Editorial Type:
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Article Type:
Research Article

Tropical Cyclone Potential Size

Danyang Wang aMinistry of Education Key Laboratory for Earth System Modeling, Department of Earth System Science, Tsinghua University, Beijing, China
bPurdue University, West Lafayette, Indiana

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Yanluan Lin aMinistry of Education Key Laboratory for Earth System Modeling, Department of Earth System Science, Tsinghua University, Beijing, China

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Daniel R. Chavas bPurdue University, West Lafayette, Indiana

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Online Publication:
02 Nov 2022
Print Publication:
01 Nov 2022
DOI:
https://doi.org/10.1175/JAS-D-21-0325.1
Page(s):
3001–3025
Corrigendum to this article
Free access

Abstract

A model for tropical cyclone (TC) potential size (PS), which is capable of predicting the equilibrium outer radius of a TC solely from environmental parameters, is proposed. The model combines an updated Carnot cycle model with a physical model for the wind profile, which serve as energetic and dynamic constraints, respectively, on the minimum pressure. Physically, the Carnot cycle model defines how much the surface pressure can be dropped energetically, and the wind profile model defines how large the steady-state storm needs to be to yield that pressure drop for a given maximum wind speed. The model yields an intrinsic length scale VCarnot/f, with f the Coriolis parameter, VCarnot similar to the potential intensity Vp, but without a dependence on the surface exchange coefficients of enthalpy Ck and momentum Cd. Analytic tests with the theory varying outflow temperature, sea surface temperature (SST), and f demonstrate that the model predictions are qualitatively consistent with the Vp/f scaling for outer size found in past work. The model also predicts a weak dependence of outer size on Cd, Ck, and horizontal mixing length lh of turbulence, consistent with numerical simulation results. Idealized numerical simulation experiments with varied tropopause temperature, SST, f, Cd, Ck, and lh show that the model performs well in predicting the simulated outer radius. The VCarnot/f scaling also better captures the dependence of simulated TC size on SST than Vp/f. Overall, the model appears to capture the essential physics that determine equilibrium TC size on the f plane.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yanluan Lin, yanluan@tsinghua.edu.cn

Abstract

A model for tropical cyclone (TC) potential size (PS), which is capable of predicting the equilibrium outer radius of a TC solely from environmental parameters, is proposed. The model combines an updated Carnot cycle model with a physical model for the wind profile, which serve as energetic and dynamic constraints, respectively, on the minimum pressure. Physically, the Carnot cycle model defines how much the surface pressure can be dropped energetically, and the wind profile model defines how large the steady-state storm needs to be to yield that pressure drop for a given maximum wind speed. The model yields an intrinsic length scale VCarnot/f, with f the Coriolis parameter, VCarnot similar to the potential intensity Vp, but without a dependence on the surface exchange coefficients of enthalpy Ck and momentum Cd. Analytic tests with the theory varying outflow temperature, sea surface temperature (SST), and f demonstrate that the model predictions are qualitatively consistent with the Vp/f scaling for outer size found in past work. The model also predicts a weak dependence of outer size on Cd, Ck, and horizontal mixing length lh of turbulence, consistent with numerical simulation results. Idealized numerical simulation experiments with varied tropopause temperature, SST, f, Cd, Ck, and lh show that the model performs well in predicting the simulated outer radius. The VCarnot/f scaling also better captures the dependence of simulated TC size on SST than Vp/f. Overall, the model appears to capture the essential physics that determine equilibrium TC size on the f plane.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yanluan Lin, yanluan@tsinghua.edu.cn
Keywords: Hurricanes/typhoons; Thermodynamics; Energy budget/balance

1. Introduction

Theoretical models have produced analytical solutions for the upper bound of tropical cyclone (TC) intensity, known as potential intensity (PI; Vp), solely from environmental parameters (Emanuel 1986, 1988; Emanuel and Rotunno 2011, hereafter ER11; Rousseau-Rizzi and Emanuel 2019). PI theory, derived from several assumptions, such as the assumption of slantwise neutrality of the eyewall structure, generally states that the PI of a TC is influenced by air–sea enthalpy disequilibrium, temperature difference of surface and tropopause air, and sea surface exchange coefficients of enthalpy Ck and momentum Cd. However, there does not currently exist a theory to predict the overall size of a TC, defined as the outer radius where near-surface winds vanish (ra), solely from environmental parameters. In fact, ra is an input parameter in several physical models of TC wind structure (ER11; Chavas et al. 2015, hereafter C15).

Idealized numerical simulations in previous studies indicate that the quasi-equilibrium state of a TC on the f plane can be maintained for tens of days or longer (Khairoutdinov and Emanuel 2013; Zhou et al. 2014; Chavas and Emanuel 2014, hereafter CE14; Cronin and Chavas 2019; Rousseau-Rizzi et al. 2021). CE14 showed that Vp/f, with f the Coriolis parameter, is a natural length scale for the equilibrium value of ra for mature TCs on an f plane. A similar radial length scale χ/f, where χ is similar to Vp but without the Ck/Cd factor (or simply taken to be 1), has also been proposed as the intrinsic TC length scale in the theory of Emanuel (1989, 1995). However, there lack explicit mechanistic explanations for why equilibrium TC size should scale with Vp/f. Khairoutdinov and Emanuel (2013) and Emanuel (2022) derived another scaling Lυqb/f for equilibrium TC size with qb the boundary layer specific humidity, based on energy and entropy budgets. In nature, though, median TC size appears to scale with the Rhines scale (Chavas et al. 2016; Chavas and Reed 2019; Lu and Chavas 2022), which depends on β−1/2, as this scale is much smaller than the f−1 scale in Earth’s tropics where f is relatively small and β is relatively large. However, there is observational evidence that the upper bound of TC size does indeed scale with Vp/f (Fig. 5c of Chavas et al. 2016), even if most storms are substantially smaller than this length scale likely owing to the effects of β.

In this study, we propose a model for the maximum potential TC size on an f plane defined solely from environmental parameters. The model yields an intrinsic scaling for equilibrium TC size and explains why equilibrium TC size appears to follow a Vp/f length scale on an f plane when Ck/Cd is held fixed as found in past work. The scaling is similar to Vp/f but without the Ck/Cd factor; hence, it is closely related to χ/f and Lυqb/f. Since the Ck/Cd factor is nondimensional, dimensional analysis alone cannot determine whether this factor should or should not exist in Vp/f length scale, so the distinction between Vp/f and χ/f is ambiguous. As Ck/Cd is often taken as approximately 1, their magnitudes will also be very similar. We test this theory, including the dependence on Ck and Cd, against idealized numerical simulations.

The basic idea of our model is as follows. It is known that steady-state TC size is constrained by the energetic cycle (i.e., Carnot cycle) of its air parcels, as is explicitly pointed out in Emanuel (1991): a larger size decreases the inner-core surface pressure deficit by requiring a larger work in the outflow to restore angular momentum lost to surface friction (though we will discuss in the end that this outflow work is not required to constrain TC size). On the other hand, through the TC wind profile (e.g., C15 model), TC size is constrained dynamically by the relationship between storm size and the surface pressure deficit via gradient wind balance (Chavas et al. 2017): given the same intensity, a larger size increases the inner-core surface pressure deficit. As the Carnot cycle model and C15 model are effectively independent of each other, a TC satisfying the two models may have a deterministic size. Thus, our hypothesis is that TC size (outer radius) can be determined by satisfying both energetic (Carnot cycle) and dynamical (C15 model) constraints simultaneously. Physically, the pressure drop experienced by inflowing parcels is limited by its energetic cycle, whereas the inner-core pressure deficit must increase as ra increases via gradient wind balance; hence, there is a maximum potential size that satisfies both energetic and dynamical constraints.

The paper is organized as follows. In section 2, we introduce an updated Carnot cycle model and briefly review the C15 wind profile model. We then show how combining the updated Carnot cycle and C15 model predicts a unique equilibrium TC size. We show that this model can qualitatively reproduce the Vp/f scaling found in past work except for the dependence on Ck and Cd, and we propose a new length scale intrinsic to our model that can also capture this dependence on exchange coefficients. We refer to our approach as a model for TC potential size (PS) since it predicts a theoretical upper limit of TC outer radius and provides a new intrinsic length scale for this size. The TC outer size in this study is specifically the TC size at equilibrium state, and the word “equilibrium” will be omitted for brevity hereafter. In section 3, we test our TC PS model against idealized numerical simulations on an f plane by varying different environmental parameters. In section 4, we test whether our modifications to the Emanuel’s Carnot cycle model are essential to the TC PS model predictions. In section 5, we summarize and discuss our findings.

2. Description of the model

a. An updated Carnot cycle model

As reviewed by Rousseau-Rizzi et al. (2021), the conceptual model of a closed thermodynamic cycle to describe a steady state TC was presented by Eliassen and Kleinschmidt (1957). The same concept was developed further by Emanuel (1986, 1988, 1991). Figure 1 shows a hypothetical path of an air parcel in an updated Carnot cycle in a fully developed TC. The updated Carnot cycle is qualitatively similar to the Emanuel’s Carnot cycle (Emanuel 1986, 1988, 1991), but with three main modifications: 1) the parcel loses nearly all the water during the eyewall ascent and experiences a complete resaturation during the inflow leg rather than simply starting from a rather high relative humidity in the inflow without an explicit remoistening process; 2) radiative cooling is acting throughout both the outflow and descent legs rather than only acting in the upper troposphere at the outflow temperature; and 3) exchanges of Gibbs free energy of water species in the hydrological cycle are considered rather than neglected. The modifications are believed to more realistically describe a hypothetical thermodynamic cycle of a TC, as it now includes an explicit treatment of the water budget and its energetics as part of the closed cycle, though the modifications are not critical to our final conclusions as shown in section 4. The complete resaturation tends to increase the net heat in the cycle but this effect is compensated by a longer path of radiative cooling and the inclusion of the exchanges of Gibbs free energies of water species.

Fig. 1.
Fig. 1.

Schematic path of a parcel in a updated Carnot cycle in a fully developed TC. Point a marks the radius of vanishing near surface wind and contains environmental relative humidity. Point m marks the radius of maximum wind. Point o marks the radially farthest point the parcel could reach in the outflow. Point o′ has the same radius as o, but with a lower altitude. Point a′ is spatially close to a where the parcel has near-zero relative humidity. The arrows mark the direction of the path and the path is a closed cycle. See text for different processes in the path. Inside the cycle is a schematic T–s diagram of the cycle.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

We assume the parcel is able to complete a closed cycle, as it could in an idealized simulation (Hakim 2011; Rousseau-Rizzi et al. 2021), which does not have mass exchanges of dry air with the environment. Starting from the outer radius ra near the surface (point a in Fig. 1), the parcel flows cyclonically inwards with constant temperature Ts, until reaching the radius of maximum wind rm (point m in Fig. 1) where it begins to ascend. From point a to m, the parcel experiences boundary layer turbulent friction and loses most of its absolute angular momentum M. At the same time, via surface enthalpy fluxes, the parcel increases its entropy s from its environmental value at point a (sa) to the highest value at point m (sm). At point a, the relative humidity H of the parcel is equal to the environmental value He; at point m, H is 1. From point m to o, the parcel experiences eyewall ascent to the tropopause and flows outwards to the farthest point o, where the parcel may achieve maximum anticyclonic velocity. The parcel loses nearly all of its water by precipitation in the eyewall. Note that M is conserved during this process and s is also nearly conserved during the eyewall ascent. From point o to o′, the parcel descends below the upper-tropospheric anticyclone by restoring M to its original value at point a through turbulent mixing with the environment. From point o′ to a, the parcel experiences inviscid descent conserving M. From o to a′, the parcel subsides with no remoistening. From a′ to a, the parcel is remoistened to ambient relative humidity. Hence, a′ marks the top of boundary layer. For simplicity we do not distinguish the differences of temperature and dry air partial pressure between a′ and a, as will be assumed below. During the outflow leg from m to o and the subsidence leg from o to a′, radiative cooling plays the dominant role in gradually reducing the parcel entropy from sm to its lowest value sa′. This results in a triangular shape instead of a rectangle in the temperature–entropy (T–s) phase space of a thermodynamic cycle in a TC and, hence, an efficiency that is less than Carnot efficiency (Hakim 2011; Pauluis and Zhang 2017). The hydrological cycle is also associated with exchanges of Gibbs free energy (aka Gibbs potential), which is a Legendre transformed function of internal energy (Callen 1985), of water species. During the inflow, the parcel increases its water mass and imports low Gibbs free energy of water vapor; during the eyewall ascent, the parcel loses its water mass and exports relatively high Gibbs free energy of liquid water (Pauluis and Zhang 2017). Both processes decrease the available mechanical work done by the parcel in the cycle (Pauluis 2011; Pauluis and Zhang 2017).

We next formulate the updated Carnot cycle. The procedure is very similar to the original one in Emanuel (1991). Following an air parcel in a TC, the Bernoulli equation may be written as (see appendix A, following Romps 2008)
d[(1+qt)12|V|2]=−αddp+F⋅dl+ϕdqt−d[(1+qt)ϕ],
where V is three-dimensional velocity vector, qt the total water mixing ratio, αd the specific volume of dry air, p the pressure, F the frictional force per unit mass of dry air, dl a finite displacement of the parcel, and ϕ = Γz the geopotential, with Γ the gravitational acceleration, and z the altitude. In Eq. (1) the diffusive fluxes of kinetic energy have been neglected.
The Gibbs relation may be written as
dh−αddp=Tds+∑gwdqw,
where h is the enthalpy of moist air per unit mass of dry air, T the temperature, s the moist entropy per unit mass of dry air, gw the specific Gibbs free energy of water species, qw the mixing ratio of water species, and subscript w = υ, l, s represents water vapor, liquid water, and solid water, respectively. Here entropy s is defined as
s≈cpdln⁡TTtrip−Rln⁡pdp0+qυLυT−qυRυln H,
where cpd is the specific heat of dry air at constant pressure, R the gas constant of dry air, Rυ the gas constant of water vapor, Lυ the latent heat of vaporization, H the relative humidity, pd the partial pressure of dry air, Ttrip and p0 the triple-point temperature and a reference pressure, respectively. Ice phase has been neglected in this expression. The heat capacity of liquid water cl and that of water vapor at constant pressure cpυ have also been neglected to be consistent with our treatment of Gibbs free energy below. This assumption also leads to a temperature independent Lυ, taken as its value at the reference temperature.
Substituting Eq. (2) into Eq. (1) and integrating along a closed cycle gives
∮Tds+∮∑gwdqw=−∮F⋅dl+∮dWp,
which states that heat added to the system is used to do mechanical work. Equation (4) is an updated version of the Emanuel Carnot cycle that now includes the exchanges of Gibbs free energy of water species ∮∑gwdqw and the work to lift water Wp=∮dWp=∮Γqtdz (Pauluis et al. 2000; Pauluis and Zhang 2017), which is transformed from the third term on the rhs of Eq. (1) (see also appendix A). Note that ∮∑gwdqw is typically negative (with properly chosen reference states of enthalpy and entropy) so that it reduces the mechanical work performed by the air parcel in a cycle (Pauluis and Zhang 2017). We may for simplicity symbolically define Qs=∮Tds and QGibbs=∮∑gwdqw. Following the Emanuel Carnot cycle model, frictional dissipation only occurs in the inflow and at large radii in the outflow. Thus, −∮F⋅dl=WPBL+Wout may be defined, with WPBL and Wout denoting work against friction in the inflow and outflow, respectively. With the symbols defined above, Eq. (4) may be rewritten as
Qs+QGibbs=WPBL+Wout+Wp.
We provide expressions for each term in Eq. (5) as follows. Qs may be expanded:
Qs=∮Tds=ηϵC∫a′mTds≈ηϵCTs(sm−sa′)≈ηϵC(RTslnpdapdm+Lυqυ m),
where ϵC=(Ts−To)/Ts is the Carnot efficiency, and η is a coefficient representing the efficiency relative to Carnot efficiency. For example, η=1/2 for a triangle parcel trajectory in T–s phase space as noted above (e.g., Fig. 6a of Pauluis and Zhang 2017). Zero relative humidity has been assumed at point a′. The dry air partial pressure and temperature at a′ have also been approximated as environmental values pda and Ts, respectively; these errors are small as the second term in the bracket on the rhs of Eq. (6) is dominant.
As shown by Figs. 6d–f of Pauluis and Zhang (2017), ∮∑gwdqw (with the reference states chosen therein) is dominated by the import of negative Gibbs free energy of water vapor (gυ) in the near-surface inflow branch of the cycle, which may be written as
gυ≈RυTlnH=RυTlnee*≈RυT[ln(RυRpdaqυ)−lne*]=RυT(ln⁡qυ+c1),
where the first line is defined on p. 94 of Pauluis (2011), e is the partial pressure of water vapor, e* the saturation vapor pressure, and c1=−ln⁡e*+ln⁡[(Rυ/R)pda]=−lnqυ a*. The dry air partial pressure is set to pda, and all remoistening is assumed to occur at this pressure to make the integration below tractable. We have used the relation that qυ=(R/Rυ)(e/pd). Thus, with Eq. (7), QGibbs=∮∑gwdqw may be expanded:
QGibbs≈∫a′−mgυdqυ        =∫qυa'qυmgυdqυ         =∫qυa′qυmRυTs(lnqυ+c1)dqυ         =RυTs∫qυa′qυmlnqυ dqυ+RυTsc1(qυm−qυa′)         ≈RυTs(−qυ m+qυ m lnqυ m)+RυTsc1qυm         =−RυTsqυm+RυTsqυm lnqυ mqυ a*        =−RυTsqυm+RυTsqυm lnpdapdm,
where es* is the near surface saturation vapor pressure and we have taken the limit of qυ a′≈0 (i.e., air reentering the boundary layer has been almost fully dehydrated). This assumption follows that fact that tropical tropopause saturation water vapor mixing ratio should be on the order of 10−5 kg kg−1. For the m–o leg, the exchange of Gibbs free energy is zero under assumption of cl ≈ 0. For the a′–a leg, from Eq. (8) we have ∫a′−agυdqυ≈−RυTsqυa+RυTsqυalnHa, which constitutes the majority of QGibbs.
Following the Emanuel Carnot cycle model (e.g., Emanuel 1991), work against friction −∮F⋅dl is done principally at the surface during the inflow, where absolute angular momentum M is lost, and in the free atmosphere in the outflow where M is restored from the environment. M is defined by
M=rυ+12fr2,
where r is radius from the TC center, υ the tangential velocity, and f the Coriolis parameter. By assuming hydrostatic balance in Eq. (1), assuming vanishing qt, and using Eq. (9), the work done against friction in the outflow may be written:
Wout=∫oo′−F⋅dl=−12(υo′ 2−υo 2)=−12[Mo′2−Mo2ro2−f(Mo′−Mo)+14f2(ro′2−ro2)]=12[f(Mo′−Mo)−Mo′2−Mo2ro2]=12[f(Ma−Mm)−Ma2−Mm2ro2]=14f2ra2−12fMm−12Ma2−Mm2ro2,
As there is no explicit theory for the size of the outflow ro, we may for simplicity assume ro to be infinitely large, which means M is restored in the environment at some large radius well beyond the storm, so that the last term in the last line of Eq. (10) can be ignored. Thus, we may have an approximation
Wout≈14f2ra2−12fMm,
The approximation of Eq. (11) overestimates Wout. Making use of Eq. (1), the work done against friction in the inflow branch may be written:
WPBL=∫am−F⋅dl≈∫am−αddp+∫am−d[(1+qt)12|V|2]≈RTslnpdapdm−12Vmax2,
where the ideal gas law has been used assuming isothermal inflow at Ts and Vmax is the maximum tangential wind of a TC near the top of boundary layer. The effect of water content on WPBL has been neglected. Note that a complete expression for WPBL can be obtained by substitution of Eq. (2) again to eliminate −αddp. This will yield two additional terms RυTsqυ alnHe+∫amgυdqυ, on the order of −102 J kg−1. As the moistening process is weak from a to m compared to that from a′ to a, these terms are neglected for simplicity here. Additionally, Vmax is typically located at the top of the boundary layer instead of near surface; the work against friction from surface to the location of Vmax at rm, would also be on the order of 102 J kg−1 assuming tangential frictional acceleration ∼10−2 m s−2. The contribution of this as well as of the vertical change of total velocity, which both depend on detailed inner-core turbulence, is not considered here.
Finally, we parameterize Wp by taking
Wp=(β−1)(WPBL+Wout),
where β is a coefficient which may be taken as 5/4 by referring to Fig. 7a of Pauluis and Zhang (2017). This value means that the mechanical work to lift water is approximately 25% of the work against friction (or frictional dissipation) in the eyewall cycle. Note that this ratio would be much larger (∼250%) for irrotational convection in tropics (e.g., Pauluis et al. 2000; Pauluis and Held 2002). A physical prediction for Wp in a cycle requires further study.
Substituting Eqs. (6), (8), (10), (12), and (13) into Eq. (5) gives
ηϵCLυqυ m−RυTsqυ m+RυTsqυ mlnpdapdm=(β−ηϵC)RTslnpdapdm+β(−12Vmax⁡2+14f2ra2−12fMm−12Ma2−Mm2ro2).
By making use of qυ=(R/Rυ)(e/pd) to relate qυ to pd and defining
y=pda/pdm,
which is the ratio of the surface ambient dry air pressure to its value at the radius of maximum wind, we may write a succinct relation for y given by
y=exp(Ay+Byln⁡y+C),
where
A=es*pdaηϵCLυ/Rυ−Ts(β−ηϵC)Ts,
B=es*pda(β−ηϵC),
C=−β(−12Vmax⁡2+14f2ra2−12fMm−12Ma2−Mm2ro2)(β−ηϵC)RTs.
Once pdm is obtained from Eq. (16), pm is obtained by pm=pdm+es*. Taking the natural logarithm of Eq. (16) shows that the lhs is directly associated with WPBL [via the RTsln(pda/pdm) term in Eq. (12)], and thus, the rhs corresponds to the available work for frictional dissipation (net heating minus Wp) in the cycle minus frictional work in the outflow. Vmax is taken to be either environmentally or externally defined. Equation (16) is directly analogous to Eq. (22) of Emanuel (1988). The roots of Eq. (16) are the intersections of the two curves represented by the lhs and rhs of Eq. (16), respectively. It can be shown that the second derivative of the rhs of Eq. (16) is always positive where y > 0. Thus, there may be one, two, or no roots for Eq. (16), similar to Emanuel (1988). In Eq. (16), A is completely determined by the thermodynamic environment corresponding to the net heating of the updated Carnot cycle. Increasing Ts or decreasing To individually or simultaneously would increase A (not shown). B arises from an approximation of taking pda as the mean dry air partial pressure in the inflow leg for integration when calculating QGibbs [Eqs. (7) and (8)]. Since B would be of the same order as A and lny would be one order less than y if y≲1.1, then the By lny term in Eq. (16) can be neglected. The primary effect from QGibbs is the −Ts term in A. Finally, C is primarily associated with work against friction in the outflow. C becomes negative for a sufficiently large TC.

We test whether Eq. (16) is able to produce reasonable pressure drops given some environmental parameters and mature TC parameters taken from our numerical simulations below (section 3) in tropical conditions. We take ambient surface pressure pa = 1015 hPa, sea surface temperature SST = 300 K, air–sea temperature difference ΔT = 1 K (thus Ts = SST − ΔT = 299 K), environmental near-surface relative humidity He=0.9 (thus pda = 985 hPa), the outflow temperature To = 200 K, Coriolis parameter f = 5 × 10−5s−1, the maximum wind of the TC Vmax = 83 m s−1, radius of maximum wind rm = 64 km, and TC outer radius ra = 2193 km. Note that the specific value taken for ra is simply to be consistent with idealized simulations of a TC and it is the same as the value obtained in section 2c. We take two tunable parameters η = 1/2 and β = 5/4 as discussed above. We also take the limit of ro = +∞. With these parameters, we have A = 0.062, B = 0.031, and C = 0.008. The two curves represented by the lhs and rhs of Eq. (16) are shown in Fig. 2. We note that there are two roots for Eq. (16) with the given values of parameters. One is 1.081, and the other is ∼18. The red line [lhs of Eq. (16)] in Fig. 2 can be interpreted as the actual work performed in the boundary layer, and the blue curve [rhs of Eq. (16)] is interpreted as the available work to be performed in the boundary layer. Thus, we observe that the smaller root corresponds to a physically stable state of a TC while the larger root corresponds to an unstable state (see also Emanuel 1988). Thus, here and hereafter, we take the smaller root of Eq. (16) for the dry air partial pressure drop determined by the updated Carnot cycle. The smaller root y = 1.081, which corresponds to an 8.1% surface dry air pressure drop, gives pm=pdm+es*=944hPa, which is a reasonable value as will be seen in simulations later.

Fig. 2.
Fig. 2.

A representative example of how the roots of Eq. (16) are obtained. Here A = 0.062, B = 0.031, and C = 0.008. See text for more details.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

One notable property of the solution for pdm, which will be necessary to understand the TC PS predictions below, is that increasing A (e.g., via increasing Ts) will decrease the Carnot prediction of pdm, while decreasing C (e.g., via increasing ra) will increase the Carnot prediction of pdm.

We note that the Carnot model assumes the loss of M in the inflow is restored solely in the upper troposphere by some turbulent mixing. The maximum anticyclonic wind speed will be ∼(1/2)fra, giving tens of meters per second. Previous studies show that though the near-surface flow beyond ra can exhibit anticyclonic flow, the wind speeds are quite small (CE14; Cronin and Chavas 2019), indicating that much of the M is restored in the free troposphere. Thus, the outflow work term in the Carnot cycle model is a reasonable though highly idealized representation of this process.

b. A brief review of C15 model

The C15 complete wind profile model numerically merges the inner-core structure model of ER11 and the outer-region structure model of Emanuel (2004, hereafter E04). The input parameters of the C15 model are outer radius ra, maximum gradient wind Vgm, lower-tropospheric subsidence velocity in the subsidence region wcool (positive downward), the surface exchange coefficients of enthalpy Ck and momentum Cd and Coriolis parameter f. The C15 model yields the tangential (gradient) wind profile V as a function of radius r, including radius of maximum wind rm. Following Wang and Lin (2021), it is convenient to use a nondimensionalized C15 model, where V is nondimensionalized by Vgm and r is nondimensionalized by ra. Thus, we have nondimensional wind V˜=V/Vgm and nondimensional radius r˜=r/ra. The nondimensional ER11 and E04 model may be written, respectively:
(2Vgmfrar˜V˜+r˜22Vgmfrar˜m+r˜m2)2−Ck/Cd=2(r˜/r˜m)22−(Ck/Cd)+(Ck/Cd)(r˜/r˜m)2,
∂(r˜V˜)∂r˜=CdVgmwcool2r˜2V˜2(1−r˜2)−fraVgmr˜.
Equation (20) indicates that the ER11 model is determined by three nondimensional parameters: Vgm/(fra), r˜m, and Ck/Cd. Equation (21) indicates that the E04 model is determined by two nondimensional parameters: Vgm/(fra) and CdVgm/wcool. Matching Eqs. (20) and (21) requires three of the four parameters to be externally defined leaving one parameter, r˜m in our case, to be internally determined. The nondimensional wind profile V˜(r˜) can be reduced to V(r) given Vgm and ra. By gradient wind balance, one can obtain pm and the whole surface pressure profile given environmental near-surface pressure pa (Chavas et al. 2017). Following ER11, Bister and Emanuel (1998), and Rousseau-Rizzi and Emanuel (2019), Vgm may be predicted solely from environmental parameters by
Vgm2=Vp2≅CkCdTs−ToTo(k0*−ka),
where Vp is the potential intensity, k0* the ambient saturation enthalpy at SST, and ka the ambient near-surface enthalpy at Ts.

c. A TC PS model combining C15 and updated Carnot cycle model

The updated Carnot cycle model predicts pm given environmental parameters Ts, To, pa, He (for calculation of pda), f, and TC intensity/structure parameters Vmax, rm, and ra. The C15 model predicts pm given environmental parameters SST, Ts, To, pa, wcool, f, Cd, Ck, He (for the calculation of Vgm), and TC structure parameter ra. For the updated Carnot cycle model, Vmax may be related to environmental parameters by introducing a coefficient γsg for supergradient wind so that
Vmax⁡=γsgVgm,
where γsg may be taken as 1.2 to match simulated TCs [also within the range of 1.1–1.25 found by Kepert and Wang (2001)]. Note that Vmax (or Vgm) need not be tied to Vp but can be defined as any value, as done in the analytic tests and axisymmetric simulations below to allow for the role of radial turbulence in modifying simulated Vmax.

Hence, with the environment specified, the C15 model prediction for pm requires only the overall storm size ra. Similarly, the updated Carnot cycle predicts pm given ra, Vmax, and rm, where Vmax is provided by Eq. (23) and rm is provided by C15 model. Hence, to predict pm from the C15 and updated Carnot cycle models, the only external parameter not defined environmentally is outer storm size ra. Moreover, pm predicted by the updated Carnot cycle is larger with larger ra (Emanuel 1991), while pm predicted by the C15 model is smaller with larger ra (Chavas et al. 2017). Thus, there will be a specific ra that allows pm predicted by both models to be equal—this specific ra is the TC PS in a given environment.

Physically, the characteristic wind structure of the TC (C15 model) would predict a lower inner-core pressure as ra increases by gradient wind balance, but the pressure drop experienced by inflow parcels is limited by its energetic cycle (updated Carnot cycle model). The potential pressure drop is reduced as ra increases, which prevents infinite expansion of the TC (also discussed in CE14). The above logic is the basic hypothesis of the TC PS model that we present here. A schematic summary of the TC PS model is shown in Fig. 3. The model itself is applied by numerically converging iteratively to the solution for ra that satisfies both constraints in a given environment.

Fig. 3.
Fig. 3.

A schematic summary of the basic idea of TC PS model. Blue boxes and arrows show logic and submodels and green boxes show necessary steps in the iterative solution.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

Taking the parameter settings in section 2a with wcool = 0.002 m s−1, He=0.9, γsg = 1.2, and ro = + ∞, Fig. 4a illustrates how the method would yield a unique solution for ra in a given environment. We test ra over a range of values from 200 to 5000 km and find the solution for both models that yields the same pm for each case. With all other parameters held fixed, increasing ra causes the C15 model to predict a monotonically decreasing pm through gradient wind balance. Meanwhile, increasing ra causes the updated Carnot cycle model to predict a monotonically increasing pm through increasing the work done in the outflow. Thus, there is a single unique solution that satisfies both models, given by ra = 2193 km, with pm = 944 hPa (Fig. 4a). As noted earlier, the TC PS represents the equilibrium upper bound on TC size and hence is expected to be larger than typical TC sizes on Earth. Indeed, the observed median TC outer size is ∼900 km and the vast majority of TCs are substantially smaller than the traditional upper-bound scaling Vp/f (Chavas et al. 2016). Observations further suggest that the largest TCs do scale with Vp/f over a reasonable range of values of this length scale from 800 to 1600 km (their Fig. 5c). As a specific example, Super Typhoon Tip (1979) was observed to produce a radius of 15 m s−1 surface winds over 1100 km during its mature stage (Dunnavan and Diercks 1980); thus, it is plausible that Tip could have an outer radius ∼2000 km.

Fig. 4.
Fig. 4.

An example of how TS PS model predicts a specific ra in a given environment. (a) pm (hPa) predicted by updated Carnot cycle (blue) and by C15 model (red) with different ra (km). The intersection point marks the specific ra and pm predicted by TC PS model. (b) Different terms (J kg−1) in the updated Carnot cycle model with different ra. The gray line shows the difference of WPBL (J kg−1) between Carnot model and C15 model.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

The behavior of the updated Carnot cycle model is understood through Fig. 4b. Increasing ra only affects parameter C in Eq. (16). As ra increases, f2ra2 significantly increases and C significantly decreases. By Eq. (16), pm increases. Physically, increasing ra causes Wout to increase, leaving less energy available for WPBL (Fig. 4b), which directly links pm through Eq. (12). When ra is over ∼3000 km, WPBL becomes negative. This outcome indicates that there is zero work available to power the near-surface circulation, and hence, an equilibrium TC cannot exist (i.e., zero surface pressure drop) at such sizes under the given environment. Qs gradually decreases as ra increases (Fig. 4b), caused by an increasing pdm through Eq. (6). QGibbs is approximately constant as ra increases (Fig. 4b).

Back to Fig. 4a, a further physical interpretation of how TC size is determined may be provided. The interpretation is based on results from idealized simulations (section 3; also Martinez et al. 2020) and in nature (Maclay et al. 2008) that TCs typically continue to expand slowly (ra increases) after its intensification. We assume the C15 wind structure model is valid at all stages (C15). Thus, we may define the value of pm predicted by the C15 model as the present pm. On the other hand, the value of pm predicted by the Carnot cycle model is interpreted as the potential pm if the TC were to maintain its present ra. When the potential pm is smaller than the present pm (small but intense TC), there is more energy available for WPBL (gray line in Fig. 4b above zero) and thus the TC may deepen and expand, because for most circumstances the only way a TC decreases its present pm is by expansion (note that the model assumes intensity is determined solely from the environment). As the TC expands, the present pm decreases and the potential pm increases as more work has to be done in the outflow, until a specific size is reached when the present pm and the potential pm are equal. Further expansion ceases above this equilibrium size, as the available energy for WPBL would be less than the present value (gray line in Fig. 4b below zero) and as a result the TC would shrink back down—hence, it is a stable equilibrium. Note that this “surplus available work” argument is the energetic explanation for intensification theory (Vmax) in Tang and Emanuel (2010, 2012); in our case the argument is that if Vmax cannot change, then the extra work can be used for expansion rather than intensification.

d. An intrinsic length scale

We next demonstrate that an intrinsic TC PS length scale, VCarnot/f, emerges from a simplified version of the updated Carnot and C15 model equations. The surface pressure deficit corresponding to C15 model may be approximated by (Chavas et al. 2017)
pda−pdm=pda(1−1/y)≈DVgm+Efra+F,
where D, E, and F are regression constants. The updated Carnot cycle model [Eq. (16)] may be linearized (neglecting the B term) by
y≈(1+C)/(1−A),
noting that lny ≈ y − 1 [similar to Eq. (24) of Emanuel 1986]. Equation (25) also gives an approximate relation ∂pdm,Carnot/∂ra∼ra/y2, with pdm,Carnot=pm,Carnot−es* (subscript “Carnot” denoting Carnot model predictions).
Eliminating y by combining Eqs. (24) and (25), assuming intensity alone does not have a strong effect on potential size, and making further simplifications (see appendix B) gives a length scale:
ra∼VCarnot/f,
where VCarnot is defined as
VCarnot2=(ηϵCLυ−RυTs)qυ a*,
with qυa*=R/Rυ(es*/pda) the near-surface saturation water vapor mixing ratio of the environment. Here VCarnot is a velocity scale from the updated Carnot cycle. The rhs of Eq. (27) corresponds to the available work of a mixed steam–Carnot cycle (Pauluis 2011) with input latent heat Lυqυa* and input sensible heat −RυTsqυa* to a dehydrated parcel.1 Hence, VCarnot differs from Vp in that Vp reflects air–sea thermodynamic disequilibrium whereas VCarnot reflects the net heat that is obtained in a thermodynamic cycle. VCarnot also lacks a dependence on Ck/Cd. VCarnot may be interpreted physically as the largest possible anticyclonic velocity that can be reached in the outflow at radius ra of the TC [Eq. (B5)], which is also a physical outcome if all the available mechanical work in the mixed steam–Carnot cycle is used against friction in the outflow [Eq. (B5)].

e. PS model and Vp/f scaling

It is important that the TC PS model should be able to qualitatively reproduce the Vp/f scaling for ra as found in idealized numerical simulations (CE14). We vary individually f, To, and SST (thus Ts) as base cases of varying Vp/f. Varying To is by far the cleanest means of varying Vp as it does not alter the thermodynamic structure of the lower troposphere (CE14). Furthermore, we vary Cd and Ck respectively to see if the Vp/f scaling still holds; in this case the thermodynamic properties (To and Ts) in the updated Carnot cycle do not change. Last, we fix all environmental parameters but only vary Vgm to see how the TC PS model would predict a size change. This test corresponds to varying the horizonal mixing length in a numerical simulation: past works (Bryan and Rotunno 2009; Rotunno and Bryan 2012; CE14) have shown how increasing the radial mixing length acts to smooth the inner core M field radially outwards, which results in a larger rm and smaller Vmax as M(rm) stays relatively constant. In the following analysis, other environmental parameters are fixed as in section 2a and 2c. As η is somewhat tunable, we also test the sensitivity of the TC PS model to η for values of 0.4, 0.5, and 0.6. Finally, we test the assumption of ro = +∞ by taking ro = 1.2ra for comparison. We then compare results for Vp/f against the length scale VCarnot/f defined above to examine which length scale best represents equilibrium size.

First, f is varied from 1.25 × 10−5 to 2 × 10−4 s−1 and thus 1/f is varied from 5 × 103 to 8 × 104 s (Figs. 5a,b). The predicted ra increases linearly with increasing 1/f, indicating that the TC PS model correctly yields an f−1 scaling. For η = 0.5 and ro = +∞, predicted ra increases from slightly over 500 to ∼8700 km. Notably, when 1/f is varied, the predicted pm remains nearly constant, indicating that equilibrium pressure intensity (inner-core surface pressure) does not depend on f. Larger η consistently increases the predicted ra and decreases the predicted pm, indicating that a more efficient system yields a larger and deeper storm (with constant maximum wind speed); this behavior will also be seen in later tests. Taking ro finite (dashed lines) does not change the qualitative behavior but does yield a larger predicted ra and lower pm. Physically this response occurs because when ro is finite, the storm has to do less work in the outflow [Eq. (10)]. This behavior will be seen in later tests as well. Physical explanation of the 1/f scaling seen in Fig. 5a is provided in the next subsection.

Fig. 5.
Fig. 5.

TC PS model prediction of (left) ra (km) and (right) pm (hPa) as (a),(b) f varied from 1.25 × 10−5 to 2 × 10−4 s−1, (c),(d) To varied from 170 to 230 K, and (e),(f) SST varied from 280 to 310 K. The x axis is the corresponding 1/f (104 s) in (a) and (b) and the corresponding Vp (m s−1) in (c)–(f). Predictions with different values of η are in different colors noted by the legend in (a). Thick solid lines are with ro = +∞ and thin dashed lines are with ro = 1.2ra. The light gray lines in the left column show Vp/f (km) and the dark gray lines show VCarnot/f (km) with η = 0.5.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

Second, To is varied from 170 to 230 K (Figs. 5c,d). The corresponding Vp varies from ∼86 to ∼54 m s−1. Overall, the predicted ra increases with increasing Vp. This is qualitatively consistent with the Vp/f scaling but with a slightly larger magnitude dependence. For η = 0.5 and ro = +∞, predicted ra increases from ∼1400 to ∼2800 km while Vp/f increases from ∼1100 to ∼1800 km. VCarnot/f scales closely with Vp/f and hence is also captured by the TC PS model. pm decreases rapidly with increasing Vp, which is associated with both the increased Vp (thus Vgm) and the increased predicted ra.

Third, SST is varied from 280 to 310 K (Figs. 5e,f). The corresponding Vp varies from ∼37 to 96 m s−1, with a proportional increase of 159% for Vp/f. We see a slightly stronger increase of predicted ra. For η = 0.5 and ro = +∞, predicted ra increases from ∼900 to ∼3300 km, with a proportional increase of ∼267%. VCarnot/f behaves similarly to Vp/f but with a steeper slope (a proportional increase of ∼197%), closer to TC PS predictions. Specifically, VCarnot increases from 76.2 to 106.7 m s−1 as SST increases from 300 to 310 K, indicating ∼4% K−1 increase of size or ∼60 km K−1 at f = 5 × 10−5 s−1. There is also a significant increase of predicted ra as η increases at constant Vp, as the system is more efficient and the additional power is used to expand the storm. Note the effect is stronger with larger Vp. Similar to varying To, we also see a significant decrease of pm as Vp increases.

The different dependence of ra on To versus SST may be understood considering their quantitatively different roles in the updated Carnot cycle [see Eq. (16)]. While To only enters Eq. (16) through ϵC, Ts enters Eq. (16) additionally as its own term, and es*.

For our base-case tests, the TC PS model is able to qualitatively reproduce the Vp/f scaling. This indicates that the TC PS model likely captures several important physical processes determining TC size. Next we distinguish Vp/f and TC PS model (VCarnot/f) with two further tests.

Next, we test how TC PS would predict a response of ra to Cd and Ck, which changes Vp but not VCarnot. Unlike To or Ts, Cd and Ck do not influence the thermodynamic property of the system, i.e., through parameter A in Eq. (16), but only influences the parameter C in Eq. (16) by modifying Vp (thus Vmax). We see from Figs. 6a–d that Vp is varied from ∼50 to ∼90 m s−1, but the predicted ra remains approximately constant, even though there is a significant decrease of predicted pm. We will see that these predictions are also qualitatively consistent with idealized simulations in section 3, indicating that the standard Vp/f scaling is not applicable in this case. Instead, VCarnot/f follows the TC PS predictions.

Fig. 6.
Fig. 6.

As in Fig. 5, but with (a),(b) Cd varied from 0.0009 to 0.003, (c),(d) Ck varied from 0.000 75 to 0.002 55, and (e),(f) Vgm varied from 40 to 90 m s−1. The x axis is the corresponding Vp (m s−1) in (a)–(d) and the corresponding Vgm (m s−1) in (e) and (f).

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

All together, then, analytical tests suggest that qualitative difference between the TC PS model and Vp/f scaling is mainly the dependence on Ck/Cd. The intrinsic scaling VCarnot/f, independent of Ck/Cd, compares well with TC PS predictions. VCarnot/f also resembles χ/f, where χ is similar to Vp but without the Ck/Cd factor, of previous TC theory (Emanuel 1995) that employs specifically χ/f as the relevant nondimensional length scale; note though that dimensional arguments alone cannot determine whether the nondimensional factor Ck/Cd should or should not be included.

Finally, we further test how TC PS predicts a response of ra to horizontal mixing length lh in a numerical model. It is known that increasing lh acts to decrease Vgm (Bryan and Rotunno 2009; CE14). Thus, to mimic the effect of lh, which is not a parameter in either updated Carnot cycle or C15 models, we vary Vgm instead. We see from Figs. 6e and 6f that the predicted ra only slightly decreases as Vgm increases, while pm decreases significantly, which is to be expected from gradient wind balance as the maximum wind speed increases. This is consistent with the observed behavior of TCs in which intensity can vary substantially while the size of the outer circulation remains fixed (Frank 1977; Merrill 1984; Chavas and Lin 2016) and reproduced by the C15 model. Here the TC PS model also predicts that inner-core radial mixing will not change outer storm size, and hence, this behavior is both dynamically and energetically consistent. We will also see that these predictions are qualitatively correct in idealized simulations in section 3.

We have also tested the sensitivity of TC PS model to γsg and β. Predicted ra increases with a higher γsg and decreases with a higher β. The former relation is because higher γsg gives a larger Vmax and hence a larger parameter C in Eq. (16), which will shift the blue curves down in Fig. 6 to yield a larger predicted ra. The latter relation is physically intuitive as a higher water loading is expected to reduce the work available against frictional dissipation, which is related to TC size.

f. Further understanding of TC PS model results

Details of how the predictions for pm from both the Carnot cycle and C15 models as ra is varied yield a single unique solution for the six tests above is shown in Fig. 7. For simplicity, only cases with η = 0.5 and ro = +∞ are shown. In each test, the basic responses of pm predicted by the updated Carnot cycle (pm,Carnot) and C15 model (pm,C15) with increasing ra are qualitatively the same as Fig. 4a: larger ra (a larger low-level storm) yields a higher pm (lower pressure intensity) for Carnot cycle and a lower pm (higher pressure intensity) for C15 model. However, the final TC PS prediction for the response of ra in each test is determined in a complicated way: the solution depends on the sensitivity of pm,Carnot to some parameter (X), ∂pm,Carnot/∂X, which corresponds to how quickly the blue curve shifts downward as a function of ra, relative to the pm,C15 sensitivity, ∂pm,C15/∂X, which corresponds to how quickly the red curve shifts downward as a function of ra. When ∂pm,Carnot/∂X>∂pm,C15/∂X, the PS will typically decrease (Figs. 7a,b). When ∂pm,Carnot/∂X<∂pm,C15/∂X, the PS will typically increase (Fig. 7c). When ∂pm,Carnot/∂X=∂pm,C15/∂X, then the PS will not change (Figs. 7d–f).

Fig. 7.
Fig. 7.

pm predicted by the updated Carnot cycle model (blue) and the C15 model (red) over a range of values of ra induced by variations in (a) f, (b) To, (c) Ts (SST), (d) Cd, (e) Ck, and (f) Vgm. We set η = 0.5. More opaque line means higher values of the variable varied. The TC PS–model-predicted pm and ra are intersections of blue and red curves, marked by black dots.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

When f is varied (Fig. 7a), we observe that pm,C15 decreases faster with ra as f increases. This directly corresponds to gradient wind balance. Correspondingly, pm,Carnot increases faster with ra as f increases. This is solely a reflection of the increase of parameter C in Eq. (16). Note that parameter A and B are constant when f is varied. The ra ∼ 1/f scaling is produced by the fact that f and ra only appear as their product fra in the TC PS model if Mm is neglected and assuming ro approaches infinity (appendix B). Thus, the TC PS model is invariant when ra ∼ 1/f. The final outcome is the prediction that pm will remain constant as f is varied [Eqs. (24) and (25)].

Comparing Figs. 7b and 7c, varying Ts has a larger impact on pm,Carnot than varying To. Increasing Ts directly increases the potential of heat received in the inflow [Eq. (6)] by increasing es* and the net heat in the cycle by increasing ϵC, whereas decreasing To only increases ϵC. The same is also seen in VCarnot.

The budget terms from Eq. (5) in the updated Carnot cycle for each of the six TC PS model tests are shown in Fig. 8. When varying f (Fig. 8a), Qs, QGibbs, and WPBL are approximately constant because pm is approximately constant and Vp (thus Vgm) is constant. Thus, from Eq. (5), Wout (and fra) must remain approximately constant too. For tests varying To and Ts, WPBL and Wout both increase with Vp (Figs. 8b,c), which is associated with a decrease of pm and increase of ra. Comparing Figs. 8b and 8c, Ts has a larger impact on Qs than To. For tests varying Cd, Ck, and Vgm (Figs. 8d–f), WPBL increases with Vp and Vgm, respectively, because of the decrease of pm; Wout is found to have similar variation as ra. QGibbs is approximately constant except for variations in Ts that induce substantial increases in water mass loading with warming.

Fig. 8.
Fig. 8.

WPBL, Wout, Qs, and −QGibbs (J kg−1) in the updated Carnot cycle with η = 0.5 corresponding to the TC PS–model predictions in the six tests where (a) f, (b) To, (c) Ts (SST), (d) Cd, (e) Ck, and (f) Vgm are varied.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

3. Evaluation of the TC PS model against idealized numerical simulations

a. Model configuration and experimental design

The numerical model used is Cloud Model 1 (CM1; Bryan and Fritsch 2002) release 19.2, which has three-dimensional (3D) and axisymmetric configurations for idealized simulations. We use a 3D configuration in a small domain without background rotation to obtain a radiative–convective equilibrium (RCE) state, which is then taken as the base state for TC simulations using axisymmetric configuration with background rotation, similar to the approach of CE14. The following model settings are the same for 3D and axisymmetric simulations, except otherwise noted. The horizontal resolution is 4 km and the vertical resolution is 100 m below 3 km of altitude and stretches gradually to 500 m at 12 km of altitude and remains constant above this altitude. The domain height is 25 km. The microphysics scheme is the Morrison double-moment scheme (Morrison et al. 2005). Turbulence is parameterized for mesoscale modeling with horizontal mixing length lh set to 1500 m and asymptotic vertical mixing length set to 100 m. A Rayleigh damping zone is applied in the upper 3 km of altitude for both 3D and axisymmetric simulations and the outer 100 km of radius for axisymmetric simulations. To effectively control the tropopause temperature Ttpp, which approximates To, radiation is represented using a simple constant cooling scheme following Chen and Chavas (2020). Ttpp is given as an external parameter. When temperature T is higher than Ttpp, then a constant cooling rate Qcool (defined positive here) is applied to the potential temperature θ; otherwise, θ is relaxed to the corresponding value to Ttpp with a relaxation time τ, i.e.,
∂θ∂t={−Qcool,T>Ttpp,θ(p,Ttpp)−θ(p,T)τ,T<Ttpp.
Here Qcool is set to 1 K day−1 and τ set to 12 h.
The surface fluxes are calculated using the classic aerodynamic formulas with constant surface exchange coefficient for momentum Cd and enthalpy Ck. Following CE14, a “background” wind speed usfc is added as a modification to the surface-flux calculations to represent the enhancement of surface fluxes from unresolved gustiness. The surface enthalpy flux Fk and surface stress τs are thus formulated as
Fk=Ckρd(|u|+usfc)(k0*−k),
τs=Cdρd(|u|+usfc)u,
where ρd is the near-surface dry-air density, u the near-surface horizontal wind vector, and |u| the near-surface horizontal wind speed. Equations (29) and (30) are applied to all 3D and axisymmetric simulations, with usfc set to 5 m s−1.

We define our Control experiment (CTL) with the following parameters: Ttpp = 200 K, SST = 300 K, f = 5 × 10−5 s−1, Cd = Ck = 0.0015, and lh = 1500 m. The initial vortex in CTL is defined following Eq. (37) of Rotunno and Emanuel (1987).

We conduct six sets of axisymmetric TC experiments (Table 1) corresponding to section 2d, with each set varying f (Ex_F), Ttpp (Ex_TTPP), SST (Ex_SST), Cd (Ex_CD), Ck (Ex_CK), and lh (Ex_LH), respectively. The initial vortices are the same as in CTL. For Ex_TTPP and Ex_SST, the base state is defined from their corresponding 3D RCE states. For other experiments, the base state is the same as the CTL simulation. The 3D simulations are run for 100 days and the average sounding calculated from the last 15 days are used as the base state for axisymmetric TC simulations. The horizontal domain size for 3D simulations is 240 × 240 km2. The domain height is 25 km for all experiments except 28 km when SST = 305 K and Ttpp = 163 K. The axisymmetric TC simulations are run for 100 days in all experiments except 150 days for f = 0.000 012 5 s−1 to achieve a sufficiently long quasi-steady state. The domain for TC simulations is 6000 km for all experiments except for cases where TC size is especially large: 8000 km for Ttpp = 163 K, 10 000 km for SST = 305 K, and 15 000 km for f = 0.000 012 5 and 0.000 025 s−1.

Table 1

Experiment sets for the axisymmetric CM1 TC simulations. CTL parameters are shown in boldface.
Table 1

Time series of TC intensity and sizes in CTL are shown in Fig. 9. After day 30, intensity remains quasi steady within the 100-day period, similar to CE14. Outer size increases rapidly to a peak at approximately 40 days and then fluctuates about a quasi-steady long-term value through 100 days. Thus, a quasi-equilibrium/steady state for tens of days can be reached in our experimental setup. We note that the environment is also quasi steady during most of the simulation period after the first few days (not shown) with no additional nudging applied to the environment. Note that the use of constant cooling for representing radiation [Eq. (28)] ensures that deep convection is persistent in the environment to balance this radiative cooling, which resupplies moisture to the free troposphere lost to precipitation. This approach avoids the long-term storm decay that occurs when using Newtonian cooling toward an initial state in the absence of an artificial moisture relaxation (Rousseau-Rizzi et al. 2021).

Fig. 9.
Fig. 9.

Time series of (a) Vmax (m s−1), (b) rm (km), (c) radius of 4 m s−1 tangential wind at 1 km height r4 (km), and (d) ra by fitting E04 model to r4 (see text). In (c) and (d), a 24 h running average has been applied. The horizontal lines in (c) and (d) mark the 40–100 day average values. Solid and dashed gray lines show Large and Small experiments (see text), respectively.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

As the true ra can be noisy in simulations owing to noise at the very outer edge of the storm, we estimate ra from r4, the radius of 4 m s−1 tangential wind at 1 km, using the E04 model, which is the outer circulation component of the C15 model. At each output time step (every 4 h), the E04 model is fit to r4 (see also Fig. 9d), with diagnosed wcool, following CE14 from the equation for radiative–subsidence balance:
wcool=Qcool/∂θ¯∂z,
where the overbar denotes time- and area-weighted horizontal radial average and pressure-weighted vertical space average over the steady state (defined below, 40–100 days for CTL) and the outer 900 km (outside the Rayleigh damping zone) of the domain and 1.5–4 km of altitude.

Simulated size is generally larger than those in CE14. This difference may be due to differences in model setup: the different microphysics schemes, difference values of usfc (3 m s−1 in CE14), a coarser vertical resolution in CE14. Additionally, CE14 chose to analyze the wind profile at z = 1.56 km, which is likely well above the boundary layer height especially at large radii. As a result, this choice will likely yield a wind profile that decays more quickly with radius in the outer circulation. Our simulations have much higher vertical resolution at low levels and hence we can select a more appropriate height (1 km) for our analysis.

The TC PS model predicts equilibrium outer size from environmental parameters alone, and hence, it should be insensitive to initial vortex size as was found in CE14. We perform two sensitivity experiments (Fig. 9) varying initial rm (∼100 km in CTL) while holding initial intensity unchanged: ∼60 km (“Small”) and ∼140 km (“Large”). Indeed, varying the initial rm does not induce systematic variations in equilibrium storm properties relative to CTL, particularly when compared to the effects of varied environment (see Fig. 10). Thus, we confirm that equilibrium outer size is largely environmentally defined. Note that initial vortex size can modify storm size within the first few days of the simulation as shown in Martinez et al. (2020) and Tao et al. (2020).

Fig. 10.
Fig. 10.

TC PS–model predictions (blue line) of (left) ra (km) and (right) pm (hPa) as a function of Vmax (m s−1) or 1/f (104 s) compared to the corresponding steady-state averaged values of CM1 results (red line) for (a),(b) Ex_F, (c),(d) Ex_TTPP, and (e),(f) Ex_SST. Background dots are 4-hourly CM1 output during the steady state. Cold to warm colors mean increased values of the parameter changed in each experiment. The light gray lines in the left column show the corresponding Vp/f (km) and the dark gray lines show the corresponding VCarnot/f (km). “Hi” and “Lo” mark high and low values of the parameter varied, respectively.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

b. Comparison of TC PS predictions with numerical simulation results

Based on Fig. 9, steady state is taken as the average of days 40–100 for simulated TCs in all experiments except for two experiments with very small f (days 100–150 for f = 0.000 012 5 s−1 and days 60–100 for f = 0.000 025 s−1). In axisymmetric simulations, Vgm is not well constrained by Vp because of a dependence of Vgm on lh (Rotunno and Bryan 2012; CE14). Thus, we use the steady-state average of simulated Vmax and Vgm (by pressure field at the lowest model level) for the TC PS model, which together define γsg (1.27 for CTL). The value of rm for Mm is predicted internally by the C15 model.

We define ra,cm1 as the steady-state average of the estimated outer radius from the E04 model, which we then compare with the TC PS prediction of ra. Similarly, we calculate steady-state averages of simulated surface pressure at rm, pm,cm1, to compare with the TC PS model prediction.

The simulated TCs have a fully developed upper-tropospheric anticyclone core, and thus, we may retain the term −(1/2)(Ma2−Mm2)/ro2 in the calculation of Wout [Eq. (19)]. ro is estimated from simulated values ro,cm1 in TC PS–model predictions. ro,cm1 is assumed to be steady-state average of the radius of the maximum anticyclonic wind in the simulations, same as the definition in section 2a. Because the upper-level anticyclone typically extends beyond the surface cyclone, we assume ro is some factor larger than ra, i.e., ro=(ro,cm1/ra,cm1)×ra (this factor is ∼1.27 for CTL).

We also provide a comparison with the Vp/f scaling. Vp [Eq. (22)] is calculated from the simulations as the steady-state mean of the value calculated from radial mean k0*−ka and radial mean tropopause temperature over the outermost 900 km (excluding the Rayleigh damping zone) of the domain. The corresponding values of Vp in each experiment is shown in Table 2. As in CE14, this is not calculated within the storm inner core (and without any iteration to account for the effect of reduced pressure on enthalpy variables) but instead is calculated from the far-field environment so that it may be considered a purely environmental parameter.

Table 2

The corresponding Vp (m s−1) in each experiment in Table 1. CTL simulation is shown in boldface. Order of experiments within each set matches that shown in Table 1.
Table 2

For direct comparison, we define the environmental parameter values in the TC PS model directly from the simulations. Specifically, SST and f are taken as their values in each CM1 experiment and To is taken as Ttpp in each CM1 experiment. As Ts (near-surface air temperature) is not rigorously defined in the TC PS model, for simplicity we take Ts to be the steady-state mean of radial mean temperature of lowest model level over the outer 900 km (excluding the Rayleigh damping zone) of the domain; pa and He are obtained in the same manner. Note that pa ∼ 1013 hPa in CTL, smaller (larger) when the atmosphere is cold (warm), e.g., ∼1004 hPa at SST = 280 K and 1024 hPa at Ttpp = 241 K. He∼0.88 in CTL, smaller (larger) when SST (or Ck) is low, e.g., 0.81 at SST = 280 K and 0.91 when SST = 305 K. The temperature difference between SST and Ts decreases from ∼1.7 K at SST of 280 to ∼0.4 K at SST of 305 K.

For a better fit to simulation results, η = 0.4 is chosen, which is a slightly lower efficiency than a pure triangle. This likely compensates for the assumption of complete resaturation, which may be too strong of an assumption and thus yields a larger energy input for thermodynamic cycle than that in a simulated TC; such detailed analysis is left for future work. We set β = 5/4 as was done in section 2. Across all simulations, ro,cm1/ra,cm1 is found to be approximately 1.25 and γsg is found to be approximately 1.27. Note that they are all allowed to vary as given in Tables 3 and 4. The variations of each are necessary for the quantitative match of the TC PS model predictions to simulations. The value of wcool is approximately 0.0014 m s−1 when SST = 300 K; it increases to 0.003 m s−1 when SST = 280 K.

Table 3

The corresponding ro,cm1/ra,cm1 in each experiment in Table 1. CTL simulation is shown in boldface. Order of experiments within each set matches that shown in Table 1.
Table 3
Table 4

The corresponding γsg in each experiment in Table 1. CTL simulation is shown in boldface. Order of experiments within each set matches that shown in Table 1.
Table 4

Figure 10 compares TC PS–model predictions of ra and pm with corresponding steady-state values of CM1 simulations for the sets of experiments varying f, Ttpp, and SST. The TC PS model does well in reproducing the simulated values of both ra and pm. The lone quantitative exception is a slightly stronger dependence of TC PS ra on Ttpp than is simulated in Ex_TTPP. Vp/f and VCarnot/f are almost identical for varied f and Ttpp and also scale reasonably well with the full TC PS model prediction. For varying SST, VCarnot/f better captures the strong dependence of simulated size on SST than Vp/f. This is because VCarnot/f does not depend on air–sea thermodynamic disequilibrium as Vp/f, so that the increase of environmental relative humidity and decrease of air–sea temperature difference at larger SST, which reduces the air–sea enthalpy disequilibrium, will not influence VCarnot/f.

Figure 11 compares TC PS–model predictions of ra and pm with corresponding steady-state values of CM1 simulations for the sets of experiments varying Cd, Ck, and lh. Equilibrium size does not vary strongly with varying Cd, which is also captured by TC PS model (Figs. 11a,b). For Ex_CK, the air–sea enthalpy disequilibrium decreases as Ck is increased, such that intensity (not shown) and Vp (Table 2) varies only weakly across the experiments in Ex_CK (Fig. 11c). This behavior is consistent with the fact that in RCE surface heat fluxes must balance column radiative cooling (CE14). Since the radiative cooling rate per unit mass is fixed in our setup, if tropospheric mass remains relatively constant then surface heat fluxes must also remain constant; hence, Vp will also remain constant. Importantly, we hold Ck fixed at its CTL value for the ER11 inner-core structure in the TC PS model, which is required to match the simulation results in Ex_CK in Fig. 11. Including variations in Ck in the ER11 model yields an unrealistically small rm when Ck is small compared to CM1 results, which yields a value of pm,C15 that is too small and hence too small of a potential size. This outcome suggests a possible shortcoming of the Ck dependence in the ER11 model. Results of Ex_LH also confirms that equilibrium size has a very weak dependence on intensity in a fixed environment, which is also captured by the TC PS model.

Fig. 11.
Fig. 11.

As in Fig. 10, but for (a),(b) Ex_CD, (c),(d) Ex_CK, and (e),(f) Ex_LH. Note that in (c) and (d), Ck is held to CTL value in TC PS model; see more details in text. Also, the x axis is Ck in (c) and (d) because Vmax does not separate different simulations in Ex_CK.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

Overall, when given simulated Vmax and Vgm, the TC PS model does a good job in reproducing the simulated ra and pm across all experiments. The VCarnot/f scaling also captures the qualitative variation of simulated ra across all experiments. Considering the idealization and simplicity of TC PS model, these results demonstrate that the TC PS model captures the essential physical processes that determine equilibrium TC size. The VCarnot/f scaling is similar to Vp/f in reproducing variations in TC PS variation and simulation but better captures variations due to SST and Cd. Finally, we note that VCarnot/f still underestimates the strong dependence of PS on SST, this should result from neglecting the role of ra in pm,C15, which should be relatively larger when PS is small (appendix B).

4. TC PS model with a simpler Carnot cycle model

In this section, we repeat the same analysis as in section 2e with a simpler Carnot cycle essentially the same as that in Emanuel (1991), and test whether the modifications (half Carnot efficiency, complete resaturation, Gibbs free energy, and work to lift water) to the Emanuel’s Carnot cycle are essential to the qualitative TC PS predictions in section 2e.

By neglecting QGibbs, Wp, and the energetic effects of remoistening back to environmental relative humidity He, and applying a Carnot efficiency ϵC to the cycle, we may repeat the same procedure as in section 2a to obtain an expression for y, similar to Eq. (16):
y=exp(A′y+C′),
with
A′=ϵC1−ϵCLυRυTses*pda,
C′=−HeA′+ϵc1−ϵces*pdaHelnHe−1(1−ϵc)RTs×(−12Vmax⁡2+14f2ra2−12fMm−12Ma2−Mm2ro2).
Note that in A′, there is no −Ts term associated with QGibbs and in C′ there is an additional −HeA′+C′(ϵc/1−ϵc) (es*/pda)HelnHe term associated with resaturation of the parcel.
Following a similar procedure in the appendix B, we may obtain a length scale corresponding to the simpler Carnot cycle:
ra∼VCarnot′/f,
where
VCarnot′2=ϵCLυ(1−He)qυ a*,
with qυ a*=(R/Rυ)(es*/pda). VCarnot′ has both features of Vcarnot and Vp as it combines the net heat in a cycle and environmental relative humidity (contributing to air–sea thermodynamic disequilibrium), though VCarnot′ is more similar to Vp. Furthermore, as VCarnot′ does not include Ck/Cd, it has an even closer relation with χ.

Next, we apply the same analysis in section 2e with the same environmental parameters. As we do not have η in the simple Carnot cycle model, we vary γsg instead, though the physical effects of γsg and η are not equivalent. Results are qualitatively similar (Figs. 12 and 13) for the six tests as those found in section 2e. A higher γsg yields a larger predicted ra and a lower predicted pm as also discussed in section 2d. The results indicate that none of the modifications to the Emanuel’s Carnot cycle as introduced in section 2a is essential to the TC PS model predictions.

Fig. 12.
Fig. 12.

As in Fig. 5, but for the TC PS model with a simple Carnot cycle. Predictions with different values of γsg are in different colors noted by legend (“sgw”). The dark gray lines show 2VCarnot′/f (km).

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

Fig. 13.
Fig. 13.

As in Fig. 6, but for a TC PS model with a simple Carnot cycle. Predictions with different values of γsg are in different colors noted by legend (“sgw”). The dark gray lines show 2VCarnot′/f (km).

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

However, these modifications do affect the hypercane condition, which refers to the condition where only one or no solutions exist for the Carnot cycle and implies very low inner-core surface pressure. If B term in the updated Carnot cycle is neglected, then the hypercane condition is 1 + lnA + C ≥ 0 for the updated Carnot cycle, where as it is 1 + lnA′ + C′ ≥ 0 for the simple Carnot cycle. Emanuel (1988) suggested a hypercane is possible with SST of 311–313 K. We get a range of 310–313 K for ra less than ∼3400 km if none of the noted modifications are included in the Carnot cycle (Fig. 14), agreeing with Emanuel (1988). This temperature increases to ∼327 K for the updated Carnot cycle (Fig. 14), suggesting that the current climate is farther from supporting a hypercane. The result comes from a much lower A than A′ (related to half of Carnot efficiency and Gibbs free energy) and a much higher C than C′ (related to complete resaturation) in the updated Carnot cycle.

Fig. 14.
Fig. 14.

Critical SST for the hypercane condition as a function of ra for the updated Carnot cycle (blue) and the simple Carnot cycle (red). In the calculation ro is taken as infinity and −(1/2)fMm is neglected in C and C′.

Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-21-0325.1

5. Summary and discussion

This study proposed a model to predict tropical cyclone (TC) potential size (PS), i.e., the equilibrium upper-bound outer radius ra on an f plane, solely from environmental parameters. The TC PS model combines an updated Carnot cycle model, which includes an explicit treatment of the water budget and its energetics as part of the closed cycle, with a physics-based model for the complete radial structure of the wind field. The model yields a deterministic value of the TC outer radius for which both model components yield the same predicted value of the surface pressure under the eyewall (pm). This model is first tested analytically against Vp/f scaling by varying different environmental parameters and then tested against numerical simulations. A new Carnot cycle–based length scaling is proposed for equilibrium upper-bound TC size. The key findings of our work are as follows:

  • Physically, the TC PS is determined as follows: the Carnot model defines how much the pressure can be dropped energetically, and then the C15 model defines, for a given intensity, how large a steady-state storm needs to be to yield that pressure drop.

  • Overall, the TC PS model predictions closely match most CM1 simulation results with f, Ttpp, SST, Cd, Ck, and lh varied. This demonstrates that the TC PS model captures the essential physical processes that determine equilibrium TC size on an f plane. Additionally, the TC PS model is able to capture the strong dependence of TC size with increasing SST, and the insensitivity to Ck and Cd, that Vp/f fails to quantitatively capture.

  • The 1/f scaling of TC PS is produced by the fact that f and ra (PS) only appear as their product fra in the TC PS model; thus, with all other environmental parameters unchanged, TC PS model will be unchanged when ra ∼ 1/f.

  • The TC PS model suggests a novel Carnot cycle–based scaling VCarnot/f for equilibrium TC size [Eq. (26)]. A fundamental difference between VCarnot and Vp is that VCarnot is based on the net heat in a thermodynamic cycle whereas Vp is based on air–sea thermodynamic disequilibrium. This may explain why VCarnot/f better captures the variation of TC size with SST in CM1 simulations. VCarnot/f suggests a 4% K−1 increase of size from SST of 300 to 310 K.

  • None of the modifications (half Carnot efficiency, complete resaturation, Gibbs free energy, and work to lift water) to the Emanuel’s Carnot cycle is essential to the qualitative behavior of the TC PS model. However, these modifications do yield a much higher critical SST (∼327 K) for the hypercane condition than that of the Emanuel’s Carnot cycle (∼310 K), suggesting that the current tropics is far from supporting a hypercane.

  • A simpler Carnot cycle (corresponding to Emanuel’s) yields a scaling of VCarnot′/f, where VCarnot′ more closely resembles Vp. This may explain why Vp/f can perform well as a scaling for TC size for certain parameters.

One potential significant limitation of the present TC PS model is the assumption of a closed cycle of a parcel in a TC in which the storm restores its own angular momentum. This would inhibit application of TC PS model to real TCs, where outflow parcels often follow outflow jets that travel far from the storm rather than directly recirculating into the storm in the subsidence region near the outer edge of the storm. However, the fact that the dynamical condition of C15 model also imposes the ra ∼ 1/f scaling, as noted in section 2d, would explain why this scaling does not depend on the outflow work term and hence the outflow work term (i.e., the assumption that the storm must restore its own angular momentum) is not essential. Note also that axisymmetry is not strictly required for a Carnot cycle (see section 4 of Emanuel 1988). Examination of how the TC PS model might need to be modified to be applicable to real TCs is an important subject of future work.

A key constraint for equilibrium upper-bound TC size in this study is the outflow work needed to dissipate the kinetic energy of the anticyclone in the upper troposphere. This process introduces f, the inclusion of which has lacked a clear physical basis in previous studies (Khairoutdinov and Emanuel 2013; Emanuel 2022). This outcome emphasizes the need for a better understanding of the descent leg of the cycle, including the processes by which absolute angular momentum is restored in addition to entropy.

Finally, Lu and Chavas (2022) demonstrated using a nondivergent barotropic model that a vortex on a β plane tends to shrink to its vortex Rhines scale as a result of radiation of planetary Rossby waves. The vortex Rhines scale is typically ∼200–400 km in their experiments, much smaller than the observed TC outer radius ∼ 900 km (Chavas et al. 2016) and the PS ∼ 2000 km in the present study. Hence, on the real Earth, the Rhines effect acts as an additional time-dependent sink of kinetic energy that opposes expansion toward the PS. A time-dependent model for TC size that incorporates this effect is a worthy avenue of future work.

1

A corresponding heating process (with moistening and cooling) will be at constant temperature Ts and constant dry-air partial pressure pda (so that also at constant volume).

Acknowledgments.

This work was supported by the National Natural Science Foundation of China (42130603, 41975127). DC was supported by NSF Grants 1945113 and 1826161. The first author thanks Wenchao Chu for maintaining the server on which the simulations were run.

Data availability statement.

Description of CM1 model is available at https://www2.mmm.ucar.edu/people/bryan/cm1/. The specific CM1v19.2 model code with the noted modifications and the namelist for CTL simulation used in this study have been uploaded to figshare with DOI: 10.6084/m9.figshare.17134400.

APPENDIX A

Derivation of Eq. (1)

Following Romps (2008), we also define
∂α(uαX)=∂X∂t+∇⋅(uX),
where X is any variable and u the three-dimensional velocity equivalent to V in Eq. (1). Note with continuity equation ∂α(uαρd) = 0, a property of Eq. (A1) is ∂α(ρduαX)=ρd(dX/dt). The kinetic energy equation is given by Eq. (17) of Romps (2008) and may be equivalently written as
∂α[(1+qt)ρduα|u|22]=−∇[|u|22(dυ+dc)]+(1+qt)ρdΓ⋅u−u⋅∇p+u⋅(ρdF).
Note Romps (2008) worked with specific humidity rather than mixing ratio here. Also note that although Romps (2008) did not include background rotation (f = 0), it does not influence the energy equations as Coriolis force is perpendicular to velocity vector and does no work. The potential energy equation may be written:
∂α[(1+qt)ρduαϕ]=ϕ∂α[(1+qt)ρduα]−(1+qt)ρdΓ⋅u,
where ∂α[ρd(1+qt)uα]=ρd(dqt/dt)=−∇⋅(dυ+dc) using continuity equations [see Eqs. (6)–(9) of Romps 2008] with dυ and dc the diffusion (rainfall) fluxes of water vapor and condensates per unit area, respectively. Equation (1) is then obtained by summing Eqs. (A2) and (A3), neglecting the first term on the rhs of Eq. (A2), assuming steady state and using the relation ∂α(ρduαX)=ρd(dX/dt).
Wp is obtained by integrating the third term on the rhs of Eq. (1) but with a negative sign:
Wp=−∮ϕdqt=∮qtdϕ=∮Γqtdz.

APPENDIX B

Derivation of Intrinsic Length Scale

Rearranging Eq. (25) gives
1−1y=A+C1+C≈A+C.
Substituting Eq. (B1) into Eq. (24) gives
(A+C)pda≈DVgm+Efra+F.
As we see in sections 2e and 3b, Vgm alone only has a small effect on PS; it is reasonable to assume that all terms associated with Vgm in Eq. (B2), i.e., DVgm + F, cancel. Thus, we have
Apda−{14(fra)2/[(1−ηϵC)RTs]}pda=Efra.
Note Mm has been neglected and ro taking as infinity in parameter C as well as β = 1 being used in A and C, meaning M is restored well beyond the storm and the work to lift water is neglected.
Rearranging Eq. (B3) gives
14(1−ηϵC)(fra)2+RTspdaEfra−RTsA=0.
To obtain an approximate solution, the second term of Eq. (B4) may be neglected when fra is large. This gives a solution:
fra=2VCarnot,
with VCarnot given by Eq. (27). In Eq. (B5) we are neglecting the size effect in C15 model while retaining the Vgm effect, merely in order to give a simple solution as it is suggested (Fig. 7) that the quadratic effect of the Wout term puts a stronger constraint on TC PS. This approximation will give an overestimation of the PS. Equation (B5) indicates that VCarnot is the anticyclonic velocity that can be reached in the outflow at radius ra of the TC given Mm negligible compared to (1/2)fra2.
A similar procedure to the above but for the simple Carnot cycle in section 4 will give a similar solution:
fra=2VCarnot′,
with VCarnot′ given by Eq. (36).

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  • Tang, B., and K. Emanuel, 2010: Midlevel ventilation’s constraint on tropical cyclone intensity. J. Atmos. Sci., 67, 1817–1830, https://doi.org/10.1175/2010JAS3318.1.

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  • Tang, B., and K. Emanuel, 2012: A ventilation index for tropical cyclones. Bull. Amer. Meteor. Soc., 93, 1901–1912, https://doi.org/10.1175/BAMS-D-11-00165.1.

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  • Tao, D., M. Bell, R. Rotunno, and P. J. Van Leeuwen, 2020: Why do the maximum intensities in modeled tropical cyclones vary under the same environmental conditions? Geophys. Res. Lett., 47, e2019GL085980, https://doi.org/10.1029/2019GL085980.

  • Wang, D., and Y. Lin, 2021: Potential role of irreversible moist processes in modulating tropical cyclone surface wind structure. J. Atmos. Sci., 78, 709–725, https://doi.org/10.1175/JAS-D-20-0192.1.

    • Search Google Scholar
    • Export Citation

  • Zhou, W., I. M. Held, and S. T. Garner, 2014: Parameter study of tropical cyclones in rotating radiative–convective equilibrium with column physics and resolution of a 25-km GCM. J. Atmos. Sci., 71, 1058–1069, https://doi.org/10.1175/JAS-D-13-0190.1.

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    • Search Google Scholar
    • Export Citation

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    • Search Google Scholar
    • Export Citation

  • Zhou, W., I. M. Held, and S. T. Garner, 2014: Parameter study of tropical cyclones in rotating radiative–convective equilibrium with column physics and resolution of a 25-km GCM. J. Atmos. Sci., 71, 1058–1069, https://doi.org/10.1175/JAS-D-13-0190.1.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Schematic path of a parcel in a updated Carnot cycle in a fully developed TC. Point a marks the radius of vanishing near surface wind and contains environmental relative humidity. Point m marks the radius of maximum wind. Point o marks the radially farthest point the parcel could reach in the outflow. Point o′ has the same radius as o, but with a lower altitude. Point a′ is spatially close to a where the parcel has near-zero relative humidity. The arrows mark the direction of the path and the path is a closed cycle. See text for different processes in the path. Inside the cycle is a schematic T–s diagram of the cycle.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    A representative example of how the roots of Eq. (16) are obtained. Here A = 0.062, B = 0.031, and C = 0.008. See text for more details.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    A schematic summary of the basic idea of TC PS model. Blue boxes and arrows show logic and submodels and green boxes show necessary steps in the iterative solution.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    An example of how TS PS model predicts a specific ra in a given environment. (a) pm (hPa) predicted by updated Carnot cycle (blue) and by C15 model (red) with different ra (km). The intersection point marks the specific ra and pm predicted by TC PS model. (b) Different terms (J kg−1) in the updated Carnot cycle model with different ra. The gray line shows the difference of WPBL (J kg−1) between Carnot model and C15 model.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    TC PS model prediction of (left) ra (km) and (right) pm (hPa) as (a),(b) f varied from 1.25 × 10−5 to 2 × 10−4 s−1, (c),(d) To varied from 170 to 230 K, and (e),(f) SST varied from 280 to 310 K. The x axis is the corresponding 1/f (104 s) in (a) and (b) and the corresponding Vp (m s−1) in (c)–(f). Predictions with different values of η are in different colors noted by the legend in (a). Thick solid lines are with ro = +∞ and thin dashed lines are with ro = 1.2ra. The light gray lines in the left column show Vp/f (km) and the dark gray lines show VCarnot/f (km) with η = 0.5.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    As in Fig. 5, but with (a),(b) Cd varied from 0.0009 to 0.003, (c),(d) Ck varied from 0.000 75 to 0.002 55, and (e),(f) Vgm varied from 40 to 90 m s−1. The x axis is the corresponding Vp (m s−1) in (a)–(d) and the corresponding Vgm (m s−1) in (e) and (f).

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    pm predicted by the updated Carnot cycle model (blue) and the C15 model (red) over a range of values of ra induced by variations in (a) f, (b) To, (c) Ts (SST), (d) Cd, (e) Ck, and (f) Vgm. We set η = 0.5. More opaque line means higher values of the variable varied. The TC PS–model-predicted pm and ra are intersections of blue and red curves, marked by black dots.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    WPBL, Wout, Qs, and −QGibbs (J kg−1) in the updated Carnot cycle with η = 0.5 corresponding to the TC PS–model predictions in the six tests where (a) f, (b) To, (c) Ts (SST), (d) Cd, (e) Ck, and (f) Vgm are varied.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Time series of (a) Vmax (m s−1), (b) rm (km), (c) radius of 4 m s−1 tangential wind at 1 km height r4 (km), and (d) ra by fitting E04 model to r4 (see text). In (c) and (d), a 24 h running average has been applied. The horizontal lines in (c) and (d) mark the 40–100 day average values. Solid and dashed gray lines show Large and Small experiments (see text), respectively.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    TC PS–model predictions (blue line) of (left) ra (km) and (right) pm (hPa) as a function of Vmax (m s−1) or 1/f (104 s) compared to the corresponding steady-state averaged values of CM1 results (red line) for (a),(b) Ex_F, (c),(d) Ex_TTPP, and (e),(f) Ex_SST. Background dots are 4-hourly CM1 output during the steady state. Cold to warm colors mean increased values of the parameter changed in each experiment. The light gray lines in the left column show the corresponding Vp/f (km) and the dark gray lines show the corresponding VCarnot/f (km). “Hi” and “Lo” mark high and low values of the parameter varied, respectively.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    As in Fig. 10, but for (a),(b) Ex_CD, (c),(d) Ex_CK, and (e),(f) Ex_LH. Note that in (c) and (d), Ck is held to CTL value in TC PS model; see more details in text. Also, the x axis is Ck in (c) and (d) because Vmax does not separate different simulations in Ex_CK.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    As in Fig. 5, but for the TC PS model with a simple Carnot cycle. Predictions with different values of γsg are in different colors noted by legend (“sgw”). The dark gray lines show 2VCarnot′/f (km).

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    As in Fig. 6, but for a TC PS model with a simple Carnot cycle. Predictions with different values of γsg are in different colors noted by legend (“sgw”). The dark gray lines show 2VCarnot′/f (km).

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    Critical SST for the hypercane condition as a function of ra for the updated Carnot cycle (blue) and the simple Carnot cycle (red). In the calculation ro is taken as infinity and −(1/2)fMm is neglected in C and C′.

Fig. 1.

Schematic path of a parcel in a updated Carnot cycle in a fully developed TC. Point a marks the radius of vanishing near surface wind and contains environmental relative humidity. Point m marks the radius of maximum wind. Point o marks the radially farthest point the parcel could reach in the outflow. Point o′ has the same radius as o, but with a lower altitude. Point a′ is spatially close to a where the parcel has near-zero relative humidity. The arrows mark the direction of the path and the path is a closed cycle. See text for different processes in the path. Inside the cycle is a schematic T–s diagram of the cycle.
Fig. 2.

A representative example of how the roots of Eq. (16) are obtained. Here A = 0.062, B = 0.031, and C = 0.008. See text for more details.
Fig. 3.

A schematic summary of the basic idea of TC PS model. Blue boxes and arrows show logic and submodels and green boxes show necessary steps in the iterative solution.
Fig. 4.

An example of how TS PS model predicts a specific ra in a given environment. (a) pm (hPa) predicted by updated Carnot cycle (blue) and by C15 model (red) with different ra (km). The intersection point marks the specific ra and pm predicted by TC PS model. (b) Different terms (J kg−1) in the updated Carnot cycle model with different ra. The gray line shows the difference of WPBL (J kg−1) between Carnot model and C15 model.
Fig. 5.

TC PS model prediction of (left) ra (km) and (right) pm (hPa) as (a),(b) f varied from 1.25 × 10−5 to 2 × 10−4 s−1, (c),(d) To varied from 170 to 230 K, and (e),(f) SST varied from 280 to 310 K. The x axis is the corresponding 1/f (104 s) in (a) and (b) and the corresponding Vp (m s−1) in (c)–(f). Predictions with different values of η are in different colors noted by the legend in (a). Thick solid lines are with ro = +∞ and thin dashed lines are with ro = 1.2ra. The light gray lines in the left column show Vp/f (km) and the dark gray lines show VCarnot/f (km) with η = 0.5.
Fig. 6.

As in Fig. 5, but with (a),(b) Cd varied from 0.0009 to 0.003, (c),(d) Ck varied from 0.000 75 to 0.002 55, and (e),(f) Vgm varied from 40 to 90 m s−1. The x axis is the corresponding Vp (m s−1) in (a)–(d) and the corresponding Vgm (m s−1) in (e) and (f).
Fig. 7.

pm predicted by the updated Carnot cycle model (blue) and the C15 model (red) over a range of values of ra induced by variations in (a) f, (b) To, (c) Ts (SST), (d) Cd, (e) Ck, and (f) Vgm. We set η = 0.5. More opaque line means higher values of the variable varied. The TC PS–model-predicted pm and ra are intersections of blue and red curves, marked by black dots.
Fig. 8.

WPBL, Wout, Qs, and −QGibbs (J kg−1) in the updated Carnot cycle with η = 0.5 corresponding to the TC PS–model predictions in the six tests where (a) f, (b) To, (c) Ts (SST), (d) Cd, (e) Ck, and (f) Vgm are varied.
Fig. 9.

Time series of (a) Vmax (m s−1), (b) rm (km), (c) radius of 4 m s−1 tangential wind at 1 km height r4 (km), and (d) ra by fitting E04 model to r4 (see text). In (c) and (d), a 24 h running average has been applied. The horizontal lines in (c) and (d) mark the 40–100 day average values. Solid and dashed gray lines show Large and Small experiments (see text), respectively.
Fig. 10.

TC PS–model predictions (blue line) of (left) ra (km) and (right) pm (hPa) as a function of Vmax (m s−1) or 1/f (104 s) compared to the corresponding steady-state averaged values of CM1 results (red line) for (a),(b) Ex_F, (c),(d) Ex_TTPP, and (e),(f) Ex_SST. Background dots are 4-hourly CM1 output during the steady state. Cold to warm colors mean increased values of the parameter changed in each experiment. The light gray lines in the left column show the corresponding Vp/f (km) and the dark gray lines show the corresponding VCarnot/f (km). “Hi” and “Lo” mark high and low values of the parameter varied, respectively.
Fig. 11.

As in Fig. 10, but for (a),(b) Ex_CD, (c),(d) Ex_CK, and (e),(f) Ex_LH. Note that in (c) and (d), Ck is held to CTL value in TC PS model; see more details in text. Also, the x axis is Ck in (c) and (d) because Vmax does not separate different simulations in Ex_CK.
Fig. 12.

As in Fig. 5, but for the TC PS model with a simple Carnot cycle. Predictions with different values of γsg are in different colors noted by legend (“sgw”). The dark gray lines show 2VCarnot′/f (km).
Fig. 13.

As in Fig. 6, but for a TC PS model with a simple Carnot cycle. Predictions with different values of γsg are in different colors noted by legend (“sgw”). The dark gray lines show 2VCarnot′/f (km).
Fig. 14.

Critical SST for the hypercane condition as a function of ra for the updated Carnot cycle (blue) and the simple Carnot cycle (red). In the calculation ro is taken as infinity and −(1/2)fMm is neglected in C and C′.
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