Trying to Understand the Coriolis Effect: Why West-to-East Air Parcels Go South

[nitin_zilch]

I am trying to understand the Coriolis effect from some time but i am unable to conceive the idea why the air parcel flowing from west to east on Earth would go southward.

what I have understood till now is:

" From the perspective of observer in inertial frame of reference, when the air parcel goes from South to North in Northern hemisphere, the direction of net velocity of the parcel is given as the resultant of its tangential velocity, and the vertical component of velocity. If we ignore friction, the parcel should continue to move in the direction specified by the resultant velocity, which is to the East of North.

Similarly, the air parcel rushing from North to South would move in the direction of the resultant velocity, which is to the West of South."

When I think the same way for the air parcel from West or East, it is obvious that there should be no apparent deflection in the parcel because there is no vertical component of velocity.

Please tell me what is wrong with my interpretation.

 

[jbriggs444]

When I think the same way for the air parcel from West or East, it is obvious that there should be no apparent deflection in the parcel because there is no vertical component of velocity.

By "vertical", I assume that you mean "radial" (i.e. toward or away from the North pole).

Let's pretend for the moment that we are dealing with rotation in a plane, looking down from above the North pole at the Earth. We can use an inertial frame of reference and see the Earth as rotating counter-clockwise below us like a carousel. Or we can adopt a frame where we are standing on the carousel rotating with it. Let's stay with the inertial point of view.

As your reference points out, as a parcel of air moves north (toward the center) with respect to the Earth, it is actually moving north-east in a spiral path. Its eastward velocity stays constant, but as it goes farther North the eastward velocity of the Earth beneath it decreases. So it acquires an earth-relative velocity that is eastward.

Similarly for a southward-moving (away from the center) parcel. But you are apparently comfortable with this part already.

Now let's do the east/west part...

Take an eastward moving parcel. It is circling the center of rotation faster than its neighboring parcels. In order for this to happen, it would have to be experiencing greater centripetal acceleration. But it is actually subject to the same net centripetal force as its neighboring parcels of air. So it does not have enough centripetal force to maintain a circular path around the pole at its current radius. It must deviate southward (away from the center).

Similarly for a westward moving parcel. It is circling the center of rotation more slowly than its neighboring parcels. In order for this to happen, it would have to be experiencing a lesser centripetal force. It actually experiences the same centripetal force as its neighbors. So it has too much centripetal force to maintain a circular path. It must deviate northward (toward the center).

You might worry about the source of the centripetal force needed to keep all of this air circling a pole. That source is gravity. The Earth bulges at the equator. If the Earth were to stop rotating and still maintain the bulge, it would be a downhill slope all the way from equator to pole. But the Earth is rotating, and the slope is, on average, just enough to provide the needed centripetal acceleration. The result is that the ground appears to be exactly level.

 

[nitin_zilch]

By "vertical", I assume that you mean "radial" (i.e. toward or away from the North pole).

You might worry about the source of the centripetal force needed to keep all of this air circling a pole. That source is gravity. The Earth bulges at the equator. If the Earth were to stop rotating and still maintain the bulge, it would be a downhill slope all the way from equator to pole. But the Earth is rotating, and the slope is, on average, just enough to provide the needed centripetal acceleration. The result is that the ground appears to be exactly level.

If gravity is the source of the centripetal acceleration, what would happen to the parcel moving east or west along the equator. I am assuming that the gravitational force experienced by the parcel is normal to the surface of Earth, and is directed along its center. In such a case there is no component of gravitational force which can move the parcel North or South respectively.

Also if you could elaborate on how gravity make the parcel going east to west turn southwards?

 

[dauto]

That's correct, there is no Coriolis effect at the equator. Imagine you are somewhere in the Northern hemisphere. Because of the spinning of the earth, there is a centrifuge effect that pushes away from the axis of rotation. that centrifuge effect deforms the Earth forcing it to bulge out at the equator and flatten at the poles until an equilibrium is established. The centrifuge effect pushes towards the equator and a component of gravity pushes towards the poles (keep in mind that the equator has budged out so it is higher ground than the poles explaining why gravity would create a poleward force. that force is just the force that makes things roll from higher ground to lower ground).

Now when the air moves eastwards it is actually spinning faster around the Earth creating a stronger centrifuge effect breaking the pre-established balance and generating a net southward force. Conversely, when the air moves westwards it is actually spinning more slowly around the Earth creating a weaker centrifuge effect breaking the pre-established balance and generating a net northward force. Either way, that net north/south force is considered part of the Coriolis effect along with the east/west force that was generated by the entirely different process that you described. In sum, the Coriolis effect actually consists of two entirely different processes that together produce the familiar behavior of the Coriolis effect.

 

[jbriggs444]

If gravity is the source of the centripetal acceleration, what would happen to the parcel moving east or west along the equator. I am assuming that the gravitational force experienced by the parcel is normal to the surface of Earth, and is directed along its center. In such a case there is no component of gravitational force which can move the parcel North or South respectively.

Yes, that is correct. Nor is there any component of centripetal acceleration that is northward or southward. East or west movement at the equator results in neither a northward nor a southward deflection.

In addition, movement north or south at the equator has no component toward the axis of rotation. So there is also no Coriolis deflection resulting from north/south movement there.

This matches real world experience. [If you want to get technical, an upward movement of air at the equator would result in a westward deflection and a downward movement of air would result in an eastward deflection, but I am not sure whether that has any significant effect on the weather].

Also if you could elaborate on how gravity make the parcel going east to west turn southwards?

The ground at the equator is about 21 km higher than the ground at the poles (measured in terms of radius from the Earth's center). http://en.wikipedia.org/wiki/Equatorial_bulge

Since it is downslope from the equator to the pole the vector sum of gravity (centerward) and the supporting normal force from the Earth's surface (slightly north of straight outward in the northern hemisphere) will tend to push the air toward the poles.

In simpler terms, the air has a tendency to flow downhill and downhill is toward the poles.

Or at least the air would flow toward the poles if the bulge were there and if the Earth were not spinning. But the Earth is spinning. The poleward force on the air is just enough to keep each little parcel of air spinning right along with the ground below it with no tendency to drift off in any particular direction. A carpenter's spirit level will indicate that the surface of the Earth is level. A marble placed on flat ground will roll neither north nor south. A north/south canal can be filled with water without all of the water pouring out of one end. Things are in balance.

Gravity is not making any given parcel of air go north or south. It is just holding it in place against the spin of the earth.

If you have a parcel of air going east, it would be rotating around the Earth's axis more rapidly than once every 24 hours. In order to remain in place it would need to be subject to more centripetal force than the down-slope provides. But the poleward force from the down-slope is fixed. So that parcel of air will not be forced north strongly enough to circle the pole along with the rocks and trees and other parcels of air. It will deflect southward relative to them.

Going for an analogy, it is like a bunch of stock cars going counter-clockwise on a banked track with the "north pole" taken to be somewhere in the infield. If one of the cars goes too fast, he will no longer be able to hold the turn and will deflect rightward (aka southward), out of the track. If one of the cars goes too slow and still steers leftward as hard as the rest, he will deflect leftward (aka northward) into the infield.

 

[nitin_zilch]

Yes, that is correct. Nor is there any component of centripetal acceleration that is northward or southward. East or west movement at the equator results in neither a northward nor a southward deflection.

In addition, movement north or south at the equator has no component toward the axis of rotation. So there is also no Coriolis deflection resulting from north/south movement there.

This matches real world experience. [If you want to get technical, an upward movement of air at the equator would result in a westward deflection and a downward movement of air would result in an eastward deflection, but I am not sure whether that has any significant effect on the weather].


The ground at the equator is about 21 km higher than the ground at the poles (measured in terms of radius from the Earth's center). http://en.wikipedia.org/wiki/Equatorial_bulge

Since it is downslope from the equator to the pole the vector sum of gravity (centerward) and the supporting normal force from the Earth's surface (slightly north of straight outward in the northern hemisphere) will tend to push the air toward the poles.

In simpler terms, the air has a tendency to flow downhill and downhill is toward the poles.

Or at least the air would flow toward the poles if the bulge were there and if the Earth were not spinning. But the Earth is spinning. The poleward force on the air is just enough to keep each little parcel of air spinning right along with the ground below it with no tendency to drift off in any particular direction. A carpenter's spirit level will indicate that the surface of the Earth is level. A marble placed on flat ground will roll neither north nor south. A north/south canal can be filled with water without all of the water pouring out of one end. Things are in balance.

Gravity is not making any given parcel of air go north or south. It is just holding it in place against the spin of the earth.

If you have a parcel of air going east, it would be rotating around the Earth's axis more rapidly than once every 24 hours. In order to remain in place it would need to be subject to more centripetal force than the down-slope provides. But the poleward force from the down-slope is fixed. So that parcel of air will not be forced north strongly enough to circle the pole along with the rocks and trees and other parcels of air. It will deflect southward relative to them.

Going for an analogy, it is like a bunch of stock cars going counter-clockwise on a banked track with the "north pole" taken to be somewhere in the infield. If one of the cars goes too fast, he will no longer be able to hold the turn and will deflect rightward (aka southward), out of the track. If one of the cars goes too slow and still steers leftward as hard as the rest, he will deflect leftward (aka northward) into the infield.

I haven't heard anything that you say here, can you give me a link or something that can give me a detailed description for the same. I want to know this but with bits of information, it is harder to make sense out of it.

 

[A.T.]

there is no Coriolis effect at the equator.

There is, if something moves vertically to the surface, like hot air masses rising up.

the Coriolis effect actually consists of two entirely different processes

The Coriolis effect is simply the velocity dependent part of the inertial forces that appear in rotating frames. Talking about a "process" or even "two entirely different processes" is rather misleading. There is no physical process behind this effect. It's just a coordinate transformation, basically a mathematical "trick".

 

[nitin_zilch]

There is, if something moves vertically to the surface, like hot air masses rising up.


The Coriolis effect is simply the velocity dependent part of the inertial forces that appear in rotating frames. Talking about a "process" or even "two entirely different processes" is rather misleading. There is no physical process behind this effect. It's just a coordinate transformation, basically a mathematical "trick".

well, this is misleading, one answer considers the role of gravity while other says its just the effect of change in frame of reference. Can you please clear out the situation, and justify for what you say?

Thanks in advance

 

[A.T.]

one answer considers the role of gravity

Gravity is not related to the Coriolis effect. Anything that moves in a rotating frame is subject to the Coriolis effect (unless it moves parallel to the frame rotation axis). It doesn't matter why it moves. The air rising above the equator was just an example. If you fire a cannon vertically at the equator the projectile will also be deflected westwards.

while other says its just the effect of change in frame of reference.

Yes, it exists only in rotating frames of reference. See the below video for a good visualization:

https://www.youtube.com/watch?v=49JwbrXcPjc

 

[nitin_zilch]

Gravity is not related to the Coriolis effect. Anything that moves in a rotating frame is subject to the Coriolis effect (unless it moves parallel to the frame rotation axis). It doesn't matter why it moves. The air rising above the equator was just an example. If you fire a cannon vertically at the equator the projectile will also be deflected westwards.

Yes, it exists only in rotating frames of reference. See the below video for a good visualization:

https://www.youtube.com/watch?v=49JwbrXcPjc

can you explain why an eastward flowing wind (west to east) in the northern hemisphere would deflect south?

 

[Cosmic Al]

If you use Newton's 2nd Law (via Euler-Lagrange Equation) for a rotating reference system, the Coriolis and centrifugal accelerations "pop out" of the equations.

 

[olivermsun]

can you explain why an eastward flowing wind (west to east) in the northern hemisphere would deflect south?

In short, if you watch the movie you notice that everything on the rotating table is being deflected toward its right, no matter which way it's traveling. So west-to-east winds get deflected south, and north-to-south winds get deflected toward the west (hence the northeasterly Trade Winds).

 

[Arfur Bryant]

Here is an attempt to visualise Coriolis Effect without using mathematics or a 2D graphic (because you really need 3D to get a proper idea).

Remember the key things; 'Frame of Reference' and 'the Earth is a spinning sphere' (not a true sphere I know).

Imagine a (really big) giant standing over Bordeaux, France (45 deg N) facing East. Your left foot is very near the true North Pole and your Right foot very close to the Equator. As the Earth rotates, the giant's right foot will have to travel further distance per unit time than the left foot. Therefore, when viewed from space, the giant appears to twist left (anticlockwise). However, now lift the giant a few feet off the surface. Viewed from the surface (underneath the giant) the giant appears to twist clockwise (right) because your frame of reference is moving and the giant is no longer attached to the surface.

You can do the same for the southern hemisphere except the effect is opposite.
You can do the same at the Equator, with the giant's feet equidistant either side of the Equator. Now there is no 'twisting' effect.

This is how I visualise the Coriolis effect, as a twisting effect, not a turning effect. Every non-point in the northern hemisphere has a twisting (to the right) effect, irrespective of the direction it was originally heading.

A cannon shell fired will therefore have an apparent right-hand deflection as it is merely heading straight for a point in space after firing. Even when fired East or West, as the 'twisting moment' depends on latitude, not direction and, once fired, the shell will still veer right as the frame of reference (firing point) continues its way around the surface of the planet.

As an aside, the sister 'pseudoforce' to Coriolis is the Euler force, which is (if you like) the vertical aspect of something being fired from the surface to space. The Euler force is maximum at the equator and zero at the poles. However the Euler force is extremely weak compared to gravity.

I'd appreciate anyone's feedback on this explanation. Does it work for you?

 

[A.T.]

As an aside, the sister 'pseudoforce' to Coriolis is the Euler force, which is (if you like) the vertical aspect of something being fired from the surface to space.

The "Euler force" would appear only if the angular velocity of the Earth would change.
http://en.wikipedia.org/wiki/Euler_force

Maybe you mean the "Eötvös effect", which is just a component of the total Coriolis force vector?
http://en.wikipedia.org/wiki/Eötvös_effect

 

[Arfur Bryant]

A.T,

Good point, I'll look at it. But isn't the fired shell subject to an acceleration (angular and vertical) due to the nature of the spinning sphere? Maybe its a combination of both...

 

[A.T.]

But isn't the fired shell subject to an acceleration (angular and vertical) due to the nature of the spinning sphere?

Euler force is related to the angular acceleration of the reference frame, not of the object itself. If you stick with the Earth as your reference frame, there is no Euler force (or is completely negligible due to changes of the spin axis over many years).

 

[Arfur Bryant]

A.T.

Ok, I used the wrong force in that aside. Thanks for the heads up.