Low-Level Circulation and Its Coupling with Free-Tropospheric Variability as a Mechanism of Spontaneous Aggregation of Moist Convection

Tomoro Yanase; Seiya Nishizawa; Hiroaki Miura; Tetsuya Takemi; Hirofumi Tomita

doi:10.1175/JAS-D-21-0313.1

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Fig. 1.

Schematic of the quasi-three-dimensional streamfunction Ψ and moisture distance S/o.

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Fig. 2.

Schematic of the vorticity-like quantity η in the moisture–height space.

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Fig. 3.

(right) Circulation field in the moisture–height space. Initial 10 days for H1000L384 (scattered case). The contour is a quasi-three-dimensional function normalized by the domain area: solid, dotted, and dashed contours are positive, zero, and negative, respectively, with an interval of 2 × 10⁻⁴ kg m⁻² s⁻¹. The shaded color is horizontal wind speed traversing the PW isolines, $\hat{υ}$ . (left) The vertical profile is the averaged normalized quasi-three-dimensional function.

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Fig. 4.

As in Fig. 3, but for various domain sizes, and the shaded color is the tendency of water vapor content due to adiabatic and SGS transport. The vertical domain is limited to up to 4 km height. Final 10 days for (a) H1000L96, (b) H1000L192, (c) H1000L384, and (d) H1000L560. Consecutive 10 days of (e)–(h) H1000L960. H1000L96, H1000L192, and H1000L384 are scattered cases, and H1000L560 and H1000L960 are aggregated cases. The purple point indicates the switching location of horizontal wind at the ground surface. Only the lowest 4 km are shown.

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Fig. 5.

Time–height cross section of the horizontal variance of water vapor content and its tendency terms. (a),(e) Horizontal variance of water vapor content. (b),(f) Summation of tendencies of horizontal variance of water vapor content due to microphysics, adiabatic, and SGS terms. (c),(g) As in (b) and (f), but for adiabatic and SGS terms. (d),(h) As in (b) and (f), but for adiabatic term. (a)–(d) L384 case and (e)–(h) L960 case.

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Fig. 6.

Kinematic metric plotted in the horizontal grid spacing–domain size space. The colors of markers are normalized quasi-three-dimensional function averaged over planetary boundary layer depth (final 10 days). Refer to the main text for the definition of the planetary boundary layer depth z_pbl. Based on Fig. 2b of Yanase et al. (2020), the markers of the crosses (circles) are scattered (aggregated) cases, and the arrows labeled by roman numerals are regime boundaries.

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Fig. 7.

Buoyancy (shaded) in the moisture–height space. Cases and times are as in Fig. 4. Microphysical heating rate (green contours) and radiative heating rate (purple contours) are also shown for values not greater than −1.5 kJ kg⁻¹ day⁻¹ with an interval of 1.0 kJ kg⁻¹ day⁻¹.

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Fig. 8.

Torque for PBL circulation in the moisture–height space. The tendency terms in Eq. (4) of the main text are shown: the total (black; multiplied by a factor of 3 for visibility), buoyancy gradient (blue), w-acceleration gradient (green), and $\hat{υ}$ -acceleration gradient (red). The values are the averages over the planetary boundary layer depth z_pbl. Cases and times are as in Fig. 4.

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Fig. 9.

PW percentile–time cross section of Q3D-SF, vorticity, and horizontal gradient of buoyancy averaged over PBL depth. (a),(d) Q3D-SF normalized by the domain size, Ψ_i_,_z/L². (b),(e) Vorticity η. (c),(f) Horizontal gradient of buoyancy projected onto the direction of the horizontal gradient of the PW, $\partial \bar{ρ} B / \partial \hat{y}$ (a)–(c) L384 case, and (d)–(f) L960 case. The dashed and dotted lines indicate the zero contours for the vorticity and the horizontal gradient of buoyancy, respectively.

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Fig. 10.

WTG assessment of vertical velocity in the dry region during the initial 5 days (dashed) and the initial 10 days (solid). Average over the area with PW lower than the median. (a) Actual vertical velocity, (b) WTG vertical velocity diagnosed by the total radiative heating and convective heating, (c) WTG vertical velocity diagnosed by the radiative heating, and (d) WTG vertical velocity diagnosed by the convective heating. Convective heating is defined as the total of microphysical heating and vertical eddy-flux convergence of dry static energy.

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Fig. 11.

Semiempirical vertical mode decomposition of vertical velocity during the initial 10 days. (a) Vertical profiles of three basis functions [see Eq. (5) of the main text]. (b) Coefficients derived by least squares fitting for each 5-percentile interval. (c) Top-heaviness: the ratio of −o₂ to o₁. (c) PBL circulation strength: the ratio of −o₃ to o₁.

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Fig. 12.

The relationships between PW and diabatic heating during the initial 10 days. (a) Radiative heating vs PW. (b) Convective heating vs PW. (c) As in (b), but focused on the lower heating range. Mass-weighted averages from the ground surface to about 6 km height. Crosses are average over the 10-percentile intervals, and triangles are averages over 50-percentile intervals.

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Fig. 13.

Horizontal power spectral density of specific humidity in the wavelength–height space. The vertical profiles show the horizontal variance of specific humidity. Initial 10 days for (a) H1000L96, (b) H1000L192, (c) H1000L384, and (d) H1000L560, and consecutive 10 days for (e)–(h) H1000L960.

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Fig. 14.

Horizontal power spectral density of specific humidity at about 2.5 km height.

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Fig. 15.

Moisture distance. The ratio of area S to the perimeter o, as a function of PW percentiles during the initial 5 days and the initial 10 days. See section 3a of the main text for the definition.

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Fig. 16.

Nearest convective core distance. (a),(c) Snapshots of the distance divided by the domain size for an instantaneous time on day 5 of the L384 and L960 cases, respectively. (b),(d) Histogram of the distance on day 5 of the L384 and L960 cases, respectively; probability density and cumulative density are shown on the blue left axes and black right axes, respectively. (e) Schematic of the relationship among the horizontal distribution of convective core, subsidence region, and nearest convective core distance. (f) The 99th-percentile value of nearest convective core distance on days 3, 5, and 10 as a function of the domain size.

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Fig. 17.

As in Fig. 14, but for (a) horizontal velocity and (b) vertical velocity.

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Fig. 18.

Schematic diagram of a mechanism of CSA onset, focusing on moisture variability and the diabatic heating and atmospheric motions associated with it.

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Fig. A1.

The schematic diagram for treating an isoline, its length, and traversing flow in a discrete field. (a) Filtered two-dimensional field. (b) Binarized two-dimensional field. (c) Examples of peripheral grids are shown at the center of each panel; the solid red line is the cell edge belonging to the isoline; the red number is the number of cell edges belonging to the isoline for each peripheral grid; the blue arrow is the traversing flow across the isoline. This example is made from a snapshot on day 50 of H2000L384 case, with a PW threshold value of 10 kg m⁻².

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Fig. A2.

As in Fig. 4, but the contour is the effective streamfunction normalized by the domain area, $Ψ_{r, z}^{BBK 05} / L^{2}$ , the shaded color is omitted, and the horizontal axis is the PW rank normalized by the number of ranks, r/r_max.

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Fig. A3.

Q3D-SF, vorticity, and horizontal gradient of buoyancy in the moisture–height space. (a),(d) Q3D-SF normalized by the domain size, Ψ_i_,_z/L². (b),(e) Vorticity η. (c),(f) Horizontal gradient of buoyancy projected onto the direction of the horizontal gradient of the PW, $\partial \bar{ρ} B / \partial \hat{y}$ . (a)–(c) Final 10 days of L384 case and (d)–(f) final 10 days of L960 case.

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Fig. B1.

Moisture budget during the initial 10 days for the L384 case in the moisture–height space. (a) Total tendency. (b) Microphysical tendency. (c) Summation of adiabatic and SGS tendencies calculated as residual (i.e., total tendency minus microphysical tendency). (d) Vertical flux convergence component of adiabatic tendency. (e) Horizontal flux convergence component of adiabatic tendency. (f) SGS tendency as a summation of the SGS turbulence component and the surface flux component.

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Fig. B2.

As in Fig. B1, but for the L960 case.

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Editorial Type:: Article

Article Type:: Research Article

Low-Level Circulation and Its Coupling with Free-Tropospheric Variability as a Mechanism of Spontaneous Aggregation of Moist Convection

Tomoro Yanase

Tomoro Yanase ^aRIKEN Cluster for Pioneering Research, Kobe, Japan
^bRIKEN Center for Computational Science, Kobe, Japan
^cDisaster Prevention Research Institute, Kyoto University, Kyoto, Japan
^dGraduate School of Science, Kyoto University, Kyoto, Japan

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https://orcid.org/0000-0003-2788-9092

Seiya Nishizawa

Seiya Nishizawa ^bRIKEN Center for Computational Science, Kobe, Japan

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Hiroaki Miura

Hiroaki Miura ^eGraduate School of Science, The University of Tokyo, Tokyo, Japan
^bRIKEN Center for Computational Science, Kobe, Japan

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Tetsuya Takemi

Tetsuya Takemi ^cDisaster Prevention Research Institute, Kyoto University, Kyoto, Japan

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Hirofumi Tomita

Hirofumi Tomita ^bRIKEN Center for Computational Science, Kobe, Japan
^aRIKEN Cluster for Pioneering Research, Kobe, Japan

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Online Publication:: 12 Dec 2022

Print Publication:: 01 Dec 2022

DOI:: https://doi.org/10.1175/JAS-D-21-0313.1

Page(s):: 3429–3451

Received:: 29 Nov 2021

Accepted:: 12 Aug 2022

Published Online:: 12 Dec 2022

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

The organization of clouds has been widely studied by numerical modeling as an essential problem in climate science. Convective self-aggregation (CSA) occurs in radiative–convective equilibrium when the model domain size is sufficiently large. However, we have not yet reached a comprehensive understanding of the mechanism of CSA onset. This study argues that low-level circulation is responsible for horizontal moisture transport and that its coupling with variabilities of diabatic heating and moisture in the free troposphere is essential. We simulated scattered and aggregated convection by varying the domain size as a control parameter constraining the horizontal scale associated with the CSA onset. Based on a new analysis method quantifying the circulation spanning dry and moist regions, we found that 1) the upgradient moisture transport in the aggregated cases is associated with low-level circulation development, amplifying the horizontal moisture contrast; 2) the horizontal buoyancy gradient due to strong radiative cooling in the dry region intensifies the low-level circulation; 3) the free-tropospheric subsidence intrudes into the boundary layer in the dry region preceding the intensification of low-level circulation; and 4) the subsidence intrusion is due to a weakening of convective heating in the free troposphere associated with the moisture variability at a larger horizontal scale. This study provides new insights into the organization mechanism of clouds unifying the different mechanisms impacting CSA: the free-tropospheric moisture, radiation, convection, and low-level circulation.

Corresponding author: Tomoro Yanase, tomoro.yanase@riken.jp

Abstract

Corresponding author: Tomoro Yanase, tomoro.yanase@riken.jp

Keywords: Atmospheric circulation; Clouds; Deep convection; Radiative-convective equilibrium; Cloud resolving models; Water vapor

1. Introduction

Radiative and convective processes are among the most important components shaping climate. Radiative–convective equilibrium (RCE) is a thermally balanced state of the atmosphere without net inflow in the lateral direction. It is one of the most fundamental pictures of the climate system. RCE was originally studied by modeling the atmosphere as a vertical column (Manabe and Strickler 1964; Ramanathan and Coakley 1978), where convection was considered to play a major role in the vertical redistribution of heat to yield realistic tropospheric lapse rates. However, convection itself is a phenomenon with a wide variety of horizontal structures. What kinds of atmospheric structures would be formed spontaneously if we explicitly treat the atmospheric motion in RCE? Through what mechanisms does this occur? These have been essential questions in climate science for a long time (Tompkins and Craig 1998), and many researchers have investigated their hypotheses.

Nakajima and Matsuno (1988) were the first to explicitly examine the behavior of moist convection under RCE by using a two-dimensional cloud-resolving model (CRM). They found a hierarchical structure between small-scale individual clouds and a large-scale cloud cluster, analogous to a real tropical atmosphere (Nakazawa 1988). As we can learn from this study, research on the spontaneous organization of moist convective clouds in RCE helps provide insights into how clouds interact with their environment.

The spontaneous organization of clouds under RCE is generally referred to as convective self-aggregation (CSA) and has been intensively studied in recent years (e.g., Wing et al. 2017; Muller et al. 2022). Various aspects of CSA in RCE have already been investigated. This includes the general characteristics in time and space (Wing and Cronin 2016; Yang 2018a, 2021; Patrizio and Randall 2019; Yanase et al. 2020, hereafter Y20), onset and maintenance mechanisms (Bretherton et al. 2005; Muller and Held 2012; Wing and Emanuel 2014; Haerter 2019), similarities to tropical disturbances such as the Madden–Julian Oscillation and tropical cyclones (Arnold and Randall 2015; Carstens and Wing 2020; Khairoutdinov and Emanuel 2013, 2018; Muller and Romps 2018; Nolan et al. 2007), and implications for the effect on climate sensitivity (Bony et al. 2016; Cronin and Wing 2017; Mauritsen and Stevens 2015; Ohno and Satoh 2018; Wing 2019). These viewpoints complement each other for an understanding of CSA. However, a more comprehensive understanding of CSA could be gained by addressing the two following questions: What numerical conditions (e.g., domain size) does CSA onset prefer? What mechanisms contribute to the formation of CSA? To address these questions, this study particularly adopts one of the most fundamental numerical frameworks of the RCE experiment; that is, using three-dimensional CRM on square domains with side lengths ranging from approximately 100 to 1000 km with a fixed sea surface temperature (SST) at the lower boundary. Even though this time-honored framework has been used for decades in RCE research, there are still unresolved issues as explained below.

Regarding the conditions favorable for the onset of CSA, previous numerical studies have suggested that the SST must be higher than a threshold value (e.g., Khairoutdinov and Emanuel 2010; Wing and Emanuel 2014). On the other hand, counterexamples of CSA occurring even at lower SSTs have been reported (e.g., Abbot 2014; Coppin and Bony 2015; Holloway and Woolnough 2016; Wing and Cronin 2016; Yang 2018a). Thus, we have not yet reached a consensus on the effect of SST on CSA onset. In addition to the SST condition, another essential factor influencing the onset of CSA is the size of the domain. Conceptually, the RCE experiment itself does not specify it, while its configuration in numerical models may strongly affect the onset of CSA. However, this experimental configuration problem can be inversely utilized to determine the characteristic length of CSA (Bretherton et al. 2005; Jeevanjee and Romps 2013; Patrizio and Randall 2019). In addition, the model resolution is also an important factor in the CSA onset from the viewpoint of numerical simulation. Muller and Held (2012) showed that the critical domain size depends on horizontal resolution based on a systematic numerical survey. Recently, we found that the critical length exists at approximately 500 km by simulations in a sufficiently wide domain with high resolution, revising the regime diagram of the CSA onset on resolution and domain size (see Fig. 2b of Y20).

Various CSA onset mechanisms have been proposed in the past. Here, we review them from the following two perspectives: the deep mechanisms that emphasize the role of the entire troposphere and the shallow mechanisms that focus the low layer from the ground surface to the cloud base.

When considering deep mechanisms, the moist static energy (MSE) budget equation is often used at the starting point. The temporal tendency of the local MSE is determined by radiative heating, transport by large-scale circulation, and small-scale convection. Under a weak temperature gradient (WTG) approximation [e.g., Emanuel et al. 2014, Eq. (1)], the MSE tendency is equivalent to the moisture tendency. Bretherton et al. (2005) built an ordinary differential equation of precipitable water (PW) under appropriate assumptions and discussed the conditions responsible for the emergence of instabilities in the RCE, that is, the requirement for the CSA onset. Based on their theory, when the anomalous radiative heating and surface flux in the moist region surpasses the lateral MSE export in the moist region, CSA occurs. To test their hypothesis, they also conducted several sensitivity experiments using CRM. The surface enthalpy flux and column radiative heating rate were horizontally homogenized. The results showed that CSA did not occur under such conditions with homogenized forcings, which was consistent with their theory.

Wing and Emanuel (2014) also used the MSE budget framework, focusing on the horizontal variance of the MSE. They pointed out the importance of feedback concerning the surface enthalpy flux and radiation in the CSA onset phase. Using a mass-flux parameterization for convection and the assumption that the boundary layer is in quasi-equilibrium (Raymond 1995), Emanuel et al. (2014) showed that it is important that the lower layer is opaque to longwave radiation to promote the growth of water perturbations. Furthermore, Beucler and Cronin (2016) investigated the instability mechanism of moisture perturbation by using a radiative transfer model, focusing on the effect of moisture on radiative cooling. Their results showed that unstable conditions sometimes appeared in real tropical atmospheric soundings, and the time scale of the growth of instabilities was approximately 1 month.

Craig and Mack (2013) discussed the temporal evolution of the horizontal moisture pattern based on the budget equation of the free troposphere (FT) portion of PW as another deep mechanism. They modeled the time tendency of moisture as the summation of subsidence drying, convective moistening, and horizontal transport that was approximated by a diffusion term. In their framework, the horizontal pattern formation corresponding to CSA occurred when moisture–convection feedback amplifying moisture perturbation surpassed the effect of diffusion damping the perturbation. Using a similar approach, Windmiller and Craig (2019) showed that there was a universality in the time scale of the initial development of horizontal moisture patterns, regardless of the difference in the formulations of reaction terms (i.e., source terms as functions of local moisture content) based on previous studies (Bretherton et al. 2005; Craig and Mack 2013; Emanuel et al. 2014). They also discussed the domain size dependence of CSA onset and argued that the upscale growth was inhibited in the small-size domain because the homogenizing effect of diffusion covered the entire area. Their study also suggested that it is still important to understand how the horizontal exchange of moisture is achieved in nature (or in a more complicated physically based model such as CRM explicitly including the advective process), because it was treated only as a diffusive process in the simple model.

In addition, shallow mechanisms have been proposed, mainly focusing on the processes related to the planetary boundary layer (PBL) and the low cloud overlying it. Bretherton et al. (2005) and Muller and Held (2012) showed that radiative cooling by low-level clouds in the dry regions would induce subsidence and lead to FT air intruding into the boundary layer. The subsidence induced near-surface divergent circulation that caused upgradient moisture transport. This mechanism due to the relationship between low cloud and radiative cooling has been validated in another study and discussed in terms of the resolution dependence of the CSA onset (Muller and Bony 2015; Yanase and Takemi 2018; Y20). Coppin and Bony (2015), based on RCE experiments by a general circulation model with parameterized convection, similarly showed that a radiatively driven cold pool was formed in the lower level of the dry region, which contributed to dry patch expansion. Moreover, the importance of the virtual effect on the dynamics of CSA has also been revealed by several studies (Yang 2018a,b). Recently, Yang and Tan (2020) conducted a budget analysis of the virtual potential temperature in a dry patch to show that it is important that the drying effect of subsidence surpassed the warming effect of subsidence for the positive feedback causing the dry patch expansion.

In addition to radiative cooling, the evaporation of precipitation may also have an important impact on CSA. Jeevanjee and Romps (2013) found that moisture homogenization through an evaporatively driven cold pool in the PBL inhibited CSA onset. The mechanism was based on the fact that CSA always occurred regardless of the domain size in the numerical experiments as long as the evaporation of rain was turned off in the PBL. Furthermore, Yang (2018b) confirmed the sensitivity to the evaporation process and added a new interpretation that evaporative cooling weakens the buoyancy in the moist region of PBL from a dynamic viewpoint.

Thus, in terms of shallow mechanisms, it is understood that radiatively driven divergent flow and evaporatively driven cold pools have positive and negative effects on CSA onset, respectively. Recently, we further examined horizontal moisture transport near the surface and argued that CSA onset was determined by the competition between radiatively driven divergent flow and the evaporatively driven cold pool (Y20). We explained the connection between CSA in RCE experiments and the domain size as follows: 1) the horizontal size of the dry subsidence region increases with the domain size (even without CSA), 2) the radiatively driven circulation is strengthened as the domain size increases, and 3) the radiatively driven circulation competes with the evaporatively driven circulation when the domain size exceeds a critical size. However, Yang (2018b) pointed out that it is necessary to discuss the PBL process and its coupling with the FT process for a deeper understanding of the mechanism. The coupling would be achieved by a nonlocal radiative interaction between the lower and upper atmosphere and a local advective interaction near the top of the PBL. The essence of this issue is to determine how the horizontal variability of moisture and convection is related to the dynamics of low-level circulation and FT processes. This would unify the two viewpoints of deep and shallow mechanisms. One straightforward method is to investigate how the domain size affects the horizontal scales of the distributions of moisture and convective elements. In this case, the domain size can be treated as a controllable parameter in which the horizontal scale is constrained.

In this study, we provide a unified view of the main mechanisms leading to CSA onset by focusing on the coupling between PBL and FT processes. We investigated the domain size dependence of RCE experiments using CRM in more detail than that of Y20. Because the domain size constrains the maximum horizontal scale of atmospheric variability in the simulation, we treat it as a controllable parameter for the atmospheric horizontal scale. The domain size dependence of the CSA onset provides insight into the physical mechanism controlling convective organization. The simulation dataset used in this study is briefly described in section 2. It was observed that the random atmospheric motions spontaneously become organized into moist-ascending and dry-descending regions once CSA occurs. In section 3, we propose a new analysis method to quantify the kinematics and dynamics of circulation across dry and moist regions. Based on this method, we analyze the evolution of organized circulation associated with the CSA onset to clarify the difference between the scattered and aggregated convection regimes (section 4a). Next, to understand the dynamics of the development of circulation spanning dry and moist regions, we examine the horizontal component of vorticity concerning the circulation.

Regarding the coupling between the PBL and FT processes, we investigate the change in the vertical velocity distribution with respect to the increase in the domain size (section 4b) because we consider that the intrusion of FT air into the PBL is a key triggering process of CSA onset (Hung and Miura 2021; Yang and Tan 2020). In particular, the FT vertical velocity can be diagnosed by diabatic heating under the WTG approximation, and diabatic heating can be related to moisture. Finally, we examine the horizontal distributions of moisture, convective elements, and atmospheric motion to associate their horizontal scale with the domain size and propose a mechanism for the CSA onset involving the horizontal scale of atmospheric variability (section 4c). In section 5, we provide an overall picture of how low-level circulation and the FT variability couple. We also discuss the mechanisms of CSA in the development and triggering phases. Finally, conclusions are provided in section 6.

2. Data

This study used a series of simulation data from Y20. These idealized numerical experiments were conducted using a nonhydrostatic atmospheric model called SCALE-RM constructed based on the Scalable Computing for Advanced Library and Environment (SCALE; Nishizawa et al. 2015; Sato et al. 2015) version 5.3.3.

The three-dimensional computational domain had lateral boundaries with doubly periodic conditions and an upper boundary at a height of 24 km, and was discretized vertically by 64 layers. The physical parameterizations used were as follows: the six-class single-moment bulk-type scheme (Tomita 2008) for the cloud microphysics, the Smagorinsky–Lilly-type first-order closure scheme (Brown et al. 1994; Scotti et al. 1993) for the subgrid-scale (SGS) turbulence, the bulk method using a universal function (Beljaars and Holtslag 1991; Wilson 2001) for the surface fluxes, and the MSTRNX radiative transfer model (Sekiguchi and Nakajima 2008).

The controlled parameters of the simulation set were horizontal grid spacing (H) and domain size (L), which varied in the ranges of 500–4000 m and 96–960 km, respectively. We assigned names to each experiment based on its horizontal grid spacing (in m) and domain size (in km); for instance, an experiment with a grid spacing of 1000 m and domain size of 96 km called H1000L96. In this study, we mainly focus on the five cases with the same horizontal grid spacing of H1000 and with different domain sizes of L{96, 192, 384, 560, 960}, with integration durations of 50 days and 40 days for the former three and the latter two cases, respectively. Y20 showed that H1000L{96, 192, 384} were scattered cases while H1000L{560, 960} were aggregated cases.

3. New analysis method

a. Quasi-three-dimensional streamfunction and moisture distance

Previous studies have often used the effective streamfunction proposed by Bretherton et al. (2005) to analyze circulation in CSA. Although this streamfunction is useful for examining the relationship between moisture and other variables, it does not necessarily reflect the actual circulation in the physical space. This is because the information of the topology in the atmosphere (e.g., how the atmospheric columns are distributed in the horizontal space) is lost owing to the sort operation for the vertical atmospheric columns. To compensate for this shortcoming, we propose a new method that retains the virtue of the method of Bretherton et al. (2005) without a sort operation. This method makes it possible to quantify the intensity of circulation between dry and moist regions while retaining the information of the horizontal scale related to the moisture field. We introduce a quasi-three-dimensional streamfunction (Q3D-SF) as follows:

The Q3D-SF can be derived from the integration of the mass conservation equation over a specific volume by neglecting the falling terms of the hydrometeors and assuming a steady state:

\oint_{C} \int_{0}^{z} ρ \hat{υ} d z^{'} d \hat{x} + \begin{matrix} \iint_{D} {(ρ w)}_{z} d S \end{matrix} = 0,

(1)

where z is the height at the top of the integral volume, C and D are horizontally closed curves for the line integral and regions for the surface integral, respectively;

d \hat{x}

is the line element of the curve C; dS is the surface element of the plane D; ρ is the total air density;

\hat{υ}

is the horizontal velocity component perpendicular to curve C; and w is the vertical velocity. Here, we assume that the vertical mass flux is zero at the ground surface. In this study, we use a contour line of PW for the curve C. PW is defined as follows:

PW \equiv \int_{0}^{z_{t}} ρ q_{υ} d z,

where z_t is the height at the upper boundary of the computational domain, and q_υ is the specific humidity. Hence, plane D is the subdomain where PW is lower than the value on the curve C. Q3D-SF refers to the second term on the left-hand side of Eq. (1) and is defined as Ψ_i_,_z in a discrete form:

Ψ_{i, z} \equiv \sum_{j = 1}^{i} {(ρ w)}_{j - 1 / 2, z} d S_{j - 1 / 2}, i = 1, 2, \dots, i_{max},

(2)

where i is the index of C and D, and j − 1/2 is the index of the area dS between C_j₋₁ and C_j. In this study, we use percentiles values, from 5th to 95th with the interval of 5, for the PW_i values corresponding to the curves C_i. Hence, i_max is 19. Figure 1 schematically shows the integral volume of Eq. (1), and Q3D-SF defined in Eq. (2). The horizontal domain is divided into dry and moist regions by isolines of PW (e.g., the curves C_i and C_i₊₁). The surface D_i denotes the region enclosed by the curve C_i. The Q3D-SF Ψ_i_,_z denotes the vertical mass flux integrated over the surface D_i at height z.

Equation (1) can be written in an alternative form:

o_{i} U_{i, z} + S_{i} W_{i, z} = 0 .

(3)

Here, o_i is the perimeter of a target domain on a horizontal cross-section (i.e., length of C_i), U_i_,_z is the vertically integrated peripherally averaged horizontal mass flux, S_i is the horizontal area of the subdomain (i.e., area of D_i), and W_i_,_z is the vertical mass flux averaged over the subdomain. They are defined as follows:

o_{i} \equiv n_{i} Δ \hat{x}, U_{i, z} \equiv \frac{\sum_{m = 1}^{n_{i}} {(\int_{0}^{z} ρ \hat{υ} d z^{'})}_{m} Δ \hat{x}}{o_{i}},

S_{i} \equiv \sum_{j = 1}^{i} d S_{j - 1 / 2}, and W_{i, z} \equiv \frac{\sum_{j = 1}^{i} {(ρ w)}_{j - 1 / 2, z} d S_{j - 1 / 2}}{S_{i}},

where n_i is the total number of cell edges belonging to isoline C_i, m is the index of the cell edges, and

Δ \hat{x}

is the discrete form of

d \hat{x} .

This allows one to define a new length scale S_i/o_i, which we name “moisture distance” since it is related to the horizontal length scale between the driest region and a certain moister region (Fig. 1), although it more generally depends on the shape and division number of a dry region. Appendix A describes how to calculate the perimeter practically.

b. Vorticity in moisture–height space

The Q3D-SF defined above is useful for quantifying the kinematics (i.e., on the state of fluid motion) of the circulation in the moisture–height space. On the other hand, we need another way to quantify the circulation’s dynamics (i.e., on the time change of fluid motion and the force that causes it). To gain a mathematical intuition on dynamic factors related to Q3D-SF, we first consider the time evolution equation of vorticity-like quantity under an idealized situation with a simple steady-state two-dimensional (one horizontal and one vertical) spatial configuration.

Figure 2 shows the schematic of the idealized situation where the spatial configuration is essentially two-dimensional denoted by

\hat{y}

and z axes. Here,

\hat{x}

\hat{y}

, and z make up a local Cartesian coordinate system. In that situation, we may choose the direction of the horizontal gradient of PW as the

\hat{y}

direction without loss of generality. At the same time, the

\hat{x}

direction is parallel to PW isolines. Now, we can consider the time evolution of the

\hat{x}

-direction component of the vorticity (more precisely, the rotation of momentum) as follows:

\frac{\partial η}{\partial t} = \frac{\partial \bar{ρ} B}{\partial \hat{y}} + \frac{\partial F_{z}}{\partial \hat{y}} - \frac{\partial F_{\hat{y}}}{\partial z},

(4)

where η is the vorticity; B is the buoyancy; and F_z and

F_{\hat{y}}

are the momentum flux convergence for the momentum in the z direction and

\hat{y}

direction, respectively. They are defined as follows:

η \equiv \frac{\partial \bar{ρ} w}{\partial \hat{y}} - \frac{\partial \bar{ρ} \hat{υ}}{\partial z}, B \equiv - g \frac{ρ^{'}}{\bar{ρ}},

F_{z} \equiv - \frac{\partial \bar{ρ} w \hat{υ}}{\partial \hat{y}} - \frac{\partial \bar{ρ} w w}{\partial z} + F_{SGS, z}, and F_{\hat{y}} \equiv - \frac{\partial \bar{ρ} \hat{υ} \hat{υ}}{\partial \hat{y}} - \frac{\partial \bar{ρ} w \hat{υ}}{\partial z} + F_{SGS, \hat{y}},

where

\bar{ρ}

is the horizontal mean air density under hydrostatic balance; ρ′ is the horizontal deviation of air density; and F_SGS,_z and

F_{SGS, \hat{y}}

are SGS components of F_z and

F_{\hat{y}}

, respectively. These latter terms, F_SGS,_z and

F_{SGS, \hat{y}}

, consist of momentum tendencies due to the SGS turbulence and the surface flux schemes. At the same time, Q3D-SF can be written as

Ψ = \iint ρ w d \hat{x} d \hat{y} = L \int ρ w d \hat{y} and Ψ = - \iint ρ \hat{υ} d \hat{x} d z = - L \int ρ \hat{υ} d z,

where L is the domain size (i.e., the length of one side of the square domain) and Q3D-SF can be related to the vorticity as

\frac{1}{L} (\frac{\partial^{2}}{\partial {\hat{y}}^{2}} + \frac{\partial^{2}}{\partial z^{2}}) Ψ = η .

Considering the wavelike spatial structure, a relationship −Ψ ∝ η holds. Thus, the vorticity and its time tendency are expected to have reverse signs compared with Q3D-SF and its time tendency, at least in this idealized situation with the two-dimensional spatial configuration.

In the actual data analysis, the $\hat{y}$ -direction component of each physical quantity is calculated by projecting the horizontal vector onto the direction of the horizontal gradient of the PW at each location and time without a strict coordinate transformation. This simplification is one of the limitations of this method and may affect the quantitative understanding of the time evolution of vorticity, but we assume that it is sufficient for qualitative understanding. In addition, it is not obvious to what extent the relationship obtained from the simple situation is able to explain the full unsteady three-dimensional fields that we will analyze. Hence, as described in appendix A, we first compared the spatial characteristics of Q3D-SF and vorticity fields (Fig. A3). Based on the comparison, the relationship between Q3D-SF and vorticity basically agreed with those expected from the steady-state two-dimensional situation described above. Thus, the vorticity framework can be useful for the investigation of the circulation’s dynamics.

4. Results

a. Low-level circulation in scattered and aggregated regimes

Figure 3 shows the overturning circulation in the moisture–height space for a scattered case (H1000L384). The shade is the horizontal wind velocity traversing the PW isolines. The contour is the Q3D-SF [Eq. (2)] normalized by the computational domain area. From this figure, we see the anticlockwise circulation (i.e., negative Q3D-SF) in the lower to upper FT (about 1–13 km height). The lower branch of the circulation is the inflow, and the upper branch is the outflow in the moist region. We also observed clockwise circulation (i.e., positive Q3D-SF) in the PBL below approximately 1 km height, characterized by the outflow from the moist region near the surface.

To observe the difference in the scattered and aggregated regimes, the circulation fields of the final 10 days in the different size domains were compared (Figs. 4a–d,h). The final 10-day fields of the scattered cases have already reached a statistical equilibrium state while that of the aggregated cases are in a transitional phase to a fully aggregated state. This can be confirmed by the time evolution of the horizontal variance of water vapor shown in the following part (Fig. 5). In contrast to the scattered cases (Figs. 4a–c), the anticlockwise circulation enters the PBL from the FT in the dry region in the aggregated cases (Figs. 4d,h). Focusing on the time evolution of the aggregated case (Figs. 4e–h), the FT circulation intrudes into the PBL in the driest part at earlier times, but concerns regions of increasing PW as time progresses. The front, indicated by the purple points, migrates toward the moister side as time passes. We additionally compared the scattered and aggregated cases by using Bretherton et al. (2005)’s streamfunction in Fig. A2. Since the qualitative features of the circulation fields by two streamfunctions are the same, we consider that the advantage of our streamfunction is not in differences in appearance, but in the clarity in the physical concept. Based on the physical concept of Q3D-SF, we can further examine the dynamical mechanism of the circulation and the horizontal scale related to the circulation, which are addressed in sections 4b and 4c.

In relation to the difference in circulation patterns, we also examined the role of moisture transport from the moisture budget viewpoint. The change of the water vapor content can be written as follows:

\frac{\partial ρ q_{υ}}{\partial t} = {(\frac{\partial ρ q_{υ}}{\partial t})}_{MP} + {(\frac{\partial ρ q_{υ}}{\partial t})}_{AD} + {(\frac{\partial ρ q_{υ}}{\partial t})}_{SGS},

where (∂ρq_υ/∂t)_MP, (∂ρq_υ/∂t)_AD, and (∂ρq_υ/∂t)_SGS are the tendencies of water vapor content due to cloud microphysics, gridscale adiabatic transport, and SGS transport. SGS transport consists of tendencies due to the SGS turbulence and the surface flux schemes. The spatial distribution of the sum (∂ρq_υ/∂t)_AD + (∂ρq_υ/∂t)_SGS is compared in Fig. 4. A feature common to all cases is that there is a drying (moistening) tendency below (above) approximately 1 km height. This upper and lower contrast mainly reflects the vertical transport of moisture by deep convective motion; moisture is transported from the PBL to the FT. This moisture tendency due to transport is generally balanced by microphysical tendencies (see Figs. B1 and B2 for the spatial distribution of the microphysical term). In addition, regarding the contrast between the dry and moist regions, there are differences among the cases. Comparing the scattered cases (Figs. 4a–c), the horizontal contrast between the dry and moist regions in the adiabatic tendency increases as the domain size increases although the overall effect is drying (moistening) in the PBL (FT) at any PW percentile. In contrast to the scattered cases, the aggregated cases (Figs. 4d,h) show that the area with the drying tendency over the entire vertical layer from the PBL to the FT appears in the dry region. Focusing on the time evolution of an aggregated case (Figs. 4e–h), the FT drying exists even in the initial 10 days, and the area with the drying tendency over the entire vertical layers spread to the moister side with the development of low-level circulation. This adiabatic drying throughout the atmospheric column is considered to play an essential role in the dry patch formation associated with the CSA onset (Yang and Tan 2020). The adiabatic drying in the dry region in the aggregated case is observed as well in our analysis as described in appendix B. Note that the large magnitude of drying tendency in the dry region is located around 1–3 km altitudes.

Based on the moisture budget viewpoint above, we further evaluate the role of moisture transport in amplifying the horizontal variance of moisture following a framework recently developed by Yao et al. (2022). Instead of the budget of horizontal variance of MSE (Yao et al. 2022), we focus on the horizontal variance of water vapor to distinguish moisture from heat (i.e., the dry static energy part of MSE). The change of the horizontal variance of water vapor content can be written as follows:

\frac{1}{2} \frac{\partial}{\partial t} \bar{{[{(ρ q_{υ})}^{'}]}^{2}} = \bar{{(ρ q_{υ})}^{'} {(\frac{\partial ρ q_{υ}}{\partial t})}^{'}_{MP}} + \bar{{(ρ q_{υ})}^{'} {(\frac{\partial ρ q_{υ}}{\partial t})}^{'}_{AD}} + \bar{{(ρ q_{υ})}^{'} {(\frac{\partial ρ q_{υ}}{\partial t})}^{'}_{SGS}},

where

\bar{A}

and A′ are the horizontal mean and deviation of a quantity A, respectively. Figure 5 shows the time evolution of the horizontal variance of water vapor content at each altitude and its production terms for the L384 and L960 cases. In contrast to the scattered case (Fig. 5a), the horizontal variance generally increases with time in the aggregated case (Fig. 5e). The growth rate of the horizontal variance appears to be the largest at the lower FT. This result is consistent with the horizontal variance of MSE examined by Yao et al. (2022). The sum of adiabatic and SGS transport terms contributed to increasing the horizontal variance of moisture in the lower FT while contributed to slightly decreasing the variance in the PBL (Fig. 5g). The adiabatic transport contribution is largely negative in the PBL (Fig. 5g). In other words, the SGS transport and microphysical terms contributes to produce the horizontal variance of moisture in the PBL. Even though the scattered case has similar patterns in adiabatic and SGS transport terms (Figs. 5c,d), the magnitude is much smaller and canceled by the microphysical term (Fig. 5b). Hence, the development of low-level circulation plays a role in amplifying the horizontal moisture contrast in the lower FT.

To confirm the above circulation patterns between the scattered and aggregated regimes, we employ a new kinematic metric (KM) defined as follows:

KM \equiv \frac{1}{L^{2} i_{max} z_{pbl}} \sum_{i = 1}^{i_{max}} \int_{0}^{z_{pbl}} Ψ_{i, z} d z,

where z_pbl is the PBL top height diagnosed as the height where the virtual potential temperature first exceeds the value in the first model layer above the surface. Hence, KM is the Q3D-SF averaged over the PBL and normalized by the horizontal domain area. Figure 6 shows KM for all Y20 experiments in the H–L parameter space. The scattered and aggregated cases have higher and lower values, respectively. It is observed that all the values of the aggregated cases are lower than those of the scattered cases. Thus, we can say that the difference in circulation patterns between the scattered and aggregated cases is robust.

Next, to understand the mechanism leading to the formation and development of anticlockwise circulation in the PBL, we examine the dynamical factors that explain such a mechanism based on the vorticity framework introduced in section 3b.

Figure 7 shows buoyancy, microphysical cooling, and radiative cooling fields in moisture–height space from the ground surface to 4 km height. The scattered cases (Figs. 7a–c) are characterized by negative buoyancy in the moist region and positive in the dry region. We also see that the sign of buoyancy is reversed in the thin layer at around 1 km height. Considering the effect of the buoyancy field on the flow field [cf. Eq. (4)], in the lowest layer below about 500 m height, negative buoyancy gradient ( $\partial B / \partial \hat{y} < 0$ ) appears to drive the negative vorticity (η < 0). Looking at the microphysical cooling field, this circulation dynamics due to the negative buoyancy field well corresponds to evaporatively driven cold pools. On the other hand, the aggregated cases (Figs. 7d,h) have negative buoyancy also in the driest region in the lowest layer below about 500 m height. The existence of negative buoyancy both in the drier and moister region is a remarkable characteristic, different from the scattered case. This negative buoyancy expands to the moist side as time passes, consistent with the time evolution of the circulation field. The positive buoyancy gradient ( $\partial B / \partial \hat{y} > 0$ ) appears to act as a driving force for positive vorticity (η > 0) in the dry region. This circulation forcing is related to radiative cooling (purple contour in Fig. 7). It should be noted here that the horizontal extent of evaporative and radiative cooling appears to differ significantly as the domain size changes. This is due to the horizontal axis of the PW percentile. It will be shown later that the absolute water vapor content in the dry region decreases as the domain size increases. This drying may influence the shrinkage or expansion of the horizontal extent of strong evaporative or radiative cooling.

To evaluate the driving force of low-level circulation, Fig. 8 shows the vorticity tendency terms in Eq. (4) and averaged over the PBL. Here, the terms $\partial F_{z} / \partial \hat{y}$ and $- \partial F_{\hat{y}} / \partial z$ are referred to as the w-acceleration gradient and $\hat{υ}$ -acceleration gradient, respectively.

In the scattered cases (Figs. 8a–c), the buoyancy gradient is negative. The $\hat{υ}$ -acceleration gradient is positive throughout the PW space. The total has the lowest negative value in the moistest region and approaches zero toward the dry region. However, the characteristics are ambiguous in the L96 case. The w-acceleration gradient was small compared with the other terms. Physically, the buoyancy gradient caused by the enhanced evaporation of falling rain in the moist region produces negative vorticity. In contrast, the $\hat{υ}$ -acceleration gradient caused by the enhanced deceleration of momentum near the surface dampens the negative vorticity.

Similar characteristics can be observed in the scattered cases explained above at the initial stage of an aggregated case (H1000L960; Fig. 8e). However, this characteristic changes over time. In the dry region, when the buoyancy gradient becomes positive, the $\hat{υ}$ -acceleration gradient is negative, and the total positive, as shown in Fig. 8f. The area with a positive value expands to the moist side with time (Figs. 8f–h). Physically, the buoyancy gradient caused by enhanced radiative cooling in the dry region produces a positive vorticity.

To see the relationship among the time evolution of Q3D-SF, vorticity, and horizontal gradient of buoyancy more closely, Fig. 9 shows their PBL averages in the PW percentile–time cross section for the L384 and L960 cases. The comparison of Q3D-SF and vorticity (Figs. 9a,b,d,e) indicates that overall they have opposite signs (see also their spatial distributions in the PW percentile–height cross-section in Fig. A3). In the scattered case, negative vorticity is predominant (Fig. 9b). Although the positive vorticity transiently appears in the dry region in a spike-like manner, it does not develop continuously (Fig. 9b). The buoyancy gradient always has a negative vorticity tendency (Fig. 9c). On the other hand, in the aggregated case, the positive vorticity continuously develops and spreads toward the moister side with time since approximately day 10 (Fig. 9e). Besides, the positive horizontal gradient of buoyancy similarly spreads toward the moister side with time (Fig. 9f). Importantly, the horizontal gradient of buoyancy propagates several days ahead of the vorticity (see dotted and dashed lines in Figs. 9e,f that indicate the zero contours of buoyancy gradient and vorticity, respectively). Hence, it is suggested that the horizontal gradient of buoyancy plays an essential role in the development of low-level circulation.

The results obtained above are summarized as follows: 1) the CSA onset is accompanied by the low-level circulation development; 2) the moisture transport associated with the low-level circulation contributes to amplifying the horizontal contrast of water vapor content; and 3) the low-level circulation development is driven by the horizontal gradient of buoyancy caused by the enhanced radiative cooling in the dry region. These results well describe the evolution of CSA on time scales of several tens of days. However, it should be noted that the positive vorticity production by the buoyancy gradient explains the development mechanism, but does not explain the discontinuous transition from scattered to aggregated states, that is, the triggering phase of CSA onset. In other words, a question arises here: what mechanisms did make a difference between the scattered and aggregated cases in the initial 10 days or shorter time scale? Considering that the low-level circulation development starts from the dry region, we need to clarify why FT circulation can intrude into the PBL.

b. Triggering mechanism of CSA

In the previous subsection, we showed that FT air intrusion into the PBL is a key trigger from the scattered to the aggregated. In this subsection, from the viewpoint of competition between FT descent and PBL ascent, we attempt to understand the triggering mechanism by examining vertical velocity profiles. First, looking at the vertical profiles of vertical velocity in the drier side region, two types of analyses were conducted: 1) diagnostics of the vertical velocity in the FT under a weak temperature gradient approximation (Yanai et al. 1973; Ruppert and Hohenegger 2018) and 2) evaluation of the characteristics of the vertical profile of vertical velocity by empirical mode decomposition (Masunaga and L’Ecuyer 2014). Figure 10a shows the vertical velocity profiles in the region where PW is lower than the median (i.e., drier half region). As the domain size increases, the descent flow in the FT strengthens, while the ascent flow in the PBL weakens. Exceeding a critical domain size, the FT descent overwhelms the PBL ascent. As a result, the FT air entered the PBL.

The vertical velocity was diagnostically obtained using the WTG approximation to investigate the factor responsible for strengthening subsidence in the dry region. This approximation corresponds with the following balance equation between the vertical advection of dry static energy and diabatic heating:

w_{WTG} \frac{d s}{d z} = Q_{1},

where w_WTG is the vertical velocity required to satisfy the WTG balance, s is the dry static energy, and Q₁ is the apparent heat source which consists of radiative heating and convective heating (e.g., Yanai et al. 1973; Ruppert and Hohenegger 2018). Figure 10b shows the vertical profiles of the diagnosed WTG velocities. As the domain size increases, the WTG velocity (Fig. 10b) reproduces the strengthened subsidence as seen in the actual vertical velocity profile (Fig. 10a), especially in the lower to middle FT (1–6 km height). As shown in Figs. 10c and 10d, we can separate the WTG velocity into contributions from convection and radiation by calculating the velocity based on each heating rate. Although both velocities decrease as the domain size increases, the change in the convective velocity is more significant than that of the radiative velocity.

To quantify this change associated with the increase in the domain size, a vertical velocity profile is approximated as a summation of three vertical modes (the first baroclinic, second baroclinic, and PBL modes) according to Masunaga and L’Ecuyer (2014) as follows:

ρ w = o_{1} Ω_{1} + o_{2} Ω_{2} + o_{3} Ω_{3},

(5)

where Ω₁ ≡ sin(πz/D_TRP), Ω₂ ≡ sin(2πz/D_TRP), Ω₃ ≡ sin(πz/d), D_TRP is the depth of circulation throughout the troposphere, and d is the depth of PBL circulation. Based on Figs. 3 and 10a, D_TRP is fixed to 13 km, whereas d is obtained by least squares fitting along with o₁, o₂, and o₃. In the fitting, the upper limit of d is set to 2 km. As shown in Fig. 11a, the base functions are set to zero above the D_TRP for the first and second modes and above d for the PBL mode. We first fit the function to the vertical velocity profile in each PW percentile range during the initial 10 days and obtained the coefficients (Fig. 11b). Figures 11c and 11d show the average ratio of coefficients between the modes. Both the ratios of the second mode to the first mode (−o₂/o₁) and the ratios of the PBL mode to the first mode (−o₃/o₁) have positive values for all the cases shown, and their magnitude gradually decreases as the domain size increases. This result indicates that, as the domain size increases, the vertical velocity profile becomes less top-heavy throughout the tropospheric depth, and the shallow PBL circulation becomes weaker relative to the deep circulation.

Based on the results, we can understand that the CSA is triggered by the strengthening of the FT subsidence due to the decrease in FT diabatic heating in the dry region. However, it is still unclear why FT diabatic heating decreases as the domain size increases. Hence, we should clarify how the horizontal contrast of diabatic heating between dry and moist regions is formed as the domain size increases.

c. Horizontal variability of radiation, convection, and moisture

The result from the WTG diagnosis showed the importance of the change in diabatic heating in the dry region. In this subsection, we relate the diabatic heating in the dry region to the horizontal spatial scale of atmospheric variability that is constrained by the domain size. First, examining the relationship between diabatic heating and the amount of water vapor, we conducted two types of analyses: 1) examination of water vapor variability by horizontal power spectra and the moisture distance, and 2) examination of convection and circulation variabilities by convective core detection and horizontal power spectra of atmospheric motion.

Previous studies have investigated relationships between diabatic heating (convection and radiation) and water vapor from various viewpoints (e.g., Tompkins 2001; Takemi et al. 2004; Bretherton et al. 2004, 2005; Holloway and Neelin 2009; Beucler and Cronin 2016; Tompkins and Semie 2017). To see how such relationships are represented in our numerical experiments, Fig. 12 shows mass-weighted averages of convective heating and radiative heating below 6 km height during the initial 10 days as functions of PW. The heating rate increased as the amount of water vapor increased in all cases, although there were variations in the magnitude among the cases. Significantly, the range of the amount of water vapor increased when the domain size increased. Considering that the magnitude of diabatic heating is largely determined by the local PW throughout the cases, we can interpret that the broadening of the heating range is caused by the broadening of PW.

To examine the variability of water vapor more closely, Fig. 13 shows the power spectral density (PSD) of specific humidity in the horizontal wavelength–height space. We followed the procedure of Durran et al. (2017) to obtain a horizontal one-dimensional PSD. Note that the PSD is multiplied by the horizontal wavenumber k for consistency with the logarithmic horizontal axis (e.g., Craig and Selz 2018). As the domain size increases, more power can be stored at longer wavelengths, especially in the lower FT, increasing the horizontal variance. Figure 14 shows the PSD at approximately 2.5 km height (an altitude to be used later for convection core detection) in each case’s initial 10 days and final 10 days. The growth of the PSD at long wavelengths is suppressed in the scattered cases. Solid (dashed) lines are for the final (initial) 10 days.

To obtain further insights into the difference in water vapor variability among the cases, the spatial scale of the water vapor field was quantified by the moisture distance introduced in section 3a. As shown in Fig. 15, at any PW percentile, the larger the domain size, the greater the moisture distance. This means an increase in the horizontal length scale of the dry patch with the domain size.

Furthermore, we quantify another spatial scale, focusing on the horizontal distribution of convective elements rather than water vapor. After detecting convective cores at each time point, we evaluated the distances to the nearest convective core from each horizontal location. A convective core is defined as a horizontally connected grid with an updraft velocity exceeding 1 m s⁻¹ at approximately 2.5 km height (e.g., Tompkins and Semie 2017). Examples of the spatial distribution of the nearest convective core distances on Day 5 for the L384 and L960 cases are shown in Figs. 16a and 16c, respectively. We can see the cellular structures in the horizontal distribution. We also examined the histogram of the nearest convective core distance normalized by domain size (Figs. 16b,d). We can see the similarity across the cases in that the probability density decreases over long distances. According to the hypothesis proposed by Y20, the horizontal size of the subsidence region increases as the domain size increases. As schematically shown in Fig. 16e, the largest value of the nearest convective core distances reflects the horizontal size of the subsidence region regardless of the geometry. To quantify this horizontal scale, Fig. 16f shows the distance at which the cumulative distribution function reaches 99% for different domain size cases. The distance monotonically increases with the domain size, which supports the argument by Y20. This characteristic indicates that air parcels travel a longer horizontal distance from the subsidence region to the convection core at the inflow level of convection. At the same time, the increase in the size of the subsidence region also indicates that the cold pools need to travel a longer horizontal distance to homogenize PBL moisture, as Jeevanjee and Romps (2013) suggested. Notably, both the moisture distance and nearest convective core distance increase with the domain size even among scattered cases, which implies that the increase in these metrics is not just a consequence of CSA but a change prior to the CSA onset.

Finally, we analyzed the horizontal variability of atmospheric motion. Figure 17 shows the PSD of the wind speed at approximately 2.5 km height (an altitude used in the previous analyses). A large portion of the power of the vertical velocity is concentrated in short wavelengths in any case, which corresponds to the individual convective-scale motion. As the domain size increased, the power in short wavelengths decreased, whereas in long wavelengths it slightly increased. This indicates that the small-scale convective motion becomes weaker as a whole.

On the other hand, similar to the spectrum of water vapor, the horizontal velocity has a larger power at longer wavelengths as the domain size increases. Comparing the horizontal component with the vertical component, the ratio of the horizontal component increases as the domain size increases. This fact leads to the understanding that the change from a scattered regime to an aggregated regime is the transition from a state where small-scale vertical motion is dominant to a state where large-scale horizontal motion is dominant.

5. Discussion

In this section, we propose an overall picture of the onset mechanism of CSA based on the results so far and discuss two perspectives: 1) the development mechanism of CSA, which is an important process from about 10 days after the start of the simulation to several tens of days; 2) the triggering mechanism of CSA, which is an important process within approximately 10 days from the start of the simulation.

Figure 18 shows the essential processes for CSA onset in the vertical cross section along the horizontal gradient of the column water vapor. When CSA occurs, a low-level circulation is responsible for horizontal moisture transport, amplifying the horizontal contrast of the moisture field. The buoyancy gradient due to radiative cooling near the surface of the dry region drives the circulation only in the aggregated cases. In contrast, the buoyancy gradient due to evaporative cooling near the surface of the moist region drives the circulation in both the scattered and aggregated cases. Before the negative buoyancy due to radiative cooling in the dry region becomes significant, the FT air intrudes into the PBL. Successive intrusion is key to the triggering process of discontinuous transition from scattered to aggregated states. The occurrence of intrusion is determined by the competition between the downwelling motion in the FT and the upwelling motion in the PBL. The magnitude of the FT subsidence is almost determined by diabatic cooling under the WTG approximation. In our RCE experiments, we can see that the weakening of convective heating largely contributes to strengthening the FT subsidence magnitude. At the same time, the magnitude of convective heating is associated with the amount of water vapor, which is analogous to a real tropical atmosphere. As revealed by spectral analyses, the amplitude of the horizontal variation of the water vapor amount strongly depends on how large-scale components of moisture variability are allowed in the simulation. Hence, increasing the domain size as a control parameter increases the moisture distance between the dry and moist regions. The increase in the moisture distance results in the dry region becoming drier. Thus, we can understand that the CSA onset mechanism unifies the two viewpoints: one is from low-level circulation, and the other is from FT variability.

Although previous studies have pointed out the importance of low-level circulation, its role and the reason for its development are not well understood. This study deepens the understanding of low-level circulation by introducing a new method for evaluating its kinematics and dynamics. Comparing the low-level Q3D-SF between scattered and aggregated cases, we found that the low-level circulation pattern is an essential characteristic of the macroscopic behavior of self-aggregation. We showed the usefulness of low-level circulation patterns as a metric of CSA for the first time, although previous studies have proposed several metrics focusing on the moisture and convection distributions (Cronin and Wing 2017; Tompkins and Semie 2017). At the same time, we enabled the evaluation of the time evolution of low-level Q3D-SF based on the vorticity equation in the moisture–height space. It is beneficial to deepen our understanding of the dynamic causes of CSA onset. Previous studies have discussed the role of evaporatively and radiatively driven cold pools from an energetic perspective (Coppin and Bony 2015; Yang 2018b). Our results support their view that evaporatively and radiatively driven cold pools have negative and positive effects on CSA, respectively. Furthermore, our results add a new perspective on how these cold pools, as local phenomena, drive more macroscopic circulation fields. Several studies highlighted the importance of the horizontal contrast of the radiative cooling in the PBL (e.g., Muller and Bony 2015; Yang 2018b). Our results not only support their results but also provide new insights into how the horizontal gradient of buoyancy produced by strong radiative cooling in the dry region drives the low-level circulation.

Regarding the triggering mechanism of discontinuous transition from scattered to aggregated regimes, this study suggests that the competition between the downwelling motion in the FT and upwelling motion in the PBL in the dry region is a key point. To understand the triggering mechanism more deeply, we should clarify the positive feedback that causes continuous subsidence intrusion and dry patch growth. Yang and Tan (2020) indicated the existence of dry subsidence feedback. This feedback indicates that the subsidence drying produces negative buoyancy due to the virtual effect (Yang 2018a,b), and the strengthened subsidence due to the negative buoyancy further dries the atmosphere. For this positive feedback to be valid, the virtual effect of subsidence drying producing negative buoyancy must surpass the warming effect, producing a positive buoyancy. Considering that the FT is statically stable while the PBL is near neutral in general, this feedback may work more effectively in the PBL than in the FT. For this reason, whether FT air intrudes into the PBL is critically important for triggering CSA. This view is also important for understanding the coupling between the FT and PBL processes. Bretherton et al. (2005) pointed out the importance of the vertical profile of vertical velocity in the context of CSA onset. They observed negative gross moist stability in the dry region (i.e., net MSE divergence in the low-MSE region) when the dry patch expanded. They further interpreted that the bottom-heavy vertical velocity profile caused the negative gross moist stability due to bottom-heavy radiative cooling. Consistent with their results, we found that the vertical velocity profile in the dry region was bottom-heavy, compensating for radiative cooling when CSA occurred.

On the other hand, we add new insights into this bottom-heaviness from the relationship with horizontal variability. In terms of the change in the vertical velocity due to the change in the domain size, the change in the convective heating was greater than that of radiative cooling, according to the WTG diagnosis in this study. Convective heating plays a major role in terms of the horizontal contrast of heating coupled with large-scale ascent and descent motion. In contrast, radiative cooling plays a minor role. However, we do not deny the importance of the horizontal variability of radiative cooling. According to sensitivity experiments by Muller and Held (2012), for the CSA onset, the radiative effect of low clouds is especially important compared with the effects of high clouds and moisture. Hence, we speculate that the bottom-heavy radiative cooling decreased convective heating by increasing the static stability at the low-cloud level. This radiative mechanism is not caused by the direct path through the WTG velocity but by the indirect path through the environment for convection. Additionally, according to Tompkins and Semie (2017), the horizontal variability of convection is highly sensitive to the representation of the lateral entrainment process of convective updrafts because the entrainment affects the relationship between the FT water vapor and convection. From these facts, we again recognize that the FT vertical velocity profile, a key for triggering CSA, is determined by delicate interactions among the horizontal variability of convection, radiation, water vapor, and clouds.

6. Conclusions

This study investigated the spontaneous onset mechanism of CSA by analyzing scattered and aggregated convection in RCE experiments simulated with small and large numerical domains, respectively. We first introduced a new method for quantifying the kinematics and dynamics of circulation in the moisture–height space. Based on this method, we showed that reverse low-level circulations predominate in scattered and aggregated cases. In scattered cases, evaporatively driven cold pools in the moist region drive the low-level circulation, characterized by the outflow from the moist region near the surface. In the aggregated cases, radiatively driven cold pools in the dry region drive low-level circulation, characterized by the outflow from the dry region near the surface. Preceding this dynamic development of low-level circulation in the aggregated cases, the FT’s subsidence intruded into the dry region’s boundary layer. This intrusion plays a triggering role in the transition from scattered to aggregated states. By examining the free-tropospheric vertical velocity, we found that the vertical subsidence profile in the dry region became less top-heavy as the domain size increased, and the strengthening of subsidence was mainly due to the weakening of convective heating. We focused on horizontal variabilities of water vapor and atmospheric motion based on the fact that diabatic (i.e., radiative and convective) heating is associated with the amount of water vapor. As the domain size increases, the amount of water vapor varies at a large horizontal scale, resulting in a large difference in diabatic heating between the dry and moist regions. From the energetics viewpoint of atmospheric motion, the spontaneous aggregation of convection is understood as the horizontal symmetry breaking associated with a transition from a predominant state of small-scale vertical convection (several tens of kilometers) to a predominant state of large-scale horizontal circulation (several hundreds of kilometers).

This study highlighted the importance of the horizontal variability of atmospheric motion and moisture for CSA onset. In contrast, several studies have proposed simple models of self-aggregation without explicitly representing moisture. For example, Haerter (2019) and Nissen and Haerter (2021) constructed mathematical models that represent the convection replication process by cold pool interaction to show that horizontally nonuniform structures spontaneously emerge only through such a process. As another example, Yang (2021) simulated CSA by using a shallow water model with a triggered convection parameterization. That study emphasized the importance of gravity waves in describing the dynamics of CSA. Although we showed that water vapor varies at a large horizontal scale within several days by focusing on moisture distance and spatial power spectrum, why the atmosphere prefers such a large-scale variability remains unclear. It would be promising for addressing this issue to investigate how cold pools interaction and gravity wave–convection interaction play roles in the time evolution of the moisture field in the initial phase. In particular, while we quantified the horizontal scale of moisture variability by moisture distance from the isoline of PW, we can obtain additional statistical information such as the division number of dry patches and the geometrical shape. From this perspective, Beucler et al. (2020) pointed out the usefulness of such geometrical information in quantifying the time evolution of convective aggregation by focusing on the merging process of the dry patches and the eccentricity of the patch [see Fig. 4 of Beucler et al. (2020)]. By further developing the method proposed in this study, we can deepen our understanding of the relationship between cold pool interactions, geometrical information of dry patches, macroscopic moisture variability, and circulations.

Acknowledgments.

The RIKEN Junior Research Associate Program and the RIKEN Special Postdoctoral Researcher Program supported Tomoro Yanase. JSPS Scientific Research 21H01591 and 19H01974 partly supported this study. The numerical simulations were conducted using the K-computer, RIKEN Center for Computational Science through the HPCI System Research project (Project ID: hp170323), the supercomputer of ACCMS, Kyoto University and Oakbridge-CX, The University of Tokyo. We thank Dr. Tristan Abbott and two anonymous reviewers for their helpful suggestions to improve our manuscript. The authors are grateful to Team SCALE for providing SCALE.

Data availability statement.

The source code of SCALE version 5.3.3 is freely available under the 2-Clause BSD license (https://scale.riken.jp/archives/scale-5.3.3.tar.gz). The simulation data were locally stored at the RIKEN R-CCS.

APPENDIX A

Detailed Considerations of the Quasi-Three-Dimensional Streamfunction

Here we describe some practical considerations of Q3D-SF: 1) how to define PW isoline for calculation of Q3D-SF, 2) how Q3D-SF is different from the streamfunction introduced by Bretherton et al. (2005), and 3) how the relationship between Q3D-SF and vorticity holds.

a. How to define a PW isoline, its length, and horizontal flow traversing the isoline

To calculate the perimeter of Eq. (3) in the main text, an isoline and its length should be defined in a discrete field. First, two-dimensional raw data are spatially smoothed by a moving average (Fig. A1a) and binarized by a threshold value (Fig. A1b). In this study, the area of moving average is (48 km)². Then, the peripheral grids are detected based on whether a cell has at least one edge in contact with another type of cell. A line connecting the edges is defined as the isoline, and its length is the perimeter. The number of edges for each peripheral grid was determined based on the horizontal pattern of four-neighbor grids (solid red lines in Fig. A1c). The traversing flow across an isoline is evaluated on the cell edges belonging to the isoline (blue arrows in Fig. A1c).

b. Comparison of Q3D-SF with the effective streamfunction of Bretherton et al. (2005)

Here we compare the Q3D-SF newly introduced in this study with the effective streamfunction of Bretherton et al. (2005). The effective streamfunction

Ψ_{r, z}^{BBK 05}

is defined as follows:

Ψ_{r, z}^{BBK 05} \equiv \sum_{j = 1}^{r} {(ρ w)}_{j - 1 / 2, z} \frac{L^{2}}{r_{max}}, r = 1, 2, \dots, r_{max},

(A1)

where r is the PW rank. The PW rank is the rank of horizontal square subdomains, with the area of L²/r_max, ordered from low PW to high PW (i.e., from dry to moist). (ρw)_j_−1/2,_z is the vertical mass flux at the height z averaged over the subdomain indexed by j − 1/2. In this study, L²/r_max is (48 km)² for L96, L192, L384, and L960 cases, and (56 km)² for L560 case. Note that the effective streamfunction defined by Eq. (A1) differs from the original version (e.g., Bretherton et al. 2005; Muller and Held 2012) in that the factor L²/r_max is multiplied in our formulation. By doing so, we retain the consistency that both the effective streamfunction and Q3D-SF have the unit of the area integrated mass flux (e.g., kg s⁻¹).

Figure A2 shows the circulation fields quantified by the effective streamfunction, as a counterpart of Fig. 4. Overall, the effective streamfunction and Q3D-SF have similar distribution in the moisture–height space (Figs. A2 and 4). Hence, the main benefit of using Q3D-SF is in that the length scale related to the horizontal distribution of moisture (i.e., “moisture distance”) can be obtained, as explained in section 3a. On the other hand, our results also indicate that the effective streamfunction, widely used in previous studies, is useful for quantifying the circulation across dry and moist regions even though it originally has been unknown to what extent the streamfunction reflects the actual circulation fields in physical space.

c. Relationship between Q3D-SF and vorticity

Here we examine the relationship between Q3D-SF and vorticity fields to see whether the mathematical intuition that their signs are opposite (see section 3b) holds true. Figure A3 shows the spatial distribution of Q3D-SF, vorticity, and horizontal gradient of buoyancy in moisture–height space for L384 (scattered) and L960 (aggregated) cases. Comparing Q3D-SF and vorticity fields in the L384 case (Figs. A3a,b), we see that Q3D-SF is positive below (negative above) approximately 1 km height, while vorticity is negative below approximately 1 km height and positive from approximately 1 km to a few hundred meters above. In the L960 case, Q3D-SF is positive in the moist region from the ground surface to approximately 0.5 km in height and negative elsewhere (Fig. A3d), while vorticity is negative in the moist region from the ground surface to approximately 0.5 km height and positive elsewhere at altitudes below approximately 1 km (Fig. A3e). Hence, Q3D-SF and vorticity have opposite signs overall when considering the contrast between the moist and dry regions and the contrast between the lower and upper altitudes below approximately 1 km height.

On the other hand, in more detail, there are places where the relationship of the opposite signs does not hold; for example, the horizontal region from 35th to 55th PW percentile range near the ground surface in the L960 case. Thus, we should be careful about simply interpreting the detailed spatial structure of Q3D-SF from the local correspondence with the vorticity field. The mismatch of their signs is also the mathematical property expected from the general Poisson’s equation form, where vorticity affects the Q3D-SF nonlocally. Even where the inverse sign relationship between the Q3D-SF and vorticity is not satisfied, the correspondence between the clockwise (counterclockwise) flow pattern in the Q3D-SF field and the negative (positive) vorticity is also characteristics consistent with the original Poisson’s equation (e.g., near 1 km altitude in the L960 case).

The horizontal gradient of buoyancy generally has same sign with vorticity below approximately 1 km height (Figs. A3c,f). On the other hand, considering detailed structures of vorticity and buoyancy gradient in the L960 case (Figs. A3e,f), the switching location of sign of buoyancy gradient near the surface (approximately on 40th PW percentile) is on the moister side than that of vorticity (approximately on 35th PW percentile). This gap between the phases of vorticity and buoyancy gradient is related to the time evolution of vorticity and discussed in section 4a.

APPENDIX B

Moisture Budget Analysis in PW Percentile–Height Space

Following the discussion in section 4a, here we examine the spatial distribution of the moisture budget terms. Figures B1 and B2 show the moisture budget terms in the moisture–height space during the initial 10 days for the L384 and L960 cases, respectively. Basically, the adiabatic and SGS terms transport moisture from the PBL to the FT (Figs. B1c and B2c). The microphysics term dries the FT through conversion to the liquid and solid water and moistens the PBL through evaporation (Figs. B1b and B2b). In the scattered regime (Fig. B1a), they are almost in balance and the total tendency vanishes. On the other hand, in the aggregated regime (Fig. B2a), they are out of balance, and the total tendency exhibits a drying tendency in the lower FT of the dry region. This net drying tendency is considered to be produced because the drying tendency by large-scale subsidence surpasses the moistening tendency by small-scale upward moisture transport. This view can be confirmed by observing that the spatial patterns of the vertical and horizontal flux convergence terms in the dry region differs between the scattered and aggregated case (Figs. B1d,e and B2d,e).

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Yang, D., 2018a: Boundary layer height and buoyancy determine the horizontal scale of convective self-aggregation. J. Atmos. Sci., 75, 469–478, https://doi.org/10.1175/JAS-D-17-0150.1.
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Yang, D., 2018b: Boundary layer diabatic processes, the virtual effect, and convective self-aggregation. J. Adv. Model. Earth Syst., 10, 2163–2176, https://doi.org/10.1029/2017MS001261.
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Yang, D., 2021: A shallow-water model for convective self-aggregation. J. Atmos. Sci., 78, 571–582, https://doi.org/10.1175/JAS-D-20-0031.1.
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Yao, L., D. Yang, and Z.-M. Tan, 2022: A vertically resolved MSE framework highlights the role of the boundary layer in convective self-aggregation. J. Atmos. Sci., 79, 1615–1631, https://doi.org/10.1175/JAS-D-20-0254.1.
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Journal of the Atmospheric Sciences

Abstract
1. Introduction
2. Data
3. New analysis method
- a. Quasi-three-dimensional streamfunction and moisture distance
- b. Vorticity in moisture–height space
4. Results

Sections

References

Figures

Metrics

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Abstract

Abstract

1. Introduction

2. Data

3. New analysis method

a. Quasi-three-dimensional streamfunction and moisture distance

b. Vorticity in moisture–height space

4. Results

a. Low-level circulation in scattered and aggregated regimes

b. Triggering mechanism of CSA

c. Horizontal variability of radiation, convection, and moisture

5. Discussion

6. Conclusions

Acknowledgments.

Data availability statement.

APPENDIX A

Detailed Considerations of the Quasi-Three-Dimensional Streamfunction

a. How to define a PW isoline, its length, and horizontal flow traversing the isoline

b. Comparison of Q3D-SF with the effective streamfunction of Bretherton et al. (2005)

c. Relationship between Q3D-SF and vorticity

APPENDIX B

Moisture Budget Analysis in PW Percentile–Height Space

REFERENCES

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