1. Introduction
As a heat engine, Earth’s atmosphere transports energy from the tropics (warm reservoir) to the pole (cool reservoir) with the reservoirs being maintained by the spatially uneven radiative heating. The equilibration between the energy transport and the differential energy source eventually determines the temperature distribution in the atmosphere. This simple portrayal of the climate system has often been formulated by an energy balance model (EBM) under the constraint of the energy conservation for each individual column of Earth’s atmosphere, with the heat flux between adjacent columns parameterized in a simple diffusive form (North and Stevens 2006, and references therein). Since its incipience more than half a century ago (Budyko 1968; Sellers 1969), EBMs have accomplished plenty in the conceptual understanding of the energetics and temperature distribution in Earth’s climate and their responses to climate perturbations. To name a few, 1) EBMs can capture the meridional distribution of the poleward atmospheric energy transport and the associated surface temperature (e.g., Sellers 1969; North 1975), and 2) EBMs help reveal the ice-albedo feedbacks and the associated ice cap instability (Budyko 1968; Sellers 1969; Held and Suarez 1974; North 1975; Lindzen and Farrell 1977). Furthermore, 3) a statistical EBM has also been constructed (Leung and North 1991) to capture the internal variability of the zonally symmetric surface temperature; 4) more recently, EBM has been also employed to parse the atmospheric heat transport and the polar-amplified surface warming in response to greenhouse gas forcing (Hwang and Frierson 2010; Merlis and Henry 2018; Armour et al. 2019; Russotto and Biasutti 2020). 5) An EBM has even been used to identify the key factors that control the different storm track seasonalities between the Northern and Southern Hemispheres (Barpanda and Shaw 2020); 6) it has also been used to provide an energetic framework to understand the shift of the ITCZ and monsoon rainbands in response to remote energy perturbations (e.g., Kang et al. 2008, 2009; Bischoff and Schneider 2014; Schneider et al. 2014) or internal modes of climate variability (Boos and Korty 2016; Lu et al. 2021). Finally, 7) EBMs have proved to be illuminative in deciphering the remote impacts of climate feedbacks on a regional scale (e.g., Roe et al. 2015).
It is not obvious why EBMs work so well, given their crude representation of the effect of the atmospheric circulation. For example, the overturning Hadley cell is an important circulation structure for exporting energy out of the deep tropics and it is difficult to justify a diffusive representation of this effect. Eddy diffusion is more fitting to describe the energy transport in the extratropics as the midlatitude storm track is dominated by eddying motions and the typical scale of the energy-containing eddies is smaller than the width of the baroclinic zone, somewhat satisfying the scale separation requirement of the mixing-length theory (e.g., Held 1999). Several studies (Barry et al. 2002; Frierson et al. 2007; Liu et al. 2017) tested the scaling theories in the literature against the midlatitude diffusivity directly diagnosed from atmospheric general circulation model (AGCM) simulations. While they all agreed that the Rhines scale is the energy-containing scale for diffusivity, they disagreed on how exactly the diffusivity should be formulated. Given the distinct mechanisms for the poleward energy transport between the tropics and the midlatitude, one can hardly justify a uniform diffusivity for the EBM, which is often the case in most of the studies. As diagnosed from both the observation (e.g., Kushner and Held 1998) and numerical simulations with models of different complexities (Mbengue and Schneider 2018; Mooring and Shaw 2020; Peterson and Boos 2020), and as will be shown later in this study, the diffusivity can be a strong function of space and time, even in an idealized aquaplanet climate model. Since the diffusivity can be thought of as encapsulating all the atmospheric motions in a single parameter, the circulation response to climate change must manifest in the change of the diffusivity. Assuming perfect compensation between the dry static energy and latent energy transport response under a uniform +4-K sea surface temperature (SST) forcing, Shaw and Voigt (2016) inferred a reduction of the midlatitude MSE diffusivity under the constraint that the change of the column moisture follows the Clausius–Clapeyron (CC) relation. Further analysis by Mooring and Shaw (2020) of the moist static energy (MSE) transport and diffusivity response to systematically cooling and warming the global mean SST in both an idealized and a realistic AGCMs indicates a poleward shift of the MSE diffusivity for both configurations and an increase of diffusivity maximum (located at the poleward flank of the westerly jet) for the aquaplanet configuration. However, the dynamical underpinning of the changes of diffusivity as climate warms remains largely elusive.
One central task of this study is to justify the diffusive treatment of the poleward MSE transport in the EBM, especially for the midlatitude atmosphere. To this end, we first show that the diffusive form of the MSE transport can be derived from the original column-integrated energy balance equation. This theoretical derivation results in a tensor-form diffusivity for MSE, allowing us to fit for the spatially dependent diffusivity in AGCM as an inverse problem by developing a simple optimization algorithm. The preliminary success of the fitting serves as a proof of concept for retrieving the MSE diffusivity tensor on the horizontal plane for the realistic atmosphere with more sophisticated machine learning techniques in the future. We then test several scaling theories for the diffusivity in the literature against the seasonal variation of the midlatitude diffusivity directly diagnosed from an aquaplanet AGCM simulation. A fitting theory should not only lend some support to the diffusive treatment of the MSE transport, but also help reveal the underlying dynamical underpinning for the diffusivity.
The success of one of these scaling theories further prompts us to parameterize the diffusivity interactively as a function of temperature and its gradient in the EBM. With the parameterized diffusivity as a dynamical constraint and the CC relation as a thermodynamic one, the EBM is subject to dual constraints in its response to external perturbations. The resultant EBM serves as a minimum modeling framework to investigate the issue of the polar-amplified temperature response in the absence of the polar-amplifying feedbacks (e.g., sea ice and clouds), with the result pointing to the nonlinearity in the CC relation as one of the root causes for the polar amplification (PA) (see also Merlis and Henry 2018).
The paper is structured as follows. Section 2 introduces the AGCM for simulating an aquaplanet climate and its response to varying CO2 forcing, as well as an EBM with an interactive diffusivity for the investigation of polar amplification in the absence of polar-amplifying feedbacks. Section 3 begins with offering a structural justification for the diffusive form of MSE transport, followed by diagnosing the diffusivity from the AGCM simulations and testing six different scaling arguments for the diffusivity. For readers interested in the background literature on the eddy mixing scale and eddy diffusivity, a synergistic review on the existing theories for the eddy diffusivity, including some developments by the authors, is provided in appendix A to bring the readers up to speed on the subject. Section 4 offers a heuristic explanation for the quantitative difference between the diffusivity of moisture and that of temperature estimated from the aquaplanet AGCM simulations. The EBM with an interactive diffusivity as inspired by the promising scaling theory serves as a useful conceptual tool for testing the effect of the circulation change in the polar amplification of the surface temperature response, a task will be carried out in section 5. Finally, this study concludes with a summary and a discussion on the possible directions of further development of the diffusive EBM. A nomenclature for all the diffusivities discussed under different contexts in this study is provided in appendix B for reference.
2. Modeling tools
a. AQUA-AM2.1
The workhorse simulations in this study are performed with the AM2.1 atmospheric model of Geophysical Fluid Dynamics Laboratory (GFDL) coupled to a motionless slab ocean with an equivalent heat capacity of 30 m. It will be referred to as AQUA-AM2.1. The atmospheric component uses a cubed-sphere grid at C48 resolution (approximately 2° resolution). The vertical grid is a 24-level hybrid coordinate grid, with sigma surfaces near the ground continuously transforming to pressure surfaces above 250 hPa and the lowest level at ∼30 m above the surface. The hydrostatic, finite-difference dynamical core is developed from the models described in Mesinger et al. (1988) and Wyman (1996). The shortwave radiation scheme follows Freidenreich and Ramaswamy (1999) but with modifications to improve the computational efficiency. The longwave radiation algorithm follows the simplified exchange approximation and is also developed and tested with the benchmark computations (Schwarzkopf and Ramaswamy 1999). Moist convection is represented by a modified relaxed Arakawa–Schubert (RAS) formulation of Moorthi and Suarez (1992). Cloud fraction is treated as a prognostic variable of the model following the parameterization of Tiedtke (1993). Surface fluxes are computed according to the Monin–Obukhov similarity theory. More details about the AM2.1 physics can be found in Anderson et al. (2004).
Both the seasonal and diurnal cycles of the solar radiation are considered, but the eccentricity is set to be 0 for simplicity. No sea ice is formed in the model, even though the slab temperature is permitted to drop below freezing. Following Bordoni and Schneider (2008), an interhemispherically symmetric, latitude-dependent q-flux representing the divergence of the meridional ocean energy transport is prescribed to the slab so that an Earth-like reference tropical circulation can be produced by the AQUA-AM2.1. Note that, due to the choice of the 30-m slab ocean depth, the phase of the seasonal evolution of precipitation and the midlatitude storm activity is shifted by approximately 2 months compared to that in the realistic climate models [e.g., supplemental Fig. 2 in Song et al. (2020)]. See Barpanda and Shaw (2020) for a detailed discussion of the dependence of the storm track seasonality on the slab mixed layer depth. To investigate the response of the diffusivity to climate warming, we vary systematically the concentration of the CO2 by a factor of 1/4, 1/2, 2, and 4, with respect to the reference concentration level, which is set to be 348 ppm. All simulations are run for 30 years, and the last 20 years are used for analysis.
b. EBM and interactive diffusivity
Latitudinal distribution of (a) Ts, (b) cosϕKZ, and (c) AHT simulated by EBM (from green to orange) under control forcing and systematic increases of a uniform solar radiative perturbation by 10 and 20 W m−2. The dashed red lines are the corresponding result from the control AQUA-AM2.1 simulation.
Citation: Journal of Climate 35, 21; 10.1175/JCLI-D-21-0721.1
Parameters for interactive MSE diffusivity in EBM.
The results of the EBM simulations from the control and solar radiation perturbation experiments are displayed in Fig. 1 for reference. The EBM control run characteristically captures both the mean and the equator-to-pole difference of the surface temperature and diffusivity in the AQUA-AM2.1 control run (compare the green lines to the dashed red lines in Figs. 1a,c), although it is not our intent to replicate the results of the AQUA-AM2.1. The EBM tends to overestimate the midlatitude diffusivity and underestimate the midlatitude meridional temperature gradient in the AQUA-AM2.1. However, these two features compensate one another, resulting in an energy transport profile consistent with that of AQUA-AM2.1. Note also that the meridional profile of the diffusivity from the AQUA-AM2.1 control bears considerable resemblance to that diagnosed from the aquaplanet configuration of the Community Earth System Model version 2 (CESM2) (see Fig. 6a in Peterson and Boos 2020).
3. Diffusivity of MSE and its seasonality
Given the complex three-dimensional flow in the real atmosphere, it is not obvious why the poleward MSE transport can be formulated as a downgradient diffusion. In this section, we first offer a rationale for the diffusive depiction of the poleward MSE transport, followed by an optimization approach to fit the monthly MSE diffusivity in a set of AQUA-AM2.1 simulations described in section 2. We then discuss the key spatial and temporal characteristics of the MSE diffusivity throughout the annual cycle, linking them to the leading seasonal modes of the circulation features. Last, we will show that the seasonality of MSE diffusivity in the midlatitude storm track can be explained by the prevailing scaling arguments reviewed in appendix A.
a. Rationalization for diffusive MSE transport
b. Behaviors of MSE diffusivity in AQUA-AM2.1
There is no theoretical basis for quantitative specification of these diffusivity coefficients in Eqs. (14) and (15) (although some scaling arguments exist for midlatitude eddy diffusivity; see appendix A), but their values can nonetheless be estimated by solving an inverse optimization problem. An algorithm is developed here to find the values of KZ that minimize the residual of the energy balance equation, Eq. (15), given the information of the column-integrated MSE (
From the conventional wisdom that Hadley circulation is efficient in transporting energy poleward, one might infer that KZ is greater in the tropics than in the midlatitude. This speculation is not borne out in the actual estimation of KZ [see the black line in Fig. 2a for the annual mean cosϕKZ and Fig. 6a for the diffusivity estimated from an aquaplanet CESM2 in Peterson and Boos (2020)], which shows a minimum at the equator near the intertropical convergence zone (ITCZ), indicating that the upward portion of the Hadley circulation is not an efficient mechanism for exporting energy. This somewhat corroborates the notion by Neelin and Held (1987) that the large time-mean precipitation in the ITCZ tends to be associated with a small gross moist stability, an indicator for low efficiency in exporting MSE per unit overturning motion. The subtropical maximum of cosϕKZ is located at a subtropical latitude (35°), instead of the latitude of the maximum EKE (at 40°) or the westerly jet (at 45°). The magnitude (1.85 × 106 m2 s−1) of the subtropical peak also matches the earlier estimates for the aquaplanet models well (Frierson et al. 2007; Shaw and Voigt 2016; Mooring and Shaw 2020). In the vertically averaged sense, this is consistent with the understanding that the meridional mixing/diffusion is more efficient on the flanks of the jet where wave breaking takes place, rather than at the core of the jet where the wave motion is more reversible (e.g., Haynes and Shuckburgh 2000; Chen and Plumb 2014). Without the cosϕ factor, the meridional structure of KZ has two peaks, one in the subtropics and another at the poleward flank of the jet. This agrees qualitatively with the MSE diffusivity arrived at in Shaw and Voigt (2016; see Fig. 2a in their supporting information) and that in Mbengue and Schneider (2018, their Fig. 1a).
The first four EOF patterns of the seasonally varying diffusivity cosϕKZ (a) and their corresponding normalized principal components (b). The bold black line displayed in (a) is the annual-mean cosϕKZ, whose values are indicated on the right y axis.
Citation: Journal of Climate 35, 21; 10.1175/JCLI-D-21-0721.1
Throughout the annual cycle, the diffusivity undergoes a substantial variation, reaching the highest value in the wintertime. This can be seen from the leading principal component (blue line in Fig. 2a) as the corresponding leading EOF projects most onto the peak intensity (blue line in Fig. 2b). The first four EOFs of the diffusivity account for >97% of the total variance of monthly variability of cosϕKZ, evincing two annual and two semiannual modes. The first EOF characterizes a meridional dipole, indicating a systematic weakening and poleward shift of the diffusivity peak as the season advances from the coldest to the warmest in each hemisphere. The third EOF captures another annual mode featuring an interhemispheric dipole within the tropics, which reflects the seasonal advance of the ITCZ and Hadley circulation. The second EOF is a semiannual one (see the orange lines in Figs. 2a,b), consistent with the fact that the maximum solar irradiance crosses the equator twice a year at the equinoxes, driving a larger than normal diffusivity in the tropics. The second semiannual mode (the fourth EOF) is weak, accounting for <5% of the variance. It appears to represent a semiannual cycle in the strength of the midlatitude baroclinicity and storm track, but the underlying physical basis remains obscure. It is worth noting that the principal component analysis of the troposphere-integrated EKE from the meridional component of the eddies shows similar temporal characteristics as the principal components of KZ throughout the annual cycle. This echoes the proportionality between the EKE and the proxy diffusivity found in the experiments with an aquaplanet model and a full AGCM with prescribed SST forcings (Mooring and Shaw 2020).
Figure 3 shows the sensitivity of the fitted annual mean MSE diffusivity to varying CO2 forcing for quarter, half, doubling, and quadrupling levels of the control. It exhibits a systematic reduction with increasing CO2 forcing, consistent with the finding of Shaw and Voigt (2016). The tropical reduction of diffusivity may be related to the weakening of tropical circulation under CO2-induced warming (e.g., Lu et al. 2007; Vecchi and Soden 2007); the subtropical reduction should be related also to the poleward shift of the storm track and eddy activity (e.g., Yin 2005; Chen et al. 2008; Mooring and Shaw 2020). As a result of the shift, the diffusivity in the polar regions is intensified, though only marginally. This overall weakening of MSE diffusivity with warming is in keeping with the notion that the work output in the atmosphere as a heat engine is constrained by the power necessary to maintain the atmospheric hydrological cycle and the demand of the latter increases more rapidly with warming than does the total heat supply to the heat engine such that the atmosphere’s ability to generate work is compromised (Laliberté et al. 2015). This sensitivity is what we intend to capture with an interactive MSE diffusivity developed for EBMs in section 5.
Sensitivity of the MSE diffusivity cosϕKZ to the concentration levels of CO2 in AQUA-AM2.1.
Citation: Journal of Climate 35, 21; 10.1175/JCLI-D-21-0721.1
c. Test of the scaling for the midlatitude diffusivity
We here use the AGCM-simulated seasonal cycle to test four interrelated scalings for the midlatitude eddy diffusivity, referred to as Rhines (Rhines 1975; Frierson et al. 2007), Green70 (Green 1970), HL96 (Held and Larichev 1996), and Barry02 (Barry 2002), plus the variants of HL96 and Barry02. Table 2 summarizes the formulas of these scalings and their underlying assumptions. Readers who are interested in the theoretical rationale behind these scalings and their historical lineage are referred to appendix A.
Summary of scaling laws for eddy diffusivity.
The “ground truth” diffusivity is the direct estimate of the monthly eddy diffusivity of temperature averaged over a 30°–60° latitudinal window (denoted by KT); it is computed by dividing the troposphere-integrated eddy heat flux by the meridional gradient of the integrated temperature for each calendar month. The eddy diffusivities for moisture and MSE are computed in similar fashion. To compute these eddy fluxes, 6-hourly model outputs on native model grids are needed. For computing the Rhines scale
Results of comparing six scaling theories against the monthly temperature diffusivity KT directly estimated from the AQUA-AM2.1 simulations: (a) Green70 scaling, (b) HL96 scaling, (c) moist HL96 scaling, (d) Rhines scaling, (e) Barry02 scaling, and (f) modified Barry scaling. Each dot corresponds to the monthly values (averaged between 30° and 60° latitude) from scaling and model estimate. Red (blue) dots are for the Northern (Southern) Hemisphere.
Citation: Journal of Climate 35, 21; 10.1175/JCLI-D-21-0721.1
As found in earlier studies for models with similar complexity (Barry et al. 2002; Frierson et al. 2007; Liu et al. 2017), the Rhines scaling (Fig. 4d) produces the closest match to the diffusivity directly computed from the model. This result strongly supports the Rhines scale as the mixing scale, and brings us one step closer to a predictive scaling theory for the seasonally varying diffusivity. Since the mixing scale and the velocity scale are linked via the relation
The Barry02 scaling (Fig. 4e) demonstrates similar skill as the moist HL96 (Fig. 4c). It may not be surprising given the fact that the two share the same theoretical lineage, as explained in appendix A. It is also worth noting that Barry02 scaling and its variant can also capture the overall decrease of KT with increasing CO2 forcing (not shown). While Barry scaling is not a predictive theory for diffusivity because the diagnosed meridional energy transport rate must be used, the HL96 scaling and its moist variant are not ready for the closure of the MSE transport either, until a viable theory for
4. The diffusivity of moisture versus temperature
To understand the diffusivity of MSE, one must understand the diffusivity of moisture as well. As mentioned above, the diffusivity of temperature and that of moisture follow the same scaling law within a constant, making a unified scaling for the diffusivity of MSE possible. This is verified by the tight unity slope between the midlatitude diffusivity of moisture (KQ) and that of temperature (KT) on a log–log plot (see Fig. 5b). However, the magnitude of KQ is larger than that of KT by approximately a factor of 2.6 (Fig. 5a). This qualitatively reflects the fact that the eddy motion in the lower troposphere is much less reversible (more turbulence-like) compared to the upper troposphere eddies (more wavelike) and irreversible motions can mix more efficiently. A passive tracer diagnosis of effective diffusivity by Chen and Plumb (2014) has shown that the upper-level diffusivity near the jet level is several times weaker than that near the surface (see their Fig. 8). Since moisture is more concentrated near the surface than the temperature distribution, it is more heavily influenced by the low-level diffusivity than the upper-level one. Further, other factors, especially those related to the diabatic processes, can influence the actual rendering of the diffusivity as well.
Temperature diffusivity KT vs column moisture diffusivity KQ in the AQUA-AM2.1 control run. (a) The meridional profiles of the annual-mean KT (orange) and KQ (green), and (b) their monthly values averaged in the NH (red) and SH (blue) midlatitudes (30°–60°).
Citation: Journal of Climate 35, 21; 10.1175/JCLI-D-21-0721.1
Here we provide a heuristic (in keeping with the level of complexity of the EBM) on the differing diffusivity for moisture versus temperature as follows, accounting for the diabatic effects in maintaining the heat or moisture budget. For the latter budget, the “diabatic” effects refer to the nonconservative processes that are responsible for depleting or moistening the atmosphere (such as evaporation and precipitation). Readers interested in the more process-oriented treatment of the diffusivity of moisture may refer to Caballero and Hanley (2012).
In summary, the exact magnitude of the diffusivity of a tracer is not only determined by the kinematics of the turbulent motion; it is also dependent on the competition between the rate of decay of the tracer property carried by air parcels and the rate of the restoration of the tracer field. As discussed by Caballero and Hanley (2012), everything else being equal, if the nonconservative processes are rapid enough to alter the tracer concentration during the ballistic motion, the effective mixing length should be significantly shortened, and the diffusivity reduced. On the other hand, as discussed above, if the mean gradient of the tracer concentration is more rapidly restored, the tracer carried by the air parcels will be mixed with a more contrasting environment, to the effect of enhanced diffusivity. Last, one should not confuse the quantitative difference between the two diffusivities here with different scaling behaviors, the latter being more about the scale-invariant relations up to a multiplicative constant.
5. Effect of diffusivity on polar amplification
Before considering the role of diffusivity and CC scaling on PA, particularly within the context of the simple EBM, a clarification is needed regarding whether PA is intrinsic to aquaplanet AGCMs. It has been mysterious why the polar region warms more than the tropics in response to a uniform radiative forcing, even in the absence of polar-amplifying feedbacks such as those associated with the changes in albedo and lapse rate (e.g., Alexeev et al. 2005; Russotto and Biasutti 2020). Recent findings, however, by Kim et al. (2018), showed nearly identical surface temperature warming in the polar region compared to the tropics in their CO2 doubling experiments using the same aquaplanet AGCM used here with seasonally varying solar insolation and interactive cloud radiative effect. PA persists when a perpetual equinox solar forcing is imposed in the baseline run [consistent with Alexeev et al. (2005)], suggesting the seasonal solar forcing can suppress PA in aquaplanet experiments lacking other polar-amplifying feedbacks. This suppression effect on PA of the seasonal cycle was confirmed by Shaw and Smith (2022), who further showed that a strong polar amplification re-emerges when sea ice is present in the aquaplanet AGCM used therein. Kim et al. (2018) argued that it is the cloud radiative feedback phase-locked with the seasonal cycle that acts to cool the summertime surface temperature over the polar region, largely cancelling out the PA warming in wintertime. This nuanced (nonlinear) interaction between polar cloud and seasonal cycle is beyond the level of complexity that our first principle-based reasoning can handle. This is in keeping with the study of Feldl and Merlis (2021), which found no apparent seasonal suppression of the annual mean PA from the seasonal solar insolation in both an EBM and an idealized moist AGCM without cloud. Therefore, the discussion below is mainly focused on the idealized, aquaplanet world without seasonal cycle and interactive cloud.1
a. A reductio ad absurdum for PA
Inspired by dynamical constraint due to the interactive diffusivity as formulated by (A13), we first offer a reductio ad absurdum argument for the polar-amplified temperature response in an ice-free and cloud-free aquaplanet atmosphere. Let us start the chain of reasoning with an increase of diffusivity in response to a uniform radiative forcing in a climate model under a thermodynamic constraint of the CC relation and a dynamical constraint of the diffusivity closure (A13). The CC relation decrees a significant increase in the meridional gradient of moisture even with a uniform temperature warming. Acted upon by the increased diffusivity, this would lead to a large increase in poleward moisture transport, and hence a large increase in latent heat release associated with the increase of precipitation in the polar region. In the absence of any strong negative feedback over the polar region, a larger high-latitude diabatic heating must be compensated by a reduced poleward sensible energy flux (
One scenario corresponds to a sharp reduction of diffusivity at a rate even greater than the CC-regulated increase of the moisture gradient. For this case, the poleward latent energy transport is decreased, and this must be compensated by an increase of sensible energy transport (Held and Soden 2006; Shaw and Voigt 2016). Since diffusivity is very much reduced, this can only be achieved by a large increase in the poleward temperature gradient. Turning to the formulation of the diffusivity, this is also impossible: KZ cannot decrease with a large increase of Ty. Finally, the only plausible scenario left is a modest decrease in KZ (relative to the increase in the moisture gradient) together with a polar-amplified warming, and this is exactly what we found in both the AQUA-AM2.1 experiments under the forcing of increasing CO2 and the EBM experiments below.
b. Sensitivity of PA to diffusivity versus CC nonlinearity
To elucidate the point above explicitly, we force the baseline EBM (i.e., no rescaling on the CC relation and the interactive diffusivity) with uniform 10 and 20 W m−2 radiation perturbations and the results are presented in Fig. 1. Note that a linear representation of the OLR is used and hence no polar-amplifying feedback is considered in the EBM here, but we still obtain slightly larger warming in the high latitudes than the tropics under a uniform radiation perturbation (to be quantified later). Consistent with the thought experiment above, in response to a uniform positive forcing, diffusivity decreases (see Fig. 1b) to largely offset the effect of increasing moisture gradient, the latter being a result of the CC nonlinearity since the warming itself is almost uniform. The compensation is almost perfect, as evinced by the overlap of the three EBM profiles of AHT in Fig. 1c. Frierson et al. (2007) first discovered similar compensation between the latent energy transport and the dry static energy transport under larger perturbations to the atmospheric water vapor content, although in a somewhat different EBM with a different parameterization of diffusivity. To facilitate comparison with their result, we have also tried their parameterization for diffusivity [proportional to the 3/2 power of the temperature gradient; see their Eq. (A6)], the compensation still holds well and polar amplification under uniform forcing persists. Notwithstanding the substantially different configurations between the EBM here and the gray radiation model used in Frierson et al. (2007), the essence of the energy compensation in these models resides in the radiative constraint, the simple treatments of radiation in both models inhibit the creation of any significant anomalous spatial structure in the total moist static energy transport.
The nonlinear relationship between the surface MSE m and temperature TS and its linearization. The black line is the relationship based on the original CC relation with fixed relative humidity; the red dashed line is the polynomial fit for the m–TS relationship; the dashed brown and orange lines correspond to the relationship wherein the nonlinear component is rescaled by a factor of 1/2 and 1/4, respectively; and the dashed purple line is the full linearization of the m–TS relationship.
Citation: Journal of Climate 35, 21; 10.1175/JCLI-D-21-0721.1
The logarithm of the PAF simulated in the EBM by systematically rescaling the nonlinear component in the m–TS relationship (x axis) and the coefficient of the diffusivity (y axis) under a uniform 10 W m−2 radiative forcing, where (a) the diffusivity is interactive and responds to the forcing perturbation and (b) the diffusivity is set to the rescaled baseline values and not allowed to respond to the radiative perturbation. The colored dashed lines correspond to those shown in Fig. 6 for different CC nonlinearities. The baseline diffusivity profile for the control case (corresponding to scale factor 1) is shown by the green line in Fig. 1b.
Citation: Journal of Climate 35, 21; 10.1175/JCLI-D-21-0721.1
The PAF is color-coded in such a way in Fig. 7 that the white color corresponds to no polar amplification. For weak polar amplification, PAF approximates the fractional factor of the polar amplification. In the EBM with interactive diffusivity, if one halves the nonlinearity in the TS–m relation, the TS response turns from weakly polar amplified (∼8% more warming in the high latitudes compared to the tropics) to not polar amplified; further linearization even turns the response to be equatorward amplified (Fig. 7a). A factor of 8 rescaling of diffusivity, on the other hand, only leads to relatively marginal change in the PAF, while the magnitude of the amplification, either poleward or equatorward, decreases with the systematic diffusivity doubling. This is expected from the fact that the EBM with a stronger diffusivity tends to maintain a climatology state with a much smaller meridional temperature gradient and hence a smaller potential for CC nonlinearity to act on.
We must note that the magnitudes of the polar amplification for the interactive diffusivity case (Fig. 7a) are rather weak; this is to large extent due to the strong sensitivity of diffusivity to the temperature gradient (
c. Circulation effects on PA and midlatitude extremes
The key advantage of the EBM formulation is its encapsulation of the effect of the atmospheric motion in a single scalar, namely diffusivity, allowing a clean isolation of the dynamical effect from the thermodynamical one. A successful closure on the diffusivity would lead to a system that can capture both the dynamic and thermodynamic response to external climate perturbations. Provided that the diffusivity scaling here [(A13)] is as fundamental a physical principle as the CC relation governing the response of the EBM, the temperature response would be predestined to be polar amplified as required jointly by these two constraints. Even if we are not confident in the exact formulation of the diffusivity, all theories and empirical evidence so far point to a positive relationship of diffusivity to the temperature gradient, and therefore the aforementioned notion still holds qualitatively. Guided by this insight, we assert that the eddy circulation in the midlatitude storm track tends to overall weaken in the face of PA in response to a uniform climate forcing, thus playing a negative role in PA, at least in this very idealized setting. This has an important implication for midlatitude extremes.
The midlatitude extremes are mostly driven by the eddy advection on the background gradient, in the form of
6. Summary and concluding remarks
EBMs have been extensively utilized for a range of climate problems, from the tropical to the polar and from climate feedbacks to the atmospheric hydrological cycle. The MSE diffusivity in EBMs is often taken to be constant or prescribed (even spatially varying) such that the circulation response and its impact on the subject in question are not considered within the EBM framework. In this study, we first demonstrate that the diffusive formulism of the MSE transport is mathematically justifiable under a few reasonable assumptions. Thus, the diffusivity can be diagnosed directly or indirectly from the AGCM simulations conducted in this study, with the result showing a strong dependence of the diffusivity on space and season. In addition, a rich body of literature exists relating the eddy diffusivity in the midlatitude storm track to the environmental physical parameters, laying the basis for parameterizing the diffusivity. We have tested the prevailing scaling arguments for the eddy diffusivity against the seasonal variation of the diffusivity diagnosed in an AGCM. The evidence strongly supports that the Rhines scale is the energy-containing scale and hence behaves as the mixing length for the heat and moisture transport in the storm track. Despite the disparate variety of the scaling arguments, a synergy may be formed between the classic baroclinic instability point of view and the turbulence cascade one, both pointing to a cubic law relationship with the mean meridional temperature gradient (e.g., the HL96 scaling). To apply this scaling relationship to the moist atmosphere, the static stability parameter therein must be adjusted to account for the reduced stability felt by the moist air parcel. Notwithstanding, the modest success of this scaling largely offers a possible explanation for the seasonal variation of the eddy diffusivity and its response to climate warming.
Under the dual constraints of the equipartitioning between the kinetic and potential components of the eddy energy and the Rhines scale being the mixing length, the HL96 scaling and its variants show a much stronger sensitivity to the meridional gradient than the vertical gradient of the potential temperature. From summer to winter, both gradients are intensified proportionally so that the midlatitude isentropic slope stays roughly unchanged in accordance with the baroclinic adjustment theory (Stone 1978). This inevitably gives rise to a sharp increase in diffusivity from summer to winter. Under a uniform radiative forcing or a uniform increase of CO2 concentration, midlatitude eddy diffusivity reduction is also found in a very idealized EBM used here and an aquaplanet AGCM, respectively.
The scaling exercise above gives us a working candidate for constructing an interactive MSE diffusivity in EBM, whose behavior is now controlled by the following two physical constraints: the thermodynamic CC relation between moisture and temperature, and the parametric relation of the MSE diffusivity with the global mean air temperature and the meridional temperature gradient (A13). An EBM governed by these two constraints can only produce a polar-amplified temperature response even in the absence of any polar-amplifying radiative feedbacks; the response otherwise would violate one of or both the constraints. The related decrease in the temperature gradient and the eddy advection speed also has an important bearing on the temperature extremes through the reduction of the variance of the temperature variability. As such, it may not be just a coincidence that the lower-tropospheric temperature variance with respect to the warmer mean state decreases over northern mid- to high latitudes in sophisticated climate models under greenhouse gas forcing (Schneider et al. 2015), even with the Arctic amplification suppressed (Dai and Deng 2021). The interplay between the nonlinear CC relation and the temperature dependent diffusivity may potentially underpin this robust feature of climate change response. In addition, with the active polar-amplifying feedbacks, especially those related to sea ice, it would be even harder to escape the consequence of the weakening of eddy activity and diffusivity (Stuecker et al. 2018; Shaw and Smith 2022).
The first principle-based reasoning may not be sufficient to infer how the polar hydrological cycle and extremes would respond to a climate warming forcing even in an idealized aquaplanet setting. In addition to the competition among the mean warming (which acts to increase the gradient of moisture), the reduced meridional temperature gradient (which acts to decrease the moisture gradient), and the reduced advection velocity, the polar hydrological response is also acutely affected by the shift of the midlatitude storm track. Even if the moisture flux intensity stayed unchanged, a poleward shift of the peak would converge the same amount of moisture to a reduced polar area, resulting in more intense precipitation minus evaporation. To understand the hydrological response, further governing principles must be developed for the key structures of the atmospheric circulation, especially the HC and the midlatitude storm track. A commendable effort has been made by Mbengue and Schneider (2018) to construct a predictive EBM for the location of the storm track by parameterizing the terminus of the Hadley cell and the energy flux out of the tropics. Another promising direction may be to account for the compensation between the sensible energy and the latent energy transport under the radiative constraint (Shaw and Voigt 2016; see also Isaac M. Held’s blog at https://www.gfdl.noaa.gov/blog_held/62-poleward-atmospheric-energy-transport/).
Our understanding of the robust climate change response so far has been mostly posited on the basic thermodynamics and radiative constraints; circulation change has been construed to be the main source of uncertainties for the regional climate projections (Xie et al. 2015; Shepherd 2014). Our study here demonstrates that the dynamical principle-based understanding of the eddy diffusivity may potentially serve as a source of certainty for some of the broadest regional features of climate change response. We hope this work can stimulate further development of first principle-based conceptual models for understanding not only the globally averaged, but also the spatially dependent, climate change response.
Such as the aquaplanet model and forcing configurations used in Alexeev et al. (2005) and the ANNFC and SEAFC cases in Kim et al. (2018).
An alternative interpretation of the Rhines scale as the energy containing scale is that rather than representing the cascade halting scale, it reflects the width of the zonal jet (Huang and Robinson 1998; Chemke and Kaspi 2015), which itself is set to be
This equipartitioning can extend to the inertial range of the downscale enstrophy cascade at scales smaller than the Rossby radius according to the geostrophic turbulence theory of Charney (1971).
Acknowledgments.
The authors thank Sandro Lubis and Jianhua Lu for their helpful suggestions during the formative stage of the manuscript. The constructive reviews from three anonymous reviewers helped improve the manuscript substantially. The correspondence with Todd Mooring helped resolve the apparent discrepancy in the profile of the MSE diffusivity with Mooring and Shaw (2020). This work is supported by Office of Science, U.S. Department of Energy Biological and Environmental Research as part of the Regional and Global Model Analysis program area. The Pacific Northwest National Laboratory (PNNL) is operated for DOE by Battelle Memorial Institute under contract DE-AC05-76RLO1830.
Data availability statement.
The dataset of the AQUA-AM2.1 simulations on which this paper is based can be accessed via data portal https://portal.nersc.gov/cfs/m1867/jianlu/GFDL_AM21_C32_datasharing/. The code for the EBM with parameterized MSE diffusivity can be made available upon request by contacting the corresponding author.
APPENDIX A
Scaling Theories for the Midlatitude Eddy Diffusivity
The concept of turbulence diffusivity is rooted in the mixing length theory (Prandtl 1925), which is applicable to the condition that the mean gradient is varying on a spatial scale that is larger than the mixing length. This condition is weakly satisfied in the midlatitude atmosphere, offering some justification for the diffusive closure for the midlatitude eddy transport (Corrsin 1975; Held 1999). The purpose of scaling is not for a quantitative, complete theory, but to seek for scale-invariant relations (often in the form of power law) with the environmental parameters up to a multiplicative constant. Here we try to recapitulate and integrate the prevailing scaling arguments for the eddy diffusivity along two somewhat parallel but interrelated threads: one is the classic thread, based on the textbook depiction of the midlatitude eddies from the perspective of baroclinic instability and baroclinic adjustment; the other is the turbulence thread, based on the advances in the understanding of the quasigeostrophic (QG) and 2D turbulence. The dichotomy of the historical development of the diffusivity theories along the two paths are illustrated in two different colors in the diagram in Fig. A1.
Relationships among the existing scaling theories for the eddy diffusivity. The open circle indicates the starting point for the derivation of the scaling theory. The filled circles connected by a thin line represents a switch under the condition specified under the line. The scaling theories based on eddy energy equipartitioning is shown in blue, and that based on turbulence thinking in purple. L indicates the mixing scale or the energy containing scale.
Citation: Journal of Climate 35, 21; 10.1175/JCLI-D-21-0721.1
a. Turbulence-based theories for diffusivity
Later studies have challenged the classic interpretation of the Rhines scale as the crossover between turbulence and Rossby wave ranges, arguing that Rossby wave can coexist with turbulence on scales smaller than LR (Galperin et al. 2006; Sukoriansky et al. 2007). Another complexity is that on a rotating sphere, the energy of the upscale cascade seeks for the largest possible scale in the zonal direction, resulting in zonal jet formation (e.g., Vallis and Maltrud 1993). These jets can in turn organize the instability so that the eddies form a “storm track” on each jet (Williams 1979; Panetta 1993). This eddy-jet coexistence can change the eddy characteristics substantially (Jansen et al. 2015; Kong and Jansen 2017). Moreover, the −5/3 inertial slope has not been observed in Earth’s atmosphere (e.g., Boer and Shepherd 1983; Nastrom and Gage 1985; Chemke and Kaspi 2015). Notwithstanding these complexities, there has been considerable success with the Rhines scale in capturing the energy-containing scale of turbulence under a wide range of choice of parameters, even for realistic atmosphere with moisture and adjustable static stability (Barry et al. 2002; Frierson et al. 2006; O’Gorman and Schneider 2007; Liu et al. 2017). This makes (A2) an appealing starting point for more predictive scalings for eddy diffusivity.
b. Classic approaches for scaling the eddy diffusivity
The underlying dynamic mechanisms aside (regardless of whether it is due to the energy arrest of the inverse cascade, triads wave–wave interaction, or the wave-mean wind interaction), the Rhines scale does behave as the energy containing scale not only in 2D β-plane turbulence under a wide range of parameters, but also in Earth-like environment with (Barry et al. 2002; Frierson et al. 2006; O’Gorman and Schneider 2007; Chemke and Kaspi 2015; Liu et al. 2017) or without moisture (Chan et al. 2021). As discussed above, the same level of empirical support can also be found for the equipartitioning of eddy energies. Taken together, these two conditions constitute the physical and empirical basis for the relationship (A8) or (A12) as a viable candidate for scaling the diffusivity in Earth’s atmosphere.
APPENDIX B
Nomenclature for Diffusivity in Different Contexts
KZ |
Zonal-mean MSE diffusivity (in general context) |
Km |
MSE diffusivity in one-dimensional EBM described in Eq. (4) |
KHC |
Parameterized tropical MSE diffusivity following the scaling for Hadley cell strength |
KST |
Parameterized midlatitude MSE diffusivity following the modified Barry scaling |
K |
Two-dimensional diffusivity tensor due to the eddy component of the MSE transport |
KS |
Symmetric component of K |
KA |
Anti-symmetric component of K |
KS |
Scalar coefficient for the symmetric component KS |
KA |
Scalar coefficient for the antisymmetric component KA |
Kχ |
Scalar coefficient for the symmetric component of the two-dimensional diffusivity tensor due to the divergent mean wind |
Kψ |
Scalar coefficient for the antisymmetric component of the two-dimensional diffusivity tensor due to the rotational mean wind |
KT |
Diffusivity for midlatitude atmospheric temperature |
KQ |
Diffusivity for midlatitude atmospheric moisture |
Midlatitude eddy diffusivity used in scaling theories |
REFERENCES
Alexeev, V. A., P. L. Langen, and J. R. Bates, 2005: Polar amplification of surface warming on an aquaplanet in “ghost forcing” experiments without sea ice feedbacks. Climate Dyn., 24, 655–666, https://doi.org/10.1007/s00382-005-0018-3.
Anderson, J. L., and Coauthors, 2004: The new GFDL global atmosphere and land model AM2-LM2: Evaluation with prescribed SST simulations. J. Climate, 17, 4641–4673, https://doi.org/10.1175/JCLI-3223.1.
Armour, K. C., N. Siler, A. Donohoe, and G. Roe, 2019: Meridional atmospheric heat transport constrained by energetics and mediated by large-scale diffusion. J. Climate, 32, 3655–3680, https://doi.org/10.1175/JCLI-D-18-0563.1.
Barpanda, B., and T. A. Shaw, 2020: Surface fluxes modulate the seasonality of zonal-mean storm tracks. J. Atmos. Sci., 77, 753–779, https://doi.org/10.1175/JAS-D-19-0139.1.
Barry, L., G. C. Craig, and J. Thuburn, 2002: Poleward heat transport by the atmospheric heat engine. Nature, 415, 774–777, https://doi.org/10.1038/415774a.
Bischoff, T., and T. Schneider, 2014: Energetic constraints on the position of the intertropical convergence zone. J. Climate, 27, 4937–4951, https://doi.org/10.1175/JCLI-D-13-00650.1.
Boer, G. J., and T. G. Shepherd, 1983: Large-scale two-dimensional turbulence in the atmosphere. J. Atmos. Sci., 40, 164–184, https://doi.org/10.1175/1520-0469(1983)040<0164:LSTDTI>2.0.CO;2.
Boos, W. R., and R. L. Korty, 2016: Regional energy budget control of the intertropical convergence zone and application to mid-Holocene rainfall. Nat. Geosci., 9, 892–897, https://doi.org/10.1038/ngeo2833.
Bordoni, S., and T. Schneider, 2008: Monsoons as eddy-mediated regime transitions of the tropical overturning circulation. Nat. Geosci., 1, 515–519, https://doi.org/10.1038/ngeo248.
Budyko, M. I., 1968: The effect of solar radiation variations on the climate of the Earth. Tellus, 21, 611–619, https://doi.org/10.3402/tellusa.v21i5.10109.
Caballero, R., and J. Hanley, 2012: Midlatitude eddies, storm-track diffusivity, and poleward moisture transport in warm climates. J. Atmos. Sci., 69, 3237–3250, https://doi.org/10.1175/JAS-D-12-035.1.
Chan, P. W., P. Hassanzadeh, and Z. Kuang, 2021: Eddy-length-scale response to static stability change in an idealized dry atmosphere: A linear response function approach. J. Atmos. Sci., 78, 2619–2626, https://doi.org/10.1175/JAS-D-21-0044.1.
Charney, J. G., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, 1087–1095, https://doi.org/10.1175/1520-0469(1971)028<1087:GT>2.0.CO;2.
Chemke, R., and K. Kaspi, 2015: The latitudinal dependence of atmospheric jet scales and macroturbulent energy cascades. J. Atmos. Sci., 72, 3891–3907, https://doi.org/10.1175/JAS-D-15-0007.1.
Chen, G., and A. Plumb, 2014: Effective isentropic diffusivity of tropospheric transport. J. Atmos. Sci., 71, 3499–3520, https://doi.org/10.1175/JAS-D-13-0333.1.
Chen, G., J. Lu, and D. M. W. Frierson, 2008: Phase speed spectra and the latitude of surface westerlies: Interannual variability and global warming trend. J. Climate, 21, 5942–5959, https://doi.org/10.1175/2008JCLI2306.1.
Cohen, J., and Coauthors, 2014: Recent Arctic amplification and extreme mid-latitude weather. Nat. Geosci., 7, 627–637, https://doi.org/10.1038/ngeo2234.
Corrsin, S., 1975: Limitations of gradient transport models in random walks and in turbulence. Adv. Geophys., 18, 25–60, https://doi.org/10.1016/S0065-2687(08)60451-3.
Dai, A., and J. Deng, 2021: Arctic amplification weakens the variability of daily temperature over northern middle-high latitudes. J. Climate, 34, 2591–2609, https://doi.org/10.1175/JCLI-D-20-0514.1.
Feldl, N. and T. M. Merlis, 2021: Polar amplification in idealized climates: The role of ice, moisture, and seasons. Geophys. Res. Lett., 48, e2021GL094130, https://doi.org/10.1029/2021GL094130.
Flannery, B. P., 1984: Energy balance models incorporating transport of thermal and latent energy. J. Atmos. Sci., 41, 414–421, https://doi.org/10.1175/1520-0469(1984)041<0414:EBMITO>2.0.CO;2.
Francis, J. A., and S. J. Vavrus, 2012: Evidence linking Arctic amplification to extreme weather in mid-latitudes. Geophys. Res. Lett., 39, L06801, https://doi.org/10.1029/2012GL051000.
Freidenreich, S. M., and V. Ramaswamy, 1999: A new multiple-band solar radiative parameterization for general circulation models. J. Geophys. Res., 104, 31 389–31 409, https://doi.org/10.1029/1999JD900456.
Frierson, D. M. W., I. M. Held, and P. Zurita-Gotor, 2006: A gray-radiation aquaplanet moist GCM. Part I: Static stability and eddy scale. J. Atmos. Sci., 63, 2548–2566, https://doi.org/10.1175/JAS3753.1.
Frierson, D. M. W., I. M. Held, and P. Zurita-Gotor, 2007: A gray-radiation aquaplanet moist GCM. Part II: Energy transport in altered climates. J. Atmos. Sci., 64, 1680–1693, https://doi.org/10.1175/JAS3913.1.
Galperin, B., S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, 2006: Anisotropic turbulence and zonal jets in rotating flows with a β-effect. Nonlinear Processes Geophys., 13, 83–98, https://doi.org/10.5194/npg-13-83-2006.
Gao, Y., L. R. Leung, J. Lu, and G. Masato, 2015: Persistent cold air outbreaks over North America in a warming climate. Environ. Res. Lett., 10, 044001, https://doi.org/10.1088/1748-9326/10/4/044001.
Green, J. S. A., 1970: Transfer properties of large scale eddies and the general circulation of the atmosphere. Quart. J. Roy. Meteor. Soc., 96, 157–185, https://doi.org/10.1002/qj.49709640802.
Haynes, P. H., and E. F. Shuckburgh, 2000: Effective diffusivity as a diagnostic of atmospheric transport: 2. Troposphere and lower stratosphere. J. Geophys. Res., 105, 22 795–22 810, https://doi.org/10.1029/2000JD900092.
Held, I. M., 1978: The vertical scale of an unstable baroclinic wave and its importance for eddy heat flux parameterizations. J. Atmos. Sci., 35, 572–576, https://doi.org/10.1175/1520-0469(1978)035<0572:TVSOAU>2.0.CO;2.
Held, I. M., 1982: On the height of the tropopause and the static stability of the troposphere. J. Atmos. Sci., 39, 412–417, https://doi.org/10.1175/1520-0469(1982)039<0412:OTHOTT>2.0.CO;2.
Held, I. M., 1999: The macroturbulence of the troposphere. Tellus, 51A, 59–70, https://doi.org/10.3402/tellusa.v51i1.12306.
Held, I. M., and M. J. Suarez, 1974: Simple albedo feedback models of the ice caps. Tellus, 26A, 613–629, https://doi.org/10.3402/tellusa.v26i6.9870.
Held, I. M., and A. Y. Hou, 1980: Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. J. Atmos. Sci., 37, 515–533, https://doi.org/10.1175/1520-0469(1980)037<0515:NASCIA>2.0.CO;2.
Held, I. M., and V. D. Larichev, 1996: A scaling theory for horizontally homogeneous, baroclinically unstable flow on a β-plane. J. Atmos. Sci., 53, 946–952, https://doi.org/10.1175/1520-0469(1996)053<0946:ASTFHH>2.0.CO;2.
Held, I. M., and B. J. Soden, 2006: Robust response of the hydrological cycle to global warming. J. Climate, 19, 5686–5699, https://doi.org/10.1175/JCLI3990.1.
Huang, H. P., and W. A. Robinson, 1998: Two-dimensional turbulence and persistent jets in a global barotropic model. J. Atmos. Sci., 55, 611–632, https://doi.org/10.1175/1520-0469(1998)055<0611:TDTAPZ>2.0.CO;2.
Hwang, Y.-T., and D. M. W. Frierson, 2010: Increasing atmospheric poleward energy transport with global warming. Geophys. Res. Lett., 37, L24807, https://doi.org/10.1029/2010GL045440.
Jansen, M. F., A. J. Adcroft, R. Hallberg, and I. M. Held, 2015: Parameterization of eddy fluxes based on a mesoscale energy budget. Ocean Modell., 92, 28–41, https://doi.org/10.1016/j.ocemod.2015.05.007.
Kang, S. M., I. M. Held, D. M. W. Frierson, and M. Zhao, 2008: The response of the ITCZ to extratropical thermal forcing: Idealized slab-ocean experiments with a GCM. J. Climate, 21, 3521–3532, https://doi.org/10.1175/2007JCLI2146.1.
Kang, S. M., D. M. W. Frierson, and I. M. Held, 2009: The tropical response to extratropical thermal forcing in an idealized GCM: The importance of radiative feedbacks and convective parameterization. J. Atmos. Sci., 66, 2812–2827, https://doi.org/10.1175/2009JAS2924.1.
Kim, D., S. M. Kang, and Y. Shin, 2018: Sensitivity of polar amplification to varying insolation conditions. J. Climate, 31, 4933–4947, https://doi.org/10.1175/JCLI-D-17-0627.1.
Kong, H., and M. F. Jansen, 2017: The eddy diffusivity in barotropic β-plane turbulence. Fluids, 2, 54, https://doi.org/10.3390/fluids2040054.
Kraichnan, R., 1967: Inertial ranges in two-dimensional turbulence. Phys. Fluids, 10, 1417–1423, https://doi.org/10.1063/1.1762301.
Kuo, H. L., 1949: Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere. J. Meteor., 6, 105–122, https://doi.org/10.1175/1520-0469(1949)006<0105:DIOTDN>2.0.CO;2.
Kushner, P. J., and I. M. Held, 1998: A test, using atmospheric data, of a method for estimating oceanic eddy diffusivity. Geophys. Res. Lett., 25, 4213–4216, https://doi.org/10.1029/1998GL900142.
Laliberté, F., J. Zika, L. Mudryk, P. J. Kushner, J. Kjellsson, and K. Doos, 2015: Constrained work output of the moist atmospheric heat engine in a warming climate. Science, 347, 540–543, https://doi.org/10.1126/science.1257103.
Lapeyre, G., and I. M. Held, 2004: The role of moisture in the dynamics and energetics of turbulent baroclinic eddies. J. Atmos. Sci., 61, 1693–1710, https://doi.org/10.1175/1520-0469(2004)061<1693:TROMIT>2.0.CO;2.
Leung, L. R., and G. R. North, 1991: Atmospheric variability on a zonally symmetric land planet. J. Climate, 4, 753–765, https://doi.org/10.1175/1520-0442(1991)004<0753:AVOAZS>2.0.CO;2.
Lindzen, R. S., and B. Farrell, 1977: Some realistic modifications of simple climate models. J. Atmos. Sci., 34, 1487–1501, https://doi.org/10.1175/1520-0469(1977)034<1487:SRMOSC>2.0.CO;2.
Liu, X., D. S. Battisti, and G. H. Roe, 2017: The effect of cloud cover on the meridional heat transport: Lessons from variable rotation experiments. J. Climate, 30, 7465–7479, https://doi.org/10.1175/JCLI-D-16-0745.1.
Lorenz, D. J., and E. T. DeWeaver, 2007: The tropopause height and the zonal wind response to global warming in the IPCC scenario integrations. J. Geophys. Res., 112, D10119, https://doi.org/10.1029/2006JD008087.
Lu, J., G. Vecchi, and T. Reichler, 2007: Expansion of the Hadley cell under global warming. Geophys. Res. Lett., 34, L06805, https://doi.org/10.1029/2006GL028443.
Lu, J., K. Sakaguchi, Q. Yang, R. Leung, G. Chen, C. Zhao, and E. Swenson, 2017: Examining the hydrological variations in an aquaplanet world using wave activity transformation. J. Climate, 30, 2559–2576, https://doi.org/10.1175/JCLI-D-16-0561.1.
Lu, J., D. Xue, L. R. Leung, F. Liu, F. Song, B. Harrop, and W. Zhou, 2021: The leading modes of Asian summer monsoon variability as pulses of atmospheric energy flow. Geophys. Res. Lett., 48, e2020GL091629, https://doi.org/10.1029/2020GL091629.
Maltrud, M. E., and G. K. Vallis, 1991: Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence. J. Fluid Mech., 228, 321–342, https://doi.org/10.1017/S0022112091002720.
Mbengue, C., and T. Schneider, 2018: Linking Hadley circulation and storm tracks in a conceptual model of the atmospheric energy balance. J. Atmos. Sci., 75, 841–856, https://doi.org/10.1175/JAS-D-17-0098.1.
Merlis, T. M., and M. Henry, 2018: Simple estimates of polar amplification in moist diffusive energy balance models. J. Climate, 31, 5811–5824, https://doi.org/10.1175/JCLI-D-17-0578.1.
Mesinger, F., Z. I. Janjic, S. Nickovic, D. Gavrilov, and D. G. Deaven, 1988: The step-mountain coordinate: Model description and performance for cases of Alpine lee cyclogenesis and for a case of an Appalachian redevelopment. Mon. Wea. Rev., 116, 1493–1518, https://doi.org/10.1175/1520-0493(1988)116<1493:TSMCMD>2.0.CO;2.
Mooring, T. A., and T. A. Shaw, 2020: Atmospheric diffusivity: A new energetic framework for understanding the midlatitude circulation response to climate change. J. Geophys. Res. Atmos., 125, e2019JD031206, https://doi.org/10.1029/2019JD031206.
Moorthi, S., and M. J. Suarez, 1992: Relaxed Arakawa–Schubert: A parameterization of moist convection for general circulation models. Mon. Wea. Rev., 120, 978–1002, https://doi.org/10.1175/1520-0493(1992)120<0978:RASAPO>2.0.CO;2.
Nastrom, G. D., and K. S. Gage, 1985: A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci., 42, 950–960, https://doi.org/10.1175/1520-0469(1985)042<0950:ACOAWS>2.0.CO;2.
Neelin, J. D., and I. M. Held, 1987: Modeling tropical convergence based on the moist static energy budget. Mon. Wea. Rev., 115, 3–12, https://doi.org/10.1175/1520-0493(1987)115<0003:MTCBOT>2.0.CO;2.
Nelder, J. A., and R. Mead, 1965: A simplex method for function minimization. Comput. J., 7, 308–313, https://doi.org/10.1093/comjnl/7.4.308.
North, G. R., 1975: Theory of energy-balance climate models. J. Atmos. Sci., 32, 2033–2043, https://doi.org/10.1175/1520-0469(1975)032<2033:TOEBCM>2.0.CO;2.
North, G. R., and M. J. Stevens, 2006: Energy-balance climate models. Frontiers of Climate Modeling, J. T. Kiehl and V. Ramanathan, Eds., Cambridge University Press, 52–72.
North, G. R., J. G. Mengel, and D. A. Short, 1983: Simple energy balance model resolving the seasons and the continents: Application to the astronomical theory of the ice ages. J. Geophys. Res., 88, 6576–6586, https://doi.org/10.1029/JC088iC11p06576.
O’Gorman, P. A., and T. Schneider, 2007: Weather-layer dynamics of baroclinic eddies and multiple jets in an idealized general circulation model. J. Atmos. Sci., 65, 524–535, https://doi.org/10.1175/2007JAS2280.1.
Oort, A. H., and J. P. Peixoto, 1983: Global angular momentum and energy balance requirements from observations. Adv. Geophys., 25, 355–490, https://doi.org/10.1016/S0065-2687(08)60177-6.
Panetta, R. L., 1993: Zonal jets in wide baroclinically unstable regions: Persistence and scale selection. J. Atmos. Sci., 50, 2073–2106, https://doi.org/10.1175/1520-0469(1993)050<2073:ZJIWBU>2.0.CO;2.
Peterson, H. G., and W. R. Boos, 2020: Feedbacks and eddy diffusivity in an energy balance model of tropical rainfall shifts. npj Climate Atmos. Sci., 3, 11, https://doi.org/10.1038/s41612-020-0114-4.
Prandtl, L., 1925: Bericht über untersuchungen zur ausgebildeten Turbulenz. Zeitschr. Angew. Math. Mech., 5, 136–139, https://doi.org/10.1002/zamm.19250050212.
Rhines, P., 1975: Waves and turbulence on a beta-plane. J. Fluid Mech., 69, 417–443, https://doi.org/10.1017/S0022112075001504.
Rhines, P., 1977: The dynamics of unsteady currents. The Sea, Vol. 6, E. A. Goldberg, et al., Eds., John Wiley and Sons, 189–318.
Roe, G. H., N. Feldl, K. C. Armour, Y.-T. Hwang, and D. M. W. Frierson, 2015: The remote impacts of climate feedbacks on regional climate predictability. Nat. Geosci., 8, 135–139, https://doi.org/10.1038/ngeo2346.
Russotto, R. D., and M. Biasutti, 2020: Polar amplification as an inherent response of a circulating atmosphere: Results from the TRACMIP aquaplanets. Geophys. Res. Lett., 47, e2019GL086771, https://doi.org/10.1029/2019GL086771.
Salmon, R., 1978: Two-layer quasi-geostrophic turbulence in a simple special case. Geophys. Astrophys. Fluid Dyn., 10, 25–52, https://doi.org/10.1080/03091927808242628.
Salmon, R., 1980: Baroclinic instability and geostrophic turbulence. Geophys. Astrophys. Fluid Dyn., 15, 167–211, https://doi.org/10.1080/03091928008241178.
Schneider, T., 2004: The tropopause and the thermal stratification in the extratropics of a dry atmosphere. J. Atmos. Sci., 61, 1317–1340, https://doi.org/10.1175/1520-0469(2004)061<1317:TTATTS>2.0.CO;2.
Schneider, T., 2006: The general circulation of the atmosphere. Annu. Rev. Earth Planet. Sci., 34, 655–688, https://doi.org/10.1146/annurev.earth.34.031405.125144.
Schneider, T., and C. C. Walker, 2006: Self-organization of atmospheric macroturbulence into critical states of weak nonlinear eddy–eddy interactions. J. Atmos. Sci., 63, 1569–1586, https://doi.org/10.1175/JAS3699.1.
Schneider, T., T. Bischoff, and G. H. Haug, 2014: Migrations and dynamics of the intertropical convergence zone. Nature, 513, 45–53, https://doi.org/10.1038/nature13636.
Schneider, T., T. Bischoff, and H. Płotka, 2015: Physics of changes in synoptic midlatitude temperature variability. J. Climate, 28, 2312–2331, https://doi.org/10.1175/JCLI-D-14-00632.1.
Schwarzkopf, M. D., and V. Ramaswamy, 1999: Radiative effects of CH4, N2O, halocarbons and the foreign-broadened H2O continuum: A GCM experiment. J. Geophys. Res., 104, 9467–9488, https://doi.org/10.1029/1999JD900003.
Sellers, W. D., 1969: A climate model based on the energy balance of the earth–atmosphere system. J. Appl. Meteor., 8, 392–400, https://doi.org/10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.
Shaw, T. A., and A. Voigt, 2016: What can moist thermodynamics tell us about circulation shifts in response to uniform warming? Geophys. Res. Lett., 43, 4566–4575, https://doi.org/10.1002/2016GL068712.
Shaw, T. A., and Z. Smith, 2022: The midlatitude response to polar sea ice loss: Idealized slab-ocean aquaplanet experiments with thermodynamic sea ice. J. Climate, 35, 2633–2649, https://doi.org/10.1175/JCLI-D-21-0508.1.
Shepherd, T. G., 2014: Atmospheric circulation as a source of uncertainty in climate change projections. Nat. Geosci., 7, 703–708, https://doi.org/10.1038/ngeo2253.
Smith, K. S., G. Boccaletti, C. C. Henning, I. Marinov, C.-Y. Tam, I. M. Held, and G. K.