What happens to the "curvature term" in the equations of motion for a rotating fluid near Earth's poles?

Question

For a rotating fluid in spherical geometry, one of the terms of the equations of motion is the "curvature term". For example, for the zonal component of velocity (corresponding to eastward motion), we have:

$$\frac{Du}{Dt} =\frac{uv}{r}\tan \phi + \mathrm{Other \ terms}.$$

Here, $r$ is the radius from the center of Earth to the point of air we are examining, $\phi$ is the latitude and $v$ is the meridional component of velocity. Near the poles (north pole for simplicity), $\phi \to \pi/2$ and therefore $\tan \phi \to \infty$. This would imply that this curvature term dominates over all other terms. Does this correspond to some real phenomena that air experiences near the poles?

Answer 1

I will get to the question, but first I would like to point out that youtube has a 'Copy video URL at current time' feature.

Rightclicking with the mouse in the video view opens a menu. The third option in that menu is 'Copy video URL at current time' The video you linked to is 47 minutes long, and the expression that you ask about appears at 40:41

There is a phenomenon that is related to the expression for meridional displacement: the Eötvös effect

I recommend that you absorb the information in that wikipedia article.

About the video:

Quote:

but we also obtain a second term which you'll notice does not include the rotation rate and this exists because of curvature of the underlying [...] the fluid parcel exists on the surface of the sphere

So:
The intention is for the geometric term to represent an effect that would occur equally on a non-rotating planet.

For the time being we simplify the problem, and we think about the case of a non-rotating planet.

The Eötvös effect wikipedia article gives the following expression for a dynamic effect that has to be taken into account when performing gravimetric measurement with a gravimeter onboard an aircraft.

$$ a_r = 2 \Omega u \cos \phi + \frac{u^2 + v^2}{R} \tag{1} $$

The counterpart of the geometric term is this one:

$$ a = \frac{u^2 + v^2}{r} \tag{2} $$

With:
a acceleration perpendicular to the Earth surface
r radius of the Earth

The expression for that correction comes from the expression for required centripetal force. When an object is in circular motion, with the motion in a plane, then (the magnitude of) the required centripetal acceleration is given by:

$$ a_c = \frac{v^2}{r} \tag{3} $$

The factor $u^2 + v^2$ in (2) is the square of the resultant velocity of the two velocity components u and v.

Example:
If the aircraft with the gravimetric equipment is flying over the north pole then the rotation-of-Earth effect is negligable. But it is still the case that the aircraft is not flying in a straight line, it is following the curvature of the Earth. The effect is exceedingly small. The radius of the Earth is a far larger number than the velocity of even an aircraft. So it's a very, very small correction. Notice especially that it is independent of the latitude

Differences

Now to the differences between the cases of measurements of gravity, and geophysical fluid dynamics.

In the context of GFD: generally you are mainly interested in how the fluid will move in the direction parallel to the surface.

• Gravimetry: the measurement probes the direction perpendicular to the local surface
• GFD: main interest is tendency to be moved in the direction parallel to the local surface

So the for the two cases the expressions will not be quite the same, but they are related.

For instance: take the case where a fluid parcel tends to rise. Let's say the fluid parcel is prevented from rising, but it has freedom to move parallel to the local surface. The tendency to rise will then result in a some tendency to move in meridional direction. That conversion effect will be at its strongest at 45 degrees latitude, and will be weaker at latitudes away from 45 degrees.

By now it should be clear that the term presented in the video cannot possibly be correct. It makes no sense.

$$ \frac{uv}{a} \tan \phi \tag{4} $$

So the name of the lecturer is Paul A. Ullrich. Notice that he is hesitating a lot. It would appear that he is not on top of it. I think he is just reading what it says on the slide. I don't think he understands what the expression is saying.

And even if you derive the correct form, the term is so small it will be swamped anyway. I am confident that in any other GFD book that geometric term is not discussed.

The slides of used in the video are also available as PDF. Introduction to atmospheric dynamics, Ch 1.

Around 29:53 into the video Paul Ullrich start talking about angular momentum conservation. That is superfluous because the equation of motion is already covering that. All he needs to do is to set up the appropriate equation of motion. Also, angular momentum is conserved only when the force that is exerted is in the radial direction only. But later Paul Ullrich applies the results in general equation that also feature pressure gradient force, which can act in any direction.

So my recomendation has to be that you seek out another resource. It would appear that you cannot trust this one.