# Why Icicles Are Long And Thin Mathematical Physics Explains How Icicles Grow

February 1, 2007 — When droplets of melted snow drip down an icicle, they release small amounts of heat as they freeze. Heated air travels upwards and helps slow down the growth of the icicle's top, while the tip is growing rapidly. Knowledge of the mathematical equations that govern icicle growth -- the same that apply to stalactites -- could help in the prevention of icicle formation on power lines.

Icicles can be dangerous and deadly, yet they can create some of the most amazing winter scenes. And for scientists, those winter scenes are playgrounds for discovery.

It's on those playgrounds that experts in physics and mathematics are building their theories on what it takes to create an icicle.

We all know icicles form when melting snow begins dripping down a surface. But what scientists didn't know is how their shape is formed. What makes each icicle different?

University of Arizona Physicist Martin Short turned to mathematics to find out.

"Icicles have a certain mathematical shape, and this mathematical shape is universal among icicles," Short tells DBIS.

So what is the math behind an icicle?

"Here I've drawn the profile of an icicle. Here is the height, and here's the radius ... Here's the profile here, and I've written the formula here. The height is proportional to the radius to the four-thirds," he says.

What does the formula have to do with an icicle's shape? "It kind of looks like a carrot," says Short. "It starts out flat and then sort of up as you go."

As water drips onto an icicle and freezes, it releases heat. The warm air rises up the sides of the icicle. Short says that warm air layer acts like a blanket that's an insulator, and so the blanket is very thin near the tip and thick at the top. That allows the top to grow very slowly and the tip to grow rapidly -- creating a long, thin icicle.

It's the same equation scientists use to study stalactites in caves, but instead of water, stalactites are formed by the buildup of calcium left after the water evaporates.

"If we know the mechanisms by which stalactites form, well, we could better preserve our natural caves that we have here, and try to stop them from eroding," Short says.

And now that scientists know how icicles are made, it could lead to breakthroughs to prevent them from forming on power lines and trees.

BACKGROUND: Researchers at the University of Arizona have found that the same mathematical formula used to describe the shape of stalactites that form in caves also describes the shape of icicles. This is surprising because the physical processes that form icicles are very different from those that form stalactites. Both have a unique underlying shape, resembling a kind of elongated carrot. This sheds light into the physics of how drips of icy water can swell into long, skinny spikes (icicles).

HOW THEY FORM: Stalactites are formations that hang from the ceilings of caves, formed when water erodes limestone and taking the calcium carbonate. As the water drips inside the cave and evaporates, it leaves behind the calcium, which forms a stalactite. The continued diffusion of carbon dioxide gas fuels the growth of a stalactite. In contrast, heat diffusion and a rising air column are keys to an icicle's growth. Icicles form when melting snow begins dripping down from a surface such as the edge of a roof. There must be a constant layer of water flowing over the icicle in order for it to grow. The growth is caused by the diffusion of heat away fro the icicle by a thin fluid layer of water, and the resulting updraft of air traveling over the surface. That updraft occurs because the icicle is generally warmer than its surrounding environment, and thus convective heating causes the surrounding air to rise. As the rising air removes heat from the liquid layer, some of the water freezes, and the icicle grows thicker and elongates.

PUT TO THE TEST: To compare the predicted shape to real icicles, the researchers compared pictures of actual icicles with their predicted shape. They found that it doesn't matter how big or small the actual icicles were, they could all fit the shape generated by the mathematical equation. The next step is to solve the problem of how ripples are formed on the surfaces of both stalactites and icicles.

ICE, ICE, BABY: Ice is the frozen form of liquid water. The same substance will behave differently at various temperatures and pressures. Water (H2O) is the most familiar example. It can be a solid (ice), a liquid (water), or a gas (steam), but it is still made up of molecules of H2O, so its chemical composition remains unchanged. At sea level, water freezes at 32 degrees Fahrenheit (0 degrees Celsius) and boils at 212 degrees Fahrenheit (100 degrees Celsius), but this behavior changes at different altitudes because the atmospheric pressure changes. In fact, get the pressure low enough and water will boil at room temperature. The critical temperature/pressure point at which H2O changes from one form to another is called a phase transition.

The American Meteorological Society and the American Geophysical Union contributed to the information contained in the video portion of this report.

Note: This story and accompanying video were originally produced for the American Institute of Physics series Discoveries and Breakthroughs in Science by Ivanhoe Broadcast News and are protected by copyright law. All rights reserved.

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