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Tropical convection is expected to decrease with warming, in a variety of ways. Specific incarnations of this idea include the “stability‐iris” hypothesis of decreasing anvil cloud coverage, as well as the decrease of both tropospheric and cloud‐base mass fluxes with warming. This paper seeks to encapsulate these phenomena into three “rules,” and to explore their interrelationships and robustness, using both analytical reasoning as well as cloud‐resolving and global climate simulations. We find that each of these rules can be derived analytically from the usual expression for clear‐sky subsidence, so they all embody the same essential physics. But, these rules do not all provide the same degree of constraint: the stability‐iris effect is not entirely robust due to unconstrained microphysical degrees of freedom, and the decrease in cloud‐base mass flux is not entirely robust due to unconstrained effects of entrainment and detrainment. Tropospheric mass fluxes on the other hand are shown to be well‐constrained theoretically, and when evaluated in temperature coordinates they exhibit a monotonic decrease with warming at all vertical levels and across a hierarchy of models.

Tropical convection and high cloudiness are expected to decrease with global warming, in a variety of ways. This phenomenon has a few different manifestations in the literature, whose relationships are unclear. We show analytically that these explanations all embody the same essential physics, but are not equally robust and thus do not all have the same predictive power.

Three constraints on tropical convection can all be derived analytically from clear‐sky subsidence

Tropospheric mass fluxes at a fixed isotherm should robustly decrease under global warming

Decreases in anvil cloud area and cloud‐base mass flux are not as robust

This article was corrected on 10 APR 2023. See the end of the full text for details.

There is a sense in the literature that tropical convection should “decrease” with global warming, in various ways. Perhaps the earliest incarnation of this idea is the reduction in the tropical overturning circulation first hypothesized by A. K. Betts and Ridgway (1989), and later demonstrated in global climate models (e.g., Knutson & Manabe, 1995; Vecchi & Soden, 2007). Another, seemingly related manifestation of this idea is that cloud‐base convective mass fluxes should decrease with warming, again first noted by Betts (A. Betts, 1998), and later reiterated by Held and Soden (2006). Chadwick et al. (2013) and Jenney et al. (2020) then found a weakening of convective mass fluxes *throughout* the troposphere, potentially generalizing these earlier results. Finally, Bony et al. (2016) argued via moist thermodynamics and mass conservation that tropical anvil cloud areas should decrease with warming, an argument known as the “stability‐iris” hypothesis. Bony et al. (2016) found evidence for the stability‐iris in GCMs, with further evidence found in observations (Ito & Masunaga, 2022; Saint‐Lu et al., 2020, 2022) as well as cloud‐resolving models (Beydoun et al., 2021; Cronin & Wing, 2017).

A lingering question about all these phenomena, however, is the degree to which they are related. Are they all equivalent somehow, or do their underlying physics differ? For example, the weakening of the tropical circulation and the decrease in tropospheric convective mass flux are governed by changes in the clear‐sky subsidence velocity [Equation 1 below], whereas the decrease in *cloud‐base* mass flux is governed by the bulk atmospheric water/energy budget [Equation 6 below]. These constraints look different superficially, but at the same time both ultimately depend on the difference in how atmospheric radiative cooling and atmospheric moisture scale with global warming. This suggests a potential equivalence between the mechanisms, which has not been pursued or made precise.

Beyond equivalence, there is also the question of whether these phenomena are equally robust. While decreases in circulation strength and convective mass flux with warming seem to occur with few exceptions, the same is not true of the stability‐iris effect: some earlier studies with cloud‐resolving models found an *increase* of anvil cloud area with warming (Singh & O’Gorman, 2015; Tsushima et al., 2014), with the more recent RCEMIP intercomparison finding a similar increase in roughly 1/3 of participating models (Stauffer & Wing, 2022; Wing et al., 2020). This diversity amongst models leads to a correspondingly large uncertainty in the associated “tropical anvil cloud area feedback,” whose magnitude and uncertainty range rival those of all other cloud feedbacks (Sherwood et al., 2020).

Given this state of affairs, it seems worthwhile to more closely scrutinize these different manifestations of decreasing convection, assessing both their inter‐relatedness and robustness. We attempt this here by encapsulating these phenomena into three “rules,” showing mathematically that they are indeed closely related, and in some cases equivalent. In fact, all three phenomena spring from a common origin, namely the well‐known expression [Equation 1 below] for subsidence vertical velocity. The specific mathematical forms of these rules suggest varying degrees of robustness, however, which we evaluate with both global and cloud‐resolving simulations.

We focus here on the stability‐iris effect (Section 2) and the decrease of convective mass fluxes both throughout the troposphere and at cloud base (Sections 3 and 4). The implications of these phenomena for the large‐scale circulation are important and require a paper in their own right; some discussion of the relevant issues is given in the conclusions. The simulations utilized here are primarily cloud‐resolving simulations performed with GFDL's FV^{3} dynamical core, run in doubly periodic radiative‐convective equilibrium (RCE) over a range of surface temperatures with non‐interactive radiation and a simplified, warm‐rain only microphysics scheme. These idealized simulations are supplemented by more comprehensive cloud‐resolving simulations with DAM (Romps, 2008), which include full complexity microphysics as well as interactive radiation. We also test some of our conclusions using 1pct_CO2 GCM simulations conducted with GFDL's CM4 (Held et al., 2019). All diagnostics reported here are time and domain mean unless otherwise stated. Further details of both sets of cloud‐resolving simulations are given in the Appendices.

The expression for the subsidence vertical velocity is derived (e.g., Jenney et al., 2020) by considering the thermodynamic energy equation in regions where there is no condensation heating (such regions are typically clear‐sky, but can also include cloudy, non‐ascending regions such as anvil clouds). The only diabatic heat sources are then radiative and evaporative cooling, denoted *w*_{sub} < 0 is given by_{d} and Γ have their usual meanings as the dry and actual lapse rates, respectively. The difference Γ_{d} − Γ is of course due to the presence of moisture, in a sense we will make precise below, so the Equation 1 indeed combines information about radiation and moisture. Equation 1 will be the starting point for each of our rules going forward. Note that evaporative cooling *w*_{sub} (e.g., Bony et al., 2016), despite the fact that precipitation efficiencies can be 0.5 or less and hence

We begin with the stability‐iris hypothesis of Bony et al. (2016). The subsidence vertical velocity in Equation 1 is not uniform in the vertical, and thus has a nonzero divergence which must be balanced by a horizontal clear‐sky convergence CSC = *∂*_{z}*w*_{sub}, or

**(Stability‐iris)**: Clear‐sky convergence, net convective detrainment, and anvil cloudiness should decrease together with warming.

Figure 1 tests this rule by showing profiles of cloud fraction (diagnosed as the fraction of grid cells with condensate mixing ratios greater than 10^{−5}), CSC [diagnosed via Equation 2], and net convective detrainment −*∂*_{z}(*M*/*ρ*) (where *M* is the convective mass flux diagnosed via conditional sampling of convecting grid cells, in kg/m^{2}/s; see Appendix A for details). These profiles are all drawn from our FV^{3} RCE simulations, using temperature as a vertical coordinate since CSC and anvil cloud peaks are well known to follow isotherms much more closely than isobars under global warming (i.e., the “Fixed Anvil Temperature” hypothesis, Hartmann & Larson, 2002; Hartmann et al., 2019). Figures 1a and 1b explicitly confirm, for the first time to our knowledge, that the independently diagnosed profiles of clear‐sky convergence and net convective detrainment are roughly the same (as they should be by mass continuity). Furthermore, these profiles both show a decrease in their upper‐level maxima with warming, as do the cloud fraction profiles in panel c. These results are all consistent with Rule 1 above.

But, Figure 1 does not show a straightforward proportionality between CSC/detrainment and cloud fraction; to the contrary, the CSC/detrainment profiles actually *change sign* in the vertical, as net entrainment in the lower troposphere gives way to net detrainment in the upper troposphere. The cloud fraction profiles are meanwhile positive definite, so the relationship between CSC/detrainment and cloud fraction cannot be a direct proportionality. Indeed, a more complex relationship was recently derived by Beydoun et al. (2021), who showed that to first order, cloud fraction _{h}*l* is a horizontal finite‐difference in log cloud condensate across the anvils, and *τ* is an inverted sum of microphysical and vertical advection time tendencies for log cloud condensate. This relationship shows that cloud fraction is determined not only by CSC, which must obey Equation 2, but also by microphysical degrees of freedom which are largely unconstrained. Indeed, these extra degrees of freedom explain how CSC can change sign, yet still be tied via Equation 3 to positive‐definite cloud fraction; all that is needed is an accompanying sign change in *τ*. Such a sign change might even be anticipated, as microphysical processes transition from being a source of cloud condensate in the lower troposphere (via condensation) to a sink in the upper troposphere (via sedimentation).

To emphasize that the microphysical degrees of freedom in Equation 3 prevent a 1‐1 relationship between cloud fraction and CSC, we re‐run our simulations with not only warm‐rain autconversion (the default setting) but also an additional, widely used accretion process which converts cloud condensate to rain (Y.‐L. Lin et al., 1983, Equation 51). Profiles of CSC and cloud fraction from these simulations are shown in Figure 2. These profiles demonstrate explicitly that similar CSC profiles do not imply similar cloud fraction profiles.

These results suggest that even if CSC is a leading‐order control on cloud fraction, the presence of largely unconstrained microphysical degrees of freedom limits the predictive power of Equation 3. In particular, CSC decreases with warming might *typically* lead to anvil area decreases with warming, but this is not guaranteed to be the case. This is consistent with the aforementioned RCEMIP result that roughly 2/3 of models exhibit a stability iris‐effect, but 1/3 do not. Similarly, Beydoun et al. (2021) found an overall stability‐iris effect in analyzing RCE simulations over a large SST range, but found the connection between CSC and anvil area to be non‐monotonic *within* their SST range. These results from the literature, along with the results shown here, suggest that Rule 1 is a general tendency of models, but is not entirely robust.

Another formalism for cloud fraction was introduced by Seeley et al. (2019, hereafter S19), who expressed cloud fraction as a product of *gross* detrainment and a positive‐definite cloud lifetime. Gross detrainment is not as easily constrained as net detrainment/CSC, but S19's cloud lifetime can be more simply interpreted as a positive‐definite lifetime of detrained cloud condensate. Regardless of these differences, however, the implication of the S19 formalism is similar: microphysical timescales play a leading‐order role along with detrainment, so changes in detrainment alone may be insufficient to predict changes in anvil area.

One can obtain another view of decreasing convection with warming by again beginning with Equation 1 and noting that *w*_{sub} should decrease with warming throughout the troposphere due to the thermodynamically constrained increase in the denominator (Knutson & Manabe, 1995). Invoking the fact that the convective mass flux *M* must be equal and opposite to the subsidence mass flux *ρw*_{sub} (assuming the subsidence area fraction is very close to 1), this then also implies that *convective mass fluxes should decrease throughout the troposphere with warming*. This is a straightforward consequence of the arguments of Knutson and Manabe (1995), but has not been emphasized in the literature and has only been sporadically studied (Chadwick et al., 2013; Jenney et al., 2020).

Before formalizing this rule and testing it here, we make it somewhat more precise by rewriting *w*_{sub} in a more convenient form. We first rewrite *F* is the net upward radiative flux) and *e* (and later *c*) is the domain‐mean evaporation (condensation) in kg/m^{3}/s. Then multiplying the numerator and denominator in Equation 1 by *C*_{p}/Γ and applying the chain rule, we obtain after some manipulation*L*(*c* − *e*) = *∂*_{z}*F* [see also Equation 10 below], and we also define a local “conversion efficiency” *α* ≡ (*c* − *e*)/*c*. Combining these relations, one can rewrite Equation 4 as*∂*_{T}*F*)(*T*) is “*T*_{s}‐invariant,” that is, the profile does not depend on *T*_{s} (this was also shown across cloud‐resolving models in Stauffer & Wing, 2022). In contrast, the factor of *T*_{s}; indeed its upper‐tropospheric peak near *T* = 220 K decreases at almost a halving for every 10 K of surface warming, approximately equal to Clausius‐Clapeyron scaling (Figure A1b). Thus, barring significant changes in the conversion efficiency *α* (which we do not find, Figure A1c), we expect the stability‐related decreases in

Convective mass flux profiles *M*(*T*) should decrease at all isotherms with surface warming.

This prediction of Equation 5 is confirmed for our simulations in Figure 3a, for both the subsidence mass flux − *ρw*_{sub} diagnosed via Equation 1 as well as the conditionally sampled convective mass flux *M*. Similarly to Figure 1, this panel confirms the equality of *M* and ‐*ρw*_{sub}, and thus confirms that the latter can be used to constrain the former (slight discrepancies between the two may be explained by the sensitivity of *M* to the thresholds used in its diagnosis—Appendix A). As per Rule 2, the profiles in Figure 3a are plotted in temperature coordinates, in which they exhibit a clean decrease with warming at essentially all levels (profiles are, however, cut‐off near cloud base for clarity; the behavior of cloud‐base *M* is discussed in the next section). Figure 3b, plotted in pressure coordinate, shows on the other hand that on isobars, the decrease of *M* and *ρw*_{sub} with warming fails in the upper troposphere. Thus, the decrease of upper‐tropospheric *M* with warming depends on the choice of vertical coordinate.

The strong theoretical foundation and encouraging validation of Rule 2 make it a candidate for a robust response of tropical convection to global warming. But, this validation has so far only taken place in the context of an idealized, limited‐area cloud‐resolving model, so further validation across the model hierarchy is required (Jeevanjee et al., 2017). To this end, we first reproduce Figures 3a and 3b but using DAM simulations; the results are shown in Figures 3c and 3d. These simulations feature interactive radiation and comprehensive microphysics, yet still show a clean decrease of *M* at virtually all isotherms (Figure 3c), again in contrast to the picture in pressure coordinates (Figure 3d). The rough equality of *M* and *ρw*_{sub} is also evident.

Next, we validate Rule 2 in a GCM. We use a 1pct_CO2 run of GFDL's CMIP6‐generation coupled model CM4, from which monthly mean parameterized convective mass flux profiles *M*_{c} were saved (see Zhao et al., 2018, for details of CM4's “double‐plume” convective parameterization). Figure 4 shows a map of time‐averaged *M*_{c} evaluated at 850 hPa, as well as tropical mean (20°S–20°N) profiles averaged in both pressure and temperature coordinates over years 1–20, 60–80, and 130–150. The map shows the marked spatial heterogeneity of *M*_{c}, similar to the pattern of tropical rainfall. Despite this complexity, however, the tropical mean profiles behave similarly to those from our RCE simulations: *M*_{c} decreases with warming throughout the troposphere, and this decrease occurs at all levels in temperature coordinates but not in pressure coordinates. In fact, the insets show that in the upper troposphere, the use of pressure coordinates actually *changes the sign* of the *M*_{c} response to warming, further underscoring the importance of the choice of vertical coordinate.

The fact that upper tropospheric *M* (on fixed isotherms) decreases robustly with warming can actually be seen as the basis for Rule 1: if we know that upper‐tropospheric mass fluxes decrease, then it follows fairly naturally that their detrainment should also decrease. Indeed, the changes in stability which drive the decrease in *M* [cf. Equation 5] are the same changes which are thought to drive the changes in the CSC peak under the stability‐iris hypothesis (Section 2).

A third perspective on decreasing convection with warming, popularized by Held and Soden (2006) and stemming from a slightly different formulation in A. Betts (1998, see Appendix C), begins by noting that the cloud‐base (or lifting condensation level) convective latent heat flux should equal the mean precipitation, or equivalently the column‐integrated free tropospheric radiative cooling *Q*_{ft} (W/m^{2}). Mathematically, this is expressed as

From this it follows that the cloud‐base mass flux *M*|_{LCL} should decrease with warming, because *Q*_{ft} increases by 1%–3%/K (e.g., Jeevanjee & Romps, 2018) whereas

**(Betts's rule)**: Cloud‐base convective mass fluxes *M*|_{LCL} should decrease with surface warming.

This rule was confirmed in a particular GCM by Held and Soden (2006) (although they used mass fluxes evaluated at 500 hPa rather than cloud‐base). On the other hand, Schneider et al. (2010) also evaluated the accuracy of Equation 6 (their Equation 8b), and found only middling agreement with their GCM simulations. Here, we can make a quick and qualitative evaluation using the mass flux profiles already shown: the FV^{3} profiles in Figure 3b seem consistent with Betts's rule, but the DAM profiles in Figure 3d do not, instead exhibiting non‐monotonic changes in *M* with warming below 800 hPa or so.

These mixed results suggest that Betts's rule is not robust. But, how can the simple argument leading to Equation 6 fail? And how does Betts's rule connect to our previous rules? We argue here that Betts's rule may not be robust because it assumes that all water vapor lofted above cloud‐base both condenses and precipitates to the surface. In other words, Equation 6 ignores detrainment (and entrainment) of water vapor, and also assumes unit precipitation efficiency. We will analytically derive a generalization of Betts's rule from our fundamental Equation 1 which accounts for these effects, and show that the associated terms are poorly constrained and plausibly lead to the behavior seen in Figure 3.

We begin by rewriting Equation 1 in terms of a flux divergence in *z* coordinates:

Next, we note that for a saturated, convecting parcel experiencing fractional entrainment per unit distance *ϵ* (m^{−1}), its saturated moist static energy (MSE) *h** evolves as (Singh & O’Gorman, 2013)*Lc*/*M*. Since the difference Γ_{d} − Γ from the left‐hand side of Equation 9 also appears in Equation 7, we may substituting and rearrange, recovering our statement of local energy balance*c* and *e* are related to the precipitation efficiency as*z*_{LCL} to the tropopause height *z*_{tp}, and noting that *∂*_{z}*M* = *ϵM* − *δM* where *δ* is gross (not net) fractional detrainment then yields finally^{3} mass‐flux profiles in Figures 3a and 3b, which tend to increase somewhat with height through the lower‐mid troposphere, to the DAM mass‐flux profiles in Figures 3c and 3d, which tend to decrease with height rather markedly below the freezing point, suggests that entrainment and detrainment even in RCE are not easily constrained.

One might hold out hope, however, that the entrainment/detrainment term might be negligible compared to the cloud‐base term; if true, this would yield a more viable constraint of*PE* ≠ 1, can predict changes in *M*|_{LCL} with warming. This will require diagnosis of all factors in Equation 13 besides *M*|_{LCL}, from both our FV^{3} and DAM simulations. We diagnose *M*|_{LCL} as an average of *M* between 800 and 850 hPa, diagnose *q*_{v} at the 2nd lowest model level (this is characteristic of the boundary‐layer values and thus of saturated parcels at cloud‐base), diagnose *Q*_{ft} as the radiative cooling integrated from cloud‐base to the tropopause, and diagnose precipitation efficiency according to Equation 11.

With these diagnostics in hand, Figure 5 compares the directly diagnosed *M*|_{LCL} to *M*|_{LCL} as estimated from both Equation 13 and Equation 6 by solving for *M*|_{LCL}. Consistent with the results of Schneider et al. (2010), this figure shows that while the theoretical estimates gives reasonable ballpark values for *M*|_{LCL}, especially when non‐unit PE is accounted for, they predict a decreasing trend which is only very roughly obeyed by the models. FV^{3} does show a decreasing trend, but its slope is less than half of that estimated from Equation 13. Meanwhile DAM shows a non‐monotonic change of *M*|_{LCL} with *T*_{s}, as indicated earlier. Thus, the entrainment/detrainment terms in Equation 12 seem to play a non‐negligible role in the change of *M*|_{LCL} with warming, inhibiting the robustness of Betts's rule. Inclusion of PE helps obtain more accurate values overall, but *changes* in PE with warming are small (varying between 0.56 and 0.51 in FV^{3}, and 0.26–0.3 in DAM) and thus do not impact the response of *M*|_{LCL}.

It is worth noting that in the original formulation of A. Betts (1998), the constraint on *M*|_{LCL} is formulated with an additional RH‐dependent term. This RH term captures the effects of entrainment/detrainment discussed above, as well as that of non‐unit precipitation efficiency, but is similarly difficult to constrain. This formulation of Betts's rule is derived and discussed further in Appendix C.

This paper has shown that.

Three rules for the decrease of convection with warming can be formulated, each of which spring from Equation 1 and thus embody the same physics

The stability‐iris effect is not entirely robust because clear‐sky convergence and cloud fraction are not directly proportional, but rather are connected by loosely constrained microphysical process [Equation 3 and Figures 1 and 2]

The decrease in tropospheric mass flux on isotherms (Rule 2) does seem to be potentially robust, based on its theoretical foundation as well as validation across a hierarchy of models [Equation 5 and Figures 3 and 4]

Betts's rule is not entirely robust, due to the loosely constrained effects of entrainment and detrainment [Equation 12 and Figure 5].

Our three rules, along with the analytical constraints from which they are deduced, are summarized in Table 1 and illustrated schematically in Figure 6. As depicted in the schematic, the three constraints are all related to Equation 1 and each other by integration/differentiation: Equation 2 is obtained by differentiation of Equation 1, Equation 5 is simply a re‐arrangement of Equation 1, and Equation 13 is obtained from Equation 1 via integration by parts.

What are the broader implications of these findings? The lack of robustness of the stability‐iris hypothesis as a potential mechanism for the tropical anvil cloud area feedback has been noted before (Sherwood et al., 2020). But, our emphasis on the microphysical degrees of freedom suggests that uncertainties in this feedback may not be easily remedied, as microphysical complexity is daunting (e.g., figure 1 of Morrison et al., 2020) and high clouds appear to be sensitive to many aspects of this complexity (e.g., evolution of various ice species, sedimentation, sub‐grid scale saturation adjustment; Ohno & Satoh, 2018; Ohno et al., 2020, 2021).

As for the decrease of mass flux profiles with warming: this is a straightforward consequence of decreasing *w*_{sub} with warming, which is well‐known, but has not been emphasized in the literature. Here, we have also emphasized the importance of temperature coordinates, and leveraged the *T*_{s}‐invariance of *∂*_{T}*F* to put the decrease of *M* on a stronger theoretical footing [Equation 5]. We have also confirmed that domain‐mean *M* = *ρw*_{sub} in cloud‐resolving simulations. Future work could investigate the degree to which this is true over the tropics in GCMs.

As for Betts's rule (Rule 3), this has long been invoked as a mechanism behind the weakening of tropical circulations, particularly the Walker circulation, as in Vecchi and Soden (2007). But, the lack of accuracy of Betts's rule found here and in Schneider et al. (2010), as well as the fact that its prior validation in Held and Soden (2006) and Vecchi and Soden (2007) relied on a single GCM and used *M*_{c} evaluated at 500 hPa rather than cloud base, suggests that a firmer basis for reasoning about the large‐scale circulation is required.

However, even with a firm grasp on how convective mass fluxes change with warming, as potentially provided by Rule 2, there are still overlooked and unanswered questions about how exactly such changes should affect the large‐scale circulation. For instance, Held and Soden (2006) point out that changes in *M*_{c} do not necessarily determine changes in the large‐scale circulation, as the latter is also profoundly influenced by the *spatial distribution* of convection. Indeed, in the unorganized RCE simulations analyzed here, there is no large‐scale circulation at any SST, yet *M*_{c} decreases with warming according to Rule 2. These points notwithstanding, Vecchi and Soden (2007) argue that a weakening of *M*_{c} should lead to a proportional weakening of large‐scale upward pressure velocities *ω*_{up}. This argument seems to assume that *M*_{c} is closely related to the large‐scale gross mass flux *σ*_{up} is the fractional area occupied by ascending grid cells), *and* that *σ*_{up} does not change significantly with warming. Future work could test these assumptions and make the connection between *M*_{c} and other measures of the large‐scale circulation (such as *ω*_{up}) more precise.

Finally, it is worth reflecting on the essential physics behind our rules. The physics behind Betts's Rule is straightforward enough: cloud‐base moisture increases faster than column‐integrated radiative cooling, so less mass flux is required. But are there analogous statements for Rules 1 and 2? The driving force there seems to be the increasing difference between Γ(*T*) and Γ_{d}, particularly in the upper troposphere, as *T*_{s} increases (Figures A1a and A1b). What causes this? Even at a fixed upper‐tropospheric isotherm *T*, *T*_{s} because the *pressure* and hence ambient air density are going down, even if the vapor pressure is not changing. This actually causes a quasi‐exponential increase of *T*_{s}, even though the isotherm *T* is fixed (see detailed discussion in Romps, 2016). This increase of _{d}. Meanwhile, the −*∂*_{T}*F* factor in Equation 5 is *T*_{s}‐invariant. Thus, Rule 2 (and also Rule 1, as a derivative of Rule 2) is again driven by a mismatch between the scalings of radiative cooling and moisture with *T*_{s}. This is reminiscent of Mapes's “two scale‐heights” argument which contrasts the thermodynamic and radiative behaviors of water vapor (Mapes, 2001), but here applied to global warming rather than our base climate.

The atmospheric model used here is the non‐hydrostatic version of GFDL's FV^{3} (Finite‐Volume Cubed‐Sphere Dynamical Core, Harris & Lin, 2013; S.‐J. Lin, 2004). The simulations analyzed here are very similar to those of Jeevanjee and Zhou (2022), so we describe some salient aspects of the simulation below, and refer the reader to Jeevanjee and Zhou (2022) for complete details.

We simulate doubly periodic radiative‐convective equilibrium (RCE) over fixed sea surface temperatures of *T*_{s} = 280, 290, 300 and 310 K. Our particular FV^{3} codebase is not equipped with interactive radiation, so radiative cooling must be otherwise parameterized. To emulate the *T*_{s}‐dependence of interactive radiation, we parameterized it as a fit to the invariant divergence of radiative flux *F* found by Jeevanjee and Romps (2018):*T*_{tp} = 200 K is the tropopause temperature. Above the tropopause temperatures are relaxed to *T*_{tp}, so the stratosphere is roughly isothermal. The invariance of (−*∂*_{T}*F*)(*T*) profiles with respect to *T*_{s} was shown on both theoretical grounds and with cloud‐resolving simulations in Jeevanjee and Romps (2018), and also confirmed across cloud‐resolving models in Stauffer and Wing (2022).

No boundary layer or sub‐grid turbulence schemes are used. Microphysical transformations are performed with a warm‐rain version of the GFDL microphysics scheme (Chen & Lin, 2013; Zhou et al., 2019), which in its default configuration models only water vapor *q*_{v} (kg/kg), cloud condensate, and rain, with the only transformations being condensation/evaporation of condensate and autoconversion of cloud condensate to rain (rain evaporation is disabled). The horizontal grid has 96 points in both *x* and *y* with a resolution of 1 km, and the 90‐level vertical grid has a stretched grid spacing of 50 m near the surface up to 5,000 m near model top at 68 km. Each simulation ran for 120 days, with domain‐mean statistics drawn from the last 5 days.

Actively convecting (updraft) grid cells are identified as having cloud condensate mixing ratios greater than 10^{−5} as well as vertical velocities *w* > 0.7 m/s, and convective mass fluxes are then defined at each level as *M* ≡ *ρw*_{up}*σ*_{up} (kg/m^{2}/s) where *w*_{up} is *w* conditionally averaged over updraft grid cells, and *σ*_{up} is the fractional area occupied by updraft grid cells (this *σ*_{up} should not be confused with the large‐scale *σ*_{up} introduced in the conclusion). Cloud‐base is defined as the lower‐level maximum in cloud fraction, and the tropopause is defined as the lowest model within 0.5 K of *T*_{tp} = 200 K. Figure A1 shows three key diagnostics for the arguments presented in this paper: the lapse rate Γ, inverse stability parameter *α* = (*c* − *e*)/*c*.

Our second set of cloud‐resolving RCE simulations use Das Atmosphärische Modell (DAM, Romps, 2008), a fully compressible, non‐hydrostatic cloud‐resolving model, coupled to radiation via the comprehensive Rapid Radiative Transfer Model (RRTM, Mlawer et al., 1997). DAM employs the six‐class Lin‐Lord‐Krueger microphysics scheme (Y.‐L. Lin et al., 1983; Lord et al., 1984; Krueger et al., 1995), and in contrast to its original formulation in Romps (2008) employs no explicit sub‐grid scale turbulence scheme, relying instead on “implicit LES” for sub‐grid scale transport (Margolin et al., 2006).

These simulations ran on a square doubly periodic domain of horizontal dimension *L* = 72 km, with a horizontal grid spacing of *dx* = 1 km. The 76 level vertical grid has a spacing which stretches smoothly from 50 m below 1,000 m–250 m between 1,000 m and 5,000 m, and then to 500 m up to the model top at 30 km. We calculated surface heat and moisture fluxes using simple bulk aerodynamic formulae, and used a pre‐industrial CO_{2} concentration of 280 ppm with no ozone. Our SSTs are the same as for the FV^{3} simulations, and all our DAM runs branched off the equilibrated runs described in Romps (2014) and were run for 60 days to iron out any artifacts from changing the domain and resolution. All vertical profiles are time‐mean and domain‐mean, averaged over the last 5 days of each run. All diagnostics are constructed identically to their FV^{3} counterparts, except the vertical velocity threshold for conditional sampling of convective mass flux is taken to be 1 m/s.

The original formulation of Betts's rule (A. Betts, 1998, his equation 1) was written in terms of surface evaporation, subsidence mass flux, and the difference in *q*_{v} between the boundary layer mean and the dry air sinking at boundary layer top. To write this in terms of the variables used here, we note the equivalence between surface evaporation and precipitation, and further assume that precipitation equals *Q*_{ft} (O'Gorman et al. 2012). We also invoke the equivalence between subsidence mass flux and convective mass flux [Equation 4 and Figure 3], and note that boundary layer mean *q*_{v} equals *α* equals 1 − *α* in Romps (2016)]. Adding these equations and noting that *∂*_{z}*M* = *M*(*ϵ* − *δ*), one can rewrite the result as*α* = (*c* − *e*)/*c* rather than PE yields

It is remarkable that this version of Betts's rule does not neglect entrainment/detrainment or evaporation of condensate, unlike Equation 6. One must conclude that these effects are then entirely encapsulated in the value of RH|_{LCL}. Given that the simplified form Equation 6 is not entirely robust, one must further conclude that RH|_{LCL} is relatively unconstrained and varies in our simulations. Indeed, both our FV^{3} and DAM simulations show RH|_{LCL} increases of 0.2 over our SST range, yielding significant decreases in the (1 − RH) factor appearing in Equation C1.

Linjiong Zhou assisted with configuring cloud‐resolving simulations with FV^{3} and developed the warm‐rain option of the GFDL microphysics package. Jake Seeley, Zhihong Tan, Linjiong Zhou, and Andrew Williams provided valuable feedback on drafts of this work. Yi Ming provided encouragement and suggested the use of GFDL CM4 output. Huan Guo assisted with location of this output on the GFDL archive. Catherine Raphael produced the schematic in Figure 6. Two anonymous reviewers provided encouragement as well as useful critical feedback.

Data and analysis and visualization scripts used in generation of the figures in this paper are available at

The originally published version of this article contained a few typographical errors. In the sixth line of the fourth paragraph of Section 2, the phrase “the mean anvil” has been removed. In the fifth and eighth lines of the fourth paragraph of Section 3, the word “increases” should be “decreases.” In addition, in the caption to Figure A1, the word “increase” should be “decrease.” The errors have been corrected, and this may be considered the authoritative version of record.