Chopard, B. and Droz, M.: Cellular Automata Modeling of Physical Systems, Collection Alea-Saclay: Monographs and Texts in Statistical Physics, Cambridge University Press,
https://doi.org/10.1017/CBO9780511549755, 1998.
a,
b,
c
Clough, S. A., Iacono, M. J., and Moncet, J.: Line‐by‐line calculations of atmospheric fluxes and cooling rates: Application to water vapor, J. Geophys. Res.-Atmos., 97, 15761–15785,
https://doi.org/10.1029/92JD01419, 1992.
a
DeWitt, T. D., Garrett, T. J., Rees, K. N., Bois, C., Krueger, S. K., and Ferlay, N.: Climatologically invariant scale invariance seen in distributions of cloud horizontal sizes, Atmos. Chem. Phys., 24, 109–122,
https://doi.org/10.5194/acp-24-109-2024, 2024.
a
Diao, M., Zondlo, M. A., Heymsfield, A. J., Avallone, L. M., Paige, M. E., Beaton, S. P., Campos, T., and Rogers, D. C.: Cloud-scale ice-supersaturated regions spatially correlate with high water vapor heterogeneities, Atmos. Chem. Phys., 14, 2639–2656,
https://doi.org/10.5194/acp-14-2639-2014, 2014.
a
Florio, B. J., Fawell, P. D., and Small, M.: The use of the perimeter-area method to calculate the fractal dimension of aggregates, Powder Technol., 343, 551–559,
https://doi.org/10.1016/j.powtec.2018.11.030, 2019.
a
Fusina, F., Spichtinger, P., and Lohmann, U.: Impact of ice supersaturated regions and thin cirrus on radiation in the midlatitudes, J. Geophys. Res.-Atmos., 112,
https://doi.org/10.1029/2007JD008449, 2007.
a,
b
Gettelman, A., Fetzer, E. J., Eldering, A., and Irion, F. W.: The Globa
l Distribution of Supersaturation in the Upper Troposphere from the Atmospheric Infrared Sounder, J. Climate, 19, 6089 – 6103,
https://doi.org/10.1175/JCLI3955.1, 2006.
a
Gierens, K. and Spichtinger, P.: On the size distribution of ice-supersaturated regions in the upper troposphere and lowermost stratosphere, Ann. Geophys., 18, 499–504,
https://doi.org/10.1127/0941-2948/2002/0011-0083, 2000.
a,
b,
c,
d,
e,
f
Haywood, J. M., Allan, R. P., Bornemann, J., Forster, P. M., Francis, P. N., Milton, S., Rädel, G., Rap, A., Shine, K. P., and Thorpe, R.: A case study of the radiative forcing of persistent contrails evolving into contrail-induced cirrus, J. Geophys. Res.-Atmos., 114,
https://doi.org/10.1029/2009JD012650, 2009.
a
Hersbach, H., Bell, B., Berrisford, P., Hirahara, S., Horányi, A., Muñoz‐Sabater, J., Nicolas, J., Peubey, C., Radu, R., Schepers, D., Simmons, A., Soci, C., Abdalla, S., Abellan, X., Balsamo, G., Bechtold, P., Biavati, G., Bidlot, J., Bonavita, M., Chiara, G., Dahlgren, P., Dee, D., Diamantakis, M., Dragani, R., Flemming, J., Forbes, R., Fuentes, M., Geer, A., Haimberger, L., Healy, S., Hogan, R. J., Hólm, E., Janisková, M., Keeley, S., Laloyaux, P., Lopez, P., Lupu, C., Radnoti, G., Rosnay, P., Rozum, I., Vamborg, F., Villaume, S., and Thépaut, J.: The ERA5 global reanalysis, Q. J. Roy. Meteor. Soc., 146, 1999–2049,
https://doi.org/10.1002/qj.3803, 2020 (data available at:
https://apps.ecmwf.int/data-catalogues/era5/?class=ea, last access: 20 June 2024).
a,
b
Köhler, D., Reutter, P., and Spichtinger, P.: Relative humidity over ice as a key variable for Northern Hemisphere midlatitude tropopause inversion layers, Atmos. Chem. Phys., 24, 10055–10072,
https://doi.org/10.5194/acp-24-10055-2024, 2024.
a,
b
Krämer, M., Rolf, C., Luebke, A., Afchine, A., Spelten, N., Costa, A., Meyer, J., Zöger, M., Smith, J., Herman, R. L., Buchholz, B., Ebert, V., Baumgardner, D., Borrmann, S., Klingebiel, M., and Avallone, L.: A microphysics guide to cirrus clouds – Part 1: Cirrus types, Atmos. Chem. Phys., 16, 3463–3483,
https://doi.org/10.5194/acp-16-3463-2016, 2016.
a
Lamquin, N., Stubenrauch, C. J., Gierens, K., Burkhardt, U., and Smit, H.: A global climatology of upper-tropospheric ice supersaturation occurrence inferred from the Atmospheric Infrared Sounder calibrated by MOZAIC, Atmos. Chem. Phys., 12, 381–405,
https://doi.org/10.5194/acp-12-381-2012, 2012.
a,
b
Lee, D., Pitari, G., Grewe, V., Gierens, K., Penner, J., Petzold, A., Prather, M., Schumann, U., Bais, A., Berntsen, T., Iachetti, D., Lim, L., and Sausen, R.: Transport impacts on atmosphere and climate: Aviation, Atmos. Environ., 44, 4678–4734,
https://doi.org/10.1016/j.atmosenv.2009.06.005, 2010.
a,
b
Lee, D., Fahey, D., Skowron, A., Allen, M., Burkhardt, U., Chen, Q., Doherty, S., Freeman, S., Forster, P., Fuglestvedt, J., Gettelman, A., De León, R., Lim, L., Lund, M., Millar, R., Owen, B., Penner, J., Pitari, G., Prather, M., Sausen, R., and Wilcox, L.: The contribution of global aviation to anthropogenic climate forcing for 2000 to 2018, Atmos. Environ., 244, 117834,
https://doi.org/10.1016/j.atmosenv.2020.117834, 2021.
a
Mandelbrot, B.: The Fractal Geometry of Nature, W. H. Freeman and Co., ISBN 0-7167-1186-9, 1982. a
Mandelbrot, B. B., Passoja, D. E., and Paullay, A. J.: Fractal character of fracture surfaces of metals, Nature, 308, 721–722,
https://doi.org/10.1038/308721a0, 1984.
a
Marenco, A., Thouret, V., Nédélec, P., Smit, H., Helten, M., Kley, D., Karcher, F., Simon, P., Law, K., Pyle, J., Poschmann, G., von Wrede, R., Hume, C., and Cook, T.: Measurement of ozone and water vapor by Airbus in-service aircraft: The MOZAIC airborne program, An overview, J. Geophys. Res.-Atmos., 103, 25631–25642,
https://doi.org/10.1029/98JD00977, 1998.
a
Murphy, D. M. and Koop, T.: Review of the vapour pressures of ice and supercooled water for atmospheric applications, Q. J. Roy. Meteor. Soc., 131, 1539–1565,
https://doi.org/10.1256/qj.04.94, 2005.
a
Peitgen, H.-O., Jürgens, H., and Saupe, D.: Chaos and fractals – new frontiers of science, 2nd edn., Springer, ISBN 978-0-387-20229-7,
https://doi.org/10.1007/b97624, 2004.
a,
b
Petzold, A., Thouret, V., Gerbig, C., Zahn, A., Brenninkmeijer, C. A. M., Gallagher, M., Hermann, M., Pontaud, M., Ziereis, H., Boulanger, D., Marshall, J., Nédélec, P., Smit, H. G. J., Friess, U., Flaud, J.-M., Wahner, A., Cammas, J.-P., Volz-Thomas, A., and Team, I.: Global-scale atmosphere monitoring by in-service aircraft–current achievements and future prospects of the European Research Infrastructure IAGOS, Tellus B, 67, 28452,
https://doi.org/10.3402/tellusb.v67.28452, 2015.
a,
b,
c
Petzold, A., Neis, P., Rütimann, M., Rohs, S., Berkes, F., Smit, H. G. J., Krämer, M., Spelten, N., Spichtinger, P., Nédélec, P., and Wahner, A.: Ice-supersaturated air masses in the northern mid-latitudes from regular in situ observations by passenger aircraft: vertical distribution, seasonality and tropospheric fingerprint, Atmos. Chem. Phys., 20, 8157–8179,
https://doi.org/10.5194/acp-20-8157-2020, 2020.
a
Petzold, A., Khan, N. F., Li, Y., Spichtinger, P., Rohs, S., Crewell, S., Wahner, A., and Kramer, M.: Most long-lived contrails form within cirrus clouds with uncertain climate impact, Nat. Commun., 16,
https://doi.org/10.1038/s41467-025-65532-2, 2025.
a,
b
Rees, K. N., Garrett, T. J., DeWitt, T. D., Bois, C., Krueger, S. K., and Riedi, J. C.: A global analysis of the fractal properties of clouds revealing anisotropy of turbulence across scales, Nonlin. Processes Geophys., 31, 497–513,
https://doi.org/10.5194/npg-31-497-2024, 2024.
a
Reutter, P., Neis, P., Rohs, S., and Sauvage, B.: Ice supersaturat
ed regions: properties and validation of ERA-Interim reanalysis with IAGOS in situ water vapour measurements, Atmos. Chem. Phys., 20, 787–804,
https://doi.org/10.5194/acp-20-787-2020, 2020.
a,
b,
c,
d
Richardson, L. F.: Atmospheric diffusion shown on a distance-neighbour graph, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 110, 709–737,
https://doi.org/10.1098/rspa.1926.0043, 1926.
a
Sakellariou, M., Nakos, B., and Mitsakaki, C.: On the fractal character of rock surfaces, in: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol. 28, 527–533, Pergamon,
https://doi.org/10.1016/0148-9062(91)91129-F, 1991.
a
Sanchez, N., Alfaro, E., and Pérez, E.: The fractal dimension of projected clouds, Astrophys. J., 625, 849–856,
https://doi.org/10.1086/429553, 2005.
a
Schuh, H. Z.: hzschuh/Fractal_Paper: Fractal Characteristics of Ice-Supersaturated Regions in the Tropopause Region of the northern midlatitudes (Version fractal_code_v1), Zenodo [software],
https://doi.org/10.5281/zenodo.21158240, 2026.
a
Snow, A. D., Whitaker, J., Cochran, M., Miara, I., den Bossche, J. V., Mayo, C., Lucas, G., Cochrane, P., de Kloe, J., Karney, C., Shaw, J. J., Anh, T. Q., Filipe, Ouzounoudis, G., Dearing, J., Lostis, G., Couwenberg, B., Hoese, D., de Bittencourt, H., Little, B., May, R., Itkin, M., McDonald, D., Schneck, C., Gohlke, C., Jurd, B., Raspaud, M., Brett, M., and Meulien, M.: pyproj4/pyproj: 3.7.1 Release, Zenodo [code],
https://doi.org/10.5281/zenodo.14876934, 2025.
a
Spichtinger, P. and Leschner, M.: Horizontal scales of ice-supersaturated regions, Tellus B, 68, 29020,
https://doi.org/10.3402/tellusb.v68.29020, 2016.
a,
b,
c,
d,
e,
f,
g
Spichtinger, P., Gierens, K., Leiterer, U., and Dier, H.: Ice supersaturation in the tropopause region over Lindenberg, Germany, Meteorol. Z., 12, 143–156,
https://doi.org/10.1127/0941-2948/2003/0012-0143, 2003a.
a,
b,
c,
d,
e,
f
Spichtinger, P., Gierens, K., and Read, W.: The global distribution of ice-supersaturated regions as seen by the Microwave Limb Sounder, Q. J. Roy. Meteor. Soc., 129, 3391–3410,
https://doi.org/10.1256/qj.02.141, 2003b.
a
Spichtinger, P., Gierens, K., Smit, H. G. J., Ovarlez, J., and Gayet, J.-F.: On the distribution of relative humidity in cirrus clouds, Atmos. Chem. Phys., 4, 639–647,
https://doi.org/10.5194/acp-4-639-2004, 2004.
a
Stuber, N., Forster, P., Rädel, G., and Shine, K.: The importance of the diurnal and annual cycle of air traffic for contrail radiative forcing, Nature, 441, 864–867,
https://doi.org/10.1038/nature04877, 2006.
a
von Savigny, C., Brinkhoff, L. A., Bailey, S. M., Randall, C. E., and Russell III, J. M.: First determination of the fractal perimeter dimension of noctilucent clouds, Geophys. Res. Lett., 38,
https://doi.org/10.1029/2010GL045834, 2011.
a,
b
Wegener, A.: Über die Eisphase des Wasserdampfes in der Atmosphäre (On the ice phase of water vapor in the atmosphere), Meteor. Z., 27, 451–459, 1910. a
Wegener, A.: Thermodynamik der Atmosphäre, J. A. Barth, Leipzig,
https://archive.org/details/bub_gb_kWtUAAAAMAAJ (last access: 5 August 2026), 1911. a
