Vortexes are common in our atmosphere.
Hurricane, or typhoon, is one type of vortexes found in our atmosphere.
A non-rotating frame is not rotating with
respect to an observer who is at rest.
A rotating frame is rotating with a
constant angular velocity
and the observer follows this rotation.
We are all standing on a rotating frame,
which is the Earth.
Balance of different forces produces vortexes in a rotating frame.
Two important forces involved here
are Coriolis Force and Centrifugal Force.
As the observer is rotating with the frame,
a moving objects so have curved motion due to the constant angular velocity
of the frame the observer is standing on.
This is called Coriolis Force.
Also, when we are on a rotating frame, we don't experience centripetal force.
Instead, we experience a force in opposite direction.
This apparent force is called centrifugal force.
Conservation of angular momentum makes the tangential velocity much larger
at the centre of the vortex then far away
from the vortex.
We are going to illustrate the balance of forces in a vortex in both rotating frame and non-rotating frame.
First, prepare a cylindrical plastic buckets of about 30 centimetres  in diameter
and 50 centimetres high with a one centimetre hole in the centre.
Plug the hole with a stopper not too tightly,
so that it can be released easily later.
Then place three upside down 100
millilitre beakers
in a square plastic container on a rotating table.
Put the bucket on top of the beakers.
Reposition the bucket so that the hole in the bucket and the centre of the square tank
are coincident with the axis of the rotation.
After that, fill the bucket with water until it is seven over tenth full.
Dyke to water to produce a better visualisation of the experiment.
Set the table into rotation with a rate of about 10 revolutions per minute.
Wait until the system reaches solid body rotation.
After the system reaches solid body rotation,
add several white paper dots at different radii from the centre.
Unplug the hole carefully. Make sure to reduce unnecessary disturbance to the system itself.
We can observe that the paper dots swirl around many more times
before exiting from the hole at the center
Paper dots far away from the centre move much more slowly
than the paper dots near the centre.
Now, repeat the experiment with no rotation.
We can observe that the paper dots move directly to the centre of the bucket with no swirling motion.
Compare to two situations.
From the above experiment, we can see that the trajectories of the paper dots are different,
when the bucket is rotating and not rotating.
It means that the flowing direction of the air parcels are different in a rotating frame and a non-rotating frame.
Let's consider the bucket as the lower atmosphere and the paper dots as air parcels.
When there is no rotation, pressure gradient force causes the air parcels
moved towards to the centre in a radial direction.
However, in a rotating frame, such as our earth,
pressure gradient force balances with centrifugal force and Coriolis force.
Rossby number describes the ratio of centrifugal acceleration to the Coriolis acceleration.
At the location far away from the centre of the vortex, Coriolis force balances with the pressure gradient force.
This is called geostrophic balance.
The value of the Rossby number is much smaller than 1.
Now, we look at location near the centre of the tank.
Due to the conservation of angular momentum,
the tangential velocity of the air parcel is much larger.
The angular momentum is proportional to the product of radius and tangential speed.
Since it is much closer to the centre, which means the radius is smaller,
the speed would increase correspondingly so that angular momentum would stay the same.
That's why the tangential velocity of the air parcel is much larger near to the centre of the vortex.
The pressure gradient force balances with the very last centrifugal force
due to the large tangential velocity.
This is called cyclostrophic balance.
The value of the Rossby number is much larger than 1.
And upon the moderate distance from the centre,
the contribution of Centrifugal force and Coriolis force are similar.
Both of them together balance with the pressure gradient force.
This is called Gradient Wind Balance.
The value of the Rossby number is approximately 1.
Besides that, in all of these cases,
it's more frictional force bends the direction of the movement of the air parcel,
just forming a swirling motion.
In conclusion, when the air parcel is closer to the centre of the vortex,
the tangential velocity is larger.
The Rossby speed number is also larger when air parcel is closer to the centre.
Pressure gradient force balances with the Coriolis force away from the centre
and the centrifugal force near the centre.
Our experiment demonstrates the phenomena in the northern hemisphere.
Can you figure out the situation in the south and hemisphere?
