Synoptic-scale variability of the polar and subpolar tropopause: Data analysis and idealized PV inversions

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1 Q. J. R. Meteorol. Soc. (2002), 128, pp doi: /qj Synoptic-scale variability of the polar and subpolar tropopause: Data analysis and idealized PV inversions By G. ZÄNGL 1 and V. WIRTH 2 1 Meteorological Institute, University of Munich, Germany 2 Institute for Physics of the Atmosphere, University of Mainz, Germany (Received 10 May 2001; revised 10 January 2002) SUMMARY The synoptic-scale variability of the polar and subpolar tropopause is investigated based on radiosonde and European Centre for Medium-Range Weather Forecasts Reanalysis data in combination with idealized potential vorticity (PV) inversions. A regression analysis is performed to examine the relationship between the relative vorticity at tropopause level, the tropopause displacement, the static stability above the tropopause, and the anomalies of tropopause temperature and potential temperature. The results are compared with regression coef cients computed from a large number of PV inversions. Generally, a cyclonically in uenced tropopause is lower, warmer and potentially colder than average, and the static stability above the tropopause is reduced. The opposite is true under anticyclonic in uence. Typically, a cyclonic vorticity of 10 5 s 1 is associated with a decrease of the tropopause height by m. Correspondingly, the mean vertical temperature gradient in the lowermost 2 km above the tropopause decreases by K km 1. Moreover, an increase of the tropopause height by 1 km is associated with a temperature decrease by 3 5 K and an increase of the potential temperature by 6 8 K. Except for the last one, these values do not depend signi cantly on the strength and the sign of the tropopause anomaly. Distinct annual cycles are observed in parts of the polar regions. Most of the relations found in the data can be reproduced with the PV inversions, except for the observed linearity for large tropopause displacements which is not always reproduced very well. The annual cycles can be partly explained with the aid of the PV inversions. KEYWORDS: Cyclone anticyclone asymmetry Regression analysis Tropopause dynamics Tropopause height 1. INTRODUCTION The tropopause is one of the key elements of atmospheric structure. It separates the troposphere from the stratosphere, two volumes of air that differ signi cantly in static stability, humidity, ozone concentration and several other properties (e.g. Brewer 1949; Dobson 1956). Soon after its discovery about 100 years ago (see Hoinka 1997 for a historical review), it was found that the height and the temperature of the tropopause exhibit large differences between the Tropics and the extratropics and that there is a distinct annual cycle in most parts of the extratropics (e.g. Teisserenc de Bort 1909). A tropopause climatology for the northern hemisphere has been constructed by Flohn (1947), and Defant and Taba (1957) demonstrated that the height variation of the tropopause between the Tropics and the polar regions is stepwise rather than continuous. In recent years, climatological knowledge about the tropopause has further increased, thanks to the availability of operational numerical weather prediction (NWP) analyses (or reanalyses) with global coverage (e.g. Hoinka 1998, 1999). It has been known for a long time that the extratropical tropopause has substantial variability on the synoptic scale. It is higher and colder than average under anticyclonic in uence, while it is lower and warmer than average under cyclonic in uence (Dines 1911). Moreover, the tropopause is known to be quite sharp in upper-tropospheric anticyclones, but it can be rather indistinct in upper-level cyclones. The latter has attracted scienti c interest because of a potential contribution to the mass exchange between the stratosphere and the troposphere (e.g. Price and Vaughan 1993; Wirth 1995). Corresponding author: Meteorologisches Institut der Universität München, Theresienstraße 37, D München, Germany. guenther@meteo.physik.uni-muenchen.de c Royal Meteorological Society,

2 2302 G. ZÄNGL and V. WIRTH Recently there has been a renewed interest in the synoptic-scale variability of the tropopause region focusing on the marked asymmetry between cyclonic and anticyclonic disturbances. Observations indicate that anticyclonic anomalies typically have a larger horizontal extent and a smaller vertical extent than cyclonic anomalies (e.g. Muraki and Hakim 2001). While such an asymmetry is absent in quasi-geostrophic theory (Juckes 1994), Muraki and Hakim (2001) showed that a next-order correction to the quasi-geostrophic equations is suf cient to capture the asymmetry. Similar asymmetries are obtained with axisymmetric potential-vorticity (PV) inversions based on the primitive equations (Wirth 2001). Most notably, a given potential-temperature anomaly is related to a substantially larger tropopause-height anomaly for cyclonic disturbances than for anticyclonic disturbances. As an indirect consequence one obtains on average larger differences between the thermal and the dynamical tropopause altitude in cyclones than in anticyclones. Overall, these studies suggest that a number of relevant features of synoptic-scale tropopause behaviour are related to fundamental properties of upper-tropospheric balanced ow. In view of these developments it appears unfortunate that none of the previous observational studies did speci cally focus on the synoptic-scale tropopause variability; they restricted their attention to the standard deviation of the tropopause pressure from its monthly or seasonal mean (Hoinka 1998; Zängl and Hoinka 2001, hereafter ZH01). The current paper is a rst attempt to ll this gap, drawing on the wealth of data available through modern NWP centres. Two goals will be pursued. First, we analyse and document the synoptic-scale variability of the polar and subpolar tropopause in considerably more detail than was done previously. To this end, we examine the relationship between the relative vorticity at tropopause level, the height and temperature perturbations of the tropopause, and the perturbations of the vertical temperature gradient above the tropopause, using a regression analysis. This analysis is based on a combined set of radiosonde data and European Centre for Medium-Range Weather Forecasts (ECMWF) Reanalysis (ERA) data. Second, we relate the results of the data analysis to the previously mentioned concepts concerning the fundamental role of upper-tropospheric balanced dynamics. PV inversions of idealized tropopause disturbances are performed and compared with the data, quantifying the extent to which the PV inversions capture the observed synoptic-scale variability of the tropopause. As it turns out, a considerable portion of the observed features can be reproduced and explained by invoking the simple dynamical concept of PV inversion. The outline of the paper is as follows. The existing criteria to de ne the tropopause are summarized in section 2. In section 3, the datasets and analysis methods are described. The set-up of the PV inversions is described in section 4. Section 5 presents the results, and conclusions are drawn in section TROPOPAUSE DEFINITIONS Several different criteria have been proposed to de ne the tropopause. The simplest criterion, allowing one to determine the location of the tropopause from a single radiosonde ascent, is the so-called thermal tropopause criterion de ned by WMO (1957). The thermal tropopause is taken to be the lowest level at which the vertical temperature =@z exceeds 2 K km 1, provided that the =@z between this level and all higher levels within 2 km does not fall below this value again. Another widely used criterion, rst proposed by Reed (1955), is based on the PV P D 1 ½.r v C 2Ä/. rµ; (1)

3 SYNOPTIC-SCALE VARIABILITY OF THE TROPOPAUSE 2303 where ½; v; Ä and µ are the density, the (three-dimensional) velocity vector, the vector of the earth s rotation and the potential temperature, respectively. The dynamical tropopause (or PV-tropopause) is de ned through a threshold value of P, like for instance P D 1.6 PVU (1 PVU D 10 6 K m 2 kg 1 s 1 ) as proposed by WMO (1986). However, most authors use higher threshold values, ranging from 2 PVU (e.g. Wirth 2000) to 3.5 PVU (e.g. Hoinka 1998). There is strong evidence that a threshold value of 1.6 PVU is too small for a meaningful tropopause de nition at least at higher latitudes (Hoerling et al. 1991; ZH01). In contrast to the vertical temperature gradient, P is materially conserved for adiabatic and frictionless motions. Thus, P is more appropriate =@z to separate stratospheric from tropospheric air masses, particularly when synoptic-scale disturbances are considered. Besides this, an attempt has been made to use the ozone mixing ratio to determine the tropopause (Bethan et al. 1996). As ozone is also a materially conserved quantity, it is in principle appropriate for distinguishing tropospheric from stratospheric air masses. However, the tropopause criterion developed by Bethan et al. is rather complicated, consisting of three consecutive steps with empirically determined threshold values, and requires a very high vertical resolution of the ozone data. Since ozone soundings providing a suf cient data quality are very sparsely distributed around the world, an ozone-based tropopause criterion is not useful for a climatological study like the present one. Throughout this paper, only the dynamical tropopause de nition will be used. 3. DATA AND METHOD OF ANALYSIS The data analysis performed in this study is based on a combination of daily (12 UTC) radiosonde and ERA data, covering the regions poleward of 55 ± latitude and the time period of The temperature pro les are extracted from the radiosonde data. Since we intend to use a dynamical tropopause de nition, and since the radiosonde network is too sparse to compute the relative vorticity in most parts of the polar regions, additional vorticity data are needed. The latter are taken from the ERA data and interpolated horizontally to the locations of the radiosonde stations. This combined dataset has been used before by ZH01. They found that the tropopause pressures computed from the ERA data alone agree very well with those computed from the combined dataset, monthly mean differences being below 3 hpa with very few exceptions. However, the vertical resolution of the ERA data (which is around 1 km at tropopause level) has been found to be too coarse to resolve the vertical temperature gradients in the tropopause region. Since changes in static stability are an important aspect of synoptic-scale tropopause variability and, therefore, need to be included in this study, the combined dataset described above is used. The threshold value to de ne the dynamical tropopause is 3.5 PVU. The dynamical tropopause is computed from the combined ERA/radiosonde dataset as described in ZH01. It is pointed out that a thickness criterion similar to the second part of the thermal WMO criterion is employed: the mean PV between the tropopause and all levels within 2 km above is required to be >3.5 PVU. This excludes thin stratospheric intrusions from our analysis. Moreover, it ensures that small-scale stability structures probably not representative for the scales resolved by the ERA data do not affect the determination of the tropopause from the combined dataset. Based on these dynamical tropopause data, a regression analysis is performed to investigate the relationship between the relative vorticity at tropopause level, tropopause height and temperature anomalies, and the static stability above tropopause level. Since the variability of the static stability below the tropopause turned out to be much smaller

4 2304 G. ZÄNGL and V. WIRTH than that above the tropopause, the former is not considered in this study. Speci cally, the regression coef cients b.h TP I TP /, b.dt =dz C2 I TP /, b.t TP I H TP / and b.µ TP I H TP / are calculated. In these expressions, the index TP denotes the tropopause, H,, T and µ the height, relative vorticity, temperature and potential temperature, respectively, and dt =dz C2 denotes the mean vertical temperature gradient between the tropopause and 2 km above. Tropopause height is considered instead of tropopause pressure because b.p TP I TP /, where p is pressure, has been found to be proportional to the mean tropopause pressure while b.h TP I TP / does not depend on the mean tropopause height. To compute the regression coef cients, the data points lying within certain geographical regions are grouped together, and each month is considered separately. The regions have been chosen such as to combine stations with similar tropopause characteristics (see ZH01). This paper focuses on four regions: [50 ± W 175 ± W, 55 ± N 65 ± N], [50 ± W 175 ± W, >65 ± N], [50 ± E 100 ± E, >65 ± N] and the Antarctic coast. In the following, the longitude bands in the northern hemisphere will be referred to as Canada and western Siberia, respectively. The regression coef cients are computed according to P.xi x/.y i y/ b.yi x/ D P.xi x/ 2 ; (2) where an overbar denotes the mean value. Thus, the regression coef cient b.yi x/ denotes the slope of the regression line y.x/, which could also be expressed The correlation coef cients obtained for the various pairs of quantities considered here are fairly high. Their absolute values range between 0.75 and 0.90 for T TP H TP, between 0.85 and 0.93 for µ TP H TP, between 0.7 and 0.85 for H TP TP, and between 0.55 and 0.8 for dt =dz C2 TP. Thus, computing regression coef cients is considered to be meaningful. Moreover, the scatterplots in Figs. 1, 3 and 5 suggest that a linear regression is appropriate since the data points may be well approximated by a straight line. 4. SET-UP OF THE PV INVERSIONS An important property of potential vorticity is its invertibility (Hoskins et al. 1985). From a given PV distribution, the temperature and wind elds can be obtained. This involves solving an elliptic partial differential equation for which proper boundary conditions have to be speci ed and some kind of dynamical balance (e.g. geostrophic or gradient wind balance) must be assumed. As shown by Thorpe (1985, 1986), the essential features of upper-level (anti)cyclones can easily be reproduced by inverting a positive (negative) PV anomaly located at tropopause level. In particular, the tropopause anomalies associated with these disturbances are captured very well, indicating that they are related through the dynamical constraints of balanced motion. The most important parameter controlling the structure of a balanced PV anomaly is the ratio between its vertical scale 1Z and horizontal scale 1X (Hoskins et al. 1985; Wirth 2000). This aspect ratio is appropriately de ned by D N f 1Z 1X ; with N and f being the Brunt Väisälä frequency and the Coriolis parameter, respectively. For 1, corresponding to a shallow anomaly, the PV anomaly is primarily manifested as an anomaly in static stability while relative vorticity contributes little to the PV anomaly. On the other hand, for À 1 the PV anomaly is related to strong relative vorticity and a weak thermal anomaly. In the intermediate range, which is relevant

5 SYNOPTIC-SCALE VARIABILITY OF THE TROPOPAUSE 2305 for synoptic-scale disturbances, both relative vorticity and the temperature anomaly are signi cant. In this study, PV inversions of idealized tropopause anomalies are performed in order to complement the data analysis described above. A number of different tropopause displacements are considered, and from these the same regression coef cients are computed as above. The numerical algorithm used for the inversions is described in Wirth (2000). PV is speci ed as a function of log p so that the displacement of the tropopause can be speci ed explicitly. The shape of the tropopause anomalies is taken to be axisymmetric. The vertical coordinate is z D H ln.p=p 0 / with H D 7 km and p 0 D 1000 hpa. Based on this coordinate, the model domain extends vertically from 0 to 20 km. In the horizontal, the model domain covers four times the radius of the tropopause anomaly (see later). The number of grid points is , corresponding to a vertical resolution of z D 39 m. At x D 0, where the centre of the tropopause anomaly is located, a symmetry condition is employed. At the remaining boundaries, von Neumann boundary conditions are used. The basic state is de ned by the surface temperature T sfc, the tropospheric vertical temperature gradient.@t =@z/ T, the height of the basic state tropopause Z TP and the stratospheric vertical temperature gradient.@t =@z/ S. Note that the vertical temperature gradients refer to the geometric height z D z.t =T 0 /, T 0 being the reference temperature corresponding to H D 7 km (T 0 ¼ 240 K). This is necessary as we intend to compare with data later on. Basic state potential vorticity is computed according to P D RT C g c p R=cp p0 : (3) In (3), g, R and c p are the gravitational acceleration, the gas constant for air and the speci c heat capacity of air, respectively. The Coriolis parameter f is set to 1: s 1, corresponding to a latitude of 63 ± N. As in the data analysis, the tropopause is de ned by P D 3.5 PVU. The tropopause anomaly is de ned through z TP.x/ D 1 C cos.¼x=r/ Z TP C 1z (4) 2 in the range x 6 r, with 1z and r being the maximum displacement and the radius of the anomaly, respectively. Within the anomaly (i.e. between z TP and Z TP ), a constant potential vorticity is assumed. For cyclonic anomalies (1z < 0), the stratospheric basic state PV at Z TP is used. Likewise, the tropospheric basic state PV at Z TP is prescribed within anticyclonic anomalies. Outside the anomaly, the basic state PV remains unchanged. Note that this is a substantial simpli cation as real tropopause anomalies extend somewhat into the troposphere and the stratosphere, with surfaces of constant PV bulging upwards or downwards, respectively, in the lower stratosphere and upper troposphere (e.g. Muraki and Hakim 2001). Experiments with more realistic anomalies and comparison with data revealed that the simple anomaly shape is suf cient when the interest is restricted to the region around the centre of the anomaly. In the outer part of anticyclonic disturbances, however, the present approach appears to be an oversimpli cation introducing signi cant errors. To keep the analysis as simple as possible, we decided to retain the simple anomaly shape and to concentrate on the anomaly centre. To compute the regression coef cients, we vary the key parameter 1z while keeping all other parameters xed. More speci cally, PV inversions for 16 different p

6 2306 G. ZÄNGL and V. WIRTH 12 a) Canada, > 65N, July 12 b) Antarctica, Aug/Sept Tropopause height (km) Relative Vorticity at tropopause level (10-5 s -1 ) Relative Vorticity at tropopause level (10-5 s -1 ) Figure 1. Scatterplots showing the relation between the relative vorticity at tropopause level and the tropopause height. Each symbol represents one radiosonde ascent. The displayed regions are (a) Canada, >65 ± N, July and (b) Antarctica, August and September. In (b), vorticity values are multiplied by 1. tropopause displacements between 1z D 1500 m and 1z D 1500 m are performed. Additional experiments are carried through with 0 6 1z m and 3000 m 6 1z 6 0 in order to assess systematic differences between cyclones and anticyclones. In these experiments, the maximum displacements have been chosen to be asymmetric because data show a similar behaviour (see Fig. 1). All regression coef cients discussed in this study refer to the anomaly centre (x D 0). As for the vertical temperature gradients, the tropopause displacements are converted to geometric height differences by taking 1H TP D 1z T.Z TP / C T.z TP / 2T 0 before the regression analysis is performed. Unless stated otherwise, the parameters de ning the basic state are set to T sfc D 270 K,.@T =@z/ T D 6:5 K km 1 and Z TP D 10 km;.@t =@z/ S is varied from 1 to 3 K km 1 in steps of 0.5 K km 1. It is noted that.@t =@z/ S D 1 K km 1 is typical for Antarctic winter while values around 2.5 K km 1 are reached in Antarctic summer. In the Arctic,.@T =@z/ S ranges between 0:5 and 2 K km 1 (ZH01; see also Fig. 7(a)). Sensitivity experiments are presented for.@t =@z/ T D 7:5 K km 1 ; changes of T sfc and Z TP have been found to be unimportant. The radius of the tropopause anomaly is set to 500 km. This last choice will be discussed later. 5. RESULTS The discussion of the results is structured according to the pair of quantities considered in the regression analysis. Data and PV inversions are discussed together. (a) Tropopause height vorticity at tropopause level For illustration, we rst show two selected scatterplots of the relation between tropopause height and relative vorticity at tropopause level in Fig. 1. In this and the

7 SYNOPTIC-SCALE VARIABILITY OF THE TROPOPAUSE 2307 TP height change per vorticity (10 8 m s) a) Canada, 55N-65N Canada, > 65N W-Siberia, > 65N Antarctica b) r = 500 km r = 750 km dt/dz(trop) = -7.5 K/km (r = 500 km) Month Stratospheric temperature gradient (K km -1 ) Figure 2. Regression coef cients b.h TP I TP / computed from (a) data and (b) PV inversions. Dash patterns are explained in the gures. See text for further explanation. following gures, vorticity values from Antarctica are multiplied by 1 so as to allow for easier comparison with the northern hemisphere regions. For both regions, it is clearly evident that the tropopause is higher under anticyclonic in uence than under cyclonic in uence. This relation is approximately linear. Moreover, it is obvious that cyclonic disturbances can be much more intense than anticyclonic ones, resulting in a signi cant asymmetry. As mentioned in the introduction, this has already been found in earlier studies (e.g. Muraki and Hakim 2001). For j TP j s 1, the distribution is fairly symmetric, but the stronger anomalies are concentrated on the cyclonic side. The extrema are twice as strong for cyclones as for anticyclones. The regression analysis (Fig. 2(a)) reveals that b.h TP I TP / typically ranges between 1:5 and m s, which means that a relative vorticity of 10 5 s 1 is related to a height change of m. In all regions, jb.h TP I TP /j is substantially larger in winter than in summer (note that seasons are shifted by six months in Antarctica). Thus, a given relative vorticity at tropopause level is related to a larger tropopause height anomaly in winter than in summer (see also Fig. 1). An interpretation of this annual cycle will be given below. The regression coef cients derived from the PV inversions (Fig. 2(b)) can rst be used to nd an appropriate anomaly radius. As discussed in the previous section, the aspect ratio between the vertical and the horizontal scale of the anomaly determines the partitioning of a PV anomaly into a thermal and a dynamical anomaly. For a given tropopause displacement, a large (small) anomaly radius implies weak (strong) relative vorticity and thus high (low) values of jb.h TP I TP /j. A comparison between Figs. 2(a) and (b) reveals that an anomaly radius of 500 km yields b.h TP I TP / values very close to the observations. However, a radius of 750 km is clearly too large. A visual analysis of tropopause-height elds computed from the ERA data revealed that these values are not unrealistic. It must be admitted, however, that a signi cant number of the tropopause disturbances are elongated shear lines rather than circular vortices. Apart from the anomaly radius,.@t =@z/ T and.@t =@z/ S have a signi cant impact on b.h TP I TP /. As evident from Fig. 2(b), jb.h TP I TP /j decreases with increasing

8 2308 G. ZÄNGL and V. WIRTH Temperature gradient above tropopause (K km -1 ) Figure 3. a) Antarctica, Aug/Sept Canada, > 65N, July Relative Vorticity at tropopause level (10-5 s -1 ) b) Antarctica, Aug/Sept Canada, > 65N, July Relative Vorticity at tropopause level (10-5 s -1 ) Same as Fig. 1, but for the relation between the relative vorticity at tropopause level and the mean vertical temperature gradient between the tropopause and (a) 2 km above, (b) 500 m above..@t =@z/ S as well as with decreasing.@t =@z/ T. This can be explained with the fact that the magnitude of the PV jump across the tropopause increases with increasing stratospheric stability as well as with decreasing tropospheric stability. Thus, the tropopause anomaly becomes, for a given displacement, more intense and is associated with stronger relative vorticity. Correspondingly, jb.h TP I TP /j decreases. This also explains why the observed values of jb.h TP I TP /j are lower in summer than in winter. In most parts of the polar regions,.@t =@z/ S is signi cantly higher in summer than in winter (ZH01, see also Figs. 3 and 7(a)). Moreover, in the Arctic, the troposphere is more stably strati ed in winter than in summer (see Fig. 7(b)). Finally, we would like to mention that the observed linearity of the H TP TP relation is reproduced very well by the PV inversions. It is interesting to note that quasi-geostrophic theory for sinusoidal tropopause disturbances (Juckes 1994) shows essentially the same behaviour as the PV inversions. Juckes s theory implies b.h TP I TP / D fk.n S N T /g 1, with k being the horizontal wave number and N T (N S ) the tropospheric (stratospheric) Brunt Väisälä frequency. Thus, b.h TP I TP / is expected to be proportional to the wavelength of the disturbance (or the anomaly radius). As can be seen from Fig. 2(b), this is indeed the case to a very good approximation. Moreover, the 1=.N S N T / dependence is consistent with the dependence on.@t =@z/ T and.@t =@z/ S found for the PV inversions. Let, for example,.@t =@z/ T D 6:5 K km 1, corresponding to N T ¼ 10 2 s 1, and let.@t =@z/ S vary from 1 to C3 K km 1, corresponding to N S between and 2: s 1. The quasi-geostrophic relation then predicts a change by a factor of 1.4, which almost exactly matches the ratio obtained from the PV inversions. The horizontal wavelength needed to match the results for an axisymmetric disturbance with r D 500 km is about 1250 km. (b) Vertical temperature gradient above the tropopause vorticity at tropopause level To give an impression of the observed range of values, the discussion starts again with scatterplots. Figure 3(a) displays the mean vertical temperature gradient between the tropopause and 2 km above (i.e. dt =dz C2 ) against TP for the same regions

9 SYNOPTIC-SCALE VARIABILITY OF THE TROPOPAUSE 2309 Change of dt/dz per vorticity (K km -1 (10-5 s -1 ) -1 ) a) Canada, 55N-65N Canada, > 65N W-Siberia, > 65N Antarctica b) 2 km, standard setup 2 km, cyclonic only 2 km, anticyclonic only 500 m, anticyclonic only Month Stratospheric temperature gradient (K km -1 ) Figure 4. Same as Fig. 2, but for regression coef cients b.dt =dz C2 I TP /. The dash-dotted line in (b) refers to b.dt =dz C0:5 I TP /. See text for explanation of symbols. and months as shown in Fig. 1. Mean vertical gradients in the lowermost 500 m above the tropopause (i.e. dt =dz C0:5 ) are displayed in Fig. 3(b). As expected, the static stability above the tropopause is much higher under anticyclonic in uence than under cyclonic in uence. Moreover, the winter summer difference already mentioned is clearly evident. The relation between dt =dz C2 and TP is close to linear although some indications of an asymmetry between cyclones and anticyclones are visible for the Antarctic data points. There, dt =dz C2 appears to approach a saturation value for strong cyclonic vorticities above 10 4 s 1. A comparison between Figs. 3(a) and (b) reveals a characteristic feature of anticyclonically in uenced tropopauses. In many cases a thin, very stable layer is present immediately above the tropopause. In summer, dt =dz C0:5 may exceed C15 K km 1. However, at 1 km above the tropopause, the static stability is already quite close to average stratospheric values. Unlike dt =dz C2, the relation between dt =dz C0:5 and TP is signi cantly asymmetric. The increase of dt =dz C0:5 under anticyclonic in uence is markedly stronger than its decrease under cyclonic in uence since, in the latter case, the static stability above the tropopause is more homogeneous. Moreover, the variability of dt =dz C0:5 is quite large so that its correlation with TP is very poor. Thus, regression coef cients are computed for dt =dz C2 only. The annual cycles of b.dt =dz C2 I TP / are displayed in Fig. 4(a). They typically range between 0:2 and 0:3 K km 1 per 10 5 s 1, indicating that a relative vorticity of 10 5 s 1 is related to a decrease of dt =dz C2 by K km 1. In contrast to b.h TP I TP /, no universal annual cycle is found. It is only in Canada (and eastern Siberia, not shown) where jb.dt =dz C2 I TP /j is signi cantly higher in winter and spring than in summer and autumn. The results of the PV inversions (Fig. 4(b)) are shown for the standard set-up as well as for purely cyclonic and purely anticyclonic vorticity (see section 4). In addition, b.dt =dz C0:5 I TP / is shown for purely anticyclonic vorticity. Comparison with Fig. 4(a) reveals that b.dt =dz C2 I TP / obtained from the PV inversions lies within the observed range. However, its dependence on.@t =@z/ S does not agree with the data as it would imply a maximum of jb.dt =dz C2 I TP /j in summer. Moreover, the approximate linearity

10 2310 G. ZÄNGL and V. WIRTH 350 a) Antarctica, Aug/Sept Canada, > 65N, July -30 b) Antarctica, Aug/Sept Canada, > 65N, July Potential tropopause temperature (K) Tropopause temperature ( o C) Tropopause height (km) Tropopause height (km) Figure 5. Same as Fig. 1, but for the relation between the tropopause height and (a) the potential tropopause temperature, (b) the tropopause temperature. of the relation dt =dz C2 TP is not reproduced by the PV inversions. For purely cyclonic vorticity, jb.dt =dz C2 I TP /j is substantially lower than for the standard set-up, and for purely anticyclonic vorticity it is larger than for cyclonic vorticity but still slightly lower than for combined cyclonic anticyclonic vorticity. The most rapid change of dt =dz C2 is obtained for weak cyclonic vorticity. Despite these minor inconsistencies, the typical vertical structure of anticyclonic tropopause anomalies is captured well by the PV inversions. As evident from Fig. 4(b), jb.dt =dz C0:5 I TP /j is about 2.5 times as large as jb.dt =dz C2 I TP /j. This is related to the rapid decay of anticyclonic anomalies in the stratosphere (see Wirth 2000, Figs. 5 and 8). It is only immediately above the tropopause that strong negative vorticity requires very stable strati cation to yield stratospheric PV values. Cyclonic disturbances decay more slowly and do not show comparable small-scale features (see Wirth 2000, Fig. 2). (c) (Potential) tropopause temperature tropopause height Finally, the relation between (potential) temperature anomalies and height anomalies of the tropopause is discussed. Despite some redundancy, both the temperature and the potential temperature of the tropopause are considered because both quantities have their own importance. From a dynamical point of view, potential temperature is more relevant since isentropic advection of potential vorticity is a key mechanism in the development of synoptic-scale disturbances. On the other hand, chemical processes depend on actual temperature rather than on potential temperature. For example, as evident from the scatterplots shown in Fig. 5, the Antarctic winter tropopause may become cold enough to allow for the formation of polar stratospheric clouds (e.g. Hanson and Mauersberger 1988). In extreme cases, this may also happen in the Arctic polar vortex region (not shown). As is well known from earlier investigations, a high tropopause is absolutely cold and potentially warm while the opposite is the case for a low tropopause. The relation between actual tropopause temperature and tropopause height is very close to linear (Fig. 5(b)). However, the increase of potential temperature with height is somewhat

11 SYNOPTIC-SCALE VARIABILITY OF THE TROPOPAUSE 2311 Potential temperature change per height change (K km -1 ) a) Canada, 55N-65N Canada, > 65N W-Siberia, > 65N Antarctica b) standard setup cyclonic only anticyclonic only dt/dz(trop) = -7.5 K/km (mixed vorticity) Temperature change per height change (K km -1 ) c) d) Month Stratospheric temperature gradient (K km -1 ) -5 Figure 6. Same as Fig. 2, but (a) (b) for regression coef cients b.µ TP I H TP /, and (c) (d) b.t TP I H TP /. stronger for a high tropopause than for a low tropopause (Fig. 5(a)). This difference can be traced back to the dependence of potential temperature on.p 0 =p/ R=cp. The annual cycles of the regression coef cients are shown in Fig. 6(a) (b.µ TP I H TP /) and Fig. 6(c) (b.t TP I H TP /); b.µ TP I H TP / typically ranges between 6 and 8 K km 1 while b.t TP I H TP / takes values between 5 and 3 K km 1. Not surprisingly, the corresponding curves in Figs. 6(a) and (c) have similar shapes. It is just the relative magnitude of the extrema that differs due to the different annual cycles of the mean tropopause pressure (see ZH01). In all northern hemisphere regions, b.µ TP I H TP / and b.t TP I H TP / attain their primary maximum in autumn and a local minimum in July. In winter, both quantities exhibit a pronounced minimum in western Siberia (and also in northern Europe) while the values remain relatively high in Canada and eastern Siberia, particularly equatorwards of 65 ± N. Finally, Antarctica has a broad minimum in winter and spring and a maximum in summer and early autumn for both quantities (note again the reversed seasons). It is mentioned that these patterns bear some similarity to the patterns found for the annual cycle of the mean tropopause pressure (ZH01).

12 2312 G. ZÄNGL and V. WIRTH The results of the PV inversions are displayed in Fig. 6(b) for b.µ TP I H TP / and in Fig. 6(d) for b.t TP I H TP /. Comparison with data indicates that the simulated values of the regression coef cients agree very well with observations for b.t TP I H TP /. However, they are slightly too large for b.µ TP I H TP / when the standard set-up is used. More signi cant discrepancies are found for the dependence of the correlation coef cients on the sign of the vorticity. Although observations show that b.µ TP I H TP / is larger for anticyclonic than for cyclonic disturbances (Fig. 5(a)), the simulated cyclone anticyclone asymmetry is substantially too strong. Correspondingly, b.t TP I H TP / obtained from the PV inversions is also markedly asymmetric while observations do not indicate any asymmetry (Fig. 5(b)). It is noted that this discrepancy can be reduced by specifying a more realistic anomaly structure (i.e. a PV anomaly that does not end abruptly at the displaced tropopause). In addition, the agreement between the simulated and the observed regression coef cients can be further improved by taking account of the fact that cyclonic anomalies are more intense than anticyclonic ones (see Fig. 1). However, this is not attempted here because it would not yield any additional insight. For completeness, it is mentioned that both b.µ TP I H TP / and b.t TP I H TP / decrease with increasing anomaly radius, corresponding to a stronger anomaly in actual temperature. Yet, this dependence is rather weak so that the choice made for the anomaly radius in section 5(a) has no signi cant impact on the results presented here. Moreover, a possibly larger radius for the anticyclonic anomalies cannot compensate the unrealistically strong cyclone anticyclone asymmetry of b.t TP I H TP / shown in Fig. 6(d). Despite these inconsistencies, the dependence of b.µ TP I H TP / or b.t TP I H TP / on the tropospheric and stratospheric vertical temperature gradients provides an explanation for the observed annual cycles. Evidently, the regression coef cients increase with both.@t =@z/ T and.@t =@z/ S. Moreover, for small displacements, b.µ TP I H TP / agrees well with an analytical result derived by Juckes (1994) from quasi-geostrophic theory: b.µ TP I H TP / : S Since.@µ=@z/ S À.@µ=@z/ T, a given absolute change of.@µ=@z/ T (or.@t =@z/ T ) is associated with a larger relative change of this quantity than an equal change of.@µ=@z/ S (.@T =@z/ S ). Consequently, b.µ TP I H TP / depends more strongly on.@µ=@z/ T than on.@µ=@z/ S. As can be seen from Figs. 6(b) and (d), this remains valid for nite tropopause displacements. A change of.@t =@z/ T by 1 K km 1 has almost as much impact on b.µ TP I H TP / as a change of.@t =@z/ S by 4 K km 1. In the following, (5) will be checked against data to see how well the observed annual cycles of b.µ TP I H TP / follow this simple relation. Observations of.@µ=@z/ T, computed between 500 and 350 hpa, and of.@µ=@z/ S, computed between 150 and 100 hpa, are displayed in Figs. 7(a) and (b), respectively. The square-root of the product of both quantities, to be compared with the observed annual cycles of b.µ TP I H TP / (Fig. 6(a)), is added in Fig. 7(c). Despite different amplitudes and generally larger absolute values, the corresponding curves indeed show some qualitative similarity. Antarctica exhibits a maximum in summer and a minimum in winter, western Siberia has maxima in spring and autumn and minima in summer and winter, and the summer minimum in Canada is also evident in both cases. Thus, it can be concluded that tropospheric and stratospheric stability have a signi cant impact on b.µ TP I H TP / (and b.t TP I H TP /). From Figs. 7(a) and (b), the relative importance of.@µ=@z/ S and.@µ=@z/ T can be inferred. In Antarctica, both quantities have a minimum in winter, the amplitude of.@µ=@z/ S being much larger than that of.@µ=@z/ T. However,

13 SYNOPTIC-SCALE VARIABILITY OF THE TROPOPAUSE 2313 Potential temperature gradient (K km -1 ) a) 20 Mean gradient 150/100 hpa b) Mean gradient 500/350 hpa c) Canada, 55N-65N Canada, > 65N W-Siberia, > 65N Antarctica sqrt [(a)*(b)] Month Month Figure 7. Observed mean potential-temperature gradients between (a) 150 and 100 hpa, and (b) 500 and 350 hpa. (c) Square-root of (a) (b) according to Eq. (5), to be compared with Fig. 6(a). Dash patterns are explained in (c). since the relative amplitudes of both quantities are similar, they are of similar importance for the observed annual cycle. In the Arctic regions,.@µ=@z/ T and.@µ=@z/ S do not run parallel, and the impact of.@µ=@z/ T dominates. Finally, the similarity between the annual cycles of b.µ TP I H TP / and the mean tropopause pressure mentioned above will be discussed brie y. As shown by ZH01, the tropopause pressure (or height) is closely related to the temperature difference between the middle troposphere and the lower stratosphere, e.g. between 500 and 100 hpa. In accordance with simple geometrical considerations, a large temperature difference is found to be associated with a high tropopause. However, a large temperature difference might also be expected to be associated with reduced stability within the troposphere and the stratosphere. This is, indeed, the case. In Antarctica, for example, the temperature difference between 500 and 100 hpa is much larger in winter than in summer (ZH01). Consequently, the tropopause is higher in winter than in summer (ZH01), and.@µ=@z/ T and.@µ=@z/ S are larger in summer than in winter (Figs. 7(a) and (b)). Together with (5), this explains the qualitative agreement between the annual cycles of tropopause height and b.µ TP I H TP /. 6. SUMMARY AND CONCLUSIONS The present study explores the synoptic-scale variability of the tropopause at polar and subpolar latitudes by performing data analysis in combination with idealized PV inversions. The data analysis is based on a combined set of radiosonde data and ECMWF Reanalysis (ERA) data. From this dataset, regression coef cients have been computed quantifying the relation between the tropopause height and the relative vorticity at tropopause level, between the mean vertical temperature gradient in the lowermost 2 km above the tropopause and the relative vorticity at tropopause level, and between the (potential) tropopause temperature and the tropopause height. The same regression coef cients are computed from a large number of idealized PV inversions and compared with those obtained from the data. The main results can be summarized as follows. Under cyclonic in uence, the tropopause is lower than average, and the strati - cation above the tropopause is less stable than average. The opposite is true for anticyclonic in uence. The decrease of tropopause height corresponding to an increase of

14 2314 G. ZÄNGL and V. WIRTH relative vorticity by 10 5 s 1 ranges between 150 and 200 m, the lower values generally occurring in summer. This annual variation can be traced back to the fact that the PV jump across the tropopause is larger in summer than in winter, which in turn is due to the more stable strati cation of the lower stratosphere in summer. Thus, for a given tropopause displacement, the PV anomaly is more intense in summer, leading to stronger relative vorticity. The relation between tropopause height and vorticity remains close to linear even for large tropopause displacements. This linearity is reproduced well by the PV inversions. The mean vertical temperature gradient between the tropopause and 2 km above decreases by K km 1 per 10 5 s 1 relative vorticity. In contrast to above, no universal annual cycle is found, but the relation is again approximately linear. However, the linearity is lost when a thinner layer above the tropopause is considered. In that case, the increase of the stability under anticyclonic in uence is signi cantly stronger than its decrease under cyclonic in uence. This asymmetry is related to the rapid decay of anticyclonic disturbances above the tropopause. In a thin layer above the tropopause the strong negative vorticity requires a very stable strati cation to yield stratospheric PV values. This characteristic structure of anticyclonic anomalies is reproduced well by the PV inversions. A high tropopause is associated with low actual temperatures and high potential temperatures. For each kilometre height change, the tropopause temperature decreases by about 3 5 K while the potential temperature of the tropopause increases by 6 8 K. This relation is very close to linear for tropopause temperature but slightly nonlinear for potential temperature. The increase of the latter quantity with tropopause height becomes stronger for high tropopauses. Different annual cycles are found depending on the region considered. The PV inversions and an analytical formula by Juckes (1994) reveal that these annual cycles can be traced back to the annual cycles of the mean tropospheric and stratospheric vertical temperature gradients. A large static stability in the troposphere and/or the stratosphere implies a large increase of tropopause potential temperature with height. As a caveat we note that during the analysis of the ERA data, a phenomenon was occasionally encountered that is not entirely consistent with our PV inversions. According to the data, it sometimes happens that upper-tropospheric shear lines are associated with negative absolute vorticity, implying negative potential vorticity (in the northern hemisphere). Thus, there are regions not far below the tropopause where atmospheric motion is not balanced. Yet, these localized regions do not appear to have a dominant impact on the overall ow. Despite this, the success in reproducing the synoptic-scale variability of the tropopause with simpli ed PV inversions encourages us to conclude that this variability is governed by upper-tropospheric balanced dynamics to a large degree. Moreover, our results further increase our con dence in PV-based methods used to analyse synopticscale weather systems (e.g. Morgan and Nielsen-Gammon 1998) or to investigate the contribution of poorly represented upper-tropospheric disturbances in NWP analyses to forecast errors (Fehlmann and Davies 1997). ACKNOWLEDGEMENTS This study is part of the rst author s PhD thesis which was written at the University of Munich under the guidance of Joe Egger. The work was supported by the German Science Foundation (Deutsche Forschungsgemeinschaft, DFG).

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