3.1 Example simulation

In this paragraph we shortly describe the flow/vortex evolution of an example simulation over several minutes. For this, Fig. 3 displays 3D contour plots of vorticity magnitude (the time and the contour surface level are indicated in each panel). We choose the default simulation, yet for illustration purposes with an enlarged domain length along flight direction ( $L_y=792\textrm{m}$ instead of 132m, the simulation is listed in the last row of Table 2). In the first panel at $t=2$ s, the four vortex tubes are straight and the relative positions between them reflect the geometric initialisation as outlined before in Fig. 1. Moreover, coloured contours are plotted in four slices perpendicular to the flight direction. Those reveal the enhanced turbulence that was superimposed in a rectangular box around the two IVs during the initialisation. The panel for $t=16$ s shows that secondary vorticity structures (SVSs) have emerged around the two IVs. Moreover, those two vortices move towards the right OV. Section 3.2 will explain in detail the reasons for this lateral propagation. Soon after, the right OV $V_{4}$ (as labelled in Fig. 1) captures $V_{3}$ . Those two now form a vortex pair and move away from the other IV $V_{2}$ . After 60s, the vortex pair $V_{3} \& V_{4}$ effectively moved upwards, many SVSs shape up around them and high vorticity values are not any longer confined to two concentrated tubes (as is obvious from the contour slice in front). On the other hand, the vortices $V_{1}$ and $V_{2}$ still feature straight vortex tubes. A closer inspection shows that vortex $V_{1}$ is stronger than $V_{2}$ , as the red area with vorticity magnitude $\omega > 1.5$ /s is larger. Moreover, a small-amplitude small-wavelength meandering of $V_{2}$ is apparent. After 3min, the vortex pair $V_{3} \& V_{4}$ has dissolved and only a few patches with elevated $\omega$ -values occur. Contrarily, the two other vortices can be still tracked as they still feature fairly straight vortex tubes and only weak SVSs are present. After 5min, the weaker $V_{2}$ -vortex has dissolved and only $V_{1}$ remains. Another 1.5min later at $t=6.5\min$ , mostly only irregular patterns of enhanced vorticity have survived. Most simulations are carried out until 10min, where we can safely assume that wake vortices induced dynamics has ceased.

3.2 Tracking of the vortex centres

In the classical case of a single aircraft, a pair of counter-rotating vortices rolls up and the vortices mutually induce a downward movement at an initial speed of

(4) \begin{equation} w_{0}=\Gamma_{t=0}/(2\pi b_{0}). \end{equation}

At cruise conditions, the final vertical displacement and the decay of the vortices is mainly governed by the strength of thermal stratification e.g.^{(Reference Unterstrasser, Paoli, Sölch, Kühnlein and Gerz35,Reference Spalart63)} and the well-known Crow-instability may occur^{(Reference Crow64)}. The formation flight simulations show a much more diverse behaviour and their early evolution is strongly affected by an interplay of the two close-by IVs. This will be demonstrated next by analysing the initial flow field; in particular, the velocity at the four VCs is analysed which gives a first hint of where each vortex travels in the beginning.

Figure 4 shows the initial positions of the four vortices (marked by asterisks) together with the velocities induced by the surrounding vortices at these positions (indicated by the arrows) for eight different cases. The arrows indicate the direction and strength of this velocity induction; the individual contributions of the vortices $V_{1}$ (red), $V_{4}$ (green) or the vortex dipole $V_{2}/V_{3}$ (blue) and their combined effect (black) are depicted. The stream function of an ordinary vortex has circular contour lines and the arrows are tangentials. Hence, the green arrow originating from the red VC, e.g. is perpendicular to the line connecting the red VC with the green one. The contributions of the two IVs (blue arrows), which are counter-rotating, largely cancel out each other far from their centres. Hence, only their combined (and small) effect on the OVs $V_{1}$ and $V_{4}$ is depicted. On the other hand, the blue arrows that originate from the IVs depict the velocity induction only from the adjacent IV (there is no self-induction). It follows that those blue arrows are perpendicular to the line connecting the two IVs.

Figure 4.

Initial positions of the vortex centres (red asterisk: $\textit{V}_{\text{1}}$ ; blue asterisk: $\textit{V}_{\text{2}}$ , $\textit{V}_{\text{3}}$ ; green asterisk: $\textit{V}_{\text{4}}$ ) and its approximate impact on the surrounding vortices. Each panel shows a different simulation (see title on top each panel). The arrows indicate the direction and strength of the velocity induction of the neighbouring vortices. The contribution of a specific vortex is plotted in the same colour as the asterisk labelling its centre. Note that the blue arrows originating from the OVs $\textit{V}_{\text{1}}$ and $\textit{V}_{\text{4}}$ show the combined effect of both IVs $\textit{V}_{\text{2}}$ and $\textit{V}_{\text{3}}$ , whereas the blue arrows originating from $\textit{V}_{\text{2}}$ and $\textit{V}_{\text{3}}$ depict the velocity induction of the adjacent IV. At each vortex, the combined effect of all surrounding vortices is given by the black arrow. The lengths of the grey arrows inserted in each panel represent a wind speed of 2m/s. We use different scales for the $\textit{V}_{\text{1}}$ and $\textit{V}_{\text{4}}$ -arrows on the one hand and the $\textit{V}_{\text{2}}$ and $\textit{V}_{\text{3}}$ -arrows on the other hand reflecting the fact that the IVs move faster than the OVs.

The classical single aircraft case is introduced as a reference in panel (a). At both VCs, the velocity is given by $u=(0,-w_{0})$ . With $\Gamma_{t=0}=520$ m²/s and $b_{0}=47.3\textrm{m}$ (as in the formation flight scenarios), $w_{0}=1.75\textrm{m}/\textrm{s}$ follows.

Moreover, the inserted legend introduces the angle $\phi$ which defines the propagation direction.

Panel (b) shows the default formation flight simulation. The velocity at the two OVs has a dominant downward component with a small lateral component to the right (black arrows). The downward transport is mainly induced by the other OV (green or red arrow, resp.). As noted before, the combined effect of the IVs on the OVs is rather small (blue arrows) and leads to a small counterclockwise rotation of the resulting velocities at $V_{1}$ and $V_{4}$ . Compared to the single aircraft case, the downward velocities are around a factor of two smaller, as the separation distance between the OVs (given by $b_{0} + DX$ ) is roughly twice as large as in the single aircraft case (vortex separation $b_{0}$ ). The IVs are transported downward by both OVs. In sum, the downdraught is twice as large as in the single aircraft case. Yet, the strongest induction comes from the adjacent IV. The IVs mutually induce a strong transport to the right. In total, the IVs travel with more than $5\textrm{m}\textrm{/s}$ in $\phi\approx 110^\circ$ direction. The second row shows scenarios with a smaller (panel c)) and larger (panel d)) lateral separation DX compared to the default scenario. We find similar velocity inductions at the OVs. Most notably, the mutual induction of the two IVs changes. In all cases, a strong lateral transport to the right remains. The vertical component, however, might be positive (for $DX > b_{0}$ ) or negative (for $DX < b_{0}$ ). In total, we find an initial transport of the IVs at a speed of $6.5\textrm{m}/\textrm{s}$ in $\phi=135^\circ$ -direction for $DX=45\textrm{m}$ . For $DX=60\textrm{m}$ , the vertical velocities cancel out each other and the IVs travel purely horizontally with $u=3\textrm{m}/\textrm{s}$ . A change in the vertical offset DZ (third row) has similar effects as a DX-variation. Again, the mutual induction of the IVs is affected the most and changes their initial speed and direction. The two remaining cases in the fourth row will be discussed later. We can conclude that in all treated cases the early vortex evolution is characterised by a fast movement of the IVs towards the OV $V_{4}$ . In general, the movement is always towards the OV of the lower vortex pair, which is produced by the LAC.

Figure 5.

Contour plot of axial vorticity $\omega_\textit{y}$ (averaged along the axial direction) for four different simulations (from top to bottom, see label on the right) and four times (from left to right, see time label inside each panel). The black and grey lines indicate the trajectories of the vortex centres (determined by local extrema of vorticity) from time $\textit{t} \,=\, \text{0}$ up to the displayed point in time. A sequence of symbols (see box in each panel) highlights the vortex positions every 30s, starting with a black plus sign for $\textit{t} \,=\, \text{0}$ s.

Figure 5 shows the axial vorticity field averaged along flight direction at different vortex ages. The positions of the VCs are tracked by applying a simple vortex identification method based on finding local vorticity extrema.

Four simulations with different $DX-DZ$ combinations are selected to demonstrate the diversity and complexity of possible vortex evolutions. The first row shows vorticity fields of the default simulation at $t=16\textrm{s},\ 1,\ 3$ and $7.5\textrm{min}$ . In the beginning, both IVs approach $V_{4}$ . Soon, $V_{3}$ gets closest to $V_{4}$ and those two vortices feature a strong mutual induction. The $V_{3}\& V_{4}$ vortex pair moves quickly away; $V_{3}$ propagates more than 300m in the first minute. The $V_{3}$ , $V_{4}$ -trajectories bend upwards in counterclockwise fashion. After 1min both vortices are around $50\textrm{m}$ above flight level, and more than $60\textrm{m}$ above their creation level. Soon after, they slow down and seem to dissolve, at least our simple method based on longitudinally averaged vorticity evaluation is not able to identify them any longer. However, the 3D representation of vorticity in Fig. 3 has already revealed that the two vortices indeed have dissolved at that stage.

The vortices $V_{1}\ V_{2}$ also form a pair and mutually induce a slow downward transport. After 3min, the vertical displacement of $V_{1}$ and $V_{2}$ is only $70\textrm{m}$ and 110m, respectively. Both vortices are long-living and, in particular, $V_{1}$ can be tracked over 9min (the latest time shown here is 7.5min, though). Whereas $V_{2}$ more or less ceases to move and the VC resides around $z=-100\textrm{m}$ , the direction of the $V_{1}$ -motion reverses at some point. Then $V_{1}$ starts to move upwards and eventually reaches its initial level $z=0\textrm{m}$ .

The second row shows the simulation where DZ is raised from $14\textrm{m}$ to $25\textrm{m}$ . The vorticity distribution after 16s looks similar to the latter case. The relative positions of the vortices $V_{2}$ , $V_{3}$ and $V_{4}$ are only slightly different. Again, $V_{3}$ and $V_{4}$ build a vortex pair. However, the small initial differences qualitatively change its trajectory and the vortex pair travels sideways. After 1min, $V_{3}$ is displaced by nearly 200m in lateral direction. Soon after, the motion slows down. After 4.5 min the vortex pair propagated only another 100m in lateral direction and moved slightly upwards. Eventually the vortices dissolve right above their creation level. As in the latter case, the vortices $V_{1}\& V_{2}$ build a vortex pair and travel straight downward during the first 2min. Contrary, two the latter case, both vortices slow down, remain at around $z=-100\textrm{m}$ and move slightly to the left.

The third row shows the simulation where the original DZ is retained and DX is increased by $10\textrm{}$ to $60\textrm{m}$ . The trajectories are qualitatively similar to case 2. Compared to this case, the final position of the $V_{3}\& V_{4}$ vortex pair is, however, only 200m to the right (instead of 300m in case 2) and around $80\textrm{m}$ lower. Moreover, the $V_{1}\& V_{2}$ vortex pair travels further down and ends up $150{-}200\textrm{m}$ below the creation level.

The forth row shows a case with smaller DX ( $45\textrm{m}$ instead of $50\textrm{m}$ in case 1). Despite this small shift in lateral separation, the vortex evolution differs largely and features new aspects not seen in cases 1 to 3. At $t=16$ s, the vortex $V_{2}$ is in between of $V_{3}$ and $V_{4}$ . $V_{2}$ and $V_{4}$ have the same sense of rotation, the weaker vortex ( $V_{2}$ ) merges in a turbulent way into the strong vortex ( $V_{4}$ ), and $V_{2}$ cannot be tracked beyond $t=30$ s. Then, $V_{3}$ spins around the strong vortex $V_{4}$ in counterclockwise sense of rotation. After 70s, the $V_{3}$ trajectory completed a full cycle around $V_{4}$ and approaches $V_{1}$ . $V_{1}$ and $V_{3}$ have the same sense of rotation. And again, the weaker vortex ( $V_{3}$ ) merges into the stronger vortex ( $V_{1}$ ) with the effect that $V_{3}$ can be tracked for less than 2min. After 2min, only $V_{1}$ and $V_{4}$ are present at an altitude of around $z=-100\textrm{m}$ . They continue to descend, though the descent slows down and comes basically to a halt after 3min. The final vertical displacement is less than 150m.

The four simulations exhibit qualitatively different vortex evolutions despite their rather small ( $<10\textrm{m}$ ) initial differences. To sum up: What are new aspects compared to the classical vortex descent behind a single aircraft?

We find a strong initial interaction of the two IVs and their lateral propagation (towards the OV of the lower vortex pair generated by the LAC). Afterwards, different phenomena can be observed in the various simulations: a strong lateral transport of the LAC vortex pair, its dissolution far above the flight level, a merging of co-rotating inner and outer vortices, or a reduced descent of the FAC vortex pair. In one case, a single vortex could be tracked over 9min, much longer than in the single aircraft case. Notably, the wake vortex evolution depends very sensitively on the initial lateral and vertical offset.

3.3 Plume dimensions

In this section, the spatial distribution of ice crystal number concentrations is analysed. Figure 6 juxtaposes the four FF simulations discussed before and the classical SA (single aircraft) case taken from U2014. For the moment, the ice crystals can be regarded as a passive tracer, i.e. changes in the contrail structure are only due to transport and not due to microphysical loss processes. Hence, we use the terms plume and exhaust instead of contrail and ice crystals in the following. We shortly recall the spatial plume initialisation. It consists of two circular discs with a uniform concentration field where the centres are collocated with the initial LAC VCs. The plumes of the FAC are given by Gaussian distributions. Their centres and peaks, respectively, are aligned with the lateral engine position and are inboard of the FAC VCs. The exhaust gets entrained into the nearby vortices within the first 10s see e.g. Fig. 33.2 in^{(Reference Unterstrasser, Sölch and Gierens65,Reference Paoli, Nybelen, Picot and Cariolle66)} . Then, most of the exhaust is trapped inside the vortices and hence the plume expansion is roughly along the vortex trajectories. In the single aircraft case, the vortex descent leads to a substantial vertical plume expansion. In a thermally stable atmosphere, baroclinic instabilities occur around the vortices and exhaust is continuously detrained form the vortex system^{(Reference Holzäpfel, Hofbauer, Darracq, Moet, Garnier and Gago67)}. The detrained exhaust moves up again due to buoyancy and makes up a curtain between the actual and the initial vortex position^{(Reference Gerz and Holzäpfel68,Reference Gerz and Ehret69)} . After 5min the highest plume concentrations can be found around the original emission altitude. Moreover, the lower part of the plume broadens between $t=3$ and $5\min$ since the vortex tubes start to meander along flight direction (over which is averaged here) as a result of the Crow instability^{(Reference Crow64)}. In this example, the plume is eventually 480m deep and 330m broad. After 5min the vortices have dissolved and the simulation ends (hence, no panel for $t=9\min$ exists for the single AC case). The further plume expansion is dominated by atmospheric processes and its analysis is not part of this study.

Figure 6.

Contour plot of ice crystal number concentrations (averaged along axial (=flight) direction) after 3, 5 and 9min. The simulations shown on columns 2–5 are the same as in Fig. 5 (see title on top). Additionally, the first column shows results from a single aircraft scenario adapted from Fig. 1 of^{(Reference Unterstrasser1)}, only for $\textit{t} \,=\, \text{3}$ and 5min. For each point in time, the according colour bar is plotted on the right. To make the contrail dimensions of all panels visually comparable, all axes use the same scale. The two red values in each panel provide the contrail width and height (the thresholds used for their evaluation are smaller than the smallest contour level at display; hence the determined numerical values may be bigger than what the plot suggests). The FORMIC simulations use $R{H_i} = {\mkern 1mu} {\rm{110}}\% $ , whereas the SA simulation uses $R{H_i} = {\mkern 1mu} {\rm{120}}\% $ .

In the FF scenarios, the vortices live longer and the third row shows the panels for $t = 9\min$ . Moreover, all plot axes (including the single AC cases) use the same scale. From the different panel sizes used in the SA case on the one hand and FF cases on the other hand, it becomes directly apparent, that the plume dimensions are qualitatively different. For $t=3$ and $5\min$ , the displayed FF plume structures in Fig. 6 are consistent with the vortex patterns found in Fig. 5. The vertical/lateral expansion is smaller/larger than in the SA case. After the vortices have dissolved and parts of the exhaust moved upwards due to buoyancy ( $t=9\min$ ), the plumes are at most 280m deep (cf. to 480m in the SA case) and $500{-}700\textrm{m}$ broad (cf. to 330m in the SA case).

3.4 Sensitivity to lateral and vertical offset and relative humidity

Sections 3.2 and 3.3 discussed in detail several selected simulations. Contrarily, this section will give a systematic overview over all simulations and summarises physical sensitivities to the lateral offset DX, vertical offset DZ and relative humidity $RH_{i}$ . Whereas DX and DZ strongly affect the vortex evolution (as seen before), $RH_{i}$ affects only ice microphysical processes in the contrail.

The contrail structure is analysed by presenting profiles of ice crystal number in transverse and vertical direction. From this the contrail width and depth can be derived. The contrail depth, in particular, increases very quickly due to the wake vortex descent, much faster than by sedimentation or turbulent diffusion in later stages.

Figure 7 shows vertical profiles of ice crystal number for various values of DX (1st column), DZ (2nd column) and $RH_{i}$ (3rd column) for three different instances of times (3, 5 and 7–8min). The discussion of the 4th column will be deferred to the appendix. The vertical profiles are computed by integrating ice crystal number concentrations over the transverse direction and averaging them along flight direction, ending up with units per m². The area left of the curves is proportional to the total ice crystal number (units: per meter of flight path). The $RH_{i}$ -column additionally shows vertical profiles of the classical SA case for the same range of $RH_{i}$ -values (simulation data taken from U2014).

Figure 7.

$\textit{DX}, \textit{DZ}, \textit{RH}_{\textit{i}}$ and IV sensitivity experiment: Profiles of ice crystal number for various times (indicated on the right of each row) in vertical direction. The simulation series are described in Table 2. The colours are defined in Fig. 9; the black solid curve in each column shows the same default simulation.

The most obvious aspect in the Fig. 7 is the difference between the FF scenarios on the one hand and the SA scenarios on the other hand. As already noted for the exemplary case in Fig. 6, the vortex pair in the single AC cases descends further down than the vortex system in the FF cases. After 3min, the SA plume extends from flight level down to 340m below flight level. In any FF scenario, the plume does not extend beyond 170m below flight level and much more ice crystals are present above flight altitude. After 5min, the SA plumes reach $z=-400\textrm{m}$ and extend only moderately into regions above flight altitude, yielding a contrail vertical extent of 480m in the maximum case. In any FF scenario, no substantial further downward movement of the plumes is apparent and all plumes lie in the region between $z=-190$ and 170m. The thickest contrail is about 300m thick and several thinner contrails are barely thicker than 200m.

The $RH_{i}$ -column shows a strong $RH_{i}$ -effect on contrail ice crystal number which has been long known for the classical case^{(Reference Sussmann and Gierens31,Reference Lewellen and Lewellen33)} . Downward moving air expands and heats adiabatically. Thereby the saturation vapour pressure increases, the local relative humidity decreases and leads to sublimation and loss of ice crystals. The weaker the ambient supersaturation (i.e. the closer $RH_{i}$ is at to 100%), the more ice crystals get lost during vortex descent. For the classical SA case, all ice crystals in the lower part of the plume vanish for $RH_{i} \le 110\%$ (red, green, black curves) and the contrail vertical dimension is reduced compared to the moister cases $RH_{i}=120\%$ or 140% (blue and brown curve). In the FF scenarios, again fewer ice crystals survive, the lower the ambient $RH_{i}$ is. Yet, the $RH_{i}$ -effect is not as pronounced as in the SA cases and the contrail depth is about the same for all $RH_{i}$ -values. This is due to the fact that the vertical plume displacement is less than in the SA case and the plume adiabatic heating is not as strong.

Figure 8.

$\textit{DX}, \textit{DZ}, \textit{RH}_{\textit{i}}$ and IV sensitivity experiment: Profiles of ice crystal number for various times (indicated on the right of each row) in transverse direction. The simulation series are described in Table 2. The colours are defined in Fig. 9; the black solid curve in each column shows the same default simulation.

Next, we discuss the sensitivity to DX and DZ (1st and 2nd column). Unlike to $RH_{i}$ , however, no systematic trends can be found in the sense that e.g. contrails become deeper for larger DX or DZ. For example, after 3min the contrail top varies from $z=0$ to $z=150\textrm{m}$ among the four DX-simulations and from $z=50$ to $z=150\textrm{m}$ among the four DZ-simulations. The contrail with the highest top (at $z=150\textrm{m}$ ) occurs for the intermediate values $DX=50\textrm{m}$ (out of the interval [ $45,60\textrm{m}$ ]) and $DZ = 7\textrm{m}$ (out of the interval [ $0,25\textrm{m}$ ]). Another example of non-systematic results can be seen in the DZ-panel for $t = 5\min$ . The vertical distributions for $DZ=0$ and $25\textrm{m}$ are nearly the same. Moreover, the $DZ=14\textrm{m}$ -contrail has a similar vertical extent, yet with somewhat lower ice crystal numbers. The contrail with $DZ=7\textrm{m}$ differs qualitatively from the three other DZ-cases, as the contrail top is located around 100m higher. The vertical profiles after $7-8\min$ show that in all FF cases, the contrails are around 170 to 250m deep; the variation within the DX-series is even less than $30\textrm{m}$ . This small spread is remarkable as we have discussed and pinpointed the quite different vortex evolutions for small changes in DX or DZ.

Next, transverse profiles for the same set of simulations are presented in Fig. 8. Again no trends with DX and DZ are apparent and we only report and shortly discuss results for $t=5\text{min}$ and $t=7-8\text{min}$ . In the SA setup the single vortex pair has decayed after $<5$ min and simulations stop around $t=5\min$ . Hence, the upper row of the figure enables the comparison between the FF and SA scenario, whereas the lower row reports the contrail dimensions after vortex decay for the FF scenarios. At initialisation, the FF contrails are around $130{-}140\textrm{m}$ (depending on DX) and SA contrails are around $90\textrm{m}$ broad. Within the first minute, a strong lateral transport occurs in some FF setups and the FF contrail width ranges from 200 to 320m (at $t=1\min$ , not shown). Due to the different strength of the lateral transport in the individual FF cases, the transverse distribution differs strongly from case to case. We find distributions with one or two distinct peaks. The strongest peaks can occur at around $x=0\textrm{m}$ in one case or at $x=300\textrm{m}$ in some other case. At $t = 5\min$ , the contrail width varies from 350m for narrow FF contrails to 570m for the broadest FF contrail. A variation of $RH_{i}$ changes the FF contrail width only weakly, whereas for the SA contrails the width ranges from 130 to 290m due to a Crow instability triggered vortex meandering in the lower plume part. Hence, typical SA contrails are usually not as broad as such behind a formation. FF contrails continue to spread after $t=5\min$ and for $t = 7 - 8 \min$ contrail width ranges from 400 to 680m.

Integrating the vertical/transverse profiles over the vertical/transverse dimension gives the total ice crystal number $\mathcal{N}$ per meter of flight path. Figure 9 shows the temporal evolution of the normalised total ice crystal number $f_\mathcal{N}(t)=\mathcal{N}(t)/\mathcal{N}(t=0)$ . Note that in the FF scenarios $\mathcal{N}(t=0)$ is twice as large as in the SA scenarios. The quantity $f_\mathcal{N}$ diminishes over time due adiabatic heating induced ice crystal loss. The curves flatten out after several minutes indicating that the vortices stopped descending and pushing the local plume $RH_{i}$ below the saturation level. In the chosen classical SA case, this happens after 3 to 4min (the time of vortex break-up depends on the AC type and atmospheric stability, though, see UG2014). In some FF scenarios, the ice crystal loss gets negligible after around 6min. But in many cases crystal loss does not fully vanish and occurs over the complete simulation period, though at a reduced rate towards the end. Again no trends with DX and DZ can be observed and the fraction of surviving ice crystals $f_{\mathcal{N},\text{s}}$ lies between 0.4 and 0.6 (surviving here refers to the $f_\mathcal{N}$ -value at the end of the simulation). $RH_{i}$ is the dominant parameter for $f_{\mathcal{N},\text{s}}$ . Generally, a larger fraction of ice crystals survives in the FF than in the SA scenarios for a given $RH_{i}$ -value. When $RH_{i}$ is reduced from 140% to 120% and further down to 110%, $f_{\mathcal{N},\text{s}}$ strongly decreases in the SA case (from 0.92 to 0.67 and further down to 0.28). In the FF scenarios, on the other hand, a reduction of $RH_{i}=140\%$ down to 120% (the supersaturation $RH_{i}-100\%$ is actually halved) has only a moderate effect and $f_{\mathcal{N},\text{s}}$ goes down from 0.96 to 0.89. For both $RH_{i}$ -values, the vortex descent does not suffice to produce a substantial crystal loss. Only, a further reduction down to 110% makes the atmospheric buffer effect so small that the reduced vortex descent of FF cases produces local subsaturations that are sufficient to lead to a substantial loss of ice crystals ( $f_{\mathcal{N},\text{s}}= 0.62$ ).

Figure 9.

Temporal evolution of normalised ice crystal number $\textit{f}_\mathcal{N}$ for the $\textit{DX}, \textit{DZ}, \textit{RH}_{\textit{i}}$ and IV sensitivity experiments. The simulation series are described in Table 2.

3.5 Robustness of findings

Various further experiments and a model comparison have been carried out in order to check the robustness of the above findings. These efforts are shortly summarised here and a full description is deferred to the appendix. In a grid sensitivity study, where the resolution and domain size along flight direction was varied, we find that the grid choices have a minor impact on the quantities of interest. Moreover, we perform four different realisations of the same simulation by simply shifting the turbulent background fields in lateral direction. We find small turbulence induced uncertainties implying the significance of the physically induced differences seen in the results section. Section 2.2.2 described various versions of IV initialisations. The initialisation choices have a non-negligible impact on the simulation results. Reducing this uncertainty will be an important future task and the way forward will be laid out in the discussion section. Finally, MGLET simulations are repeated for different DX-DZ combinations. In all three MGLET simulations we find patterns consistent with those of the EULAG simulations. Keeping in mind the diversity and complexity of the observed vortex evolution for small variations of DX or DZ, it is very convincing that the patterns are similar in both models and this boosts the confidence in the robustness of the simulation results.