We calculate the PVG tropopause from four recent atmospheric reanalysis datasets which span the global troposphere and stratosphere: ERA-Interim and ERA5 produced by the European Centre for Medium-Range Weather Forecasts (ECMWF), MERRA-2 from the National Aeronautics and Space Administration (NASA), and JRA-55 produced by the Japan Meteorological Agency (JMA). All of these datasets are full-input reanalyses, meaning they combine observations from satellites as well as from surface stations and upper-air measurements. These observations are successively blended with short-range model forecasts in an assimilation scheme, generating best estimates of meteorological parameters in the past.

ERA5 is the most recent reanalysis produced by ECMWF, succeeding ERA-Interim in 2019 . ERA5 is generated by means of a four-dimensional variational assimilation (4D-Var) scheme  in conjunction with the Integrated Forecasting System (IFS) cycle 41r2, version 2016. ERA5 extends from 1950 to the present day, providing hourly estimates. The spatial grid has a finer resolution and reaches up to higher vertical levels than ERA-Interim, with a horizontal grid spacing of 31 km (TL639); in the vertical, 137 levels span the surface to the top level of 0.01 hPa . We use the updated version ERA5.1, where a low-temperature bias in the lower stratosphere has been corrected in the data between 2000 and 2006 . For comparison purposes, we also consider the following, older reanalyses.

ERA-Interim precedes ERA5 and was initially introduced by ECMWF in 2008 . Constructed through the IFS model cycle 31r2, version 2007, and a 4D-Var scheme, this dataset covers the satellite era from 1979 to 2019 with 6-hourly analysis time steps. ERA-Interim adopted the linear reduced Gaussian grid N128 (wavenumber truncation TL255) with a horizontal grid spacing of approximately 79 km. It comprises 60 vertical levels in hybrid σ–p coordinates, extending from the surface to the top level at 0.1 hPa .

The Japanese 55-year Reanalysis (JRA-55) was launched by the Japan Meteorological Agency in 2013 ; it extends from the start of global radiosonde observations in 1958 to the end of January 2024 and has been succeeded by the reanalysis JRA-3Q . JRA-55 was computed with JMA's operational data assimilation system and global spectral forecast model (GSM) from December 2009, comprising 6-hourly datasets. Horizontally, a linear reduced Gaussian grid (N160, equivalent to TL319) was employed with a spacing of 55 km. Vertically, the dataset encompasses 60 hybrid σ–p vertical levels extending from the surface up to 0.1 hPa .

The Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2), introduced by NASA's Global Modeling and Assimilation Office (GMAO) in 2015, provides 6-hourly datasets spanning the satellite era from 1980 to the present . Data assimilation is conducted using the Goddard Earth Observing System model, version 5.12.4 (2015), and a 3D-FGAT (“first guess at the appropriate time”) assimilation system . For its spatial representation, MERRA-2 employs a regular latitude–longitude grid with a spacing of 0.5° latitude and 0.625° longitude. The vertical grid comprises 72 σ–p levels, ranging from the surface up to 0.01 hPa .

The vertical spacing of the native model grids around the tropopause is approximately 1 km for ERA-Interim, JRA-55 and MERRA-2 and 0.3 km for ERA5, transformed from pressure to log-pressure altitude (, their Fig. 2.1). For this study, all reanalysis datasets have been interpolated vertically from the native model levels to potential temperature surfaces (θ) using pressure and temperature fields. The reanalysis data are interpolated in steps of 10 K for ERA-Interim, MERRA-2 and JRA-55, while the finer-resolved ERA5 data are interpolated in 5 K steps. ERA5 data are used with a 1° ×1° latitude–longitude grid as provided by the ECMWF to reduce computational and data storage effort. For the climatological studies, as presented here, this reduced horizontal grid spacing likely has a negligible effect. The PVG tropopause is calculated based on the zonal and meridional wind speeds (u and v), potential vorticity (PV), and potential temperature (θ) from 6-hourly reanalysis data. A computationally faster alternative using monthly climatologies of reanalyses is detailed in Sect. . We consider the period from 1980 to 2017, as it provides the largest overlap of available postprocessed reanalysis datasets at the time of this study.

This section summarizes the underlying dynamical concepts and determination method of the PVG tropopause as established by . Our contribution applies the methodology to four different reanalyses and incorporates additional criteria to mitigate outliers and noise. The location of the PVG tropopause is calculated on isentropes, i.e., surfaces of constant potential temperature. Under the hydrostatic assumption, PV can be expressed, depending on potential temperature θ as a vertical coordinate e.g.,, as 1PV=N2θρgζθ+f. Here, ρ denotes the density, g the gravitational acceleration and f the Coriolis parameter which represents Earth's rotation-induced vorticity. N refers to the Brunt–Väisälä frequency, and its square is called the buoyancy frequency: 2N2=gTΓd-Γ, with the temperature T; the temperature lapse rate Γ=-∂T∂z, wherein z denotes the altitude; and the dry adiabatic lapse rate Γd. The buoyancy frequency is a measure of static stability; if N2 is positive, the ambient air is stably stratified. Transitioning from the troposphere to the stratosphere, static stability (as represented by N2) rapidly increases, which contributes to pronounced meridional PV gradients across the tropopause (see Eq. ). The variable ζθ refers to the relative isentropic vorticity, i.e., the vertical component of the rotation of the horizontal wind speed v=(u,v,0) calculated on an isentropic plane : 3ζθ=∂v∂xθ-∂u∂yθ. As apparent from Eq. (), relative vorticity is based on gradients of zonal and meridional wind speed. In regions of strong horizontal wind shear – e.g., at the flanks of the subtropical jets – ζθ changes noticeably, which further enhances meridional PV gradients near the subtropical jets and tropopause break. In summary, both the rapid changes in static stability across the tropopause and the strong horizontal wind shear at the subtropical jet streams contribute to steep PV gradients near the tropical tropopause break, laying the foundation for the PV-gradient tropopause in this region.

On each isentrope, horizontal coordinates of equivalent latitude are introduced. Equivalent latitude (φe) is a one-to-one mapping of potential vorticity, determined from contours of equal PV on an isentropic surface . For each PV value between -30 and +30 PVU in steps of 0.1 PVU, the corresponding PV isoline is found, and the area A(PV) poleward of this isoline is computed. A circle with the same area A(PV) is centered at the North Pole or South Pole, and the radius of this circle in degrees of latitude is defined as the equivalent latitude: 4φe=arcsin1-A(PV)2πRE2, where RE is the Earth's radius. On each isentrope, we determine the equivalent latitude of the tropopause break, φeTP(θ), from the potential vorticity gradient and horizontal wind speed. For this, the potential vorticity gradient is computed numerically as the derivative of the preset PV values with respect to equivalent latitude ∂PV(φe,θ)∂φe. As the subtropical jet streams correlate with sharp PV changes across the tropopause break, the highest PV gradient in the vicinity of the subtropical jet stream is searched for to determine the PVG tropopause. To account for the subtropical jets, we calculate the horizontal wind velocity from zonal (u) and meridional wind speeds (v) in each reanalysis. First, zonal and meridional winds are averaged along each PV contour in order to obtain the mean along equivalent latitudes, from here on called the “equivalent latitude zonal mean”. The average horizontal wind speed at each equivalent latitude is calculated from the average zonal and meridional winds as vh=u2+v2. In order to also obtain the conventional latitude ϕ as a coordinate, the latitude values along each PV contour are averaged similarly to the winds. Since at the tropopause break strong horizontal winds and sharp PV gradients occur, we multiply the PV gradient by the horizontal wind velocity, defining this product as the quantity Q: 5Q(φe,θ)=∂PV(φe,θ)∂φevh(φe,θ). The algorithm then searches for maxima of Q. Therefore, the PVG tropopause break on each isentrope θ is defined as the equivalent latitude where the highest local maximum of Q occurs, i.e., 6φeTP(θ)=φe(θ)|max(Q), upon certain conditions for the maxima, which are described below. For a visual representation of the PVG tropopause break on certain isentropes, see . In this method, the tropopause break was searched for in an equivalent latitude range of 5°<|φe|<85° in each hemisphere and is defined as the absolute maximum of Q. However, in seasons with a strong polar jet – especially during austral winter – the Q maximum corresponding to the polar jet occasionally exceeds that of the subtropical jet and is falsely recognized by the algorithm as the tropical tropopause break. As a consequence, the method is altered to preferentially select the subtropical jet maximum such that multiple local maxima are detected and the most equatorward maximum is chosen as the PVG tropopause. In order to exclude the polar jets, the maximum search is empirically confined to a poleward limit in equivalent latitude of ±80° between the 320 and 350 K isentropes and ±55° above 350 K. Furthermore, the minimum equivalent latitude is set to ±15° in order to exclude equatorward outliers. The equivalent latitude intervals in this study are therefore 15°<|φe|<80° between 320 and 350 K and 15°<|φe|<55° above 350 K. To ensure that only substantial maxima in Q are chosen, a threshold is set for the prominence of the maxima. The prominence is defined as the height difference between a local maximum and the highest local minimum next to it. In the search for Q maxima, the relative prominence is set to amount to at least 0.2 times the absolute maximum of Q on each isentrope.

On every isentropic level ranging from 320 to 380 K and for each time step in the original reanalysis datasets, a comprehensive profile of the variables relevant to the PVG tropopause is computed. This profile includes latitude (ϕ), the combined variable (Q), potential vorticity (PV), zonal wind speed (u) and horizontal wind speed (vh). All these variables are averaged over potential vorticity contours and are dependent on the equivalent latitude (φe) as a dimension. Time series for the PVG tropopause are then generated for each isentropic level. These time series encapsulate the values of the aforementioned variables at the PVG tropopause, specifically the equivalent latitude (φeTP), potential vorticity (PV^TP), zonal wind speed (uTP), horizontal wind speed (vhTP) and mean latitude (ϕTP).

For the purpose of comparing the PVG tropopause with conventional tropopause definitions, the thermal tropopause has been determined according to the World Meteorological Organization WMO;. The WMO thermal tropopause marks the rapid increase in static stability N2 that occurs at the transition from troposphere to stratosphere (see Eq. ) and is based on characteristics of the temperature lapse rate Γ. More precisely, the WMO thermal tropopause is defined as the lowest altitude where Γ falls below 2 K km⁻¹, provided the average lapse rate between this level and all higher levels within 2 km remains below 2 K km⁻¹. In our study, the WMO tropopause is determined from the temperature in each reanalysis dataset. Transformed to potential temperature coordinates, this yields the potential temperature of the WMO tropopause dependent on latitude, θ(ϕ).

In order to disentangle long-term changes and variability in the PVG tropopause such as the seasonal cycle, El Niño–Southern Oscillation (ENSO) and quasi-biennial oscillation (QBO), we apply a multilinear regression method to the time series of tropopause latitude and potential vorticity on each isentropic level. The monthly-mean time series for each hemisphere of tropopause latitude ϕθTP(t) and potential vorticity PVθTP(t) (generalized as f in Eq. ) and for the latitude also the hemispheric difference (NH-SH, Southern Hemisphere) are fit to the multilinear regression model in Eq. () using least-squares approximation. Regressors include the time t for long-term trends, the mean seasonal cycle S, two orthogonal QBO indices represented by the monthly-mean zonal wind speeds near the Equator at pressure levels of 30 hPa (QBO₃₀) and 50 hPa (QBO₅₀) following , and the ENSO index: 7fθTP(t)=a0+alin⋅t+aseas⋅S(t)+aqbo30⋅QBO30(t)+aqbo50⋅QBO50(t)+aenso⋅ENSO(t)+R(t). Here, R(t) denotes the residual of the fit and a0 the vertical offset. The regressors S, QBO₃₀, QBO₅₀ and ENSO are determined in advance of the multilinear fit: the mean seasonal cycle S(t) is computed directly from the monthly-mean time series by averaging each monthly value over the climatological period. Regarding the QBO, two orthogonal indices are considered, i.e., monthly-mean zonal wind speeds near the Equator on isobars 30 and 50 hPa, which have been observed at a Singapore weather station (1° N, 104° E) . The bimonthly multivariate ENSO index version 2, MEI.v2, combines five variables: sea level pressure, sea surface temperature, zonal and meridional surface wind speeds, and outgoing longwave radiation in the tropical Pacific region (30° N–30° S and 100° E–70° W). MEI.v2 is provided by the National Oceanic and Atmospheric Administration (NOAA) .

To ensure that the regression coefficients ai yield the amplitudes of each variability, the regressors S, QBO₃₀, QBO₅₀ and ENSO – furthermore generalized as y(t) – are normalized before fitting, i.e., divided by their maximum amplitude, which is calculated as half the difference between absolute maximum and minimum of the regressor (y(t)max-y(t)min)/2. The amplitude aqbo of the combined QBO variability factor, i.e., at both 30 and 50 hPa, is computed as follows: first, the time series of latitude and PV are fitted to the 30 and 50 hPa QBO indices separately, as indicated in Eq. (). The two resulting terms are added (fqbo(t)=aqbo30⋅QBO30(t) +aqbo50⋅QBO50(t)), and the maximum amplitude aqbo is computed as 8aqbo=12max(fqbo)-min(fqbo).

Since determining the PVG tropopause from subdaily global reanalysis data is computationally expensive, we test simplifications of the algorithm to enhance usability. Specifically, we examine the influence of using monthly or zonal climatologies of reanalysis data and omitting the subtropical jet wind criterion. The following additional analyses are carried out using ERA5 data and compared with the standard method (i.e., 6-hourly, global reanalysis data with wind criterion):

monthly reanalysis climatologies

For analyses (3) and (4), ERA5 data are also averaged over longitude. In these cases, the computation of equivalent latitude is omitted, and the meridional gradient of PV is computed with respect to geographical latitude. In analyses (2) and (4), the wind criterion is omitted, and only the PV gradient is computed, excluding vh from Eq. (). To compare the results, the seasonal multiyear climatologies, mean seasonal cycle and trends computed with the aforementioned methods are discussed in Sect. .

Several studies indicate strong regional variations in long-term trends in tropical width, specifically the tropopause break and the subtropical jets . Therefore, a similar regional analysis of the PVG tropopause break latitude is conducted here using ERA5 reanalysis data in the time range from 1980–2017. Since the PVG tropopause is computed using equivalent latitude, which directly corresponds to PV, the tropopause on each isentrope aligns with a specific PV value. By identifying this PV contour on each isentrope within the reanalysis data, we can map the latitude–longitude contour of the tropopause on each isentropic surface.

These PV contours are used to create global tropopause surfaces θTP (ϕ,λ). In order to obtain the lowest potential temperature for the tropopause, we iterate through the isentropes from top to bottom, starting with the highest potential temperature level of 380 K down to 320 K in steps of 5 K. Within the loop, the PV field on each isentrope is obtained from the reanalysis. As PV fields can be noisy, outliers are masked using a 3σ filter which iterates through latitude, computes the average μ and standard deviation σ of PV within each latitude circle, and masks all PV values outside of the interval [μ-3σ,μ+3σ] such that these outliers are ignored in the subsequent computation. The tropopause contour on each isentrope is found by searching for the corresponding value PV^TP in the PV field. The geographical area where PV is more extreme than at the tropopause, |PV|≥|PVTP|, is then set to the current potential temperature θ. This procedure is repeated until reaching the lowest isentrope: 320 K.

Originally, the PVG tropopause is only defined near the tropopause break and the subtropical jet. In order to create a global tropopause approximating a physical transport barrier, we extend the tropopause in the extratropics with a PV isosurface, similar to the traditional dynamical tropopause: beyond the 320 K contour – e.g., in the area where PV is more extreme than at the 320 K tropopause – we continue the tropopause with the isosurface corresponding to PV at the 320 K tropopause. This PV surface is obtained from a piecewise-linear one-dimensional interpolation of potential temperature over PV at each point, i.e., θ(ϕ,λ,PV).

The result of this iterative algorithm is a global field of tropopause height θTP (ϕ,λ). Lastly, a median filter with a circular kernel of 4° radius is run over the whole field in order to reduce noise and outliers.

To examine regional long-term trends, the PVG tropopause contour is determined on each isentrope from monthly-mean ERA5 data as detailed above. The intersections of the tropopause with every 10th meridian are saved as time series ϕTP (λ,θ,t) and analyzed using multilinear regression as described in Sect. .

The climatological structure of the PVG tropopause is examined based on 1980–2017 climatologies of the tropopause latitude and potential vorticity in winter and summer of each hemisphere. To investigate the location of the tropopause in comparison to the PV distribution and subtropical jets following , Fig.  illustrates seasonal climatologies of the PVG tropopause plotted as a function of latitude and potential temperature, along with the WMO thermal tropopause. All shown quantities are calculated as zonal means in an equivalent latitude-based coordinate system and subsequently transformed into actual latitudes, as described in the previous section. Since the results are similar for all four considered reanalyses, only the most recent dataset, ERA5, is visualized and discussed in this section. A detailed comparison of the results for different reanalyses and a discussion of robustness are presented in Sect. .

According to its definition, the estimated PVG tropopause coincides with the regions of the highest PV gradient, visible as the areas with the highest PV contour density in Fig. , as well as the u maxima representing the subtropical jet cores. In agreement with , the dynamical tropopause is not well represented by any constant PV value but instead crosses several PV isolines, while PV tends to increase with potential temperature (see Eq. ). Hence, the common practice of defining the dynamical tropopause as a PV isosurface (e.g., 2 PVU) for studies of stratosphere–troposphere exchange  is generally an oversimplification. Figure  further shows that the PVG tropopause agrees well with the WMO lapse-rate tropopause up to ∼ 360 K throughout different seasons and hemispheres.

A comparison of the December–February climatology in Fig. a to the June–August climatology in Fig. b reveals a distinct seasonal cycle in the PVG tropopause, characterized by substantial shifts into the summer hemisphere due to the seasonal ITCZ movement. As observed in Fig. b, this poleward shift in summer is especially prominent in the Northern Hemisphere between the 320 and 340 K surfaces, where the PVG tropopause extends far poleward until 70° N, while in austral summer (SH in Fig. a), the PVG tropopause reaches only until 60° S at 320 K. We hypothesize that these differences in the PVG tropopause latitude between austral and boreal summer stem from the seasonal variations in the subtropical and eddy-driven jet streams. In the summer of both hemispheres, the subtropical jets weaken , which is evident regarding the u contours in Fig. , exhibiting substantially lower wind speeds in the summer hemisphere (i.e., SH in Fig. a, NH in Fig. b) than in the winter hemisphere. Additionally, the subtropical jet often coalesces with the eddy-driven jet e.g.,. Especially during phases of lower jet wind speeds in summer, the subtropical and eddy-driven jets merge at lower levels, resulting in a poleward-displaced PVG tropopause. Also during summer at the lower isentropes, the poleward-displaced PVG tropopause agrees well with the thermal tropopause. In boreal summer, as shown in the right half of Fig. b, the subtropical jet is even weaker than in austral summer (left half of Fig. a). Additionally, show that the eddy-driven jet extends farther poleward in boreal summer than in austral summer, reaching latitudes northward of 70° N, which corresponds to the high latitudes of the PVG tropopause observed at 320–330 K in Fig. b. In brief, the weakening of the subtropical jet in summer, coalescence of subtropical and eddy-driven jets, and farther poleward extension of the NH eddy-driven jet are likely attributed to the far poleward displacement of the tropopause in boreal summer. Furthermore, the variability in the PVG tropopause (measured in terms of standard deviation) is larger in summer than in winter in each hemisphere, which is probably due to the aforementioned subtropical jet weakening in summer, creating wider, less pronounced wind maxima and therefore larger fluctuations in the PVG tropopause (Eq. ).

Figure  reveals that the PVG tropopause corresponds well with the WMO definition up to about 360 K, with the thermal tropopause largely ranging within the standard deviation of the PVG tropopause. During boreal summer (right half of Fig. b), the thermal tropopause is located somewhat more poleward on the 320 K isentrope and lies above the PVG tropopause between 60 and 80° N. This weak deviation of the thermal and PVG tropopauses is possibly caused by cyclonic systems in the upper troposphere, which often shift the 2 PVU surface below the thermal tropopause. As the PVG tropopause closely corresponds to the 2 PVU surface on lower isentropes, this could explain the different locations of both tropopauses there .

Above 360 K, the thermal WMO tropopause narrows until reaching maximum heights of 370 to 380 K between 20 and 40° latitude and drops slightly near the Equator. The PVG tropopause, on the other hand, narrows to 30° latitude around the 350 K isentrope and widens poleward above, reflecting variations in the location of maximum PV gradients and jet streams. As the PVG tropopause corresponds both to the PV gradient and to zonal wind (u) maxima of the subtropical jets, lower PV gradients lead to a stronger influence of u maxima, causing the PVG tropopause to follow the subtropical jets more closely on higher isentropes. This can be observed in both panels of Fig. , where the PVG tropopause coincides with the regions of maximum PV gradient up to 360 to 370 K, recognizable by closely spaced PV isolines, while above, the PVG tropopause tends to align with the horizontal wind contours. As the subtropical jet streams mark a transport barrier, this relation to the PVG tropopause is an asset in transport studies, providing a valuable complement to other definitions.

Near the Equator, however, the PVG tropopause cannot be computed, as PV is ill-defined there. Additionally, PV gradients and jet streams weaken above 380 K, leading to weak Q maxima (see Eq. ), which frequently results in an undefined PVG tropopause. Below 380 K, the PVG tropopause is mostly well-defined. As a consequence, the PVG tropopause can be considered a reliable definition up to 370 to 380 K in the subtropics and midlatitudes.

This section examines the potential vorticity at the PVG tropopause PV^TP and how PV^TP varies between hemispheres and seasons. To visualize in which range and frequency PV occurs at the tropopause, Fig.  displays probability density functions (PDFs) of PV^TP between 1980 and 2017. As an example, results for the newest reanalysis ERA5 are shown here; comparisons between different reanalyses can be found in Sect. .

On the left-hand side of Fig. , panels a–d display the PV climatologies of December–February (DJF), while panels e–h on the right-hand side present the season June–August (JJA). To this end, the PVG tropopause and corresponding PV values are determined through the method described in Sect.  on each isentropic level between 320 and 380 K for four times daily from 1980 to 2017. The results are then binned into each of the four seasons, of which only DJF and JJA are shown here. From each seasonal dataset of PV^TP, the probability density functions are computed.

The upper row in Fig.  (panels a, b, e, f) displays two-dimensional PDFs in the PV–θ plane, which are calculated as follows: for each hemisphere, PV^TP is binned into two-dimensional histograms with a bin size of 0.1 PVU horizontally and 5 K vertically for ERA5 and 10 K for other reanalyses. These histograms are normalized to compute the two-dimensional PDFs. In order to illustrate further statistical properties of the PV distributions, Fig.  (a, b, e, f) delineates the means, medians and standard deviations, as well as 5th and 95th percentiles, on top of the two-dimensional PDFs.

To highlight the PV^TP distribution on selected isentropes, one-dimensional PDFs are shown in the bottom row of Fig.  (panels c, d, g and h) which are computed as one-dimensional normalized histograms of PV^TP with bin sizes of 0.5 PVU on each isentropic level. In order to visualize the PV distribution along the tropopause across several isentropes, a vertically averaged one-dimensional PDF is computed as the arithmetic mean of all one-dimensional histograms between 320 and 380 K.

The two-dimensional PDFs for December–February in Fig. a and b indicate that between 320 and 380 K, the absolute value of potential vorticity at the tropopause, |PVTP|, increases with height. Extremes of PV^TP reach ± 1.5 PVU at 320 K to ±8 PVU at 380 K. The variability, outlined by the standard deviation and 5th and 95th percentiles, amounts to 1 to 2 PVU, maximizing between the 340 and 360 K isentropes. In austral summer (Fig. a), the variability tends to be larger than in boreal winter (Fig. b). The mean and median of PV^TP range from -2 to -7 PVU in austral summer and 2 to 6 PVU in boreal winter. The mean and median closely align, except for 340 to 350 K in Fig. a where the largest variability occurs.

The one-dimensional PDFs for December–February displayed in Fig. c and d confirm the increase in |PVTP| with height. The mode (i.e., peak) of each one-dimensional PDF, which represents the most probable PV value at the tropopause, increases from ca. ± 2.5 PVU on the 320 K isentrope to around ±6 PVU near the tropical tropopause at 380 K. Comparing the one-dimensional PDFs in austral summer (Fig. c) to boreal winter (Fig. d), the modes are similar on the 320 K isentropes but are offset considerably towards larger |PVTP| on isentropes over 320 K. Additionally, the distribution at 320 K exhibits a taller, sharper peak in austral summer.

Furthermore, the one-dimensional distribution's peak widths reflect the vertical change in variability observed in the two-dimensional PDFs above. Figure c and d exhibit relatively narrow peaks corresponding to lower variability at 320 and 380 K and wider peaks corresponding to higher variability at 340 and 360 K. Regarding the one-dimensional PDF variabilities in austral summer (Fig. c) in contrast to boreal winter (Fig. d), the distribution on the 360 K isentrope is substantially wider and exhibits much larger PV variance, reflecting the larger variance observed in the two-dimensional PDF for austral summer (Fig. a). The larger PV variance in summer is likely due to the weakening of the subtropical jet and coalescence with the eddy-driven jet .

The vertical mean PV distributions across the tropopause in December–February (Fig. c and d) exhibit a wide peak structure, which manifests in two peaks in austral summer (Fig. c) and only one peak in boreal winter (Fig. d). The two-peak structure in austral summer is related to the pronounced shift towards higher |PVTP| on the 340, 360 and 380 K isentropes, while the mode at 320 K stays roughly the same, as described above.

The right-hand side of Fig.  (panels e–h) shows the potential vorticity PDFs for June–August. Both the one- and two-dimensional distributions confirm that |PVTP| increases with height. Regarding the two-dimensional PDFs in the upper panels, Fig. e and f, an overall offset of PV^TP towards more positive values can be observed in June–August compared to December–February, with extremes ranging from -1 to -7 PVU in austral winter (Fig. e) and 2 to 8 PVU in boreal summer (Fig. f). This seasonal offset towards positive PV likely corresponds to the northward shift in the ITCZ in June–August. The variance and distance between the 5th and 95th percentiles are larger in boreal summer (Fig. f) than in austral winter (Fig. e), which leads to the conclusion that PV variance is generally larger in the summer of each hemisphere than in winter. Moreover, the PV PDFs for the summer of each hemisphere (austral summer: Fig. a; boreal summer: Fig. f) differ more strongly from each other than the winter seasons (boreal winter: Fig. b; austral winter: Fig. e), where the PV distributions appear almost symmetric. This larger variability in summer is also apparent from the PVG tropopause climatology in Fig.  and is likely related to the weakening of the summer hemisphere STJ, leading to wider Q maxima (see Eq. ).

The one-dimensional PDFs for June–August shown in the lower right panels, Fig. g and h, provide evidence that the most prominent seasonal changes in potential vorticity occur on the isentropes between 340 and 360 K, where the mode of |PVTP| shifts by roughly 2 PVU, as was already observed in the one-dimensional PDFs for December–February (Fig. c and d). The large PV changes between 340 and 360 K account for the second peak around ±6 PVU observed in the vertically averaged one-dimensional PDFs in each summer hemisphere (Fig. c and h).

To summarize, Fig.  provides evidence that the potential vorticity at the tropopause PV^TP covers the range ±1.5 to ± 8 PVU with negative values in the Southern Hemisphere and positive values in the Northern Hemisphere, reaching larger absolute values with height. Seasonal changes include an ITCZ-related offset towards more positive PV during June–August, which is especially prominent on the 340 to 360 K isentropes, as well as larger PV variability in the tropopause in the summer hemisphere, which is likely linked to the general variability in the PVG tropopause in summer due to STJ weakening. These findings are similar in the different reanalyses: ERA5 (displayed, e.g., in Fig. ), ERA-Interim, MERRA-2 and JRA-55 (not shown here). In agreement with , our results confirm that potential vorticity spans a wide range across the tropopause, which is not reflected in common dynamical tropopause definitions consisting of a single 1.5 to 4 PVU isosurface. The PVG tropopause accounts for this variability in potential vorticity, suggesting an advantage of this definition over the PV-monosurface dynamical tropopause.

After consideration of the tropopause shape and potential vorticity distributions for the single reanalysis ERA5 in Sect.  and , this section examines the robustness of the representation of the PVG tropopause in different meteorological reanalyses (ERA5, ERA-Interim, MERRA-2 and JRA-55). To assess robustness among the reanalyses, the latitudes and potential vorticity distributions of the PVG tropopause are shown in Figs.  and  as climatological means. The respective figures show the 1980–2017 climatological mean latitude ϕTP and potential vorticity PV^TP at the PVG tropopause in the solstice seasons in both hemispheres for all four reanalyses. To visualize the tropopause variability, the standard deviations determined from ERA5 are also shown.

Figure  reveals that the PVG tropopause latitude ϕTP is consistent among different reanalyses. Moreover, the seasonal variability in the tropopause location, with substantial shifts into the summer hemisphere as observed in Fig. a and d, turns out to be very similar in the different datasets (discussed in more detail in Sect. ). In particular, latitudinal differences between different reanalyses are on the order of 1°, i.e., 1 order of magnitude smaller than the seasonal variability which is on the order of 10°. Note that the reanalysis differences are larger in summer (Fig. a and d), occasionally reaching up to 3°, with one exceptionally large deviation in boreal summer (Fig. d) at 380 K, where JRA-55 exceeds the other reanalyses by around 5°. These stronger differences between reanalyses in summer are potentially related to the larger PVG tropopause variability observed in summer, which is likely due to the weakening of the STJ and related widening of Q maxima (see Eq. ) as described in Sect. .

Figure  manifests the increase in potential vorticity |PVTP| at the PVG tropopause with increasing potential temperature, as observed for ERA5 in Fig. , robustly in all four reanalyses, as well as the related seasonal variability. Overall the differences in PV^TP between the four reanalyses are smaller than the natural variability reflected in the standard deviation of ERA5 values. However, in some seasons and levels, part of the reanalyses exhibits noticeable differences in the means of PV^TP. For instance, in austral summer (Fig. a) around 340 K, differences between ERA5 and MERRA-2 amount to 0.5 PVU. In boreal summer (Fig. d) between 320 and 350 K, differences between the reanalyses even reach up to 1 PVU. Overall, the PV change with height and seasonality is qualitatively similar in different reanalyses. Quantitatively, however, the PVG tropopause PV values differ between the reanalyses and are less robust than the tropopause latitude. ERA5 tends to yield the largest and MERRA-2 and JRA-55 the smallest absolute values of PV^TP. Examination of the potential vorticity fields reveals considerable variation in PV values and PV gradients among the reanalyses. ERA5, for instance, features the most extreme PV values and steepest PV gradient, which contributes to the observed PV^TP differences between the reanalyses. However, the latitude of the strongest PV gradient remains similar in all reanalyses. As a result, the latitude of the PVG tropopause ϕTP is more robust than the corresponding PV value PV^TP.

Given the differences in PV values among the four reanalyses, we advise using the same reanalysis for comparisons between the PVG tropopause and other variables. In particular, the tropopause PV values need to be consistently calculated from one dataset. This mitigates the risk of introducing errors stemming from variations in the PV fields across different reanalyses. However, comparing multiple reanalyses or models is important to represent the range of uncertainty in the state of the atmosphere.

To examine the variability and long-term changes in the PVG tropopause, we analyze the monthly-mean time series of the tropopause latitude ϕθTP(t) and potential vorticity PVθTP(t) between 1980 and 2017 on every isentropic level by means of multilinear regression. The regression function f (see Eq. ) includes regressors for the seasonal cycle, quasi-biennial oscillation (QBO), El Niño–Southern Oscillation (ENSO) and a long-term linear trend.

We examine potential simplifications of the PVG tropopause determination by comparing the original method (utilizing 6-hourly reanalysis data, zonally resolved, including the STJ wind speed) to three alternative methods applied to the ERA5 reanalysis, which employ monthly climatologies of the input data, employ zonal means of the input data while computing gradients with respect to geographical latitude instead of equivalent latitude and omit the STJ criterion, as detailed in Sect. . To evaluate how well these alternatives reproduce the PVG tropopause results from the original method, this section compares the seasonal multiyear climatologies, mean seasonal cycle and trends computed with the aforementioned methods.

Figure  shows that creating monthly averages of reanalysis data before applying the PVG tropopause algorithm does not substantially change the climatological latitude of the PVG tropopause between 340 and 370 K but leads to some deviations on the lower and higher isentropes. Omitting the subtropical jet wind criterion introduces noise, as previously observed by in the polar vortex. Since the PV gradient can be enhanced in various regions – such as the edge of the Monsoon anticyclone  – the subtropical jet wind speed acts as an additional constraint helping to identify the maximum in the PV gradient that corresponds to the PVG tropopause. Averaging the reanalysis zonally before the PVG tropopause algorithm and simply computing the PV gradient with respect to common latitude instead of equivalent latitude alter the results drastically, as the PV-gradient method is originally intended to work with equivalent latitudes, i.e., on contours of PV, based on concepts by , , and . We therefore continue the following analyses only with the monthly-mean, global reanalysis data including the wind criterion.

Figure  displays the climatological seasonal cycle of the PVG tropopause on three isentropic levels. On all isentropes, the seasonality of equatorward and poleward shifts is qualitatively similar in both datasets, but monthly climatologies tend to yield a more equatorward tropopause than the original 6-hourly reanalysis data. This deviation is especially noticeable in the NH on the 330 K isentrope (Fig. a) and the 370 K isentrope in boreal autumn (Fig. e), where the tropopause computed from monthly means is located 5 to 10° equatorward of the 6-hourly tropopause product.

The long-term trends of the PVG tropopause computed from monthly climatologies are presented in Fig. . The variation with altitude is qualitatively similar for monthly climatologies and subdaily data, indicating widening from 320 to 340 K, narrowing from 340 to 370 K and widening above. However, using monthly climatologies yields smaller widening trends from 320 to 340 K but larger narrowing trends from 340 to 370 K than those obtained from subdaily data.

As a conclusion, we do not recommend using zonal mean reanalyses, since computing the PV gradient with respect to geographical latitude instead of equivalent latitude fundamentally changes the method. In addition, the STJ wind criterion in Eq. () is necessary because the redundancy of the horizontal wind and PV-gradient maxima at the tropopause break stabilizes the PVG tropopause calculation.

Computing the PVG tropopause from monthly averaged data yields long-term climatologies similar to the original 5-hourly product in the range of 340 to 370 K, but the results may deviate noticeably on lower and higher isentropes. In addition, using monthly-mean data tends to result in a more equatorward tropopause. Comparing long-term trends, the monthly means qualitatively match the 6-hourly data but result in much stronger (almost double) narrowing trends. Overall, using monthly averages efficiently reduces computational effort and yields similar climatologies in a certain range but may not be sufficiently accurate in describing variabilities and trends. Based on these results, we advise using subdaily to daily data for variability and trend analysis of the PVG tropopause.

Given the significant regional variations in long-term trends observed for tropical width, the tropopause break and the subtropical jets , this study conducts a similar regional analysis of the PVG tropopause break. The global tropopause surface, θTP (ϕ,λ), was determined by identifying the corresponding PV contour on each isentropic level as detailed in Sect. . The results are presented in Fig.  as climatologies for the solstice seasons from 1980 to 2017, while the equinoctial seasons are shown in Fig. . These global fields reveal noticeable zonal variations in tropopause height and latitude, which can be attributed to undulations of the subtropical jet streams caused by Rossby waves, as well as baroclinic instability forming high- and low-pressure areas, leading to zonal variability in the PV field. The tropopause break is predominantly located between 30 and 40° latitude and is notably steeper in each winter hemisphere – visible in the Northern Hemisphere in Fig. a and in the Southern Hemisphere in Fig. b. As the PVG tropopause is originally defined in the subtropics, the tropopause was extended poleward with the PV isosurface corresponding to the 320 K contour, which generally falls between ±1 and ±4 PVU, aligning well with traditional definitions of the dynamical tropopause e.g.,.

Regional long-term trends of PVG tropopause latitude were computed at the tropopause intersections with each isentrope and every 10th meridian. As an example for the subtropical tropopause, the trends on the 330 and 360 K isentropes in both hemispheres are presented in Fig. , resolved by longitude. On the 330 K isentrope, trends range between 0 and 0.75° per decade in the Northern Hemisphere and -0.25 and 0.1° per decade in the Southern Hemisphere, indicating an overall tropical widening at lower levels. On the 360 K isentrope, trends vary between -0.5 and 0.25° per decade in the Northern Hemisphere and 0 and 0.8° per decade in the Southern Hemisphere, which corresponds to an overall equatorward narrowing of the upper troposphere. The zonal structure of the tropopause trends at 360 K shows the strongest equatorward narrowing trends (negative in the NH, positive in the SH) at longitudes above the Pacific between 90 and 180° W. Comparing these PVG tropopause trends on the 360 K isentrope with the results of concerning trends in the tropopause break shows a similar longitudinal pattern: the strongest narrowing trends are found over the east Pacific in both studies.