Explaining coriolis when the relevant pressure systems are at the same latitude

[hugh flung_dung]

I've been asked (another) awkward question during a groundschool session. I started to answer but then confessed that I was in danger of waffling so would stop and report back later with the answer. Fundamentally the question was that the Coriolis effect/force is easy to understand when the H/L pressure systems are at different latitudes, but what happens when they are on the same latitude?

HFD

[Heston]

Oh dear. I think you've been caught out by the simplified explanation of Coriolis which works as you say, but isn't really an accurate explanation. Read the Wikipedia entry - Coriolis force appears because a body on the surface of the earth is being accelerated by gravity into the circular path that the earth's surface is following (without gravity it would fly off in a straight line into space - Newton's laws, inertia and all that) and when itmoves with respect to the rotating earth it appears to have an additional force acting on it - the Coriolis force.

Lattitude doesn't really come into it.

It really is a swine to understand without the maths...

H

[BillieBob]

Try this - the best I can do off the top of my head and without the aid of a whiteboard:

Consider two circular portions of the earth's surface, one centred at the pole and the other centred at the equator. The circle at the pole, when viewed from above in free space will be rotating (at about 15 degrees per hour) whilst the circle at the equator will have no rotation. A similar circle anywhere on the earth's surface away from the equator will exhibit a degree of rotation (proportional to the cosine of the latitude).

Now, apply your normal explanation for coriolis, starting from the centre of an imaginary circle (which you have established exhibits a degree of rotation) and you will find that it works equally well irrespective of the geographic direction of the pressure gradient.

A mathematical solution, whilst undoubtedly more accurate, is neither necessary nor desirable for the average student.

[hugh flung_dung]

Heston: guilty as charged! However, I must be being dense because if the particle is moving along the plane of rotation (along a parallel of latitude) I can't see how gravity would cause it to appear to change direction.
The wikipedia page is interesting (especially the Eötvös effect, which I hadn't heard of) but sadly the days when I could intuitively understand matrix determinants are long gone.

BillieBob: that's pretty good but I think the average bod would struggle with the idea that the locus of any point on the surface has a degree of circularity - in fact I'm struggling a little. Do you have a convincing explanation, or just one based on the limits?

HFD

[rsuggitt]

Easy. It's all to do with the velocity of a parcel of air in the direction of a line of latitude.

If the parcel of air has come from the equator, said velocity will be high. If it's come from the poles, said velocity will be low.

The apparant track of each parcel of air over the surface of the earth depends on the velocity of the parcel of air compared with the velocity of the surface of the earth where the air has ended up.

[Black Jake]

How about this:

Air in motion in the atmosphere will try to go the shortest way from high to low pressure. Thus it will follow a great circle track across the Earth. With the exception of the equator, all great circles have a North/South component. So air moving from e.g. 50N60W to 50N20W will not be moving due East, but will have a bit of East and a bit of North. Thus the argument that Coriolis "force" is a phenomenon resulting from air moving North or South over a rotating Earth (Coreolis being proportional to V Sin latitude) still applies.

(it might be bull**it, but it works for me)

BJ

PS, I hope in true instructor style you said something like, "excellent question, very well put. Lets have a coffee break and we'll continue this discussion later."

[Heston]

Going back to HFD's original question - the key is that even when the motion is between points at the same lattitude this is still motion on the plane from those points perpendicular to the axis of rotation of the earth. Imagine the earth as consisting of many thin discs - each one is like a rotating turntable. Its motion on this plane that results in the coriolis force.

The coriolis force on moving air masses is zero at the equator because the force acts in the plane of the disc: at the equator this is vertical with respect to the earth's surface.

Explanations that talk about simply the tangential velocity being different at different lattitudes (as in most PPL text books) are not complete enough to answer HFD's student's question.

my brain hurts now...

H

Edit to add: agree that the maths isn't appropriate for students (or me for that matter...)

[BillieBob]

Do you have a convincing explanation, or just one based on the limits?Well, the one I gave has convinced every student I've ever used it on, although it's more convincing when backed up by the proper diagrams.

I am a great believer in the KISS principle and avoid getting into mathematical explanations whenever possible, particularly when the student may not even have passed maths at GCSE. Clearly, that is not always possible but in this case, it is.

As for terms like circularity and locussszzzzzzzzzzz!

[rsuggitt]

The coriolis effect is dependent on the latitude an airmass has come from, not where the air mass is now.

[Pull what]

Gosh you guys really know how to complicate something really simple.

Watch this video it may help you

The Coriolis Force - YouTube

[xrayalpha]

Great video, just shows it.

Saw a TV documentary about weather where top weather person - female - collapsed into giggles when asked to explain Coriolis. Colleague stuck his head round the door, and heard the word Coriolis and sort of muttered; over my pay grade!

Basically, they accepted it happens and didn't worry too much about being able to explian (understand?) why. And they were weather forecasters!

What real relevance is it to us? Are we training pilots or weather forecasters?

Because if pilots, why?

If to help us understand the weather - rather than to test ability to absorb and regurgitate difficult concepts, or test our ability to answer tricky questions - then why teach stuff that weather forecasters don't seem too fussed by!

[hugh flung_dung]

Wow, thanks everyone.

BJ: that's nice and simple and unless someone points out a flaw it might well become part of my explanation, but clearly it's not the whole story

BillieBob: is there any chance you could pop a sketch on here because I still can't get my head around what it might look like (KISS is my middle name when teaching;), but not when trying to understand)

Heston: so are you talking about the angle between the plane of rotation and the Earth's surface? I hadn't thought of that, but it might be tricky to explain.

rsuggitt: could you explain your point a bit more? (we all had to learn the formula during CPL/ATPLs, the question now is for a PPL and is about "why")

Everyone else: thanks but we all know that it happens - the question was about a simple *explanation* for a PPL student that is better than the overly-simple one I've used until now

HFD

[Heston]

Yes part of the problem with the childrens' roundabout explanation - which is correct of course as far as it goes - is that the earth's surface is not the plane of rotation (whereas the childrens' roundabout is the plane of rotation). The coriolis force appears in the plane of rotation - so the resultant force parallel to the earth's surface will be different at different lattitrudes. I suppose it is analogous to the variation in magnetic dip at different lattitudes - well sort of. Anyway visualising it is a 3D problem, not a 2D one.

H

[Mr Optimistic]

Coriolis Forces (http://www.animations.physics.unsw.edu.au/jw/coriolis.html)

This only works if you take your feet off the ground.

Never that comfortable with coriolis myself (should have tried harder). Think Heston has it. Presume you imagine that you are happily standing at some latitude and the earth is wizzing you round in a small circle. What fun. However here at latitude 54 degrees north my vertical is making an angle of 36 degrees with the polar axis. My reference is a plane I regard as horizontal, and I could draw north and east directions on it. If I look at the earth's angular velocity vector in relation to this, this vector is aligned with my north mark but stands at an angle of 54 degrees off the plane -tilted up to me. The component of that vector lying in the plane pointing north sure enough takes my feet round to the my east. However the component wich points up through my vertical is going to rotate me clockwise looking up. So if I was initially looking east my sightline would move left (northwards). An object moving east would appear to me to be bent to the right.

Wish I hadn't started this.

Not sure if this is correct, but it may buy time to make your escape.

[24Carrot]

I think if I were trying to explain this, I would not spend much time on the Coriolis Effect, I would concentrate on what is actually happening in the atmosphere.

Also I think people understand Centrifugal Force better the Coriolis Effect, so I would try to base my argument on that.

I would start with a parcel of air, e.g. an imaginary balloon.
Air may not weigh much, but gravity does act upon the balloon, and yet it does not fall.
This is an opportunity to explain that the air pressure pressing on the bottom of the balloon is greater than the pressure on the top, and the pressure difference provides a force which balances gravity.
Digressions into altimetry, beer bubbles, and why hot air rises are in order here.

Next I would talk about Centrifugal Force, conkers on strings, etc.
In particular, the CF gets stronger as the rotation gets faster.

Locating our imaginary balloon over London, ideally with a globe to spin, or at least a diagram, I would explain that CF acts on the balloon as well as gravity.
The balloon is not moving, so there must be yet another pressure difference balancing it.
It can also be noted that the CF acts Southwards and upwards.
Digressions about the atmosphere being thicker at the Equator are in order here.

If our balloon is in some "stationary" air, i.e. there is zero wind over London, its true rotational speed is due to the earth's rotation, and the consequent CF must exactly balance a pressure gradient.

Now we are set up to explain the "same latitude Coriolis Effect".

If our balloon is in some air which is moving Eastwards i.e. a West wind, its actual rotational speed is a little bigger than the average air for its latitude.
So the CF is a little bigger, and so this air parcel will move South and up compared to the average air around it, and the Earth below.

Conversely, an East wind will lower the CF, and the air parcel will move North and down.

There is not much scope for vertical air movement, and so the main apparent effect is for moving air to "turn right" in the Northern Hemisphere.

If you work through the maths on this, it produces the whole Coriolis Effect.


MATHS:

Briefly, the CF (more precisely, the centripetal acceleration) is:
V^2 / [ R cos(Lat) ] where:
Lat is the latitude, v is the surface's rotational speed at that latitude, and R is the earth's radius.

If the rotational speed 'v' is increased by a wind speed 'u', the difference is:
(v+u)^2 - v^2 = 2vu + u^2 (but u^2 is small enough to ignore).

So the increase in CF is: 2vu / [ R cos(Lat) ]
but v / [ R cos(Lat) ] is Rot, the Earth's rate of rotation,
so we have: 2 u Rot

Finally, noting that the component along the earth's surface is sin(Lat)
we get the full Coriolis Effect: 2 u Rot sin(Lat).


FOOTNOTE:

Incidentally, I believe the common "Easy Explanation" for air moving North only describes half the Coriolis effect.

Imagine you are in a balloon in some air moving North, clutching a DI.
You would see your track move increasingly East, according to the DI, because of the "wind shear" with latitude.
(Or conservation of angular momentum, if you prefer spinning ice skaters).
This is half the Coriolis Effect.

You would also see the Geographic North Pole move increasingly west according to the DI.
This is the other half of the Coriolis effect.

[Craggenmore]

Haha. When I used to instruct I would say, "Good question Bloggs, research it and tell me all about it all when you're next in."

They never forgot..!

V^2 / [ R cos(Lat) ] where:
Lat is the latitude, v is the surface's rotational speed at that latitude, and R is the earth's radius.

If the rotational speed 'v' is increased by a wind speed 'u', the difference is:
(v+u)^2 - v^2 = 2vu + u^2 (but u^2 is small enough to ignore).

So the increase in CF is: 2vu / [ R cos(Lat) ]
but v / [ R cos(Lat) ] is Rot, the Earth's rate of rotation,
so we have: 2 u Rot

Finally, noting that the component along the earth's surface is sin(Lat)
we get the full Coriolis Effect: 2 u Rot sin(Lat).


An excellent explanation - almost all my students had this equation easily to hand on short finals ;)

[24Carrot]

An excellent explanation - almost all my students had this equation easily to hand on short finals

OK, I included the maths to reconcile my words with the required result - it wasn't part of the explanation!

[gfunc]

I hope you'll excuse my intrusion, I'm not a flying instructor, but I am a research meteorologist.

The key to understanding the Coriolis effect (I avoid the term force deliberately) is to recognise that it is essentially a correction for our frame of reference. If you were perfectly stationary high above the Earth in space (i.e. standing off the roundabout in the video), all air parcels would travel in straight lines in the direction of the force applied.

However, since we are looking at everything from the Earth's surface, which is a rotating frame of reference the air parcels appear to peel off left or right (depending on your hemisphere) as if a mystical force were acting. It's simply because we (on the ground) are moving relative to free space - essentially sitting on some shifting goal posts.

I think that lots of confusion arises from the name 'force' as it implies something is doing work to force the air parcels to curve, hence I favour Coriolis 'effect' or 'correction'.

Hope this helps!

Gareth.

[oggers]

...it doesn't matter if low B happens to be at the same latitude as high A because from the centre of high A there is actually lower pressure in every direction. Therefore the wind swirls out in every direction from the centre and not simply towards the nearest identifiable centre of low pressure.

Notwithstanding valid explanations of the coriolis effect.