Coriolis frequency explained
.
The rotation rate of the Earth (Ω = 7.2921 × 10-5 rad/s) can be calculated as 2π / T radians per second, where T is the rotation period of the Earth which is one sidereal day (23 h 56 min 4.1 s).[2] In the midlatitudes, the typical value for
is about 10
-4 rad/s.
Inertial oscillations on the surface of the Earth have this
frequency. These
oscillations are the result of the
Coriolis effect.
Explanation
Consider a body (for example a fixed volume of atmosphere) moving along at a given latitude
at velocity
in the Earth's rotating reference frame. In the local reference frame of the body, the vertical direction is parallel to the radial vector pointing from the center of the Earth to the location of the body and the horizontal direction is perpendicular to this vertical direction and in the
meridional direction. The Coriolis force (proportional to
), however, is perpendicular to the plane containing both the earth's angular velocity vector
(where
|\boldsymbol{\Omega}|=\Omega
) and the body's own velocity in the rotating reference frame
. Thus, the Coriolis force is always at an angle
with the local vertical direction. The local horizontal direction of the Coriolis force is thus
. This force acts to move the body along
longitudes or in the meridional directions.
Equilibrium
Suppose the body is moving with a velocity
such that the centripetal and Coriolis (due to
) forces on it are balanced. This gives
v2/r=2(\Omega\sin\varphi)v
where
is the radius of curvature of the path of object (defined by
). Replacing
, where
is the magnitude of the spin rate of the Earth, to obtain
f=\omega=2\Omega\sin\varphi.
Thus the Coriolis parameter,
, is the angular velocity or frequency required to maintain a body at a fixed circle of latitude or zonal region. If the Coriolis parameter is large, the effect of the Earth's rotation on the body is significant since it will need a larger angular frequency to stay in equilibrium with the Coriolis forces. Alternatively, if the Coriolis parameter is small, the effect of the Earth's rotation is small since only a small fraction of the centripetal force on the body is canceled by the Coriolis force. Thus the magnitude of
strongly affects the relevant dynamics contributing to the body's motion. These considerations are captured in the nondimensionalized
Rossby number.
Rossby parameter
In stability calculations, the rate of change of
along the meridional direction becomes significant. This is called the
Rossby parameter and is usually denoted
where
is the in the local direction of increasing meridian. This parameter becomes important, for example, in calculations involving
Rossby waves.
See also
Notes and References
- Book: Vallis, Geoffrey K.. Atmospheric and oceanic fluid dynamics : fundamentals and large-scale circulation. 2006. Cambridge University Press. Cambridge. 978-0-521-84969-2. Reprint..
- Book: Goldstein, Herbert . Charles P. Poole . John L. Safko . Classical Mechanics . Addison Wesley . 1980 . 2nd . 0-201-02918-9. 178. Herbert Goldstein . Classical Mechanics (Goldstein book) .
