On the Diffusivity of Moist Static Energy and Implications for the Polar Amplification Response to Climate Warming

Jian Lu aPacific Northwest National Laboratory, Richland, Washington

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Wenyu Zhou aPacific Northwest National Laboratory, Richland, Washington

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Hailu Kong bThe University of Chicago, Chicago, Illinois

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L. Ruby Leung aPacific Northwest National Laboratory, Richland, Washington

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Bryce Harrop aPacific Northwest National Laboratory, Richland, Washington

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Fengfei Song cOcean University of China, Qingdao, China
dQingdao National Laboratory for Marine Science and Technology, Qingdao, China

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Abstract

Energy balance models (EBMs) have been widely used in a range of climate problems, but the assumption of constant diffusivity in the parameterization of the moist static energy (MSE) flux can hardly be justified. We demonstrate in this study that the diffusive MSE flux can be derived from the basic energy balance equation with a few tolerable assumptions. The estimated diffusivity is both spatially and seasonally dependent, and its midlatitude average is then tested against several scaling theories for the midlatitude eddy diffusivity. The result supports the diffusivity theory of Held and Larichev modified for the moist atmosphere, affording a dynamics-based parameterization of MSE diffusivity. The implementation of the parameterization in an EBM leads to an interactive MSE diffusivity that accounts for the midlatitude eddy response to climate forcing perturbations. Under a uniform radiative forcing, the EBM with a diffusivity so parameterized produces a weakening of the midlatitude diffusivity and a modestly polar-amplified surface temperature response as an inevitable outcome under the dual constraints of the nonlinear Clausius–Clapeyron relation and the temperature gradient-dependent diffusivity, even in the absence of any poleward-amplifying radiative feedbacks. As the consequence of more isothermal temperature and reduced diffusivity, the variance of the midlatitude surface temperature also decreases with warming.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

H. Kong’s current affiliation: American Express Company, New York, New York

Corresponding author: Jian Lu, jian.lu@pnnl.gov

Abstract

Energy balance models (EBMs) have been widely used in a range of climate problems, but the assumption of constant diffusivity in the parameterization of the moist static energy (MSE) flux can hardly be justified. We demonstrate in this study that the diffusive MSE flux can be derived from the basic energy balance equation with a few tolerable assumptions. The estimated diffusivity is both spatially and seasonally dependent, and its midlatitude average is then tested against several scaling theories for the midlatitude eddy diffusivity. The result supports the diffusivity theory of Held and Larichev modified for the moist atmosphere, affording a dynamics-based parameterization of MSE diffusivity. The implementation of the parameterization in an EBM leads to an interactive MSE diffusivity that accounts for the midlatitude eddy response to climate forcing perturbations. Under a uniform radiative forcing, the EBM with a diffusivity so parameterized produces a weakening of the midlatitude diffusivity and a modestly polar-amplified surface temperature response as an inevitable outcome under the dual constraints of the nonlinear Clausius–Clapeyron relation and the temperature gradient-dependent diffusivity, even in the absence of any poleward-amplifying radiative feedbacks. As the consequence of more isothermal temperature and reduced diffusivity, the variance of the midlatitude surface temperature also decreases with warming.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

H. Kong’s current affiliation: American Express Company, New York, New York

Corresponding author: Jian Lu, jian.lu@pnnl.gov

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

A simple EBM is employed to investigate the effect of interactive MSE diffusivity on the polar-amplified temperature response. It is adopted from the EBM version in Hwang and Frierson (2010), which diffuses a surface MSE defined as m = CpTs + Lυq, where Ts is the surface temperature and q the corresponding specific humidity, Cp is the specific heat for dry air, and Lυ is latent heat for vaporization. The specific humidity is calculated based on Ts, assuming a constant 80% relative humidity. Like the traditional EBMs (e.g., Budyko 1968; North et al. 1983), the EBM here also adopts a linear-form OLR: OLR = aTsb. This choice allows us to only focus on the nonlinearity embedded in the CC relation and its interplay with the interactive diffusivity. The equation of the EBM is expressed as
mt{Kmm}=aTs+b+Fs+S0(1α),
where a = 1.49 W m−2 K−1, representing the Planck feedback at 295.5 K following the Stefan–Boltzmann law; b = 200.6 W m−2, chosen to maintain a temperature around 295.5 K in the tropics; S0 is the latitudinally dependent, annual mean incoming solar irradiance; α = 0.31 is the planetary albedo; Fs represents the net surface flux to the atmosphere and is set to be the same as the q-flux prescribed in the AQUA-AM2.1; and Km is the MSE diffusivity. Note that this choice of parameter a also gives a similar global mean surface temperature response to a similar radiative perturbation as in AQUA-AM2.1.
A distinct feature of the EBM here is that the MSE diffusivity is interactive and latitudinally dependent. The interactive MSE diffusivity is devised for the EBM in a self-contained manner; all the parameters used in the closure are either prescribed constants or computed in the EBM itself. Leveraging the modest success of the modified Barry scaling (to be detailed in section 3c and appendix A; see Fig. 3f), we scale the magnitude of the extratropical diffusivity using Eq. (A13b) within an adjustable constant (k˜):
KST=k˜(μCpT0)3/2β2(Tsy)3,
where both β and Ts/y are averaged between 25° and 55°, and T0 indicates the global mean of the model-generated Ts (other parameters are listed in Table 1). Parameterized as such, KST increases sharply with the temperature gradient and decreases with polar-amplified global warming. However, this is only valid for the diffusivity in midlatitudes dominated by the eddies. For the tropics, energy transport is governed by a fundamentally different dynamical process. Here we introduce a tropical diffusivity KHC and assume it to be proportional to the HC streamfunction (ψHC), which is in turn assumed to follow the scaling of Held and Hou (1980) for the strength of the HC (see also Schneider 2006):
KHCc0ψHCc0ρ0a2H¯τ(LrΔhT0)3/2ΔhΔυ,
where Lr=gH¯/(Ω2a2), with H¯ being the tropical mean tropopause height, set to be constant; τ is the Newtonian relaxation time scale, Δh is the temperature difference between equator and 30° latitude, and T0 is the same as defined above. For simplicity, only Δh and T0 vary in the EBM model; other parameters are fixed to be constant. Coincidentally, KHC also has a 3/2-power dependence on T01. In both parameterizations for KHC and KST, there is an adjustable constant (c0 and k˜, respectively), which is chosen to maintain the main characteristics of the diffusivity diagnosed from the AQUA-AM2.1 (Fig. 1b). Other constants are adopted from Schneider (2006) and all constants are enumerated in Table 1.
Fig. 1.
Fig. 1.

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

Table 1

Parameters for interactive MSE diffusivity in EBM.

Table 1
To mimic the meridional structure of the diffusivity diagnosed from the AGCM (Fig. 3), the latitude-dependent diffusivity in each hemisphere comprises two continuously connected segments, one between the equator and the peak latitude ϕp, another between ϕp and the pole, as follows:
Km(ϕ)={KSTKSTKHC2(1+cosϕΔϕ1π),if |ϕ|ϕp,KSTcosϕϕp2Δϕ2π,if |ϕ|>ϕp,,
where ϕp = 35° is the peak latitude of the diffusivity; Δϕ1 = 35° and Δϕ2 = 55°.

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

Let us start with the vertically integrated energy balance equation for the global atmosphere:
m*t+v*m*=F,
where m* denotes MSE, v* is the horizontal velocity vector, and the angle brackets indicate the density-weighted vertical integration; F indicates the net energy input to the column, estimated from the net energy flux from both the top of the atmosphere and the surface (Neelin and Held 1987). By introducing an MSE-weighted velocity
vv*m*/m*,
Eq. (5) can be rewritten as
mt+(vm)=F,
with m representing the vertically integrated MSE. We define the climatology as the multiyear time average for a given point of the year and the deviation therefrom as the perturbation. The seasonally evolving energy balance equation can then be written as
m¯t+(v¯m¯+vm¯)=F¯,
where the overline indicates climatology for a given point of the year.
The mixing length treatment of the eddy component allows us to express the eddy flux term vm¯ in a general diffusive form:
vm¯≈ Km¯,
with K being the two-dimensional diffusivity tensor, consisting of a symmetric and an antisymmetric component:
K=KS+KA=KS[1001]+KA[0110].
An assumption of eddy isotropy has been made in the second equality above. As a result, only two scalar coefficients are needed to fully represent the eddy diffusive effect. Expressed as such, the scalar KA can be physically interpreted as the streamfunction of the advection velocity due to eddies, or the eddy-induced part of the transformed Eulerian-mean velocity on the horizontal plane.
In the midlatitudes, the MSE transport is dominated by synoptic eddies and downgradient diffusion treatment in the form of (9) is justifiable. However, this is not the case in the tropics, where the overturning Hadley circulation is the dominant mechanism for poleward energy transport (e.g., Oort and Peixoto 1983; Neelin and Held 1987). How can a diffusive form energy transport for the tropics be rationalized then? The Helmholtz decomposition theorem allows us to decompose velocity vector v into rotational (vψ) and divergent (vχ) components. Ignoring the advection effect of the divergent component, we may decompose the MSE divergence by the mean circulation as follows:
(v¯m¯)=[(v¯ψ+v¯χ)m¯]v¯ψm¯+m¯v¯χ.
Expressing the rotational mean wind with a streamfunction Kψ, then the first term above can be written in a tensor form:
v¯ψm¯=(Kψ[0110]m¯).
If one assumes that the divergent mean wind blows down the gradient of the mean MSE in the form vχ≈  Kχlnm¯, an ad hoc assumption reflecting qualitatively the behavior of the tropical mean circulation, one may write the second term as m¯v¯χm¯(Kχlnm¯), which can be further approximated to be
m¯v¯χ(Kχm¯)
under the assumption that Kχ varies more rapidly than m¯ in space—a weak “temperature” gradient of sorts in the MSE context [in the same spirit of the assumption made by Neelin and Held (1987) to obtain their Eq. (2.12)]. Substituting (12) and (13) into (11), one can arrive at a diffusive expression for the divergence of the mean MSE transport. Organizing all the terms together, we finally yield a tensor diffusive energy balance model:
m¯t{((KS+Kχ)[0110]+(KAKψ)×[0110])m¯}=F¯.
Note here that the scalar coefficients KS + Kχ and KAKψ are dependent on the geographic locations. The diffusivity coefficients in a two-dimensional space can be estimated as an inverse problem via optimization if the fields of MSE and the net energy input are known either from observation or model experiments [see Lu et al. (2021) for an example]. This allows us to build diffusive EBMs for energy transport on a two-dimensional, horizontal plane, with more real-world applications.
Zonally averaging (14) would result in the familiar one-dimensional EBM with a scalar but latitudinally varying diffusivity (e.g., Budyko 1968; Sellers 1969; North 1975):
m¯¯t{KZm¯¯}=F¯¯,
where the double overbars (=) represent the zonal and climatological mean and a subscript Z is used to denote the zonal-mean attribute of the diffusivity. Despite the assumptions made in deriving (15), from its expression one can see that KZ can be uniquely diagnosed from m¯¯ and F¯¯.

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 (m¯¯) and the column net energy input (F¯¯). The simplex method for function minimization of Nelder and Mead (1965) is used to find the global minimum of the cost function that is defined as the Euclidian distance between the right- and left-hand sides of Eq. (15). To avoid overfitting and instability of the algorithm, all quantities are represented by 42 Legendre polynomials. Since it is the seasonal evolution of the MSE diffusivity that is of interest here, we utilize the optimization algorithm to fit for the diffusivity KZ for each month using the monthly climatological values of m¯¯ and F¯¯ computed from the 20-yr-long simulation of AQUA-AM2.1. At a monthly time scale, the atmospheric energy is not in equilibrium, and the tendency term m¯¯/t must be accounted for in computing the cost function.

To verify the fitted monthly diffusivity above, we also compute the MSE diffusivity directly for each month using
KZ=(cosϕm¯¯ϕ)1   −π/2ϕdϕ(F¯¯m¯¯t)a2cosϕ,
where ϕ is latitude and a is the radius of Earth [following Frierson et al. (2007)]. An animation of the resultant monthly evolving, latitude-dependent diffusivity for the AQUA-AM2.1 control simulation from both approaches can be viewed in the online supplemental material. The agreement between the fitted and the directly computed KZ is to our satisfaction, so the following discussion of the behavior of KZ will be mainly focused on the former, unless explicitly stated. The advantage of the former approach is that it overcomes the problem of singularity from division by the near-zero m¯¯/ϕ in the direct approach. For the two-dimensional case, the diffusion coefficients KS + Kχ and KAKψ are difficult to compute directly without accurate estimation of the horizontal energy transport, so an optimization-based approach can be a useful alternative for their estimation.

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).

Fig. 2.
Fig. 2.

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.

Fig. 3.
Fig. 3.

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.

Table 2

Summary of scaling laws for eddy diffusivity.

Table 2

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 LR=2V/β, where V is the root-mean-square wind, only the meridional component of the eddy wind is used. The eddy component is defined as the deviation from the monthly climatological and zonal mean. Note that since the eddy moisture flux and eddy heat flux are related via the Clausius–Clapeyron relation, the diffusivity of moisture (denoted by KQ) follows the same scaling law as KT. (This has been confirmed by the direct computation shown in Fig. 5b.) Therefore, only the scaling results for KT are reported in the log–log diagrams in Fig. 4.

Fig. 4.
Fig. 4.

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 LR=2V/β, a predictive scaling theory for diffusivity would be one that can predict either one of these two scales. The equipartition-based Green70 scaling with a fixed domain scale underestimates the rate of the increase of KT from summer to winter (Fig. 4a); it also predicts disparate values for spring and autumn months, owing mostly to the different static stability values between the two seasons. The original (dry) HL96 scaling inherits similar dependence on the static stability to the Green70 scaling, whereas it tends to overestimate the rate of the increase of the diffusivity due to its cubic relationship with the wind shear (Fig. 4b). Similar results were found in Barry et al. (2002) in their aquaplanet model simulations. However, the moist HL96 scaling accounting for the effect of moisture through the effective static stability [N*2=[(1L)/(1+CL)]N2; see appendix A] shows a much-improved agreement with the “ground truth” of KT (Fig. 4c). It is thus not the assumption of the Rhines scale, but the use of the dry static stability in the original scaling, that is responsible for the overprediction of the scaling slope. The conversion factor (1L)/(1+CL) from N2 to N*2 is smallest in summer and largest in winter, thus bringing the moist HL96 scaling to a much better conformity with the actual KT.

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 N*2 can be developed. The middle way between the Barry02 and the moist HL96 scalings is the formula (A13b) (see Table 2 and appendix A), which is derived from the Barry02 scaling, whereas maintaining the same power-law relationships with β and ∂U/∂z as in HL96. The result of scaling [Eq. (A13b); see also Eq. (2)] is shown in Fig. 4f. Although not as skillful as either moist HL96 or Barry02, it serves as a working starting point for possible closure of the MSE transport in EBMs. Particularly, all the parameters in Eq. (A13b) can be simulated in the EBM. Another appeal of this formula is that it can capture the overall reduction of the diffusivity with climate warming, a key element to consider in understanding the origin of the polar-amplified warming to a uniform forcing in the simple EBM here (to be detailed in section 5).

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.

Fig. 5.
Fig. 5.

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).

For the air mass poleward of the interfacial latitude ϕI, equatorward of which the air is being heated and poleward of which the air is being cooled, the net loss of heat is balanced by the poleward heat flux:
KTdT˜dy=a(π2ϕI)T˜eT˜τT,
where the tilde denotes the areal average from latitude ϕI to the pole, and the diabatic cooling has been expressed as a relaxation toward a target value T˜e The corresponding moisture balance equation is similarly written as
KQdQ˜dy=a(π2ϕI)Q˜eQ˜τQ.
In the expressions above, T˜e and Q˜e can be interpreted as the values at radiative-convective equilibrium; in the absence of the poleward transport, they must be smaller than their corresponding values with the transport (symbols without the subscript e). The relaxation time scale for moisture and temperature can differ substantially, as they are governed by different physical processes. Note that the equation pair above can also be derived from the entropy budget for a latitudinal range enclosed between the peak latitude of the heat/moisture transport and the pole. Thus, for a given macroscopic temperature difference between the warm and cool reservoirs, diffusivity KT also signifies the rate of entropy production.
Assuming that the column moisture in radiative convective equilibrium and that realized both follow the CC relation, we can rewrite (16b) as
KQγCCdT˜dy=a(π2ϕI)γCCT˜eT˜τQ,
where γCC=[(rLυQ˜0)/(RwυT˜02)]eLυ/(T˜0Rwυ) is the CC rate, the rate of change of saturation specific humidity with temperature at reference temperature T˜0 and reference specific humidity level Q˜0, and r is relative humidity, with other variables assuming their conventional meanings. Dividing (17) by (16a) leads to the following proportionality:
KQKTτTτQ
with an immediate insight that the ratio between the diffusivity of moisture versus temperature is set by the ratio between the relaxation time scale for temperature versus moisture. Physically, this implies that the mixing effect of a certain tracer over the distance of a mixing length is greater if the gradient of the tracer is more rapidly restored.

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 (υT¯). According to the diffusivity formulation (A13), the only possible change of diffusivity compatible with the reduction of υT¯ is a reduction in diffusivity, contradicting the starting point of the reasoning. This leaves the following two possible scenarios.

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.

To isolate the nonlinear effect in the CC relation on the PA, we first fit a third-order polynomial to the nonlinear relationship between TS and m:
Ts=(9.7297×108m+4.9287)+(5.9506×1027m3+8.1028×1017m22.8006×107m+286.6861)
Here we deliberately break up the polynomial into to a linear part and the remainder therefrom. The linear part fits the original TSm relationship well for m ≤ 2.5 × 109 J, its extrapolation to larger m values leads to the linearization of the TSm relationship (purple line in Fig. 6), which is equivalent to ignoring the nonlinearity in the CC relation. Between the polynomial fit and the linearized relation, we rescale the nonlinear component (i.e., terms in the second pair of parentheses) by a factor of 1/2 and 1/4 for two weakened nonlinear relationships between TS and m. These four different portrayals of the TSm relationship allow us to investigate systematically the effect of the CC nonlinearity on the PA of the surface temperature response. Further, to investigate the sensitivity to MSE diffusivity, for each set of the CC relation modifications we also scale the interactive diffusivity by a factor of 1/2, 1, 2, and 4, respectively. The results of the polar amplification factor (PAF) are presented in Fig. 7, computed as the natural logarithm of the ratio of the TS warming over polar region (60°–90°) to that over the tropics (0°–30°) in response to a uniform 10 W m−2 solar radiative forcing.
Fig. 6.
Fig. 6.

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 mTS 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 mTS relationship.

Citation: Journal of Climate 35, 21; 10.1175/JCLI-D-21-0721.1

Fig. 7.
Fig. 7.

The logarithm of the PAF simulated in the EBM by systematically rescaling the nonlinear component in the mTS 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 TSm 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 (Ty3), which acts to cushion the effect of whatever source of spatial inhomogeneity. If the diffusivity is not allowed to respond while being forced by the 10 W m−2 radiative perturbation, the magnitude of PAF is very much boosted (by up to a factor of 5 on average) for the cases of full CC nonlinearity (leftmost column in Fig. 7b; note the change of the scale of the color bar). Reducing the CC nonlinearity can markedly reduce the PAF, and a fully linearized mTS relationship leads to the complete disappearance of the PA (see the rightmost column in Fig. 7b). Taken together, the results in Fig. 7 indicate that the CC nonlinearity is the root cause for the polar-amplified response in temperature in the absence of any spatially varying feedbacks. A very similar argument has been put forward by Merlis and Henry (2018), but from a different angle. They recast the downgradient MSE diffusion as the diffusion of dry static energy, with the effective diffusivity thereof being spatially varying as a function of the background temperature to account for the effect of the latent energy diffusion. The effective diffusivity so expressed increases exponentially with temperature as required by the CC relation, and thus a warmer climate would imply an enhanced effective diffusivity for the dry static energy transport and hence a more isothermal atmosphere (see also Flannery 1984). The physical insight into the essential role of the CC nonlinearity in PA gained from the two approaches is the same.

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 υ(T¯/y) for temperature extremes and υ(Q¯/y) for hydrological extremes. Under the conditions of a more isothermal atmosphere and weakened eddy motion, the temperature fluctuations can only be reduced, corroborating the results from analyzing the realistic climate models (e.g., Schneider et al. 2015; Gao et al. 2015; Dai and Deng 2021). Any argument for more extreme midlatitude cold events (e.g., Francis and Vavrus 2012; Cohen et al. 2014) under climate warming must overcome not only the warmer background mean temperature but also the reduction of the temperature variance under the dual dynamic-thermodynamic constraints discussed above. For the hydrological extremes, the outcome will be the result of the tug-of-war between the increased background moisture gradient (despite the weakened temperature gradient) and the decreased eddy activity; it will be further obfuscated by the poleward shift of the jet stream and storm track (e.g., Lorenz and DeWeaver 2007; Lu et al. 2017; Smith et al. 2021).

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.

1

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).

A1

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 (Umax/β)1/2 by the Rayleigh–Kuo stability criterion (Kuo 1949) and organizes a storm track in such a way that the eddy scales are confined by the width of the jet.

A2

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.

Fig. A1.
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

It proves convenient to lay out the premise for the turbulence argument for diffusivity in the context of a two-layer QG turbulence model for a Boussinesq fluid on a β plane, consisting only of the barotropic and the first baroclinic mode. In this model, the energy source of eddies arises from baroclinic instability, which injects the eddy kinetic energy (EKE) to the system at the scale of the baroclinic Rossby radius, the same scale at which energy is transferred to the barotropic mode in the form of EKE (e.g., Vallis 2006). However, the barotropic EKE does not stay at the scale of the injection; the net effect of nonlinear eddy–eddy interaction in 2D turbulence that governs the barotropic dynamics is to cascade the EKE to larger spatial scales to be finally dissipated by surface friction. According to the Kolmogorov–Kraichnan theory for the inertial range of 2D turbulence (Kraichnan 1967), the EKE for the scales larger than the injection scale must cascade upscale along a k5/3 spectral slope. The upscale cascade must be halted somewhere; the classic theory of Rhines (1975) predicts the halting scale to be where the eddies are large enough to begin to feel the β effect. This crossover scale between the turbulence cascade regime and the Rossby wave regime, known as the Rhines scale, can thus be determined by matching the frequency satisfying the turbulence dispersion relationship (ω = Vk) with that of the Rossby wave dispersion relationship [ω = −β/(2k), assuming isotropy] (Vallis and Maltrud 1993):
LR=(2V/β)1/2.
Here V is the eddy velocity in the root-mean-square sense. Since any spectrum steeper than k1 will cause the bulk of the energy to be concentrated at the halting scale (Held 1999), the Rhines theory, if valid, would determine an energy containing scale—the dominant scale for the eddy diffusivity. Another important property of the two-layer QG model on a β plane is that for scales significantly larger than the Rossby radius, the baroclinic potential vorticity (PV, which is proportional to the thickness/temperature in the two-layer model) behaves like a passive tracer, advected by the flow associated with the barotropic PV without significantly modifying the flow by which it is being advected (Rhines 1977; Salmon 1978, 1980). As a result, the eddy flux of the baroclinic PV, or equivalently the thickness in two-layer QG model, can be described as a “downgradient diffusion,” engendered by the barotropic flow at the energy containing scale. In the two-layer model, the diffusivity can be regarded as being vertically uniform, applicable to both the baroclinic PV and buoyancy (Held 1978). In the continuously stratified atmosphere, this depiction of diffusive flux is more suitable for the lower troposphere, especially around the steering level (where the intrinsic phase speed approaches the background mean zonal wind).
With the backdrop above, one may proceed with the scaling arguments for diffusivity. Dimensional consideration leads to the scaling of diffusivity as the product of a length scale and a velocity scale (e.g., Kushner and Held 1998). Letting the Rhines scale be the energy containing length scale, a scaling for the eddy diffusivity can be readily obtained from (A1):
DVLV3β.
Since V can be computed directly from the observations or model simulations, (A2) can be used to validate the Rhines theory for the diffusivity (see Fig. A1 for its relationship with other scaling arguments). This is also the scaling adopted by Frierson et al. (2007) to produce a good fit to the effective diffusivity in a moist AGCM under a wide range of perturbations of the moisture content.

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.

A complete closure for eddy heat flux must parameterize both the length scale and the velocity scale. This can be achieved through simple dimensional analysis for 2D turbulence on a β plane. For a turbulence regime where friction remains only necessary to dissipate kinetic energy, but not to interfere the turbulence properties within the inertial range, the only relevant quantities for eddy scales are the parameter β and the energy cascade rate across the inertial range, which, in equilibrium, must be the same as the energy dissipation rate (ϵ), and therefore
L=L(ϵ,β);V=V(ϵ,β)
Since friction is not involved in determining these scales, the only dimensionally feasible scaling for the length and velocity scales are, respectively,
Lβϵ1/5β3/5 and
Vβϵ2/5β1/5,
where subscript β denotes that they correspond to the β scale in the geostrophic turbulence literature when turbulence resides in the so-called β regime (e.g., Vallis and Maltrud 1993; Smith et al. 2002). The product of the two scales leads to the celebrated scaling for the diffusivity in the turbulence literature:
Dϵ3/5β4/5,
which serves a pivot point for many (e.g., Barry et al. 2002; Held and Larichev 1996; Maltrud and Vallis 1991; Vallis and Maltrud 1993; Panetta 1993; Spall 2000) to develop a working scaling argument for geostrophic turbulence under different external parameters (see Fig. A1).
For example, treating the midlatitude storm track as a heat engine that obtains its energy from transporting heat from a warm to a cold reservoir, Barry et al. (2002) proposed a scaling relation in which the production/dissipation of kinetic energy ϵ is proportional to the entropy production of the storm track: μ(δT/T0)q, where q is the energy transport rate from the warm reservoir to the cold one; δT and T0 indicate the temperature difference and the mean of the two reservoirs, respectively; the multiplicative constant μ is a tunable parameter, representing a utilization coefficient. Assuming δTaT/∂y, where a is the radius of Earth, (A4) can then be written as
D(μaqT0Ty)3/5(2β)4/5.
This final form of scaling for heat diffusivity was shown by Barry et al. (2002) to have the best skill in capturing the variation of the diffusivity under a range of perturbations of solar radiation, planetary radius, rotation rate, surface temperature gradient, and mean surface temperature (see their Fig. 3). Formula (A5) is referred to as the Barry02 scaling here as well as in the main text. Note that (A5) has yet to be a predictive scaling, since it still requires information about the poleward energy transport rate q or the differential heating across the storm track, a point we will return to later.
Building on the relationship between baroclinic production and kinetic energy dissipation, Held and Larichev (1996) proposed an alternative scaling for diffusivity. For the storm track governed by QG dynamics, the only significant energy source is the baroclinic production through downgradient heat fluxes, and energy is predominantly dissipated at large scales through friction. Thus, it is feasible to assume a proportionality between the kinetic energy dissipation and the baroclinic production:
ϵϵp=υb¯B/yB/z.
Here, B is the zonal-mean buoyancy, υb¯ is the eddy buoyancy flux, and angle brackets denote an average over latitude and height. Substituting a downgradient form of buoyancy flux υb¯=DB/y into (A6) yields
ϵD(By)2(Bz)1=DT2,
where T1 is an inverse time scale proportional to the Eady growth rate f Ri−1/2, with Ri = N2(∂U/∂z)−2 being the Richardson number. Using (A4) and (A7) to eliminate ϵ, one obtains another scaling relation for D:
Dβ2T3=β2λ3U3,
where λ = NH/f is the Rossby deformation radius and U is the mean thermal wind, and all quantities here should be understood in a tropospherically averaged sense. This scaling argument can also be derived from generalizing the PV diffusion in a homogeneous, baroclinically unstable flow from an f plane to a β plane, assuming the dominance of the temperature component in the background PV gradient. Relation (A8) is referred to as HL96 scaling in this study. It is noted that for a two-layer QG model the lower-level PV flux can be well approximated by the eddy buoyancy flux and hence the diffusivity for the lower-level PV and buoyancy are interchangeable. To be detailed below, (A8) can also be derived from the Green (1970) scaling by equating the length scale therein to the Rhines scale. The HL96 scaling is appealing as 1) it is rooted in the PV flux closure of geostrophic turbulence and 2) there is no explicit dependence on any eddy term, only the dependence on the external parameters and background mean fields. Barry et al. (2002) tested scaling relation (A8) against their aquaplanet model simulations, but found it gave too steep a fitting slope compared to the Barry02 scaling (see also Fig. 4 herein).
An apparent caveat of the HL96 scaling is that it is based on the energy production and PV closure for dry dynamics. In the moist atmosphere, the condensation in the updraft portion of the eddy motion acts to reduce the static stability felt by the eddies. Therefore, an effective static stability accounting for the moist condensation effect is more appropriate in estimating λ = NH/f in (A8). This has been achieved by Lapeyre and Held (2004), who applied a modification to the static stability
N*2=1L1+CLN2
to allow for the interpretation of the diffusivity properties for moist atmosphere with the equivalent dry scaling theory. Here, C is a nondimensional, linearization parameter of the CC relation, and it can be interpreted as the absolute value of the ratio of the slope of the isolines of the mean mixing ratio to the mean isentropic slope in the y–z plane; L is the nondimensionalized latent heat. According to these authors, C is estimated to be 2 for the typical midlatitude atmosphere, and L to increase linearly from 0.2 in winter to 0.35 in summer. Scaling (A8) with the modified static stability N* will be referenced as moist HL96 scaling.

b. Classic approaches for scaling the eddy diffusivity

An existential challenge to the scaling arguments above is the fact that the −5/3 inertial range for inverse cascade does not pan out in Earth’s atmosphere (Straus and Ditlevsen 1999; Schneider and Walker 2006). Thus, despite the fact that the Rhines scale coincides with the energy-containing scale for fluid suitable for midlatitude dynamics, the physical meaning of the Rhines scale should be interpreted with caution.A1 Through simulations of continuously stratified atmosphere over a range of parameters, Schneider (2004) and Schneider and Walker (2006) found that the static stability adjusts to prevent the supercriticality from significantly exceeding one and hence to inhibit a strong inverse cascade and the −5/3 inertial range, consistent with the baroclinic adjustment theory (Stone 1978; Held 1982). In the absence of a strong upscale cascade in the real atmosphere, an alternative path toward a closure for the eddy heat flux is to fall back to the classic linear instability thinking. For the linearly most unstable waves, the eddy kinetic energy and eddy available potential energy are equipartitioned.A2 Therefore, the velocity scale is set by the eddy available energy, that is,
VbN=LeByN=LefUz.
Note that the thermal wind relationship has been applied in the second equality. Setting the length scale Le in (A9) to be Rossby radius leads to the classic scaling of Stone (1972):
DVλNH2fUz.
On the other hand, specifying Le in (A9) to be a fixed domain size, we arrive at a diffusivity equivalent to that of Green (1970):
DVLdLd2fNUz,
where Ld denotes the domain size. One can see that the Green (1970) and Stone (1972) scaling show opposite sensitivity to the static stability. We must point out that the linear relationship with N in the Stone (1972) scaling has not borne out in many modeling studies (e.g., Barry et al. 2002; Schneider 2004); it is therefore very unlikely that the Rossby radius is the mixing length relevant to the diffusivity. An interesting turn of fortune follows if the mixing scale in (A9) is set to be the Rhines scale and it is in turn determined by the equipartitioning velocity. Expressing D as the product of the velocity scale times the corresponding Rhines scale gives
DVLRβ2f3N3(Uz)3,
which, intriguingly, restores the HL96 scaling [cf. Eq. (A8)]!

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.

Finally, even the Barry02 scaling can be linked to HL96. Going back to formula (A5) and expressing the eddy heat transport in a downgradient diffusion form:
qa1CpυT¯=a1CpDTy,
we arrive at the following relationship after a simple manipulation:
Dκ5/2(μCpT0)3/2(2β)2Ty3
or
D4κ5/2(μCpT0f2H2R2)3/2β2(Uz)3
via the thermal wind relation. The latter expression recovers the power-law relationships with β and ∂U/∂z of the HL96 scaling. Its equivalence to the moist HL96 scaling can be achieved simply by assuming a proportionality μCp/T0R2/(N*2H2), in which R is the gas constant of the dry air. This is equivalent to assuming the normalized utilization coefficient (by the mean temperature) of the storm track as a heat engine is set by the inverse of the effective static stability. Although the derivation of (A13) is less dynamical compared to HL96, the absence of the static stability factor makes it more readily to be implemented in a diffusive EBM (as done in section 5).

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

D

Midlatitude eddy diffusivity used in scaling theories

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