This study is based on the reanalysis data set ERA5 of the European Centre for Medium-Range Weather Forecasts (ECMWF), which is available from 1950 to the present . We use data for the period of 1 January 1979–31 December 2019, remapped from the original T639 spectral resolution to a regular latitude–longitude grid. For the identification of upper-tropospheric PVAs in the quasi-Lagrangian approach, we select ERA5 model-level data for maximum possible vertical resolution, with a temporal resolution of 3 h and a horizontal grid spacing of 0.5∘. We use spatially coarser data (1.0∘) for the PV inversion, also with 3-hourly resolution and with 17 pressure levels (1000, 950, 925, 900, 850, 800, 700, 600, 500, 400, 300, 250, 200, 150, 100, 70, and 50 hPa). Mean temperature and wind tendencies at model levels from ERA5 short-range forecasts serve to estimate non-conservative processes, with a spatial resolution of 0.5∘ in the horizontal and a temporal resolution of 1 h. ERA5 provides these tendencies accumulated over the previous hour, from which we calculate a 3-hourly mean around the analysis time (e.g., taking the mean of data valid at 02:00, 03:00, and 04:00 UTC for the analysis at 03:00 UTC). We distinguish here between tendencies from all parameterizations and non-radiative parameterizations.

This work uses Ertel's PV as q=σ-1(ζθ+f) in its hydrostatic approximation on isentropic levels, where ζθ is the component of relative vorticity perpendicular to an isentropic surface, f the Coriolis parameter, and σ=-g-1(∂p/∂θ) the isentropic layer density with gravity g, pressure p, and potential temperature θ. The PV tendency equation is given by isentropic advection and non-conservative PV modification (N): 1∂q∂t|θ=-v⋅∇θq+N, with v=(u,v,0) the horizontal isentropic wind vector and ∇θ the gradient operator along an isentropic surface. The non-conservative PV modification is given by 2N=-θ˙∂q∂θ+1σ∇θ×v+fk⋅∇θθ˙+k⋅∇θ×v˙, with k=(0,0,1) the unit vector perpendicular to an isentropic surface, θ˙ the non-conservative heating rate, and v˙ values the sources and sinks of non-conservative momentum (e.g., friction or gravity wave drag). We estimate the non-conservative PV modification term N with the mean 3-hourly temperature and wind tendencies introduced in Sect. .

The advection term in Eq. () is further separated into different processes . PVAs (q′) are defined as deviations from a climatological background state q0 such that q′=q-q0 is valid. For each time step within the year (recall Δt = 3 h), averages are constructed based on the period of 1980–2019. This is used for the climatological background state q0 defined as a 30 d running mean climatology centered on the respective time to ensure smooth transitions in q0 between consecutive time steps. Following the basics of PV thinking, the three-dimensional distribution of PVAs can be further separated into upper-tropospheric and lower-tropospheric PV (and temperature) anomalies. Here, the separation level of upper-tropospheric and lower-tropospheric PV anomalies lies between 600 and 650 hPa. Piecewise PV inversion with the nondivergent wind field under nonlinear balance is performed on pressure levels between 25 and 80∘ N and yields the wind fields vup′ and vlow′ associated with the upper-tropospheric and lower-tropospheric PV anomalies, respectively. It is important to note that our quantitative analysis is performed for the evolution of upper-tropospheric anomalies. From the PV perspective of midlatitude dynamics, we may thus consider the impact of the upper-tropospheric anomalies on themselves (mediated by vup′) as quasi-barotropic dynamics and the impact of the lower-tropospheric anomalies on the upper-tropospheric anomalies (mediated by vlow′) as baroclinic dynamics. The piecewise PV inversion thus provides the possibility to consider the influence of the dynamics in the lower troposphere and the influence of the wave on itself separately from each other. The background wind field v0 is obtained in the same way as the background PV field q0 as a 30 d running mean climatology (1980–2019) centered on the respective calendar day and is approximately balanced. Per definition, the wind fields vup′ and vlow′ from piecewise PV inversions are nondivergent and will hence be further complemented by the divergent flow vdiv′. All wind fields are interpolated to isentropic levels. Following , we select an isentropic level average around 320 K (namely 315, 320, and 325 K) for the EuBL in March 2016. The full wind field can finally be divided into 3v=v0+v′=v0+vdiv′+vrot′=v0+vdiv′+vup′+vlow′+vres′.

We introduce here the residual vres′, which arises due to (i) characteristics inherent in piecewise PV inversion, e.g., nonlinearities and imperfect knowledge of boundary conditions; (ii) numerical inaccuracies; and (iii) the interpolation of wind fields from pressure to isentropic levels. A more detailed discussion of the PV partitioning, the piecewise PV inversion technique, and the residual is given in .

With the partitioning of PV into a background state and anomalies thereof, and the associated partitioning of the wind field, a PV-anomaly tendency equation can be written following Eq. () as 4∂q′∂t=∂q∂t-∂q0∂t=∂q′∂t|qb+∂q′∂t|bc+∂q′∂t|div+∂q′∂t|eddy+∂q′∂t|noncons+∂q′∂t|res=-vup′⋅∇q0+v0⋅∇q′-vlow′⋅∇q0-vdiv′⋅∇q-∇⋅(vrot′q′)+N-vres′⋅∇q0+∂q0∂t+v0⋅∇q0, where we have included in the residual term the (very small) tendencies due to our use of a slowly varying background state.

The first term (in square brackets) describes the PV thinking of (upper-tropospheric, linear) barotropic Rossby waves , hereafter referred to as quasi-barotropic PV tendency. For a linear wave in uniform background flow, both individual contributions to this tendency are in quadrature with the PVAs but with opposite signs. The first contribution (vup′⋅∇q0) represents intrinsic (phase and group) propagation, including the downstream development and amplification of anomalies at the leading edge of a Rossby wave packet. The second contribution (v0⋅∇q′) represents the Doppler shift, i.e., the advection of the wave pattern by the background flow. In a background flow with horizontal shear, this term contributes to the deformation of PVAs. The second term in Eq. () (vlow′⋅∇q0) describes baroclinic interaction with lower-tropospheric PVAs, which leads on average to baroclinic growth, i.e., the amplification of upper-tropospheric ridge and trough anomalies e.g.,. This term will hereafter be referred to as baroclinic PV tendency. The third term describes the impact of the divergent flow (hereafter referred to as divergent PV tendency). It is difficult to accurately attribute this divergent flow to individual processes, e.g., dry balanced dynamics vs. moist processes. It is usually most reasonable, however, to attribute large values of the divergent PV tendency near the tropopause to latent heat release, which invigorates mid-tropospheric ascent and hence divergent outflow aloft. A more detailed discussion of this issue can be found in and . In the current study, we verify this relationship explicitly using trajectory calculations. The fourth term in Eq. () describes the nonlinear redistribution of PVAs in terms of the convergence of PVA flux by the (anomalous) rotational wind (∇⋅(vrot′q′)), hereafter referred to simply as eddy flux convergence. Note that eddy flux convergence may change PVAs locally but may neither generate new nor amplify existing PVAs in a globally averaged sense (because the flux vanishes at the boundary of the global domain). Furthermore, eddy flux convergence may not change the area-integrated amplitude of PVAs that are defined by a boundary at which q′=0 . The fifth term (N) depicts the direct non-conservative PV modification and contains all non-conservative processes, like latent heat release, friction, and radiation.

In the later Sects.  and , our description of the evolution of PVAs associated with the dynamics of blocked weather regimes in the Eulerian and quasi-Lagrangian perspectives will build on the tendency in Eq. ().

In this study, we use the year-round definition of seven weather regimes in the North Atlantic–European region (NAE; 80∘ W–40∘ E, 30–90∘ N; dashed gray in Fig. a) by adapted to ERA5. Geopotential height anomalies are calculated based on a 90 d running mean climatology from 1979–2019 at 500 hPa with a temporal resolution of 6 h. Anomalies are additionally filtered by a 10 d low-pass filter Lanczos filter; to exclude high-frequency signals. After normalization of the anomalies for a year-round definition, k-means clustering is performed for the expanded phase space of the leading seven empirical orthogonal functions that describe 74.4 % of the variability. A weather regime is then defined as the cluster mean of one of seven clusters, which was shown to be the optimal number in the year-round definition. The seven weather regimes consist of three cyclonic regime types (zonal regime – ZO, Scandinavian trough – ScTr, Atlantic trough – AT) and four anticyclonic regime types (Atlantic ridge – AR, European blocking – EuBL, Scandinavian blocking – ScBL, Greenland blocking – GL).

In this study, we apply different methods to quantify weather regime dynamics to a EuBL regime life cycle. The mean year-round regime pattern for EuBL is shown in Fig. a. A positive geopotential height anomaly over the eastern North Atlantic and Europe dominates the regime pattern and is flanked by two areas of negative geopotential height anomalies upstream over Greenland and downstream over the Mediterranean and East Asia.

To make a quantitative statement about the similarity of an instantaneous pattern to the seven weather regimes, we use the weather regime index (IWR) , defined as 5IWR(t)=PWR(t)-PWR‾1NT∑t=1NT[PWR(t)-PWR‾]2, withPWR(t)=∑(λ,φ)∈NHΦL(t,λ,φ)ΦWRL(λ,φ)cos⁡φ∑(λ,φ)∈NHcos⁡φ, where NT is the total number of time steps within a climatological sample (all times in 1979–2019), PWR‾ the climatological mean of the projection PWR, ΦL(t,λ,φ) the low-frequency geopotential height anomaly at 500 hPa, ΦWRL the low-frequency geopotential height pattern that defines the weather regime, and (λ,φ) the respective latitude–longitude in the Northern Hemisphere (NH).

Objective weather regime life cycles are derived based on the IWR for each regime and time step. Following , a regime life cycle is defined as a persistent IWR above 1.0 for more than 5 consecutive days that shows for at least one time step the highest IWR in all seven regimes. A weather regime life cycle is called active if the IWR lies above 1.0, and the first time step at which IWR > 1.0 is defined as the onset of the life cycle. The decay is then set as the first time at which the IWR is below 1.0 again. The course of the weather regime index around the EuBL life cycle in March 2016 is shown in Fig. .

We are interested in the processes governing the dynamics of the EuBL in March 2016 from a PV perspective. Because EuBL is defined in terms of a low-frequency anomaly pattern (Sect. ), we here apply the same 10 d low-pass filter to the PVAs, hence considering low-frequency PVAs qL. Following, e.g., and , who studied the stream function evolution of low-frequency modes like the North Atlantic Oscillation, we define a normalized projection of qL and its tendencies onto the low-frequency PV pattern of the weather regime, qW RL, as 6PqL(t)=∑(λ,φ)qL(t,λ,φ)qW RL(λ,φ)cosφ∑(λ,φ)qW RL2(λ,φ)cosφandP∂qL/∂t(t)=∑(λ,φ)∂qL(t,λ,φ)/∂tqW RL(λ,φ)cosφ∑(λ,φ)qW RL2(λ,φ)cosφ.

Note that the projection of the tendency is equivalent to the tendency of the projection PqL because qW RL is constant with time, and the operator ∂/∂t commutes with the summation operator. The normalized projection is performed over the Northern Hemisphere in the latitudinal band between 25 and 80∘ N, since the PV tendencies are limited to this domain. The low-pass-filtered anomaly qL is averaged between 315 and 325 K (Sect. ). The weather regime PV pattern qW RL is defined as the mean of the low-frequency PVAs vertically averaged between 500 and 150 hPa on active life-cycle days (shown for EuBL in Fig. b), consistent with the definition used in the quasi-Lagrangian perspective to track PVAs (Sect. ). Analogous to the definition of the IW R in Sect. , we subtract the climatological background value and do a standardization. PqL then closely resembles the IW R and describes how similar a certain PV pattern is to one of the weather regimes (Eq. ). The similarity of PqL and IW R is shown for the EuBL regime life cycle in March 2016 in Fig. . Due to the large qualitative agreement, we conclude that the dynamics underlying the evolution of IW R can be interpreted in terms of the dynamics of PqL.

The individual contributions governing the evolution of PqL (namely the quasi-barotropic term – QB, the baroclinic term – BC, the divergent term – DIV, the convergence of the eddy flux term – EDDY, the non-conservative term – NON-CONS, and the residual term – RES) are obtained by applying our 10 d low-pass filter to each term in Eq. () and inserting that term into the RHS (right-hand side) of Eq. (). This leads to the projection defined as 7P∂qL/∂t(t)|j=∑(λ,φ)∈NH∂qL(t,λ,φ)∂t|jqW RL(λ,φ)cosφ∑(λ,φ)∈NHqW RL2(λ,φ)cosφ,j∈QB, BC, DIV, EDDY, NON-CONS, RES, which describes a normalized pattern correlation between the low-frequency PV tendencies and the regime pattern. If a projection is positive, the associated process favors a given regime pattern. If a projection is negative, the associated process works against a regime pattern. The observed temporal evolution of PqL agrees very well with the diagnosed evolution, ∑jP∂qL/∂t|j (Fig. ). There is a near-constant difference of 0.07 d-1 between the diagnosed and observed evolution (a relative difference of 12 % at onset time), which increases during the decay stage of the regime. However, the diagnosed tendencies still capture the overall evolution of PqL very well. In combination with the relative smallness of the difference, we thus assert that the assessment of the relative importance of individual processes is not compromised.

We complement the Eulerian perspective with a quasi-Lagrangian perspective that follows upper-tropospheric anticyclonic PVAs in blocked weather regime life cycles and quantifies the processes in their PVA amplitude evolution. To identify contributing PVAs, we first identify and track negative upper-tropospheric PVAs as the vertical average between 500 and 150 hPa based on ERA5 model-level data and then define the spatial overlap of each of the identified PVAs with the defining weather regime PV pattern. The identification and tracking of negative PVAs closely follow the blocking diagnostics of , with the following modifications: (i) PVAs are calculated as deviations from a 30 d running mean climatology (1980–2019) centered on the respective calendar day, (ii) PVAs are not smoothed in time but smoothed spatially over a scale of 150 km, (iii) a weaker threshold of -0.8 PVU is used which enables early detection, and (iv) the spatial overlap criterion for the tracking of the PVAs is reduced to a minimum (overlap > 0 %) to enable the tracking of fast-moving, transient PVAs (compared to quasi-stationary PVAs in ). The fixed threshold of -0.8 PVU is used exclusively in this case study and captures approximately the 35 % strongest negative PVAs in the Northern Hemisphere in terms of area for the period of 1980–2019. In this context, the threshold value fulfills both conditions required: (1) it is close enough to q′ = 0 such that the budget of the integrated PVA amplitude of can be closed as well as possible, and (2) it is far enough away from q′ = 0 such that the grouping of PVAs over the entire Northern Hemisphere by single thin filaments is avoided, which enables the investigation of single distinct PVAs. Furthermore, the algorithm is modified to identify splitting and merging events. For further details about the identification and tracking the interested reader is referred to the Appendix (Appendix , Fig. ). In the remainder of this study, the negative upper-tropospheric PVAs identified and tracked in the quasi-Lagrangian perspective are referred to as PVAqL-'s in the following. The assignment of PVAqL-'s to active weather regime life cycles is based on the spatial overlap with a predefined regime mask. The regime mask is defined as the area encapsulated by the area where the values of the weather regime pattern are smaller than -0.3 PVU (bright green contour line in Fig. b). PVAqL-'s that exhibit at least a 10 % overlap with the regime mask at the time of an active regime are attributed to that specific regime life cycle. In general, more than one PVAqL- may contribute to a given regime life cycle.

We apply the PV framework of to PVAqL- tracks and consider the importance of processes that contribute to the amplitude evolution. This allows for deeper insights into the associated dynamics. Our amplitude metric is the spatial integral of q′ over the area A of the PVAqL-, i.e., ∫Aq′dA. Using the Leibniz integral rule, the tendency equation for this amplitude metric following their Eq. 6 is given by 8ddt∫Aq′dA=∫A∂q′∂tdA+∮Sq′(vs⋅n)dS, with the normal vector n in outward direction of the boundary S of A, and vs is the motion of S. The first term on the right describes the contributions to the amplitude strength, whereas the second term represents the contribution to the area change. We expand the first term of the RHS according to the partitioning introduced in Eq. (), which yields 9ddt∫Aq′dA=∫A-vup′⋅∇q0dA︸QB+∫A-vlow′⋅∇q0dA︸BC+∫A-vdiv′⋅∇q0+q′(∇⋅vdiv′)dA︸DIV+∫ANdA︸NON-CONS+∫A-v0⋅∇q0dA︸BGA+∫A-vres′⋅∇q0dA︸RES+∫A-∂q0∂tdA︸∂q0/∂t+∫A-∇⋅(vq′)dA+∮S(t)q′(vs⋅n)dS︸BND, where we have rearranged terms, used the nondivergence of the balanced flow (v0, vup′, vlow′), and assumed the residual wind vres′ to be nondivergent too

Nonlinearly balanced flow is nondivergent, and we consider the residual wind to largely stem from the decomposition of the nondivergent flow by piecewise PV inversion. We find, however, that the individual balanced flow components exhibit small but non-negligible values of horizontal divergence when evaluating on isentropic levels. We attribute this spurious divergence to numerical inaccuracies when interpolating from pressure to isentropic levels and the simplification of using the horizontal wind only for the interpolation and when evaluating divergence. For the sum of v0 + vup′ + vlow′ + vres′, however, the divergence vanishes. Therefore, we now do not further consider the spurious individual terms in our PV budget.

We complement the Eulerian and quasi-Lagrangian perspectives with a Lagrangian perspective that focuses on the diabatic history of air parcels that end up in the PVAqL- associated with a blocked regime. We employ the Lagrangian analysis tool LAGRANTO , using three-dimensional wind fields from ERA5 model-level data. The diabatic history of air parcels is investigated with a set of 3 d backward trajectories that end up in the PVAqL- defined in the quasi-Lagrangian perspective (Sect. ). Following , trajectory calculations are launched for each grid point within a PVAqL- (Δx = 0.5∘) on nine pressure levels between 500 and 150 hPa (Δp = 50 hPa) for each time step. In accordance with and , we trace potential temperature θ along the trajectories, and the 3 d backward trajectories are classified based on their net change of θ along the trajectory. Trajectories are classified as diabatically “heated” if Δθh,max > 2 K is fulfilled.

In addition, we create an additional set of trajectories to detect WCBs as trajectories that ascend by at least 600 hPa in 48 h, based on a similar methodology to . Analogous to , we distinguish different stages of the WCB and assign all WCB trajectory parcels that are located above 400 hPa to the WCB outflow stage. For this purpose, 2 d backward trajectories are started 3-hourly in the Northern Hemisphere at an equidistant grid of Δx = 100 km and at 13 equidistant vertical levels between 400 and 100 hPa. This is not the traditional way to determine WCB outflow by forward trajectories, but it offers the advantage of calculating trajectories directly from the PVAqL-'s. The filtering of trajectories is omitted here to avoid double counting as well as the criterion that the ascent must take place in the vicinity of an extratropical cyclone.

We consider the onset, maintenance, and decay of the EuBL regime over the North Atlantic–European region from the Eulerian perspective by focusing on the processes that locally contribute to a certain regime pattern. Low-pass-filtered PV tendencies are projected onto the regime pattern to quantitatively determine the mechanisms that govern the evolution of the regime pattern. Further insight into the dynamics of the pattern evolution can be gained by considering the positive and negative PVAs separately, i.e., by projecting the individual tendencies only onto the positive (trough) and negative (ridge) parts of the pattern, respectively. The individual contributions of different processes to the evolution of the EuBL regime pattern are shown as tendencies in Fig. . Positive values indicate that an individual mechanism contributes to the onset of the regime pattern, and negative values imply that a particular process contributes to the decay of the regime pattern. The time series of the tendencies is complemented with spatial information on PV and PV tendencies for selected times in Fig. .

The most dominant contribution to the regime pattern evolution around the onset arises from linear wave dynamics as described by the quasi-barotropic PV tendency (Fig. a). As discussed in Sect. , this PV tendency term describes the residual of the (westward) intrinsic phase propagation and the eastward advection of PVAs by the background flow. During the considered period, the latter term dominates, and thus the downstream advection of PVAs is of crucial importance for the onset of the regime pattern. We have already shown in Sect.  that the negative PVA (PVAqL-), which later represents the block over Europe, forms upstream. Considering negative and positive PVAs separately, the propagation of this PVAqL- from upstream to the target region is reflected in large values of the quasi-barotropic tendency in the projection (Fig. c) and amplifying quasi-barotropic tendencies in the ridge of the regime pattern (Fig. d, f). A consistently positive contribution from the quasi-barotropic term for the cyclonic part of the regime pattern underlines the propagation of positive PVAs, but the contribution is much smaller (Figs. b and d, f). Baroclinic PV tendencies are predominantly associated with the maintenance of the regime pattern but do not play a leading role in its onset (Fig. a). However, they become important in counteracting other PV tendency terms towards the end of the life cycle. The contributions to the full regime pattern arise predominantly from the anticyclonic part of the regime pattern, suggesting a contribution to the amplification of the ridge over Europe (Fig. c). However, a closer look at the spatial pattern reveals that the baroclinic term contributes to the secondary anticyclonic part of the regime pattern located over the US East Coast and not to that related to the block itself (Fig. f and h). The divergent PV tendencies are of further importance in the onset stage of the EuBL regime pattern and show their maximum positive contribution to the onset of the pattern around 9 March, 18:00 UTC (Fig. a). Divided into the anticyclonic and cyclonic parts of the regime pattern, the divergent term almost exclusively contributes to the former and weakens the latter, with a large contribution to the regime decay in the second half of the regime life cycle (Figs. b, c and c, e, g). Negative divergent PV tendencies overlap with the cyclonic regime pattern over the central North Atlantic at that time (Fig. g), which could be related to the onset of the subsequently established Atlantic ridge regime pattern (Fig. ).

Nonlinear processes as diagnosed by the convergence of nondivergent eddy fluxes have been the focus of many previous studies. In this case, they are negative, such that they support neither the onset nor the maintenance of the pattern in the current case and are consistently associated with the decay of the regime pattern (Fig. a). A clear minimum is visible in the separate consideration of the anticyclonic part of the pattern around the onset, where nonlinear processes such as wave breaking are associated with a decay of the regime pattern (Fig. c). Note, however, that the eddy fluxes may still help to maintain the regime pattern by reducing the strength of the westerly flow upstream . A dipole pattern associated with the eddy fluxes that indicates such a reduction is found in the average over many cases of Greenland blocking .

When the two diabatic PV tendency terms – radiative and non-radiative tendencies – are considered together, their effect on the regime onset and decay cancels out almost completely (Fig. a). The radiative tendency strengthens the cyclonic part of the regime pattern and weakens the anticyclonic part (Fig. b and c). Thereby, the projected tendency is almost constant and changes only slowly, indicating that the radiative diabatic tendency is not closely linked to the mechanisms governing the regime evolution. While non-radiative diabatic tendencies are associated with a decay of the cyclonic regime part, they strengthen the anticyclonic part of the regime, suggesting that these tendencies are dominated by latent heat release.

The Eulerian perspective elaborates on the importance of the advection of existing PVAs by the background flow in the onset stage of the EuBL regime pattern. From this perspective, divergent PV tendencies have a non-negligible effect in building up the anticyclonic part of the regime. However, the Eulerian perspective misses the processes associated with the development of PVAs advected into the region. Further insights into the evolution of these PVAs can be gained by tracing the PVAs that are advected into the region, especially the negative PVAs associated with the anticyclonic part of the pattern over northern Europe (PVAqL-'s).

The quasi-Lagrangian perspective focuses on traced anticyclonic PVAs in the Northern Hemisphere (PVAqL-'s) that contribute to a blocked regime life cycle and their amplitude evolution. In contrast with the low-frequency Eulerian perspective above, here we consider the instantaneous PV evolution. We will first discuss the track of the main PVAqL- (already indicated in light-green contours in Fig. ), based on the center of mass coordinates and the splitting and merging events. Subsequently, we quantify the individual contributions to the amplitude evolution of the main PVAqL-, consider direct diabatic impacts separately, and finally conclude with a summary.

Here we investigate (i) the importance of WCB outflow and, more generally, (ii) the importance of latent heat release by backward trajectories starting in the PVAqL- from the quasi-Lagrangian perspective, following and . Importance is assessed by the fraction of backward trajectories that fulfill specified criteria. The criteria that define WCB outflow are given in Sect. . For (ii), diabatically heated trajectories are defined as trajectories that experience a Δθ > 2 K. Fig. a compares the divergent PV tendencies from the quasi-Lagrangian perspective, integrated over the PVAqL- area (red) with the fraction of heated trajectories (black) and the fraction of WCB outflow (shading).

All three quantities are positively correlated over the lifetime of the PVAqL-, with a correlation factor between divergent PV tendencies and the fraction of heated trajectories of 0.44, between divergent tendencies and the fraction of WCB outflow of 0.57, and between the fraction of heated trajectories and the fraction of WCB outflow of 0.73. These positive correlations support the common expectation that the modification of the tropopause by upper-tropospheric divergent flow is enhanced by latent heat release in WCBs and represent a direct quantitative link between WCBs and reinforcing divergent PV tendencies within the PVAqL-. Most importantly, all prominent peaks of the divergent PV tendency (> 3⋅107 PVUm2s-1, gray horizontal line in Fig. a) are associated with a WCB outflow fraction of at least 20 %, and vice versa. For the presented case, we can thus demonstrate with a high degree of certainty that the divergent PV tendencies are indeed an indirect moist impact and that the WCB outflow dynamically modifies the tropopause. In some periods within the PVAqL- life cycle (e.g., 21–23 March), we find a high fraction of heated trajectories, but the divergent PV tendency is relatively low or even contributes to amplitude weakening (Fig. a). One explanation is that the timing and, in particular, the exact location where diabatic heating occurs along the 3 d trajectory is essential, as this ultimately shapes the effect on the PV distribution within the PVAqL-. Thus, if the trajectory experiences the ascent and thus the period of maximum heating so early that it reaches the upper troposphere outside of the PVAqL- and migrates nearly horizontally into the PV anomaly, the effect on the amplitude amplification is much smaller than if the trajectory experiences its ascent directly in the immediate vicinity of the PVAqL-.

We further follow and investigate the distribution of the maximum heating and cooling rates along the 3 d backward trajectories that are started from the PVAqL- ± 1 d around the onset and decay of the EuBL (Fig. b). Additionally, we analyze the spatial origin of the diabatically heated and non-heated backward trajectories separately (Fig. ). As in , we see a broad heating regime with values as high as Δθ > 20 K within 3 d and a narrow non-heated regime with values as low as Δθ < -5 K. Most 3 d backward trajectories (53 % around the onset and 76 % around the decay) experience diabatic cooling before they arrive in the PVAqL-, most probably due to longwave radiative cooling. Around the EuBL onset, a high fraction of these air parcels are located upstream of the PVAqL- in the mid or upper troposphere and probably reach the PVAqL- by adiabatic advection (Fig. b). In comparison, we see in Fig. d in addition to the adiabatic advection of air parcels from upstream some recirculating air parcels in the upper troposphere that are already located within or near the PVAqL-.

The substantially increased fraction of heated trajectories around onset (47 %) compared to around decay (24 %) in Fig. b demonstrates the importance of diabatic heating for the development and strengthening of the PVAqL- in an earlier stage of the life cycle from the Lagrangian perspective. Concerning the spatial origin of the heated trajectories, we identify the southern North Atlantic as a key source region around the EuBL onset and decay (Fig. a and c). The position of air parcels upstream of the PVAqL- in the lower troposphere suggests that air parcels will most probably experience latent heat release on their ascent to the upper-tropospheric PVAqL- in the following 3 d.

The Lagrangian perspective thus complements what was learned from the quasi-Lagrangian perspective and links the amplitude-enhancing divergent PV tendency contribution from the quasi-Lagrangian perspective to the occurrence of latent heating (mostly) associated with the occurrence of WCBs, which represents an important contribution, especially around the EuBL onset.