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Article Type:
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An Analytical Model for Tropical Cyclone Outer-Size Expansion on the f Plane

Danyang Wang aDepartment of Earth, Atmospheric, and Planetary Sciences, Purdue University, West Lafayette, Indiana

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https://orcid.org/0000-0001-7142-014X
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Daniel R. Chavas aDepartment of Earth, Atmospheric, and Planetary Sciences, Purdue University, West Lafayette, Indiana

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Online Publication:
09 Jul 2024
Print Publication:
01 Jul 2024
DOI:
https://doi.org/10.1175/JAS-D-23-0088.1
Page(s):
1097–1125
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Abstract

Tropical cyclones are known to expand to an equilibrium size on the f plane, but the expansion process is not understood. In this study, an analytical model for tropical cyclone outer-size expansion on the f plane is proposed. Conceptually, the storm expands because the imbalance between latent heating and radiative cooling drives a lateral inflow that imports absolute vorticity. Volume-integrated latent heating increases more slowly with size than radiative cooling, and hence, the storm expands toward an equilibrium size. The predicted expansion rate is given by the ratio of the difference in size from its equilibrium value rt,eq to an environmentally determined time scale τrt of 10–15 days. The model is fully predictive if given a constant rt,eq, which can also be estimated environmentally. The model successfully captures the first-order size evolution across a range of numerical simulation experiments in which the potential intensity and f are varied. The model predictions of the dependencies of lateral inflow velocity and expansion rate on latent heating rate are also compared well with numerical simulations. This model provides a useful foundation for understanding storm size dynamics in nature.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Danyang Wang, wang5571@purdue.edu

Abstract

Tropical cyclones are known to expand to an equilibrium size on the f plane, but the expansion process is not understood. In this study, an analytical model for tropical cyclone outer-size expansion on the f plane is proposed. Conceptually, the storm expands because the imbalance between latent heating and radiative cooling drives a lateral inflow that imports absolute vorticity. Volume-integrated latent heating increases more slowly with size than radiative cooling, and hence, the storm expands toward an equilibrium size. The predicted expansion rate is given by the ratio of the difference in size from its equilibrium value rt,eq to an environmentally determined time scale τrt of 10–15 days. The model is fully predictive if given a constant rt,eq, which can also be estimated environmentally. The model successfully captures the first-order size evolution across a range of numerical simulation experiments in which the potential intensity and f are varied. The model predictions of the dependencies of lateral inflow velocity and expansion rate on latent heating rate are also compared well with numerical simulations. This model provides a useful foundation for understanding storm size dynamics in nature.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Danyang Wang, wang5571@purdue.edu
Keywords: Atmosphere; Hurricanes/typhoons; Tropical cyclones; Entropy

1. Introduction

Tropical cyclone (TC) outer size is known to expand with time toward an equilibrium size in idealized simulations on the f plane (Chan and Chan 2014, 2015; Chavas and Emanuel 2014; Martinez et al. 2020). Expansion with time is also seen in reanalysis or simulations on spherical geometry (Schenkel et al. 2018, 2023). Reanalysis data show that the median expansion rate of TC outer radius (of 8 m s−1 near-surface wind) is tens of kilometers per day, with extreme cases being hundreds of kilometers per day (Schenkel et al. 2023). Although TC intensity and intensification have been understood with the help of some relatively well-established analytical theories (Emanuel 1986, 2012; Emanuel and Rotunno 2011; Wang et al. 2021a,b), a conceptual understanding of tropical cyclone size and size expansion is not as complete. Although theoretical models link inner size (radius of maximum wind) to outer size (Emanuel and Rotunno 2011; Chavas and Lin 2016), the mechanism of the changes in inner and outer sizes is not the same (Weatherford and Gray 1988; Chavas and Knaff 2022); the present study will focus on the TC outer-size-expansion mechanism.

Recently, Wang et al. (2022) proposed a model for tropical cyclone potential size (TC PS) on the f plane that explains equilibrium TC size and is solely dependent on environmental parameters. The model yields a new scaling that is similar to the length scale Vp/f, where Vp is the potential intensity and f is the Coriolis parameter, proposed in prior work (Chavas and Emanuel 2014). The TC PS model combines the Carnot cycle model for the energetics of a TC (Emanuel 1988, 1991) and a model for the complete low-level TC wind field (Chavas et al. 2015) to solve for an equilibrium size based on the most efficient thermodynamic cycle. However, such a method does not provide a description of how other parts of the TC are working, without which the potential size may not be achieved at all. It is also unsatisfying that the thermodynamic cycle is formulated in steady state so that it does not mechanistically explain how and why a TC expands toward equilibrium. Although the model suggests that an energy surplus exists when a TC is smaller than its potential size, it cannot explain how this energy surplus might drive expansion.

Previous studies on TC size expansion consistently note the importance of low-level inflow for bringing environmental absolute angular momentum inward to drive expansion (Hill and Lackmann 2009; Bui et al. 2009; Wang 2009; Chan and Chan 2014, 2015, 2018; Martinez et al. 2020; Wang and Toumi 2022), which is a direct reflection of the spinup of the outer-core wind field. The TC size-expansion rate has been further found to depend on initial vortex size (Xu and Wang 2010; Chan and Chan 2014; Martinez et al. 2020) and rainband activity (Hill and Lackmann 2009; Wang 2009; Fudeyasu et al. 2010; Martinez et al. 2020), as well as cloud radiative forcing (Bu et al. 2014, 2017). Simulations have also shown that TC size is able to continue expanding substantially long after intensity becomes quasi-steady (Hill and Lackmann 2009; Chan and Chan 2014, 2015; Martinez et al. 2020). However, a simple universal understanding of why a TC should expand, how fast, why size should approach an equilibrium, and how this behavior depends on environmental parameters is still lacking. This is partly because the lateral inflow or import of absolute angular momentum has yet to be fully and quantitatively linked to environmental parameters and internal processes. Such a quantitative link, either direct or indirect, is necessary for a predictive model for size. Indeed, if given an inflow velocity, then size expansion may be predicted as shown in Wang and Toumi (2022). However, the inflow velocity varies significantly with height, from larger values within the boundary layer to near zero at some height above the top of the boundary layer. Thus, one needs to consider the integrated inflow mass flux instead of picking one single height.

In this study, we propose a model for size expansion toward equilibrium on the f plane, in terms of the outer radius of a certain tangential wind speed at the top of the boundary layer.1 We seek a model that

  • is predictive and analytic;

  • yields a characteristic expansion rate from the environmental/external parameters;

  • explains the physical process that drives TC expansion and why this expansion vanishes such that there exists an upper bound of size.

We test these model outcomes via comparison of model predictions with sets of numerical simulation experiments.

Our model for TC size expansion is presented in section 2. Basic predictions of the model and its comparison to numerical simulations are provided in sections 3–5. Further physical interpretations of the model are provided in section 6. A summary of key conclusions and discussion is given in section 7.

2. Theory: An analytical outer-size-expansion model on the f plane

a. Basic model structure

Below, we present a theory for TC expansion toward equilibrium that is summarized conceptually as follows: 1) in radiative–convective equilibrium (RCE) without a TC, net condensational heating equals net radiative cooling; 2) when a TC forms, the TC volume is shifted substantially out of RCE, such that condensational heating substantially exceeds radiative cooling (consistent with enhanced surface fluxes); and 3) the TC expands in response as a result of strong low-level inflow as part of the overturning circulation that exports excess heat. As it expands, area-integrated radiative cooling increases faster than net condensational heating until low-level inflow is weak enough so that surface friction prevents any further expansion of wind field. The storm has reached its equilibrium size. A schematic plot is shown in Fig. 1.

Fig. 1.
Fig. 1.

A schematic plot of the expansion model presented in section 2. See the text for details.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

We define rt as the radius of a fixed tangential velocity υt (e.g., r8 is the radius of υt = 8 m s−1 tangential wind) at the top of the boundary layer in the TC outer-core region. As TC size expansion is basically low-level spinup of TC outer core, the expansion rate of rt can be given by
drtdt=∂υ∂t/(−∂υ∂r),r=rt,
which is obtained by taking dυ/dt = 0 with υ = υ[r(t), t], where υ is the tangential wind, r is the radius, and t is the time.2 This relation is also presented in Tsuji et al. (2016).
The local spinup tendency for an axisymmetric TC at any height on the f plane is given by
∂υ∂t=−u(f+ζ)−w∂υ∂z+F,r=rt,
where ζ=(∂υ/∂r)+(υ/r) is the relative vorticity, u is the radial velocity, w is the vertical velocity, and F≈(1/ρd)(∂τυ/∂z) is the turbulence frictional force, with ρd being the dry air density and τυ being the turbulence stress in the azimuthal direction. Equation (2) is simplified by choosing rt sufficiently far from the center so that ζ can be neglected compared to f. Further integrating Eq. (2) from the surface to hw, some height in the lower troposphere below which the inward mass flux constitutes the majority of the total lateral inward mass flux, and neglecting vertical advection, gives
∂υ∂t≈−fut−Cd(μυt)2/hw,
where the aerodynamic formula for surface stress τυ,sfc=ρdCd|V10|υ10 is applied, with Cd being the surface exchange coefficient for momentum, V10 being the 10-m surface horizontal velocity, and υ10 being the 10-m tangential velocity, ut is the vertical mean radial velocity, υt is the corresponding tangential velocity at the top of the boundary layer at rt, and μ is a surface wind reduction factor (i.e., basically the ratio of υ10 to υt). An implicit assumption made moving from Eq. (2) to (3) is that ∂υ/∂t and ζ are approximately constant in the vertical below hw.3 In Eq. (3), the second term on the RHS can be taken as a constant, a key conceptual benefit since we are following the radius of a fixed wind speed. Substituting Eq. (3) into Eq. (1) gives
drtdt=1(−∂υ∂r)[−fut−Cd(μυt)2/hw].
There are two quantities ∂υ/∂r and ut that are not specified and must be linked to internal processes or environmental parameters. The slope of the wind profile ∂υ/∂r can be obtained from the wind profile solution for the outer wind field from Emanuel (2004, hereafter E04) model (see appendix A), which depends only on the environmental parameters. The E04 model does not have a simple analytical solution, but we will provide an analytic approximation for ∂υ/∂r based on this model in section 2b. An expression for ut is derived next.
We propose ut to be driven principally by the energetics of the TC, i.e., latent heating and radiative cooling, which can be described by a dry-entropy budget. Here, dry entropy is loosely defined by sd=cpln⁡(θ/Ttrip), where cp is the specific heat of dry air at constant pressure, θ is the potential temperature, and Ttrip is the triple-point temperature; sd thus defined is a close approximation of the true dry entropy and is more convenient for budget analysis in numerical simulations. The budget of the dry entropy sd within the TC volume from the center to rt is written as
∂S∂t=QlatTe,lat−QradTe,rad+S˙res+Fr+Fu,
where S is the mass-integrated dry entropy within the volume; Qlat and Qrad are the net condensational heating (latent heat) and the total radiative cooling (defined positive), respectively, with Te,lat and Te,rad being their respective effective temperatures; S˙res represents other sources of dry entropy, such as surface sensible heating, diffusion of sensible heat, and dissipative heating; and Fr and Fu are the fluxes of dry entropy into the volume from the lateral (at rt) and vertical directions (at the upper extent of the volume), respectively.4 The first two terms on the RHS are dry-entropy sources due to latent heating and radiative cooling, respectively. See appendix D for detailed expressions. To achieve a simple expression for size expansion, we neglect ∂S/∂t,S˙res,Fu. This assumes that the dominant terms are sources/sinks from latent heating (source), radiative cooling (sink), and net lateral transport into the TC from the environment (supported by simulations in appendix D). Doing so yields the balance equation
QlatTe,lat−QradTe,rad≈−Δsd2πrtρiuthw,
where we have rewritten the lateral flux term in terms of a bulk free-tropospheric dry static stability given by
Δsd=Fr/(2πrtρiuthw).
Though Δsd must also depend on the vertical profile of lateral flow (for which we lack a clear constraint), the physical meaning of Δsd can be better understood in the ideal case where the inflow is confined to near the surface and the outflow is confined to near the tropopause level: In this case, Δsd represents the difference of sd between the surface and tropopause. Appendix D shows that this is a reasonable assumption for TCs; discussion of the meaning of Δsd in general is also provided in appendix D. The parameter ρi is an effective inflow air density corresponding to ut so that 2πrtρiuthw is the lateral mass flux at rt below hw. A reference of Δsd is the difference between moist entropy and dry entropy near the sea surface, which is equivalent to the difference of sd between tropopause and surface. A corresponding sufficient condition5 is the eyewall being in slantwise neutrality, which applies to the later stage of TC intensification and peak intensity (Bryan and Rotunno 2009; Peng et al. 2018; Wang et al. 2021b), which is the principal period for size expansion to occur (e.g., Martinez et al. 2020). For near-surface air with water vapor mixing ratio qυ = 0.018 kg kg−1, temperature T = 300 K, and relative humidity 80% (tropical value; see Dunion 2011), this gives a reference Δsd of Lυqυ/T ≈ 150 J K−1 kg−1, where Lυ = 2.501 × 106 J kg−1 is the latent heating of vaporization. Thus, Δsd can be taken as primarily determined by sea surface temperature.
Latent heating is assumed to be principally produced in the eyewall (see appendix D), which is largely driven by boundary layer frictional convergence as found in both observations of vertical velocities (Stern et al. 2016) and implicit in the slantwise neutrality assumption of potential intensity theory (Emanuel 1986, 1995; Khairoutdinov and Emanuel 2013). Hence, Qlat may be written as
Qlat=ϵp,ewαpQc,ew≈ϵp,ewαpLυqυbMew,
where qυb is the boundary layer water vapor mixing ratio just outside of the eyewall corresponding to Mew; ϵp,ew is the precipitation efficiency in the eyewall region, defined as the ratio of condensation to the mass of water vapor imported upward into the eyewall (see appendix D for practical diagnosis); Qc,ew is the latent heating rate due to the total condensation in the eyewall; αp is the ratio of net latent heating in the eyewall region to that within rt; and Mew is the eyewall updraft mass flux. Given that the eyewall updraft is driven by boundary layer frictional convergence, Mew is also equal to the friction-induced inflow mass flux into the eyewall. Thus, Mew/ρw (which will appear shortly) should be strongly controlled by the inner-core size and TC intensity, where ρw is the effective dry air density for the boundary layer inflow under the eyewall (close to ρi; see appendix C for calculation). Here, ρw becomes implicit, as in the boundary layer momentum equations, only the gradient wind matters and air density will not explicitly appear (Kuo 1982; Kepert 2001).
The radiative cooling Qrad may be written as (Chavas and Emanuel 2014)
Qrad=πrt2cpΔpgQcool,
where
Δp=p0R/cp+1[(psp0)1+R/cp−(ptp0)1+R/cp].
In the above, Qcool is a constant radiative cooling rate for potential temperature, Δp/g is the effective mass obtained by the vertical integration over a pressure layer, with g being the gravitational acceleration, p0 = 1000 hPa is the reference pressure, R is the gas constant of dry air, and cp is the heat content of dry air at constant pressure. Taking the surface pressure ps = 1000 hPa and the tropopause pressure pt = 100 hPa with Qcool = 1 K day−1 yields a value of 88 W m−2, close to the 100 W m−2 value in tropics (Pauluis et al. 2000).
An expression for the inflow velocity is obtained by first rearranging Eq. (6):
−ut=12πrthwρi(QlatTe,latΔsd−QradTe,radΔsd),
and then substituting for Qlat using Eq. (8) and Qrad using Eq. (9) to yield
−ut=12πhwϵp,ewαpLυqvbTe,latΔsd(Mewρw)1rt−12hwcpΔpρigQcoolTe,radΔsdrt,
where we take ρi ≈ ρw. We may write this more compactly as
−ut=1hwA(Mewρw)1rt−1hwBrt,
where we define two thermodynamic parameters
A=12πϵp,ewαpLυqvbTe,latΔsd and
B=12cpΔpρigQcoolTe,radΔsd.
Parameter A is nondimensional and is related to the latent heating that drives expansion, while B is a velocity and is related to radiative cooling that suppresses expansion.
The size-expansion model is obtained by substituting Eq. (10c) into Eq. (4) to yield
drtdt=rt,eq−rtτrt.
Here, rt,eq is the equilibrium size when drt/dt = 0 is achieved [Eq. (12)] and τrt is the time scale for expansion. Equation (12) states that the expansion rate is given by the difference in size from equilibrium divided by a time scale τrt.
Quantity rt,eq in Eq. (12) is given by
rt,eq=[fA(Mewρwrt)−Cd(μυt)2]/(fB),
which can be expressed explicitly by Qlat and Qcool as [using Eqs. (8) and (11)]
rt,eq=[f12πrtρiQlatTe,latΔsd−Cd(μυt)2]/[12fcpΔpρigQcoolTe,radΔsd].
Equations (13a) and (13b) indicate that rt,eq may vary with rt (and thus time), but here, we will take it to be a constant in order to seek an analytical solution of Eq. (12); this assumption is later tested in section 4b. A useful form of rt,eq is obtained by writing Eq. (13a) at equilibrium (rt = rt,eq):
rt,eq=[fA(Mewρw)eq1rt,eq−Cd(μυt)2]/(Bf),
where the subscript “eq” means equilibrium. Before solving for rt,eq, we first define the equilibrium radius of zero net source of dry entropy rRCE,eq, inside of which the system is in RCE, by taking the LHS of Eq. (6) to be zero [using Eqs. (8) and (9)] and solving for rt at equilibrium:
rRCE,eq=AB(Mewρw)eq.
Thus, rRCE,eq scales with (Mew/ρw)eq (this relationship will be revisited later). Note that rRCE,eq cannot be obtained by directly taking υt = 0 in Eq. (14) because υt = 0 implies ut = 0 in equilibrium [Eq. (4)], but Eq. (7) does not allow ut = 0. Substituting Eq. (15) into Eq. (14) and solving for rt,eq, we have
rt,eq=−Cd(μυt)21Bf+Cd2(μυt)41B2f2+4rRCE,eq22,
which can be written compactly as
rt,eq=−ξυt2+ξ2υt4+4rRCE,eq22
by defining
ξ=Cdμ2Bf.
Equations (16a) and (16b) imply that rt,eq scales with 1/f if rRCE,eq scales with 1/f. Equation (16b) also shows that rRCE,eq > rt,eq.6
The time scale for expansion τrt in Eq. (12) is given by
τrt=(−∂υ∂r)hwfB.
Here, τrt is proportional to 1/f and ∂υ/∂r, meaning that the time scale is larger if f is smaller or the local slope of the wind profile is larger in magnitude. Note that τrt exists independent of the specific parameters for eyewall dynamics, as it depends on B but not A.

Conceptually, the model links the expansion to the radial velocity ut induced by the dry-entropy imbalance within the TC volume. A stable equilibrium size rt,eq independent of time and current size is assumed to exist [Eq. (12)]. The following parameters of the model are taken as constants: ϵp,ew, αp, hw, f, Lυ, qυb, Te,lat, Te,rad, Δsd, Δp, and ρi, and thus A, B, and ξ. Doing so simplifies the problem enough to make it analytically tractable. Simulations also indicate that taking parameters ϵp,ew, αp, Δsd, Δp (implicit in Fig. D1), ρi (not shown, also Te,lat and Te,rad) as constant is reasonable (see appendix D).7 Note a constant rt,eq also implies a constant Mew/(ρwrt) [Eq. (13a)]. In this manner, Qlat is proportional to rt [Eq. (8)] and Qrad is proportional to rt2 [Eq. (9)]. Hence, ut [Eq. (10)] monotonically decreases in magnitude with expansion so that TC size will approach an equilibrium.

If, in addition to rt,eq, τrt is also time invariant, the solution of Eq. (12) with initial condition rt(t0) = rt0 is given by
rt(t)=(rt0−rt,eq)e−(t−t0)/τrt+rt,eq.
As τrt is positive definite, rt will exponentially approach the equilibrium size rt,eq, where τrt is the e-folding time scale. Moreover, rt,eq is a stable equilibrium, as size approaches rt,eq for rt0 < rt,eq (expansion) and rt0 > rt,eq (shrinking). Equation (19) gives an exponential solution, similar to Wang and Toumi (2022) for absolute size, though this solution is exponential in the decay of the distance from equilibrium and hence allows for size to reach an equilibrium value as is known to exist on the f plane.

Up to this point, ∂υ/∂r in τrt [Eq. (18)] is not yet defined analytically, which is needed for a full analytical solution of Eq. (12). Moreover, rt,eq [Eq. (13)] is not yet defined in terms of environmental parameters, which requires an expression for (Mew/ρw)eq. In the following subsections, we will resolve these issues and obtain a full analytical solution of Eq. (12).

b. Analytical expression of ∂υ/∂r

Equation (16) provides an expression for the equilibrium radii of different wind speeds, which has the exact same form as the E04 model (see appendix A). The slope of the equilibrium wind profile ∂υt/∂rt,eq can be obtained by taking the derivative of rt,eq with respect to υt in Eq. (16) in a fixed environment:
(∂υt∂rt,eq)−1=∂rt,eq∂υt=−2υt(12υtξ′+ξ)rt,eq2rt,eq+ξυt2,
where ξ′ = dξ/dυt. Equation (20) has the same form as the E04 model (see appendix A). Note that ξ is a constant with respect to rt in a given environment when υt is fixed but may vary with υt. For example, closer to the center (larger υt), the absolute vorticity is larger, so ξ should decrease accordingly (though above we have approximated the absolute vorticity by f). Here, as ξ′ should also be a constant with respect to rt at fixed υt, to simplify the math, we take the approximation (1/2)υtξ′+ξ=σξ, with σ being a constant fitting parameter. Here, σ is set to a constant value of 0.7 (for υt = 8 m s−1, shown below).8 Note that σ > 0 is presumed so that the RHS of Eq. (20) is negative, corresponding to a TC wind profile in which the azimuthal wind speed decreases with radius. With this, we rewrite Eq. (20) in general by dropping the subscript “eq” and write ξ as ξ0 to mark that it is only associated with ∂υ/∂r:
(∂υ∂r)−1|υ=υt=∂r∂υ|υ=υt=−2rtυtσξ02rt+ξ0υt2.
The assumption implicitly made to move from Eqs. (20) to (21) is that (∂υ/∂r)|υ=υt at given rt and υt for slowly evolving wind fields can be approximated by equilibrium values. This assumption follows the fairly nice performance of the E04 model, which is derived for steady state, of matching observed TC outer wind profiles for storms that are not necessarily in steady state (Chavas et al. 2015). This assumption will be shown to work nicely in section 5. Equation (21) provides an analytical approximation of ∂υ/∂r. Compared to the full E04 model, Eq. (21) does not need numerical integration but still contains similar physics to the E04 model. Additional discussion of the properties of Eq. (21) is provided in appendix B. In the next subsection, we will substitute Eq. (21) into Eq. (18) to yield an analytical solution of Eq. (12).

Now, we demonstrate that σ = 0.7 is useful for ∂υ/∂r [Eq. (21)] at υt = 8 m s−1 (this specific υt will be used for the evaluation of the model in sections 3 and 4). We define a baseline environment of ξ0 = 35 105 s2 m−1 with f = 5 × 10−5 s−1, Cd = 0.0015, and surface air temperature Ts = 300 K for demonstration (note that a complete parameter setting in this baseline environment is given in section 3). The radiative-cooling-induced subsidence velocity wcool = 0.0027 m s−1 is set (positive downward) for the E04 model in the baseline environment. The quantity ∂υ/∂r at r8 from the E04 model (solid) and in Eq. (21) (dashed) in the baseline environment is shown in Fig. 2a. The parameter σ is varied from 0.1 to 1.1 to show the sensitivity of Eq. (21) to this quantity. Indeed, Eq. (21) with σ = 0.7 does very well in reproducing ∂υ/∂r for any value of r8 and over a wide range of values of f (Fig. 2b), compared to the E04 model.

Fig. 2.
Fig. 2.

(a) The value ∂υ/∂r at r8 from the E04 model (solid) and the expansion model Eq. (21) (dashed) with σ varied from 0.1 to 1.1 (light to dark, with an interval of 0.1) at f = 5 × 10−5 s−1 and wcool = 0.0027 m s−1. (b) As in (a), but with σ = 0.7 and with different f (10−5 s−1; see the legend); the E04 model in solid lines and Eq. (21) in dashed lines. (c) As in (b), but for different wcool (m s−1; see the legend) for the E04 model (solid) and TSST (286, 293, 300, 307 K; warmer color means lower TSST) for Eq. (21) (dashed). See the text for parameter settings.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

In addition, we test whether Eq. (21) performs reasonably when wcool changes. Note the present expansion model is not framed to have wcool but have B playing the same role in Eq. (21) as (1/2)wcool in the E04 model (see appendix A). In addition, as wcool≈Qcool/(∂θ/∂z) and ∂θ/∂z can be considered mainly determined by sea surface temperature TSST, thus wcool may be considered as a function of Qcool and TSST, both of which enters Eq. (21) through ξ0 by B [Eq. (11b)]. To vary wcool, we pick four values 0.0042, 0.0032, 0.0027, and 0.0022 m s−1 for the E04 model, which corresponds to TSST = 286, 293, 300, and 307 K,9 respectively, in the set of numerical experiments sea surface temperature TSST (ExSST) (see appendix C), for Eq. (21). Quantity Qcool = 1 K day−1 is set. The ∂υ/∂r at r8 is shown in Fig. 2c. It is seen that Eq. (21) works qualitatively the same as the E04 model: Higher SST corresponds to lower magnitude of slope, except that Eq. (21) produces a larger variation of ∂υ/∂r than the E04 model. This is associated with the fact that B basically follows with the Clausius–Clapeyron (C–C) scaling while wcool appears to vary more slowly. It will be shown in section 5 that Eq. (21) turns out to match nicely with simulations.

As discussed above and demonstrated by Fig. 2, we use Eq. (21) with σ = 0.7 for (∂υ/∂r)|υ=υt in the expansion model.

Combining Eqs. (18) and (21) gives a final expression for τrt:
τrt=2rt+ξ0υt22rtυtσξ0hwfB.

c. Analytical solution for size evolution

Substituting Eq. (22) into Eq. (12) yields
drtdt=2fBhwσξ0υtrt(rt,eq−rt)2rt+ξ0υt2,
which gives an explicit form of the expansion rate (drt/dt > 0 when 0 < rt < rt,eq and drt/dt < 0 when rt > rt,eq). Equation (23) again indicates that rt,eq is a stable equilibrium, and drt/dt=0 when rt = rt,eq (and for rt = 0). The maximum expansion rate occurs at a size given by
rt,expmax=−ξ0υt2+ξ02υt4+2rt,eqξ0υt22,
where it is seen 0 < rt,expmax < rt,eq/2. When rt > rt,eq, (∂/∂rt)(drt/dt)<0, meaning that size shrinks faster toward rt,eq when rt is farther from rt,eq.
Solving Eq. (23) with the initial condition rt = rt0 at t = t0 gives
t−t0=12ξ0υtfBσhw[−(2+ξ0υt2rt,eq)ln⁡(rt,eq−rtrt,eq−rt0)+ξ0υt2rt,eqln⁡(rtrt0)],rt>0 and rt≠rt,eq.
Equation (25) is the analytical solution of the full size-expansion model in section 2a. The solution is expressed by time t as a function of rt, which is an implicit function of t; an analytic solution for rt(t) is not tractable. The input parameters are all external or environmentally defined (presently rt,eq can be either external or environmentally defined by TC PS). A method for determining rt,eq [Eq. (16b)] from environmental parameters is provided next.

d. Formulation for updraft mass flux

An environmentally defined rt,eq will be obtained through Eqs. (15) and (16b) if (Mew/ρw)eq is environmentally defined. In this subsection, we parameterize (Mew/ρw)eq by using a combination of theory and empirical estimation based on numerical simulation results.

The parameterization may be derived directly from mass continuity: The eyewall updraft mass flux is balanced by a constant subsidence velocity, which is usually assumed to be driven by radiative cooling (e.g., E04). The streamfunction is given by ∂ψ/∂r=2πρdrw, where ρd is the dry air density and w is the vertical velocity. Integrating radially over the subsidence region at the altitude of hw yields
rψ02≈ψmaxπρwwcool+rψmax2,
where rψ0 and rψmax are the radii of ψ = 0 and maximum ψ (or ψmax) at hw, respectively, and wcool is the environmental clear-air subsidence velocity (positive downward). The inner radius term rψmax2 may be neglected as it is more than an order of magnitude smaller than rψ02. Hence,
ψmax≈πρwwcoolr02,
where r0, the radius of vanishing wind, should be equivalent to rψ0 in E04. TC PS shows that equilibrium r0 (or r0,eq) scales with VCarnot/f, which does not depend on Cd. Following our assumption that most of the upward mass flux occurs within the eyewall so that Mew ≈ ψmax, thus, we propose that
(Mew/ρw)eq∝πwcoolCdνVCarnot/f
a relationship we test with numerical simulations, with ν being a constant coefficient. Here, a role of Cd is tested as it may influence the eyewall upward mass flux by influencing surface friction. Following Eq. (27) of Wang et al. (2022), VCarnot is defined as
VCarnot2=(ηϵCLυ−RυTSST)qvs*,
where ϵC=(TSST−Ttpp)/TSST is the Carnot efficiency, Ttpp is the tropopause temperature, Rυ is the gas constant of water vapor, qvs* is the saturation water vapor mixing ratio at TSST, and η = 0.4 is a coefficient accounting for the “triangle” shape of the thermodynamic cycle following section 3b of Wang et al. (2022).
The exact relationship is not known, and thus, we seek the relation in Eq. (28) via linear regression from equilibrium states of simulated TCs (see appendix C). There is a tight linear relationship between the two quantities in Eq. (28). We estimate the coefficient based on the linear fit to the experiment sets varying Ttpp, Cd, Ck, and TSST to avoid overfitting to the experiments varying f, whose slope deviates slightly, but the result holds reasonably well for those experiments too. The result is shown in Fig. 3. A best-fit estimate of ν = −0.07 is obtained, which suggests that Cd has nearly zero effect on (Mew/ρw)eq, consistent with the finding in Wang et al. (2022). As a final result of the fitting, we have
(Mewρw)eq=0.79πwcoolCd−0.07VCarnot/f.
The environmentally defined rt,eq is obtained by substituting Eq. (30) into Eq. (15) and substituting the resulting rRCE,eq into Eq. (16b). As discussed in section 2b, rt,eq will scale with 1/f and additionally increase with Cd−0.07VCarnot. The expansion model is now capable of being fully predictive and analytic.
Fig. 3.
Fig. 3.

(a) The term (Mew/ρw)eq  (m1.5 s−0.5) as a function of πwcoolCdνVCarnot/f (m1.5 s−0.5) in TTPP (red), CD (green), CDTTPP (blue), CK (purple), FCOR (gray), and ExSST (orange) during equilibrium periods. Data of TTPP, CD, CDTTPP, CK, and ExSST are used to determine ν by linear regression. Note that only ensemble index 0 in TTPP and ExSST is used, to be consistent with the sample sizes of CD, CDTTPP, and CK, which do not contain ensemble experiments. Fitted ν is shown on the upper right of the plot. Data are first processed by a 120-h running average. The black line visualizes the equation shown in the figure. In the equation, y=(Mew/ρw)eq and x=πwcoolCdνVCarnot/f. See appendix C for experimental design. (b) As in (a), but zoomed in without FCOR. Equilibrium periods for TTPP, FCOR, and ExSST are defined in section 4, and those for CD, CDTTPP, and CK are the same as TTPP.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

e. Model summary and implementation

To summarize, this section has proposed a model for the expansion of TC size in which expansion is driven by latent heating, which is dominated by heating in the eyewall and suppressed by radiative cooling. The model can be fully predictive and analytic if there is an environmentally defined size-independent rt,eq. The steps to put the model into practice are as follows:

  • Assuming known rt,eq, the evolution of outer radius rt is given by Eq. (25). Parameter B is given in Eq. (11b), and parameter ξ0 = ξ is given in Eq. (17). For υt = 8 m s−1, ξ0 = 35 105 s2 m−1 for Cd = 0.0015, TSST = 300 K and f = 5 × 10−5 s−1 and σ = 0.7.

  • The equilibrium size rt,eq is predicted from the environmental parameters by using Eq. (30) for(Mew/ρw)eq, plugging the result into Eq. (15) to calculate rRCE,eq and plugging the result into Eq. (16) for rt,eq.

Once given rt,eq, Qlat as a proportional function of rt is calculated through Eqs. (15), (16), and (8), or directly by Eq. (13b). The radial velocity ut is given by Eq. (10); the local tangential wind acceleration ∂υ/∂t (at r = rt) is given by Eq. (3).

In section 3, we examine the basic properties of the analytical solution for the evolution of storm size as well as the underlying physical processes of the model. In section 4, we use numerical simulations to test the model predictions of both expansion rate (section 4a) and rt,eq (section 4b). Section 5 provides detailed tests of the expansion model against numerical simulations in terms of simplifying assumptions and prediction of intermediate variables. Section 6a explores the physical meaning of rt,eq. Section 6b examines the model’s representation of the dependence of the inflow velocity, the local tangential wind spinup, and the size-expansion rate on the latent heating rate. Section 6c discusses the sensitivity of the model to Δsd and other parameters.

3. Behavior of theoretical model

a. Parameter settings in baseline environment

We next discuss the basic behavior of the expansion model solution in an idealized baseline environment. We take υt = 8 m s−1 as our outer-size wind speed, and hence we use r8 in lieu of rt at times. We define an idealized baseline environment for analysis with parameter values representative of tropical cyclones in the present-day tropical atmosphere. We set as constants: f = 5 × 10−5 s−1; Qcool = 1 K day−1; hw = 2.5 km (depth that captures the majority of the lateral inflow mass flux); ρi = 1.1 kg m−3; Cd = Ck = 0.0015; μ = 0.92 (to match CM1 simulations, where a surface gustiness has been added, described in appendix C); Lυ = 2.501 × 106 J kg−1; ϵp,ew = 1, αp = 0.8 (indicating that the eyewall dominates the net latent heating in a TC, estimated from simulations in appendix C); and σ = 0.7. VCarnot [Eq. (29)] is defined with TSST = 300 K, Ttpp = 200 K, and environmental surface pressure ps = 1015 hPa, which collectively yields VCarnot = 66 m s−1. These values correspond to the Control (CTL) simulation. We further set qυb=qυs*=0.022kgkg−1, with qυs* being the environmental saturation mixing ratio of the surface air temperature.10 The parameter Δsd is set to Δsd=Lυqvs*/Ts=187.1JK−1Kg−1, and we set Te,rad = Te,lat = Ts without loss of generality.11 Tropopause pressure pt is set to 100 hPa [forQrad, Eq. (9), which gives 89.3 W m−2]. These settings yield A = 0.16, B = 0.0007 m s−1, and ξ = 35 105 s2 m−1, and we set ξ0 ≡ ξ throughout, except in section 6c. Finally, we set wcool = 0.0027 m s−1, which will only be used to calculate rt,eq from (Mew/ρw)ew [Eq. (30)] in section 4b. Unless otherwise noted, these values are held constant throughout so that the use of the analytic model is in as simple a setup as possible.

These constants are complete for the fully predictive size-expansion model and will be used for tests below unless otherwise noted. In particular, we will at times arbitrarily set rt,eq for certain examinations in sections 3–6, which will also be noted.

b. Behavior of theoretical model

In this subsection, we examine the basic behavior of our TC size-evolution model and the underlying physical processes.

First, the basic evolution of size predicted from our model [Eq. (25)] is shown in Fig. 4. Two representative cases are shown from an initial size of rt0 = 250 km at the initial time t0 = 0 day: expansion toward a larger equilibrium size of rt,eq = 1200 km (blue curve) and shrinking toward a smaller equilibrium size of rt,eq = 100 km (red curve). For both cases, the model predicts a reasonable time scale of 10–20 days. The rate of expansion/shrinking vanishes as size approaches its equilibrium (rt,eq). The maximum expansion rate occurs during the first half of the expansion process at a radius rt,expmax of approximately 500 km [Eq. (24)].

Fig. 4.
Fig. 4.

Analytical solution of size evolution (rt, km vs t, day; solid lines) in Eq. (25) for two cases: rt expanding toward a larger rt,eq (blue) and rt shrinking toward a smaller rt,eq (red). Horizontal dashed lines mark rt,eq. Dots mark the initial condition t0 and rt0. The triangle marks the location (rt,expmax) of the maximum expansion rate.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

Next, we show the solution for expansion toward a range of equilibrium sizes (rt,eq). The size evolution rt(t), expansion rate drt/dt, and time scale τrt are shown in Fig. 5. The radius r8 increases with time and approaches rt,eq after day 20. Larger expansion rate corresponds to larger rt,eq (Fig. 5b). This is because the time scale τrt is the same across all experiments [Eqs. (12), (18), and (21), Fig. 5c], as this quantity is a function of size alone in this example. Time scale τrt monotonically decreases with size, with a first rapid decrease when r8 < 500 km and slowly decrease afterward (note τrt = 22.4, 13.2, 10.2, 7.9 days at r8 = 250, 500, 750, 1200 km, respectively, Fig. 5c). Physically, this is because the E04 model is flatter (smaller slope) for larger storm (longer tail) so from Eq. (18) τrt becomes smaller too. Note that τrt is greater than 15 days when the TC is small (r8 ≈ 400 km) and decreases below 10 days as size approaches rt,eq. The variation of τrt is determined by the variation of ∂υ/∂r at r8 [Eq. (18), Fig. 2). Correspondingly, the expansion rate peaks at tens of kilometers per day (Fig. 5b), a similar order of magnitude to that seen in observations (Schenkel et al. 2023). Finally, the radius of the maximum expansion rate increases with rt,eq, following Eq. (24).

Fig. 5.
Fig. 5.

Idealized expansion model prediction. (a) Time evolution of r8 (km; solid) with rt,eq (km; dotted); (b) dr8/dt (km day−1) as a function of r8 (km); (c) τrt (day) as a function of r8 (km); in (a)–(c), thicker and more opaque lines mark higher values of rt,eq. The dot marks the initial condition, and triangles mark the location of the maximum expansion rate.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

We next explore physically why the TC expands in the first place and why it eventually reaches equilibrium, following the conceptual diagram in Fig. 1. We examine the budget terms in the expansion equation [Eq. (4)] and the equation for the dependence of the inflow velocity on latent heating and radiative cooling [Eq. (10a)]. The five underlying processes/terms in the schematic (Fig. 1) are the latent heating per unit area Qlat/(πrt2), the radiative cooling per unit area Qrad/(πrt2), ut, turbulent friction −Cd(μυt)2/hw, and ∂υ/∂t at rt. Our model prediction of each of these terms is shown in Fig. 6.

Fig. 6.
Fig. 6.

As in Fig. 5, but for (a) latent heating per unit area Qlat/(πrt2) (W m−2; red) and radiative cooling per unit area −Qrad/(πrt2) (W m−2; blue), (b) radial velocity ut (m s−1), (c) −Cd(μυt)2/hw(m s−2), and (d) local spinup rate ∂υ/∂t (m s−2) at rt, The dots in (b) and (d) mark equilibrium.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

During the expansion stage, latent heating is significantly larger than radiative cooling. The latent heating rate can exceed 900 W m−2 when the TC is small (r8 ≈ 250 km), which is also supported by the simulations in appendix C with f = 5 × 10−5 s−1 (not shown). This amount of heating would induce ∼8 K day−1 temperature increase of the atmosphere (assuming constant pressure) without lateral energy/entropy exchange, whereas the actual average temperature change rate is on the order of 10−1 K day−1 in a TC (estimated from the CTL simulation with Ttpp = 200 K and f = 5 × 10−5 s−1 in appendix C). Thus, the overturning circulation is needed to export excess latent heating in the TC by exporting higher-entropy air aloft and importing lower-entropy air at low levels. The induced low-level inflow by this overturning circulation may be strong enough so that the local spinup at rt is achieved and TC starts to expand. Quantity ut linearly increases with rt [Eq. (10)] with an equilibrium value of about −0.65 m s−1 (Fig. 6b), which corresponds to zero local spinup. Friction at rt is a constant by design at a value of ∼−3.25 × 10−5 m s−2 (Fig. 6c). As a result, ∂υ/∂t at rt linearly decreases with size [Eq. (3)], such that an equilibrium is guaranteed.

It follows from Eq. (13) that larger rt,eq corresponds to larger Qlat (Fig. 6a), and thus, the quantity Qlat/(πrt2)∼1/rt. Meanwhile, the quantity Qrad/(πrt2) is a constant ∼89 W m−2. Thus, there exists a TC size at which there is zero net heating in the TC, and thus the expansion rate must vanish before this radius is reached. At equilibrium itself, the area-integrated latent heating inside of rt,eq still slightly exceeds radiative cooling because the nonzero surface friction also exists [Eqs. (4) and (10)].

Finally, taking ∂υ/∂t (Fig. 6d) and ∂υ/∂r (Fig. 2) together, the expansion rate (Fig. 5c) peaks in the middle of expansion rather than the beginning because of the larger slope of the wind profile when the TC is small.

4. Comparison of theoretical prediction against simulations

a. Evaluating modeled expansion assuming known equilibrium size

In this subsection, we test our model’s prediction for the time-dependent evolution of size against numerical simulations for the case where rt,eq is known. To do this, we set rt,eq constant and equal to the ensemble-mean equilibrium size of the simulated TC in each experiment to evaluate how well the analytical expansion model solution [Eq. (25)] can capture the first-order structure of expansion and its variations across experiments. We define τrt using the parameters of the base environment in section 3a. Three sets of numerical simulations (see appendix C) are taken for comparison: one set varying the tropopause temperature Ttpp (TTPP), which modulates the potential intensity and VCarnot, one varying f Coriolis parameter (FCOR), and one varying the ExSST, which also modulates the potential intensity and VCarnot. CTL experiment is defined at Ttpp = 200 K, TSST = 300 K, and f = 5 × 10−5 s−1, corresponding to TCs on real Earth. In simulations, r8 is defined at 950 m of altitude.

1) Comparison with TTPP

The ensemble size evolutions across experiments for TTPP are shown in Fig. 7a (solid line and shading). The ensemble-mean time series of r8 (and other variables of interest) is calculated in the following manner in order to exclude the effect of possible different start times of expansion in different ensemble members. Time series of r8 (and other variables of interest) in each ensemble member is shifted in time so that day 0 is the first day when r8 exceeds an estimated value of half-equilibrium size, which is taken as half of the equilibrium (days 40–50) r8 of ensemble index 0. Cases with Ttpp = 174, 187 K are neglected (here and throughout, except in section 6b) because their size evolution is very similar to Ttpp = 163 K. It is seen that TC size increases with time for about 20 days and approaches an equilibrium, similar to the qualitative behavior of the ideal expansion model prediction in section 3a. We take the average of ensemble-mean r8 during the last ten days of ensemble-mean r8 time series in TTPP as rt,eq for our expansion model; in this manner, the expansion model only needs τrt, which is the same as in section 3a (Fig. 5b) as determined by the same parameters as in section 3a.

Fig. 7.
Fig. 7.

(a) Temporal evolution of a 120-h running averaged ensemble-mean r8 (km) in TTPP (solid; cases with Ttpp = 241, 227, 214, 200, 163 K are shown) and analytical prediction of the expansion model (dashed) taking rt,eq equal to equilibrium sizes of TTPP (see the text). The shading marks 1 standard deviation from ensemble mean. Dots mark the initial condition for the expansion model. (b) The corresponding expansion rate dr8/dt (km day−1) in TTPP (solid lines are ensemble mean of the 24-h expansion rate as a function of the 120-h running averaged ensemble-mean r8; shading marks one standard deviation) and in the expansion model (dashed). (c),(d) As in (a) and (b), but for FCOR; (e),(f) As in (a) and (b), but for ExSST. Warmer color means higher values of the variable being varied (see the legend).

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

To compare theory and simulation, we set rt0 and t0 in our expansion model [Eq. (25)] to be the first ensemble-mean r8 above rt,eq/2 and the corresponding time in simulations, respectively, for all cases. This approach was used to compare intensification theory against simulations in Ramsay et al. (2020). The analytical model predictions of size evolution [Eq. (25)] are shown in dashed lines (Fig. 7a). The corresponding expansion rate in TTPP and that predicted by the expansion model is shown in Fig. 7b. Overall, there is a very good match between TTPP and the expansion model prediction. Higher Ttpp corresponds to smaller equilibrium r8 and smaller peak expansion rate (Figs. 7a,b). The lone case that matches a bit less well is the 163 K case, which expands a bit more slowly than the theory predicts in the first half of expansion.

2) Comparison with FCOR

An equivalent comparison as in Fig. 7a is performed with FCOR (Fig. 7b). First, we note that equilibrium size in FCOR scales approximately with a 1/f scaling, with the time scale of expansion longer with lower f. Here, we define equilibrium size as the average ensemble-mean r8 during the last 20 days for f = 1.25 × 10−5 and f = 2.5 × 10−5 s−1, the last 10 days for f = 5 × 10−5 s−1, the 20-day period ending 10 days before the end of the ensemble time coordinate for f = 10 × 10−5 s−1, and the 5-day period ending 30 days before the end of the ensemble time coordinate for f = 15 × 10−5 s−1. An earlier period is chosen for high f cases to capture their peak sizes. These equilibrium sizes are used as rt,eq for the expansion model. Analogous to our analysis for TTPP in Fig. 7a, the expansion model only needs τrt then, which is determined by the same parameters in section 3a. The lone exception is that τrt depends on f, which is set to the corresponding value in FCOR in each experiment.

The analytical solution of size evolution [Eq. (25)] is then compared with FCOR (Fig. 7c). Integration constants rt0 and t0 are set in the same manner as Fig. 7a. The expansion rate in FCOR and that predicted by the expansion model is shown in Fig. 7d. Overall, the expansion model again compares reasonably well with the experiments in FCOR. The lone case that does not match as well is for f = 1.25 × 10−5 s−1, which expands more gradually than predicted by the theory. Otherwise, the expansion model does reasonably well for f = 2.5 × 10−5 s−1 (to a lesser extent for the early stage of expansion) and f = 5 × 10−5 s−1 (and larger f), which are the principal latitudes (10° and 20°N) of TC development on Earth.

3) Comparison with ExSST

An equivalent comparison as in Figs. 7a and 7b is performed with ExSST (Figs. 7c,d). It is first noted that in ExSST when TSST is higher, both the equilibrium size and expansion rates are higher (solid lines and shading of Fig. 7c). We define rt,eq as the average ensemble-mean r8 during the 10-day period ending 15 days before the end of the ensemble time coordinate for TSST = 286 K, the last 10 days for TSST = 293 and 300 K, and the 10-day period ending 10 days before the end of the ensemble time coordinate for TSST = 307 K. Earlier periods are chosen for two cases to capture their peak sizes. Unlike comparison with TTPP and FCOR, to compare the expansion model prediction with ExSST, more parameters need to be set according to different TSST. In particular, Δsd, Te,rad, Te,lat, and ρi are set corresponding to TSST,12 so that we have the corresponding B in Eq. (25).

The expansion model predictions of r8 evolution in comparison with ExSST are shown in dashed lines of Fig. 7e. The expansion model works nicely, except that the model struggles to fully capture the high expansion rate when TSST = 307 K, though the expansion model does qualitatively correct predictions (Fig. 7f). We note that the peak ensemble-mean expansion rate r8 in ExSST is ∼25, 35, 55, 95 km day−1 when TSST = 286, 293, 300, 307 K, respectively.

Overall, the expansion model prediction compares fairly well with the simulation experiments in Fig. 7. We conclude the following:

  1. Quantity τrt provides a reasonable time scale for expansion (10–15 days for 20°N).

  2. Quantity rt,eq can be assumed constant with respect to rt (here r8) given an environment defined by Vp (and VCarnot) and f (to a lesser extent for 5° and 10°N).

  3. The expansion model predicts a reasonable size evolution (and expansion rate) when Ttpp, TSST, and f change.

b. Prediction of equilibrium size

The above subsection indicates that the expansion model works reasonably well given a known value of rt,eq taken from the simulations. We next test how well the model can predict rt,eq from environmental parameters based on the parameterization of (Mew/ρw)eq [Eq. (30)] in section 2d. The quantity VCarnot in TTPP, FCOR, and ExSST is directly calculated with TSST, Ttpp, and surface pressure (about 1012 hPa in CTL) from the initial state.

The resulting predictions for rt,eq are compared with the ensemble-mean equilibrium r8 in TTPP (Fig. 8a), FCOR (Fig. 8b), and ExSST (Fig. 8c). The predictions for rt,eq reasonably follow the simulated values in TTPP, FCOR, and ExSST, with a closer match in TTPP and ExSST. Specifically, rt,eq increases with VCarnot and is proportional to 1/f, though the latter dependence is a bit weaker than a pure linear dependence on 1/f. Thus, rt,eq can in principle be estimated from environmental parameters.

Fig. 8.
Fig. 8.

(a) Ensemble-mean equilibrium sizes of TTPP simulations (solid) and the predicted rt,eq (dashed). (b) As in (a), but for FCOR. (c) As in (a), but for ExSST. Dots mark different cases in TTPP, FCOR, and ExSST. In (a)–(c), colors have the same meaning as Fig. 7.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

5. Evaluation of model foundation

a. Simplifying assumptions: constant Qlat/(2πrt) and constant rt,eq

The size-expansion model assumes constant rt,eq, which comes from the assumption of constant Qlat/(2πrt) [see Eq. (13)]. In this subsection, we first assume whether Qlat/(2πrt) and rt,eq are approximately constant in simulations.

1) Constant Qlat/(2πrt)

The diagnosed Qlat/(2πr8) in TTPP, FCOR, and ExSST is shown in Fig. 9 (solid lines and shading). It is seen that Qlat/(2πr8) generally increases with expansion, but an assumption of that being constant may not be considered unreasonable considering the relative variations of Qlat/(2πr8) and r8 itself. To evaluate how good the assumption of constant Qlat/(2πrt) is, we seek Qlat∝rta at r8 and see whether a is close to 1, compared to 2 (the Qrad scaling), with a obtained by linear regression. As a result, for TTPP, a ≈ 1.2 for Ttpp = 163, 200, 214, 227 K and a ≈ 1.1 for Ttpp = 241 K; for FCOR, a ≈ 1.4, 1.2, 1.2, 1.0, 1.2 for f = 1.25 × 10−5, 2.5 × 10−5, 5 × 10−5, 10 × 10−5, 15 × 10−5 1/s, respectively; for ExSST, a = 1.1, 1.2, 1.2, 1.1 for TSST = 286, 293, 300, 307 K, respectively. Values of a are all close to 1. Thus, the assumption of a constant Qlat/(2πrt) is verified for r8 in TTPP, FCOR (to a lesser extent for f = 1.25 × 10−5 s−1), and ExSST.

Fig. 9.
Fig. 9.

(a) The term Qlat/(2πr8)(108 W m−1) in TTPP (solid lines: ensemble mean; shaded: one standard deviation) and those correspondingly predicted by the expansion model (dashed; see the text for details). (b) As in (a), but for FCOR. (c) As in (a), but for ExSST. Colors have the same meaning as Fig. 7.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

In addition, also shown in Fig. 9 is the Qlat/(2πr8) predicted by Eq. (13b) given rt,eq as in section 4a. It is seen that the expansion model [Eq. (13b)] reasonably reproduces both the qualitative dependence of Qlat/(2πr8) on Ttpp, f, and TSST and values of Qlat/(2πr8) themselves as well.

2) Constant rt,eq

Though in the above analyses rt,eq has been treated as a time-independent constant, it can vary with time and size following Eq. (13b). The time-dependent rt,eq determined by Eq. (13b) using the ensemble mean of simulated Qlat/(2πr8) and all other parameters same as in section 3a (except that f is set to the corresponding value in FCOR and the same treatment of parameters for ExSST as in section 4a) are examined; the results are shown in Fig. 10. It is seen that the time-dependent rt,eq generally increases with expansion, and the magnitude of the increase can be as large as ∼60% in TTPP and ExSST. However, considering that the magnitude of size expansion of r8 itself can be 300% of its initial value in TTPP and ExSST, then the assumption of rt,eq being constant may be considered reasonable in TTPP and ExSST.

Fig. 10.
Fig. 10.

Time-dependent rt,eq (km) as a function of r8 (km) for (a) TTPP, (b) FCOR, and (c) ExSST. Colors in (a)–(c) have the same meaning as Fig. 7. See the text for details.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

In FCOR, this assumption holds reasonably for f = 5 × 10−5 s−1 and larger values but does not hold for f = 2.5 × 10−5 and f = 1.25 × 10−5 s−1, in which the time-dependent rt,eq even approaches zero and negative values at the beginning of the expansion. This may explain why the expansion rate is overestimated by the expansion model in the early stage of the f = 2.5 × 10−5 s−1 case and in the whole expansion period of the f = 1.25 × 10−5 s−1 case (Fig. 7b). Notably, negative rt,eq is obviously incorrect. This simply indicates that, when f is small, the expansion model misses some important process (likely eddy momentum flux in the tangential wind budget) other than latent heating in favor of TC expansion while overestimating latent heating. Additionally, it is reasonable that Eq. (13b) does not produce the exact equilibrium sizes in simulations because of the simplifications of the expansion model.

b. Intermediate model predictions

In this subsection, the intermediate variables predicted by the expansion model underlying the final predictions in section 4a are given, along with their comparison with simulations. The intermediate variables are ∂υ/∂r, ∂υ/∂t, ut at r8, and τrt.

For TTPP, ∂υ/∂r, ∂υ/∂t, and ut as well as those predicted by the expansion model are given in Fig. 11. It is seen that ∂υ/∂r at r8 predicted by the expansion model nicely matches TTPP, except that when r8 < 300 km, ∂υ/∂r decreases more rapidly in TTPP. There is no clear systematic dependence of ∂υ/∂r on Ttpp, consistent with the expansion model, in which ∂υ/∂r is a single curve. The quantities ∂υ/∂t and ut at r8 predicted by the expansion model also nicely match TTPP (Figs. 11b,c), which is systematically higher in magnitude with lower Ttpp and has similar slopes with respect to r8. This traces back to Eq. (10c), where ∂ut/∂rt = B/hw, which is a constant when Ttpp changes. Following Eq. (3), (∂/∂rt)(∂υ/∂t|υ=υt)=−f∂ut/∂rt; thus, given f a constant (approximated absolute vorticity), then (∂/∂rt)(∂υ/∂t|υ=υt) is a constant. Quantity τrt for TTPP is the same as in Fig. 5.

Fig. 11.
Fig. 11.

(a) The term ∂υ/∂r (s−1) at r8 in TTPP (solid lines: ensemble mean; shading: one standard deviation) and predicted by the expansion model (dashed) directly underlying the prediction in Fig. 7. Abscissa is r8 (km). (b) As in (a), but for ∂υ/∂t (m s−2) at r8. (c) As in (a), but for ut (m s−1) at r8. Warmer color means higher Ttpp (see the legend).

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

For FCOR, τrt, ∂υ/∂r, ∂υ/∂t, and ut predicted by the expansion model as well as simulated values (other than τrt) are given in Fig. 12.

Fig. 12.
Fig. 12.

(a) The τrt (day) for FCOR predicted by the expansion model directly underlying the prediction in Fig. 7. Abscissa is r8 (km). (b)–(d) As in Figs. 11a–c, but for FCOR. Warmer color means higher f (see the legend).

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

It is seen that τrt, which is proportional to 1/f and −∂υ/∂r at r8 [Eq. (18)], increases as f decreases. Changing f also changes ∂υ/∂r at r8 (Fig. 12b). The wind profile is steeper (larger slope) for larger f, which is also observed in FCOR. Hence, the observation that τrt increases monotonically as f decreases indicates that the effect of 1/f on τrt dominates that of (−∂υ/∂r). Note that the difference of τrt will vanish for very large rt as expected by the property of [−(∂υ/∂r)] in appendix B. The longer expansion time scale τrt with lower f in the expansion model is also consistent with the experiments in FCOR.

The expansion model also qualitatively correctly predicted the weaker dependence of ∂υ/∂t at r8 on r8 when f is smaller (Fig. 12c) and the identical dependence of ut at r8 on r8 when f varies (Fig. 12d). This also traces back to Eq. (10c). As B does not depend on f, thus ∂ut/∂rt is a constant; then, (∂/∂rt)(∂υ/∂t|r=rt) will be smaller when f is smaller [Eq. (3)]. A smaller magnitude of ut in FCOR than predicted when f = 2.5 × 10−5 and f = 1.25 × 10−5 s−1 is consistent with the weaker than predicted latent heating rate in these cases (Fig. 9).

The same analysis is repeated for ExSST (Fig. 13). First, it is seen that τrt increases with TSST. From Eq. (18), τrt is proportional to 1/B and [−(∂υ/∂r)] at rt. From Eq. (11b), it follows that 1/B closely follows the C–C scaling when TSST varies. However, from Eq. (21), it is inferred that [−(∂υ/∂r)] should decrease, but with a speed more slowly than the inverse C–C scaling, with TSST, because 1/B appears in both numerator and denominator. This explains that τrt increased with increased TSST. Correspondingly, the expansion model correctly predicts the decrease of the magnitude of ∂υ/∂r at r8 when TSST increases (Fig. 13b). The prediction is also quantitatively reasonable.

Fig. 13.
Fig. 13.

(a) The τrt (day) for ExSST predicted by the expansion model directly underlying the prediction in Fig. 7. Abscissa is r8 (km). (b)–(d) As in Figs. 11a–c, but for ExSST. Warmer color means higher TSST (see the legend).

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

For ∂υ/∂t at r8 in ExSST (Fig. 13c), it is seen to start from similar values when r8 is small but decreases more slowly with r8 when TSST is larger, which is seen qualitatively predicted by the expansion model. This is in contrast with TTPP, where (∂/∂rt)(∂υ/∂t) is approximately the same (Fig. 11). Qualitatively speaking, Qlat/(2πr8) and Δsd both (roughly for the former) follow the C–C scaling, and then, the first term of the RHS of Eq. (10) should not change much with TSST. This indicates that when rt is small, ut and thus ∂υ/∂t should be approximately the same when TSST varies. Similar to the analysis for TTPP and FCOR, ∂ut/∂rt = B/hw and B decreases with TSST following the inverse C–C scaling; thus, assuming absolute vorticity roughly the same (compared to the C–C scaling), then ∂υ/∂t at r8 decreases more slowly with r8 when TSST is larger. For ut, it is first noted that the predicted ut starting from similar values and decreasing more slowly with r8 with increasing TSST is supported by ExSST to some extent. However, it is noticed that the equilibrium values of ut increase with TSST, whereas they are identical as predicted by the expansion model. This may be explained by the relative vorticity at r8, which increases with TSST (it can be inferred from Fig. 13b), with the same f. Thus, from Eq. (2), the equilibrium ut should increase with TSST. Such behavior is not captured by the expansion model because the relative vorticity is neglected (assumed much smaller than f) in Eq. (3).

Finally, comparing ∂υ/∂t and ∂υ/∂r at r8, it is concluded that the dependence of ∂υ/∂r on TSST contributes considerably to the faster peak expansion rate with higher TSST in ExSST (Fig. 7).

6. Further physical interpretation of the model

a. Physical meaning of rt,eq

Equation (13a) provides an expression for equilibrium size, which is derived independently of TC PS (Wang et al. 2022) but shares many similar properties. In this subsection, we further quantitatively discuss the physical meaning of rt,eq.

In essence, rt,eq principally depends on (or is reflected by) latent heating rate, radiative cooling rate, Δsd, and f [Eq. (13b)].13 All else being equal, rt,eq scales directly with Qlat/(2πrt) [Eq. (13b)], and hence, a larger equilibrium size should be associated with a larger Qlat/(2πrt) throughout the expansion. This is evident in the TTPP experiments: Qlat/(2πr8) is indeed systematically higher with lower Ttpp, corresponding to larger equilibrium sizes (Fig. 9a, solid lines), which is also correctly predicted by the expansion model, demonstrating that the larger latent heating rate during the expansion leads to larger equilibrium size. For TTPP, the enhanced latent heating rate arises principally because of the enhanced overturning mass flux [Eq. (8)], which is larger at higher potential intensity [Eq. (30)]. And this also leads to a higher expansion rate of TCs with lower Ttpp, consistent with the analysis in section 3a.

Similar dependencies are evident when f is varied. Since rt,eq scales as 1/f [Eqs. (15), (16), and (30)], Qlat/(2πrt) should also scale with 1/f [Eq. (13b)]. This is found to be qualitatively true in both FCOR (Fig. 9b, solid lines) and the expansion model prediction (Fig. 9b, dashed lines). The f = 1.25 × 10−5 s−1 simulation case deviates more strongly in that this quantity is substantially smaller than the model predicted value during the main expansion stage when f = 1.25 × 10−5 s−1; this is consistent with its deviation from the model prediction of size itself discussed in section 3a.

When TSST increases, Δsd increases. As a result, Qlat needs to increase (roughly following the C–C scaling) to achieve an increase of rt,eq in ExSST (Fig. 9c). As Δsd also varies with TSST, rt,eq cannot be inferred from Qlat alone when TSST is changing.

To summarize, in the expansion model, equilibrium size is effectively modulated by Qlat/(2πrt), and the expansion rate is modulated by the equilibrium size. For a given initial size and environment, a larger Qlat/(2πrt) translates to a larger rt,eq and thus a larger expansion rate. In practice, if a TC moves to a more favorable environment for convection, then its expansion rate would be expected to increase and its expected equilibrium size also increases, as quantitatively described by Eq. (13b). This interpretation is consistent with the behavior of observed storms, which have been found to expand when convection is enhanced outside of the storm inner core across a variety of distinct forcing mechanisms (Maclay et al. 2008).

The dependence of rt,eq on the latent heating rate [Qlat/(2πrt)] is complementary to the TC PS model, with the former being a reflection of the volume (or mass)-integrated processes of the system instead of a single parcel’s cycle. As noted above, the larger latent heating rate produces a larger rt,eq (with the same f and TSST) as it is associated with a larger Vp, consistent with VCarnot/f scaling (Wang et al. 2022).14 Meanwhile, the effect of Cd in the size scaling is rather small because an increase of Cd reduces the storm intensity but increases the inflow angle and the inflow depth under the eyewall at the same time, such that Mew remains relatively constant.

b. Expansion mechanism

The model [Eq. (12)] physically assumes that TC size expansion is driven principally by latent heating, which drives the low-level lateral inflow that imports absolute vorticity to expand the storm. In this section, we test how the model predicts the dependencies of the inflow velocity ut, the local spinup rate ∂υ/∂t, and the expansion rate drt/dt on the latent heating rate in a given TSST and how they are compared with simulations. In this subsection, TTPP will be used as a demonstration as TTPP provides the cleanest test; FCOR will be supplementary; ExSST does not serve as a material for the test because Qlat and Δsd, which have compensating effects, are both changing.

While the final version of the model is predictive based on the environmental parameters alone, the model also provides a quantitative dependence of a response of expansion rate to a change of latent heating Qlat in a given environment. Equations (1), (3), and (10) indicate that all else being equal, an increase of latent heating leads to an increase in the lateral inflow magnitude and thus the expansion rate. This understanding is useful when a TC experiences an inner-core structural variation such as the secondary eyewall formation (e.g., Kossin and Sitkowski 2009), which may lead to a size expansion.

Manipulating Eq. (10) gives that for a given rt, ∂ut/∂(Qlat/πrt2)∝rt. It follows that [∂/∂(Qlat/πrt2)](∂υ/∂t)∝frt and [∂/∂(Qlat/πrt2)](drt/dt)∝[−(∂υ/∂r)]−1frt at that given rt. The linear relations are compared with TTPP, shown in Fig. 14. The r8 is first taken as 550 km, and data in TTPP are collected with r8 from 500 to 600 km (Figs. 14a–c). This size corresponds to the relatively large (above median ∼400 km; Schenkel et al. 2023) TCs on Earth. We see an overall nice match of both the slope (sensitivity) and the absolute value between the expansion model prediction (baseline environment setting) and TTPP for all of ut, ∂υ/∂t, and dr8/dt. TCs with lower Ttpp in TTPP are associated with larger latent heating rate, which leads to stronger inflow velocity, local spinup rate, and expansion rate, consistent with the evolution of r8 in Fig. 7. This also supports the overall hypothesis of the expansion model that latent heating drives expansion. Note that the expansion rate is rather sensitive to latent heating. For Qlat/(πr82)=500 Wm−2 in Fig. 14c (dr8/dt ≈ 40 km day−1; see the dotted fitting line), a 20% change of latent heating may either double the expansion rate [dr8/dt ≈ 75 km day−1 with Qlat/(πr82)=600 Wm−2] or terminate expansion [with Qlat/(πr82)=400Wm−2].

Fig. 14.
Fig. 14.

(a) Colored dots show ut (m s−1) and Qlat/(πr82) (W m−2) when r8 is above 500 km and below 600 km in TTPP (all ensemble members); warmer color means higher Ttpp. The black line shows the expansion model prediction (baseline environment setting) with r8 = 550 km, and black dots mark cases in Figs. 5 and 6. The dotted line is a linear regression fitting to the centroids of each cluster (each Ttpp value). (b) As in (a), but for ∂υ/∂t (m s−2) at r8. (c) As in (a), but for dr8/dt (km day−1), shown as the 24-h expansion rate of the 120-h running averaged r8. (d)–(f) As in (a)–(c), but with r8 = 350 km in the expansion model and r8 from 300 to 400 km in TTPP. Note all cases in TTPP are shown.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

The same analysis is repeated but with r8 = 350 km in the expansion model and r8 from 300 to 400 km in TTPP (Figs. 14d–f). This size corresponds to relatively small (below median) TCs on Earth. The predictions of ut and ∂υ/∂t as linear functions of Qlat/(πr82) are still generally valid, except that the discrepancy for dr8/dt appears larger compared to r8 = 550 km (Figs. 14a–c). This discrepancy likely arises from a bias in representing the tangential wind budget.

The dependence of ut on the latent heating rate can also be tested against FCOR, as (also at constant rt) ∂ut/∂[Qlat/(πrt2)] does not depend on f {but ∂(∂υ/∂t)/∂[Qlat/(πrt2)] and ∂(dr8/dt)/∂[Qlat/(πrt2)] depend on f}. Analogous to the comparison for TTPP in Fig. 14, the comparison is shown in Fig. 15. The expansion model matches the simulations in both the absolute value and the slope of the relations reasonably well. Notably, the latent heating rate is substantially larger with smaller f, consistent with Fig. 9b.

Fig. 15.
Fig. 15.

(a) Colored dots show ut (m s−1) and Qlat/(πr82)(Wm−2) when r8 is above 500 km and below 600 km in FCOR (all ensemble members); warmer color means higher f. The black line shows the expansion model prediction with r8 = 550 km, and black dots mark cases in Figs. 5 and 6. The dotted line is a linear regression fitting to the centroids of each cluster (each f value). (b) As in (a), but with r8 above 300 km and below 400 km for FCOR and r8 = 350 km for the expansion model.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

Physically, the correspondence of high Qlat to low Ttpp and low f may be explained by the corresponding high frictionally induced eyewall updraft mass flux. For low Ttpp, it is the high intensity that is likely most responsible (Xi et al. 2023); for low f, it is the low boundary layer inertial stability that is likely most responsible (Smith et al. 2015; Li et al. 2023).

Overall, then, the expansion model can predict the dependence of ut, ∂υ/∂t, and dr8/dt on the latent heating rate, especially for ut. This provides experimental evidence that the lateral inflow velocity and its resulting spinup are indeed driven principally by latent heating in the TC, with their quantitative dependence described by Eq. (10).

c. Model parameters

1) Sensitivity to Δsd

In this subsection, we test the sensitivity of the expansion model (section 2) to Δsd, an important parameter that modulates both rt,eq [Eq. (16)] and τrt [Eq. (18)] through B. Increasing Δsd decreases B, which decreases rt,eq and increases τrt, which both cause the expansion rate to decrease [Eq. (12)].

Physically, the overturning acts to export entropy because the inflow brings in lower entropy air while the outflow takes out higher entropy air. This depends on Δsd. If Δsd is larger, then the overturning circulation can be less intense to achieve the same net export, which means smaller magnitude of the inflow velocity and hence slower expansion. The expansion model works reasonably partly because in a mature TC, the main portion of the inflow mass flux is confined to low levels. Hence, overturning circulations are very efficient at exporting entropy; this behavior can be characterized as having a strongly positive (i.e., stable) gross moist stability (Raymond et al. 2009).

To perform the sensitivity test, we vary Δsd about the base value (187.1 J K−1 kg−1) in section 3 by multiplying it by 0.5, 0.75, 1.0, 1.25, and 1.5. In the calculation, ∂υ/∂r is not modified (and hence ξ0 remains fixed to the value in section 3, though A, B, and ξ vary with Δsd) for simplicity. The quantity rt,eq is determined by the (Mew/ρw)eq [Eq. (30)] in the base environment through Eqs. (15) and (16).

The results are shown in Fig. 16. When Δsd varies from 50% to 150% of its base value, rt,eq decreases from 2000 to 1300 km; the quantity τrt increases by a factor of 3. The expansion rate decreases from about 250 km day−1 to below 50 km day−1. Though the overall values of the expansion rate rt,eq and τrt are still reasonable, it is evident that the expansion process can be directly modulated by Δsd. This implies that in a warmer climate, a likely increase of Δsd would partially offset the effect of an increase of the latent heating rate that drives faster expansion (this is already reflected by an increase of τrt with higher TSST in Fig. 13). More realistic simulations of TC evolution in a warmer climate are an important avenue of future work.

Fig. 16.
Fig. 16.

As in Fig. 5, but with Δsd varied from 50% to 150% of its base value with an interval of 25%; see the text for details. In (a)–(c), thicker and more opaque lines mark higher values of Δsd.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

2) Sensitivity to other parameters

In this subsection, we provide a discussion of model sensitivity to other parameters. The expansion rate is proportional to (rt,eq–rt) and 1/τrt [Eq. (12)]. The quantity τrt is proportional to [−(∂υ/∂r)] at rt, hw, 1/f, and 1/B. The magnitude of [−(∂υ/∂r)] decreases with ξ0 (=ξ) (see appendix B), with a specific example given in Fig. 2. The quantity hw = 2.5 km is considered a fixed value and should not vary because it is tightly linked to the value of Δsd (see appendix D). Parameter B is inversely proportional to Δsd, which follows the C–C scaling.

If rt,eq needs to be predicted solely from environmental parameters, then Eq. (16b) indicates that rt,eq increases with rRCE,eq and decreases with ξ. The quantity rRCE,eq depends on A/B and (Mew/ρw)eq, and Eq. (15) yields that ∂ln⁡ rRCE,eq/∂ln⁡A=1/2 and ∂ln⁡ rRCE,eq/∂ln⁡B=−1/2; the quantity (Mew/ρw)eq is proportional to VCarnot2/f2 [Eq. (28)]. Figure 16 shows how rt,eq depends on ξ. Figure 8 provides examples of how environmentally predicted rt,eq depends on Ttpp, TSST, and f.

7. Summary and discussion

In this paper, a predictive analytic model for the tropical cyclone size expansion on the f plane is proposed and its overall behavior is tested against numerical simulations varying tropopause temperature (TTPP), Coriolis parameter (FCOR), and sea surface temperature TSST (ExSST). The expansion rate is described by a simple kinematic relation and is equal to the ratio of the local spinup rate of the tangential wind (∂υ/∂t) at outer radius rt to the negative slope of the wind profile at that radius. The model predicts that size expansion is driven by latent heating (dominated by the eyewall) and suppressed by radiative cooling. This prediction is achieved by combining the tangential velocity budget at rt with the volume-integrated entropy (heat) budget inside of rt, with the two linked via a simple relationship between outer storm size and the upward mass flux of the overturning circulation. Area-integrated latent heating is proportional to rt, while area-integrated radiative cooling is proportional to rt2, such that the storm size eventually reaches an equilibrium (rt,eq). The size-expansion rate is the ratio of the difference between rt,eq and the present size to a time scale τrt, and both parameters may be defined from environmental parameters alone. Key takeaways are as follows:

  • The model yields a predictive, analytic solution for the evolution of storm size toward its equilibrium size (both larger or smaller than the present size) given the environmental parameters and an external set rt,eq. This solution performs well in predicting the simulated size evolutions and expansion rates in simulations across a range of values of tropopause temperature, sea surface temperature (and hence potential intensity), and Coriolis parameter.

  • The model successfully produces a characteristic expansion rate for r8 of tens of kilometers per day with reasonable environmental parameters (Fig. 5), in line with past work using data for historical storms.

  • The model predicts that the local spinup rate decreases quasi-linearly with expansion and that ∂υ/∂r at r8 decreases in magnitude with expansion, both consistent with simulations. The results taken together explain why the expansion rate peaks during the early-to-mid stages of expansion rather than at the beginning, as seen in simulations.

  • The model predicts the reasonable time scale τrt of 10–15 days when f = 5 × 10−5 s−1, TSST = 300 K, and the radiative cooling rate 1 K day−1. The time scale is constant when varying Ttpp and larger when f is smaller, consistent with simulations.

  • The model predicts that the equilibrium size rt,eq increases withwcoolCd−0.07VCarnot/f, which is a scaling for the square root of equilibrium eyewall updraft mass flux and r0,eq. The quantity rt,eq is directly linked to the latent heating rate within the TC volume and is complementary to the TC potential size of Wang et al. (2022) while providing clearer mechanistic insight into the process of expansion.

  • The model predicts that enhanced latent heating will cause the storm to expand faster (in a given environment), consistent with past observational work, finding that storms tend to expand significantly after convection is forced outside of the eyewall regardless of the forcing mechanisms (Maclay et al. 2008).

  • The model provides a mechanistic understanding of why TC size expands toward an equilibrium in the first place: The overturning circulation exports excess latent heating in the TC when it is sufficiently small, and the resulting induced low-level inflow that imports absolute vorticity into the volume in excess to surface friction spins up the tangential wind in the outer circulation. As a result, the TC expands.

Here, we have taken a number of parameters as constant across our tests for simplicity to arrive at an analytically tractable model that appears to capture the first-order behavior of the dynamics of TC size. In reality, such parameters may not be perfectly constant (both in time and across experiments), a topic that could be more carefully examined within the simulations presented in this paper. Here, we have evaluated this assumption for a few key parameters, but additional detailed tests of the model assumptions as well as quantifying model parameters in simulations could be tackled in a future study.

This model offers a valuable foundation for better understanding and predicting changes in storm size on Earth. For example, it is known that TCs will shrink in the absence of convection with nonzero β (meridional gradient of f) due to radiation of planetary Rossby waves (Chavas and Reed 2019; Lu and Chavas 2022). How this shrinking effect alters the evolution of TC size on the f plane is an important question for predicting changes in TC size in nature. Incorporation of the β effect in the expansion model is a valuable avenue for future work.

One key assumption of the present model is the existence of an equilibrium TC outer size rt,eq that depends on environmental parameters on the f plane. Though a short-term equilibrium of size does exist in our simulations, a long-term existence/maintenance of equilibrium size is not conclusive in literature: Persing et al. (2019) pointed out that long-term (longer than tens of days) maintenance in limited domains may require an artificial source of relative angular momentum. However, in the present expansion model, rt,eq is valid and well defined at any particular instance and does not require that this radius remains constant for all time (it is able to maintain for about 10 days or potentially longer in our simulations). Nonetheless, an understanding of rt,eq in the context of long-term TC maintenance in limited domains remains an open question.

One open mechanistic question that we did not analyze in detail is the connection between potential intensity and updraft mass flux. In our simulations, the expansion rate increases with Vp (by lowering Ttpp) consistent with our theory, and the majority of the condensational heating within r8 does occur within the eyewall (appendix D). Hence, it is likely that Vp modulates the strength of overturning circulation in the eyewall by modulating TC intensity, which in turn modulates boundary layer frictional convergence. However, we do not explicitly quantify the eyewall dynamics in the present model.15 This also simplifies the problem a bit because potential intensity theory exists to describe the eyewall structure and successfully predicts a characteristic maximum wind speed, whereas the rainband activity is much less well understood and is much harder to predict. Correspondingly, although the rainband activity has been reported to effectively modulate TC size expansion, it is not actively predicted in the present model and is only parameterized by a constant αp.

Finally, we note that the proposed expansion model predicts that the expansion rate is a function of present size [Eq. (23)], and hence, it is independent of the preceding history of storm size including initial vortex size. This is supported by an additional set of experiments varying initial vortex size and intensity (see the online supplemental material), suggesting that Vp (VCarnot) may be a more effective factor than the initial vortex structure to modulate the size-expansion rate.

1

For TC size, one may consider a single outer wind radius because the wind field structure is fully specified from a single input size (Chavas et al. 2015).

2

Technically, rt is understood as rt = r(υt, t, ϑ), where ϑ represents a series of environmental parameters and υt is a time-independent tangential velocity. Since the main focus for the expansion rate is with a fixed υt in a given environment (fixed ϑ), we write drt/dt instead of ∂rt/∂t.

3

A diagnostics of ensemble simulations of the CTL experiment (appendix C) shows that this assumption is generally reasonable except that it deviates more from simulations at the beginning stage of the expansion process, suggesting a potentially lower predictive capability of the expansion model at the beginning of stage of the TC size expansion.

4

A dry static energy budget is also viable, and the effective temperatures will not appear so that sensible and latent heat need not be separated. However, we use dry-entropy budget because it is more tractable for comparison with numerical simulations.

5

This specific assumption is common but is not critical to the present theory since it is a storm-integrated theory.

6

Actually, rRCE,eq would be equivalent to r0,eq, the equilibrium radius of vanishing wind, if lim⁡υt→0ξυt2=0. A close relation between rRCE,eq and r0,eq is indeed seen in numerical experiments (not shown).

7

Diagnosed qυb from the CTL simulation (appendix C) increases ∼15% during expansion (not shown), but this size dependence is secondary because the expansion model eventually depends on equilibrium size rt,eq. Note also that the qυb increase is not explained by a corresponding surface pressure drop, which is only ∼2.5%.

8

Fitting parameter σ accounts for dln⁡ξ/dln⁡υt=−dln⁡(ζ+f)/dln⁡υt+dln⁡(Cdμ2/B)/dln⁡υt=−dln⁡ζa/dln⁡υt+dln⁡(Cdμ2/B)/dln⁡υt=2(σ−1). Note f is originally ζa in Eq. (2); if ζa ≈ f were not applied, the derivation up to Eq. (18) will be the same except replacing f by ζa. For ∂υ/∂r, ζa cannot be approximated by f as in Eq. (3); ζ must be retained here for a proper understanding. Qualitatively, ζa increasing with υ is generally supported by the E04 solution. Quantitatively, d lnζa/d lnυt at r8 is found in the E04 solution to generally increase from 0.2 to 0.4 with decreasing r8 when r8 < 1000 km and about 0.1 when r8 > 2000 km (not shown). This translates to σ ranging from 0.8 to 0.95. The deviation from 0.7 should be accounted for by the assumptions made in Eq. (3) and by d ln(Cdμ2/B)/d lnυt, which the present model cannot predict.

9

The corresponding B is 0.0016, 0.0011, 0.0007, 0.0005 m s−1, about half of (1/2)wcool. This quantitative difference by itself does not indicate that Eq. (21) is wrong; rather, it raises a question whether the E04 model is correct as a result of compensating errors. As shown in appendix A, the E04 model is a mass-balance derivation in the same framework as for Eq. (21), which is an energy (entropy)-balance derivation. As both are physical, the error should arise from simplifying assumptions. For the E04 model, the uncertainty seems to be whether the actual subsidence velocity is indeed a radially constant value given by wcool. However, an analysis toward a more complete mechanistic understanding of ∂υ/∂r is out of the scope of this study.

10

This can be a ∼20% overestimate of qυb as it does not account for the vertical profile of boundary layer qυ. This would lead to a ∼200-km overestimate of rt,eq in section 6c. Taking a vertical average from surface to 2 km of altitude (approximately the inflow depth associated with Mew) of the analytical saturation mixing ratio qυ* profile of Romps [2016, his Eqs. (8) and (11)] appears to resolve this issue, though here we keep it simple and not adopting the Romps (2016) model.

11

In simulations with TSST = 300 K, Te,lat and Te,rad are about 275 K, though here we avoid specifying these values based on simulations for simplicity. Our approach yields Te,latΔsd = Te,radΔsd = Lυqυb, i.e., a characteristic difference of the dry static energy between tropopause and the surface. This ∼10% overestimation Te,lat and Te,rad would induce a ∼10% overestimation of ξ and a ∼55-km underestimate of rt,eq in section 6c.

12

Quantity ρi is simply set proportional toTSST−1.

13

Quantity Qlat/rt is completely determined by rt,eq through Eq. (13b). Thus, the present model does not allow Qlat to deviate from its “expected” value, though it can happen in nature.

14

When TSST changes, Eq. (13b) can only predict the minimum increase of Qlat to maintain the same rt,eq. In this sense, the TC PS model is more powerful.

15

Thus, we do expect Vp to represent a characteristic intensity of the TC (see the supplemental material), though the exact relation between Vp and maximum wind speed is not considered here (see the discussion of superintensity in Persing and Montgomery 2003; Li et al. 2020).

16

Quantity wcool was incorrectly calculated to be about half of the correct value in Wang et al. (2022).

Acknowledgments.

We thank four anonymous reviewers for their constructive comments that help greatly enhance the manuscript. This work is supported by the National Science Foundation (NSF) AGS Grant 1945113.

Data availability statement.

Description of the 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 https://doi/org/10.6084/m9.figshare.22674361.

APPENDIX A

E04 Model and Its Relation to Expansion Model

The E04 model provides the near-surface wind profile through a slab boundary layer model in the subsidence region where the net vertical velocity is typically negative and is expressed as
∂υ∂r=2Cdrυ2wcool(r02−r2)−f−υr,
where wcool is a constant clear-air subsidence velocity (positive downward) induced by radiative cooling. In the model, ∂υ/∂r is determined locally by the inflow mass flux and friction and the inflow mass flux is determined by the accumulated subsidence mass flux outward.
Rearranging Eq. (A1), we have ζa=ζ+f=2Cdrυ2/[wcool(r02−r2)]. Solving for r gives
r=−ξE04υ2+ξE042υ4+4r022,
with
ξE04=2Cdwcoolζa.
It is seen that Eq. (16b) has the same form as Eq. (A2), with ξ playing the same role as ξE04, B playing the same role as(1/2)wcool, f playing the same role as ζa, and rRCE,eq playing the same role as r0. In addition, B, having the dimension of velocity, is already in a form of Qcool divided by stability; thus, B is also physically consistent with (1/2)wcool. Taking the derivative of r with respect to υ in Eq. (A2) will give the same form as Eq. (20).
Below, we show that the E04 model can show up in the same framework as in section 2, with a mass balance derivation. Considering the approximate mass balance in a cylinder from the TC center to rt and from the surface to hw:
1αpMew−πrt2wcoolρw=−2πrtρiuthw.
Rearrangement gives
−ut=1hw12π1αp(Mewρw)1rt−1hw12wcoolrt=1hwAmb(Mewρw)1rt−1hwBmbrt,
where
Amb=12παp,
Bmb=12wcool,
where the subscript “mb” means “mass balance.” It is noticed that Eq. (A5) has the equivalent form as Eq. (10c), with A and B replaced by Amb and Bmb, respectively. The following derivation for drt/dt will be the same as in section 2. In particular, the equivalent counterpart of ξ will be ξmb=2Cdμ2/(fwcool), which more closely resembles ξE04.

APPENDIX B

Further Discussion on ∂υ/∂r [Eq. (21)]

Equation (21) indicates that ∂υ/∂r is negative definite and it tends to −∞ when rt tends to 0; ∂υ/∂r tends to −1/(σξ0υt) when rt tends to + ∞. Specifically, Eq. (21) gives
∂∂ξ0(∂υ∂r|υ=υt)=1σξ02υt>0,
which indicates that the magnitude of ∂υ/∂r decreases with increased ξ0 and vice versa. Physically, when Cd is increased or f is decreased, the magnitude of ∂υ/∂r will decrease. Additionally, when 2rt≫ξ0υt2, then (∂υ/∂r)−1|υ=υt≈−2rtυtσξ0/2rt=−υtσξ0, giving a proportional dependence of ∂υ/∂r on fB/Cd. When 2rt≪ξ0υt2, then (∂υ/∂r)−1|υ=υt≈−2rtσ/υt, proportional to rt and independent of ξ0.

APPENDIX C

Experimental Design and Processing

Numerical experiments are performed with Cloud Model 1 (CM1; Bryan and Fritsch 2002), which is a nonhydrostatic model mainly designed for idealized simulations. The model configuration is essentially the same as Wang et al. (2022). The radiation is represented by applying a constant cooling rate Qcool of potential temperature θ where the temperature is above a prescribed tropopause temperature Ttpp. Where the temperature is lower than Ttpp, the θ is relaxed to the value corresponding to Ttpp with a time scale τ = 12 h. A surface gustiness usfc = 5 m s−1 is added to 10-m wind in the aerodynamic formula for surface drag and enthalpy fluxes. The Morrison double-moment microphysics scheme (Morrison et al. 2005) is used.

We set the environment in the CTL simulation as Ttpp = 200 K, TSST = 300 K, f = 5 × 10−5 s−1, Cd = Ck = 0.0015, and Qcool = 1 K day−1. In the first set of experiments (TTPP; Table C1), we vary the tropopause temperature to vary Vp. This method has an advantage to isolate structural changes in the TC owing to changes of Vp without substantially affecting the lower-tropospheric properties, such as wcool. To further test the role of Cd in modulating Mew, two more sets of experiments CD and CDTTPP are designed (Table C1). In CD, Cd is varied, while in CDTTPP, Cd and Ttpp are both varied in a manner that Vp does not change (assuming the same air–sea enthalpy disequilibrium). The parameter Ck is further varied in CK (Table C1) to test whether Mew is dominantly friction driven, as the boundary layer thermodynamic property is expected to change in CK. In the fifth set of experiments (FCOR, Table C1), we vary the Coriolis parameter f. In the sixth set of experiments (ExSST, Table C1), we vary TSST. Note that CD, CDTTPP, and CK will only be used for parameterizing (Mew/ρw)eq (section 2d), with their equilibrium periods all set to 40–50 days.

Table C1.

Parameters in experiments.
Table C1.

TCs are simulated using the axisymmetric configuration of CM1, and the base state of the atmosphere is generated by three-dimensional simulations in radiative–convective equilibrium without the background rotation, same as Wang et al. (2022). In FCOR, CD, and CK, the base state of the atmosphere is all the same as the case with Ttpp = 200 K in TTPP. TCs are simulated for 50 days in all simulations except 150 days for f = 1.25 × 10−5 s−1 and 100 days for f = 2.5 × 10−5 s−1 as the TCs with low f take longer to reach size equilibrium. The initial vortex for all experiments is the same and defined as in Rotunno and Emanuel (1987). The initial vortex maximum wind is about 13 m s−1 at a radius of about 100 km; see also the supplemental material. For each value of the parameter being varied in TTPP, FCOR, and ExSST, we perform four ensemble simulations. We denote ensemble-index-0 as the ensemble member where the initial sounding outside of the vortex is not perturbed. For the other three ensemble members, the initial state of potential temperature over the whole domain is randomly perturbed with a maximum amplitude of 2.5 K (“irandp” = 1 in CM1 namelist) based on that of ensemble-index-0.

The eyewall upward mass flux is approximated by the inflow under the eyewall:
Mew(t)=−2πrew∫0hρdudz,
where rew is some radius not far from the eyewall (here chosen as two times rumin, the radius of the minimum radial velocity in the boundary layer), u is the radial velocity, and h is the height of the inflow layer, taken as the height where the radial velocity u is greater than 0.1 times the minimum u at rew (=2rumin) following Zhang et al. (2011). The Mew is processed by a 120-h running average. And ρw is calculated such that −2πrewh_ρwuavg=Mew in Eq. (C1), where uavg is the vertical mean radial velocity of the inflow after 120-h running average and where h¯ is h after 120-h running average.
To evaluate Eqs. (1), (3), and (4), a spatial and temporal average is applied to remove noise in CM1 outputs:
[X]¯=1z2−z1∫z1z2[2(rt+Δr)2−(rt−Δr)2×∫rt−Δrrt+Δr(1P∫t−P/2t+P/2Xdt)rdr]dz.
This average [⋅]¯ will apply to each term of Eq. (3): ∂υ/∂t and ut in Figs. 11–13, ut in Figs. 14 and 15; Δr = 100 km, P = 120 h, z1 = 0, and z2 = hw = 2.5 km are set. Note that one exception is that when applied to ∂υ/∂r (Figs. 11–13): We take the central difference using two radii rt − Δr and rt + Δr of the 120-h temporal running average of υ, at a fixed height 950 m, for smoother results. As ∂υ/∂t and ∂υ/∂r are very noisy, they are further applied to a 24-h running average after applying [⋅]¯ in Figs. 11–13 for clearer visualization.

Quantity wcool is calculated in the same manner as Wang et al. (2022). A typical value of wcool is 0.0027 m s−1 in the simulations.16 Quantity Qlat from simulations (Figs. 9, 14, and 15) is also needed: Qlat is calculated as the 120-h running average of the mass integration (radially within rt) of cpΠθ˙pc, with θ being the potential temperature, Π being the Exner function, and θ˙pc being the potential temperature source due to phase changes (the tendency from the microphysics section provided by CM1 subtracted by the energy fallout term; see below). The height of the volume for the integration is defined as follows. First, the location of the maximum outflow velocity umax of 120-h running averaged fields is found. The height of the volume is defined as 1 km above the height where the outflow velocity is less than 0.1 umax.

APPENDIX D

Support for Eq. (6) and Some Parameter Diagnosis and Interpretation

First, we provide a support of the dry-entropy balance equation Eq. (6). Recall that we define sd=cpln⁡(θ/Ttrip). Each term in Eq. (5) may be given as follows [see θ budget in Bryan and Rotunno (2009) and CM1 governing equations in the CM1 homepage], defining volume integration∫υ=∫0zl∫0rt2πrdrdz:
∂S∂t=∫υ∂ρdsd∂t,
QlatTe,lat=∫υ{ρd[−cυcυm(LυTq˙gl+LfTq˙ls+LsTq˙gs)+(cυcυm−RRm)Rυ(q˙gl+q˙gs)]},
−QradTe,rad=∫υ(ρdcpθθ˙rad),
S˙res=∫υ−ρd(cυcυmRm−R)∇u+∫υ−cυcυm∇JT+∫υcυcυm1T(WT+ϵ),
Fr=∫0zl−2πrt(ρdusd)|r=rtdz,
Fu=∫0rt−2πr(ρdwsd)|z=zldr,
where cυ is the specific heat of dry air at constant volume; cυm = cυ + cvvqυ + clql + csqs, with cυυ, cl, and cs being the specific heat of water vapor, liquid water, and solid water at constant volume, respectively, and ql and qs are the mixing ratios of liquid water and solid water, respectively; R is the gas constant of dry air, Rm = R + qυRυ is the gas constant of moist air, with Rυ being the gas constant of water vapor; Lυ, Lf, and Ls are the latent heat of vaporization, freezing, and sublimation, respectively; q˙gl and q˙gs are the qυ source from phase changes between gas and liquid water and gas and solid water, respectively; q˙ls is the ql source from phase changes between liquid and solid water; θ˙rad is the θ source due to radiative cooling; vector u is the velocity; vector J is the sensible heat flux per unit area; WT is the heating/cooling rate per unit volume due to falling hydrometeors; and ϵ is the dissipative heating. In particular, WT = −cυυdυ∇T − cldl∇T − csds∇T + g ⋅ (dυ + dl + ds), where dυ, dl, and ds are the diffusion (fall out) fluxes of water vapor, liquid water, and solid water per unit area, respectively (see also appendix A of Romps 2008; Wang and Lin 2021), and g is the gravitational acceleration.

In practice, Eq. (D1) in a CM1 simulation is obtained by its automatic output of θ budget. Specifically, Qlat/Te,lat is obtained from the CM1 output “ptb_mp” (θ source from the microphysics section) subtracted by the effect of WT using the CM1 output of the terminal fall speed of different hydrometeors. Note, in particular, that ptb_mp itself can be a close approximation of θ source due to phase changes because a simple scale estimation gives that the dry-entropy source due to WT [Eq. (D1d)] is only a few percent of that due to phase changes [Eq. (D1b)], assuming ∂T/∂z = −7 K km−1 and a mean condensation height (for the vertical distance of falling) of 3 km (this estimation is supported by explicit diagnosis not shown).

Figures D1a and D1b show the support of the approximation given by Eq. (6) by representative cases of CTL and Ttpp = 163 K case in TTPP. It is evident that the local tendency ∂S/∂t, dry-entropy source S˙res, and vertical flux Fu are indeed negligible with dominant terms Qlat/Te,lat,Fr and −Qrad/Te,rad, which is important when TC is large. The crucial assumption/approximation that Qlat/Te,lat scales with rt and Qrad/Te,rad scales with rt2 is more clearly (than in the main text) supported by Figs. D1c–f for TTPP and FCOR (similar for ExSST, not shown).

Fig. D1.
Fig. D1.

(a),(b) Terms in the dry-entropy budget Eq. (D1) (J K−1 s−1) in CTL and the Ttpp = 163 K case in TTPP; legend shows (from top to bottom) terms Fr,Fu,Qlat/Te,lat,−(Qrad/Te,rad),S˙res,∂S/∂t as the sum of these terms and directly calculated (solid lines: ensemble mean; shading: one standard deviation). (c),(d) The equation [(Qlat/Te,lat)+S˙res+Fu]/r8 (J K−1 s−1 m−1) in TTPP and FCOR (solid lines: ensemble mean; shading: one standard deviation), respectively. Warmer color means higher values of Ttpp or f; dashed lines mark the expansion model predictions in section 4. (e),(f) As in (c) and (d), but for −(Qrad/Te,lat)/r8 (J K−1 s−1 m−1).

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

In section 3, we set αp = 0.8, ϵp,ew = 1; here, some support for this setting is shown in Fig. D2. Consistent with the definition in section 2a, αp is calculated as the ratio between the latent heating within two times the radius of maximum wind rm and Qlat. Two times of rm is to account for the slope of the eyewall, which could also include some inner rainband according to Wang (2009). It is seen that αp in TTPP and ExSST is approximately constant being about 0.7–0.8, suggesting the dominant contribution of latent heating in the eyewall to total latent heating within r8. In FCOR, however, it is seen that αp can be substantially smaller when f = 2.5 × 10−5 and f = 1.25 × 10−5 s−1, indicating that latent heating outside of the eyewall is important in driving the expansion of these cases. It is also noted that when f = 2.5 × 10−5 s−1, αp is still mainly above 0.6 when r8 is smaller than 1000 km, a radius more relevant to TCs on Earth.

Fig. D2.
Fig. D2.

(a)–(c) Diagnosed αp (solid lines: ensemble mean; shading: one standard deviation) as a function of r8 in TTPP, FCOR, and ExSST, respectively. (d)–(f) As in (a)–(c), but for ϵp,ew. Warmer colors mean higher values of the variable being varied. See the text for details.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

Here, for convenience, ϵp,ew is diagnosed/estimated as the ratio of 120-h running averaged precipitation to the 120-h running averaged sum of surface water vapor and lateral water (vapor and hydrometeors) fluxes within (at) two times of rm. Note ϵp,ew thus defined is similar to the large-scale precipitation efficiency in Sui et al. (2005). It is seen in Figs. D2d–f that ϵp,ew is about 1 during the whole expansion stage. This indicates a negligible local accumulation of water in the atmosphere, and nearly all water vapor input to the eyewall changes phase.

We also show Δsd diagnosed by Eq. (7) in simulations in Fig. D3. It is seen that the diagnosed Δsd is approximately constant with time in TTPP, FCOR, and ExSST. The diagnosed Δsd is very close in different experiments of TTPP and FCOR; the value is also not far from the 187.1 J K−1 kg−1 estimation in the ideal base environment in section 3. The diagnosed Δsd in ExSST also matches well with the expansion model, following the C–C scaling. This supports our assumption of a constant Δsd and directly supports the interpretation of Δsd. Our assumption that Δsd is mainly driven by sea surface temperature is verified. Note the diagnosed Δsd slightly increases with decreasing Ttpp; this also means the faster expansion rate with lower Ttpp is not caused by a Δsd sensitivity.

Fig. D3.
Fig. D3.

Diagnosed Δsd (J K−1 kg−1; solid lines: ensemble mean; shading: one standard deviation) in (a) TTPP, (b) FCOR, and (c) ExSST using Eq. (7). Colors have the same meaning as Fig. D2. Note in the calculation, the numerator and denominator of Eq. (7) are first processed by a 120-h running average. Dashed lines in (c) show the expansion model predicted Δsd=Lυqυs*/Ts, which are also used in section 4.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0088.1

The meaning of Δsd may be further understood in a more ideal picture where the inflow at rt occurs only from the surface to the height zi and the outflow only occurs from zi to zl and where zi≲zl so that the outflow is well confined to the tropopause level. To simplify the math, we also assume that the buoyancy frequency N is a constant so that ∂sd/∂z=(cp/g)N2 is a constant (not a bad approximation as seen in CTL; see the supplemental material). We denote sd at the surface to be sd0 so that sdi=sd0+(cp/g)N2zi is the sd at height zi; we denote sd,tpp the sd at height zl. Thus, Fr [Eq. (D1e)] is written as
Fr=∫0zi−2πrt(ρdusd)dz+∫zizl−2πrt(ρdusd)dz
The first term on the RHS is written as
∫0zi−2πrt(ρdusd)dz=−2πrt∫0ziρdu(sd0+cpgN2z)dz=sd0(−2πrt∫0ziρdudz)−2πrtcpgN2∫0ziρduzdz=sd0ψi+cpgN2∫0zi∂ψ∂zzdz=sd0ψi+cpgN2[(ψz)|0zi−∫0ziψdz]=sd0ψi+cpgN2ψizi−cpgN2∫0ziψdz=sdiψi−cpgN2∫0ziψdz,
where ψ is the mass streamfunction. The second term on the RHS of Eq. (D2) is written as
∫zizl−2πrt(ρdusd)dz=∫zizl∂ψ∂zsddz≈sdi(−ψi).
Thus, we have Fr=−(cp/g)N2∫0ziψdz=−(cp/g)N2ψ¯zi. Note also that 2πrtρiuthw=−ψhw. Then, Δsd [Eq. (7)] is
Δsd=Fr/(−ψhw)=cpgN2ziψ¯ψhw≈(sd,tpp−sd0)ψ¯ψhw.
Thus, we see that Δsd will represent the difference of sd between tropopause and the surface if ψhw is close to the vertical mean of ψ in the whole inflow layer. A structure with the inflow confined near the surface satisfies this condition, but other vertical profiles of the inflow can also be valid.

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  • Rotunno, R., and K. A. Emanuel, 1987: An air-sea interaction theory for tropical cyclones. Part II: Evolutionary study using a nonhydrostatic axisymmetric numerical model. J. Atmos. Sci., 44, 542–561, https://doi.org/10.1175/1520-0469(1987)044<0542:AAITFT>2.0.CO;2.

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  • Schenkel, B. A., N. Lin, D. Chavas, G. A. Vecchi, M. Oppenheimer, and A. Brammer, 2018: Lifetime evolution of outer tropical cyclone size and structure as diagnosed from reanalysis and climate model data. J. Climate, 31, 7985–8004, https://doi.org/10.1175/JCLI-D-17-0630.1.

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  • Schenkel, B. A., D. Chavas, N. Lin, T. Knutson, G. Vecchi, and A. Brammer, 2023: North Atlantic tropical cyclone outer size and structure remain unchanged by the late twenty-first century. J. Climate, 36, 359–382, https://doi.org/10.1175/JCLI-D-22-0066.1.

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  • Smith, R. K., G. Kilroy, and M. T. Montgomery, 2015: Why do model tropical cyclones intensify more rapidly at low latitudes?. J. Atmos. Sci., 72, 1783–1804, https://doi.org/10.1175/JAS-D-14-0044.1.

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  • Stern, D. P., G. H. Bryan, and S. D. Aberson, 2016: Extreme low-level updrafts and wind speeds measured by dropsondes in tropical cyclones. Mon. Wea. Rev., 144, 2177–2204, https://doi.org/10.1175/MWR-D-15-0313.1.

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  • Sui, C.-H., X. Li, M.-J. Yang, and H.-L. Huang, 2005: Estimation of oceanic precipitation efficiency in cloud models. J. Atmos. Sci., 62, 4358–4370, https://doi.org/10.1175/JAS3587.1.

    • Search Google Scholar
    • Export Citation

  • Tsuji, H., H. Itoh, and K. Nakajima, 2016: Mechanism governing the size change of tropical cyclone-like vortices. J. Meteor. Soc. Japan, 94, 219–236, https://doi.org/10.2151/jmsj.2016-012.

    • Search Google Scholar
    • Export Citation

  • 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

  • Wang, D., Y. Lin, and D. R. Chavas, 2022: Tropical cyclone potential size. J. Atmos. Sci., 79, 3001–3025, https://doi.org/10.1175/JAS-D-21-0325.1.

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  • Wang, S., and R. Toumi, 2022: An analytic model of the tropical cyclone outer size. npj Climate Atmos. Sci., 5, 46, https://doi.org/10.1038/s41612-022-00270-6.

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  • Wang, Y., 2009: How do outer spiral rainbands affect tropical cyclone structure and intensity? J. Atmos. Sci., 66, 1250–1273, https://doi.org/10.1175/2008JAS2737.1.

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  • Wang, Y., Y. Li, and J. Xu, 2021a: A new time-dependent theory of tropical cyclone intensification. J. Atmos. Sci., 78, 3855–3865, https://doi.org/10.1175/JAS-D-21-0169.1.

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  • Wang, Y., Y. Li, J. Xu, Z.-M. Tan, and Y. Lin, 2021b: The intensity dependence of tropical cyclone intensification rate in a simplified energetically based dynamical system model. J. Atmos. Sci., 78, 2033–2045, https://doi.org/10.1175/JAS-D-20-0393.1.

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  • Weatherford, C., and W. Gray, 1988: Typhoon structure as revealed by aircraft reconnaissance. Part I: Data analysis and climatology. Mon. Wea. Rev., 116, 1032–1043, https://doi.org/10.1175/1520-0493(1988)116<1032:TSARBA>2.0.CO;2.

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  • Xi, D., S. Wang, and N. Lin, 2023: Analyzing relationships between tropical cyclone intensity and rain rate over the ocean using numerical simulations. J. Climate, 36, 81–91, https://doi.org/10.1175/JCLI-D-22-0141.1.

    • Search Google Scholar
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  • Xu, J., and Y. Wang, 2010: Sensitivity of the simulated tropical cyclone inner-core size to the initial vortex size. Mon. Wea. Rev., 138, 4135–4157, https://doi.org/10.1175/2010MWR3335.1.

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  • Zhang, J. A., R. F. Rogers, D. S. Nolan, and F. D. Marks, 2011: On the characteristic height scales of the hurricane boundary layer. Mon. Wea. Rev., 139, 2523–2535, https://doi.org/10.1175/MWR-D-10-05017.1.

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  • Schenkel, B. A., N. Lin, D. Chavas, G. A. Vecchi, M. Oppenheimer, and A. Brammer, 2018: Lifetime evolution of outer tropical cyclone size and structure as diagnosed from reanalysis and climate model data. J. Climate, 31, 7985–8004, https://doi.org/10.1175/JCLI-D-17-0630.1.

    • Search Google Scholar
    • Export Citation

  • Schenkel, B. A., D. Chavas, N. Lin, T. Knutson, G. Vecchi, and A. Brammer, 2023: North Atlantic tropical cyclone outer size and structure remain unchanged by the late twenty-first century. J. Climate, 36, 359–382, https://doi.org/10.1175/JCLI-D-22-0066.1.

    • Search Google Scholar
    • Export Citation

  • Smith, R. K., G. Kilroy, and M. T. Montgomery, 2015: Why do model tropical cyclones intensify more rapidly at low latitudes?. J. Atmos. Sci., 72, 1783–1804, https://doi.org/10.1175/JAS-D-14-0044.1.

    • Search Google Scholar
    • Export Citation

  • Stern, D. P., G. H. Bryan, and S. D. Aberson, 2016: Extreme low-level updrafts and wind speeds measured by dropsondes in tropical cyclones. Mon. Wea. Rev., 144, 2177–2204, https://doi.org/10.1175/MWR-D-15-0313.1.

    • Search Google Scholar
    • Export Citation

  • Sui, C.-H., X. Li, M.-J. Yang, and H.-L. Huang, 2005: Estimation of oceanic precipitation efficiency in cloud models. J. Atmos. Sci., 62, 4358–4370, https://doi.org/10.1175/JAS3587.1.

    • Search Google Scholar
    • Export Citation

  • Tsuji, H., H. Itoh, and K. Nakajima, 2016: Mechanism governing the size change of tropical cyclone-like vortices. J. Meteor. Soc. Japan, 94, 219–236, https://doi.org/10.2151/jmsj.2016-012.

    • Search Google Scholar
    • Export Citation

  • 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

  • Wang, D., Y. Lin, and D. R. Chavas, 2022: Tropical cyclone potential size. J. Atmos. Sci., 79, 3001–3025, https://doi.org/10.1175/JAS-D-21-0325.1.

    • Search Google Scholar
    • Export Citation

  • Wang, S., and R. Toumi, 2022: An analytic model of the tropical cyclone outer size. npj Climate Atmos. Sci., 5, 46, https://doi.org/10.1038/s41612-022-00270-6.

    • Search Google Scholar
    • Export Citation

  • Wang, Y., 2009: How do outer spiral rainbands affect tropical cyclone structure and intensity? J. Atmos. Sci., 66, 1250–1273, https://doi.org/10.1175/2008JAS2737.1.

    • Search Google Scholar
    • Export Citation

  • Wang, Y., Y. Li, and J. Xu, 2021a: A new time-dependent theory of tropical cyclone intensification. J. Atmos. Sci., 78, 3855–3865, https://doi.org/10.1175/JAS-D-21-0169.1.

    • Search Google Scholar
    • Export Citation

  • Wang, Y., Y. Li, J. Xu, Z.-M. Tan, and Y. Lin, 2021b: The intensity dependence of tropical cyclone intensification rate in a simplified energetically based dynamical system model. J. Atmos. Sci., 78, 2033–2045, https://doi.org/10.1175/JAS-D-20-0393.1.

    • Search Google Scholar
    • Export Citation

  • Weatherford, C., and W. Gray, 1988: Typhoon structure as revealed by aircraft reconnaissance. Part I: Data analysis and climatology. Mon. Wea. Rev., 116, 1032–1043, https://doi.org/10.1175/1520-0493(1988)116<1032:TSARBA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Xi, D., S. Wang, and N. Lin, 2023: Analyzing relationships between tropical cyclone intensity and rain rate over the ocean using numerical simulations. J. Climate, 36, 81–91, https://doi.org/10.1175/JCLI-D-22-0141.1.

    • Search Google Scholar
    • Export Citation

  • Xu, J., and Y. Wang, 2010: Sensitivity of the simulated tropical cyclone inner-core size to the initial vortex size. Mon. Wea. Rev., 138, 4135–4157, https://doi.org/10.1175/2010MWR3335.1.

    • Search Google Scholar
    • Export Citation

  • Zhang, J. A., R. F. Rogers, D. S. Nolan, and F. D. Marks, 2011: On the characteristic height scales of the hurricane boundary layer. Mon. Wea. Rev., 139, 2523–2535, https://doi.org/10.1175/MWR-D-10-05017.1.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    A schematic plot of the expansion model presented in section 2. See the text for details.

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

    (a) The value ∂υ/∂r at r8 from the E04 model (solid) and the expansion model Eq. (21) (dashed) with σ varied from 0.1 to 1.1 (light to dark, with an interval of 0.1) at f = 5 × 10−5 s−1 and wcool = 0.0027 m s−1. (b) As in (a), but with σ = 0.7 and with different f (10−5 s−1; see the legend); the E04 model in solid lines and Eq. (21) in dashed lines. (c) As in (b), but for different wcool (m s−1; see the legend) for the E04 model (solid) and TSST (286, 293, 300, 307 K; warmer color means lower TSST) for Eq. (21) (dashed). See the text for parameter settings.

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

    (a) The term (Mew/ρw)eq  (m1.5 s−0.5) as a function of πwcoolCdνVCarnot/f (m1.5 s−0.5) in TTPP (red), CD (green), CDTTPP (blue), CK (purple), FCOR (gray), and ExSST (orange) during equilibrium periods. Data of TTPP, CD, CDTTPP, CK, and ExSST are used to determine ν by linear regression. Note that only ensemble index 0 in TTPP and ExSST is used, to be consistent with the sample sizes of CD, CDTTPP, and CK, which do not contain ensemble experiments. Fitted ν is shown on the upper right of the plot. Data are first processed by a 120-h running average. The black line visualizes the equation shown in the figure. In the equation, y=(Mew/ρw)eq and x=πwcoolCdνVCarnot/f. See appendix C for experimental design. (b) As in (a), but zoomed in without FCOR. Equilibrium periods for TTPP, FCOR, and ExSST are defined in section 4, and those for CD, CDTTPP, and CK are the same as TTPP.

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

    Analytical solution of size evolution (rt, km vs t, day; solid lines) in Eq. (25) for two cases: rt expanding toward a larger rt,eq (blue) and rt shrinking toward a smaller rt,eq (red). Horizontal dashed lines mark rt,eq. Dots mark the initial condition t0 and rt0. The triangle marks the location (rt,expmax) of the maximum expansion rate.

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

    Idealized expansion model prediction. (a) Time evolution of r8 (km; solid) with rt,eq (km; dotted); (b) dr8/dt (km day−1) as a function of r8 (km); (c) τrt (day) as a function of r8 (km); in (a)–(c), thicker and more opaque lines mark higher values of rt,eq. The dot marks the initial condition, and triangles mark the location of the maximum expansion rate.

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

    As in Fig. 5, but for (a) latent heating per unit area Qlat/(πrt2) (W m−2; red) and radiative cooling per unit area −Qrad/(πrt2) (W m−2; blue), (b) radial velocity ut (m s−1), (c) −Cd(μυt)2/hw(m s−2), and (d) local spinup rate ∂υ/∂t (m s−2) at rt, The dots in (b) and (d) mark equilibrium.

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

    (a) Temporal evolution of a 120-h running averaged ensemble-mean r8 (km) in TTPP (solid; cases with Ttpp = 241, 227, 214, 200, 163 K are shown) and analytical prediction of the expansion model (dashed) taking rt,eq equal to equilibrium sizes of TTPP (see the text). The shading marks 1 standard deviation from ensemble mean. Dots mark the initial condition for the expansion model. (b) The corresponding expansion rate dr8/dt (km day−1) in TTPP (solid lines are ensemble mean of the 24-h expansion rate as a function of the 120-h running averaged ensemble-mean r8; shading marks one standard deviation) and in the expansion model (dashed). (c),(d) As in (a) and (b), but for FCOR; (e),(f) As in (a) and (b), but for ExSST. Warmer color means higher values of the variable being varied (see the legend).

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

    (a) Ensemble-mean equilibrium sizes of TTPP simulations (solid) and the predicted rt,eq (dashed). (b) As in (a), but for FCOR. (c) As in (a), but for ExSST. Dots mark different cases in TTPP, FCOR, and ExSST. In (a)–(c), colors have the same meaning as Fig. 7.

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

    (a) The term Qlat/(2πr8)(108 W m−1) in TTPP (solid lines: ensemble mean; shaded: one standard deviation) and those correspondingly predicted by the expansion model (dashed; see the text for details). (b) As in (a), but for FCOR. (c) As in (a), but for ExSST. Colors have the same meaning as Fig. 7.

  • <sc>Fig</sc>. 10.
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    Fig. 10.

    Time-dependent rt,eq (km) as a function of r8 (km) for (a) TTPP, (b) FCOR, and (c) ExSST. Colors in (a)–(c) have the same meaning as Fig. 7. See the text for details.

  • <sc>Fig</sc>. 11.
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    Fig. 11.

    (a) The term ∂υ/∂r (s−1) at r8 in TTPP (solid lines: ensemble mean; shading: one standard deviation) and predicted by the expansion model (dashed) directly underlying the prediction in Fig. 7. Abscissa is r8 (km). (b) As in (a), but for ∂υ/∂t (m s−2) at r8. (c) As in (a), but for ut (m s−1) at r8. Warmer color means higher Ttpp (see the legend).

  • <sc>Fig</sc>. 12.
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    Fig. 12.

    (a) The τrt (day) for FCOR predicted by the expansion model directly underlying the prediction in Fig. 7. Abscissa is r8 (km). (b)–(d) As in Figs. 11a–c, but for FCOR. Warmer color means higher f (see the legend).

  • <sc>Fig</sc>. 13.
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    Fig. 13.

    (a) The τrt (day) for ExSST predicted by the expansion model directly underlying the prediction in Fig. 7. Abscissa is r8 (km). (b)–(d) As in Figs. 11a–c, but for ExSST. Warmer color means higher TSST (see the legend).

  • <sc>Fig</sc>. 14.
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    Fig. 14.

    (a) Colored dots show ut (m s−1) and Qlat/(πr82) (W m−2) when r8 is above 500 km and below 600 km in TTPP (all ensemble members); warmer color means higher Ttpp. The black line shows the expansion model prediction (baseline environment setting) with r8 = 550 km, and black dots mark cases in Figs. 5 and 6. The dotted line is a linear regression fitting to the centroids of each cluster (each Ttpp value). (b) As in (a), but for ∂υ/∂t (m s−2) at r8. (c) As in (a), but for dr8/dt (km day−1), shown as the 24-h expansion rate of the 120-h running averaged r8. (d)–(f) As in (a)–(c), but with r8 = 350 km in the expansion model and r8 from 300 to 400 km in TTPP. Note all cases in TTPP are shown.

  • <sc>Fig</sc>. 15.
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    Fig. 15.

    (a) Colored dots show ut (m s−1) and Qlat/(πr82)(Wm−2) when r8 is above 500 km and below 600 km in FCOR (all ensemble members); warmer color means higher f. The black line shows the expansion model prediction with r8 = 550 km, and black dots mark cases in Figs. 5 and 6. The dotted line is a linear regression fitting to the centroids of each cluster (each f value). (b) As in (a), but with r8 above 300 km and below 400 km for FCOR and r8 = 350 km for the expansion model.

  • <sc>Fig</sc>. 16.
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    Fig. 16.

    As in Fig. 5, but with Δsd varied from 50% to 150% of its base value with an interval of 25%; see the text for details. In (a)–(c), thicker and more opaque lines mark higher values of Δsd.

  • <sc>Fig</sc>. D1.
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    Fig. D1.

    (a),(b) Terms in the dry-entropy budget Eq. (D1) (J K−1 s−1) in CTL and the Ttpp = 163 K case in TTPP; legend shows (from top to bottom) terms Fr,Fu,Qlat/Te,lat,−(Qrad/Te,rad),S˙res,∂S/∂t as the sum of these terms and directly calculated (solid lines: ensemble mean; shading: one standard deviation). (c),(d) The equation [(Qlat/Te,lat)+S˙res+Fu]/r8 (J K−1 s−1 m−1) in TTPP and FCOR (solid lines: ensemble mean; shading: one standard deviation), respectively. Warmer color means higher values of Ttpp or f; dashed lines mark the expansion model predictions in section 4. (e),(f) As in (c) and (d), but for −(Qrad/Te,lat)/r8 (J K−1 s−1 m−1).

  • <sc>Fig</sc>. D2.
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    Fig. D2.

    (a)–(c) Diagnosed αp (solid lines: ensemble mean; shading: one standard deviation) as a function of r8 in TTPP, FCOR, and ExSST, respectively. (d)–(f) As in (a)–(c), but for ϵp,ew. Warmer colors mean higher values of the variable being varied. See the text for details.

  • <sc>Fig</sc>. D3.
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    Fig. D3.

    Diagnosed Δsd (J K−1 kg−1; solid lines: ensemble mean; shading: one standard deviation) in (a) TTPP, (b) FCOR, and (c) ExSST using Eq. (7). Colors have the same meaning as Fig. D2. Note in the calculation, the numerator and denominator of Eq. (7) are first processed by a 120-h running average. Dashed lines in (c) show the expansion model predicted Δsd=Lυqυs*/Ts, which are also used in section 4.

Fig. 1.

A schematic plot of the expansion model presented in section 2. See the text for details.
Fig. 2.

(a) The value ∂υ/∂r at r8 from the E04 model (solid) and the expansion model Eq. (21) (dashed) with σ varied from 0.1 to 1.1 (light to dark, with an interval of 0.1) at f = 5 × 10−5 s−1 and wcool = 0.0027 m s−1. (b) As in (a), but with σ = 0.7 and with different f (10−5 s−1; see the legend); the E04 model in solid lines and Eq. (21) in dashed lines. (c) As in (b), but for different wcool (m s−1; see the legend) for the E04 model (solid) and TSST (286, 293, 300, 307 K; warmer color means lower TSST) for Eq. (21) (dashed). See the text for parameter settings.
Fig. 3.

(a) The term (Mew/ρw)eq  (m1.5 s−0.5) as a function of πwcoolCdνVCarnot/f (m1.5 s−0.5) in TTPP (red), CD (green), CDTTPP (blue), CK (purple), FCOR (gray), and ExSST (orange) during equilibrium periods. Data of TTPP, CD, CDTTPP, CK, and ExSST are used to determine ν by linear regression. Note that only ensemble index 0 in TTPP and ExSST is used, to be consistent with the sample sizes of CD, CDTTPP, and CK, which do not contain ensemble experiments. Fitted ν is shown on the upper right of the plot. Data are first processed by a 120-h running average. The black line visualizes the equation shown in the figure. In the equation, y=(Mew/ρw)eq and x=πwcoolCdνVCarnot/f. See appendix C for experimental design. (b) As in (a), but zoomed in without FCOR. Equilibrium periods for TTPP, FCOR, and ExSST are defined in section 4, and those for CD, CDTTPP, and CK are the same as TTPP.
Fig. 4.

Analytical solution of size evolution (rt, km vs t, day; solid lines) in Eq. (25) for two cases: rt expanding toward a larger rt,eq (blue) and rt shrinking toward a smaller rt,eq (red). Horizontal dashed lines mark rt,eq. Dots mark the initial condition t0 and rt0. The triangle marks the location (rt,expmax) of the maximum expansion rate.
Fig. 5.

Idealized expansion model prediction. (a) Time evolution of r8 (km; solid) with rt,eq (km; dotted); (b) dr8/dt (km day−1) as a function of r8 (km); (c) τrt (day) as a function of r8 (km); in (a)–(c), thicker and more opaque lines mark higher values of rt,eq. The dot marks the initial condition, and triangles mark the location of the maximum expansion rate.
Fig. 6.

As in Fig. 5, but for (a) latent heating per unit area Qlat/(πrt2) (W m−2; red) and radiative cooling per unit area −Qrad/(πrt2) (W m−2; blue), (b) radial velocity ut (m s−1), (c) −Cd(μυt)2/hw(m s−2), and (d) local spinup rate ∂υ/∂t (m s−2) at rt, The dots in (b) and (d) mark equilibrium.
Fig. 7.

(a) Temporal evolution of a 120-h running averaged ensemble-mean r8 (km) in TTPP (solid; cases with Ttpp = 241, 227, 214, 200, 163 K are shown) and analytical prediction of the expansion model (dashed) taking rt,eq equal to equilibrium sizes of TTPP (see the text). The shading marks 1 standard deviation from ensemble mean. Dots mark the initial condition for the expansion model. (b) The corresponding expansion rate dr8/dt (km day−1) in TTPP (solid lines are ensemble mean of the 24-h expansion rate as a function of the 120-h running averaged ensemble-mean r8; shading marks one standard deviation) and in the expansion model (dashed). (c),(d) As in (a) and (b), but for FCOR; (e),(f) As in (a) and (b), but for ExSST. Warmer color means higher values of the variable being varied (see the legend).
Fig. 8.

(a) Ensemble-mean equilibrium sizes of TTPP simulations (solid) and the predicted rt,eq (dashed). (b) As in (a), but for FCOR. (c) As in (a), but for ExSST. Dots mark different cases in TTPP, FCOR, and ExSST. In (a)–(c), colors have the same meaning as Fig. 7.
Fig. 9.

(a) The term Qlat/(2πr8)(108 W m−1) in TTPP (solid lines: ensemble mean; shaded: one standard deviation) and those correspondingly predicted by the expansion model (dashed; see the text for details). (b) As in (a), but for FCOR. (c) As in (a), but for ExSST. Colors have the same meaning as Fig. 7.
Fig. 10.

Time-dependent rt,eq (km) as a function of r8 (km) for (a) TTPP, (b) FCOR, and (c) ExSST. Colors in (a)–(c) have the same meaning as Fig. 7. See the text for details.
Fig. 11.

(a) The term ∂υ/∂r (s−1) at r8 in TTPP (solid lines: ensemble mean; shading: one standard deviation) and predicted by the expansion model (dashed) directly underlying the prediction in Fig. 7. Abscissa is r8 (km). (b) As in (a), but for ∂υ/∂t (m s−2) at r8. (c) As in (a), but for ut (m s−1) at r8. Warmer color means higher Ttpp (see the legend).
Fig. 12.

(a) The τrt (day) for FCOR predicted by the expansion model directly underlying the prediction in Fig. 7. Abscissa is r8 (km). (b)–(d) As in Figs. 11a–c, but for FCOR. Warmer color means higher f (see the legend).
Fig. 13.

(a) The τrt (day) for ExSST predicted by the expansion model directly underlying the prediction in Fig. 7. Abscissa is r8 (km). (b)–(d) As in Figs. 11a–c, but for ExSST. Warmer color means higher TSST (see the legend).
Fig. 14.

(a) Colored dots show ut (m s−1) and Qlat/(πr82) (W m−2) when r8 is above 500 km and below 600 km in TTPP (all ensemble members); warmer color means higher Ttpp. The black line shows the expansion model prediction (baseline environment setting) with r8 = 550 km, and black dots mark cases in Figs. 5 and 6. The dotted line is a linear regression fitting to the centroids of each cluster (each Ttpp value). (b) As in (a), but for ∂υ/∂t (m s−2) at r8. (c) As in (a), but for dr8/dt (km day−1), shown as the 24-h expansion rate of the 120-h running averaged r8. (d)–(f) As in (a)–(c), but with r8 = 350 km in the expansion model and r8 from 300 to 400 km in TTPP. Note all cases in TTPP are shown.
Fig. 15.

(a) Colored dots show ut (m s−1) and Qlat/(πr82)(Wm−2) when r8 is above 500 km and below 600 km in FCOR (all ensemble members); warmer color means higher f. The black line shows the expansion model prediction with r8 = 550 km, and black dots mark cases in Figs. 5 and 6. The dotted line is a linear regression fitting to the centroids of each cluster (each f value). (b) As in (a), but with r8 above 300 km and below 400 km for FCOR and r8 = 350 km for the expansion model.
Fig. 16.

As in Fig. 5, but with Δsd varied from 50% to 150% of its base value with an interval of 25%; see the text for details. In (a)–(c), thicker and more opaque lines mark higher values of Δsd.
Fig. D1.

(a),(b) Terms in the dry-entropy budget Eq. (D1) (J K−1 s−1) in CTL and the Ttpp = 163 K case in TTPP; legend shows (from top to bottom) terms Fr,Fu,Qlat/Te,lat,−(Qrad/Te,rad),S˙res,∂S/∂t as the sum of these terms and directly calculated (solid lines: ensemble mean; shading: one standard deviation). (c),(d) The equation [(Qlat/Te,lat)+S˙res+Fu]/r8 (J K−1 s−1 m−1) in TTPP and FCOR (solid lines: ensemble mean; shading: one standard deviation), respectively. Warmer color means higher values of Ttpp or f; dashed lines mark the expansion model predictions in section 4. (e),(f) As in (c) and (d), but for −(Qrad/Te,lat)/r8 (J K−1 s−1 m−1).
Fig. D2.

(a)–(c) Diagnosed αp (solid lines: ensemble mean; shading: one standard deviation) as a function of r8 in TTPP, FCOR, and ExSST, respectively. (d)–(f) As in (a)–(c), but for ϵp,ew. Warmer colors mean higher values of the variable being varied. See the text for details.
Fig. D3.

Diagnosed Δsd (J K−1 kg−1; solid lines: ensemble mean; shading: one standard deviation) in (a) TTPP, (b) FCOR, and (c) ExSST using Eq. (7). Colors have the same meaning as Fig. D2. Note in the calculation, the numerator and denominator of Eq. (7) are first processed by a 120-h running average. Dashed lines in (c) show the expansion model predicted Δsd=Lυqυs*/Ts, which are also used in section 4.
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