## 1. Introduction

Simple diffusive energy balance climate models that predict the distribution of zonal mean surface temperatures on Earth have a long history and have been used to study a variety of questions, the abrupt transition to snowball Earth as the climate cools being a classic example (e.g., North 1975 and references therein). There has been a resurgence of interest in this kind of simple model in recent years, in part to rationalize the changes in poleward heat flux in comprehensive models as parameters or external forcings are varied (e.g., Frierson et al. 2007; Hwang and Frierson 2010; Hwang et al. 2011; Rose et al. 2014; Roe et al. 2015; Armour et al. 2019; Mooring and Shaw 2020). Most of these models diffuse moist enthalpy, rather than dry enthalpy (temperature) in isolation, following the original suggestion of Flannery (1984). In some cases, simple theories for the diffusivity are proposed (e.g., Frierson et al. 2007), but more commonly the diffusivity is simply assumed to be a constant that is empirically determined or taken from a control simulation (e.g., Hwang and Frierson 2010; Roe et al. 2015).

There remains a lack of consensus on the theoretical understanding of the optimal diffusivity for use in such models. Here we examine a path toward a theory for the diffusivity that couples a dynamical constraint borrowed from two-dimensional turbulence theories, relating the diffusivity to kinetic energy dissipation, with an entropy budget that connects kinetic energy dissipation and the temperature difference between regions in which the atmosphere is heated and cooled by the circulation. This approach is motivated by two specific existing theories, Held and Larichev (1996, hereafter HL96) and Barry et al. (2002, hereafter BCT02). One of our goals is to better understand the relationship between these two theories and see if they can be improved. Another goal is to shed some light on when the assumption of constant diffusivity is adequate.

We use a dry idealized general circulation model on the sphere, that of Held and Suarez (1994), to test diffusivity theories. We are interested in the HL96 and BCT02 theories in part because the way in which they bring in the entropy budget suggests a generalization to a moist atmosphere. However, here we analyze a dry general circulation model (GCM) only. In addition, for the simulations described in the main text, we vary only one parameter in this model, the meridional temperature gradient in radiative equilibrium. The HL96 and BCT02 theories differ in their dependence on the static stability, and it is known from previous work (Schneider and Walker 2006; Zurita-Gotor 2008; Chai and Vallis 2014) that large changes in static stability are generated when the model’s meridional temperature gradient is forced to increase or decrease. Therefore, varying the meridional temperature gradient in radiative equilibrium seems to be a logical place to focus in trying to evaluate these theories. In the spirit of connecting to the simplest diffusive models, we focus on an overall global scaling for the diffusivity rather than a local, spatially varying diffusivity. This is a useful starting point because the meridional structure of the eddy and total heat transport do not change markedly with this particular change of parameter. The model and the diffusivity that it produces are described in section 2.

The HL96 and BCT02 theories can be thought of as consisting of two parts. The first part, shared by both theories, is an expression for the diffusivity as a function of the kinetic energy dissipation. The second part, in which the two theories differ, relates the kinetic energy dissipation to the global entropy production due to the irreversible frictional heating and, therefore, to radiative entropy destruction, and then back to the energy transport and the diffusivity. The first part is introduced in section 3. The second part of the argument is presented in section 4 in such a way as to generalize HL96 and BCT02. The resulting diffusivity theory is then presented in section 5. The GCM simulations are compared with these theoretical relationships within each of these sections. Section 6 presents a possible extension of the theory to moist atmospheres, and describes qualitatively how diffusivities in moist and dry atmospheres should differ. Finally, our conclusions are given in section 7, where we also return to problems that prevent these arguments from being definitive.

It will also become clear in the following that the planetary vorticity gradient is central to the theory presented in this study. The planetary vorticity gradient is introduced into the theory from two-dimensional turbulence theories. To better understand its effect on the validity of the proposed theory, a set of GCM simulations with variations in the planetary rotation rate has been discussed in the first author’s thesis (Chang 2019). These results raise a variety of additional issues on the uncertainty of extending two-dimensional turbulence theories to a broader parameter space, which seem to be better tackled in models with a lower complexity. Therefore, we choose not to address them here, but include some noteworthy parts of the results in appendix A for reader’s reference. The calculations for the model’s global entropy budget are also described in appendix B.

## 2. The model and its diffusivity

We conduct our numerical simulations using the Geophysical Fluid Dynamics Laboratory idealized global spectral atmospheric model. Its dynamical core solves for the primitive equations on the sphere with no surface topography, and it is run at T85 resolution with 30 equally spaced *σ* levels in the vertical (where *σ* is the model’s vertical coordinate defined as the ratio of the pressure at a given level and the surface pressure). This model’s idealized physics is identical to that described in Held and Suarez (1994). We review some features of the setup that are important for this work.

*Q*in the model is parameterized as Newtonian relaxation that damps the temperature

*T*back to the prescribed radiative equilibrium value

*T*

_{eq}, i.e.,

*k*is the inverse of diabatic forcing time scale and

_{T}*c*is the specific heat capacity at a constant pressure. The standard setting for

_{p}*T*

_{eq}is

*p*is pressure,

*κ*=

*R*/

*c*, and

_{p}*R*is the ideal gas constant. The constants are chosen as

*p*

_{0}= 1000 hPa, Δ

*= 60 K, and Δ*

_{y}*= 10 K. Thus, the radiative equilibrium state is statically stable in the tropics and slowly approaches to neutral in the extratropics. Any further stabilization is caused by the large-scale dynamics. In addition,*

_{z}*k*is defined as

_{T}*σ*= 0.7,

_{b}*k*is designed to represent radiative cooling, while a shorter time scale

_{a}*k*is imposed in the nominal boundary layer (

_{s}*σ*≥

*σ*) to mimic the surface heat fluxes and to avoid an unrealistically strong inversion at the surface. There is no diurnal or seasonal cycle in diabatic heating, and both the radiative equilibrium and the statistically steady-state climates are zonally symmetric. A Rayleigh drag is implemented near the surface to account for the effect of bottom friction. For numerical stability, ∇

_{b}^{8}hyperdiffusion is also operated on horizontal momentum (vorticity and divergence) and temperature fields in the model. To conserve global energy, Rayleigh frictional heating is directly added back to the local temperature budget, and numerical energy loss is compensated by a spatially uniform temperature correction at each model time step.

We refer to the simulation with the standard setting in Held and Suarez (1994) as our control simulation. A series of simulations where we vary the meridional temperature gradient at radiative equilibrium, i.e., Δ* _{y}* in Eq. (2), from 10 to 180 K is examined. All the diagnostics presented are calculated from the last 1000 days in a 1500-day integration.

Our main interest is a bulk diffusivity *D* that can represent the globally averaged poleward heat transport, but there is ambiguity on how best to evaluate this quantity. We calculate *D* in the following analysis as the hemispherically averaged column-integrated meridional dry static energy flux, divided by hemispherically averaged column-integrated mean meridional dry static energy gradient. The hemispherical average is performed with surface area weighting. The column mass-weighted integration is taken from *σ* = 0.25 to *σ* = 1. While the details of this choice of averaging are somewhat arbitrary, and they have some effect on the magnitude of the diffusivity in the control, they have little effect on the scaling with Δ* _{y}*, in large part because these global means primarily reflect midlatitude values where the peaks in the flux and the meridional gradient occur.

Figure 1 shows the diffusivity *D* (in units of length^{2} time^{−1}) from the idealized GCM as a function of the mean meridional temperature gradient ∂* _{y}T*|

*, both computed using the choice of hemispherical and vertical averaging described above. We show the result when computing*

_{p}*D*with the total flux of dry static energy (indicated by the subscript “total”;

*D*

_{total}) and the transient eddy flux only (indicated by the subscript “eddy”;

*D*

_{eddy}). Their difference is primarily due to the transport by the Ferrel cell rather than the Hadley cell (note that there are no stationary eddies in this idealized GCM). The Ferrel cell compensates part of the transient poleward flux and increases more rapidly with the temperature gradient than the transient flux, so the total flux increases more slowly than the transient flux. Both diffusivities increase roughly proportional to the 3/2 power of ∂

*|*

_{y}T*for the smaller values of Δ*

_{p}*(corresponding to a flux proportional to the 5/2 power of the gradient), but with much weaker dependence at the larger values of Δ*

_{y}*examined. A diffusivity that predicts the total flux is the most relevant for use in the simplest diffusive energy balance models, but we will need to return to the appropriateness of comparing different theories to the transient eddy or the total flux in the following.*

_{y}## 3. Rhines scaling

*β*plane. An attractive scaling theory proposed by earlier work (e.g., Smith et al. 2002) assumes that the key dimensional parameters are the planetary vorticity gradient

*β*(in units of length

^{−1}time

^{−1}) and the rate of kinetic energy dissipation

*ϵ*(in units of length

^{2}time

^{−3}). If there is a substantial classical inverse energy cascade,

*ϵ*can also be thought of as the strength of the energy cascade rate or the rate of dissipation of the kinetic energy on large scales in the surface boundary layer. Dimensional analysis then yields for the kinematic diffusivity,

*V*and

*L*, respectively, and then set

*D*∼

*VL*:

*L*to

*V*, or equivalently the eddy kinetic energy,

*V*

^{2}, as in Rhines (1975). Using these scales we can also write

*β*dependence in these scaling relations, we have examined an additional set of dry GCM simulations with varying planetary rotation rates and indeed identified some aspects of their limitations when applying to different parameter regimes (appendix A). For an Earthlike environment with strong horizontal inhomogeneity, a modest inverse cascade and few vortices, the quantitative relevance of these relations also remains unclear. Despite these reservations, we view evaluations with full Earthlike GCMs, as in BCT02, as encouraging, and as will be disclosed in the following, they indeed explain our simulations with varying meridional temperature gradient very well. For conciseness, we refer to these relations between diffusivity, kinetic energy dissipation, eddy kinetic energy, and

*β*in Eqs. (4)–(7) as

*Rhines scaling*.

We now formally check the extent to which these scaling arguments hold in the dry GCM when Δ* _{y}* is varied. We start in Fig. 2 with the relation in Eq. (5) between the eddy kinetic energy

*V*

^{2}and the kinetic energy dissipation

*ϵ*. Here we define

*V*

^{2}as the global mean eddy kinetic energy averaged over the troposphere (from

*σ*= 0.25 to

*σ*= 1) while

*ϵ*is the global mean total (zonal mean plus eddy) kinetic energy dissipation. The underlying picture justifying the latter choice is that some of the kinetic energy cascaded by the eddies is transferred to the mean flow before being dissipated by friction, with the remainder dissipated directly from the eddy energy, so the sum of the dissipation in the mean flow and in eddies should correspond better to the energy cascaded by the eddies. The small-scale dissipation by the models subgrid momentum diffusion is negligible. The value of

*β*in the predictions is kept constant. An alternative choice defining

*β*at the latitude of maximum column-integrated eddy kinetic energy indicates that the effect of this choice on the theoretical fit is negligible, a simplification due to our restriction to variations in Δ

*, which do not result in large shifts in latitude in the eddy kinetic energy or poleward energy flux.*

_{y}Validating Eq. (5), the Rhines scaling for the eddy kinetic energy (*V*^{2}; multiplied by mass per unit area to be plotted in units of J m^{−2}). The value of the control simulation is used to rescale the prediction for all simulations. The value of *β* is set to be constant.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating Eq. (5), the Rhines scaling for the eddy kinetic energy (*V*^{2}; multiplied by mass per unit area to be plotted in units of J m^{−2}). The value of the control simulation is used to rescale the prediction for all simulations. The value of *β* is set to be constant.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating Eq. (5), the Rhines scaling for the eddy kinetic energy (*V*^{2}; multiplied by mass per unit area to be plotted in units of J m^{−2}). The value of the control simulation is used to rescale the prediction for all simulations. The value of *β* is set to be constant.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

The close agreement with this Rhines scaling for the eddy kinetic energy encourages further comparison between the diffusivity scaling and the GCM. We show the comparison with the scaling in Eq. (4) for the diffusivities defined by both the total flux and eddy flux in Figs. 3a and 3b. A comparison with Eq. (7) would show very similar behavior given the tight fit to the Rhines scaling between *V*^{2} and *ϵ* in Fig. 2. The choice of the eddy flux, i.e., *D*_{eddy}, results in the better fit to the theoretical scaling. While it is possible that the arbitrariness in how best to define a globally averaged diffusivity may be affecting this result, experimentation with different definitions suggests to us that the difference between the two panels in Figs. 3a and 3b is larger than that due to different choices of averaging. As mentioned in the discussion of Fig. 1, the Ferrel cell cancels a modest portion of the eddy flux in the simulations, and this cancellation increases with increasing Δ* _{y}*, so the total flux increases less rapidly than the eddy flux. The scaling for the eddy flux is also shown in Fig. 3c in a log–log plot of

*D*

_{eddy}versus

*ϵ*. One can read off the slope in this form, confirming the 3/5 slope predicted by Eq. (4).

Validating Eq. (4), the Rhines scaling for the diffusivity defined by (a) total flux (*D*_{total}) and (b) eddy flux (*D*_{eddy}). (c) The log–log plot of the diffusivity for eddy flux (*D*_{eddy}) vs kinetic energy dissipation (*ϵ*; multiplied by mass per unit area to be plotted in units of W m^{−2}). The dotted line indicates an *ϵ*^{3/5} power-law dependence, as predicted by Eq. (4).

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating Eq. (4), the Rhines scaling for the diffusivity defined by (a) total flux (*D*_{total}) and (b) eddy flux (*D*_{eddy}). (c) The log–log plot of the diffusivity for eddy flux (*D*_{eddy}) vs kinetic energy dissipation (*ϵ*; multiplied by mass per unit area to be plotted in units of W m^{−2}). The dotted line indicates an *ϵ*^{3/5} power-law dependence, as predicted by Eq. (4).

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating Eq. (4), the Rhines scaling for the diffusivity defined by (a) total flux (*D*_{total}) and (b) eddy flux (*D*_{eddy}). (c) The log–log plot of the diffusivity for eddy flux (*D*_{eddy}) vs kinetic energy dissipation (*ϵ*; multiplied by mass per unit area to be plotted in units of W m^{−2}). The dotted line indicates an *ϵ*^{3/5} power-law dependence, as predicted by Eq. (4).

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

## 4. Constraining kinetic energy dissipation by global entropy budget

### a. The model entropy budget

*ϵ*, in Eq. (4) can be constrained due to the fact that the dissipative heating associated with

*ϵ*generates entropy. The strength of this entropy generation in turn is limited by the strength of global entropy destruction. We consider here the

*material*entropy of the molecular degrees of freedom of the atmosphere, excluding the entropy in the radiation itself, treating radiative transfer as external (e.g., Bannon 2015). At the statistically steady state, we start with the global entropy budget (e.g., Peixoto et al. 1991),

*Q*is the dissipative heating rate,

_{d}*Q*is the nondissipative heating rate,

*S*is the sources of entropy other than dissipative heating,

*T*is the temperature, an overline denotes the time average, and brackets the mass-weighted average over the global atmosphere. The small-scale diffusion of temperature and moisture must also be included in

*S*in general, and the latter in particular can be important in moist atmospheres (e.g., Pauluis and Held 2002a,b). Yet, in the GCM analyzed in this paper, the former is insignificant and the latter is not present, so that

*S*can be neglected.

*S*, to relate the lhs of Eq. (8) to

*ϵ*we need to define a characteristic temperature

*T*at which the dissipative heating occurs on average:

_{ϵ}*Q*and

*T*is negligible so that

*η*> 0 is used to denote the strength of the entropy destruction due to spatial covariation of the time mean diabatic heating and time mean temperature. Equations (8)–(10) together imply that the kinetic energy dissipation,

*ϵ*, is proportional to

*η*, if we assume

*T*can be approximated as a constant characteristic mean temperature

_{ϵ}*T*

_{0}for the parameter space explored,

*η*to the time mean temperature structure taking advantage of an additional argument, which is then the last step before we close the scaling theory for the diffusivity in section 5. For now, we first examine the extent to which Eq. (11) holds in the simulations, based on the values of

*η*calculated from the simulated values of

In Fig. 4, we see that *η* tends to overestimate the increase of *ϵ* with increasing Δ* _{y}*. This is partially because the simulated value of

*T*, that is, the averaged dissipative temperature, decreases with increasing Δ

_{ϵ}*(Fig. B1b). Since the frictional dissipation is all near the surface and in midlatitudes, we can think of the averaged dissipative temperature as an average midlatitude lower-tropospheric temperature. This temperature decreases with Δ*

_{y}*because of an increase in static stability. Given the form of the radiative forcing, this increase in stability cools the surface. Additionally, part of this discrepancy can be attributed to entropy production due to the temporal covariation of diabatic heating and temperature. The size of the eddy entropy production due to the covariance in time is roughly 10%–20% of the mean entropy destruction in our simulations. While small, this term also increases moderately with increasing Δ*

_{y}*(Fig. B1a). We have confirmed that the proportionality between*

_{y}*ϵ*and

*η*is very good if the variation in

*T*and the temporal radiative heating covariation with temperature are taken from the model simulations. While it might be possible to construct approximate theories for these two minor effects, the resulting theories would likely be specific to the form of the radiative forcing used here and not directly transferable to other models. For now, we assume simply that

_{ϵ}*ϵ*∝

*η*, ignoring the modest discrepancy in Fig. 4. Yet we will need to return to this difference in section 5.

Validating Eq. (11), the entropy budget scaling for the kinetic energy dissipation (*ϵ*; multiplied by mass per unit area to be plotted in units of W m^{−2}). The value of *T*_{0} is set to be constant.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating Eq. (11), the entropy budget scaling for the kinetic energy dissipation (*ϵ*; multiplied by mass per unit area to be plotted in units of W m^{−2}). The value of *T*_{0} is set to be constant.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating Eq. (11), the entropy budget scaling for the kinetic energy dissipation (*ϵ*; multiplied by mass per unit area to be plotted in units of W m^{−2}). The value of *T*_{0} is set to be constant.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

### b. Estimating entropy destruction from the mean thermal structure

*η*, the global entropy destruction, in a flux-times-gradient form through an integration by parts. All quantities in the following are time averages and we drop the overline notation. We assume here that the time mean temperatures are zonally symmetric. Following Held (2007), we have

*F*,

_{y}*F*) are the meridional and vertical components of the mean adiabatic dry static energy flux. The subscript

_{p}*m*denotes the vertical coordinate on the sloping surface along which the energy flux is aligned, that is, the

*mixing slope*in the sense that

*η*using Eq. (12) depends on the details of the mean thermal structure and heat flux’s spatial distribution. For this expression to be useful for the type of global scaling relations that we are hoping for, we need to further assume that it can be written in terms of characteristic domain averages,

*c*and mass to have units of length

_{p}^{3}time

^{−3}) and ∂

*ln*

_{y}*T*|

*should all be interpreted as characteristic global values. The flux*

_{m}*ln*

_{y}*T*|

*can be thought of proportional to the effective temperature difference between where the atmosphere is heated and cooled. In writing Eq. (14), we also assume that the characteristic mean temperature here is proportional to the characteristic temperature of kinetic energy dissipation,*

_{m}*T*, and can be assumed as a constant

_{ϵ}*T*

_{0}in Eq. (11).

*η*in terms of

*Q*over the region where

*Q*> 0 (

*T*

_{h}_{,}

*are defined as*

_{c}*T*

_{h}_{,}

*as any reasonable approximation to characteristic temperatures of the heated and cooled regions. Setting*

_{c}*η*of the form in Eq. (14).

*ln*

_{y}*T*|

*, the effective temperature gradient oriented along the mixing slope. If the mixing slope can be thought of as approximately horizontal, then ∂*

_{m}*ln*

_{y}*T*|

*reduces to ∂*

_{m}*ln*

_{y}*T*|

*. If the mixing is instead assumed to be along an isentrope, then one can show that locally, using hydrostatic balance and the definition of potential temperature*

_{p}*θ*,

*N*

^{2}is the static stability and

*ln*

_{y}*p*|

*. More generally, if the ratio of the mixing slope to the isentropic slope is*

_{θ}*μ*= 0) and the isentropic (

*μ*= 1) limits:

*μ*, ∂

*ln*

_{y}*T*|

*depends on the temperature gradients both in the meridional and vertical directions, and their relative importance to ∂*

_{m}*ln*

_{y}*T*|

*can be thought of as determined by*

_{m}*μ*, which characterizes the displacement of the total heat flux relative to the isentropic surface.

*ϕ*≡

*y*/

*a*(

*a*is the radius of planet), and pressure of the heated and cooled regions, [

*θ*,

_{h}*θ*,

_{c}*σ*,

_{h}*σ*], just as the centroid defined in Eq. (16) but with simple instead of inverse weighted average. The characteristic mixing slope can then be estimated as

_{c}*μ*using the values of potential temperatures at these same centroid points:

For our control simulation, this calculation illustrated in Fig. 5a yields the value *μ* ≈ 0.33. If one ignores the centroids and simply looks at the heating distribution and the isentropic surfaces in this figure, one is more likely to guess that the mixing slope is not that different from the isentropic slope. The discrepancy arises in part from the fairly complex distribution of heating and cooling, resulting in the centroids being displaced from the maxima in the heating and cooling patterns.

Time-mean thermal structure of the control simulation: (a) the total diabatic heating diagnosed from Eq. (1) (or equivalently the divergence of total heat fluxes) and (b) the heating computed as the divergence of eddy heat fluxes. In both panels, isentropes are plotted as gray contours. The two black dots denote for the centroids of heating and cooling, respectively, computed from the diabatic heating shown in the corresponding panel, and the dashed line linking them is the bulk mixing surface defined accordingly. The detailed definitions are described in the text.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Time-mean thermal structure of the control simulation: (a) the total diabatic heating diagnosed from Eq. (1) (or equivalently the divergence of total heat fluxes) and (b) the heating computed as the divergence of eddy heat fluxes. In both panels, isentropes are plotted as gray contours. The two black dots denote for the centroids of heating and cooling, respectively, computed from the diabatic heating shown in the corresponding panel, and the dashed line linking them is the bulk mixing surface defined accordingly. The detailed definitions are described in the text.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Time-mean thermal structure of the control simulation: (a) the total diabatic heating diagnosed from Eq. (1) (or equivalently the divergence of total heat fluxes) and (b) the heating computed as the divergence of eddy heat fluxes. In both panels, isentropes are plotted as gray contours. The two black dots denote for the centroids of heating and cooling, respectively, computed from the diabatic heating shown in the corresponding panel, and the dashed line linking them is the bulk mixing surface defined accordingly. The detailed definitions are described in the text.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

If we assume the spatial distribution of heating and cooling is set primarily by eddy fluxes, the small value *μ* would seem surprising, given the expectation that time averaged eddy fluxes are expected to be mostly adiabatic. In baroclinic instability theory, the range 0 < *μ* < 1 is referred to as the *wedge of instability* (Eady 1949; Green 1960). The mixing slope must be larger than zero to generate kinetic energy, but smaller than unity, as most easily understood from a potential temperature variance budget. A typical value for the most unstable wave is *μ* ≈ 0.5, where the production of potential temperature variance by the downgradient horizontal eddy flux is partly compensated by the destruction of variance by the countergradient vertical flux, the remainder balanced by the growth of variance in the amplifying wave. However, in a statistically steady state, the transient term is not present, and any imbalance due to *μ* being smaller than unity must be balanced by the diabatic destruction of variance in a globally averaged sense. Inspection of the global potential temperature variance budget confirms that the diabatic destruction is weak in our simulation, corresponding to *μ* ≈ 0.7. Locally, the values of *μ* estimated from the local eddy fluxes and local isentropic slopes are noisy but mostly larger than 0.5 (not shown).

The larger value of *μ* obtained from the eddy variance budget has prompted us to repeat the mixing slope calculation using the heating and cooling distribution from the eddy fluxes only (Fig. 5b). We however find that it does not make much difference from the slope computed from the total heating (Fig. 5a). Instead, the only difference between the two seems to be the equatorward extension of the heating centroid (or equivalently the dynamically cooled region) in the case with the total heating, associated with the Hadley cell. The Hadley cell in this model is mostly eddy driven (Kim and Lee 2001), with vertical motion that is comparable or slower than the radiative relaxation time, potentially helping to explain the small value of *μ* found in both the total heating and the eddies only calculations.

That said, we still find these mixing slope calculations puzzling as it remains difficult for us to reconcile them with the fact that this model’s atmosphere is indeed a very efficient heat engine based on our analysis of global entropy budget, which would otherwise suggest a *μ* value that is close to unity (appendix B). Fortunately, the uncertainty surrounding the mixing slope and the value of *μ* is somewhat a minor issue when trying to determine the relevant ∂* _{y}*ln

*T*|

*in the entropy budget scaling, Eq. (14), for our varying Δ*

_{m}*simulations. Specifically, we see that Eq. (14) with ∂*

_{y}*ln*

_{y}*T*|

*and*

_{m}*μ*estimated by the proposed approach works out nicely in predicting

*η*. In the cases with smaller Δ

*, the results using higher values of*

_{y}*μ*also give reasonable predictions, in large part owing to the specific way of the changes of the time mean temperature structure with Δ

*as we describe further in the following.*

_{y}We start with validating Eq. (14) using ∂* _{y}*ln

*T*|

*computed from Eq. (19) and*

_{m}*μ*computed from Eq. (21). We first note that

*μ*varies relatively little as a function of Δ

*, as shown in Fig. 6a. Using these values of*

_{y}*μ*, estimating ∂

*ln*

_{y}*T*|

*from the expression in Eq. (19) (Fig. 6b), and then taking*

_{m}*η*in Fig. 7. The comparison with the simulated values of

*η*is very good when scaled by the control value despite the series of approximations, and the result is nearly unchanged if we use the constant value of

*μ*= 0.3 or 0.4 in this computation (not shown).

Calculations required to obtain ∂* _{y}*ln

*T*|

*from Eq. (19): (a) the estimated ratio of the mixing slope and the isentropic slope (*

_{m}*μ*) based on Eq. (21), (b) the estimated

*a*∂

*ln*

_{y}*T*|

*,*

_{m}*a*∂

*ln*

_{y}*T*|

*,*

_{p}*aκ*∂

*ln*

_{y}*p*|

*(where*

_{θ}*a*is the planetary radius), and (c) the estimated

*a*∂

*ln*

_{y}*T*|

*,*

_{m}*a*∂

*ln*

_{y}*T*|

*,*

_{p}*aκ*∂

*ln*

_{y}*p*|

*, normalized by the simulated value of the control simulation.*

_{θ}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Calculations required to obtain ∂* _{y}*ln

*T*|

*from Eq. (19): (a) the estimated ratio of the mixing slope and the isentropic slope (*

_{m}*μ*) based on Eq. (21), (b) the estimated

*a*∂

*ln*

_{y}*T*|

*,*

_{m}*a*∂

*ln*

_{y}*T*|

*,*

_{p}*aκ*∂

*ln*

_{y}*p*|

*(where*

_{θ}*a*is the planetary radius), and (c) the estimated

*a*∂

*ln*

_{y}*T*|

*,*

_{m}*a*∂

*ln*

_{y}*T*|

*,*

_{p}*aκ*∂

*ln*

_{y}*p*|

*, normalized by the simulated value of the control simulation.*

_{θ}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Calculations required to obtain ∂* _{y}*ln

*T*|

*from Eq. (19): (a) the estimated ratio of the mixing slope and the isentropic slope (*

_{m}*μ*) based on Eq. (21), (b) the estimated

*a*∂

*ln*

_{y}*T*|

*,*

_{m}*a*∂

*ln*

_{y}*T*|

*,*

_{p}*aκ*∂

*ln*

_{y}*p*|

*(where*

_{θ}*a*is the planetary radius), and (c) the estimated

*a*∂

*ln*

_{y}*T*|

*,*

_{m}*a*∂

*ln*

_{y}*T*|

*,*

_{p}*aκ*∂

*ln*

_{y}*p*|

*, normalized by the simulated value of the control simulation.*

_{θ}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating Eq. (14), the entropy budget scaling for the entropy destruction (*η*; multiplied by mass per unit area to be plotted in units of W m^{−2} K^{−1}), with ∂* _{y}*ln

*T*|

*estimated from the simulations.*

_{m}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating Eq. (14), the entropy budget scaling for the entropy destruction (*η*; multiplied by mass per unit area to be plotted in units of W m^{−2} K^{−1}), with ∂* _{y}*ln

*T*|

*estimated from the simulations.*

_{m}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating Eq. (14), the entropy budget scaling for the entropy destruction (*η*; multiplied by mass per unit area to be plotted in units of W m^{−2} K^{−1}), with ∂* _{y}*ln

*T*|

*estimated from the simulations.*

_{m}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

The seemingly insensitivity to *μ* in this result encourage further simplification on *μ* by considering its two limits, i.e., the horizontal (*μ* = 0) and the isentropic (*μ* = 1) limits. These limits correspond to the theories of BCT02 and HL96: BCT02 assumes that the mixing is horizontal (*μ* = 0) when coupling the entropy budget to Rhines scaling, while HL96 assumes that it is isentropic (*μ* = 1). The consequences of these assumptions into the estimate of *η* are larger deviations from the simulated *η* values. The BCT02 limit overestimates the sensitivity of *η* to Δ* _{y}* for all values of Δ

*(Fig. 8a). The HL96 approximation works well for values of Δ*

_{y}*(and*

_{y}*η*) smaller than the control and then underestimates the sensitivity of

*η*to Δ

*for larger values (Fig. 8b).*

_{y}As in Fig. 7, but for the *η* predictions based on different approximations on ∂* _{y}*ln

*T*|

*: (a)*

_{m}*μ*= 0 proposed by BCT02, (b)

*μ*= 1 proposed by HL96, and (c)

*μ*= 1 with a constant isentropic slope (∂

*ln*

_{y}*p*|

*= cons), or equivalently, a constant ∂*

_{θ}*ln*

_{y}*T*|

*.*

_{m}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

As in Fig. 7, but for the *η* predictions based on different approximations on ∂* _{y}*ln

*T*|

*: (a)*

_{m}*μ*= 0 proposed by BCT02, (b)

*μ*= 1 proposed by HL96, and (c)

*μ*= 1 with a constant isentropic slope (∂

*ln*

_{y}*p*|

*= cons), or equivalently, a constant ∂*

_{θ}*ln*

_{y}*T*|

*.*

_{m}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

As in Fig. 7, but for the *η* predictions based on different approximations on ∂* _{y}*ln

*T*|

*: (a)*

_{m}*μ*= 0 proposed by BCT02, (b)

*μ*= 1 proposed by HL96, and (c)

*μ*= 1 with a constant isentropic slope (∂

*ln*

_{y}*p*|

*= cons), or equivalently, a constant ∂*

_{θ}*ln*

_{y}*T*|

*.*

_{m}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

It may be surprising that given the small value of *μ*, the HL96 isentropic mixing limit works as well as it does for small Δ* _{y}*. Indeed, it is found in these simulations that the contribution of the vertical temperature difference to the relevant temperature difference oriented along the mixing slope, ∂

*ln*

_{y}*T*|

*, is much larger than the horizontal contribution. This is illustrated in Fig. 6b, where the two terms in Eq. (19), i.e., the expression for the estimate of ∂*

_{m}*ln*

_{y}*T*|

*, are compared: ∂*

_{m}*ln*

_{y}*T*|

*and*

_{p}*κ*∂

*ln*

_{y}*p*|

*. The latter corresponds to the temperature difference along the isentrope [Eq. (17)] and is seen to dominate over the range of Δ*

_{θ}*that is smaller than the control. Therefore, even with a large value of*

_{y}*μ*, ∂

*ln*

_{y}*T*|

*ends up being mostly determined by*

_{m}*κ*∂

*ln*

_{y}*p*|

*. Most of the effect of using an incorrect value of*

_{θ}*μ*is then removed when rescaling by the control value, as in Fig. 6c. The

*μ*= 0 limit, in contrast, results in fractional changes in the temperature gradient that are too large.

Given the dominance of *κ*∂* _{y}*ln

*p*|

*in Eq. (19), we can further simplify Eq. (19) using the result of Schneider and Walker (2006), who found that the mean isentropic slope is nearly constant in a similar model as Δ*

_{θ}*is varied. If we assume*

_{y}*μ*≈ 1 and a fixed ∂

*ln*

_{y}*p*|

*when Δ*

_{θ}*is varied, Eq. (19) suggests the effective temperature gradient along the mixing slope, ∂*

_{y}*ln*

_{y}*T*|

*, is also fixed. This provides a way of thinking about the decent prediction for*

_{m}*η*obtained by assuming the constancy of ∂

*ln*

_{y}*T*|

*and*

_{m}*η*∝

*F*

_{total}in Fig. 8c for small Δ

*. The quality of the fit for large Δ*

_{y}*, on the other hand, may be due to some compensation of errors. The implications of these simplifications, both*

_{y}*μ*≈ 1 and a fixed ∂

*ln*

_{y}*p*|

*, on the prediction for the diffusivity will be discussed further in the next section.*

_{θ}## 5. Theoretical predictions for the bulk diffusivity

*c*) given the meridional mean temperature gradient:

_{p}T*c*and other constants (e.g., mass) for conciseness.

_{p}Figure 9a compares the simulated diffusivity values for eddy flux (*D*_{eddy}) with the expression in Eq. (23), estimating the value of ∂* _{y}*ln

*T*|

*from the simulations (as shown in Fig. 6b). This results in a diffusivity, when normalized by the control value, that is reasonably well satisfied for values of Δ*

_{m}*smaller than the control, but too sensitive to Δ*

_{y}*for larger values. In Fig. 9b, we plot instead the explicit form Eq. (24). Even though this seems physically equivalent, the departures from the simulated diffusivity values are magnified. This arises from the stronger temperature gradient dependence exposed when eliminating the flux in favor of the diffusivity, roughly multiplying these departures by a factor of 5/2 = (3/2)/(3/5).*

_{y}Validating the scaling predictions for *D*_{eddy} assuming

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating the scaling predictions for *D*_{eddy} assuming

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Validating the scaling predictions for *D*_{eddy} assuming

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

*ϵ*and

*η*in Eq. (11), the implications of which we can examine by setting

*γ*≡

_{η}*ϵ*/

*η*. Relaxing both of these approximations we would have instead,

*γ*and

_{F}*γ*from the simulations. These move the results in the right direction, with

_{η}*γ*and

_{F}*γ*roughly comparable in importance for the improvement. The exercise also suggests that a better fit for the larger values of Δ

_{η}*could result from these corrections, and possibly others, without altering the essence of the argument based on Rhines scaling and the entropy budget.*

_{y}As in Fig. 9, but with the corrections by *γ _{F}* =

*γ*=

_{η}*ϵ*/

*η*, i.e., Eqs. (25) and (26).

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

As in Fig. 9, but with the corrections by *γ _{F}* =

*γ*=

_{η}*ϵ*/

*η*, i.e., Eqs. (25) and (26).

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

As in Fig. 9, but with the corrections by *γ _{F}* =

*γ*=

_{η}*ϵ*/

*η*, i.e., Eqs. (25) and (26).

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

*ln*

_{y}*T*|

*estimated from the simulations. As in section 4, to better understand potential simplifications on this effective temperature gradient or difference, it is interesting to study the limits proposed by BCT02 and HL96. The diffusivity expression proposed by BCT02 can be obtained from Eq. (22) by assuming that the mixing is horizontal (*

_{m}*μ*= 0), and setting

*q*is the magnitude of the vertically averaged tropical dynamical cooling or extratropical heating:

*μ*= 0 approximation through to the explicit form then yields a predicted diffusivity proportional to the third power of the horizontal temperature gradient. This is a clear overestimation as we have seen that the simulated diffusivity instead depends much more weakly on the horizontal temperature gradient (Fig. 1b). The prediction is also plotted in Fig. 11a to confirm the

*η*result in Fig. 8a and to show the implications of the errors in

*η*for the diffusivity.

As in Fig. 9b, but with different ∂* _{y}*ln

*T*|

*approximations based on (a)*

_{m}*μ*= 0 proposed by BCT02 and (b)

*μ*= 1 proposed by HL96. Note that the smallest and largest Δ

*cases in (a) are not shown because their predicted values are out of the range displayed.*

_{y}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

As in Fig. 9b, but with different ∂* _{y}*ln

*T*|

*approximations based on (a)*

_{m}*μ*= 0 proposed by BCT02 and (b)

*μ*= 1 proposed by HL96. Note that the smallest and largest Δ

*cases in (a) are not shown because their predicted values are out of the range displayed.*

_{y}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

As in Fig. 9b, but with different ∂* _{y}*ln

*T*|

*approximations based on (a)*

_{m}*μ*= 0 proposed by BCT02 and (b)

*μ*= 1 proposed by HL96. Note that the smallest and largest Δ

*cases in (a) are not shown because their predicted values are out of the range displayed.*

_{y}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

*μ*= 1 and thus ∂

*ln*

_{y}*T*|

*is proportional to the isentropic slope ∂*

_{m}*ln*

_{y}*p*|

*[Eq. (17)]. Taking*

_{θ}*/∂*

_{y}b*, while ∂*

_{z}b*|*

_{y}T*can also be replaced by the horizontal buoyancy gradient, ∂*

_{p}*, so this expression reduces to*

_{y}b*D*∼

*β*

^{−2}

*τ*

^{−3}, where

*τ*

^{−1}is the

*Eady growth rate*,

*τ*∼ ∂

*/*

_{y}b*N*, as in HL96. When compared to the BCT02 approximation (Fig. 11a), the diffusivity predicted by Eq. (28) (Fig. 11b) looks more similar to the one from the simulated ∂

*ln*

_{y}*T*|

*(Fig. 9b), which is again consistent with the results for*

_{m}*η*(Figs. 7, 8a,b).

The HL96 limit, Eq. (28), requires some constraints on the static stability in order to obtain a diffusivity that is a function of horizontal gradients only, so that it can be easily incorporated into the simplest diffusive energy balance models. We can take advantage of the result that the isentropic slope hardly changes as Δ* _{y}* is varied in a very similar model (Schneider and Walker 2006), which we have confirmed to be a good approximation for the estimate of entropy destruction in Fig. 8c. In Fig. 12, we show the results for the diffusivity estimated from this approximation, which resemble the results in Figs. 9b and 10b well for Δ

*larger than the control value. For smaller Δ*

_{y}*, the departures from the simulated diffusivity in Fig. 12a are reduced when compared with Fig. 9b, but the departures from the simulated diffusivity when corrected by the*

_{y}*γ*and

_{F}*γ*factors in Fig. 12b are magnified when compared with Fig. 10b. Therefore, the apparently nice fit for smaller Δ

_{η}*in Fig. 12a involves some error cancellation between different approximations.*

_{y}As in Figs. 9b and 10b, but assuming *μ* = 1 with a constant isentropic slope (∂_{y}ln*p*|* _{θ}* = const.), or equivalently, a constant ∂

_{y}ln

*T*|

*.*

_{m}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

As in Figs. 9b and 10b, but assuming *μ* = 1 with a constant isentropic slope (∂_{y}ln*p*|* _{θ}* = const.), or equivalently, a constant ∂

_{y}ln

*T*|

*.*

_{m}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

As in Figs. 9b and 10b, but assuming *μ* = 1 with a constant isentropic slope (∂_{y}ln*p*|* _{θ}* = const.), or equivalently, a constant ∂

_{y}ln

*T*|

*.*

_{m}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Despite these subtleties, the result in Fig. 12a remains helpful for us to understand the diffusivity dependence on the horizontal temperature gradient. In Eq. (28), if the isentropic slope can be assumed constant, the result is a diffusivity with no explicit dependence on the static stability, but proportional to the 3/2 power of the horizontal (meridional) temperature gradient, ∂* _{y}T*|

*. This is the dependence that we have seen in Fig. 1b for our simulations if Δ*

_{p}*is not too large. The importance of the static stability response for this result is emphasized by comparing to the BCT02 theory, in which the mixing slope is approximated to be horizontal and the resulting diffusivity is proportional to the third power, rather than 3/2 power, of ∂*

_{y}*|*

_{y}T*as discussed above. The BCT02 limit is clearly not accurate for the dry GCM considered here, but we discuss its potential implication on moist atmospheres in the next section.*

_{p}It is also useful to compare the horizontal temperature gradient dependence derived here with the results in Frierson et al. (2007). When mimicking the behavior of their moist GCM with a diffusive energy balance model, Frierson et al. (2007) also obtained a diffusivity proportional to the 3/2 power of the temperature gradient. Their justification is related to that discussed here in that they avoid discussion of the static stability by also invoking a constraint on the isentropic slope. The detailed theory is however different. In particular, the dependence on rotation rate they proposed is *f*^{−3/2}*β*^{−1/2} for a given temperature gradient (where *f* is the Coriolis parameter), rather than *β*^{−2}, although this distinction is not important if the latitudinal structure of the flux does not change significantly. More importantly, as we describe in the following, our development of the theory suggests that it requires modification before application to a moist atmosphere.

## 6. Implications for moist atmospheres

We discuss the additional complications that a moist atmosphere introduces into this scaling argument, to help orient future model tests. The scaling theory we have described is based on the constraints of Rhines scaling and the global entropy budget. The former does not depend on the composition of the air, so this part of the theory, resulting in the relation between diffusivity and kinetic energy dissipation, i.e., Eq. (4), does not require any modification when applying to moist atmospheres. The analysis of the entropy budget and its connection to kinetic energy dissipation is more dependent on moist processes.

*h*. If it is still reasonable to ignore the energy transport by the mean flow, we can consider only the eddy flux of moist static energy. Finally we need to assume that

*D*in Eq. (22) can be used to diffuse the moist static energy, an assumption that seems plausible given the desire to diffuse the quantities that are more conservative, but can also be questioned (e.g., O’Gorman and Schneider 2008). Given these assumptions, we can solve for

*D*to obtain the explicit expression that replaces (24):

As discussed in section 5, BCT02 effectively used the implicit form, Eq. (23), assuming that the mixing slope is horizontal (*μ* = 0) to obtain the resulting diffusivity rewritten in Eq. (27). They provided support for this scaling comparing to simulations with a comprehensive atmospheric GCM, as did Liu et al. (2017) using a comprehensive coupled model. At face value, these fits suggests that the role of vertical stratification may not be particularly important for the diffusivity scaling in moist atmospheres. While purely horizontal mixing cannot be strictly correct, since there must be some kinetic energy generation regardless of whether an atmosphere is dry or moist, it could still be a useful approximation for some systems and parameter variations in which vertical temperature gradients do not change substantially, so that the effective temperature gradient relevant for the entropy budget is proportional to the horizontal gradient (e.g., in the simulations with varying rotation rates discussed in appendix A). Perhaps there is an argument that this is a good approximation for moist atmospheres.

On the other hand, it is interesting to speculate, based on Eq. (29), as to how one might be able to justify the assumption of a constant diffusivity in the global warming context, that is, why changes in diffusivity might be small compared to the changes in the poleward heat flux (e.g., Hwang and Frierson 2010; Roe et al. 2015; Mooring and Shaw 2020). The meridional gradient of moist static energy is typically simulated to increase with warming due to the temperature dependence of water vapor saturation, despite the decrease in lower-tropospheric temperature gradient (e.g., Armour et al. 2019). This increase would need to be balanced by a decrease in the effective temperature difference relevant for the entropy budget [as suggested by Knietzsch et al. (2015), for example] to justify a constant diffusivity. One might start with the assumption that the mixing slope in moist atmospheres follows a moist adiabat. The temperature difference between the surface and a fixed pressure along this slope will then decrease with warming to balance the increase in moisture gradient. This provides a starting point in thinking about the potential for compensation, but is presumably an upper bound since penetration to higher levels would moderate this decrease in temperature difference, and the likelihood that the mixing slope lies between moist and dry isentropes, given that poleward/upward motion tends to follow the former and equatorward/downward motion the latter. In any case, the perspective outlined above does suggest that the competition between the changes in meridional moist static energy gradient and in the effective temperature difference along the mixing slope for the moist entropy budget of the large-scale flow is critical to this issue.

## 7. Concluding remarks

In this study, we present a scaling theory for the poleward eddy heat transport, assuming that it can be characterized by a bulk diffusivity (hereafter diffusivity). This theory can be regarded as a generalization of the theories proposed by HL96 and BCT02. The theory is compared to simulations with an idealized dry general circulation model (GCM), in which we vary the equator-to-pole temperature gradient by modifying the radiative forcing. We also discuss how the theory can potentially be extended to moist atmospheres.

The theory consists of two constraints between the kinetic energy dissipation *ϵ* and the diffusivity *D*. One constraint, shared by the HL96 and BCT02 theories, is provided by Rhines scaling [i.e., Eq. (4)] in which it is claimed that *β* and *ϵ* are the only dimensional quantities from which a kinematic diffusivity, with units of length^{2} time^{−1}, can be constructed. While most easily justified for tracer transport by two-dimensional turbulence, the utility of this scaling appears to be much more general, as best illustrated, perhaps, by the comparison to simulations with a full atmospheric model by BCT02. Yet there are known limitations based on work with quasigeostrophic models, especially regarding how to incorporate the effect of the strength of surface friction (Thompson and Young 2006, 2007; Kong and Jansen 2017; Chang and Held 2019, 2021; Gallet and Ferrari 2020, 2021). An assessment on the simulations where the rotation rate is varied confirm the relevance of the above uncertainty over a broader parameter space of this idealized dry GCM (appendix A). Despite the uncertainty, we verify that Rhines scaling works very well for the diffusivity associated with the eddy flux of temperature (or dry static energy) as the meridional temperature gradient is varied in the dry GCM examined.

The second constraint on the relationship between *ϵ* and *D* comes from the global entropy budget, in which the entropy production due to frictional dissipation, which is approximately proportional to *ϵ*, is assumed to be balanced by the destruction of entropy by the radiative forcing [i.e., Eq. (11)]. This balance can be upset by other sources of entropy, the radiative damping, or covariance in time between the radiative heating and temperature, being the key additional source in the dry GCM. While small in absolute terms, its variation across model simulations does distort the simplest scaling for the diffusivity. Additionally, the distortion can also result from the variation in the averaged dissipative temperature, that is, the ratio of *ϵ* and the entropy production due to frictional dissipation. Because most frictional dissipation occurs near the surface in the midlatitudes, this temperature generally scales with midlatitude surface temperature, which also varies with the meridional temperature gradient in the GCM.

*mixing slope*[i.e., Eq. (14)]. The resulting theory for the diffusivity [as described in Eq. (24) and restated below assuming a constant

*β*] has the dependence on mean temperature gradients of the form

*s*is the dry static energy and the subscript

*m*refers to the mixing slope. The BCT02 theory is obtained by assuming that this mixing slope is horizontal [i.e., Eq. (27)]. The HL96 theory is obtained by assuming that the mixing slope is the isentropic slope [i.e., Eq. (28)]. The GCM’s behavior is shown to be intermediate between these two limits, but with the HL96 scaling of more quantitative relevance in this model, primarily because vertical temperature differences are much larger than horizontal differences.

The introduction of the mixing slope, or the isentropic slope in particular, introduces dependence on the static stability, which complicates the incorporation into simple diffusive climate models. Yet, if in addition to assuming that the mixing is aligned on average along isentropes, one assumes following the study of Schneider and Walker (2006) with a very similar model, that the static stability increases proportionally to the horizontal temperature gradient in this model, so that the isentropic slope is relatively unchanged as the gradients increase, one finds that the temperature gradient along the isentrope is unchanged, leading to a diffusivity proportional to (∂* _{y}T*|

*)*

_{p}^{3/2}. The diffusivity simulated in the GCM has this 3/2 power-law dependence on horizontal temperature gradient when this gradient is smaller or comparable to that in the control, but is less sensitive to the gradient for larger gradients.

In a moist atmosphere, analogous arguments lead to the same scaling but with dry static energy replaced by the moist static energy [i.e., Eq. (29)]. The dependence of the diffusivity on the product of a moist static energy gradient and an effective temperature gradient oriented along the mixing slope is a distinctive result from our arguments. How quantitatively useful it is for interpreting energy transport in moist atmospheres remains to be seen. An interesting feature of this scaling is that compensating changes between the two gradients have the potential to result in relatively moderate changes in the diffusivity, which could help us better understand why diffusivity changes are generally found to be less important than the change in energy transport in warming climates (e.g., Hwang and Frierson 2010; Roe et al. 2015).

Another key issue about the presented theory is that Rhines scaling seems more relevant for the eddy flux while the entropy budget is more simply connected to the total flux (eddy plus mean; there are no stationary eddies in the idealized GCM). This distinction is a source of discrepancies between the theory and model simulations. In particular, it appears to be responsible for a significant part of the discrepancy for large temperature gradient, along with the intricacies of the distinction between kinetic energy dissipation and entropy destruction. Consistent with the transformed Eulerian mean formalism, one might hope that a focus on the diffusion of potential vorticity (PV) would provide a more unified treatment of eddy and mean fluxes. Some two-layer quasigeostrophic studies do suggest that diffusion of lower-layer PV, using similar scaling arguments, may be superior to diffusing temperature (e.g., Zurita-Gotor 2007). In continuous quasigeostrophic theory the horizontal flux of temperature near the surface is best thought of as part of the PV flux distribution, and analogous to the lower-layer PV in the two-layer system, so in a dry model this would lead back to a focus on a diffusive approximation for the lower-tropospheric eddy temperature flux [see Held (1999) for further discussion]. Inclusion of the mean meridional circulation into the theory then requires an additional treatment of upper tropospheric PV fluxes. A more unified treatment of eddy and mean fluxes, and extensions to moist atmospheres, are the two key directions in which these arguments need to be developed further and improved.

## Acknowledgments.

We thank Nick Lutsko and Pablo Zurita-Gotor for the help of model setup on GFDL RDHPCS. An earlier version of this work is presented in CYC’s doctoral dissertation, which was kindly read by Pablo Zurita-Gotor, Steve Garner, and Bob Hallberg. Our appreciation is also given to the two anonymous reviewers for their valuable comments. CYC acknowledges support by NSF Grant AGS-1733818.

## Data availability statement.

The Geophysical Fluid Dynamics Laboratory idealized global spectral atmospheric model is available at https://www.gfdl.noaa.gov/idealized-spectral-models-quickstart/.

## APPENDIX A

### Simulations Varying the Planetary Rotation Rate

In addition to the simulations with varying Δ* _{y}*, we have conducted a set of simulations where we vary the planetary rotation rate Ω to study the proposed theory in a broader range of parameter space. Readers can refer to Chang (2019) for a more complete discussion on these simulations. We here highlight parts of the results that provide extra implications on the theory’s validity and limitations.

We first note that the distinction between *D*_{total} and *D*_{eddy} is less important (<±10%) in these simulations, so we present the results of *D*_{eddy} only for an easier comparison with the Δ* _{y}* cases. The simulated

*D*

_{eddy}is shown in Fig. A1a as a function of Ω, which varies from 0.25 to 7 times the control rotation rate, i.e., the rotation rate of Earth Ω

*. A substantial reconfiguration of the circulation is seen over this range of Ω, from superrotation in the slowly rotating atmospheres to the presence of multiple jets in fast rotating atmospheres. The definition of a single globally relevant bulk diffusivity is, therefore, also more ambiguous given the substantial changes of the flow’s meridional structure. Despite that,*

_{e}*D*

_{eddy}defined accordingly seems to be well explained by the eddy kinetic energy based Rhines scaling, Eq. (7), suggesting

*D*

_{eddy}may still capture some essential dynamics in the simulations (Fig. A1b).

Results of the varying rotation-rate simulations: (a) *D*_{eddy} vs Ω, (b) validating the Rhines scaling, Eqs. (7) and (4), and (c) validating the entropy budget scaling, Eq. (14). Plotting conventions are the same as in the previous figures. The value of *β* is computed assuming a fixed latitude and is only a function of Ω, as we find the variation in the latitudes of secondary importance.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Results of the varying rotation-rate simulations: (a) *D*_{eddy} vs Ω, (b) validating the Rhines scaling, Eqs. (7) and (4), and (c) validating the entropy budget scaling, Eq. (14). Plotting conventions are the same as in the previous figures. The value of *β* is computed assuming a fixed latitude and is only a function of Ω, as we find the variation in the latitudes of secondary importance.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Results of the varying rotation-rate simulations: (a) *D*_{eddy} vs Ω, (b) validating the Rhines scaling, Eqs. (7) and (4), and (c) validating the entropy budget scaling, Eq. (14). Plotting conventions are the same as in the previous figures. The value of *β* is computed assuming a fixed latitude and is only a function of Ω, as we find the variation in the latitudes of secondary importance.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

The success of Eq. (7) is somewhat surprising given that the *ϵ*-based Rhines scaling, Eq. (4), works less ideally in comparison. The breakdown of Eq. (4) for the lower rotation rates is likely related to there not being an inverse cascade in this regime or that the *β* effect is no longer the primary mechanism that halts the inverse cascade. Some modest deviations of the theoretical prediction from the simulation can also be noticed for the cases with Ω much higher than Ω* _{e}* (Fig. A1b). As we mentioned in section 3, the theoretical basis for Eqs. (7) and (4) appears to be an area of active research and awaits a better understanding. Furthermore, we notice that the subgrid-scale dissipation in our model tends to take out significant kinetic energy in our higher Ω simulations (even at T213 resolution). While some deviations are reduced if we include subgrid-scale dissipation in the definition of

*ϵ*, a justification for such a practice remains unclear to us. Therefore, the physical significance of these discrepancies requires more investigation, but we interpret these results to be qualitatively supporting the Rhines scaling, at least for the higher Ω cases.

It is interesting to note that, in the rapidly rotating cases, *D*_{eddy} scales approximately to Ω^{−4/5} in Fig. A1a, which is consistent with Fig. A1b if the contribution from the variations in *ϵ* is neglected. Indeed, with a 7-times increase in Ω and an about-6-times decrease in *D*_{eddy}, *ϵ* is found to roughly halve when the rotation rate increases in the simulations. While a distinction between *ϵ* and *η* exists (due primarily to the subgrid-scale dissipation in these simulations), we expect the entropy budget scaling, Eq. (14), to predict a nearly invariant *η* as well. The results are encouraging considering the subtlety of the calculations involved (Fig. A1c), confirming that Eq. (14) with the simulated ∂* _{y}*ln|

*not only describes the sensitivity of*

_{m}*η*to Δ

*(Fig. 7) but also a lack thereof in these varying Ω cases.*

_{y}Finally, another intriguing aspect of the Ω results is that the BCT02 approximation appears to better predict the Ω dependence of *η* (Fig. A1c), in contrast to the advantage of HL96 approximation for the Δ* _{y}* simulations (Fig. 8). This does not imply the mixing slope being more horizontal as assumed by BCT02 in the Ω cases, since the simulated value of

*μ*still falls between 0.3 and 0.5. Decomposing the simulated ∂

*ln*

_{y}*T*|

*based on Eq. (19) reveals that this is instead related to the relatively small variations in static stability with Ω, and the variations in horizontal temperature gradient alone capture most of the variations in ∂*

_{m}*ln*

_{y}*T*|

*. Our varying Ω simulations thus demonstrate how the limit of BCT02 theory may be sufficient when interpreting some parameter variations. This agrees with the apparent relevance of BCT02 theory reported by Liu et al. (2017), which is based on the examination of a set of varying Ω simulations in a full GCM as well.*

_{m}## APPENDIX B

### Calculations of Global Entropy Budget

*Q*,

*Q*,

_{d}*Q*

_{h}_{,}

*, and*

_{T}*T*denote for the heating due to Newtonian relaxation [defined in Eq. (1)], heating due to kinetic energy dissipation (from both Rayleigh drag and hyperviscosity), heating due to temperature hyperdiffusion, and temperature, respectively. The lhs of the equation is the time mean diabatic entropy destruction, that is,

*η*defined in Eq. (10), which is the only destruction term. The other terms in the rhs are production terms. They are, from left to right, the eddy diabatic entropy production, the mechanical entropy production, and the entropy production of temperature diffusion (Fig. B1a).

Global entropy budget for the varying Δ* _{y}* simulations: (a) different terms in the global entropy budget described in appendix B and (b) the averaged dissipative temperature (

*T*) defined in Eq. (9). The entropy production of temperature diffusion is not shown as it is generally negligible.

_{ϵ}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Global entropy budget for the varying Δ* _{y}* simulations: (a) different terms in the global entropy budget described in appendix B and (b) the averaged dissipative temperature (

*T*) defined in Eq. (9). The entropy production of temperature diffusion is not shown as it is generally negligible.

_{ϵ}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

Global entropy budget for the varying Δ* _{y}* simulations: (a) different terms in the global entropy budget described in appendix B and (b) the averaged dissipative temperature (

*T*) defined in Eq. (9). The entropy production of temperature diffusion is not shown as it is generally negligible.

_{ϵ}Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0242.1

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