D Coppin

Romain Roehrig
email: romain.roehrig@meteo.fr

Convection self-aggregation in CNRM-CM6-1: equilibrium and transition sensitivity to surface temperature

Convection self-aggregation in CNRM-CM6-1:

Equilibrium and transition sensitivity to surface temperature

Plain Language Summary

In idealized configurations of the Earth, convective clouds can spontaneously organize into large clusters: this is convective self-aggregation. We investigate the sensitivity of this process to surface temperature in the atmospheric component of the stateof-the-art global climate model CNRM-CM6-1. For surface temperatures spanning a typical range of tropical conditions (295-305 K), the model exhibits an aggregated state when equilibrium is reached. As the surface gets warmer, convection is more aggregated, the troposphere gets drier, high clouds get less frequent in dry regions and low-cloud cover increases. When starting from homogeneous conditions, an initial rapid phase of self-aggregation occurs in all experimented SSTs. Radiative processes are instrumental in leading to selfaggregation. For warm surface temperature above approximately 298 K, a second, slower, transition occurs and leads to higher levels of self-aggregation. It is associated with an adjustment of the large-scale circulation, in which shallow circulations in the lower troposphere (surface-700 hPa) and between dry and moist regions strengthen. The radiative loss of energy within the boundary layer, and its unbalanced state between dry and moist regions after the initial transition is argued to the main process at play.

Introduction

Tropical deep convection organizes across a wide range of scales, driven by a variety of physical processes. It can be forced by equatorial waves [START_REF] Kiladis | Convectively coupled equatorial waves[END_REF], topography or surface temperature gradients, either above ocean (Shamekh et al., 2020a), land [START_REF] Becker | Climate and climate sensitivity to changing CO2 on an idealized land planet[END_REF][START_REF] Hohenegger | The role of the permanent wilting point in controlling the spatial distribution of precipitation[END_REF] or at the boundaries between both types of surface (Coppin & Bellon, 2019a, 2019b). At mesoscale, convection is able to generate its own sources of organization as it is the case for Mesoscale Convective Systems [START_REF] Houze | Mesoscale convective systems[END_REF] or squall lines [START_REF] Rotunno | A theory for Strong, Long-Lived Squall Lines[END_REF]. At larger scales, large convective envelopes such as the Madden-Julian Oscillation [START_REF] Madden | Observations of the 40-50-day tropical oscillation -a review[END_REF] or various forms of organization along the equator are also able to modify the average zonal or meridional circulations [START_REF] Bellenger | Impact of organized intraseasonal convective perturbations on the tropical circulation[END_REF].

One type of organization that arises in idealized numerical simulations, such as under the Radiative-Convective Equilibrium (RCE) hypothesis, is self-aggregation (e.g., [START_REF] Wing | Self-Aggregation of Deep Convection and its Implications for Climate[END_REF]. This spontaneous organization of deep convection has been studied in a wide range of models, from small-domain large-eddy or cloud-permitting simulations (Bretherton et al., 2005;[START_REF] Muller | Detailed investigation of the self-aggregation of convection in cloud-resolving simulations[END_REF][START_REF] Tompkins | Organization of tropical convection in low vertical wind shears: Role of updraft entrainment[END_REF] to global, Earth-scale simulations with general circulation models (GCM - [START_REF] Popke | Climate and climate change in a radiative-convective equilibrium version of ECHAM6[END_REF][START_REF] Coppin | Physical mechanisms controlling the initiation of convective self-aggregation in a General Circulation Model[END_REF][START_REF] Becker | Imprint of the convective parameterization and sea-surface temperature on large-scale convective selfaggregation[END_REF], and under a wide range of surface boundary conditions: from fixed and uniform surface temperature [START_REF] Khairoutdinov | Rotating radiative-convective equilibrium simulated by a cloud-resolving model[END_REF][START_REF] Wing | Physical mechanisms controlling self-aggregation of convection in idealized numerical modeling simulations[END_REF][START_REF] Cronin | Clouds, Circulation, and Climate Sensitivity in a Radiative-Convective Equilibrium Channel Model[END_REF] to an interactive surface, based on an ocean mixed-layer model [START_REF] Coppin | Internal variability in a coupled general circulation model in radiative-convective equilibrium[END_REF][START_REF] Yang | Boundary Layer Diabatic Processes, the Virtual Effect, and Con[END_REF][START_REF] Shamekh | Self-Aggregation of Convective Clouds With Interactive Sea Surface Temperature[END_REF]. These models share the same drying of the free troposphere as convection aggregates and the subsequent increase in outgoing longwave radiation to space (Bretherton et al., 2005;[START_REF] Holloway | Observing Convective Aggregation[END_REF][START_REF] Wing | Convective Self-Aggregation in Numerical Simulations: A Review[END_REF]. This atmospheric response to convective aggregation is also consistent with observations [START_REF] Tobin | Observational evidence for relationships between the degree of aggregation of deep convection, water vapor, surface fluxes, and radiation[END_REF][START_REF] Tobin | Does convective aggregation need to be represented in cumulus parameterizations[END_REF][START_REF] Stein | Observed relationships between cloud vertical structure and convective aggregation over tropical ocean[END_REF]. In contrast, models do not agree on the sensitivity of aggregation to sea surface temperature (SST) nor on the details of the various mechanisms controlling the initiation, maintenance or inhibition of convec-tive aggregation. For example, in contrast to Cloud Permitting Models (CPM), aggregation almost always increases with SST in GCMs (Becker & Wing, 2020). Such a difference critically limits our ability to understand and quantify the impact of convective aggregation on the climate system. Therefore efforts to better characterize the robustness and dependency of self-aggregation to the surface temperature and to better understand the underlying mechanisms recently culminated in the RCE Model Intercomparison Project (RCEMIP, [START_REF] Wing | Radiative-convective equilibrium model intercomparison project[END_REF]: using a coordinated setup of RCE simulations, RCEMIP aims at clarifying the discrepancies between CPMs and GCMs, as well as among the numerous GCMs that took part in the exercise.

Even though the mechanisms leading to self-aggregation differ among models, most of them indicate that feedbacks between longwave (LW) radiation, water vapor and clouds (Bretherton et al., 2005;[START_REF] Muller | Detailed investigation of the self-aggregation of convection in cloud-resolving simulations[END_REF][START_REF] Craig | A coarsening model for self-organization of tropical convection[END_REF][START_REF] Wing | Physical mechanisms controlling self-aggregation of convection in idealized numerical modeling simulations[END_REF][START_REF] Coppin | Physical mechanisms controlling the initiation of convective self-aggregation in a General Circulation Model[END_REF] favors the initiation and maintenance of self-aggregation while the surface flux feedback alternates from being positive in the early stages to being negative later on [START_REF] Tompkins | Radiative-convective equilibrium in a three-dimensional cloud-ensemble model[END_REF][START_REF] Wing | Physical mechanisms controlling self-aggregation of convection in idealized numerical modeling simulations[END_REF][START_REF] Coppin | Physical mechanisms controlling the initiation of convective self-aggregation in a General Circulation Model[END_REF][START_REF] Holloway | The sensitivity of convective aggregation to diabatic processes in idealized radiative-convective equilibrium simulations[END_REF][START_REF] Wing | Self-aggregation of convection in long channel geometry[END_REF]. Other identified processes appear more model-dependent: the relative importance of clear-versus cloudy-sky radiative processes, the relative contribution of direct (diabatic) or indirect (i.e. through the atmospheric circulation) radiative effects in the evolution of convective aggregation, the role of moist static energy (MSE) horizontal advection and the role of the shallow circulation that develops in a number of CPM simulations between convectively-active and convectively-suppressed regions [START_REF] Muller | What favors convective aggregation and why[END_REF][START_REF] Shamekh | Self-Aggregation of Convective Clouds With Interactive Sea Surface Temperature[END_REF].

The latter point has been the focus of several studies pointing out the crucial role of either the free troposphere or the boundary layer (BL) in the establishment of this shallow circulation and its potential role in the initiation of convective self-aggregation. Bretherton et al. (2005) find that enhanced radiative cooling in the lower troposphere of the dry regions leads to the formation of a shallow circulation transporting MSE upgradient, from low-MSE to already high-MSE regions, thereby favoring self-aggregation through the increase of MSE gradients and the MSE variance. This has been confirmed by several CPM studies, although the nature of the radiative feedbacks driving this shallow circulation depends on the model and its configuration. [START_REF] Muller | What favors convective aggregation and why[END_REF] suggest that the BL differential radiative cooling rate between dry and moist regions is the main driver. The BL-centric framework of [START_REF] Yang | Boundary Layer Diabatic Processes, the Virtual Effect, and Con[END_REF] confirms the key role of BL di-abatic processes and further suggests that an additional buoyancy effect is necessary to establish a horizontal pressure gradient able to drive convective self-aggregation. This hypothesis has been verified by conceptual bulk models for both the dry and moist BL structures [START_REF] Naumann | A conceptual model of a shallow circulation induced by prescribed low-level radiative cooling[END_REF][START_REF] Naumann | A moist conceptual model for the boundary layer structure and radiatively driven shallow circulations in the trades[END_REF], which show that heterogeneous radiative BL cooling is able to produce pressure gradients between areas of strong and weak BL cooling. The strength of the induced shallow circulation is comparable to that caused by surface temperature differences of a few kelvins, emphasizing the potential first-order effect of spatial differences in BL radiative cooling for self-aggregation.

The strength of the shallow circulation has also been linked to the speed of selfaggregation. Using a CPM coupled with interactive SSTs, [START_REF] Shamekh | Self-Aggregation of Convective Clouds With Interactive Sea Surface Temperature[END_REF] underline that larger surface pressure anomalies, which result from both BL radiative cooling and positive SST anomalies in the dry regions, strongly modulate how fast convection self-aggregates. But, in these simulations with interactive SSTs as well as in those more commonly using fixed SSTs, the larger radiative cooling in the BL and lower troposphere strongly depends on the free-tropospheric drying induced by the large-scale deep circulation that emerges with self-aggregation. The respective role and balance between these two circulations in convective self-aggregation remains unclear, as well as how this balance can shift with surface warming. The existence and role of such BL differential radiative cooling and associated shallow circulation has yet to be shown in GCMs.

In this paper, we document and investigate the mechanisms responsible for convection self-aggregation in the CNRM-CM6-1 GCM [START_REF] Voldoire | Evaluation of CMIP6 DECK Experiments With CNRM-CM6-1[END_REF][START_REF] Roehrig | The CNRM Global Atmosphere Model ARPEGE-Climat 6.3: Description and Evaluation[END_REF]. This analysis thus focuses on the equilibrium states reached by the model under various SSTs, but also on the paths taken by the model to aggregate convection. For some SSTs, the path involves multiple phases of self-aggregation, with different timescales.

After describing the CNRM-CM6-1 model, the experiments performed with it and the diagnostics used to study self-aggregation in Section 2, we investigate the equilibrium states of the model in Section 3. In particular we assess whether different metrics characterizing self-aggregation consistently evolve with increasing SST. Section 4 then investigates the transient response and the feedbacks driving the different phases of convection self-aggregation. Section 5 summarizes and discusses our main findings.

Methods

The CNRM-CM6-1 atmospheric component

We use the atmospheric component of the CNRM-CM6-1 climate model [START_REF] Voldoire | Evaluation of CMIP6 DECK Experiments With CNRM-CM6-1[END_REF], namely the global atmospheric model ARPEGE-Climat 6.3 [START_REF] Roehrig | The CNRM Global Atmosphere Model ARPEGE-Climat 6.3: Description and Evaluation[END_REF]. This model version contributed to the RCEMIP initiative [START_REF] Wing | Radiative-convective equilibrium model intercomparison project[END_REF][START_REF] Wing | Clouds and Convective Self-Aggregation in a Multimodel Ensemble of Radiative-Convective Equilibrium Simulations[END_REF]. ARPEGE-Climat is a spectral model derived from cycle 37 of the ARPEGE/IFS (Integrated Forecast System) numerical weather prediction model developed jointly by Météo-France and the European Center for Medium-range Weather Forecast. It uses a linear triangular truncation T127 with a corresponding reduced Gaussian grid [START_REF] Hortal | Used of Reduced Gaussian Grids in Spectral Models[END_REF]. The model horizontal resolution is about 1.4 • . Along the vertical the model encompasses 91 vertical levels, following a progressive hybrid σ-pressure coordinate. The first and last model levels are near 10 m and 80 km, respectively, and the vertical resolution ranges from 20 to 200 m in the boundary layer, while being around 400-500 m in the free troposphere.

The dynamical core is based on a two-time level semi-Lagrangian numerical integration scheme. It resolves the vorticity and divergence form of the primitive equations, with temperature and surface pressure logarithm being the thermodynamic state variables. It also computes the advection of specific humidity and eight microphysical species (four for the large-scale microphysics scheme, four for the convection scheme). Horizontal diffusion, which intensity depends on the wave length, the altitude and the diffused variable, is used to stabilize the model and allows, together with the semi-Lagrangian scheme, to keep rather long model time steps (15 minutes).

Longwave radiation calculations follow the GCM version of the Rapid Radiation

Transfer Model [START_REF] Mlawer | Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the longwave[END_REF] while the shortwave radiation calculations are based on the six-band scheme of [START_REF] Fouquart | Computations of solar heating of the Earth's atmosphere-A new parameterization[END_REF] and [START_REF] Morcrette | Impact of a New Radiation Package, McRad, in the ECMWF Integrated Forecasting System[END_REF]. The stratiform microphysics scheme treats cloud liquid water, cloud ice crystals, rain and snow, and accounts for autoconversion, sedimentation, icing-melting, precipitation evaporation, and collection processes [START_REF] Lopez | Implementation and validation of a new prognostic large-scale cloud and precipitation scheme for climate and data-assimilation purposes[END_REF]. The turbulence is solved by the 1.5-order turbulent kinetic energy scheme of [START_REF] Cuxart | A turbulent scheme allowing for mesoscale and large-eddy simulations[END_REF] using the mixing length of Bougeault and Lacarrere (1989). Finally, dry, shallow and deep convection regimes are represented using the unified, bulk, mass-flux framework described in [START_REF] Piriou | A subgrid convection scheme for representing dry, moist and precipitating convection in largescale models[END_REF]. It follows the ideas of [START_REF] Gueremy | A continuous buoyancy based convection scheme: Oneand three-dimensional validation[END_REF] for the convective profile and closure, and those of [START_REF] Piriou | An approach for convective parameterization with memory: Separating microphysics and transport in grid-scale equations[END_REF] for an explicit separation between the convective vertical transport and the convective microphysical processes. The convective microphysical processes are thus treated in the same way as the large-scale, resolved microphysical processes [START_REF] Lopez | Implementation and validation of a new prognostic large-scale cloud and precipitation scheme for climate and data-assimilation purposes[END_REF], considering only that they occur in the convective environment. As a result, convective microphysical species mirror those in the convection environment, thereby allowing entrainment and detrainment of the condensates. Entrainment and detrainment processes depend on the prognostic updraft vertical velocity and follow the buoyancy sorting approach of Bretherton et al. (2004). The scheme closure is based on the relaxation of the dilute Convective Available Potential Energy.

RCEMIP simulations

CNRM-CM6-1 is run in the RCE configuration without rotation, following RCEMIP guidelines [START_REF] Wing | Radiative-convective equilibrium model intercomparison project[END_REF]: fixed and uniform SSTs of 295 K, 300 K and 305 K, constant and uniform incoming solar radiation at the top of atmosphere and zenith angle (551.58 W m -2 and 42.05 • , respectively). The simulations are uniformly initialized from the equilibrium profiles obtained from single-column experiments with the same model and for the same SSTs. The three CNRM-CM6-1 RCE simulations show different degrees of convection aggregation as emphasized by the patterns of Column Relative Humidity (CRH -ratio of precipitable water to saturated precipitable water) and the aggregation indices indicated in Figure 1 (see also [START_REF] Wing | Clouds and Convective Self-Aggregation in a Multimodel Ensemble of Radiative-Convective Equilibrium Simulations[END_REF].

Since the timing and strength of convective self-aggregation may depend on the initial state, we designed ensembles of five simulations for each of the 295-K, 300-K and 305-K SSTs. Each member of the ensemble is initialized with a globally-averaged instantaneous state taken from the equilibrium phase of the first member at the same SST (i.e. the RCEMIP simulation described above). However, the sensitivity of our results to the initial state was found weak (not shown) and therefore most of the following results are shown for only the first member of each ensemble. Besides, in order to further investigate the aggregation sensitivity to SSTs, additional experiments are performed at each SST between 295 K and 305 K by increment of 1 K. These simulations are initialized as in [START_REF] Wing | Radiative-convective equilibrium model intercomparison project[END_REF], adjusting the moisture initial profile of each corresponding singlecolumn model simulation to keep a relative humidity of 80% near the surface. All the simulations last three years. The equilibrium values are averages over the last year.

Moist static energy framework

The traditional framework to analyse self-aggregation of deep convection is based on the frozen moist static energy (FMSE), which is conserved under adiabatic processes including the phase change of water. When integrated over the column, its variance increases as convection organizes: the FMSE increases in moist regions and decreases in dry regions. In the CNRM-CM6-1 model, the FMSE h follows the definition of [START_REF] Wing | Radiative-convective equilibrium model intercomparison project[END_REF]:

h = c p T + gz + L v q v -L f q i (1)
where c p denotes the specific heat of moist air, T the temperature, g the gravity acceleration, z the geopotential height, L v and L f the latent heat of vaporization and fusion at the water triple point, respectively, and q v and q i the specific humidity and the ice specific mass, respectively (including convective and large-scale components of cloud ice crystal and precipitating snow).

The FMSE range strongly depends on the SST, which renders the comparison of indices or budgets based on FMSE difficult for different SSTs. To account for this dependency, we follow [START_REF] Pope | Cloudradiation interactions and their contributions to convective self-aggregation[END_REF] and define the normalized vertically-integrated FMSE h n between theoretical upper and lower limits using the formula:

h n = h -h min h max -h min (2)
where hats ( ) denote a density-weighted vertical integrals, and h min and h max the lower and upper limits of h for a given SST, respectively. h min is defined as the vertically-integrated FMSE of a dry adiabatic profile with zero moisture in the troposphere, plus the integrated FMSE of the initial profile above the tropopause. h max corresponds to the vertically-integrated FMSE of a fully saturated moist pseudo-adiabatic profile from the surface to the tropopause, plus the integrated FMSE of the initial profile above the tropopause. The tropopause is defined as the lowest level in the initial profile at which the lapse rate decreases be-

low 2 K km -1 .
To investigate the relative importance of different processes impacting the variance of the normalized vertically-integrated FMSE h n , we use the same budget equation derived from [START_REF] Wing | Physical mechanisms controlling self-aggregation of convection in idealized numerical modeling simulations[END_REF] but replace the vertically-integrated FMSE h by its normalized counterpart (see also [START_REF] Pope | Cloudradiation interactions and their contributions to convective self-aggregation[END_REF]:

1 2 ∂ h 2 n ∂t = h n SEF n + h n NetSW n + h n NetLW n -h n ⁄ ∇ h • (uh n ) (3)
with SEF the surface enthalpy flux (sum of sensible and latent heat fluxes), NetSW and NetLW the net atmospheric column shortwave (SW) and longwave radiative heating sources, and ⁄ ∇ h • (uh n ) the vertically-integrated horizontal divergence of the normalized FMSE.

Primes ( ) denote the local anomalies from the instantaneous domain mean. This enables us to better compare the strength of the feedbacks driving self-aggregation for different SSTs.

Characterization of CRH distributions

The eters are able to qualitatively describe the shape of the full CRH distribution and render its changes, in particular the weight between its dry and moist components when they exist, as well as their peak position and narrowness. As an interpretation of the lognormal parameters, higher expected value µ corresponds to a broader distribution while higher standard deviation σ means the distribution is more skewed towards one of its extremes (e.g., Text S2 and Figure S2). In addition to these parameters, we also estimate the CRH value at the peak of each lognormal distribution (CRH d and CRH m for the dry and moist distributions, respectively).

f (x) = 1 -α xσ d √ 2π e - (ln x -µ d ) 2 2σ 2 d + α xσ m √ 2π e - (ln x -µ m ) 2 2σ 2 m ( 4
3 Convection self-aggregation equilibrium in CNRM-CM6-1

Quantifying the level of convection self-aggregation

While convection is mostly organized along bands of high CRH, the main difference between SSTs is the larger dry areas and increasing contrasts at high SSTs (Fig- ure 1). To objectively quantify self-aggregation, a wide range of indices is used in the literature. Figure 2 illustrates some of those that are easily applicable to coarse-resolution GCMs for all explored SSTs as well as for all members of the 295-K, 300-K and 305-K ensembles. The correlations between the same variables as those on Figure 2 are provided in Figure S3.

All indices using vertical integrals of variables associated with humidity, i.e. the variances of vertically-integrated FMSE (var( h)) and normalized FMSE (var( h n )), precipitable water (var(PRW)) and CRH (var(CRH)), show a gradual increase of self-aggregation with warming, with simulations between 298 K and 301 K having a similar equilibrium.

Since var( h n ) is well correlated with var(CRH) and var(PRW) (correlations above 0.92, see Figure S3), and facilitates the comparison of the aggregation mechanisms across SSTs, we now use it as our main index to quantify convective aggregation. Note that given the fast damping of temperature anomalies by gravity waves in the tropical free troposphere (e.g., [START_REF] Sobel | The weak temperature gradient approximation and balanced tropical moisture waves[END_REF], such high correlations between FMSE-based indices and moisturebased variances is expected (see also [START_REF] Becker | Imprint of the convective parameterization and sea-surface temperature on large-scale convective selfaggregation[END_REF].

The shallow circulation efficiency η (see appendix A for details) is a dynamical index which quantifies the fraction of mass transport between dry and moist regions done by the shallow circulation [START_REF] Shamekh | Self-Aggregation of Convective Clouds With Interactive Sea Surface Temperature[END_REF]. It is highly correlated with var( h n ) (correlation of 0.98, see Figure S3). This suggests a direct link between self-aggregation and the strength of the shallow circulation. The variance of normalized FMSE and η are also well correlated with the surface pressure difference between moist and dry regions (∆p s ) and the net radiative boundary-layer warming difference between moist and dry regions (∆∂ t T | rad , positive when radiative cooling is stronger in the dry regions), with correlation above 0.95 in absolute value (Figure S3). The latter difference mainly results from differences in the LW clear-sky temperature tendencies (second to last column in Figure 2). This suggests that, as proposed by [START_REF] Naumann | A conceptual model of a shallow circulation induced by prescribed low-level radiative cooling[END_REF] and [START_REF] Naumann | A moist conceptual model for the boundary layer structure and radiatively driven shallow circulations in the trades[END_REF], this heterogeneous radiative boundary-layer cooling is consistent with positive surface pressure anomalies in dry regions, which thereby strengthen the shallow circulation. In turn, the latter positively feeds back on self-aggregation as it enhances the FMSE import in moist regions and thus the variance of h n .

In contrast to the previous indices, the subsiding fraction (SF), i.e. the fraction of the domain where subsidence occurs at 500 hPa (noted SF500), as well as those fraction computed using the 850-hPa vertical velocity (SF850) or the vertically-averaged vertical velocity (SF), increase from 295 K to 298-299 K and then decrease up to 305 K, with a rate depending on the SF index version. This behavior strongly contrasts with the other indices and indicates that a maximum subsiding fraction does not always relate to maximum aggregation as quantified with the h n variance (see also [START_REF] Wing | Clouds and Convective Self-Aggregation in a Multimodel Ensemble of Radiative-Convective Equilibrium Simulations[END_REF]. The rather high sensitivity of the SF indices to the level used in their calculation questions the way self-aggregation should be quantified.

Because of the bi-modal property of the CRH distribution (e.g., Figure S1 -also true for h n or PRW, not shown), the use of a variance metric is limited. We therefore explore a more detailed approach to characterize the CRH distribution and its sensitivity to SSTs (Section 2.4) and identify which components of the CRH distribution most likely control the changes in the variance of h h .

The weight α of the moist PDF decreases with SST until 298 K and then saturates, with a distinct minimum at 298-299 K (Figure 3). A similar pattern is found for µ m and CRH m , further emphasized by the strong correlations between these three parameters.

σ m is also maximum at 298-299 K but decreases back to low-SST levels at higher SST.

This underlines that, for SSTs up to 299 K, the moist component of the CRH distribu-tion becomes moister and narrower, while its area decreases. For higher SSTs, the moist PDF moves back to slightly lower CRH values while maintaining a similar fraction of the full PDF.

In contrast, µ d and CRH d decrease with SST and strongly correlate with the h n variance. Therefore, as SST increases, the dry component of the CRH distribution becomes drier and narrower. This suggests that the monotonic increase of the h n variance is primarily driven by the further drying of the dry regions, especially above 301-302 K.

The distinct maximum of σ m at 298-299 K and its relationship with var( h n ) mirrors that between SF indices and var( h n ) in Figure 2. The high correlation of α with µ m and CRH m also suggests that the moistest regions play a role in the fraction of the domain covered by subsidence (or large-scale ascent) at equilibrium. The relationship between SF indices and the moist PDF parameters is however more complex and SF indices only weakly correlate with µ m and CRH m (not shown). This hints that SF indices and the large-scale deep circulation are weakly explained by the CRH level in moist regions.

To summarize, SF indices and α consistently vary with SST, thus characterizing self-aggregation as a balance between the areas occupied by moist/convective and dry/subsiding regions, while var( h n ) primarily change with SST following the evolution of the moisture distribution shape, especially that of its dry component.

Atmospheric vertical structure at equilibrium

Convective self-aggregation is generally associated with a drier free troposphere (Bretherton et al., 2005;[START_REF] Tobin | Observational evidence for relationships between the degree of aggregation of deep convection, water vapor, surface fluxes, and radiation[END_REF][START_REF] Tobin | Does convective aggregation need to be represented in cumulus parameterizations[END_REF][START_REF] Stein | Observed relationships between cloud vertical structure and convective aggregation over tropical ocean[END_REF]. This is true in CNRM-CM6-1, particularly in the lower free troposphere, below 650 hPa, where relative humidity (RH)

decreases gradually with SST (Figure 4a). This decrease is primarily driven by the dry regions getting drier and by their spatial expansion at SSTs below 298 K and their slight reduction at higher SSTs (see Figure S4 and the evolution of α with SST on Figure 3).

The structure of this dry free troposphere varies with SST, from having a single minimum around 500 hPa at 295 K to having a mostly uniform RH profile with two local mimima at 800 hPa and 300 hPa at 305 K. The vertical structure of the RH profile continues to evolve at intermediate SSTs (298-302 K), yet at a slower pace, while aggre-gation indices only weakly evolve. This continuous evolution of the RH profile might reflect changes in convective activity (e.g., more detrainment between 600 and 400 hPa due to increased dilution of convective updrafts), which still do not impact the aggregated state, but can help its further transitions at higher SSTs. The cloud fraction also gradually decreases with SST between 850 hPa and 300 hPa (Figures 4b and5). In contrast, the low-level cloud fraction increases with SST, mainly reflecting the low-cloud changes in the dry regions but also to some extent the spatial expansion of dry regions (Figure S5). A slight downward shift of the low-cloud fraction is also observed from 298 K on.

In the upper troposphere, as expected from thermodynamical considerations [START_REF] Hartmann | An important constraint on tropical cloud -climate feedback[END_REF][START_REF] Bony | Thermodynamic control of anvil cloud amount[END_REF], high clouds rise with increasing SST. The high-cloud fraction decreases from 295 K to 298 K, and then increases from 298 K to 305 K, albeit at a slower rate. In moist regions, it increases from 295 K to 298 K, then slightly decreases until 305 K (Figure S5), while in dry regions, it slightly decreases (Figure S5, see also Figure 5). Thus, for high SSTs, the model behavior at global scale contrasts with the high-cloud fraction decrease with SST predicted by the stability-iris effect [START_REF] Bony | Thermodynamic control of anvil cloud amount[END_REF]. Although the cloud fraction monotonically reduces in dry regions, convective clouds become thicker in moist regions, possibly also more frequent, thereby compensating the iris effect contraction of the anvil-type high clouds.

In terms of large-scale circulation, Figure 5 emphasizes changes from a large area of shallow convection at moderate CRH and a strong lower tropospheric subsidence at low CRH (Figure 5a) to an extended yet weaker subsidence area in the lower troposphere at moderate CRH, near layers with high low-cloud fractions (Figure 5d). In the moistest region, the circulation evolves from top-heavy to mid-or bottom-heavy ascents, consistently with the enhancement of the shallow circulation between dry and moist regions.

Energy budget at equilibrium and climate feedbacks

The Top-Of-Atmosphere (TOA) and surface energy budgets at equilibrium of the 295-K to 305-K simulations are documented with Figures S6 and S7 and Tables S1 andS2. Their evolution with SST is consistent with previous studies (e.g., [START_REF] Wing | Clouds and Convective Self-Aggregation in a Multimodel Ensemble of Radiative-Convective Equilibrium Simulations[END_REF].

At first order, the TOA budget is dominated by an increase of clear-sky outgoing longwave radiation, consistent with the increase in column water vapor partly compensated by a mean lapse-rate intensity decrease (Table S3). Clouds, most probably their low-level component (Figure 4), slightly reduces the enhanced longwave cooling at high SSTs. At the surface, the net energy input into the ocean decreases with increasing SSTs, mostly as a result of the changing balance between the latent heat and clear-sky longwave radiative fluxes. At high SSTs, shortwave water vapor absorption and cloud radiative effects significantly increase, contributing to the reduced net energy input into the ocean.

To further quantify the effects of cloud and convective aggregation changes onto the climate system radiative budget, the Cess-type net climate feedback parameter λ = ∆RTOA/∆SST where RTOA is the net radiative flux at TOA (Cess & Potter, 1988) and associated longwave/shortwave clear/cloudy-sky contributions are computed and plotted as a function of the change of h n variance (Figure 6 and Table S4). The net climate feedback parameter is negative and is at first order well anticorrelated with the change of h n variance (correlation of -0.77 -note there is no clear relationship between λ and the equilibrium aggregated state as quantified with var( h n )). The feedback is the strongest for the transition from 297 K to 298 K (-10.3 W m -2 K -1 ), corresponding to the largest change in convective aggregation. While the climate feedback parameter is mostly driven at low SSTs by its clear-sky longwave component through the precipitable water increase (Table S3), it results from a more complex balance at higher SSTs, involving the clearand cloudy-sky longwave and cloudy-sky shortwave components. For SSTs above about 299 K, the cloud feedback is mostly negative (around -1 W m -2 K -1 ) except for the 304-K-to-305-K transition. It is consistent with an increased low-level cloud fraction, for which the negative shortwave feedback is found to exceed their positive longwave feedback (Figure 4).

Mechanisms leading to convection self-aggregation

Whatever the SST, a first phase of convection self-aggregation occurs during the first 20 days of the simulations (Figure 7a). For SST above approximately 298 K, a second phase of self-aggregation involves longer timescales, from about 100 days at 305 K to 400 days at 298 K (Figure S8). Note that the model behavior is consistent across the ensemble members (not shown). We first focus on the first phase of self-aggregation, common to all the SSTs explored in the present work.

First phase of convection self-aggregation

The mechanisms driving the first phase of self-aggregation are investigated using the budget of the h n variance (Equation 3) to highlight the involved feedbacks (Figure 8, see also Figure S9 for the separation between clear-and cloudy-sky feedbacks).

At all SSTs, the initial self-aggregation is driven by the LW cloud feedback, with an additional contribution, yet weaker from the latent heat flux feedback. The latter decreases with SST and, after a few days, becomes negative. The SW and LW clear-sky feedbacks also contribute to enhance self-aggregation, though with a slight delay. The amplitude of the surface flux and SW feedbacks is larger at 295 K and 300 K than at 305 K, which likely explains why convection self-aggregates slightly faster at these SSTs.

The sensible heat flux feedback is always positive and weak, and slightly larger at low SSTs. Finally, the advection feedback is always negative, except around day 10 at 305 K. Its intensity slightly increases with SSTs. Though the feedback amplitude varies with SSTs, their time evolution over the first 20 days and their relative contribution are mostly similar across SSTs. Therefore, we investigate hereafter the 295-K simulation in more detail to identify the regions where the feedbacks are the most active (Figure 9). Similar diagnostics for 300 K and 305 K are provided in Figures S10 andS11.

Following 2-3 days of spin-up, convection rapidly self-aggregates between days 5

and 10 (black line on Figure 9). The diabatic feedback, dominated by the cloudy-sky longwave feedback, is maximum in the dry regions (Figures 9c andS12). The shortwave (mostly its clear-sky component, see Figure S12) and the surface flux feedbacks in the dry regions also weakly contribute when self-aggregation starts. In contrast, the advection feedback is mostly negative, except in the driest and moistest regions.

This first phase results in a rapid initial drying visible in CRH and precipitable water (black lines in Figure 10a,e, respectively) and the apparition of a relatively low proportion of very dry columns.

At 295 K, the CRH distribution stops evolving after the first 15 days (Figure 10a).

For SSTs above approximately 298 K, a second phase of self aggregation occurs (Fig-

ure 10b-d). The CRH distribution becomes fully bi-modal as the proportion of dry columns increases and becomes similar to or larger than that of their moist counterpart.

Second phase of convection self-aggregation

When it exists, the second phase of self-aggregation, which ends when the simulation reaches its final equilibrium given in Figure 2, involves much longer timescales than the first phase of self-aggregation (Figure 7). It is characterized by the progressive drying of the free troposphere (Figure 11a-d), particularly in the dry regions (not shown).

The second phase of aggregation consists in a first period when aggregation indices remain approximately constant (Figure 7). It is shorter at high SSTs. It is followed by a second period during which self-aggregation accelerates until its final equilibrium. This Then, from day 50, σ d sharply increases, µ d (and CRH d , not shown) sharply decreases, while µ m decreases at a much more slower pace. This suggests a driving role of the dry regions in initiating the second self-aggregation phase. The delayed increase of precipitable water in the moist regions is also consistent (Figure 10h). The transition is concomitant with the slow strengthening of the shallow circulation, which becomes as intense as the deep circulation near day 70 (η = 0.5, Figure 7b). This change in the large-scale overturning circulation is further illustrated in Figure 12 The shallow circulation continuously strengthens from day 20 onwards in the 305-K simulation, consistently with the increase of the boundary-layer geopotential height and surface pressure differences between dry and moist regions (Figures 7g,m) and the opposite trend, albeit weaker, in the low and mid troposphere (Figures 7e-f). Around day 60-70, self-aggregation accelerates, at the same time when the shallow circulation efficiency η exceeds 0.5 (Figure 7b), that is when the shallow circulation becomes stronger than its deep counterpart. This is also true at 302 K (around day 150) and 300 K (around day 400), while it clearly does not append at 295 K. The acceleration also coincides with a period of time when moist convective regions become less mobile on average over the globe, with convection suddenly staying over the same area for 10 to 20 days (Figure 7c, see appendix B for the diagnostic computation). The enhanced shallow circulation efficiency is likely able to support a positive net import of FMSE within moist regions thereby favoring their maintenance at the same location for longer time periods (e.g., [START_REF] Raymond | The Mechanics of Gross Moist Stability[END_REF].

Feedback analysis of the second self-aggregation phase

To further understand the processes at play during the second phase of self-aggregation,

we now analyze the feedbacks involved in the h n variance budget at 305 K (Figures 13 and14). Similar results, but with different timings, are found for SSTs above 298 K (e.g., Figure S13 for 302 K).

After the initial phase of self-aggregation, most feedbacks do not evolve much, especially between day 20 and day 50. Then, from day 60, while the feedback magnitudes remain similar, except for the advection feedback, the CRH ranks they impact vary. The The adjustment of the circulation thus drives a spatial decoupling between the deep and shallow circulations, which is key to weaken globally the negative advection feedback and constrain its negative values to remain close to the moist regions. This thereby allows the positive LW feedback to further increase self-aggregation. In the dry regions, the positive vertical advection feedback further enhance self-aggregation, most probably through the further drying of the atmospheric columns (see also Figure 10).

Sensitivity of the second self-aggregation phase processes to SST

The previous sections suggest an important role of the shallow circulation strengthening during the second self-aggregation phase. Therefore, we now analyze the potential temperature budget contrast between dry and moist regions, to better understand which process might explain its sensitivity to SST. Figures 15 and16 show the detail of the potential temperature budget in dry and moist regions, respectively, as a function of the degree of self-aggregation (variance of h n ), for the 295-K, 300-K, 302-K and 305-K simulations, and for three layers of the atmospheric column, namely the boundarylayer (1000-925 hPa), the lower free troposphere (850-700 hPa) and the mid free troposphere (600-400 hPa). The layers are chosen according to the tendency vertical profiles, but the following results weakly depends on the exact pressure levels chosen to define these layers.

In the 305-K simulation, only the boundary-layer tendency significantly evolves beyond the first phase (Figure 15). As convection continues to self-aggregate, the boundarylayer heating by turbulent processes increases and is slightly enhanced by the weakly increasing cloudy-sky LW radiative heating, and weakened by the increasing cooling by convective and large-scale microphysical processes (i.e. condensation and evaporation).

The total effect of diabatic processes is balanced by a weak, slightly increasing, advective cooling. The boundary-layer potential temperature budget thus depicts an increased mixing within the boundary layer, most probably due to both an increased of the buoyancy surface flux and the free troposphere air entrainment at the boundary-layer top, together with more low-level cloudiness at its top (see also Figure 12) and enhanced evaporation of weakly-precipitating cumulus or stratocumulus.

In contrast, in the boundary layer of the wet regions (Figure 16k-o), the turbulent and cloudy-sky LW radiative heating rates weakly evolve after the first self-aggregation phase, while the heating by convective and large-scale condensation significantly increases.

It is slightly reinforced by the reducing clear-sky LW radiative cooling. Above, the potential temperature budget is mainly controled by the convection and large-scale microphysical heating, which is consistent with diabatic heating profiles becoming more bottomheavy. This is counterbalanced by the advective cooling. Thus, the boundary-layer temperature contrast evolution, thereby generating geopotential horizontal gradients which can enhance the shallow circulation (Figure 7b,g,m), mostly rely on turbulent processes within dry regions and convection or large-scale condensation within moist regions. Above, in the lower and mid free troposphere, the increasing condensational heating in moist regions also favors the strengthening of the shallow circulation upper branch.

Nevertheless, after the first self-aggregation phase, most potential temperature tendencies approximately follow the same trajectory across the four SSTs displayed on Fig-

ures 15 and 16, with only a few exceptions. The clear-sky LW radiative tendency within dry regions exhibits a clear sensitivity to SST at the end of the first self-aggregated state.

In the free troposphere, the advective tendency mostly mirrors this clear-sky LW radiative tendency, consistently with a large-scale subsidence mostly driven by radiative processes. In the boundary layer, turbulent mixing processes partially compensate the destabilization of the lower part of the column by clear-sky LW radiative cooling.

As a result, the following picture of the second self-aggregation phase is suggested.

After the first phase of self-aggregation, higher SSTs drive higher LW radiative cooling in the dry regions, both within the boundary layer and mid free troposphere (around 600-400 hPa). In the boundary layer, though the destabilization increased by radiative processes is partially balanced by an enhanced turbulent mixing, the temperature contrast with moist regions enhances the dry-to-moist region pressure gradient and thus the lower branch of the shallow and deep circulations. Above, the enhanced radiative cooling strengthens the large-scale subsidence, drying further the free troposphere and thereby providing a drier environment for convective updrafts. Their dilution is enhanced at upper levels, which thus leads to more bottom-heavy diabatic heating profiles. This further enhances the shallow circulation, driving a positive feedback on deep convection. As convective/moist regions become less mobile, radiative feedbacks can reinforce their local effect, i.e. enhancing the drying effect of the radiatively-driven large-scale subsidence and enhancing the boundary-layer pressure gradient between dry and moist regions. This occurs until a new equilibrium is achieved. Cloud processes further feeds back positively during this second phase of self-aggregation.

Low-level clouds have been pointed out by several studies as potential driver to convection self-aggregation (e.g., Wing et al., 2017, and references therein), and they do contribute here as a positive feedback during the second phase of self-aggregation. However, after the first phase of self-aggregation in CNRM-CM6-1, their LW radiative heating within the boundary layer only marginally depends on SST (Figure 15 and 16), at least com-pared to the clear-sky LW radiative cooling. In contrast, Figure 4b indicates a significant increase in low-level cloud fraction with SST, especially from SSTs having a single phase of self-aggregation to those having two phases. This thus remains unclear whether these clouds play an active role in the second-phase transition occurrence or whether they are only the footprint of this transition. Sensitivity experiments will be conducted in the future to further understand their role.

Conclusions and discussions

In this study, we investigate convective self-aggregation in the CNRM-CM6-1 general circulation model and assess its dependence to sea surface temperature (SST) in the non-rotating radiative-convective equilibrium (RCE) framework as defined within the RCEMIP exercise [START_REF] Wing | Radiative-convective equilibrium model intercomparison project[END_REF]. We use the three simulations run for this project (homogeneous SST of 295, 300 and 305 K), supplemented by 5-member ensembles at the RCEMIP SSTs and additional experiments exploring intermediate SSTs between 295 K and 305 K. In all numerical experiments, self-aggregation occurs within the first 20 days, at a slightly faster pace at lower SST. As SST increases, the self-aggregated equilibrium gets drier, and the large-scale circulation between dry and moist regions exhibits a strengthening shallow component. Low-cloud cover also increases, mostly in the dry regions. As expected from thermodynamical arguments, the top of high clouds rises with increasing SSTs. In contrast to the iris effect found with other models [START_REF] Bony | Thermodynamic control of anvil cloud amount[END_REF], highcloud fraction does not exhibit any clear monotonic shrinking tendency with increasing SSTs, except below 298 K. High-cloud fraction does diminish in dry region, but high clouds become thicker or more frequent in moist convectively-active regions. This behavior may be consistent with the high equilibrium sensitivity in the CNRM-CM6-1 and the role of cloudy-sky longwave feedbacks in driving it [START_REF] Saint-Martin | Tracking Changes in Climate Sensitivity in CNRM Climate Models[END_REF], as a weak or absent iris effect as found here would remove a negative feedback on the climate system.

For all experimented SSTs, CNRM-CM6-1 exhibits a rapid initial phase of self-aggregation similar to that found in other models (e.g., [START_REF] Wing | Convective Self-Aggregation in Numerical Simulations: A Review[END_REF] This transition is slower, lasting from 150 days at 305 K to more than a year at 298 K.

At the beginning of this new transition, a first adjustment of the large-scale geopotential horizontal gradients between moist and dry regions, and thus of the associated circulation, occurs, mostly within the mid-troposphere. Its origin remains so far elusive and requires further work in the future. Then, a progressive shift from a top-heavy circulation to a more bottom-heavy circulation occurs. This clearly does not happen at low SSTs.

Thus, at high SSTs, a shallow circulation settles and become even more efficient than its deep counterpart. The degree at which self-aggregation stabilizes seems in particular related to the relative importance between the shallow and the deep circulations (the η metric). The speed of this second phase of self-aggregation also appears connected to that of the shallow circulation efficiency enhancement, similarly to what is found in [START_REF] Shamekh | Self-Aggregation of Convective Clouds With Interactive Sea Surface Temperature[END_REF].

The second phase of self-aggregation occurs simultaneously to several notable changes.

As mentioned above, a shallow circulation settles and becomes stronger than the deep circulation. Convective moist regions become less mobile. Dry regions get significantly drier and occupy wider areas, while moist regions only marginally get moister. Positive advection feedbacks appears in the driest regions. The occurrence of this second phase seems primarily driven by clear-sky radiative processes in dry regions, both within the boundary layer and the mid free troposphere. As discussed in [START_REF] Naumann | A conceptual model of a shallow circulation induced by prescribed low-level radiative cooling[END_REF][START_REF] Naumann | A moist conceptual model for the boundary layer structure and radiatively driven shallow circulations in the trades[END_REF], Shamekh et al. (2020a) and [START_REF] Yang | Boundary Layer Diabatic Processes, the Virtual Effect, and Con[END_REF], the enhanced differential radiative cooling in the boundary layer at higher SSTs increases the pressure gradient between dry and moist regions, which thus strengthens the lower branch of the shallow and deep circulations.

Above, in the dry regions neighboring moist regions, the enhanced radiative cooling enhances the large-scale subsidence, drying further the free troposphere, and thereby providing a drier environment for convective updrafts. Their dilution is likely enhanced at mid and upper levels, thereby leading to more bottom-heavy diabatic heating profiles.

This further enhances the shallow circulation, which positively feeds back on deep con-vection. Besides, the fact that convection is less mobile allows the strengthening of all previous mechanisms, as they can act on the same place for a longer period of time. Cloud processes also act as another positive feedback during this transition. This schematic remains an hypothesis, albeit consistent with the diagnostics provided in this manuscript.

It will be further tested in the future through dedicated experiments.

In addition to more classical metrics of self-aggregation, we propose in this work a more detailed, albeit qualitative, framework to characterize the CRH spatial distribution and its temporal evolution: the CRH distribution, when bi-modal, can be well approximated by two log-normal distributions capturing the properties of the dry and moist regions. The associated diagnostics emphasize that transition to self-aggregation and selfaggregated states in CNRM-CM6-1 are mostly driven by adjustments within the dry regions, both in terms of level of dryness and of covered area. Applying these diagnostics to the RCEMIP ensemble might help better link self-aggregation levels and the CRH distribution and understand why self-aggregation usually increases with SST in GCMs but not necessarily in CPMs.

Finally, the long timescale of self-aggregation in CNRM-CM6-1 (150 to 400 days depending on SSTs) questions the way GCM and CPM RCE simulations are compared, as within the RCEMIP framework. GCMs are run over about 3 years while CPM simulations only last 100 days. The latter may be enough for small-domain CPMs to achieve equilibrium (e.g., [START_REF] Silvers | Radiative convective equilibrium as a framework for studying the interaction between convection and its large-scale environment[END_REF][START_REF] Patrizio | Sensitivity of convective self-aggregation to domain size[END_REF][START_REF] Matsugishi | Sensitivity of the horizontal scale of convective self-aggregation to sea surface temperature in radiative convective equilibrium experiments using a global nonhydrostatic model[END_REF], but this remains to be investigated for the largest CPM domains [START_REF] Patrizio | Sensitivity of convective self-aggregation to domain size[END_REF]. In particular, such long self-aggregation timescales may explain some of the strong differences between GCM and CPM RCE states and their sensitivity to SSTs. This calls for further investigation in the future, to assess whether CNRM-CM6-1 has a peculiar, unusual behavior, or whether large-domain CPMs do further self-aggregate on longer timescales.

style simulations and for each member of the 295-K, 300-K and 305-K ensembles, used in the present paper, are publicly available at https://thredds-su.ipsl.fr/thredds/ catalog/rcemip/catalog.html. A permanent identifier will be created if the present paper is accepted. Hourly output for the 302-K RCEMIP-style simulation, the ARPEGE-Climat software (Version 6.3) used for running the simulations, and the scripts used in the analysis are available upon request to the corresponding author. ).

Both regions are defined according to CRHc. The boundary-layer values are calculated as the weighted average between 1000 hPa and 925 hPa. Each dot corresponds to the time average over the last year of the corresponding 3-year simulation. same as (a-e) but for the 850-700-hPa and 1000-925-hPa atmospheric layers, respectively. Note that all panels horizontally span a 2 K day -1 range to help compare the tendency variations. 295K d50-365 300K d0-20 300K d20-50 300K d50-365 302K d0-20 302K d20-50 302K d50-365 305K d0-20 305K d20-50 305K d50-365 Figure 16. Same as Figure 15 for moist regions.Note that all panels horizontally span a 4 K day -1 range to help compare the tendency variations, except panels c,e,h,j which have a wider range of 7 K day -1 .

dVar( h n )/dT [10 -3 K -1 ] -2 0 2 λ cld 0 1 dVar( h n )/dT [10 -3 K -1 ] -2 0 2 λ cld,LW 0 1 dVar( h n )/dT [10 -3 K -1 ] -2 0 2 λ cld,SW 296K-295K 297K-296K 298K-297K 299K-298K 300K-299K 301K-300K 302K-301K 303K-302K 304K-303K 305K-304K Figure

  Figure S1 (supplemental material).

  [START_REF] Wing | Clouds and Convective Self-Aggregation in a Multimodel Ensemble of Radiative-Convective Equilibrium Simulations[END_REF] computed similar feedback parameters for the RCEMIP simulations, but using simulations only at 295, 300 and 305 K. Besides, for the CNRM-CM6-1 model as for all model, they included the second phase transition in their computation (see next sections). Our results are nevertheless overall consistent with[START_REF] Wing | Clouds and Convective Self-Aggregation in a Multimodel Ensemble of Radiative-Convective Equilibrium Simulations[END_REF], even though we emphasize here that the use of large SST intervals may hide rather abrupt changes of the model equilibrium state and feedbacks occurring within smaller SST intervals. Also, the CNRM-CM6-1 model shows opposite cloud feedback components to most of the RCEMIP models (CPM or GCM), which have in general a positive shortwave and negative longwave cloud feedbacks. This likely reveals a different balance between low-level and high-level cloud changes, which needs further analysis in the RCEMIP ensemble.

  Figure10a-d).

  in a CRH rank-altitude diagram(following Bretherton et al., 2005, see appendix A for the streamfunction computation). Compared to 295 K where the streamfunction is maximum in the upper troposphere and does not vary after the initial 20-day self-aggregation, the streamfunction at 305 K evolves from a top-heavy circulation, similar to that at 295 K, albeit weaker, to a more bottom-heavy circulation, especially after 150 days. The shallow circulation is clearly visible, mostly confined near the margins of moist convective regions. At 295 K, a shallow circulation similarly exists but remains weak compared to the deep one.

  LW radiation positive feedback, which remains the dominant positive feedback, mostly occur in moderately-dry CRH columns, thus close to the margins of the moist convectivelyactive regions. It also remain significant, yet weaker, in the driest regions. This LW feedback is mainly driven by its cloudy-sky component (Figure14b,e). In contrast, the SW (mostly its clear-sky component), sensible heat flux and latent heat flux feedbacks do not evolve much over the second phase period (Figures13b,d-e and 14a,d).Finally, the advection feedback is strongly modified during the self-aggregation acceleration. From day 60, it becomes positive in the driest columns, significantly impacting at day 110 about one third of the domain. There, its positive vertical component dominates its negative horizontal counterpart (Figures14c,f-note that an advective form of the last term of Equation 3 is used to introduce horizontal and vertical advective feedback component). The opposite occurs in the transition zone between dry and moist regions (around the grey line on Figures13 and 14), where the negative horizontal advection feedback dominates. On average over the whole domain, the advection feedback is weak, which thus allows the positive LW feedback to enhance self-aggregation during this second phase. This contrasts with what occurs during the first 60 days, when the vertical and horizontal advection feedbacks are mostly collocated: the total advection feedback is significantly negative and can partly counterbalance the positive LW feedback.

  : it primarily involves positive radiative feedbacks, especially in the cloudy-sky longwave and clear-sky shortwave components. At the lowest SSTs, the latent heat flux feedback also favors self-aggregation initiation, but rapidly becomes a strongly negative feedback. Sensible heat fluxes only marginally contribute to self-aggregation at all SSTs, slightly more at colder SSTs. The use of the normalized frozen moist static energy framework of Pope et al. (2021) allows us to more appropriately compare the weights of the various self-aggregation feedbacks at different SSTs. It emphasizes that the clear-sky shortwave and surface enthalpy flux feedbacks are notably weaker at 305 K than at lower SSTs. The stronger feedbacks at low SSTs are thus consistent with a faster initial self-aggregation.Following this first phase of self-aggregation, simulations with surface temperature above approximately 298 K exhibits a second transition towards a new state of self-aggregation.

Figure 1 .

 1 Figure 1. Snapshots of column relative humidity (CRH) at day 240 of the CNRM-CM6-1 RCE simulations at (a) 295 K, (b) 300 K and (c) 305 K. Different aggregation indices used to characterize convective aggregation are noted in the bottom left corner of each panel (SF850: subsiding fraction considering the pressure vertical velocity at 850 hPa; η: shallow circulation efficiency parameter (see text for detail); var(PRW): spatial variance of precipitable water; var(CRH): spatial variance of CRH; var( hn): spatial variance of normalized vertically-integrated FMSE).

Figure 2 .

 2 Figure 2. Matrix of the relationships between various indices computed for each SST equilibrium: var( h) (in 10 15 J 2 m -4 ), var( hn) (in 10 -3), var(PRW) (in

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Matrix of the relationships across SSTs between the hn variance (in 10 -3 ) and several parameters describing the CRH distribution at equilibrium: α, µ d , σ d , µm, σm, CRH d and CRHm (see Section 2.4 for their definition). Each panel shows the relationship between two indices as indicated along the matrix diagonal for all SSTs (colors) and for all members of each 295-K, 300-K and 305-K ensemble (same color within each ensemble). Each dot corresponds to the time average over the last year of the corresponding 3-year simulation.

Figure 7 .FTmidFigure 8 .Figure 9 .Figure 10 .Figure 11 .Figure 13 .Figure 14 .Figure 15 .

 7891011131415 Figure 6. Top-of-atmosphere (TOA) radiative feedbacks (in W m -2 K -1 ) computed for consecutive SSTs as a function of the change in hn variance (in 10 -3 K -1 ). Radiative feedbacks are computed as the change in the corresponding TOA radiation field divided by the change in SST (1 K). λ: net radiative feedback; λLW: longwave radiative feedback; λSW: shortwave radiative feedback. The cs and cld subscripts indicate to clear-and cloudy-sky components, respectively.

  Bougeault, P., & Lacarrere, P. (1989). Parameterization of orography-induced turbulence in a mesobeta-scale model. Monthly weather review , 117 (8), 1872-1890.

	960	vective Self-Aggregation.	Journal of Advances in Modeling Earth Systems,
	961	doi: 10.1175/1520-0493(1989)117 1872:POOITI 2.0.CO;2 10 (9), 2163-2176. doi: 10.1029/2017MS001261 a) 295 K
		Bretherton, C. S., Blossey, P. N., & Khairoutdinov, M. (2005). An energy-balance
		analysis of deep convective self-aggregation above uniform SST. Journal of the
		Atmospheric Sciences, 62 (12), 4273-4292. doi: 10.1175/JAS3614.1 Bretherton, C. S., McCaa, J. R., & Grenier, H. (2004). A new parameterization var(h n )=0.001 var(CRH)=0.032 var(PRW)=0.064 =0.454 SF850=0.543
		for shallow cumulus convection and its application to marine subtropical
		cloud-topped boundary layers. Part I: Description and 1D results.	Monthly
		Weather Review , 132 (4), 864-882. ANPFSC 2.0.CO;2 b)	doi: 10.1175/1520-0493(2004)132 0864: 300 K
		Cess, R. D., & Potter, G. L.	(1988).	A methodology for understanding and in-
		tercomparing atmospheric climate feedback processes in general circulation models. Journal of Geophysical Research: Atmospheres, 93 (D7), 8305-var(h n )=0.003 var(CRH)=0.041 var(PRW)=0.1 =0.468 SF850=0.646
		c)
		var(h n )=0.005 var(CRH)=0.058 var(PRW)=0.171 =0.572 SF850=0.679

manuscript submitted to Journal of Advances in Modeling Earth Systems (JAMES)

Acknowledgments

The authors thank two anonymous reviewers for their insightful comments, which help improve the manuscript. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 820829. We thanks the German Climate Computing Center (DKRZ) for hosting the standardized RCEMIP data.

Appendix A

In this paper, the large-scale circulation is characterized through the streamfunction within a rank of CRH-pressure plan. To compute the streamfunction Ψ, the 32768 columns are ordered from the lowest to the highest CRH and averaged by groups of 32 columns. The 1016 groups of columns are given an index i = 1, 2, . . . , 1024. Then, Ψ is calculated as a horizontal integral of the vertical velocity averaged over each of these groups, starting from the driest column (i = 1):

with Ψ i=0 (z) = 0 for all z, w the vertical velocity and ρ the mean density profile. Thus, Ψ i (z) can be interpreted as the net upward mass flux at height z accumulated over the i driest blocks.

The vertical structure of the streamfunction shows two cells: a shallow circulation with a maximum below 750 hPa and a deeper cell with a maximum above 500 hPa. To calculate the contribution of the shallow circulation to the total circulation (shallow + deep), [START_REF] Shamekh | Self-Aggregation of Convective Clouds With Interactive Sea Surface Temperature[END_REF] define the circulation efficiency η as:

with Ψ max , the maximum of the shallow circulation, Ψ max,deep , the maximum of the deep circulation and Ψ min , the local streamfunction minimum between them.

The numerator is the net boundary-layer mass divergence out of dry regions into moist regions, which returns to the dry regions below the height of the minimum, around 600 hPa. The denominator quantifies the overall large-scale circulation strength, measured by the total mass transport from dry to moist regions. Thus η (between 0 and 1) measures the fraction of mass transport from dry to moist regions performed by the shallow circulation.

Appendix B

To quantify how much moist/convectively-active regions are mobile, we calculate the correlation between the CRH map of a given day and that of each of the following days (noted ρ CRH ). We then identify the lead time (in days) when the correlation goes below 0.5 (noted d(ρ CRH =0.5)). This quantifies how long the CRH map remains approximately similar. Results remains qualitatively similar when using precipitable water or correlation thresholds of 0.3 and 0.8.

Open Research

Hourly output of the 295-K, 300-K and 305-K RCEMIP CNRM-CM6-1 simulations are part of the RCEMIP dataset, publicly available at http://hdl.handle.net/21.14101/ d4beee8e-6996-453e-bbd1-ff53b6874c0e. Daily output for the 295-305-K RCEMIP-