1. Introduction

One-dimensional (1D) models of radiative–convective equilibrium (RCE) have played an important role in our understanding of Earth’s equilibrium climate sensitivity (Schlesinger 1986) from the early days of Manabe and Wetherald (1967) through to the recent work of Kluft et al. (2019). Over the past two decades, it has become possible to replace those 1D models with 2D or 3D cloud-resolving models (CRMs), allowing for a more realistic treatment of clouds and convection. Unfortunately, the use of CRMs to study the response of RCE to altered CO₂ has been limited to only a handful of studies (Bretherton 2007; Romps 2011; Khairoutdinov and Yang 2013; Bretherton et al. 2014; Singh and O’Gorman 2015; Romps 2019), likely due to 1) the high computational cost of CRMs and 2) the multiple years required to fully equilibrate to a new concentration of CO₂ (Cronin and Emanuel 2013). To help overcome these barriers, this paper describes a technique that accelerates by a factor of 30 the equilibration of sea surface temperature (SST) and CO₂ in small-area cloud-resolving simulations.

This 30× speedup is relative to a simulation in which the slab ocean has negligible thickness (e.g., ≪1 m); this is often called a swamp ocean. Compared to simulations with a thick (e.g., ≳1 m) slab ocean, the speedup obtained with this technique is even greater than a factor of 30. Although the focus here is on CO₂, this technique can be applied to the study of any radiative forcing, be it from variations in some other greenhouse gas or in the concentration of aerosols as in Khairoutdinov and Yang (2013). And, while we will study only standard small-domain CRM simulations of RCE here, the technique is equally applicable to any other model with a surface that is effectively 0D (e.g., a slab ocean with an infinite horizontal conductivity). Furthermore, as will be discussed in section 8, it should be possible to further extend the method to global climate models with a slab ocean.

But, again, the focus here is on CO₂-induced warming in cloud-resolving simulations of RCE, and the great benefit of a technique for rapid equilibration is that it makes it computationally feasible to run many simulations. This has implications for the study of equilibrium climate sensitivity (ECS) and the direct effects of CO₂.

With regards to ECS, global climate models (GCMs) exhibit a curious behavior: when run to high combinations of temperature and CO₂, their ECS is found to have a pronounced maximum at SSTs in the range of 300–320 K (Russell et al. 2013; Wolf et al. 2018). But, since GCMs are expensive to run, the existing studies have included only about three simulations within that 20-K temperature range, giving ECS as only a coarsely resolved function of SST. Furthermore, it has been argued that the peak in ECS is caused by cloud feedbacks (Popp et al. 2016; Wolf et al. 2018), which motivates studying the phenomenon with cloud-resolving models. Here, the acceleration technique allows the ECS to be calculated in RCE as a smooth function of SST (with 1-K intervals) using cloud-resolving simulations.

In the study of the direct effects of carbon dioxide, global climate models have been used to quantify the rapid adjustment of clouds to a sudden change in CO₂. This is typically done with a small handful of GCM experiments that simulate the effect of a sudden doubling or quadrupling of CO₂ (Gregory and Webb 2008; Andrews and Forster 2008; Colman and McAvaney 2011; Wyant et al. 2012; Zelinka et al. 2013; Kamae and Watanabe 2013). To get a fuller picture, at least in the context of RCE, the acceleration technique allows the direct effect to be explored over a wide range of SSTs and CO₂ concentrations in a CRM, where clouds are resolved rather than parameterized. This provides a bird’s-eye view of the direct effects of CO₂ relative to the effects of the resulting warming.

In the following sections, we will derive the relevant equilibration time scales (section 2), introduce the technique for rapidly approaching equilibration between the atmosphere, ocean, and CO₂ (section 3), and describe the cloud-resolving simulations to test the technique (section 4). Thanks to the speedup facilitated by the equilibration technique, 36 simulations are used to explore CO₂ concentrations ranging from below 1 ppmv to nearly 10 000 ppmv (section 5). Those simulations reveal an ECS that peaks prominently at warm temperatures (section 6). An additional set of simulations are then used to explore the direct effect of CO₂ on temperature, humidity, clouds, tropopause height, convective available potential energy (CAPE), and precipitation rate (section 7). Conclusions are presented in section 8.

2. Time scales

Let us derive the time it takes for an atmosphere to equilibrate to a new RCE state in two different scenarios: 1) an atmosphere over a zero-heat-capacity ocean (i.e., a slab ocean of infinitesimal thickness) being subjected to a sudden change in the top-of-atmosphere (TOA) forcing (say, from a sudden change in the concentration of CO₂), and 2) an atmosphere over an infinite-heat-capacity ocean (i.e., an ocean with a fixed temperature) being subjected to a sudden change in its sea surface temperature. These time scales were derived by Cronin and Emanuel (2013) by solving an eigenvalue problem and then approximating the results. Here, we take shortcuts to the answers and relegate some of the details to the appendix.

3. The equilibration technique

The traditional method for equilibrating the atmosphere, ocean, and CO₂ is to fix the desired CO₂ concentration and let the atmosphere and ocean evolve until they equilibrate. This is accomplished by running the atmosphere over a slab ocean that obeys a simple prognostic equation relating its temperature tendency to the surface enthalpy-flux imbalance. That approach takes hundreds of days to equilibrate for the reason described in section 2a.

To take advantage of the much faster equilibration described in section 2b, we describe here a different approach. Rather than fixing the CO₂ concentration and evolving a prognostic equation for the SST, we will fix the SST and evolve a prognostic equation for the CO₂ concentration. The two methods are compared in Fig. 1. For example, imagine we are interested in obtaining the ECS as a function of T_s. To obtain this, we can run a set of simulations, each with a different T_s, and let them equilibrate their atmosphere and their CO₂ concentration G. With the resulting pairs of T_s and G, we can then construct, by interpolation, the functions T_s(G) and G(T_s). The ECS, as a function of T_s, would then be given by

$\text{ECS}\left(T_s\right)=T_s\left[2\times G\left(T_s\right)\right]-T_s.$(5)

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(left) Standard slab-ocean simulations fix the CO₂ concentration and let the SST vary naturally according to its heat budget. This approach takes years of model time to equilibrate. (right) The technique proposed here is to fix the SST and let the CO₂ vary in a way that closes the TOA energy budget. This approach takes about one month of model time to equilibrate.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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4. Simulations

To evaluate this equilibration technique and to employ it in a study of the direct effects of CO₂, we will use three different sets of simulations, which we will refer to as SlabO, ProgC, and FixC. The SlabO simulations are standard slab-ocean simulations of the type depicted on the left in Fig. 1; these are run with three different CO₂ concentrations. The ProgC simulations use fixed SSTs and (8) to prognose CO₂, as depicted on the right in Fig. 1; there are 36 of these simulations spanning SST values from 285 to 320 K in 1-K increments. The FixC simulations use fixed SSTs (again, 36 simulations covering 36 SSTs) and a CO₂ concentration that is set to 280 ppmv; these are the “fixed-carbon” warming experiments most often performed with cloud-resolving models. These three sets of simulations are summarized in Table 1.

Table 1.

The three types of simulations: slab ocean (SlabO), prognostic CO₂ (ProgC), and CO₂ fixed at 280 ppmv (FixC).

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The cloud-resolving model used here is Das Atmosphärische Modell (DAM; Romps 2008), which has a finite-volume and fully compressible dynamical core coupled to a six-class single-mode microphysics scheme (Lin et al. 1983; Lord et al. 1984; Krueger et al. 1995) and the Rapid Radiative Transfer Model for General Circulation Models (RRTMG; Clough et al. 2005; Iacono et al. 2008) to represent shortwave and longwave radiation. In DAM, no explicit diffusion is added to the numerical diffusion already generated by the third-order advection scheme. The total solar irradiance and solar zenith angle are held fixed throughout the simulations. The cosine of the solar zenith angle θ is set to the insolation-weighted—that is, cos(θ)-weighted—average of cos(θ) during the diurnal cycle at the equator on 1 January (Romps 2011). This gives a zenith angle of 43.75°; this would be 43.91° on the equator at the solstice, and both of these are similar to the value of arccos(2/3) ≈ 48° that would be appropriate for a global average (Cronin 2014). Using a total solar irradiance (TSI) of 1366 W m⁻² (somewhat higher than the modern estimate of 1361 W m⁻²; Kopp 2016), the daily averaged insolation at the equator on 1 January is 413.3 W m⁻². The model’s TSI is set to that mean insolation (413.3 W m⁻²) divided by cos(43.75°), which equals 572.1 W m⁻². In similar configurations, DAM has been shown to broadly match conditions observed in the tropics, although it produces a higher CAPE and relative humidity (Romps 2011) and a lower cloud fraction (Seeley et al. 2019).

By volume, N₂ and O₂ are set to 79% and 21% of dry air, respectively, and those percentages are then reduced proportionately to accommodate the specified concentration of CO₂. The radiation scheme sees nitrogen, oxygen, carbon dioxide, water vapor, cloud drops, and cloud ice. Ozone is excluded from these runs because DAM does not have a chemistry module that would allow the ozone profile to evolve in a mechanistic way in the warming experiments. Fortunately, stratospheric ozone has only a small impact on equilibrium climate sensitivity (Dacie et al. 2019).

All of the simulations have their domain-averaged horizontal wind damped to zero on a 1-h time scale. Surface fluxes of mass, momentum, and energy are calculated using bulk-aerodynamic formulas with a fixed 5 m s⁻¹ wind speed and a transfer coefficient of 1.5 × 10⁻³. All domains have a model top at a height of 61 km. Unless specified otherwise, the model domain is square and 108 km wide; this domain is used for all of the SlabO, ProgC, and FixC simulations, and is small enough to avoid convective aggregation. The horizontal grid spacing is set to a uniform 1 km, and a stretched grid is used in the vertical with 134 levels whose spacing smoothly transitions from a uniform Δz = 50 m below an altitude of 500 m to a uniform Δz = 500 m between altitudes of 5 and 48 km. The effect of the Brewer–Dobson circulation on water vapor is emulated by relaxing (on a 5-day time scale) the water vapor mass fraction at each height in the stratosphere to the smaller of 1) the value at the tropopause (where there is zero net radiative cooling) and 2) the smallest value between the tropopause and that height.

The timeline of the simulations is shown in Fig. 2. The first step is to identify the appropriate TOA flux imbalance, which will be applied as a cooling of the ocean for the SlabO simulations and also used as the value of N₀ in (8) for the ProgC simulations. To this end, the first cloud-resolving simulation (labeled “Standard RCE” in Fig. 2) is run over a 300-K ocean with a CO₂ concentration of 280 ppmv. This simulation is restarted from a similar RCE profile, but is run for 100 days to ensure equilibration. Because this is a fixed-SST run, there is an imbalance in the enthalpy fluxes at the top of the atmosphere (TOA) and at the ocean surface. Once equilibrated, those imbalances are equal because there can be no net convergence of energy into the atmosphere. Averaging over the last 50 days of the simulation, the TOA net downwelling radiative flux is found to be 112 W m⁻². As noted by Romps (2011), this matches the ~100 W m⁻² of net downwelling radiative flux observed at the top of the atmosphere over the western Pacific warm pool (Tian et al. 2001).

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Timeline indicating how each set of simulations is initialized. The thick solid lines denote the last 50 days of the ProgC and FixC simulations, over which quantities are averaged to obtain equilibrated values.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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The SlabO simulations are restarted directly from the end of that Standard RCE simulation. The three SlabO simulations are run with 1×, 4×, and 16× the original CO₂ concentration (i.e., 280, 1120, and 4480 ppmv). The slab ocean is given a depth of 20 cm, is initialized to 300 K, and is cooled at a rate of 112 W m⁻² to guarantee that the 280-ppmv simulation equilibrates to an SST of 300 K. Each SlabO simulation is run for four years.

Before starting the ProgC and FixC simulations, a set of 36 simulations are run on a small 16-km-wide square domain with SSTs ranging from 285 to 320 K in 1-K increments. These simulations are labeled “Small domain” in Fig. 2, are restarted from the Standard RCE simulation, and are run for 100 days. The point of these small-domain simulations is to bring the atmosphere to rough equilibration with its new SST in a computationally inexpensive manner. After 100 days of those small-domain simulations, each of the ProgC and FixC simulations is restarted from the small-domain simulations with the matching SST. ProgC uses (8) with N₀ = 112 W m⁻² and τ = 1 week. FixC uses a fixed CO₂ concentration of 280 ppmv. The ProgC and FixC simulations are run for 100 days and equilibrated properties are obtained by averaging over the last 50 days.

5. Results

The bottom panel of Fig. 3 shows how the SlabO simulations approach equilibrium. The 280-ppmv slab-ocean simulation does nothing interesting, as expected, because it is restarted with the same SST and CO₂ concentration as the already equilibrated RCE simulation. The 4×CO₂ (1120 ppmv) simulation reaches its equilibrated SST in about 3 years, while the 16×CO₂ (4480 ppmv) simulation is still not equilibrated after 4 years.

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(top) Time series of surface air temperature for the small-domain simulations (first 100 days) followed by the ProgC simulations (next 100 days). (middle) As in top, but for CO₂. (bottom) Time series of SST for the SlabO simulations.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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In contrast, the ProgC simulations equilibrate in a couple months. The top and middle panels of Fig. 3 show the time series of surface air temperature and CO₂ concentration, respectively, for these simulations. Although the simulations are run for a total of 200 days (split into 100 days of the fixed CO₂ on a small domain followed by 100 days of ProgC on the standard domain), that is more time than is needed to reach equilibration. In the first span of 100 days, the simulations fully equilibrate to the new SST in about 30 days, which is as expected: this is several multiples of the 5-day e-folding time calculated in section 2b. In the second span of 100 days, the simulations equilibrate their TOA flux in about 30 days: this is a few multiples of the 1-week CO₂ adjustment time scale used in (8).

As a sanity check, we can confirm that the ProgC simulations have the desired net TOA flux. Figure 4 plots the time series of each simulation’s net TOA downwelling radiation anomaly (i.e., net downwelling TOA minus 112 W m⁻²), which are offset in the vertical so that each time series can be seen. Each thin horizontal line corresponds to a zero TOA anomaly for the SST value that it intercepts on the ordinate. Within a month of switching to the prognostic equation for CO₂, the simulations have adjusted their TOA flux anomaly to zero. Although this is a useful sanity check, it is important to note the TOA flux anomaly is forced to go to zero by Eq. (8) on whatever time scale we choose. Had we chosen a 1-min time scale, this TOA flux anomaly would have gone to zero within a few minutes from the 100-day mark. (Recall that we choose a 1-week time scale to prevent the CO₂ concentration from undergoing wild swings.) The more compelling evidence that a simulation has equilibrated is that its CO₂ concentration has reached a steady value. As we saw in Fig. 3, this occurs after about one month, implying that any remaining stratospheric temperature adjustment that continues beyond a month (on the order of 1–10 K) has no consequential impact on the atmosphere’s energy balance.

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Time series of net downwelling TOA radiative-flux imbalance (TOA downwelling flux minus 112 W m⁻²) for the experiments using fixed-SST/fixed-CO₂ on the small domain for the first 100 days and fixed-SST/prognostic-CO₂ ProgC simulations for the next 100 days. The time series are shifted in the vertical by dividing the TOA imbalance by 30 W m⁻² K⁻¹ and adding the SST; therefore, every 1 K of displacement on the vertical axis corresponds to 30 W m⁻² of net TOA imbalance. The thin horizontal lines mark the 36 SST values, which correspond to zero TOA imbalance.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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Since the ProgC simulations equilibrate after 30 days, we can average the CO₂ concentration over the last 50 days of the 100-day simulations to get the equilibrated CO₂ concentrations (denoted by G). This gives us 36 pairs of T_s and G (one pair for each T_s from 285 to 320 K in 1-K increments). We can linearly interpolate between these 36 pairs to produce T_s(G) as a piecewise-linear function. This is plotted in Fig. 5. Note that the low computational cost of these simulations—made possible by the fixed-SST/prognostic-CO₂ methodology—has allowed us to evaluate this function at 1-K intervals in a cloud-resolving model.

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The function SST(CO₂) created by linearly interpolating the 36 equilibrated fixed-SST/prognostic-CO₂ simulations, which are plotted as black dots.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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To confirm that the ProgC simulations are giving the correct T_s(G) relationship, we can compare them to the SlabO simulations. Since the 16×CO₂ simulation has not fully equilibrated after four years, Fig. 6 shows the Gregory plots (Gregory et al. 2004) for each of the runs. Each gray dot represents an hourly average of the slab-ocean simulation, and the solid black lines follow the monthly average. The diamonds show the predictions from the T_s(G) function of Fig. 5, derived from the ProgC simulations. Regardless of how one fits a line to the SlabO data (e.g., over the last 6 months, the last 24 h, or any length of time in between), the Gregory plot data predict equilibrated SSTs that are within 0.1 to 0.3 K of the values obtained from Fig. 5.

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Gregory plot of the three slab-ocean simulations with 1×, 4×, and 16× the reference CO₂ concentration of 280 ppmv. The gray dots are hourly averages and black curves are monthly averages. The diamonds mark the final SST predicted by the SST(CO₂) curve in Fig. 5.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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6. Equilibrium climate sensitivity

Since Fig. 5 uses a logarithmic axis for the CO₂ concentration, a constant equilibrium climate sensitivity (ECS; the change in SST per doubling of CO₂) would correspond to a straight line in this plot. We see departures from this behavior at very low CO₂ concentrations (at and below 1 ppmv) and at concentrations above 280 ppmv. The departure from a constant ECS at low CO₂ concentrations is due to the fact that CO₂ loses its efficacy as a greenhouse gas at concentrations just below 1 ppmv. If the curve were continued to the left, it would approximate a horizontal line at an SST of around 284–285 K.

At high temperatures and CO₂ concentrations, there is a remarkable increase in the ECS, which is caused by variations in both the CO₂ forcing and the feedback parameter. The left panel of Fig. 7 shows the instantaneous top-of-atmosphere radiative forcing from doubling CO₂. We see that the radiative forcing increases from 0 W m⁻² at 0 ppmv, to about 4 W m⁻² at 280 ppmv, to over 5 W m⁻² at concentrations exceeding 1600 ppmv. This is consistent with studies using global climate models (Hansen et al. 2005; Colman and McAvaney 2009; Caballero and Huber 2013; Wolf et al. 2018) and a simple RCE model (Kluft et al. 2019), which have all found that a doubling of CO₂ causes greater radiative forcing at higher CO₂ concentrations. As we can confirm by comparing to the FixC runs, this variation in forcing is due to the change in the CO₂ concentration, not the warming.

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As a function of SST, the (left) instantaneous top-of-atmosphere forcing from doubling the CO₂ concentration, (middle) inverse magnitude of the feedback parameter (a.k.a. the “climate sensitivity”), and (right) product of the two, which is the equilibrium climate sensitivity (ECS).

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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The middle panel of Fig. 7 shows the inverse magnitude of the feedback parameter, which is often referred to as the climate sensitivity (with units of K W⁻¹ m², not to be confused with the equilibrium climate sensitivity). Here, the feedback parameter is defined as the ECS, as defined by (5), divided by the instantaneous top-of-atmosphere doubled-CO₂ forcing. We see that the climate sensitivity peaks around 310 K, matching the location of the peak at around 300–320 K found in global climate models (Leconte et al. 2013; Wolf and Toon 2015; Popp et al. 2016; Wolf et al. 2018). Finally, the last panel shows the product of the forcing and the climate sensitivity, which is the ECS. The ECS exhibits a peak around 310 K, consistent with the behavior found in both a 1D model (Meraner et al. 2013) and global climate models (Russell et al. 2013; Wolf et al. 2018).

7. Direct effects of CO₂

When trying to achieve an equilibrated RCE with the desired TOA net radiative flux, we have seen that it is much faster, and just as accurate, to evolve the CO₂ concentration instead of the SST. But, when using RCE simulations to study greenhouse gas warming, is it really necessary to enforce a constant net TOA flux? Why not hold the CO₂ concentration constant and simply step up the SST as has been done many times before in studies with both CRMs (e.g., Muller et al. 2011; Cronin and Wing 2017) and GCMs (e.g., Cess and Potter 1988)?

The simulations performed in that way (i.e., with varied SST, but constant CO₂) will have different values of the net TOA flux, but they may be adequate for answering many types of questions. Of course, if the goal is to accurately calculate the climate sensitivity, there is no substitute for equilibrating two or more simulations (with different CO₂ concentrations) to the same TOA flux. For everything else, however, the relevant question is whether the “direct effect” of CO₂ (i.e., the impact of varying CO₂ while holding SST fixed) matters for the phenomenon being studied (Sherwood et al. 2015; Kamae et al. 2015). The new equilibration technique allows us to explore the direct effects of CO₂ as continuous functions of surface temperature.

As a reminder, FixC is identical to ProgC, except that the CO₂ concentration is held fixed at 280 ppmv; therefore, a variable in ProgC minus the same variable in FixC is the direct effect of CO₂ on that variable. The four rows of Fig. 8 show profiles of temperature, specific humidity, cloud fraction, and net radiative heating in the ProgC and FixC simulations (left and middle columns, respectively), along with their differences (right column). Each simulation is color-coded, ranging from the darkest blue at an SST of 285 K, through blue–green at 300 K, and up to the darkest red at 320 K. By eye, we see that the profiles in the left column (ProgC) are very similar to the profiles in the middle column (FixC). This tells us that the effect of warming on these four variables is much larger than the direct effect of CO₂. (Note that the differences in the third column are plotted over a smaller range to make visible the relatively small direct effect.)

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Profiles of (top) temperature, (second row) specific humidity, (third row) cloud fraction, and (bottom) radiative heating in the (left) ProgC and (middle) FixC simulations. The simulations are color-coded from dark blue to dark red as the SST increases from 285 to 320 K, as can be seen from where the temperature profiles intercept the abscissa. (right) The difference in the profiles between the ProgC and FixC simulations at the same SST; this is the direct effect of CO₂ becoming higher (red) and lower (blue) than 280 ppmv.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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Although the direct effects of CO₂ are relatively small in magnitude, they are robust. Focusing on the right column of Fig. 8, we see that an increase in CO₂ concentration (red colors; vice versa for blue) while holding SST fixed tends to 1) warm the troposphere, 2) humidify the troposphere, 3) decrease cloud cover (Gregory and Webb 2008; Andrews and Forster 2008; Colman and McAvaney 2011; Zelinka et al. 2013; Kamae and Watanabe 2013), and 4) decrease radiative cooling (Newell and Dopplick 1970), which will then cause a reduction in precipitation (Mitchell et al. 1987).

To quantify the relative impacts of the direct effect and the warming effect, we can calculate mean absolute changes for both of these. In particular, let us calculate the mean absolute change in temperature, specific humidity, cloud fraction, and radiative cooling over all SST values and from heights up to 12 km (which is in the troposphere for all SST values), once for the direct effect and once again for the warming effect. For X equal to one of temperature, specific humidity, cloud fraction, and radiative cooling, Fig. 9 gives the value of

${\displaystyle \frac{1}{35\text{K}}}{\displaystyle {\int }_{285\mathrm{K}}^{320\mathrm{K}}d{T}_s{\displaystyle \frac{1}{12\mathrm{km}}}{\displaystyle {\int }_{0\mathrm{km}}^{12\mathrm{km}}dz\left|\text{Δ}X\right|}},$(9)

where Δ is the difference either between ProgC and FixC (for the direct effect) or between FixC and FixC at 300 K (for the warming effect). We see that the warming effect is at least an order of magnitude larger than the direct effect for tropospheric temperature, specific humidity, and cloud fraction. For the radiative cooling, the magnitude of the direct effect is about one-third that of the warming effect.

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The direct effect and warming effect, as calculated using expression (9) using the appropriate definition of Δ, for temperature, specific humidity, cloud fraction, and radiative cooling. The numbers next to each diamond give the values calculated by expression (9) and the position of the diamonds on the logarithmic ordinate illustrate the magnitude relative to the warming effect (e.g., for the direct effect on temperature, 0.25/16 = 0.016).

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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Although Fig. 9 suggests that the changes in cloud cover from the direct effect are an order of magnitude smaller than from warming, changes calculated using Eq. (9) include vertical shifts in clouds that may not be as relevant to the atmosphere’s energy balance as changes in the magnitudes of cloud-fraction maxima. To tease apart these two effects, we can track the changes in the location and areal fraction of the lower and upper peaks in cloudiness. The left panel of Fig. 10 shows the profile of cloud fraction from the base simulation (300 K, 280 ppmv). The lower and upper peaks in cloudiness are identified as the maxima of quadratic fits to the three closest points, and they are marked in the left panel by colored lines. Those four colors correspond to the four other panels, which show their variations with SST in the (dashed) FixC and (solid) ProgC simulations. (The CO₂ axes are shown for the ProgC simulations; the FixC simulations all use 280 ppmv.) The dashed curves show how the clouds change in response to an altered SST. The solid curves show how the clouds change in response to both an altered SST and the altered CO₂ required to generate that SST in an energetically balanced way. Therefore, the difference between the dashed and solid curves is the direct effect of CO₂ on the clouds. Since the dashed and solid curves are nearly identical, we see that the direct effect is small. Nevertheless, there is a noticeable and consistent direct effect on the shallow-cumulus peak: an increase in CO₂ tends to lower the height of the shallow clouds [consistent with Wyant et al. (2012), Kamae and Watanabe (2013), and Andrews and Ringer (2014)] and decrease their total area.

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(left) The profile of cloud fraction in the base simulation with a 300-K ocean and a 280-ppmv CO₂ concentration. The cloud fractions and locations of the lower (cumulus) peak and the upper (anvil) peak are indicated by four different colors, which correspond to the colors used in the other panels. (middle top) Anvil temperature in the FixC simulations (dashed) and ProgC simulations (solid) as a function of SST (the CO₂ axis is for the ProgC simulations only; the FixC simulations all use 280 ppmv). (right top) As in middle top, but for anvil cloud fraction. (middle bottom) As in middle top, but for cumulus pressure. (right bottom) As in right top, but for cumulus cloud fraction.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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Despite the fact that RCE has no large-scale circulation (e.g., no subtropical stratus), the magnitude of the direct effect of CO₂ on shallow clouds found here in RCE is similar to what is seen in global climate models. Zelinka et al. (2013) studied the effect of quadrupling CO₂ while holding SSTs fixed in five climate models. Their Fig. 7 shows the profiles of cloud fraction in the lower troposphere (averaged equatorward of 45° in regions of high lower-tropospheric stability) from the five models for two different experiments: one with climatological SSTs and one with climatological SSTs and quadrupled CO₂. Extracting the data directly from the vector graphics and applying a quadratic fit to identify the location and value of the lower-tropospheric peak, the effect of quadrupling CO₂ is found to change the peak cloud cover by a fractional amount of +1%, −7%, −3%, −1%, and −1% for CanESM2, CCSM4, MIROC5, MRI-CGCM3, and HadGEM2-A, respectively. (Note that these are fractional changes in the cloud cover; e.g., for a cloud cover of 10%, a −1% fractional change means that the cloud cover becomes 9.9%.) The average fractional change in cloud cover among these five GCMs is −2% for a quadrupling of CO₂. In the RCE simulations, we can calculate the fractional change in the peak in lower cloud cover from a quadrupling of CO₂, and we can apportion that change to the warming (i.e., the change in SST) and the direct effect (i.e., the change in CO₂). This is accomplished by comparing three cases: the FixC simulation at 300 K and 280 ppmv, the ProgC simulations interpolated to 1120 ppmv (with a corresponding SST of 305.6 K), and FixC simulations (with 280 ppmv) interpolated to an SST of 305.6 K. The result is that the warming associated with a quadrupling of CO₂ leads to a −4% fractional change, while the direct effect of the quadrupled CO₂ generates an additional −2% fractional change, matching what is found in the data of Zelinka et al. (2013). Likewise, the large-eddy simulations of Wyant et al. (2012) also generated a −2% fractional change in low cloud cover in response to a quadrupling of CO₂.

Before we conclude, let us look at three more quantities—the tropopause height, CAPE, and the precipitation rate—to see if they are affected directly by CO₂. We might expect CO₂ to affect the tropopause height through its direct impact on the balance between shortwave O₃ heating and longwave CO₂ cooling. Recall, however, that there is no ozone in these simulations, so the stratosphere is relatively cold and relatively insensitive to CO₂. Indeed, Fig. 11a shows that the tropopause height in these simulations is virtually unaffected by the CO₂ concentration. The tropopause is defined here as the height where the net radiative cooling is zero; due to convective cooling from overshooting updrafts, there is a layer of compensatory radiative heating atop the troposphere and, therefore, a well-defined height at which the net radiative cooling is exactly zero. In this panel and the others of Fig. 11, there are two curves: one for the equilibrated-CO₂ simulations (solid) and one for the 280-ppmv simulations (dashed). In Fig. 11a, the two curves are nearly identical, indicating that the direct effect of CO₂ on the tropopause height is negligible in these simulations.

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(a) The tropopause height as a function of SST for simulations in which CO₂ is equilibrated (solid) and simulations in which CO₂ is held constant at 280 ppmv (dashed). (b) As in (a), but for CAPE. (c) As in (a), but for the precipitation rate.

Citation: Journal of Climate 33, 9; 10.1175/JCLI-D-19-0682.1

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As for CAPE, recent work has shown that CAPE is set by the difference between the adiabatic lapse rate and the entraining lapse rate (Singh and O’Gorman 2013; Romps 2016; Seeley and Romps 2015). Given the entrainment rate and the temperature and humidity profiles, we can calculate these lapse rates and, therefore, CAPE. We have already seen in Fig. 9 that the direct effect of CO₂ on temperature and humidity is an order of magnitude less than the effect from sea surface warming, so a natural hypothesis is that CAPE is not much affected directly by CO₂. Figure 11b confirms this: CAPE as a function of SST is largely unaffected by variations in CO₂.

Finally, let us consider the precipitation rate. Since the precipitation rate is closely pegged to the net radiative cooling of the troposphere, and since an addition of CO₂ reduces that cooling (as noted in the discussion of Fig. 8), we know that a direct effect of CO₂ must be to cause a decrease in precipitation (Mitchell et al. 1987; O’Gorman et al. 2012). Figure 11c confirms such an effect, albeit a modest one. The FixC simulations (in which net upwelling TOA flux increases with SST) generate an average rate of precipitation increase of 3.0% K⁻¹. The ProgC experiments (in which the net upwelling TOA flux is constant) generate an increase of 2.3% K⁻¹, in line with GCMs subjected to CO₂-induced warming (Held and Soden 2006; Lambert and Webb 2008; Stephens and Ellis 2008). We see that the direct effect cuts into the precipitation increase by about one quarter, which is consistent with the direct effect on radiative cooling being about a third as large as the warming effect, as seen in Fig. 9.

8. Summary and discussion

In section 2, we derived the time scales for an atmosphere and ocean to equilibriate under two scenarios: a zero-heat-capacity ocean and an infinite-heat-capacity ocean. In either case, the equilibration time scale is given by the heat capacity of the atmosphere divided by the sensitivity to temperature of an enthalpy flux. In the case of an ocean with zero heat capacity, the relevant flux is the net radiative flux at the top of the atmosphere, which has a weak sensitivity to the deviation of the near-surface air temperature from its equilibrium value (~1.5 W m⁻² K⁻¹). In the case of an ocean with infinite heat capacity, the relevant flux is the net enthalpy flux at the surface, which has a strong sensitivity to the deviation of the near-surface air temperature from its equilibrium value (~40 W m⁻² K⁻¹). By virtue of these different sensitivities, an atmosphere over a fixed sea surface temperature will approach equilibrium with the SST about 30 times faster than an atmosphere over a thin slab ocean will approach a zero TOA flux anomaly.

This motivated the equilibration technique described in section 3, which reduces by 30× the computational time required to conduct greenhouse gas warming experiments in limited-area cloud-resolving models. The idea is to apply a fixed temperature increment to the surface and then allow both the atmosphere and the carbon dioxide concentration to evolve prognostically. The equation used to evolve the CO₂ concentration is one that drives the net TOA radiative flux anomaly to zero on a reasonably short time scale, chosen here to be one week.

This technique was used in sections 4 and 5 to equilibrate a cloud-resolving model to 36 different concentrations of carbon dioxide. This produced the sea surface temperature as a continuous function of the CO₂ concentration, as shown in Fig. 5. This revealed a peak in the equilibrium climate sensitivity (ECS) at warmer temperatures as seen in the right panel of Fig. 7. There are two contributions to this peak in ECS. The first is that the forcing from a doubling of CO₂ increases with the CO₂ concentration from 0 to over 5 W m⁻². The second is that the magnitude of the feedback parameter (climate sensitivity) has a trough (peak) around 310 K.

We then asked whether there was any use for this equilibration technique other than for calculating the ECS. This question is identical to asking whether the direct effects of CO₂ are important. In section 7, we calculated the direct effect of CO₂ (right column of Fig. 8) and the effect of warming (middle column of Fig. 8) for all SST values from 285 to 320 K in 1-K increments. Averaging over height and SST, the direct effect of CO₂ on cloud fraction is an order of magnitude smaller than the effect of warming on cloud fraction, and the relative effects of CO₂ on tropospheric temperature and specific humidity are even smaller (see Fig. 9). Nevertheless, the small direct effect on shallow clouds matches what has been reported in previous studies of global climate models, namely the lowering of the cloud heights and the decrease in their areal fraction (see Fig. 10). In contrast to those small effects, the direct effect of CO₂ on net radiative cooling of the troposphere is about one-third as large as the effect of warming (see Fig. 9), implying a substantial direct effect on precipitation. This was confirmed in Fig. 11c, where we see that the direct effect of CO₂ lowers the mean precipitation-rate increase from 3.0% to 2.3% K⁻¹. Consistent with recent theoretical developments about convective available potential energy (CAPE), the direct effect of CO₂ on CAPE was found to be quite negligible (Fig. 11b).