2.1 ECMWF operational analysis data

For the detection of cyclone tracks we use operational analysis fields from the integrated forecast system (IFS) from the ECMWF, for August to October from 2010 until 2014. The spatial extent of the area covers the North Atlantic from 60^∘ W to 20^∘ E and from 20 to 75^∘ N, and therefore encompasses the autumn maximum of Atlantic storm tracks (Wernli and Schwierz, 2006). Moreover, the choice of the time period as well as the region of the data was motivated by the preparation of the airborne measurement campaign WISE (Wave driven isentropic exchange) which took place over the North Atlantic in autumn 2017.

We use 6-hourly analysis fields, with a grid spacing of 0.25^∘ in the horizontal, a vertical level count of L91 for the years 2010 until 2012, and L137 for 2013 and 2014. The abbreviation L91 (L137) describes the 91 (137) vertical level of the IFS's native hybrid sigma coordinates ranging from the Earth's surface up to 0.01 hPa atmospheric pressure. We decided to use the operational analysis data as opposed to, for example, a more consistent reanalysis data set like ERA-Interim, due to the finer vertical resolution in the tropopause region. While ERA-Interim, with 60 model levels, has a vertical grid spacing of about 1 km in the UTLS, the operational analysis has a vertical grid spacing of about 300–400 m, depending on the tropopause location and on the vertical grid spacing of L91 and of L137. In particular, this leads to a much better representation of the static stability in the lower stratosphere. The formation of the TIL in numerical models is known to be sensitive to the horizontal and vertical resolution as well as their ratio (e.g. Birner, 2006; Wirth and Szabo, 2007; Son and Polvani, 2007; Erler and Wirth, 2011).

We use analysis data on model levels which provide the best vertical resolution in the UTLS of roughly 300 m. Many of the desired variables such as the temperature T, the three-dimensional wind (u, v, ω), the cloud ice water content ciwc, and relative vorticity ζ_rel are directly provided by the ECMWF, while other quantities have to be derived from the primary fields, such as static stability N², potential vorticity PV, vertical wind shear S², and the Richardson number Ri.

We define the strength of the TIL as the maximum in static stability within 3 km above the lapse rate tropopause. The lapse rate tropopause is defined as the lowest level where the temperature lapse rate falls below 2.0 K km⁻¹ and its average between this level and all higher levels within 2 km above this level remains below this value (WMO, 1957). We use this specific TIL strength definition, since the high-resolution data show large variability in the UTLS region, with several maxima often evident above the tropopause. Therefore, we find this definition of the TIL strength to be preferable over, for example, the first maximum in static stability above a threshold ($\mathrm{4}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻², e.g. Gettelman and Wang, 2015). Another way to describe the static stability above the tropopause is to calculate the mean value of N² over a predefined vertical extent relative to the tropopause (e.g. Kunkel et al., 2014). While such a TIL strength definition achieves very similar results on a synoptic scale to the ones we will describe in Sect. 3, it also filters a lot of small-scale variability and therefore partly neutralises the advantage of analysing high-resolution data.

2.2 Cyclone tracking

A major goal of this study is to analyse the evolution of dynamical features in the UTLS in life cycles of baroclinic waves and link these with the evolution of the static stability N² above the tropopause. Baroclinic life cycles up to the point of breaking are often associated with surface cyclones (e.g. Thorncroft et al., 1993), and the flow in the UTLS above these cyclones is an important region with regard to the enhancement in static stability above the tropopause. Several methods are available to trace cyclones, using for example the associated maximum in relative vorticity on lower- or middle-tropospheric pressure levels, or the minimum in mean sea level (MSL) pressure. The IMILAST experiment (Neu et al., 2013) showed that many of these methods achieve comparable results. We tested several methods with short time periods and with comparable results. Ultimately, we decided to use the MSL pressure field due to its smoothness compared to, for example, the relative vorticity, which makes it easier to identify cyclone centres. Our algorithm based on Hanley and Caballero (2012) therefore identifies surface cyclones from local minima in the MSL pressure field and traces them in time and space. Since our data have a fine horizontal grid spacing and are limited to the North Atlantic, we had to partly adapt the tracking algorithm to our data set. The major steps are (1) smoothing of the MSL pressure field, (2) identification of all local minima at all time steps, and (3) the connection of the local minima from consecutive time steps to cyclone tracks. The following paragraph will give more details.

A local minimum in a gridded MSL pressure field is defined as a grid point that has a lower value than its surrounding eight grid points. To reduce the amount of local MSL pressure minima found at each time step, a Cressman filter (Cressman, 1959) is applied, averaging each grid point in the field with its neighbouring grid points within a radius r<r₀ (r₀ being 500 km), using weights of $(r_{\mathrm{0}}^{\mathrm{2}}-r^{\mathrm{2}})/(r_{\mathrm{0}}^{\mathrm{2}}+r^{\mathrm{2}})$. The smoothed field exhibits fewer local minima, reducing the amount of criteria needed to define a cyclone centre, without altering the tracks of the cyclones fundamentally. After applying the Cressman filter and following Hanley and Caballero (2012) once more, the MSL pressure field at each time step is projected onto an area-preserving Lambert projection centred at the North Pole, to counteract the bias in zonal resolution caused by the convergence of the meridians on the native latitude–longitude grid. The projected MSL pressure field is then interpolated onto a regular equidistant grid with 28 km grid spacing, which corresponds to the 0.25^∘ horizontal resolution at the Equator. The algorithm then searches and saves every local minimum in the MSL pressure field, with two extra criteria being applied: (1) an upper threshold of p_t=1007.25 hPa, and (2) the neglection of all minima located over orography higher than 1500 m. The first criterion replaces the pressure gradient criterion applied in Hanley and Caballero (2012), since the limitation to a regional domain makes it difficult to calculate consistent pressure gradients. The value for p_t was determined by testing several values below 1013.25 hPa, with 1007.25 hPa being the largest value of minimum MSL pressure where our algorithm was able to connect the local minima to coherent cyclone tracks. Our algorithm therefore neglects very weak minima in the pressure field, but since weak cyclones or cyclones in very early or very late stages of their life cycles are often not strongly connected to features in the upper tropospheric flow, it is sufficient for this study to not account for these stages of the cyclone life cycle. Also, we are focussing on the time periods around the mature stage of the baroclinic waves, i.e. when the MSL pressure reaches its lowest values. The second criterion is another result of the IMILAST experiment (Neu et al., 2013) and is applied due to the error associated with reducing the surface level pressure to sea level from such altitudes.

In the next step, the algorithm connects minima from consecutive time steps by searching in a given radius for the nearest minimum. For minima associated with a newly formed cyclone, the search radius in the second time step is 720 km from the position where the cyclone first appears. For minima already existing for two or more time steps, the algorithm follows Wernli and Schwierz (2006) with a “first-guess” approach, where the first-guess location of the cyclone is a linear continuation of the track in latitude–longitude coordinates: ${\mathit{x}}^{*}\left(t_{n+\mathrm{1}}\right)=\mathit{x}\left(t_n\right)+\mathrm{0.75}\left[\mathit{x}\right(t_n)-\mathit{x}(t_{n-\mathrm{1}}\left)\right]$. Wernli and Schwierz (2006) introduce the factor of 0.75 because cyclone movement tends to get slower during a cyclone's life cycle. The corresponding MSL pressure minimum is then defined as the nearest minimum from ${\mathit{x}}^{*}\left(t_{n+\mathrm{1}}\right)$ within a radius of 840 km. For more information concerning the values of the search radii and the first-guess approach see Hanley and Caballero (2012) and Wernli and Schwierz (2006). Following another result from the IMILAST experiment (Neu et al., 2013) only cyclones with a lifetime of at least 24 h are considered, which translates to at least five 6-hourly time steps in the IFS analysis data. The algorithm furthermore neglects cyclones with less than two time steps before and/or after the global minimum in MSL pressure along their path to make sure that the actual intensification period is covered by the data. We want to emphasise that due to these two criteria only extratropical cyclones are selected. Tropical cyclones in extratropical transition emerging from the western edge of our data region which might have a strong signal in the MSL field but no real intensification period are neglected by the algorithm. Figure 1 shows the cyclone paths tracked by the algorithm after applying all criteria. The distribution of tracks matches well with the climatological cyclone frequencies described in Wernli and Schwierz (2006). Aside from the large accumulation over the North Atlantic there are also several tracks located over northern Africa and the Mediterranean Sea. The relatively small number of tracks over the Mediterranean Sea can be explained by the p_t=1007.25 hPa upper limit criterium and the fact that these Mediterranean cyclones hardly exhibit strong minima in the MSL pressure field. This study focusses on Atlantic storm tracks, and therefore the cyclones over northern Africa and the Mediterranean Sea are sorted out by a geographical criterion.

Figure 1Mercator projection of the area covered, with all 139 cyclone tracks found by the algorithm. Crosses indicate the location of minimum sea level pressure along each track.

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Composites of extratropical cyclones

Ultimately, we want to analyse the variability of the tropopause inversion layer within extratropical baroclinic waves. For this, we compute composites of the cyclones at the time of maximum intensity, which we define as the occurrence of the global minimum surface pressure along the track. We select a subset of the gridded data as provided by the ECMWF for each cyclone by rotating the pole of a spheric polar coordinate system (Θ, ϕ) onto the centre of the cyclone and interpolated the original data onto that new grid. The horizontal resolution of the new coordinate system is set to 0.25^∘ and the spherical cap radius to ${\mathrm{\Theta }}_{max}=\mathrm{15}$^∘. The radius is chosen such that all relevant features around the cyclone centre are covered. Figure 2 illustrates the interpolation of the data from the original latitude–longitude grid provided by the ECMWF (light grey area for the whole sphere, dark grey for the limited area used in this study) onto an arbitrarily placed new spherical polar coordinate system (orange area). The rotation of the new coordinate system onto the cyclone centre is performed following Bengtsson et al. (2007), who provide a detailed description of the rotation matrix in their appendix. The original ECMWF IFS variables in latitude–longitude coordinates are interpolated onto a column covered by the new coordinate system, with the vertical coordinate using the lapse-rate-based tropopause as a reference altitude (Birner, 2006). The tropopause height is defined as zero with negative (positive) height values below (above) and a vertical grid spacing of Δz=100 m. The interpolation onto the new coordinate system based on the centre of the surface cyclone does not account for a potential vertical tilt of the cyclone. However, since this study focusses on the point in time of maximum cyclone intensity when the vertical structure tends to align, we find a tilt to be negligible. The ensemble of tropopause-based columns of variables from each cyclone is then averaged to create a three-dimensional composite of the flow in the vicinity of the cyclones. Subsequently, the mean absolute height coordinate is recovered as follows. Each horizontal coordinate (Θ, ϕ) of each individual cyclone within the new coordinate system still carries the information about the absolute tropopause height at that location, and therefore a mean tropopause height can be calculated and assigned to that horizontal location within the composite.

Figure 2Schematic of the size of the analysed region (dark grey) in comparison to the full sphere, as well as the size of a rotated spheric polar coordinate system (orange) with ${\mathrm{\Theta }}_{max}=\mathrm{15}$^∘ radius. The dark grey semi-transparent upright plane illustrates (with exaggerated vertical extent) the vertical cross section aligned from north to south as displayed in Fig. 9.

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In the special case of composites of horizontal or quasi-horizontal fields like PV on isentropes or the TIL strength, we first calculate the fields for each cyclone and then afterwards the mean. This method preserves more information because the three-dimensional tropopause-based averaging still smoothes vertical information due to the variability of, for example, the height of the maximum in N² above the tropopause. In contrast to other studies analysing cyclone composites (e.g. Bengtsson et al., 2007; Catto et al., 2010), we keep the orientation of each individual cyclone instead of rotating them depending on their path of migration. In our case this approach leads to a better representation of the dynamical and thermodynamical features in the UTLS.

3.1 Baroclinic wave with LC2 characteristics

Figure 3 shows three consecutive time steps of the evolution of the first baroclinic wave with the time of maximum intensity depicted in the second row. We focus our analysis and discussion on the flow features inside the 15^∘ radius around the cyclone centre. Furthermore, we focus our discussion on sea level pressure, isentropic potential vorticity (IPV) at 330 K, strength of the tropopause inversion layer as defined in the beginning of Sect. 2, and relative vorticity at the lapse rate tropopause (from left to right in Fig. 3).

Figure 3Evolution of a baroclinic wave-breaking event as seen in ECMWF IFS analysis data. The middle row represents 14 October 2014 at 06:00 UTC (the point in time of maximum cyclone intensity), and the upper and lower row show the situation 24 h prior and past the maximum intensity. Column (a) shows the pressure at mean sea level (solid lines p_msl≤1013 hPa, dashed lines p_msl>1013 hPa, in steps of 5 hPa), (b) the IPV on 330 K, (c) the maximum of static stability $N_{max}^{\mathrm{2}}$ within 3 km above the thermal tropopause as an indicator for the TIL strength, and (d) the relative vorticity at lapse rate tropopause height. Blue lines in the middle row show 40 and 50 m s⁻¹ horizontal wind speed isolines at 200 hPa. The dashed black line shows the path of migration of the tracked cyclone, with the position of the cyclone centre at the point in time of the meteorological field displayed marked by the black ×. Black circles show the 15^∘ radius used for the composites. Note the deformation of the circles due to the mercator projection.

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A total of 24 h before the cyclone reaches maximum intensity the MSL pressure already falls below 975 hPa. The IPV shows a baroclinic wave centred above the tracked surface cyclone centre, with large IPV values coming from high latitudes and tilting in the northwest–southeast direction into the jet. During the intensification of the surface cyclone, the upper air wave enters what Thorncroft et al. (1993) call the cyclonic wrap-up phase, which in this case study exhibits the distinct features of an LC2. During the wrap-up, which is vertically aligned with the strong surface cyclone, the trough stays on the northern cyclonic shear side of the jet streak as indicated by the 200 hPa horizontal wind maximum in Fig. 3d. The wave-breaking event is meridionally confined by the jet streak and only a very thin streamer of enhanced IPV air turns anticyclonically on the last day depicted in Fig. 3.

The relative vorticity at lapse rate tropopause height correlates with the IPV where cyclonic (anticyclonic) flow exhibits large (low) values of IPV. The regions with relatively small (large) values of IPV and anticyclonic (cyclonic) flow exhibit enhanced (reduced) values of static stability above the tropopause. This correlation especially holds true inside the 15^∘ radius area and at the point in time of maximum cyclone intensity. This agrees well with the anticipated role of balanced dynamics (Wirth, 2003, 2004) and idealised baroclinic life cycle simulations with a focus on the evolution of the TIL (e.g. Erler and Wirth, 2011). It furthermore agrees with the measurement-based study by Pilch Kedzierski et al. (2015) concerning the correlation between the upper tropospheric relative vorticity and the static stability above the tropopause within troughs and ridges. The regions with the largest values of static stability up to $N^{\mathrm{2}}=\mathrm{10}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻² are located inside the ridge in the first time step and later in the second time step in the tropospheric IPV air mass which is wrapped up around the underlying cyclone centre. In the last time step, the TIL strength especially within the 15^∘ radius reverts to lower values. We want to highlight the fact that the enhancement of the TIL strength during the wave-breaking event is not uniform, but rather shows a large spatial variability and wave-like patterns on different horizontal scales. The regions with the strongest enhancement of static stability above the tropopause in the second time step depicted in Fig. 3, as well as inside the deformed ridge in the third time step, are associated with the regions commonly affected by the warm conveyor belt (Madonna et al., 2014). The connection between the warm conveyor belt outflow and the regions of enhanced static stability above the tropopause is also a feature of the baroclinic life cycle simulations by Kunkel et al. (2016).

Figure 4Vertical cross section over the North Atlantic on 14 October 2014 at 06:00 UTC (a) at 42^∘ N and (b) at 60^∘ N. The filled contour as well as the thin solid black contour lines show static stability N² in steps of $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻², dashed black lines show isentropes. The bold solid black line indicates the lapse rate tropopause, the dotted red line shows the 2 pvu isoline of potential vorticity.

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The TIL strength at lower latitudes in the quasi-horizontal illustration (Fig. 3c) sometimes exhibits strong gradients or jumps which appear non-coherent. These strong gradients are an artefact of the evaluation, due to (1) the restriction of the TIL strength to the global maximum in static stability within 3 km above the lapse rate tropopause, (2) the high variability of the tropopause defining lapse rate in the high-resolution data, (3) the occurrence of sharp tropopause jumps and double tropopauses (especially during wave-breaking events), and (4) the strongly structured TIL itself. The occurrence of such features should be kept in mind when analysing such synoptic situations in a high-resolution data set. We compared such prominent structures in the quasi-horizontal illustration of the TIL with a variety of vertical cross sections, to ensure that the statements we make about the TIL evolution hold true and are in fact not a numerical evaluation artefact. Figure 4a shows an example for a vertical cross section corresponding to the second time step depicted in Fig. 3 and at 42^∘ N. It illustrates the large variability of the three dependent variables PV, static stability N², and the lapse rate tropopause height. The jump of the lapse rate tropopause at about 37^∘ W as well as the small-scale stratospheric PV intrusion at about 12^∘ W illustrate the complexity of the tropopause location and the challenge to account for this by the different definitions of the tropopause and the TIL. The vertical cross section in Fig. 4b shows the TIL structures at 60^∘ N on the same day and north of the cyclone centre. It illustrates that the regions of enhanced static stability exhibit less variability at this latitude. They are located above the ridge between 40 and 10^∘ W, with the wave-like horizontal pattern that is also visible in Fig. 3c.

3.2 Baroclinic wave with LC1 characteristics

Figure 5 shows the subsequent baroclinic wave-breaking event 3 d later over the North Atlantic. The MSL pressure on 16 October still shows two distinct minima: the decaying northern one associated with the previous wave-breaking event, and a newly formed minimum. The background state of the UTLS is still significantly distorted due to the LC2 wave-breaking event described in Sect. 3.1. Similar to the previous case, a relatively small scale baroclinic wave is evident in the IPV field above the underlaying surface cyclone centre. In contrast to the previous case, as the surface cyclone grows stronger and the upper air wave enters the wrap-up phase (Fig. 5, middle row), the initially cyclonically tilted trough with enhanced values of IPV turns anticyclonically and later on a substantial part of the trough penetrates the jet in its excursion southwards. The jet splits into two jet streaks (Fig. 5d, middle row) and the trough gets thinned, eventually producing a cut-off (Fig. 6). These wave-breaking characteristics meet the definition of an LC1 as described by Thorncroft et al. (1993). During the evolution of the cyclone the relations between tropospheric IPV, anticyclonic relative vorticity at tropopause height, and an enhancement in static stability above the tropopause are all evident. The regions of maximum static stability with values up to $N^{\mathrm{2}}=\mathrm{10}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻² are again located inside the ridge of low IPV air wrapping up around the underlaying cyclone centre at about the time of maximum cyclone intensity. The thinning and southward-moving trough itself exhibits low values of static stability N² above the tropopause, in agreement with its positive relative vorticity, but the flow around the trough shows no distinct strong signal in the quasi-horizontal static stability distribution, especially when compared to the cyclonic wrap-up 24 h earlier.

Figure 5As in Fig. 3, but for the subsequent cyclone with maximum intensity on 17 October 2014 at 18:00 UTC, depicted in the middle row. Top row shows 24 h earlier and bottom row 24 h later.

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Figure 6As in Fig. 3, but only for one point in time, showing the tracked cyclone associated with the upper tropospheric cut-off from the previous baroclinic wave breaking.

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Figure 6 shows a comparable evolution about two days later for the secondary cyclone associated with the cut-off which formed from the thinning streamer. This cyclone was also tracked by the algorithm, and, while it is weaker and exhibits less horizontal extent than the two previous cases, the region of maximum static stability with values of up to $N^{\mathrm{2}}=\mathrm{10}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻² still evolves inside the wrapped-up low-IPV air. The wrapped-up cut-off exhibits maximum static stability values of about $N^{\mathrm{2}}=\mathrm{4}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻². The regions of high static stability above the tropopause are located north of the cyclone centre inside the flow with negative relative vorticity, as well as inside the flow that turns anticyclonically northwest of the cyclone centre and towards the jet maximum (as indicated by the blue contour lines in Fig. 6d).

In conclusion, we analysed two subsequent baroclinic life cycles, the first resembling an LC2, and the second an LC1 when compared with idealised baroclinic life cycle simulations. In both cases the regions of strongest enhancement in static stability above the tropopause are located inside the ridge of low-IPV air moving northward from low latitudes and wrapping up around the underlaying primary cyclone about the time of maximum cyclone intensity. The flow around the southward excursion of the trough in the LC1 wave exhibits no distinct signal, until a secondary cyclone associated with the cut-off from the trough evolves. This cyclone-linked behaviour can also be seen in the idealised baroclinic life cycle simulations from Erler and Wirth (2011). Their Fig. 4 shows a comparable evolution of the TIL strength with consideration that there are two northern surface cyclone intensification periods and one cut-off related cyclone forming at low latitudes.

Based on the analysis of these two case studies, we further motivate the analysis of the evolution of the TIL during baroclinic life cycles based on the flow above surface cyclones. We recognise that surface cyclones exist which are not linked to baroclinic wave-breaking events and also baroclinic wave-breaking events (in the sense of meridional irreversible redistribution of isentropic potential vorticity) exist which are not linked to surface cyclones. Nevertheless, we expect a composite of strong surface cyclones to give a characteristic representation of the flow where a strong TIL forms during the subset of those baroclinic wave-breaking events which are in fact linked to surface cyclones.