On a non-rotating earth: In the absence of friction or any other force, a current of water set into motion in the ocean on a horizontal pressure surface (i.e., no pressure gradient force) would simply continue forever in the same direction at the same speed (i.e., at constant velocity). On our (rotating) earth: In the same situation, that water will feel the continual pull of the Coriolis force at right angles to the velocity, and will begin to move in a circle. This is called inertial motion. Inertia means an object in motion to remain in motion, while an object at rest will remain at rest unless acted on by an external force. It's a fundamental property of mass. The motion we are describing here is not truly inertial, of course, because the earth (and all objects on it) has an angular acceleration and the earth's gravity field is acting on the water. But in our relative coordinate frame, spinning along on the earth, this is as inertial as it gets. Because the path of an inertial current repeats itself with time, inertial motions are often referred to as inertial oscillations. The acceleration of "Inertial motion" in the ocean can be described by the following equations:

Where f is the Coriolis parameter, u is the eastward component of velocity and v is the northward component. u at any given time t or u(t) is given by u(t-1) + du (the velocity just a moment before plus an incremental change caused by the Coriolis force acting at right angles). These equations can be solved to find that:

Where V is the initial speed (remember that speed is a scalar and V= sqrt(u^2+v^2)). The period of the motion (length of time until it repeats itself) is determined by the latitude and is given by T = 2p/f. The radius of the circle is r = V/f . (note: for browsers that don't handle symbols well, p = 3.14159). Remember that the Coriolis parameter, f, varies with the sine of latitude and is equal to 2*Omega sin(latitude).

This balance of forces can be an important one for oceanic features on the scale of  10 km  to 100 km and a few days' time, for example around features like fronts. This balance might be combined with motion due to the tide to produce velocities that have characteristics of inertial motion but which may include other components as well.

For example, if the wind blows along the continental shelf-slope front, then dies away,  inertial motion will be observed and might contribute to strong currents or meandering along the front. The strength of maximum inertial currents in a situation like this can be three to four times the average frontal velocities, and the amount of energy in inertial motions is equivalent to the amount in the semi-diurnal tide +(Mooers, Flagg and Boicourt, 1978). And currents, as we know, produce mixing and nutrient transport and all sorts of good things like that.

1. Given these equations, please plot the velocity of an inertial current on the continental shelf at a latitude 42N, initially moving northeastward at 50cm/s.
2. What is the radius r and period T of its inertial motion?
3. What would r and T be if the initial speed was only 20cm/s?
4. What would be different if the initial velocity was 50cm/s to the west? Quantify your answer.
5. What would be different if this motion happened at 30S instead of 42N? Quantify your answer.