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If I throw a ball straight up, it deflects slightly to the west due to Coriolis forces. If instead I watch a bubble float up in water, is the bubble deflected west, east, or neither?

I think the bubble also moves west, but am not sure. My reasoning is that the air in the bubble must feel a Coriolis force to the west because it is rising. Further, as the bubble rises, the water around it moves, and the net motion of the water is down. Water moving down experiences a Coriolis deflection to the east. Bubbles move the opposite direction of water, so if the water is moving east, then the bubble should move west.

I don't have any reasonable-sounding counter-arguments, but I'm not completely convinced by my argument, either.

Edit: One reason I'm a unsure about this problem is that a bubble does not get thrown outward by a centrifugal force. Imagine a plastic ball filled with water and spinning fast. A bubble is sitting in the water in the equatorial place, half way to the edge of the ball. There's centrifugal force out towards the wall, but the bubble does not move that way. The bubble moves up the pressure gradient. The water gets thrown to the wall, and thus there is higher pressure at the wall than at the center. The bubble moves towards the center. If a bubble moves counter to the centrifugal force, I should be careful before claiming it moves with the Coriolis force.

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3 Answers

up vote 2 down vote accepted

It does.

To convince yourself, remember that rising hot air does experience a Coriolis force, so I am quite sure that your bubble does too.

Also, think of what the Coriolis acceleration is - it is an apparent acceleration due to the fact that you, the observer, are in a rotating reference frame, so your definition of "straight up" is actually a curve. When the bubble goes up you see it as having a curved trajectory - but this has to do with your rotating definition of straight up. Therefore this acceleration applies to it.

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Thanks, Sklivvz. Thinking about this problem in the context you set, the question becomes, "Does a bubble move straight up when viewed in an inertial frame?" (Or something like that - I guess to be more accurate we'd ask if it conserves momentum in the direction of Earth's rotation in an inertial frame.) I'll add a little detail to the question explaining some of my reservations. –  Mark Eichenlaub Nov 20 '10 at 22:44
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Here is one case in which a bubble will move against the Coriolis force, although it's not exactly the same case as addressed in the question.

This reference claims that Coriolis forces used to come into play in ship navigation by deflecting the bubble in a level, but doesn't say which way.

Consider the bubble in a level. If you are in the northern hemisphere and are moving further north, the Coriolis force is to the west. The Coriolis force acts on the air in the level's bubble, but also on the water in the level. The force is proportional to mass, so by the equivalence principle we can think of it as a small gravitational force pointing west. A bubble in water moves opposite gravity, so the bubble moves east, against the Coriolis force.

The difference between this scenario and a bubble simply floating up in water is that in the level, the bubble and water move in the same direction, but not so for a bubble floating up.

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This is the situation that first came to my mind, too. But here you have the force acting on the whole system (liquid and bubble), while in the case of the rising bubble there is no force on the static liquid to form the pressure gradient. –  dmckee Nov 23 '10 at 19:17
    
Yeah, I think that is the important distinction, thanks! –  Mark Eichenlaub Nov 23 '10 at 19:19
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Summary:
The bubble will be deflected to the East.

The two cases that you present here are different in more ways than meets the eye.

Throwing something straight up

Treating air friction as negligable we see that as soon as the ball has left your hand it follows a ballistic trajectory. For the duration of the flight the ball is in a keplerian orbit, be it an orbit that intersects the surface of Earth again in a matter of seconds. Nonetheless, the mechanics is orbital mechanics.

When a ball is thrown exactly vertically its horizontal velocity component is the velocity of co-rotating with the Earth.

As we know, an object in a keplerian orbit slows down as it ascends to the highest point of it's orbit. At the highest point the angular velocity is slowest. So: when a ball is thrown exactly vertically then along the ascend it will increasingly lag behind the Earth's angular velocity. That is, relative to the Earth the ball will deflect to the West.

In orbital mechanics the changes in angular velocity are not attributed to coriolis effect. If anything it's referred to as conservation of angular momentum. Ascending to the highest point of an orbit the angular velocity decreases, descending the angular velocity builds up again.

A bubble rising up

Let me first examine the following case: a flexible tube is submerged vertically. At the surface is a pump that forces water down. So you have vertical flow. To get the simplest case let's say that tube is located somewhere along the equator. To descend to a greater depth is to move towards the center of rotation. As the water descends to a greater depth you have a coriolis effect: the descending water will tend to pull ahead of the Earth's angular velocity. Hence the vertical tube will flex to the East

For a bubble to rise surrounding water must descend. I assume the ascending bubble will simply co-move with any deflection of the descending flow that surrounds it. The descending flow will be deflected to the East.

General remarks

There is a big difference between the case of throwing a ball and a rising bubble, so there is no direct comparison.

Once more the example of the vertical tube: the fluid in the vertical tube is carried by a buoyancy force from the water column underneath. But in the case of the ball there is no buoyancy; the trajectory is ballistic.

In your original question you hint at that difference; you raise the question whether some centrifugal effect must be taken into account

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The baseball can either be treated as a Kepler orbit in an inertial frame or as the Coriolis force pushing on it from an accelerated frame. They are both fine ways to analyze the problem. What I don't follow is the assumption that the bubble moves the same way as the fluid around it. Why should it do that? –  Mark Eichenlaub Dec 28 '12 at 2:07
    
@ Mark EichenLaub - The water that surrounds the bubble is much heavier than the bubble. I treat the mass of the air in the bubble as negligably small as compared to the mass of the water that flows down around the bubble. –  Cleonis Dec 28 '12 at 22:10
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