Editorial Type:
Article
Article Type:
Research Article

Mean Cold Pool Size of Quasi-Equilibrium Convection. Part I: Why Do Cold Pools Collide?

Hao Fu Department of Earth System Science, Stanford University, Stanford, California
Department of the Geophysical Sciences, The University of Chicago, Chicago, Illinois

Search for other papers by Hao Fu in
Current site
Google Scholar
PubMed Close
https://orcid.org/0000-0002-9524-0001
and
Morgan E O’Neill Department of Earth System Science, Stanford University, Stanford, California
Department of Physics, University of Toronto, Toronto, Ontario, Canada

Search for other papers by Morgan E O’Neill in
Current site
Google Scholar
PubMed Close
Online Publication:
23 Jan 2025
Print Publication:
01 Feb 2025
DOI:
https://doi.org/10.1175/JAS-D-23-0143.1
Page(s):
237–250
Companion to this article
Full access

Abstract

Precipitation-driven cold pools play an important role in organizing tropical convection. Previous studies of tropical convection in the radiative–convective equilibrium (RCE) setup found that cold pools tend to collide with each other and trigger new convection. It remains unclear why most cold pools do not have enough space to dissipate without collision. We explain it as the smaller mean cold pool radius Req compared to its maximum potential radius Rmax. The latter denotes the radius needed for a cold pool’s buoyancy deficit to be dissipated by surface heating. Applying an energy balance constraint leads to an analytical solution for their ratio Rmax/Req, which depends on the Bowen ratio, surface precipitation–evaporation ratio, and rain sedimentation efficiency. The theory predicts that in the regime of marine tropical convection where the Bowen ratio is much smaller than one, Req cannot reach Rmax, and cold pools must collide frequently. This prediction is supported by large-eddy simulations using varying rain evaporation rates. In Part II, we combine the energy balance constraint with a convective life cycle model to obtain a theory of the mean cold pool radius Req.

© 2025 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: Hao Fu, haofu736@gmail.com

Abstract

Precipitation-driven cold pools play an important role in organizing tropical convection. Previous studies of tropical convection in the radiative–convective equilibrium (RCE) setup found that cold pools tend to collide with each other and trigger new convection. It remains unclear why most cold pools do not have enough space to dissipate without collision. We explain it as the smaller mean cold pool radius Req compared to its maximum potential radius Rmax. The latter denotes the radius needed for a cold pool’s buoyancy deficit to be dissipated by surface heating. Applying an energy balance constraint leads to an analytical solution for their ratio Rmax/Req, which depends on the Bowen ratio, surface precipitation–evaporation ratio, and rain sedimentation efficiency. The theory predicts that in the regime of marine tropical convection where the Bowen ratio is much smaller than one, Req cannot reach Rmax, and cold pools must collide frequently. This prediction is supported by large-eddy simulations using varying rain evaporation rates. In Part II, we combine the energy balance constraint with a convective life cycle model to obtain a theory of the mean cold pool radius Req.

© 2025 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: Hao Fu, haofu736@gmail.com
Keywords: Cold pools; Deep convection; Radiative–convective equilibrium

1. Introduction

Isolated deep convection is an important component of the tropical atmospheric circulation (Emanuel 1994). In a shear-free environment, a deep convective updraft typically has a radius of a few kilometers and dissipates in an hour due to the onset of precipitation (e.g., Byers and Braham 1949; Ogura and Takahashi 1971; Ferrier and Houze 1989). The evaporation of raindrops produces a gust of near-surface cold air called a cold pool. A cold pool usually appears as a density current, a kind of fluid motion with a sharp horizontal density contrast (e.g., Ungarish 2009). As a cold pool propagates away from the precipitation region, its volume is raised by entraining environmental air, and its temperature is raised by surface heating (Ross et al. 2004; Grant and van den Heever 2016; Romps and Jeevanjee 2016). The lateral leading edge of the cold pool is called a gust front, which aggregates and lifts the moist air in the mixed layer to the level of free convection and triggers future convection (e.g., Tompkins 2001; Grandpeix and Lafore 2010; Feng et al. 2015; Langhans and Romps 2015; Torri et al. 2015; Haerter et al. 2019; Fuglestvedt and Haerter 2020; Meyer and Haerter 2020; Yang et al. 2021; Falk and van den Heever 2023; Sakaeda and Torri 2023). Cold pools play a critical role in organizing convection in such a back-and-forth convective triggering process. Figure 1 shows a population of densely packed cold pools in the tropical west Pacific captured by the NOAA-20 satellite. The width of individual cold pools is roughly 10–50 km, with white cloud arcs marking their edges and concentrated cumulus clouds appearing at their intersections. It remains unclear what sets a basic length scale in tropical convection: the cloud spacing, which critically depends on the cold pool size (e.g., Tompkins 2001).

Fig. 1.
Fig. 1.

A snapshot of the corrected reflectance (true color) NOAA-20/VIIRS satellite product on 14 Jul 2021, downloaded from the NASA Worldview website. The longitudinal width is 331 km, and the latitudinal width is 222 km. The down-to-the-hour timestamp is not available.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

To understand the role of cold pools in organizing convection, a starting point is to study the radiative–convective equilibrium (RCE) state, which is a simplified representation of the tropical atmosphere with a sea surface of uniform temperature (e.g., Riehl and Malkus 1958; Manabe and Strickler 1964; Held et al. 1993; Robe and Emanuel 1996; Tompkins 2001; Parodi and Emanuel 2009; Romps 2014a,b; Nissen and Haerter 2021). Radiative cooling is mainly balanced by the condensation heating of water vapor, and the water vapor is supplied by surface evaporation. We specifically focus on a mesoscopic RCE setup, which is large enough to accommodate a large number of clouds and small enough to exclude convective self-aggregation (e.g., Jeevanjee and Romps 2013). In this setup, the cloud population is statistically homogeneous and isotropic in the horizontal directions.

When in or close to an RCE state, various stochastic factors lead to a wide range of cold pool sizes (Tompkins 2001; Schlemmer and Hohenegger 2014; Feng et al. 2015; Gentine et al. 2016; Haerter et al. 2019). The first step to understanding the cold pool size histogram is to understand its expectation value: the mean cold pool radius Req. The Req has been studied carefully, but what key factors control Req remains unclear and lacks a theoretical model. Nissen and Haerter (2021) obtained some clues from large-eddy simulations (LES): the cold pool size increases as the rain evaporation rate is raised from an artificially small value, but this trend still awaits a theory to explain. Though a theory of mean cold pool size is still unavailable, Romps and Jeevanjee (2016) analytically solved the radius by which the buoyancy of a cold pool fully recovers to the environmental value. We refer to this radius as the “maximum potential radius” Rmax. Is the distance Rmax ever realized in nature? Tompkins (2001) found that a cold pool usually triggers new convection when its buoyancy deficit is significantly recovered yet still has some residual momentum. Torri and Kuang (2019) carefully tracked cold pool collisions and found that for each cold pool, the median time of the first collision is around 10 min, and most cold pools can experience a collision in their lifetime. The cold pools in Fig. 1 are also densely packed and should be colliding. Cold pool collision is so ubiquitous that the maximum potential radius proposed by Romps and Jeevanjee (2016) is unlikely to be realized.

Why is it seemingly inevitable that cold pools must collide with each other? The question can probably be rephrased as why Req cannot reach Rmax, as demonstrated in Fig. 2. Romps and Jeevanjee (2016) showed that Rmax depends on the initial volume of a cold pool and, therefore, the cloud radius. Schlemmer and Hohenegger (2014) showed that the cloud size statistically depends on the cold pool size. Any attempt to solve Req or Rmax seems trapped in circular reasoning. How can we escape the deadlock? To solve the two unknowns, this two-paper series proposes two constraints: a macroscopic constraint from the energy balance of RCE and a microscopic constraint from the survival competition between cold pools.

Fig. 2.
Fig. 2.

A schematic diagram for illustrating the two important length scales studied in this two-paper series: the mean radius Req and the maximum potential radius Rmax of cold pools in quasi-equilibrium convection. The Req represents half the deep convective spacing at a snapshot. The Rmax is the radius by which a cold pool’s buoyancy fully recovers to the environmental value.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

Part I presents the macroscopic energy balance theory to constrain Req and Rmax, which utilizes the relation between surface sensible heating rate, surface evaporation rate, and surface rainfall rate. Our theory does not require a strict RCE where all properties of the atmosphere, particularly the water vapor content, reach a steady state. By introducing a variable surface precipitation–evaporation ratio, the theory applies to any state where the atmospheric humidity and the mean cold pool size evolve slowly. This state is called “quasi-equilibrium convection,” which is close to RCE yet not strictly in RCE (Arakawa and Schubert 1974; Yano and Plant 2012). Fu and O’Neill (2025; hereafter Part II) studies the microscopic constraint that results from the survival competition between cold pools: what is the cold pool radius responsible for the most intense next-generation cold pool? We will present a convective life cycle model to quantify the trade-off between mechanical lifting (e.g., Droegemeier and Wilhelmson 1985), moisture aggregation (e.g., Langhans and Romps 2015), and updraft entrainment (e.g., Böing et al. 2012) in determining the cold pool intensity. It is then combined with the energy balance constraint to solve Req and Rmax.

As for the structure of Part I, section 2 diagnoses Req and the cold pool activity in LES with varying rain evaporation rates. Section 3 uses assumptions validated in LES to build the energy balance constraint between Req and Rmax. Section 4 concludes the paper.

2. The equilibrium cold pool radius in LES

a. Simulation setup

In this section, we perform a series of LES with the Bryan Cloud Model 1 (CM1) version 19.10 (Bryan and Fritsch 2002). Varying rain evaporation rates are used to generate a population of simulated cold pools. The experimental method closely follows Nissen and Haerter (2021), who used the University of California, Los Angeles (UCLA) large eddy simulator rather than CM1. Our LES uses a uniform sea surface temperature of 300 K and a zero Coriolis parameter in a 96 × 96 km2 doubly periodic domain, free of initial background wind. The mesh has 480 × 480 × 130 grid points, with a uniform horizontal grid spacing of 200 m, and a vertically nonuniform grid with 15 grid points within the lowest 1 km. The height of the domain is 28 km, with Rayleigh damping above 20 km to suppress the reflection of gravity waves. The model uses the TKE scheme as the subgrid model (Deardorff 1980), the “sfcmodel = 3” surface layer model based on the similarity theory (Jiménez et al. 2012), the RRTMG radiation transfer scheme (Clough et al. 2005) [with a fixed zenith angle of 50.5° and a fixed solar constant of 650.83 W m−2, following the setting of Bretherton et al. (2005)], and Morrison double-moment microphysics scheme (Morrison et al. 2005). We initialized the model by adding random potential temperature perturbation on a sounding generated by a coarse-resolution RCE simulation with a doubled vertical grid spacing and a 2-km horizontal grid spacing. The coarse-resolution simulation is not an LES but uses the simple planetary boundary layer scheme by Bryan and Rotunno (2009) to parameterize turbulence. It has a 120 × 120 km2 domain. The horizontally averaged potential temperature and water vapor mixing ratio at t = 100 days are used to generate the initial sounding for all the LES.

For the reference test where the rain evaporation rate is normal, the domain-averaged precipitable water (PW) oscillates within the first 2 days and then slowly increases (the yellow line in Fig. 3a, see more details below). This indicates that the coarse-resolution initialization is close to yet still deviates from the RCE state of the high-resolution LES setup. However, both the standard deviation of PW (Fig. 3b) and the diagnosed mean cold pool radius (Fig. 3c, which will be introduced) do not systematically change after 2 days. The above evidence indicates that the atmosphere enters a quasi-equilibrium state after 2 days. We will use the data between t = 3 days and t = 5 days to analyze quasi-equilibrium statistics, including the mean cold pool radius.

Fig. 3.
Fig. 3.

(a) The time evolution of the domain-averaged column PW (m) for Eυ = 0.2 (blue line), Eυ = 0.5 (red line), Eυ = 1.0 (the reference test, yellow line), and Eυ = 2.0 (purple line). (b) As in (a), but for the spatial standard deviation of column PW with a logarithmic ordinate. (c) As in (a), but for the time evolution of the mean cold pool radius Req. The overshooting of Req for the Eυ = 2.0 test between t = 3.5 days and t = 4 days is judged as an occasional event, which does not appear in the larger-domain (144 × 144 km2) simulation (Fig. S2). In all panels, the shading shows the period between t = 3 days and t = 5 days, during which the quasi-equilibrium statistics are calculated.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

We performed 12 experiments where the inverse of rain evaporation relaxation time (represented by the parameter EPSR in “morrison.F” file) is multiplied by a constant coefficient Eυ, with Eυ = 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.2, 1.5, 1.8, and 2.0 for EXP 1-12. The Eυ = 1.0 test is the reference test. The rain evaporation rate is amplified in Eυ > 1.0 tests and suppressed in Eυ < 1.0 tests. To be aligned with Nissen and Haerter (2021), the rain evaporation at all heights is modified, rather than just in the mixed layer as adopted by some previous studies of cold pools (e.g., Jeevanjee and Romps 2013; Grant et al. 2018; Wang et al. 2019; Fu and O’Neill 2024).

First, we compare the convective pattern of different tests, as shown in Fig. 4. As Eυ increases, the mixed layer gets moister and colder due to the stronger rain evaporation. At t = 4 days, by which the quasi-equilibrium state has been reached, all experiments exhibit densely packed cold pools, and there is a visible increase of the cold pool size as Eυ increases, in agreement with the LES of Nissen and Haerter (2021). For the 12 tests, no convective self-aggregation (CSA) is observed within the first 5 days. This differs from Nissen and Haerter (2021), who observed a clear CSA pattern by t = 2 days in their Eυ = 0.1 and 0.2 tests. In an additional Eυ = 0.1 test we performed (not shown), there is indeed a signal of CSA at t = 5 days. This different behavior of CSA may be due to the model difference between CM1 and UCLA large eddy simulator. Two additional experiments with a domain size of 144 × 144 km2 at Eυ = 1.0 and 2.0 show no significant difference in the convective pattern (Fig. S1 in the online supplemental material) and mean cold pool radius (Fig. S2), so the 96 × 96 km2 domain size should be large enough for studying the mean cold pool radius.

Fig. 4.
Fig. 4.

(top) The water vapor mixing ratio vertically averaged within the lowest 551 m of the domain at t = 4 days. (a)–(d) The Eυ = 0.2, 0.5, 1.0, and 2.0 tests, respectively. (bottom) The z = 12.5 m (near-surface) potential temperature at the same time. In each plot, the white lines denote the 10−2 g kg−1 contours of cloud liquid water mixing ratio at z ≈ 4 km, which visualizes the cloud position. The cold pool sizes in the Eυ = 2 test are visually larger than those in the Eυ = 1, which seems to oppose the stagnancy behavior diagnosed with the time average. We argue that a single snapshot is insufficient to confirm the stagnancy of Req with Eυ because the cold pool size fluctuates over time. The LES movies can be downloaded from online (https://doi.org/10.5281/zenodo.13785498).

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

b. The trend of mean cold pool radius

Figure 4 shows that the clouds preferentially occur at the intersection of gust fronts, which is visualized by the mixed-layer water vapor mixing ratio (vertically averaged within the lowest 551 m, i.e., the lowest 11 model levels, without density weighting), similar to the observed tropical convection shown in Fig. 1. Cold pools in a snapshot have various sizes because they have different initial intensities and are at different ages of their evolution. Because a cold pool typically has a moisture anomaly at the gust front (e.g., Langhans and Romps 2015), the water vapor mixing ratio vertically averaged in the mixed layer is used to diagnose the cold pool radius. The mean cold pool radius Req is diagnosed in two steps, as illustrated in Fig. 5a:

  1. The spatial autocorrelation function of the mixed layer vertically averaged water vapor mixing ratio is calculated. Due to the quasi-isotropy of the setup, the 1D x-direction and y-direction profiles are extracted from the 2D autocorrelation function and then averaged over to yield an approximation to the (1D) axisymmetric component of the autocorrelation function.

  2. The distance where the averaged 1D autocorrelation function first drops to 0.1 from above is marked. The Req is taken as twice the marked autocorrelation distance. This choice roughly corresponds to the autocorrelation distance at the antiphase point.

Fig. 5.
Fig. 5.

(a) The temporally averaged (a 2-day-long time series between t = 3 days and t = 5 days) spatial autocorrelation of the mixed layer vertically averaged water vapor mixing ratio for Eυ = 0.2 (blue line), Eυ = 0.5 (red line), Eυ = 1.0 (yellow line), and Eυ = 2.0 (purple line). An additional dashed black line denotes the 0.1 autocorrelation value used to diagnose the mean cold pool radius. (b) As in (a), but for the temporal autocorrelation of the mixed layer vertically averaged water vapor mixing ratio, using the time series between t = 3 days and t = 5 days. The time series have been detrended with a linear fit. The temporal autocorrelation is calculated on each of the 480 × 480 grid points and then averaged. Panel (b) is discussed in appendix A.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

This approach is close to the cold pool diagnostic method of Haerter et al. (2017) but with two major differences. First, they used the vertically integrated moisture convergence within the lowest 2 km. We choose not to follow their approach because we argue that the water vapor mixing ratio averaged in the mixed layer is more relevant to cold pools. Second, they let the cold pool radius be the minimum point of the spatial autocorrelation function. Our autocorrelation function is too flat near the minimum point, which makes the result sensitive to fluctuations. Thus, we use the more clear-cut 0.1 threshold instead. Note that the shapes of cold pools are not circular, so Req is not strictly a “radius.” It should only be understood as a characteristic length scale roughly equivalent to half the deep convective cloud spacing (Fig. 2).

Figure 6a shows that the diagnosed Req indeed increases with Eυ, but the curve flattens significantly for Eυ≳0.8. This stagnant phenomenon is similar to Fig. 2B of Nissen and Haerter (2021). They performed four different Eυ tests (Eυ = 0.1, 0.2, 0.6, 1.0) and studied the first 18 h since precipitation onset, which is probably not in a quasi-equilibrium state. They used a gust front tracking method to diagnose the cold pool radius. The diagnosed radius reaches an asymptotic value at large cold pool ages, which might be considered Req. Their Fig. 2B shows that the diagnosed cold pool radius grows steadily between their Eυ = 0.1, 0.2, 0.6 tests but remains roughly the same for their Eυ = 0.6 and Eυ = 1 tests. They identified the growing trend but did not comment on the stagnancy reflected by their Eυ = 0.6 and Eυ = 1 tests.

Fig. 6.
Fig. 6.

Some statistics averaged in the slot between t = 3 days and t = 5 days. (a) The blue dots denote the mean cold pool radius Req diagnosed from the LES for different Eυ. The shading denotes the ±1 standard deviation range of the Req time series. (b) The domain standard deviation of the near-surface (z = 12.5 m, the lowest model level) potential temperature. The shading denotes the ±1 standard deviation range of the time series. (c) The standard deviation of the near-surface buoyancy vs the square of the domain-averaged near-surface wind speed uc (black dots). In addition, we plot the domain-averaged CIN as red crosses.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

c. The trend of cold pool intensity

The maximum potential radius Rmax is an unrealized theoretical quantity that is hard to diagnose. Thinking in another way, Rmax has been shown to depend on the total buoyancy deficit of a cold pool upon formation (Romps and Jeevanjee 2016), so a cold pool with a higher Rmax may have a stronger buoyancy deficit averaged in its lifetime, a feature we refer to as a “higher intensity.” Because cold pools in the domain are at different life cycle stages, sampling in time should be equivalent to sampling in space. We use the spatial standard deviation of a near-surface (z = 12.5 m) potential temperature, std(θ′), to infer the trend of Rmax with Eυ.

While Req has an upper bound with Eυ (Fig. 6a), std(θ′) keeps rising with Eυ (Fig. 6b). The std(θ′) represents the strength of buoyancy fluctuation in the mixed layer, indicating that the cold pools are getting more negatively buoyant despite the stagnation of their mean radius. The mean near-surface wind speed uc also rises with Eυ (Fig. 6c). Furthermore, when interpreting std(θ′) as the potential temperature contrast between the cold pool and the environment, we find that uc and std(θ′) obey the formula of the unsheared density current propagation speed (e.g., von Kármán 1940; Benjamin 1968; Wakimoto 1982; Ross et al. 2004; Torri and Kuang 2019):
uc2∼gstd(θ′)Hmθ0.
Here, g = 9.8 m s−2 is the gravitational acceleration, Hm = 500 m is a rough estimate of the mixed-layer depth that characterizes the cold pool thickness (Benjamin 1968; Emanuel 1994), and θ0 = 300 K is a reference potential temperature. The formula describes the balance between the acceleration and the pressure gradient force in the cross-front momentum equation, which predicts that a more negatively buoyant cold pool propagates faster. The alignment to the density current formula indicates that the horizontal acceleration within cold pools dominates the near-surface wind field. This provides a physical justification, which supplements the visual observation of Fig. 4, that cold pools are densely packed and interactive. Combining this finding with the observation that Req eventually plateaus with increasing Eυ while std(θ′) keeps rising, we may judge that cold pools get more intense and interact more strongly as Eυ increases. In section 3, we explain it as the farther deviation of Req from Rmax.
Before wrapping up this section, we would like to clarify a possible misunderstanding. One may suppose that a higher uc makes convective triggering more efficient and possibly causes a runaway amplification of the convective activity. This is not the case because Fig. 6c shows that the domain-averaged convective inhibition (CIN) energy, which denotes the mechanical work required to lift an air parcel from the surface to its level of free convection, also increases with uc as
CIN∼uc2.
The presence of CIN makes it hard for convection to develop spontaneously without an external trigger factor (e.g., Mapes 1997). CIN is higher when the mixed-layer air parcel is colder and drier, or the lower part of the free troposphere is warmer. The increase of CIN with Eυ is likely due to the more negatively buoyant cold pools, which reduces the buoyancy of the air parcel. In other words, as cold pools get colder and drive stronger upward accelerations along their edges, these upward accelerations are counteracted by increasingly more negative buoyancy related to larger CIN. The scaling relation in Eq. (2) possibly implies an invariant convective triggering efficiency, defined as the fraction of parcels that can overcome CIN and turn into a deep convective updraft. Mapes (2000) heuristically derived such a quantity by drawing an analogy of the subgrid-scale variability to “temperature” defined in the sense of Maxwell–Boltzmann distribution and named it “K.” He let the deep convective heating be proportional to exp(−CIN/K), an integral of the perturbation’s distribution function over CIN that can be interpreted as the convective triggering probability or efficiency. In our setup, cold pools are the dominant sources of variability in the mixed layer, so we let K∼uc2. As a result, exp(−CIN/K) should remain invariant as Eυ increases. This property will be used in Part II, which allows us to parameterize the convective mass flux of individual clouds without caring for the trend of CIN.

3. Theory

a. The maximum potential cold pool radius Rmax

In this section, we constrain the Req–Rmax relation with a scaling theory based on the energy balance of quasi-equilibrium convection. The maximum potential radius Rmax is the radius by which an initial cold air cylinder loses all its buoyancy anomalies with respect to the environmental air, as illustrated in Fig. 7. The expression of Rmax is derived by Romps and Jeevanjee (2016) in a box model, assuming axisymmetry and uniform potential temperature in the cold pool:
Rmax=(92πCθΔQeρcpΔθ)1/3.
Here, ρ is the air density (unit: kg m−3) and cp is the air isobaric specific heat (unit: J kg−1 K−1). The quantity ΔQe (unit: J) is the total subcloud evaporative cooling per convective event. The Δθ is the temperature difference between the sea surface and the mixed layer. The nondimensional parameter Cθ is the surface sensible heat exchange coefficient. The parameters ρ, cp, Δθ, and Cθ are set as constants. The dependence of Rmax on ΔQe indicates that Rmax is an overall measure of cold pool intensity.
Fig. 7.
Fig. 7.

A schematic diagram of the single cold pool evolution problem studied by Romps and Jeevanjee (2016), using a box model. The cold pool is produced by a total evaporative cooling of ΔQe (J). The initial cold air is assumed to be a cylinder (volume V0, radius R0, and height H0) with a uniform potential temperature anomaly θ′|R=R0. As the cold pool collapses, it keeps a cylindrical shape with a radius of R and height Hc, and its volume V grows by entrainment. The color depth of the shading denotes the magnitude of the cold pool’s potential temperature deficit, which is assumed to be uniform within the cold pool. The cold pool dissipates under the surface sensible heat flux Fsurf,s (W m−2). The cold pool radius by which the potential temperature deficit reduces to zero is the maximum potential radius Rmax.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

First, we review the derivation of Eq. (3) for readers unfamiliar with the theory. The buoyancy is expressed in potential temperature formulation rather than density formulation as adopted by Romps and Jeevanjee (2016). Is the water vapor buoyancy important? Figure 4 shows that the characteristic potential temperature difference across a gust front is O(1) K, and the water vapor mixing ratio difference is O(2) g kg−1. Their contribution to buoyancy perturbation has a ratio of around 0.04, so the water vapor buoyancy is neglected for simplicity. Initially, there is a cylinder of uniformly cold air in an infinitely large domain with neutral stratification. The cylinder collapses due to the pressure gradient force. We let its radius be R(t), which increases with time. The cylinder remains axisymmetric and expands with a radial velocity ur:
dRdt=ur|r=R.
The ur is assumed to be proportional to the radial distance r and ur|r=R is the radial velocity at the cold pool’s edge. The cold pool keeps entraining environmental air and increasing its volume V. Let ε (unit: m−1) be the cold pool’s fractional entrainment rate. The volume obeys
dVdt=ε ur|r=RV.
The height of the cold pool Hc can be calculated with V and R:
Hc=VπR2.
With the above geometric information, a governing equation of the cold pool’s potential temperature anomaly θ′ with respect to the environmental value is derived. The buoyancy anomaly is diluted by entrainment and eliminated by the surface sensible heat flux Fsurf,s (unit: W m−2). The Fsurf,s obeys the bulk aerodynamic formula:
Fsurf,s≈ρcpCθΔθur.
Note that Eq. (7) uses the temperature difference between the mixed layer and the sea surface, an approximation to the temperature difference between the cold pool and the sea surface. Using Eqs. (6) and (7), we obtain the governing equation of θ′:
dθ′dt=−εur|r=Rθ′+1V∫0RFsurf,sρcp2πrdr=−ε ur|r=R θ′+23CθΔθHcur|r=R.
Here, the volumetric average of the surface flux term has been used, and the radial velocity has been assumed to increase linearly with radius. The 2/3 factor is associated with the cylindrical geometry.
Romps and Jeevanjee (2016) solved Eqs. (4)–(8) with a set of initial conditions: initial radius R0, initial height H0, initial volume V0≡πR02H0, and initial potential temperature anomaly:
θ′|R=R0=−ΔQeρcpV0.
The final expression of θ′ is
θ′=e−ε(R−R0)[θ′|R=R0+2π9CθΔθV0(R3−R03)]≈e−εR[θ′|R=R0+2π9CθΔθV0R3]=θ′|R=R0e−εR(1−R3Rmax3).
A key step is applying Eq. (4) to transform the time coordinate to the radius coordinate: dR = ur|r=Rdt, which eliminates ur|r=R. The derivation of the second line assumes the cold pool initial radius R0 to be much smaller than R because we are concerned with a fully developed cold pool. The solution of Rmax [Eq. (3)] is obtained by letting θ′ = 0 K in Eq. (10). The Rmax has two key properties:

  • The Rmax does not depend on details of the momentum evolution.

  • The Rmax does not depend on the fractional entrainment rate ε because entrainment dilutes the cold pool air but does not change the total heat needed to eliminate the cold anomaly. However, ε influences the decaying process of θ′ and appears as a dilution factor e−εR in Eq. (10).

Finally, we remark that the Rmax expression [Eq. (3)] is not a special result of the Romps–Jeevanjee model. The cold pool model of Ross et al. (2004), which neglects entrainment and assumes a strict balance between momentum advection and pressure gradient force at the gust front, also arrives at Rmax ∝ (ΔQe/Cθ)1/3 when the background wind is negligible compared to the cold pool wind. See Eq. (13) of Ross et al. (2004) for more details.

b. The bulk energy balance constraint for Req–Rmax

The theory of Rmax is based on the balance between the surface sensible heating and the cold pool’s buoyancy deficit. The gust front wind also induces surface latent heat flux, which controls the buoyancy deficit of the next-generation cold pool. Can we close the loop with a bulk energy balance constraint?

The total evaporative cooling of a convective event ΔQe can be linked to the net condensation heating of a convective event ΔQc (unit: J) via the sedimentation efficiency (SE; Langhans et al. 2015), which denotes the fraction of rainwater that survives evaporation and falls to the surface as precipitation:
ΔQe=ΔQc1−SESE.
To link ΔQc with the bulk energy balance, a key step is finding the spatiotemporal number density of cold pools or, equivalently, convective clouds. The LES diagnostic result in appendix A, Req ∼ ucΔt [Eq. (A1)], shows that the cold pool propagates at a mean speed of uc and triggers a new convection in Δt time after covering a distance of the mean cold pool radius Req. Here, Δt denotes half of the convective interval time, briefed as the convective half-period. This spatiotemporal relation yields a spatiotemporal number density of cold pools Nc (unit: m−2 s−1):
Nc=1ΔtReq2=ucReq3,
as illustrated in Fig. 8. We define the domain-averaged condensation heating power as Fsurf,p (unit: W m−2):
Fsurf,p≡NcΔQc.
The subscript “surf,p” expresses the fact that the condensation heating power is proportional to the surface precipitation flux.
Fig. 8.
Fig. 8.

A schematic diagram for the organization of convection in space and time. The horizontal axis denotes a certain spatial direction x. The vertical axis denotes time t. The blue shading denotes the trajectories of gust fronts in space and time. The equilibrium cold pool radius Req is half the deep convective spacing at a snapshot. It takes a cold pool Δt time to cover Req distance at a mean speed of uc ∼ Req/Δt.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

Not all the surface-evaporated water vapor converts to precipitation, so the surface latent heat flux Fsurf,e (unit: W m−2) does not necessarily balance the condensation heating power Fsurf,p. We let the ratio of the condensation heating power to the surface latent heat flux be the surface precipitation–evaporation ratio χpe:
χpe≡Fsurf,pFsurf,e.
In a strict RCE state, the atmospheric moisture content is fixed, yielding χpe = 1. The sign of χpe can be judged from the trend of the domain-averaged column precipitable water (PW¯, shown in Fig. 3a for our LES), based on the budget equation of PW¯:
dPW¯dt=1ρlLυ(Fsurf,e−Fsurf,p)+ADV,
where ρl is the density of liquid water, Lυ is the specific vaporization heat of water, and ADV denotes the lateral moisture advective transport by the mesoscale or synoptic-scale background circulation, with ADV = 0 in our setup. When PW¯ increases with time, as is in most of our LES, there is less surface precipitation than surface evaporation, yielding χpe < 1. In the real atmosphere, the background circulation makes ADV ≠ 0. If the circulation converges moisture (ADV > 0), such as in the intertropical convergence zone (e.g., Peixóto and Oort 1984), a steady state with dPW¯/dt≈0 yields χpe > 1.
Substituting Eqs. (11)–(14) into the expression of Rmax [Eq. (3)], we get
Rmax=(92πΔQeρcpCθΔθ)1/3=(92πΔQcρcpCθΔθ1−SESE)1/3=(92πFsurf,p/NcρcpCθΔθ1−SESE)1/3=(92πFsurf,pρcpCθΔθuc1−SESE)1/3Req=(92πFsurf,pFsurf,s1−SESE)1/3Req=(92πFsurf,pFsurf,eFsurf,eFsurf,s1−SESE)1/3Req.
We have used the domain-averaged surface sensible heat flux Fsurf,s = ρcpCθΔθuc, an adaptation of Eq. (7), in deriving the fifth line of Eq. (16). This formulation assumes the characteristic surface wind speed for heat uptake as the gust front propagation speed, supported by the LES diagnosis in section 2c. Figure S3 shows a snapshot of the surface evaporative flux in the Eυ = 1 test, confirming that surface evaporation is significantly higher inside cold pools where the wind speed is significantly higher, though substantial evaporation also occurs outside. Note that uc bridges the surface flux and the spatiotemporal pattern of convective organization and is canceled out in Eq. (16). Introducing the Bowen ratio,
B≡Fsurf,sFsurf,e,
which is the ratio between the surface sensible heat flux to the surface latent heat flux (Bowen 1926), we get a very simple Req–Rmax relation:
Rmax=Req(92π1−SESEχpeB)1/3︸Φ.
The ratio of Rmax to Req is named Φ, which only depends on the SE, the Bowen ratio B, and the precipitation–evaporation ratio χpe. Equation (18) predicts the following:

  1. A smaller sedimentation efficiency increases Φ and makes Req deviate more from Rmax. This is because a stronger rain evaporative cooling, associated with a smaller SE, increases the cold pool’s buoyancy deficit and requires a longer Rmax to recover.

  2. A smaller Bowen ratio increases Φ and makes Req deviate more from Rmax. This is because the elimination of a cold pool’s buoyancy deficit is mostly by surface sensible heating. Weaker surface sensible heating slows the cold pool recovery and increases its Rmax.

  3. A higher χpe increases Φ because heavier precipitation produces stronger cold pools. This indicates that a steady PW region with moisture convergence (ADV > 0) should have more densely packed cold pools.

The B over the tropical ocean is around 0.1 (Jo et al. 2002). The typical SE of tropical deep convection is SE ≈ 0.6 (Langhans et al. 2015; Lutsko and Cronin 2018). Substituting these into Eq. (18), and assuming χpe = 1, we get Φ ≈ 2. Figure 9 further illustrates how Φ changes with 1 − SE. For B = 0.1, our theory predicts that Req cannot reach Rmax unless SE≳0.9. This scenario has very weak rain evaporation, where less than 10% of rainwater evaporates. Thus, cold pools of quasi-equilibrium marine tropical convection unavoidably collide unless the rain evaporation rate is unrealistically low. A possible future work is sketching the phase space of atmospheric convection over the sea and land and marking the regime where cold pools frequently collide.

Fig. 9.
Fig. 9.

Plots of Φ ≡ Rmax/Req for B = 0.05 (blue line) and B = 0.1 (red line) predicted by the energy balance constraint. Both plots use χpe = 1. The Φ < 1 regime is unphysical, but we retain it in the plots.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

Readers might be curious about the role of radiative cooling rate in the Req–Rmax relation, which is not explicit in Eq. (18). In marine tropical convection, radiative cooling has been shown to be minor in the boundary layer moist static energy budget (Thayer-Calder and Randall 2015) and, therefore, possibly plays a minor role in cold pool dynamics. The radiative cooling in the free troposphere is mainly balanced by condensation heating (Emanuel 1994). Thus, the free-tropospheric radiative cooling rate sets the condensation heating power Fsurf,p (e.g., Robe and Emanuel 1996). How the free-tropospheric radiative cooling rate influences the surface sensible heat flux Fsurf,s is less obvious. Thus, we cannot draw any inference now and suggest investigating how B, χpe, and SE respond to the radiative cooling rate.

c. Comparison with LES

In appendix B, we apply the energy balance theory [Eq. (18)] to calculate Rmax/Req in experiments with different Eυ, using the diagnosed B, χpe, and SE. Most of the change of Eυ is reflected in the change of SE. The calculated Rmax/Req increases with Eυ. This Req–Rmax relation solved from the bulk energy balance cannot be directly verified because Rmax is a theoretical quantity. However, it can be indirectly inferred from our LES result in section 2. A higher buoyancy fluctuation and wind speed indicate that a higher Eυ produces more intense cold pools in a convective event, which qualitatively indicates a higher Rmax. The higher Rmax and the stagnant Req correspond to a higher Rmax/Req, consistent with the energy balance theory. In Part II, the whole Req theory, which combines the energy balance and survival competition constraints, is quantitatively tested with LES.

4. Conclusions

The radiative–convective equilibrium (RCE) setup is probably the simplest setup for understanding convective organization. The evaporation of rainwater produced by deep convection induces cold pools that propagate in the mixed layer and trigger future convection, organizing the cloud population. This two-paper series presents a theory of the mean cold pool radius Req in RCE. The condition can be relaxed to a quasi-equilibrium state with slow-varying atmospheric humidity. The Req is a key length scale that determines the deep convective spacing and informs how the convective mass flux is distributed to each cloud. The challenge is that Req depends on the cloud radius and the cold pool intensity, which are also functions of Req. To break the deadlock, we hypothesize that Req obeys two constraints: the bulk energy balance that imposes a macroscopic constraint and the cold pool survival competition that imposes a microscopic constraint. This Part I introduces the energy balance constraint.

We find that the energy balance constraint poses a relationship between Req and the maximum potential radius of a cold pool Rmax. As a cold pool expands, it dissipates due to surface sensible heating. The Rmax is the radius by which a cold pool’s buoyancy deficit fully recovers, as has been analytically studied by Romps and Jeevanjee (2016). A critical quantity linking Req and Rmax is the gust front propagation speed, which is well represented by the domain-averaged near-surface wind speed uc. It not only determines the surface heat flux but also links the organization of convection in space and time via Req ∼ ucΔt. We find that the ratio Φ ≡ Rmax/Req only depends on the Bowen ratio B, the surface precipitation–evaporation ratio χpe, and the rain sedimentation efficiency (SE). A smaller B means a weaker surface sensible heat flux and, therefore, a larger radius for a cold pool to recover from its buoyancy deficit, increasing Φ. A smaller χpe means the atmosphere produces less precipitation than surface evaporation, yielding weaker cold pools and reducing Φ. A smaller SE means stronger rain evaporation and a higher initial buoyancy deficit of the cold pool, increasing Φ. For the tropical marine condition of B ∼ 0.1 and χpe ∼ 1, the theory predicts that Req cannot reach Rmax, explaining why most cold pools have to collide.

The theory is compared to LES. We perform LES with the rain evaporation amplified by a factor of Eυ, following the method of Nissen and Haerter (2021). A higher Eυ corresponds to stronger rain evaporation, roughly equivalent to a lower SE. We perform more experiments than Nissen and Haerter (2021) and find that a higher rain evaporation rate generally increases Req, but Req reaches an upper bound for Eυ≳0.8. Meanwhile, quantities reflecting the cold pool intensity, e.g., the near-surface mean wind speed and buoyancy fluctuation, still rise with Eυ. Because Rmax represents the overall intensity of a cold pool in its lifetime, these results qualitatively indicate that Rmax/Req increases with Eυ. Then, we calculate Rmax/Req with the energy balance theory, using the diagnosed B, χpe, and SE. The calculated Rmax/Req robustly increases with Eυ, consistent with the qualitative inference.

Part I only points out a constraint between Req and Rmax based on the energy balance. It neither solves Req nor Rmax. Despite this, the finding of Rmax/Req≳1 indicates that cold pools strongly interact. More quantitatively, a higher Rmax/Req means clouds are more likely to be triggered by the intersection of multiple cold pools than an isolated cold pool. Feng et al. (2015) used a cloud-permitting simulation to show that the clouds triggered by intersecting cold pools are usually wider and taller than those triggered by isolated cold pools. Based on this, they called for incorporating the cold pool packing density in convective parameterization schemes—an issue our theory exactly aims to address. We further hypothesize that when cold pools are more densely packed, there might be fewer opportunities for gravity waves to trigger convection. In Part II, we introduce a constraint between Req and Rmax based on a survival competition hypothesis between cold pools. Combining the energy balance constraint and the survival competition constraint yields a solution of Req that agrees with LES and explains why it is capped by an upper bound.

B1

The original theory of Torri and Kuang (2016) uses specific humidity, which has a different denominator (total air mass rather than dry air mass) from mixing ratio. Because the unit of water species output in CM1 is the mixing ratio, we use the mixing ratio as an approximation in Eq. (B1).

Acknowledgments.

This paper series is adapted from chapter 2 of the first author’s Ph.D. thesis. The first author is now supported by the T. C. Chamberlin Fellowship from the University of Chicago. We thank Hugh Morrison, Bowen Zhou, Junfei Li, Zhihong Tan, and Zhaohua Wu for their helpful comments. We thank Stanford Research Computing Center and NCAR CISL for providing the computational resources. We thank George Bryan at NCAR for maintaining and distributing the Bryan Cloud Model 1 for research. We thank the NASA WORLDVIEW team for making the satellite product available. We thank three anonymous reviewers for their insightful comments that greatly improved the scientific quality, structure, and language of the paper series.

Data availability statement.

The supplemental figures, LES movies, a math derivation note, the LES namelist file, and all the figure plotting codes are deposited in Zenodo (https://doi.org/10.5281/zenodo.13785498). The LES data can be obtained by contacting the corresponding author.

APPENDIX A

The Organization of Convection in Space and Time

In this appendix, we analyze the organization of convection in space and time, i.e., the relationship between convective spacing and time interval. The Hövmoller diagram (Fig. A1) confirms that the triggering of most convective events is associated with the passage of at least one active gust front. Convection is a highly intermittent event that only takes a small fraction of space and time, with downdraft following an updraft burst.

Fig. A1.
Fig. A1.

The Hovmöller diagram of the z = 12.5 m water vapor mixing ratio (filled map; g kg−1), which is near the surface, and z = 825 m vertical velocity, which is above the mixed-layer top (white line for −0.7 m s−1 and black line for 1 m s−1 contour). The data use the y = 48-km intercept of the reference test. Only the data between x = 0 km and x = 48 km from t = 4 days to t = 5 days are displayed. The red circles highlight a few convective events.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

We diagnose the convective period using temporal autocorrelation of the mixed layer vertically averaged water vapor mixing ratio with the data between t = 3 days and t = 5 days. The temporal autocorrelation function at each grid point is calculated and then averaged over. Before calculating the autocorrelation, the time series at each grid point has been detrended by subtracting a linear fit. Like the method of determining Req, twice the time lag where the temporal autocorrelation drops to 0.1 is taken as the convective half-period Δt. The diagnostic method is illustrated in Fig. 5b. Figure A2a shows that Δt drops as Eυ increases. Figure A2b shows that Req and Δt are linked by the mean near-surface wind speed uc, which has been shown to represent the cold pool propagation speed in section 2c:
Req∼ucΔt.
Thus, Δt represents the time needed for a cold pool to propagate a distance of Req, an important relation for deriving Rmax/Req in section 3b. Haerter et al. (2019) were probably the first to conceptually propose the kinematic relation between the basic length scale, time scale, and cold pool propagation speed in a self-organized convection. Our diagnosis based on autocorrelation provides a support.
Fig. A2.
Fig. A2.

(a) The convective half-period Δt diagnosed from the temporal autocorrelation function of the mixed layer vertically averaged water vapor mixing ratio. (b) The “*” denotes the relation between Δt and Req/uc, where uc is the domain-averaged near-surface wind speed (z = 12.5 m level). In the plot, Δt, Req, and uc are normalized with their value at Eυ = 1. The dashed line is a 1-to-1 reference.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

APPENDIX B

Theoretical Prediction of Rmax/Req as a Function of Eυ

In this appendix, we calculate Rmax/Req in the LES with the energy balance theory [Eq. (18)], using the diagnosed B, χpe, and SE as the input parameters.

The Bowen ratio B in the LES increases from around 0.06 to 0.1 as Eυ increases from 0.15 to 2.0 (Fig. B1a). The increase of B with Eυ is likely due to the lower subcloud saturation deficit (Fig. B1b) that suppresses the surface evaporation.

Fig. B1.
Fig. B1.

Diagnostic quantities of the LES averaged between t = 3 days and t = 5 days. (a) The Bowen ratio B, which is calculated as the domain-averaged sensible heat flux divided by the domain-averaged latent heat flux. (b) The saturation deficit qυs − qυ in the mixed layer (vertically averaged within the lowest 551 m and then horizontally averaged in the domain). (c) The surface precipitation–evaporation ratio χpe, which is calculated as the domain-averaged surface rainfall rate divided by the domain-averaged surface evaporation rate. (d) The rainwater mixing ratio qr averaged in the downdraft region of the mixed layer.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

The surface precipitation–evaporation ratio χpe in the LES decreases from around 1 to 0.85 as Eυ increases from 0.15 to 2.0 (Fig. B1c). The diagnosed χpe is consistent with the gradual climbing of the domain-averaged column precipitable water (PW, Fig. 3a), which is more prominent for a higher Eυ test.

Because our LES did not output the rain evaporation rate variable, the sedimentation efficiency (SE) is diagnosed indirectly. It is estimated with the rainwater evaporation equation. According to Torri and Kuang (2016), the material derivative of the rainwater mixing ratio qr approximately obeysB1
DqrDt∝−Eυ(qυs−qυ)qr1/3,
where D/Dt is the substantial derivative, qυs is the saturation vapor mixing ratio, and qυs − qυ is the saturation deficit. Figure B1b shows that the saturation deficit averaged in the mixed layer drops by around 25% as Eυ increases from 0.15 to 2.0. This results from the stronger rain evaporative moistening of the mixed layer. The qr averaged in the mixed-layer portion (lowest 551 m) of the downdraft region increases with Eυ (Fig. B1d). The downdraft region is defined as the horizontal zone where the rainwater mixing ratio at z = 551 m is higher than 10−2 g kg−1 and the vertical velocity at this level is downward. The increase of qr with Eυ results from multiple factors, which includes the direct effect of the higher evaporation rate, the smaller saturation deficit that suppresses evaporation, and the wider clouds (to be shown in Part II) that enhances the updraft and likely enhances the rainwater production rate in convective towers. We need to set a prescribed reference value SEref for the Eυ = 1 case and then use Eq. (B1) and the diagnosed qυs − qυ and qr to infer other cases. We let (qυs − qυ)|ref and qr|ref be the diagnosed saturation deficit and rainwater mixing ratio of the Eυ = 1 test, respectively. The SE is diagnosed with
1−SE∝Eυ(qυs−qυ)qr1/3⇒1−SE1−SEref≈Eυ1qυs−qυ(qυs−qυ)|ref(qrqr|ref)1/3⇒SE≈1−Eυ(1−SEref)qυs−qυ(qυs−qυ)|ref(qrqr|ref)1/3.
Using the diagnosed B, χpe, qυs − qυ, and qr shown in Fig. B1, and choosing SEref = 0.55 and SEref = 0.65, we calculate Φ with Eqs. (18) and (B2) and plot it in Fig. B2. These two values of SEref roughly represent the SE of two simulated deep convective clouds reported by Langhans et al. (2015), one for a normal case (SE = 0.62) and the other in a humidified environment (SE = 0.55). The diagnosed Φ rises from around 1 at Eυ = 0.15 to around 3 at Eυ = 2.
Fig. B2.
Fig. B2.

The Φ calculated with Eq. (18), using the diagnosed B, qυs − qυ, and χpe in Fig. B1. The SE is calculated with Eq. (B2). The red dots denote the SEref = 0.55 case, and the yellow dots denote the SEref = 0.65 case.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-23-0143.1

REFERENCES

  • Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci., 31, 674–701, https://doi.org/10.1175/1520-0469(1974)031<0674:IOACCE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Benjamin, T. B., 1968: Gravity currents and related phenomena. J. Fluid Mech., 31, 209–248, https://doi.org/10.1017/S0022112068000133.

    • Search Google Scholar
    • Export Citation

  • Böing, S. J., H. J. J. Jonker, A. P. Siebesma, and W. W. Grabowski, 2012: Influence of the subcloud layer on the development of a deep convective ensemble. J. Atmos. Sci., 69, 2682–2698, https://doi.org/10.1175/JAS-D-11-0317.1.

    • Search Google Scholar
    • Export Citation

  • Bowen, I. S., 1926: The ratio of heat losses by conduction and by evaporation from any water surface. Phys. Rev., 27, 779–787, https://doi.org/10.1103/PhysRev.27.779.

    • Search Google Scholar
    • Export Citation

  • Bretherton, C. S., P. N. Blossey, and M. Khairoutdinov, 2005: An energy-balance analysis of deep convective self-aggregation above uniform SST. J. Atmos. Sci., 62, 4273–4292, https://doi.org/10.1175/JAS3614.1.

    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 2917–2928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., and R. Rotunno, 2009: The maximum intensity of tropical cyclones in axisymmetric numerical model simulations. Mon. Wea. Rev., 137, 1770–1789, https://doi.org/10.1175/2008MWR2709.1.

    • Search Google Scholar
    • Export Citation

  • Byers, H. R., and R. R. Braham, Jr., 1949: The Thunderstorm: Final Report of the Thunderstorm Project. U.S. Government Printing Office, 287 pp.

  • Clough, S. A., M. W. Shephard, E. J. Mlawer, J. S. Delamere, M. J. Iacono, K. Cady-Pereira, S. Boukabara, and P. D. Brown, 2005: Atmospheric radiative transfer modeling: A summary of the AER codes. J. Quant. Spectrosc. Radiat. Transfer, 91, 233–244, https://doi.org/10.1016/j.jqsrt.2004.05.058.

    • Search Google Scholar
    • Export Citation

  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18, 495–527, https://doi.org/10.1007/BF00119502.

    • Search Google Scholar
    • Export Citation

  • Droegemeier, K. K., and R. B. Wilhelmson, 1985: Three-dimensional numerical modeling of convection produced by interacting thunderstorm outflows. Part I: Control simulation and low-level moisture variations. J. Atmos. Sci., 42, 2381–2403, https://doi.org/10.1175/1520-0469(1985)042<2381:TDNMOC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Emanuel, K. A., 1994: Atmospheric Convection. Oxford University Press, 580 pp.

  • Falk, N. M., and S. C. van den Heever, 2023: Environmental modulation of mechanical and thermodynamic forcing from cold pool collisions. J. Atmos. Sci., 80, 375–395, https://doi.org/10.1175/JAS-D-22-0020.1.

    • Search Google Scholar
    • Export Citation

  • Feng, Z., S. Hagos, A. K. Rowe, C. D. Burleyson, M. N. Martini, and S. P. de Szoeke, 2015: Mechanisms of convective cloud organization by cold pools over tropical warm ocean during the AMIE/DYNAMO field campaign. J. Adv. Model. Earth Syst., 7, 357–381, https://doi.org/10.1002/2014MS000384.

    • Search Google Scholar
    • Export Citation

  • Ferrier, B. S., and R. A. Houze Jr., 1989: One-dimensional time-dependent modeling of gate cumulonimbus convection. J. Atmos. Sci., 46, 330–352, https://doi.org/10.1175/1520-0469(1989)046<0330:ODTDMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and M. E O’Neill, 2024: The small-amplitude dynamics of spontaneous tropical cyclogenesis. Part I: Experiments with amplified longwave radiative feedback. J. Atmos. Sci., 81, 381–399, https://doi.org/10.1175/JAS-D-23-0170.1.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and M. E O’Neill, 2025: Mean cold pool size of quasi-equilibrium convection. Part II: The survival competition hypothesis. J. Atmos. Sci., 82, 251–265, https://doi.org/10.1175/JAS-D-24-0052.1.

    • Search Google Scholar
    • Export Citation

  • Fuglestvedt, H. F., and J. O. Haerter, 2020: Cold pools as conveyor belts of moisture. Geophys. Res. Lett., 47, e2020GL087319, https://doi.org/10.1029/2020GL087319.

    • Search Google Scholar
    • Export Citation

  • Gentine, P., A. Garelli, S.-B. Park, J. Nie, G. Torri, and Z. Kuang, 2016: Role of surface heat fluxes underneath cold pools. Geophys. Res. Lett., 43, 874–883, https://doi.org/10.1002/2015GL067262.

    • Search Google Scholar
    • Export Citation

  • Grandpeix, J.-Y., and J.-P. Lafore, 2010: A density current parameterization coupled with Emanuel’s convection scheme. Part I: The models. J. Atmos. Sci., 67, 881–897, https://doi.org/10.1175/2009JAS3044.1.

    • Search Google Scholar
    • Export Citation

  • Grant, L. D., and S. C. van den Heever, 2016: Cold pool dissipation. J. Geophys. Res. Atmos., 121, 1138–1155, https://doi.org/10.1002/2015JD023813.

    • Search Google Scholar
    • Export Citation

  • Grant, L. D., T. P. Lane, and S. C. van den Heever, 2018: The role of cold pools in tropical oceanic convective systems. J. Atmos. Sci., 75, 2615–2634, https://doi.org/10.1175/JAS-D-17-0352.1.

    • Search Google Scholar
    • Export Citation

  • Haerter, J. O., P. Berg, and C. Moseley, 2017: Precipitation onset as the temporal reference in convective self-organization. Geophys. Res. Lett., 44, 6450–6459, https://doi.org/10.1002/2017GL073342.

    • Search Google Scholar
    • Export Citation

  • Haerter, J. O., S. J. Böing, O. Henneberg, and S. B. Nissen, 2019: Circling in on convective organization. Geophys. Res. Lett., 46, 7024–7034, https://doi.org/10.1029/2019GL082092.

    • Search Google Scholar
    • Export Citation

  • Held, I. M., R. S. Hemler, and V. Ramaswamy, 1993: Radiative-convective equilibrium with explicit two-dimensional moist convection. J. Atmos. Sci., 50, 3909–3927, https://doi.org/10.1175/1520-0469(1993)050<3909:RCEWET>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Jeevanjee, N., and D. M. Romps, 2013: Convective self-aggregation, cold pools, and domain size. Geophys. Res. Lett., 40, 994–998, https://doi.org/10.1002/grl.50204.

    • Search Google Scholar
    • Export Citation

  • Jiménez, P. A., J. Dudhia, J. F. González-Rouco, J. Navarro, J. P. Montávez, and E. García-Bustamante, 2012: A revised scheme for the WRF surface layer formulation. Mon. Wea. Rev., 140, 898–918, https://doi.org/10.1175/MWR-D-11-00056.1.

    • Search Google Scholar
    • Export Citation

  • Jo, Y.-H., X.-H. Yan, J. Pan, M.-X. He, and W. T. Liu, 2002: Calculation of the Bowen ratio in the tropical Pacific using sea surface temperature data. J. Geophys. Res., 107, 3134, https://doi.org/10.1029/2001JC001150.

    • Search Google Scholar
    • Export Citation

  • Langhans, W., and D. M. Romps, 2015: The origin of water vapor rings in tropical oceanic cold pools. Geophys. Res. Lett., 42, 7825–7834, https://doi.org/10.1002/2015GL065623.

    • Search Google Scholar
    • Export Citation

  • Langhans, W., K. Yeo, and D. M. Romps, 2015: Lagrangian investigation of the precipitation efficiency of convective clouds. J. Atmos. Sci., 72, 1045–1062, https://doi.org/10.1175/JAS-D-14-0159.1.

    • Search Google Scholar
    • Export Citation

  • Lutsko, N. J., and T. W. Cronin, 2018: Increase in precipitation efficiency with surface warming in radiative-convective equilibrium. J. Adv. Model. Earth Syst., 10, 2992–3010, https://doi.org/10.1029/2018MS001482.

    • Search Google Scholar
    • Export Citation

  • Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with a convective adjustment. J. Atmos. Sci., 21, 361–385, https://doi.org/10.1175/1520-0469(1964)021<0361:TEOTAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mapes, B. E., 1997: Equilibrium vs. activation control of large-scale variations of tropical deep convection. The Physics and Parameterization of Moist Atmospheric Convection, Springer, 321–358.

  • Mapes, B. E., 2000: Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model. J. Atmos. Sci., 57, 1515–1535, https://doi.org/10.1175/1520-0469(2000)057<1515:CISSTE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Meyer, B., and J. O. Haerter, 2020: Mechanical forcing of convection by cold pools: Collisions and energy scaling. J. Adv. Model. Earth Syst., 12, e2020MS002281, https://doi.org/10.1029/2020MS002281.

    • Search Google Scholar
    • Export Citation

  • Morrison, H., J. A. Curry, and V. I. Khvorostyanov, 2005: A new double-moment microphysics parameterization for application in cloud and climate models. Part I: Description. J. Atmos. Sci., 62, 1665–1677, https://doi.org/10.1175/JAS3446.1.

    • Search Google Scholar
    • Export Citation

  • Nissen, S. B., and J. O. Haerter, 2021: Circling in on convective self-aggregation. J. Geophys. Res. Atmos., 126, e2021JD035331, https://doi.org/10.1029/2021JD035331.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and T. Takahashi, 1971: Numerical simulation of the life cycle of a thunderstorm cell. Mon. Wea. Rev., 99, 895–911, https://doi.org/10.1175/1520-0493(1971)099<0895:NSOTLC>2.3.CO;2.

    • Search Google Scholar
    • Export Citation

  • Parodi, A., and K. Emanuel, 2009: A theory for buoyancy and velocity scales in deep moist convection. J. Atmos. Sci., 66, 3449–3463, https://doi.org/10.1175/2009JAS3103.1.

    • Search Google Scholar
    • Export Citation

  • Peixóto, J. P., and A. H. Oort, 1984: Physics of climate. Rev. Mod. Phys., 56, 365–429, https://doi.org/10.1103/RevModPhys.56.365.

  • Riehl, H., and J. S. Malkus, 1958: On the heat balance in the equatorial trough zone. Geophysica, 6, 503–538.

  • Robe, F. R., and K. A. Emanuel, 1996: Moist convective scaling: Some inferences from three-dimensional cloud ensemble simulations. J. Atmos. Sci., 53, 3265–3275, https://doi.org/10.1175/1520-0469(1996)053<3265:MCSSIF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2014a: An analytical model for tropical relative humidity. J. Climate, 27, 7432–7449, https://doi.org/10.1175/JCLI-D-14-00255.1.

    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2014b: Rayleigh damping in the free troposphere. J. Atmos. Sci., 71, 553–565, https://doi.org/10.1175/JAS-D-13-062.1.

  • Romps, D. M., and N. Jeevanjee, 2016: On the sizes and lifetimes of cold pools. Quart. J. Roy. Meteor. Soc., 142, 1517–1527, https://doi.org/10.1002/qj.2754.

    • Search Google Scholar
    • Export Citation

  • Ross, A. N., A. M. Tompkins, and D. J. Parker, 2004: Simple models of the role of surface fluxes in convective cold pool evolution. J. Atmos. Sci., 61, 1582–1595, https://doi.org/10.1175/1520-0469(2004)061<1582:SMOTRO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Sakaeda, N., and G. Torri, 2023: The observed effects of cold pools on convection triggering and organization during DYNAMO/AMIE. J. Geophys. Res. Atmos., 128, e2023JD038635, https://doi.org/10.1029/2023JD038635.

    • Search Google Scholar
    • Export Citation

  • Schlemmer, L., and C. Hohenegger, 2014: The formation of wider and deeper clouds as a result of cold-pool dynamics. J. Atmos. Sci., 71, 2842–2858, https://doi.org/10.1175/JAS-D-13-0170.1.

    • Search Google Scholar
    • Export Citation

  • Thayer-Calder, K., and D. Randall, 2015: A numerical investigation of boundary layer quasi-equilibrium. Geophys. Res. Lett., 42, 550–556, https://doi.org/10.1002/2014GL062649.

    • Search Google Scholar
    • Export Citation

  • Tompkins, A. M., 2001: Organization of tropical convection in low vertical wind shears: The role of cold pools. J. Atmos. Sci., 58, 1650–1672, https://doi.org/10.1175/1520-0469(2001)058<1650:OOTCIL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Torri, G., and Z. Kuang, 2016: A Lagrangian study of precipitation-driven downdrafts. J. Atmos. Sci., 73, 839–854, https://doi.org/10.1175/JAS-D-15-0222.1.

    • Search Google Scholar
    • Export Citation

  • Torri, G., and Z. Kuang, 2019: On cold pool collisions in tropical boundary layers. Geophys. Res. Lett., 46, 399–407, https://doi.org/10.1029/2018GL080501.

    • Search Google Scholar
    • Export Citation

  • Torri, G., Z. Kuang, and Y. Tian, 2015: Mechanisms for convection triggering by cold pools. Geophys. Res. Lett., 42, 1943–1950, https://doi.org/10.1002/2015GL063227.

    • Search Google Scholar
    • Export Citation

  • Ungarish, M., 2009: An Introduction to Gravity Currents and Intrusions. 1st ed. CRC Press, 512 pp.

  • von Kármán, T., 1940: The engineer grapples with nonlinear problems. Bull. Amer. Math. Soc., 46, 615–683, https://doi.org/10.1090/S0002-9904-1940-07266-0.

    • Search Google Scholar
    • Export Citation

  • Wakimoto, R. M., 1982: The life cycle of thunderstorm gust fronts as viewed with Doppler radar and rawinsonde data. Mon. Wea. Rev., 110, 1060–1082, https://doi.org/10.1175/1520-0493(1982)110<1060:TLCOTG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Wang, Y., C. A. Davis, and Y. Huang, 2019: Dynamics of lower-tropospheric vorticity in idealized simulations of tropical cyclone formation. J. Atmos. Sci., 76, 707–727, https://doi.org/10.1175/JAS-D-18-0219.1.

    • Search Google Scholar
    • Export Citation

  • Yang, Q., L. R. Leung, Z. Feng, F. Song, and X. Chen, 2021: A simple Lagrangian parcel model for the initiation of summertime mesoscale convective systems over the central United States. J. Atmos. Sci., 78, 3537–3558, https://doi.org/10.1175/JAS-D-21-0136.1.

    • Search Google Scholar
    • Export Citation

  • Yano, J.-I., and R. S. Plant, 2012: Convective quasi-equilibrium. Rev. Geophys., 50, RG4004, https://doi.org/10.1029/2011RG000378.

Supplementary Materials

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci., 31, 674–701, https://doi.org/10.1175/1520-0469(1974)031<0674:IOACCE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Benjamin, T. B., 1968: Gravity currents and related phenomena. J. Fluid Mech., 31, 209–248, https://doi.org/10.1017/S0022112068000133.

    • Search Google Scholar
    • Export Citation

  • Böing, S. J., H. J. J. Jonker, A. P. Siebesma, and W. W. Grabowski, 2012: Influence of the subcloud layer on the development of a deep convective ensemble. J. Atmos. Sci., 69, 2682–2698, https://doi.org/10.1175/JAS-D-11-0317.1.

    • Search Google Scholar
    • Export Citation

  • Bowen, I. S., 1926: The ratio of heat losses by conduction and by evaporation from any water surface. Phys. Rev., 27, 779–787, https://doi.org/10.1103/PhysRev.27.779.

    • Search Google Scholar
    • Export Citation

  • Bretherton, C. S., P. N. Blossey, and M. Khairoutdinov, 2005: An energy-balance analysis of deep convective self-aggregation above uniform SST. J. Atmos. Sci., 62, 4273–4292, https://doi.org/10.1175/JAS3614.1.

    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 2917–2928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., and R. Rotunno, 2009: The maximum intensity of tropical cyclones in axisymmetric numerical model simulations. Mon. Wea. Rev., 137, 1770–1789, https://doi.org/10.1175/2008MWR2709.1.

    • Search Google Scholar
    • Export Citation

  • Byers, H. R., and R. R. Braham, Jr., 1949: The Thunderstorm: Final Report of the Thunderstorm Project. U.S. Government Printing Office, 287 pp.

  • Clough, S. A., M. W. Shephard, E. J. Mlawer, J. S. Delamere, M. J. Iacono, K. Cady-Pereira, S. Boukabara, and P. D. Brown, 2005: Atmospheric radiative transfer modeling: A summary of the AER codes. J. Quant. Spectrosc. Radiat. Transfer, 91, 233–244, https://doi.org/10.1016/j.jqsrt.2004.05.058.

    • Search Google Scholar
    • Export Citation

  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18, 495–527, https://doi.org/10.1007/BF00119502.

    • Search Google Scholar
    • Export Citation

  • Droegemeier, K. K., and R. B. Wilhelmson, 1985: Three-dimensional numerical modeling of convection produced by interacting thunderstorm outflows. Part I: Control simulation and low-level moisture variations. J. Atmos. Sci., 42, 2381–2403, https://doi.org/10.1175/1520-0469(1985)042<2381:TDNMOC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Emanuel, K. A., 1994: Atmospheric Convection. Oxford University Press, 580 pp.

  • Falk, N. M., and S. C. van den Heever, 2023: Environmental modulation of mechanical and thermodynamic forcing from cold pool collisions. J. Atmos. Sci., 80, 375–395, https://doi.org/10.1175/JAS-D-22-0020.1.

    • Search Google Scholar
    • Export Citation

  • Feng, Z., S. Hagos, A. K. Rowe, C. D. Burleyson, M. N. Martini, and S. P. de Szoeke, 2015: Mechanisms of convective cloud organization by cold pools over tropical warm ocean during the AMIE/DYNAMO field campaign. J. Adv. Model. Earth Syst., 7, 357–381, https://doi.org/10.1002/2014MS000384.

    • Search Google Scholar
    • Export Citation

  • Ferrier, B. S., and R. A. Houze Jr., 1989: One-dimensional time-dependent modeling of gate cumulonimbus convection. J. Atmos. Sci., 46, 330–352, https://doi.org/10.1175/1520-0469(1989)046<0330:ODTDMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and M. E O’Neill, 2024: The small-amplitude dynamics of spontaneous tropical cyclogenesis. Part I: Experiments with amplified longwave radiative feedback. J. Atmos. Sci., 81, 381–399, https://doi.org/10.1175/JAS-D-23-0170.1.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and M. E O’Neill, 2025: Mean cold pool size of quasi-equilibrium convection. Part II: The survival competition hypothesis. J. Atmos. Sci., 82, 251–265, https://doi.org/10.1175/JAS-D-24-0052.1.

    • Search Google Scholar
    • Export Citation

  • Fuglestvedt, H. F., and J. O. Haerter, 2020: Cold pools as conveyor belts of moisture. Geophys. Res. Lett., 47, e2020GL087319, https://doi.org/10.1029/2020GL087319.

    • Search Google Scholar
    • Export Citation

  • Gentine, P., A. Garelli, S.-B. Park, J. Nie, G. Torri, and Z. Kuang, 2016: Role of surface heat fluxes underneath cold pools. Geophys. Res. Lett., 43, 874–883, https://doi.org/10.1002/2015GL067262.

    • Search Google Scholar
    • Export Citation

  • Grandpeix, J.-Y., and J.-P. Lafore, 2010: A density current parameterization coupled with Emanuel’s convection scheme. Part I: The models. J. Atmos. Sci., 67, 881–897, https://doi.org/10.1175/2009JAS3044.1.

    • Search Google Scholar
    • Export Citation

  • Grant, L. D., and S. C. van den Heever, 2016: Cold pool dissipation. J. Geophys. Res. Atmos., 121, 1138–1155, https://doi.org/10.1002/2015JD023813.

    • Search Google Scholar
    • Export Citation

  • Grant, L. D., T. P. Lane, and S. C. van den Heever, 2018: The role of cold pools in tropical oceanic convective systems. J. Atmos. Sci., 75, 2615–2634, https://doi.org/10.1175/JAS-D-17-0352.1.

    • Search Google Scholar
    • Export Citation

  • Haerter, J. O., P. Berg, and C. Moseley, 2017: Precipitation onset as the temporal reference in convective self-organization. Geophys. Res. Lett., 44, 6450–6459, https://doi.org/10.1002/2017GL073342.

    • Search Google Scholar
    • Export Citation

  • Haerter, J. O., S. J. Böing, O. Henneberg, and S. B. Nissen, 2019: Circling in on convective organization. Geophys. Res. Lett., 46, 7024–7034, https://doi.org/10.1029/2019GL082092.

    • Search Google Scholar
    • Export Citation

  • Held, I. M., R. S. Hemler, and V. Ramaswamy, 1993: Radiative-convective equilibrium with explicit two-dimensional moist convection. J. Atmos. Sci., 50, 3909–3927, https://doi.org/10.1175/1520-0469(1993)050<3909:RCEWET>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Jeevanjee, N., and D. M. Romps, 2013: Convective self-aggregation, cold pools, and domain size. Geophys. Res. Lett., 40, 994–998, https://doi.org/10.1002/grl.50204.

    • Search Google Scholar
    • Export Citation

  • Jiménez, P. A., J. Dudhia, J. F. González-Rouco, J. Navarro, J. P. Montávez, and E. García-Bustamante, 2012: A revised scheme for the WRF surface layer formulation. Mon. Wea. Rev., 140, 898–918, https://doi.org/10.1175/MWR-D-11-00056.1.

    • Search Google Scholar
    • Export Citation

  • Jo, Y.-H., X.-H. Yan, J. Pan, M.-X. He, and W. T. Liu, 2002: Calculation of the Bowen ratio in the tropical Pacific using sea surface temperature data. J. Geophys. Res., 107, 3134, https://doi.org/10.1029/2001JC001150.

    • Search Google Scholar
    • Export Citation

  • Langhans, W., and D. M. Romps, 2015: The origin of water vapor rings in tropical oceanic cold pools. Geophys. Res. Lett., 42, 7825–7834, https://doi.org/10.1002/2015GL065623.

    • Search Google Scholar
    • Export Citation

  • Langhans, W., K. Yeo, and D. M. Romps, 2015: Lagrangian investigation of the precipitation efficiency of convective clouds. J. Atmos. Sci., 72, 1045–1062, https://doi.org/10.1175/JAS-D-14-0159.1.

    • Search Google Scholar
    • Export Citation

  • Lutsko, N. J., and T. W. Cronin, 2018: Increase in precipitation efficiency with surface warming in radiative-convective equilibrium. J. Adv. Model. Earth Syst., 10, 2992–3010, https://doi.org/10.1029/2018MS001482.

    • Search Google Scholar
    • Export Citation

  • Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with a convective adjustment. J. Atmos. Sci., 21, 361–385, https://doi.org/10.1175/1520-0469(1964)021<0361:TEOTAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mapes, B. E., 1997: Equilibrium vs. activation control of large-scale variations of tropical deep convection. The Physics and Parameterization of Moist Atmospheric Convection, Springer, 321–358.

  • Mapes, B. E., 2000: Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model. J. Atmos. Sci., 57, 1515–1535, https://doi.org/10.1175/1520-0469(2000)057<1515:CISSTE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Meyer, B., and J. O. Haerter, 2020: Mechanical forcing of convection by cold pools: Collisions and energy scaling. J. Adv. Model. Earth Syst., 12, e2020MS002281, https://doi.org/10.1029/2020MS002281.

    • Search Google Scholar
    • Export Citation

  • Morrison, H., J. A. Curry, and V. I. Khvorostyanov, 2005: A new double-moment microphysics parameterization for application in cloud and climate models. Part I: Description. J. Atmos. Sci., 62, 1665–1677, https://doi.org/10.1175/JAS3446.1.

    • Search Google Scholar
    • Export Citation

  • Nissen, S. B., and J. O. Haerter, 2021: Circling in on convective self-aggregation. J. Geophys. Res. Atmos., 126, e2021JD035331, https://doi.org/10.1029/2021JD035331.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and T. Takahashi, 1971: Numerical simulation of the life cycle of a thunderstorm cell. Mon. Wea. Rev., 99, 895–911, https://doi.org/10.1175/1520-0493(1971)099<0895:NSOTLC>2.3.CO;2.

    • Search Google Scholar
    • Export Citation

  • Parodi, A., and K. Emanuel, 2009: A theory for buoyancy and velocity scales in deep moist convection. J. Atmos. Sci., 66, 3449–3463, https://doi.org/10.1175/2009JAS3103.1.

    • Search Google Scholar
    • Export Citation

  • Peixóto, J. P., and A. H. Oort, 1984: Physics of climate. Rev. Mod. Phys., 56, 365–429, https://doi.org/10.1103/RevModPhys.56.365.

  • Riehl, H., and J. S. Malkus, 1958: On the heat balance in the equatorial trough zone. Geophysica, 6, 503–538.

  • Robe, F. R., and K. A. Emanuel, 1996: Moist convective scaling: Some inferences from three-dimensional cloud ensemble simulations. J. Atmos. Sci., 53, 3265–3275, https://doi.org/10.1175/1520-0469(1996)053<3265:MCSSIF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2014a: An analytical model for tropical relative humidity. J. Climate, 27, 7432–7449, https://doi.org/10.1175/JCLI-D-14-00255.1.

    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2014b: Rayleigh damping in the free troposphere. J. Atmos. Sci., 71, 553–565, https://doi.org/10.1175/JAS-D-13-062.1.

  • Romps, D. M., and N. Jeevanjee, 2016: On the sizes and lifetimes of cold pools. Quart. J. Roy. Meteor. Soc., 142, 1517–1527, https://doi.org/10.1002/qj.2754.

    • Search Google Scholar
    • Export Citation

  • Ross, A. N., A. M. Tompkins, and D. J. Parker, 2004: Simple models of the role of surface fluxes in convective cold pool evolution. J. Atmos. Sci., 61, 1582–1595, https://doi.org/10.1175/1520-0469(2004)061<1582:SMOTRO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Sakaeda, N., and G. Torri, 2023: The observed effects of cold pools on convection triggering and organization during DYNAMO/AMIE. J. Geophys. Res. Atmos., 128, e2023JD038635, https://doi.org/10.1029/2023JD038635.

    • Search Google Scholar
    • Export Citation

  • Schlemmer, L., and C. Hohenegger, 2014: The formation of wider and deeper clouds as a result of cold-pool dynamics. J. Atmos. Sci., 71, 2842–2858, https://doi.org/10.1175/JAS-D-13-0170.1.

    • Search Google Scholar
    • Export Citation

  • Thayer-Calder, K., and D. Randall, 2015: A numerical investigation of boundary layer quasi-equilibrium. Geophys. Res. Lett., 42, 550–556, https://doi.org/10.1002/2014GL062649.

    • Search Google Scholar
    • Export Citation

  • Tompkins, A. M., 2001: Organization of tropical convection in low vertical wind shears: The role of cold pools. J. Atmos. Sci., 58, 1650–1672, https://doi.org/10.1175/1520-0469(2001)058<1650:OOTCIL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Torri, G., and Z. Kuang, 2016: A Lagrangian study of precipitation-driven downdrafts. J. Atmos. Sci., 73, 839–854, https://doi.org/10.1175/JAS-D-15-0222.1.

    • Search Google Scholar
    • Export Citation

  • Torri, G., and Z. Kuang, 2019: On cold pool collisions in tropical boundary layers. Geophys. Res. Lett., 46, 399–407, https://doi.org/10.1029/2018GL080501.

    • Search Google Scholar
    • Export Citation

  • Torri, G., Z. Kuang, and Y. Tian, 2015: Mechanisms for convection triggering by cold pools. Geophys. Res. Lett., 42, 1943–1950, https://doi.org/10.1002/2015GL063227.

    • Search Google Scholar
    • Export Citation

  • Ungarish, M., 2009: An Introduction to Gravity Currents and Intrusions. 1st ed. CRC Press, 512 pp.

  • von Kármán, T., 1940: The engineer grapples with nonlinear problems. Bull. Amer. Math. Soc., 46, 615–683, https://doi.org/10.1090/S0002-9904-1940-07266-0.

    • Search Google Scholar
    • Export Citation

  • Wakimoto, R. M., 1982: The life cycle of thunderstorm gust fronts as viewed with Doppler radar and rawinsonde data. Mon. Wea. Rev., 110, 1060–1082, https://doi.org/10.1175/1520-0493(1982)110<1060:TLCOTG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Wang, Y., C. A. Davis, and Y. Huang, 2019: Dynamics of lower-tropospheric vorticity in idealized simulations of tropical cyclone formation. J. Atmos. Sci., 76, 707–727, https://doi.org/10.1175/JAS-D-18-0219.1.

    • Search Google Scholar
    • Export Citation

  • Yang, Q., L. R. Leung, Z. Feng, F. Song, and X. Chen, 2021: A simple Lagrangian parcel model for the initiation of summertime mesoscale convective systems over the central United States. J. Atmos. Sci., 78, 3537–3558, https://doi.org/10.1175/JAS-D-21-0136.1.

    • Search Google Scholar
    • Export Citation

  • Yano, J.-I., and R. S. Plant, 2012: Convective quasi-equilibrium. Rev. Geophys., 50, RG4004, https://doi.org/10.1029/2011RG000378.

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

    A snapshot of the corrected reflectance (true color) NOAA-20/VIIRS satellite product on 14 Jul 2021, downloaded from the NASA Worldview website. The longitudinal width is 331 km, and the latitudinal width is 222 km. The down-to-the-hour timestamp is not available.

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

    A schematic diagram for illustrating the two important length scales studied in this two-paper series: the mean radius Req and the maximum potential radius Rmax of cold pools in quasi-equilibrium convection. The Req represents half the deep convective spacing at a snapshot. The Rmax is the radius by which a cold pool’s buoyancy fully recovers to the environmental value.

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

    (a) The time evolution of the domain-averaged column PW (m) for Eυ = 0.2 (blue line), Eυ = 0.5 (red line), Eυ = 1.0 (the reference test, yellow line), and Eυ = 2.0 (purple line). (b) As in (a), but for the spatial standard deviation of column PW with a logarithmic ordinate. (c) As in (a), but for the time evolution of the mean cold pool radius Req. The overshooting of Req for the Eυ = 2.0 test between t = 3.5 days and t = 4 days is judged as an occasional event, which does not appear in the larger-domain (144 × 144 km2) simulation (Fig. S2). In all panels, the shading shows the period between t = 3 days and t = 5 days, during which the quasi-equilibrium statistics are calculated.

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

    (top) The water vapor mixing ratio vertically averaged within the lowest 551 m of the domain at t = 4 days. (a)–(d) The Eυ = 0.2, 0.5, 1.0, and 2.0 tests, respectively. (bottom) The z = 12.5 m (near-surface) potential temperature at the same time. In each plot, the white lines denote the 10−2 g kg−1 contours of cloud liquid water mixing ratio at z ≈ 4 km, which visualizes the cloud position. The cold pool sizes in the Eυ = 2 test are visually larger than those in the Eυ = 1, which seems to oppose the stagnancy behavior diagnosed with the time average. We argue that a single snapshot is insufficient to confirm the stagnancy of Req with Eυ because the cold pool size fluctuates over time. The LES movies can be downloaded from online (https://doi.org/10.5281/zenodo.13785498).

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

    (a) The temporally averaged (a 2-day-long time series between t = 3 days and t = 5 days) spatial autocorrelation of the mixed layer vertically averaged water vapor mixing ratio for Eυ = 0.2 (blue line), Eυ = 0.5 (red line), Eυ = 1.0 (yellow line), and Eυ = 2.0 (purple line). An additional dashed black line denotes the 0.1 autocorrelation value used to diagnose the mean cold pool radius. (b) As in (a), but for the temporal autocorrelation of the mixed layer vertically averaged water vapor mixing ratio, using the time series between t = 3 days and t = 5 days. The time series have been detrended with a linear fit. The temporal autocorrelation is calculated on each of the 480 × 480 grid points and then averaged. Panel (b) is discussed in appendix A.

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

    Some statistics averaged in the slot between t = 3 days and t = 5 days. (a) The blue dots denote the mean cold pool radius Req diagnosed from the LES for different Eυ. The shading denotes the ±1 standard deviation range of the Req time series. (b) The domain standard deviation of the near-surface (z = 12.5 m, the lowest model level) potential temperature. The shading denotes the ±1 standard deviation range of the time series. (c) The standard deviation of the near-surface buoyancy vs the square of the domain-averaged near-surface wind speed uc (black dots). In addition, we plot the domain-averaged CIN as red crosses.

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

    A schematic diagram of the single cold pool evolution problem studied by Romps and Jeevanjee (2016), using a box model. The cold pool is produced by a total evaporative cooling of ΔQe (J). The initial cold air is assumed to be a cylinder (volume V0, radius R0, and height H0) with a uniform potential temperature anomaly θ′|R=R0. As the cold pool collapses, it keeps a cylindrical shape with a radius of R and height Hc, and its volume V grows by entrainment. The color depth of the shading denotes the magnitude of the cold pool’s potential temperature deficit, which is assumed to be uniform within the cold pool. The cold pool dissipates under the surface sensible heat flux Fsurf,s (W m−2). The cold pool radius by which the potential temperature deficit reduces to zero is the maximum potential radius Rmax.

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

    A schematic diagram for the organization of convection in space and time. The horizontal axis denotes a certain spatial direction x. The vertical axis denotes time t. The blue shading denotes the trajectories of gust fronts in space and time. The equilibrium cold pool radius Req is half the deep convective spacing at a snapshot. It takes a cold pool Δt time to cover Req distance at a mean speed of uc ∼ Req/Δt.

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

    Plots of Φ ≡ Rmax/Req for B = 0.05 (blue line) and B = 0.1 (red line) predicted by the energy balance constraint. Both plots use χpe = 1. The Φ < 1 regime is unphysical, but we retain it in the plots.

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    The Hovmöller diagram of the z = 12.5 m water vapor mixing ratio (filled map; g kg−1), which is near the surface, and z = 825 m vertical velocity, which is above the mixed-layer top (white line for −0.7 m s−1 and black line for 1 m s−1 contour). The data use the y = 48-km intercept of the reference test. Only the data between x = 0 km and x = 48 km from t = 4 days to t = 5 days are displayed. The red circles highlight a few convective events.

  • <sc>Fig</sc>. A2.
    View in gallery
    Fig. A2.

    (a) The convective half-period Δt diagnosed from the temporal autocorrelation function of the mixed layer vertically averaged water vapor mixing ratio. (b) The “*” denotes the relation between Δt and Req/uc, where uc is the domain-averaged near-surface wind speed (z = 12.5 m level). In the plot, Δt, Req, and uc are normalized with their value at Eυ = 1. The dashed line is a 1-to-1 reference.

  • <sc>Fig</sc>. B1.
    View in gallery
    Fig. B1.

    Diagnostic quantities of the LES averaged between t = 3 days and t = 5 days. (a) The Bowen ratio B, which is calculated as the domain-averaged sensible heat flux divided by the domain-averaged latent heat flux. (b) The saturation deficit qυs − qυ in the mixed layer (vertically averaged within the lowest 551 m and then horizontally averaged in the domain). (c) The surface precipitation–evaporation ratio χpe, which is calculated as the domain-averaged surface rainfall rate divided by the domain-averaged surface evaporation rate. (d) The rainwater mixing ratio qr averaged in the downdraft region of the mixed layer.

  • <sc>Fig</sc>. B2.
    View in gallery
    Fig. B2.

    The Φ calculated with Eq. (18), using the diagnosed B, qυs − qυ, and χpe in Fig. B1. The SE is calculated with Eq. (B2). The red dots denote the SEref = 0.55 case, and the yellow dots denote the SEref = 0.65 case.

Fig. 1.

A snapshot of the corrected reflectance (true color) NOAA-20/VIIRS satellite product on 14 Jul 2021, downloaded from the NASA Worldview website. The longitudinal width is 331 km, and the latitudinal width is 222 km. The down-to-the-hour timestamp is not available.
Fig. 2.

A schematic diagram for illustrating the two important length scales studied in this two-paper series: the mean radius Req and the maximum potential radius Rmax of cold pools in quasi-equilibrium convection. The Req represents half the deep convective spacing at a snapshot. The Rmax is the radius by which a cold pool’s buoyancy fully recovers to the environmental value.
Fig. 3.

(a) The time evolution of the domain-averaged column PW (m) for Eυ = 0.2 (blue line), Eυ = 0.5 (red line), Eυ = 1.0 (the reference test, yellow line), and Eυ = 2.0 (purple line). (b) As in (a), but for the spatial standard deviation of column PW with a logarithmic ordinate. (c) As in (a), but for the time evolution of the mean cold pool radius Req. The overshooting of Req for the Eυ = 2.0 test between t = 3.5 days and t = 4 days is judged as an occasional event, which does not appear in the larger-domain (144 × 144 km2) simulation (Fig. S2). In all panels, the shading shows the period between t = 3 days and t = 5 days, during which the quasi-equilibrium statistics are calculated.
Fig. 4.

(top) The water vapor mixing ratio vertically averaged within the lowest 551 m of the domain at t = 4 days. (a)–(d) The Eυ = 0.2, 0.5, 1.0, and 2.0 tests, respectively. (bottom) The z = 12.5 m (near-surface) potential temperature at the same time. In each plot, the white lines denote the 10−2 g kg−1 contours of cloud liquid water mixing ratio at z ≈ 4 km, which visualizes the cloud position. The cold pool sizes in the Eυ = 2 test are visually larger than those in the Eυ = 1, which seems to oppose the stagnancy behavior diagnosed with the time average. We argue that a single snapshot is insufficient to confirm the stagnancy of Req with Eυ because the cold pool size fluctuates over time. The LES movies can be downloaded from online (https://doi.org/10.5281/zenodo.13785498).
Fig. 5.

(a) The temporally averaged (a 2-day-long time series between t = 3 days and t = 5 days) spatial autocorrelation of the mixed layer vertically averaged water vapor mixing ratio for Eυ = 0.2 (blue line), Eυ = 0.5 (red line), Eυ = 1.0 (yellow line), and Eυ = 2.0 (purple line). An additional dashed black line denotes the 0.1 autocorrelation value used to diagnose the mean cold pool radius. (b) As in (a), but for the temporal autocorrelation of the mixed layer vertically averaged water vapor mixing ratio, using the time series between t = 3 days and t = 5 days. The time series have been detrended with a linear fit. The temporal autocorrelation is calculated on each of the 480 × 480 grid points and then averaged. Panel (b) is discussed in appendix A.
Fig. 6.

Some statistics averaged in the slot between t = 3 days and t = 5 days. (a) The blue dots denote the mean cold pool radius Req diagnosed from the LES for different Eυ. The shading denotes the ±1 standard deviation range of the Req time series. (b) The domain standard deviation of the near-surface (z = 12.5 m, the lowest model level) potential temperature. The shading denotes the ±1 standard deviation range of the time series. (c) The standard deviation of the near-surface buoyancy vs the square of the domain-averaged near-surface wind speed uc (black dots). In addition, we plot the domain-averaged CIN as red crosses.
Fig. 7.

A schematic diagram of the single cold pool evolution problem studied by Romps and Jeevanjee (2016), using a box model. The cold pool is produced by a total evaporative cooling of ΔQe (J). The initial cold air is assumed to be a cylinder (volume V0, radius R0, and height H0) with a uniform potential temperature anomaly θ′|R=R0. As the cold pool collapses, it keeps a cylindrical shape with a radius of R and height Hc, and its volume V grows by entrainment. The color depth of the shading denotes the magnitude of the cold pool’s potential temperature deficit, which is assumed to be uniform within the cold pool. The cold pool dissipates under the surface sensible heat flux Fsurf,s (W m−2). The cold pool radius by which the potential temperature deficit reduces to zero is the maximum potential radius Rmax.
Fig. 8.

A schematic diagram for the organization of convection in space and time. The horizontal axis denotes a certain spatial direction x. The vertical axis denotes time t. The blue shading denotes the trajectories of gust fronts in space and time. The equilibrium cold pool radius Req is half the deep convective spacing at a snapshot. It takes a cold pool Δt time to cover Req distance at a mean speed of uc ∼ Req/Δt.
Fig. 9.

Plots of Φ ≡ Rmax/Req for B = 0.05 (blue line) and B = 0.1 (red line) predicted by the energy balance constraint. Both plots use χpe = 1. The Φ < 1 regime is unphysical, but we retain it in the plots.
Fig. A1.

The Hovmöller diagram of the z = 12.5 m water vapor mixing ratio (filled map; g kg−1), which is near the surface, and z = 825 m vertical velocity, which is above the mixed-layer top (white line for −0.7 m s−1 and black line for 1 m s−1 contour). The data use the y = 48-km intercept of the reference test. Only the data between x = 0 km and x = 48 km from t = 4 days to t = 5 days are displayed. The red circles highlight a few convective events.
Fig. A2.

(a) The convective half-period Δt diagnosed from the temporal autocorrelation function of the mixed layer vertically averaged water vapor mixing ratio. (b) The “*” denotes the relation between Δt and Req/uc, where uc is the domain-averaged near-surface wind speed (z = 12.5 m level). In the plot, Δt, Req, and uc are normalized with their value at Eυ = 1. The dashed line is a 1-to-1 reference.
Fig. B1.

Diagnostic quantities of the LES averaged between t = 3 days and t = 5 days. (a) The Bowen ratio B, which is calculated as the domain-averaged sensible heat flux divided by the domain-averaged latent heat flux. (b) The saturation deficit qυs − qυ in the mixed layer (vertically averaged within the lowest 551 m and then horizontally averaged in the domain). (c) The surface precipitation–evaporation ratio χpe, which is calculated as the domain-averaged surface rainfall rate divided by the domain-averaged surface evaporation rate. (d) The rainwater mixing ratio qr averaged in the downdraft region of the mixed layer.
Fig. B2.

The Φ calculated with Eq. (18), using the diagnosed B, qυs − qυ, and χpe in Fig. B1. The SE is calculated with Eq. (B2). The red dots denote the SEref = 0.55 case, and the yellow dots denote the SEref = 0.65 case.
All Time Past Year Past 30 Days
Abstract Views 1374 1374 63
Full Text Views 530 530 335
PDF Downloads 268 268 72