1. Introduction
Tropical cyclone (TC) or hurricane potential intensity (PI) theory is the maximum TC intensity that an environment can sustain (Emanuel 1986, 2003). PI is expressed either as a minimum surface pressure or maximum surface wind speed that is determined from the thermodynamic environment. Though most TCs do not reach their PI (≈75 m s^{−1} wind speed in Earth’s tropics), PI has been widely used to interpret the climatology, climate variability, and future climate changes of TC activity (e.g., Emanuel et al. 2004; Camargo et al. 2007; Emanuel et al. 2008; Knutson et al. 2010; Sobel et al. 2016).
There have been critiques of PI theory based on its assumptions of axisymmetric TC structure and boundary layer thermodynamics (e.g., Persing and Montgomery 2003; Smith et al. 2008). In spite of these known limitations, PI is central to a ventilation index that is a useful predictor of intensification in individual tropical cyclones (Tang and Emanuel 2012). Furthermore, PI accounts for the simulated TC intensity increase in TC forecast simulations with warmed temperatures from climate change projections (Knutson and Tuleya 2004) and the sensitivity of TC intensity in singlestorm convectionpermitting simulations to temperature changes (Nolan et al. 2007; Wang et al. 2014; Ramsay et al. 2020). There are also climaterelevant idealized TC simulations (Merlis and Held 2019) with multiple TCs where PI accounts for changes in the TC intensity of the most intense TCs under varied sea surface temperature (Zhou et al. 2014; Merlis et al. 2016). Given these results, it is fair to consider PI a useful perturbation or scaling theory for intensity changes of the most intense TCs. It is PI’s temperature sensitivity—where PI has proven useful—that motivates this research, rather than the detailed dynamics of individual TCs—where PI has limitations.
PI has been assessed in future climate warming scenarios by Emanuel (1987) and subsequent generations of climate model simulations have been thoroughly examined (Vecchi and Soden 2007; Sobel and Camargo 2011; Emanuel 2013; Sobel et al. 2016). The tropicalmean PI (assessed over tropical oceans) typically increases in proportion to the tropical surface warming at a rate of ≈1 m s^{−1} K^{−1}. This increase in PI, from a climatological value of ≈75 m s^{−1}, corresponds to fractional sensitivity of about 1.5% K^{−1}. Superimposed on this tropicalmean increase in PI are geographic variations that are substantial in magnitude (∼5× larger than the tropicalmean change with some regional decreases) and uncertain as a result of their dependence on regional climate projections (Vecchi and Soden 2007; RousseauRizzi and Emanuel 2021).
PI theory has a physical interpretation in terms of a Carnot cycle and a corresponding approximate formula (described below) that accounts for the tropicalmean PI increase under global warming (Emanuel 1987, 2003; Sobel et al. 2016). However, published assessments of observed PI trends or future climate projections of PI in the recent generations of Coupled Model Intercomparison Project (CMIP) simulations have exclusively made use of a definition of PI that is a difference in convective available potential energy (CAPE), which is implemented as an iterative, numerical algorithm for PI (Bister and Emanuel 2002). Bister and Emanuel (2002) described the algorithm, and a numerical implementation of it has been publicly disseminated by K. Emanuel (ftp://texmex.mit.edu/pub/emanuel/TCMAX/). Given the importance of PI in scientific and public discourse about climate change’s effects on TC intensity, there is a need for a better understanding of the relationship between the quantitative analyses that use the publicly disseminated PICAPE code and the Carnot PI formula, which offers a physical understanding of the origin of the tropicalmean increase in PI under warming. Here, we present an analysis of the assumptions under which the CAPEbased and Carnot PI expressions are equivalent and derive a new approximate PI formula from the CAPEbased PI that is evaluated analytically using the Romps (2016) theory for CAPE.
Our analysis of the equivalence of the existing PI forms considers the standard thermodynamic cycle associated with PI theories (e.g., Emanuel 1988a, Fig. C1), where the mechanical work done is equal to a CAPE difference. We allow for phase disequilibrium in moist thermodynamics, consistent with the standard physical picture of increasing relative humidity along the surface inflow branch of the TC, and find that there is an irreversible entropy production term in the TC cycle that has previously been ignored. This term needs to be included to reconcile PI forms. Further, we find that unless an additional correction term is introduced to account for heat capacity changes in reversible thermodynamics formulations, the Carnot PI will overestimate the amount of work the system can produce and will substantially exceed the CAPEdefined PI. To quantitatively assess the success of these newly described correction terms, we numerically compute PI over a wide range of environmental surface relative humidities and find that they succeed in bridging the differences in the existing forms of PI.
To shed light on the physical picture underlying the connection between the CAPE definition of PI and the Carnot form, we also derive a new, approximate form of PI starting from the CAPE definition. This derivation builds on recent progress in understanding moist convection in the tropical atmosphere by viewing deep convection as an entraining plume that is neutrally buoyant with respect to the environment (Singh and O’Gorman 2013). This line of research has explained the increase in CAPE with warming (Singh and O’Gorman 2013; Seeley and Romps 2015), which is simulated by both cloudsystemresolving models and general circulation models (Singh and O’Gorman 2013; Sobel and Camargo 2011). It has also formed the basis of new theories for the relative humidity and thermal stratification of the tropical atmosphere (Romps 2014, 2016). In particular, Romps (2016) gives an approximate form of CAPE that can be evaluated analytically. In what follows, this CAPE theory is used to derive an approximate PI formula. For Earthlike conditions, the newly derived CAPEbased PI formula is nearly identical to the wellknown Carnot PI formula.
We review PI in section 2, assess the conditions under which the CAPE and Carnot PI formulations are equivalent in section 3, present physical intuition for how the CAPE PI and Carnot formula are connected and the results of a systematic derivation of a CAPEbased approximate PI formula in section 4, compare the CAPEbased PI to the results of numerical CAPE–PI algorithm in section 5, and conclude in section 6.
2. Potential intensity
a. Carnot cycle form
The PI theory developed by Emanuel (1986) assumes axisymmetric structure, angular momentum conserving flow and thermal wind balance away from the boundary layer, and a wellmixed boundary layer. Here only velocity PI, denoted PI, is considered, though results can be translated to pressure PI with a suitable TC structure model (e.g., Chavas et al. 2017).
Though there has been substantial discussion of upper troposphere and lower stratosphere temperature changes—affecting T_{o}—on PI (Emanuel et al. 2013; Vecchi et al. 2013; Gilford et al. 2017), these changes do not dominate the observed trends in recent decades (Wing et al. 2015). Rather, the air–sea disequilibrium increase with warming largely accounts for the tropicalmean PI increase.
b. CAPEbased PI algorithm
The algorithm of Bister and Emanuel (2002), which makes use of this PI form, iterates to adjust the parcel pressure used in these two CAPE calculations to that of the TC eyewall, taken to be the pressure PI. Because this pressure change relative to the environment is common to the two CAPEs used to determine the velocity PI, it has a modest
This CAPE algorithm has been used for all quantitative analyses of PI changes in CMIP simulations of climate change (e.g., Vecchi and Soden 2007; Sobel et al. 2019). Yet, it is not straightforward to identify why (3) increases as the climate warms, in contrast to (1), which has increases that typically scale with evaporation increases. One of the contributions of this research is to develop this understanding. For example, one might ask if the numerically evaluated (3) increase is related to the tropical environment’s projected increase in CAPE (e.g., Sobel and Camargo 2011). Our derivation shows that PI changes determined via the CAPE formula are not, in fact, related to environmental stratification changes (see also Garner 2015 for discussion of the limited role of environmental CAPE in a given climate).
3. When are Carnot and CAPEbased PI equal?
To establish the conditions required for the equivalence between PI forms, we define a PI thermodynamic cycle that has two isothermal and two isentropic legs, illustrated in Fig. 1. The isothermal surface inflow (i) leg is assumed to occur at constant total pressure, which does not account for the surface pressure gradient within a mature storm. Entropy is constant along the ascent (a) leg, which is assumed to be saturated throughout. The outflow (o) leg is assumed to match an isothermal and saturated environment, but pressure is not constant. The descent (d) leg is isentropic, but not saturated throughout, and in the unsaturated part of the descent, the water vapor mixing ratio is constant. For reversible thermodynamics, the total water mixing ratio is constant along both the ascent and the descent legs and varies along the isothermal inflow and outflow legs. We note that this cycle is designed to satisfy the assumptions of various PI forms and, in its integrated form, to establish a comparison between these forms, but as discussed previously (Emanuel 1988b; RousseauRizzi and Emanuel 2019; RousseauRizzi et al. 2021), it need not represent the actual cycle an air parcel undergoes along the TC secondary circulation. Hence the labels of the cycle legs are meant to give a sense of the direction of integration of the thermodynamic cycle, more so than to establish a direct comparison to the secondary circulation. We make no claim, for example, that the air in TCs is actually saturated at the surface before ascending in the eyewall, even though this occurs in the thermodynamic cycle presented here. In this context, taking pressure to be constant at the surface is an approximation to the PI model of Emanuel (1988b), which accounts for the environmental surface conditions, but not to the PI model of Bister and Emanuel (1998), which depends on local conditions in the eyewall.
In this section, we first review the basis of the CAPE PI definition by showing that the difference in CAPE is equal to the mechanical work produced by this thermodynamic cycle. Then, we integrate suitable thermodynamic equations for TCs—allowing for evaporation in the subsaturated air—for reversible and pseudoadiabatic cases over the cycle. Here, we cannot use the simplified differential forms of moist thermodynamic equations that assume phase equilibrium [e.g., reversible thermodynamics, Emanuel (1994) or those of Bryan and Rotunno (2009) for pseudoadiabatic thermodynamics] because a PI sufficiently large to sustain a real TC requires evaporation of liquid water into unsaturated air. (Substantial values of PI could, in theory, exist for dry or fully saturated reversible moist thermodynamics, though this requires large ≈10K air–sea temperature differences, which is unlike conditions observed over tropical oceans.) The results of these integrals relate existing PI forms and allows us to present “correction” terms that provide a precise connection between the existing PI equations.
a. CAPE and the work of the TC thermodynamic cycle
b. Work integrals around the PI thermodynamic cycle
1) Integration for reversible thermodynamics
2) Integration for pseudoadiabatic thermodynamics
c. Equivalence of PI forms
1) Analytic formulas
2) Numerical results
To test the analytic results of the previous section, we numerically evaluate the forms of PI and then add the correction terms to show that they do indeed lead to equivalence. The appearance of the irreversible entropy production term Δs_{irr} in all of the corrections suggests that a fruitful path to evaluate the analytic results is to consider a wide range of surface air relative humidity, as this term grows with subsaturation. Here, we perform calculations with no air–sea temperature difference, to clarify the differences between the entropy and enthalpy Carnot PI formulas in the simplest thermodynamic cycle that is consistent with the PI assumptions.
Figure 2a shows the comparison between the uncorrected PI formulations. If we take CAPE PI as a reference due to the close relation between the CAPE difference and the mechanical work produced by the cycle, we can see that a blunt application of the Carnot PI formulas, either based on reversible entropy or enthalpy leads to outlandishly high values of PI due to the effects of the changes in heat capacity on entropy or enthalpy. Conversely, pseudoadiabatic PI formulations only depart due to the irreversible entropy production term, which is small for Earthlike conditions. For
(a) PI vs relative humidity for enthalpy Carnot PI [PI_{k}; black, (1)], entropy Carnot PI [PI_{s}; red, (2)] and CAPE PI [PI_{C}; blue, (3)] as a function of relative humidity, for both reversible (dashed lines) and pseudoadiabatic (solid lines) thermodynamics. (b) PI vs relative humidity for corrected enthalpy Carnot PI [PI_{k}; black, (10), (12)], corrected entropy Carnot PI [PI_{s}; red, (11), (13)] and CAPE PI [PI_{C}; blue, (3)] as a function of relative humidity, for both reversible (dashed lines) and pseudoadiabatic (solid lines) thermodynamics. The correction terms include the effects of both heat capacity and irreversible entropy production for the moist reversible thermodynamics formulation, and the effects of irreversible entropy production only for pseudoadiabatic thermodynamics. (c) PI vs relative humidity with a 1K air–sea temperature contrast, for the algorithm of Bister and Emanuel (2002) (solid blue), the algorithm without iterating on surface pressure (dash–dotted blue), the algorithm without iterating on surface pressure and neglecting virtual effects (dotted blue), the pseudoadiabatic enthalpy Carnot formula [solid black, (1)], and the newly introduced analytic formula based on an entraining plume CAPE formulation [solid magenta, (17)].
Citation: Journal of Climate 35, 3; 10.1175/JCLID210360.1
(a) PI vs relative humidity for enthalpy Carnot PI [PI_{k}; black, (1)], entropy Carnot PI [PI_{s}; red, (2)] and CAPE PI [PI_{C}; blue, (3)] as a function of relative humidity, for both reversible (dashed lines) and pseudoadiabatic (solid lines) thermodynamics. (b) PI vs relative humidity for corrected enthalpy Carnot PI [PI_{k}; black, (10), (12)], corrected entropy Carnot PI [PI_{s}; red, (11), (13)] and CAPE PI [PI_{C}; blue, (3)] as a function of relative humidity, for both reversible (dashed lines) and pseudoadiabatic (solid lines) thermodynamics. The correction terms include the effects of both heat capacity and irreversible entropy production for the moist reversible thermodynamics formulation, and the effects of irreversible entropy production only for pseudoadiabatic thermodynamics. (c) PI vs relative humidity with a 1K air–sea temperature contrast, for the algorithm of Bister and Emanuel (2002) (solid blue), the algorithm without iterating on surface pressure (dash–dotted blue), the algorithm without iterating on surface pressure and neglecting virtual effects (dotted blue), the pseudoadiabatic enthalpy Carnot formula [solid black, (1)], and the newly introduced analytic formula based on an entraining plume CAPE formulation [solid magenta, (17)].
Citation: Journal of Climate 35, 3; 10.1175/JCLID210360.1
(a) PI vs relative humidity for enthalpy Carnot PI [PI_{k}; black, (1)], entropy Carnot PI [PI_{s}; red, (2)] and CAPE PI [PI_{C}; blue, (3)] as a function of relative humidity, for both reversible (dashed lines) and pseudoadiabatic (solid lines) thermodynamics. (b) PI vs relative humidity for corrected enthalpy Carnot PI [PI_{k}; black, (10), (12)], corrected entropy Carnot PI [PI_{s}; red, (11), (13)] and CAPE PI [PI_{C}; blue, (3)] as a function of relative humidity, for both reversible (dashed lines) and pseudoadiabatic (solid lines) thermodynamics. The correction terms include the effects of both heat capacity and irreversible entropy production for the moist reversible thermodynamics formulation, and the effects of irreversible entropy production only for pseudoadiabatic thermodynamics. (c) PI vs relative humidity with a 1K air–sea temperature contrast, for the algorithm of Bister and Emanuel (2002) (solid blue), the algorithm without iterating on surface pressure (dash–dotted blue), the algorithm without iterating on surface pressure and neglecting virtual effects (dotted blue), the pseudoadiabatic enthalpy Carnot formula [solid black, (1)], and the newly introduced analytic formula based on an entraining plume CAPE formulation [solid magenta, (17)].
Citation: Journal of Climate 35, 3; 10.1175/JCLID210360.1
Figure 2b verifies this reasoning by comparing all three forms of PI, with the added correction terms, for both reversible and pseudoadiabatic thermodynamics. The three forms of PI collapse onto two distinct profiles: one for reversible thermodynamics and one for pseudoadiabatic thermodynamics. Those two profiles do not depart much from one another due to the neglect of the effects of water species on density in the CAPE computation. The relative difference between corrected PI_{k} and PI_{s} is about 10^{−12}, while the relative difference between PI_{C} and corrected PI_{k} is about 0.002. This second value is larger, and may arise from inaccuracies in the numerical computation of CAPE, but it is still fairly small. For reference, merely choosing between different commonly used empirical relations to compute
d. List of equivalence conditions
To establish the requirements for the equivalence between PI forms, we started by integrating both reversible and pseudoadiabatic differential forms of thermodynamic equations around the PI cycle to obtain (6)–(9). From there, we obtained correction terms that establish precisely why the computations of PI from different forms depart from one another. In the pseudoadiabatic case, the only correction needed is to account for irreversible entropy production, which is negligible in Earthlike conditions, but not at low surface relative humidity. In the reversible case, an additional and much more important correction term arises from accounting for the variations of heat capacity. Our equations are valid if the following conditions are met:

the surface air–sea temperature difference is small,

pressure is constant along the inflow leg of the PI cycle,

we neglect the effects of water vapor on density, and

the outflow leg of the PI cycle matches an isothermal environment.
For Earthlike conditions, relaxing condition 1 does not have a large influence on the comparison of PI forms. Condition 2 is only an assumption with respect to certain PI theories (e.g., Emanuel 1988b), but is consistent with other forms (e.g., Bister and Emanuel 1998). Condition 3 has a nonnegligible effect, which we will quantify in section 5, and the validity of condition 4 depends on the uppertropospheric stratification. In contrast to most literature on PI, we did not assume phase equilibrium in our derivations, which leads to the appearance of the irreversible entropy production correction term.
4. An approximate Carnot PI formula from CAPE definition
By integration over thermodynamic cycles, the previous section enumerated the conditions under which Carnot and CAPE PI are equal. But the physical connection between CAPE differences and the Carnot formulas remains enigmatic. This section utilizes the theory of Romps (2016) to shed light on this, first in a backoftheenvelope fashion (section 4a) and then in greater detail (section 4b).
a. A napkin derivation
This straightforward physical picture connecting the CAPE formula and the Carnot expression has not previously been described. In what follows, we present a complete version of this result (derived in appendix B), show an intuitive graphical representation of where the buoyancy contrast between the two parcels’ profiles arises, describe the magnitude of the terms in the more complete expression, and evaluate the approximations.
b. Complete approximate PI formula from CAPE definition
Here, we review the CAPE theory of Romps (2016), apply it to derive an approximate PI formula, and evaluate the magnitude of its terms to reconcile it with (1).
1) Review of Romps (2016) CAPE theory
Romps (2016, hereafter R16) developed an analytical theory for CAPE by first deriving analytical formulas for moist adiabatic temperature and humidity profiles, allowing for the effects of dilution by entrainment. He then invoked the “zero buoyancy” plume model of Singh and O’Gorman (2013), which says that a tropicalmean temperature profile can be obtained as the temperature profile of a dilute moist adiabat (the corresponding entraining plume thus has zero buoyancy relative to the mean profile). This led R16 to a general analytic expression for tropical CAPE. Here, however, we are interested in a difference of CAPEs, which is independent of the environmental profile; we thus need not invoke the zerobuoyancy model, or make any other assumptions about the environmental profile. Instead, we simply apply R16’s undilute, moistadiabatic temperature and humidity profiles to the saturated “hurricane” and environmental parcels to calculate their CAPE difference.
2) Evaluation of PI with CAPE theory
The detailed derivation is presented in appendix B. The essence is that we evaluate the CAPE definition of PI (3) using the R16 profiles for our undilute parcels. The common environmental soundings cancel, leaving only the integrated buoyancy differences of the undilute parcels.
To gain an intuition for that analysis, we plot these heights in Fig. 3a (black lines) for representative conditions: T_{s} = 300 K, T_{a} = 299 K,
(a) Height and its components [dry in red and humidity in blue; (B1)] vs temperature for the two nonentraining parcels that determine the PI (B3) for representative values of relative humidity, surface, surface air, and outflow temperatures. Potential intensity is proportional to the integral of the difference between the black solid and dashed curves. (b) Temperature vs height over a range of surface relative humidities with
Citation: Journal of Climate 35, 3; 10.1175/JCLID210360.1
(a) Height and its components [dry in red and humidity in blue; (B1)] vs temperature for the two nonentraining parcels that determine the PI (B3) for representative values of relative humidity, surface, surface air, and outflow temperatures. Potential intensity is proportional to the integral of the difference between the black solid and dashed curves. (b) Temperature vs height over a range of surface relative humidities with
Citation: Journal of Climate 35, 3; 10.1175/JCLID210360.1
(a) Height and its components [dry in red and humidity in blue; (B1)] vs temperature for the two nonentraining parcels that determine the PI (B3) for representative values of relative humidity, surface, surface air, and outflow temperatures. Potential intensity is proportional to the integral of the difference between the black solid and dashed curves. (b) Temperature vs height over a range of surface relative humidities with
Citation: Journal of Climate 35, 3; 10.1175/JCLID210360.1
3) Magnitude of terms
To reconcile this new analytic formula with (1), we examine the magnitude of the terms in the expression (17) for Earthlike conditions.
First, the subcloud humidity contributions are small compared to those of the freetroposphere (B4). This can be quantified by considering the magnitude of an upper bound on the subcloud term. Replacing the temperaturedependent saturation specific humidity
The appearance of
5. Comparison to CAPEbased PI algorithm
This section evaluates the impacts of the equivalence conditions enumerated in section 3 on the CAPEbased PI algorithm, and also compares these results to the new approximate PI expression (17). Although CAPE PI should be largely independent of the environmental temperature profile, the CAPE–PI algorithm requires one as an input. For simplicity we use dilute moist adiabats obtained from the R16 formalism, consistent with the zerobuoyancy plume model. This is akin to using moist adiabats as approximate tropical temperature profiles, but including entrainment.
First, we use a representative Earthlike tropical sounding (R16 dilute adiabat plus representative surface RH) in the PI algorithm to assess the quantitative importance of factors neglected in both the derivation of the correction terms in section 3 and the new analytic formula in section 4. In particular, both assume constant surface pressure and omit virtual effects. Second, we consider a range of soundings with varying surface relative humidity. Not only was this a useful demonstration of the role of irreversible entropy production in the correction terms described in section 3, but it is also a climate variation that possibly distinguishes the new approximate PI formula (17) (with its dependence on both LCL and surface temperature) from the Carnot approximation (dependent on surface temperature).
The R16 dilute adiabats are specified by the surface air temperature T_{a}, the outflow temperature T_{o}, the surface relative humidity
a. Assessment of derivations’ approximations
We compare the results of the standard CAPEbased PI algorithm with altered algorithms that bring the numerical algorithm toward the R16 theory by using the same approximations. For the Earthlike sounding, the CAPEbased PI algorithm has a velocity PI of 94.3 m s^{−1} (Table 1, standard).
Results of the numerical CAPEbased PI algorithm (3) for the velocity PI (m s^{−1}) for an Earthlike sounding with T_{s} = 300 K (see section 5 for other sounding details) in the top row and the percentage increase in velocity PI in response to 1 K surface warming in the bottom row. The columns are variants of the algorithm to assess the magnitude of the approximations used in the derivation of (17), with the full description of the altered algorithms in section 5a.
In the derivation of the equivalence conditions in section 3, the surface pressure was assumed constant, and in the numerical evaluations of thermodynamic quantities shown in Figs. 2a and 2b, it was chosen to be 10^{5} Pa. Likewise, in the derivation of the new approximate form of PI from the R16 CAPE theory, we did not consider the effect of the TC pressure on CAPE. To assess the neglect of the TC pressure drop relative to the environment, we alter the CAPEbased PI algorithm by not iterating the parcel pressures to that of the pressure PI. In the numerical algorithm, we perform a single iteration, so that the parcel pressures of the CAPE calculations remain equal to that of the environment. Comparison of the values in Table 1 shows that this decreases the PI by ≈4.7% (no iteration).
We note that this is a much smaller effect of TC pressure on PI than that reported in some of the literature, starting with Emanuel (1988a). This occurs because the model of Emanuel (1988a) implies that the central pressure of the TC only influences the saturation parcel mixing ratio, while the model of Bister and Emanuel (2002) considers that the TC central pressure influences both the saturation and the environmental parcels, leading to a large cancellation of the pressure change effect in Bister and Emanuel (2002) and in the associated PI algorithm, but not in the model of Emanuel (1988a).
One of the assumptions in section 3 and in the R16 theory for CAPE is to neglect the virtual (water vapor) effect on density. To assess the omission of the virtual effect, we alter the buoyancy calculation, replacing virtual temperature with temperature, in the algorithm’s CAPE subroutine. Table 1 shows that this decreases the PI by ≈3% (no virtual effect).
The R16 theory also assumes that the parcel (subscript p) buoyancy can be approximated by the ratio of the temperature difference relative to the environment (subscript e) and the tropospheric average temperature: b ∝ (T_{p} − T_{e})/T_{avg}. The PI algorithm computes CAPE as an integral in pressure, rather than altitude. Therefore, we replace the pressure of the R16 sounding—obtained by hydrostatic integration of the vertically varying temperature—with an approximate pressure that is a hydrostatic integration using the tropospheric average temperature T_{avg}. Table 1 shows that this increases the PI by ≈0.1% (buoyancy approximation). When all three approximations are used simultaneously, the PI decreases by ≈6.9% (Table 1, all), suggesting that these small approximations add close to linearly.
Table 1 also shows the percentage change in PI when the surface temperature is warmed by 1 K, holding the surfacetoair temperature difference fixed. One might take this to be a starting point for the magnitude of the sensitivity of PI to global warming; however, energetically consistent climate change simulations typically have decreases in the surfacetoair temperature difference and increases in surface relative humidity (e.g., Richter and Xie 2008), which would reduce the PI increase. The standard algorithm has a 4.1% increase in PI for this simple warming case and all of the algorithms have comparable sensitivities (Table 1, bottom row). For this perturbation, the new formula (17) has a 3.5% increase and the Carnot formula (1) has a 3.2% increase. This shows that the assumptions used in the derivation are modest not only in terms of the climatological PI, but also for the response to climate perturbations.
In summary, the conditions used to derive the equivalence of PI forms and the approximations used in the derivation of (17) modestly alter the PI for Earthlike conditions, when they are used in the numerical CAPEbased PI algorithm. This shows that climatological values of PI can be recovered with the simplifications used in the derivations. Furthermore, the sensitivity to a simple warming case is little changed by these approximations.
b. Application to surface relative humidity changes
The Carnot approximate PI formula (1) and the newly derived CAPEbased approximate PI formula (17) have similar dependence on outflow temperature T_{o} and sea surface temperature T_{s}. Therefore, we turn again to changes in surface relative humidity
The CAPEbased approximate formula (17) will have increasing PI from increasing surface–air disequilibrium, like the Carnot formula, but it also has a dependence on
The CAPEbased PI algorithm varies from velocity PI near 190 to 55 m s^{−1} across this range of soundings (Fig. 2c, blue solid line). Taking the Earthlike
We note that since we effectively enforce different near–surface thermodynamic disequilibrium for all
In summary, both the Carnot formula and new approximate formula slightly overestimate the sensitivity to systematically varied surfaceair relative humidity compared to a numerical CAPEbased calculation with the same assumptions and underestimate the sensitivity compared to the CAPE–PI algorithm. Nevertheless, these differences only emerge for fairly low surface relative humidity, rather than in Earthlike situations.
6. Conclusions
Potential intensity (PI) theory plays an important role in climate change discourse about tropical cyclones (e.g., Sobel et al. 2016). For example, there is confidence in the expectation that the intensity of the most intense tropical cyclones will increase as a result of warming because it is found in both simulations (e.g., Knutson and Tuleya 2004) and PI theory (Emanuel 1987). The tropicalmean PI increase is robustly simulated and has previously been interpreted in terms of the Carnotcycle based approximate PI formula that depends on the air–sea enthalpy disequilibrium, which increases with warming. However, quantitative assessments of PI changes in climate models use the iterative numerical CAPEbased algorithm, where it is less clear why PI increases with warming.
Here, we presented a new analysis of when the Carnot and CAPE PI definitions are equivalent. The CAPE PI definition is equal to the mechanical work produced by the TC cycle, under conditions we enumerated. Independent of the thermodynamic formulas, there is an irreversible entropy production term in the TC cycle that has previously been ignored, and this term must be added to the Carnot definition of PI to make it equivalent to the CAPE definition. For reversible thermodynamics, an additional correction term that accounts for heat capacity changes is needed to reduce the Carnot PI to the CAPE PI. Our numerical assessment of the analytic formulas that connect the PI forms successfully captures the PI dependence on surface relative humidities, where factor of ≈4 variation in PI provides a stringent test.
We also used the CAPEbased definition of PI to provide a physical interpretation for the Carnot form by building on recent advances in the understanding of CAPE. The essence is that the buoyancy difference between the two parcels that determine the PI’s CAPE difference result from surface moist static energy contrasts, which become buoyancy contrasts as latent heat is released over the temperature depth of the troposphere. This can be formalized using the CAPE theory of R16 in the CAPE definition of PI. The resulting approximate PI formula and its sensitivity to warming are comparable to the previously discussed approximate Carnot form of PI, though the new formula’s PI is ≈10% higher. The derivation uses approximations that lead to modest ≈5% changes when the CAPEbased algorithm is modified to use the same approximations (Table 1), suggesting no quantitatively important errors are introduced in our derivation of an approximate PI formula.
The research presented here connects the numerical CAPE PI algorithm, based on the CAPE definition of PI, to the more physically intuitive Carnot definitions, and provides a new approximate formula derived from the CAPE definition that is quite similar to the existing Carnot approximate formula. This bridges the gap between the quantitative technique used to assess future climate model projections and the physical explanation for the increase in PI under warming as the result of nearsurface thermodynamic changes.
Acknowledgments.
The authors thank Kerry Emanuel for providing the PICAPE code (ftp://texmex.mit.edu/pub/emanuel/TCMAX/). Two anonymous reviewers and Dan Chavas provided helpful feedback. We thank Stephen Garner, Paul O’Gorman, Adam Sobel, and Wenyu Zhou for motivating discussions, and Stephen Garner and Hiroyuki Murakami for comments on a draft of the manuscript. TM was supported by a NSERC Discovery grant and Canada Research Chair (Tier 2).
Data availability statement.
The code to reproduce the figures is available at https://web.meteo.mcgill.ca/∼tmerlis/code/rousseaurizzi_etal21_cape_pi_code_rev1.tgz.
APPENDIX A
Irreversible Entropy Production Integral
APPENDIX B
Derivation of an Approximate PI Formula Using R16
In this appendix, we will substitute the R16 theory for CAPE (16) into the CAPE PI formula (3) to derive a new approximate PI formula (17) that is similar to the Carnot expression.
There are differences in the formula that we presented above and R16. First, there are changes in some variables to avoid potential confusion in the context of TCs and PI. More importantly, the subcloud layer is ignored in R16, which eliminates the step function from (B1) and reduces the LCL quantities to surface quantities (T_{LCL} → T_{s} and
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