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21 March 2003

Howling Wind

We moved our DMS school bike ride to next Friday because the wind was howling last week. Wind is just moving air, but think about where all that air is moving from. There's no sudden vacuum over there, so there must be more air moving into that place from somewhere else. And so on... Does that mean that if it's windy anywhere, it must be windy everywhere? Of course not, but why? What if a biker rides there and back, into and with that wind. If biker and wind both keep a steady pace, will the round trip time be just the same as if there were no wind, since it all cancels out?

The winds of war are howling as well, so this picture of our
tattered prayer flags at my school is appropriate. They sent
lots of our prayers last week with wind gusts above 50 mph.

This is an example of an argument by infinite regress. It's windy, so think about where all that air is moving from. There's no sudden vacuum over there, so there must be more air moving in from somewhere else. But there's no sudden vacuum there either, so there must be even more air moving in from somewhere else. And so on... ad infinitum around the world. So must it be windy everywhere? Of course something's wrong with this argument, but what?

I thought of another "howling wind" stumper: What do we actually hear when the winds howl? Why does the wind sound so dramatic in the pines?

Wind is moving air, but there's always an end to it because air moves up and down as well as sideways to make closed convection cells. Air rises over land as the sun warms it up every day. This pulls in cool air from the ocean to make our usual afternoon sea breezes. That rising air then cools and settles back down over the ocean to complete the loop. Biking there and back into and with the wind is always slower than riding with no wind. Even if steady winds help and hinder your speed equally while coming and going, the headwinds slow you down for a longer time.


Our usual afternoon sea breeze that blows inland from the ocean is the classic example of a convection cell. It doesn't have to be windy everywhere and there is no infinite regress because winds usually move in closed loops. Warm air rises over land during the day, which creates thermal low pressure and pulls air in from the relatively cool ocean.

A reverse land breeze can happen at night when the land cools down more than the ocean. That reverses the air flow to bring cool air down from the mountains back to the now relatively warm ocean. I don't usually notice it since I live in the mountains, and I don't get up any earlier than I have to. But I remember those great sunrise offshore winds when I was a surfer kid in high school!

Pre-sunrise offshore land breeze at the Santa Barbara Harbor pulling fingers of cloud over the top of the Santa Ynez Mountains. Sailors have depended on these early morning offshore winds since the time of Homer and beyond.

Hurricanes, cyclones, and tornadoes are a different kind of closed circulation loop horizontally around a circle. Maybe the high jet stream winds and the trade winds that circle the earth are another closed loop that moves some air around the planet without disturbing the rest of the atmosphere.

Air circulates in closed loops on a global scale as well. Warm air rises from the tropics and settles closer to the poles, turning to the right as it moves due to the Coriolus effect of the Earth rotating from west to east. The circulation is divided in different Hadley Cells that are responsible for persistent global climate features like trade winds, doldrums, the intertropical convergence zone, and the distribution of deserts and Mediterranean Climates. The fractal combination of local winds on top of storm fronts on top of global circulation patterns reminds me of the classic Japanese Hokusai painting of "The Great Wave" which shows waves on waves on waves...

Scientists are just beginning to understand and model this big picture of how the Earth works powered by the sun and the seasons. Because the atmosphere is moving in closed cells, it doesn't have to be windy everywhere if it is windy somewhere. An application of the Brouwer Fixed Point Theorem known as the hairy ball theorem (really!) proves that it's not even possible to be windy everywhere!

The great mathematician and philosopher L.E.J. Brouwer (1881-1966) proved the impossibility of an "everywhere nonzero-tangent vector field on the 2-sphere". Imagine that you are combing a long-haired tennis ball trying to get all the hairs to lay flat. You can't do it. You can brush everything up from the "south pole" to the "north pole", but you will end up with a tuft and a hole. It's the same with any other strategy. Since the Earth is a ball, and the wind always has a direction, like a hair, there must always be (at least) one high-pressure and another low-pressure cyclone somewhere. The combed hairy ball must have at least two bald spots. So somewhere on earth there must always exist two distinct regions of dead calm. I think this means that hair cowlicks and parts and swirls are inevitable! Try to visualize combing long hair on a torus or doughnut shape. An everywhere-hairy doughnut is possible if you always comb around the loop. What about dogs and giraffes? That's topology!

Another application of the Fixed Point Theorem is that you can take any two maps of the same region, but one is larger than the other, and you randonly put one on top of the other, there must always be at least one point on the small map that lies on the exact same point on the large map. This is interesting math that is not intuitive!

I have no more time this weekend to explain riding into and with the wind, but see my previous stumpers on There and Back Again (29 September 2000), Piece of Cake (5 November 1999), and Up This Hill and Down (22 January 1999). I'll get back to this if I find time. The sound of the wind in the pines can wait for another stumper.

I'm short on time again, but here are a few starting web links for your own research on this stumper.

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