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The Coriolis Effect and Geostrophic Motion

The coriolis effect isn’t the easiest thing to get your head around, but it is an essential element in understanding the large scale motions of the atmosphere and the oceans.

If you roll a ball along a flat frictionless surface it keeps going in the same direction. This is because objects that have no forces on them continue in the same direction at the same speed. (The combination of direction and speed is known as velocity, which is a vector. A vector consists of a magnitude (e.g. speed) and a direction).

Well, that statement was not strictly true – because it wasn’t specific enough.

If you get onto a merry go round and launch your same ball in one direction you observe it move away in a curved arc. But someone above the merry go round, perhaps someone who had climbed up a pole and was looking down, would observe the ball moving in a straight line.

It’s all about frames of reference.

Now we live on planet that is rotating so we have to consider the “merry go round” effect.

There are two approaches for a mathematical basis (and we will keep the maths separated):

• consider everything from an inertial frame – as if all motion was viewed from space (note 1)
• consider everything from the surface of the planet

If we considered everything from space then the problem would actually be more difficult. On the plus side thrown balls would go in a straight line (as normal). On the minus side the boundaries of the oceans, mountains and everything else important would be constantly on the move and we would need mathematical trickery beyond most people’s comprehension.

So everyone goes for option b – consider motion from the surface of the planet. This means the frame of reference is constantly on the move.

Coriolis

The excellent Atmosphere, Ocean and Climate Dynamics by Marshall & Plumb (2008) comes with a number of accompanying web pages most of which have some videos.

See GFDLab V: Inertial Circles – visualizing the Coriolis force for some detail and the video link, or click on the image below for the video link:

Figure 1 – Click for the video

• the left hand video is the inertial frame of reference – stationary camera
• the right hand video is the rotational frame of reference – the camera is moving with the turntable

This is the best video I have found for making clear what happens in a rotating frame.

With some relatively simple maths, the equations of motion in an inertial frame get transformed into a rotating frame of reference.

Two new terms get introduced:

• the Coriolis acceleration = “stuff appears to veer off to the side as far as I can tell” effect
• centrifugal acceleration = “things get thrown outwards like on a merry-go-round that goes very fast” effect

The centrifugal acceleration is not so significant, just a slight modifier of magnitude and direction to the very strong gravitational effect. But the Coriolis effect is very significant.

Now the Coriolis effect is easy to demonstrate on a rotating table, but we live on a rotating sphere and so there are some complexities that require the use of vector maths to calculate.

Mathematically it is easy to show that the Coriolis effect is modified by a factor relating to latitude. Specifically the effect is multiplied by the sine of the latitude, which means that at the equator the Coriolis effect is zero (sin 0° = 0), and at 30° it is half the maximum (sin 30°=0.5) and at the poles it has the full effect (sin 90° = 1.0).

I found it difficult to come up with a conceptual model which helps readers see why this is so. Readers who have had to think about the effect of resolving forces and rotations into orthogonal directions might be able to provide a conceptual picture – so please add comment if you think so. (Note 2).

Some Maths

The Coriolis effect has to be seen in the light of the other terms in the equation of motion.

The intimidating version, for those not used to the equations of motion for fluids in a Lagrangian formulation (note 3):

Du/Dt + 1/ρ.∇p∇φ + fz x u = Fr …..[1]

where bold characters are vectors, z is the unit vector in the upward direction, u = velocity vector (u,v,w), φ = gravitational potential modified by the centrifugal force, ρ = density, p = pressure and f = Coriolis parameter.

And in not-quite-plain English, the change in velocity with time (following a moving parcel of fluid) plus pressure force plus gravitional force plus the coriolis force equals the frictional force (note that the terms are effectively for unit mass).

The Coriolis parameter:

f = 2Ω sinφ …..[2]

where Ω = the rotational speed of the earth (in radians/sec) = 2 π / (24*3600) = 7.3 x 10-5 /s

And the simpler version in each local x,y,x direction with some simplifications applied (like the hydrostatic equilibrium approximation):

Du/Dt + 1/ρ . ∂p/∂x –  f.v = Fx ….(local x-direction) …[3a]

Dv/Dt + 1/ρ . ∂p/∂y + f.u = Fy ….(local y-direction) …[3b]

1/ρ . ∂p/∂z  + g = 0 ….(local z-direction) …[3c]

Geostrophic Balance and the Magnitude of the Coriolis Effect

Analysis of fluid flows is often carried out via non-dimensional ratios.

The Rossby number is the ratio of acceleration terms to the Coriolis force, and in the atmosphere at mid-latitudes is typically 0.1.

Another way of saying this is that the acceleration terms in equation 3 are a lot smaller than the Coriolis term. And in the free atmosphere (away from the boundary layer with the earth’s surface) the friction terms are negligible. This simplifies equation 3:

ug = – 1/fρ . ∂p/∂y ….[4a]

vg =   1/fρ . ∂p/∂x ….[4b]

With ug, vg defining the solution – geostrophic balance – to these simplified equations. This tells us that the E-W wind speed is proportional to the pressure change in the N-S direction, and the N-S wind speed is proportional to the pressure change in the E-W direction.

From Marshall & Plumb (2008)

Figure 2 – Colored text added

What might be surprising is the instead of the wind flowing from high to low pressure, it flows at right angles – along the lines of constant pressure.

So of course we have to ask whether these simplifications are justified..

Here is a sample of the 500 mbar wind and geopotential height:

From Marshall & Plumb (2008)

Figure 3

We can see that the wind at 5oo mbar (about 5km high) is quite close to geostrophic balance.

By contrast, if we look at surface winds:

From Marshall & Plumb (2008)

Figure 4

Here we see that the wind is flowing more across the pressure field from high to low pressure – this is because of the effect of friction at the surface. The friction term in equation 3 cannot be ignored when we want to calculate the motion near boundary layers.

Conclusion

This is just an interesting part of climate science. The large scale atmospheric and oceanic motion is fascinating and also necessary for understanding the science of climate.

Notes

Note 1: Even watching the planet from space is not an inertial frame of reference as the earth is rotating around the sun, and the sun is rotating around the center of the galaxy, etc, etc.. To avoid this article being a 100 page unfathomable treatise on rederiving the equations of motion, there are necessarily many simplifications, offered without caveat or explanation.

Note 2: The components of the Coriolis force on the surface of a sphere are calculated from Ω x u (where the “x” is the vector cross product, not “times”).

Ωu = (0,  Ωcosφ,  Ωcosφ) x (u,  v,  w)

= (Ωcosφ.w – Ωsinφ.v,   Ωsinφ.u,  -Ωcosφ.u)

w is the vertical component of wind and is generally very small compared with horizontal components. So when at the equator (φ=0°), then:

Ωu = (Ωcosφ.w,   0,  -Ωcosφ.u)

the u-direction (W-E) is very small because w is very small, and the w-direction (vertical) is not important because it competes with the much larger gravity term

Note 3: The term D/Dt has a specific meaning that might be new to many people. This is the Lagrangian differential, which is the change in the property of a fluid following that element of fluid. Rather than the change in property of a fluid at a fixed point in space.

D/Dt ≡ ∂/∂t + u∂/∂x + v∂/∂y + w∂/∂z, where u = (u,v,w) is the velocity vector

26 Responses

1. A good article.

[“Mathematically it is easy to show that the Coriolis effect is modified by a factor relating to latitude…

I found it difficult to come up with a conceptual model which helps readers see why this is so.”]

This may be of help with the ‘conceptual problem’ bit:

Imagine a giant standing with his (or her) left foot on the North Pole and right foot 500 miles South along the Greenwich Meridian. The giant is facing East.
Viewed from space (and ignoring galactic swirl etc), the giant will be twisting anti-clockwise due to the Earth’s rotation (because the right foot is travelling faster than the left foot). Disregard movement in any other plane.

Now place the same giant with his/her feet 250 miles either side of the equator, still facing East. Viewed from space, the giant is not twisting at all. He/she is merely moving in space at the same velocity as any point on the equator.

Conceptually, it may be better to think of coriolis as a twisting motion.

Obviously, if you lift the giant’s feet off the surface at the North pole, he will be twisting clockwise when viewed from the surface.

Hope this helps.

• Arfur: If I understand correctly, the Coriolis effect only applies to objects with a meridional component of motion. Your giants are stationary and presumably therefore not subject to a Coriolis force. Furthermore, you are looking at motion from an inertial frame of reference – space – rather than a non-inertial frame of reference, the earth’s surface.

SOD: I’m under the impression that the Coriolis psuedoforce may be responsible for circular motion around highs and lows and for the jet stream(s). I can see how winds moving meridionally towards or away from a low or high experience a latitudinal force, but I can’t see how those blowing latitudinally acquire experience the (equal) meridional force needed to complete circular motion. On the other hand, I can picture how the poleward component of the Hadley circulation in the midlatitudes produce midlatitude jet stream(s) at high altitudes where friction is low and equatorial trade winds near the surface where friction is high. If you chose to explore these phenomena in depth, I would be interested.

• Frank,

My understanding is that the Coriolis effect is not dependent on direction. Coriolis will affect a artillery shell whichever direction it is fired. I do not believe that there is a direction factored into the equation.

As to my giants, they were purely to help visualise the effect. Yes, the visualisation comes from viewing them from space, in which case they are NOT stationary. That is why I wrote the last sentence about ‘viewing from the surface’. Viewed from the surface when his feet are in the air and he is no longer tied to the surface, the giant will be seen to be twisting clockwise (his right foot is ‘turning’ to the right) due to the observers movement through space. Either way, it helps me to visualise the phenomenon as a twisting motion.

As to winds, again ALL movement of air (as with any other body) is subject to Coriolis, irrespective of direction of pressure gradient. The Pressure Gradient Force initiates the movement and Coriolis (also sometimes called Geostrophic) effect acts on the moving air (body) at right angles until the C(F) balances the PGF and the wind is then at right angles to the initial movement and to the right in the Northern hemisphere (along isobars). In the Northern hemisphere therefore, someone standing with his back to the wind will always have the area of low pressure to his left (Buys Ballot’s Law).

I repeat, the giant analogy was purely an attempt to help anyone visualise the concept.

2. For my mental model of why the Coriolis force is effectively zero at the equator..

the Coriolis force swings objects off to the side, so imagine someone moving from a pole down to the equator.. which, “projected” onto a 2d disk, is further and further out to the edge of the spinning disk.. at the edge of the disk they are actually right over on their side so the Coriolis force is directed outward with respect to the 2d disk, but this is actually upwards or downwards for the person on the real 3d sphere at the equator.

And as the force then competes with gravity it is insignificant.

• Yes, that works for someone moving from North to South but it may be difficult to conceive of the same effect when travelling say from New York to London, whereas a twisting effect helps (personally) my mental picture. But whatever works for different people!

I believe the ‘force’ you are talking about in the vertical plane at the Equator is called the Euler force and, yes, it is insignificant when compared to gravity.

ps, Folk just need to remember that ‘Coriolis force’ is not a force at all but a pseudo force.

3. on January 16, 2012 at 2:14 pm | Reply Alexander Harvey

Hi scienceofdoom,

If you haven’t so far but get the chance, watch the 1958 Bell Labs production “The Unchained Goddess” just to see how they tried to get across the fundamentals of meteorology way back when. Some of it is excruciating cringe-making, but mostly it hits its marks.

It deals with Coriolis Effect and much else. It is only really remembered because of a miunte or two segment dedicated to the effects of CO2 and the warming Earth. This segment has been highlighted as perhaps the first popular attempt at communicating the CO2 issue. Given that the film was made at the end of the IGY (1957-1958) and the start of the Keeling Graph and all that, it may well be.

The whole fim is ~55 minutes and is a bit of a treasure. Given the interest in such matters it seems a little surprising that the complete version does not seem to be easily found on the web, by me at least. It was/is out there somewhere for I have watched it.

Alex

4. SoD – please can you offer some comment on the role of conservation of angular momentum wrt Coriolis? I have always thought that it was the physical manifestation of this principle which causes (in both northern and southern hemispheres) meridional movements of particles towards the poles to accelerate eastwards in order to preserve their angular momentum wrt the axis of rotation?

• curious,

Try this article: http://www.aos.princeton.edu/WWWPUBLIC/gkv/history/Persson98.pdf . If I understand it correctly, conservation of angular momentum explains only half the Coriolis force.

• DeWitt,

Angular momentum and the Coriolis force are two separate entities.

Apply a rotating frame of motion to the equations of motion and we get two new terms:

2Ω x u + Ω x Ω x u

Coriolis acceleration + Centrifugal acceleration

The article is pretty interesting.
This graphic from the article helps me visualize Coriolis force vs latitude a lot better:

• Now I have read your comment again, I realize I mis-read it the first time.

It still seems wrong but I will think some more, as with my comment to curious, the subject is more confusing than it first appears.

5. curious,

In broad brush it is angular momentum and not the Coriolis force that is the cause of the W->E (=westerly) winds in mid-latitude..

However, the mechanisms are complex and inter-related, in a way that I have not untangled in my mind.

Conservation of angular momentum implies that slow moving surface winds will have increasing W->E velocity as the latitude increases – to infinity at 90′ – so clearly there must come a latitude when this process breaks down.

The increase in W->E wind velocity with height is a result of the equator-pole temperature differential – the “thermal wind”.

Movement of air poleward is turned “to the side” by the coriolis force.

More on all of this in the next article.

• SoD, DeWitt – thanks for the follow up. This is a subject that has been on my mind for a while and I have not found a definitive treatment. Personally I have found many of the explanations and descriptions confusing and lacking coherent treatment wrt to the frame of reference used to illustrate their views. I did find a couple of links which helped but I don’t have them on this current machine. FWIW it came back to mind thinking of balancing of rotating machinery issues when discussing Anastassia’s work on “Where do winds come from?” over at tAV. IMO it should be possible to arrive at a definitive view on the angular momentum effects and when I get chance I’ll do some more digging.

• curious,

at the basic level, the W-E winds are easily explained by angular momentum.

At the equator convection carries air up, and then it flows out to the subtropics and down.

W-E wind velocity = u
Angular momentum = L
Angular rotation = Ω
Radius of earth = a
Perpendicular distance from axis of rotation to surface = r
Latitude = φ

So from simple trigonometry, r=a cosφ

Angular momentum of wind, L = Ωr2 + ur (= ang. mom. due to rotation of earth plus contribution from wind)

Suppose that the W-E wind, u=0 at the equator.

So L0 = Ωa2

As the wind moves towards the pole it conserves angular momentum (of course some will be dissipated due to friction with other air masses, but this is the simple version).

So at latitude φ, L=Ωa2cos2φ + ua cosφ = L0 = Ωa2

Solving, this gives u = Ωa sin2φ/cosφ

So at φ=30′, u = 135 m/s (W-E direction)

We observe that the subtropical jet at around 10km has a maximum average of about 30m/s. (Instantaneous maximum is more like 50m/s):

So the “export” of angular momentum easily explains the W->E zonal movement.

6. SoD – thanks for the calculation on January 17, 2012 at 8:54 pm.

I agree with those numbers but I think you have minor glitch in the algebra with “a squared” appearing in your solution instead of “a”?

For info: this morning I found references to some very relevant and interesting papers:

Mintz (1951), Palmen (1951), Palmen and Alaka (1952), Wiin-Nielsen (1967), Starr (1968), Lorenz (1969), Piexoto and Crisi (1965). I’ve not had time to follow up on the sources but Palmen and Alaka looks particularly relevant with an angular momentum budget for latitudes 20-30deg North.

• curious, thanks I fixed the mistake

7. Here is the implied W-E wind speed vs latitude for conservation of angular momentum (from the equation in the earlier comment):

Clearly this is not possible, and not what we see once we get to mid-latitudes. More on this in the next article.

8. The most intuitive explanation for me of Coriolis being stronger at the poles is that Coriolis results from the fact that a parcel of initially S-moving air (say) in the N hemisphere (say) is moving from regions where the ground’s Eastward rotation is slow, to regions where the ground’s Eastward rotation is faster. So the parcel appears to be deflected to the West relative to the ground because – well, because it’s in faster company, and losing ground.

But why is the ground’s Eastward rotation faster as you go South?
Simply because, in the N hemisphere, going further South takes you further from Earth’s spin axis (a line between the poles). You have to go faster to go around a big circle every 24 hours than around a smaller circle every 24 hours.

But why does this explain the effect being stronger near the poles?
Now that we’ve established that Coriolis depends on moving toward or away from the spin axis, let’s go back to our South-traveling air parcel: When it’s near the (North) Pole, travelling 1 mile S takes it nearly 1 mile further from the Earth’s spin axis. But if it’s just N of the Equator, going 1 mile further South barely takes it further from the axis at all (maybe it goes from 4,000 miles to 4000.003 miles from the axis – whatever, it’s a tiny fractioni of a mile). The bigger changes in spin-axis distance near the Pole account for the stronger Coriolis effect there.

Fine point (harking back to high school trig): More generally and more precisely, going 1 mile South in the N hemisphere takes you about (sin L) miles further from the Earth’s rotation axis, where L is your latitude. This is the “sine of the latitude” factor mentioned by scienceofdoom.

9. Conceptualising the Coriolis effect on a North/South heading is simple compared with an East/West heading. This is why I prefer to visualise it as a ‘twisting’ effect (see my first comment). At every point along the East/West trajectory, the off-surface moving object suffers a twisting effect. There is no factor of ‘direction’ written into the Coriolis equation. Essentially, Coriolis is derived from a combination of conservation of angular momentum and gravity.

Just sayin’…

10. […] Comments « The Coriolis Effect and Geostrophic Motion […]

11. […] about angular momentum. I always knew what it was, but in the small discussion that followed The Coriolis Effect and Geostrophic Motion I found myself wondering whether conservation of angular momentum was something independent of, or […]

12. Hi SoD
Don’t know if your up for a side question, but i was looking for information on the coriolis effect in seawater when i came across your blog [added to favs]. What i specifically wanted to understand was, if you have a current moving north [say from 70-75degN] constrained from moving east how does that energy get expressed, and if in temperature how much +C per m3 or km3.
john

13. […] In The Coriolis Effect and Geostrophic Motion we saw that on a rotating planet winds get deflected off to the side  (from the point of view of someone on the rotating planet). This means that winds flowing from the tropics to the north pole will get deflected “to the right”. […]

14. what about the winds which starts to move from high to low from west to east purely as they are moving in direction of the rotation of earth, will it be effected by geostrophic force?

15. Alexander Harvey,

Google “the unchanged goddess you tube”

55 minutes of Frank Copra

I watched this as young man of 9 or 10 years old in ’58

AL

16. […] is a bit of a problem in practice. The Navier-Stokes equations in a rotating frame can be seen in The Coriolis Effect and Geostrophic Motion under “Some […]

17. Reblogged this on Hypergeometric and commented:
Compact explanation of Coriolis forces and rotating frames. Comments are well-done, too. What is not indicated or conveyed is why geostrophic flow causes fluid flow in rotating frames to have structure rather than diffuse uniformly. And, ironically, this “sorting out” produces “emergent structure” where students might expect there to be more chaos.