Inertial reference frames in an atmospheric science context

Question

A Hadley cell cycles air meridionally between the equator and 20 ◦ latitude. Assuming no zonal wind at the equator, what is the mean speed and direction of the trade winds at 20 ◦ ?

My instructor insists that this problem can be solved via inertial reference frames only, without directly using the Coriolis "force". I'm not sure how, though.

I understand the physical behavior of the system: the Hadley circulation moves air from tropical latitudes toward the equator. Those winds are deflected by the Earth's rotation beneath the wind such that the wind arrives southwest of the direction that the wind's velocity vector originally pointed. That's the directional component.

1) The velocity of the air parcel at 20 degrees North latitude. I'm completely lost as to how to derive this from the information given.

2) The direction of the deflection. This comes from the inertial reference frame. I understand the physics here, but am not sure how to set up the math.

Thanks for any help anyone can provide. I'm happy to clarify any other information as well.

Answer 1

First off:
I'm aware that it is physics.stackexchange policy that answers to homework questions are deleted.

In defence of this answer:
Here, a general underlying principle is discussed. That is, this answer is totally unsuitable for copy/paste-and-blindly-hand-over-to-the-instructor.

---

It is indeed the case that assessing this geophysical effect does not require using a rotating reference frame.

The thing to recognize is that all air mass has an angular velocity relative to the Earth's axis.

For the sake of simplification, assume that the air mass movement of the Hadley cell circulation does not experience any force (or force component) in the direction parallel to the equator. If that is the case then there is conservation of angular momentum.

At the equator air mass that is stationary relative to the Earth is moving at a velocity of about 1500 meters per second (if I remember correctly, you'll have to check that.)

As it moves away from the equator the air mass moves closer to the Earth's axis. Applying conservation of angular momentum you can readily work out what the velocity of the air mass will be when it arrives at 20 degrees latitude.

---

Incidentally, the Hadley cells were named after Hadley because he was the first to suggest this explanation for the existance of the trade winds. However, there was one thing that Hadley got wrong. Hadley reasoned in terms of conservation of linear momentum. That is, Hadley suggested that as air mass migrates from the equator to a latitude away from the equator it keeps the same linear velocity.

Of course, in Hadley's time conservation of angular momentum had not yet been recognized as a general principle of mechanics. It would be unfair to Hadley to hold this error against him.

Actually, to this day there are atmospheric physics textbooks in which this error is repeated.

General note:
The assumption that there is no force in the direction parallel to the equator is probably a fatal oversimplification. Probably there is a drain of kinetic energy from the trade winds to surrounding slower moving air mass. So the correct calculation, using conservation of angular momentum, may arrive at a higher velocity for the trade winds than the actual velocity. Any discrepancy must be due to forces unaccounted for. This is a case of rotational mechanics, and the principle that applies here is conservation of angular momentum, not conservation of linear momentum.

Anyway, the thing to recognize is that you can use a conservation law.