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The Coriolis effect is the apparent deflection of moving objects from a straight path when they are viewed from a Rotating Frame Of Reference . The effect is named after Gaspard-Gustave Coriolis , a French scientist who described it in 1835, though the mathematics appeared in the Tidal Equations of Pierre-Simon Laplace in 1778. One of the most notable examples is the deflection of winds moving along the surface of the Earth to the right of the direction of travel in the Northern Hemisphere and to the left of the direction of travel in the Southern Hemisphere . This effect is caused by the rotation of the Earth and is responsible for the direction of the rotation of large Cyclones : winds around the center of a cyclone rotate counterclockwise on the northern hemisphere and clockwise on the southern hemisphere.

The Coriolis effect is caused by the ''Coriolis force'', which appears in the Equation Of Motion in a rotating frame of reference. Sometimes this force is called a Fictitious Force (or ''pseudo force''), because it does not appear when the motion is expressed in an Inertial Frame Of Reference . In such a frame, the motion is explained by the real impressed forces, together with Inertia . In a rotating frame, the Coriolis and Centrifugal forces are needed in the equation to correctly describe the motion.

Contrary to popular belief, the Coriolis effect is not the determining factor in the rotation of water in toilets or bathtubs (see the Draining Bathtubs And Toilets section below).


FORMULA


In non-vector terms: at a given rate of rotation of the observer, the magnitude of the Coriolis acceleration of the object is proportional to the velocity of the object and also to the sine of the angle between the direction of movement of the object and the axis of rotation.

The vector formula for both the magnitude and direction the Coriolis acceleration is

: ec a_C = -2 \, ec \omega imes ec v

where (here and below) ec v is the velocity of the particle in the rotating system, and ec \omega is the angular velocity vector (which has magnitude equal to the rotation rate and is directed along the axis of rotation) of the rotating system. The equation may be multiplied by the mass of the relevant object to produce the Coriolis force:

: ec F_C = -2 \, m \, ec \omega imes ec v.

See Fictitious Force for a derivation.

The × symbols represent Cross Product s. (The cross product does not Commute : changing the order of the vectors changes the sign of the product.)

The ''Coriolis effect'' is the behavior added by the ''Coriolis acceleration''. The formula implies that the Coriolis acceleration is perpendicular both to the direction of the velocity of the moving mass and to the rotation axis. So in particular:
  • if the velocity is parallel to the rotation axis, the Coriolis acceleration is zero

  • if the velocity is straight inward to the axis, the acceleration is in the direction of local rotation

  • if the velocity is straight outward from the axis, the acceleration is against the direction of local rotation

  • if the velocity is in the direction of local rotation, the acceleration is outward from the axis

  • if the velocity is against the direction of local rotation, the acceleration is inward to the axis


For example, consider a location with latitude arphi on sphere that is rotating around the north-south axis. A local coordinate system is set up with the x axis horizontally due east, the y axis horizontally due north and the z axis vertically upwards. The rotation vector, velocity of movement and Coriolis acceleration expressed in this local coordinate system are:

: ec \omega = \omega \begin{pmatrix} 0 \ \cos arphi \ \sin arphi \end{pmatrix},     ec v = \begin{pmatrix} v_e \ v_n \ v_u \end{pmatrix},     ec a_C = 2\,\omega\, \begin{pmatrix} v_n \sin arphi-v_u \cos arphi \ -v_e \sin arphi \ v_e \cos arphi\end{pmatrix}\,

When considering atmospheric or oceanic dynamics, both the velocity and the Coriolis acceleration are small in the vertical direction, due to the short vertical length scale. The restriction of the above to the horizontal plane is:

: ec v = \begin{pmatrix} v_e \ v_n\end{pmatrix}\,,     ec a_c = f \begin{pmatrix} v_n \ -v_e\end{pmatrix}\, ,   where f = 2 \omega \sin arphi \, is called the '' Coriolis Parameter ''.

From this it can be immediately seen that (for positive arphi and \omega\,) a movement due east results in a force due south and a movement due north in a force due east — both turned 90° to the right.


CAUSES

The Coriolis effect exists only when using a rotating reference frame. It is mathematically deduced from the law of Inertia . Hence it does not correspond to any actual acceleration or force, but only the ''appearance'' thereof from the point of view of a rotating system.

The Coriolis effect exhibited by a moving object can be interpreted as being the sum of the effects of two different causes of equal magnitude.

The first cause is the change of the velocity of an object in time. The same velocity (in an inertial frame of reference where the normal laws of physics apply) will be seen as different velocities at different times in a rotating frame of reference. The apparent acceleration is proportional to the angular velocity of the reference frame (the rate at which the coordinate axes changes direction), and to the velocity of the object. This gives a term -\boldsymbol\omega imes\mathbf{v}.
The minus sign arises from the traditional definition of the cross product ( Right Hand Rule ), and from the sign convention for angular velocity vectors.

The second cause is change of velocity in space. Different points in a rotating frame of reference have different velocities (as seen from an inertial frame of reference).
In order for an object to move in a straight line it must therefore be accelerated so that its velocity changes from point to point by the same amount as the velocities of the frame of reference. The effect is proportional to the angular velocity (which determines the relative speed of two different points in the rotating frame of reference), and the velocity of the object perpendicular to the axis of rotation (which determines how quickly it moves between those points). This also gives a term -\boldsymbol\omega imes\mathbf{v}.


What the Coriolis effect is not

  • The Coriolis effect is not a result of the curvature of the Earth, only of its rotation. (However, the value of the Coriolis parameter, f \ , does vary with latitude, and that dependence ''is'' due to the Earth's shape.)

  • The fact that ballistic missiles and satellites appear to follow curved paths when plotted on common world maps is mainly due to the fact that the earth is spherical and the shortest distance between two points on the earth's surface (called a Great Circle ) is usually not a straight line on those maps. Every two-dimensional (flat) map necessarily distorts the earth's curved (three-dimensional) surface in some way. Typically (as in the commonly used Mercator Projection , for example), this distortion increases with proximity to the poles. In the northern hemisphere for example, a ballistic missile fired toward a distant target using the shortest possible route (a great circle) will appear on such maps to follow a path north of the straight line from target to destination, and then curve back toward the equator. This occurs because the latitudes, which are projected as straight horizontal lines on most world maps, are in fact circles on the surface of a sphere, which get smaller as they get closer to the pole. Being simply a consequence of the sphericity of the Earth, this would be true even if the Earth didn't rotate. The Coriolis effect is of course also present, but its effect on the plotted path is much smaller.

  • The Coriolis effect cannot sustain the rotation of objects about a fixed axis (relative to the Rotating Reference Frame ), as there is no displacement involved.

  • The Coriolis force should not be confused with the Centrifugal Force given by m \boldsymbol\omega imes(\boldsymbol\omega imes\mathbf{r}). A Rotating Frame Of Reference will always cause a Centrifugal Force no matter what the object is doing (unless that body is Particle-like and lies on the axis of rotation), whereas the Coriolis force requires the object to be in motion relative to the rotating frame with a velocity that is not parallel to the rotation axis. Because the Centrifugal Force always exists, it can be easy to confuse the two, making simple explanations of the effect of Coriolis in isolation difficult. In particular, when \mathbf{v} is tangential to a circle centered on and perpendicular to the axis of rotation, the Coriolis force is parallel to the centrifugal force. It is then possible to construct a rotating reference frame of a different rotational speed, where \mathbf{v} is zero and there is no Coriolis force



VISUALIZATION OF THE CORIOLIS EFFECT