## 1. Introduction

A recent observational study by Birner et al. (2002, hereafter BDS02) investigated the static stability in the extratropical tropopause region. The authors analyzed 10 yr (1990–99) of twice-daily radiosonde profiles from two stations in southern Germany (Munich and Stuttgart) with a vertical resolution of approximately 50 m. They produced a climatology by composing a large number of profiles with the themal tropopause as common reference level. The resulting buoyancy frequency squared *N*^{2} is depicted schematically in Fig. 1 (solid line). The key features are a jump in *N*^{2} right at the height *z*_{tp} of the thermal tropopause and a sharp maximum just above *z*_{tp}. Immediately beneath *z*_{tp} the values of *N*^{2} are somewhat increased with respect to typical tropospheric values further below. While this result is likely to be representative for southern Germany, it remains for future studies to find out whether one obtains a similar picture throughout the midlatitudes.

These observations motivated us to investigate the thermal structure of the extratropical tropopause region from a more theoretical point of view. We ask the following question: what is the impact of conservative balanced dynamics on the profiles of static stability in the tropopause region? Our approach is similar to Wirth (2001, hereafter W01). Essentially, we specify axisymmetric distributions of potential vorticity (PV), perform PV inversion in order to obtain the related balanced wind and temperature fields (Hoskins et al. 1985), and analyze those fields paying special attention to the tropopause region. In particular we compute profiles of *N*^{2} obtained for different anomalies and compose them in the same way as was done by BDS02, that is, with the thermal tropopause as common reference level. The anomalies are specified such that they can be interpreted as the result of a formation process implying adiabatic advection across the gradients of a smooth background atmosphere (cf. Fig. 2 later). Clearly this is an idealized approach that emphasizes fundamental properties of balanced flow. Rather than trying to be as realistic as possible we prefer to keep the setup simple in order to facilitate the interpretation. Nevertheless, the model turns out to qualitatively reproduce the essential features obtained by BDS02. This encourages us to carry through a detailed analysis yielding insight into the underlying mechanisms within the current framework.

In the analysis we shall consider “related” cyclones and anticyclones, which are treated identically except for the sign of their tropopause potential temperature anomaly Δ*θ.* It was shown in W01 that this leads—despite the (anti-) symmetry in Δ*θ*—to a pronounced *asymmetry* between cyclones and anticyclones regarding the height of the thermal and the dynamical tropopause. Not surprisingly, this cyclone–anticyclone asymmetry turns out to be relevant in our current problem, too.

The paper is organized as follows. Section 2 briefly reviews the model and discusses how it is used in the present study. It is then applied in section 3 in order to successively unravel various mechanisms that contribute to shaping the thermal structure of the tropopause region. Finally, a summary and conclusions are provided in section 4.

## 2. Model

The model setup is essentially as in W01, which the reader may consult for the details. Here, only the main features will be presented as well as some modifications.

*z*= −

*H*ln (

*p*/

*p*

_{0}) as vertical coordinate, where

*p*denotes pressure;

*H*= 7 km is a constant mean scale height; and

*p*

_{0}= 1000 hPa is a constant reference pressure. With this vertical coordinate the natural expression for the buoyancy frequency squared reads

*T*is temperature,

*g*denotes the acceleration due to gravity,

*κ*=

*R*/

*c*

_{p}= 2/7 is the ratio of the gas constant

*R*and the specific heat

*c*

_{p}at constant pressure for dry air, and where

*T*

_{s}is defined through

*RT*

_{s}=

*gH*(see Andrews et al. 1987). We will refer to

*N*

^{2}as static stability throughout this text. Potential vorticity is given by

*ρ*

_{0}=

*p*/(

*RT*

_{s}), potential temperature

*θ*=

*T*exp(

*κz*/

*H*), and absolute vorticity

*ζ*_{a}. On the level of approximation of the primitive equations the latter is given by

*ζ*_{a}=

*f*

**k**+ ∇ ×

**v**with the horizontal wind

**v**and the unit vector

**k**in the vertical.

*s*(Fig. 2). The variable

*s*is defined as the deviation of the tropopause potential temperature

*θ*

^{tp}from the constant value

*θ*

^{tp}

_{0}

*f*

_{0}= 10

^{−4}s

^{−1}is the Coriolis parameter, and its profile at

*s*= 0 will be called reference PV profile

*Q*

_{ref}(

*z*) =

*Q̂*(0,

*z*). Since the background atmosphere is assumed to be shaped by processes that are not modeled explicitly, it implies some freedom, which we draw on by rendering the profile of

*N*

^{2}as simple as possible. It is characterized by two constant values, one for the troposphere (

*N*

^{2}

_{t}

^{−4}s

^{−2}) and one for the stratosphere (

*N*

^{2}

_{s}

^{−4}s

^{−2}), and a smooth transition over a few hundred meters mediated by a hyperbolic tangent with a vertical scale of

*δ*= 250 m (Fig. 3). This profile, with the transition centered at the respective reference level, will be referred to as

*N*

^{2}

_{ref}

*z*) in the following. The (approximately) piecewise constant nature of

*N*

^{2}

_{ref}

*z*) makes it easy to focus on the changes brought about by the dynamics of the anomaly formation, which we intend to investigate in this study. Note that in W01 the lapse rate ∂

*T*/∂

*z*rather than

*N*

^{2}was chosen to be the piecewise constant.

*f*-plane PV anomaly from the background atmosphere is meant to emulate the formation of a cutoff cyclone or anticyclone implying a strongly nonlinear development from an initially zonal background flow (e.g., Palmén 1949). It is indicated in Fig. 2 through the arrows and explained in detail in W01. At the outer boundary of the computational domain we specify the PV profile to be equal to

*Q*

_{ref}(

*z*) and the profile of buoyancy frequency squared to be equal to

*N*

^{2}

_{ref}

*z*). In the interior we obtain an axisymmetric PV distribution

*r*denotes radius,

*ζ̃*

*r*

^{−1}∂(

*rυ*)/∂

*r*is the vertical component of relative vorticity in isentropic coordinates, and

*υ*is the tangential wind. The PV anomaly is essentially characterized by two parameters, namely, the radial scale Δ

*R*and the tropopause potential temperature anomaly Δ

*θ*in the center of the votex. Two examples are displayed in Figs. 4a,b. The dynamical tropopause is defined through

*Q*= 2.5 PV units (PVU) and is delineated as the interface between the white and the dark shading in Fig. 4. The two anomalies in Figs. 4a,b differ by the sign of Δ

*θ,*where the negative sign represents a cyclonic anomaly and the positive sign an anticyclonic anomaly. Such pairs of anomalies will be referred to as related cyclones and anticyclones in the following. Note that in this example the change in PV across the dynamical tropopause in the vortex center is less pronounced for cyclones than for anticyclones (see Figs. 4a,b). A modification to the background state, which we describe later in this paper, will investigate the impact of this particular feature on the results.

Assuming hydrostatic and gradient wind balance the PV anomaly is “inverted” to obtain the wind and pressure field in isentropic coordinates. The boundary conditions are specified as described in W01. Unless noted otherwise, our results do not sensitively depend on that specification. Finally, transformation to *z* as vertical coordinate yields the axisymmetric *f*-plane vortices, which will be analyzed in the following. The result for our two sample vortices is displayed in Figs. 4c,d. The basic features in the tropopause region correspond well to those of the idealized vortices of Thorpe (1986), although in the latter work the PV anomaly is more localized. More importantly, they correspond well in the case of the cyclone to the features of observed cutoff cyclones (e.g., Fig. 10.8 in Palmén and Newton 1969; or Fig. 1.4.d in Newton and Holopainen (1990); and similar cross sections are obtained for upper-tropospheric shear lines (Appenzeller et al. 1996), which would correspond to plane symmetry rather than circular symmetry (cf. Wirth 2000). There is a noticeable asymmetry between the cyclonic and the anticyclonic vortex in Figs. 4c,d regarding the displacement of the dynamical tropopause in the vortex center and the strength of the tangential wind, which is reminiscent of the observed difference between strong upper-tropospheric cyclones and anticyclones (Muraki and Hakim 2001). Is was shown in W01 that this cyclone–anticyclone asymmetry has implications for the difference in altitude between the thermal tropopause (thick solid line in Figs. 4c,d) and the dynamical tropopause. As we shall see later, this difference between the two tropopauses plays a significant role for the composite profiles of *N*^{2}, too.

Vertical resolution is an important aspect in the current work. It is effectively determined by the smooth transition of *N*^{2}_{ref}*δ* of a few hundred meters (see Fig. 3). The vertical grid spacing (about 70 m) is much smaller by comparison, which is desirable, as otherwise the results would strongly depend on the numerical resolution. The sensitivity with respect to *δ* will be addressed in section 4.

## 3. Results

The model is used in order to successively unravel the mechanisms that contribute to shaping the thermal structure in the tropopause region. The anomalies investigated in sections 3a through 3d have an amplitude of Δ*θ* = ±20 K. Since earlier studies have indicated the crucial role of the anomalies' aspect ratio (Wirth 2000), we consider two distinct values for the radial scale Δ*R* so as to investigate one case of a flat aspect ratio (Δ*R* = 2500 km) and one case of an intermediate aspect ratio (Δ*R* = 500 km). Very tall aspect ratios (Δ*R* → 0) are not considered explicitly, because the result is known a priori: in this limit the PV anomaly is realized entirely in terms of vorticity, and the thermal structure is determined by the outer boundary, which is specified from the reference atmosphere. As a consequence the profile of *N*^{2} would be identical to the reference profile *N*^{2}_{ref}*R* and different values for Δ*θ,* and we retrieve profiles not only from the votex center but from different radii in order to test the robustness of the results.

### a. Effect of PV advection

By construction, the reference profile *N*^{2}_{ref}*z*) is a piecewise constant with a smooth transition over a distance of a few hundred meters (see Fig. 3). However, in the center of the anomaly the tropopause region consists of air that has been advected quasi-horizontally some finite distance across the background atmosphere. This makes the PV profile at the vortex center deviate from the reference profile *Q*_{ref}(*z*), because the background PV at the tropopause level varies with horizontal distance. To the degree that PV is materialized in terms of static stability, this modifies the profile of *N*^{2} in comparison with *N*^{2}_{ref}

*N*

^{2}

_{stat}

*N*

^{2}yields the “static buoyancy frequency squared”

*N*

^{2}if a given profile of

*Q*(

*z*) were realized in a static atmosphere. It expresses the PV profile in terms of a buoyancy frequency squared, which is the relevant quantity in the present context. The fact that the vortex center is, indeed, not part of a static atmosphere but rather embedded in a balanced flow will be considered further later.

The long dashes in Fig. 5 depict the profiles of *N*^{2}_{stat}*z*) at the vortex center for cyclonic and anticyclonic anomalies with Δ*R* = 2500 km and Δ*R* = 500 km, respectively. In each case there is a sharp increase of *N*^{2}_{stat}*z*) with altitude approximately at the location of the dynamical tropopause, which is located at *z* = 5.9, 12.6, 6.1, and 11.7 km in Figs. 5a–d, respectively. This is consistent with (7), because *N*^{2}_{stat}*z*) is proportional to *Q*(*z*), and the latter features a sharp increase at the dynamical tropopause by construction. A quantitative comparison with *N*^{2}_{ref}*z*) indicates that generally *N*^{2}_{stat}*z*) > *N*^{2}_{ref}*z*) for cyclonic conditions (left column in Fig. 5) and *N*^{2}_{stat}*z*) < *N*^{2}_{ref}*z*) for anticyclonic conditions (right column in Fig. 5). Note that the profile *N*^{2}_{stat}*z*) is a result of the rearrangement of PV during the generation process, that is, horizontal advection on isentropes in combination with implied vertical displacement of the isentropes (cf. Figs. 2 and 4c for the cyclonic case). The net result depends on the balance between horizontal and vertical advection, which in turn depends on both the background PV and the details of anomaly construction. In the present case the implied lower-stratospheric downward displacement in the case of cyclones in combination with the positive vertical PV gradient in the background atmosphere is responsible for the increased values of *N*^{2}_{stat}*N*^{2}_{stat}

None of the profiles of *N*^{2}_{stat}*N*^{2} in Fig. 1 are related to the physical process encapsulated in *N*^{2}_{stat}

### b. PV partitioning

Our hypothetical quantity *N*^{2}_{stat}*z*) deviates from the actual static stability *N*^{2}(*z*), because a given profile of PV materializes partly in terms of a thermal anomaly (static stability) and partly in terms of a dynamical anomaly (vorticity). A key parameter determining the partitioning between thermal and dynamical anomaly is the aspect ratio of the anomaly (cf. Wirth 2000).

*N*

^{2}(

*z*), and the short dashes denote

*N*

^{2}

_{dyn}

*N*

^{2}

*z*

*N*

^{2}

_{sat}

*z*

*Q,*cyclonic flow with

*ζ̃*

*σ*and, according to (5), smaller

*N*

^{2}than a hypothetical static situation with

*ζ̃*

*N*

^{2}

_{dyn}

*N*

^{2}

_{dyn}

*N*

^{2}

_{stat}

*N*

^{2}(solid line) generally shows smaller values than

*N*

^{2}

_{ref}

*N*

^{2}

_{ref}

There are systematic differences between the four cases in Fig. 5 regarding *N*^{2}_{dyn}*N*^{2}_{dyn}*R* is consistent with the concept of “Rossby height” (Hoskins et al. 1985), which relates the vertical penetration to the horizontal extension in the case of a localized PV anomaly. The difference between cyclones and anticyclones can be traced back partly to the nonlinearity of the PV inversion [Eqs. (6) and (7) in W01] and partly to the transformation from *θ* to *z* as vertical coordinate, which both contribute in the same sense to enhance the sharpness of the peak in the case of anticyclones, but lessen it in the case of cyclones. As noted before, PV inversion is performed iteratively. The solution from the first step in the iteration, which corresponds to the quasigeostrophic approximation (see Hoskins et al. 1985), shows only a minor cyclone–anticyclone asymmetry regarding the shape of the wind field *υ*(*r,* *θ*), but as the iteration proceeds the solution becomes more peaked for anticyclones and less peaked for cyclones. This asymmetry is further enhanced through the vertical coordinate transformation owing to the fact that the spacing between isentropes above and below the tropopause is smaller in comparison with the reference atmosphere for anticyclones, but larger for cyclones (cf. Figs. 4d and 4c).

*N*

^{2}

_{stat}

*N*

^{2}

_{dyn}

*f*

_{0}

*N*

^{2}

_{stat}

*f*

_{0}

*ζ̃*

*N*

^{2}

*f*

_{0}

*ζ̃*

*N*

^{2}

_{stat}

*N*

^{2}

_{dyn}

*F*is given by

*N*

^{2}

_{dyn}

*N*

^{2}and

*N*

^{2}

_{stat}

*ζ̃*

*υ.*Figure 6 shows

*F*as a function of

*z*in the vortex center for the four cases corresponding to Figs. 5a–d. The features pointed out before regarding

*N*

^{2}

_{dyn}

*F.*Since our balanced vortices satisfy

*ζ̃*

*f*

_{0}everywhere, the factor

*F*is negative for cyclonic flow and positive for anticyclonic flow. The extremum of

*F*is close to the dynamical tropopause (although small deviations turn out to be important). The stratospheric penetration is smaller for taller anomalies (Δ

*R*= 500 km) than for shallow anomalies (Δ

*R*= 2500 km), and

*F*is significantly more peaked for anticyclones than for the corresponding cyclones. The latter feature is rendered plausible by examining the behavior of

*F*in the limit of very strong cyclones (

*ζ̃*

*ζ̃*

*f*

_{0}), respectively. In the cyclonic case one obtains

*F*→ −1, whereas in the anticyclonic case one obtains

*F*→ +∞. In other words, the asymmetry in the sharpness of the peak of

*N*

^{2}

_{dyn}

Adding the dynamical contribution *N*^{2}_{dyn}*N*^{2}_{stat}*N*^{2}(*z*) depicted by the solid line in Fig. 5. For the anticyclonic vortices (Figs. 5b,d) the peak of *N*^{2} just above the dynamical tropopause is much more pronounced than that of *N*^{2}_{stat}*z*), while for the cyclonic vortices *N*^{2}(*z*) is either close to the reference profile (Fig. 5a) or features a more gradual transition from low-tropospheric values to high-stratospheric values (Fig. 5c). Clearly, the dynamical contribution substantially helps to produce a sharp peak in the profile of *N*^{2}(*z*) for anticyclonic vortices. Note also that in Fig. 5d the dynamical contribution *N*^{2}_{dyn}*N*^{2} in a layer of a few kilometers below the peak.

As a next step we average related cyclonic and anticyclonic profiles from the vortex center using the dynamical tropopause as a common reference level. The dynamical tropopause as the reference level appears most natural, since this is approximately the location where *Q,* *N*^{2}_{stat}*N*^{2} have their sharpest increase with altitude. Figures 7a and 7b show the results for Δ*R* = 2500 km and Δ*R* = 500 km, respectively. Both panels depict the average (solid line) of the cyclonic (long dashes) and the anticyclonic (long–short dashes) profiles. For comparison, the short dashes depict the reference profile *N*^{2}_{ref}*R* the average profile is characterized by a distinct peak just above the reference level. For Δ*R* = 500 km one obtains increased values of *N*^{2} (with respect to *N*^{2}_{ref}*N*^{2}_{dyn}

BDS02 used the thermal tropopause rather than the dynamical tropopause as the common reference level, because the former is much easier to diagnose from the observations. This is the source for another important asymmetry between cyclones and anticyclones, which was discussed in detail in W01. While the two tropopauses generally are rather close together for anticyclones, one expects substantial deviations for cyclonic anomalies of intermediate aspect ratio (cf. Figs. 4c and 4d), such that on average the thermal tropopause is located above the dynamical tropopause. If one chooses to define the reference level through the thermal tropopause, one thus obtains on average increased values of *N*^{2} below the reference level. Figure 8 shows the composite profiles in the same format as in the previous figure, except that now the thermal tropopause is used as the common reference level. As before, the average profiles for both values of Δ*R* are characterized by a distinct peak above the reference level, but now the increase of *N*^{2} (with respect to *N*^{2}_{ref}*R* = 500 km the peak is displaced slightly upward. Both features are associated with the displacement between the thermal and the dynamical tropopause and can, in this sense, be considered as artifacts. Nevertheless, even with the thermal tropopause as reference level the strong peak from the anticyclonic dynamical contribution survives the averaging procedure. Unless stated otherwise we will use the thermal tropopause as the reference level from now on.

We conclude that in our model the dynamical contribution relating a static PV profile to the actual static stability in a dynamical vortex is a key ingredient for shaping the thermal structure of the tropopause region.

### c. β effect

So far we considered a background atmosphere with no horizontal variation of PV except across the tropopause (Fig. 2). However, if the background state were to represent a zonal mean atmosphere with latitude as horizontal coordinate, it would be more consistent to include an overall increase of PV with increasing latitude in the spirit of the *β* effect. We simulate this through replacing *f*_{0} in (3) by *f*(*s*) = *f*_{0}(1 − *Bs*) with *B* = (60 K)^{−1}. The modified background atmosphere is shown in Fig. 9a. Note that the modification leaves both the reference temperature and *N*^{2}_{ref}*z*) unchanged, but it changes the PV distribution leading to a nonzero horizontal gradient in both the troposphere and the stratosphere. With the chosen value of *B* the isolines of PV in the tropopause region are approximately parallel to the tropopause. As a consequence, given a thin layer above the tropopause, PV hardly varies with horizontal distance in that layer. This behavior contrasts with the situation considered so far (and depicted in Fig. 2), where the PV values in the lowest part of the stratosphere are lower in “polar latitudes” (with a low tropopause) than in “subtropical latitudes” (with a high tropopause).

The resulting composite profiles are displayed in Fig. 10 and should be compared with those in Fig. 8. The two cases (*β* and no-*β*) are broadly similar, but some differences are noteworthy. First, consider Δ*R* = 2500 km. Including the *β* effect, even the cyclonic anomaly (long dashes in Fig. 10a) features a broad maximum of *N*^{2}(*z*) above the tropopause, while for the anticyclonic anomalies (long–short dashes) the maximum is less pronounced than without *β.* The analysis (no figure shown) reveals that the cyclonic maximum is a feature of the static profile *N*^{2}_{stat}*z*). The modified background atmosphere combined with our method of anomaly construction implies a layer of approximately constant PV above the tropopause in the cyclonic anomaly. Owing to the decaying exponential in (7) this leads to a decrease of *N*^{2}_{stat}*H*/(*κ* + 1) = 7*H*/9 yielding a relative maximum in *N*^{2}_{stat}*β* effect renders the displacement between thermal and dynamical tropopause in the cyclonic case less pronounced than previously. As a consequence, there is less of a “misplacement” of the reference level, which explains why in Fig. 10a one observes purely tropospheric values of *N*^{2} up to the reference level, in contrast to the situation in Fig. 8a.

For anticyclonic flow the *β* effect works in the opposite direction: the static profile *N*^{2}_{stat}*z*) (not shown) features a significantly smoother transition from tropospheric to stratospheric values such that the dynamical contribution (being very similar as before) is just able to produce a weak (yet noticeable) peak in Figs. 10a,b (long–short dashes, to be compared with the same lines in Fig. 8). We conclude that our *β* effect primarily modifies *N*^{2}_{stat}*z*) through changes in advected PV. The difference in behavior between cyclones and anticyclones is less pronounced than before.

### d. Subtropical tropopause region

In the real atmosphere, temperature profiles in low latitudes are characterized by a sharp increase from a negative to a positive vertical gradient across the tropopause (e.g., Defant and Taba 1957), while in middle and high latitudes the lower stratosphere tends to be approximately isothermal. In other words, the lower-stratospheric static stability is higher in low latitudes than in high latitudes. This feature presumably results from processes (such as radiation) that operate on timescales longer than the dynamical timescales considered here. In the context of the present model it should be accounted for as a property of the background atmosphere. It is implemented through a modification of the background atmosphere “south” of the reference latitude (i.e., modifying those profiles for which the tropopause is higher than 10 km) by multiplying *N*^{2}_{s}*s.* The result is shown in Fig. 9b. For *s* = 20 K the lower-stratospheric temperature gradient is roughly equal to +10 K km^{−1}, which is at the upper end of the observed range. At first glance the subtropical distributions of PV and *θ* in Fig. 9b may appear somewhat unrealistic. Note, however, that *s* is proportional to latitude only as long as the zonal mean tropopause temperature increases linearly with latitude, which is not the case in the neighborhood of the subtropical jet. We believe that our simple modification is meaningful in the current framework, as it focuses on the essential aspect, which is the advection of high values of stratospheric PV during the formation of anticyclones. Note that the *β* effect is included in this experiment as well.

By construction, cyclonic vortices are not affected by this modification; but for anticyclonic vortices the hypothetical profile *N*^{2}_{stat}*N*^{2}_{dyn}*R* the sharp peak from the anticyclonic vortex (long–short dashes) is strong enough to yield a pronounced peak even after averaging with the related cyclonic profile. Broadly speaking, the increased static stability in the lowest part of the subtropical stratosphere overcompensates the smoothing resulting from the *β*-effect in the case of anticyclonic vortices.

### e. Average effect

So far we restricted our analysis to profiles taken from the center of four different vortices, which were characterized by two different values of Δ*R* and Δ*θ,* each. It is not straightforward to compare these profiles with the observed composite sketched in Fig. 1, since a stationary observer encounters a wide spectrum of vortices of different sizes and amplitudes. In addition, it is highly unlikely that all the measurements are done at the very time when the vortex center is overhead. In this study we are not trying to simulate realistic station statistics corresponding to Fig. 1, because this would require a statistical analysis regarding the size and strength of upper-tropospheric vortices above southern Germany, which is beyond the scope of this paper. In particular, we do not make any assumption about the asymmetry between cyclones and anticyclones regarding their distribution in terms of amplitude |Δ*θ*|, but rather continue to compose related cyclones and anticyclones differing only by the sign of Δ*θ.*

Nevertheless we want to assure the robustness of our results by investigating whether (or to what extent) they remain valid when key parameters are changed. First, we vary Δ*R* over the range 500–2500 km with |Δ*θ*| = 20 K kept fixed. More specifically we use Δ*R* = 500 km + (*k* − 1) × 250 km (*k* = 1, 2, … , 9). The lower part of this range is meant to cover strongly nonlinear coherent structures in the tropopause region. A 33-yr climatology of such structures in the extratropics compiled by Hakim (2000) revealed typical radial scales of 500–800 km, which translates to Δ*R* = 750–1200 km in our notation, while the study of Zängl and Wirth (2002), which focuses on higher latitudes, suggests smaller values (Δ*R* ≈ 500 km). The high end of our range for Δ*R* is meant to account for the effect of large-amplitude synoptic-scale waves. Both the *β* effect and the increased static stability in the subtropical stratosphere are included as described above, and again only profiles from the vortex center are used. Averaging the 2 × 9 profiles thus obtained one arrives at the composite shown in Fig. 12a (solid line). Apparently, the essential features pointed out before are also present in this average, most notably the peak in *N*^{2} above the tropopause. The analysis reveals that it is mostly the anticyclonic profiles contributing to this peak, while the cyclonic profiles weaken and slightly smooth the step of *N*^{2} across the tropopause. However, the anticyclonic profiles dominate the average.

The analysis is repeated with the dynamical tropopause as the common reference level, since (as pointed out before) this would be a more natural choice. The result is shown in Fig. 12b. Indeed, the peak is somewhat narrower than before and the values of *N*^{2} are less enhanced below the reference level.

The composite from Fig. 12a cannot be compared yet with the observed composite sketched in Fig. 1, because all the profiles inherent in the average of Fig. 12 are taken from the centers of various different vortices. When each of these vortices is used to extract six different profiles at six different radii *r*_{k} = (*k* − 1)Δ*R*/6 (*k* = 1, 2, … , 6), we obtain the composite shown in Fig. 13a (solid line). The peak above the tropopause as well as the enhanced values below the tropopause (with respect to *N*^{2}_{ref}*θ*| = 10 K instead of |Δ*θ*| = 20 K. The resulting profile is qualitatively very similar to that shown in Fig. 13a, but the amplitude of the peak is reduced by roughly 50%.

It is also interesting to compute average profiles of tangential wind *υ* with the same composing technique as for *N*^{2}, that is, using the thermal tropopause as common reference level. The result is shown in Fig. 13b (solid line). In contrast to *N*^{2}(*z*), the wind in the tropopause region sensitively depends on the upper and lower boundary conditions for *υ* when performing PV inversion. If we use zero Dirichlet instead of our standard treatment (which is zero Neuman), we obtain the dashed profile in Fig. 13b. Despite the strong sensitivity with respect to the boundary conditions, the qualitative features are very similar in both profiles, with a peak slightly below the tropopause and a stronger decay above the tropopause than below. Both features are present in the observations of BDS02 (see their Fig. 2a), which strengthens our case.

## 4. Summary and conclusions

This study is motivated by the observations of BDS02, who showed that the buoyancy frequency squared *N*^{2} in the extratropical tropopause region has a characteristic profile with a sharp peak just above the tropopause and slightly enhanced values below the tropopause in comparison with typical tropospheric values further below. A major aspect of that study is the method of composing different profiles, which implies an averaging procedure with the thermal tropopause used as common reference level. Here we ask the following question: given a background atmosphere with a well-defined tropopause, what is the impact of conservative balanced dynamics on the profiles of static stability in the tropopause region, and may the observed composite be at least partly due to balanced dynamics? In our analysis we assume a background atmosphere that is characterized by a sloping tropopause and a simple reference profile *N*^{2}_{ref}*θ.* The analysis focuses on the profiles of *N*^{2}(*z*) as a function of altitude in the tropopause region. When composing profiles from different vortices including both cyclones and anticyclones, we follow BDS02 using the thermal tropopause as the common the reference level. The profiles of *N*^{2} thus obtained show characteristic deviations from the reference profile *N*^{2}_{ref}

In the analysis it proves helpful to distinguish two mechanisms. First, specific properties of the background atmosphere have an impact, since they are advected during the anomaly formation. This mechanism materializes in the form of a local PV anomaly and is quantified through a hypothetical profile *N*^{2}_{stat}*N*^{2}_{dyn}

Using the preceding terminology we summarize our analysis as follows.

A major systematic impact in our model arises from the dynamical contribution

. The latter is positive and has a sharp peak right around the dynamical tropopause for anticyclones; it is negative and much less peaked in the case of cyclones. Owing to this asymmetry the anticyclonic peak dominates the composite.*N*^{2}_{dyn}The ratio

*F*between and*N*^{2}_{dyn} can be expressed in terms of isentropic relative vorticity*N*^{2}_{stat} as*ζ̃**F*= /*N*^{2}_{dyn} = −*N*^{2}_{stat} /(*ζ̃**f*_{0}+ ), where*ζ̃**f*_{0}is the Coriolis parameter. It follows that the asymmetry in between cyclones and anticyclones is related to the asymmetry in*N*^{2}_{dyn} , which is bounded below by −*ζ̃**f*_{0}for anticyclonic balanced flow, but which has no upper bound for cyclonic balanced flow.The advection of PV across the background atmosphere may have a significant impact on the profile of

, but the related modification generally depends on the details of the background atmosphere as well as the anomaly formation. We identified two aspects of the background atmosphere (next two list items) that are likely to contribute in a systematic way to the profile of*N*^{2}_{stat}*N*^{2}through the advective mechanism.The

*β*effect, that is, the fact that the absolute value of planetary vorcitity is larger in regions where the tropopause is lower, leads to slightly increased values of*N*^{2}just above the reference level in the case of strong cyclones, but it is unlikely to produce a very sharp peak in a climatological composite.Advection of high static stability from the subtropical lower stratosphere may significantly contribute to the peak of

*N*^{2}above the reference level in the case of anticyclones.Enhanced values of

*N*^{2}below the reference level may be due to both cyclones and anticyclones of intermediate aspect ratio. As is known from previous work, there may be a displacement between the dynamical and the thermal tropopause. This displacement is largest in the case of cyclonic anomalies of intermediate aspect ratio, for which the thermal tropopause is located significantly above the dynamical tropopause (cf. Fig. 4c). In the present context, where the reference level is defined through the thermal tropopause, this yields increased values of*N*^{2}in the upper troposphere. Likewise one obtains enhanced values of*N*^{2}for anticyclones of intermediate aspect ratio, since in that case is positive and extends a few kilometers below the peak (Fig. 5d).*N*^{2}_{dyn}We computed average profiles drawn from different radii

*r*_{k}and using vortices with different radial scale Δ*R.*The resulting composite profiles of*N*^{2}agrees very well with the observed profile in a qualitative sense, although the peak is less pronounced in our model. Repeting this analysis for the tangential wind*υ*we obtained, again, good qualitative agreemend regarding the key features observed by BDS02.The dynamical tropopause would be a more natural choice as the reference level than the thermal tropopause, because the former is related to a strong increase in PV with altitude, which is reflected in similar behaviour of

*N*^{2}. As a consequence, the peak in*N*^{2}above the reference level would be more pronounced for the composite profile when the reference level is defined through the dynamical tropopause.

With the present numerical technique it is difficult to properly treat very large gradients and near discontinuities. This is the rationale behind the smooth transition between tropospheric and stratospheric values of *N*^{2} in the background atmosphere (cf. Fig. 3), and it may be part of the reason why in our model neither the observed sharpness of the peak nor the near discontinuity of *N*^{2} across the tropopause are borne out very well. We did a sensitivity study repeating the computations shown in Fig. 12a for different values of *δ,* which is the transition scale between *N*^{2}_{t}*N*^{2}_{s}*N*^{2}_{ref}*z*). The result is shown in Fig. 14. Subsequently halving the value of *δ* from 800 m over 400 m to 200 m increases the maximum value of *N*^{2} at the peak by about 0.5 × 10^{−4} s^{−2} each time. In addition, for the smaller values of *δ* the peak is located closer to the reference level (which is at *z* = 0 in Fig. 14). We conclude that in our model the characteristic features of the composite *N*^{2} like the sharp increase across the tropopause and the peak just above the tropopause are the more pronounced the more *N*^{2}_{ref}

The present study does not answer the question of what eventually determines the structure of static stability in the real atmosphere. In particular we did not address the question regarding the mechanism that produces a tropopause in the reference atmosphere; the near discontinuity in *N*^{2}_{ref}*N*^{2} in the final profiles. We only considered one side of the coin focusing on the conservative generation of synoptic-scale anomalies and their effect on *N*^{2}. In the long run there must be competing nonconservative processes so as to establish a statistically stationary climatology. It is interesting in this context to note that there may be a pronounced asymmetry between cyclones and anticyclones regarding the dominant decay mechanism (cf. Wirth 1995; Zierl and Wirth 1997). A more complete model is needed to describe the competition between these different processes. Such a model should also produce a realistic spectrum of anomalies in a self-consistent manner, which is not possible with the current approach. Haynes et al. (2001) presented a study in which baroclinic life cycles compete against Newtonian relaxation toward a baroclinically unstable equilibrium state. It would be interesting to reanalyze the results from such a model using our findings as working hypotheses, and to test whether, or to what extent, our dynamical mechanisms can be identified and prove important in a climatological sense.

In the light of the above caveats and the idealized nature of our approach, the results should be viewed as suggestive. However, regarding the fact that our basic mechanism is related to fundamental properties of balanced flow and encouraged by the good qualitative agreement between our synthetic composites and corresponding observations, we conclude that nonlinear balanced dynamics is likely to play a role in producing the characteristic shape of the profile of static stability in the extratropical tropopause region, and especially in producing the sharp peak in *N*^{2} right above the tropopause.

## Acknowledgments

The author wants to thank T. Birner and two anonymous referees for constructive criticism.

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