Tuesday, October 9, 2007

Mogul's Happen

(Oops, I made a calculation error. It turns out that moguls should form at much shallower slopes than I initially estimated. See the Mathy Bits section for an explanation. Thanks for catching the mistake, MC!)

People seem to have a love-hate relationship with moguls - they're lots of fun if your timing and knees are good. They can be a real pain if you prefer a more casual descent down the slopes and you happen to get stuck above a gnarly mogul field.

I've hit the moguls on occasion, and regretted it more than once. I can recall sitting on the snow nursing my hyper-extended knee and wondering why on earth they make these things.

After looking into the science behind them, I found out that no one actually makes moguls at all, at least not on purpose. Moguls happen all by themselves, with a little unintentional help from snowboarders and skiers.

Moguls are an example of something called a self-organizing structure. All you need is a snow-covered slope and some folks to slide down it, and soon they will push the snow into lumps that form stunningly organized moguls. There's no way to stop it from happening. All you can do is run over the slope with a groomer to cut the things down from time to time.

Typically, self-organizing structures arise whenever patterns are formed as a result of many simple parts that follow a few basic rules. Common examples you've probably seen include crystals (salt, diamond, and snowflakes to name just a few), sand dunes, ocean waves, flocking birds, schooling fish, and the patterns in bubbles rising through a glass of champagne.

It's hard to predict in advance how the rules might lead to patterns. The best we can do, in most cases, is to look at the pretty structures and then work backward to figure out what rules might have made it happen. Even then, it's not always clear what the underlying rules really are, so physicists sometimes make up rules that seem to explain things reasonably well. This is what we call empirical science.

As far as I can tell from researching the scientific journals, nobody truly understands moguls. That hasn't stopped them from coming up with theoretical models that work adequately, even if they're probably not completely true.

One of the scientists who has looked into moguls is Regis University computer science professor and extreme skier Dave Bahr. He's made time lapse movies of moguls and shown - now get this - that moguls slowly migrate up hill over the course of a winter. Check out his videos. They're eerie. It's like the moguls are alive.

Bahr explains how this happens with a little cartoon, and has promised to write a formal paper about it eventually. Basically, as you make your way through the moguls, you end up scraping some of the snow from the downhill side of one bump and depositing it on the uphill side of the next one down. So even though you're pushing snow downhill, the moguls themselves move up.

There isn't much else in the scientific literature about moguls, but a recent physics paper analyzing the washboard ripples that often form on dirt roads seems to offer some insight into bumps on the slopes. The physicists who wrote the paper filled a tray with sand, and then ran a wheel across it to see when and how ripples would form. And ripples almost always developed - unless they rotated the tray very, very slowly. They concluded that all dirt and gravel roads would develop washboard ripples if traffic moved any faster than a few miles an hour. That is, you would have to keep traffic to speeds slower than the rate at which most people walk if you want to stop ripples from forming on your dirt driveway. Check out this short video showing their experiment in action.

The thing that makes this research relevant to snowboarding and skiing is that they also tried dragging a non-rotating square block along the sand. That too caused ripples to form. When you think about it, it's a lot like what you're doing when you slide down the slopes on a board or skis. Mogul fields are essentially giant washboard roads that are created by snowboard and ski traffic rather than cars and trucks.

If you apply the theory behind road ripples to moguls on snow, it turns out that moguls won't form if the slope is mellow enough. It has to be VERY mellow though. If I did the math right (you can check it in the mathy bits below), only slopes with less than 7 degrees 2.3 degrees incline (about a 12% 4% grade) will stay mogul-free on their own - that's bunny slope territory darned near flat! Anything steeper will have to be groomed to keep the moguls down.

The bottom line is there's no one to blame for moguls but ourselves and physics. So you either have to learn to ride them, or stick to groomed trails, or (best and hardest of all) find a fresh back bowl where the snow is still untouched and nobody's had a chance to pile up those bumps.

The Mathy Bits

(As you will see from the portions crossed out, I initially calculated that moguls form on slopes steeper than 7 degrees. But after someone pointed out that I seemed to have made a math error, I found that the actual answer is probably closer to 2.3 degrees. It turns out that I put the factor for the snowboard width in the numerator rather than the denominator at one point when I made the substitution for v^2. I don't show all these details because it's hard to do math in the Blogger interface. But if you want to talk about the details, drop me a note at "buzzskyline at gmail.com".)

In their Physical Review Letters paper about washboard roads, physicists Nicolas Taberlet, Stephen W. Morris, and Jim N. McElwaine suggested that there's something called the Froude number, which predicts when ripples will form on a dirt or gravel road. They write the Froude number equation essentially like this


v is velocity
g is the acceleration due to gravity
p is the density of the sand (or snow)
m is the mass of the wheel (or snowboarder)
w is the width of the wheel (or snowboard)

Whenever Fr is greater than one, washboard ripples form. Of course, the only thing that isn't constant in the equation is v, which is just the speed of the wheel over the sand. All the rest of the components are fixed.

It's pretty clear that all you need to do to keep Fr below one is to slow down.

To apply this to snow, I estimated that the density of packed snow is about half the density of liquid water (500 kg per cubic meter), the mass of the average snowboarder is about 70 kg, and the width of the average snowboard is about 20 centimeters.

If you recall from my post about Speed Snowboarding, you can solve for the terminal velocity of a snowboarder pretty easily by rearranging this equation

m*g*sin(theta)= u*m*g*cos(theta) + (p*A*Cd*v^2)/2

The terminal velocity is the speed limit on the hill. Not that it's illegal to speed at the slopes, it's just that physics won't let you go faster than the terminal velocity.

If I plug the terminal velocity from the Speed Snowboarding post in the equation for the Froude number, I can see that Fr<1 whenever the angle theta is less than about 7 degrees 2.3 degrees.

You should bear in mind that the washboard road paper uses some pretty shaky logic to come up with their Froude number, and that I am just blindly applying their calculation to snow, so take this all with a grain of salt.

Still, it seems reasonable that there should be some slope that's too mild for moguls to form. Although I don't know if anyone has tested it, I bet you'd never see moguls on a bunny slope with a 7 degree 2.3 degree or less drop, even if it were never groomed.

On the other hand, even a gentle beginner slope would be steep enough to develop moguls if left ungroomed for long.