The Coriolis effect is a phenomenon that is observed by

observers in rotatating

frames of reference.

Let S be an inertial frame of reference, and S' a frame that is rotating with respect to S. Let **r** denote a position vector. This will be the same in both frames. Further, let (**r**')_{S}, (**r**')_{S'}, (**r**'')_{S}, (**r**'')_{S'} denote the velocity and acceleration of **r** as measured in S and S' respectively.

As a consequence of Euler's theorem there is an angular velocity **w** such that (**r**')_{S} = (**r**')_{S'} + **w** x **r**, where x denotes a vector product. Using this no less than three times we get

(**r**'')_{S} =
((**r**')_{S}')_{S} =
((**r**')_{S}')_{S'} + **w** x (**r**')_{S} =
((**r**')_{S'} + **w** x **r**)')_{S'} + **w** x ((**r**')_{S'} + **w** x **r**) =
(**r**'')_{S'} + 2**w** x (**r**')_{S'} + **w** x (**w** x **r**)

(where we have assumed **w** constant, since this holds in most of the applications we are interested in). If **r** is the position vector of particle of mass m, acted upon by a force **F**, then

**F** = m(**r**'')_{S} =
m(**r**'')_{S'} + 2m**w** x (**r**')_{S'} + m**w** x (**w** x **r**)

which can be rewritten as

m(**r**'')_{S'} =
**F** - 2m**w** x (**r**')_{S'} - m**w** x (**w** x **r**)

Thus to an observer in S'' there appear two pseudo forces: the centrifugal force -m**w** x (**w** x **r**) and the Coriolis force -2m**w** x (**r**')_{S'}

With all the cross products in can be a bit difficult to interpret those expressions. The result is that if **r**_{w}, is the component of **r** perpendicular to **w**, and **r**, **w** have magnitudes r, w then the centrifugal force is mw^{2}r_{w} directed outwards from the axis of rotation, and the Coriolis force is 2mwr', directed at a 90° angle to the direction of motion.

Earth itself is a rotating frame of reference, with an angular velocity w = 2π/24h. The centrifugal force is not particularly interesting, since it is practically constant and is 'absorbed' by gravitation. The Coriolis effect is noticable though.

Suppose an object is moving at speed v on the surface of the Earth on the northern hemisphere at latitude L. Then component of the coriolis force acting on the object along the surface of the Earth is 2mwv*sin L, and it is directed to the right with respect to the direction of motion. On the southern hemisphere sin L is negative, so the object is deflected to the left instead.

On the equator sin L = 0, so an object moving along the surface of the Earth there does not experience any Coriolis force. This is not the same thing as saying that the Coriolis effect vanishes there; a falling onject is deflected to east by a force 2mwv*cos L, and this effect is in fact greatest at the equator!

The Coriolis effect is particularly noticable in eg the weather. If there is an area of low pressure air moves in towards the centre. The motion is deflected to the right (on the northern hemisphere), and the result is that the air moves counterclockwise around the origin. If there is a high pressure the effect is reversed.

The impact of the Coriolis effect on a fluid flow depends on the dimensionless Rossby number R_{0} = U/fL, where U is the velocity scale of the system, L is the length scale and f is the Coriolis parameter. Coriolis effects are significant for R_{0} << 1 and insignificant for R_{0} >> 1. The reason why the weather but not your sink is affected by the Coriolis effect is therefore not that the sink is smaller, but that the ratio of length to velocity is smaller in the sink.