a. Tracer tagging

Equation (13) demonstrates that if the tracer field c is positive, the tag fractions c_i are also positive because $\left(1-{\chi }_i\right)\left(S_i^{+}/c\right)$ will not increase χ_i above 1, and $-{\chi }_i{\displaystyle {\sum }_{j\ne i}\left(S_j^{+}/c\right)}$ will not decrease χ_i below 0, so 0 ≤ χ_i ≤ 1. Since c_i = χ_ic, then 0 ≤ χ_i ≤ 1.

Since the {c_i} are positive and, via (2) and (11), sum to c for all time, they remain bounded by the greatest value of c. This ensures that the system will not diverge even under long time integration, making this formulation useful for equilibrium climate integrations.

While using an initialization tag gives some additional information, the decay time scale can be very long for some cases. In this case, it is advantageous to set the tag which takes the longest time to be produced to be c(t₀), and the other tags to be 0. This can significantly reduce the amount of time for the model to spin up to an equilibrium time. The specifics of the initialization for heat tags is discussed at the end of this section, where we utilize this second approach to reduce the amount of spinup time needed.

b. A conceptual example

As an example of such a calculation, we restrict the sources of c, $S_i^{+}$, to only be allowed to have positive values in the equatorial half of the domain y ∈ [0, L/2]. Thus, tags can only be produced in this region, whereas they can be transported and destroyed over the whole domain. Now consider a situation in which S₁ is sourced only for poleward flow and S₂ is sourced only for equatorward flow. That is, $S_1^{+}>0$ for υ > 0 and $S_2^{+}>0$ for υ < 0, again for y ∈ [0, L/2]. Then c in the polar half of the domain would consist solely of c₁, i.e., χ₁ = 1, χ₂ = 0 for y ∈ [L/2, L], and the entire poleward flux of c at y = L/2 would be from c₁. Although this example seems somewhat contrived, the fictional c₁ is not that far from the convective tag, as shown in section 3.

From this example it can be seen that the tags provide a way to attribute the tracer distribution to processes far from the source regions. In our simplified system it might seem that the tags are not necessary; however, this is because the circulation is greatly simplified. For more realistic three-dimensional circulations the mass transport can be highly complicated, and so the connection between tracer content in remote regions and their sources is difficult to determine without the tracer tags.

c. Numerical implementation in an idealized moist general circulation model

We use the implementation of the class of idealized moist general circulation model developed in Frierson et al. (2006), Frierson (2007), and O’Gorman and Schneider (2008) that is found in the Isca framework (Vallis et al. 2018). This model, coded with the NOAA GFDL Flexible Modeling System, solves the moist hydrostatic primitive equations on the sphere using a pseudospectral method. The Isca framework provides a set of standard physical parameterizations, including updated radiative transfer schemes. The horizontal resolution of the spectral core is T85 and in the vertical there are 30 levels evenly spaced in the model’s sigma coordinate. A fourth-order hyperviscosity with a base coefficient of 0.1 day⁻¹ that is applied to the highest resolved wavenumber is used. The atmosphere is situated above a global slab ocean with a depth of 100 m. The model is run with annual average insolation and without a diurnal cycle. The boundary conditions and boundary forcing are zonally homogeneous and hemispherically symmetric about the equator. The long-term statistics are expected to respect these related symmetries such as hemispheric symmetry and no stationary waves.

Potential temperature and tracers, including water vapor, are advected using the model’s standard spectral advection scheme. For water vapor solely, a hole-filling scheme is used to preserve nonnegativity. Although a spectral advection scheme does not guarantee nonnegativity (Rasch and Williamson 1990), negative values were mitigated by reducing the time step of the model to 150 s for both the dynamics and physical parameterizations.

Of the various combinations of sub-grid-scale physical parameterizations and radiative transfer available within the Isca framework, we select the following:

1. A two-stream, two-plus-one band, gray radiation scheme, described in section 4.2 of Vallis et al. (2018). The scheme parameterizes the optical depth in terms of a nominal CO₂ concentration value and of the water vapor field in the model. It includes a shortwave window, a longwave band that treats the infrared window region of the spectrum, and another longwave band that treats the rest of the spectrum. The radiative scheme used in our version of the model has a nominal CO₂ value of 360 ppm [documented in section 4.2 in Vallis et al. (2018)]. In section 4.6, this value is doubled to explore the response to climate change.

2. A vertical diffusion that parameterizes turbulent boundary layer fluxes using simplified Monin–Obukhov similarity theory (Frierson et al. 2006). The effect of this parameterization is to spread the surface fluxes of enthalpy, moisture, and momentum over the model planetary boundary layer. The surface fluxes appear in the atmospheric tendencies through this parameterization.

3. A large-scale condensation scheme that removes water vapor and adds latent heat to a grid cell if the grid cell reaches saturation. The water vapor is assumed to be immediately removed from the atmosphere without allowing for reevaporation. This is different from Frierson et al. (2006), but is the same formulation that was used in O’Gorman and Schneider (2008).

4. A subgrid-scale convection scheme that is the “shallower” option documented in Frierson (2007). The convection scheme checks if there is convective available potential energy in a column and, if there is, relaxes the column toward a moist adiabatic profile in a manner that conserves total column enthalpy.

The tracer tagged here is potential temperature θ, which evolves according to

${\displaystyle \frac{\partial \theta }{\partial t}}=-\nabla \cdot \left(\mathbf{v}\theta \right)+D_H\left(\theta \right)+\Pi Q,$(14)

where the Exner function Π = (p₀/p)^κ and Q represents the sum of the model’s parameterized heating rates. The development of the heat tags follows by applying the generic tracer tagging methods in section 2a to Eq. (14). Table 1 summarizes this correspondence, and the rest of the section develops the idea in full.

Table 1.

Tagged tracer symbols for the generic tracer introduced in section 2a and for the heat tags introduced in section 2c.

View Table

We use Eq. (10) as follows: we identify θ with c, set N_S = 4 tags, and following the correspondence in Table 1,

$\theta ={\displaystyle \sum _{i=1}^{N_S}}{\theta }_i={\theta }_{\text{conv}}+{\theta }_{\text{cond}}+{\theta }_{\text{diff}}+{\theta }_{\text{radi}},$(15)

and split the heating terms into θ sources and sinks,

$\Pi Q=\Pi Q^{+}+\Pi Q^{-},$(16)

with

$\Pi Q^{+}=\Pi Q_{\text{conv}}^{+}+\Pi Q_{\text{cond}}^{+}+\Pi Q_{\text{diff}}^{+}+\Pi Q_{\text{radi}}^{+}$(17)

and

$\Pi Q^{-}=\Pi Q_{\text{conv}}^{-}+\Pi Q_{\text{diff}}^{-}+\Pi Q_{\text{radi}}^{-}.$(18)

Here, the subscripts conv, cond, diff, radi indicate, respectively, heating from the subgrid-scale convection scheme, the large-scale condensation, the vertical diffusion in the boundary layer parameterization, and the shortwave and longwave radiative transfer schemes. This is summarized in Table 2. Note the absence of $Q_{\text{cond}}^{-}$ in (18). This is zero by design, because the model precipitation scheme has no reevaporation of falling water vapor (cooling and hence a θ sink). Applying Eq. (10) to θ yields a set of tracers that evolve according to

${\displaystyle \frac{\partial {\theta }_{\text{conv}}}{\partial t}}=-\nabla \cdot \left(\mathbf{v}{\theta }_{\text{conv}}\right)+D_H\left({\theta }_{\text{conv}}\right)+\Pi \left(Q_{\text{conv}}^{+}+{\displaystyle \frac{{\theta }_{\text{conv}}}{\theta }}\hspace{0.17em}{Q}^{-}\right),$(19a)

${\displaystyle \frac{\partial {\theta }_{\text{cond}}}{\partial t}}=-\nabla \cdot \left(\mathbf{v}{\theta }_{\text{cond}}\right)+D_H\left({\theta }_{\text{cond}}\right)+\Pi \left(Q_{\text{cond}}^{+}+{\displaystyle \frac{{\theta }_{\text{cond}}}{\theta }}\hspace{0.17em}{Q}^{-}\right),$(19b)

${\displaystyle \frac{\partial {\theta }_{\text{diff}}}{\partial t}}=-\nabla \cdot \left(\mathbf{v}{\theta }_{\text{diff}}\right)+D_H\left({\theta }_{\text{diff}}\right)+\Pi \left(Q_{\text{diff}}^{+}+{\displaystyle \frac{{\theta }_{\text{diff}}}{\theta }}\hspace{0.17em}{Q}^{-}\right),$(19c)

${\displaystyle \frac{\partial {\theta }_{\text{radi}}}{\partial t}}=-\nabla \cdot \left(\mathbf{v}{\theta }_{\text{radi}}\right)+D_H\left({\theta }_{\text{radi}}\right)+\Pi \left(Q_{\text{radi}}^{+}+{\displaystyle \frac{{\theta }_{\text{radi}}}{\theta }}\hspace{0.17em}{Q}^{-}\right).$(19d)

The sources and sinks are separated at every grid point and time step in the model. For instance, in a convectively unstable column the convection scheme heats higher model levels, contributing locally to $Q_{\text{conv}}^{+}$, and cools lower model levels, contributing locally to $Q_{\text{conv}}^{-}$ and hence to Q⁻ [in Eqs. (18) and (19a)]. Apart from small numerical errors discussed in the next subsection, the tag sum should satisfy (15) at every time step and grid point.

Table 2.

Heat tag names used throughout the paper.

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Initialization and equilibration of the tag tracers are represented in Fig. 1. Experimentation (not shown) reveals that the radiative tag radi, which we will find to be sourced and largely confined to the stratosphere (section 3), takes several years to equilibrate, which is an order of magnitude longer than found for the other tags, owing to the relatively long lifetime of stratospheric air masses (Hall and Plumb, 1994). In the experiments reported here, the heat tag is initialized by setting θ_radi to θ and all other tags to zero, which satisfies total tag conservation (15) by construction, and which is designed to bring the full set more quickly into equilibrium. Figure 1 shows the time evolution of the globally averaged tracer tag fractions {χ_i}, which reveals the relatively fast equilibration of χ_conv, χ_cond, and χ_diff compared with χ_radi.

Time evolution of the globally averaged tracer tag fractions χ_i during the tag spinup.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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Time evolution of the globally averaged tracer tag fractions χ_i during the tag spinup.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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Time evolution of the globally averaged tracer tag fractions χ_i during the tag spinup.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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After approximate statistical equilibrium is reached (e.g., after day 2000 for the simulation in Fig. 1), the following 2500 days of simulation is used as an analysis period. We denote by an overbar the zonal mean and time mean (for this analysis period). The error associated with total tag conservation is measured by

$R={[\hspace{0.17em}{\displaystyle \frac{1}{T}}{{\displaystyle \int }}_0^T{\left(1-{\displaystyle {\Sigma }_i}{\chi }_i\right)}^2\hspace{0.17em}dt]}^{1/2}.$(22)

The error is less than 1% throughout most of the troposphere, and less than 2% everywhere below 200 hPa (Fig. 2).

The root-mean-squared residual between θ and the sum of the tagged tracers, given by Eq. (22). The gray lines show $\overline{\theta }$ with a contour interval of 30 K.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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The root-mean-squared residual between θ and the sum of the tagged tracers, given by Eq. (22). The gray lines show $\overline{\theta }$ with a contour interval of 30 K.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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The root-mean-squared residual between θ and the sum of the tagged tracers, given by Eq. (22). The gray lines show $\overline{\theta }$ with a contour interval of 30 K.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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a. Distribution of tracer sources and sinks

The climatological distribution of tag sources for θ_i, $\left\{\Pi Q_i^{+}\right\}$, is shown in Figs. 3a–d and the climatological distribution of tag sinks for θ_i, $\left\{\Pi Q_i^{-}\right\}$, is shown in Figs. 3e–h. The distribution of convection and condensation sources ($\Pi Q_{\text{conv}}^{+}$ and $\Pi Q_{\text{cond}}^{+}$, Figs. 3a,b) shows the partitioning between large-scale and subgrid-scale latent heating. In the tropical midtroposphere, most of the latent heating happens through the subgrid-scale convective parameterization. There is a strong isolated maximum in condensation source in the tropical upper troposphere that is approximately half the maximum of the convection source. This reflects saturation of air that is explicitly resolved by the model, which occurs at the top of the Hadley cell’s region of strong vertical ascent. A small amount of shallow convection heat tag source also extends into the subtropics and midlatitudes. In the midlatitudes, condensation becomes the dominant tag source of latent heating (Fig. 3b). In this region, regions of vertical ascent associated with baroclinic eddies create conditions for large-scale saturation. Cooling in the convective parameterization ($\Pi Q_{\text{conv}}^{-}$) only occurs weakly in a location near the tropical surface (Fig. 3e), reflecting the rare occasions when the convective parameterization acts to cool the lower troposphere. By definition, ($\Pi Q_{\text{cond}}^{-}\equiv 0$). Overall, latent heating is the greatest source of heating in the model, and plays a very minor role in cooling.

(a) Zonal mean climatological positive θ tendency from Q_conv (contour interval: 1 K day⁻¹). (b)–(d) As in (a), but for Q_cond, Q_diff, and Q_radi. (e)–(h) As in (a)–(d), but for the negative θ tendency. Extra contours are marked in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Zonal mean climatological positive θ tendency from Q_conv (contour interval: 1 K day⁻¹). (b)–(d) As in (a), but for Q_cond, Q_diff, and Q_radi. (e)–(h) As in (a)–(d), but for the negative θ tendency. Extra contours are marked in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Zonal mean climatological positive θ tendency from Q_conv (contour interval: 1 K day⁻¹). (b)–(d) As in (a), but for Q_cond, Q_diff, and Q_radi. (e)–(h) As in (a)–(d), but for the negative θ tendency. Extra contours are marked in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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The diffusive heat tag source, $\Pi Q_{\text{diff}}^{+}$ (Fig. 3c), is greatest near the surface, and its distribution reflects the height of the planetary boundary layer in the model atmosphere (Frierson et al. 2006). There is also strong diffusive heating near the tropical surface, where the hot surface provides large sensible heat fluxes into the atmosphere. There is cooling present throughout the lower and midtroposphere from the diffusive parameterization ($\Pi Q_{\text{diff}}^{-}$, Fig. 3g). This cooling is due to boundary layer mixing that occurs when the boundary layer extends relatively deeper into the troposphere. This pattern of diffusive heating below the boundary layer and diffusive cooling above the boundary layer is also seen in the storm tracks (Ling and Zhang, 2013).

Radiative heating is the only tag source term in the stratosphere, visible in these plots only toward the equator. This heating reflects the parameterized shortwave absorption by ozone. As expected, the cooling from the radiative scheme (Fig. 3h) extends throughout the atmosphere. The greatest cooling rates are in the tropical upper troposphere, where the thin optical depth above allows for the upwelling radiation to leave the atmosphere.

b. Globally integrated tag fractions

The well-mixed values for the four heat tags, shown in Table 3, suggests that in this model just under one-third of the θ content of the atmosphere should be generated by each of subgrid-scale convection, large-scale convection, and vertical diffusion. The remaining small fraction of heat tag arises from shortwave radiation in the stratosphere. The differences between the well-mixed estimates and the actual values reflect the extent to which tags are prevented from being well mixed (visible in Fig. 4). The well-mixed estimate of ⟨χ_conv⟩, ⟨χ_cond⟩, and ⟨χ_diff⟩ are within 20% of their actual values, suggesting that these heat tags, especially the convective tag, are indeed rapidly mixed throughout the atmosphere. The ⟨χ_cond⟩ and ⟨χ_conv⟩ are somewhat lower than their well-mixed estimates, suggesting that diabatic cooling acts more efficiently on these tracers, i.e., ⟨(χ_i)*(ΠQ⁻)*⟩ < 0. The radiative tag fraction is a factor of 4 greater than its well-mixed estimate, suggesting strong isolation and relatively little loss from radiation, i.e., ⟨(χ_i)*(ΠQ⁻)*⟩ > 0.

Table 3.

Well-mixed estimate, globally averaged tag fractions with respect to θ, ⟨χ_i⟩, and the globally averaged tag fractions with respect to θ_e, ⟨θ_i/θ_e⟩.

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(a) Zonal mean climatological equivalent potential temperature θ_e (contour interval: 20 K). (b) As in (a), but for the potential temperature θ. (c) As in (a), but for the humidity normalized to temperature units (L_υ/C_p)q (contour interval: 10 K). (d) Zonal mean climatologies of θ_conv (contour interval: 20 K). (e)–(g) As in (a), but for θ_cond, θ_diff, and θ_radi. The dashed white line in (g) is the estimate of the dynamical tropopause, defined as the 1.5 potential vorticity unit surface calculated using climatological mean quantities. Extra contours are marked in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Zonal mean climatological equivalent potential temperature θ_e (contour interval: 20 K). (b) As in (a), but for the potential temperature θ. (c) As in (a), but for the humidity normalized to temperature units (L_υ/C_p)q (contour interval: 10 K). (d) Zonal mean climatologies of θ_conv (contour interval: 20 K). (e)–(g) As in (a), but for θ_cond, θ_diff, and θ_radi. The dashed white line in (g) is the estimate of the dynamical tropopause, defined as the 1.5 potential vorticity unit surface calculated using climatological mean quantities. Extra contours are marked in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Zonal mean climatological equivalent potential temperature θ_e (contour interval: 20 K). (b) As in (a), but for the potential temperature θ. (c) As in (a), but for the humidity normalized to temperature units (L_υ/C_p)q (contour interval: 10 K). (d) Zonal mean climatologies of θ_conv (contour interval: 20 K). (e)–(g) As in (a), but for θ_cond, θ_diff, and θ_radi. The dashed white line in (g) is the estimate of the dynamical tropopause, defined as the 1.5 potential vorticity unit surface calculated using climatological mean quantities. Extra contours are marked in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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c. Zonal mean tag distribution

We begin the discussion of the global distribution of tags by showing the climatological distribution of $\overline{{\theta }_e}$ (Fig. 4a) and its decomposition into dry and moist components using θ_e ≈ θ + (L_υ/C_p)q (Figs. 4b,c). The moist stratification is unstable in the tropics, near neutral in the midlatitudes, and stable at the poles. The moisture content in the atmosphere is primarily confined to the tropical surface, which explains the unstable stratification in the tropics. The potential temperature can be further subdivided using the heat tags, so that Fig. 4b is the sum of Figs. 4d–g.

The climatological distribution of the heat tags is shown in Figs. 4d–g. The distribution of the heat tags reflects their creation and loss by diabatic heating and cooling (Fig. 3) as well as atmospheric transport. For instance, θ_diff is produced only below about 700 hPa, but because there is insufficient cooling near the surface to balance the tag source, θ_diff is distributed through a larger region of the atmosphere.

The convective heat tag θ_conv (Fig. 4d) has a maximum in the deep tropics where the greatest source is located (Fig. 3a). The large transfer of energy from the surface to the atmosphere in the deep tropics generates convection, which creates large amounts of θ_conv, and drives the Hadley circulation, which transports θ_conv from the lower troposphere to the upper troposphere. These effects combined create a large amount of θ_conv in the deep tropics with a strong vertical gradient from the surface to the upper troposphere. Thus, the tropical upper-tropospheric gradient of $\overline{\theta }$ (Fig. 4b) is attributable to this convective process. Between 15° and 30° the vertical gradient is considerably reduced. This region is in the downwelling branch of the Hadley cell, which transports θ_conv from the upper troposphere to the lower troposphere.

Outside the tropics, $\overline{{\theta }_{\text{conv}}}$ has a large meridional extent, with values as high as 60–90 K over the poles. In the extratropics $\overline{{\theta }_{\text{conv}}}$ is greatest in the upper troposphere.

The condensation tracer tag θ_cond has a maximum in the midlatitudes, near 45°. Since this is a region of descent in the climatological zonal mean, the vertical uplift required to bring the parcels to saturation comes from regions of ascent inside midlatitude eddies (Schemm et al. 2013; Pfahl et al. 2015). The maximum of $\overline{{\theta }_{\text{cond}}}$ is above and poleward of the midlatitude maximum of $\Pi Q_{\text{cond}}^{+}$, indicative of the transport away from the source. Thus the baroclinic eddies act not just to generate latent heating in the midlatitudes, but also to transport θ_cond away from its source region.

The diffusive heat tag $\overline{{\theta }_{\text{diff}}}$ is greatest near the surface. The strong vertical near-surface gradient in the tropics and midlatitudes reflects the near-surface diffusive heat source and the diffusive cooling sink immediately above. In the tropics, there is an intrusion of high-θ_diff air that extends into the midtroposphere and that compensates for the reduced contribution of θ_conv in this region. This leads to weak meridional gradients of θ in this region (Fig. 4b), and suggests a dynamical compensation between diffusive and convective processes in the deep tropics that will be further discussed below. In the midlatitudes, there is a strong vertical gradient over the regions where $Q_{\text{diff}}^{+}$ is nonzero. Poleward of this, the vertical gradient is reduced, suggesting vertical mixing of θ_diff as air masses are advected poleward. There is a tongue of relatively low-θ_diff air extending from the midlatitude upper troposphere into the subtropics, suggesting that this air mass has not recently been in contact with the surface. This is similar to the results of (Galewsky et al. 2005), who show that the air mass in the subtropics primarily comes from the isentropic descent from the upper troposphere.

Because there is little shortwave heating in the troposphere, θ_radi is primarily found in the stratosphere and dominates the other heat tags there. The strong gradients of θ_radi effectively trace out the dynamical tropopause in the extratropics (dashed white line in Fig. 4g). In this model, stratospheric air intrusions into the troposphere are too infrequent and/or weak to maintain large values of $\overline{{\theta }_{\text{radi}}}$ in the presence of radiative cooling (θ_radi sink) in the troposphere.

The spatial distribution of the tag fractions (Figs. 5a–d) give complementary information to the zonal means of the heat tags. While the θ_i express a balance between the sources $Q_i^{+}$ and sinks Q⁻ [Eqs. (19a)–(19d)], the χ_i express tradeoffs between the different sources $Q_i^{+}$ [Eq. (13)]. We thus expect χ_i to be largest in regions where $Q_i^{+}$ is large compared to the other $Q_{j\ne i}^{+}$. Thus $\overline{{\chi }_{\text{conv}}}$ is dominant in the upper troposphere where there is strong convective heating and $\overline{{\chi }_{\text{radi}}}$ (Fig. 4h) is reduced in the troposphere where other tag sources dominate.

(a)–(d) As in Figs. 4d–g, but for the corresponding tag fractions χ_i (contour interval: 0.05). Extra contours are marked in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a)–(d) As in Figs. 4d–g, but for the corresponding tag fractions χ_i (contour interval: 0.05). Extra contours are marked in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a)–(d) As in Figs. 4d–g, but for the corresponding tag fractions χ_i (contour interval: 0.05). Extra contours are marked in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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The convective tag fraction $\overline{{\chi }_{\text{conv}}}$ (Fig. 5a) peaks in the tropics and accounts for just under half the total θ content in that region. The χ_conv tag fraction extends into the polar upper troposphere, where it remains as much as 30% of the total, consistent with its well-mixed estimate. This complements the analysis of Orbe et al. (2015), who show that a large amount of tropical air mass sourced at the tropical surface and undergoing conservative advection is then transported to the tropical upper troposphere and then to the Arctic upper troposphere. Similarly χ_conv, which is sourced in the tropics, extends throughout the upper troposphere, which demonstrates a transport pathway between the tropical and polar upper troposphere. Additionally, for the heat tags as formulated here, the identity of convective tag is robust even in the presence of diabatic cooling from radiation.

The condensation tag fraction $\overline{{\chi }_{\text{cond}}}$ is greatest in the extratropics, and accounts for about 30% of the θ content of midlatitude free troposphere and polar tropospheric air (Fig. 5b), which is, again, consistent with its well-mixed estimate. The maximum θ_cond in the Arctic is located in the midtroposphere, below the upper-tropospheric maximum of Arctic θ_conv. These characteristics resemble air sourced at the midlatitude surface and undergoing conservative advection (Orbe et al. 2015). The large contribution to θ in the polar column could also be anticipated from Fajber et al. (2018), who showed that the poleward transport of warm air was associated with latent heating in the midlatitudes. It should be noted that $\overline{{\chi }_{\text{cond}}}+\overline{{\chi }_{\text{conv}}}$ amounts to more than 60% throughout most of the polar troposphere, despite the lack of latent heating in the polar atmosphere.

The diffusive tag makes up at least 40% of the potential temperature in the boundary layer, and this fraction increases from the equator to the pole (Fig. 5c). This follows the change in the ratio of the surface sensible to surface moisture fluxes, which also increases from the equator to the pole. As the boundary layer air gets colder it also becomes drier at a faster rate. This increases the fraction of the diffusive tracer tag near the polar and midlatitude boundary layer. Low amounts of χ_diff are seen in the subtropical upper troposphere. This is similar to the results of Galewsky et al. (2005), who show that air in the subtropics is transported there from the upper troposphere, where the air mass has relatively little θ_diff.

The distribution of $\overline{{\chi }_{\text{radi}}}$ (Fig. 5d) is similar to the distribution of $\overline{{\theta }_{\text{radi}}}$. Since the tropospheric radiative heating is small compared with the other sources of tropospheric heating, $\overline{{\chi }_{\text{radi}}}$ is small compared to the other tag fractions there.

d. Dynamic variability of the heat tags

The distributions of the individual heat tags $\left\{\overline{{\theta }_i}\right\}$ (Figs. 4d–g) feature more spatial structure than their sum ($\overline{\theta }$, Fig. 4b), suggesting that the distributions are not independent. Such relationships are also evident in snapshots like Fig. 6, which shows, at a typical time in the simulation, the vertical mean of ${\chi }_{\text{conv}}^{\prime }$, ${\chi }_{\text{cond}}^{\prime }$, ${\chi }_{\text{diff}}^{\prime }$, and ${\chi }_{\text{radi}}^{\prime }$, where the prime indicate the deviation from the time and zonal mean. In the tropics and midlatitudes, there are regions with coincident opposite-signed anomalies of χ_cond and χ_conv. Anomalies of χ_cond and χ_radi are relatively weak in the tropics and strong in the extratropics, and also feature coincident opposite signed anomalies.

(a) Representative snapshot of the vertically integrated ${\chi }_{\text{conv}}^{\prime }$. (b)–(d) As in (a), for ${\chi }_{\text{cond}}^{\prime }$, ${\chi }_{\text{diff}}^{\prime }$, and ${\chi }_{\text{radi}}^{\prime }$.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Representative snapshot of the vertically integrated ${\chi }_{\text{conv}}^{\prime }$. (b)–(d) As in (a), for ${\chi }_{\text{cond}}^{\prime }$, ${\chi }_{\text{diff}}^{\prime }$, and ${\chi }_{\text{radi}}^{\prime }$.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Representative snapshot of the vertically integrated ${\chi }_{\text{conv}}^{\prime }$. (b)–(d) As in (a), for ${\chi }_{\text{cond}}^{\prime }$, ${\chi }_{\text{diff}}^{\prime }$, and ${\chi }_{\text{radi}}^{\prime }$.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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Thus motivated to further explore these relationships, we analyze a decomposition of the potential temperature variance:

$\mathrm{var}\left({\theta }_i\right)={\displaystyle \sum _{i=1}^{N_S}}\hspace{0.17em}{\displaystyle \sum _{j=1}^{N_S}}\mathrm{cov}\left({\theta }_i,{\theta }_j\right)={\displaystyle \sum _{i=1}^{N_S}}\left[\mathrm{var}\left({\theta }_i\right)+{\displaystyle \sum _{j\ne i}}\mathrm{cov}\left({\theta }_i,{\theta }_j\right)\right],$(27)

where $\mathrm{cov}\left({\theta }_i,{\theta }_j\right)\equiv \overline{{\theta }_i^{\prime }{\theta }_j^{\prime }}$ and var(θ_i) ≡ cov(θ_i,θ_i). The decomposition formally involves 10 terms; representative results from the decomposition are shown in Fig. 7.

(a) Zonal mean, climatological mean variance of θ_conv (contour interval: 20 K²). (b)–(c) As in (a), but for θ_cond and θ_diff. (d) Two times the covariance between θ_conv and θ_conv and θ_diff. Extra contours are in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Zonal mean, climatological mean variance of θ_conv (contour interval: 20 K²). (b)–(c) As in (a), but for θ_cond and θ_diff. (d) Two times the covariance between θ_conv and θ_conv and θ_diff. Extra contours are in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Zonal mean, climatological mean variance of θ_conv (contour interval: 20 K²). (b)–(c) As in (a), but for θ_cond and θ_diff. (d) Two times the covariance between θ_conv and θ_conv and θ_diff. Extra contours are in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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The greatest variability in θ_conv is in the tropical lower troposphere (Fig. 7a), reflecting variability in convection near the surface and the mixing of high-θ_conv air masses from the free troposphere into the boundary layer. There is also strong variability that extends through the subtropics and into the midlatitudes. This variability might be associated with variability near the source and variability in the transport.

Variability in θ_cond (Fig. 7b) is much smaller than in θ_conv. It is greatest at the subtropical edge of the Hadley cell, the midlatitude upper troposphere, and the midlatitude near surface. We note that the greatest variability in the subtropics is not near regions of large $Q_{\text{cond}}^{+}$ (Fig. 3b), again highlighting the role of transport and dynamics in creating variability in the heat tags.

In contrast to θ_conv and θ_cond, θ_diff appears to be locally controlled in the boundary layer; variability is greatest where most of the tag is produced (Figs. 7c, 3c). This could be due to the alternating pattern of $Q_{\text{diff}}^{+}$ and $Q_{\text{diff}}^{-}$ (Figs. 3c,f). As θ_diff leaves the boundary layer it passes through a region of high $Q_{\text{diff}}^{-}$, which removes it from the atmosphere, and variability in this process seems to be reflected in Fig. 7c.

In the troposphere, these individual tag variances are much larger than, and have a spatial structure distinct from, the total variance of θ (Fig. 8a). This reflects compensation (large negative covariances) between the tags that is summarized by showing the covariance of the most variable tropospheric tag, θ_conv and the sum of the other tropospheric tags, θ_cond and θ_diff, in Fig. 7d. The compensation is most obvious in the deep tropics where the large variances from θ_conv and θ_diff are almost perfectly cancelled by twice the covariance between the tropospheric heat tags [the factor of 2 is introduced to compare the variance terms to the covariance terms in the double sum in Eq. (27)].

(a) Zonal mean, climatological mean variance of θ (contour interval: 5 K²). (b) The zonal mean, climatological mean covariance between θ_conv and θ (contour interval: 5 K²). (b)–(d) As in (a), but for θ_cond and θ_diff. Extra contours are in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Zonal mean, climatological mean variance of θ (contour interval: 5 K²). (b) The zonal mean, climatological mean covariance between θ_conv and θ (contour interval: 5 K²). (b)–(d) As in (a), but for θ_cond and θ_diff. Extra contours are in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Zonal mean, climatological mean variance of θ (contour interval: 5 K²). (b) The zonal mean, climatological mean covariance between θ_conv and θ (contour interval: 5 K²). (b)–(d) As in (a), but for θ_cond and θ_diff. Extra contours are in white.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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While the covariance is negative throughout the troposphere, in the stratosphere there is positive covariance between all of the tropospheric tags (as partially shown in Fig. 7d). The variance of θ_radi is much greater than the variance of θ in the stratosphere (not shown), and has large negative covariability with the tropospheric tags in this region, in contrast to the positive covariability of the tropospheric tags in this region. We therefore interpret the positive covariability of the tropospheric tags in the stratosphere as being indicative of intrusions of tropospheric air into the stratosphere, carrying into the stratosphere air that is enriched with a mixture of tropospheric tags.

Poleward of 65°, cov(θ_conv, θ) is important in the upper troposphere, but cov(θ_diff, θ) is the dominant contribution to var(θ) (Fig. 8c) in the lower troposphere. Near the surface the air is isolated from the rest of the atmosphere (Orbe et al. 2015), while above the planetary boundary layer in the polar region the variability of θ is still dominated by cov(θ_conv, θ). This suggests that although local dynamics are important for the variability of the surface temperature, the variability of the mid- and upper troposphere in polar regions is controlled by remote contributions (Fajber et al. 2018).

In the stratosphere, the dominant contribution to stratospheric θ variance is θ_radi (not shown). This is suggested by Fig. 6 and consistent with the results and analysis of Fig. 7 discussed above.

e. Airmass analysis

While the covariance analysis of the previous subsection is useful for describing how the variability of the tags relates to the variability of θ, it is hard to compare between different regions, since θ and var(θ) varies greatly with latitude and pressure. However, 0 < χ_i < 1, and so χ_i can be compared between different regions more easily. In this subsection we therefore use the joint probability distribution function (pdf) of χ_i to characterize the variation between the tags and show that they can be used to characterize different air masses in the atmosphere.

Airmass characteristics can be inferred from the mass-weighted joint pdfs of the tag fractions. Figure 9 shows the joint pdf of χ_cond and χ_conv for the tropical midtroposphere (Fig. 9a), the extratropical midtroposphere (Fig. 9b), the tropical lower troposphere (Fig. 9c), and the extratropical lower troposphere (Fig. 9d). Notice that almost the entire troposphere occupies the range 0.20 < χ_conv < 0.50 and 0.2 < χ_cond < 0.40, which bracket the well-mixed estimates (Table 3). For over 95% of the mass of the regions considered here, χ_radi < 0.05 (not shown). In these regions, then, the diffusive tag is constrained by the two other tags via [Eq. (12)]:

${\chi }_{\text{diff}}\approx 1-{\chi }_{\text{conv}}-{\chi }_{\text{cond}}.$(29)

To aid in our analysis, we will analyze two populations, represented by the lines

${\chi }_{\text{cond}}^1={\chi }_{\text{conv}}^1+0.10\Rightarrow {\chi }_{\text{diff}}^1\approx 0.90-2{\chi }_{\text{conv}}^1$

(shown in Fig. 9a) and

${\chi }_{\text{cond}}^2=-{\displaystyle \frac{2}{3}}\hspace{0.17em}{\chi }_{\text{conv}}^2+0.56\Rightarrow {\chi }_{\text{diff}}^2\approx 0.44-{\displaystyle \frac{1}{3}}\hspace{0.17em}{\chi }_{\text{conv}}^2$

(shown in Fig. 9d). The relationships between χ_conv and the other two tropospheric heat tag fractions for these populations are sketched in Fig. 9f. We note that these relationships are for illustrative purposes and do not represent a rigorous separation of the data into distinct populations.

(a) Double tag pdfs for 0°–30° latitude and 400–700 hPa, or the tropical midtroposphere. (b) As in (a), but for the extratropical midtroposphere. (c) As in (a), but for the tropical lower troposphere. (d) As in (a), but for the extratropical lower troposphere. The solid gray lines in (a) and (d) describe two different populations of points, as defined and discussed in the text. (e) The values of χ_conv, χ_cond, and χ_diff [implied by Eq. (29)] for the gray lines shown in (a) and (b).

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Double tag pdfs for 0°–30° latitude and 400–700 hPa, or the tropical midtroposphere. (b) As in (a), but for the extratropical midtroposphere. (c) As in (a), but for the tropical lower troposphere. (d) As in (a), but for the extratropical lower troposphere. The solid gray lines in (a) and (d) describe two different populations of points, as defined and discussed in the text. (e) The values of χ_conv, χ_cond, and χ_diff [implied by Eq. (29)] for the gray lines shown in (a) and (b).

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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(a) Double tag pdfs for 0°–30° latitude and 400–700 hPa, or the tropical midtroposphere. (b) As in (a), but for the extratropical midtroposphere. (c) As in (a), but for the tropical lower troposphere. (d) As in (a), but for the extratropical lower troposphere. The solid gray lines in (a) and (d) describe two different populations of points, as defined and discussed in the text. (e) The values of χ_conv, χ_cond, and χ_diff [implied by Eq. (29)] for the gray lines shown in (a) and (b).

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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The first population is found mainly in the tropics. It is characterized by a long tail of high χ_conv values being replaced by χ_cond and, in a ratio of 2:1, by χ_diff. This air features high values of χ_conv in the upper and mid–tropical troposphere, and is then mixed with air of lower χ_conv. Integrating the pdfs over the region χ_conv > 0.30, an approximate boundary for this population, we find that this population contains about 33% of the mass of the atmosphere.

The second population is mainly found near the surface. Integrating the pdfs over the region χ_conv ≤ 0.26 we find that this population makes up about 41% of the atmosphere. It is characterized by high χ_diff values compared to the first population. The contributions of χ_conv and χ_cond increase at the same rate, although χ_cond is higher. This population lies near or in the planetary boundary layer.

Between the populations, in the region 0.25 < χ_conv < 0.30 the air has χ_cond ≈ 0.36. We interpret the air in this region as resulting from the mixing of the two different populations. This mixed population is found primarily in the extratropical midtroposphere (Fig. 9b), where there are no diabatic sources in the atmosphere. In this region the fractionation between χ_conv and χ_cond is set by the mixing of air masses created in different environments. The mixed population also exists in the tropical mid- and lower troposphere. This is the signature of air mixing from either the surface or polar latitudes into the tropics. Integrating the pdfs over the region 0.25 < χ_cond < 0.30 we find that this mixed population makes up about 26% of the mass of the atmosphere.

Taken together, this diagnostic suggests thinking of the troposphere as composed of an air mass with high χ_conv values (intermediate χ_diff and low χ_cond values), residing in the tropics, and an air mass with low χ_conv values (intermediate χ_cond and high χ_diff values), residing near the surface. The air mass in the lower tropical troposphere and extratropical midtroposphere come from the mixing of these two air masses.

f. Vertically integrated transport

Total θ transport (Fig. 10f shows the tropospheric contribution to $H_{\theta }^T$; $H_{\text{radi}}^T$ is small but composed of large cancellations, as discussed below) is dominated by a poleward flux of magnitude 6 PW from $H_{\text{conv}}^T$ (solid curve in Fig. 10a) from the tropics into the midlatitudes. In the tropics, this is dominated by the Hadley circulation term $H_{\text{conv}}^M$ (dotted line in Fig. 10a) which is then taken over by eddy transport $H_{\text{conv}}^E$ (dashed line in Fig. 10b) in the extratropics. Air tagged as convectively generated is transported out of the deep tropics through the Hadley cell, which is then transported further poleward by the midlatitude eddies, explicitly demonstrating the relation implied by Trenberth and Stepaniak (2003).

Vertically integrated transport for (a)–(d) the heat tags, (e) L_υq, (f) θ − θ_radi (see text for details), (g) θ_e − θ_radi, and (h) θ_e. The solid line is H^T, the dotted line is H^M, and the dashed line is H^E. The units for all are PW.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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Vertically integrated transport for (a)–(d) the heat tags, (e) L_υq, (f) θ − θ_radi (see text for details), (g) θ_e − θ_radi, and (h) θ_e. The solid line is H^T, the dotted line is H^M, and the dashed line is H^E. The units for all are PW.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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Vertically integrated transport for (a)–(d) the heat tags, (e) L_υq, (f) θ − θ_radi (see text for details), (g) θ_e − θ_radi, and (h) θ_e. The solid line is H^T, the dotted line is H^M, and the dashed line is H^E. The units for all are PW.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0290.1

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