1); $q^{*}=\rho^{*}_{v}/\rho_{d}$,
$q_{l}=\rho_{l}/\rho_{d}$, and $q_{t}=q^{*}+q_{l}$ are the mixing ratio for
saturated water vapor, liquid water, and total water, respectively;
$\rho_{d}$, $\rho^{*}_{v}$, and $\rho_{l}$ 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 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; $p_{0}$
and $T_{0}$ are reference air pressure and temperature.

The ideal gas law for the partial pressure $p_{v}=p-p_{d}$ of water vapor is

$p_{v}=N_{v}RT,\quad N_{v}=\frac{\rho_{v}}{M_{v}},$ (B2)

where $M_{v}$ and $\rho_{v}$ are the molar mass and density of water vapor,
respectively. Using Eq. (B2) with $p_{v}=p_{v}^{*}$ in the definition of
$q^{*}$

$q^{*}\equiv\frac{\rho^{*}_{v}}{\rho_{d}}=\frac{M_{v}}{M_{d}}\frac{p_{v}^{*}}{p_{d}}\equiv\frac{M_{v}}{M_{d}}\gamma_{d}^{*},\quad\gamma_{d}^{*}\equiv\frac{p_{v}^{*}}{p_{d}},$
(B3)

and applying the Clausius-Clapeyron law

$\frac{dp_{v}^{*}}{p_{v}^{*}}=\xi\frac{dT}{T},\quad L\equiv
L_{v}M_{v},\quad\xi\equiv\frac{L}{RT},$ (B4)

we obtain from Eq. (B1)

$\displaystyle Tds^{*}$
$\displaystyle=d(L_{v}q^{*})-\alpha_{d}dp+(c_{pd}+c_{l}q_{t})dT+c_{l}T\ln\frac{T}{T_{0}}dq_{t}$
(B5a)
$\displaystyle=L_{v}dq^{*}-\alpha_{d}dp+c_{p}dT+c_{l}T\ln\frac{T}{T_{0}}dq_{t},$
(B5b)

where $c_{p}\equiv c_{pd}+q^{*}c_{pv}+q_{l}c_{l}$ and $\alpha_{d}\equiv
1/\rho_{d}$. Integrating Eq. (B5a) with $T_{0}=T_{\rm b}$ yields

$-\oint\alpha_{d}dp=\oint Tds^{*}-c_{l}\oint q_{t}dT-c_{l}\oint
T\ln\frac{T}{T_{\rm b}}dq_{t}.$ (B6)

Along the isotherms $dT=0$, i.e., the paths $\rm B^{\prime}b$ and $\rm
cC^{\prime}$ are characterized by temperatures $T_{\rm b}$ and $T_{\rm c}$,
respectively, while $ds^{*}=0$ along the adiabats $\rm bc$ and $\rm
C^{\prime}B^{\prime}$, i.e., $s^{*}_{\rm b}=s^{*}_{\rm c}$, $s^{*}_{\rm
B^{\prime}}=s^{*}_{\rm C^{\prime}}$ both for reversible ($dq_{t}=0$) and
pseudoadiabatic ($q^{*}=q_{t}$) cases. The first term on the right-hand side
of Eq. (B6) can be written as follows

$\oint Tds^{*}=\int\limits^{\rm b}_{\rm B^{\prime}}Tds^{*}+\int\limits^{\rm
C^{\prime}}_{\rm c}Tds^{*}=\left(1-\frac{T_{\rm c}}{T_{\rm b}}\right)\int_{\rm
B^{\prime}}^{\rm b}Tds^{*}.$ (B7)

Here we took into account the fact that integral over the colder isotherm can
be expressed in terms of integral over the warmer isotherm (Pauluis, 2011):

$\int\limits_{\rm c}^{\rm C^{\prime}}Tds^{*}=T_{\rm c}\int\limits_{\rm c}^{\rm
C^{\prime}}ds^{*}=T_{\rm c}s^{*}\bigg{|}_{\rm c}^{\rm C^{\prime}}=T_{\rm
c}(s^{*}_{\rm B^{\prime}}-s^{*}_{\rm b})=-\frac{T_{\rm c}}{T_{\rm
b}}\int\limits^{\rm b}_{\rm B^{\prime}}Tds^{*}.$ (B8)

In the reversible case (i.e., $q_{t\rm b}=q_{t\rm c}$ and $q_{t\rm
B^{\prime}}=q_{t\rm C^{\prime}}$), the second and third circulation integrals
on the right-hand side of Eq. (B6) are evaluated as follows

$\displaystyle\oint q_{t}dT=-\oint Tdq_{t}=\int\limits^{\rm c}_{\rm
C^{\prime}}Tdq_{t}-\int\limits^{\rm b}_{\rm B^{\prime}}Tdq_{t}=\Delta
q_{t}(T_{\rm c}-T_{\rm b}),$ (B9) $\displaystyle\oint T\ln\frac{T}{T_{\rm
b}}dq_{t}=T_{\rm c}\ln\frac{T_{\rm c}}{T_{\rm b}}\int\limits_{\rm c}^{\rm
C^{\prime}}dq_{t}=-\Delta q_{t}T_{\rm c}\ln\frac{T_{\rm c}}{T_{\rm b}},$ (B10)

where $\Delta q_{t}\equiv q_{t\rm b}-q_{t\rm B^{\prime}}=q_{t\rm c}-q_{t\rm
C^{\prime}}$.

Using Eqs. (B7), (B9) and (B10), integral (B6) can be cast into the following
form

$\displaystyle-\oint\alpha_{d}dp$ $\displaystyle=\left(1-\frac{T_{\rm
c}}{T_{\rm b}}\right)\int_{\rm B^{\prime}}^{\rm b}Tds^{*}+c_{l}\Delta
q_{t}\left[T_{\rm b}-T_{\rm c}+T_{\rm c}\ln\frac{T_{\rm c}}{T_{\rm b}}\right]$
$\displaystyle=\varepsilon_{C}\int_{\rm B^{\prime}}^{\rm b}Tds^{*}+c_{l}T_{\rm
b}\Delta
q_{t}\left[\varepsilon_{C}+(1-\varepsilon_{C})\ln(1-\varepsilon_{C})\right].$
(B11)

Taking into account that for $\varepsilon_{C}=1-T_{\rm c}/T_{\rm b}\lesssim 1$

$\varepsilon_{C}+(1-\varepsilon_{C})\ln(1-\varepsilon_{C})\simeq\frac{\varepsilon_{C}^{2}}{2},$
(B12)

we obtain Eq. (23) from Eq. (Appendix B). For the largest observed
$\varepsilon_{C}=0.35$, approximation (B12) is a $13\%$ underestimate.

For an infinitely narrow cycle, with $\rm B^{\prime}\to b$, Eqs. (B7) and
(B9)–(Appendix B) are valid even if $\rm B^{\prime}b$ and $\rm cC^{\prime}$
are not isotherms, due to the smallness of temperature change on these paths
as compared to the finite difference $T_{\rm b}-T_{\rm c}$.

In the pseudoadiabatic case (i.e., $q_{t}=q^{*}$), the second and third
circulation integrals on the right-hand side of Eq. (B6) are evaluated along
the isothermal interval ${\rm B^{\prime}b}$ and unclosed contour ${\rm
bcC^{\prime}B^{\prime}}$ (denoted by symbol $\curvearrowright$):

$\displaystyle-\oint q_{t}dT=\oint Tdq^{*}=\int\limits^{\rm b}_{\rm
B^{\prime}}Tdq^{*}+\int\limits_{\curvearrowright}Tdq^{*}=\Delta q^{*}(T_{\rm
b}-T_{P}),$ (B13) $\displaystyle-\oint T\ln\frac{T}{T_{\rm
b}}dq_{t}=-\int\limits_{\curvearrowright}T\ln\frac{T}{T_{\rm
b}}dq^{*}\approx-\ln\frac{T_{P}}{T_{\rm
b}}\int\limits_{\curvearrowright}Tdq^{*}=\Delta
q^{*}T_{P}\ln\frac{T_{P}}{T_{\rm b}}.$ (B14)

Here $\Delta q^{*}\equiv q^{*}_{\rm b}-q^{*}_{\rm B^{\prime}}$, $T_{P}$
defined in Eq. (28) designates the mean temperature at which water vapor
condenses and precipitates. In Eq. (B14) the first equality implies that the
integral over the warmer isotherm gives zero contribution. Then we estimate
the integral by assuming that $\ln T/T_{\rm b}$ is a slowly varying function
with respect to $T$, so that it can be taken out of the integral at the point
$T=T_{P}$. With the use of Eqs. (B13) and (B14) the water warming term in the
pseudoadiabatic case is

$c_{l}\Delta q^{*}\left(T_{\rm b}-T_{P}+T_{P}\ln\frac{T_{P}}{T_{\rm
b}}\right)=c_{l}T_{\rm b}\Delta
q^{*}\left[\varepsilon_{P}+(1-\varepsilon_{P})\ln(1-\varepsilon_{P})\right]\simeq
c_{l}T_{\rm b}\Delta q^{*}\frac{\varepsilon_{P}^{2}}{2},$ (B15)

where $\varepsilon_{P}\equiv 1-T_{P}/T_{\rm b}$. 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):

$\displaystyle\frac{dq^{*}}{q^{*}}$
$\displaystyle=\frac{d\gamma^{*}_{d}}{\gamma^{*}_{d}}=(1+\gamma^{*}_{d})\left(\frac{dp_{v}^{*}}{p_{v}^{*}}-\frac{dp}{p}\right)=(1+\gamma^{*}_{d})\left(\xi\frac{dT}{T}-\frac{dp}{p}\right),$
(B16) $\displaystyle dT$
$\displaystyle=\left(\frac{dq^{*}}{q^{*}}+\frac{dp}{p_{d}}\right)\frac{T}{\xi(1+\gamma^{*}_{d})}=\left(\frac{dq^{*}}{q^{*}}+\alpha_{d}\frac{M_{d}}{RT}dp\right)\frac{T}{\xi(1+\gamma^{*}_{d})}.$
(B17)

Putting Eq. (B17) into Eq. (B5b) with $T_{0}=T$ and grouping the $dq^{*}$ and
$dp$ terms yields Eq. (40). Combining Eqs. (36), (34) and (44) we obtain for
$\beta$ in Eq. (36)

$\beta=\frac{1-\varepsilon_{C}(1-\varkappa_{2})-K_{2}}{(1-K_{2})(1+\varkappa_{1})-K_{1}(1-\varkappa_{2})}.$
(B18)

###  Appendix C Water lifting: Deriving Eq. (27)

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

$\displaystyle\oint\alpha q_{t}dp$ $\displaystyle=\int_{\rm B^{\prime}}^{\rm
b}\alpha q^{*}\frac{\partial p}{\partial r}dr-\int_{\rm b}^{\rm
c}q_{t}d\left(\frac{V^{2}}{2}+gz\right)-\int_{\rm c}^{\rm
C^{\prime}}q_{t}gdz-\int_{\rm C^{\prime}}^{\rm
B^{\prime}}q_{t}d\left(\frac{V^{2}}{2}+gz\right)$ (C1a)
$\displaystyle=\int_{\rm B^{\prime}}^{\rm b}\alpha q^{*}\frac{\partial
p}{\partial r}dr-q_{t}\frac{V^{2}}{2}\bigg{|}_{\rm b}^{\rm
c}-q_{t}\frac{V^{2}}{2}\bigg{|}_{\rm C^{\prime}}^{\rm B^{\prime}}+\int_{\rm
b}^{\rm c}\frac{V^{2}}{2}dq_{t}+\int_{\rm C^{\prime}}^{\rm
B^{\prime}}\frac{V^{2}}{2}dq_{t}-\int_{\curvearrowright}gq_{t}dz$ (C1b)
$\displaystyle=\int_{\rm B^{\prime}}^{\rm b}\alpha q^{*}\frac{\partial
p}{\partial r}dr+q^{*}\frac{V^{2}}{2}\bigg{|}_{\rm B^{\prime}}^{\rm
b}+q_{t}\frac{V^{2}}{2}\bigg{|}_{\rm c}^{\rm C^{\prime}}+\int_{\rm b}^{\rm
c}\frac{V^{2}}{2}dq_{t}+\int_{\rm C^{\prime}}^{\rm
B^{\prime}}\frac{V^{2}}{2}dq_{t}-\int_{\curvearrowright}gq_{t}dz$ (C1c)
$\displaystyle=\int_{\rm B^{\prime}}^{\rm b}q^{*}\left(\alpha\frac{\partial
p}{\partial r}+\frac{1}{2}\frac{\partial V^{2}}{\partial r}\right)dr+\int_{\rm
B^{\prime}}^{\rm b}\frac{V^{2}}{2}\frac{\partial q^{*}}{\partial
r}dr+q_{t}\frac{V^{2}}{2}\bigg{|}_{\rm c}^{\rm C^{\prime}}+\int_{\rm b}^{\rm
c}\frac{V^{2}}{2}dq_{t}+\int_{\rm C^{\prime}}^{\rm
B^{\prime}}\frac{V^{2}}{2}dq_{t}-\int_{\curvearrowright}gq_{t}dz$ (C1d)
$\displaystyle=\int_{\rm B^{\prime}}^{\rm b}q^{*}\left(\alpha\frac{\partial
p}{\partial r}+\frac{1}{2}\frac{\partial V^{2}}{\partial r}\right)dr+\int_{\rm
B^{\prime}}^{\rm b}\frac{V^{2}}{2}\frac{\partial q^{*}}{\partial
r}dr+\int_{\rm c}^{\rm C^{\prime}}\frac{q_{t}}{2}\frac{\partial
V^{2}}{\partial
z}dz+\int_{\curvearrowright}\frac{V^{2}}{2}dq_{t}-\int_{\curvearrowright}gq_{t}dz.$
(C1e)

Here it is assumed that $q_{l}=0$ at the isotherm ${\rm B^{\prime}b}$. In Eq.
(C1a), we have used the Bernoulli equation for the two streamlines, $\rm bc$
and $\rm B^{\prime}C^{\prime}$, and the hydrostatic equilibrium equation (1)
for the vertical path $\rm cC^{\prime}$. The unclosed contour ${\rm
bcC^{\prime}B^{\prime}}$ is denoted by symbol $\curvearrowright$. Note that
$z_{\rm B^{\prime}}=z_{\rm b}$ and $r_{\rm c}=r_{\rm C^{\prime}}$.

Writing the last integral in Eq. (C1e) as

$\int_{\curvearrowright}q_{t}dz=q^{*}z\bigg{|}_{\rm b}^{\rm
B^{\prime}}-\int_{\curvearrowright}zdq_{t}\equiv(z_{P}-z_{\rm b})\Delta
q^{*},$ (C2)

we obtain

$\oint\alpha q_{t}dp=-\int_{\rm b}^{\rm
B^{\prime}}q^{*}\left(\alpha\frac{\partial p}{\partial
r}+\frac{1}{2}\frac{\partial V^{2}}{\partial r}\right)dr-\int_{\rm b}^{\rm
B^{\prime}}\frac{V^{2}}{2}\frac{\partial q^{*}}{\partial
r}dr-\left[g(z_{P}-z_{\rm b})+\frac{V^{2}_{P}}{2}\right]\Delta q^{*}+\int_{\rm
c}^{\rm C^{\prime}}\frac{q_{t}}{2}\frac{\partial V^{2}}{\partial z}dz.$ (C3)

Here $\Delta q^{*}\equiv q^{*}_{\rm b}-q^{*}_{\rm B^{\prime}}$ and $z_{P}$ and
$V_{P}^{2}$ are defined in Eq. (28).

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

$\int_{\rm b}^{\rm B^{\prime}}(1+q^{*})\left(\alpha\frac{\partial p}{\partial
r}+\frac{1}{2}\frac{\partial V^{2}}{\partial
r}\right)dr=-\varepsilon_{C}\int_{\rm b}^{\rm B^{\prime}}T\frac{\partial
s^{*}}{\partial r}dr-c_{l}T_{\rm b}\frac{\varepsilon^{2}_{P}}{2}\int_{\rm
b}^{\rm B^{\prime}}\frac{\partial q^{*}}{\partial r}dr\\\ -\int_{\rm b}^{\rm
B^{\prime}}\frac{V^{2}}{2}\frac{\partial q^{*}}{\partial
r}dr-\left[g(z_{P}-z_{\rm b})+\frac{V^{2}_{P}}{2}\right]\Delta q^{*}+\int^{\rm
C^{\prime}}_{\rm c}(1+q_{t})\frac{1}{2}\frac{\partial V^{2}}{\partial z}dz.$
(C4)

In the limit $r_{\rm B^{\prime}}\to r_{\rm b}$ and $z_{\rm C^{\prime}}\to
z_{\rm c}$ 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
$\left(\partial V_{\rm c}/\partial z\right)dz|_{r=r_{\rm c}}=\left(\partial
V_{\rm c}/\partial r\right)dr|_{z=z_{\rm b}}$.

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:

$\displaystyle X_{P}\Delta q^{*}$
$\displaystyle\equiv-\int_{\curvearrowright}Xdq^{*}=\overline{X}_{\rm
bc}(q^{*}_{\rm b}-q^{*}_{\rm c})+\overline{X}_{\rm cC^{\prime}}(q^{*}_{\rm
c}-q^{*}_{\rm C^{\prime}})+\overline{X}_{\rm C^{\prime}B^{\prime}}(q^{*}_{\rm
C^{\prime}}-q^{*}_{\rm B^{\prime}}),$ (C5a) $\displaystyle\overline{X}_{\rm
kj}$ $\displaystyle\equiv-\frac{1}{q^{*}_{\rm k}-q^{*}_{\rm j}}\int_{\rm
k}^{\rm j}Xdq^{*},$ (C5b)

where $\Delta q^{*}\equiv q^{*}_{\rm b}-q^{*}_{\rm B^{\prime}}$,
$X=\\{V^{2},T,z\\}$ and $\overline{X}_{\rm kj}$ is the average value of $X$
over path $\rm kj$.

In the limit $r_{\rm B^{\prime}}\to r_{\rm b}$ and $z_{\rm C^{\prime}}\to
z_{\rm c}$, when the two streamlines coincide, we have from Eq. (C5)

$X_{P}=\overline{X}_{\rm bc}+\frac{\partial\overline{X}_{\rm bc}}{\partial
q^{*}_{\rm b}}(q^{*}_{\rm b}-q^{*}_{\rm c})+(X_{\rm c}-\overline{X}_{\rm
bc})\frac{q^{*}_{\rm c}-q^{*}_{\rm C^{\prime}}}{q^{*}_{\rm b}-q^{*}_{\rm
B^{\prime}}},$ (C6)

where $X_{\rm c}=\overline{X}_{\rm cC^{\prime}}$. When $(q^{*}_{\rm
c}-q^{*}_{\rm C^{\prime}})/(q^{*}_{\rm b}-q^{*}_{\rm B^{\prime}})\sim
q^{*}_{\rm c}/q^{*}_{\rm b}\ll 1$ (which is always the case when $z_{\rm c}$
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 $\xi^{-1}\sim 0.05$ of the relative change of $p_{v}^{*}$ and, hence, of
$q^{*}$: $q^{*}_{\rm b}\partial\overline{T}_{\rm bc}/\partial q^{*}_{\rm
b}\sim\overline{T}_{\rm bc}/\xi$. Therefore, for $X=T$ the second term on the
right-hand side of Eq. (C6) can be neglected, and $T_{P}=\overline{T}_{\rm
bc}$, which is the mean temperature of precipitation along $\rm bc$.

With the mean temperature of precipitation approximately independent of
$q^{*}_{\rm b}$, the mean precipitation height $H_{P}\equiv\overline{z}_{\rm
bc}$ scales as $H_{P}\sim T_{P}/\Gamma\sim q^{*}_{\rm b}$, because for
sufficiently large $q^{*}$ the moist adiabatic lapse rate $\Gamma$ is
approximately inversely proportional to $q^{*}$ (e.g., Makarieva and Nefiodov,
2021, Eq. (A10)). Therefore, for $X=z$ the second term in Eq. (C6) is
approximately equal to the first one, such that

$z_{P}=aH_{P}$ (C7)

with $a\sim 2$. For more accurate estimates, we evaluated hydrostatic
saturated reversible and pseudoadiabatic profiles ($ds^{*}=0$) with surface
pressure $p_{s}=950$ hPa, $z_{\rm b}=1$ km and variable $T_{\rm b}$ and
$z_{\rm c}$ (Table C1). For the same $z_{\rm c}$, the values of $T_{\rm c}$
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 $T_{s}$. For $T_{\rm b}=300$ K and $z_{\rm c}=18$ km we
have $a=1.6$. The same calculations show that the approximate Eq. (B14) for
the pseudoadiabatic water warming constitutes about $60\%$ of the exact value
(this is due to neglected correlation between $T$ and $dq^{*}$ when taking
$\ln(T/T_{\rm b})$ out of the integral in Eq. (B14)). As the water warming
term in the pseudoadiabatic case is an order of magnitude lower than the water
lifting term, this inaccuracy is inconsequential for our estimate of $K_{1}$
(29).

Table C1: Estimates of $H_{P}=\overline{z}_{\rm bc}$, $X_{P}=\\{T_{P},z_{P}\\}$ (C6), $a$ (C7), $\beta$ (B18), and $K_{1}$ (29) for hydrostatic pseudoadiabatic (in bold) and reversible profiles with surface temperature $T_{s}$, surface pressure $p_{s}=950$ hPa, and $z_{\rm b}=1$ km.  $T_{s}$, K |  $T_{\rm b}$, K |  $z_{\rm c}$, km |  $T_{\rm c}$, K | $\varepsilon_{C}$ |  $z_{P}$, km |  $T_{P}$, K | $\varepsilon_{P}$ |  $H_{P}$, km | $a$ | $\beta$ | $K_{1}$ | $\beta K_{1}/\varepsilon_{C}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathbf{303}$ | $\mathbf{300}$ | $\mathbf{18}$ | $\mathbf{197}$ | $\mathbf{0.34}$ | $\mathbf{11}$ | $\mathbf{275}$ | $\mathbf{0.08}$ | $\mathbf{6.8}$ | $\mathbf{1.6}$ | $\mathbf{0.65}$ | $\mathbf{0.042}$ | $\mathbf{0.08}$
303 | 300 | 18 | $201$ | $0.33$ | $18$ | $201$ | $0.33$ |  |  | $0.65$ | $0.044$ | $0.09$
$\mathbf{303}$ | $\mathbf{300}$ | $\mathbf{10}$ | $\mathbf{262}$ | $\mathbf{0.13}$ | $\hphantom{0}\mathbf{7}$ | $\mathbf{283}$ | $\mathbf{0.06}$ | $\mathbf{5.3}$ | $\mathbf{1.3}$ | $\mathbf{0.77}$ | $\mathbf{0.026}$ | $\mathbf{0.16}$
303 | 300 | 10 | $261$ | $0.13$ | $10$ | $261$ | $0.13$ |  |  | $0.78$ | $0.036$ | $0.22$
$\mathbf{289}$ | $\mathbf{284}$ | $\mathbf{11}$ | $\mathbf{211}$ | $\mathbf{0.26}$ | $\hphantom{100}\mathbf{5.5}$ | $\mathbf{266}$ | $\mathbf{0.06}$ | $\mathbf{4.1}$ | $\mathbf{1.3}$ | $\mathbf{0.77}$ | $\mathbf{0.020}$ | $\mathbf{0.06}$
289 | 284 | 11 | $212$ | $0.25$ | $11$ | $266$ | $0.25$ |  |  | $0.77$ | $0.028$ | $0.09$

###  Appendix D Reply to the Editor and reviewers

Oct 11, 2021

Ref.: JAS-D-21-0172777Original submission JAS-D-21-0172 with line numbers can
be found at https://bioticregulation.ru/ab.php?id=mpi4

Editor Decision

Dear Dr. Makarieva,

I am now in receipt of two reviews of your manuscript "Water lifting and
outflow gain of kinetic energy in tropical cyclones", and an editorial
decision of Minor Revision has been reached. I apologize for the delay in
decision. We had lost the help of one reviewer and therefore had to seek out
an additional Associate Editor to review your manuscript. The reviews are
included below. As you will see, although the decision is Minor Revision, the
reviewers seek some vital clarification on key equations and the writing.
Please address these as thoroughly and succinctly as possible in revision.

We invite you to submit a revised paper by 10 December 2021. If you anticipate
problems meeting this deadline, please contact me as soon as possible at
<EMAIL_ADDRESS>
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Sciences. I look forward to receiving your revision.

Sincerely,

Dr. Christopher Rozoff

Editor

Journal of the Atmospheric Sciences

December 2, 2021

Ref.: JAS-D-21-0172

Resubmission of revised manuscript

Dear Dr. Rozoff,

Thank you for your consideration of our work and for your positive evaluation
of it. We have re-organized the manuscript following our reviewers’
suggestions. We have added a new figure, which summarizes the main equations
and assumptions, as suggested by our first reviewer.

The new framework that we outline has diverse ramifications, which cannot be
all considered in one paper. We appreciate this opportunity to present the
entire framework in a comprehensive and explicit form, such that its
implications could be further developed by anyone interested.

Sincerely,

Anastassia Makarieva

(on behalf of the authors)

RESPONSE TO THE REVIEWERS

We sincerely appreciate the reviewers’ constructive comments and suggestions.

Reviewer #1:

Summary: The authors start from a more general set of basic equations and
tested different assumptions for deriving MPI. I think this study is
beneficial for critical thinking of MPI under different assumptions and
enables the quantitative comparison among the corresponding results. I can
generally follow the derivations with additional effort in some places which
need more clarification. My comments are mainly about these unclear/uneasy-to-
follow places.

Minor comments:

1\. L163: Eq(8) used Vb=VB’ and Vc=VC’, doesn’t it mean an infinitely narrow
bcC’B’b?

We clarified in the end of section 1.a as follows: Equation (8) describes an
infinitely narrow cycle $\rm B^{\prime}bcC^{\prime}B^{\prime}$ with $\partial
V^{2}/\partial z=0$ at $\rm cC^{\prime}$ and $\partial V^{2}/\partial r=0$ at
$\rm B^{\prime}b$. It is also valid for a special case of $\rm
B^{\prime}bcC^{\prime}B^{\prime}$ being a closed streamline with $V_{\rm
b}=V_{\rm B^{\prime}}$ (Emanuel, 1988).

2\. Eq(17), Eq(25) and L709: the formula partial(Vc)/partial(r) is tricky at
the first glance, I think extra explanation is needed. E.g., ’it is the result
of the right hand side of Eq(16) after taking the limit’.

We have added a clarification after Eq. (27).

3\. Eq(21): ’=’ -$>$ ’$\simeq$’

Corrected.

4\. L255: ’in the right-hand’ -$>$ ’on the right-hand’

Corrected everywhere.

5\. L281-L282: Why Tw=(Tb+Tc)/2?

This relationship is approximate ($=$ replaced with $\simeq$ in the revised
text) and valid for small $\varepsilon_{C}$, as explained in the Appendix. It
is intended as a possible way to understand the physical meaning of Eq. (23).
We have clarified as follows: "Warming the water removed at the colder
isotherm with temperature $T_{\rm c}$ and returned at the warmer isotherm with
$T_{\rm b}$ requires extra heat $c_{l}(T_{\rm b}-T_{\rm c})\Delta q^{*}$. As
the moist air ascends and cools from $T_{\rm b}$ to $T_{\rm c}$, the mean
temperature at which the water loses heat is, in the linear approximation,
$T_{\rm w}\simeq(T_{\rm b}+T_{\rm c})/2$. Accordingly, the maximum efficiency
with which this heat can be converted to work is $(T_{\rm b}-T_{\rm w})/T_{\rm
b}\simeq\varepsilon_{C}/2$."

6\. Eq(26) vs Eq(27): In appendix, A29, A33 and A34 -$>$ A30, A38 and A39 is
not straightforward enough.

We have now presented these derivations in a more detailed and explicit form,
see Eqs. (C5)-(C7). This more accurate consideration produced a somewhat
greater estimate of $z_{P}$ (28) for the pseudoadiabatic case (C7) and brought
the pseudoadiabatic and reversible values of $K_{1}$ closer to each other.
Thank you for emphasizing this point in our derivations.

7\. L330: need to mention where such point is: ’e.g., at/near RMW’

Clarified as suggested.

8\. L341-L342: it’s better to point out the place/assumption in Emanuel (1986)
that leads to the incorrectness.

We clarified in a footnote as follows. In contrast, Emanuel (1986, p. 602)
incorrectly concluded that the outflow term becomes significant if the outflow
radius is very large. Footnote: This conclusion stemmed from Emanuel (1986)’s
Eq. (18), where an outflow term proportional to a large squared radius first
appeared. While deriving this equation for a cycle with finite $\rm
B^{\prime}b$, Emanuel (1986, p. 588), on the one hand, used the conservation
of angular momentum along streamlines $\rm bc$ and $\rm B^{\prime}C^{\prime}$
and, on the other hand, assumed $r_{\rm B^{\prime}}$ (interpreted as “the
radial extent of the storm near the sea level”) to be large enough for
$\partial p/\partial r$ to vanish and, at the same time, small enough for
$r\partial p/\partial r|_{\rm b}\gg r\partial p/\partial r|_{\rm
B^{\prime}}$(for details, see Makarieva et al., 2019, Appendix A). With
$r\partial p/\partial r\sim\rho v^{2}$, ignoring this term at $\rm B^{\prime}$
means that $V^{2}_{\rm b}-V^{2}_{\rm B^{\prime}}\sim V^{2}_{\rm b}$, while at
the point of maximum wind this difference is zero.

9\. Eq(33): this is a general form for both dry and moist situations, right?
If so, L361-L362 is confusing. Only sensible heat input? I think under
isothermal assumption, there can still be latent heat input if the boundary
layer is not saturated.

10\. L379-L383: as the authors mentioned, (40)

Former Eq. (33) (now Eq. 34) is indeed a general form for both dry and moist
situations. It says that at the point of maximum wind, work equals
dissipation. By itself, it does not say anything about heat input. However,
under isothermal conditions, work $-\alpha dp$ corresponds to sensible heat
input. This is what was stated at L361-L362.

The latent heat is indeed present (former Eq. (40), now Eq. 44). Work (and
dissipation at the point of maximum wind) equals total heat input times
efficiency. Total heat input is the sum of latent and sensible heat.
Efficiency is less than unity. So there is a possibility for the relationship
$-\alpha dp=\varepsilon(-\alpha dp+L_{v}dq)$ to hold. However, it cannot be
generally valid. This is discussed in section 3.

11\. L526-L528: it is equivalent to Vc*rc-1/2f*rb2 $>=0$ given Mb=Mc. Does
this constrain the selection of c point? Maybe I missed it, is this c or C’
point arbitrary picked? Can the authors comment on this c selection?

Generally, the selection of point $\rm c$ is completely arbitrary. Moreover,
as we now clarify after Eq. (16), the obtained results do not depend on the
orientation of $\rm cC^{\prime}$ (horizontal, vertical or tilted). Expression
for $K_{2}$ (33) is written taking into acount Eq. (32), which is based on
$M_{\rm b}=M_{\rm c}$. Therefore, the condition $v_{\rm c}=0$ specifies the
outflow radius $r_{\rm c}$ according to the above relationship. It is about
$r_{\rm c}\lesssim 10r_{\rm b}$, as discussed in section 2.d.

12\. In the appendix: are A21-A26 for reversible process? If so, please
explicitly state it. It’s easier for readers to get a sense quickily.

We clarified after Eq. (B6) that Eqs. (B7)-(B8) are general, while before Eq.
(B9) we noted that Eqs. (B9)-(B12) are for reversible processes.

13\. L688: T=Tp: I think it is not very straight-forward to get this, can the
authors provide some estimation about the error as for A26?

This error estimation (about $60\%$ but not consequential for the main
conclusions) is provided after Eq. (C7).

14\. This paper is not easy for readers to go though given the dense
equations. I would suggest the authors emphasize a little more on the key
equations and the comparisons with the previous studies. A table and a figure
like table 1 and Figure 2 but for comparison of assumptions / equations will
be of great help.

We have added a new figure (Fig. 3), where we emphasize the key assumptions
and equations. In the text, we emphasize that there are three distinct
assumptions (Eqs. (10), (20) and (45)) that lead two different resulting
formulae for $V_{\rm max}$.

Reviewer #3:

Brief summary:

This study examines the role of water lifting in tropical cyclones in a
theoretical framework and finds that the water lifting’s effect on reducing
the storm intensity is generally small. The framework is energetically-based
and axisymmetric. The relationship between the theory and Emanuel’s maximum
potential intensity theory is also described.

Recommendation:

The manuscript is well written and provides a nice theoretical framework for
understanding the role of water lifting in ideal tropical cyclones. My overall
recommendation is to accept, but I encourage the authors to consider the
general comments below to improve the manuscript.

General comments:

1) The manuscript is a bit dense and difficult to follow at times. Part of
this is unavoidable due to the theoretical nature of the manuscript. However,
I would recommend some rewriting of the conclusions section to better bring
out the main points. For example, in first paragraph of the conclusions, a
main summary of the important findings is not given. Rather, first equations
are given without any segue to provide context. Additionally, the conclusions
section contains reviews of past work, which would be better located in the
introduction or main body. Many readers of JAS will first look at the abstract
and conclusions to determine interest in the rest of the paper. It is
important to distill the main points in the manuscript into these sections so
that the paper will be of more general use to a broader audience.

We have re-organized the manuscript as follows. In the Introduction, we
presented a clear structure of the main findings of the paper (the last three
paragraphs). In the concluding section, we have merged the material from the
first three subsections ("How significant is the water lifting term?", "The
estimate of Sabuwala et al. (2015)", "The outflow term") to the corresponding
sections in the main body of the manuscript (now sections 2.c "Water lifting"
and 2.e "Estimates of maximum velocity"). The flow of arguments in the
Conclusions is now facilitated by a graphical abstract of the main findings in
the new Figure 3 suggested by the first referee.

2) A critical aspect of any theory is to demonstrate either through
observational analysis or numerical simulations that the theory is valid. This
manuscript contains no internal evaluation of the theory, although comparisons
to other studies’ estimates of water lifting power are given in Table 2. I
understand the manuscript is already quite long, but such analysis would
strengthen the manuscript. In the end, perhaps such analysis needs to be in a
follow-on manuscript, but I do believe it is needed at some point. I suggest
using the axisymmetric CM1 model to validate the theory in ideal conditions.
For example, the water lifting term could be calculated in the model
simulations and its significance could be assessed in relation to the theory.

In this work, we have outlined the theoretical framework as comprehensively
and explicitly as possible. (We believe that it could be a certain lack of
clarity that contributed to the corresponding issues having remained
unsettled.) This has indeed made the manuscript quite long. However, as we
state in the revised Conclusions: "The developed analytical framework can be
used to evaluate corresponding magnitudes for different scenarios in numerical
models." This can be done in a follow-on study.

Specific minor comments:

L449: typo "overall"  Corrected.

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