Idealized Simulations of the Tropical Climate and Variability in the Single Column Atmosphere Model (SCAM): Radiative‐Convective Equilibrium

To explore the interactions among column processes in the Community Atmosphere Model (CAM), the single‐column version of CAM (SCAM) is integrated for 1000 days in radiative‐convective equilibrium (RCE) with tropical values of boundary conditions, spanning a parameter or configuration space of model physics versions (v5 vs. v6), vertical resolution (standard and 60 levels), sea surface temperature (SST), and some interpretation‐driven experiments. The simulated time‐mean climate is reasonable, near observations and RCE of a cyclic cloud‐resolving model. Updraft detrainment in the deep convection scheme produces distinctive grid‐scale structures in humidity and cloud, which also interact with radiative transfer processes. These grid artifacts average out in multi‐column RCE results reported elsewhere, illustrating the nuts‐and‐bolts interpretability that SCAM adds to the hierarchy of model configurations. Multi‐day oscillations of precipitation arise from descent of warm convection‐capping layers starting near the tropopause, eventually reset by a burst of convective deepening. Experiments reveal how these oscillations depend critically on an internal parameter that controls the number of neutral buoyancy levels allowed for determining cloud top and computing dilute convective available potential energy in the deep convection scheme, and merely modified a little by disabling cloud‐base radiation (heating of cloud base). This strong dependence of transient behavior in 1D on this parameter will be tested in the second part of this work, in which SCAM is coupled to a parameterized dynamics of two‐dimensional, linearized gravity wave, and in the 3D simulations in future study.

(1) Figure 1. Diagram of the hierarchy of configurations for evaluation of physical parameterization suites (upper branch) and dynamical cores (lower branch, see details in Reed and Jablonowski [2012]) in CAM/CESM. Orange-shaded blocks are the new configurations built and studied in the lead author's dissertation (I.-K. Hu, 2020) and the present work.
In the above equations, overbars denote horizontal averages over a "filter" scale, customarily the size of a GCM grid cell (∼100 km) but for a single column model (SCM) this may imply the entire Earth. c p is the specific heat capacity at constant pressure, s ≡ c p T + gz the dry static energy, T the temperature, g the acceleration due to gravity, z the geopotential height, L v the specific latent heat of vaporization, v the horizontal velocity, ω the vertical pressure velocity, and q v the water vapor mixing ratio; c p and L v are both taken as constants. All the source terms on the right hand sides of both equations are filter-scale averages: Q R is radiative heating rate, while Q 1c and Q 2c are, respectively, "apparent" heat source and moisture sink, respectively. Q 1c and Q 2c are sums of the collective effects of subgrid vertical motions, manifested at the filter scale where they interact with "apparent" advection of filtered gradients by filtered winds on the left hand sides. Those collective effects include both net condensation (c − e) and ice phase changes (neglected symbolically here for clarity), plus the convergence of sub-filter scale vertical eddy fluxes: where ′ denotes the deviation from a horizontal average. The sub-filter-scale horizontal eddy fluxes are assumed to be negligible (Yanai et al., 1973). In an atmospheric general circulation model (AGCM) like CAM, the advection terms and right-hand sides of Equations 1 and 2 are handled by the dynamical core and physical parameterizations, respectively. The time derivative is solved by a discrete time-integration scheme that is in some ways the core of the model (counted as part of the "dynamical" core, customarily). Parameterizations are the locus of impacts from boundary conditions, including surface fluxes of momentum as well as heat and moisture.
The most popular utilization of SCAM (and other SCMs too) is to exploit it as a column in strongly forced weather situation (the white box in the upper left corner in Figure 1), by running SCAM with prescribed, time-evolving advective tendencies estimated from observations (as the replacement of the advection generated by the dynamical core) and comparing the simulated variables to the observed ones. In this approach, however, those prescribed advective tendencies may conceal genuine interactions among physical parameterizations, limiting the transferability of lessons learned from SCAM to its parent AGCM (CAM). For example, the prescribed vertical velocity in an SCM can strongly constrains the simulated precipitation, if the lapse rate and radiative cooling are both relatively time-invariant (Sobel & Bretherton, 2000). By neglecting the advection terms of Equations 1 and 2, and running the model for a long period of time (so the time-derivative terms become zero), SCAM can reach its RCE, a state that is achieved by the column physics itself. In the second part of our work (the larger orange-shaded block in Figure 1), SCAM is coupled to PLSD (i.e., parameterized advection terms) of the tropics, in the hope of bridging the gap between the SCAM and CAM (3D) RCE configurations. Because the SCAM-PLSD coupling is formulated as deviations around an assumed "environment," and is most intellectually sensible if that assumed environment is a time-mean state of the column (i.e., an RCE), the RCE of SCAM needs to be established and documented first (this paper).
Three main dimensions of model parameter space are explored in this study: model physics, vertical resolution, and sea surface temperature (SST). For model physics, specifically, this work mainly focuses on the standard physics suites of CAM version 5 and 6, with a few internal parameters modified for sensitivity tests. The motivations behind these parametric dimensions will be briefly described in turn below.
Convective parameterizations are often rightly considered as first among many sources of atmosphere model uncertainty (e.g., Sanderson et al., 2008;Sherwood et al., 2014). Deep convection is not the only challenge: unsaturated turbulence in the subcloud layer and elsewhere, shallow cumulus convection, and even grid-resolved overturning circulations also qualify as "convection." Cloud impacts on radiative transfer comprise a challenging set of processes at both climatic and intraseasonal time scales, but often are causally downstream of convection. For instance, Sherwood et al. (2014) found that the differences in climate sensitivity among 43 climate models, 10.1029/2021MS002826 4 of 30 ascribable to the low-cloud radiative feedback, can largely be further attributed to differences in the simulated strength of "mixing" between the lower and middle tropical troposphere, by both parameterized and resolved motions. Recognizing these subtleties of the term "convection," in this study we find that many SCAM behaviors are traceable mainly to deep convection, handled by a single scheme or algorithm (the Zhang-McFarlane scheme (G. J. Zhang & McFarlane, 1995), as elaborated further below).
The numerics of parameterization are sometimes overlooked as yet another source of difficulties and compromises in both global climate and tropical weather (Gross et al., 2018). For example, solely by increasing the vertical resolution in a GCM, Inness et al. (2001) managed to improve the simulations of middle-topped congestus by a convection scheme, and thereby the Madden-Julian Oscillation (MJO; Madden & Julian, 1971. Similar changes of the simulated convective cloud depths with increased vertical resolution, and its influence on the simulations of the Intertropical Convergence Zone and MJO, are also reported in Ruti et al. (2006) and Retsch et al. (2017).
Climate sensitivity estimation is an important function of climate models, and can be obliquely addressed even in a single column RCE by prescribing a range of SST and measuring the change imparted on top-of-atmosphere radiation (Cess & Potter, 1988). Without taking such estimates too literally, we do probe a terrestrial range of SSTs below, following an established protocol (Wing et al., 2018). All our results and sensitivity tests, but especially these SST experiments, link this paper's results to multi-column RCE simulations (Reed et al., 2021), the next rung up on the CAM/CESM model hierarchy ( Figure 1).
The reasons above explain the parameter space chosen for this study's sensitivity tests. After a methodology section about model and configuration details, results are reported along each dimension of that parameter space.

Single Column Atmosphere Model (SCAM)
SCAM (Gettelman, Truesdale, et al., 2019) is the single-column version of Community Atmosphere Model (CAM), which is the atmospheric component of the Community Earth System Model (CESM) developed and maintained at the National Center of Atmospheric Research (NCAR). The SCAM implemented in the current study has two numerical differences from its parent AGCM (CAM). First, SCAM uses a hybrid sigma-pressure vertical coordinate (Collins et al., 2004, their Figure 3.1), which is different from the floating Lagrangian coordinate used in CAM. Second, SCAM uses the Eulerian dynamical core for reading the profiles of initial conditions, as well as the computation of vertical advection, while CAM uses the Finite Volume dynamical core (Neale et al., 2010). In our RCE simulations, the computation of vertical advection is void because vertical velocity is assumed to be zero for the domain the SCM is taken to represent (the whole Earth). Nonetheless, the dynamical core is still required for model initialization.
This study uses two standard-version physics packages of CAM: CAM5 and CAM6 physics, which are the default atmosphere components of the versions of CESM used for CMIP5 (Taylor et al., 2012) and CMIP6 (Eyring et al., 2016), respectively. Both physics packages were configured and run in the CESM version 2.1.0, and their individual parameterized processes are summarized below. In both versions those process tendencies are applied in a "time-split" (sequential) time integration method. A process-split integration, in which each parameterization scheme works based on the same state, can be another option. Compared to the time-split mode, the simulations using process splitting might result in more condensation (Williamson, 2013b) and stronger convective tendencies due to multiple counting of available moisture and instability, as well as different amounts and lifetimes of hydrometeors due to substantial differences in processes of formation of hydrometeors (e.g., Pithan et al., 2016). Parameterizations associated with aerosol are ignored in this study (interested readers are referred to Bogenschutz et al. [2018] and Gettelman, Truesdale, et al. [2019]).

(S)CAM5 Physics
In the default parameterizations of CAM5, deep convection is treated with the mass flux parameterization of G. J. Zhang and McFarlane (1995, hereafter ZM), with the modifications of dilute convective available potential energy ("dCAPE") by Neale et al. (2008) and convective momentum by Richter and Rasch (2008). Shallow convection is parameterized as in Park and Bretherton (2009, hereafter PB). Large-scale condensation and stratiform cloud fraction are addressed by the Sunqdvist-type cloud macrophysics scheme of Park et al. (2014, hereafter PBR). Stratiform cloud microphysics is represented by a two-moment parameterization for both liquid and ice (Morrison & Gettelman, 2008, hereafter MG1), with the ice closures as described in Gettelman et al. (2010). The planetary boundary layer (PBL) scheme is based on down-gradient diffusion of moist conserved variables (Bretherton & Park, 2009, hereafter BP). Shortwave and longwave radiative transfer calculations are performed using the Rapid Radiative Transfer Model for General Circulation Model (RRTMG; Iacono et al., 2008). More details can be found in Neale et al. (2010).

(S)CAM6 Physics
From CAM5 to CAM6, many changes had been made. While the ZM deep convection scheme is still used, some of its parameters were adjusted for model tuning. One such parameter that stands out in this study is the number of negative buoyancy layers allowed below cloud top (variable name is zmconv_num_cin in the model code; ZMnumcin for short hereafter) in the simple lifted-parcel computation of dCAPE. In addition to the retuned ZM scheme, the shallow convection, planetary boundary layer, and warm cloud macrophysics schemes are replaced by the Cloud Layers Unified By Binormals (CLUBB) parameterization Larson et al., 2002), which uses a higher-order closure approach, with several terms closed by integration over an assumed subgrid probability density function. Other changes include an upgrade from a diagnostic precipitation scheme of MG1 to a prognostic precipitation scheme (Gettelman, 2015, hereafter MG2;Gettelman & Morrison, 2015), a new ice nucleation scheme (Shi et al., 2015;Y. Wang et al., 2014) to better represent mixed-phase and cirrus ice nucleation, and a fix to the energy formulation in CAM (Williamson et al., 2015). Furthermore, many parameters were retuned, such as maximum threshold humidity over ice in the cold cloud macrophysic scheme (which is the same as in the CAM5 physics suite). More details of CAM6 are described in Gettelman, Truesdale, et al. (2019).

System for Atmospheric Modeling (SAM)
As a reference to spotlight possible artifacts of a single column (with no air motions, such as clear-air subsidence), we also performed cyclic convection-resolving simulations with SAM version 6.11.2. SAM is a nonhydrostatic cloud model built and described by Khairoutdinov and Randall (2003). Briefly, SAM solves anelastic equations of motion, and uses the liquid water/ice static energy, total nonprecipitating water (sum of water vapor, cloud water, and ice), and total precipitating water (sum of rain, graupel, and snow) as the prognostic thermodynamic variables. The model integrates the momentum equations in time using the third-order Adams-Bashforth scheme and in space using second-order centered differences in flux form and with enforced conservation of kinetic energy. All scalars are transported using a positive definite and monotonic scheme. In this study, we use the original SAM's single-moment microphysics scheme for the cloud microphysics, a Smagorinsky-type prognostic closure for the effect of subgrid-scale turbulence, the bulk aerodynamic formula with constant exchange coefficients and surface wind speed for computing the surface fluxes, and the RRTMG scheme for interactive radiation. The domain is 128 × 128 km in the horizontal with a grid spacing of 1 km and doubly periodic lateral boundaries, and has 74 vertical levels with a model top at 33 km. The time step is 8 s.

RCE Configuration
This work uses the RCE configurations designed for the Radiative-Convective Equilibrium Model Intercomparison Project (RCEMIP; Wing et al., 2018), with a few minor modifications listed below. This adherence allows us to compare and connect our results to CAM5 and CAM6 simulations of RCE on the sphere, as described in Reed et al. (2021). Briefly, a model following the RCEMIP protocols is run on a water-covered (i.e., no land and sea ice), non-rotating (i.e., the Coriolis parameter, or Earth's angular velocity, is zero) planet, with spatially uniform sea surface temperature (SST) and insolation imposed as a forcing. Three SSTs are tested: 295, 300, and 305 K. The insolation is adjusted to be a value of 409.6 Wm −2 that is constant in time (i.e., no seasonal variation and diurnal cycle) by using a reduced solar constant of 551.58 Wm −2 and a fixed zenith angle of 42.05°. Global mean dry-air surface pressure is set to 1014.8 hPa. The surface albedo is fixed at a value of 0.07. Trace gases are prescribed as climatological (time invariant) values as follows: the CO 2 concentration is set to 348 ppmv, CH 4 is set to 1,650 ppbv, and N 2O is set to 306 ppbv. The ozone climatology is prescribed to be an analytic approximation of the horizontally uniform equatorial profile derived from the Aqua-Planet Experiment ozone climatology. In SCAM, direct and indirect aerosol effects are removed by excluding aerosol from the radiative transfer calculation (direct) and specifying the cloud droplet concentration (N c = 1.0 × 10 8 m −3 ) and ice crystal number concentration (N i = 1.0 × 10 5 m −3 ) within the microphysics parameterization (indirect). In SAM, direct aerosol effects are removed in the same way as in SCAM, and there are no indirect aerosol effects as information about cloud particle number concentration is not involved in the single-moment microphysics scheme.
Modifications of the RCEMIP configuration are listed as follows. First, the radiation scheme is called at every time step, rather than the default hourly interval, to avoid possible numerical instability resulting from infrequent calculation of radiation (Pauluis & Emanuel, 2004). Second, this study initializes the model with a moist adiabat that is appropriate to a given SST and patched to a 200-K isothermal stratosphere; the initial moisture profile are generated using the initial temperature profile and a specified relative humidity (RH) of 70%. We chose these relatively idealized forms of initial profiles, rather than the analytical profiles of the RCEMIP, for a better flexibility of testing the model sensitivity to initial conditions (although the result is that we find no important sensitivity of RCE to initial conditions). Third, the wind speed for the calculation of surface flux differs from that used in RCEMIP. A uniform U = 5 ms −1 wind profile is specified in our configuration, while in RCEMIP, winds are allowed to evolve freely, with a minimum wind speed of 1 ms −1 enforced in the surface flux scheme. We chose the time-invariant, 5 ms −1 wind speed because it is a typical value used in many previous studies on idealized simulations of tropical phenomena (e.g., convectively coupled waves; Kuang, 2008), to which our companion study can have a more direct comparison. From the observational perspective, the mean SST and near-surface wind speed are, respectively, ≈301 K and ≈3.9 ms −1 over all the available data from the Tropical Warm Pool International Cloud Experiment (TWP-ICE; around Darwin from 21 January to 13 February 2006; May et al., 2008), and ≈299 K and ≈2.7 ms −1 over the oceanic points within the 30°S-30°N belt spanning 1979-2020 from ERA5 reanalysis (the fifth generation of atmospheric reanalysis produced by European Centre for Medium-Range Weather Forecasts; Hersbach et al., 2020).
There are a few incidental changes in parameter settings between the L30/L32 and L60 runs, mostly for the vertically propagating gravity wave scheme, but no significant differences were seen when L30/L32 settings are used in the L60 runs and vice versa (not shown).
Our SCAM-RCE model time step is chosen to be 600 s, shorter than the default time step 1,800 s. Mean thermodynamic profiles are consistent with those obtained using that longer time step. This choice sharpens time variability, and adds numerical consistency with the second prong of our work, where SCAM is coupled to a Damped Gravity Wave (DGW) version of parameterized large-scale dynamics (PLSD) whose advection scheme uses 600 s. When using even shorter time steps, such as 60 and 300 s, the SCAM6 RCE runs show some undesired artifacts that might be related to the subcycling between the CLUBB and MG2 schemes. More details can be found in Appendix B.1 of I.-K. Hu (2020), where the sensitivity of model behaviors to time step was extensively explored.
Except for some additional mechanism-denial experiments in Section 4.1.2, each basic SCAM-RCE simulation is integrated for 1000 days, with the first 100 days excluded from analysis. The SAM-RCE simulations are integrated for 100 days with the first 35 days discarded. All apparent-advection forcing fields that traditional SCMs use, customarily derived from a field campaign's Intensive Observation Period, are set to zero. For expression convenience, our simulations will sometimes be called as PHYS_NL_SST (or a shorter version, i.e., PHYS_NL or PHYS_SST), where PHYS can be "SCAM5" or "SCAM6," NL can be "L30," "L32," or "L60," and SST can be "295," "300," or "305 K." For example, a SCAM6 RCE simulation using the L32 vertical grid and 305-K SST can be called as SCAM6_L32_305 K.

Column-Integrated Budgets in RCE Framework
The mass-integrated budgets of moist and dry static energy and their difference (column moisture) are useful analysis tools for investigating the relationship between tropical convection and the associated large-scale circulations. In this study, they can be useful for examining the relationship between the parameterized physics and its associated column state in SCAM. On both sides of Equations 1 and 2, ignoring the horizontal filtering denoted by overbars, we take a mass-weighted vertical integral from the surface (p s ) to top pressure p t (here taken to be the model's highest level): where X is any given variable. The results are: where P, H, and E are horizontally averaged precipitation, surface sensible heat flux, and surface latent heat flux, respectively. Note that in deriving Equations 6 and 7, the relation P ≃ 〈c − e〉 is used, and both horizontal and vertical advection terms are neglected in RCE framework (in which the horizontal advection term and ω are exactly zero from the perspective of a global average and approximately negligible from the perspective of a broad-tropical average). The column-integrated moist static energy (MSE; h ≡ s + L v q v ) budget is then obtained by summing Equations 6 and 7:

Column Energetics
Time-mean column energetics of RCE simulations across model physics suites, vertical resolution, and SST are summarized in Table 1. For reference, the next rung of the CAM hierarchy (3D global RCE) may be viewed in Reed et al. (2021, their Table 1). In these 900-day averages, close balance is achieved between c p 〈Q R 〉 and H + L v E (energy conservation; time mean of Equation 6) and between L v E and L v P (moisture conservation; time mean of Equation 7). As expected for models of a stable climate, all versions feature larger values of column-integrated radiative cooling and surface total heat fluxes with increasing SST. At a given SST, however, the RCE energetics differ in unanticipated ways across model physics and vertical resolution. On the standard levels (L30 and L32), SCAM5 has smaller OLR and larger precipitable water than SCAM6, a result that generally agrees with the corresponding 3D RCE runs (except for SST = 305 K because the 305-K run of CAM6 has strong artifacts; Note. Columns are 900-day means of surface precipitation rate (L v P, in energy flux unit), precipitable water (PW), outgoing longwave radiation (OLR), column-integrated radiative heating (c p 〈Q R 〉), surface total heat flux (SRFLX ≡ H + L v E), and surface latent heat flux (L v E).  Table 1 of Reed et al., 2021). In the runs with 300-and 305-K SSTs, SCAM5 features smaller atmospheric cooling and precipitation than SCAM6; while the corresponding 3D RCE runs tend to have larger precipitation in CAM5 than in CAM6 (excluding the 305-K run of CAM6; Table 1 of Reed et al., 2021). Increasing vertical resolution to L60 yields opposite effects on SCAM5 versus on SCAM6: compared to the runs using the standard vertical grid, SCAM5 has larger OLR and stronger column-integrated radiative cooling (probably due to reduced high clouds and the associated longwave cloud effects as shown in Figures 3b and 3c below), larger precipitation, and smaller precipitable water; except for precipitable water, the opposite occurs for SCAM6 (for which no easy mnemonic springs to mind). One philosophical ideal for a well-tuned and "scale-aware" climate model of any dimensionality would be that numerical details have no effect on the solutions; clearly CESM remains far from that (e.g., Herrington & Reed, 2020).
Values are somewhat comparable to those in 3D RCE runs (Reed et al., 2021). As in those runs, with a single prescribed tropics-like SST the net radiation is wildly imbalanced, as too-little low cloud allows excessive insolation to the surface. For this reason, climate sensitivity estimates around this imbalanced state would miss the point, and the SST dependence is simply of interest in its own right as a factor in tropical dynamics.

Simulations With 300-K SST
To make some contact with observations, we analyze the SCAM RCE simulations with SST set to 300 K, which is a typical tropical SST that is in the range between the mean SST values of broad and the deep tropics, represented by ERA5 and TWP-ICE respectively.
Mean climate profiles of temperature and humidity are broadly comparable to those estimated from the ERA5 reanalysis and measured from the TWP-ICE observational campaign, as well as that of the SAM RCE simulation ( Figure 2). Much of the similarity simply follows from lapse rates being near a moist adiabat (pseudoadiabat equal to SAM's mean temperature near 850 hPa; solid gray curve), combined with some fundamental processes that constrain relative humidity (Romps, 2014). For visual convenience, the temperature profiles of SCAM are shown as their deviations from the SAM temperature profile. Both the SCAM5 profiles (panel a) are slightly warmer than SAM through the free troposphere, suggesting that perhaps their updrafts are less diluted by entrainment (Singh & O'Gorman, 2013;Zhou & Xie, 2019), despite their drier mean humidity profiles (panel b). The opposite occurs for SCAM6 (panel d), except for mean humidity profiles being drier than that of SAM as well (panel e). The mean moisture profiles from the SCAM (and SAM) RCE simulations lie between those from the ERA5 reanalysis and TWP-ICE observational data sets. This may be expected, as the dry subtropical areas substantially shape ERA5's mean humidity over the broad tropics (when averaging over 10°S-10°N, the mean moisture profile resembles those SCAM profiles), while the TWP-ICE possesses a slightly warmer SST and mean ascent. At tropopause level, the altitude of minimum temperature from all the SCAM RCE simulations is lower (and so the minimum temperature is warmer) than the observations and SAM RCE simulation. It is unclear whether to invoke radiative or convective processes to understand this difference. These SCAM5 and SCAM6 values appear comparable to those in 3D RCE (Reed et al., 2021).
Some oscillatory structures in q v (panels 2b and 2e) suggest internal mixed layers, but the way they are tied to grid levels (shown as tick marks at right) seems quite numerically artificial. RH, which is computed over water and ice where the temperature is greater than 273.15 K and less than 253.15 K (250.15 K for ERA5), respectively, with a ramping function between those temperatures, shows these resolved-grid-level variations of moisture more clearly in panels (c and f). It is shown below that these relate to updraft detrainment levels in the ZM deep convection scheme. SCAM RCE also exhibits time-mean saturation over ice (RH ≥ 100) near the tropopause. While ice supersaturation is frequently observed by aircraft (e.g., Patnaude et al., 2021) as well as in satellite data (e.g., Gettelman et al., 2006), and occurs in the RCE simulations of other single column models, cloud-resolving models, and large eddy simulations (Figure 8 in Wing et al. [2020]), its prevalence in the grand mean is an artifact of the strict single column geometry: in 3D RCE with CAM5 and CAM6, dry regions make the space-time mean unsaturated ( Figure 3 in Reed et al. [2021]). In the lower-middle troposphere, mean RH profiles all hover near the 60%-80% range, with SAM being the outlier, on the humid side.
Radiation and cloud profiles are shown in Figure 3, with SCAM5 on top and SCAM6 on bottom, all at SST = 300 K. Across simulations, the mean radiative heating profiles are negative (solid curves repeated in panels a, b, d, and e), as longwave cooling (dashed curves in panels b and e) predominates over shortwave warming (dashed curves in panels a and d). Cloud radiative effects (CRE = cloudy sky minus clear sky) are shown in dotted curves, in both the shortwave and longwave panel columns, and can be qualitatively appreciated in terms of condensate mass and cloud fraction profiles at right in panels (c and f).
The general profiles of radiative heating are broadly similar, and realistic based on textbook radiative transfer results. SAM has stronger radiative cooling overall, apparently because it lacks the tropopause-level saturation and near overcast of SCAM (seen in right hand panels of Figures 2 and 3). That tropopause cloud layer causes local solar absorption near 200 hPa (dotted curves in panels a and d), along with longwave absorption there and reduced longwave cooling to space everywhere below (both contributing to deep layers of positive Q CRE,LW , dotted curves in panels b and e). Compared to the simulations of SCAM, SAM (black dotted) has similar shortwave CRE (SWCRE) but weaker longwave CRE (LWCRE). Low clouds also absorb some sunshine at their tops, shade the layer below, and emit from their tops as seen in the CRE profiles. L30 for SCAM5 or L32 for SCAM6; red: L60). Grid levels are denoted with ticks on the right edge of each panel. Timemean profiles from the ERA5 reanalysis over the oceanic points within 30°S-30°N spanning 1979-2020 (golden line) and the TWP-ICE (green line) field experiment over its period of available data, as well as from the SAM RCE simulation over days 35-100 (black line), are overlain for comparisons. In panels (a and d), the solid gray line is the pseudoadiabat ("pa") equal to SAM's mean temperature near 850 hPa, and the dashed gray line denotes the equality between the SCAM and SAM temperature profiles.
The oscillations tied to grid levels are prominent in the SCAM5 runs (top row). Near the surface, spikes of cooling and heating are contributed by LWCRE (dotted in panel b). That in turn is ascribable to spikes in liquid water content (dashed curves in panel c) since cloud fraction (solid in panel c) is smooth and modest. These grid-level spikes are SCM geometry artifacts: spatial means of 3D RCE results do not exhibit them (Reed et al., 2021, although they also show similar cloud liquid peaks below 900 hPa), nor do SAM results (black curves). In the upper troposphere, the oscillatory cooling on L30 is also contributed by LWCRE (blue dotted curves in panel b). However, on L60 (red curves), SCAM5's upper-level cooling oscillations are a puzzle: they are not in the CRE profile (red dotted), while the humidity oscillations ( Figure 2c) seem rather modest in amplitude and fewer in number.
These discrete grid artifacts are of smaller amplitude and less prominent in SCAM6 (bottom row), but still the curves are not smooth. This rugged structure is robust to longer time averages of SCAM, but again is smoothed away by spatial averaging over 3D RCE runs of CAM6 (Reed et al., 2021).
SCAM6 has near 100% cloud fraction at the tropopause, while SCAM5 has 60%-80% (comparing solid curves in Figures 3c and 3f). These large values of cloud fraction are barely seen in the RCE simulations of 3D CAM and SAM, whose cloud fractions are in the range of 20%-30% (with the CAM6 305-K SST run being an exception, which has cloud fraction ∼80%). Nevertheless, since SCAM5 and SCAM6 use an almost identical algorithm of cold cloud macrophysics for calculating ice stratus fraction (the predominant contributor to the upper tropospheric cloud fraction in panels c and f), this difference is notable. From Gettelman et al. (2010), the diagnostic ice stratus fraction (a i,st ) is In Equation 9, RHi ≡ (q v + q i )/q sat , where q v and q i are the water vapor and ice mass specific humidities, respectively, and q sat is the saturation specific humidity over ice. RHi max and RHi min are prescribed maximum and minimum threshold humidities with respect to ice. Although RHi max has been retuned in SCAM6, changing RHi max back to SCAM5's value does not alter the mean cloud fraction profile of SCAM6 much (not shown  Convective moisture sinks Q 2c (middle panels b and e) add up to zero in this 900-day average, which means moisture is conserved in the long-term as expected from Equation 2. To first order, non-deep convection carries the surface evaporation into the the 1,000-600 hPa layer in SCAM5 and a thinner 1,000-800 hPa layer in SCAM6. As we shall see later in Section 4.1.1, moistening is constrained below 800 hPa in SCAM6 because deep convection is frequently capped at 800 hPa (Figure 10k below).
The right panels (c and f) of Figure 4 show updraft detrainment ( ) and entrainment ( ) , as the decomposed terms of vertical gradient of updraft mass flux, based on ∕ = − . The time-mean detrainment profile (solid lines) feature several peaks above 800 hPa, which are associated with the various discretized cloud tops of multiple cloud types in the ZM scheme. Between 400 and 700 hPa, the zig-zag levels of detrainment roughly match the zig-zag levels of relative humidity (dotted lines, which are identical to the solid lines in Figure 2c), indicating that deep convective detrainment is a primary moisture source in this layer. The importance of deep convective detrainment for the midtropospheric moisture is more obvious in SCAM5 than in SCAM6, because in SCAM6 non-deep convective processes also involve in the production and consumption of moisture, as discussed in Section 4.1.1 below (shown as colored fills in Figure 10h).

Dependence on SST
The RCE simulations of SCAM5, SCAM6, and SAM (serving as a baseline) across 10 K of SST starting from 295 K are examined in Figure 5. The time-mean profiles of temperature deviation from the 300-K SST run display top-heavy warming with increasing SST for all the tested models, but SAM warms more than both SCAM5 and SCAM6 do, particularly in the upper troposphere. When using the standard L30 or L32 vertical grid, the profile of SCAM's temperature anomaly against the 300-K SST run exhibits a "dent" in the upper troposphere (probably due to discretization of relatively coarse resolution), and the altitude of this "dent" increases with increasing SST.
As SST increases, the q v profile of SCAM tends to have more near-vertical steps (uniform, perhaps indicative of mixing of this conserved tracer), which are manifested as zig-zag levels in the RH profile. This might be because the updraft ensemble plume in the deep convection scheme becomes more buoyant with increasing SST and is more likely to detrain at multiple levels. SCAM5 with both the L30 and L60 vertical grids feature a well-mixed boundary layer near the surface, and generally a less stairstep-punctuated structure of moisture compared to SCAM6. In some profiles q v increases with height, perhaps suggesting that detrainment from deep convection rather than mixing (like by shallow convection) is the reason for these overhanging step-like structures.
The profile of cloud fraction (fourth column) resembles the profile of cloud condensates (fifth and sixth columns). In both SCAM and SAM, the maxima of upper tropospheric cloud fraction and cloud ice content shift upward with increasing SST. When plotting as a function of mean temperature rather than pressure (Figure 6), for each model, the peak of ice stratus appears at approximately the same temperature across SST. This is consistent with the Fixed Anvil Temperature hypothesis (Hartmann & Larson, 2002).  Figure 3, but for the decomposed terms of (a and d) convective heating Q 1c , (b and e) convective drying Q 2c , and (c and f) vertical gradient of mass flux overlapped with relative humidity. The superscripts DC, RES, and MIC denote the contribution of deep convection scheme, sum of schemes of non-deep convective processes ("residual"), and microphysics scheme, respectively, to the total field. The variables and represent updraft detrainment and entrainment, respectively, contributed from the deep convection scheme. In panels (c and f), the top axes correspond to the scale of relative humidity. Note that unlike Figure 3, this figure focuses only on the fields from the SCAM_300 K RCE simulation.

Temporal Evolution
The first 100 days have been discarded from all the RCE analyses above. We have examined this "spinup" period in various cases, and sought evidence for possible initial condition dependence or hysteresis, but found none. RCE is established within about a month, a few times the characteristic timescale which can be understood as either a vertical turnover time of the troposphere (Tompkins & Craig, 1998) or as a bulk residence time for water vapor. Column water vapor of about 50 mm divided by a 3 mm/d rain rate is about 17 days, the same as a potential temperature θ stratification of about 50 K (set by the difference between near-surface θ e and θ) divided by a 3 K/d radiative cooling rate (a reasonable value after converting Q R from Figure 3 to θ units) and is consistent with a long time scale (15 days) governed by radiative-subsidence velocity in Tompkins and Craig (1998). This "overturning" of the troposphere in nature gets mimicked by SCM column processes, with no explicit clear-air subsidence, illustrating nicely the implicit "compensating subsidence" interpretation of deep convective tendencies implied in the mass flux diagnosis of Figure 4 above. Figure 5. Time-mean profiles of (from left to right) temperature deviation from the 300-K SST run, specific humidity, relative humidity, cloud fraction, cloud liquid content, and cloud ice content from the RCE simulations of (from up to bottom) SCAM5_L30 (blue lines), SCAM5_L60 (red lines), SCAM6_L32 (blue lines), and SCAM6_L60 (red lines) with SST = 295 K (long-dashed lines), 300 K (solid lines), and 305 K (short-dashed lines) over days 100-1,000. In each panel, the profiles of SAM RCE simulations (black lines) are overlapped, and the distributions of vertical grid of SCAM are denoted on the right edge.
Shorter than that characteristic timescale, transient water or energy storage is possible in an atmospheric column, making RCE a statistical rather than a truly steady equilibrium. Figure 7 shows 20-day time series samples of precipitation (in energy flux unit, black) and column-integrated radiative cooling (〈Q R 〉; orange) for RCE simulations with SAM at three SSTs. The variability of precipitation increases with SST, as in Reed et al. (2021, their Figure 2), even as surface fluxes and 〈Q R 〉 are more nearly constant for all tested SSTs, suggesting that water vapor storage and discharge (conversion between L v q v and s, since their sum (i.e., MSE) is conserved given these near-constant diabatic terms) is the main mechanism of the vacillations.
The lower panels in Figure 7 show mean power spectra of surface precipitation over two 35-day segments with a 5-day overlap (i.e., 65 days in total for analysis as the first 35 days of a 100-day run have been discarded) from the SAM simulations with SST set to (from left to right) 295, 300, and 305 K. The spectra of the first-order autoregressive (AR1) model and its corresponding upper and lower bounds of 99% confidence level are built following Gilman et al. (1963) and Wilks (2019), and the statistical significance is interpreted on a posteriori basis (i.e., we did not expect a spectral peak at a specific frequency in advance). No spectral features are evident in Figure 7, indicating that no periodic contributions to the SAM's precipitation time series are detected at a statistically significant level within 65 days.  Figure 5, but with time-mean profiles of relative humidity, cloud fraction, cloud liquid content, and cloud ice content plotted as a function of time-mean temperature, and with the sets of simulations using the same model physics but two different vertical grids merged together.
The same analysis is applied to SCAM-RCE simulations across model physics, vertical resolution, and SST in Figure 8. Random 20-day samples at top suffice to show behavior, while spectra estimated from the full 900-day time series (separated into twenty 50-day segments with a 10-day overlap) are much smoother, with tighter AR1 error bounds than Figure 7. SCAM precipitation variance also increases with SST, but its absolute value is not directly comparable to SAM; a larger SAM domain would have smaller variance in its area-mean time series. Strikingly, SCAM-RCE exhibits radiative cooling variations almost as large as the precipitation variations, governed mainly by how the single column's cloudiness fluctuates in time. In a strict single column, s can be stored and discharged in these vacillations, as the weak temperature gradient constraint doesn't apply, as we shall see below.
Spectra all have that RCE-governed roll-off at periods longer than 1-2 weeks, but some SCAM versions manage to have spectral peaks with periods up to 10 days. Multi-day features in the precipitation waveforms tend to have sharp attack and gradual decay (a saw-tooth pattern), which shows up as a harmonic-overtone series of peaks in the power spectra. The mechanisms of these multi-day oscillations will be elaborated below. Many but not all spectra also exhibit a broad humps at periods near 0.5-1 day, perhaps indicative of another shorter characteristic timescale in the physics packages, perhaps stemming from the few-hour CAPE relaxation timescale parameter in the ZM deep convection scheme. Another possibility is boundary-layer stabilization and recovery, but the surface chilling from the ZM scheme is extremely shallow as seen above (Figure 4).
The temporal variability of simulated precipitation in SCAM RCE depends on the choice of model physics, as well as on the vertical grid. Regardless of the prescribed SST value, the set of SCAM5_L30 simulations (top row, panels a-c) exhibits weaker and less red (high frequency only) variance compared to the other three configurations. When using the L60 vertical grid, SCAM5-RCE (second row, panels d-f) shows 5-to 6-day oscillations with a sawtooth (sharp-attack) waveform. The CAM6 physics (bottom two rows, panels g-l) has even stronger variability, with ragged upward spikes in wet epochs and sometimes zero precipitation. That variability becomes even more extreme on 60 levels (bottom row), where the half-day and 10-day vacillations become very well separated spectrally.

Column-Integrated Budgets and Time-Pressure Cross Sections
To further investigate the mechanism of multi-day oscillations in the simulations using CAM6 physics and/or L60 grid, we analyze representative samples from SCAM5_L60_300 K and SCAM6_L60_300 K runs as the forms of  (black) and column-integrated radiative heating (orange) from days 80 to 100 and (lower panels) mean spectral analysis of surface precipitation over two 35-day segments (black) for the SAM-RCE simulations with SST set to (from left to right) 295, 300, and 305 K. In the lower panels, the spectra of the first-order autoregressive model and its corresponding upper and lower bounds of 99% confidence level are shown in gray, red, and blue, respectively, and logarithmic scale is used for both axes. mass-integrated budgets (Section 2.5) and time-pressure cross sections, which facilitate visual interpretation (as pressure is proportional to mass).
Nine-day series from the SCAM5_L60_300 K simulation are shown in Figure 9. Left-column panels illustrate state variables, middle-column panels illustrate convective tendencies, and right-column panels illustrate radiative tendencies. The time derivative of 〈h〉 (black curve) and sawtooth precipitation cycle (gray curve) are repeated in all three columns in the top panels.  Figure 7, but for the control sets of SCAM-RCE simulations comprising the choices of (from top to bottom) the SCAM5_L30, SCAM5_L60, SCAM6_L32, and SCAM6_L60 with SST set to (from left to right) 295, 300, and 305 K. Each subplot shows the time series of precipitation from days 800 to 820 in the upper panel and the averaged spectra over twenty 50-day segments of precipitation in black in the lower panel. The green shaded windows denote the selected samples for cross-section plots of buoyancy as shown in Figure 11.
The multi-day oscillation is seen in panel a to involve gradual buildup and sudden discharge of 〈h〉. Time derivatives of both 〈s〉 (red curve) and latent energy 〈L v q〉 (blue curve) contribute to this 〈h〉 buildup and discharge, in roughly equal proportions. In other words, both heating processes (radiative as well as convective), and moistening/drying processes (the other facet of deep moist convection's impact) are involved. Furthermore, these are coupled intimately, through the strong impact of vapor and cloud (left-column state variables) on radiation (right-column tendencies). . Column energetics (top) and vertically resolved cross sections of the 5-day cycle in the SCAM5_L60_300 K run. Left-column sections display state variables, middle-column sections display convective tendencies, and right-column sections display radiative heating tendencies. In the top-row panels, panel titles are a color-coded legend, and gray dashed lines denote zero energy flux. In all other panels, the left-justified quantity is the color-filled contours, while the right-justified quantity describes the contours (black for positive values, purple for negative values, with units as indicated in the title). The contour interval for panels (d-i) is the same as the color bar scale in panels (e, f, h, and i). Positive-only fields in the bottom row are contoured in cyan, with interval denoted in the title. In panels (j and l), cyan contours start at 2 mg·kg −1 . δx is temporal anomaly of any given variable x against its 900-day mean; CWAT is total cloud condensate (= liquid + ice); M is convective mass flux; other fields and notations are described earlier in the paper.
Visually, the temporal variations of 〈s〉 and 〈L v q〉 are vertical integrals of the values represented by the color-filled blue-to-red ink in panels d and g respectively, where δX denotes the temporal anomaly of any given variable X from its 900-day mean. The 〈s〉 variability is contributed at both 600 hPa and 150-300 hPa (colored fills in panel d), while moisture variations are fairly coherent over 300-700 hPa, albeit with about three interleaved sub-layers of descending anomalous moisture (red fill in panel g) and dryness (blue fill in panel g). Time differentiation reveals the tendencies (open contours on top of those color fills), showing that these multi-day descending-layer variations are propagated and evolved by pulsatile convective sources and sinks. Moistening and drying tendencies are especially multi-layered, suggesting deep convective detrainment at various levels (shown directly as cyan contours in panel k). The time-mean moisture oscillations in the vertical in Figure 2 may be obliquely related to the nearly horizontal striping patterns of colored fills in panels g (although these are temporal anomalies) and j.  Figure 9, but for the SCAM6_L60_300 K simulation. Note the different (from Figure 9) and irregular scales are used for both color fills and contours in panels (d-i).
More details are offered in the diagram than would be illuminating to verbally describe here, but the attentive reader can fruitfully glimpse various relationships between state, convective, and radiative variables, and even test some alternate mental hypotheses about the mechanisms at play.
To focus reader attention, we begin from the sharp increase of precipitation late in day 804, as deep convection's top abruptly deepens from 300 to 150 hPa (mass flux in panel k, heating in panel e). This jump-up followed the gradual cooling (purple contours in panel d) of an anomalously warm buoyancy-capping layer near the 300 hPa level (red fill in panel d). Why was that 300-150 hPa warm layer cooling with time over the earlier parts of day 804? By radiative emission (blue fill in panel f), enabled by the disappearance of high clouds above the 300 hPa level (cyan contours in panel j) and the vanishing of their radiative heating effect (Q CRE , red fill in panel i), leaving only smooth vapor cooling.
Those high clouds had previously been maintained in a layer of large ambient RH (dark gray in panel j) by pulses of intermittent non-deep convection (a close zoom of panels e and h reveals filaments of Q 1c and Q 2c fill colors with near red-blue balance, but no contours indicating the deep convective contribution). That non-deep convection was presumably driven by cloud-base radiative heating (red fill in panel i) and by the radiative cooling (blue fill in panel f) of that humid layer above the capping inversion (red fill in panel g), between these brief cloud-refreshing non-deep convection pulses of increasing depth. A time-mean signature of these SCAM5 near-tropopause, non-deep-convective cloud processes (phase changes and fall of ice) can be noticed in Figures 4a and 4b.
On around day 805, when deep convection has surmounted well above the 200 hPa level, the intense sensible heat flux convergence at its top (fills and contours matching in panel e) creates the next anomalous warm layer (red fill in panel d), whose gradual descent comprises the next 5-day cycle, terminated by the next convective burst on around day 809.5. What makes that warm convection-capping layer descend? The positive tendency (black contours in panel d), in leading quadrature with the anomalous layer of warmth (red fill), resembles Q 1c in panel e, but with a few features that also implicate mostly longwave Q CRE (red fills without matching contours in panel i). In summary, convective heating and radiative heating in the upper troposphere appear to drive this 5-day cycle of gradual descent of deep convection's top after sudden jumps.
A weaker thermal anomaly near 600 hPa is also seen in panel d, but this appears to be a faint, low-pass filtered (more sinusoidal) echo of the sawtooth cycle in the upper troposphere, as its positive and negative tendencies (contours in panel d) are contributed by subtle interplays of convective and radiative heating processes that do not appear to affect precipitation importantly. Layered moisture anomalies (color fill in panel g), partly grid-levellocked and partly descending, are also cyclic on the 5-day timescale. These are governed by complex layering in deep convective detrainment, as shown by the collocated Q 2c (colored fills) and 2 (contours) in panel h and (cyan contours) in panel k. Again, these appear to be passive responses to the oscillation in the upper troposphere, not drivers, as demonstrated in the mechanism denial experiments shown next.
The stranger, longer-period (10-day) oscillation in SCAM6_L60_300 K simulation can also be illustrated and analyzed within the same graphical conventions (Figure 10). To cover at least one full cycle of oscillation, the analysis is applied to a longer time window than that in Figure 9. In this case, the upper tropospheric warm anomalies that cap convection keep descending through the middle and even lower tropospheric layer (red fill in panel d), explaining the oscillation's longer period and larger amplitude in column-integrated energetics.
Consider again the time progression through the precipitation burst. Deep convection, before its outburst on around day 806.5, is capped below 700 hPa (no contours above that level in panels e and h), resulting in uncompensated radiative cooling of the free troposphere (blue fill in panel d). As deep convection fails to develop and consume moisture in the free troposphere, P < E and moisture strongly accumulates in the column (blue curve is well above zero in panel a). Within the column, this accumulating moisture is stored largely in the 600-800 hPa layer (indicated by predominance of saturated red fill in the vertical integral of panel g). Evidently the ZM scheme dumps moisture around 800 hPa, while periodic outbreaks of non-deep convection distribute it further upward (indicated by the maxed-out purple contours of 2 in panel h, allowing a view through to the color-filled total Q 2c ). Surprisingly, this capping of the ZM deep convection scheme is accomplished by the small-amplitude and thin positive temperature anomaly at 800 hPa (light orange filled-contour sliver in panel d)! This anomalously warm cap layer is maintained almost constant in the presence of strong, fluctuating convective heating and radiative cooling (red and blue fills in panels e and f, respectively), a remarkable and presumably nonrandom (that is, intimately interactive) balancing act of processes that mocks any attempt to unconditionally explain time variability using these tendency terms in a multi-term governing equation.
Unlike the relatively invariant surface fluxes in the SCAM5_L60 run, surface flux H + L v E in the SCAM6_L60 run (green curve in panel c) evolves significantly with time, despite the constant wind speed in the flux formula. Before the outburst of deep convection, latent heat from the ocean, the dominant component of total surface flux, is relatively smaller (while sensible heat from the ocean remains unchanged, not shown), indicating that the column's moist anomaly is expressed even in the surface layer. The opposite occurs during and after the deep convection, as the near surface air (like the column mean) becomes drier. These surface variations are faint and more sinusoidal, perhaps another example of passive signatures of weak, smoothed downward propagation of a predominantly upper-tropospheric cycle, like the midtroposphere anomalies of SCAM5_L60 RCE discussed above in Figure 9d.
By day 806.5, even the peculiar capping assumptions of the ZM scheme cannot prevent deep convection in the wildly, unrealistically unstable profile with very moist low levels and cold upper troposphere. Convection explodes, likely hitting its numerical limiters in ways that is model code-specific with limited generalities to be worth diagnosing in detail. After days of this deep convection, with a few time gaps, the troposphere is sufficiently warmed aloft in a descending wedge (red fill in panel d), and dried (blue fill in panel g), to the point that the ZM scheme is again capped in the lower troposphere. Even the stratosphere is warmed during this convective burst, up to 50 hPa. It is notable that the stratospheric overshoot of convective impacts up to 50 hPa may also be seen in the behavior of CAM6 in 3D RCE at sufficiently warm SST, the 305-K case in Reed et al. (2021). Perhaps these unrealistic SCM behaviors may actually have some relevance or bearing further up the model hierarchy, even if only on model numerics or pathologies rather than mainstream performance metrics.
The above analyses indicate that the multi-day oscillations hinge on the capping of the ZM deep convection scheme. To illustrate how the scheme determines the deep convective cloud top, Figure 11 shows cross sections of lifted-parcel buoyancy (filled color), annotated with time series curves of neutral-buoyancy levels (NBLs, gray) and the final convective cloud top height (CCTH, black), from 300 K RCE simulations over selected periods (green-shaded boxes in Figure 8). For definiteness, hourly instantaneous output data is used here, rather than the hourly average output data used in all previous figures. Discretization of NBLs at model levels results in a slight visual mismatch, with the diagnosed NBLs slightly above the warm-to-cold color transition in the filled-contour buoyancy display.
A minor yet influential difference between the CAM5 and CAM6 physics suites is the value of ZMnumcin (see Section 2.1.2), which was changed from 5 in (S)CAM5 to 1 in (S)CAM6, in the closure of the ZM deep convection scheme. This closure's value of dCAPE and top altitude are used in a yes-no decision (a "trigger" step), and also as inputs to the more computationally expensive "cloud model" algorithm the ZM scheme uses to create its contributions to Q 1c and Q 2c tendency profiles. By setting ZMnumcin to 5, SCAM5 allows up to five NBLs below cloud top (gray curves, top row) and determines the CCTH to be the NBL that gives a maximum dCAPE from up to five tentative NBLs. In contrast, with ZMnumcin set to 1, SCAM6 always uses the lowest NBL as the CCTH (bottom panels have only a black curve), explaining how it can be capped near 800 hPa (or even near the surface sometimes) by such a tiny, weak warm anomaly in its exotic and noisy 10-day oscillations.
In the SCAM5_L30 time sample (panel a of Figure 11, just a few hours), buoyancy exhibits three nearly constant NBLs in the free troposphere (gray curves). However, the CCTH (black line) jumps discretely from the highest NBL to the lowest NBL near 520 hPa at the central time, due to an imperceptible decrease of the part of the buoyancy integral above that 520 hPa level. In the SCAM5_L60 sample (panel b, centered on a convective jump), the five tentative NBLs (with the lowest NBL sitting at the lowest model level at all times) and the CCTH flicker with time prior to the jump. At the central time, the CCTH jumps upward to about 150 hPa, as lifted-parcel buoyancy switches from negative to positive in the 300-150 hPa layer. Updraft-top heating (Figure 9e) leads to a slightly lower NBL and CCTH a few hours later, and the process repeats, with CCTH then descending gradually with time. Heating from both deep convection and LWCRE combine to warm the layer, making lifted-parcel buoyancy negative in that deepening (descending) warm layer of Figure 9d. The uppermost troposphere remains warm (making a lifted parcel negatively buoyant, blue in panel b) for another 60 hr until a new cycle begins. In the SCAM6 runs (bottom row of Figure 11), because the CCTH is the lowest NBL (black and gray curves are the same), the extent of deep convection varies more drastically with time than in the SCAM5 runs. Even a very slight warming of a single model layer can totally prevent deep convection, even in the presence of enormous lifted-parcel buoyancy (panel d).
Although this ZM capping mechanism by slight warm layers is debatable in its realism, it is robust, so these multi-day oscillations are robust across the tested SSTs for example, (not shown), except for slightly different periods and layer depths above the variable and descending capping warm layers. Multi-day oscillations also appear in the CAM-RCE simulations (the raw data is not shown; the domain-mean precipitation time series with a 10-day running mean applied are shown in Figure 2 of Reed et al. [2021]), but have substantial differences from those observed in this study in terms of period, regularity, distribution of variance, and dependence on model physics. Investigating the differences of multi-day oscillations between CAM and SCAM RCE simulations will be a focus of future work.

Mechanism-Denial Experiments
To further confirm the mechanistic interpretations above, we have run four sets of mechanism-denial experiments. Table 2 explains the aliases to these experiments, and time series and spectra of the run with SST set to 300 K are shown in Figure 12. Again, slight differences were seen in the sets of experiments with different SSTs; nevertheless, the lessons learned from the 300-K SST set of experiments generally hold across experiments with different SSTs (not shown).  Figure 12) to the control simulations (middle column in Figure 8), we found that the SCAM5_L60 and SCAM6_L60 runs swap their features almost perfectly, except for different predominant low frequencies (i.e., ∼10 days in Figure 8k vs. ∼7 days in Figure 12e). The features of SCAM5_L30 and SCAM6_L32 roughly swap as well, particularly in terms of the magnitude of variance, statistically significant peak, and overall spectral pattern. This set of experiments suggests that under the SCM-RCE framework, the parameter ZMnumcin alone modulates the temporal variability of column energetics primarily, despite the wholesale differences in other parameterization schemes between SCAM5 and SCAM6.
Based on the analysis in Section 4.1.1 and the results from the ZMnumcinSWAP experiments, we conducted a set of experiments in which the strict constraint of cloud top determined in the triggering/closure part is disabled. Instead, we let the "cloud model" determine the cloud top as the level where the MSE of updraft ensemble is equal to the saturation MSE of the environment (h*). The setup of ZMnblh* experiment exactly follows that of the NBL_h* experiment in M. Wang and Zhang (2018), and the SCAM-RCE results are shown in the secondfrom-the-left column in Figure 12. Compared to the control set of simulations, the variability with period longer than 5 days diminishes in all the ZMnblh* experiments, particularly evident for the SCAM6_L60 run whose 9-day oscillation of the control run completely disappears. One exception is the SCAM5_L30 run (panel b), whose ∼6-day variability is enhanced, and further investigations are required for a better understanding. Despite a seemly outlier (the SCAM5_L30 run), the AR1 curves of the rest three ZMnblh* runs tend to be flattened, indicating an overall depletion of variability at low frequencies when the restriction of dCAPE-determined cloud top is removed.
To test the role CREs play for the RCE variability, we made clouds transparent to radiation by setting cloud fraction to zero in the RRTMG radiation scheme. This setup is similar to, if not identical with, the cloud-off counterpart of Clouds On-Off Klimate Model Intercomparison Experiment (Stevens et al., 2012). In this set of experiments, the CREs are removed, while the interaction between water vapor and radiation, or clear-sky radiation ("clrRAD"), is still allowed. When comparing to the control set of simulations, we found that the variability magnitudes, particularly those with period longer than 5 days, reduce in the SCAM5_L60 and SCAM6_L32 runs. However, the distributions of spectral peaks are similar, as their AR1 curves still emphasize low frequencies. Furthermore, the longer-than-5-day variances increase and become statistically significant in the SCAM5_L30 run (∼7-day peak in panel c), and shift toward higher frequencies in the SCAM6_L60 run (∼9 days in Figure 8k vs. ∼5 days in Figure 12o). Hence, CREs may partly contribute to the temporal variability in the RCE of SCAM, but are secondary, unlike in some previous studies on variability of RCE simulations (e.g., Q. Hu & Randall, 1994).
Finally, we ran a set of experiments in which both the ZMnblh* and clrRAD setups are used at the same time (right-most column in Figure 12). In general, the precipitation is invariant in time for all the four runs. Even though there are some high-frequency peaks that are statistically significant, their variances are very small. Cross comparison among the ZMnblh*, clrRAD, and ZMnblh* + clrRAD sets of experiments reveal that capping of deep convection's top primarily modulates the multi-day vacillations of the SCAM RCE, particularly when a finer-resolved (e.g., L60) grid is used. Cloud-radiative interactions, although not serving as a driver, can be influential for the variability of relatively low frequencies as well.
To sum up, the key results are: (a) The large amplitude multi-day oscillations substantially diminish when we remove artificial limitations on the number of NBLs allowed below CCTL, allowing deep convection up ZMnblh* + clrRAD Activating ZMnblh* and clrRAD at the same time to the highest NBL. (b) The oscillations generally remain (although different in details) when cloud-radiative interactions are disabled by making all clouds transparent in the radiation scheme. Again, these results confirm the mechanism of descending ZM-scheme capping layers starting in the upper troposphere, punctuated by sudden deepenings, as seen in the section plots (not shown) to explain the sharp-attack waveforms in these figures. Even without cloud-base radiative heating, convection itself causes the capping layers to descend, as cloud top heating by eddy heat flux convergence (detrainment) warms the base of the descending capping layer, so the oscillations are radiation-modified but primarily convective.  Table 2 and the text for the designs of experiments. Note the different scales of variance are used in the spectral sub-panels across all the simulations.

Discussion and Conclusions
As motivated in Section 1, the SCAM-RCE framework studied here embodies a base level in the atmospheric model hierarchy of the CESM, illustrating in a pure way the interactions among convection of all kinds, temperature and vapor and cloud, and radiation process schemes, as implemented numerically on a vertical grid-but without advection, not even in the vertical. These processes form the hardest nut of uncertainty at the heart of climate modeling. They suffice to create a plausible time mean climate, with near moist adiabatic stratification and reasonable relative humidity profiles. A cyclic cloud-resolving model, SAM, provides a reference in which the vertical overturnings envisioned in the convection schemes are explicit, if distorted, and SCAM performs adequately relative to this reference, except perhaps for an unrealistic tropopause saturation and overcast problem.
Top-of-atmosphere radiative balance is not well obeyed by any RCE runs with tropics-like constant SST (Reed et al., 2021), so in this sense SST sensitivity experiments may be more suited for investigations of clouds and circulation coupling, and less so for climate sensitivity.
Beyond the relatively straightforward mean-profile partial success, time variations more starkly exposed the oddities of strict determinism in a single column's process interactions. Perhaps surprisingly, small parameter tweaks within the nominally same deep convection scheme are predominant in SCAM5-SCAM6 differences, overshadowing much larger codebase changes to the non-deep convection packages. Diagnosis and interpretation in full-complexity configurations, which are much more computationally costly and ambiguous to interpret, might never have revealed that fact. For instance, ZMnumcin is not among the stepwise increments of the many compensating and reinforcing process differences along the path from official CAM5 to CAM6 versions, in Supporting Information of Gettelman, Hannay, et al. (2019), used to reveal reasons for the latter's high climate sensitivity.
The next step up in complexity from one-dimensional (1D) RCE, currently, is the 3D RCE configuration of Reed et al. (2021). That work also briefly compares SCAM-RCE and AMIP simulations using the standard CAM5 and CAM6 packages and settings. However, with the cost of simulation experiments, and the blurring effects of averaging over many columns in very different climate regimes (self-aggregated dry and moist areas), the differences documented there have necessarily speculative interpretations. To bridge this 1D-3D gap, our companion paper (in preparation) attempts to find some robust version of an intermediate rung: a single column with parameterized (rather than disabled) vertical advection, perhaps with stabilizers that may be viewed as notional representations of horizontal advection. While its focus is on the nature of time variability (weather), that variability might also help soften the strict determinism of a single column, smoothing out some of the time-mean grid-scale artifacts seen above.
The power or utility of the model hierarchy may be queried from either end: reductionist (stepping down the hierarchy from realism) and constructionist (stepping up the hierarchy from fundamentals).
1. Reductionist: Is some phenomenon in 3D simulations able to be explained, thanks to its continued existence as dimensionality is reduced to a single column? A mean state near moist adiabatic is one example. However, that phenomenon already has explanations rooted even deeper, in pure conceptual principles (Earth's strong gravity makes convection neutralize instability on a much shorter timescale than evolution toward pure radiative equilibrium). The power of SCMs seems closer to a mere debugging tool than to this sort of clinchingly strong use of reductionist reasoning. 2. Constructionist: Are the process interactions seen in a SCM recapitulated, perhaps in forms that are distorted by advection or masked by spatial as well as temporal averaging, in some higher-dimensional setting? This seems a more fruitful approach scientifically. The physically grounded heuristics and imagination required to postulate how SCM results may scale up is almost exactly what we mean by "understanding" in a complex science like climate. Formally, SCM results could motivate hypotheses, which for integrity should be pre-registered at a site like openscience.org. These should be crafted to be cleanly testable in 3D simulations.
The oscillations of Section 4, governed by varying deep convective top capping, seem physically plausible, and should be possible in natural air columns, although perhaps they may be rather brittle artifacts of too-strict determinism. Specifically, the oscillations may be associated with deep overshooting convection in the tropical tropopause layer (Fueglistaler et al., 2009), whose climate balances are weakly constrained, such that it can decouple for substantial periods from the troposphere and its main deep convective process. Especially in CAM6, the ZM scheme's artificial capping by the lowest non-positive lifted-parcel buoyancy level-no matter how slight!-leads to unrealistic behaviors, such as sounding profiles that would never be observed in Earth's tropics. Despite its unrealism as a literal prediction, can this behavior nonetheless help us imaginatively hypothesize (or explain) anything about the performance of CAM6?
The weakest prediction is merely that disabling this restriction, as in the ZMnblh* experiment of Table 2, will make some statistically significant difference to the multi-day weather variance in, and therefore the mean state of, the tropical belt of CESM. For example, by using the exact ZMnblh* setup in CAM5, M. Wang and Zhang (2018) found that the simulated tropical convective cloud top heights are improved and more comparable to CloudSat observations. Xie et al. (2018) also explored the effect of ZMnumcin in the Energy Exascale Earth System Model Atmosphere Model version 1, which shares many parameterization schemes with CAM6. They found the reduction of ZMnumcin (from 5 to 1) acts to suppress the simulated deep convection over tropical oceans (their Figures  17a and 17c), in consistency with the capped convection found in the SCAM6 RCE simulations in Section 4.1. Suppression of deep convection due to changing ZMnumcin from 5 to 1 is hypothesized to occur in the 3D RCE simulations of CAM (Reed et al., 2021) as well, as the convective-to-stratiform precipitation ratio decreases from CAM5 to CAM6. Perhaps that suppression of deep convection could be enabled in some more physically satisfying way, such as through entrainment profile structure.
We can go a little further and hypothesize that the removal of cloud top restriction will reduce variability, especially of the kinds that are modulated by density capping: the high frequency convectively coupled waves (Kiladis et al., 2009). With a slightly different mechanism in mind (storage and discharge of moisture), we might extend this prediction to intraseasonal variability, whose general increase with diverse forms of convective inhibition and triggering mechanisms was documented in W. Wang and Schlesinger (1999) or in prior ZM scheme work (G. J. Zhang & Mu, 2005). Experiments to test this hypothesis are left to future proposals and projects, but these can be designed to guarantee a clear result, whether positive or negative. In the second part of our work, experiments with parameterized large-scale dynamics (PLSD) also bear on the hypothesis as discussed here, and so must be noted here as additional reasons for its specification.

Appendix A: RCEMIP Versus Its Modified Version in SCAM
The RCE configuration used in this paper is slightly different from that of the official RCEMIP configuration, with modifications described in Section 2.3. To test how robust our conclusions hold for various RCE setups, and to have a reference that may have a more direct comparison to Reed et al. (2021, in which the SCAM RCEMIP mean profiles are used as the initial conditions for 3D CAM RCEMIP runs), we examined the SCAM results using the standard RCEMIP protocols. The discrepancies in the SCAM results using the standard RCEMIP configuration and its modified version (presented earlier) are briefly described in this auxiliary section. Table A1 shows the time-mean column energetics as in Table 1, but for the simulations using the standard RCEMIP setup. The RCEMIP values, except for OLR, across model physics suites, vertical resolution, and SST are smaller than their corresponding RCE runs shown in Table 1. This is expected, because freely simulated wind speed in the RCEMIP runs is much smaller than 5 m/s and essentially provides smaller surface fluxes than the modified RCE runs. However, for the RCEMIP simulations using the standard vertical grid (L30 and L32), SCAM5 has larger precipitation, larger precipitable water, and larger column-integrated radiative cooling than SCAM6. The smaller values of precipitation and precipitable water in the simulations using the CAM6 physics are observed in the corresponding 3D RCEMIP simulations (Reed et al., 2021, their Table 1), suggesting consistency between the SCAM-RCEMIP and CAM-RCEMIP sets of simulations to some degree. The results shown in Table 1 agree how precipitable water change across the two model physics suites, but lack the monotonic changes in both precipitation and column-integrated radiative cooling. In the L60 set of RCEMIP simulations, SCAM5 has larger precipitation and column-integrated radiative cooling, but smaller precipitable water than SCAM6, broadly consistent with The results shown in Table 1 except for precipitable water.
Compared to the modified RCEMIP runs, the simulated mean state of the RCEMIP runs is much cooler and drier (not shown), which, again, is probably due to the smaller surface flux. Moreover, SCAM using the standard RCEMIP setup tends to produce a smaller time-mean cloud ice content at lower altitude and a smaller cloud liquid content, except for the cases of SCAM5_L30 (which has a larger cloud liquid content) and SCAM6_L32 (which has a larger cloud ice content). However, the maximum cloud fraction in the upper troposphere does not change much, probably because the temperature, moisture and cloud ice content are reduced at a similar scale so that RH i in Equation 9 barely changes. In general, besides these noted differences and differences in details such as vertical oscillatory structures of moisture and tendencies of radiation and convection, the characteristics of time-mean profiles are robust across the examined RCE setup and SST.
Temporal variability of the RCEMIP set of simulations is analyzed and shown in Figure A1, and its differences from that of the runs using the modified version of RCEMIP are described as follows. When using the standard RCEMIP setup, (a) the SCAM5_L30 has increased variability with period longer than 1 day; (b) the SCAM5_L60 features increased variability with period less than 1 day, and the 5-day oscillation is statistically significant only in the run with 305-K SST; (c) in the SCAM6_L32 set of runs, variability with period longer than 7 days become more statistically significant, but the corresponding magnitude does not necessarily increases; (d) the SCAM6_L60 set of runs remain roughly the same, with slightly different spectral peaks. Notwithstanding these discrepancies, the features of temporal variability of precipitation (and column energetics) are broadly similar across the two RCE sets of simulations.  Figure A1. Same as Figure 8, but for the RCEMIP sets of simulations. Note a different scale of variance is used for panels (a-c).
A series of sensitivity tests consisting of transition from the standard RCEMIP configuration to the modified version used for this study was conducted (not shown). The results suggest that prescribed wind speed for surface flux computation and time step interval both contribute to the differences between the two RCE configurations explored. However, despite the differences described here, the conclusions drew from the modified RCEMIP set of simulations robustly hold for the standard RCEMIP set of simulations (not shown): the multi-day oscillations of the SCAM RCE are primarily modulated by capping of deep convection's top, and are only partially modulated by CRE.

Data Availability Statement
Codes for replicating the results described in this work are available in a Zenodo repository: https://zenodo.org/ record/5765683.