# Periodic Motions and the Definition of Scales

In physics, angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital frequency, and radian frequency) is a scalar measure of rotation rate. Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity. The term angular frequency vector $\vec{\omega}$ is sometimes used as a synonym for the vector quantity angular velocity.[1]

Since one revolution (or oscillation) is equal to 2π radians, then the the angular speed ω can be expressed as the function of the time it takes the system to undergo one cycle.

$\omega = {{2 \pi} \over T} = {2 \pi f} = \frac {|v|} {|r|} ,$

where

ω is the angular frequency or angular speed (measured in radians per second),
T is the period (measured in seconds), and is the reciprocal fo the frequency (measured in oscillations per second).
f is the ordinary frequency (measured in hertz) (sometimes symbolised with ν),
v is the tangential velocity of a point about the axis of rotation (measured in meters per second),
r is the radius of rotation (measured in meters).

If is often useful to consider the time it takes for an oscillating system to make one complete revolution or to pass through one wavelength, or to return to its starting point (for a pendulum or spring) for various physical systems whose oscillations have a characteristic frequency.. For example, in a stable atmosphere the oscillation period of an air parcel about its initial location will be related to the Brunt-Vaisalla Frequency,

which relates to the static stability parameter:

The Brunt-Vaisalla Frequency merely states the obvious--for a stable parcel, the greater the vertical displacement relative to its initial position and more stable the parcel, the more quickly the air parcel will "attempt" to return to its initial position. By its own momentum it will overshoot that initial position and then will oscillate around it.

For synoptic scale weather systems, the period of oscillation relates to the Coriolis parameter. The determination of whether the characteristic period relates to one or the other can be made on the basis of the Rossby Radius of Deformation.

The Rossby Radius of Deformation can be used as a scaling factor to determine what simplifications can be made to the governing equations of motion. For conditions in the middle latitutdes, the Rossby Radius of Deformation is roughly 1000 km or so...meaning that physical circulation systems need to be at least this size for the geostrophic approximation to be valid and the period of oscillation relates to Coriolis accelerations. For physical systems with a radius less than 1000 km or so, then Coriolis accelerations are relatively unimportant and other factors, such as the oscillations around a vertical plane in a statically stable atmosphere are most important.

Solving these equations for the period of synoptic scale systems yields a period of roughly 1 day or more for synoptic scale systems (meaning the time it takes an air parcel to pass through the system) and 10 minutes or so for microscale systems. Thus, the mesoscale can be considered to consist of atmospheric motions whose oscillations can be characterized between these liimits.