the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 18 Jul 2025
Fractal Characteristics of Ice-Supersaturated Regions in the Tropopause Region of the northern midlatitudes
Abstract. Ice supersaturated regions (ISSRs) are air masses in the upper troposphere and lower stratosphere (UTLS) where relative humidity with respect to ice (RHi) exceeds 100 %, i.e. regions with enhanced water vapor concentrations. These are potential formation regions of cirrus clouds and contrails. While the impact of cloud free regions of enhanced water vapor on the planetary radiation balance is small to negligible, thin cirrus clouds and aircraft induced contrail cirrus formed within them might have a large radiative impact. Understanding the characteristics of ISSRs, including their geometry and seasonal variability, is essential for improving atmospheric models in representing ice clouds correctly. While ISSR’s path-length statistics, i.e. 1D characteristics, have been studied, their geometric properties, particularly fractal properties as self-similarity, and their seasonal variability remain largely unexplored. We identify ISSRs using ERA5 reanalysis data spanning from 2010 to 2020 at three pressure levels. An area-perimeter method is employed to compute fractal dimensions. The results reveal slopes equaling fractal dimensions with high coefficients of determination, strongly suggesting that ISSRs in the UTLS exhibit fractal behavior. A seasonal cycle in both dimension and total count of observed ISSRs was found. We hypothesize that this is caused by the seasonal variation of convective and frontal activity. We further analyzed the latitudinal and longitudinal spans of ISSRs and the path lengths of modeled flights along commercial flight routes. The results of the horizontal extensions are consistent with the fractal properties, and suggest distinct formation processes for ISSRs.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2025-2498', Anonymous Referee #1, 01 Aug 2025
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RC2: 'Comment on egusphere-2025-2498', Anonymous Referee #2, 12 Aug 2025
Review of “Fractal Characteristics of Ice-Supersaturated Regions in the Tropopause Region of the northern midlatitudes”
In this paper, the authors present the first ever analysis of fractal characteristics of ice-supersaturated regions (ISSRs). They use ERA5 data from 2010 to 2020 in their study, focusing on the 80N to 30N band, and using three pressure levels to study the upper troposphere and the tropopause. They use the area-perimeter relation to calculate the fractal dimension of ISSR objects. The authors also study the horizontal span of ISSRs as well as simulated aircraft pathlengths through the identified objects.
The authors demonstrate that ISSRs are fractals and discover a strong seasonal cycle in the fractal dimension of ISSRs, with smaller values in the summer and larger values in the winter. They suggest that different ISSR formation processes could be the cause of the seasonal cycle. They also find a seasonal cycle in pathlengths, as well as a scale break in their Weibull plots of pathlengths which would support the possibility of two different formation processes.
This study presents a novel approach to studying ISSR characteristics by demonstrating that ISSRs are fractal objects and analysing their fractal properties. The authors nicely link this fundamental analysis to their study of pathlengths in the second part of the paper, which is more directly relevant to commercial flights and contrail formation. Overall, the paper is very relevant and should be published after major revisions.
General comments:
The authors present a very extensive analysis of the fractal properties of ISSRs, and provide a lot of details which makes the study reproducible. However, I think the manuscript is too long and needs to be more concise. There are several subsections that should be removed or significantly shortened. For example, most of the method is standard methodology and is currently overexplained. Sections 2.2 to 2.5 can be shortened significantly, with a focus on keeping key information for reproducibility but removing explanations / justifications of standard methodological steps. Most of these subsections should be removed or moved into an appendix to provide detailed information for readers who are unfamiliar with object detection and the area-perimeter method. Figures 1, 2, 3 and 4 take up too much space, and should be moved to the appendix alongside the detailed technical explanations or should be condensed into one figure to provide a concise, graphical overview of the methodology. The rest of the paper needs similar re-working to enable readers to grasp and better appreciate the extensive analysis and novel results.
Language should also be clearer and more concise. Long sentences should be shortened, split, and / or re-ordered as necessary for improved clarity. There are small but frequent language and grammar issues throughout the paper which impact the clarity of the presented study. A non-exhaustive list of examples is provided in “Technical corrections” below.
Specific comments:
There are some issues with your calculation of area and perimeter of ISSR objects.
- The Lovejoy (1982) paper that introduced the area-perimeter relation, as well as recent work using it (e.g., Christensen & Driver, 2021; Rees et al., 2024), define it as P ~ A^(D/2). It is confusing that you do not follow this convention in your analyses, and I am not convinced that “with Eq. (5) the dimensional relation between length and area, as well as the deviation from the usual dimension, is more obvious”. To avoid confusion, I suggest you follow the conventional definition. As a result, Figure 8 and D1 should be re-plotted with area on the x-axis, and perimeter on the y-axis.
- L. 152 implies that you are counting the perimeter of holes in the ISSR islands towards the total perimeter of the ISSR object. This leads to an overestimation of the fractal dimension. As explained by Rees et al. (2024), including the perimeter of interior holes introduces a strong resolution-dependence of your results, as interior holes tend to fill at coarser resolutions, which would not be desirable. Brinkhoff et al. (2015) study this issue in-depth and suggest removing objects that include holes. Fig. 7 suggests that a lot of ISSRs have holes, so I would not suggest removing these objects from your analyses but instead fill interior holes and do not add their circumference to the object’s perimeter. This is standard practice (see e.g., Christensen et al., 2021 & Rees et al. 2024).
- Single pixel ISSRs / ISSRs with too few pixels are not valid fractal objects. As you explain in LL. 251-252, pixels are approximated as rectangles, so these objects will be assigned dimension alpha = 2 due to the low resolution of the data. Since these measurements represent neither the area nor the perimeter of these small ISSRs correctly, they are invalid measurements of fractal dimension. Objects below a certain pixel threshold cannot be studied with your dataset and should thus be removed. This is standard practice, Brinkoff (2015) discard objects that are smaller than 15 pixels while Christensen & Driver (2021) discard objects smaller than 24 pixels; the exact cutoff for ISSRs should be chosen based on some simple sensitivity tests.
Section 3.2 introduces three “regimes” of ISSRs. However, regime (a) is not a valid regime because, as explained above, ISSRs with a size below a certain number of pixels are not valid measurements and should thus be removed. I also find regime (c) somewhat problematic, as there is a fundamental upper limit to the size of ISSRs, on the one hand due to the size of our planet, and on the other because you only study the 80 N - 30 N band and remove ISSRs that cross the edge of this band. I thus wonder if what looks like a scale break between "regimes" (b) and (c) instead simply demonstrates that the size of ISSRs is not unbounded. I would suggest removing this part (LL 244 – 259) entirely, or at least removing regime (a), shortening this section significantly, and mentioning that the change in fractal dimension shows that there is an upper limit to the size of ISSRs.
Technical corrections:
As mentioned above, there are typos, language and grammar issues throughout the paper. I provide some examples here but suggest that the authors carefully edit their manuscript.
- Please avoid colloquial terms:
- 59, 111, 126, 220 & 488, and footnote on p. 5: remove the word “actually”
- 174 & 255: remove the phrase “kind of”
- 235: remove “quite”
- Grammar / language:
- g. LL. 7, 180: the word “as” is used incorrectly, please change it to either “such as” or “like”.
- 44-46: The logical structure of these sentences is unclear due to multiple contradictions in a row; please rephrase them.
- 50: Starting this sentence with “However” implies that the following statement contradicts the previous sentence, which is not the case. Please rephrase this sentence.
- 66 & 96: please remove “respectively”.
- 76: please rephrase the sentence “Applying a definition … connected areas.” as it is unclear.
- Citations:
- I would suggest removing “see” from the citations (e.g., lines 67 and 69), as it appears unnecessary.
- I am unsure whether “cf.” is used appropriately (e.g., lines 57 and 113). I recommend either removing it or verifying that it accurately conveys the intended meaning.
- Conciseness
- 94: the brackets (i.e. full 11 years of data) can be removed as this is duplicate information.
- 162: please remove “One has to mention here that”
- 165 – 172: please shorten this explanation, it suffices to say that the Hausdorff dimension cannot be calculated for gridded data, include a relevant citation, and then explain the method you used.
- Typing errors:
- 77: “asses” -> “assess”
- 173 “two-dimension” -> “two-dimensions”
- 199 “can not” -> “cannot”
- 203 missing full stop at end of sentence
- 235: “wide spread” -> “widespread”
Citation: https://doi.org/10.5194/egusphere-2025-2498-RC2
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- 1
Review of
Fractal Characteristics of Ice-Supersaturated Regions in the Tropopause Region of the northern midlatitudes
by H. Schuh et al.
Synopsis and recommendation
The present paper provides for the first time an analysis of fractal properties of ice supersaturated regions (ISSRs). The data for this excercise are obtained from full 11 years of ERA5 reanalyses in 0.25°x0.25° spatial and 6h temporal resolution. Pressure levels 300, 250 and 200 hPa in the NH zonal belt extending from 30° to 80° are considered. The tropopause is often in this pressure regime, such that the data belong to the UTLS, the upper troposphere and lowermost stratosphere. "Fractal dimension" in this paper is determined as the exponent in the Area-Perimeter relation A=C*Pα. It turns out that indeed α is generally close to 1.5, which signifies fractal properties of ISSRs.
Additionally the authors count ISSRs on a monthly basis and determine the pathlength statistics through ISSRs in the N-S and W-E directions as well as along great circles that represent flight paths between city pairs connecting Europe with America.
The central result of the paper, apart from the fractal nature of ISSRs itself, is the seasonal variation of the fractal dimension with higher values in winter and lower values in summer. This finding corresponds consistently to seasonal variations found in the other quantities (numbers and pathlengths). It is suggested that the seasonal variation could result from differing ISSR formation mechanisms: Convection in summer vs. frontal activity in winter, but a definite solution to this question is beyond the scope of this paper.
The paper provides a novel aspect to the study of ice supersaturation and clearly deserves a publication in ACP. However, there are some more or less serious aspects that should be considered before the paper is eventually accepted for publication.
Major comment:
Appendix A leads to an important discussion, that it is not completed in the present version of the paper. First, neither circles nor ellipses are fractal in plane geometry. They start to be fractal only in their imperfect numerical representation as gridded objects. So there is probably an impact of the spatial resolution on the value of alpha. Probably there is also an impact of the orientation of the ellipse relative to the x and y directions, but perhaps a small one. So the main question is here, is the fractal dimension something with physical significance or is it a numerical artefact due to the gridding. If the resolution of the grid (in your numerical experiment) would be increased for a fixed ellipse, the dimension should approach 2. So this is a serious problem. What is real and what is just a numerical artefact. For the ERA5 data it is not a priori clear what a change of the resolution would result in. But I think, this is an important question. Perhaps one could test this with a subset of the data (say four months in different seasons) by artificially making the resolution coarser, perhaps 1°x1° or so. It might turn out that the actual result of the paper is that there is a fractal dimension of about 1.5 FOR A RESOLUTION OF 0.25°x0.25°.
The experiment with the frayed square is perhaps better suited to test the approach for ISSRs, because ISSRs are frayed as well, while ellipses and circles are not.
I suggest to transfer this important discussion to the discussion section of the main text and not to hide it in the appendix.
LL 74 ff (and in the conclusions): "we may refine atmospheric models ...". Unfortunately this is an empty promise. Please indicate some possibilities and directions how the knowledge of a fractal nature of ISSRs could be exploited to refine their modelling. Moreover, I have a strange feeling, since the fractal nature of ISSRs is here derived just from a model. So what needs to be refined, for instance in the ECMWF model, since it already produces the fractal dimensions?
Minor comments:
Introduction:
L 31, 32: The sentence "making OLR ... senstitive to absolute changes ..." is a weak statement. What characterises "absolute changes"? Every change is absolute, isn't it? To my opinion, the important issue is that OLR is sensititve to RELATIVE changes of H2O concentration. This can be seen in an old paper by Clough et al. (1992). Absolute changes in H2O concentration are much larger in the lower troposphere than in the UTLS. If only the absolute size of a change would count, the H2O in the UTLS would be totally unimportant.
LL 46 ff: It is a bit surprising that the potential fractal nature of ISSRs is quasi presupposed although it is stated just before, that the 2D/3D properties of ISSRs have never been studied. Later in this paragraph follows a lot that would support the hypothesis that ISSRs could be fractal. I suggest to reorganise this paragraph a bit, so that a kind of logic appears, e.g.: First state that many studies have shown a fractal nature of clouds. Then it is natural to assume that also their "birth regions", that is, ISSRs, have fractal properties, and this will indeed be shown in the paper and the fractal properties will be studied.
Section 2:
Section 2.2: As far as I know, RH is an output field in ERA5 pressure level data. Why don't you use it directly? Or do you use model levels with interpolation on the mentioned PLs? Another question that becomes essential in section 3.1 when ISSRs are counted is, whether and how do you treat cloud-clearance. q as a grid-mean is an average over the cloud-free and cloudy part of a grid box and the resulting RHi or Si is such an average as well. How is this treated?
L 115, 116: I would argue that the orientation of the grid is a mathematical necessity that has no physical correspondence and that therefore the 8-neighbourhood is more appropriate to the physics of the situation.
L 118: Do I understand it correctly, when I interprete this in a way that only grid cells are counted that are completely surrounded by (8) other ISSR cells? Before, I thought that any of these 9 points counts as ISSR, and that in principle two or more points connected by a diagonal would count as an ISSR island with slant orientation? Please provide more explanation, since this is central to understanding.
Section 2.3 can be drastically shortened and figure 2 can be deleted. It is not necessary to explain these basic things to the reader. Mentioning the python routine should suffice. I suggest to combine it with Section 2.4, which treats a less trivial problem.
L 155: "Analytical manifold". I suggest to modify the discussion a bit. I assume, you are interested in physical objects, not in analytical manifolds. I agree that there is a problem with analytical manifolds, as demonstrated in the appendix. However, ISSRs aren't analytical manifolds, they have internal structure that is not resolved in the ERA5 data and so the determination of their "true" area and perimeter may be an ill-posed problem anyway. To me, it seems not necessary to invoke a special geometric problem that only occurs in the mathematical or numerical treatment, if the real object is represented anyway in a quite crude fashion in the gridded data.
Section 3:
Section 3.1: Instead of or additionally to the number of ISSRs you could provide the total area of all ISSRs, or the mean fractional area of ISSRs per month (that is, take for each output the total ISSR area, divide it by the area of the 30°-80° zone and average the result for each month), beause this quantity may have more relevance to other topics like contrail avoidance. Cf. also section 4.1 where you show that ISSRs seem to be larger in winter than in summer. My feeling here is that the number is important for checking statistical significance, but areas are physically more significant. Otherwise, I am a bit surprised by the result here, because to my knowledge there are more contrails in winter than in summer which seems to be in conflict with your result. Moreover, my state of knowledge was that there is more ISSR in winter than in summer (see eg. Spichtinger et al. 2003, MetZ, Analysis of the Lindenberg data, or Spichtinger et al.2003, QJ, Analysis of MLS data, Fig. 5). Lamquin et al. (2012, their fig 10) show a more differentiated picture and your results should be discussed with these papers in mind. A comment on this would be welcome.
Section 3.2: The appearance of very small and even one-grid ISSRs is surprising since I thought they were sorted out by your 8-connectivity criterion (c.f. my comment above to L 118).
LL 262 ff: Since it may be that the resulting α depends on the spatial resolution of the data, please indicate here which spatial resolution underly the α determinations in the quoted studies.
Section 4:
LL 288 ff: Although it seems to be quite tempting to invoke the Weibull distribution as one of the extreme-value distributions here, it is probably a wrong application. You are not considering, say, monthly populations of pathlengths and then consider for every month the maximum. Perhaps if would be ok, if you would put the maximum pathlenght of each single ISSR island into a pool and then study the distribution of these pooled maxima. Such excercises would be an application ground for extreme-value statistics. But the fact, that ISSRs have extreme humidity values does not justify to use extreme value statistics for their pathlengths.
Section 4.1,2: Please check! North-south is latitude and west-east is longitude. The first sentence is hard to understand, the title says "along longitude" and in L 301 it says "North-South span". I am a bit lost.
LL 318 ff: In the present paper the exponents of the Weibull distribution are relatively close to 1 (which would be a simple exponential distribution). The early study by Gierens and Spichtinger (2000) found an exponent of 0.5, which is significantly different from the present result. Do you have any ideas what could cause this difference?
Section 5:
Point 1: If you determine the areas as suggested above, please add the results as well. The word "heights" should be replace with "altitudes" or "higher levels".
Miscellaneous:
L 80: "horizontal span" is a strange expression here, later it becomes clear what you mean. I suggest to use pathlength here and you may later use span as well.
Eq. 4: Check the units or at least write in the text that the formula needs T in K and gives p_si in Pa or hPa. This is unclear.
L 143: "entail" should be replaced by "include".
L 162: please add a comma after "consequence" and add "s" to "become".
L 172: The statement that two dimensional objects have a dimension between 1 and 2 sounds a bit strange. I suggest to replace "two-dimensional" with "plane" or "planar".
Figure 5: check the grammar of the caption.
Tables 1 - 3: Are the values actually precise to the order of 10 metres? (As well in the corresponding figures).
L 334: "showcase"? Perhaps just "show"?
Section 6:
L 431: formation
REFERENCES:
Clough, S., et al., 1992: Line by line calculations of atmospheric fluxes and cooling rates: application to water vapour. J. Geophys. Res., 97, 15761-
15786.
Gierens, K., P. Spichtinger, 2000: On the size distribution of ice-supersaturated regions in the upper troposphere and lowermost stratosphere. Ann. Geophys. 18, 499-504
Lamquin, N., et al., 2012: A global climatology of upper-tropospheric ice supersaturation occurrence inferred from the Atmospheric Infrared Sounder calibrated by MOZAIC. Atmos. Chem. Phys., 12, 381-405.
Spichtinger, P., K. Gierens, U. Leiterer, H. Dier, 2003: Ice supersaturation in the tropopause region over Lindenberg, Germany. Meteorol. Z., 12, 143-156.
Spichtinger, P., K. Gierens, W. Read, 2003: The global distribution of ice-supersaturated regions as seen by the Microwave Limb Sounder.
Q. J. R. Meteorol. Soc., 129, 3391-3410.