SIAM News Blog

The Turbulent Tale of Icy Clouds

By Matthew R. Francis

Clouds are one of the most important influencers of climate on Earth as well as other worlds that have atmospheres. They comprise a major component of the hydrologic cycle, bounce light back into outer space, and trap radiation that is emitted or reflected from the ground. On Earth, clouds are made of water in the form of droplets and ice crystals; these components sometimes exist separately and sometimes form a mixture, depending on the elevation and atmospheric conditions.

Despite the importance of clouds and humanity’s millennia-long interest in the sky, accurate mathematical descriptions of clouds remain elusive due to their internal complexity and complicated interactions with the atmosphere. Ice presents a particular challenge, as the shape and size of the grains raise difficulties in theory as well as experiment.

Figure 1. Light pillars—including these examples over Cambridge Bay in Nunavut, Canada—form when light shines through atmospheric ice crystals. The crystals align as they fall through turbulent air, which allows for coherent light scattering. Figure courtesy of Eric Van Lochem and Flickr under the Attribution-ShareAlike (CC BY-SA 2.0) Generic license.
During the Ed Lorenz Lecture at the 2023 Fall Meeting of the American Geophysical Union (AGU)—which took place in San Francisco, Calif., in December 2023—physicist Alain Pumir of École Normale Supérieure de Lyon described new theoretical and experimental approaches that explain the self-organization of ice and other particles under turbulence. “All of the processes [within clouds] that you can think of are affected by turbulence,” Pumir said, noting that cloud modeling requires an understanding of collective atmospheric phenomena. “When you have many droplets, you care about collective effects. That’s also the case for little ice crystals.”

The way in which light coherently reflects and refracts from atmospheric ice—e.g., as sun dogs, light pillars (see Figure 1), lunar haloes, and other beautiful phenomena—indicates that the microscopic crystals align with each other under certain conditions. Like snowflakes, these ice crystals exhibit hexagonal symmetry down to the micron scale, while the clouds that contain them can stretch for many kilometers both horizontally and vertically. Modeling efforts must therefore treat the relevant physical properties while identifying any superfluous aspects. “The worst greenhouse gas is water [vapor],” Pumir said, adding that the largest uncertainties in climate change models come from clouds, which contribute to reflection and radiation. “How [ice crystals] align or don’t align has consequences in terms of reflection, either on Earth or from above.”

And it’s not just ice. In fact, many of Pumir’s collaborators model nonspherical atmospheric particles like volcanic ash or microplastics, which also play a role in cloud formation when water or ice collects around them as nuclei. Some scientists have even suggested seeding clouds with small particles to cool the atmosphere; in this context, Pumir’s research could lend essential insight as to whether this type of geoengineering project is even feasible.

Dropping Coins, Scientifically

As so often happens in science, Pumir’s first attempts to solve the problem were unsuccessful. “We did it completely wrong,” he cheerfully admitted, adding that other cloud formation experts were happy to explain the shortcomings of his approach. Armed with new ideas and a broader web of collaborators, Pumir examined previous progress in related areas, such as the settling of volcanic ash. Water droplets and particles—whether liquid water, ice, or another form—fall under gravity but are simultaneously buoyed by air. Ice in particular forms hexagonal crystals that are wider than they are thick, which means that a physics-based treatment must consider orientation as well as motion in three dimensions.

In the absence of air resistance, a nonrotating object simply falls without tumbling. But a fall through any fluid dramatically changes the situation, spinning the crystal and ultimately yielding two preferred orientations: edge down or face down. The latter configuration is more stable, analogous to a dropped coin in a swimming pool. However, Pumir pointed out the immediate breakdown of this analogy: the density of liquid water is close to that of ice crystals, whereas ice is significantly denser than air \((\rho_\textrm{ice}/\rho_\textrm{air} \sim 1000)\).

Figure 2. High-speed camera footage of 3D-printed spheroids as they settle to the bottom of an air chamber. The two left panels depict the tumbling motion at the beginning of the fall from different viewpoints, and the two right panels show the spheroids at the end of the fall. Figure courtesy of [1].
To mitigate these problems, Pumir’s collaborators—led by Gholamhossein Bagheri at the Max Planck Institute for Dynamics and Self-Organization—constructed a device that they call the Göttingen turret. This apparatus injects thin, 3D-printed plastic disks into a chamber of air in a highly controlled manner. The team tracked the disks’ settling process with two pairs of state-of-the-art, high-speed cameras. These “million-dollar babies,” as Pumir put it, operate at 2,932 frames per second and were situated at the top and bottom of the chamber — locations that allowed them to reconstruct the disks’ tumbling motions (see Figure 2).

Larger-scale simulations with multiple particles that fall in tandem are currently beyond the capability of the simple Göttingen turret. However, Bagheri’s group has used hexagons and modified snowflake-like shapes to perform follow-up experiments for better comparison with real-world ice crystals.

Turbulence Brings Us Together

Meanwhile, a full theoretical treatment of icy clouds requires an acknowledgment of the translational and rotational inertia of crystals alongside atmospheric fluid dynamics — i.e., how the crystals fall and tumble through turbulent air. At the same time, we intuitively (and mathematically) know that too much turbulence prevents the crystals from aligning and hence disrupts coherent light scattering. As such, the goal is to find the appropriate balance.

“The temptation was to say, ‘It’s a small particle [and] the Reynolds number (Re) is small,’” Pumir said, referring to the physical parameter that measures a fluid’s smoothness and viscosity. In nonturbulent laminar flow, Re is much smaller than \(1\), while turbulence dominates a fluid at Re \(\sim 1000\). Pumir and his colleagues focused on a middle regime—Re \(\sim 10\)—where turbulence is present but not dominant. “The Re is small but not that small, and that makes a world of difference,” he said.

Accounting for all of these factors, the theorists built the following model based on Newtonian physics:


\[\frac{d\mathbf{n}}{dt}=\boldsymbol{\omega} \times \mathbf{n}\]

\[\frac{d}{dt}(\mathbb{J}(\mathbf{n}) \cdot \boldsymbol{\omega})=\mathbf{T_h}.\]

Here, \(\mathbf{v}\) is the particle velocity, \(\mathbf{n}\) is the vector that is normal to the flat face of the crystal, \(\boldsymbol{\omega}\) is the angular velocity, and \(\mathbf{g}\) is the gravity vector. \(\mathbb{J}\) is the inertia tensor for a flat ellipsoid, which matches the experimental configuration and simplifies the math. The researchers treated the hydrodynamic force \(\mathbf{F_h}\) and torque \(\mathbf{T_h}\) perturbatively:



where the zero-order terms represent objects that are falling with air resistance but with a negligible Re. The correction terms with coefficients \(\{C_F,C_T\}\) are a combination of empirical and theoretical analyses that account for small but finite turbulence [1].

In a vacuum, a coin-like shape will spin freely if it has an initial rotation. The addition of fluid resistance results in two energetically optimal orientations: unstable equilibrium when the ellipsoid is edge-down (\(\mathbf{n}\) is perpendicular to \(\mathbf{g}\)), and stable equilibrium when the flat side is facing down. This theoretical result agrees with the Göttingen turret experiment and explains why ice crystals align to produce light pillars — at least, when turbulence does not dominate the system.

Clouds From Both Sides Now

Ed Lorenz—for whom the AGU lecture is named—is best known for work that proved that even simple atmospheric models lead to unpredictable outcomes, thus demonstrating that large-scale weather control is likely impossible. Lorenz and like-minded researchers helped to revolutionize modern interest in chaos and nonlinear phenomena — including turbulence, which is intrinsically a multiscale phenomenon. Turbulence affects the overall shape of clouds, all the way down to the microscopic length scales where individual droplets and ice crystals exist. “You can’t just look at one part,” Pumir said. “You have to do the whole problem.”

Pumir noted that the system becomes a competition between turbulence and normal settling. Stronger turbulence makes particles tumble, but theoretical analysis revealed that a moderate amount of turbulence brings crystals together — possibly facilitating larger aggregations called graupels and sometimes leading to precipitation. However, the model found that the presence of larger particles, lower turbulence, or a combination of both typically causes the coin-like object to settle with its broad face downward.

As demonstrated by experiment and theory, understanding clouds requires complicated physics: the inertia of ice particles, turbulence in the air, and collisions between crystals. The results may be messy, but as with light pillars, sometimes disorder is necessary to produce beauty.

[1] Bhowmick, T., Seesing, J., Gustavsson, K., Guettler, J., Wang, Y., Pumir, A., … Bagheri, G. (2024). Inertia induces strong orientation fluctuations of nonspherical atmospheric particles. Phys. Rev. Lett., 132(3), 034101.

Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is

blog comments powered by Disqus