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Re: [Phys-L] conserved momentum thru gas...

On 12/18/22 5:56 AM, O A via Phys-l wrote:

how is the atmosphere keeping up with the rotational velocity of the
spinning earth? This would neccesarily mean the same angular velocity but a
greater linear velocity at, say 100km above solid earth. Again though,
that's assuming the earth/atmosphere 'system' is 1 frame of reference (not
2 separate).

Well, the obvious answer is, there is no "how".

In the simplified model that underlies the question, a
uniform rotational velocity satisfies all the equations
of motion.

Perhaps the intent was to ask how it got that way, i.e.
how uniform rotation was established.

-Gaseous atmosphere, where we know gases are definitionally unbonded

Unbonded yes ... but not non-interacting. There is viscosity.
If the air were a superfluid we would be having a much more
complicated discussion, but it's not.

You can estimate the magnitude of the viscosity as follows:
You know that when a hurricane is cut off from its supply
of energy, it peters out in a few days. That's a tremendous
amount of angular momentum that gets transferred via friction
between one air parcel and the next, until finally transferred
to the solid earth via friction at the surface.

In the absence of disturbances (mainly solar heating), uniform
rotational velocity would be established very quickly and would
be maintained for all time. The steady state would not require
any explanation other than "why not?".


In reality uniform rotation is a zeroth-order approximation.
We take that as the reference frame and consider motions
relative to that. Motion relative to the rotating frame
requires centrifugal and Coriolis contributions to the
equations of motion.

Uneven solar heating creates updrafts and downdrafts on a
planet-wide scale. This in turn leads to horizontal flows,
by conservation of material. These flows are heavily affected
by Coriolis forces. The result is, to first order, the
*tricellular* model. This manifests as prevailing westerlies
at temperate latitudes, trade winds, horse latitudes, jet
streams, etc. etc. etc.


How is conservation of momentum 'transferring' through scores of miles of

Two answers:

The short answer is "viscosity".

The deeper answer requires a more nuanced discussion of conservation.

The law of conservation of momentum (or energy or charge or whatever)
is a *local* law ... local in space and time. Momentum is conserved
right here, right now. This is important; otherwise the law would
be useless since any violation would be too easy to explain away.

The momentum (angular or otherwise) of a parcel of air changes only
insofar as momentum is transferred across the boundary of the parcel.
This is the correct way to state the conservation law.

Conservation itself is not "transferred". It does not "flow".
The momentum is what flows.

Angular momentum is transferred from one parcel to the next via
viscous friction at their shared boundary.

Momentum is transferred across great distances one step at a time.
-- Newton's cradle is a graphic demonstration of linear momentum
transferred from one object to the next to the next.
-- A Shive wave machine demonstrates transfer of rotational motion
from one object to the next to the next.