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  • Birner, T., , D. Sankey, , and T. Shepherd, 2006: The tropopause inversion layer in models and analyses. Geophys. Res. Lett., 33, L14804, doi:10.1029/2006GL026549.

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  • Bush, A. B. G., , and W. R. Peltier, 1994: Tropopause folds and synoptic-scale baroclinic wave life cycles. J. Atmos. Sci., 51, 15811604.

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  • Doms, G., , U. Schättler, , and C. Schraff, cited 2007: A description of the Nonhydrostatic Regional Model COSMO. Consortium for Small-Scale Modelling. [Available online at http://www.cosmo-model.org/content/model/documentation/core/default.htm.]

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  • Hakim, G. J., 2000: Climatology of coherent structures on the extratropical tropopause. Mon. Wea. Rev., 128, 385406.

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  • Son, S.-W., , and L. M. Polvani, 2007: Dynamical formation of an extra-tropical tropopause inversion layer in a relatively simple general circulation model. Geophys. Res. Lett., 34, L17806, doi:10.1029/2007GL030564.

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  • Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, 745 pp.

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  • Wirth, V., 2003: Static stability in the extratropical tropopause region. J. Atmos. Sci., 60, 13951409.

  • Wirth, V., , and T. Szabo, 2007: Sharpness of the extratropical tropopause in baroclinic life cycle experiments. Geophys. Res. Lett., 34, L02809, doi:10.1029/2006GL028369.

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  • View in gallery

    Zonally symmetric balanced initial conditions: zonal wind u (m s−1; gray shading and solid contours, 10 m s−1 spacing) and isentropes (solid lines, 10 K spacing, starting with 275 K in the lower right corner); the thick solid line marks the tropopause. Note that the thermal and the dynamical tropopause coincide in the initial state.

  • View in gallery

    Evolution of min(pMSL) (hPa) for all life cycle experiments A–K (see Table 1). Note the weak intensification of life cycle E and the delayed development of K. The reference experiment (A) is indicated by a thick solid line and the additional sensitivity tests (F–K) are plotted in gray. Crosses were used for the setup of Wirth and Szabo (2007; K), while plus signs indicate our configuration with the same resolution (life cycle H). For purely technical reasons the life cycle experiments D, I, and J terminated earlier. Also note that I and J are barely visible because they almost exactly coincide with life cycle H.

  • View in gallery

    Plots of θ (K; gray shading and contours) on the 2.5-PVU isosurface for the reference experiment A, showing days (top left) 3, (top right) 5, (bottom left) 8, and (bottom right) 12; see text for further details. For ease of viewing the domain has been doubled in the zonal dimension (the solution was only computed between 30°W and 30°E). The part of the domain used for domain averages is enclosed by thick solid lines.

  • View in gallery

    Plots of (×10−4 s−2; gray shading and contours) at the tropopause for the reference experiment A, on days (top left) 3, (top right) 5, (bottom left) 8, and (bottom right) 12; see description of Fig. 3 for further details.

  • View in gallery

    Tropopause-based zonal average of N2 (×10−4 s−2; gray shading and solid contours) for the reference experiment A, averaged from days 5 to 15. Superimposed are isentropes (thin dashed contours; spacing 10 K, beginning at 295 K), the dynamical tropopause (2.5 PVU, thick dashed line), and the thermal tropopause (WMO criterion, thick solid line). The zonal mean tropopause height has been restored. The solid contours also indicate N2, starting at 1.5 × 10−4 s−2 in the troposphere, and increasing in intervals of 0.5 × 10−4 s−2 up to 6.5 × 10−4 s−2 in the TIL.

  • View in gallery

    Tropopause-based (left) N2 profiles and (right) temperature profiles for life cycle experiments A–E, averaged over the whole domain and from days 5 to 15. The horizontal lines (light gray) indicate the average tropopause level, as determined by the WMO criterion. The initial profile (solid gray line) was added for comparison. Line styles are as in Fig. 2.

  • View in gallery

    Sensitivity experiments with different (left) life cycle configurations (F–K) and (right) tropopause definitions. All profiles were averaged as in Fig. 6. The initial profile and the reference experiment (life cycle A) have been added for comparison (solid gray lines). As indicated in the legend, some of the profiles have been shifted vertically. The N2 profiles in the right panel were averaged with respect to different dynamical and thermal tropopause definitions (only life cycle A); the PV and lapse rate threshold values are 2.5 and 3.5 PVU, and 1.4 and 2.6 K km−1, respectively. The right panel also shows an N2 profile obtained from domain-averaged temperature and pressure fields (mean-flow profile, thick solid line).

  • View in gallery

    (left) The N2 profiles for life cycle A averaged over shorter time intervals: one before wave breaking (days 1–3) and three after (days 5–7, 9–11, and 13–15); otherwise averaged as in Fig. 6. The profiles have been shifted vertically in chronological order. (right) The N2 profiles binned according to hTP and ζTP, as in Birner (2006) and Randel et al. (2007), respectively; averages are taken within each bin but otherwise as in Fig. 6. Note that the fraction of profiles with low tropopause (hTP < 9 km) is fairly small (~10%).

  • View in gallery

    (left) Plots of as a function of ζTP at the (thermal) tropopause. The distribution function was averaged over days 5–15. See text for details. (right) Domain-averaged time series of , for life cycles A–E.

  • View in gallery

    Hovmöller diagrams of zonally averaged (top) ζTP (s−1) and (bottom) (×10−4 s−2) at the thermal tropopause; each in gray shading and solid contours (the contour levels correspond to the values indicated at the color bar). Both panels show the reference experiment (life cycle A). The thick solid contour in the top panel indicates the zero vorticity contour. The wave breaking event is clearly visible at day 5.

  • View in gallery

    (left) Estimates for the PDF of ζTP at the (thermal) tropopause; the distribution function was averaged over days 5–15. The ordinate is the probability density per vorticity [(×10−4 s−1)−1]. The distribution functions were interpolated from normalized histograms of very high resolution. (right) Domain-averaged time series of , which is the width of the left-hand (anticyclonic) side of the distribution of ζTP, measured at a probability density of 0.1 × 104 s.

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The Static Stability of the Tropopause Region in Adiabatic Baroclinic Life Cycle Experiments

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  • 1 Institute for Atmospheric Physics, University of Mainz, Mainz, Germany
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Abstract

The tropopause inversion layer (TIL) is a region of enhanced static stability just above the WMO-defined thermal tropopause. It is a ubiquitous feature in midlatitudes and is well characterized by observations. However, it is still lacking a satisfactory theoretical explanation.

This study utilizes adiabatic baroclinic life cycle experiments to investigate dynamical mechanisms that lead to TIL formation. As the baroclinic wave grows, a strong TIL forms above anticyclonic anomalies, while no TIL is found above cyclonic anomalies; this is consistent with previous results. However, during the early growth phase there is no TIL in the global or zonal average: positive and negative anomalies cancel out exactly. The zonal and global mean TIL only emerges during the mature stage of the life cycle, after the onset of wave breaking. The TIL predominantly occurs equatorward of the jet and the vertical structure bears resemblance to the TIL in midlatitudes; there is no equivalent to the subpolar TIL. Life cycles without significant wave breaking develop neither a global nor a zonal mean TIL. No global mean TIL is found in any life cycle if the dynamical tropopause definition is used.

In addition, a new mechanism of dynamical TIL formation is presented, suggesting that the TIL in the global and zonal mean is linked to a strongly skewed distribution of relative vorticity after wave breaking.

Current affiliation: Department of Physics, University of Toronto, Toronto, Ontario, Canada.

Corresponding author address: Andre R. Erler, Department of Physics, University of Toronto, 60 St. George St., Toronto ON M5S 1A7, Canada. E-mail: aerler@atmosp.physics.utoronto.ca

Abstract

The tropopause inversion layer (TIL) is a region of enhanced static stability just above the WMO-defined thermal tropopause. It is a ubiquitous feature in midlatitudes and is well characterized by observations. However, it is still lacking a satisfactory theoretical explanation.

This study utilizes adiabatic baroclinic life cycle experiments to investigate dynamical mechanisms that lead to TIL formation. As the baroclinic wave grows, a strong TIL forms above anticyclonic anomalies, while no TIL is found above cyclonic anomalies; this is consistent with previous results. However, during the early growth phase there is no TIL in the global or zonal average: positive and negative anomalies cancel out exactly. The zonal and global mean TIL only emerges during the mature stage of the life cycle, after the onset of wave breaking. The TIL predominantly occurs equatorward of the jet and the vertical structure bears resemblance to the TIL in midlatitudes; there is no equivalent to the subpolar TIL. Life cycles without significant wave breaking develop neither a global nor a zonal mean TIL. No global mean TIL is found in any life cycle if the dynamical tropopause definition is used.

In addition, a new mechanism of dynamical TIL formation is presented, suggesting that the TIL in the global and zonal mean is linked to a strongly skewed distribution of relative vorticity after wave breaking.

Current affiliation: Department of Physics, University of Toronto, Toronto, Ontario, Canada.

Corresponding author address: Andre R. Erler, Department of Physics, University of Toronto, 60 St. George St., Toronto ON M5S 1A7, Canada. E-mail: aerler@atmosp.physics.utoronto.ca

1. Introduction

The tropopause inversion layer (TIL) is a feature of the extratropical tropopause region, which only recently came into the focus of active research (Birner et al. 2002). It is a layer of greatly enhanced static stability just above the tropopause, approximately 1 km deep in midlatitudes and more than 2 km deep in subpolar regions (Birner 2006). Right at the tropopause, the static stability increases almost discontinuously and then gradually decreases upward toward typical lower-stratospheric values.

The TIL was first studied in high-resolution radiosonde profiles by Birner et al. (2002). They used a tropopause-based averaging technique, where the local World Meteorological Organization (WMO)–defined thermal tropopause (rather than the sea level) is used as the reference level of the vertical coordinate. Later, Randel et al. (2007) showed that the TIL is a ubiquitous climatological feature of the extratropical tropopause using GPS radio occultation measurements. Using radiative transfer calculations, they also showed that differential heating from strong gradients in trace gas concentrations can result in a considerable temperature inversion at the tropopause. Recent studies suggest a more general association of the TIL with the extratropical tropopause transition layer (Hegglin et al. 2009), which is characterized by a transition in chemical constituents from stratospheric to tropospheric trace gas mixing ratios (Hoor et al. 2004).

The strength of the TIL is correlated with tropopause height (Birner 2006) and anticyclonic vorticity (Randel et al. 2007). Both are consistent with the notion that balanced dynamics plays an important role (Wirth 2003). The formation of a TIL was also observed in idealized modeling studies such as those of Wirth and Szabo (2007) and Son and Polvani (2007). The former used a similar approach as presented in this paper (see below); the latter employed a simple idealized general circulation model (using Newtonian relaxation to a simple reference profile) and found a pronounced TIL in the forced-dissipative statistical equilibrium state of their model.

The importance of balanced dynamics for TIL formation was first proposed by Wirth (2003). He analyzed stationary axisymmetric vortices obtained from a fully nonlinear potential vorticity (PV) inversion. In his simulations the tropopause above anticyclonic vortices is sharp with a strong inversion just above, while the tropopause above cyclonic vortices is relatively weak and indistinct with no signs of an inversion. These results are broadly consistent with the observed synoptic-scale variability of the TIL, although the observations show a weak TIL even in cyclonic regions. He also showed that cyclonic and anticyclonic anomalies do not act as “mirror images,” so that the TIL may partly be a net effect from averaging over synoptic scale vorticity anomalies of different sign.

In this work we study the net effect of more realistic ensembles of anomalies. In contrast to Wirth (2003), our ensembles are dynamically consistent with realistic statistical distributions of different vorticity anomalies with regard to their size and amplitude. The importance of realistic statistics can be gleaned from the work of Hakim et al. (2002), who considered decaying turbulence in an idealized two-dimensional model of tropopause dynamics. The model is based on an extended formulation of surface quasigeostrophy, which is second order in Rossby number.1 In first-order quasigeostrophic dynamics, positive and negative anomalies are perfectly symmetric, so that the statistical distribution of cyclones and anticyclones is identical. In the second-order model of Hakim et al. (2002), however, a pronounced asymmetry between cyclones and anticyclones emerges. While anticyclonic anomalies exhibited a spatially extended and diffuse structure, cyclones were generally smaller and more localized, clustering around a preferred horizontal length scale. Even this enhanced model represents a high degree of abstraction from the real tropopause, but it suggests that the surface area of the tropopause is dominated by large anticyclonic vortices, so that the TIL associated with anticyclonic vortices dominates in domain-averaged profiles.

To obtain dynamically consistent flow statistics, a modeling approach of intermediate complexity is used in this study. We consider adiabatic baroclinic life cycle experiments following the work of Thorncroft et al. (1993). In contrast to the idealized study of Wirth (2003), this includes full time dependence, as well as dynamical consistency. At the same time this approach allows us to focus on the contributions of conservative dynamics to TIL formation.

The approach of using adiabatic baroclinic life cycle experiments to study the TIL was pioneered by Wirth and Szabo (2007), and indeed they found a weak TIL in their simulations. Our experiments are similar to theirs, even though there are important improvements regarding the model setup (see section 2). In addition, we provide a significantly expanded analysis, placing particular emphasis on the temporal evolution of the TIL and the significance of wave breaking. We shall specifically consider domain-averaged profiles because these can most easily be compared to the idealized but fundamental work of Wirth (2003) and Hakim et al. (2002).

The material of this paper is organized as follows. In section 2 we give a brief introduction to the model and the baroclinic life cycle setup used in this study. The meridional and vertical structure of the TIL obtained in our experiments is described in section 3, along with a comparison with previous studies. An analysis of the evolution of the TIL in our model and an interpretation in terms of fundamental dynamical concepts follows in section 4. We conclude with a summary in section 5. The appendix contains details on our numerical methods, formulas, and mathematical derivations.

2. Baroclinic life cycle experiments

For our simulations we used the Consortium for Small-Scale Modeling (COSMO) model, which is a fully compressible nonhydrostatic regional weather prediction model (Doms et al. 2007). Only the dynamical core was used, with all physical parameterization removed (including radiation, moist processes, convection, and turbulence), so that the dynamics are adiabatic and frictionless except for a fourth-order horizontal diffusion and boundary effects.

The model was configured in channel geometry, with the channel aligned along constant latitude circles and ranging in latitude from 20° to 80°N. While the domain is periodic in the zonal direction, the vertical and meridional boundary conditions are fixed. The zonal domain width was chosen to match the zonal wavenumber under consideration; if not otherwise noted, it is 60° longitude corresponding to zonal wavenumber k = 6. Our highest-resolution experiments have a grid spacing of 0.2° in the horizontal and a vertical grid spacing of 65 m in the tropopause region.

We conducted several different life cycle experiments, varying the vertical and horizontal resolution, the zonal wavenumber, the barotropic shear, and the domain geometry. The relevant parameters are given in Table 1. All experiments were initialized with a zonally symmetric state with piecewise constant static stability in the troposphere and stratosphere and a corresponding wind field in thermal wind balance. Figure 1 shows the zonal wind (gray shading and solid contours) and isentropes (dashed contours) of the initial state; the initial tropopause height (thick solid line) and the surface potential temperature were prescribed with a smooth meridional gradient, such that the initial state is baroclinically unstable. In each case the evolution was triggered by superimposing a small perturbation onto the zonal mean reference state.

Table 1.

Model configurations discussed in this work. Life cycles A–E were used for the primary analysis, while F–K were used for sensitivity tests. The zonal domain length Λ follows from k: Λ = 360°/k; an exception is life cycle J, which uses a rotated pole axis and is centered around the coordinate equator. Following Wirth and Szabo (2007), Λ was multiplied by a metric factor to yield the same size as a midlatitude channel: ΛK = (360°/k) cos(45°).

Table 1.
Fig. 1.
Fig. 1.

Zonally symmetric balanced initial conditions: zonal wind u (m s−1; gray shading and solid contours, 10 m s−1 spacing) and isentropes (solid lines, 10 K spacing, starting with 275 K in the lower right corner); the thick solid line marks the tropopause. Note that the thermal and the dynamical tropopause coincide in the initial state.

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

The evolution of minimum surface pressure min(pMSL) for all life cycle experiments is plotted in Fig. 2. In all life cycle experiments except E and J wave breaking occurs between days 4 and 5; it is associated with a sharp drop in minimum surface pressure.

Fig. 2.
Fig. 2.

Evolution of min(pMSL) (hPa) for all life cycle experiments A–K (see Table 1). Note the weak intensification of life cycle E and the delayed development of K. The reference experiment (A) is indicated by a thick solid line and the additional sensitivity tests (F–K) are plotted in gray. Crosses were used for the setup of Wirth and Szabo (2007; K), while plus signs indicate our configuration with the same resolution (life cycle H). For purely technical reasons the life cycle experiments D, I, and J terminated earlier. Also note that I and J are barely visible because they almost exactly coincide with life cycle H.

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

We use life cycle experiment A (Table 1), which has the highest resolution, as our reference experiment. Figure 3 displays snapshots of its evolution at days 3, 5, 8, and 12; in order to illustrate the synoptic situation in the tropopause region, we show potential temperature θ at the dynamical tropopause, which is defined here as the 2.5-potential vorticity unit (PVU) isosurface.2 A high dynamical tropopause is generally associated with anticyclonic circulation, while a low dynamical tropopause is indicative of cyclonic circulation (Wirth 2001); also note that the thermal tropopause in cyclonic regions is higher than the dynamical tropopause.

Fig. 3.
Fig. 3.

Plots of θ (K; gray shading and contours) on the 2.5-PVU isosurface for the reference experiment A, showing days (top left) 3, (top right) 5, (bottom left) 8, and (bottom right) 12; see text for further details. For ease of viewing the domain has been doubled in the zonal dimension (the solution was only computed between 30°W and 30°E). The part of the domain used for domain averages is enclosed by thick solid lines.

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

Thorncroft et al. (1993) discuss the synoptic evolution of baroclinic life cycles in terms of two paradigmatic cases: LC1, the anticyclonic case, and LC2, the cyclonic case. The initial conditions shown in Fig. 1 result in LC1; its evolution is initially dominated by a “θ streamer” (analogous to a PV streamer on an isentropic surface) oriented from southwest to northeast (Fig. 3, day 5), which is associated with subsequent wave breaking. Later, a cutoff cyclone forms and migrates into the southern part of the domain (day 8). It is interesting to note that in our high-resolution simulations (life cycles A and B) the cutoff cyclone rejoins with the cyclonic reservoir to the north. This behavior was not reported by Thorncroft et al. (1993), nor does it occur in our low-resolution experiments (G–J, in Table 1). In the reference experiment (A) this occurs between days 10 and 12 and is associated with a secondary (cyclonic) wave breaking at day 12 [the second drop in min(pMSL); Fig. 2]. The small cyclone near 30°N at day 12 in Fig. 3 is a spinoff from the rejoining of the original cutoff cyclone; it is not directly associated with a secondary wave breaking. The k = 4 life cycle (life cycle D in Table 1; LC1 type) follows a very similar evolution as the k = 6 LC1-type life cycles.

The LC2-type life cycle evolution (not shown) was obtained by superimposing barotropic cyclonic shear on the initial conditions (cf. Thorncroft et al. 1993); only life cycle C is of that configuration (cf. Table 1). In this case no cutoff cyclone and no PV streamers form, but instead the evolution is dominated by a large cyclonic trough in the domain center; the trough axis is aligned from northwest to southeast, perpendicular to the orientation of the streamer in LC1 (cf. Thorncroft et al. 1993).

The k = 9 life cycle (E) differs substantially in that no significant wave breaking occurs; this can be understood in terms of the high wavenumber cutoff (e.g., Vallis 2006, chapter 6.6). Also note that all life cycle experiments (except E) exhibit strong tropopause folds similar to those reported by Bush and Peltier (1994).

Wirth and Szabo (2007) used an earlier version of the same model and a similar initial state. However, while in our configuration the coordinate pole coincides with the actual rotational pole, Wirth and Szabo (2007) chose to rotate the coordinate pole by 45° in order to take advantage of the more regular grid spacing near the equator of the coordinate system. The tradeoff is that in this case the channel is no longer parallel to constant latitude circles (of the physical system), which introduces a variation of the Coriolis parameter with longitude (i.e., the zonal symmetry is lost).3

The setup of life cycle K is of the type used by Wirth and Szabo (2007). We find that the growth rate of baroclinic waves in this setup is significantly smaller and wave breaking does not occur until days 5–7 (cf. Fig. 2, dotted line with crosses).

3. The tropopause inversion layer

In this section we describe the structure of the TIL in our simulations, report results from sensitivity studies, and discuss the agreement and differences with observations. Unless noted otherwise, all data were averaged with the thermal tropopause as the common reference level (cf. Birner et al. 2002). Domain averages and averaged vertical profiles are representative for the latitudinal range from 30° to 70°N and for the entire zonal dimension. The thermal tropopause is defined as the lowest level at which the lapse rate γ = −∂T/∂z falls below 2 K km−1, and the average lapse rate between this level and all higher levels within 2 km does not exceed 2 K km−1 (WMO 1957). We essentially follow the interpolation method described by Zängl and Hoinka (2001) to determine the tropopause height in gridded vertical profiles.4

As the measure for static stability we use the square of the Brunt–Väisälä frequency N. It is defined as
e1
where g is the gravitational acceleration and is potential temperature (with pressure p, reference pressure p0, the gas constant for dry air Rd, and the specific heat capacity at constant pressure cp).

To quantify TIL strength in N2 profiles, Birner et al. (2006) introduced ; it is defined as the maximum value of N2 occurring in the vertical column extending from the local tropopause up to 1.5 km above the tropopause. In this work we will also use as the primary measure of TIL strength.

Figure 4 shows snapshots of at days 3, 5, 8, and 12 (life cycle A). The horizontal structure of the TIL largely reflects the synoptic evolution shown in Fig. 3: a strong TIL forms in regions where the dynamical tropopause is high, but there is no significant TIL when the dynamical tropopause is low. Furthermore the TIL appears to be strongest just next to tropopause breaks, before it falls off sharply. This picture holds for all life cycle configurations that lead to significant wave breaking. It is also evident that the TIL evolves with time; this will be discussed in section 4.

Fig. 4.
Fig. 4.

Plots of (×10−4 s−2; gray shading and contours) at the tropopause for the reference experiment A, on days (top left) 3, (top right) 5, (bottom left) 8, and (bottom right) 12; see description of Fig. 3 for further details.

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

a. The characteristics of the TIL in adiabatic baroclinic life cycles

1) The meridional structure

The tropopause inversion layer is often displayed in the form of zonal mean N2 (Birner 2006; Son and Polvani 2007; Randel et al. 2007). Figure 5 shows the zonal mean tropopause inversion layer obtained from our reference experiment, averaged from days 5 to 15.

Fig. 5.
Fig. 5.

Tropopause-based zonal average of N2 (×10−4 s−2; gray shading and solid contours) for the reference experiment A, averaged from days 5 to 15. Superimposed are isentropes (thin dashed contours; spacing 10 K, beginning at 295 K), the dynamical tropopause (2.5 PVU, thick dashed line), and the thermal tropopause (WMO criterion, thick solid line). The zonal mean tropopause height has been restored. The solid contours also indicate N2, starting at 1.5 × 10−4 s−2 in the troposphere, and increasing in intervals of 0.5 × 10−4 s−2 up to 6.5 × 10−4 s−2 in the TIL.

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

The structure of the tropopause in Fig. 5 can be compared to the initial state displayed in Fig. 1: between 40° and 60°N the tropopause height is increased and the slope of the tropopause steepens (note the different scaling of the ordinate). The point of the steepest tropopause slope moves poleward from 50° to about 60°N and coincides with the jet maximum at all times. Note that locally the tropopause slope is even steeper, but the jet meanders, so that the structure is smoothed out in the zonal mean (cf. Fig. 3). In the region of the jet the thermal (solid line) and the dynamical (dashed line) tropopause differ significantly, while they almost coincide equatorward of the jet, where the tropopause is high and the circulation is predominantly anticyclonic. In the region of the jet the TIL becomes weaker and is almost indiscernible on the poleward side, where the circulation is predominantly cyclonic.

Only data from the reference experiment (life cycle A) are shown in Fig. 5. The other LC1-type experiments of k = 6 and k = 4 (life cycles B, D, and F–K) exhibit essentially the same structure, although the TIL becomes weaker with lower resolution. The meridional structure of the LC2-type experiments, on the other hand, is quite different: here a characteristic large, central vortex forms, which results in a lower degree of zonal symmetry, so that the tropopause break at the (strongly meandering) jet is smoothed out in the zonal mean, and the transition between midlatitudes and subpolar latitudes appears more gradual (not shown). There is also a small region poleward of the jet where the dynamical and the thermal tropopause coincide and a TIL is present; this is much more pronounced in the LC2-type simulation. Although the meridional structure is different, the correlation of TIL strength with anticyclonic vorticity and high tropopause height also holds in LC2.

2) The vertical structure

Figure 6 shows domain averaged (i.e., zonally and meridionally averaged) vertical profiles of N2 (Fig. 6, left) and temperature (Fig. 6, right) for life cycle experiments A–E, as well as the initial profile for reference. Evidently, a significant TIL forms even in the domain average. In the reference experiment the N2 enhancement in the TIL is approximately 1 × 10−4 s−2, and in all our experiments the TIL is about 500 m deep. The TIL in the wavenumber k = 4 life cycle (D) is very similar to the TIL in the k = 6 configuration at the same resolution (life cycle H; Fig. 7, left). Only life cycle E, the k = 9 configuration, shows a rather different behavior: the final state is much closer to the initial state than in all other experiments, and there is almost no TIL. Note that the TIL in averaged N2 profiles appears significantly weaker than the values in Fig. 4 would suggest. The TIL in individual profiles tends to be thinner but sharper; however, because of uncertainties in the tropopause height, the N2 maxima are placed at slightly different altitudes above the tropopause (Bell and Geller 2008). As a consequence, individual peaks are smoothed out in averaged profiles. Gravity waves also perturb the static stability profile and may therefore further aggravate this effect.

Fig. 6.
Fig. 6.

Tropopause-based (left) N2 profiles and (right) temperature profiles for life cycle experiments A–E, averaged over the whole domain and from days 5 to 15. The horizontal lines (light gray) indicate the average tropopause level, as determined by the WMO criterion. The initial profile (solid gray line) was added for comparison. Line styles are as in Fig. 2.

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

Fig. 7.
Fig. 7.

Sensitivity experiments with different (left) life cycle configurations (F–K) and (right) tropopause definitions. All profiles were averaged as in Fig. 6. The initial profile and the reference experiment (life cycle A) have been added for comparison (solid gray lines). As indicated in the legend, some of the profiles have been shifted vertically. The N2 profiles in the right panel were averaged with respect to different dynamical and thermal tropopause definitions (only life cycle A); the PV and lapse rate threshold values are 2.5 and 3.5 PVU, and 1.4 and 2.6 K km−1, respectively. The right panel also shows an N2 profile obtained from domain-averaged temperature and pressure fields (mean-flow profile, thick solid line).

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

In the profiles of Fig. 6 the static stability of the upper troposphere and the height of the tropopause are increased significantly, compared to the initial profile (solid gray profile), while the static stability of the lower stratosphere is somewhat decreased; both phenomena are also present in the forced-dissipative simulations of Son and Polvani (2007).

The left panel of Fig. 6 shows the temperature profiles corresponding to the N2 profiles in the right panel of Fig. 6. Apparently, the peak in N2 is associated with an inversion layer in the temperature gradient that extends about 500 m upward into the lower stratosphere. The upper boundary of this layer roughly coincides with the altitude where the N2 peak ends and the profile returns back to normal stratospheric values. The temperature inversion is present in all life cycles except E, but its amplitude is always less than 1 K. This is significantly smaller than in observations (e.g., Randel et al. 2007).

3) The TIL and the temperature inversion

Static stability is related to the vertical gradient of temperature; it is thus not surprising that features related to the TIL appear in both temperature and N2 profiles. In the following we show that temperature anomalies at the tropopause are proportional to the integral of N2 anomalies.

Randel et al. (2007) introduced a measure for TIL strength based on temperature profiles, which we will denote ΔTTP; it is defined as
e2
Here, T(zTP) denotes the tropopause temperature, and T(zTP + Δz) is the temperature at a distance Δz above the tropopause. Note that Δz should be chosen sufficiently high above the tropopause; Randel et al. (2007) use Δz = 2 km. As detailed in appendix B, ΔTTP can approximately be expressed as
e3
where T0 is a typical lower stratospheric temperature, and is the N2 anomaly with respect to an isothermal stratospheric background profile given by . Equation (3) shows that ΔTTP depends on both the amplitude and the thickness of the TIL, whereas only depends on the amplitude of the N2 anomaly, so that the two measures are not interchangeable and measure different physical characteristics. Equation (3) also explains the different location of the N2 peak and the temperature maximum in Fig. 6: the maximum of the N2 anomaly occurs just above the tropopause, where the lapse rate reversal occurs, but the integral in Eq. (3) keeps on growing until the N2 anomaly falls below the stratospheric background value , which happens approximately 500 m above the tropopause. In this sense the small values of ΔTTP in our simulations are consistent with the small vertical depth of the TIL.

b. Sensitivity experiments

We investigated the sensitivity of our results to several aspects of the experimental setup and procedure. Domain averaged N2 profiles are used in this context to assess the effect on the TIL. The results are displayed in Fig. 7. As mentioned before, we varied the domain geometry. In particular, we were able to reproduce the results of Wirth and Szabo (2007), as shown in Fig. 7 (left; life cycle K). The TIL obtained in this experiment is much weaker than in the conventional midlatitude setup, in particular when the enhancement of N2 with respect to stratospheric values is considered. (For comparison, a standard midlatitude life cycle of the same resolution is life cycle H.)

1) Sensitivity to model resolution

To estimate the impact of insufficient model resolution we conducted several life cycle experiments with different horizontal and vertical resolution (cf. Table 1, F–J). The N2 profiles obtained from the sensitivity tests are displayed in Fig. 7 (left); all profiles show a TIL, although the sharpness and amplitude vary. Note that some profiles have been shifted vertically, but the tropopause height is in fact not sensitive to resolution. The TIL in the higher-resolution experiments in Fig. 7 (left) is stronger and significantly sharper than the TIL in the lower-resolution experiments; this suggests that the TIL and the processes that lead to its formation are not well resolved even at our current resolution, and a fully resolved TIL will likely be even sharper (cf. Müller and Wirth 2009).

Wirth and Szabo (2007) reported an increase in TIL sharpness with increasing vertical resolution; however, Son and Polvani (2007) observed the most significant increase with increased horizontal resolution. We suggest that the aspect ratio of the horizontal and vertical resolution plays an important role. Consistent with the estimate of Birner (2006), we empirically found an aspect ratio of approximately Δzϕ ≈ 300 meters per degree to be optimal for the representation of the TIL (cf. the profiles of life cycles G, H, and J in Fig. 7). The life cycle configurations A, D, E, and H of the present study, and the T85/L80 integration of Son and Polvani (2007), as well as the Canadian Middle Atmosphere Model (CMAM; cf. Birner et al. 2006), are all very close to this optimal aspect ratio.

2) Sensitivity to tropopause definition

Two different types of tropopause definitions are commonly used: the thermal (lapse rate based) tropopause and the dynamical (PV based) tropopause. In the case of the dynamical tropopause, there is no universally accepted threshold value for PV. We therefore performed the analysis over a range of commonly used values. The thermal tropopause involves a threshold value on γ = −∂T/∂z, with the standard definition (WMO 1957) using γ = 2 K km−1. Although this definition is widely accepted, we also tested a range of threshold values for the thermal tropopause.

Figure 7 (right) displays selected N2 profiles (for life cycle A) similar to those in Fig. 6 (left), but computed with respect to different tropopause definitions: the dynamical tropopause at 2.5 and 3.5 PVU and the lapse rate–based tropopause with threshold values of γ = 1.4 and 2.6 K km−1. The profiles are representative for threshold values from 2.0 to 4.5 PVU and 1.0 to 3.0 K km−1, respectively.

In the case of the thermal tropopause a lower γ results in a stronger TIL and a higher tropopause, presumably because in regions where the tropopause is less distinct, a lower lapse rate will only be found at higher altitudes. A similar sensitivity study was conducted by Birner (2006; see his appendix A), but the dependence on the lapse rate threshold was found to be relatively weak; the higher sensitivity in our experiments may be due to the coarser resolution of our numerical experiments (cf. Bell and Geller 2008).

Figure 7 (right) also reveals a rather unexpected result: profiles averaged with respect to the dynamical tropopause exhibit no TIL at all in the domain average. This result is consistent with the vertical and meridional structure of the TIL in Figs. 5 and 6: the height of the dynamical tropopause does, on average, not change, while the thermal tropopause rises on average, so the dynamical tropopause is often below the thermal tropopause, but never above.

The reason for the different behavior is related to the adiabatic nature of the simulations: the dynamical tropopause is a material, impenetrable surface, and its dynamical evolution is hence constrained, while the thermal tropopause, which is not a material surface, can undergo irreversible changes without violating conservation properties. This result is relevant for time scales shorter than radiative time scales and may also have implications for the interpretation of profiles averaged with respect to the ozone tropopause (e.g., in Tomikawa et al. 2009), which is more akin to the dynamical tropopause than to the thermal tropopause (Wirth 2000).

3) The effect of averaging

All averaged profiles discussed in this study were first computed gridpoint-wise and then averaged horizontally. Figure 7 (right) also contains one N2 profile for the reference life cycle A, which instead was computed from tropopause-based domain averaged pressure and temperature profiles (thick solid line, labeled “mean flow”). Because of the nonlinearity in the definition of N2 [Eq. (1)], the computation of N2 from p and T profiles and the horizontal averaging formally do not commute. The TIL of the mean flow profile in Fig. 7 (right) is somewhat weaker than in the reference profile but is still relatively close. We conclude that the nonlinearity in N2 is in fact not very significant, and our results are not very sensitive to the sequence of averaging and computing N2.

c. Comparison with observations and modeling studies

In this study we only consider dynamical mechanisms that lead to TIL formation; diabatic effects are absent in our experiments. A limitation associated with the purely adiabatic approach is that we have to solve an initial value problem, and the choice of the initial state can potentially have an effect on the solution. Furthermore, because no restoring mechanisms are present, we are only looking at the dynamical tendencies due to only one baroclinic life cycle; in a forced-dissipative equilibrium state we see the tendencies of multiple baroclinic eddies that are at the same time balanced by diabatic effects (as in Son and Polvani 2007). Observational climatologies already incorporate all these effects.

1) Meridional structure

In Fig. 5 the amplitude of the TIL and the height of the tropopause change very strongly with latitude at the location of the jet. The latitudinal dependence is smoother and more continuous in zonal mean climatologies from observations (Birner 2006; Randel et al. 2007). This is plausible because climatologies are averages over a large number of life cycles of different types and wavenumbers and the jet stream typically shifts and meanders strongly.

Zonally averaged composites of LC1- and LC2-type simulations (not shown) do indeed lead to a smoother tropopause structure that bears more resemblance to the observed meridional profile; the TIL structure also becomes meridionally more homogeneous, but the maximum remains in midlatitudes. We do not find a strong TIL in subpolar regions in any of our experiments, while in observations the TIL appears to be very prominent in this region (Randel et al. 2007). We find much better agreement with observations in midlatitudes, equatorward of the jet, in particular in the winter hemisphere, where we also expect the highest baroclinic wave activity.

The small region of enhanced static stability poleward of the jet (more prominent in LC2) may be caused by an induced residual circulation as proposed by Birner (2010); it is, however, unlikely that the TIL on the equatorward side of the jet (which would be located in the upwelling branch) is related to a residual circulation.

2) Vertical structure

With regard to the vertical structure of the TIL, we find good agreement between our reference experiment and radiosonde profiles in midlatitudes. The main difference is the amplitude of the TIL, which can largely be attributed to the still insufficient resolution; with lower-resolution experiments the agreement deteriorates quickly. Note, however that even though the N2 peak is still significantly smaller than in radiosonde profiles, the amplitude of the temperature inversion ΔTTP is of the same magnitude as in observations (in midlatitudes).

Figure 8a shows N2 profiles averaged over shorter time intervals (note the artificial vertical shift). It is apparent that the strength of the TIL evolves with time to a maximum between days 9 and 11 and subsequently decays (see section 4 for details). The profile averaged over days 9–11 does in fact agree very well with radiosonde profiles from midlatitude stations, and the vertical structure is reproduced very accurately; note in particular the near reversal in the N2 gradient just beneath the tropopause in the high-resolution experiment (A).

Fig. 8.
Fig. 8.

(left) The N2 profiles for life cycle A averaged over shorter time intervals: one before wave breaking (days 1–3) and three after (days 5–7, 9–11, and 13–15); otherwise averaged as in Fig. 6. The profiles have been shifted vertically in chronological order. (right) The N2 profiles binned according to hTP and ζTP, as in Birner (2006) and Randel et al. (2007), respectively; averages are taken within each bin but otherwise as in Fig. 6. Note that the fraction of profiles with low tropopause (hTP < 9 km) is fairly small (~10%).

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

Compared to subpolar radiosonde profiles there are two main differences: the vertical depth of the TIL in our experiments is relatively small and the amplitude ΔTTP of the temperature inversion is an order of magnitude smaller than in observations. The TIL in subpolar (and polar) regions is typically several kilometers thick and the strength of the temperature inversion exceeds 5 K (Birner 2006; Randel and Wu 2010).

These results suggest that conservative dynamics can probably account for a significant part of the midlatitude TIL, but the different structure of the subpolar and polar TIL is probably caused by diabatic effects (such as radiative forcing; cf. Randel et al. 2007) or is associated with a large-scale circulation pattern that is not adequately captured by synoptic-scale dynamics (such as the Brewer–Dobson circulation; cf. Birner 2010).

3) Relationship with tropopause height and vorticity

The correlation of TIL strength with tropopause height and anticyclonic vorticity is well established in observations (Birner 2006; Randel et al. 2007) and also reproduced in our simulations. A comparison of Figs. 3 and 4 qualitatively illustrates this relationship. Figure 8 (right) shows N2 profiles divided into different categories according to tropopause height and vorticity at the tropopause, following Birner (2006) and Randel et al. (2007), respectively.

The qualitative behavior of the profiles divided according to tropopause height hTP agrees well with the corresponding profiles in Birner (2006; Fig. 9, left). Note in particular the reversal in the gradient of N2 just beneath the tropopause, which is also present in the high-TP profile (dashed line in our Fig. 8, right). In the low-TP profile (dashed–dotted line) we also obtain a small TIL, but the vertical structure differs from observations. Note, however, that the low-TP category represents less than 10% of the area in our simulations.

In profiles divided according to relative vorticity at the tropopause ζTP (solid lines with markers) we also find a TIL in cyclonic regions, although it is weaker and the tropopause is lower; this result is consistent with observations (Randel et al. 2007, their Fig. 4b) and the idealized simulations of Son and Polvani (2007, their Fig. 2). It is not consistent with the results of Wirth (2003), who found no TIL above cyclonic PV anomalies. However, the calculations of Wirth (2003) were highly idealized and only considered balanced dynamics. In our experiments we can, for example, not exclude contributions from unbalanced flow components such as gravity waves, which may also influence the temperature and vorticity profile in the tropopause region.

Figure 9 (left) shows average values as a function of ζTP for the life cycle experiments A–E.5 The plot is similar in nature to a histogram, where all profiles are divided into (a very large number of) bins, according to their value of ζTP; the difference is that here we plot the average value in each bin, and not the number of profiles (as in a proper histogram). As expected, the graph shows an inverse relationship between and ζTP: in fact, the relationship is almost linear in a region from ±1 × 10−4 s−1. This is also the region that represents the bulk of the profiles (>90%). There are essentially two points to be made here. First, the slope of as a function of vorticity increases strongly with increasing resolution. This is consistent with numerical experiments on idealized PV inversion (Müller and Wirth 2009). Second, in the high-resolution LC1-type experiments the most extreme cyclonic profiles also have enhanced average values: in the reference experiments in excess of 5 × 10−4 s−2. This result is not consistent with the idealized model of Wirth (2003); a theoretical explanation would either have to invoke a more complex balanced mechanism or unbalanced flow components such as gravity waves. Plougonven and Snyder (2007) investigated the generation of inertia–gravity waves in baroclinic life cycles and found differences between LC1 and LC2 that would be consistent with the hypothesis that enhanced values in cyclonic regions are caused by gravity waves.

Fig. 9.
Fig. 9.

(left) Plots of as a function of ζTP at the (thermal) tropopause. The distribution function was averaged over days 5–15. See text for details. (right) Domain-averaged time series of , for life cycles A–E.

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

4. Evolution of the TIL

In the following we discuss the temporal evolution of the tropopause inversion layer and the importance of wave breaking for the formation of the TIL. We will also present an analysis that points to a possible balanced dynamical mechanism for the formation of a TIL in globally averaged profiles.

a. The significance of wave breaking

The snapshots in Fig. 4 as well as the N2 profiles averaged over different (shorter) time intervals (Fig. 8, left) clearly show a temporal evolution in the TIL. The profile averaged over days 1–3 (i.e., before wave breaking) does not exhibit a TIL at all. The profile averaged over days 5–7 has a TIL, but it is not as sharp as the TIL in the profile averaged over days 9–11; the tropopause height also slightly increases between days 7 and 9 (note the artificial shift). The TIL in the profiles averaged over days 13–15 is again weaker; the TIL seems to decay. Likewise, very high values of first emerge at day 5 in Fig. 4, and the area occupied by high values of increases significantly between day 5 and day 8.

The evolution of displayed in Fig. 9 (right) confirms this picture for life cycles A–C. The formation of the TIL in the domain average coincides well with the wave breaking event at day 5. The secondary wave breaking events around day 10 may be associated with the maxima in but the evidence is not clear. Life cycle D does not have a secondary wave breaking and does not show the strong intensification that peaks around day 10; the low-resolution k = 6 life cycle (H) shows the same behavior as life cycle D. Life cycle E initially forms a slight TIL, but it quickly disappears as the wave decays.

Figure 10 shows Hovmöller diagrams of zonally averaged values of relative vorticity ζTP at the thermal tropopause level (top) together with (bottom). The former (ζTP) is a vertical average in a 1-km-deep column centered at the tropopause. The sudden change in the vorticity field at the tropopause between days 4 and 5 indicates the beginning of the nonlinear phase of the life cycle and the onset of wave breaking, consistent with Figs. 2 and 3. The pattern shown in Fig. 10 is representative for all experiments except life cycle E (k = 9) and life cycle K (the setup of Wirth and Szabo 2007). In the latter case (K), wave breaking occurs about two days delayed, and in the former (E) there is no wave breaking at all; accordingly, in life cycle K the TIL formation is delayed and averages out entirely in life cycle E. Note that the relative vorticity anomalies visible in Fig. 10 (top) before day 4 are associated with the initial jet and do not correspond to PV anomalies.

Fig. 10.
Fig. 10.

Hovmöller diagrams of zonally averaged (top) ζTP (s−1) and (bottom) (×10−4 s−2) at the thermal tropopause; each in gray shading and solid contours (the contour levels correspond to the values indicated at the color bar). Both panels show the reference experiment (life cycle A). The thick solid contour in the top panel indicates the zero vorticity contour. The wave breaking event is clearly visible at day 5.

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

Figure 10 not only illustrates the effect of the wave breaking event but also shows a positive correlation between TIL strength and anticyclonic (i.e., negative) vorticity at the tropopause after the wave breaking event. The correlation only emerges during wave breaking, and there is no signal in the zonal mean before the wave breaking event.

For the time period after day 5, Fig. 10 confirms qualitatively what has already been said in the context of Fig. 9 in section 3. Before the wave breaking event the correlation is also present in individual profiles, but the amplitude of variations is not nearly as strong, and positive and negative anomalies average out entirely in the zonal and global mean.

b. The distribution of vorticity

Up to this point we have only discussed the correlation of TIL strength with vorticity at the tropopause, but not the distribution of vorticity. Except for the appearance of a TIL over some cyclonic anomalies, our results are consistent with Wirth (2003). He found that the strength of the TIL depends on the horizontal scale and the amplitude of the PV anomaly because the partitioning into a wind (vorticity) and a static stability anomaly changes with the scale and aspect ratio (cf. Wirth 2000, appendix A). Although he showed that a TIL can be obtained in composite profiles with equal amplitudes of positive and negative anomalies, it is also well known that the distribution of cyclonic and anticyclonic vortices in the atmosphere is not symmetric (Hakim 2000). This is also consistent with Fig. 10, where positive and negative vorticity anomalies at the tropopause are evidently not distributed symmetrically (note the asymmetric shading scale): cyclonic anomalies appear to be localized and of high amplitude, while anticyclonic anomalies cover more area and have smaller amplitudes. Thus, changes in the distribution of vorticity have to be considered as well in order to understand the relationship between wave breaking and TIL formation.

Hakim et al. (2002) studied this asymmetry in highly idealized numerical experiments. Using a surface quasigeostrophic model of second order in Rossby number, the authors were able to show that a pronounced asymmetry between cyclonic and anticyclonic vortices develops in otherwise isotropic decaying 2D turbulence. The process described by Hakim et al. (2002) resembles the evolution of the vorticity field that we obtain in our experiments. Figure 11 (left) shows an estimate of the probability distribution function (PDF) of ζTP, after the wave breaking event (life cycles A–E); the initial distribution is also shown for reference (gray line). The asymmetry is clearly visible: the median of the distribution is on the anticyclonic side, but the density quickly decreases toward lower vorticity values, indicating that the surface area of the tropopause is dominated by shallow (weak) anticyclones; on the cyclonic side the density is generally lower but does not decrease as quickly toward higher values, consistent with cyclones having higher amplitudes but covering less area. Figure 11 (left) can be compared to Fig. 5b in Hakim et al. (2002): there is qualitative agreement and our results suggest that the results of Hakim et al. (2002) may be applicable to tropopause dynamics in three dimensions.

Fig. 11.
Fig. 11.

(left) Estimates for the PDF of ζTP at the (thermal) tropopause; the distribution function was averaged over days 5–15. The ordinate is the probability density per vorticity [(×10−4 s−1)−1]. The distribution functions were interpolated from normalized histograms of very high resolution. (right) Domain-averaged time series of , which is the width of the left-hand (anticyclonic) side of the distribution of ζTP, measured at a probability density of 0.1 × 104 s.

Citation: Journal of the Atmospheric Sciences 68, 6; 10.1175/2010JAS3694.1

In Fig. 11 (right) the evolution of the distribution width of anticyclonic vorticity at the tropopause is displayed; the measure is defined as the largest negative vorticity value where the distribution curve reaches a vorticity density of 0.1 × 104 s. We only consider the broadening of the anticyclonic side because in our theoretical framework (balanced dynamics), only anticyclonic anomalies are relevant for TIL formation. Figure 11 (right) should be compared with Fig. 9 (right): similar to , increases sharply during wave breaking, but does not form a pronounced peak around days 9–11. Again, only life cycle E, which exhibits no wave breaking, shows virtually no signal. A comparison of life cycle experiments A–D further shows that the relationship between and is not uniform over all life cycles and is likely affected by resolution. In particular, in the higher-resolution life cycles (A, B, and C) the initial increase in is fairly weak compared to the peak emerging two days later.

The fact that the peak in around days 9–11 has no equivalent in the distribution width of may indicate that this second enhancement in TIL strength is not related to the balanced dynamical mechanisms discussed above. More research is needed on this phenomenon; it may be related to the enhanced values found in cyclonic regions (cf. Fig. 9, left).

c. Discussion: A dynamical mechanism for TIL formation

From the evidence presented in the preceding sections, we conclude that it is the wave breaking event that leads to the formation of the TIL (and also to the rise of the thermal tropopause). The fact that life cycle E exhibits neither a TIL nor wave breaking, and that in life cycle K both occur only delayed and significantly weaker, further supports this hypothesis.

Two theoretical studies that have been instrumental in our interpretation of the balanced dynamical contribution to TIL formation: Wirth (2003) provides the fundamental mechanism for the formation of a TIL in the framework of balanced dynamics, while the work of Hakim et al. (2002) sheds light on the distribution of vorticity anomalies at the tropopause. Our experiments provide a link between those highly idealized studies, so that a more comprehensive picture of (balanced) dynamical TIL formation emerges: a disturbance at the tropopause grows as a baroclinic wave, until it breaks; the wave breaking event is associated with irreversible, chaotic mixing of PV, which leads to asymmetric changes in the vorticity field, analogous to Hakim et al. (2002). The result is a distribution of vorticity anomalies that favors large anticyclonic vortices covering most of the surface area at the tropopause level, and deep cyclonic vortices of a significantly smaller horizontal scale. Hence the static stability at the tropopause will be dominated by anticyclonic vortices, and with the results from Wirth (2003) it is straightforward to conclude that a TIL-like structure will emerge in averaged profiles.

The value of the work of Wirth (2003) and Hakim et al. (2002) lies in providing a framework that enabled us to interpret the results obtained from our experiments.

5. Summary and conclusions

The main question that we set out to answer is to what extent the observed TIL can be explained by conservative dynamics, with an emphasis on the contributions of balanced dynamics. To this end we performed baroclinic life cycle experiments in a midlatitude channel setup. The numerical experiments were initialized with a piecewise constant N2 profile with a near discontinuity at the tropopause, but without any TIL. To focus on the dynamical contributions, all diabatic processes were removed.

Similar to Wirth and Szabo (2007), we find that a TIL forms in most of our experiments. However, our TIL is significantly stronger than theirs owing to a different (more realistic) model setup and higher resolution. Although our TIL is rather weak in the jet region and at its northern flank, it is still a distinct feature of globally averaged profiles. Our results also show that the TIL is not simply a residual from averaging over synoptic-scale variability, but that nonlinear interactions during wave breaking are crucial: the appearance of the TIL in the zonal mean coincides with the wave breaking event during the mature stage of the life cycle, and a significant TIL forms in all experiments in which wave breaking occurs; no TIL forms in the experiment that does not exhibit wave breaking. Furthermore, we find that domain averaged profiles averaged with respect to the dynamical (PV based) tropopause as reference level do not exhibit a TIL at all.

Consistent with previous results, we also find that the representation of the TIL depends on the aspect ratio of vertical to horizontal grid spacing: a ratio of approximately 1:400 (300 meters per degree) appears to be optimal.

Our simulations are qualitatively consistent with observations regarding the correlation of TIL strength with anticyclonic vorticity (Randel et al. 2007) and tropopause height (Birner 2006); we even find a TIL in regions where the circulation is cyclonic. We find the best agreement with observations in the winter hemisphere, which is consistent with the higher activity of baroclinic eddies and weaker radiative forcing during that season. The vertical structure agrees well with radiosonde profiles from midlatitudes, and the highest-resolution experiment shows many fine-scale features consistent with observations. The subpolar TIL, on the other hand, is not well represented; this may be related to the absence of diabatic effects.

We suggest a possible mechanism for the dynamical formation of a TIL, drawing on concepts from Wirth (2003) and Hakim et al. (2002). During the wave breaking event, nonlinear interactions lead to a significant asymmetry in the distribution of cyclonic and anticyclonic vorticity anomalies at the tropopause. As a consequence, horizontally extended shallow anticyclones with enhanced static stability above the tropopause dominate the area and a TIL emerges in averaged profiles. Before the onset of wave breaking, the effects of positive and negative anomalies cancel out in the zonal and global mean.

The key result of this study is that conservative dynamics can lead to TIL formation and that wave breaking is crucial in this process. The results also support the case that the balanced flow component contributes significantly to TIL formation, although it may be further enhanced because of gravity wave activity. The design of the study was such as to bridge the gap between the idealized studies of Wirth (2003) and Hakim et al. (2002) and the forced-dissipative simulations of Son and Polvani (2007). From this synthesis, a new interpretation of the underlying dynamical mechanisms emerged. Remaining inconsistencies between our results and observations, in particular in subpolar regions, leave room for other processes (such as radiation) as important additional effects.

Acknowledgments

The authors thank Jörg Trentmann and Markus Tacke for assistance with the technical aspects of running the COSMO model, Tamas Szabo for providing source code of an earlier implementation of the model, Hartmut Borth for helpful discussion and valuable suggestions for the data analysis, Heini Wernli and Seok-Woo Son for interesting discussions, and Theodore G. Shepherd for discussions and helpful comments on the manuscript. Furthermore we thank Thomas Birner and two anonymous reviewers for insightful comments on an earlier version of this manuscript that led to significant improvements.

APPENDIX A

Numerical Methods

The horizontal and the maximum vertical resolution of each life cycle configuration are detailed in Table 1. In the horizontal directions the grid is evenly spaced in spherical coordinates, with the meridional domain boundaries at 20° and 80°N. Only the area between 30° and 70°N was considered for domain averages so as to avoid artifacts from boundaries. The domain extends up to 25 km in the vertical; the vertical grid is spaced unevenly with the highest resolution at an altitude between 6 and 14 km. At the lateral (northern and southern) and the upper boundaries we use Dirichlet boundary conditions with vanishing mass flux across the boundary; a weak relaxation mechanism is employed to damp wave propagation. At the bottom boundary a no-slip condition is used instead of a realistic boundary layer formulation; there is no topography.

We use a fifth-order centered finite-difference approximation in the horizontal and a third-order scheme in the vertical. A third-order Runge–Kutta scheme is used for time stepping. Numerical stability is maintained by a fourth-order numerical diffusion scheme. Fast sound and gravity waves are treated separately in a time-split scheme. We use a fully implicit scheme for the fast component so as to achieve maximum damping of these modes. This reduces the vertical advection scheme in the fast modes to first order, but it improves the long time stability significantly. Note that fourth-order numerical diffusion only acts in the horizontal, so that we solely rely on the advection scheme to stabilize vertical velocities and control wave propagation.

Numerical errors such as numerical diffusion and dispersion depend strongly on resolution in the sense that poorly resolved phenomena are subject to significant errors while well-resolved phenomena do not suffer from such effects (Lomax et al. 2003). The fact that the simulated TIL becomes more pronounced as the resolution is increased provides a strong indication that numerical errors lead, if anything, to an underestimation of the TIL. This is also consistent with the study of Müller and Wirth (2009). Therefore, it is unlikely that the formation of a TIL in our simulations is a numerical artifact.

Wirth and Szabo (2007) used an older version of the COSMO model (2.1) with a leapfrog time-stepping scheme and a different implementation of the boundary conditions. The fact that we can reproduce their results with high accuracy further indicates that our results are not sensitive to the numerical methods (except for resolution).

APPENDIX B

The Relationship between and ΔTTP

The definition of N2 is given in Eq. (1); assuming hydrostatic balance ∂p/∂z = −gp/(RdT) the pressure dependence can be eliminated, leading to
eb1
where Γd = g/cp is the dry adiabatic lapse rate; only T and N2 are functions of the vertical coordinate z. Solving for the temperature gradient ∂T/∂z yields
eb2
The amplitude of the temperature inversion at the tropopause ΔTTP, defined in Eq. (2), can be written in terms of the temperature gradient as
eb3
We decompose temperature into a constant stratospheric background temperature T0 and a perturbation T′(z), leading to
eb4
The temperature in the lower stratosphere is close to constant and is generally on the order of T0 = 220 K; the strength of the temperature inversion, which represents the largest deviation from the background temperature T0, does not exceed 5 K. Thus, with an error of at most 3% and in most cases less than 1%, we can neglect T′(z) and write
eb5
Subtracting the static stability of an idealized isothermal stratospheric profile, we define and rewrite Eq. (B5) as
eb6
The quantity ΔTTP is proportional to the vertical integral of the static stability anomaly ΔN2; in an N2 profile this corresponds to the area enclosed by the N2 peak (the TIL) and the constant stratospheric background (a vertical line).

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1

Juckes (1994) showed that, under the assumption that PV values are piecewise constant in the troposphere and stratosphere, the surface quasigeostrophic (sQG) system can be used to model the dynamics of the tropopause.

2

In the case of multiple dynamical tropopauses (e.g., tropopause folds), we choose the higher one.

3

The usual definition of zonal wavenumber (the number of wave crests encircling the pole) is not directly applicable to the rotated pole setup. Following Wirth and Szabo (2007), we chose the domain length such that the domain length along the coordinate equator of the rotated pole setup is equal to the zonal domain length at 45°N in the corresponding midlatitude setup.

4

We use centered differences on full model levels instead of forward differences on half levels, which introduces a slight smoothing of the profile.

5

Note that the domain average of is not the same as the maximum N2 value in domain-averaged profiles (such as Fig. 6) because taking the maximum does not commute with taking the mean (cf. section 3.1.3).

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