The deterministic motions of clouds and turbulence, despite their chaotic nature, have nonetheless been shown to follow simple statistical power-law scalings: a fractal dimension

The Earth system is radiatively open and materially closed. Radiatively, Earth's global mean temperature is sustained by a balance between absorption of high-intensity shortwave sunlight and the reemission at longwave frequencies to the cold of space. Materially, the total dry atmospheric mass is confined to the planet by gravity and can only be redistributed by turbulent circulations that mix air over a broad range of scales within the thin atmospheric layer. Clouds play important roles in determining the magnitude of both categories of flow. Geometrically speaking, cloud areas govern radiative fluxes

A scientific challenge is that the seemingly objective properties of cloud area and perimeter are a function of the more subjective choice of spatial resolution,

Fractal geometry is often used as a tool for characterizing the resolution-dependent complexity of shapes. The fractal dimension

Generally, we define here a geometric quantity that does not vary with the length scale as “scale-invariant” such as the scaling of

Although the initial result of ^{2}. The apparent increase in measured

Clouds have been shown to be multifractal; i.e.,

While the fractal dimension and scale invariance are intrinsically linked, their relationship with turbulent structures in the atmosphere is less clear. Two paradigms of turbulence scaling in the atmosphere have been the topic of decades of debate: split 2D and 3D isotropic scaling regimes for large and small scales

The scaling exponent of the diffusivity with respect to the length scale can be experimentally obtained from measurements of velocity fluctuations,

The Hurst exponent has various mathematical applications, but here we employ its usage in the field of fractal geometry (for the non-intermittent case) to relate the scaling of turbulent fluctuations with respect to separation distance,

The

As Sect.

Figure ^{−8} m scale, inconsistent with a description of isotropic molecular diffusion

Diagram showing the similarity of rotational motions of clouds from the planetary diameter

Specifically,

Simplifications of the first-order structure function have also been used to determine

Observations of scaling behaviors in clouds, expressed through either the fractal dimension or the turbulent structure functions, point to a robust relationship between

To explore how the cloud perimeter varies with measurement resolution

In

Illustration of the theorized cloud edge mixing engine from

After invoking mass continuity for the cloud edge circulation,

From Eq. (

Note that

Equation (

From a climatological perspective, it is instead the ensemble of clouds with total perimeter density

The distinction between

The survival function can be related to a cumulative distribution function (CDF) through

Comparing the exponents in Eqs. (

However, while

This is the difference between the spatial dimension of the domain and the fractal dimension.

Summary of main formulas.

Equation (

The polar-orbiting datasets considered are from the instruments VIIRS and MODIS, which have native pixel resolution ^{2}. For MODIS, it is 1261 ^{2}. The VIIRS and MODIS datasets include 60 and 72 cloud masks from 2 June 2021. Their respective cloud mask techniques are described by ^{2} and average image dimensions of 5048

Geostationary datasets are obtained from instruments denoted here by their more familiar satellite names, Himawari (instrument name: AHI), GOES-West (ABI), and Meteosat-11 (SEVIRI), which provide full-disk imagery of Earth with

To provide unique observations of global cloud coverage, we also include cloud masks from GEO-Ring and EPIC. GEO-Ring is a composite of geostationary satellite imagery ^{2} at

Summary of satellite datasets used in this study.

As a means to compare measurements of

In order to compare the 3D model data with 2D satellite retrievals, we define the SAM cloud masks as 3D binary arrays for mixing ratios of non-precipitating cloud condensate, ^{−1}. A 2D facsimile of a satellite cloud mask is created from a vertical projection of the 3D cloud mask that represents the view from above. The 2D binary cloud mask is assigned a cloudy pixel based on threshold value

To obtain values of

A least-squares linear regression is performed on values of

Alongside measurements of perimeter density, ^{2}.

Measured cloud fraction,

Global cloud fraction values,

This bifurcation of cloud fraction reflects that as an image of a cloud field is coarsened to a single pixel, the coarsened pixel value is determined by averaging and rounding to zero or unity the pixel values in the original image (illustrated in Sect.

The application of vertical pixel thresholding in SAM results in a wide range of native cloud fraction values of

The resolution dependence of cloud perimeter density,

Notably, the value of

For satellite datasets, values of

Note that modeled values of

To summarize the observations, global cloud perimeter density,

To account for how the dimensionality of turbulence may help explain the difference between the measured value of

As introduced in Sect.

For 2D isotropic turbulence, where vertical motions are negligible due to, e.g., stratification, the diffusivity scaling exponent can be obtained from dimensional analysis with the conserved property being enstrophy ^{−3}

The framework of generalized scale invariance ^{2} s^{−5}. To account for this anisotropy in the vertical,

Note that Eqs. (

Table

Observational values from Sect.

Theorized values (top) of

Because cloud shapes and sizes are determined by objective physical processes that are independent of subjective measurement resolution, in principle it should be possible to infer information about cloud geometries from the physical properties of the planet and its atmosphere. To this end, we examine order-of-magnitude limiting case values for

Given the turbulent nature of fractal cloud edges, the Kolmogorov microscale,

For the planetary scale (denoted with

Figure

Measured perimeter density

What is striking is how well the predicted value of

Visualization of theorized and observed

Figure

Comparing the results here with observations of vertical wind structure functions by

Because each of the satellite cloud masks considered in Fig.

Here, this complication is limited because we are averaging adjacent pixels in a binary cloud mask rather than a brightness field, leading to a more accurate measurement of the fractal dimension (

The measured relationship between the ensemble cloud perimeter density,

Global cloud measurements of

Measured values of the ensemble fractal dimension,

Values of

Globally distributed, the total perimeter of clouds has a resolution dependence in satellite and numerical datasets, one that can be tethered to physically parameterized values evaluated at the Kolmogorov microscale and the planetary diameter that points to the existence of an intermediate 2D/3D turbulence regime that applies at all conceivable tropospheric scales. Observations of clouds on other planets in the solar system could help identify whether the observed scaling is specific to present-day Earth or, in fact, general to stratified atmospheres. Any generalization of the scaling laws could prove useful for constraining predictions of cloud behaviors in a future climate state on Earth or for exoplanetary studies where – like Earth's pale blue dot – only coarse-resolution physical parameters are available.

Values of the parameters and variables used for the calculation of

The Brunt–Väisälä frequency, ^{−1}

Values of variables and parameters described in the text used to determine theoretical values of

^{−1}

^{2}s

^{−3}

^{2}s

^{−1}

^{2}s

^{−1}

The VIIRS and EPIC datasets were downloaded from NASA Earthdata (

KNR: methodology, formal analysis, and writing (original draft and review and editing). TJG: conceptualization, funding acquisition, supervision, methodology, and writing (review and editing). TDD: methodology and analysis and writing (review and editing). CB: writing (review and editing). SKK: funding acquisition and writing (review and editing). JCR: methodology and writing (review and editing).

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors thank Shaun Lovejoy and two anonymous reviewers for constructive comments. The AERIS/ICARE Data and Services Center at l'Université de Lille and the Center for High-Performance Computing at the University of Utah provided data storage and computing services.

This research has been supported by the National Science Foundation (grant nos. 2022941 and 2341274).

This paper was edited by Shaun Lovejoy and reviewed by two anonymous referees.

^{−3}to 10

^{6}m, Non-linear variability in geophysics: Scaling and fractals, Springer, Dordrecht, 111–144,