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.
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).
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 …..
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 …..
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.
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:
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:
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.
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.
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”).
Ω x u = (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:
Ω x u = (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