## 1. Introduction

The height of the tropopause is often considered to be set primarily by the combined effects of a dynamically active troposphere and a stratosphere in near-radiative equilibrium (e.g., Manabe and Strickler 1964; Held 1982; Thuburn and Craig 1997; Schneider 2007). Held (1982) introduced the concept of separating dynamical and radiative constraints to determine tropopause height. In one form of this concept, one assumes a given surface temperature and (constant) tropospheric lapse rate and obtains the height of the tropopause such that the temperature profile at and above the tropopause is in radiative equilibrium (Thuburn and Craig 2000). Provided a dynamical constraint exists that relates tropospheric lapse rate and tropopause height for given surface temperature, one has a closed problem to be solved for the tropospheric lapse rate and tropopause height simultaneously (Schneider 2007). The basic dependence of tropopause height on surface temperature and tropospheric lapse rate in this concept, among other things, provides a qualitative explanation for the observed equator-to-pole contrast in tropopause height (see above references).

In the tropics moist convection tends to constrain tropospheric lapse rates to a moist adiabat, at least in the middle to upper troposphere (Folkins and Martin 2005). The classical radiative–convective equilibrium model, which assumes a fixed tropospheric lapse rate set by convection, therefore predicts a tropopause as the upper boundary of the convectively adjusted region with an overlying stratosphere in radiative equilibrium (e.g., Manabe and Wetherald 1967). Note, however, that the tropopause, when defined as the coldest point in the temperature profile, may be well separated from the top of convection in these radiative–convective equilibrium models (Thuburn and Craig 2002). In the extratropics baroclinic eddies take over the role to adjust the tropospheric lapse rate (Haynes et al. 2001) and provide a dynamical constraint (e.g., Schneider 2004), even though convection might still play a role there (Juckes 2000; Frierson et al. 2006).

A stratosphere in radiative equilibrium clearly represents an idealization. Stratospheric dynamics, as represented by the stratospheric residual circulation, provides adiabatic cooling near the tropical tropopause and lower stratosphere, which tends to raise the tropical tropopause out of reach of convection (e.g., Highwood and Hoskins 1998) and therefore contributes to form the so-called tropical tropopause layer (TTL). Near the extratropical tropopause and lower stratosphere the stratospheric residual circulation provides adiabatic warming, which tends to lower the extratropical tropopause. That is, the equator-to-pole contrast in tropopause height is at least partially due to the stratospheric dynamics (Kirk-Davidoff and Lindzen 2000). A stronger stratospheric circulation leads to a larger equator-to-pole contrast in tropopause height and a weaker circulation leads to a smaller contrast (Thuburn and Craig 2000; Wong and Wang 2003). The present study finds that more than half of the equator-to-pole contrast in tropopause height is due to stratospheric dynamics. Moreover, the separation of the top of convection and the tropopause in the tropics, that is, the existence of the TTL, is found to be almost entirely due to stratospheric dynamics. Stratospheric dynamics also significantly impacts tropopause height variability on intraseasonal (Son et al. 2007) and interannual (e.g., Zhou et al. 2001) time scales, while stratospheric processes play a crucial role in future tropopause height trends (Son et al. 2009).

Another limitation of stratospheric radiative equilibrium (SRE) is that it fails to predict the observed lower stratospheric static stability structure. In particular, the midlatitudinal isothermal lower stratospheric stratification in SRE (cf. Thuburn and Craig 2002) stands in contrast to the recently described marked stratification maximum just above the tropopause—the tropopause inversion layer (TIL) (Birner et al. 2002; Birner 2006; Randel et al. 2007b; Bell and Geller 2008; Bian and Chen 2008). However, processes that form and maintain this TIL are presently not well understood. Birner et al. (2002) speculated that warming due to lower stratospheric subsidence could play a role in forming a TIL at midlatitudes. [They also presumed that tropospheric eddies would lead to a relative cooling at tropopause level due to tropopause lifting, which was intended to describe the observed negative correlation between tropopause height and temperature (e.g., Zängl and Wirth 2002) but is somewhat misleading given that tropospheric eddies tend to warm the upper troposphere.] Wirth (2003, 2004) suggested that the asymmetry between upper-level cyclones and anticyclones and their effects on the local stratification around the tropopause is responsible for the existence of a TIL in the climatological mean, and Wirth and Szabo (2007) and A. R. Erler and V. Wirth (2009, unpublished manuscript) tested this idea in idealized baroclinic life cycle experiments. Randel et al. (2007b) pointed out that the lowermost stratospheric structure of water vapor and ozone resulting from stratosphere–troposphere exchange (STE) events has a radiative feedback such as to enhance stratification just above the tropopause, that is, to create a TIL (see also Kunz et al. 2009). It should be noted that this radiative mechanism inherently includes transport effects by assuming that the dynamics involved in the STE eventually lead to the observed water vapor and ozone distributions in the lowermost stratosphere (e.g., Pan et al. 1997, 2000; Hoor et al. 2002; Hegglin et al. 2006, 2009). Evidence that the radiative mechanism might not be the only mechanism at work to form a TIL comes from mechanistic dry core general circulation model (GCM) experiments that spontaneously form a TIL (Son and Polvani 2007). These experiments represent forced–dissipative equilibrium scenarios where the forcing mimics radiation and is represented by simple Newtonian relaxation and only large-scale dry dynamics are considered. That is, potential radiative effects due to dynamically created structures in radiative tracers such as water vapor and ozone are excluded. Finally, GCM simulations with a comprehensive chemistry–climate model (CCM) also exhibit a TIL of realistic strength and seasonal structure, even though of somewhat unrealistic vertical location and extent, presumably due to limited vertical resolution (Birner et al. 2006) combined with uncertainties in the precise location of the tropopause (Bell and Geller 2008).

In the present study it is asked to what extent large-scale stratospheric dynamics are responsible for the lower stratospheric static stability structure and the equator-to-pole contrast in tropopause height. Our approach is based on the assumption that ultimately the net effect of large-scale dynamics is represented by the residual mean meridional circulation in the transformed Eulerian mean (TEM) sense (Andrews et al. 1987) and its meridional and vertical structure (which determines meridional and vertical residual velocities).

Results based on simulations with a comprehensive chemistry–climate model [essentially the same as in Birner et al. (2006)] will be analyzed and compared to 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) data. The effect of stratospheric dynamics on tropopause height and lower stratospheric static stability will be further investigated using offline radiative transfer calculations. Concerning the static stability structure around the tropopause and in the lower stratosphere, it will prove insightful to assess the vertical structure of the residual circulation around the tropopause. In particular, the midlatitudinal residual circulation undergoes structural changes around the tropopause. It will be shown in the present study that the corresponding vertical gradient in residual vertical velocities around the tropopause represents a positive forcing of static stability that represents a possible cause of the TIL, particularly in midlatitudes.

Section 2 describes the tools and discusses the tropopause definition used. Section 3 analyzes terms due to the residual circulation in the heat and static stability budgets. Sections 4 and 5 study the effect of stratospheric dynamics, in particular the stratospheric residual circulation, on tropopause height and lower stratospheric static stability. Finally, section 6 summarizes the main results.

## 2. Tools and tropopause definition

The main tools in this study are the Canadian Middle Atmosphere Mode (CMAM) simulations, complemented by results using ERA-40 (Uppala et al. 2005). CMAM represents a comprehensive chemistry–climate model (Beagley et al. 1997; Scinocca et al. 2008). The configuration used here corresponds to T47 spectral horizontal resolution and 71 vertical levels that extend up to ∼100 km altitude. Around the tropopause the vertical resolution amounts to about 1 km. A 3-yr integration identical to the one used in Gettelman and Birner (2007) after spinup is used (year-to-year variability is small given that interannual variability of the prescribed sea surface temperature is not allowed). For reference, ERA-40 has a T159 horizontal resolution and 60 levels, resulting in roughly the same vertical resolution around the tropopause as CMAM. Son and Polvani (2007) found that the TIL in their simple GCM simulations depends on horizontal resolution. One would therefore expect ERA-40 with its more than 3 times higher horizontal resolution compared to CMAM to exhibit a much stronger TIL. However, data assimilation as incorporated in analyses products such as ERA-40 acts to smooth the temperature structure around the tropopause and thus leads to a weaker TIL compared to CMAM (Birner et al. 2006). Further, A. R. Erler and V. Wirth (2009, unpublished manuscript) recently found an aspect ratio of horizontal to vertical resolution of ∼300–400 to be most appropriate to simulate near-tropopause dynamics (which is similar to the one discussed in Birner (2006)). The CMAM resolution aspect ratio near the tropopause is ∼400, whereas it is ∼120 for ERA-40; that is, ERA-40 may only represent well vertical structures around the tropopause corresponding to dynamical systems with horizontal scales about three times that of the model resolution.

An offline radiative scheme—the Column Radiation Model (CRM), a standalone version of the radiation model used in version 3 of the NCAR Community Climate Model (CCM3)—is employed in sections 4 and 5. A detailed description of the CCM3 radiative scheme can be found in Kiehl et al. (1996) and online (http://www.cgd.ucar.edu/cms/crm/). The CRM divides the solar spectrum into 18 discrete spectral intervals for ozone, water vapor, and carbon dioxide and uses a broadband approach in the longwave. Input profiles of water vapor and ozone are required and these are set to the CMAM seasonal mean profiles (see section 5). Carbon dioxide is assumed to be well mixed and set to a constant mixing ratio of 356 ppmv. Effects due to clouds are not considered in the present study; that is, only clear-sky radiative heating rates are computed. The diurnal cycle is switched off by calculating solar heating rates based on diurnally averaged solar zenith angles using a solar constant of 1367 W m^{−2}. Vertical levels in CRM are set exactly equal to CMAM model levels to ensure maximum consistency; however, the model top in CRM is set to 10 hPa (about 30 km).

### a. Tropopause definition

A thermal tropopause definition is applied in this study. Conventionally, the thermal tropopause is defined as the lowest level at which the atmospheric lapse rate falls below 2 K km^{−1}, provided the average lapse rate between this level and all higher levels within 2 km remains below 2 K km^{−1} (WMO 1957). In the tropics, the cold point tropopause is often considered to be more appropriate, which essentially represents an alternative thermal tropopause using 0 K km^{−1} in place of the 2 K km^{−1} threshold. In the present study the sensitivity of tropopause height to a strongly perturbed stratosphere (several tens of kelvin in temperature) is investigated (see section 4). For such a strongly perturbed climate the somewhat arbitrary thresholds of 2 K km^{−1} for the lapse rate and 2 km for the thickness of the WMO definition cannot be expected to yield robust results. A different thermal tropopause criterion, based on the assumption that the troposphere and the stratosphere can be distinguished through their different thermal stratification, is therefore adopted here. We define the thermal tropopause as the level of maximum curvature of the temperature profile, that is, the level of maximum gradient in stratification (obtained through interpolation of *z* onto ∂* _{zzz}T* = 0 in the neighborhood of the model level of maximum ∂

*). For the unperturbed climate (from CMAM or ERA-40) this maximum curvature definition yields very similar tropopause heights compared to the standard WMO definition in the extratropics and very similar tropopause heights compared to the cold point definition in the tropics. This definition is therefore also advantageous in that it represents a single appropriate tropopause definition for all latitudes. It should be noted that in this study the tropopause definition is only applied to zonal mean seasonal mean profiles; that is, robustness to, say, individual wave perturbations in the temperature profile is not critical.*

_{zz}T## 3. Residual circulation forcing of heat and static stability

*Q*denotes the diabatic heating rate of the zonal mean potential temperature (mainly due to radiation in the stratosphere),

*s*

_{Θ}≡ −∂

_{y}

_{z}

*w*

*w*

*υ*

*g*

^{−1}∂

_{z}

*N*

^{2}are not computed in tropopause-based coordinates here but simply in conventional pressure coordinates to facilitate comparison to the TEM streamfunction. Evidently, the resulting static stability structure around the tropopause (including the TIL signature) is not much affected in GCMs such as CMAM by the method of averaging [cf. Fig. 1 to Fig. 2(top) in Birner et al. (2006)]. Closer analysis of this somewhat surprising result suggests that this is due to reduced variability in tropopause height in CMAM (not shown), whereas tropopause height variability in observations or meteorological analyses is sufficiently strong to blur sharp features such as the TIL in conventional coordinates.

*w*′Θ′

*s*

_{Θ}/

*s*), with

_{m}*s*≡

_{m}*w*′Θ′

*υ*′Θ′

*s*

_{Θ}≈

*s*this cross-isentropic eddy heat flux contribution can be neglected, which, given that

_{m}*s*

_{Θ}≪ 1 in the region of interest here, requires that

*w*′Θ′

*υ*′Θ′

Under quasigeostrophic scaling, meridional advection vanishes (e.g., Andrews et al. 1987) and one obtains the simple steady-state balance *w*_{z}*Q*. Throughout most of the stratosphere ∂_{y}*w*_{z}*w**υ**υ**w**υ*

The individual contributions to the heat budget (2) associated with the vertical and meridional components of the residual velocity are shown for CMAM in Figs. 2a,c (DJF) and Figs. 3a,c (JJA) and their sum in Figs. 2e and 3e. Splitting up the total dynamical heating contribution into those due to the individual residual velocity components serves mainly to quantify deviations from the basic quasigeostrophic balance *w*_{z}*Q*, which is only expected to be accurate in the stratosphere. It is evident that the vertical (*w**υ**υ**υ*_{y}*w*_{z}

Strong localized warming occurs in the subtropical upper troposphere in winter at levels that are at or just above the midlatitude and polar tropopause (∼10–12 km altitude). This localized warming together with cooling further aloft represents a potentially important formation mechanism of double tropopause events, as are frequently observed in the winter subtropics (Randel et al. 2007a). The warming induced by the downwelling over the winter polar cap spans the entire column, exhibits significant vertical structure, and appears to be at a minimum around the tropopause. All of the above discussed characteristics are consistent with the full diabatic heating rate structure (not shown).

*w*

*υ*

*g*/

*g*

^{−1}∂

_{z}

*w*

*w*

Figures 2c and 2d (boreal winter, DJF) and Figs. 3c and 3d (boreal summer, JJA) show the contributions due to *w**υ**w**υ**w**w*

Positive forcing exists in the lowermost stratosphere at almost all latitudes, most distinctly in the winter midlatitudes (due to *w*

Similar results concerning the residual circulation contributions to the heat and static stability budgets are obtained from ERA-40 (Fig. 4). The most notable difference between CMAM and ERA-40 exists around the tropical tropopause where ERA-40 shows a characteristic dipole structure of negative static stability forcing just below and positive static stability forcing just above the tropopause (similar to the behavior in midlatitudes). CMAM, on the other hand, shows positive static stability forcing due to the residual circulation throughout the tropical upper troposphere and lower stratosphere, even though this positive forcing appears to be enhanced in the lowermost stratosphere, as in ERA-40.

## 4. Effect of stratospheric dynamics on tropopause height

Stratospheric dynamics as represented by the residual circulation induces departures from local radiative equilibrium temperatures (Θ_{rad}) with temperatures colder than Θ_{rad} within the tropical upward branch and temperatures warmer than Θ_{rad} within the extratropical downward branch. To quantify this departure and the overall effect of stratospheric dynamics on tropopause structure and lower stratospheric static stability, we first use the CRM as described in section 2 to compute stratospheric radiative equilibrium (SRE) solutions given the CMAM simulated seasonal mean water vapor and ozone distributions. These SRE solutions assume a given, quasi-fixed tropospheric climate that closely resembles the one simulated by CMAM (“quasi-fixed” refers to slight necessary adjustments in the upper troposphere to prevent superadiabatic lapse rates; see appendix A for details). That is, a hypothetical climate state with a troposphere as simulated by CMAM and a stratosphere in radiative equilibrium is computed. It should be noted that the distributions of water vapor and ozone as simulated by CMAM are shaped in part by stratospheric dynamics (e.g., ozone mixing ratios in the polar winter stratosphere would be much lower without the contribution due to the Brewer–Dobson circulation). As such, the SRE as defined here still includes the indirect effect of stratospheric dynamics due to the altered distributions of water vapor and ozone.

Figure 5 shows the temperature difference between CMAM and the SRE solution for DJF and JJA. As expected the SRE solution exhibits a warmer tropical stratosphere and a colder extratropical stratosphere, consistent with radiative warming due to the upward branch and radiative cooling due to the downward branch of the residual circulation. This temperature difference is larger during DJF than during JJA, reflecting the seasonal contrast in the strength of the circulation. Tropical temperature profiles of the SRE solution hardly show a difference between DJF and JJA (not shown). It is furthermore interesting to note that the largest temperature difference in the tropics exists just above the tropopause, consistent with a maximized radiative time scale there (Randel et al. 2002). The zero lines in Fig. 5 in the stratosphere roughly coincide with the zero heating lines due to the residual circulation in Figs. 2e and 3e, confirming that the major balance in the stratospheric heat budget exists between circulation-induced forcing and radiation.

The tropopause of the SRE solution is located much lower in the tropics and somewhat higher in the extratropics compared to CMAM. That is, the difference between the tropical and extratropical tropopause is much reduced in the SRE solution compared to CMAM with the tropics playing the dominant role in the change of the equator-to-pole contrast in tropopause height. Apparently, stratospheric dynamics provide a leading order contribution to the equator-to-pole contrast in tropopause height, consistent with Kirk-Davidoff and Lindzen (2000).

In the tropics the SRE tropopause is located just above the top of convection at ∼12 km altitude (cf. Gettelman and Birner 2007). This suggests that the thermal properties of the layer between the top of convection and the tropical tropopause—the tropical tropopause layer—are predominantly set by the stratospheric residual circulation on average (cf. Highwood and Hoskins 1998), but a more detailed investigation is required to confirm this. In contrast, Thuburn and Craig (2002) find a well-separated tropical tropopause from the top of convection in their radiative–convective equilibrium solution (i.e., without any tropical upwelling-induced cooling). Reasons for this discrepancy are presently unclear but are possibly related to the way the troposphere is treated [i.e., convective adjustment to constant lapse rate in Thuburn and Craig (2002), the CMAM height-dependent lapse rate here].

Further, the polar tropopause during winter is elevated compared to midlatitudes in the SRE solution, especially in southern polar winter where the tropopause attains altitudes as high as in the tropics. This is consistent with the largest differences in static stability between the SRE solution and CMAM in polar winter (Fig. 6, top), when the missing heating in the SRE solution due to solar radiation and dynamics causes temperatures to continue to decrease throughout the stratosphere (see also Zängl 2002). A strong vertical gradient in static stability, that is, a tropopause, still exists but is located much higher than in CMAM. Note that by definition the tropopause, as defined in this study, in general coincides with the level of maximum gradient in

The difference in tropopause height between CMAM and the SRE solution is further quantified in Fig. 7 (solid lines). Throughout the tropics the introduction of stratospheric dynamics causes an elevation of the tropopause by about 3–4 km compared to a stratosphere in radiative equilibrium. In the extratropics the change in tropopause height is not as dramatic but still ranges between 1 and 2 km, with somewhat larger values during winter, especially in the polar regions as discussed above. Overall, stratospheric dynamics roughly lead to a more than doubled equator-to-pole contrast in tropopause height (∼3 km in the SRE solution compared to ∼8 km in CMAM or ERA-40).

In the heat budget (2) it was assumed that most of the stratospheric dynamical heating is due to the residual circulation by neglecting cross-isentropic eddy heat flux contributions. To test this assumption, equilibrium solutions of Eq. (2) have been obtained for the stratosphere using CMAM’s residual velocities and clear-sky radiation, as before from the CRM, in place of *Q*. As in the case of the SRE solution, the troposphere was quasi fixed in these stratospheric circulation–radiation solutions (hereafter SCR solutions). The dashed lines in Fig. 7 show the difference in tropopause height between the SCR solution and the SRE solution, which qualitatively resembles the difference between CMAM and the SRE solution (full lines). Large differences generally occur in the subtropics where the meridional gradient in tropopause height is large. Further, the circulation-induced elevation of the tropical tropopause falls short of the total amount in CMAM, which includes all dynamical (and diabatic) contributions. Larger discrepancies also occur near the South Pole. Nevertheless, overall the SCR solution captures the main modification of tropopause height [and static stability; see Fig. 6 (bottom) and next section] that comes about due to stratospheric dynamics.

## 5. Effect of stratospheric dynamics on lower stratospheric static stability

### a. General structure

Figure 6 (top) shows the static stability structure of the SRE solution. The tropical stratosphere in the SRE solution exhibits smaller static stability compared to CMAM; in particular, the lower stratospheric static stability maximum (corresponding to the tropical TIL) is not as pronounced in the SRE solution, suggesting that dynamics play a significant role in causing this maximum. The midlatitudinal static stability maximum (TIL) almost disappears in the SRE solution during winter, especially in the Northern Hemisphere. The summer TIL in the extratropics, on the other hand, remains intact compared to CMAM, suggesting that radiation plays a dominant role in the formation of the TIL there. Lower stratospheric static stability is generally larger in the extratropics of the SRE solution compared to CMAM (outside the winter polar regions).

Figure 6 (bottom) shows the static stability structure of the SCR solution, which qualitatively resembles the CMAM static stability structure. The strongest difference from the SRE solution is the existence of a strong static stability maximum (TIL) in midlatitudes during winter in the SCR solution. This suggests that stratospheric dynamics, as represented by the residual circulation, represents the main cause of the TIL in midlatitudes during winter, at least in CMAM. Closer inspection of the strength of the static stability maximum in the SCR solution reveals that it is stronger than in CMAM. This is largely due to missing vertical diffusion (both eddy and numerical) in the SCR solution, which plays an important role in limiting static stability around the tropopause in CMAM (not shown). The tropical TIL is much more pronounced in the SCR solution than in the SRE solution, suggesting that the residual circulation represents a main forcing for the TIL there as well.

### b. Extratropical tropopause inversion layer

As discussed above, the residual circulation represents a strong positive forcing of static stability, in particular in the lowermost extratropical stratosphere. We will therefore now discuss this feature in more detail.

#### 1) Vertical structure of residual circulation

The residual circulation near the extratropical tropopause in CMAM exhibits a characteristic vertical structure (Fig. 1): in particular, residual vertical velocities undergo a rather strong transition from near-zero values in the extratropical upper troposphere to negative values in the extratropical lower stratosphere. The above behavior is much more pronounced in winter than in summer (cf. Fig. 1), reflecting mainly the seasonal contrast in the stratospheric part of the residual circulation.

*w*

_{z}

_{max}= 0 (one might also more generally interpret this expression as the tendency of an initial value problem starting with constant stratospheric

_{z}

*w*

_{z}

*w*

_{z}

*w*

_{z}

*w*

*w*

*w*

_{z}

*w*

*w*

#### 2) Annual cycle

The strength of the stratospheric residual circulation varies strongly with season. The strength of the dynamical forcing term _{z}*w*_{max} at northern midlatitudes together with the annual cycles of the maximum of the zonal mean *N* ^{2} (i.e., *N* ^{2} for CMAM and ERA-40. The forcing, approximately −∂_{z}*w*_{max}, maximizes in December (CMAM) and January (ERA-40), whereas _{z}*w*_{max} and _{z}*w*_{max}) ≈ 0.014 day^{−1} ≈ (70 days)^{−1}]. The 2–4-month offset with −∂_{z}*w*_{max} leading *N* ^{2} follow a somewhat different annual cycle from

The same relationship between the annual cycles of −∂_{z}*w*_{max} and _{z}*w*_{max} still provides a positive forcing during winter but is less than half as strong as in midlatitudes. Furthermore, −∂_{z}*w*_{max} is around zero (negative for CMAM) during summer when

#### 3) Simple arguments

*Q*tends to be dominated by radiative heating, which is often approximated by simple Newtonian damping:

*τ*

_{rad}represents the radiative damping time scale (typically ∼30–60 days) and Θ

_{rad}represents the (hypothetical) radiative equilibrium potential temperature. The corresponding diabatic forcing term in the static stability budget (3) then becomes

_{rad})/Θ

_{rad}≪ 1 and

*τ*

_{rad}independent of altitude yields the simple radiative static stability forcing

*τ*

_{dyn}= (−∂

_{z}

*w*

_{max})

^{−1}(

*τ*

_{dyn}has units of time but, strictly speaking, is not equal to the dynamical time scale of the problem that in steady-state balance would equal

*τ*

_{rad}). For constant

*τ*

_{dyn}(say, corresponding to the annual mean value) this yields a time scale of adjustment to equilibrium of

*τ*

_{equ}=

*τ*

_{rad}(1 −

*τ*

_{rad}/

*τ*

_{dyn})

^{−1}and an equilibrium solution of

*τ*

_{dyn}∼ 120 days for CMAM (northern midlatitudes, see Fig. 9); that is,

*τ*

_{dyn}/

*τ*

_{rad}∼ 2–4 using

*τ*

_{rad}∼ 30–60 days. In this case an equilibration time scale of

*τ*

_{dyn}) than in CMAM, which would lead to a larger

For northern midlatitudinal winter (DJF) *τ*_{dyn} ∼ 80 days; that is, *τ*_{rad} ∼ 30–60 days). This yields a range for the equilibration time scale of *τ*_{equ} ∼ 50–240 days, whose lower half is consistent with the 60–120-day offset of _{z}*w*_{max} (Fig. 9). In this case, ^{−4} s^{−2} during March–April) but tends to predict too large

In the polar regions radiation seems to represent the dominant cause of the TIL, especially during summer (cf. Figs. 1 and 6). During winter dynamical forcing might play a role: *τ*_{dyn} ∼ 200 days; that is, *τ*_{dyn}/*τ*_{rad} ∼ 10/3 − 20/3. Using

The SCR solution discussed above and shown in Fig. 6 represents an equilibrium between the forcing due to the residual circulation and clear-sky radiation. A similar solution can also be obtained by replacing the radiative forcing in Eq. (2) by Newtonian damping, Eq. (5), using a given radiative time scale *τ*_{rad} and (hypothetical) radiative equilibrium distribution Θ_{rad}. Figure 11 (top) shows such hypothetical Θ_{rad} distributions along with corresponding *τ*_{rad}). Evidently, and by construction, this hypothetical radiative equilibrium distribution does not contain a TIL.

Initialized with the radiative equilibrium distribution *t* = 0, *φ*, *z*) = Θ_{rad}(*φ*, *z*), Eq. (2) is integrated forward in time applying the Newtonian damping approximation (5) until equilibration (≳100 days). Figure 11 (bottom) shows the resulting distributions of

## 6. Summary and discussion

The effect of large-scale dynamics as represented by the residual mean meridional circulation in a TEM sense, in particular its stratospheric part, on lower stratospheric static stability and tropopause structure has been investigated using a chemistry–climate model, reanalysis data, and simple idealized modeling. Upwelling in the tropics induces cooling and therefore lifts the tropopause, whereas downwelling in the extratropics induces warming that lowers the tropopause (cf. Thuburn and Craig 2000; Wong and Wang 2003). These circulation-induced changes in tropopause height are consistent with corresponding forcing contributions in the static stability budget that constitute a strong localized positive forcing of static stability just above the tropopause, most pronounced in the winter midlatitudes. This strong positive forcing due to the stratospheric residual circulation (i.e., due to stratospheric eddies) causes a local maximum in static stability just above the tropopause, especially in the winter midlatitudes, corresponding to the so-called tropopause inversion layer (TIL). Strong negative static stability forcing due to the residual circulation is diagnosed in the extratropical upper troposphere (i.e., due to tropospheric eddies). This negative forcing is consistent with the tendency of tropospheric eddies to lift the tropopause, opposing the tendency due to the positive lower stratospheric forcing. The resulting dipole forcing structure effectively sharpens the extratropical tropopause. In fact, the extratropical tropopause roughly coincides with zero forcing of static stability due to the residual circulation.

Strong negative static stability forcing due to the residual circulation is also found in the subtropical upper troposphere in a region of strong meridional temperature and static stability gradients, which helps to maintain low static stability and a high and cold tropopause there. Near the subtropical edge of the tropical tropopause this negative static stability forcing overlies positive forcing due to the circulation around the level of the extratropical tropopause. This favors the frequent formation of a double tropopause in the subtropics, especially during winter.

The results summarized above rely on a TEM perspective that assumes cross-isentropic (diabatic) eddy heat fluxes to be negligible in the heat budget; see discussion around Eq. (2). This allows the residual circulation to be interpreted as the diabatic circulation; that is, the dynamical heating rates are given by the advection of

Theories for the height of the tropopause conventionally assume a stratosphere in (local) radiative equilibrium (e.g., Schneider 2007 and references therein). Different tropical and extratropical tropopause heights result from different tropospheric lapse rates and surface temperatures in these theories. To test the impact of stratospheric dynamics on tropopause height, a (hypothetical) stratospheric radiative equilibrium (SRE) solution has been obtained using the CCM distributions of water vapor and ozone and by constraining the troposphere to remain quasi fixed to the CCM troposphere (i.e., a realistic tropospheric lapse rate structure is prescribed, in contrast to conventional radiative–convective equilibrium studies, which typically assume a constant tropospheric lapse rate, that may depend on latitude). The tropopause in this SRE solution is strongly reduced in the tropics (by 3–4 km). The cold point tropopause and the top of convection do not appear to be well separated in the tropical SRE solution (i.e., a TTL does not exist), in contrast to the results in Thuburn and Craig (2002) based on radiative–convective equilibrium solutions that assume a constant tropospheric lapse rate. The reasons for this discrepancy are currently unclear and deserve further investigation. The extratropical tropopause in the SRE solution is located higher than in the CCM (by 1–2 km, around 11–12 km consistent with radiative–convective equilibrium estimates; e.g., Fig. 1b in Thuburn and Craig 2000), resulting in a strongly reduced equator-to-pole contrast (less than half of that of the CCM). Kirk-Davidoff and Lindzen (2000) obtained a similarly strong influence of stratospheric dynamics on the equator-to-pole contrast in tropopause height based on a simple energy balance model with a fixed tropospheric isentropic potential vorticity gradient. In general, the present analysis finds tropopause height modifications due to stratospheric dynamics on the order of the seasonal cycle or longer. On a global mean the mass of the troposphere is reduced in the SRE solution compared to the control run mainly due to the large reduction in tropical tropopause height. When the residual-circulation-induced heating is reintroduced [the stratospheric circulation–radiation (SCR) solution] as the dominant stratospheric dynamical contribution, the tropopause structure of the CCM is qualitatively recovered. Small discrepancies exist in the tropics and in the southern polar region.

It is important to note that the above findings do not diminish the role of tropospheric dynamics in setting the height of the tropopause. The equator-to-pole contrast in tropopause height is still on the order of a few kilometers in the SRE solution, which by definition cannot be due to stratospheric dynamics. Moreover, the change in tropopause height when going from a pure radiative equilibrium to a radiative–convective equilibrium is on the order of the tropopause height modifications due to stratospheric dynamics or larger (e.g., Manabe and Wetherald 1967). Furthermore, simple constraints that assume a stratosphere in radiative equilibrium, such as based on tropospheric lapse rate and surface temperature (Thuburn and Craig 1997) or on surface temperature and its meridional gradient (Schneider 2004), may still determine the tropopause height response to external perturbations (as shown by those authors in idealized GCM experiments). To what extent the stratospheric circulation may play a modifying role in these constraints, for example, in determining the tropopause height response to climate change (Son et al. 2009), remains an open question.

Concerning the lower stratospheric static stability structure, it is shown in the present study that the SRE solution does not contain a tropopause inversion layer (TIL) in the winter extratropics and has a much less pronounced tropical TIL than in the control run. The SCR solution, on the other hand, contains a TIL at all latitudes that closely resembles that of the CCM. This confirms the validity of the approximation in the heat budget, Eq. (2), in the present context. Stratospheric dynamics in the form of the residual circulation seem to play a dominant role in forming a TIL except for the polar regions during summer. The vertical structure of the vertical residual velocity is identified as the dominant forcing term [as confirmed by Miyazaki et al. (2010) in a high-resolution GCM simulation]. In midlatitudes, the annual cycle of maximum static stability just above the tropopause _{z}*w*_{max} with a time lag of 2–4 months. Simple arguments for this time lag are provided based on approximating radiative heating rates by simple Newtonian damping. Finally, it is shown that a midlatitudinal TIL can be obtained by replacing radiation in the SCR solution by simple Newtonian damping.

The TIL formation mechanism discussed here is consistent with the one speculated about in Birner et al. (2002) for midlatitudes in that subsidence within the downward branch of the stratospheric residual circulation plays a crucial role. It is furthermore interesting to note that the pivotal forcing term in the static stability budget, −∂_{z}(*w**w**above* the tropopause) the mechanism discussed in Wirth (2004), Wirth and Szabo (2007), and A. R. Erler and V. Wirth (2009, unpublished manuscript) might be part of the mechanism discussed here. More research is required to clarify this.

The results of the present study highlight that, even though the TIL is a global phenomenon, its formation mechanisms involve different processes at different latitudes. In the polar region radiation seems to represent the dominant cause of the TIL, along the lines of Randel et al. (2007b) and Kunz et al. (2009) [although note that recent findings by Grise et al. (2010) suggest a significant stratospheric dynamical impact on interannual TIL variability in the polar regions during winter]. In midlatitudes large-scale dynamics, as represented by the residual circulation, seem dominant. In the tropics both radiation and circulation seem important. Similar balances between radiation and residual circulation are seen in the CMAM and ERA-40 with some minor differences, mainly on a quantitative level. This suggests that mechanisms as found in CMAM carry over to reanalysis data and probably the real atmosphere.

## Acknowledgments

Fruitful discussions with Michaela Hegglin and Ted Shepherd during the early stages of this work are gratefully acknowledged. James Anstey and Stephen Beagley provided technical help with the CMAM runs. Thanks to the thoughtful comments and constructive criticism by V. Wirth and two anonymous reviewers, as well as to comments by Anne Kunz, Paul Konopka, and Kevin Grise on an earlier version of the manuscript. TB’s funding at the University of Toronto came through the Natural Sciences and Engineering Research Council and the Canadian Foundation for Climate and Atmospheric Sciences. Free access to ERA-40 data, originally generated by ECMWF, was provided through the National Center for Atmospheric Research.

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## APPENDIX A

### Tropospheric Setup for Stratospheric Radiative Equilibrium Calculation

*z*

_{0}(typically located around 500 hPa in the extratropics and around 300 hPa in the tropics). Then, an idealized profile of the vertical temperature gradient Γ(

*z*) = ∂

*between*

_{z}T*z*

_{0}and the tropopause is computed based on the assumption that the functional form of the mid to upper tropospheric profile of Γ(

*z*) can be approximated by a second-order polynomial:

*a*(

_{i}*i*= 0, 1, 2) are obtained through the boundary conditions Γ̃(

*z*

_{0}) = Γ(

*z*

_{0}) and Γ̃(

*z*

_{TP}) = Γ(

*z*

_{TP}), and the condition that Γ̃(

*z*

_{0}) represents a minimum (i.e., 0 = 2

*a*

_{2}

*z*

_{0}+

*a*

_{1}). The tropopause

*z*

_{TP}as used here is obtained by a two-step calculation. First, an interim tropopause is calculated as the level of maximum curvature in the temperature profile. The average lower stratospheric temperature gradient Γ

_{str}is then computed (between the interim tropopause and ∼25 km) and the final instantaneous tropopause is determined as the level above which Γ first exceeds Γ

_{str}. This modified tropopause proved to represent a more robust estimate for the purpose of adjusting the tropospheric temperature profile at each time step. Finally, the temperature profile above

*z*

_{0}is adjusted such that Γ does not exceed tilde Γ at any level. Figure A1 shows the resulting profiles of Γ over the tropics and northern midlatitudes for the SRE solution and CMAM (for DJF only). Evidently, the functional form of Γ as obtained from the SRE solution closely resembles the one from CMAM.

## APPENDIX B

### Idealized Functional Relationships for Θ_{rad} and τ_{rad}

Here Θ_{rad} is specified through a corresponding idealized distribution of _{rad}(*φ*, *z* = *z*_{BC}) = *φ*, *z* = *z*_{BC}), where *z*_{BC} is arbitrarily set to the model level just below 5 km, which represents a compromise between lying well below the tropopause but still above ground everywhere in the model.

*τ*

_{rad}are specified as idealized steplike functions of the general form

*z*

_{TP}(

*φ*) is the tropopause height taken from the model and Δ

_{TP}is the thickness of the transition from the value

*B*−

*A*far below the tropopause to

*A*+

*B*far above the tropopause. Here we set Δ

_{TP}= 1 km, roughly corresponding to the model’s vertical resolution around the tropopause. Then

*A*(

*φ*) and

*B*(

*φ*) determine tropospheric and stratospheric background values (denoted by subscripts

*t*and

*s*, respectively):

*N*

_{t}

^{2}=

*B*−

*A*and

*N*

_{s}

^{2}=

*B*+

*A*in the case of

*τ*

_{t}=

*B*−

*A*and

*τ*

_{s}=

*B*+

*A*in the case of

*τ*

_{rad}. For simplicity the tropospheric background values are set constant:

*N*

_{t}

^{2}= 1 × 10

^{−4}s

^{−2}and

*τ*

_{t}= 5 days (these tropospheric background values are not crucial in this study as the focus is on the region just above the tropopause).

*N*

_{s}^{2}and

*τ*in latitude of the form

_{s}*φ*

_{0s,n}and Δ

*φ*

_{s,n}represent seasonally dependent tropics–extratropics transition latitudes and corresponding widths,

*φ*

_{0s,n}are taken to be the latitudes of maximum ∂

_{φ}z_{TP}in the Northern (n) and Southern (s) Hemisphere, and Δ

*φ*

_{s,n}are arbitrarily set to 10° in the winter hemisphere and 15° in the summer hemisphere:

*C*= 1 × 10

^{−4}s

^{−2}and

*D*= 5 × 10

^{−4}s

^{−2}in the case of

*N*

_{s}^{2}, and

*C*= 10 days and

*D*= 40 days in the case of

*τ*.

_{s}*f*

_{pn}that is 1 outside the polar stratosphere and reduces

*z*

_{pn}= 15 km, Δ

*z*

_{pn}= 1 km, and

*φ*

_{pn}= 60°N for DJF (

*φ*

_{pn}= 60°S for JJA) and Δ

*φ*

_{pn}= 10°.