For a projectile going through thin air over some spot on the Earth, there is essentially no Coriolis force. The projectile passes over a particular spot on the Earth at t=0. A coordinate system, whose origin is at that particular spot, is locked to the Earth surface with its x-y plane tangent to the surface. This coordinate system rotates about its z-axis through an angle

t omega sin(latitude)

in a small time interval t. Omega is the angular rotation rate of the Earth about its polar axis. The projectile is moving at velocity v, so its lateral displacement from the point where it would have been in a non-rotating coordinate system is

(v t) [t omega sin(lat)].

Differentiating this offset with respect to time gives

2 v t omega sin(lat)

as the velocity transverse to the projectile's path in the rotating coordinate system. Calling 2 omega sin(lat) the Coriolis parameter f, the transverse velocity is

f v t

and the transverse acceleration in the rotating coordinate system is f v. This apparent acceleration is the result of a coordinate transformation, and is not the result of any physical force acting on the projectile.

The situation is different in the case of a surface current of some depth passing through an ocean basin. The basin water is rotating parallel to the plane of the surface at the angular rate of

omega sin(lat).

There is an interference between the rotating basin water and the water of the stream moving through it. In the northern hemisphere the basin water piles up on the right-hand side of the current, so the surface of the stream is tilted. The same Coriolis parameter f is involved here, and we have the geostrophic equilibrium relation

f v = g (transverse surface slope of the stream)

where v is the stream velocity and g is the acceleration of gravity. The above relation holds where the flow is in geostrophic equilibrium. That is, where an actual Coriolis force is balanced by the component of the gravitational acceleration g which acts parallel to the surface. Where the stream flow is in geostrophic equilibrium, the velocity is parallel to the sea surface height contours that are derived by processing satellite-borne radar altimeter data.

Geostrophic equilibrium is not maintained where the stream is guided by the bathymetry of the ocean basin. For example, the Loop Current is guided out of the Gulf of Mexico by the Florida Straits. Here, the steep coast exerts forces on the flowing water, and the sea surface height contours often intersect the Cuba shore line. The flow is essentially parallel to the shore, so it must cross the sea surface height contours that intersect the shore, and is not in geostrophic equilibrium in these places.

So in the case of a projectile passing through thin air we have a Coriolis effect, and in the case of an ocean current we can have a real Coriolis force.

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