We use two of the three CMIP6-era models analyzed by that produce an inverted-V-shaped Nd–L correlation: the US Department of Energy Exascale Earth System Model (E3SM) and NASA Goddard Institute for Space Studies (GISS) ModelE3. These models have different turbulence schemes; as entrainment-mediated ACI mechanisms must involve at least the turbulence and microphysics parameterizations, it was desirable to include model diversity in this study. ModelE3 and E3SM are used in single-column mode (Sects.  and ). E3SM is also used in 3D atmosphere runs (Sects. , , and ).

In part 1, we used E3SMv1; here, we use E3SMv2 instead because it is significantly more efficient at archiving the large, high-frequency fields required for the entrainment diagnostics. E3SMv2 differs from v1 largely in the parametric tuning rather than in changes to the physics formulation. The Nd–L relationship documented in E3SMv1 persists in E3SMv2 for Sc clouds (Fig. S1).

In part 1, we used a ModelE3 parameter tuning derived from machine learning; here, we use the default tuning from the ModelE3 development team, which produces a very similar Nd–L relationship but does not produce the oscillations in surface precipitation at the lowest Nd values that result from an assertive subgrid-scale multiplier tuning of autoconversion in nonturbulent layers.

As the process understanding of entrainment-mediated RA_ℒ is based chiefly on a small number of canonical subsidence Sc cases, our main focus is on understanding this cloud type in the GCMs as well. Thus, while the eventual goal is to understand the full spectrum of clouds that occur in the real atmosphere, for now we apply a restrictive set of criteria to maximize the similarity between model clouds and Sc:

warm (cloud-top temperature warmer than freezing and zero ice water path) and overcast (cloud fraction of f>0.9) columns,

We use an extensively studied, idealized subsidence Sc-like case specification to construct single-column experiments that are designed to investigate the mechanisms underlying the response of entrainment and L to Nd in GCM physics.

The models analyzed in part 1 produced RAL<0 in the default model configuration, presumably because of precipitation suppression. To disentangle the opposing RA_ℒ of precipitation suppression and a potential entrainment mechanism (RAL>0), we turn off the precipitation suppression in E3SM by setting the exponent on Nd in the autoconversion parameterization to zero, removing the explicit Nd dependence of the autoconversion process. To maintain a climate state similar to the default model, we increase the autoconversion scale factor . This is equivalent to presenting autoconversion with a globally constant Nd≈50 cm⁻³ and results in present-day top-of-atmosphere radiative flux and cloud radiative effect changes of <1 W m⁻² compared with the default configuration. The warm-cloud over-ocean mean, calculated using present-day (PI) and preindustrial (PD) values, is log⁡LPD-log⁡LPI =2.0×10-3 in this configuration compared with 3.2×10-2 in the default configuration. This indicates that switching off the precipitation suppression mechanism eliminates the strong negative RA_ℒ of the default model configuration but does not expose a strong positive RA_ℒ in its stead. The Nd–L correlation becomes more negative in Sc clouds when precipitation suppression is turned off (Fig. S2) compared with when precipitation suppression is active.

As the causal ACI mechanism hypothesized to lead to RAL>0 is enhanced entrainment with increasing Nd, we make it a focus of this paper to understand how entrainment behaves in the models. To this end, we use an entrainment diagnostic that calculates the mixing between the free troposphere (FT) and well-mixed boundary layer as a residual term in the mixed-layer budgets of water and temperature. Note that we do not include budget terms for nonlocal transport through the PBL top by parameterized convection schemes. This limits the applicability of the diagnostics to strictly stratocumulus-topped boundary layers (excluding, e.g., the stratocumulus-to-cumulus transition), which the subsidence-based and stability-based regional selection (Sect. ) is intended to enforce.

Let qv and ql be water vapor and liquid mixing ratio and θ and T potential temperature and temperature. In adiabatic expansion and condensation, the total water mixing ratio of 1qt=qv+ql and liquid water potential temperature, θl, are conserved; we approximate 2θl=θ-LvcpθTql, with Lv being the latent heat of the evaporation of water (which, for simplicity, we take as temperature-independent using its value at 273 K) and cp the isobaric specific heat of dry air. Budget equations for θl, qt, and total mass, vertically integrated over the PBL, involve fluxes of water, dry air, and heat across the boundaries of the PBL . Crucially for our purposes, this includes the entrainment flux into the boundary layer. We express the budget equations following but modify the notation to highlight the similarity with source and sink terms in a prognostic equation in Lagrangian form: 3hD^(ρθl)D^t=-ΔFcp+LvRcp+SHcp+Eθ(θl+-〈θl〉),4hD^(ρqt)D^t=-R+LHLv+Eq(qt+-〈qt〉),5ρ|z=h∂h∂t+v⋅∇Hh=Eh-ω/g, where ρ is the density, ΔF the radiative cooling, R the precipitation mass flux at the surface, LH and SH the latent and sensible heat fluxes at the surface, h the PBL geometric depth, v the wind vector, ω=Dp/Dt the large-scale pressure velocity, and g the gravitational acceleration. The operator ∇H is the horizontal gradient. Quantities with a superscript plus (i.e., qt+ and θl+ in Eqs. –) are evaluated just above the inversion. Quantities in angular brackets (〈A〉) are mass-weighted vertical averages of a quantity (A), evaluated at a model-level midpoint k between the lowermost atmosphere level, ksfc, and the uppermost level below the inversion, kpbl: 6〈A〉=1〈ρ〉h∑k=ksfckpblρkΔzkAk. The PBL-averaged material derivative of a 3D quantity A is defined as 7D^(ρA)D^t=〈ρ〉∂〈A〉∂t+〈ρv⋅∇HA〉+〈ρ〉(A|z=h--〈A〉)v⋅∇Hh. The PBL-averaged material derivative differs from the conventional material derivative in two ways. First, vertical advection is absent as the motion of the boundary between PBL and free troposphere does not affect the within-PBL average. Second, the final term in Eq. () accounts for the horizontal advection of the PBL top; the A|z=h- notation indicates the field is to be evaluated just below the inversion.

Each of the budgets of θl, qt, and h (Eqs. –) can be solved for an entrainment mass flux to get Eθ, Eq, and Eh, respectively. Physically, the following applies: 8Eθ=Eq=Eh. However, models do not necessarily respect this equality. Therefore, we retain the freedom to diagnose Eθ, Eq, and Eh separately; in the following, we use the degree of equality between these fluxes as a criterion for model fidelity to the physical system.

In a numerical model, the spatial discretization can potentially alter the behavior of the PBL in a qualitative way. The physical Sc-topped PBL entrains free-tropospheric air by O(1m)-scale turbulent exchange through a sharp buoyancy barrier and references therein. In the model, the static stability due to thermodynamic jumps across the inversion at PBL top is less localized and weaker due to the finite vertical resolution. Depending on the model resolution, the resolved-scale advection scheme, and the turbulence parameterization, vertical mixing across the poorly resolved inversion may be too strong. This can occur because the stability reported to the turbulence scheme is underestimated or because fluctuations in the resolved-scale vertical velocity mix the air masses instead of moving the boundary between them. Collectively, we term these behaviors model artifacts. The problem with model artifacts in mixing is that the effect of such mixing on the PBL temperature and humidity need not have the correct susceptibility to the multitude of anthropogenic perturbations – forcing by and adjustments to both aerosol and greenhouse-gas forcings as well as feedback mechanisms in response to anthropogenic global warming – that are hypothesized to influence cloud-top entrainment by changing the atmospheric state (Nd, temperature, and humidity in the PBL and FT and FT emissivity).

Therefore, our task is to determine whether the mixing in the model behaves more like physical entrainment or more like artificial mixing. If we calculate Eθ, Eq, and Eh in Eqs. ()–() as residuals, then they describe the mixing between PBL and FT, including both entrainment and model artifacts. We then apply three criteria that help us make that determination:

In the real atmosphere, Eθ=Eq=Eh all describe the same entrainment mass flux that comes about due to turbulent processes at the boundary-layer top. In a numerical model, however, the equality of the entrainment fluxes is not a given. For one, models treat θl and qt differently, for example, to ensure nonnegative definite qt. For another, the length scales at which entrainment occurs reach below 1 m, far beyond the resolved dynamics of most types of models. Mixing between the boundary layer and FT in a model, therefore, results from a combination of resolved advection and parameterizations. Having multiple independent measures of the entrainment mass flux affords us the ability to ask both whether the model-diagnosed entrainment estimates are consistent and whether they are physical. Consistent fluxes are highly correlated, with Eq, Eθ, and Eh close to a 1:1 regression slope. (In an Eulerian model, Eh is difficult to diagnose when the advective tendency of h over a model time step is small compared to the vertical resolution, which is the case in GCMs. We restrict our analysis to Eq and Eθ.)

Under Sc conditions (as defined in Sect. ), E3SM produces vertical profiles of qt and θl consistent with a fairly well-mixed PBL capped by a fairly sharp thermodynamic jump. Figure  shows composite vertical profiles stratified by PBL depth.

The effective entrainment qualitatively agrees very well with physical understanding of the Sc-topped PBL. The fluxes derived from the separate budgets agree well with each other, yielding a close relationship with a slope of near 1 (see joint probability in Fig. ). Furthermore, the sign of the fluxes is consistent with physical entrainment from the FT into the PBL (E>0; see the marginal cumulative distribution functions in Fig. ) rather than showing a distribution including both positive and negative values, which would be consistent with numerical diffusion. The instantaneous entrainment also qualitatively responds in the expected way to instantaneous variability (as opposed to climatological spatial variability, seasonal temporal variability, etc.) in properties of the atmospheric column (Fig. ). Entrainment increases with surface heat fluxes and cloud-top radiative cooling, consistent with increased turbulence production leading to increased entrainment; decreases with the magnitude of the θl jump at the inversion, consistent with a stronger buoyancy barrier suppressing entrainment (the negative qt jump, which reduces the buoyancy barrier, is strongly correlated with the positive θl jump); and is independent of the instantaneous grid-scale vertical velocity, consistent with large-scale subsidence moving the boundary between air masses (i.e., the FT and the PBL) rather than mixing them.

In summary, the entrainment behavior of the GCM, according to the criteria tested for, appears free of numerical artifacts that are due to coarse model resolution. While not exhaustive (e.g., the effect of mixing on the model levels above the inversion is not examined), these entrainment diagnostics allow us to compare the GCM physics to the foundational LES modeling studies on Sc entrainment e.g.,, which diagnose entrainment from the entrainment velocity, i.e., the PBL mass budget. We next focus on the effects of anthropogenic perturbations on the modeled entrainment and on how the parameterized model physics affects those entrainment responses.

Figure  shows that entrainment depends on numerous properties of the atmospheric column. This creates a bewildering web of possible causal and covariability effects by which entrainment internal variability could be correlated with aerosol internal variability in 3D atmosphere runs. SCM runs, in contrast, provide a clean way to diagnose cause and effect in GCM column physics, which is a reason they are widely used during model development. First, the SCM allows us to hold any combination of boundary conditions on a single column fixed. Thus, we can switch off any effects mediated by the grid-scale horizontal circulation. These effects include synoptic-scale confounders of the type discussed by . Second, and relatedly, we are free to explore the effects of model physics on ACI without having to retune the model to the global-mean energy balance. Thus, we avoid the difficult problem of whether to attribute changes in model behavior to the physics changes under investigation versus the nuisance changes required to restore the energy balance that might also affect the ACI behavior e.g.,.

A welcome side effect (and a raison d'être) of SCM use is that the experiment setup closely matches the LES runs that inform so much of our process understanding. Like , , and , we can focus on well-understood subtropical-subsidence-region Sc and vary one boundary condition – the aerosol concentration – at a time (and, if desired, one model-physics mechanism at a time). This does not address the question of whether the model results are representative of the global-mean effective radiative forcing, which is best addressed with global runs, but it answers the question of whether LES and GCM column physics respond similarly to perturbations around an as-near-as-possible identical base state.

The E3SM SCM uses the two-mode prescribed aerosol concentration profile specified for RF02 , while the ModelE3 SCM uses a single accumulation mode aerosol size distribution as described above. We then modify the amplitude of this profile to elucidate the causal effect of Nd change on L. In the ModelE3 SCM, we scan the aerosol number concentration Na={20,30,40,60,80,120,160,320,640} cm⁻³. In the E3SM SCM, we scale the prescribed aerosol concentration up and down by a factor of 8: Na=Na0×{1/8,1/4,1/2,1,2,4,8}. Prescribing aerosol eliminates ACI mechanisms in which clouds affect the aerosol state, such as potential effects of aerosol scavenging on the Nd–L relationship . This simplifies the attribution of L responses to Nd perturbations by removing one class of processes from consideration.

Figures  and show the ModelE3 and E3SM SCM time series. In both models, the PBL deepens, indicating entrainment in excess of the subsidence rate; the PBL in both models also deepens by a similar amount when native longwave radiative transfer is used (see Appendix). With the exception of a short duration before and after steps in the discretized PBL depth in E3SM, both models maintain an overcast cloud; the loss of cloud cover in E3SM's SCM when the PBL top jumps by a model level is clearly a model artifact, so these periods are excluded from further analysis by requiring f>0.9 as in the 3D model analysis both here and in part 1. Furthermore, the discrete PBL depth increases are associated with discontinuities in the entrainment diagnostics: a dependence of E on how long the top model level of the PBL has been subject to entrainment in E3SM and a reversion to a constant E once the PBL has deepened in ModelE3. We may be mitigating the E3SM artifact by averaging over two full deepening cycles, effectively averaging over the dependence of E on position in the deepening cycle. We attempt to mitigate the ModelE3 artifact by only averaging E until the first PBL deepening occurs. This choice is subjective, and we note that the other obvious choice, which is to average the entrainment over the same time period (hours 2–12) as the other variables, produces no clear relationship between E and Nd. However, the problem with this averaging method is that earlier deepening in the more entraining cases means that these cases spend more time in the apparent low-entrainment state that characterizes the entrainment-diagnostics artifact, which most strongly lowers the average apparent entrainment in the highest-entrainment cases. On balance, it appears to us that the averaging method we choose better summarizes the behavior of ModelE3 – namely, that the greater magnitude of the entrainment flux with increasing Nd is consistent with the earlier deepening.

As expected, varying the aerosol concentration strongly affects the droplet concentration. From this response, the two SCMs then diverge in the details of their behavior, but they reach the same behavioral endpoint at sufficiently large Nd: enhanced entrainment leading to a loss of L as Nd increases. In ModelE3, L after spin-up starts out with a monotonically increasing Nd dependence; after spin-up, the low-Nd runs experience an increase in L, while the high-Nd runs experience a decrease, leading to a time-average L that first increases with Nd and then decreases (Figs.  and ). The increasingly negative L tendency as a function of Nd accompanies an increasingly strong entrainment warming and drying (Fig. ). The increase in entrainment with increasing Nd competes with a decrease in precipitation (Fig. ). ModelE3 with native longwave radiation, however, maintains a 3 times greater L after 24 h duration (see Appendix).

In E3SM, L after spin-up has a monotonically decreasing relationship with Nd, which remains true at each point in time throughout the runs. As all runs experience fairly rapid L loss with time, time averages only show weak dependence on Nd (Fig. S3). We can instead quantify susceptibilities by scanning across the different aerosol experiments at each time step; this is shown for L in Fig.  and for E in Fig. . Precipitation in E3SM largely ceases after the first time step even though drizzle was measured in the DYCOMS-II RF02 observations. In ModelE3 with native LW radiation, precipitation experiences sharp peaks with a periodicity similar to PBL depth increases.

Whether the effect of varying the aerosol concentration on L is expected depends on our Bayesian prior. The agreement on RAL<0 across CMIP5-era GCMs , which persists in the newer-generation models examined in part 1, and the absence of enhanced entrainment physics in some GCMs e.g., have led to the notion that GCMs may be structurally incapable of representing the enhanced entrainment mechanism known from LES e.g.,. This thinking is in tension with a long line of parameterization work e.g.,, indicating that it may well be possible to represent this mechanism in GCMs either through direct parameterization or as emergent behavior. If our expectation is based on the LES-based process understanding, then we would predict the causal effect of Nd to be a decrease in L. The surprising result is that the SCM sides with the LES (see Appendix) rather than the 3D GCM with which the SCM shares its model physics. Recall that the E3SM 3D GCM run, in contrast to the E3SM SCM run, showed negligible response of climatological L to the anthropogenic Nd increase (Sect. ).

While the SCM behavior is consistent with our mechanistic understanding of entrainment-mediated drying, we need to point out several caveats. First, the details of what entrainment-mediated drying entails are different in the two models as discussed above, and we find in Sect.  that the behavior in the 3D E3SM run is more consistent with the ModelE3 SCM than with the E3SM SCM. Second, the E3SM SCM entrainment fluxes reach the equivalent of several centimeters per second in entrainment velocity, which is significantly stronger than the fluxes that LES or the ModelE3 SCM produces for this case and stronger than the fluxes that the E3SM 3D run produces for Sc on average. Third, if the E3SM physics had not serendipitously produced very low precipitation rates, the decrease in L with increasing Nd would probably have been overwhelmed by the precipitation suppression signal. None of these caveats negate the finding that the GCM physics appears capable of producing entrainment-mediated L loss qualitatively consistent with LES findings, and significant intermodel diversity is to be expected in SCM studies . They do, however, indicate that there is ample further process investigation to be performed in future work.

We conduct several additional E3SM SCM experiments with perturbed physics. These experiments further test that the causal effect of aerosols on L in SCM mode not only has the same sign as in LES but also proceeds via the same physical mechanisms and show that the entrainment-mediated drying can be tuned to agree quantitatively with LES.

Sedimentation–entrainment feedback is the source of entrainment enhancement. Size-dependent sedimentation is one of the processes by which higher-Nd clouds lose liquid relative to lower-Nd clouds in LES under sufficiently dry overlying air . The microphysics parameterizes size-dependent sedimentation. showed through process denial experiments that L loss only occurs when this process is included in the SCM version of the Geophysical Fluid Dynamics Laboratory (GFDL) AM3 model. This is true in the E3SM SCM as well; Fig.  shows that ∂log⁡L/∂log⁡Nd, which is negative when the size-dependent sedimentation is active, becomes ≈0 when we switch off the sedimentation flux and thus its size dependence.

If the entrainment-mediated adjustment of L is evident in the SCMs, what becomes of it in the full 3D model atmosphere? From , we know that RAL<0; that is, L under PD emissions is greater than L under PI emissions, opposite in sign to what is expected from the entrainment-mediated mechanisms and from the SCM. To understand why this happens, we use diagnostics targeted at entrainment mechanisms and perform model experiments designed to isolate entrainment mechanisms.