Constraining orientation statistics of ice crystals in clouds with observations from deep space

Ice crystals in clouds are often modeled as chaotically oriented despite frequent in situ and remote observations of horizontally oriented crystals. Zenith-pointing ground-based and nadir-pointing space-borne lidars often encounter intense specular reflections (glints), attributed to horizontally oriented particles (HOPs). When the size and shape of these ice crystals are just right, they appear to fall in precisely horizontal orientation with remarkable accuracy. Here, we attempt to constrain the relative contributions, wobbling amplitudes, and sizes of HOPs. Although there is an extensive literature on the topic, our discussion renders orientation randomness more precise and includes several additional considerations: (i) deep space optics of the Earth polychromatic imaging camera (EPIC)/Deep Space Climate Observatory (DSCOVR) observations of angular sizes for cloud glints are brought to bear on the problem; (ii) exponential decay of glint reflectance with angles is observed; (iii) a dimensionless moment of inertia constraint is considered to further constrain sizes; (iv) the dependence of air kinematic viscosity $\nu$ is introduced into the argument in tandem with the one on the Reynolds number.

1 Introduction

Atmospheric clouds contain ice crystals of all kinds of sizes, shapes, and orientations. Among these, thin horizontally oriented particles (HOPs) are of special interest as these often dominate remote sensing, such as saturating ground-based lidar echoes near nadir incidence. A tangible manifestation of HOPs is the subsun phenomenon, often observable from an airplane window while flying above clouds (Katz, 1998). Thus, it is of no surprise that interest in HOPs shows no sign of subsiding. Just how ubiquitous and horizontal are they? Here, we aim to bring deep space (taken from $\sim 1.5\times 10^6$km) observations to bear on the problem. Without much ado, let us begin with a simple observation that the common assumption in the models of ice crystals is that of chaotic or random distribution (Field et al., 1997; Zhuravleva, 2025), and such a distribution would presumably include HOPs. Does the notion of “preferential orientation” so often mentioned in the extensive literature on HOPs refer to all HOPs, the random fraction, or to the excess of HOPs beyond the random fraction? To the best of our knowledge, this question has not been posed until now. Geometric probability is a subtle subject; for example, the Buffon needle paradox (Hamming, 2018). Therefore, we begin by revisiting the notion of perfect orientation randomness.

By considering the notion of orientation randomness rigorously and examining reflectance data from deep space, we argue that the excess of HOPs beyond the random fraction is substantial. However, before proceeding, note that, for any reasonable probability density function (pdf), the probability of exact horizontal orientation is zero, and one must associate some angular spread $\mathrm{\Delta }\theta$ with the notion of HOPs. Here we propose the angular size of the Sun, $0.5°$ for such a characteristic spread because the angles of incidence, reflection, etc., are defined meaningfully only within such accuracy.

We delve into the mathematics, guided by the following intuitive expectation: a perfectly “random” orientation distribution of otherwise identical ice crystals should result in idealized uniformly gray and evenly bright “Lambertian” clouds, akin to matte Lambertian surfaces, while brighter cloud regions, such as subsuns (Katz, 1998), would require excess HOPs and, conversely darker spots associated with deficit or dearth of HOP concentration, relative to the random fraction. Fluctuations in optical depth are not a part of this picture, as only specular reflection (a single scattering phenomenon) is considered. If this intuitive anticipation holds, there must be a connection between the notion of Lambertian scatterer and the probabilistic geometry of random orientation. This is indeed the case, as shown in the next section, where we revisit the notion of perfect orientation randomness in the context of cloud ice crystals. Note that in contrast to formal mathematical treatments of perfect orientation randomness in terms of the Euler angles (Goldstein et al., 1980; Mishchenko and Yurkin, 2017; Li et al., 2023), our development is in terms of traditional spherical coordinate system angles.

2 Orientation randomness and stochastic derivation of the Lambert cosine law

Let the normal to the ice crystal plane, denoted as $\widehat{n}$, make an angle $\theta$ with the z-axis, the latter in the zenith direction (along gravity). No loss of generality is entailed as $\widehat{n}$ need not be a symmetry axis as its assignment is entirely arbitrary. We then require that $\widehat{n}$, positioned at the origin of a spherical coordinate system, pierce the surrounding unit sphere at any point with uniform probability. In other words, the probability density function $p(\theta ,\varphi )$ is

$p\left(\theta ,\varphi \right)=\frac{1}{4\pi }.$

This is the statement of perfect orientation randomness, and a rigorous formal proof can be found in Miles (1965). What does this imply in terms of the joint probability density $p(\theta ,\varphi )$, with $\varphi$ being the azimuth angle?

The statement of perfect orientation randomness implies statistical independence of the two angles, $\theta$ and $\varphi$, and therefore the joint pdf factorizes: $p(\theta ,\varphi )=p(\theta )p(\varphi )$. Using the definition of a solid angle $d\mathrm{\Omega }=\sin \theta d\theta d\varphi$, we obtain $p(\theta )p(\varphi )d\mathrm{\Omega }={(4\pi )}^{-1}\sin \theta d\theta d\varphi$, yielding

$p\left(\varphi \right)=\frac{1}{2\pi };p\left(\theta \right)=\frac{1}{2}\sin \left(\theta \right).(1)$

Aside from the ice cloud context, Equation 1 is not new (Fisher et al., 1993), yet many find the “non-uniform” $\theta$-dependence surprising in this context. Moreover, what appears to have gone unnoticed is the link to the so-called cosine law, also variously called Knudsen cosine law in vacuum science (Greenwood, 2002) or Lambert’s cosine law in optics and radiative transfer (Pedrotti et al., 2017). In the former case, it is the evaporating molecules that exhibit perfect randomness of flight direction. Misunderstanding the cosine law occasionally causes serious confusion in diverse fields, such as vacuum science and various vapor deposition devices; see the well-cited Greenwood (2002) or Palau et al. (2023) for ultramicroscopy.

In the case of radiative transfer, interpretation of the cosine law usually appeals to the apparent brightness perceived by an imaginary observer, regardless of the viewing angle. More precisely, as shown in Figure 1, the reflected radiant intensity obeys Lambert’s cosine law, which makes the reflected radiance the same in all directions. Despite the ubiquity of applications (Ross and Marshak, 1989), all derivations of the cosine law, to the best of our knowledge, are semi-empirical. Here, we contribute to this topic with a simple (in retrospect) but subtle new observation that geometric probability alone suffices to establish the equivalence between perfect orientation randomness (hence, uniform brightness when applied to scattered photons) and the cosine law.

Figure 1

Figure 1. Diagram of Lambertian diffuse reflection. The black arrow shows incident radiance, and the red arrows show the reflected radiant intensity in each direction. When viewed from various angles, the reflected radiant intensity and the apparent area of the surface both vary with the cosine of the viewing angle, so the reflected radiance (intensity per unit area) is the same from all viewing angles. This figure and its caption text are from: https://en.wikipedia.org/wiki/Lambertian_reflectance. See also https://creativecommons.org/licenses/by-sa/3.0/.

To that end, guided by the anticipated equivalence of perfect orientation randomness to the cosine law, consider a new random variable $\zeta =\cos \theta$ and inquire about its distribution, $p(\zeta )$. To find $p(\zeta )$, we use the transformation law for random variables: $p(\zeta )d\zeta =p(\theta )(-d\theta )$, where the minus is due to the mapping $f(\theta )=\cos \theta$ decreasing monotonically as $\theta$ increases from 0 to $\pi$ (Goodman, 2015). The transformation law reflects the fact that the probability contained in a differential area must be invariant under a change of variables. Therefore, $p(\zeta )=|\frac{d\theta }{d\zeta }|d\theta$ and for our mapping $\zeta =\cos \theta$ yields the uniform pdf for $\zeta =\cos \theta$: $p(\zeta )=1/2$, on the (−1,1) interval. Hence,

$p\left(\zeta \right)=\frac{1}{2}(2)$

on the (−1,1) interval. Another way to see this is to note that by Equation 1, $p(\theta )d\theta =(1/2)\sin \theta d\theta$, which must equal $p(\zeta )(-d\zeta )=p(\zeta )(-d\cos \theta )=p(\zeta )\sin \theta d\theta$. Hence, Equation 2.

We argue that this uniformity of the cosine pdf is a manifestation of the Lambert cosine law, that is, perfectly “diffuse” scattering, as shown in Figure 1. This can also be viewed as a uniform distribution of a projection onto the unit normal (zenith) or a uniform distribution of a scalar product of a unit vector with a z-axis. Alternatively, one can interpret this as the Lambert law of brightness, hence “Lambertian clouds.” Another metaphor for this could be an observer embedded in a “whiteout” snowstorm. In summary, Lambert’s cosine law is equivalent to the perfectly random orientation of scattered photon paths, be it clouds, surfaces, or anything else.

To tighten the stochastic interpretation of Lambert’s law, we note that the above equivalence theorem can be readily generalized. Namely, a cosine between or scalar product of any two randomly oriented and statistically independent unit vectors is also uniformly distributed via Equation 2. All that is needed for the extension is the statistical independence of the two vectors. For example, in the classic example of diffuse scattering from a rough surface, it is the local normal to the rough surface that is statistically independent of the scattered photon direction. As an illustration of the broad novelty of this argument, we cite an example from fluid turbulence in the first figure in Iyer et al. (2022), which explains the high Reynolds number (Re) curve of the figure. There, uniformity of the cosine between fluid velocity and vorticity at the sample point signifies not only the random orientation but also a complete statistical decoupling of the two vectors at high Re, an insight not contained in the turbulence literature. In summary, the uniform distribution of the ice plate projections is a robust characterization of the “chaotic” part of the distribution, and the perfectly random orientation of scattered light ray directions is the stochastic basis behind the Lambert cosine law for a perfectly diffuse scatterer. Thus, if one were to plot the joint probability distribution of reflected photon directions rather than “reflected radiant intensity” in Figure 1, all arrows pointing anywhere in the upper hemisphere would have equal length. The present authors feel that this derivation is simple enough to enter texts on optics and radiative transfer, for example, when describing uniformly glowing emitters. All that is needed is the uniform probability distribution of emitted photon directions.

Returning to the context of perfect orientation randomness of ice crystals, we ask: what is the fraction of horizontally oriented ice particles within the perfectly random orientation distribution? As already mentioned, to answer this question, one must assign a spread of angles about the horizontal orientation. What spread to use? We shall let the observations described in the next section answer this question. For the sake of numerical illustration, consider the $0.5°$ for the angular size of the Sun and similar values for diffraction spreading (Kostinski et al., 2021). We assign $\mathrm{\Delta }\theta \approx 1°$.

The fraction of HOPs within the chaotic distribution is then the area (integral) under the uniform pdf of $\zeta =\cos \theta$, $p(\zeta )=\frac{1}{2}$ over the $(\pm )\mathrm{\Delta }\theta =1°$ range. Expanding $\cos \theta$ in Taylor series about the HOP value $\theta =0$ and keeping the leading term in $\mathrm{\Delta }\theta$, $\epsilon \equiv {(\mathrm{\Delta }\theta )}^2/2$, we evaluate the fraction as follows.

${\int }_{-1}^{-1+\epsilon }\frac{1}{2}d\zeta +{\int }_{1-\epsilon }^1\frac{1}{2}d\zeta \approx 2\times \left(1/2\right){\theta }^2={\left(\pi /180\right)}^2\approx 3.05\times 10^{-4}.(3)$

This share of ice crystals, fewer than one crystal in 3,000, is a relatively small fraction of the total ice crystal concentration because $\epsilon$ is so small. Tripling $\mathrm{\Delta }\theta$ to account for the $3°$ full width of the specular excess curve in Figure 2d, discussed below, yields a ninefold increase in $\epsilon$ and therefore in the corresponding fraction of HOPs for a random collection. Specular excess of reflectance (above the Lambertian value) is due to the concentration of HOPs in excess of these calculated fractions for the perfectly oriented collection of particles, and it is this excess that defines preferential concentration of HOPs.

Figure 2

Figure 2. Glint geometry and orientation statistics of ice crystals. (a) An example of excess reflectance at a specular (glint) spot, observed by the DSCOVR/EPIC camera CCD detector on 15 June 2018 (at 6:31:55 UTC); see https://epic.gsfc.nasa.gov. It is caused by horizontally oriented ice crystals in clouds; (b) definition of a glint angle $\delta$ as a deviation from exact specular geometry; (c) color-coded image of time-averaged reflectance versus the CCD detector position. Reflectance values of approximately 0.3 are in a rough agreement with the Earth albedo for SW broadband; (d) specular excess reflectance, averaged over all 2017 Level 1A images that satisfy following conditions: specular spot is over land, that is, less than 1% of the specular pixel contains surface water as judged by MODIS MCD12C1 surface cover product (DOI:10.5067/MODIS/MCD12C1.006); images are available for all 10 EPIC bands. This yielded 1,967 of 5,531 EPIC images taken in 2017 (35.5%). No cloud detection or any other data screening. Exponential decay $\exp (-\delta /\bar{\delta })$ of the specular excess reflectance with glint angle $\delta$ is observed, with the $1/e$ spreads $\bar{\delta }$s shown in the legend.

We note that the above argument considers only single scattering, whereas reflectance from most clouds includes a substantial multiple scattering component. Multiple scattering offers an additional layer of randomization for Lambertian clouds and, therefore, our estimate of the chaotic fraction of cloud ice crystals must be viewed as approximate. Statistics of glinting clouds with respect to optical depth, in particular, are discussed in this special issue (Várnai et al., 2025).

We shall next present cloud ice observations from deep space to argue that HOPs are present in cold and mixed clouds in excess of the random fraction. This is how the statement “ice crystals in clouds exhibit preferential orientation” is understood here. We shall also confirm that, in agreement with an extensive literature on the subject, the angular spread of the excess HOPs is small, on the order of $\mathrm{\Delta }\theta \sim 1°$. This is, in fact, zero spread within the measurement uncertainty. This conclusion of perfect alignment is rather perplexing from the fluid-mechanical point of view, and this apparent conflict of optics with fluid mechanics will be our focus because observations from deep space have the promise of contributing to fundamental science. So let us now turn to the observations from deep space and use these to constrain the orientation of micron-sized objects from beyond 1.5 million km.

3 The optics of cloud glints as observed from deep space

The Deep Space Climate Observatory (DSCOVR) spacecraft drifts about the Lagrangian point $\approx 1.4-1.6\times 10^6$ km from Earth, where its Earth polychromatic imaging camera (EPIC) observes the entire sunlit face of the Earth every 1–2 h. The EPIC camera onboard DSCOVR is a 10-channel spectroradiometer (317–780 nm). The detector is a 2048 × 2048 pixel charge-coupled device (CCD), and images are obtained for 10 narrow-band spectral channels. The typical pixel size $D$ near the image center is $\approx 10$ km by 10 km, and the pixel angular spread from EPIC is $\approx (3\times 10^{-4})°$ (Kostinski et al., 2021; Marshak et al., 2018). Because of a rotating wheel of color filters inside the EPIC telescope, there is a time lag between the images at different wavelengths: 3 min difference between blue (443 nm) and green (551 nm), 4 min between blue and red (680 nm), and 54 s between green and red. The RGB images are collected every 1–2 h, generated daily, and are made available to the general public at http://epic.gsfc.nasa.gov. One such image is shown in panel (a) of Figure 2. The corresponding data are level 1B (L1B), publicly available at https://asdc.larc.nasa.gov (Atmospheric Science Data Center at the NASA Langley archive). More details about EPIC camera spatial and spectral resolution can be found at https://epic.gsfc.nasa.gov.

Panel (b) of Figure 2 illustrates the geometry of the specular reflection and positioning of the glint and the definition of the glint angle as the deviation from the exact specular direction. The primary purpose of panel (b) is to help understand panel (c), the CCD image of reflectance, and to realize that the extent of the excess specular reflectance spot on the CCD is directly related to the angular spread of the HOP orientations. Panel (c) shows that the average reflectance registered by the CCD detector is approximately 0.3, in broad agreement with typically measured SW albedo values. The peak reflectance is within a single pixel of the specular point, a fact used by Kostinski et al. (2021) to test geolocation. The average excess of specular reflectance, related to HOP orientation distribution, is shown in panel (d). Our data are reflectance measured by EPIC/DSCOVR vs. distance from the specular spot in pixels (see Figure 2c), and we can convert pixel distance to glint angle by using $\approx 0.1°$ turn of the normal (zenith direction) across the $\approx$10 km pixels.

For each wavelength in panel (d), the average excess specular reflectance $R$ plotted along the Y-axis vs. the glint angle $\delta$ was calculated as $R(\delta )-R(\delta =3°)$, where $R(\delta )$ is the mean reflectance of all pixels in all of the 1,967 input EPIC images that have vegetated land as surface type and are observed at a given glint angle $\delta$. Pixel reflectances are taken from the EPIC Level 1A product, and glint angles are calculated using the solar and view zenith and azimuth angles provided in that product. Panel (d) shows exponential decay with the deviation of the glint angle from the exact specular direction as defined in panel (b). This result is somewhat surprising, as this exponential is in contrast to the conventional wisdom, for example, Saito and Yang (2019) study the optics of HOPs by following the traditional Gaussian distribution of tilt angles. These empirical exponential distributions, $E(\delta )\propto exp(-\delta /\bar{\delta })$, have the $1/e$ spreads $\bar{\delta }$s shown in the legend of panel (d). As defined in 2 days, here $\delta$ is the “glint angle,” that is, the angle between the DSCOVR observation direction and the direction of the true geometrical glint from a horizontal surface. Note that all three specular excess curves drop to $1/e$ value within half a degree, and within two degrees, the reflectance is down to the Lambertian value.

The background reflectance values (mean reflectances far from the specular spot, where the glint angle is approximately 3°): at 325 nm: 0.337; at 680 nm: 0.354; at 780 nm: 0.448, whereas the peak excess reflectance values are approximately 0.035, 0.08, and 0.08, respectively. Rayleigh scattering and atmospheric attenuation vary between the three channels, and, in relative terms, the peak excess albedo is largest for the 680 nm red channel: $0.08/0.354\sim 22\%$. It is somewhat smaller for the other two channels: $\approx 10\%$ for the ultraviolent (UV) channel and $\approx 18\%$ for the infrared (IR) channel. This well-defined and localized pattern of specular excess in reflectance is a strong indication that ice crystals are not entirely randomly but rather preferentially oriented, meaning an excess of HOPs beyond the chaotic value of the relative population as given by Equation 3.

Given the exponential decay with the glint (deviation) angle in panel (d), we assume that the orientation probability distribution (HOP normals) mimics that decay and is also exponential with respect to the glint angle, as follows from panels (b) through (d) of Figure 2. This picture yields standard deviations from horizontal orientation of about a degree, and the entire angular spread of under two degrees. This is noteworthy as it suggests that most of the ice crystals responsible for the specular excess, to within measurement accuracy, fall in a nearly perfect horizontal orientation. Indeed, given the half a degree angular size of the Sun (leaving aside non-uniformity of radiance across the disk), the diffraction spread $\sim \lambda /d$ (with $\lambda$ being the wavelength and d being the relevant size of the ice crystal), of about the same for crystals under 100 microns, constrains the spread of orientations tightly. For the range of particle sizes in ice or mixed clouds, how compatible is such a highly ordered fall of an individual ice crystal with the underlying fluid mechanics?

4 The fluid mechanics of wobbling HOPs

Whatever the exact shape, HOPs can be neither too small nor too large. All suspended particles are susceptible to bombardment of air molecules, and for sub-micron particles, this Brownian motion prevents any semblance of steady orientation. Nor can the HOPs be too large, as the fall speed becomes large enough to develop a vortical wake behind the particle, thereby establishing a wobbling fall pattern. Thus, only a finite-size window is available for HOPs.

More specifically, at smaller sizes, the impact of collisions with air molecules, regarded as rotational diffusion, can be estimated by limiting the rms angle of ice crystals: see, for example, Katz (1998), p. 3360, their Equation 10. In radians, with $t$ denoting time and $D_0$ being the rotational diffusion coefficient, we require that angular dispersion obeys

$<{\theta }^2{>}^{1/2}={\left(2D_{0}t\right)}^{1/2}<0.01,$

so that wobbling of small ice plates beyond $0.6°$ seldom occurs. For disk-like thin ice plates, this occurs at radius $r=10\mu$m and disk width of $d=1.5\mu$m as discussed, for example, in Katz (1998). Loosely, HOPs can be tens of microns in “size” (often defined as the maximal dimension) and remain HOPs, but they cannot be much smaller. Can they be much larger?

Although the physics of the lower size limit is not about the Reynolds number (Re), to gain intuition for the upper size limit, we estimate Re here for the lower size limit as well. Leaving aside for a moment the precise meaning of “size” for a given shape, for the spherical case, $\mathrm{R}\mathrm{e}=UL/\nu$ where for the size $L$, we use the value of $r$ (the spherical radius), $U$ represents fall speed, and $\nu$ is the kinematic viscosity, that is, viscosity of air scaled by the air mass density, $\eta /\rho$. At the above dimensions of HOPs, a laminar (Stokes) flow value of $\mathrm{R}\mathrm{e}\sim 10^{-2}$ results when size is interpreted as the radius of the disk. The lower size limit on ice plates, implied by Brownian motion, is carefully considered by Westbrook et al. (2010) with the estimate of $\sim 10$ microns for “size.” Here, we add to these estimates by noting the constraint imposed on HOP size from below by the EPIC observations. Considering the diffraction spread $\sim \lambda /r$ and using the observed angular spread of panel (d) of Figure 2, namely, an upper bound of about a $1°\approx 1/60$ to solve for the radius of the disk for disk-like HOPs, one obtains the lower size limit of approximately $30\mu$m.

We now turn to the upper size limit as larger ice crystals fall out faster and develop a vortex street and a subsequent turbulent wake, thereby causing wobbling and flutter. As to a ballpark estimate, there is an agreement in the literature (Westbrook et al., 2010; Katz, 1998) of an upper bound for steady motion of $\mathrm{R}\mathrm{e}\approx 100$. At what crystal sizes does this occur? To gain intuition, consider the rate of Re increase with size for a sphere. For small spheres, falling in the Stokes regime, terminal speed $U\sim r^2$ so that $\mathrm{R}\mathrm{e}=Ur/\nu$, at a given kinematic viscosity $\nu$, increases cubically with $r$: $\mathrm{R}\mathrm{e}\sim r^3$. This rapid scaling of Re with size $r$ appears to be a new observation, and it pushes $r$ to approach the mm-size range for $\mathrm{R}\mathrm{e}=100$, as discussed next.

As the settling speed increases, the dependence on $r$ becomes less sensitive but remains substantial. In addition, unlike spheres, the fall speed for plates does not depend quadratically on size, even in the Stokes regime, but $UD$, the numerator of the Reynolds number, is roughly quadratic in “diameter.” A detailed numerical study of the falling planar ice crystals is provided in a recent article (Nettesheim and Wang, 2018). Measurements of fall speed are available from Westbrook (2008), based on Doppler lidar, and convenient empirical fits are also provided by the author for smaller HOPs (maximal dimension under a hundred microns) whose fall speeds do not exceed 7 cm per sec. On the other hand, Ansmann et al. (2009) report a fall speed of greater than 1 m/s and attribute lidar returns exclusively to ice particles of sizes greater than 100 microns. This is both surprising and interesting: as mentioned above, HOPs larger than 200 microns fall with a speed of 1 m/s and yield $\mathrm{R}\mathrm{e}>100$, outside the steady fall range of HOPs as typically reported for Re, for example, Katz (1998). Perhaps this disconnect can be resolved by the dendritic structure of these larger HOPs. Indeed, in a recent careful and comprehensive set of laboratory experiments by Stout et al. (2024), the authors suggest that dendrites are more stable. Their $2.5°$ experimental accuracy in orientation of the c-axis of crystals happens to broadly match our remote sensing observations from deep space as presented in panel (d) of Figure 2.

Other recent studies in both physics and atmospheric science literature suggest that ice dendrites are steadier than their solid counterparts; for example, see Sánchez-Rodríguez and Gallaire (2025), where it is argued on the basis of laboratory experiments that tumbling is eliminated by permeability. This is buttressed by the study of falling perforated disks (Tinklenberg et al., 2025), where the introduction of holes (by frilling) in the plates has a striking effect. On the other hand, Joshi and Govindarajan (2025) argue that bi-laterals (shapes with two axes of symmetry) can exhibit tumbling even in the absence of inertia.

Thus, if confined solely to the Re range, the current situation is still puzzling as the recent modeling studies (Zhuravleva, 2025; Wu et al., 2025) also estimate larger than $10^3$ micron for a size for HOPs, but the settling speed of tens of cm/sec for these large crystals implies $\mathrm{R}\mathrm{e}\approx 100$, which is in the wobbling range. Perhaps, one should not be surprised about disagreement in the literature on the range of Reynolds numbers because Re alone, aside from perfect spheres, does not suffice unless shape is specified precisely. In contrast to falling spheres, the fall of ice crystals cannot be specified by the value of Re unless the shape is fully characterized and the “size” definition is kept fixed. Another dimensionless number was chosen in early work (Willmarth et al., 1964) and then by Field et al. (1997) in their study of falling disks, and it now appears firmly established in the literature. This is the dimensionless moment of inertia, $I^{*}$

$I^{*}=I_{\mathit{disk}}/{\rho }_{\mathit{air}}{d}^5=\pi {\rho }_{\mathit{ice}}t/{\rho }_{\mathit{air}},$

where $d$ is the disk diameter, $t$ its thickness, the $\rho$s denote mass densities, and $I$ is the actual moment of inertia. Adding the range of acceptable $I$s helps to further constrain the wobble-free range. The empirical relationships of Auer and Veal (1970) (see their Figure 2) suggest $t/D\approx 0.04$, but the scatter in the data is large. Taking the ratio of ice to air mass densities $\approx 10^3$, we obtain

$I^{*}\approx 50\times \left(t/d\right).$

What is the value of the $t/d$ ratio for typical ice crystal plates? The phase diagram of Figure 2 in Field et al. (1997) shows a steady state regime for the Reynolds numbers $\mathrm{R}\mathrm{e}=Ud/\nu$ (where $U$ is the fall speed and $\nu$ is the kinematic viscosity of air) below approximately 110 and $I^{*}<0.01$. The 2nd bound yields a constraint on the plate geometry: $t/d<10^{-3}$. This is rather stringent: our plates of, say, 100 microns cannot exceed 100 nm in thickness. The Reynolds number constraint imposes $D<100(\nu /U)$. For atmospheric conditions at the Earth’s surface, this implies (in cm and sec):

$D<\left(100/7\right)\left(1/U\right)\sim O\left(0.1cm\right),$

where the kinematic viscosity of air was taken to be (1/7) cm/sec² and the fall speed on the order of tens of cm/sec.

Before proceeding with numerical estimates, another important new observation must be made, as it may help resolve the conflicting requirements discussed above. The Reynolds number depends on kinematic rather than dynamic velocity. As an example, see an otherwise thorough, comprehensive, and thoughtful study (Stout et al., 2024) where Re is defined incorrectly. Although probably a simple typo in Stout et al. (2024), the distinction between the two viscosities is particularly important in atmospheric science because of the dependence of kinematic viscosity on altitude. Indeed, while dynamic viscosity $\eta$ is independent of mass density, the kinematic viscosity $\mu$ does depend on air mass density ${\rho }_{\mathit{air}}$, and the latter decreases exponentially with altitude $z$. We digress briefly to mention that this independence of viscosity on gas density was derived by Maxwell in 1860 and so surprised him that he went on to confirm it experimentally himself (Reif, 1965, p.478).

The dependence of kinematic viscosity on altitude (density) is particularly important when comparing laboratory measurements on the ground with remote sensing measurements in the atmosphere. The kinematic viscosity: $\nu =\mu /{\rho }_{\mathit{air}}$ but ${\rho }_{\mathit{air}}={\rho }_0{e}^{-z/H}$, with $H$ being the scale height. Thus, at the altitude of 10 km, using the scale height of 7.5 km, the air is approximately four times as kinematically viscous, and the corresponding Reynolds numbers decreases by approximately a factor of four.

Taking the fall speed as $U=30$ cm/sec for a 100-micron plate yields $\mathrm{R}\mathrm{e}\approx 2$, but it is less than unity at the cirrus cloud height because of the altitude dependence. Thus, the same particles can be HOPs at ice cloud altitudes yet wobble at much lower altitudes or in the laboratory. Taking the spherical scaling for Re in the laminar regime: $\mathrm{R}\mathrm{e}\sim D^3$ yields $100^{1/3}=4.6$ for the extent of the wobble-free range of Re. Hence, the Reynolds number constraints permit up to 500-micron diameter plates, and the moment of inertia constraint also pushes toward larger crystals. Finally, the larger kinematic viscosity aloft may allow for nearly mm-size flakes to float without flutter. The surprise then would be at the cloud tops, where the rare large flakes may float. Is this consistent with the existing atmospheric and remote sensing literature?

5 A brief survey of glint observations

Insofar as the optics and fluid mechanics of falling ice crystals are universal ingredients, glints produced by mid- and high-latitude clouds are expected to be similar to ice clouds observed by EPIC. To that end, we surveyed cloud glint observations at all latitudes. As pointed out, for example, by Ceccaldi et al. (2013) on p.7, the CALIOP pointing angle was changed in November 2007, from near nadir to $3°$ off nadir, in order to avoid the ice crystal-induced glints. Most lidar systems use the $3°$ rule, and this is in remarkably good agreement with our Figure 2d, where the specular excess is seen to disappear by $3°$.

In polarization lidar studies of HOPs, Noel and Sassen (2005) and Noel and Chepfer (2010) report on observations of ice crystals in mid-latitude clouds and give the average wobble range of $1°$ in high clouds to $2°$ in lower ones. This weaker flutter at higher altitudes is consistent with the kinematic viscosity effect, discussed above. Stillwell et al. (2019) reviewed the radiative influence of HOPs for Greenland observations and reported an 8% effect on reflectance for optically thin clouds. Recent experiments on thin disks (Zhong et al., 2011; Zhong et al., 2013; Lee et al., 2013) investigate fall with very small dimensionless moments of inertia, a regime relevant here. O’Neill et al. (2021) report on contrails several km above cloud anvils. Thus, water is lofted into the stratosphere, and HOPs might conceivably be detected there because of the much lower Re (kinematic viscosity effect). Westbrook et al. (2010) estimate the ice concentration needed for an appreciable lidar signal. Specifically, a backscatter of $10^{-4}$ per meter per steradian requires only one $500\mu$m plate per cubic meter, when perfectly aligned. Westbrook et al. (2010) their Table 3 on p.266 indicates a diameter of 400 microns and the associated Re = 4, but the fall velocity of 23 cm/s is kept regardless of size.

For the sake of completeness, let us recall that HOPs do not fall through quiescent ambient air. Rather, the ambient air is often turbulent. How does this turbulence affect the fall of HOPs? In an early reference, Cho et al. (1981) considered the problem of turbulence destroying the preferred orientation of ice crystals in a cumulonimbus cloud. They concluded that “turbulence is unable to destroy the preferred orientation of falling ice crystals.” Following that work, Klett (1995) proposed an orientational model of particles in turbulence and also argued that turbulence does not destroy the preferred orientation of falling ice crystals. The most recent research on the topic considered settling spheroids, and Jeffrey torque is examined to determine when turbulence can disorient the settling particles (Jiang et al., 2021, see p.8 in particular). Again, the horizontally oriented fall of spheroidal remains intact for the range of Re and I of interest here.

6 Concluding remarks

We first remark that understanding the physics of HOPs is important beyond cloud physics. For example, geoengineering with aspherical suspended particles is likely to be considered soon. The orientation of such particles will have a strong radiative impact. This can be seen at once from a rather simple argument, based on the approximate validity of geometrical optics to estimate a scattering cross section of optically large particles, $\lambda \gg r$. Specifically, in the Mie regime for large size parameters, relevant to our observations of ice crystals, the scattering cross section of ice crystals can be approximated as twice the geometric one. In such a case, one can use the Cauchy theorem. For example, see Kostinski and Mongkolsittisilp (2013), Kostinski and Derimian (2020), and Várnai et al. (2020), who state that an orientation-averaged projection area of a convex body equals one-quarter of its surface area. For a thin disk, $t/r\ll 1$, it is $\pi r^2/2$ rather that $\pi r^2$ for the horizontal orientation. This doubling of the scattering cross section of HOPs illustrates at once the plausibility of significant radiative effects depending on the orientation statistics. The only exception would be suspended bodies possessing three equivalent axes of symmetry, such as ice cubes, 3D-printed jacks (Cole et al., 2016), the five platonic shapes (Kostinski and Mongkolsittisilp, 2013), etc. The fall speeds of such shapes are independent of orientation and, in the absence of torques, orientation statistics are perfectly random.

Returning to the glinting ice crystals, our tentative conclusion, based on the EPIC/DSCOVR data of Figure 2 and the available laboratory and lidar data on conflicting ranges of Re and the dimensionless moment of inertia $I^{*}$, leads one toward a narrow window of larger but slower ice crystal plates. Kinematic viscosity dependence on altitude contributes toward resolving the conflicting requirements but does not suffice. However, large yet slowly falling ice crystals can occur when they either have dendritic structure or are permeable or, perhaps, both. Indeed, the most recent fluid mechanics studies strongly suggest that permeability and/or dendritic structure are effective means of removing the tumbling (Tinklenberg et al., 2025; Sánchez-Rodríguez and Gallaire, 2025).

Data availability statement

The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding author.

Author contributions

AK: Conceptualization, Investigation, Supervision, Writing – original draft, Writing – review and editing, Formal Analysis, Funding acquisition, Methodology, Project administration, Resources, Validation, Visualization. AM: Conceptualization, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Validation, Visualization, Writing – review and editing. TV: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Writing – review and editing, Visualization.

Funding

The authors declare that financial support was received for the research and/or publication of this article. This work was supported in part by the NASA DSCOVR project, ACMAP program, and the National Science Foundation AGS-2217182.

Acknowledgements

The authors are grateful to Richard Eckman and to members of the NASA DSCOVR team for their help and support. ABK is grateful to D. Kestner for useful comments.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.

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