Extratropical cyclones are a primary cause of severe impacts in the mid-latitudes, repeatedly causing extensive damage and loss of life due to extreme winds and precipitation. Strong winds in extratropical cyclones can be linked to a number of distinct airstreams, often long-lived and of synoptic-scale size, such as the widely studied warm and cold conveyor belts , the descending dry intrusion . Severe gusts can also be associated with cold-frontal convection, as pointed out by . However, in Shapiro-Keyser cyclones strong surface winds can also be associated with a shorter-lived and smaller-scale airstream, called “sting jet”. A sting jet (SJ) is a mesoscale airflow that descends off the tip of the cyclone cloud head and accelerates into its frontal-fracture region. Despite their local and transient nature, SJs can lead to distinct regions of exceptionally strong and damaging near-surface winds. First identified two decades ago when re-examining the Great Storm that devastated southern England on 16 October 1987 , SJs were since identified in some of the most damaging cyclones affecting the British Isles and continental Europe . Recent examples of notable SJ cyclones include Storm Eunice , Storm Ciarán and Storm Éowyn , explosively deepening windstorms that justified the issuing of severe wind warnings, producing extreme or even record-breaking surface gusts and causing extensive damage and loss of life.

The three storms just mentioned have all in common the development of a SJ that is associated with the onset of symmetric instability (SI) in the cloud head, as illustrated in for Storm Eunice, for Storm Ciarán and by preliminary analysis of Met Office operational forecast data for Storm Éowyn (not shown). As explained in detail in , SI is essentially a form of inertial instability generalised to a baroclinic flow. It is indicated by negative values of Ertel potential vorticity (PV hereafter), which is defined as PV=ξ⋅∇θρ, where ρ is the air density, ξ the absolute vorticity vector, and θ the potential temperature. SI is derived applying the semi-geostrophic approximation to a two-dimensional base state that is in thermal wind balance and has a uniform thermal gradient. SI is released as a consequence of slantwise displacements happening at angles between the slopes of absolute geostrophic momentum and θ. The moist counterpart of SI is conditional symmetric instability (CSI, indicated by negative values of moist saturated PV, MPV*), that is defined by replacing θ with saturated equivalent potential temperature (θe∗). As surfaces of θe∗ are more sloped than those of θ, CSI is more common than SI. However, CSI is a conditional instability and requires saturation to be released, while SI does not and can be released as soon as it forms. In recent years consensus grew on the presence of SI in intense SJ cases, as explained in the following paragraph. We therefore focus our analysis on SI, and on regions of negative PV, rather than on its moist and conditional counterpart.

The hypothesis that the acceleration and descent of SJs could be at least partly associated with the release of CSI or even SI dates back to the pioneering works of . In their analysis of the Great Storm, they highlighted the banded structure of the cloud-head tip and illustrated it as an indication of multiple slantwise circulations, with descending cloud-free branches associated with the acceleration of the sting jet out of the cloud head. The associated seminal modelling work by confirmed the presence of these slantwise circulations, with the SJ appearing within their descending part. Subsequent studies added further evidence to the hypothesis that SJ descent and acceleration is associated with slantwise circulations caused by mesoscale instability in cloud-head frontal circulations or at least reduced or neutral stability, as in . The review by stresses the likelihood of a continuum of behaviours, “from balanced descent partly associated with frontolysis in the frontal-fracture region, through horizontally smaller scale and stronger frontolytic descent associated with weak stability to slantwise convective downdraughts, to multiple slantwise convective downdraughts associated with the release of CSI and even, possibly, SI.” They also highlight the evidence that this release of instability is associated with substantial acceleration, on top of the speed-up caused by the quasi-geostrophic frontolytic descent highlighted in and in a region of the cyclone that is already prone to strong winds, which would usually be aligned with cyclone propagation.

used model simulations of Storm Tini to develop a conceptual model illustrating the primary role of vorticity tilting via slantwise circulations and its close link to the generation of negative PV. This conceptual model was confirmed by the idealised simulations in , showing bands of strongly positive and negative horizontal vorticity being associated with tight frontal circulations in the cloud head. Large negative values of the tilting term in the vertical vorticity budget were diagnosed along SJ trajectories travelling towards the tip of the cloud head, where the formation of a narrow filament of negative PV just outside the bent-back warm front was observed. This formation of negative PV filaments along the SJ as it travels in a narrow region in the cloud head and near the bent-back front has also been confirmed by the most recent studies of intense sting jet storms, such as Eunice and Ciarán, as explained in Sect.  . It can thus confidently be stated that intense SJ are normally characterised by the presence of SI.

SJ cyclones are often characterised by banding in the cloud head , although caution might be exerted, as discussed in , in associating it to slantwise circulations and mesoscale instability. Slantwise circulations in the cloud head have been observed in rapidly developing cyclones, such as the FASTEX IOP16 cyclone described in and , in which the strength of the circulations led to describing them as forming “multiple cloud heads.” highlighted the importance of sublimation of ice as an effective mechanism for the maintenance of slantwise descent, thanks to the associated cooling. The role of latent cooling caused by precipitation (liquid or solid) falling from above and helping to maintain descending motions is stressed also by . Focusing on SJs, found a strong correlation between evaporative cooling and SJ descent. Subsequent studies, such as and among others, provided differing results on the importance of latent cooling for the descent and acceleration of SJs, as summarised in . It is worth noting that, given that the presence of a precipitating cloud above the airstream is a necessary condition for the cooling to take place on it, these processes become less likely as the SJ progresses in its journey off the tip of the cloud head and into the frontal-fracture region.

In summary, the link between SJ dynamics and the presence of SI, or of reduced stability, has now been established. The onset of this instability, indicated by negative PV, is driven by tight frontal circulations in the cloud head, near the bent-back warm front. This suggests the likely involvement of moist processes, that can help to initiate and maintain the descent via latent cooling. However, the distinct roles of individual diabatic processes in reducing PV along the SJ as it travels in the cloud head, before starting its descent and acceleration out of it, have not been identified yet.

In the last couple of decades two methods have been mainly used to investigate the role of individual diabatic processes in the evolution of extratropical cyclones and associated features, such as SJs, assessing the related changes in PV and/or potential temperature (θ). These are passive tracers and tendency diagnostics. We summarise the two approaches here, while a more comprehensive description, also featuring a variety of examples from the literature, can be found in Sect. 5.4 of .

The use of tracers follows from the seminal work by , in which the accumulation of PV increments caused by the different non-conservative model processes was partitioned using the routines and parameterisation schemes of the model and advected at every time step, together with the conserved component, using the model advection scheme. Since then, most diabatic tracers studies have been performed using the Met Office Unified Model (MetUM). Particularly relevant for our study are the works by and , focusing on the role of different diabatic processes in modifying PV in several regions of extratropical cyclones. The requirement of tracers to be incorporated in the model code represents an obstacle towards easy use and portability. After around three decades since pioneering works, studies implementing diabatic tracers in models other than the MetUM are still limited, one example being the analysis by of diabatic and baroclinic contributions to the evolution of intense Mediterranean cyclones using the WRF model.

An alternative approach is the use of tendency diagnostics. In this case tendencies from the different diabatic and frictional processes are included in the model output. These tendencies are then traced along trajectories, normally computed offline using the resolved wind. This is what was done by , who showed that it was possible to use the heating tendencies available in the output of the COSMO model to provide a detailed breakdown of the role of individual diabatic processes in modifying PV in a warm conveyor belt. This study was followed by further work using diabatic tendencies to reveal the role of non-conservative processes on the structure and evolution of warm conveyor belts using the IFS model and, more recently, the ICON model . Other main features of extratropical weather have also recently been analysed using diabatic tendencies in the IFS, such as the jet stream and the tropopause and, of particular relevance to our study, tropospheric fronts in extratropical cyclones . In , the analysis of diabatic tendencies in an extratropical cyclone case study showed that PV increase along the warm front was mainly generated by condensation and turbulence, with PV decrease primarily caused by snow melting and sublimation. Condensation, convection, snow melting and sublimation were found to be the most important contributors to PV increases at the cyclone centre and at the bent-back end of the warm front. The same approach was extended to a series of month-long global free-running simulations in , providing a systematic analysis of low-level PV modification along the different fronts characterising extratropical cyclones. This analysis confirms that at the warm front high PV is mainly caused by condensation (in the cold season) and turbulent mixing of momentum (in the warm season) while melting and snow sublimation play a role in the generation of low PV, particularly to the north of the front. At the bent-back front and at the cyclone centre, condensation is still the main contributor to PV increase.

Both the passive tracers and tendency diagnostics methods are based on model parameterisation schemes. Both methods also highlight the Lagrangian nature of modifications to PV and θ, as non-conservative processes are integrated (online or offline) along the flow. For this study we choose to use the tendency diagnostics methods, so that we can take advantage of an established set up that has already proven to be effective for the identification of individual diabatic processes when applied on fronts and other features characterising extratropical cyclones.

Therefore, in this study we follow the same diabatic tendency methodology as in and we apply it to a model simulation of Storm Ciarán. As described at the beginning of this introduction, Ciarán was a highly diabatic high-impact windstorm that featured a SJ displaying a clear SI signal . Hence, it is an appropriate case study to investigate the role of diabatic processes in the evolution of SJs and the related onset of SI.

The remainder of this article is structured as follows. The model simulation, use of diabatic tendencies and tracing along Lagrangian trajectories are described in Sect. . The results section (Sect. ) begins with the illustration of the presence of a symmetrically unstable SJ and a description of the evolution of PV along SJ trajectories. This is followed by a detailed analysis of the diabatic processes modifying PV along the trajectories and in the environment around them. The section ends by focusing on airstreams associated with additional regions of negative PV. Section  completes the article, as it contains a discussion of the results and the overall conclusions.

The simulation analysed in this study is performed using the Cycle 47r3 version of the Integrated Forecasting System (IFS) model of the European Centre for Medium-Range Weather Forecasts (ECMWF), which was operational from October 2021 to June 2023 at ECMWF. A 2 d forecast is initialised with ECMWF analysis field at 00:00 UTC on 1 November 2023 and it is run at a cubic-octahedral spectral transform discretisation of TCo1279 with 137 vertical levels, the same as the operational forecast. The corresponding horizontal resolution is approximately 9 km and the hourly output fields are interpolated to a regular grid at 0.1° resolution.

Apart from the standard model outputs, the temperature, moisture, and momentum tendencies associated with individual parametrised subgrid-scale processes are retrieved, similar to and . Using these tendencies, the PV framework is applied to evaluate the effect of diabatic processes on the formation of sting jets. As PV is materially conserved under adiabatic and frictionless flow, any changes in PV can be linked to potential temperature (Qi) and momentum tendencies (Fi) imposed by the non-conservative processes i, shown in Table . The material change rate of PV (PVR) can be described by 1PVRi=1ρξ⋅∇Qi+∇×Fi⋅∇θ where again ρ is the air density, ξ the absolute vorticity vector, and θ the potential temperature. Compared to , the cloud microphysical processes considered in this study are slightly modified, with a major update of the moist physics applied on Cycle 47r3 of the IFS. In particular, melting of ice and snow is grouped into a single process “melting” and the freezing and riming of cloud droplets are grouped into a single process “freezing”, as detailed in Table .

In order to evaluate the influence of diabatic processes on the evolution of the sting jet, backward trajectories are calculated with the Lagrangian Analysis Tool LAGRANTO . The release region of the backward trajectories is described in detail in Sect. . All momentum and potential temperature tendencies are traced along the trajectories of interest. With this approach we can quantify the instantaneous rate of change in PV due to every process separately (PVR) and then calculate the time-accumulated PV tendency (APV) along the flow. The net PV modification is described by the sum of large-scale microphysics, radiation, convection and turbulence, plus a residual term: 2ΔPV=APVall+RES=APVcloud+APVsw+APVlw+APVconv+APVturb+RES where RES denotes the residual. This Lagrangian method of tracing diabatic tendencies along trajectories has already been successfully applied to a range of weather features, as detailed in Sect. . A more detailed description of the method can be found in and .

Storm Ciarán was, as described in the introduction, a severe and highly diabatic windstorm that featured SJ development. In this section we show that a clear SJ is identified in the IFS simulation performed and used in this study, illustrate its properties and highlight that this SJ is associated with SI.

In this section we use the horizontal maps in Fig.  and vertical cross-sections in Fig.  to detail the evolution of PV along SJ trajectories and in the environment around them, focusing in particular on the cloud head and the bent-back warm front. The different panels of Fig.  cover the evolution of the rapidly deepening cyclone from before the start of the SJ ascent all the way towards the end of its descent. At each of these times they illustrate the structure of the storm at the pressure level that is nearest to the mean pressure of the trajectories. The two vertical cross-sections in Fig.  are generated using the transects in Fig. e. They refer to 11:00 UTC (t=-4 h), the end of the SJ ascent when PV has decreased below zero on all trajectories. They are orientated along and across the SJ and the negative PV filament associated with it.

Figure a refers to when the SJ trajectories are yet to start ascending, and are found at low levels on the outer (i.e. cold) side of the warm front, with widespread low or negative PV around them. Figs. b–d show that as the SJ trajectories ascend up to above 775 hPa by t=-4 h, the cyclone continues to evolve following the Shapiro-Keyser conceptual model, with the cold front detached from the bending-back warm front. The SJ trajectories rotate cyclonically around the warm front and at t=-4 h are located at the rear end of a narrow filament of negative PV found just on the cold side of the front and roughly parallel to it. The structure and location of this filament is consistent with the results from the idealised simulations shown in and thus with the conceptual model of SJ evolution there verified, after being introduced in , highlighting the role of vorticity tilting via slantwise circulations in the generation of negative PV (Sect. ). Other negative-PV filaments are visible throughout these times ahead of the one in which the SJ is shown. This suggests the presence of earlier SJ activity in the cloud head, in the same way as the presence of another negative-PV filament behind the trajectories points at the potential for later sting-jet activity. This potential additional SJ activity will be investigated in Sect. , using the trajectories calculated from the release points included in these panels. Additional weaker high-low PV bands (with some pockets of negative PV) are present to the outside of the warm front and the SJ negative-PV filament. These bands, also oriented along the front and sitting within the cloud head, suggest the presence of multiple secondary circulations around the main frontal zone and indicate potential for SI even in outer regions.

Figure e shows the SJ trajectories travelling as a compact group in the initial stages of their descent while leaving the tip of the cloud head. At t=-3 h the negative-PV region in which they travel is starting to be associated with the cloud gap characterising the tip of the cloud head, further confirming that descent, with its consequent decrease in relative humidity, is taking place around the trajectories. Figure f shows the trajectories entering the frontal-fracture region while in the late stages of their descent. The bent-back front is continuing its wrapping process and the trajectories are now ahead of the cloud-head tip and are reaching their maximum speed, as shown in Figs.  and  a). There is now another negative-PV filament behind the main SJ, that is reaching the tip of the cloud head and opening a gap in the cloud head as it presumably descends and accelerates as a later SJ, also investigated in Sect. .

The two vertical cross-sections in Fig.  display the environment around the SJ trajectories at t=-4 h, the end of the ascent. Section AB, in Fig. a, is oriented along the negative-PV filament and shows that the trajectories are found in an elongated area of negative PV centred between 700 and 800 hPa. Whilst the SJ trajectories are still in the final stages of their ascent, the negative-PV filament is generally oriented downward, with some regions extending down towards the surface. The area of markedly positive PV, with widespread values exceeding 3 PVU above the negative-PV filament is part of the outward-slanted bent-back warm front, intercepted by the section between 600 and 700 hPa. Section CD, in Fig. b, is oriented across the bent-back warm front and the negative-PV filament. It confirms the slanted structure of the warm front from around 750 hPa upwards, whilst the intense front is upright at lower levels. The same slanted structure is displayed by the several PV bands present in the cloud head that are located on the outer side of the most intense positive-negative PV dipole, associated with the bent-back front and of which the SJ negative-PV filament is part. This multipole PV pattern is consistent with the results in , in which a tripole PV pattern was found to be associated with a thermally direct frontal circulation (in their case associated with a single cold front), with enhanced PV at the front and reduced PV on the outer sides of the warm-sector condensational heating and cold-sector evaporative cooling. This PV pattern is also consistent with the results from the idealised simulations in , also showing strong PV bands located in correspondence with frontal circulations in the cloud head. The results from this section therefore suggests the presence of multiple thermally direct circulations in the cloud head, particularly intense at the bent-back front, driven by moist processes and by the associated heating and cooling.

Overall, the results presented up to this point give extensive evidence that the selected trajectories are associated with the evolution of a SJ displaying the development and subsequent release of SI (indicated by negative PV), as the SJ ascends into the cloud head and then descends and accelerates out of its tip, respectively. At the same time, intense frontal circulations and associated moist processes drive the evolution of PV in the environment around the trajectories. This case can therefore be used to investigate the diabatic processes at play during the decrease in PV along the ascending SJ, ultimately tackling the question of identifying the processes acting to make the SJ symmetrically unstable and thus enabling further acceleration during its descent.

Figure  contains a box-and-whisker plot displaying the change in PV and the accumulation of the different PV tendencies in the time characterised by SJ ascent, t=-6 h to t=-4 h (09:00 to 11:00 UTC). The total change in PV is shown by the leftmost box and whiskers (ΔPV), illustrating the generalised decrease in PV during SJ ascent, with both mean and median value around -1.1 PVU in 2 h. Only part of this PV decrease is captured by the tendencies, with mean and median of the sum all of PV tendencies (∑APV_all in Fig. ) showing a decrease of around -0.35 PVU (roughly a third of the actual PV decrease) and the extension of the whiskers indicating large variability across different trajectories. This substantial discrepancy between the actual change in PV and that captured by the sum of the accumulated PV tendencies can be ascribed to mainly two factors that must be acknowledged as limitations of this study.

When processes are fast-changing and non-linear, as is the case with SJ dynamics, instantaneous hourly output (used in this study, meaning that the ascent is described by only three instantaneous output values for each tendency) can differ greatly from timestep-by-timestep data.

Figure  contains maps of instantaneous PV tendencies for the four moist processes contributing to PV modification during SJ ascent. These maps are generated at 11:00 UTC (t=-4 h), the end of SJ ascent, showing the tendencies at the level nearest to the mean pressure of the trajectories (775 hPa).

Figure a shows the instantaneous PV tendency due to condensation of water vapour, PVRcond. Large values of PVRcond, both positive and negative are present throughout the extent of the warm front, mainly on its warm side, and at the bent-back warm front. Moderate values also cover part of the cloud head, on the cold side of the bent-back warm front. Some gaps exist between the large values near the front and the SJ trajectories lie in one of these gaps, characterised by near-zero PVRcond but immediately next to an area of large negative values. These features are consistent with the results in , with the caveat that their composite of more than 100 rapidly developing cyclones is much less noisy than our single-case analysis of Storm Ciarán. They show positive PVRcond throughout the length of the warm front and a PVRcond dipole, with negative values inside, at the bent-back front. As in our study, their results highlight the primary role of PVRcond in regions characterised by ascent and indicate that the pattern of PVRcond becomes less straightforward as frontal environments increase their curvature and heating gradients are no longer vertically oriented, such as in bent-back fronts. The PV tendency due to the evaporation of cloud water, PVRevc, is displayed in Fig. b. Its pattern is also characterised by a large number of local maxima and minima, mainly near the warm front and along its bent-back part, but also scattered around the cold sector and in the frontal-fracture region. SJ trajectories lie again in an area of negligible tendency, this time near multiple local maxima.

Figure c and d contain, respectively, the instantaneous PV tendencies due to melting of snow and ice (PVRmelt) and sublimation of snow (PVRsubs). Values in both panels are substantially better spatially constrained than for PVRcond and PVRevc, with non-negligible tendencies only present at the warm and bent-back warm fronts. PVRmelt is characterised by a clear along-front dipole band on the eastern part of the warm front, with positive values on the cold side, and a small elongated positive patch covering part of the bent-back warm front. This patch is separated from the warm-front dipole by a clear gap and the SJ trajectories are located just outside it, in an area of near-zero PVRmelt. The primary role of melting of snow in generating an overall negative PV anomaly at the warm front is shown in , consistent with these results, with highlighting its positive effect at the bent-back front. The dipole visible here along the warm front suggests that the height of the melting core decreases going outward, thus crossing the pressure level the map refers to. Focusing on sublimation of snow, the largest values of PVRsubs (Fig. d) only reach around ±0.5 PVU h⁻¹, much less than for all the other tendencies shown in Fig. . These limited PVRsubs values are located in a handful of small-scale dipole bands oriented along the warm and bent-back warm front, again with positive values on the cold side. This highlights the localised nature of PV-changing snowfall and is consistent with mentioning the role of sublimation only at the bent-back front. Despite the reduced magnitude and spatial extent of PVRsubs values, in this case SJ trajectories are located in area with clearly negative tendencies. This is in contrast with all the other tendencies shown in this figure, that have trajectories located in regions of negligible change.

In summary, large values of PVRcond, PVRevc and PVRmelt (with large variability and noise for the first two) are present near or along the warm and bent-back fronts. Values of PVRsubs are smaller and more localised and, as for PVRmelt, they are oriented in along-front dipoles (positive values outside). However, at this time, SJ trajectories are located in an area of non-zero (negative) tendencies only for PVRsubs. The maps shown in Fig.  also highlight the complexity of the bent-back front area, hinting at its three-dimensional nature and therefore stressing the importance of analysing its vertical structure. We do this by looking at the vertical sections in Fig. . These sections are generated following the along-front and across-front transects shown in Figs. d and , used also for the sections in Fig. . Like all those figures, they refer to 11:00 UTC (t=-4 h), when SJ trajectories are reaching the end of their ascent and PV along them is decreasing fast.

The vertical cross-sections in Fig. a–b show the instantaneous PV and theta tendencies due to the condensation of water vapour. The presence of large values of PVRcond along the front, both positive and negative, highlighted by the map in Fig. a, is confirmed by these sections. Two heating cores are present at the front, centred at around 850 and 600 hPa, leading to multiple positive and negative local extremes in PVRcond, alternating in an almost-vertical direction (Fig. b). The presence of additional frontal bands in the cloud head (see also Fig. b) is associated with a heating structure with gradients in both vertical and across-front directions and thus a complex PVRcond pattern. The heating cores shown by the across-front section are also visible in the along-front one (Fig. a), in the low levels and near the cloud top, in some places coinciding with regions particularly close to saturation. The low-level heating core is associated as expected with a PVRcond dipole that has negative values above the heating maximum and positive below it. However, due to the complex three-dimensional structure of condensational heating, the PVRcond dipole associated with the intense mid-tropospheric Qcond maximum is not oriented in a negative-above-positive direction. The region of mid-tropospheric heating is tilted with height, as shown in the across-front section (Fig. b). The area with large PVRcond values along the bent-back front, shown in Fig. a, is also tightly curved. There is thus a mix of multiple heating cores located next to each other in a fully three-dimensional pattern. In addition to this, at this time the vertical component of absolute vorticity is negative along the SJ trajectories, as explained in Sect. , and in the immediate surroundings, further complicating the relationship between heating and PV modification (Sect. ). This complex PVRcond pattern is therefore inherently associated with the local structure of the bent-back front displayed by the chosen vertical sections. Additional sections computed further north or south display a simpler dipole structure in PVRcond (not shown), markedly different from that just described. As a final remark on Fig. a–b, it is important to note that, whilst being closely surrounded by nearby large positive and negative values, the SJ trajectories lie in a localised region of small negative PVRcond. This further highlights the importance of the exact location in which the trajectories travel in their PV evolution, further motivating the presence of large variability in PVRcond across trajectories and the large discrepancy between the actual change in PV and the sum of the time-accumulated tendencies.

Vertical sections of the tendencies due to evaporation of cloud water, PVRevc and Qevc show complex patterns similar to those of PVRcond and Qcond and are not included in the article. Figure c–d display vertical cross-sections of the tendencies of PV and θ due to melting of ice and snow. The related PVRmelt map, shown in Fig. c, is characterised by a clear dipole at the warm front and a positive patch at the bent-back front, with the SJ trajectories just to the outside. These sections display a simple positive-above-negative PVRmelt dipole pattern where cooling due to melting occurs away from frontal zones and moist isentropes (i.e., θe contours) are near-horizontal. In these regions, the height of the cooling core and thus also of the dipole is set by the freezing level. The pattern becomes more complex and three-dimensional closer to the bent-back front, where moist isentropes tighten and become slanted or even, locally, vertical. This is the region in which SJ trajectories travel during their ascent, also characterised by a dip in the inward-ascending freezing level. Figure e–f show vertical cross-sections of the tendencies of PV and θ due to the sublimation of snow. As already illustrated by the related map (Fig. d), non-negligible values of PVRsubs are present only in local regions, including near the bent-back front, where SJ trajectories are located. Sublimational cooling exceeding -1 K h⁻¹ occurs on the trajectories, consistent with the presence of large snow water content along the trajectories at this time (Fig. d) and the local dip in freezing level. It is worth noting that the local cooling experienced by the SJ trajectories has also the effect of reducing their buoyancy against the surrounding environment, thus favouring the start of their descent over a continuation of their ascent. Due to tight and slanted-to-vertical moist isentropes and a curved frontal environment, the PVRsubs structure is closer to a horizontal dipole (both in the along- and across-front directions) rather than a vertical one. The trajectories lie near the core of the negative side of this dipole. While for all the other processes examined, PV modification along the trajectories was close to zero, sublimation of snow provides a clear, albeit limited, contribution to PV decrease along SJ trajectories, as they ascend in this narrow zone located just to the outside of the bent-back front and near the freezing level.

In Sect.  we use maps and vertical cross-sections to fully characterise the three-dimensional structures of the moist processes identified in Sect.  as playing a role in the PV decrease along ascending SJ trajectories. However, this characterisation is performed at a single time, 11:00 UTC (t=-4 h), near the end of the SJ ascent. To analyse the evolution in time of these processes we complement this analysis by looking at time-pressure profiles, or “Lidar plots”. These mean profiles are constructed by averaging the troposphere above and below the locations of all the trajectories at each time, hence mimicking what would be the output of an imaginary lidar or radar scanner moving with the trajectories. The large variability across trajectories and the especially complex structure of the tendencies associated with condensation and evaporation can lead to substantial inaccuracies and possibly misleading results in time-pressure profiles. This is particularly true when using the mean location of the trajectories and instantaneous hourly data, as in this study. Therefore, in this section we focus on time profiles of the tendencies associated with melting and sublimation of snow, displayed in Fig. .

Time-pressure profiles of PV and θ tendencies due to melting of ice and snow (Fig. a and c) show that until the start of their ascent the SJ trajectories travel, with no appreciable PV tendency, underneath a region of cooling and the associated positive-above-negative dipole in PV tendency. This dipole is located just below the freezing level, with the core of the cooling being at the bottom of a region particularly close to saturation. As they start ascending, the trajectories cross an area with reduced cooling and then find themselves above the cooling core as its magnitude increases again. This, together with the transition to the more complex PVRmelt structure described when looking at Fig. d, results in generally little net PV modification, but with considerable variability across trajectories (Fig. c), caused by the several regions of positive and negative PVRmelt encountered during the ascent. Time-pressure profiles of PV and θ tendencies due to sublimation of snow (Fig. b and d) confirm that sublimational cooling appears only in the second part of the ascent when substantial snowfall occurs near the tip of the bent-back front and SJ trajectories reach the freezing level (Fig. d). SJ trajectories travel close to the core of the cooling, with values down to near -1 K h⁻¹ at the end of the ascent. The PV decrease is limited but non-negligible, with the mean trajectory exceeding -0.1 PVU h⁻¹ at the same time, consistent with the pattern shown in Figs. f, d and d.

The PV maps presented in Fig.  highlight the presence of PV < 0 filaments in the cloud head, just outside the bent-back front, for an extended period of time. This continued presence of PV < 0 filaments points at the potential for additional SJ activity, before and after the formation of the main SJ airstream analysed in this study. Late SJ activity would also be consistent with the results in , highlighting the presence of a later SJ pulse after the main one, with peak 850 hPa wind speed reached at 13:00 and 17:00 UTC, in Met Office operational forecasts. To verify whether these early and late PV < 0 filaments are associated with additional SJ airstreams, we calculate four additional trajectory sets and assess their properties. These four sets of trajectories are named after their release time and pressure level, listed below.

09UTC (t=-6 h) 875 hPa (Fig. b, blue dots, 34 trajectories)

Figure  contains timeseries of wind speed, pressure, PV and relative humidity for the four additional trajectory sets. Three of these sets, released at 09:00, 10:00 and 14:00 UTC, display a substantial increase in wind speed after release time, associated with weak descent, an increase in PV back to positive values and a gradual decrease in relative humidity. The maps in Fig.  show that this evolution occurs while the trajectories, having “turned the corner” off the cloud head, become roughly aligned with the direction of propagation of the storm and move eastward, towards an incipient (for the early sets) or mature (for the late set) frontal-fracture region. All these features are commonly observed in SJs, including in the main SJ analysed in this study, despite acceleration in the two early sets and the descent in all sets being less intense than what is normally shown by well-developed SJs. This means that the two early trajectory sets display airstreams that behave as weak SJ-like airstreams, while the late set contains an airstream that is more comparable to the main SJ in terms of intensity, with its trajectories reaching 50 m s⁻¹, despite only showing a very limited descent after a clearer ascent from the boundary layer. This points at the occurrence of a well-developed SJ (which would be consistent with ), that, due to the later stage of evolution of the cyclone, is not able to fully descend towards the surface as the main SJ.

The trajectory set released at 11:00 UTC instead shows a completely different behaviour, with trajectories progressing northward while rapidly ascending from the boundary layer to the upper troposphere. These features are consistent with the behaviour of WCBs , although it is surprising to see them in trajectories released from a PV < 0 filament just north of the main PV signature of the warm front, rather than fully within the warm sector. A deeper investigation of the characteristics and evolution of these trajectories would be necessary to fully identify their origin and potential belonging to a WCB airstream. Such an investigation would go beyond the scope of this article and we therefore leave it to future studies.

The box-and-whiskers plots in Fig.  show the accumulated change in PV, in the 3 h leading up to minimum PV, displayed by the tendencies associated with the four cloud processes playing a role in the evolution of PV of the main SJ: condensation of water vapour (APVcond), evaporation of cloud water (APVevc), melting of ice and snow (APVmelt) and sublimation of snow (APVsubs). Focusing on the three SJ-like airstreams (sets released at 09:00, 10:00 and 14:00 UTC), we see large variability across trajectories in APVcond in the 14:00 UTC set and a more limited range in the 10:00 UTC and, particularly, 09:00 UTC sets. This confirms the clear role of condensation in causing large variations in PV near the warm and bent-back fronts, increasing in magnitude as the cyclone intensifies. The role of melting is also highlighted, with a negative APVmelt signal in the 10:00 UTC and, particularly clearly given the limited variability, 14:00 UTC sets. These results, and the fact that they are clearer for the late and more intense trajectory set, are consistent with those described for the main SJ. This is not the case when looking at APVsubs, as the limited but robust decrease in PV experienced by the main SJ is not shown by any of the three SJ-like airstreams. Overall, these results highlight that conditions leading to the formation of PV < 0 filaments in the cloud head and subsequent SJ-like airstream generation are present for a sustained period of time during the explosive intensification of the cyclone, with condensation and melting actively changing PV. However, the most favourable conditions, leading to a fully-developed SJ able to accelerate up to 50 m s⁻¹ and beyond while descending towards the boundary layer, are only present for a limited time. As a result, the additional airstreams identified do not reach the intensity of the main SJ, with only the latest airstream displaying comparable wind speed. The role of snow sublimation, localised in time and space, is only limited to the main SJ, which thus benefits from additional PV decrease and cooling while accelerating into the frontal-fracture region at the optimal time.