Mountains and Balancing Spin

#0610 "Wild Life"
Big storms that bring lots of snow like this requires
looking at the mountains and spin... 

Last Science Tuesday in “Angular Momentum Spins Up the Winds of Climate”, we were all in a spin about the conserved nature of angular momentum and figure skaters. Lots of wonderful nature exists as a result of living on a globe spinning on an axis tilted toward a star. The sun is our source of all energy, past, present and future.

The total angular momentum of our figure skater is actually comprised of two parts. The dominant portion that we examined last week is the large and exciting component comprised of the rapidly rotating skater on the ice. The other portion that is always in the background, results from the location of the skating rink. If the rink is at the North Pole (the northern tip of the Earth’s axis of rotation), our skater and everyone else in that arena rotates once every day. That  rotation is fast considering that we are simply standing there doing nothing. Pointing the thumb of our Coriolis Hand upward means that our fingers are curled in the same sense as the cyclonic rotation. The meteorological convention is that cyclonic rotation is positive and in the same sense as the rotation of the Earth. 

To make things easier, we had also better point out that angular momentum has two aspects – the speed of rotation (spin) and the pointing direction of the rotation axis. At the North Pole the rotation axis of the skater is aligned with the rotation axis of the Earth. The cyclonic spin of the skater is augmented by the cyclonic spin of the Earth. 

If that skating rink is gradually shifted along any line of longitude from the North Pole toward the equator, the axis of rotation of the skater becomes less aligned with that of the Earth. At the equator the rotation axis of the skater is perpendicular to that of the Earth. Imagine a stationary skater looking eastward at the equator. The skater does not spin at all as the earth does its daily rotation. The ice rink has also melted.

The total angular momentum of the skater is the sum of these two components of spin and that total is conserved in the absence of friction. The component of angular momentum that results from the location of the rink is typically called “planetary angular momentum” by meteorologists as the Earth is doing all of the work. To keep things simpler and save a dozen letters on each repitition, let's just refer to the angular momentum as "spin" from now on and remember that it is conserved. 

Now let’s replace the skater with a cylinder of air of constant mass.  As noted in “Isentropic Surfaces - Science and Art Merges”, air follows isentropic surfaces for free with no exchange of energy. Spin must also be conserved for flows following constant energy surfaces in the absence of friction. 

In this thought experiment, we constrain the lid of the cylinder of air to follow a cold and higher isentropic surface while the bottom follows an isentropic surface near the ground. What happens when we move this cylinder with the westerly mid-latitudinal winds along a line of latitude? What then happens if we place a north to south mountain range in its path? North to south mountain ranges are actually quite common on the Earth but that is another story that makes nature and the weather so very interesting. 

The isentropic surface near the ground follows the west to east terrain profile closely.  The higher isentropic surface is a smoothed out version that spreads out the sharpness of the terrain features. As one would expect, the biggest impacts on the cylinder of air are felt over the mountain but there are significant implications both upstream and in the lee of the mountain.

Between 1 and 2 in the accompanying graphic, the upper isentropic surface has already started to feel the spread out effects of the mountain but not so much at the surface. The girth of the cylinder decreases thus increasing its spin like the figure skater pulling in her arms. To maintain a constant total spin, the parcel responds by diverting a bit to the south where lower values of planetary spin occur. 

At 2 when the bottom of the cylinder first reaches the mountain, the cylinder is rapidly scrunched into a squat can. This is like a skater very quickly loosing height and spreading that weight outwards far from the axis of rotation (like what happens in a “camel spin”). There is a big decrease in the cylinder spin and the cylinder itself must take a sharp turn to the north in order to gain higher planetary spin. The total spin is still constant. 

At the mountain peak (3), the bottom of the air cylinder starts to rapidly drop following the sharp terrain. The air cylinder is rapidly stretched and the girth decreases. The cylinder spin increases dramatically. The air cylinder takes a rapid detour to the south in order to reach the lower planetary spin values at lower latitudes. The total spin is still the same as what the cylinder had when it started. 

Once the cylinder reaches the plains at 4, the stretching is reversed and the cylinder girth increases. The vertical height of the cylinder starts to decrease again as the upper level isentropic surface starts to get far enough away from the influence of the mountain. The spin of the cylinder starts to decrease and the cylinder turns again to the north to offset that loss of cylinder spin with the increased planetary spin found at higher latitudes. 

The air cylinder overshoots the original latitude. A series of ridges and troughs that gradually decay in amplitude then form downstream from the mountain. 

The dashed line mapped on the accompanying graphic (mountain barrier at 3) is the path of the cylinder with respect to the Earth as it was described above in words. The direction of motion of this air cylinder is simply the wind. The implication is that the trajectories of these cylinders or parcels of air over time must also be pressure height contours that describe the geostrophic wind.  In “The Answer Really IS Blowing in the Wind”, the wind was found to follow the pressure height contours. The diversion of air parcels to conserve spin must also change the pressure patterns around the mountain. A ridge of high pressure is created over and upstream of the mountain, while a trough of low pressure is formed downstream in the lee of the mountain barrier. And all as a result of conserved spin on a spinning Earth. Wow. 

If we look at the change in pressure along the dashed line of latitude in the image above, we see this. A high pressure over and upstream from the mountain generates a pressure gradient force (PGF) pointing eastward toward the lee trough. This PGF blows on the mountain barrier and returns westerly momentum that was picked up by the atmosphere in the tropics back to the Earth. This poleward transport of westerly momentum from the tropics to the mid-latitudes keeps the total spin of the earth biosphere in balance. 

Next week will reveal what this means for the weather… it will be worth waiting for. My friend and professor from the University of Alberta, Dr Edward Lozowski had a careful look at this week's Blog and offered some invaluable suggestions making it both simpler and better. Thank you my friend!

Warmest regards and keep your paddle in the water,

Phil the Forecaster Chadwick