The study analyzes eleven years of ERA5 reanalysis data covering the period from 1 January 2010, to 31 December 2020. Data is available at the standard synoptic times each day (00:00, 06:00, 12:00, 18:00 UTC), resulting in a total of 16 060 datasets for analysis. ERA5 provides data with a horizontal resolution of 0.25° in longitude and latitude, and a vertical resolution of 137 model levels with an approximate resolution in the tropopause region of about 300m. The data set contains variables of 3D winds (u,v,w), temperature T, and specific humidity q. For the investigation we distinguish water vapor in air masses in the state of sub or super saturation with respect to ice. The geographical area used for the analysis is narrowed down to range from 80 to 30° latitude in the Northern Hemisphere in order to exclude the tropics and polar regions. In our investigation we focus on pressure levels of 200, 250, and 300 hPa, representing the upper troposphere and tropopause region similarly to the investigation by. The data on adjacent model levels were interpolated on the respective isobaric surfaces.

Here, we briefly describe the processing of the data; for details we refer to Appendix .

Figure  shows the number of observed ISSRs by month on the 200, 250 and 300 hPa pressure levels during the period from 2010 to 2020. The total number is a sum of all ISSRs appearing at each point in time. This causes the number to be significantly higher than the ISSRs that appeared in a given month, as ISSRs are counted multiple times when having a life-span longer than 6 h. A clear dependence of the number of ISSRs with altitude is evident. At lower altitudes (i.e. higher pressures) we find significantly more ISSRs than at higher altitudes. This can be explained by the fact that in the extratropics the (thermal) tropopause is located in the pressure range between 300 and 200hPa, and ISSRs are almost exclusively located in the troposphere. Only few ISSRs seem to reach into the stratosphere . Generally, ISSRs are most frequently located close to the tropopause.

The tropopause also shows a seasonal cycle with high altitudes during summer and lower altitudes during winter e.g.,. Nevertheless, the probability to measure dry and warm air in the stratosphere at a certain pressure level in the extratropics is much higher for pressure level p=200hPa, than for p=250hPa or p=300hPa. This might explain the strong differences in the amount of ISSRs between the three pressure levels. The strong seasonal cycle in the number of ISSRs is a prominent feature, which is most pronounced at pressure level p=300hPa, i.e. in the upper troposphere. We find maxima in summer and minima in winter for all 3 pressure levels. The month with the highest observed number at pressure level p=300hPa is July 2013 with 9511 detected ISSRs, the minimum number of 3290 ISSRs at the same pressure level can be observed in February 2018. Respectively, the highest number of ISSRs at pressure level p=200hPa was recorded in July 2020 with 3462 ISSRs, whereas the minimum was observed in February 2012 with 954 ISSRs. Here, we see the substantial spread in the amount of ISSRs.

The seasonal cycle compares well with former investigations. also found this signature with highest values of ISSR numbers during summer and lowest counts in the winter's season. Furthermore, the strong variation of numbers with the vertical levels agrees well with our findings.

In addition to the total number of ISSRs we investigate the total area of the detected ISSRs. Analogous the number of ISSRs, in Fig.  the total area of observed ISSRs by month on the 200, 250 and 300 hPa pressure levels in the timespan from 2010 to 2020 is shown. As for the evaluation of the total number, the area might be enhanced by counting long-lived ISSRs multiple times. The total area is largest at the lowest altitude (i.e. pressure level p=300hPa), and is decreasing with increasing altitude. The highest total area in one month was observed in May 2019 with a summed area of 2.65×109km2 at pressure level p=300hPa. On the same pressure level the minimum of 1.32×109km2 is found in February 2018. At the highest altitude level (pressure level p=200hPa) the area is strongly reduced compared to the other two levels, which have comparable area values. At p=200hPa the month with the highest total area of supersaturation in respect to ice is July 2020, summing up to 1.12×109km2. February 2015 at this pressure level constitutes the absolute minimum observed on all pressure levels with 0.41×109km2. Like the total number, the area also shows a seasonal cycle. We find maximum values in the summer months, and minimum values in the winter months, similarly to the investigations of the total number of ISSRs. The cycle is most pronounced in the two lower levels (p=300hPa and p=250hPa), while for the highest level (p=200hPa) the variation is weak but still apparent. In summer, the total area of ISSRs at p=250hPa is approaching values comparable to (p=300hPa. This comparability only holds for the area maximum in summer, whereas the area decreases more strongly at p=250hPa compared to p=300hPa in winter.

In Fig.  examples of ISSRs at the three different pressure levels over the North Atlantic region are presented for three times in January 2010, giving a snapshot. A similar example is shown in Fig.  for one snapshot in July 2019 (3 times). In winter the ISSRs are discernibly more compact and cover a limited area (Fig. ). At the highest altitudes (i.e. at pressure level p=200hPa) there are very few ISSRs while at lower altitudes (higher pressures) more ISSRs can be seen. In winter the tropopause is located at lower altitudes, thus the pressure level p=200hPa is (most likely) completely contained in the stratosphere. In contrast, in July we find widespread and frayed ISSRs at all three pressure levels (Fig. ). Even at p=200hPa ISSRs can be found, since during summer the tropopause is located at higher altitudes and ice supersaturation is present there.

To gain a better understanding of the correlation between ISSR count and area in relation to seasons, we calculated the mean ISSR area from the corresponding number and area statistics. The results are shown as timeseries in Fig. .

We see a pronounced seasonal cycle in the mean area of ISSRs with a very distinct maximum in winter (mean values A¯∼5–6×1011m2) and a clear minimum in summer (mean values A¯∼2.5–3×1011m2). Assuming a purely spherical shape, this would translate into a factor of 2 for the radii. This behavior hints at two different formation mechanisms dominating in summer and winter respectively.

While the average area per ISSR on the 200 and 300 hPa levels show good agreement across all seasons, this ratio is noticeably larger at 250 hPa during summer. This feature does not represent an outlier but can be explained by jointly considering the number count and total area. In winter, both the number count and total area at 250 hPa are substantially lower than at 300 hPa. During summer, the total area at both levels is comparable, despite a lower number count at 250 hPa. Consequently, the ratio of total area to number count is increased at 250 hPa in summer. The underlying cause of this feature is the 250 hPa level being recurrently located within both the upper troposphere and lower stratosphere depending on the season. In contrast, 200 hPa remains predominantly above and 300 hPa predominantly below the tropopause throughout the year .

The different shape of ISSRs in winter and summer, as observed in Figs.  and , can also be seen in the (statistical) investigations of the fractal dimension, as described in the next section.

The fractal dimensions for all ISSRs in the data set are derived using the methods explained in Sect.  and Appendix . We calculate area and perimeter for each detected ISSR (i.e. each connected “island”) and determine the fractal dimension by using all data of a certain pressure level for a whole month in a double logarithmic scatter plot. In Fig.  an example is shown for January 2017, in the Supplement another example provided for July 2017).

In the scatter plots the data is confined to a linear point cloud. There is a perceptible change in slope at areas of A∼2×1012m2. As this signature is weak, we do not separate these two regimes in this evaluation. It might be related to the fact that the horizontal extent of objects on isobaric surfaces in the Earth's atmosphere is limited by the finite surface of the Earth. This can also be seen in the investigations of the horizontal extents in Sect. . A linear regression is applied to the complete dataset of each respective month, and the resulting slope is used to determine the respective fractal or self-similar dimension, as indicated in Eq. (). We find values in the range D∼1.18–1.375, i.e. values well above the value of D=1 for a geometric object.

As illustrated in the simple examples presented in Appendix , the increase of fractal dimension for larger objects might be driven either by more elongated objects (for example the systematically stretched rectangular) or by more frayed objects. Both reasons seem to be plausible from the presented snapshots in Figs.  and , but a clear attribution is not possible.

Finally, we determine the annual evolution of the fractal dimension as obtained from monthly data at different pressure levels; the results are represented in Fig. . We see a clear separation by vertical levels. For each month, the fractal dimensions are ordered, with the smallest value assigned to pressure level p=200hPa, followed by the medium value at pressure level p=250hPa and the highest value at pressure level p=300hPa. The fractal dimensions of the observed ISSRs are in the ranges D∼1.18–1.29 (p=200hPa), D∼1.22–1.36 (p=250hPa), and D∼1.26–1.375 (p=300hPa). All fractal dimensions D show a strong seasonal cycle with maxima during summer and minima during winter. The amplitude for the cycles is largest for the pressure level p=250hPa, followed by a slightly reduced amplitude for the pressure level p=300hPa. The variation of the monthly fractal dimension is smallest for the pressure level p=200hPa, but it is still well pronounced. The correlation coefficient for the linear fit indicating the quality of the fits by the linear regression are relatively high (r2≥0.95). Thus, we can conclude that these results are robust and constitute a new and pronounced feature of ISSRs. The influence of the grid resolution (i.e. grid coarsening) on the derived fractal dimension is discussed in the Supplement. Unless otherwise indicated, all plots and results are based on ERA5 data at a horizontal resolution of 0.25° in both longitude and latitude.

The large seasonal variation of D might point to different physical formation mechanisms, as discussed below.

Although this is the first study investigating the fractal nature of ISSRs, we can try to compare our results to similar studies on fractal dimensions, e.g., for high clouds observed in satellite data and models with global storm resolving resolutions. Since ice clouds form in ISSRs, there should be a comparable relationship. evaluated satellite data (spatial resolution of 1km) of (ice) clouds and found a fractal dimension of D=1.37±0.02 for high cirrus clouds, and a fractal dimension of D=1.18±0.05 for cumulonimbus clouds. These values are in the same range as values of D found in this study, showing clear fractal properties of clouds. investigated satellite data (5km grid) and high resolution model data (initialized using the ECMWF 9km analysis data, linear resolution of 2.5 to 7.8km for the core ensemble) and found fractal dimensions in the range D∼1.35–1.45 for the satellite data, and D∼1.27–1.53 for the different (high-resolution) models, which are also comparable to our findings. Since we are not distinguishing between cloudy and clear air inside ISSRs, there might be a mixture of different signatures of clouds and cloud free air massses (constituting all ISSRs). Nevertheless, these results show that ISSRs exhibit a clear signature of self-similarity.

First, we investigate the meridional slices, i.e. pathlengths along longitudes. Since the major variation of horizontal extents is between winter and summer, we exclusively show these results in all respective figures. The probability density functions for winter and summer, differentiated into three pressure levels are shown in Fig. . The mean and median values are reported in Table .

For all pressure levels p=200/250/300 hPa, the mean meridional extent of ISSRs is significantly higher in winter compared to the summer months. While the density does not differ substantially between seasons for ISSRs up to a size of approximately 103km, the disparity between summer and winter inflates with increasing measured span. The absolute mean span increases with decreasing pressure (i.e. with rising altitude levels). The relative difference comparing summer and winter slightly increases with higher pressure (i.e. lower altitudes). All values are reported in Table . The strong seasonal cycle is present in both the mean values and the extreme tails of the distributions. In winter, the season with maximum mean values, a clear enhanced probability of finding extremely large pathlengths (i.e. enhanced values in the tail of the distribution) can be seen, whereas the probability of extremely large pathlengths is reduced in summer. The histograms are not symmetric but right-skewed, so the mean is pulled upward by few large ISSRs. The median is more robust and thus more suitable to assess the typical 2D features of ISSRs. The median behaves similarly to the mean in that it generally decreases with increasing pressure, the exception being a minor increase from 200 to 250 hPa in the summer months.

The data shows two regimes with roughly linear correlation but different slopes in the Weibull plots (i.e. in the respective values of parameter p) in Fig. . Thus, there are two distinct distributions superimposed in the histograms. The transition between these regimes is roughly located at L∼2×102km. For the smaller ISSRs we find values of p∼1.5, whereas the distribution is close to an exponential distribution (p∼1) for the larger ISSRs.

The presence of two clearly separated size regimes suggests different formation mechanisms of ISSRs operating at distinct spatial scales, i.e. synoptic-scale vs. localized on mesoscales, leading to characteristic horizontal extents. This hypothesis is subject to further investigation, since an in depth analysis is out of scope of this study.

Alike investigating the meridional slices along longitudes, we now investigate horizontal extents along latitudes, i.e. slices in zonal (East–West) direction within ISSRs. The probability density functions for the two seasons (summer/winter), separated into three pressure levels, are shown in Fig. .

The distribution of measured zonal distances shows almost identical qualitative characteristics as the meridional counterpart. We see a strong decrease in frequency of occurrence for spans larger than the mean values. In contrast to the meridional distributions, there is an additional very small mode at very large spans (L∼10000km), indicating the occurrence of very few very large ISSRs in East-West direction. This difference can be explained by pathlengths in meridional direction being limited to the 30–80° N range we study, while zonal pathlengths can theoretically extend to the length of the latitude. For all pressure levels, the mean span is significantly higher in the winter months compared to the summer months. Differences in the distribution of spans by season are most clear for larger spans. The mean generally decreases slightly with a higher pressure level. The decrease in mean span with higher pressure is more evident in the summer months. Analogous to the meridional observations, the median is consistently lower than the mean. The median increases marginally from 200 to 250 hPa in winter but behaves analogous to the mean in all other instances. It is lowest at the highest pressure level. As with the meridional span, the distribution is right-skewed. All values are reported in Table . We see the same feature of enhanced probability for extremely large pathlengths during winter and reduced probability during summer.

In Fig.  we investigate the Weibull plots for the distributions. For the zonal spans, three distinct regimes are visible. There is a regime of very small spans (up to L∼102km), without a clear Weibull signature in the Weibull plot. In addition, there are two further regimes, separated at values L∼2×103km, which show linear behavior in the Weibull plots, thus indicating two different regimes clearly distinguishable by a different Weibull parameter p. The regime of intermediate spans of ISSRs results in an almost exponential distribution (Weibull parameter p∼1), whereas the regime of very large spans shows a Weibull characteristic with parameter p∼0.2. This characteristic remains virtually identical across seasons.

The regime of very small pathlengths might be an artifact of the evaluation. The real area of a pixel (or grid point) depends crucially on the latitude. Scanning high latitude circles will systematically produce small horizontal sizes for ISSRs consisting of few pixels.

The tiny mode of very large spans of ISSRs of orders up to almost latitudinal circumference might indicate the impact of synoptic and large scale weather systems, triggering atmospheric flows in the zonal direction with a persistent vertical upward motion, thus leading to a persistence of ice supersaturation in the UTLS. The upper bound of such synoptically driven ISSRs is given by the circumference of the Earth at certain latitudes.

In addition to the rather artificial longitudinal and latitudinal measurements, realistic pathlengths are investigated. In lieu of real in situ measurements, tracks along commercial aircraft routes are investigated. We exemplarily chose three transatlantic flight routes based on the flight tracks frequently found in the IAGOS data . The three tracks (i.e. direct flight routes Berlin–Chicago, Frankfurt–Atlanta, and Paris–Miami) are assumed as geodesics on a sphere (Fig. ). The pathlengths are obtained by measuring the distance along the sections of ice supersaturation on the flight track from airport to airport.

The histograms (Fig. ) show numerical artifacts for small ISSRs. We find the same features of pathlength statistics for mean values and distributions as in Sect.  and . The winter months yield the highest mean pathlengths while summer returns the shortest mean distances. For all seasons there is an increase of mean and median values with increasing altitude. The highest mean/median pathlengths can be found at the highest level (p=200hPa), and the lowest values at p=300hPa with intermediate values at the level p=250hPa. The distributions mostly agree for the two seasons, with a larger deviation for the largest pathlengths. During winter, there are more large ISSRs than during summer. The mean and median values of the pathlength distributions are represented in Table . The mean exceeds the median in all instances.

In the evaluation of the Weibull plots (Fig. ), we find two different regimes of distinct Weibull distributions with different values of p, separated at pathlengths L∼102km, analogous to the meridional extents. The difference in the regimes is more pronounced compared to the former evaluation, and the signal of the very large zonal extents is missing in this evaluation. This is expected, as the largest possible extent is the distance between airports. As before, we overlook the very small pathlengths in the linear regression. For both regimes, the seasonal cycle is not strongly pronounced. In the small pathlengths regime, we find values p∼1.95, whereas in the regime of large pathlengths, the values are close to an exponential distribution, i.e. p∼1. This signal is very robust, with almost no seasonal variation in the parameters or with respect to the different levels.

The results of the longitudinal, latitudinal and pathlength measurements agree well with another. The observation of the summer months yielding the shortest and winter the longest mean distances generally holds true across methods. This agreement applies to most results in terms of dependence on pressure level as well. The mean span is generally larger at lower pressures, i.e. higher up in the atmosphere. While this is always true for the comparison between levels p=200hPa and p=300hPa, in the intermediate level p=250hPa some variations can be seen. The mean values for zonal extents are generally larger than those for the meridional extents and the flight track pathlengths. This feature likely stems from ISSRs usually extending more in East-West direction (as shown in Figs.  and ), driven by the mean large scale flow.

The results of the simulated IAGOS flights have a larger qualitative agreement with the meridional extents. In both cases, there is a scale break at L∼102km , separating two different modes of Weibull distributions. The tiny mode of exceptionally large spans as found in the zonal slices cannot be detected in the pathlengths of realistic flight tracks. These very few, very large ISSRs are mostly elongated into zonal direction and the maximum of possible pathlength is limited by flight track length.

The distributions as well as the mean (and median) values of the simulated pathlengths compare qualitatively well with the former investigations of pathlengths as obtained from aircraft measurements . The mean (and median) values as obtained from the ERA5 data are generally larger than those of aircraft data. This is not surprising, since the aircraft data has a much higher temporal (and thus spatial) resolution. The Weibull shaped distributions can be seen in all evaluations with a varying degree of prominence of a scale break at pathlengths L∼102km. The fitted Weibull parameter of the distributions differ between aircraft data and ERA5 evaluation. The general feature of a smaller slope for pathlengths below the scale break and a higher slope for pathlengths above the scale break is present in all evaluations. The difference in slopes for aircraft vs. ERA5 data is likely due to the different resolutions of the data sets. Finally, we see that the range of the obtained distributions of pathlengths is roughly the same in all investigations (ranging from few 10 km up to ∼5000km).