# Inertial Oscillations

## What is an inertial oscillation?

Imagine you are standing the middle of a very large frozen lake in Northern Minnesota with a hockey puck and a hockey stick. You give the hockey puck a good whack and watch it go sailing off in the direction you hit it. If the lake were big enough, and perfectly frictionless, what would happen to the hockey puck? Would it keep going in a straight line?

As it turns out, because of the Coriolis effect, the hockey puck would turn to the right and eventually loop back to nearly the same spot you were standing when you gave it that initial hit (it would actually end up slightly west of where you were standing). The time it takes for the hockey puck to return can be computed with the Coriolis frequency and corresponds to a period of about 17 hours in Northern Minnesota. This motion is what we call an inertial oscillation.

More formally I would define an inertial oscillation like this:

An inertial oscillation is the motion of a frictionless point mass particle constrained to the surface of the Earth.

This problem is intimately related to geodesics, the general notion of what defines a straight line on a particular geometry.

## Why are they called inertial oscillations?

Given that the hockey puck returns to nearly the same spot and loops around indefinitely, the ‘oscillation‘ part of the name should be clear. The ‘inertial‘ part, however, isn’t necessarily quite as obvious.

## Do inertial oscillations exist in nature?

The short answer: no. The longer answer is yes, but not in isolation. Below is the path of a drifter that was…. to be continued.

## Exact Solution

The exact solutions for the path of an inertial oscillation on the Earth can be computed in closed form with Jacobi elliptic functions. The following matlab script can be used to compute their path.

### Earth

ExactInertialFrequency.m — computes the exact frequency for an inertial oscillation on the Earth.

ExactInertialSolution.m — computes the exact path, velocity, and inertial frequency for an inertial oscillation on the Earth.

MakeRotatingEarthFigures.m — creates the sample figures used in the manuscript.

### Sphere

ExactInertialSphereFrequency.m — computes the exact frequency for an inertial oscillation on a sphere

ExactInertialSphereSolution.m — computes the exact path, velocity, and inertial frequency for an inertial oscillation on a sphere.

MakeRotatingSphereFigures.m — creates the sample figures used in the manuscript.

### Orbit

ExactOrbitSolution.m — returns the time, x, y, and z position of a particle in orbit around the earth as observed in the rotating frame.

MakeRotatingOrbitFigures.m — creates the sample figures used in the manuscript.

### Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. OCE-1031286. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.