a. Supercriticality

In SW06, we showed that large-scale averages of extratropical thermal structures satisfy the constraint that the supercriticality

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does not substantially exceed one, where f₀ and β₀ are reference values of the Coriolis parameter and of its derivative. The supercriticality (1) resembles the supercriticality of the quasigeostrophic two-layer model (e.g., Pedlosky 1970; Held and Larichev 1996). However, the surface potential temperature gradient ∂_yθ_s appears in it in place of the vertically averaged potential temperature gradient in the two-layer model, and the bulk stability,

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with surface pressure p_s and tropopause pressure p_t, involves a static stability at or near the surface, −^s, in place of a vertically averaged static stability appearing in the two-layer model. Also, whereas an inviscid quasigeostrophic two-layer model is baroclinically stable if its supercriticality is less than one, S_c < 1 does not imply that an atmosphere is baroclinically stable.

As discussed in SW06, the supercriticality is a nondimensional measure of the slope of near-surface isentropes. It is also a ratio S_c ∼ 〈p_s − p_e〉/〈p_s − p_t〉 of two pressure differences: that between the surface and the level (p_e) up to which baroclinic entropy fluxes extend, and that between the surface and the tropopause. In the limit of small surface potential temperature gradients, baroclinic entropy fluxes are weak, and the extratropical thermal stratification and tropopause height are controlled by radiation and convection. In this limit, provided the radiative–convective thermal stratification is statically stable, baroclinic entropy fluxes do not extend to the tropopause, such that 〈p_e〉 > 〈p_t〉 and S_c < 1. This corresponds to the shallow-wave limit discussed by Held (1978a) in the context of quasigeostrophic theory. In the limit of large surface potential temperature gradients, baroclinic entropy fluxes modify the extratropical thermal stratification and tropopause height and extend to the tropopause. In this limit, the pressure range over which baroclinic eddies redistribute entropy defines the troposphere (Held 1982; Schneider 2004), such that 〈p_e〉 ∼ 〈p_t〉 and S_c ∼ 1. The net result is the constraint S_c ≲ 1 on extratropical thermal structures.

We found the constraint S_c ≲ 1 to be well satisfied in the idealized GCM simulations considered here, with consequences for properties of the macroturbulence (SW06; SchM). For example, Fig. 2 shows spectra of the barotropic eddy kinetic energy for two simulations with twice Earth’s radius, one with S_c < 1 (Δ_h = 22.5 K) and one with S_c ∼ 1 (Δ_h = 90 K). The spectra are typical of the simulations in general, both in that they exhibit a power-law range with an approximate n⁻³ or somewhat steeper scaling with spherical wavenumber n and also in that the Rossby wavenumbers are similar to the energy-containing wavenumbers.¹ The Rhines wavenumber is larger than the energy-containing wavenumber for S_c < 1 and similar to the energy-containing wavenumber for S_c ∼ 1. The energy-containing wavenumber is larger in the simulation with S_c < 1 than in the simulation with S_c ∼ 1, and this is also evident in instantaneous flow fields. For example, Fig. 3 shows the potential vorticity on isentropes at instants in the statistically steady states of the simulations in Fig. 2. The isentropes lie in the midtroposphere in the tropics and cross the tropopause in the extratropics. Wave breaking and potential vorticity filaments at the tropopause and the different length scales of the energy-containing eddies in the two simulations are evident. See SW06 for further discussion of characteristic eddy length scales and O’Gorman and Schneider (2007) for further discussion of the shape of the energy spectra. Our macroturbulence closure builds on the constraint S_c ≲ 1 and consequences such as those manifest in Figs. 2 and 3.

b. Relation between eddy energy and mean available potential energy

The macroturbulence closure relates eddy energies to the mean available potential energy and to the supercriticality, which both depend only on mean fields. In energy-balance models for a planet’s surface climate, it is desirable to express eddy fluxes in terms of surface quantities or quantities that can be estimated from them. We therefore approximate the mean available potential energy (MAPE) per unit area in a baroclinic zone of meridional width L_z by

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where

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is an inverse measure of the static stability at or near the surface, p₀ is a reference surface pressure, and κ = R/c_p is the adiabatic exponent. The approximate mean available potential energy (3) can be obtained from Lorenz’s (1955) quadratic expression for the mean available potential energy based on potential temperature variations on pressure or sigma surfaces by Taylor expansion and by approximating the potential temperature gradient and static stability in the troposphere by their surface values. See appendix A for details and for a demonstration of the accuracy of the approximation in the simulations. Using the supercriticality (1) and the approximate mean available potential energy (3) in a macroturbulence closure yields a closure in which eddy energies depend only on mean fields at the surface and on the tropopause pressure, which can be estimated from mean fields at the surface if one assumes, as in the approximation (3), that the static stability does not vary strongly in the vertical within the troposphere (Held 1982; Thuburn and Craig 2000; Schneider 2007).

Our macroturbulence closure relates the eddy available potential energy (EAPE) to the approximate mean available potential energy,

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where ξ(S_c) is a monotonically increasing function of the supercriticality, with normalization convention ξ(1) = 1. The constraint S_c ≲ 1 then implies ξ ≲ 1. For S_c ≪ 1, the closure (5) implies EAPE ≪ MAPE, consistent with baroclinic entropy fluxes that extend only over a fraction of the troposphere being inefficient at converting mean available potential energy into eddy energy (Held 1978a). For S_c ∼ 1, the closure (5) reduces to EAPE ∼ MAPE_a (cf. Green 1970).

Figure 4 shows the eddy available potential energy as a function of the approximate mean available potential energy in the idealized GCM simulations.² The mean fields entering the approximate mean available potential energy (3), and all other mean fields in what follows, are averaged over baroclinic zones that are determined based on the magnitude of the near-surface eddy flux of potential temperature. The length scale L_z is the meridional width of the so-determined baroclinic zones. The boundaries of the baroclinic zones for the two simulations shown in Fig. 1 are marked by the gray vertical bars in that figure. See appendix B for conventions for the determination of baroclinic zones and for the sensitivity of our results to them. Eddy available potential energies, and all other eddy fields in what follows, are averaged between 20° and 90° latitude in both (statistically identical) hemispheres. For most simulations, eddy fields could be averaged globally without substantially changing the results; restricting the averages to latitudes poleward of 20° helps to avoid singularities in low latitudes arising in the computation of the eddy available potential energy in simulations with statically near-neutral or neutral convective lapse rate. Eddy fields could also be averaged over the baroclinic zones over which mean fields are averaged without substantially changing the results.

Figure 4 shows that, for sufficiently large surface potential temperature gradients and mean available potential energies, all simulations approximately condense onto the linear relation EAPE = MAPE_a. This is consistent with the closure (5) and with the result of SW06 and SchM that, for sufficiently large surface potential temperature gradients, all simulations satisfy S_c ∼ 1. For small surface potential temperature gradients and mean available potential energies, Fig. 4 indicates EAPE < MAPE_a, consistent with the closure (5) and ξ(S_c) < 1 for S_c < 1.³ The simulations in which the supercriticality, estimated as in SW06 and SchM, is less than 90% of the value in the simulation with the largest Δ_h in the corresponding series are identified by pale plotting symbols. Figure 4 shows that, for each series of simulations, the simulations with EAPE < MAPE_a are generally the simulations in which the supercriticality is less than the O(1) saturation value for the series. Exceptions are a few simulations with a statically neutral convective lapse rate (γ = 1.0) and low surface potential temperature gradients, for which the static stability near the surface is nearly zero and EAPE and MAPE_a are poorly defined. The eddy available potential energy and mean available potential energy in Earth’s atmosphere in the annual mean (not taking moisture effects into account) are likewise comparable with each other, with a ratio EAPE/MAPE_a ≈ 1.17 similar to that in the simulations in the regime EAPE ∼ MAPE_a (Fig. 4a).

The scaling of the eddy available potential energy with the mean available potential energy thus is in two regimes: (i) A regime EAPE < MAPE_a, in which S_c < 1 and the extratropical thermal stratification and tropopause height are controlled by radiation and convection; (ii) a regime EAPE ∼ MAPE_a, in which S_c ∼ 1 and baroclinic entropy fluxes modify the extratropical thermal stratification and tropopause height.

The form of the function ξ(S_c) for S_c < 1 could in principle be determined from results such as those in Fig. 4. However, the value of the supercriticality in simulations with small surface potential temperature gradients depends too strongly on how the mean fields entering the supercriticality are averaged to allow us to determine the function ξ(S_c) reliably. In identifying simulations with supercriticalities less than 90% of the saturation value for a given series of simulations, we estimated the supercriticalities as in SW06 and SchM, although mean fields in those papers were averaged over wider baroclinic zones than in the present paper (see appendix B for details). Choosing the narrower baroclinic zones of the present paper generally gives somewhat higher supercriticalities for simulations with small surface potential temperature gradients and hence gives somewhat different functional forms for ξ. Smoothing the boundaries of the baroclinic zones, as well as the latitudes at which the reference values f₀ and β₀ are evaluated, across variations in Δ_h within each series of simulations (SW06; SchM) also affects supercriticalities for simulations with small surface potential temperature gradients. With these caveats, near the transition from the regime with S_c < 1 to the regime with S_c ∼ 1, the function ξ appears to be of the form ξ ≈ S^r_c with exponent r = 4 ± 1; however, for smaller surface potential temperature gradients and smaller supercriticalities, the dependence on the supercriticality appears to be stronger. For example, as Δ_h is increased from 30 to 60 K in the series of simulations with a = 4a_e, the eddy available potential energy increases by almost four orders of magnitude, while both the approximate mean available potential energy and the supercriticality change only by O(1) factors (see the two simulations with lowest EAPE for a = 4a_e in Fig. 4a). This points to the existence of critical potential temperature gradients for baroclinic instability, which may be an artifact of the vertical discretization in the GCM. The ambiguities in the choice of averaging regions and the questions about the influence of the vertical discretization on our results for the smallest surface potential temperature gradients prevent us from making more precise statements about the form of the function ξ.

The function ξ in the closure (5) can depend on external parameters, such as the planetary radius or rotation rate and radiative or frictional parameters. Figure 4c reveals that there is a weak dependence on the Newtonian relaxation time. The ratio EAPE/MAPE_a decreases by about a factor of 1.7 for the simulations with the largest values of Δ_h as the Newtonian relaxation time is increased from τ = 12.5 to 200 days. The saturation value of the supercriticality decreases by a similar factor (SchM), suggesting that the decrease in the ratio EAPE/MAPE_a may be related. On the other hand, the ratio EAPE/MAPE_a decreases more weakly if the mean available potential energy is averaged over wider baroclinic zones. For example, it decreases by only a factor of 1.3 if baroclinic zones are determined as in SchM—which leaves the significance of the decrease unclear. Systematic dependences on other external parameters are not evident in the figure. For sufficiently large surface potential temperature gradients and mean available potential energies, the simulations with different planetary radii and rotation rates and with reduced surface roughness length appear to collapse onto the closure (5), without explicit dependence on these parameters (or with a dependence too weak to be apparent). However, our exploration of the dependence of the closure (5) on frictional parameters obviously is not exhaustive and cannot rule out explicit dependences on them.

c. Relations among eddy energies

Closures for other eddy energies can be obtained from the closure (5) by relating the other eddy energies to the eddy available potential energy. In the linearly most unstable baroclinic waves, eddy available potential energy and barotropic and baroclinic eddy kinetic energy are equipartitioned (i.e., they scale linearly with each other). Nonlinear eddy–eddy interactions can modify eddy scales and the partitioning among eddy energies (Charney 1971; Rhines 1975; Salmon 1980, 1982), but in quasigeostrophic models, strong nonlinear eddy–eddy interactions require that the quasigeostrophic counterparts of the supercriticality satisfy S_c ≫ 1 (Pedlosky 1970, 1979; Held and Larichev 1996). The constraint S_c ≲ 1 on extratropical thermal structures suggests that nonlinear eddy–eddy interactions are weak and that macroturbulent eddies in dry atmospheres exhibit the same equipartitioning of eddy energies as the linearly most unstable baroclinic waves (SW06).

Figure 5 shows the barotropic eddy kinetic energy (EKE_bt) as a function of the eddy available potential energy. Over the entire range of climates simulated, spanning several decades of eddy energies, the simulation results indeed collapse approximately onto the linear relation EAPE = EKE_bt. Simulations with statically near-neutral or neutral convective lapse rates (γ = 0.9 and 1.0) exhibit anomalously large eddy available potential energies, particularly for small surface potential temperature gradients and eddy available potential energies, for which the stabilization of the extratropical thermal stratification by baroclinic entropy fluxes is weak (Fig. 5b). These anomalies are caused by vanishing or small static stabilities near the surface, where the parameterized mixing of dry static energy in the planetary boundary layer tends to reduce the static stability toward zero. The anomalies largely disappear when levels within the planetary boundary layer are excluded from the computation of the eddy available potential energy. The ratio of eddy available potential energy to barotropic eddy kinetic energy in Earth’s atmosphere (EAPE/EKE_bt ≈ 0.8) is similar to that in the simulations (Fig. 5a).⁴

Figure 6 shows the baroclinic eddy kinetic energy (EKE_bc), rescaled by the factor (a′Ω′)^0.3 with a′ = a/a_a and Ω′ = Ω/Ω_e, as a function of the eddy available potential energy. The baroclinic eddy kinetic energy and eddy available potential energy likewise scale linearly with each other.⁵ Their ratio exhibits a dependence on planetary radius and rotation rate, scaling roughly as EAPE/EKE_bc ∼ (a′Ω′)^0.3. Whether this ratio indeed obeys a power law in a′Ω′, and if so, with what exponent, cannot be determined unambiguously from the simulations. Power laws with exponents roughly between 0.2 to 0.5 are consistent with the simulation results.

The baroclinic eddy kinetic energy is smaller than the barotropic eddy kinetic energy and the eddy available potential energy. The simulation results collapse approximately onto the linear relation EAPE = 2.25 (a′Ω′)^0.3 EKE_bc. A few exceptions are again simulations with a statically neutral convective lapse rate (γ = 1.0), for which eddy available potential energies are anomalously large (Fig. 6b). The anomalies again disappear if levels within the planetary boundary layer are excluded from the computation of the eddy available potential energy (SW06). As for the barotropic eddy kinetic energy, the ratio of eddy available potential energy to baroclinic eddy kinetic energy in Earth’s atmosphere (EAPE/EKE_bc ≈ 2.2) is similar to that in the simulations (Fig. 6a).

The equipartitioning EAPE ∼ EKE_bt is consistent with nonlinear eddy–eddy interactions being weak. However, one would also expect this equipartitioning if eddy–eddy interactions were strong and engendered an inverse energy cascade (Held and Larichev 1996). That the equipartitioning EAPE ∼ EKE_bt is seen in the simulations may therefore not be surprising, although it is still striking that it is seen so clearly over such a wide range of eddy energies. The equipartitioning EAPE ∼ EKE_bc is likewise consistent with nonlinear eddy–eddy interactions being weak. If eddy–eddy interactions were strong and engendered a substantial inverse energy cascade, one would expect EAPE ∼ EKE_bt ≫ EKE_bc (Held and Larichev 1996; see also Vallis 2006, chapter 9.3). The equipartitioning EAPE ∼ EKE_bc therefore is evidence that nonlinear eddy–eddy interactions are weak and an inverse energy cascade beyond the scale of the linearly most unstable baroclinic waves is inhibited (SW06). This gives a self-consistent picture: in SW06, we showed that if S_c ≲ 1 and EKE_bt ≲ MAPE_a, as implied by the closure (5) and by the equipartitioning EKE_bt ∼ EAPE, and provided that the relevant baroclinic zone width L_z in the approximate mean available potential energy (3) does not exceed eddy length scales, as is reasonable to assume given that the relevant mean available potential energy is that accessible to eddies, then the Rhines scale is smaller than or at most similar to the Rossby radius (e.g., Fig. 2). Therefore, the energy-containing eddy length scale is similar to the Rossby radius, an inverse energy cascade is inhibited, and nonlinear eddy–eddy interactions are weak.

We presently do not have an explanation for the dependence of the ratios EAPE/EKE_bc ∼ EKE_bt/EKE_bc on the planetary radius and rotation rate. The dependence does not seem to be related to the ratio of domain size to the scale of the energy-containing eddies, and thus to the number of jets in each hemisphere and to the room for a potential inverse energy cascade. Like simulations with an increased planetary radius and rotation rate, some simulations with Earth’s radius and rotation rate exhibit small energy-containing eddies and multiple jets in each hemisphere, without exhibiting a significantly different EAPE/EKE_bc ratio than Earth-like simulations with larger energy-containing eddies and only one jet in each hemisphere (see also SW06; Schneider 2006).

With the scaling laws relating the eddy available potential energy to the mean available potential energy and the eddy energies to each other, we can now derive closures for mean values of eddy fluxes.

a. Regime transitions

At the transition between the regime S_c < 1, in which the extratropical thermal stratification and tropopause height are controlled by radiation and convection, and the regime S_c ∼ 1, in which baroclinic entropy fluxes modify the extratropical thermal stratification and tropopause height, the dependence of eddy fields on mean fields changes. The scaling laws for the dependence of eddy fields on mean fields involve the product MAPE_a × ξ(S_c). Restricted to the dependence on the surface potential temperature gradient and the static stability and with a power law ξ ∼ S^r_c for the function ξ, this product scales as

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where we have used S_c ∼ Γ_s〈∂_yθ_s〉. The relation (12) does not fit the simulation results quantitatively because the dependences suppressed in it, for example, on the tropopause pressure, are important to account for the simulation results. However, the relation (12) illustrates that the dependence of MAPE_a × ξ(S_c) on the surface potential temperature gradient and on the static stability changes qualitatively at the transition between the regimes. In the regime S_c < 1, MAPE_a × ξ(S_c) depends explicitly on the static stability (controlled by convection) and depends strongly nonlinearly on the surface potential temperature gradient (as mentioned in section 3b, the exponent r = 4 ± 1 is consistent with the simulation results near the transition). In the regime S_c ∼ 1, the explicit dependence on the static stability can be eliminated, and MAPE_a × ξ(S_c) depends linearly on the surface potential temperature gradient.

The change in the dependence of MAPE_a × ξ(S_c) on the surface potential temperature gradient and on the static stability implies similar changes in the dependence of eddy fields on these mean fields at the regime transition. Consider, for example, the mean eddy flux of surface potential temperature and the mean eddy momentum flux convergence. In the regime S_c < 1, they depend explicitly on the convective lapse rate and depend strongly nonlinearly on the surface potential temperature gradient and thus on the mean available potential energy (Figs. 7a′–c′ and 8a′–c′). In the regime S_c ∼ 1, they do not depend explicitly on the convective lapse rate and depend more weakly on the surface potential temperature gradient and thus on the mean available potential energy. In this regime, the mean eddy flux of surface potential temperature, following the same reasoning as in Eq. (12), scales as 〈∂_yθ_s〉^3/2. This is roughly consistent with the scaling of the column-integrated eddy flux of sensible heat with the surface potential temperature gradient over a seasonal cycle in midlatitudes of Earth’s atmosphere (Stone and Miller 1980). The mean eddy momentum flux convergence, using the expression (B2) for the Rossby radius, scales as 〈∂_yθ_s〉^1/2. Climate stability implications of the different scalings in the two regimes are discussed below.

b. General circulation

As outlined in the introduction, the eddy flux of surface potential temperature and the eddy momentum flux convergence are of fundamental importance for the surface climate of a planet and for the general circulation of its atmosphere. The closure (9) relating the mean eddy flux of surface potential temperature to mean fields can be used in an energy-balance model for a planet’s surface climate to determine the extratropical mean temperatures vertically averaged over the troposphere and, given the static stability, the extratropical mean surface temperatures. The closure (11), relating the mean eddy momentum flux convergence to mean fields, can be used to determine the extratropical mean zonal surface wind, provided that turbulent boundary layer drag can be modeled as depending only on the mean surface wind and other mean fields at the surface. For the function ξ(S_c) in these closures, a power law ξ = S^r_c with exponent r = 4 ± 1 may be a starting point for qualitative studies of the dependence of the extratropical surface climate on external parameters. The static stability required in such studies can be taken to be the greater of a convective static stability and the static stability consistent with S_c ∼ 1, such that the constraint S_c ≲ 1 is satisfied.

Beyond the extratropical surface climate, the closure (11) for the mean eddy momentum flux convergence also has implications for the tropical Hadley circulation. In Walker and Schneider (2006), we showed that eddy momentum flux divergence can strongly influence the strength of Hadley cells. In most of the simulations considered here—the principal exceptions being simulations with statically near-neutral or neutral convective lapse rates or with very small surface potential temperature gradients (see Walker and Schneider 2006, their Fig. 6)—the Coriolis force on the mass flux in the upper branches of the Hadley cells, near their center, is primarily balanced by eddy momentum flux divergence; that is, the effective Rossby number of the Hadley cells is small. The simulations in which this balance approximately holds generally are the simulations in which, as discussed in section 5, the global-mean eddy momentum flux divergence is proportional to the mean eddy momentum flux convergence that we considered. Moreover, the global-mean divergence is dominated by the divergence in the equatorward flanks of the subtropical jets—the divergence that influences the Hadley circulation (see Fig. 1 for an example). The implication is that, in those simulations, the strength of the Hadley cells scales with the mean eddy momentum flux convergence, with some modifications resulting from variations in the extent of the Hadley cells that affect the relevant value of the Coriolis parameter linking the Hadley cell mass flux and the eddy momentum flux divergence. The strength of the Hadley cells exhibits the same regime transition as the mean eddy momentum flux convergence, with the dependence of the Hadley cell strength on the surface potential temperature gradient changing at the transition (Walker and Schneider 2006). Coupled to a scaling law for the extent of the Hadley cells and thus for the relevant value of the Coriolis parameter, the closure (11) for the mean eddy momentum flux convergence gives a scaling law for the strength of a Hadley cell when its effective Rossby number is small.

c. Climate stability

The regime transition in the scaling of the mean eddy momentum flux convergence and the mean eddy flux of surface potential temperature has implications for the stability of climate with respect to changes in meridional radiative forcing gradients. Consider changes in radiative forcing gradients that are sufficiently small that their effect on the radiative–convective equilibrium static stability, if there is any, can be neglected. In a climate in which baroclinic entropy fluxes modify the extratropical thermal stratification and tropopause height (S_c ∼ 1), a small change in the radiative forcing gradient is counteracted by a proportionately weak response in the eddy flux of surface potential temperature and, if the effective Rossby number of the Hadley cell is small so that the Hadley cell strength primarily responds to changes in eddy momentum flux divergence, by a proportionately weak response in Hadley cell strength. Conversely, in a climate in which convection and radiation control the extratropical thermal stratification and tropopause height (S_c < 1), a small change in the radiative forcing gradient is counteracted by a proportionately stronger response in the eddy flux of surface potential temperature and, if the effective Rossby number of the Hadley cell is small, by a proportionately stronger response in the Hadley cell strength. That is, the proportionate response of the surface potential temperature gradient to a given fractional change in the radiative forcing gradient can be expected to be weaker in a climate in which convection controls the extratropical thermal stratification than in a climate in which baroclinic eddies modify the extratropical thermal stratification. See Zhou and Stone (1993a, b) for a demonstration of this effect for the extratropical climate in a two-layer model.

Assuming these considerations carry over to moist atmospheres, at least qualitatively, one may speculate that a warm climate with sufficiently large convective static stability (e.g., a statically very stable moist adiabat) exhibits a proportionately weaker response to changes in radiative forcing gradients (such as occur, e.g., over the course of a seasonal cycle) than a cold climate with small convective static stability (e.g., a statically neutral dry adiabat). However, effects such as water vapor feedback and changes in the radiative–convective equilibrium static stability in response to radiative forcing changes are likely to complicate any manifestation of such dynamical effects.

d. Relation to previous work

Several aspects of the scaling laws and regime transitions discussed in this paper have previously been discussed in the literature. Green (1970) presented a closure that relates eddy energies to the mean available potential energy. This closure is essentially equivalent to the closure (5) for the regime S_c ∼ 1. Stone (1972) presented a closure that relates eddy energies to the baroclinic mean kinetic energy. Such a closure does not fit our simulation results well because it gives an incorrect dependence of eddy energies on the static stability and on the planetary radius and rotation rate. Both Green and Stone, as well as several subsequent authors, used equipartitioning of eddy energies in closures for atmospheric macroturbulence, with Green arriving at a closure for the eddy flux of potential temperature similar to the closure (9); however, that eddy energies in GCM simulations of a wide range of circulations satisfy equipartitioning relations—with a striking degree of accuracy—has not been demonstrated before. It also appears not to have been noticed before that the ratio of the baroclinic eddy kinetic energy to the other eddy energies depends on the planetary radius and rotation rate.

The importance for macroturbulence closures of modifications of the extratropical thermal stratification by baroclinic entropy fluxes was discussed by Held (1978a, b). Based on quasigeostrophic theory and Stone’s (1972) analysis, Held (1978a) argued that the column-integrated eddy flux of potential temperature should depend more strongly on potential temperature gradients when a quasigeostrophic counterpart of the supercriticality satisfies S_c ≪ 1 than when it satisfies S_c ≫ 1. This is qualitatively similar to our results, albeit with quantitative differences and with the difference that the limit S_c ≫ 1 appears unattainable in atmospheric macroturbulence. Based on simulations with a two-layer model on the sphere, Held (1978b) suggested a closure similar to (5) that included an empirical function of a quasigeostrophic supercriticality analogous to our function ξ(S_c). Building on Held’s work, Zhou and Stone (1993a, b) demonstrated that modifications of the atmospheric thermal stratification by baroclinic entropy fluxes are important for the scaling of eddy potential temperature fluxes in Lorenz’s (1960) two-level model on the sphere. They showed that the scaling of eddy potential temperature fluxes with potential temperature gradients can be in two regimes, depending on whether eddies are or are not allowed to modify the thermal stratification. Held (2007) presented alternative qualitative arguments of how scaling laws of atmospheric macroturbulence can change depending on whether or not baroclinic entropy fluxes modify the extratropical thermal stratification.

e. Limitations and broader applicability

The results presented in this paper have two fundamental limitations. First, the scaling laws we presented are scaling laws for mean values of eddy energies and eddy fluxes. They do not give information on the spatial structure of the fluxes, except for the information that the eddy momentum flux varies meridionally on the energy-containing eddy length scale. For theories of a planet’s surface climate and general circulation beyond the level of scaling laws for bulk quantities, information on the horizontal and vertical structure of the eddy fluxes is required. For an energy-balance model for a planet’s surface climate, one may attempt to use diffusive closures of the eddy flux of surface potential temperature that are consistent with the scaling laws for mean values of the flux presented in this paper. We have experimented with using the surface potential temperature gradient, smoothed over eddy length scales, as a meridional structure function for the eddy flux of surface potential temperature. Although it appears adequate for qualitative explorations of the dependence of a planet’s surface climate on external parameters, such a procedure is not accurate over the entire range of climates we simulated. A more promising (albeit computationally more demanding) approach appears to be to use the weakness of eddy–eddy interactions in turbulence closures that solve for the spatial structure of eddy statistics by a suitable approximation of the hierarchy of moment equations (Marston et al. 2008; O’Gorman and Schneider 2007).

Second, the scaling laws we presented neglect the effects of latent heat release in phase changes of water. An essential question in understanding how latent heat release modifies the scaling laws for dry atmospheres is the question of how the effective static stability experienced by baroclinic eddies is modified. This is an open question (Schneider and O’Gorman 2008). The general consistency of the annual-mean statistics for Earth’s atmosphere with the scaling laws and simulation results suggests that the scaling laws may be modified to apply to moist atmospheres. Encouraging results suggesting that aspects of the scaling laws carry over directly to moist atmospheres are presented by O’Gorman and Schneider (2008b), who show that eddy kinetic energies scale with the (dry) mean available potential energy in a range of simulations with an idealized GCM that includes a hydrologic cycle. The reasons for this empirical result and the question of how other aspects of the scaling laws for dry atmospheres can be generalized to moist atmospheres remain to be investigated.