Abstract

The relative contributions of the meridional gradients in insolation and in longwave optical depth (caused by gradients in water vapor) to the equator-to-pole temperature difference, and to Earth’s climate in general, have not been quantified before. As a first step to understanding these contributions, this study investigates simulations with an idealized general circulation model in which the gradients are eliminated individually or jointly, while keeping the global means fixed. The insolation gradient has a larger influence on the model’s climate than the gradient in optical depth, but both make sizeable contributions and the changes are largest when the gradients are reduced simultaneously. Removing either gradient increases global-mean surface temperature due to an increase in the tropospheric lapse rate, while the meridional surface temperature gradients are reduced. “Global warming” experiments with these configurations suggest similar climate sensitivities; however, the warming patterns and feedbacks are quite different. Changes in the meridional energy fluxes lead to polar amplification of the response in all but the setup in which both gradients are removed. The lapse-rate feedback acts to polar amplify the responses in the Earth-like setup, but is uniformly negative in the other setups. Simple models are used to interpret the results, including a prognostic model that can accurately predict regional surface temperatures, given the meridional distributions of insolation and longwave optical depths.

1. Introduction

The meridional gradients in insolation and in longwave optical depth (due to gradients in water vapor) play central roles in Earth’s climate. Together, these gradients are responsible for the equator-to-pole temperature difference that drives the large-scale dynamics of Earth’s atmosphere: the Hadley circulation in the tropics and the baroclinic turbulence that characterizes atmospheric circulation in the midlatitudes (e.g., Held 2000; Vallis 2006). The equator-to-pole temperature difference also plays an important role in driving the circulation of the oceans, both directly through differential heating of the ocean surface, and indirectly by driving the atmospheric surface winds that force oceanic motions. However, the relative contributions of the meridional gradients in insolation and in longwave optical depth to the equator-to-pole temperature difference, and to Earth’s climate in general, are currently unknown, and are the subject of investigation here.

Many previous studies have investigated Earth-like climates with varied equator-to-pole temperature differences. For example, this temperature difference has been varied in idealized general circulation models (GCMs) to develop and test scaling laws for midlatitude dynamics (e.g., Schneider and Walker 2006; O’Gorman and Schneider 2008a; Zurita-Gotor and Vallis 2011) and to investigate the properties of tropical stationary waves (e.g., Arnold et al. 2012; Lutsko 2018). A separate line of research has examined warmer climates than today with reduced equator-to-pole temperature gradients, such as were experienced at past times in Earth’s history and may reappear in extreme future climate change scenarios (e.g., Huber and Sloan 2001; Abbot and Tziperman 2008; Caballero and Huber 2013; Popp et al. 2016).

There has also been much interest recently in simulations with comprehensive climate models with uniform sea surface temperatures, creating global radiative–convective equilibrium (RCE) worlds. These simulations, which have been performed with both prescribed surface temperatures and with slab oceans, and typically without rotation, are taken as global analogs for the tropical atmosphere. Recent studies have focused on convective organization and related phenomena such as the Madden–Julian oscillation in this configuration (e.g., Coppin and Bony 2015; Reed et al. 2015; Pendergrass et al. 2016), the internal variability of these systems (Arnold and Randall 2015; Coppin and Bony 2017), and the response of global RCE simulations to increased CO2 concentrations (Popke et al. 2013). Global RCE simulations with rotation have recently been used to study tropical cyclones (Shi and Bretherton 2014; Merlis et al. 2016), and several studies have investigated the structure of the intertropical convergence zone (ITCZ) in global, rotating RCE simulations (Sumi 1992; Kirtman and Schneider 2000; Chao and Chen 2004).

None of these studies has addressed how the meridional gradients in insolation and in longwave optical depth combine to create the equator-to-pole temperature gradient seen on Earth, however. We address this basic question here by performing simulations with a gray radiation GCM in which the gradients in insolation and in longwave optical depth are eliminated individually or jointly. Gray radiation GCMs have been shown to reproduce the main features of the atmospheric circulation on Earth (Frierson et al. 2006) and are therefore powerful tools for studying changes in the basic climate and in the large-scale circulation. Moreover, the radiation can be precisely controlled in these models. An example of this, which is relevant for our study, is that longwave optical depths are prescribed, making it simple to eliminate this gradient. Conventionally, these models also do not include clouds, further simplifying the analysis. Many topics have been investigated in gray radiation models, including atmospheric eddy length scales, meridional energy transports, eddy kinetic energy, tropical precipitation, the Hadley circulation, and the dynamics of the ITCZ (e.g., Frierson et al. 2006, 2007; O’Gorman and Schneider 2008a,b; Schneider et al. 2010; Levine and Schneider 2015; Bischoff and Schneider 2016).

We consider the effects of eliminating each of the gradients separately and of eliminating both gradients simultaneously, which produces an RCE world (rotation is still included), and focus on the temperature structure of these simulations. By comparing with an Earth-like control simulation, these simulations provide insight into the roles these two gradients play in setting up the climate that is experienced on Earth. The RCE simulation also provides context for interpreting the relevance of global RCE simulations with more comprehensive climate models for the real Earth. In addition, we test how the GCM’s response to global warming-like forcings is affected by eliminating these gradients. Comparing the tropical responses to these forcings with the high-latitude responses helps reveal the mechanisms responsible for the polar amplification of warming in this type of model. Finally, we note that our simulations are also potentially relevant for understanding the atmospheres of exoplanets, with high obliquity for instance, as well as for understanding the atmosphere of a snowball Earth, which would contain very little water vapor and so would have a much weaker longwave optical depth gradient (e.g., Pierrehumbert 2005).

In the following section we provide details on the model we have used and the experiments we have performed. After this the impacts of eliminating the gradients on the global-mean temperature of the model are discussed in section 3 and we then investigate the zonal-mean temperature structure in section 4. In section 5 we describe how the different configurations respond to global warming–like perturbations, before ending with a summary and conclusions (section 6).

2. Model and experiments

The GCM is the idealized model first described by Frierson et al. (2006), which solves the primitive equations on the sphere and is forced by a gray radiation scheme. The GCM is coupled to a slab ocean of depth 1 m, with no representation of ocean dynamics or sea ice, and the model includes the simplified Betts–Miller (SBM) convection scheme of Frierson (2007). A mixed layer depth of 1 m was used so that the model would spin up quickly, while leaving the resulting mean climate the same as for larger mixed layer depths. We show results using a convective relaxation time scale of 2 h and a reference relative humidity = 0.7, but sensitivity tests were conducted in which these two parameters were varied and the results are very similar to what is presented below (not shown). The boundary layer scheme is the one used by O’Gorman and Schneider (2008b). In every experiment the GCM was integrated at T85 truncation (corresponding to a resolution of roughly 1.4° by 1.4° on a Gaussian grid) with 30 vertical levels extending up to 16 hPa, starting from a state with uniform SSTs. The simulations lasted for 1000 days, with averages taken over the final 700 days, and the results of each simulation were symmetrized about the equator, since the Northern and Southern Hemispheres are identical in the model.

The radiative fluxes are calculated using the two-stream approximation assuming hemispheric isotropy, with a single band:

 
formula
 
formula

where U is the upward flux, τ is the optical depth, , and D is the downward flux. This is the gray-gas approximation to the full radiative transfer equations. The boundary condition at the surface is and at the top of the atmosphere is . The radiative heating in the temperature equation is

 
formula

In the control setup the incoming solar radiation takes the form

 
formula

where is the global-mean insolation (including the effect of surface albedo), determines the meridional insolation gradient, and ϕ is latitude. None of the simulations includes a diurnal cycle. The longwave optical depth is specified to approximate the effects of atmospheric water vapor (Frierson et al. 2006). At the surface this takes the form

 
formula

where is the surface value at the equator and is the surface value at the pole. The longwave optical depth is then

 
formula

where is the surface pressure and the linear term is included to reduce stratospheric relaxation times ( is set to 0.1). We note that although the distribution of longwave absorbers is held fixed, water vapor is modeled prognostically by the GCM and so it influences lapse rates independently of the structure of τ.

The insolation is constant in time (i.e., there are no seasons), and we will focus on experiments that do not include atmospheric absorption of solar radiation in order to simplify the analysis. We have repeated some experiments with the model configuration of O’Gorman and Schneider (2008b), which includes the absorption of solar radiation by the atmosphere. This is done by calculating the downward shortwave flux at a given pressure level as , where [the other model parameters are set to the same values as in O’Gorman and Schneider (2008b)]. These experiments produced qualitatively similar results to our main suite of simulations (see section 6).

We consider four configurations of the model. The “control” (Earth-like) simulation used the same parameters as listed in Table 1 of Frierson et al. (2006), with = 938.4 W m−2, = 1.4, = 6, and = 1.5. In the uniform τ experiment the meridional gradient of τ was removed by setting to its average value everywhere (i.e., = 4.5). In the uniform experiment the meridional gradient in incoming solar radiation was removed by setting to zero, so that = 938.4 W m−2 at all latitudes, while keeping the original distribution of . In a fourth experiment the gradients in τ and were both eliminated by setting both quantities to their global-mean values everywhere, which is our RCE configuration. We will refer to the latter three simulations as the “perturbation” experiments. We have also run four “global warming” experiments, in which the optical depth in each configuration is increased everywhere by 30%. Although these experiments all have the same global-mean τ, the net change in optical depth at each latitude is different in the simulations with uniform τ from the ones with meridional gradients in τ.

To estimate the radiative forcing due to these perturbations, we have repeated the perturbation experiments, but kept the SSTs fixed at their time- and zonal-mean values from the control run. The changes in the net TOA imbalance from the control simulation then define the troposphere-adjusted radiative forcings (Hansen et al. 2005), shown in the top panel of Fig. 1, with positive values where the downward TOA flux is increased. We note, however, that all of our runs start from the same initial conditions so these forcings are not actually applied to the GCM. In each of the perturbation experiments the forcing is negative in the tropics and positive at high latitudes, and the largest absolute value of the forcing is found at high latitudes. Setting τ uniform is the smallest perturbation, with a maximum local forcing of about 35 W m−2, while the maximum local forcing in the uniform experiment is about 168 W m−2 and in the RCE simulation it is about 200 W m−2 (hence the forcing induced by eliminating both gradients is slightly smaller than the sum of the forcings due to eliminating each of the gradients individually). The radiative forcings in the global warming experiments are shown in the bottom panel of Fig. 1.

Fig. 1.

(top) Troposphere- and stratosphere-adjusted radiative forcing for the three perturbation experiments (see text for description of how is calculated). The global-mean forcings are 1.87, 2.26, and 2.71 W m−2 for the uniform τ, uniform , and RCE cases, respectively. (bottom) Troposphere- and stratosphere-adjusted radiative forcing for the four global warming experiments. The forcing is positive where the downward flux at TOA is increased. The global-mean forcings are 15.74, 17.97, 17.40, and 18.02 for the control, uniform τ, uniform , and RCE configurations, respectively.

Fig. 1.

(top) Troposphere- and stratosphere-adjusted radiative forcing for the three perturbation experiments (see text for description of how is calculated). The global-mean forcings are 1.87, 2.26, and 2.71 W m−2 for the uniform τ, uniform , and RCE cases, respectively. (bottom) Troposphere- and stratosphere-adjusted radiative forcing for the four global warming experiments. The forcing is positive where the downward flux at TOA is increased. The global-mean forcings are 15.74, 17.97, 17.40, and 18.02 for the control, uniform τ, uniform , and RCE configurations, respectively.

3. Global-mean temperature

We begin by discussing how the perturbations affect the global-mean surface temperature . Before presenting the results of the simulations, we use the simplicity of the gray radiation scheme to develop some intuition for how will respond to the perturbations. We consider three idealizations of the model’s physics:

  1. an all-troposphere atmosphere,

  2. an atmosphere with a troposphere and an isothermal stratosphere, and

  3. an atmosphere with a troposphere and a stratosphere that are in local radiative equilibrium.

We will also assume that the tropospheric lapse rate is only proportional to pressure.

a. All-troposphere atmosphere

For an all-troposphere atmosphere the surface temperature can be related to the outgoing longwave radiation (OLR) and by (see section a of the  appendix)

 
formula

where γ is the exponent relating temperature and pressure (since we assume the lapse rate is only proportional to pressure):

 
formula

The dependence of on OLR, , and γ in Eq. (6) is shown in the left panels of Fig. 2. Note that increases as these parameters are increased, although it becomes less sensitive to when the optical depth is large, which corresponds to the runaway greenhouse regime. The global-mean (and hence the global-mean OLR) and are fixed in the perturbation experiments, which means that in this system can only change because of changes to the lapse rate, with an increase in the lapse rate resulting in a larger surface temperature. This is essentially the lapse-rate feedback, in which an increase in the lapse rate produces a positive feedback on the temperature response to a radiative perturbation.

Fig. 2.

Contour plots of (top) as a function of and γ and (bottom) as a function of OLR and γ, for (left) the all-troposphere atmosphere, (middle) the atmosphere with isothermal stratosphere, and (right) the atmosphere with radiative-equilibrium stratosphere. The black crosses mark the values of and OLR in the top and bottom panels, respectively, and γ in the control simulation. In the top panels OLR = 234.6 W m−2 and in the bottom panels = 4.5. In the middle and right columns the red curves are calculated with = 0.0965, which is equivalent to a tropopause height of 200 hPa and the dashed blue lines are calculated with = 0.167, which is equivalent to a tropopause height of 300 hPa. The cyan lines show the 280.5-K contour, which is the surface temperature in the control simulation (solid lines use = 0.0965 and dashed lines use = 0.167).

Fig. 2.

Contour plots of (top) as a function of and γ and (bottom) as a function of OLR and γ, for (left) the all-troposphere atmosphere, (middle) the atmosphere with isothermal stratosphere, and (right) the atmosphere with radiative-equilibrium stratosphere. The black crosses mark the values of and OLR in the top and bottom panels, respectively, and γ in the control simulation. In the top panels OLR = 234.6 W m−2 and in the bottom panels = 4.5. In the middle and right columns the red curves are calculated with = 0.0965, which is equivalent to a tropopause height of 200 hPa and the dashed blue lines are calculated with = 0.167, which is equivalent to a tropopause height of 300 hPa. The cyan lines show the 280.5-K contour, which is the surface temperature in the control simulation (solid lines use = 0.0965 and dashed lines use = 0.167).

b. Isothermal stratosphere

In an isothermal stratosphere the temperature is everywhere the same as the tropopause temperature , which is equal to , and Eq. (6) is modified to (see section b of the  appendix)

 
formula

where is the optical depth at the tropopause.

The dependence of on OLR, , and γ is shown in the middle columns of Fig. 2. Now has an additional dependence on the tropopause height and the red curves in Fig. 2 use = 0.096, which corresponds to a tropopause height of 200 hPa, while the blue curves use 0.167, which corresponds to a tropopause height of 300 hPa. Both values produce curves that are very similar to the all-troposphere limit, although lowering the tropopause cools for a given , and this cooling is larger for larger values of OLR, , or γ.

c. Stratosphere in radiative equilibrium

Finally, if the stratosphere is in radiative equilibrium the surface temperature is given by (section c of the  appendix)

 
formula

[Robinson and Catling (2012) provided a similar derivation to the one in the  appendix as part of the development of a more general analytic model for the global-mean surface temperature of planetary atmospheres in radiative–convective equilibrium (see their section 2.6)].

The new dependence of on OLR, , and γ is shown in the right columns of Fig. 2. Note that is warmer in this system than the all-troposphere system for small γ and colder for large γ, and lowering the tropopause now causes to increase for a given , although this effect weakens for larger values of OLR, , or γ.

d. Simulation results

In the GCM the global-mean surface temperature increases by 2.4 K when τ is set uniform, by 4.3 K when is set uniform, and by 5.7 K in the RCE case (Table 1). So the warming due to eliminating both gradients simultaneously is smaller than the sum of the warmings due to eliminating the gradients individually.

Table 1.

List of model configurations, with corresponding values of global-mean surface temperature (GMST) and equator-to-pole temperature difference in the control and perturbation experiments, as well as the global-mean surface temperature response to increasing τ everywhere by 30%.

List of model configurations, with corresponding values of global-mean surface temperature (GMST) and equator-to-pole temperature difference in the control and perturbation experiments, as well as the global-mean surface temperature response to increasing τ everywhere by 30%.
List of model configurations, with corresponding values of global-mean surface temperature (GMST) and equator-to-pole temperature difference in the control and perturbation experiments, as well as the global-mean surface temperature response to increasing τ everywhere by 30%.

Our theoretical analysis indicates that these warmings are due to increases in the lapse rate and/or to changes in the height of the tropopause. In Fig. 3  is plotted versus γ (black circles; note that we take to be the temperature at the lowest model level, not the SST temperature) and using the three approximations (lines). The red lines correspond to = 0.096 (i.e., a tropopause near 200 hPa) and the blue lines correspond to = 0.167 (tropopause near 300 hPa). For each simulation we calculate the average value of γ in the troposphere, with the tropopause defined as the height at which the lapse rate is −2 K km−1. The theoretical curves fit the data well, with the isothermal stratosphere curves matching the data slightly less well than the other two approximations. Increases in the global-mean tropospheric lapse rate are thus the main cause of the increases in , with changes in the height of the tropopause playing a secondary role.

Fig. 3.

vs γ for the control simulation and the three perturbation simulations (dots) and the theoretical relationships from Eq. (6) (solid), Eq. (7) (dashed), and Eq. (8) (dotted). The red curves use = 0.096 and the blue curves use = 0.167. The term γ is calculated in the simulations as the average value of γ in the troposphere, with the tropopause defined as the height at which the lapse rate is −2 K km−1. The magenta squares show the results for simulation which include solar absorption by the atmosphere and the magenta line shows the curve for Eq. (6) using the global-mean OLR and values from the experiments with solar absorption included.

Fig. 3.

vs γ for the control simulation and the three perturbation simulations (dots) and the theoretical relationships from Eq. (6) (solid), Eq. (7) (dashed), and Eq. (8) (dotted). The red curves use = 0.096 and the blue curves use = 0.167. The term γ is calculated in the simulations as the average value of γ in the troposphere, with the tropopause defined as the height at which the lapse rate is −2 K km−1. The magenta squares show the results for simulation which include solar absorption by the atmosphere and the magenta line shows the curve for Eq. (6) using the global-mean OLR and values from the experiments with solar absorption included.

The right panel of Fig. 4 demonstrates the extent to which the tropospheric lapse rates increase1 in the perturbation experiments, with the largest increase (up to about −4 K km−1) in the RCE experiment and the smallest increase (~−1 K km−1) in the uniform τ experiment, matching the increases in . The reasons for the increased lapse rates are discussed in section 4c.

Fig. 4.

(left) Global-mean temperature profiles for the control experiment and the three perturbation experiments. The black dashed line shows the emission temperature . (right) Global-mean lapse-rate profiles for the same experiments. The tropopause is defined as the altitude at which the lapse rate is −2 K km−1.

Fig. 4.

(left) Global-mean temperature profiles for the control experiment and the three perturbation experiments. The black dashed line shows the emission temperature . (right) Global-mean lapse-rate profiles for the same experiments. The tropopause is defined as the altitude at which the lapse rate is −2 K km−1.

e. Tropopause height

The global-mean height of the tropopause also varies in the perturbation experiments from its value of around 200 hPa in the control simulation (Fig. 4). In the uniform τ experiment increases slightly and the transition from troposphere to stratosphere is sharper than in the control experiment (left panel of Fig. 4) because the climate is more spatially homogeneous in this setup. The height decreases in the uniform experiment and then descends even more, to about 300 hPa in the RCE experiment.

Thompson et al. (2017) recently proposed a “thermodynamic” constraint for the height of the tropopause. Starting from the thermodynamic energy equation, Thompson et al. define as the cross-isentropic vertical pressure velocity required to balance diabatic heating for a given static stability

 
formula

where is the static stability and Q is the radiative heating defined in Eq. (2). Under the weak temperature gradient approximation, the dominant balance in the tropics is between diabatic heating and vertical motion acting on the static stability, so the tropopause can be defined as the height at which . In the extratropics horizontal temperature advection plays a more important role in balancing diabatic heating; however, horizontal temperature advection can only redistribute thermal energy and so, over a large-enough domain (e.g., in the global mean), it does not balance diabatic heating. In the global mean then, the balance of Eq. (9) can be expected to hold to a good approximation, and constitutes a useful constraint on the global-mean tropopause height.

As discussed in the previous section, the lapse rates increase in the perturbation experiments, and so N decreases as the troposphere becomes more unstable (middle panel of Fig. 5). At the same time, Q increases in the upper troposphere of the perturbation experiments (i.e., the radiative cooling is weaker; left panel of Fig. 5). Since the global-mean optical depth profiles are the same in the four experiments, the changes in Q result from differences in the structures of the temperature profiles: as the lapse rates increase, more of the net column radiative cooling comes from the lower troposphere. Compared with the control experiment, Q increases more in the upper troposphere of the uniform and RCE experiments than N does, and so the tropopause descends (right panel of Fig. 5). In the uniform τ experiment, however, the two effects roughly cancel and so the tropopause height is similar to the control experiment.

Fig. 5.

Global-mean profiles of (left) Q, (middle) N, and (right) for the upper troposphere and lower stratosphere of the control experiment and the three perturbation experiments.

Fig. 5.

Global-mean profiles of (left) Q, (middle) N, and (right) for the upper troposphere and lower stratosphere of the control experiment and the three perturbation experiments.

4. Meridional temperature structure

a. Emission temperature

The bottom panel of Fig. 6 shows the OLR as a function of latitude for the control experiment and the three perturbation experiments. In the uniform τ case the meridional OLR gradient increases, as the tropics emit more OLR and the high latitudes emit less OLR. In the uniform case the OLR gradient reverses, as the high latitudes emit more than the tropics, demonstrating that a planet with an Earth-like distribution of longwave absorbers but a much reduced equator-to-pole temperature gradient can emit more at high latitudes than from the tropics, a situation that might be relevant for planets with high obliquity. In the RCE case the OLR is essentially constant with latitude.

Fig. 6.

(top) Zonal-mean surface temperatures for the control experiment and the three perturbation experiments. (bottom) Zonal-mean outgoing longwave radiation for the same experiments.

Fig. 6.

(top) Zonal-mean surface temperatures for the control experiment and the three perturbation experiments. (bottom) Zonal-mean outgoing longwave radiation for the same experiments.

To understand these differences, consider a two-box model of the atmosphere, consisting of a tropical (30°S–30°N) box and an extratropical (everything else) box. The energy balance in each box is

 
formula
 
formula

where S is the insolation into the tropical box (subscript 2) or into the extratropical box (subscript 1), O is the outgoing radiation from the boxes, and F is the flux of energy between the boxes, defined so that positive F corresponds to an energy transport from the tropics into the extratropics. Both the S and O terms can be decomposed into a global-mean component (overbar) and a departure from that mean [Δ(⋅)]:

 
formula
 
formula

Substituting into the energy balance equations and then subtracting the extratropical equation from the tropical equation, we get

 
formula

In the uniform case is zero but the tropical box still contains more energy than the extratropical box, because of the larger optical depth in the tropics, and heat is exported from the tropics to the extratropics. So is negative in this case, and the extratropics emit more radiation than the tropics. This is analogous to what is seen in the tropical Pacific, where the warm pool region emits less OLR than the cold pool region because the higher relative humidity there makes the atmosphere optically thick in the longwave (Pierrehumbert 1995). In the uniform τ case is unchanged from the control simulation but increases because of the reduced greenhouse effect in the tropics and the increased greenhouse effect in the extratropics. This is balanced by a reduction in the magnitude of F. Finally in the RCE case , F, and are all very close to zero.

b. Surface temperature

1) Diagnostic analysis

The equator-to-pole surface temperature difference is largest in the control experiment (~54 K), decreases to about 32 K in the uniform τ experiment and to 15 K in the uniform experiment, before going to 0 in the experiment with both gradients eliminated (Table 1). The largest temperature changes are at high latitudes, which warm by more than 40 K between the control case and the case with both and τ uniform (top panel of Fig. 6), while the tropics cool by about 10 K. The equator-to-pole surface temperature difference in the control experiment is about 15% larger than the sum of the experiments in which a single gradient is eliminated.

We perform a local feedback analysis to diagnose the reasons for these surface temperature changes. The only radiative feedbacks in the GCM are the Planck feedback and the lapse-rate feedback , so we can write the local surface temperature change as (Feldl and Roe 2013; Henry and Merlis 2017, manuscript submitted to J. Climate)

 
formula

where is the radiative forcing defined in section 2, H is the vertically integrated moist static energy (MSE) flux, and is the zonal-mean temperature difference from the control experiment.

To estimate the Planck feedback we use the GCM’s radiation scheme to calculate the difference in OLR between the equilibrated temperature field in the control simulation and this field with 1 K added at all levels and latitudes; that is, . In other words, we calculate the radiative kernel for the Planck feedback (Soden et al. 2008). Figure 1 shows for the three perturbation experiments, and the lapse-rate feedback is calculated as a residual from Eq. (12), where we have calculated and directly from model output.

To understand how the different terms contribute to the total surface temperature change at each latitude, we have calculated what the surface temperature change would be if various terms were eliminated from Eq. (12). For instance, the magenta dash-dotted lines in Fig. 7 show the temperature changes that would result if (i.e., if only the Planck feedback were present). The dashed cyan lines add the meridional energy transport and the orange dotted lines show the difference between the black lines (the total surface temperature change) and the cyan lines to indicate the effects of the lapse-rate feedback.2 Note that a feedback is defined as positive if the sign of the forcing and of the associated temperature response are the same, and negative if the signs are different. Since the forcing is positive in the extratropics and negative in the tropics, a positive (negative) temperature change in the extratropics constitutes a positive (negative) feedback, and a negative (positive) temperature change in the tropics constitutes a positive (negative) feedback.

Fig. 7.

(top) for the uniform τ experiment estimated using just the Planck feedback (magenta dashed line), using the Planck feedback and the change in MSE transport (cyan dash–dotted line) and the actual (solid black line). The orange dotted line shows the actual minus estimated using just the Planck feedback and the change in the MSE transport (i.e., the black line minus the cyan dash–dotted line), a measure of the contribution to by the lapse-rate feedback. (middle) As at (top), but for the uniform experiment. (bottom) As at (top), but for the RCE experiment.

Fig. 7.

(top) for the uniform τ experiment estimated using just the Planck feedback (magenta dashed line), using the Planck feedback and the change in MSE transport (cyan dash–dotted line) and the actual (solid black line). The orange dotted line shows the actual minus estimated using just the Planck feedback and the change in the MSE transport (i.e., the black line minus the cyan dash–dotted line), a measure of the contribution to by the lapse-rate feedback. (middle) As at (top), but for the uniform experiment. (bottom) As at (top), but for the RCE experiment.

In the uniform τ case the Planck feedback alone underestimates the magnitude of the extratropical response (which is positive) by about half and overestimates the magnitude of the tropical response (which is negative) by a factor of about 2 (magenta line in the top panel of Fig. 7). The change in the MSE flux divergence counteracts this, as less MSE is exported from the tropics to the extratropics, reducing the temperature change at all latitudes (cyan line in the top panel of Fig. 7). Finally the lapse-rate feedback is weak in the tropics but positive and large in the extratropics, contributing to a polar amplification of the response.

In the uniform case the Planck feedback alone would produce a very large temperature response, almost double the actual value of at all latitudes (magenta line in the middle panel of Fig. 7). The MSE transport counteracts this again, substantially reducing the temperature change at all latitudes (cyan line in the middle panel of Fig. 7), while the lapse-rate feedback is positive in the extratropics, increasing the temperatures there, and negative in the tropics (the magnitude of is reduced in the tropics). This is because the lapse rate increases at all latitudes (see Fig. 8), and so at latitudes where the forcing is positive and where the forcing is negative. The net effect of the lapse-rate feedback is a slight polar amplification of the temperature perturbation. The balance of terms is similar in the RCE case, but the changes are larger than in the uniform case and the lapse-rate feedback is responsible for a substantial polar amplification (bottom panel of Fig. 7).

Fig. 8.

Zonal-mean lapse rates in the control experiment and the three perturbation experiments.

Fig. 8.

Zonal-mean lapse rates in the control experiment and the three perturbation experiments.

2) A prognostic model

The previous section diagnosed the causes of the zonal-mean temperature changes in the perturbation experiments. Given the simplicity of the gray radiation model, we would also like a prognostic model that can predict these changes.3 To do this, we again divide the atmosphere into an extratropical box (box 1) and a tropical box (box 2). Using the all-troposphere limit of section 3, and assuming again that temperature is only proportional to pressure, the surface temperature in each box can be calculated by substituting Eq. (A2) into Eq. (10) and rearranging:

 
formula
 
formula

This system now has five unknowns: , , F, , and , and so we will develop closures for the γ terms and for F.

The first assumption we make is that F is proportional to the surface temperature difference . The top-left panel of Fig. 9 plots and F for the eight experiments we have conducted (the control experiment, the three perturbation experiments, and the four global warming experiments) and demonstrates that this is a reasonable assumption, agreeing with previous literature that has modeled meridional atmospheric energy fluxes diffusively (Sellers 1969; Kushner and Held 1998; Barry et al. 2002; Frierson et al. 2007). So we set and estimate a by linear regression, giving a value of 2 W m−2 K−1.

Fig. 9.

(top left) Energy flux from box 2 into box 1 vs the temperature difference between box 1 and box 2 in the eight simulations with the GCM. (top right) Average tropospheric lapse rate in box 1 (crosses) and box 2 (circles) vs the temperature difference between box 1 and box 2, diagnosed from the eight simulations with the GCM. (bottom left) Temperatures in box 1 (crosses) and box 2 (circles) in the GCM vs estimates from the simple model with . (bottom right) Temperatures in box 1 (crosses) and box 2 (circles) in the GCM vs estimates from the simple model with . In the top panels the lines show linear least squares regressions of vs F in the left panel and and in the right panel. In the bottom panels the lines show the 1:1 lines.

Fig. 9.

(top left) Energy flux from box 2 into box 1 vs the temperature difference between box 1 and box 2 in the eight simulations with the GCM. (top right) Average tropospheric lapse rate in box 1 (crosses) and box 2 (circles) vs the temperature difference between box 1 and box 2, diagnosed from the eight simulations with the GCM. (bottom left) Temperatures in box 1 (crosses) and box 2 (circles) in the GCM vs estimates from the simple model with . (bottom right) Temperatures in box 1 (crosses) and box 2 (circles) in the GCM vs estimates from the simple model with . In the top panels the lines show linear least squares regressions of vs F in the left panel and and in the right panel. In the bottom panels the lines show the 1:1 lines.

Next, we assume that and are also both proportional to . This assumption is based on the idea that, for a given global-mean temperature, a larger temperature difference results in smaller lapse rates throughout the troposphere (Fig. 8). The top-right panel of Fig. 9 plots the tropical and extratropical γ in the eight simulations and again suggests that these assumptions are reasonable. So we can write equations for the two sets of γ terms:

 
formula
 
formula

Least squares linear regression gives estimates for , , , and of −0.0013 K−1, 0.244, −0.0015 K−1, and 0.239, respectively. We note that these values depend on the global-mean values of insolation and optical depth, and should not be taken as being universal. Furthermore, this model implies that there is a relationship between the lapse rates and F, with (and similarly for ). Since the α terms are negative, this means that the meridional heat flux increases as the lapse rates decrease.

We can relate and via

 
formula

and so, substituting into Eqs. (13a) and (13b),

 
formula
 
formula

and we now have a closed set of equations for . To estimate , we manipulate Eq. (14b) such that the right-hand side is zero and then find the value of that minimizes the left-hand side (note that ):

 
formula

Equation (15) can then be used to estimate , and and can be estimated from Eq. (16) and then used to estimate F. Equivalently, one can first solve for .

The bottom-left panel of Fig. 9 compares estimates of and from this system with the values diagnosed from the simulations and shows that our simple model produces an excellent fit to the data from the GCM simulations. So we can predict the mean temperatures in each box (as well as the global-mean temperature) given values of , , , and . We have not systematically explored the ability of our model to predict temperatures across other climate states (and we note again that the parameters depend on the global-mean insolation and optical depth), but this is a promising demonstration that it has predictive power. The model is able to predict the warming of global-mean temperature in the perturbation experiments and in the global warming experiments because it includes the Planck feedback and the lapse-rate feedback, which are the only feedbacks present in this GCM [Eq. (12)].

Assuming a single global-mean value of γ (i.e., ) for each of the simulations produces a very similar fit to the data (bottom right panel of Fig. 9). This is equivalent to assuming that is always small compared to γ, and indicates that the differences in the lapse rate between the tropics and the extratropics are of secondary importance for the different surface temperatures and OLR values in these regions. Fixing γ at a single value for all of the simulations does not produce a good fit to the data (not shown), even though variations in γ across the simulations are of a similar magnitude to the differences between and (top-right panel of Fig. 9). This suggests that capturing the trend of γ decreasing as increases is crucial for obtaining a good fit to the GCM data.

Given zonal-mean profiles of insolation and longwave optical depth then, the key factors determining zonal-mean surface temperatures are the energy flux from the tropics to the extratropics and the global-mean lapse rate.

c. Lapse-rate changes

As discussed in section 3d, the global-mean tropospheric lapse rate increases (becomes more negative when measured in kelvins per kilometer) as the gradients are eliminated, with the increase being weakest in the uniform τ case and strongest in the RCE experiment (Fig. 8). In the tropics this increase is easily understood because convection sets tropical temperatures in all of the experiments (Fig. 10) and so the temperature profiles move to colder moist adiabats as the size of the perturbation increases.

Fig. 10.

Zonal-mean vertically integrated convective heating rates (dashed lines) and convergence of meridional energy fluxes (solid lines) for the control experiment and the three perturbation experiments.

Fig. 10.

Zonal-mean vertically integrated convective heating rates (dashed lines) and convergence of meridional energy fluxes (solid lines) for the control experiment and the three perturbation experiments.

The changes in the extratropics are more complex. The largest lapse rates in the control case are in midlatitudes, between about 30° and 60° with weak lapse rates at high latitudes (~−2 K km−1 in the polar midtroposphere). In the uniform τ case the high-latitude lapse rates increase significantly, and the largest lapse rates are now near the poles. Figure 10 shows that in both of these experiments the high latitudes are in “radiative–advective equilibrium” (RAE; Payne et al. 2015; Cronin and Jansen 2016), with horizontal energy fluxes balancing radiative cooling. The high-latitude lapse rates increase further in the uniform case, and these regions also transition to being in radiative–convective equilibrium, and then the lapse rates decrease slightly in the RCE case.

We use a one-level energy balance model to understand this behavior (Abbot and Tziperman 2009; Payne et al. 2015). This consists of a surface level with temperature and an atmospheric level with temperature . In equilibrium the energy balances for the surface and the atmosphere are, respectively,

 
formula
 
formula

where is the solar flux absorbed at the surface, is the meridional advective heat flux by the atmosphere, is the vertical convective heat flux, and is the atmospheric emissivity (and hence absorptivity).

For a system in RAE, in which , this system can be solved for and to give

 
formula
 
formula

The left panel of Fig. 11 plots how the temperature difference varies in RAE as a function of and ε for = 97.6 W m−2, the mean insolation averaged over latitudes poleward of 60° of in the control and uniform τ simulations. The temperature difference increases as the optical depth increases and decreases when the meridional energy flux increases.

Fig. 11.

(left) The temperature difference for a system in RAE and with a mean insolation of 97.6 W m−2 as a function of the emissivity and the meridional energy flux, calculated using Eqs. (19a) and (19b). The round markers plot the emissivity and vertically integrated meridional energy flux, averaged over latitudes poleward of 60°, for the control setup, with the black marker denoting the base case and the red marker the “global warming” case. The crosses denote the same items but for the uniform τ experiments. The red contour in the bottom-right corner shows the critical temperature difference above which the system becomes unstable to moist convection. (right) The critical temperature difference as a function of emissivity, calculated using a mean insolation of 234.6 W m−2. The black markers plots the temperature differences that would arise in the uniform (round marker) and RCE (cross) base cases if their high latitudes were in RAE, assuming = 100 W m−2. The blue markers show the values for the control and uniform τ cases (i.e., using a mean insolation of 97.6 W m−2).

Fig. 11.

(left) The temperature difference for a system in RAE and with a mean insolation of 97.6 W m−2 as a function of the emissivity and the meridional energy flux, calculated using Eqs. (19a) and (19b). The round markers plot the emissivity and vertically integrated meridional energy flux, averaged over latitudes poleward of 60°, for the control setup, with the black marker denoting the base case and the red marker the “global warming” case. The crosses denote the same items but for the uniform τ experiments. The red contour in the bottom-right corner shows the critical temperature difference above which the system becomes unstable to moist convection. (right) The critical temperature difference as a function of emissivity, calculated using a mean insolation of 234.6 W m−2. The black markers plots the temperature differences that would arise in the uniform (round marker) and RCE (cross) base cases if their high latitudes were in RAE, assuming = 100 W m−2. The blue markers show the values for the control and uniform τ cases (i.e., using a mean insolation of 97.6 W m−2).

The round markers on the left panel indicate the values of ε and from the control and uniform experiments, where both these quantities are also averaged over latitudes poleward of 60° and we take to be the vertically integrated meridional heat flux. This suggests that will increase from about 20 to about 26 K, whereas in the GCM increases from 16 to 28 K (we take 600 hPa as the representative atmospheric level). While the temperature difference should not be taken as a direct measure of the lapse rate, as the representative atmospheric level can vary, this demonstrates that the increase in the high-latitude lapse rates in the uniform experiment is primarily caused by the increased optical depths there, with the slight reduction in playing a secondary role. Figure 2 of Cronin and Jansen (2016) also demonstrates how lapse rates increase in RAE atmospheres as the optical depth increases.

That these cases are in RAE means that they are stable to convection and hence we would expect the lapse rates to be higher in the uniform and RCE cases, because the high latitudes of these simulations are convecting (Fig. 10). But the high-latitude insolation also increases in these experiments, so we cannot infer this directly and instead must compare how the convecting lapse rates in these experiments compare with the RAE lapse rates in the experiments with weaker high-latitude insolation.

The right panel of Fig. 11 plots the critical temperature difference above which the model is unstable to convection for S0 = 234.6 W m−2 (assuming ). To calculate this curve we assume that the moist static energy at the surface is equal to the moist static energy at the representative atmospheric level:

 
formula

where is the latent heat of evaporation, is the surface specific humidity, is the saturation specific humidity at the atmospheric level, and is the height of the atmospheric level. Following Abbot and Tziperman (2009), we use a surface relative humidity of 80% and calculate using a scale height of 8 km. The terms , , and can be solved for by combining this equation with the one-layer model, after specifying . (The round markers in the right panel show the RAE temperature differences for the uniform and RCE cases, assuming Fa = 100 W m−2).

Comparing with the left panel confirms that the critical temperature difference in the uniform and RCE cases is larger than the (RAE) temperature differences in the control and uniform τ cases. Hence the lapse rates increase in the uniform and RCE cases compared to the control and uniform τ experiments because the increased insolation makes the high latitudes unstable to moist convection, as evidenced by the two round markers in the right panel of Fig. 11 lying above the curve of critical temperatures. This causes a transition from RAE to RCE, and the critical lapse rates for this insolation value (234.6 W m−2) are larger than the RAE lapse rates in the presence of the weaker insolation (97.6 W m−2). The right panel of Fig. 11 also shows that the critical temperature difference decreases as the optical depth increases [see also Fig. 1 of Abbot and Tziperman (2009)], explaining why the high-latitude lapse rates are larger in the uniform case than in the RCE case.

5. Temperature response to forcings

As mentioned in section 2, we also performed experiments with each of the four configurations in which τ was increased by 30%, which mimics the effects of increased CO2 concentrations in this GCM. We note again that although the global-mean change in τ is the same in all of the configurations, in the control setup and the uniform τ setup the forcing is larger in the tropics than in the extratropics, whereas in the uniform and RCE setups the forcings are homogeneous in latitude (bottom panel of Fig. 1).

Interestingly, the global-mean temperature change is insensitive to the base state, as in all four cases increases by between 4 and 4.2 K (Table 1). There are substantial differences in the latitudinal structure of this warming (top-left panel of Fig. 12), however, and the similar sensitivities in the four configurations may be a coincidence. Most importantly, in the control case there is a polar amplification of about 6 K, in the uniform τ case the polar amplification is about 3.7 K, in the uniform case the polar amplification is about 1 K, and there is no amplification in the RCE case.

Fig. 12.

(top left) Zonal-mean surface temperature responses in the four global warming experiments. (top right) Zonal-mean temperature changes due only to the radiative forcing and Planck feedback. (bottom left) Zonal-mean temperature changes due to the radiative forcing, the change in moist static energy transport, and the Planck feedback. (bottom right) The differences between the zonal-mean surface temperature responses and the changes due to the radiative forcing, the change in moist static energy transport, and the Planck feedback.

Fig. 12.

(top left) Zonal-mean surface temperature responses in the four global warming experiments. (top right) Zonal-mean temperature changes due only to the radiative forcing and Planck feedback. (bottom left) Zonal-mean temperature changes due to the radiative forcing, the change in moist static energy transport, and the Planck feedback. (bottom right) The differences between the zonal-mean surface temperature responses and the changes due to the radiative forcing, the change in moist static energy transport, and the Planck feedback.

The other panels in Fig. 12 explore the reasons for these responses, using the diagnostic framework of Eq. (12). The top-right panel shows that the forcing and Planck feedback alone result in a tropically amplified warming in the control setup (black curve). The forcing decreases away from the equator (bottom panel of Fig. 1), as does the magnitude of the Planck feedback parameter (not shown). Close to the tropics these changes cancel out so that is relatively uniform, but at higher latitudes the forcing decreases faster than the Planck feedback and decreases [see section 2 of Henry and Merlis (2017, manuscript submitted to J. Climate) for a discussion of these patterns]. This is countered by the changes in the horizontal energy flux (bottom left panel) and, to a lesser extent, by the lapse-rate feedback, which is negative in the tropics and positive at higher latitudes (bottom right panel).

The reason for the positive lapse-rate feedback at high latitudes can be seen from the left panel of Fig. 11: increasing the optical depth increases the lapse rate in RAE, although this is countered somewhat by the increased meridional heat flux. We refer the reader to Payne et al. (2015), Cronin and Jansen (2016), and Henry and Merlis (2017, manuscript submitted to J. Climate) for more in-depth investigations of why the lapse-rate feedback is positive at high latitudes and negative at low latitudes for Earth-like gray radiation models. Pithan and Mauristen (2014) also found a strong polar amplification of warming due to the lapse-rate feedback in an analysis of CMIP5 models.

The terms are similar in the uniform τ setup except that each term contributes less polar amplification. For instance, while the forcing is less tropically amplified than in the control case, also decreases less rapidly and so is roughly the same in the tropics as at the poles, with minima at midlatitudes. The bulk of the polar amplification still comes from the change in MSE transport, although this is less polar amplification than in the control case, while the lapse-rate feedback is similar to in the control case but is weaker at high latitudes. The left panel of Fig. 11 shows that changes in the high-latitude lapse rate will be small in this configuration because the emissivity is already so high that increasing it further does not affect the lapse rate substantially.

In the uniform setup has little meridional structure and the polar amplification comes entirely from the changes in the meridional heat transport. The lapse-rate feedback is negative and roughly uniform in latitude, as all latitudes are in RCE. That the polar amplification in the control setup, the uniform τ setup, and the uniform setup are all mainly due to meridional heat transport agrees with the results from our prognostic model, suggesting that meridional variations of the lapse-rate feedback are of secondary importance for capturing the polar amplification of warming in the GCM [Cai (2005) and Cai (2006) also found that meridional energy fluxes are an important driver of polar amplification]. Finally, in the RCE case all the changes are homogeneous in latitude.

The strongly negative lapse-rate feedbacks in the uniform and RCE setups are responsible for the fact that the global-mean surface temperature changes are roughly the same in all four configurations despite the global-mean forcings being substantially larger in the uniform and RCE setups (bottom panel of Fig. 1).

6. Summary and conclusions

In this study we have investigated the response of a moist, idealized GCM to eliminating the meridional gradients in insolation and in longwave optical depth. We have performed experiments in which these gradients were eliminated separately (the uniform τ and uniform experiments) and an experiment in which both were eliminated at the same time (the RCE experiment) and have used a number of simple models to interpret the differences in the climates of the model configurations.

Our first main result is that eliminating these gradients causes the global-mean surface temperature of the model to increase. A one-dimensional system consisting of an all-troposphere atmosphere with temperature proportional to pressure captures the temperature changes across these simulations, demonstrating that the increased lapse rates in the perturbation experiments are primarily responsible for the increased surface temperatures. The lapse rates increase at all latitudes in the perturbation experiments, but for a variety of reasons. In the tropics, the lapse rates increase because the tropics cool and so tropospheric temperatures move to colder moist adiabats. In the uniform τ experiment the extratropical lapse rates increase because of the increased high-latitude optical depths, whereas in the uniform and RCE experiments the extratropical lapse rates increase because of the increased high-latitude insolation, which destabilizes the high latitudes and causes a transition from RAE to RCE there [see Abbot and Tziperman (2008) for a discussion of high-latitude convection and its implications for equable climates].

In the global mean, the tropopause descends in the uniform and RCE experiments but is slightly higher in the uniform τ experiment. We have used the thermodynamic constraint of Thompson et al. (2017) to explain these variations. In the uniform and RCE experiments the tropopause descends because the radiative heating profile becomes more bottom-heavy and goes to zero lower in the atmosphere compared to the control simulation. The radiative heating profile also becomes more bottom-heavy in the uniform τ experiment, but this is overcompensated for by the reduced tropospheric stability and so the tropopause rises slightly.

Moving on to regional changes, the OLR increases in the tropics and decreases in the extratropics in the uniform τ experiment compared with the control. In the uniform experiment the OLR is largest at high latitudes, which is similar to the present-day Earth’s tropics, where regions of colder SSTs emit more radiation to space because of the optically thinner overlying atmosphere. The OLR is constant with latitude in the RCE experiment.

A linear feedback analysis shows that the Planck feedback causes a strong polar amplification of the response in all of the perturbation experiments, when compared to the control. This is damped somewhat by a reduction in the meridional moist static energy flux, while the lapse-rate feedbacks are large and positive in the extratropics and weakly positive in the tropics, contributing to the polar amplification of the responses. Complementing this diagnostic analysis, we have also presented a prognostic model of zonal surface temperatures in this GCM, which accurately predicts the tropical and extratropical temperatures across the eight simulations (the control simulation, the three perturbation experiments, and the four global warming experiments). The success of this model demonstrates that, given zonal-mean profiles of insolation and longwave optical depth, the energy flux from the tropics to the extratropics and the global-mean lapse rate are the main factors controlling zonal-mean surface temperatures. Similar box models for the temperature structures of tidally locked, rocky planets have been developed by Yang and Abbot (2014) and Koll and Abbot (2016) and provide some suggestions for how clouds could be added to our model.

To summarize these results, relative to the RCE case, adding the meridional gradient in longwave optical depth (the uniform case) produces a climate that is analogous to what is seen in the tropical Pacific, with the warmer tropics playing the role of the west Pacific and emitting less radiation than the cooler extratropics (the east Pacific) (Pierrehumbert 1995). Adding the insolation gradient without adding the optical depth gradient produces a climate that is similar to the control climate, but has weaker horizontal energy transports and more convection outside the tropics. As might be expected then, the insolation gradient has a larger influence on the model’s climate than the gradient in optical depth, although both make sizeable contributions to the equator-to-pole temperature gradient and to the GCM’s climate in general.

The global-mean surface temperature response to increasing the optical depth by 30% is the same in all four configurations; however, the effective forcing is significantly larger in the uniform and RCE configurations than in the control and uniform τ configurations, and this is countered by the stronger lapse-rate feedback in the former experiments. In the control setup the forcing and Planck feedback alone would lead to a tropical amplification of the warming. In all but the RCE case, the changes in MSE flux act to polar amplify the warming, and in the control and uniform τ configurations the lapse-rate feedback also contributes to the polar amplification. Cai (2005) and Cai (2006) also emphasized the key role meridional energy fluxes play in the polar amplification of warming. In the other experiments the lapse-rate feedback is negative at all latitudes, with little meridional structure.

Our study is an important first step for understanding the roles the meridional gradients of insolation and longwave optical depth play in setting up Earth’s climate, and future studies with comprehensive models will be able to build off the insights obtained here. The experiments with solar absorption by the atmosphere included gave qualitatively similar results to our main suite of experiments, although the quantitative agreement with our theoretical models is not as good (the magenta squares and line in Fig. 3 are an example). It is also worth noting that the coefficient of shortwave absorption is fixed in latitude and height in these experiments (see section 2) and so it does not include the effects of latitudinal variations in atmospheric water vapor (or ozone) concentrations.

This leaves clouds and the water vapor feedback as the main atmospheric processes still to be accounted for, as well as the dynamics of ice sheets, the ocean, and land surface processes [Winton (2003) explored the climate response to eliminating meridional ocean heat transport in two coupled climate models]. Our model also did not include a seasonal cycle, which would affect the mean climate states of our different configurations. For instance, winter inversions could develop at the high latitudes of planets whose insolation is globally uniform in the annual mean, insulating the surface climate there from the overlying atmosphere and inhibiting high-latitude convection. In a model capable of simulating low-level stratocumulus clouds this would likely cause a substantial cooling of high-latitude surface temperatures (Abbot 2014).

Comparing simulations with more comprehensive models to our idealized GCM results will allow the effects of these different factors to be isolated, while the simple conceptual models we have developed and used here provide a useful framework for developing a complete understanding of how Earth’s climate would be affected by eliminating the gradients in insolation and/or in longwave optical depth.

Acknowledgments

We thank Daniel Koll, Levi Silvers, and Tim Cronin for helpful discussions and for comments on earlier versions of this manuscript. We also thank Dorian Abbot, Nadir Jeevanjee, and an anonymous reviewer for thorough readings of the manuscript and productive comments. Nicholas Lutsko was partly supported by NSF Grant AGS-1623218, “Collaborative Research: Using a Hierarchy of Models to Constrain the Temperature Dependence of Climate Sensitivity.”

APPENDIX

Derivations of Eqs. (6)–(8)

a. All-troposphere atmosphere

Equation (1a) can be solved at to give

 
formula

If the lapse rate is only proportional to pressure, then

 
formula

where for a dry atmosphere and for a moist atmosphere. Since ,

 
formula

Substituting into Eq. (A1) gives

 
formula

which can be rearranged for :

 
formula

Note that if then the integral is the function with argument

 
formula

[see section 4.3.2 of Pierrehumbert (2010)].

b. Isothermal stratosphere

For an atmosphere with an isothermal stratosphere above the troposphere Eq. (A1) can be written as

 
formula

where is the optical depth at the tropopause. Using the results of the previous subsection,

 
formula

The temperature of the stratosphere is everywhere the same as the tropopause temperature , which is equal to and so the second integral can be evaluated to give

 
formula

and the surface temperature is

 
formula

If then we return to Eq. (6) of the all-troposphere limit.

c. Stratosphere in radiative equilibrium

Radiative equilibrium demands that the net divergence of the radiative flux be zero everywhere such that the net heating by the radiation vanishes

 
formula

and hence is constant. Using the boundary conditions at τ = 0, and so, adding Eqs. (1a) and (1b),

 
formula

and then integrating yields

 
formula

Substituting from Eq. (A7) results in an equation for the stratospheric temperature in radiative equilibrium

 
formula

Substituting into Eq. (A4) and rearranging then gives

 
formula

and so

 
formula

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Footnotes

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

1

We will refer to lapse rates as “increasing” when they become more negative when using height coordinates.

2

The lapse-rate feedback changes sign at some latitudes, so is not well defined.

3

We will refer to this model as “prognostic” despite tuning it with the perturbation experiments in order to differentiate it from the diagnostic framework of the previous section. The model presented in this section could be used to make predictions for other perturbations, such as damping the insolation gradient rather than reducing it. A tuning based solely on the control experiment could also be attempted.