The Physics of Snowflakes & the Spirit of Christmas

We shovel snow from our door steps because although no two snowflakes are alike, far more than two land in the same place. Rigorous shovelling helps beat the cold. and once warmed up, we can think about falling snowflakes. And if we want to be even more captivated, we can observe them.

First the thinking part. If we want to merely predict the velocity of falling snowflakes, we already run into a complication. At least raindrops begin as spheres, and then as they grow larger, their shape approximates that of a burger bun. That affects their area and drag coefficient —numbers needed in assessing to what extent air slows down the rate of falling drops. But snowflakes are formed in a countless variety of shapes and sizes. There is far more averaging out to do.

So assume that it’s been done. We subsequently write an expression for the product of air density, the flakes’ average area, their average total drag coefficient and square of their velocity. Then we subtract that expression from the force of gravity. The difference will equal to the so-called net force, which is the product of mass and rate of change of velocity with respect to time—Newton’s Second Law.

In our differential equation, velocity appears on the equation’s two sides, one of which also has the variable of time. Isolating the variables and using appropriate substitutions allows us to integrate and solve for velocity. As the time that the snowflake falls increases, exponential terms drop out of the equation, and the flake’s terminal velocity seems to depend only on the  the snowflake’s mass and the shape -influenced and gravitational constants we mentioned earlier.

8051109Now we observe. As we stated at the onset, many snowflakes land in the same place. But only a few meters above any given spot, it is apparent that many paths lead to a common destination. Some flakes tumble; some abandon the terminal velocity we took so long to calculate, and they yield themselves to whimsical eddies. How they arrive is influenced not only by shape, mass and gravity but by sheer luck—luck due to the random, pinpoint fluctuations in temperature and pressure that affect their air space.

And these unpredictable*, forgotten, dance-like movements of deviant snowflakes open our eyes and widen our mouths. They drain our minds of thoughts of shovelling and of future slush and social conflicts. For a few moments the destinies of snowflakes is all that matters, and then we are reminded of a beautiful, non-mathematical expression in which snow is equated with Christmas.

*N.B. In reality the larger snowflakes may behave like sheets of falling paper which experience aerodynamic lift, a lift dominated by the product of linear and angular velocities. Those of you interested in computer simulations of falling snow might find this link interesting: https://www.cs.rpi.edu/~cutler/classes/advancedgraphics/S08/final_projects/fermeglia_willmore.pdf

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Snow, Chemistry And The Spirit Of Christmas

Sometimes a view from an outsider is what’s needed to make us pause and reflect. Two of our students, one on an exchange program from Germany and one here from the southern U.S, have been enthralled this week at school not by the equations on the board or by the books we read but by the snow that they rarely see in their native environment.

One of them went out and ate the fluffy form of H2O, like we did when we were children,and then he fell back and made snow-angels. The other student just couldn’t wait for the week’s second snowfall.

Too accustomed to snow, as Canadians we associate it with shoveling and difficult driving conditions. We dream of escaping the cold and forget what a privilege it is to live in a land of ice and snow at a point in history when thanks to technology and science we can enjoy and understand snow. Unlike our Nordic ancestors, we are not condemned to shivering in a storm with inadequate clothing and poorly insulated shelter. In a country endowed with hydroelectricity, it is so easy to stand in warmth and to watch a cascade of transient crystals land on a window.

We know why it is truly rare for two snowflakes to be alike because their patterns depend on their immediate environment.   The slightest changes in pressure or temperature can affect the shape; as one drop of water crystallizes it releases energy, changing the destiny of an immediate neighbor. And yet all these differences represent variations upon a theme: all flakes have six sides and six needles. In the three dimensions of its solid form, snow molecules maximize the number of intermolecular attractions between themselves. Specifically, the oxygen of one H2O molecule attracts two hydrogen atoms from two different molecules.

Conversely, each of water’s hydrogen atoms bonds to an oxygen atom from two different H2O’s. A staggered hexagon results with a water molecule at each vertex. Even when, in the smallest of snowflakes, this arrangement is repeated about three billion trillion times, the basic pattern remains the same.

But enough said about chemistry. Snow is also the framework for the spirit of the holiday season. A few months ago my daughter was told there was no Santa and that parents were the ones filling the stockings on Christmas Eve. She approached me on the subject but seemed more curious than sad, so I told her a “transition-truth”. I told her that Santa is a spirit who gets into parents and makes them buy the gifts that children desire. In previous years we had written to Santa Claus together, but last week she wrote the letter, sealed the envelope, walked past me, and headed to the mailbox. I said, “Wait. Aren’t you going to tell me what you asked for?”

“No,” she replied. “You will be filled with the spirit of Santa and you’ll know what to get me.”

Let’s hope that the snow does not melt before Christmas.

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