Abstract

This paper discusses the sensitivity of the isentropic slope in a primitive equation dry model forced with Newtonian cooling when the heating is varied. This is done in two different ways, changing either the radiative equilibrium baroclinicity or the diabatic time scale for the zonal-mean flow. When the radiative equilibrium baroclinicity is changed, the isentropic slope remains insensitive against changes in the forcing, in agreement with previous results. However, the isentropic slope steepens when the diabatic heating rate is accelerated for the zonal-mean flow. Changes in the ratio between the interior and the boundary diffusivities as the diabatic heating rate is varied appear to be responsible for the violation of the constant criticality constraint in this model. Theoretical arguments are used to relate the sensitivity of the isentropic slope to that of the isentropic mass flux, which also remains constant when the radiative-equilibrium baroclinicity is changed. The sensitivity of the isentropic mass flux on the heating depends on how the gross stability changes. Bulk stabilities calculated from isobaric averages and gross stabilities estimated from isentropic diagnostics are not necessarily equivalent because a significant part of the return flow occurs at potential temperatures colder than the mean surface temperature.

1. Introduction

One of the most salient features of the tropospheric thermal structure is the robustness of the extratropical isentropic slope. Although the horizontal/vertical potential temperature gradients change significantly from winter to summer, their ratio, the isentropic slope, remains roughly constant throughout the annual cycle (Stone 1978). In both hemispheres, the (normalized) isentropic slope is on the order of 1, which implies that potential temperature differences between subtropics and Poles and between the surface and tropopause are comparable. In other words, the isentrope leaving the surface in the subtropics reaches the tropopause in polar latitudes:

 
formula

where ∂yθ, ∂pθ are characteristic meridional/vertical potential temperature gradients, Ly ∼ | f|/βa is a characteristic meridional scale (a is the radius of earth), and Hp = pspt is a characteristic pressure difference between surface and tropopause. Early theories (Stone 1978) attributed the robustness of the isentropic slope to “baroclinic adjustment,” that is, to the efficient neutralization of the mean state by the eddies. The rationale for this was that the condition ξ = 1 (with Hp the layer depth) corresponds to a stability limit for the two-layer quasigeostrophic (qg) model (Phillips 1954). For this reason, the parameter ξ has been historically called the criticality. There are, however, two important difficulties with this idea: (i) the neutrality condition ξ = 1 is artificial and unique to the two layer model and (ii) the two-layer model equilibrates at values ξ > 1 [see Zurita-Gotor and Lindzen (2007) for a review].

More recently, Schneider (2004) has confirmed Stone’s findings on the robustness of the extratropical isentropic slope using an idealized dry GCM. This is illustrated in Fig. 1, which shows results from simulations with a model similar to Schneider’s when the “radiative-equilibrium” meridional temperature gradient is varied (see section 3 for details). Figures 1a,b show the mean meridional and vertical temperature potential gradients at 48°N, and Fig. 1c shows their ratio, the isentropic slope. We can see that the isentropic slope remains very robust in these runs, much more so than the meridional and vertical gradients independently.1 Since the robustness of the isentropic slope holds at all heights, the same is true for its vertical derivative and for the (potential vorticity) PV gradient, consistent with the observations of Kirk-Davidoff and Lindzen (2000) that the PV gradient varies little throughout the seasonal cycle.

Fig. 1.

For the runs with γ = 1 and δy = 60 (thick solid), 90 (thin solid), 120 (dashed), and 180 K (dashed–dotted), the time- and zonal-mean (a) meridional potential temperature gradient, (b) vertical potential temperature gradient, and (c) isentropic slope at 48°N. (d) Same as (c) but for runs with δy = 60 K and γ = 1 (thick solid), 5 (thin solid), 20 (dashed), 40 (dashed–dotted), and 0.2 (dotted).

Fig. 1.

For the runs with γ = 1 and δy = 60 (thick solid), 90 (thin solid), 120 (dashed), and 180 K (dashed–dotted), the time- and zonal-mean (a) meridional potential temperature gradient, (b) vertical potential temperature gradient, and (c) isentropic slope at 48°N. (d) Same as (c) but for runs with δy = 60 K and γ = 1 (thick solid), 5 (thin solid), 20 (dashed), 40 (dashed–dotted), and 0.2 (dotted).

Schneider (2004) did not attribute his finding to neutrality, but proposed a completely different theoretical framework. He showed that the assumption of a vertically uniform diffusivity also leads to the constraint ξ ≈ 1. Heuristically, one can understand this result by noting that the constraint ξO(1) is equivalent to the condition that the vertically integrated positive and negative PV gradients are comparable (Zurita-Gotor and Lindzen 2007). This must be the case when the diffusivity is vertically uniform because the eddy PV fluxes (including generalized boundary contributions) integrate vertically to zero when the eddy momentum fluxes are small. There are also some subtle differences between the traditional formalism and Schneiders’s; for instance, near-surface temperature gradients are used in place of their midtropospheric counterparts.

On the other hand, some previous studies (e.g., Stone and Branscome 1992; Cehelsky and Tung 1991; Welch and Tung 1998) have argued that the two-layer qg model has its own form of baroclinic adjustment, in the sense that it also tends to display a robust isentropic slope when the forcing is varied. Since this slope is typically supercritical, the above finding cannot be attributed to neutrality. It also cannot be a general property of the system, as it conflicts with results from doubly periodic simulations. Regardless of the neutrality issue and the generality of this result, whether the isentropic slope is robust or not has consequences of its own because the isentropic slope is a measure of the basic-state PV gradient. That the isentropic slope remains robust implies that local theories like diffusion must fail because the system produces different fluxes for the same mean state (Zurita-Gotor 2007). This has led Zurita-Gotor (2007) to propose a generalized baroclinic adjustment concept as the tendency of the system to equilibrate at some preferred equilibrium states, regardless of the physics. In contrast with local theories like diffusion, in the baroclinic adjustment limit, the eddy fluxes are not determined locally but depend on the fluid redistribution throughout the domain.

Using this framework, Zurita-Gotor (2007) investigated whether local diffusion or baroclinic adjustment was more relevant for the quasigeostrophic two-layer eddy-mean closure in different parts of the parameter space. His results suggest that the local paradigm is more appropriate, even for narrow jets. Zurita-Gotor found that the model’s criticality changes slowly with the forcing, but there is no preferred equilibrium state. To arrive at these conclusions, Zurita-Gotor (2007) used a variation of the traditional Newtonian cooling formulation, in which only the forcing time scale (for the zonal mean) is changed, but the diabatic damping scale for the eddies is fixed. This allowed the author to change the forcing by several orders of magnitude, including very short time scales, for which the flow would be stabilized if the eddies were damped at the same rate. These strong variations provided a clearer picture of the basic-state sensitivity on the forcing than in previous studies.

The purpose of the present study is to repeat the analysis of Zurita-Gotor (2007) for the continuous primitive equation (PE) case. The results of Schneider (2004) described above imply that the condition ξ ≈ 1 is very robust for this model, even when the meridional temperature gradient changes strongly. On the other hand, one would expect the basic state criticality to approach its radiative equilibrium value ξRE when forcing strongly enough as in Zurita-Gotor (2007). If so, it is not clear how the transition between both limits occurs. One possibility is that, as in the qg case, the isentropic slope changes slowly but smoothly with the forcing. That would imply that there is nothing special about the observed value ξ ≈ 1, other than it is very hard to change it due to the strong dynamical feedbacks (possibly including non-qg feedbacks). Another possibility is that different regimes exist, so that the constraint of a constant criticality holds over some parts of the parameter space but fails when the forcing becomes strong enough. In either case, it is important to understand which of the assumptions leading to the constant criticality constraint is violated when the criticality changes.

This paper is organized as follows: section 2 translates the arguments of Zurita-Gotor (2007) into the isentropic framework. Section 3 introduces the numerical model used. Section 4 describes the sensitivity of the isentropic slope and eddy scale in our model on the forcing. Section 5 shows some isentropic diagnostics aimed at testing Schneider’s assumptions. Section 6 finishes with some concluding remarks.

2. Theory

Zurita-Gotor (2007) discusses the sensitivity of the isentropic slope/criticality in the two-layer qg model. Zurita-Gotor argues that the criticality regulates the net residual mass transport, based on the Transformed Eulerian Mean (TEM) momentum balance and a diffusive closure. Since this mass circulation must transport enough heat to balance the diabatic heating, the criticality must increase monotonically with the net heating (which is internally determined) and hence with the inverse diabatic time scale (which is what one actually puts into the model). However, the dependence of the criticality on the diabatic time scale is slow, for the following two reasons: (i) the strength of the circulation—or the net heating2 increases slowly (sublinearly) with the diabatic forcing rate and (ii) the steepness of the diffusive closure (Held and Larichev 1996) implies that moderate changes in criticality suffice to produce large variations in the eddy PV fluxes and in the mass circulation.

Qg theory is widely regarded as a reasonable approximation in the midlatitudes. This has motivated separating in the past the determination of the extratropical mean state into two different problems: (i) the determination of the meridional temperature gradient using a qg framework, given the observed static stability and (ii) the determination of the static stability independently, using other constraints. However, as shown by Schneider (2004) and illustrated in Fig. 1, the adjustments in the meridional temperature gradient and in the static stability may not be independent. As an alternative one can use an isentropic framework, which formally resembles the qg TEM formalism (Tung 1986) but does not assume a fixed static stability. We next generalize the above arguments to this framework.

a. Relation between isentropic mass transport and net heating

In the qg problem, the net heating and the mass transport increase proportionally because the stratification is prescribed. In contrast, the relation between the mass and energy transports is more complicated in the PE case because the stratification can change. We can make this explicit by manipulating the time-mean continuity equation in isentropic coordinates:

 
formula

We use the formalism of Tung (1986) and the surface conventions of Schneider (2005): ρθ = −g−1θpℋ (θθs) is the isentropic density (ℋ is the Heavyside function and θs the surface temperature), bars denote zonal and time averages on isentropic surfaces, and * = / ρθ represents a mass-weighted average and () deviations from this average.

Multiplying this equation by θ and integrating by parts, we obtain

 
formula

Integrating meridionally from the latitude y to the pole, and vertically between θb and θt, the lowest and highest temperature over which the isentropic circulation extends, and defining the isentropic mass flux as

 
formula

(θm is the tropospheric potential temperature at which the circulation reverses), we obtain

 
formula

where 𝒬 is the vertically integrated heating and

 
formula

is the gross stability (Neelin and Held 1987; Czaja and Marshall 2006), a measure of the energy transport per unit of mass circulation. Equation (5) shows that the maximum poleward energy transport occurs at the latitude y0 where the vertically integrated heating changes sign, and increases with the net heating. The quantity

 
formula

is thus a measure of the diabatic maintenance of the mass circulation, the isentropic equivalent to the midchannel potential momentum forcing αT(MMR) in Zurita-Gotor (2007). The main difference is that Sθ, which is prescribed in the qg case, may also play a role in the PE problem. With enhanced heating the meridional temperature gradients increase, and the results of Schneider (2004) imply that the same is true for the stratification and the gross stability Sθ. This points to a weaker sensitivity of the mass circulation on the diabatic heating rate than in the qg case, when Sθ is prescribed.

b. Relation between isentropic mass transport and isentropic slope

The second part of the argument in Zurita-Gotor (2007) relates the mass flux to the eddy PV flux via the momentum balance, and the latter to the isentropic slope by means of a diffusive closure. Similar relations exist for the PE case when an isentropic formalism is used. Following Schneider (2005), we write

 
formula

where Pf /ρθ is the potential vorticity, and we use Schneider’s convention that ρθP = f inside the surface so that P* ≈ f /ρθ. The density 0 represents the time- and zonal-mean isentropic density (y,θ), evaluated at the mean surface temperature (y) at that latitude. Schneider and Walker (2006) approximate this term as 0 ≈ −(2gs)−1. The superscript s denotes along-surface means, and the eddy heat flux in the last equation in (8) is calculated using the balanced meridional velocity only, not the full one (see Schneider 2005 for details). We used a planetary-geostrophy approximation above, which requires for consistency neglecting the eddy momentum fluxes and surface drag; the cross-isentropic momentum advection was also neglected. A final assumption implicit in Eqs. (8) is that all the poleward flow occurs in the interior troposphere [i.e., θm is above the surface layer, in the sense of Held and Schneider (1999)], so that the form drag forcing only affects isentropic layers with equatorward mean flow.

To relate these mass fluxes to the isentropic slope, we use diffusive closures:

 
formula

with average diffusivities 𝒟1, 𝒟2, and 𝒟s over the regions of poleward/equatorward flow, and at the surface, respectively. Plugging this into Eq. (8), and following a similar derivation to that in Schneider and Walker (2006), we obtain

 
formula

To integrate this equation, we use the definition of the mean isentropic density

 
formula

which allows us to express the vertical integrals of ρθ and ∂yρθ as follows:

 
formula
 
formula

The above expressions define the isentropic quantities (θ) and Ĩ = −(∂/∂y). Note that (θ) and Ĩ(θ) only coincide with the Eulerian mean pressure p(θ) and the Eulerian mean isentropic slope I(θ) of the isentropic surface θ when θ is above the surface layer [i.e., when ℋ(θθs) = 1 at all longitudes and times]. In contrast, for isentropes within the surface layer, ps is a function of θ, and likewise (θ) ≠ p(θ). See Held and Schneider (1999) for details.

Plugging these definitions into Eq. (10), we obtain

 
formula

In the last step we assumed that θm is above the surface layer, so that p(θ) ≡ (θ) and I(θ) ≡ Ĩ(θ), and neglected the isentropic slope at the tropopause. The quantity r = (pmpt)/Hp represents the fraction of the tropospheric depth occupied by the poleward flow, and the subscripts m and t indicate values evaluated at the potential temperatures θm and θt.

An analogous derivation for the equatorward flow yields

 
formula

The second term in Eq. (15) arises from the use of a diffusive closure for the surface eddy heat flux and Schneider and Walker’s (2006) approximation for 0. Equating both mass fluxes

 
formula

and setting 𝒟1 = 𝒟2 = 𝒟s, we can recover the criticality constraint of Schneider and Walker (2006)  Sc ≈ 1 (note that their criticality Sc differs from our definition of ξs by a factor of 2).

There are obvious similarities between the above expressions and their quasigeostrophic counterparts [but see Schneider (2005) for a discussion of the differences]. Equations (14)(15) suggest that the mass flux should increase with the criticality, both in the interior and at the surface. However, what makes the mass flux so sensitive to the criticality in the qg case is the additional dependence of the diffusivity on the criticality, which is very steep (Held and Larichev 1996). In contrast, there is not a well-developed turbulent theory in the non-qg case, when the stratification is free to adjust; as shown by Held (2007), this can make a big difference for the steepness of the closure.

3. Description of the model

The model used consists of the dynamical core of the Geophysical Fluid Dynamics Laboratory’s (GFDL) climate model supplemented with highly idealized physics and is very similar to that described by Held and Suarez (1994). The model is spectral, has a flat surface, and uses sigma coordinates in the vertical. A T42 truncation with 20 equally spaced vertical levels was employed in all runs presented in this paper. The model is dry, and the only forcing and dissipation included are Rayleigh friction, Newtonian heating, and hyperdiffusion. The Rayleigh coefficient for friction αM has a maximum value of 1 day−1 at the surface and decreases linearly with σ to a value of zero at the top of the boundary layer (σb = 0.7). The heating is defined through linear relaxation to a prescribed radiative-equilibrium distribution independent of longitude. The Newtonian forcing rate αT adopts a different value in the free troposphere (αTf = 1/20 day−1) and over the boundary layer, where it is also latitude-dependent (αTs = 1/4 day−1 at the equatorial surface):

 
formula

Finally, the radiative-equilibrium profile to which temperature is relaxed is defined by the following functional form:

 
formula

where Tst = 200 K is the stratospheric equilibrium temperature, Tm = 295 K is the global-mean surface temperature, δze = 10 K is the tropical static stability, p0 = 1000 mb is a reference pressure, and κ = 2/7. In our analysis, we perturb the parameter δy to change the baroclinicity. Our radiative-equilibrium profile is very similar to that originally proposed by Held and Suarez (1994), except that the mean surface temperature (as opposed to the surface temperature at the equator) stays constant as the baroclinicity is varied. This implies that the mean extratropical surface temperature and tropopause height change relatively little, in contrast with the simulations of Schneider and Walker (2006).

The main difference between our formulation and other heating formulations used in previous studies is the usage of a different time scale for the eddies and the zonal mean, following Zurita-Gotor (2007). Specifically, we add to the thermodynamic equation

 
formula

a term Q* = −αT(γ − 1)(TTR), so that we use damping rates αT [cf. Eq. (17)] for all harmonics and forcing rates γαT for the zonal-mean component T. The scalar parameter γ thus provides a simple way to change the strength of the forcing without affecting the eddy damping. In the following, we will refer to changes in γ as “changes in the diabatic forcing rate” for brevity, with the understanding that only the zonal-mean forcing rate is changed (except for the runs described in the appendix).

This choice of forcing removes the ambiguity in the interpretation of the results when the zonal-mean forcing rate and the eddy damping are changed simultaneously, as these changes affect the eddy amplitude in opposite ways. This may lead to nonmonotonic behavior depending on which effect dominates [see the appendix and Chen et al. (2007) for an example using mechanical damping]. Our modification has little impact for moderate diabatic time scales, when eddy growth rates are not very sensitive to the diabatic time scale, but makes an important difference in the strongly forced limit. In particular, one can presumably get arbitrarily close to radiative equilibrium by forcing strongly enough as in Zurita-Gotor (2007). This would not be possible with the standard Newtonian formulation because the flow becomes baroclinically stable when the diabatic time scale is too short. Nevertheless, the appendix shows that the results are very similar when using our procedure and when changing the full diabatic time scale, up to the point where eddy damping effects dominate.

We emphasize that we do not associate this forcing with any physically realizable process. It should be simply regarded as an ad hoc procedure that allows us to force the problem more strongly than with the traditional formulation. There could be alternative more physical ways to achieve the same result, for instance adding a prescribed heating term. This is supported by the results of Zurita-Gotor (2007), who shows that it is the time-mean heating that primarily determines the equilibrium state in his two-layer qg model. The climates that he gets are very similar in a run forced with enhanced zonal-mean heating as in Eq. (19) and in a run forced with prescribed heating, when the prescribed heating is taken to be the diagnosed time-mean heating from the former run. Likewise, Chang (2006) has devised in the PE case an iterative procedure to calculate the heating required for obtaining any desired equilibrium state. However, we have chosen to force the problem using the formulation in Eq. (19) rather than using a prescribed heating term because the interpretation is simpler and the connection with previous results using Newtonian heating is clearer.

We consider 10 different values of γ [γ = 0.1, 0.2, 0.5, 1 (control), 2, 5, 10, 20, 40, and 100] and 4 different values of δy [δy = 60 (control), 90, 120, and 180 K] in this study, but not in all possible combinations (for large γ and δy the flow is very energetic and a prohibitively short time step is required). The diagnostics presented below were calculated averaging the last 400 days of 600-day simulations, after discarding the first 200 days of spinup. For the weakly forced runs (γ < 1), the results were also compared with those from extended integrations (1200 days) to ensure statistical convergence.

4. Results

Figure 2 shows some diagnostics for our control run, which has δy = 60 K and γ = 1. The top panel shows the zonal-mean zonal wind. There is some hint of a split jet, but the maximum upper-level wind is observed well into the midlatitudes, roughly collocated with the maximum surface westerlies. The middle panel shows the transient eddy kinetic energy EKE = ½(2 + 2), which is a bit larger than observed. Finally, the bottom panel shows the isentropic mass circulation.

Fig. 2.

Climatology of the control run: (a) zonal-mean zonal wind [contour interval (ci) = 5 m s−1); (b) EKE (ci = 50 m2 s−2); (c) isentropic mass flux (ci = 20 Sv, 1 Sv = 109 kg s−1).

Fig. 2.

Climatology of the control run: (a) zonal-mean zonal wind [contour interval (ci) = 5 m s−1); (b) EKE (ci = 50 m2 s−2); (c) isentropic mass flux (ci = 20 Sv, 1 Sv = 109 kg s−1).

In this study we vary the strength of the forcing changing δy and γ. As discussed in section 2a, a useful parameter to compare both types of perturbations is the net diabatic heating/cooling, defined as the areal integral of the vertically integrated heating over the range of latitudes where this vertically integrated heating is one-signed. One may regard this variable as a uniform measure of the forcing for all types of perturbations. Figure 3a shows that under this measure, the range of forcings considered in this study is broader when varying γ than when varying δy, consistent with the larger impact of γ perturbations on the circulation (Figs. 3c–f). When any of these two parameters increase, the surface wind is accelerated (Figs. 3c,d) and there is an intensification and poleward expansion of the baroclinic zone (Figs. 3e,f); the eddy activity maximum also shifts poleward.

Fig. 3.

Net tropospheric heating/cooling for the values of δy indicated as a function of (a) γ and (b) the meridional temperature gradient ΔhΘ (the thick line emphasizes the points with γ = 1). Surface wind for runs with (c) γ = 1 and the values of δy indicated and (d) δy = 60 K and varying γ (solid: γ > 1, dashed: γ < 1). (e), (f) Same as (c), (d) but for the barotropic EKE.

Fig. 3.

Net tropospheric heating/cooling for the values of δy indicated as a function of (a) γ and (b) the meridional temperature gradient ΔhΘ (the thick line emphasizes the points with γ = 1). Surface wind for runs with (c) γ = 1 and the values of δy indicated and (d) δy = 60 K and varying γ (solid: γ > 1, dashed: γ < 1). (e), (f) Same as (c), (d) but for the barotropic EKE.

An important difference between both types of perturbations is their different impact on the meridional temperature gradient. Figure 3b shows that the parameter γ provides a more “efficient” way to change the strength of the forcing, in the sense that ΔhΘ increases much less for the same increase in the heating than when δy is changed. The large temperature range that ensues in the latter case for large heatings poses some technical problems. For instance, for the highest value of δy considered (180 K), the polar surface cools to temperatures that are comparable to the stratospheric temperature Tst = 200 K. This prevents increasing δy even further. One can get around this problem using a larger global-mean temperature Tm as in Schneider and Walker (2006), but this introduces some ambiguity because it also affects the mean tropospheric height. In contrast, there are no such problems when varying γ because the temperature gradient does not increase as much for the same increase in the heating.

We next investigate the sensitivity of the criticality in our model. Figure 4a shows the vertical ΔυΘ = |Hppθ| versus meridional ΔhΘ = |( f /β)∂yθ| temperature gradient varying δy, under different averaging conventions. The thick line shows the results when the temperature gradients are evaluated at 775 mb and averaged meridionally over the region where the barotropic EKE is within 25% of its maximum. We use the latitude of maximum area-weighted barotropic EKE as the reference latitude for evaluating f and β, and estimate Hp using the World Meteorological Organization (WMO) definition of the tropopause. These averaging conventions produce Sc = ½ξ = ½ΔhΘ/ΔυΘ ≈ 1 (the reference dashed–dotted line), in good agreement with the findings of Schneider and Walker (2006). However, as these authors also noted, the specific value of ξ (or Sc) is sensitive to the averaging convention used. For example, the line with square markers shows how the results change when using a pressure level of 525 mb, and the line with triangles shows the results using the latitude of maximum barotropic EKE (with no area weighting) as the reference latitude. Finally, the line with diamonds shows the results when the barotropic relative vorticity gradient is added to β [see Zurita-Gotor (2007) for the rationale of this]. All these lines have ξ = ΔhΘ/ΔυΘ ≈ constant, but only with our first averaging convention Sc ≈ 1. This suggests that at every pressure level the isentropic slope remains roughly constant as δy is varied (see also Fig. 1c). Although ξ is always on the order of one, its actual value depends on the reference level chosen.

Fig. 4.

(a) Normalized vertical vs meridional potential temperature gradients for runs with δy = 60, 90, 120, and 180 K and γ = 1 using various averaging schemes. (b) Same as (a) but also varying γ. (c) Sensitivity of the normalized isentropic slope on γ for the sets of runs with δy = 60 (circles), 90 (squares), 120 (diamonds), and 180 K (triangles). (d) For δy = 60 K and varying γ, sensitivity of the criticality (thick, circles) and its components: the meridional (pluses) and vertical (crosses) potential temperature gradient and the tropopause height (stars), all normalized by their control values.

Fig. 4.

(a) Normalized vertical vs meridional potential temperature gradients for runs with δy = 60, 90, 120, and 180 K and γ = 1 using various averaging schemes. (b) Same as (a) but also varying γ. (c) Sensitivity of the normalized isentropic slope on γ for the sets of runs with δy = 60 (circles), 90 (squares), 120 (diamonds), and 180 K (triangles). (d) For δy = 60 K and varying γ, sensitivity of the criticality (thick, circles) and its components: the meridional (pluses) and vertical (crosses) potential temperature gradient and the tropopause height (stars), all normalized by their control values.

The results presented so far only confirm previous findings by Schneider (2004) and Schneider and Walker (2006). A more interesting question is whether ξ also remains robust when the diabatic time scale is changed and, if so, how this can be compatible with the presumed approach to radiative equilibrium for large enough γ. Figures 4b,c show that, in fact, ξ does not remain robust but varies smoothly with the zonal-mean diabatic time scale. Departures from the constant criticality constraint are observed not just in the strong-forcing limit, when the adiabatic theory of Schneider (2004) is expected to break down, but for all values of γ. The sensitivity of the criticality on γ is in fact comparable when the forcing is strong (γ > 1) and when the forcing is weak (γ < 1). Qualitatively similar results are found with other averaging conventions (see Fig. 1d for changes in the vertical structure as γ is varied). This suggests that there is nothing special about the control run value of ξ, other than it varies very slowly with the forcing.

Another intriguing question is whether non-qg feedbacks make the sensitivity of ξ on γ weaker than in the qg case. Figure 4c shows that ξ varies by roughly an order of magnitude when γ changes by three orders of magnitude, which is actually faster than found by Zurita-Gotor (2007) for the two-layer qg model. However, this might be due to our choice of a radiative-equilibrium profile that is statically neutral in the midlatitudes. Additional experiments relaxing to a stably stratified profile (not shown) display a weaker sensitivity more in consonance with the qg results. The criticality also flattens out for large γ in these runs, as ξ asymptotes its finite radiative-equilibrium value.

Figure 4d describes in more detail the sensitivity of the criticality on γ for the set with δy = 60 K. We can see that ξ increases monotonically with γ with a well-defined power law, although the exponent does change somewhat for γ smaller and larger than 1. The uniform behavior of ξ is striking because the meridional/vertical temperature gradients exhibit different sensitivities in different parts of the γ space. For small γ the behavior of the flow is qg-like, in that the bulk of the ξ changes are due to changes in ∂yθ with little changes in ∂pθ. In contrast, for large γ, ∂yθ saturates near its radiative equilibrium value, and the steepening of the isentropic slope is associated with reduced stratifications ∂pθ. Changes in Hp are relatively unimportant over the full parameter space. What is remarkable is that all these sensitivities conspire to produce the uniform dependency of ξ on γ reported above. Figure 4c shows that similar results are found for other values of δy, for which ξ also depends uniformly on γ despite the more complex behavior of ∂yθ, ∂pθ apparent in Fig. 4b.

The uniform dependency of ξ on the forcing is reminiscent of previous results by Zurita-Gotor and Lindzen (2006) and Zurita-Gotor (2007) for the two-layer model. These authors found that the PV gradient varied uniformly with the heating in their model, even when the partition between its different components was affected by other external parameters. For instance, when changes in the zonal-mean frictional time scale lead to a change in the relative vorticity gradient, the baroclinic PV gradient adjusts to keep the same full PV gradient. Zurita-Gotor and Lindzen (2006) argues that the PV gradient does not change much with the frictional time scale because changes in friction have little impact on the net heating and on the strength of the circulation.

Section 2 discusses how these arguments change in the PE problem. As shown in section 2b, one can also relate heuristically the mass flux to the criticality in this case. The main difference is that changes in the heating do not necessarily imply changes in the mass flux and criticality as in the qg problem because the gross stability Sθ can now change. Depending on how Sθ changes, the criticality will exhibit a different sensitivity on the heating. This is illustrated in Fig. 5. Figure 5a shows that although the criticality increases in general with the heating, their relation is not unique. In particular, one can have significant changes in the heating with no observable changes in the criticality when only δy is varied (thick line; this corresponds to Fig. 4a). In contrast, there is much less scatter when plotting the criticality as a function of the mass flux (Fig. 5b), which supports our speculation that the criticality regulates the mass flux via the momentum balance and diffusive closure. Note that the mass flux varies very little when δy changes, keeping γ constant (markers cluster together in Fig. 5b) consistent with the constant criticality observed for these runs.

Fig. 5.

Criticality as a function of (a) the heating and (b) the isentropic mass flux; gross stability as a function of (c) γ and (d) the heating. In (a)–(d), the thick solid line joins points with γ = 1. Markers as in Fig. 4.

Fig. 5.

Criticality as a function of (a) the heating and (b) the isentropic mass flux; gross stability as a function of (c) γ and (d) the heating. In (a)–(d), the thick solid line joins points with γ = 1. Markers as in Fig. 4.

To understand the sensitivity of the isentropic slope one needs to understand first what determines the gross stability. When δy is varied, Sθ increases exactly linearly with the heating (Fig. 5d), consistent with the fact that the mass circulation and the criticality stay constant [cf. Eq. (5)]. In contrast, because the gross stability increases more slowly than the heating when γ is varied, the mass circulation must strengthen in those runs, and the criticality must increase. The question of what determines the gross stability is not an easy one. Figure 5c shows that the gross stability initially increases with γ, and flattens out for large γ. This is a very different sensitivity than displayed by the bulk stability ΔυΘ (Fig. 4c), which remains flat for small γ and drops for large γ. If anything, Sθ would appear to scale better with the meridional than with the vertical temperature gradients. Section 5 discusses in more detail the determination of the gross stability using isentropic diagnostics.

a. Sensitivity of tropopause height

Changes in tropopause height are important for keeping ξ constant in Schneider’s (2004) framework. However, they only play a minor role here because the mean extratropical surface temperature changes little by construction in our runs [see, e.g., Schneider (2007) for a discussion of the tropopause sensitivity on surface temperature]. Although the tropopause height is also affected by changes in the extratropical stratification (Held 1982), the sensitivity is weak in the sense that fractional changes in Hp resulting from changes in the stratification ∂pθ alone have less of an impact on ξ than the actual fractional changes in ∂pθ.

This is illustrated in Fig. 4d, which shows that the WMO tropopause changes much less (in relative terms) than the meridional and vertical temperature gradients. Moreover, changes in Hp enhance rather than compensate the changes in the isentropic slope. In particular, the tropopause goes down with increasing γ, which is exactly opposite to what would be required to maintain a constant criticality when I increases. The tropopause must drop as the flow approaches radiative equilibrium because the eddies always raise the tropopause relative to radiative equilibrium. Similar results would be obtained with other measures of the tropopause based on the thermal structure (or on potential vorticity), since these must also approach the radiative equilibrium tropopause for large γ. Tropopause estimates based on isentropic diagnostics (e.g., Schneider 2004) work a bit better, and the tropopause rises when the isentropic slope steepens (not shown). However, these changes are still much smaller than required to produce a constant criticality.

b. Eddy scales

The criticality parameter ξ plays a fundamental role for theories of quasigeostrophic turbulence on the beta plane (Held and Larichev 1996). This parameter measures the extent of the inverse cascade when this cascade is halted at the Rhines scale, expressed as the ratio between the eddy scale and the deformation radius (see also Held 1999). Consistent with the result that ξ ≈ 1, Schneider and Walker (2006) show that the eddy scale, deformation radius, and Rhines scale all change proportionally in their model when the parameters are varied. However, we have shown that ξ can change in our model when γ is changed, so it is interesting to see what this implies for the existence of an inverse cascade.

Figure 6a shows how the eddy length scale and Rossby radius depend on γ for the set with δy = 60 K. We estimate the eddy length scale as L = 2π/, where is the centroid of the barotropic EKE spectrum (see Frierson et al. 2006), and averaging over the latitudinal range defined above. The Rossby radius λ = NH/f is estimated using the Coriolis parameter at the reference latitude, the stratification at the reference pressure level 775 mb, and the WMO tropopause. The eddy length scale initially increases with γ, but it saturates at a value of about 6300 km. One possible explanation is that, at this scale, the size of the planet starts to compete with the Rhines scale. However, the saturation length scale does increase with other values of δy (6700 km for δy = 90 K, 7400 km for δy = 120 K, and 8000 km for δy = 180 K). The saturation of L could also be related to the saturation of the meridional temperature gradient observed in Fig. 4d.

Fig. 6.

(a) For the set with δy = 60 K and varying γ, eddy scale (pluses), Rossby radius (crosses), and their ratio (stars), all normalized with their control values. (b) Ratio between eddy scale and Rossby radius, as a function of the criticality for all runs (markers as in Fig. 4). (c) Same as (b) but using the gross stability in the definition of the criticality and Rossby radius. (d) Ratio between baroclinic EKE and eddy available potential energy as a function of the criticality for all runs. The thick lines in (b), (d) emphasize the points with γ = 1.

Fig. 6.

(a) For the set with δy = 60 K and varying γ, eddy scale (pluses), Rossby radius (crosses), and their ratio (stars), all normalized with their control values. (b) Ratio between eddy scale and Rossby radius, as a function of the criticality for all runs (markers as in Fig. 4). (c) Same as (b) but using the gross stability in the definition of the criticality and Rossby radius. (d) Ratio between baroclinic EKE and eddy available potential energy as a function of the criticality for all runs. The thick lines in (b), (d) emphasize the points with γ = 1.

The increase in the eddy length scale is consistent with Rhines’ arguments, as the EKE of the flow increases with γ. However, this eddy expansion does not necessarily imply an inverse cascade because the Rossby radius may also increase. This is what happens in Schneider’s runs, and in our model when δy is varied. However, the sensitivity of the Rossby radius is very different when γ is varied. Figure 6a shows that λ is initially fairly flat, but it soon starts decreasing. As a result, L/λ increases monotonically with γ over the full parameter space, even after L saturates. The uniform dependency of L/λ on γ suggests that we might be able to express L/λ = f (ξ), as in some theories of qg turbulence. This is tested in Fig. 6b for all runs considered. The prediction has a lot of scatter, but also some skill. The slope of the L/λ = f (ξ) relation approaches the Held and Larichev (1996) prediction (n = 1) over the region where L changes more than λ, but flattens after L saturates.

Because there is a lot of ambiguity on how to define the Rossby radius, Schneider and Walker (2006) propose eddy energy partition as an alternative measure of the inverse cascade. In the presence of an inverse cascade, eddy available potential energy grows faster with the forcing than the baroclinic eddy kinetic energy because thermal anomalies are created by barotropic advection on scales larger than the deformation radius (Held and Larichev 1996). In contrast with this prediction, Schneider and Walker (2006) show that eddy available potential energy (EAPE) and baroclinic EKE remain proportional in their model as the parameters are varied, which they interpret as the lack of an inverse cascade. Figure 6d shows the ratio between both forms of energy in our model as a function of the criticality. To compute the EAPE we use at each height the mean stratification poleward of 35° as the reference stratification. Over the parameter regime where L increases, the ratio between baroclinic EKE and EAPE decreases with γ, consistent with an inverse cascade. However, the variations are small and this tendency is reversed after L saturates. Hence, it is not clear that this diagnostic supports the existence of an inverse cascade.

It is thus possible that Schneider and Walker’s (2006) conclusion on the lack of an inverse cascade might still hold in our runs if an appropriate definition of the Rossby radius were used.3 As discussed by Schneider (2004), one expects the mean flow properties to affect the local closure, except in the strongly nonlinear limit. Moreover, Schneider’s results suggest that this limit might be unattainable. The main problem with our Rossby radius definition is that λ decreases as γ increases, while the length scale L increases or remains flat. The λ decrease is primarily due to changes in the stratification (see Fig. 4d), while tropopause changes only play a minor role. As we saw, the gross stability Sθ exhibits the opposite sensibility to the stratification (Fig. 5c), and scales more like the meridional temperature gradient. We have also examined the relation L/λθ = f (ξθ) using the gross stability as the relevant stratification in the definition of both λ and ξ. We can see (Fig. 6c) that the variability is significantly reduced from Fig. 6b when the same scale is used. This occurs because L and Sθ correlate well with ΔhΘ; in particular, they all saturate for large γ in contrast with the steep decrease in ΔυΘ. In the next section, we perform isentropic diagnostics to investigate how the gross stability and the bulk stability can exhibit such strikingly different sensitivities.

5. Isentropic diagnostics

We have calculated isentropic statistics for all runs discussed above, using two different methods. With the first method (e.g., Schneider 2004), instantaneous fields are interpolated to θ coordinates at each time step, and means are calculated directly in isentropic coordinates. With the second method (e.g., Czaja and Marshall 2006), the instantaneous θ and υ fields are interpolated to a refined pressure-coordinate mesh and ρθ and * are calculated binning values across θ categories. Both methods produce very similar results, except when γ is large. In that limit, the stratification becomes locally small or unstable with some frequency, making the first method prone to large errors. Held and Schneider (1999) and Koh and Plumb (2004) discuss how to deal with mixed layers. We prefer the second method because it does not require inverting the θ(p) relation and hence does not suffer from the same problem.

On the other hand, eddy PV fluxes are calculated using the following expression:

 
formula

where the eddy momentum fluxes and relative vorticity were neglected, so that Pf /ρθ and P* ≈ f /ρθ. We write for brevity, with the understanding that the meridional velocity υ is set to zero when an isentropic layer is not present. This term is dominated by correlations between υ and ℋ (Held and Schneider 1999; Schneider 2005) and vanishes above the surface layer when the geostrophic approximation is used for υ. It is this term that gives rise upon vertical integration to the surface contribution in the isentropic momentum balance of Schneider (2005). Using the geostrophic approximation for υ, he shows that

 
formula

so that this term produces the approximate contribution to the isentropic mass flux:

 
formula

a. Bulk stability and gross stability

Figure 7a shows the mean tropopause and surface potential temperatures as a function of γ for the set with δy = 60 K. Schneider (2004) uses the difference between these temperatures as an estimate for the extratropical bulk stability. We can see that neither temperature changes very much (the small decrease in the tropopause θ depends on the mass flux threshold used to define it), rendering this measure of the bulk stability roughly constant. This is different from what we found for ΔυΘ = Hppθ, which decreases sharply with γ for large γ (Fig. 4b). These two measures differ because most of the bulk stability comes from the upper troposphere for large γ.

Fig. 7.

(a) Tropopause (thin dashed) and surface (thin solid) potential temperatures, and mean (circulation weighted) potential temperatures for the poleward (thick dashed) and the equatorward (thick solid) isentropic mass flow at the reference latitude, as a function of γ for the set with δy = 60 K. The differences between these two sets give the bulk stability ΔυΘ and the gross stability Sθ, as indicated by the arrows. (b) As in (a) but for the set with γ = 1, as a function of δy.

Fig. 7.

(a) Tropopause (thin dashed) and surface (thin solid) potential temperatures, and mean (circulation weighted) potential temperatures for the poleward (thick dashed) and the equatorward (thick solid) isentropic mass flow at the reference latitude, as a function of γ for the set with δy = 60 K. The differences between these two sets give the bulk stability ΔυΘ and the gross stability Sθ, as indicated by the arrows. (b) As in (a) but for the set with γ = 1, as a function of δy.

Also shown in Fig. 7a are the mean (circulation weighted) temperatures for the poleward and equatorward flow. Their difference, the gross stability Sθ, increases monotonically with γ in contrast with the bulk stability sensitivity. A closer inspection of Fig. 7a reveals that the difference between them arises because the surface temperature is not representative of the mean temperature for the equatorward flow. This is not surprising because the equatorward flow occurs over isentropic layers that are frequently interrupted (Held and Schneider 1999). As γ increases an increasingly larger fraction of the equatorward flow occurs in “buried” isentropes with potential temperatures θ < s; for large enough γ the mean temperature of the equatorward flow becomes colder than the mean surface temperature s. The fraction of equatorward flow that is buried appears to be controlled by the diabatic time scale. In contrast, when only δy is varied the surface potential temperature and the mean temperature for the equatorward flow correlate well (Fig. 7b). This implies that the bulk stability provides a good estimate of the gross stability in that case.

Figure 8 examines in more detail the vertical structure of the isentropic mass flux and its components. Figure 8a shows the full isentropic mass flux and makes the point again that, as γ increases, a larger fraction of the equatorward flow is buried and its mean temperature decreases. On the other hand, Figs. 8b,c show the contributions to the isentropic momentum balance by the eddy PV flux * and the Coriolis forcing acting on the mean geostrophic flow f, respectively. To calculate the mass flux driven by each term, one should divide the corresponding term by P* = f /ρθ [cf. Eq. (20)]. It is readily apparent that (i) the Coriolis forcing term is roughly symmetric about the mean surface temperature and (ii) in contrast, the forcing by the positive eddy PV flux is almost entirely buried. This flux peaks near the lowest surface temperature, reminiscent of the delta-function of qg theory, but in contrast with this delta-function the PV flux is distributed over the whole buried atmosphere instead of concentrated at a single potential temperature. The associated mass flux, −*/P*, peaks at larger temperatures, but is still almost entirely buried (not shown). As the surface temperature gradient increases with γ, the eddy PV flux peaks at temperatures increasingly colder than the mean surface temperature and produces the bias between the mean surface temperature and the mean temperature of the equatorward flow shown in Fig. 7a.

Fig. 8.

(a) The θ-distribution of the isentropic flow ρθ* at the reference latitude as a function of γ. Same but for the (b) eddy PV flux * and (c) Coriolis forcing f. The thick lines in (a)–(c) indicate the surface potential temperature (solid), the tropopause potential temperature (dashed), and the potential temperature of max mass flux (dashed–dotted). The tropopause is defined here as the isoline in which the mass flux reduces to 15% of its maximum value. For (a)–(c), the ci is indicated in SI units in the title.

Fig. 8.

(a) The θ-distribution of the isentropic flow ρθ* at the reference latitude as a function of γ. Same but for the (b) eddy PV flux * and (c) Coriolis forcing f. The thick lines in (a)–(c) indicate the surface potential temperature (solid), the tropopause potential temperature (dashed), and the potential temperature of max mass flux (dashed–dotted). The tropopause is defined here as the isoline in which the mass flux reduces to 15% of its maximum value. For (a)–(c), the ci is indicated in SI units in the title.

b. Diffusive closure in isentropic coordinates

Schneider (2005) divides the isentropic mass flux in three different contributions: (i) the poleward flow driven by the negative eddy PV flux, (ii) the equatorward flow driven by the positive PV flux, and (iii) the mean geostrophic flow in interrupted isentropes, which is also equatorward. When integrated vertically, this last term is proportional to the surface eddy heat flux by the balanced meridional flow. Schneider uses diffusive closures for the isentropic eddy PV fluxes and for the surface eddy heat flux, and assumes that the diffusion coefficient is the same. We next assess how well these assumptions are satisfied in our runs.

A numerical difficulty when calculating these quantities is that both ρθυ̂* and ρθyP* are poorly defined at the bottom of the surface layer, where ρθ vanishes. However, their ratio—the diffusivity—remains smooth throughout the surface layer. The associated mass flux tends smoothly to zero at the bottom of the surface layer (to calculate the mass flux one divides by P*, which introduces an additional ρθ factor). It is thus convenient to define an average diffusivity for each region as the ratio between the following two terms:

 
formula

when integrated over the corresponding region of one-signed PV fluxes. We will refer to these terms as the net positive or negative eddy PV flux and mean PV gradient. A nice property of the 1/P* weighting is that it makes the integrand proportional to the mass flux, so that the net positive and negative eddy PV fluxes have the same order of magnitude when expressed in this norm (the mass flux integrates to zero). The empirical diffusivities thus obtained are very similar to the averages of the local empirical diffusivity over the corresponding regions, but the latter estimate is noisier (not shown). We also define the following equivalent surface PV flux and PV gradient:

 
formula

With these definitions, Fig. 9 shows the gradients, fluxes, and empirical diffusivities over the three regions defined above. Several aspects of this figure are noteworthy:

  • As long as γ is not too small, the PV diffusivity is very similar over the regions of positive and negative PV fluxes, consistent with Schneider’s assumptions.

  • In contrast, the surface diffusivity is typically larger; only for large γ does the interior diffusivity approach the surface diffusivity. That the surface diffusivity is larger than the interior diffusivity is consistent with the shorter time scales used at the surface (Schneider 2004, his section 4d). However, what is probably more important is that the ratio between both diffusivities changes significantly in our runs, while it was assumed to be approximately constant in Schneider (2004).

  • Setting 𝒟1 = 𝒟2 = 𝒟int and 𝒟int < 𝒟s in Eq. (16), one obtains 
    formula
    which implies that the flow is subcritical. The integrated PV gradient at the surface is smaller than beta integrated over the troposphere (Zurita and Lindzen 2001).
  • This result agrees with the predictions of Schneider and Walker (2006) that the flow can be subcritical, but not supercritical. However, the subcriticality in our runs occurs for different reasons than discussed by these authors. Assuming that the diffusivity is uniform, Schneider and Walker show that the flow is critical when the baroclinic eddies extend to the tropopause, and subcritical when they do not. Here, the eddies extend to the tropopause but the flow is still subcritical because the surface diffusivity is larger than the diffusivity in the interior. To the extent that the faster forcing at the surface implies larger diffusivities there than in the interior (Schneider 2004), departures in the vertical structure of the diffusivity should not alter the conclusion of Schneider and Walker (2006).

  • The diffusivity increases with γ, consistent with the increase in the eddy PV fluxes and isentropic mass flux. However, the implied sensitivity on the criticality (which changes by an order of magnitude over this range) is far less steep than predicted by the qg closure of Held and Larichev (1996).

  • For moderate or large γ the mass fluxes driven by the positive PV flux and the surface term are very similar, consistent with the scaling of Schneider (2005).

  • The gradients, fluxes, and diffusivities are essentially a function of γ and very little sensitive on δy. The corresponding curves for other values of δy lie on top of those displayed (not shown). In particular, the diffusivity does not change much with δy.

Fig. 9.

(a) Isentropic PV gradient, (b) PV flux, and (c) empirical diffusivity, estimated for the upper region with positive PV gradient/negative PV flux (dashed), the lower region with negative PV gradient/positive PV flux (thin solid), and at the surface (thick solid).

Fig. 9.

(a) Isentropic PV gradient, (b) PV flux, and (c) empirical diffusivity, estimated for the upper region with positive PV gradient/negative PV flux (dashed), the lower region with negative PV gradient/positive PV flux (thin solid), and at the surface (thick solid).

c. Eulerian versus isentropic means

What is disconcerting about the previous results is that they seem inconsistent with the Eulerian diagnostics presented in Fig. 4.4 The ξs varies between 1 and 10 in Fig. 4c, while the above results imply that the flow is subcritical: Sc = ξs/2 ≤ 1. Figure 10a examines the sources of this discrepancy. The thin solid curve reproduces the criticality already shown in Fig. 4c. This curve was calculated using a reference pressure level p = 775 mb, tuned to give Sc ≈ 1 for the control run as in Schneider and Walker (2006). The thick solid curve shows the criticality ξs implied by the isentropic diagnostics; this curve seems to asymptote to a value only marginally larger than 2. Finally, the thick dashed line shows the results calculated using the surface as the reference level for the Eulerian averages, which is more consistent with the isentropic diagnostics. With this last choice, the agreement between the Eulerian and isentropic diagnostics is significantly improved, although some discrepancies remain, especially for large γ. These arise from inaccuracies in the approximations of Schneider and Walker (2006) when converting from isentropic to isobaric averages. In particular, the bias between both curves for large γ is due to the different behavior of ρ0θ and (s)−1 [cf. Eq. (A3) in Schneider and Walker (2006)] in this regime. While the Eulerian mean stratification at the surface keeps decreasing with γ, the isentropic density at the mean surface temperature stabilizes at interior values (Fig. 10b). This makes the isentropic quantity ρ0θys a poor estimate of the Eulerian mean isentropic slope at the surface in this regime, and prevents in particular the large criticality implied by the vanishing stratification at the surface.

Fig. 10.

(a) Criticality based on Eulerian averages and pref = 775 mb (thin solid) or pref = ps (thick dashed) and based on isentropic diagnostics (thick solid). (b) Isentropic density (contours; ci: 25) and mean surface temperature (thick solid); the mean density ρ0θ is defined as the isentropic density at that temperature. (c) Eulerian mean pressure p(θ) (thin) and isentropic pressure (θ) (thick) for γ = 1. (d) Same as (c) but for γ = 40.

Fig. 10.

(a) Criticality based on Eulerian averages and pref = 775 mb (thin solid) or pref = ps (thick dashed) and based on isentropic diagnostics (thick solid). (b) Isentropic density (contours; ci: 25) and mean surface temperature (thick solid); the mean density ρ0θ is defined as the isentropic density at that temperature. (c) Eulerian mean pressure p(θ) (thin) and isentropic pressure (θ) (thick) for γ = 1. (d) Same as (c) but for γ = 40.

Another striking aspect of Fig. 9a is that the net positive and negative PV gradients change very little with γ—the same is true for the baroclinic and barotropic PV gradients (not shown). This stands in contrast with the behavior of the Eulerian mean isentropic slope at constant pressure, which increases with γ at all levels (e.g., Fig. 1d). Here again, one ought to be careful when comparing means on isentropic and pressure surfaces, as isentropic averages are affected by interruptions with the lower boundary (Held and Schneider 1999). In particular, the quantities that enter the isentropic PV gradient are not the Eulerian mean pressure and isentropic slope p(θ) and I(θ), but the quantities (θ) and Ĩ(θ) defined by Eqs. (12) and (13). To understand this distinction, it is illustrative to consider the case of the mean surface temperature s. While p(s) = ps, the pressure (s) is typically representative of levels higher up in the troposphere because a significant fraction of the atmosphere is buried in isentropic layers with θ < s.

Figure 10 compares these two pressures at 48°N for the control run (Fig. 10c) and for a run with the same δy and γ = 40 (Fig. 10d). The thin line shows the Eulerian mean pressure p(θ), while the thick line shows (θ). We can see that the two pressure distributions are very close for the control run, but diverge markedly with large γ. In that limit, the time-mean state represented by p(θ) is nearly neutral, while the pressure (θ) is indicative of stronger stratifications more in line with the vertical stability of the control run. This is why the isentropic PV gradient is nearly identical for both simulations despite their very different Eulerian mean isentropic slope. Figure 8c shows that, even in the control run, some fraction of the poleward transport occurs over the surface layer as in the simulations of Koh and Plumb (2004). Held and Schneider (1999) show that this is what one expects in the presence of a deep mixed layer, a limit that is approached in our simulations for large γ. In that limit, one cannot equate pmm at the potential temperature θm of flow reversal as in the last step of Eq. (14), which implies that it is not appropriate to make inferences for the Eulerian mean state based on isentropic diagnostics.5 In particular, an infinitely steep isentropic slope can still be compatible with subcriticality in isentropic coordinates. The physical picture that emerges for large γ is that of a fluid with strong meridional potential temperature gradients and weak vertical potential temperature gradients that is advected quasi-horizontally. In that scenario, the mean isentropic density at any given latitude is more representative of the frequency with which air masses at that potential temperature are advected through that latitude than of an actual vertical stratification. The surface layer expands to occupy nearly the full troposphere (Fig. 8c).

6. Concluding remarks

This paper extends the two-layer results of Zurita-Gotor and Lindzen (2006) and Zurita-Gotor (2007) to a primitive equation model. Making use of a new forcing procedure that allows us to force the flow faster than with the standard Newtonian formulation, we investigated the relevance of the generalized baroclinic adjustment paradigm for this model. The basic question that we pose is whether the model’s criticality varies smoothly with the forcing or adopts some preferred value. The answer to this question was found to depend on how the forcing is changed. When the radiative-equilibrium baroclinicity δy is changed, our results agree with those of Schneider (2004): the model equilibrates at a preferred isentropic slope, even for significant perturbations in the heating. In contrast, the criticality increases uniformly with the forcing when we change the diabatic forcing rate through the parameter γ. Appendix A shows that this result is not an artifact of the forcing procedure and also holds when changing the eddy and zonal-mean time scales simultaneously.

To understand these results, it is illuminating to consider how the gross stability changes. The momentum balance implies that the mass transport must be balanced by the eddy PV flux, while the heat balance demands that the energy transport balances the net diabatic heating. When the gross stability is constant, as in qg theory, the mass and energy transport are proportional to each other. This implies that the eddy PV flux must increase with the heating, which should also translate into an enhanced criticality based on standard diffusive arguments. In contrast, in the primitive equation case the gross stability can change: depending on how it changes, the mass circulation and the criticality will display a different sensitivity on the heating. We found that when δy is varied, the gross stability increases proportionally to the heating. As a result, only the heat transport but not the mass circulation changes, and the criticality stays constant as found by Schneider (2004). In contrast, when γ is varied the gross stability increases more slowly than the heating, which requires a strengthening of the circulation and a steepening of the isentropic slope.

When changing γ, the sensitivity of the criticality in our model is different than predicted by Schneider (2004). To investigate why this is the case, we have performed detailed isentropic diagnostics of the simulations. We found that the interior PV diffusivity has no vertical structure, as conjectured by Schneider, but differs from the potential temperature diffusivity at the surface. Since the ratio between both diffusivities changes with γ, so must the criticality. However, because the surface diffusivity is always larger than the interior diffusivity in our model (probably due to the faster forcing at the surface; see Schneider 2004), the flow must be subcritical. This result stands in contrast with the Eulerian diagnostics, which show that the isentropic slope can become infinitely steep in the limit of fast forcing. The discrepancy arises because of distinctions between averages on isobars and on isentropes. Consistent with Held and Schneider (1999), we showed that isentropic means are not representative of Eulerian averages on sigma levels when the surface layer is deep and a large fraction of the transport occurs in buried isentropes with temperatures colder than the mean surface temperature. When the diabatic forcing is strengthened, the surface layer expands and the fraction of the atmosphere that is buried increases. The implications of these results for the existence of an inverse cascade are inconclusive. The ratio between the eddy length scale and the Rossby radius increases with γ, but there is some ambiguity in the definition of the Rossby radius. The ratio between baroclinic eddy kinetic energy and eddy available potential energy changes nonmonotonically, and only displays small variations.

Although it is possible to change the isentropic slope in our model, the sensitivity is weak and large changes in the forcing are required to obtain moderate changes in the isentropic slope. This suggests that the assumption of a constant criticality might still be a reasonable one over a broad region of the parameter space, albeit only to the extent that the Newtonian formulation is appropriate. Given the sensitivity of our results to the form of heating, it cannot be discarded that the constraints on the isentropic slope could be different with other heating formulations. This might help explain the violation of the constant-criticality constraint with realistic heating (Thuburn and Craig 1997) and the sensitivity of the isentropic slope to moisture (Juckes 2000; Frierson et al. 2006). In the idealized moist GCM of Frierson et al., the isentropic slope flattens with the moistening, and the circulation weakens. The weakening of the circulation is consistent with the framework put forward in this paper because the gross moist stability increases (due to latent contributions), while the net heating changes very little with the moistening (Frierson et al. 2007). To understand how a weaker circulation translates into a flatter isentropic slope, one would need a turbulent diffusion theory that also incorporates moisture.

Fig. A1. (a) For the set with δy = 60 K, barotropic EKE when the zonal-mean diabatic heating (thin, circles) or the full diabatic heating (thick, stars) is accelerated by a factor of γ. (b) Same as in (a) but for the criticality ξ = ΔhΘ/ΔυΘ, (c) the eddy scale L, and (d) L/λ. (e) As in Fig. 8c, but changing the diabatic time scale for both eddies and mean. (f) As in Fig. 10d, but for a run in which both eddies and mean are accelerated by a factor of 10.

Fig. A1. (a) For the set with δy = 60 K, barotropic EKE when the zonal-mean diabatic heating (thin, circles) or the full diabatic heating (thick, stars) is accelerated by a factor of γ. (b) Same as in (a) but for the criticality ξ = ΔhΘ/ΔυΘ, (c) the eddy scale L, and (d) L/λ. (e) As in Fig. 8c, but changing the diabatic time scale for both eddies and mean. (f) As in Fig. 10d, but for a run in which both eddies and mean are accelerated by a factor of 10.

Acknowledgments

This work was supported by the Ministerio de Educación y Ciencia of Spain under the Ramón y Cajal program. The numerical simulations were performed on GFDL’s computer system, where the author enjoyed a UCAR visiting scientist fellowship. Thanks are due to Ed Gerber for helping me set up the primitive-equation model used in this study. I am also grateful to Dargan Frierson, Isaac Held, Dick Lindzen, Olivier Pauluis, Tapio Schneider, Geoff Vallis, and two anonymous reviewers for helpful comments at different stages of this work. Tapio Schneider shared his insights on the relations between his work and these results, particularly in what concerns the estimation of the Rossby radius and the existence of an inverse cascade, and offered many other comments that led to an improvement of the paper. The isentropic diagnostics presented in section 5 and the simulations in the appendix were added during the revision to address the concerns of an anonymous reviewer.

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APPENDIX

Impact of the Heating Formulation

As described in section 3, we use in this paper a modification of the Newtonian heating formulation, in which the zonal-mean and eddy diabatic time scales are changed independently. This prompts the question of whether differences between our results and those from previous studies are due to the broader range of time scales considered here or if they are a by-product of an artificial heating formulation that may not be physically realizable. To answer this question, we have performed a set of simulations in which both the zonal-mean and eddy diabatic heating rates are accelerated by the same factor of γ.

The results are shown in Fig. A1. Figure A1a compares the barotropic eddy kinetic energy when changing the full diabatic time scale (stars) or only its zonal-mean component (circles). Not surprisingly, the EKE is larger in the former (latter) case when γ < 1 (γ > 1). Differences between both sets are small for γ ≤ 2, but there are already significant differences for γ = 5. The EKE reaches a maximum for this value of γ when changing the full diabatic time scale, but keeps increasing thereafter when only the zonal-mean diabatic time scale is changed. Figure A1b displays the sensitivity of the criticality. The differences are also small for γ ≤ 5.

As γ is increased beyond this value, the criticality actually grows faster for the full-heating run, presumably because the weaker eddies are less efficient in competing with the forcing. Figure A1c shows that the EKE reduction when the eddy diabatic damping is strong is also accompanied by a decrease in the eddy length scale. It is for this reason that L/λ does not grow indefinitely with γ (Fig. A1d), as we found when only changing the zonal-mean diabatic time scale. Figure A1e shows the θ-distribution of the momentum forcing by the Coriolis force acting on the mean meridional flow. Despite some significant differences with Fig. 8c for γ ≥ 5, there is also some hint that the surface layer might be deepening with γ. Finally, Fig. A1f shows that the distinction between p and is not an artifact of the forcing. Overall, this comparison supports the notion that differences in the results when changing the full diabatic time scale or its zonal-mean component alone are due to the damping effect on the eddies and do not reflect a fundamental difference in the underlying dynamics.

These results suggest that the sensitivity of the isentropic slope on the forcing time scale found in this study is not an artifact of the formulation. We believe that differences between our results and Schneider’s (2004) are due to the different parameter range considered. Schneider only varies the surface diabatic time scale, and only by a factor of 10, while we vary the diabatic time scale by three orders of magnitude everywhere. Finally, we note that supercritical states have also been obtained in previous studies using alternative heating formulations. For instance, the meridional and vertical temperature gradients can be independently imposed in the idealized nonlinear storm track of Chang (2006; see also Chang and Zurita-Gotor 2007) using the appropriate prescribed heating.

Footnotes

Corresponding author address: Dr. Pablo Zurita-Gotor, Departamento de Geofísica y Meteorología, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, 28040, Madrid, Spain. Email: pzurita@alum.mit.edu

1

Schneider (2004) shows that Eq. (1) also holds when Hp changes, in which case the isentropic slope must change. The tropopause height changes relatively little in our runs because we keep the global mean surface temperature constant as the baroclinicity is varied (see section 4a for more details).

2

Zurita-Gotor (2007) quantifies the circulation using potential momentum (Zurita-Gotor and Lindzen 2006), but note that his midchannel potential momentum forcing is proportional to the net heating.

3

We thank the anonymous reviewers for pointing this out. One of the reviewers also suggested estimating the Rossby radius using the gross stability, as described below.

4

By Eulerian diagnostics, we refer to diagnostics based on the time- and zonal-mean potential temperature distribution at fixed sigma levels.

5

We emphasize that it is the Eulerian mean climate that is the primary concern of this paper. Even if we had a fully closed isentropic theory, its applicability for climate would be limited if we did not know how to convert from isentropic to pressure coordinates.