the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 18 Mar 2024
A global analysis of the fractal properties of clouds revealing anisotropy of turbulence across scales
Abstract. The deterministic motions of clouds and turbulence, despite their chaotic nature, nonetheless follow simple statistical power-law scalings: a fractal dimension D relates individual cloud perimeters p to measurement resolution, and turbulent fluctuations scale with separation distance through the Hurst exponent ℌ. It remains uncertain whether atmospheric turbulence is best characterized by split isotropy that is three-dimensional with ℌ = 1/3 at small scales and two-dimensional with ℌ = 1 at large scales, or by wide-range anisotropic scaling with an intermediate value of ℌ. Here, we introduce an “ensemble fractal dimension” De – analogous to D – that relates the total cloud perimeter per domain area 𝒫 as seen from space to measurement resolution, and show theoretically how turbulent dimensionality and cloud edge geometry are linked through ℌ =De − 1. Observationally, by progressively coarsening the resolution of cloud mask arrays from various global satellite platforms and a numerical simulation of a tropical domain we find the scaling De ~ 5/3, or ℌ ~ 2/3, a value nearly consistent with a previously proposed “23/9D” anisotropic turbulent scaling. Remarkably, the same scaling links two “limiting case” estimates of 𝒫 evaluated at the planetary scale and the Kolmogorov microscale, as separated by 11 orders of magnitude, suggesting that cloud and turbulent behaviors are constrained by basic planetary parameters.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2024-552', Anonymous Referee #1, 17 May 2024
This paper relates theoretically turbulent features and cloud geometry features (mainly perimeter). Then, estimates of the latter on numerous satellite data as well output of numerical simulation enable to confirm previously developed framework to address the issue of anisotropy of turbulence across scales. In general, I found the paper well written and conclusions relevant for the community. I believe that only minor modifications are needed, mainly to improve clarity and help the reader through the numerous equations/approximations and data sets.
General comments:
- Calling “Xi” (Greek letter) the spatial resolution is a bit confusing, because it often called “scale” with resolution being the ratio between outer scale and observation scales. On a similar point, it is not clear to me why either “l” or “xi” are used whereas it seems to me that they both represent the observation scale at which the studied geometrical set/field is studied. Could you please clarify.
- A summary with the main formulas that are first theoretically derived and then validated with data would be helpful for the reader. May be in the form of a figure or table.
- It would be interesting to discuss results of Fig. 4 in light of the scaling relation between perimeter and area which is reminded l. 38.
- For some of the analysed fields, you have 3D data. Why not trying an analysis in 3D directly instead of reconstructing a 2D field before carrying out the analysis ?
- It should be clarified better how a pixel is set to cloudy or not during the coarsening process. Indeed, many of the observed process depends on this. See detailed comments below.
Detailed comments:
- l. 54-55: please clarify what you mean.
- Eq. 2: it assumes no intermittency correction.
- Eq. 8: please provide more explanations on how is obtained.
- l. 171: please explain better the notation D_mu and how it is used.
- l. 177: the issue of the intermittency correction is briefly mentioned here. There could also be one in eq. 2. I believe that this issue and its implications on the various equations used should be clarified.
- l. 187-190: a scheme on how P and A are computed in practice would be helpful, and also how observation scale is changed.
- l. 232-233: what is done once the rounding is implemented ? This approach and its impact should be discussed with regards to a common approach when computing fractal dimension that would be a consider a coarser pixel as cloudy it at least one of the pixels is contains at higher resolution is cloudy.
- Fig. 3: how are side effect due the round shape handled ?
- In general for section 3: a table with a summary of the data used would be helpful for the reader.
- Section 4.1: it is not clear to me why this bifurcation is observed ? How sensitive is it to how a pixel is set to cloudy or not at coarser resolution, which is not very clear for me now ?
- Section 4.2 and Fig. 4.b: Can the range of scales used to perform linear be clarified ? Native scale is excluded but seems inserted in straight lines visible in Fig. 4.b. Points for high values of xi/xi_N also seem to deviate from the straight line. Indicators of the quality of the linear regression should be added and discussed. Again how sensitive are results to the way a coarser pixel is set to cloudy or not ?
Citation: https://doi.org/10.5194/egusphere-2024-552-RC1 -
RC2: 'Comment on egusphere-2024-552', Anonymous Referee #2, 27 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-552/egusphere-2024-552-RC2-supplement.pdf
- AC1: 'Comment on egusphere-2024-552', Karlie Rees, 16 Jun 2024
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2024-552', Anonymous Referee #1, 17 May 2024
This paper relates theoretically turbulent features and cloud geometry features (mainly perimeter). Then, estimates of the latter on numerous satellite data as well output of numerical simulation enable to confirm previously developed framework to address the issue of anisotropy of turbulence across scales. In general, I found the paper well written and conclusions relevant for the community. I believe that only minor modifications are needed, mainly to improve clarity and help the reader through the numerous equations/approximations and data sets.
General comments:
- Calling “Xi” (Greek letter) the spatial resolution is a bit confusing, because it often called “scale” with resolution being the ratio between outer scale and observation scales. On a similar point, it is not clear to me why either “l” or “xi” are used whereas it seems to me that they both represent the observation scale at which the studied geometrical set/field is studied. Could you please clarify.
- A summary with the main formulas that are first theoretically derived and then validated with data would be helpful for the reader. May be in the form of a figure or table.
- It would be interesting to discuss results of Fig. 4 in light of the scaling relation between perimeter and area which is reminded l. 38.
- For some of the analysed fields, you have 3D data. Why not trying an analysis in 3D directly instead of reconstructing a 2D field before carrying out the analysis ?
- It should be clarified better how a pixel is set to cloudy or not during the coarsening process. Indeed, many of the observed process depends on this. See detailed comments below.
Detailed comments:
- l. 54-55: please clarify what you mean.
- Eq. 2: it assumes no intermittency correction.
- Eq. 8: please provide more explanations on how is obtained.
- l. 171: please explain better the notation D_mu and how it is used.
- l. 177: the issue of the intermittency correction is briefly mentioned here. There could also be one in eq. 2. I believe that this issue and its implications on the various equations used should be clarified.
- l. 187-190: a scheme on how P and A are computed in practice would be helpful, and also how observation scale is changed.
- l. 232-233: what is done once the rounding is implemented ? This approach and its impact should be discussed with regards to a common approach when computing fractal dimension that would be a consider a coarser pixel as cloudy it at least one of the pixels is contains at higher resolution is cloudy.
- Fig. 3: how are side effect due the round shape handled ?
- In general for section 3: a table with a summary of the data used would be helpful for the reader.
- Section 4.1: it is not clear to me why this bifurcation is observed ? How sensitive is it to how a pixel is set to cloudy or not at coarser resolution, which is not very clear for me now ?
- Section 4.2 and Fig. 4.b: Can the range of scales used to perform linear be clarified ? Native scale is excluded but seems inserted in straight lines visible in Fig. 4.b. Points for high values of xi/xi_N also seem to deviate from the straight line. Indicators of the quality of the linear regression should be added and discussed. Again how sensitive are results to the way a coarser pixel is set to cloudy or not ?
Citation: https://doi.org/10.5194/egusphere-2024-552-RC1 -
RC2: 'Comment on egusphere-2024-552', Anonymous Referee #2, 27 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-552/egusphere-2024-552-RC2-supplement.pdf
- AC1: 'Comment on egusphere-2024-552', Karlie Rees, 16 Jun 2024
Peer review completion
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Karlie N. Rees
Thomas D. DeWitt
Corey Bois
Steven K. Krueger
Jérôme C. Riedi
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(6618 KB) - Metadata XML