# I Conceptual question about the Coriolis force and the weather

#### haushofer

Dear all,

there is something bugging me for a while now, and it's about a favourite topic of confusion: the Coriolis-force!

A fictitious force is a force which disappears if you transform to an inertial frame. At the algebraic level I have a good understanding of this, I think (Newton's 2nd law not being covariant under time-dependent rotations). Now my conceptual confusion is this: if I see those lovely satellite photos with those curly atmospheric movements like

it is often stated that this rotation is due to the Coriolis force: winds are being deflected due to the rotation of the earth in going from high to low pressure. But somehow I can't reconcile this with the statement that a fictitious force disappears if you become inertial. This would mean that if I would hang in outer space, inertial "with respect to the stars", this lovely hurricane would stop rotating in front of my eyes, and winds would simply go straight from high to low pressure without deflection. Where do I go wrong here?

Let's make a comparison to the Foucault pendulum:

From the inertial point of view outside of earth, the earth is rotating underneath the pendulum, while on earth (where we stand still with respect to the earth) we see the pendulum's plane rotating (precessing). Of course, in both case the pillars are kicked. Somehow, I don't see something similar happening for our hurricane, as in "it stops to rotate for an inertial observer hanging outside of the earth".

So what's my silly thought here? Thanks in advance! ;)

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#### A.T.

This would mean that if I would hang in outer space, inertial "with respect to the stars", this lovely hurricane would stop rotating in front of my eyes, and winds would simply go straight from high to low pressure without deflection.
And? Have you plotted the 3D-trajectories of the air-masses in the inertial frame?

The air doesn't "go straight from high to low pressure" in the inertial frame, because there are other forces acting (gravity, ground reaction). And even if there were no other forces, it wouldn't move but accelerate that way.

#### haushofer

And? Have you plotted the 3D-trajectories of the air-masses in the inertial frame?

The air doesn't "go straight from high to low pressure" in the inertial frame, because there are other forces acting (gravity, ground reaction). And even if there were no other forces, it wouldn't move but accelerate that way.
No, I haven't. But indeed, I consider an idealized case in which the atmosphere is more or less in hydrostatic equilibrium (gravity balances vertical pressure gradients) and no friction. Of course, for a hurricane that's a bit silly, so perhaps that's where my confusion arises.

#### haushofer

So my question is basically this:

if I am an inertial observer (with respect to the fixed stars) looking down on earth, will I see air masses still be deflected in cases where other forces (friction, etc.) are neglegible? How will I perceive those hurricanes?

I can try to make a simulation in Python, but before I do that, I want to make sure I understand the basic stuff first.

#### PeroK

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it is often stated that this rotation is due to the Coriolis force: winds are being deflected due to the rotation of the earth in going from high to low pressure. But somehow I can't reconcile this with the statement that a fictitious force disappears if you become inertial. This would mean that if I would hang in outer space, inertial "with respect to the stars", this lovely hurricane would stop rotating in front of my eyes, and winds would simply go straight from high to low pressure without deflection. Where do I go wrong here?
Consider a related scenario: two objects are a significant distance apart on the Earth's surface and one pulls the other directly towards it. In the Earth's frame, this is straight-line motion.

Depoending on how fast this happens, the two objects need an additional real opposite-to-Coriolis force in order to achieve their straight line motion.

If this real force is missing, then straight line motion does not take place and, in the Earth's frame, the motion becomes curved. That's what we see in weather systems: the high pressure air is trying to move directly to the low pressure region, but in the absence of a real opposite-to-Coriolis force, you get a cyclone or anticyclone. In the Earth's frame, this motion is explained by a fictitious Corliolis force.

What you see from an external inertial frame is different in two respects. Something at rest in the Earth frame is already subject to a "real" gravitational force (in this context). And, the motion of the high pressure air in this frame is explicable without the fictitious Corliolis force.

#### PeroK

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So my question is basically this:

if I am an inertial observer (with respect to the fixed stars) looking down on earth, will I see air masses still be deflected in cases where other forces (friction, etc.) are neglegible? How will I perceive those hurricanes?

I can try to make a simulation in Python, but before I do that, I want to make sure I understand the basic stuff first.
You model it using real forces only.

#### A.T.

No, I haven't. But indeed, I consider an idealized case in which the atmosphere is more or less in hydrostatic equilibrium (gravity balances vertical pressure gradients) and no friction. Of course, for a hurricane that's a bit silly, so perhaps that's where my confusion arises.
Well, fluid dynamic is complex, and maybe not a good example to understand anything basic. Maybe you should consider pucks sliding without resistance along a spinning planet. And note that gravity is not always fully balanced by the ground reaction force.

if I am an inertial observer (with respect to the fixed stars) looking down on earth, will I see air masses still be deflected in cases where other forces (friction, etc.) are neglegible? How will I perceive those hurricanes?
The inertial observer will see trajectories that are consistent with Newtons 2nd Law, without the need for introducing fictitious forces.

#### Ibix

Perhaps it would be easier to start with a simpler case? If you have two cannons back to back, facing due North and due South and far from the equator and poles, where do their shots go and why? This has the same frame-change effects as the hurricane without the fluid dynamics.

#### haushofer

The inertial observer will see trajectories that are consistent with Newtons 2nd Law, without the need for introducing fictitious forces.
Ok. So follow-up question: how would he/she perceive my hurricane?

From your answer and my understanding, I'd say: non-rotating (if friction is not an issue). Airmasses will not be deflected, because there are no fictitious forces, only a pressure gradient between the outside and the center of the hurricane. But somehow I've never seen them.

So what would you say?

#### haushofer

Perhaps it would be easier to start with a simpler case? If you have two cannons back to back, facing due North and due South and far from the equator and poles, where do their shots go and why? This has the same frame-change effects as the hurricane without the fluid dynamics.
Yes, I'll work that out explicitly; it's even simpler to do it on a disc instead of a sphere.

#### PeroK

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Ok. So follow-up question: how would he/she perceive my hurricane?

From your answer and my understanding, I'd say: non-rotating (if friction is not an issue). Airmasses will not be deflected, because there are no fictitious forces, only a pressure gradient between the outside and the center of the hurricane. But somehow I've never seen them.

So what would you say?
A hurricane is a real, physical phenomenon; the external observer would observe the same relative motion between an air mass, say, and a fixed object on the surface.

Part of your problem is this statement:

A fictitious force is a force which disappears if you transform to an inertial frame.
This is not a good definition, especially if taken too literally. For example:

A car accelerates towards a tree and crashes into it. In the car's frame, the tree accelerated as the result of a fictitious force.

Your conclusion, therefore, is that if you switch to an inertial frame, the fictitious force on the tree disappears and ... the collision never happens?

#### A.T.

how would he/she perceive my hurricane?
Perception is subjective, and not really relevant here. Objectively, the trajectories in the inertial frame will be such that the real forces suffice to explain them using Newtons 2nd Law.

#### haushofer

A hurricane is a real, physical phenomenon; the external observer would observe the same relative motion between an air mass, say, and a fixed object on the surface.

Part of your problem is this statement:

This is not a good definition, especially if taken too literally. For example:

A car accelerates towards a tree and crashes into it. In the car's frame, the tree accelerated as the result of a fictitious force.

Your conclusion, therefore, is that if you switch to an inertial frame, the fictitious force on the tree disappears and ... the collision never happens?
Yes, I think I see your point here. On a non-rotating earth, air will move from point A on earth (high pressure) to point B on earth (low pressure). But we, standing on the earth, will see that the air will actually deflect and miss B (and explain this by the Coriolis force). So an inertial observer in space will also agree that the air does not reach point B (like with the Foucault pendulum: the sticks are tipped by the pendulum, whether you're standing at the earth's surface or hanging in space).

But then my confusion is: apparently, air is not moving in a straight line, so the inertial observer also needs a force to explain this...

In this NOVA-video,

at 2:54 onwards, it's stated that the air for an inertial observer will move in a straight line. I use this popular video, because I'm a bit suprised that all these popular explanations don't really seem to address this point.

#### A.T.

...apparently, air is not moving in a straight line...
Of course not. If it would, it would fly off the Earth.
...so the inertial observer also needs a force to explain this...
Gravity, interaction with the ground and the surrounding air masses.

#### PeroK

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at 2:54 onwards, it's stated that the air for an inertial observer will move in a straight line. I use this popular video, because I'm a bit suprised that all these popular explanations don't really seem to address this point.
Yes, interesting. That's obviously wrong, as is clearly shown by the (external) view of Jupiter's Red Spot. The fluid is clearly rotating from the outside as well.

The critical difference is that fluid motion isn't of the form of an initial impulse followed by inertial motion. Even without a rotating reference frame, if a particle is already moving and it experiences a force not parallel with its velocity, then it moves in a curve.

This could simply be seen as the difference between a force on a stationary object leading to linear motion and a force on a moving object leading to curved, even circular motion.

If we look at things in an inertial frame first:

Note that both the low pressure and high pressure regions are initially moving non-inertially. Let's assume we have instantaneously created regions of high and low pressure in an otherwise stationary atmosphere (wrt the Earth's surface). As the "moving" air mass picks up speed, it gets pulled in a circular/spiral motion.

That is all explained (crudely) by the rotation of the Earth.

Now, in the Earth's frame, there is initially no relative motion, so why doesn't one air mass move directly to another? The obvious answer is to look at it from an inertial frame and see why. Alternatively, in the rotating frame you have to invent a fictitious force, which then explains the circular motion.

#### haushofer

Perception is subjective, and not really relevant here. Objectively, the trajectories in the inertial frame will be such that the real forces suffice to explain them using Newtons 2nd Law.
Well yes, it is relevant, because that's exactly my question. As I said, I understand algebraically what's going on (perform a time-dependent rotation on Newton's second law F=0 and see two extra terms appearing due to the second time derivative). My question was: will an inertial observer also see the hurricane as on that picture, or is the curling solely due to the fact that the earth is rotating and visible for rotating observers only? It's a simple yes or no question.

To put it into math: let's take a rotating disc, rotating with angular velocity $\omega$ around an axis. Let's use fixed coordinate $x^i$ for the inertial observer hanging above the disc, and $x^{'i}$ for an observer standing at the disc rotating along. Let's align $x^3 = x^{'3}$. Now we can write the rotation via an element of $SO(3)$ $R^i{}_j$ as

$$x^{'i} = R^i{}_j(\theta = \omega t) x^j$$

where for concreteness we can rotate along the $x^3=x^{'3}$ axis. Now we stand on the outside of the disc and throw a ball towards the center with initial velocity $v_0$. If we denote the coordinates of the path of this ball with $x^{i}(t)$ for our inertial observer, we have

$$\ddot x^{i} (t) = 0$$

No force is applied to the ball; it's going with constant velocity towards the center of the disc in a straight line. For the rotating observer however, we have ($R_j{}^i$ denotes the inverse of $R^i{}_i$)

$$\ddot x^{'i} (t) = \frac{d^2 }{dt^2} \Big(R_j{}^i x^j (t) \Bigr) = R_j{}^i \ddot{x}^j + 2\dot{R}_j{}^i \dot{x}^j + \ddot{R}_j{}^i x^j = 2\dot{R}_j{}^i \dot{x}^j + \ddot{R}_j{}^i x^j$$

The first term on the very right (with the factor 2) is the Coriolis-force, the second the centrifugal force (these terms can be massaged a bit by using properties of the rotation matrices to put these two inertial forces in their familiar form).

So the message is simple: our inertial observer will see the ball going in a straight line, our rotating observer will see it deflected.

If I carry on this analysis to parcels of air moving from A to B, I'd say we get the same result. No deflection/rotation for the inertial observer.

edit: here, with "straight line" I mean a geodesic, following the curvature of the earth of course, if you want to carry this analogy from the disc to a sphere.

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#### haushofer

Of course not. If it would, it would fly off the Earth.
Yes, that's what I mean with a straight line: a geodesic.

#### PeroK

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My question was: will an inertial observer also see the hurricane as on that picture, or is the curling solely due to the fact that the earth is rotating and visible for rotating observers only? It's a simple yes or no question.
Yes, the hurricane is real.

If we imagine a fixed point on the Earth's surface and a circular air flow about that point (wrt the Earth). The period of rotation might be a few minutes or an hour, say. The inertial observer will see a circular air flow, added to whatever rotational motion the fixed point has.

Perhaps an important point is that you could take this point of view:

Atmospheric air flows are caused by forces acting on a rotating air mass (inertial point of view).

Atmospheric air flows are not caused by the Coriolis force. The Coriolis force explains them in a non-inertial frame; where, among other things, other real forces need to be ignored or cancelled out by other fictitious forces.

#### A.T.

My question was: will an inertial observer also see the hurricane as on that picture,
The shape on a picture is again a different question from whether he will perceive it as "rotating" over time.

If I carry on this analysis to parcels of air moving from A to B, I'd say we get the same result. No deflection/rotation for the inertial observer.
Except the parcels of air have real forces acting on them from the surrounding air.

#### haushofer

A car accelerates towards a tree and crashes into it. In the car's frame, the tree accelerated as the result of a fictitious force.

Your conclusion, therefore, is that if you switch to an inertial frame, the fictitious force on the tree disappears and ... the collision never happens?
Well, no, I'd say the collission is real, but the path of the tree is different from both observers.

Likewise, I understand that the (existence of the) hurricane is real. But following your analogy, the question is how the flow of air inside this hurricane is perceived. On the picture, you see a curly flow, and the question is: does that curly flow result solely from us being rotating? The question is obviously not, as you remarked in your other post: the red spot on Jupiter is also perceived by inertial observers.

So then the conclusion is clear: the rotation of such hurricanes can not be solely due to the Coriolis effect, I'd say.

#### A.T.

On the picture, you see a curly flow,
On a static picture you don't see any flow. What matters are the actual 3D trajectories of the air masses in the inertial frame.

#### Ibix

So then the conclusion is clear: the rotation of such hurricanes can not be solely due to the Coriolis effect, I'd say.
It's due to the Coriolis effect if viewed from a rotating frame. It's due to something else (conservation laws in the cannon ball case, more complex in the hurricane case, and the motion of the surface of the Earth) in the inertial frame. The description is frame-dependant - so "it's due to the Coriolis force" is, indeed, only true in some frames.

#### haushofer

I'll let it sink in this evening, will come back to it tomorrow; my wife is expecting me to make diner. I'm sure I'm missing something quite obvious, but everytime I think I get it, a contradicting picture emerges in my mind, putting me back to the thinkingboard. Maybe the whole "throwing a ball on a disc"-explanation is indeed deceptive if you want to compare it with flows of air masses. Thanks everybody and apologizies for my stupidity.

#### haushofer

It's due to the Coriolis effect if viewed from a rotating frame. It's due to something else (conservation laws in the cannon ball case, more complex in the hurricane case, and the motion of the surface of the Earth) in the inertial frame. The description is frame-dependant - so "it's due to the Coriolis force" is, indeed, only true in some frames.
Yes. What I meant is that in the "throwing a ball on a disc"-example of the video I gave, the curved path of the ball perceived rotating along with the disc becomes a straight path if you go to an inertial observer. Apparently this is not a good analogy for (the airflow of) our hurricane

#### Ibix

Yes. What I meant is that in the "throwing a ball on a disc"-example of the video I gave, the curved path of the ball perceived rotating along with the disc becomes a straight path if you go to an inertial observer. Apparently this is not a good analogy for (the airflow of) our hurricane
Yeah - because the air has interactions that the ball doesn't. Might be interesting to consider a ring of cannons aimed radially inwards, firing shots of non-negligible diameter. Although the East-West shots wouldn't naturally deviate (much) from a straight line, there is a pattern to how their neighbours push them.

"Conceptual question about the Coriolis force and the weather"

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