Mountains and Weather (Part I)

Consider a spinning ice skater. To increase her rate of rotation, she brings her arms closer to her body. Why does this occur? The answer lies within conservation of angular momentum.

Now you may have seen this concept discussed in your standard mechanics course, like PHYS 211 or AP Physics. You might also recall that the magnitude of angular momentum is the product of angular speed and moment of inertia, where the moment of inertia quantifies the distribution of mass from some fixed rotation axis. Unfortunately, such a nice relationship is not so for most atmospheric phenomenon because air parcels are not solid bodies like those observed in PHYS 211. Only at infinitesimal scales can solid body rotation approximate atmospheric circulation.

Fortunately, Rossby, an atmospheric dynamist, simplified our need to look at infinitesimal scales with the concept of barotropic potential vorticity:

              D[(ζ + f)/h] /Dt = 0 

What each variable means in this equation is not of significant importance to us. For now, just consider ζ to be a measure of spin (where positive values indicate counter-clockwise flow), f to be a positive constant, and h to be height.  The physical interpretation of Rossby’s formula is the following: If a column of air compresses or expands (causes height changes), then the trajectory of that column (its spin) will be distorted.

Now consider a uniform flow of air headed eastward towards a mountain. What happens to the trajectory of the air column as it reaches the peak of the mountain?

  D[(ζ + f)/h] /Dt = 0

  (ζ + f)/h = positive constant 

Notice that, if height decreases, the numerator must decrease to balance the constant. If f is a positive constant, then ζ must be negative to decrease the numerator. Therefore, the column acquires a clockwise spin and moves southward.

So if you ever look at a weather map of wind speeds and directions, take a closer look at the flow before and after winds reach the Rocky Mountains. You may be able to see the dip southward courtesy of angular momentum conservation.