Physics Buzz: The Hidden Rule that Shapes Trees, Lightning, and Cracks in the Earth

Thursday, January 18, 2018

The Hidden Rule that Shapes Trees, Lightning, and Cracks in the Earth

Seeing bare tree branches silhouetted against a sunset sky is one of the best things about winter. Bereft of leaves, the trees reveal their intricate skeletons—almost fractal, reminiscent of neurons, or the network of blood vessels that perfuse the body. These complex patterns of growth and branching are produced by an invisible algorithm—less a blueprint than a computer program—encoded in the tree’s DNA, optimized over millions of years of evolution. Taking data on sunlight, airflow, and proximity to other branches, the tree regulates the expression of growth hormones to ensure that it’s making the most of its space. With all the care that goes into their creation, it’s no surprise that the patterns they produce come out so marvelously complex.

What is surprising, and even more marvelous, is when similarly complex patterns emerge almost out of nowhere, in the fractures running through an ice sheet, or glass, or—in the Complex Flow Laboratory at Swansea University—something as seemingly mundane as air in wet sand.

The winding cracks in this image were created naturally by compressed air injected into wet sand, with color denoting when they formed—red is earliest, violet is latest.
Image Credit: Campbell, Ozturk, & Sandnes (2017). Physical Review Applied.

How does a system of nothing more than mud and air mirror the fractal beauty of biological life? Dr. Bjornar Sandnes and the students under his direction at Swansea have spent years figuring out how these patterns arise, injecting compressed air into narrow glass cells, tightly packed with sand and saturated with water.

Image Credit: Campbell, Ozturk, & Sandnes (2017). Physical Review Applied.

These cells create a “window” that lets the researchers study how the gas works its way through the mixture—sometimes bubbling, sometimes forming fingerlike projections, or labyrinths of cracks.

This figure, from an earlier work by Sandnes and his collaborators, shows how the gas’ behavior depends on the density of the grains (on the y-axis), and the gas injection rate (on the x-axis).
Image Credit: Sandnes, et alNature Communications (2011).

“We study these flow patterns because they are important in many natural and industrial systems,” explains Sandnes, “but first and foremost because we are curious to discover how such beautiful structures can grow spontaneously…and what physical mechanisms shape their form and function.”

By measuring and analyzing the properties of these complex flow patterns, Sandnes and his students have developed mathematical tools to describe the interplay among forces that gives rise to these entrancingly organic-looking structures, sharing their findings near the end of last year in the American Physical Society’s journal Physical Review Applied.

A variety of the so-called “invasion patterns” formed by the gas, depending on how fast it gets pumped in to the cell.
Image Credit: Campbell, Ozturk, & Sandnes (2017). Physical Review Applied.

“Physically, the spatial density of the patterns are so striking that it begs to be investigated,” says Deren Ozturk, a PhD student in the Complex Flow Lab. “It’s why I joined the team—non-biological natural patterns are particularly fascinating to me.”

For all their complexity, though, the principle that governs the formation of these fractures turns out to be surprisingly simple.

As air is pushed into the system, the sand’s fractures grow in a “stick-slip” fashion, spreading in short bursts interspersed with periods where nothing seems to be happening. During those “stick” periods, though, gas is still being injected, and the air pressure inside the fractures rises. When that pressure becomes great enough, it pushes aside grains of sand, expanding into the space that they had occupied—a “slip”. But the displaced grains, forced out into the surrounding bulk, create a denser region surrounding the newly formed fracture. In that region around the fracture, called a compaction front, the extra-dense packing of the sand makes it harder for the air to create a new inroad.

When the gas manages to spread out, it tends to follow the path of least resistance, pushing aside the least-densely-packed grains to sprout a new branch of the fracture from an existing one. As a result, the compaction fronts around existing fractures create a sort of shield that causes new branches to shy away from them, to spread out into their own space instead. Since following the path of least resistance means giving other fractures a wide berth, the result is a design that naturally strikes a balance between spreading out and filling the space efficiently—a little like the branches of a tree.

If you’re a science enthusiast, you might feel a sense of déjà vu watching the fractures spread through the cell in that video—it looks a lot like a stop-motion version of a Lichtenberg figure being etched into wood, as high-voltage electricity tries to find a path to ground.

Made by applying a conductive solution to the surface of the wood, then applying an extremely high voltage, Lichtenberg figures are as dangerous to create as they are beautiful.

This similarity isn’t a coincidence—the burnt wood in a Lichtenberg figure forms a conductive channel for electrons, allowing them to flow easily into that space the same way that air flows into the fractures. But the high concentration of charge in those established channels pushes away the charges in neighboring ones, causing them to spread out from one another.

The rule also applies in air, as seen in this ultra-slow-motion shot of lightning finding its way to ground.

It’s the same story as the sand’s compaction fronts, only with voltage rather than mechanical pressure. It’s no surprise, then, that voltage in wires is commonly described as being closely analogous to fluid pressure in pipes, in the “hydraulic analogy” of electricity.

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