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Encyclopedia > Coriolis force

In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. When the equations of motion are formulated in a rotating coordinate system a term arises which looks like a force, called the Coriolis force. See also centrifugal force.

In changing from an essentially inertial coordinate system (such as the "frame of the fixed stars") to a rotating frame of reference (such as that of the Earth's surface), a term appears in the equation of motion described by the formula for Coriolis force:

where bold indicates vector quantities, m is mass, v is the velocity and ω is the angular velocity of the coordinate system. Note that this equation ignores the second-order term in ω, which in geophysical terms is small, and can in any case be absorbed into the gravitational potential term.

Hurricane Isabel east of the Bahamas on 2003-09-15. Photograph courtesy NASA.

The direction of the coriolis force is parallel to the surface, and perpendicular to the meridian. It is proportional to the component of the velocity parallel to the meridian and to the rate of rotation of the coordinate system. If an object is travelling on earth in the northern hemisphere, the Coriolis force will deflect the object to the right. In the southern hemisphere the reverse is true. The effect breaks up the atmospheric circulation from the tropics to the polar regions into a series of cells in which the surface winds have a prevailing eastward or westward component.

The Coriolis force plays a strong role in weather patterns, where it affects prevailing winds and the rotation of storms, as well as in the direction of ocean currents. Above the atmospheric boundary layer, friction plays a relatively minor role, as air parcels move mostly parallel to each other. Here, an approximate balance between pressure gradient force and Coriolis force exists, causing the geostrophic wind, which is the wind effected by these two forces only, to blow along isobars (along lines of constant geopotential height, to be precise). Thus a northern hemispheric low pressure system rotates in a counterclockwise direction, while northern hemispheric high pressure systems or cyclones on the southern hemisphere rotate in a clockwise manner, as described by Buys-Ballot's law.

The Coriolis effect must also be considered in astronomy, and stellar dynamics, where it affects phenomena such as the rotational direction of sunspots. The flight paths of airplanes, artillery shells, and missiles must account for the Coriolis effect or risk being off course by significant amounts. (See external ballistics.)

The Coriolis effect can also be observed in the motion of a simple pendulum. For instance, if a pendulum is set swinging at the North pole, the pendulum will oscillate in a fixed plane while the earth rotates beneath it. Hence for an observer on earth, the plane of oscillation would appear to rotate once a day. This effect is present at other latitudes although the oscillations are more complicated but the phenomenon is qualitatively the same. (See Foucault pendulum for more details). Another classical instance where the Coriolis force produces a measurable effect is in the deflection of falling object: a body falling freely is deflected to the east. The effect is greatest at the equator, and decreases to zero as the observer moves towards either pole.

See Taylor-Proudman theorem for a startling consequence of the Coriolis effect: in a rotating reference frame, if the flow has low Rossby number but high Reynolds number, all steady solutions to the Navier-Stokes equations have the property that the fluid velocity is uniform along any line parallel to the rotation axis. In oceanic flow, it is possible to ignore the non-vertical components of the Earth's rotation, so if the conditions of the theorem apply ( is universal but using 0.1m / s as a typical flow speed and using 4km as a depth, f = 10 - 4s - 1 gives which is marginal), the fluid velocity is identical at all points along any single vertical line (known as a Taylor column). The Taylor-Proudman theorem is widely used when considering limnological flows, astrophysical flows (such as solar and jovian dynamics) and some industrial problems such as turbine design.

Although the Coriolis force is relatively small and does not have an observable influence on small systems such as the whirlpool of a draining bathtub, toilet or sink [1] (http://www.ems.psu.edu/~fraser/Bad/BadCoriolis.html) [2] (http://math.ucr.edu/home/baez/physics/General/bathtub.html), the Coriolis effect can have a visible effect over large amounts of time and has been observed to cause uneven wear on railroad tracks and cause rivers to dig their beds deeper on one side.

A practical application of the Coriolis force is the mass flow meter, an instrument that measures the mass flow rate of a fluid through a tube. The instrument was introduced in 1977 by Micro Motion Inc. Simple flow meters measure volume flow rate, which is proportional to mass flow rate only when the density of the fluid is constant. If the fluid has varying density, or contains bubbles, then the volume flow rate multiplied by the density is not an accurate measure of the mass flow rate. The Coriolis mass flow meter works by applying a vibrating force to a curved tube through which the fluid passes. The Coriolis effect creates a force on the tube perpendicular to both the direction of vibration and the direction of flow. This force is measured to give the mass flow rate. Coriolis flow meters can also be used with non-Newtonian fluids, which tend to give inaccurate results with volume flow meters. The same instrument can be used to measure the density of the fluid, since this affects the resonant frequency of the vibrating tube. A further advantage of this instrument is that the fluid is contained in a smooth tube, with no moving parts that would need to be cleaned and maintained, and that would impede the flow. 2

Effects due to the Coriolis force also appear in atomic physics. In polyatomic molecules, the molecule motion can be described by a rigid body rotation and internal vibration of atoms about their equilibrium position. As a result of the vibrations of the atoms, the atoms are in motion relative to the rotating coordinate system of the molecule. A Coriolis force is therefore present and will cause the atoms to move in a direction perpendicular to the original oscillations. This leads to a mixing in molecular spectra between the rotational and vibrational levels.

Insects of the group Diptera use two small vibrating structures at the side of their bodies to detect the effects of the Coriolis force. These so called Halteres play an important role in these insects' ability to perform aerobatics.


Is the Coriolis force "fictitious"?

There is an aspect of the coriolis effect that is a manifestation of inertia. If that aspect is assumed to be a force, then that is a fictitious force.

Take the situation of a large rotating disk, with a box on it, and in the box a weight is suspended, it is suspended on all sides with springs. If for example the angular velocity of the disk would be increased, the weight would "lag behind" slightly, due to inertia. That is, at first the springs are deformed, and as the springs are compressed they start exerting more and more force on the weight, until the weight is brought up to speed. So if the weight momentarily lags behind, an observer monitoring the position of the weight in its suspension can infer that the angular velocity of the disk must be changing. Given the nature of inertia, only that scenario explains the displacement of the weight in its suspension.

If, given a constant angular velocity of the disk, the box is moved over a section of meridian (a meridian is a straight line from the center of the disk to the rim) and the box is moved away from the center of the disk, then the weight will be seen to "lag behind" too. An observer monitoring the position of the weight in its suspension, can infer that the disk must be rotating. If the disk is rotating, then moving away from the center means that the weight needs to be accelerated to the velocity corresponding to the circumference that goes with the new position on the disk. Only that scenario explains the displacement of the weight in its suspension.

Conservation of angular momentum

If the rotating disk is suspended frictionless, and its angular velocity is not maintained by outside control, then only the disk's own inertia maintains the angular velocity. When the box with the weight is moved over the meridian away from the hub of the disk, its angular momentum must be increased if its motion is to remain synchronized with the disk's motion. This increase in the angular momentum of the weight will be at the expense of the angular momentum of the rotating disk. In total there wil be conservation of angular momentum.

Performing calculations

Under some circumstances it is more practical to use a model in which the disk is stationary. In order to make the calculation produce exactly the same outcome as it would in a model with a rotating disk, a fictitious force is introduced. This fictitious force momentarily displaces the weight in its suspension, during the move along the meridian.

Having two alternative models available enables the physicists to choose the most practical one for performing a calculation. A well-equipped physicist can perform calculations in various reference frames.
In order to perform the calculation it is necessary to know the rotation rate with respect to the one frame with zero rotation. As long as that reference is indispensible in order to obtain a correct outcome the non-rotating frame is the actual reference frame of the calculation.

The non-rotating frame

According to physics, when it comes to rotation, there is one preferred frame: the non-rotating frame. This can be illustrated graphically with the following setup.
Take a gymbal mounted gyroscope. Before the gyroscope is spun up it can revolve in its gymbal mounting in any plane of rotation. As the gyroscope is spun up, it spontaneously ceases to revolve. If the suspension of the gyroscope is sufficiently frictionless, the gyro-axis will stay pointing at the same star, the star it happened to be pointing at when it ceased to revolve in its mounting. If you take several gyroscopes, all in virtually frictionfree gymbal mounting, you will see that they all, since they all maintain direction, are in sync. Seen from the point of view of an observer who doesn't know (of chooses to ignore) that the Earth rotates, it is clear that the gyroscopes are all tuned in to something, something is causing them to be in sync. On Earth, if the gyroscopes are suspended virtually frictionfree, they display the sidereal rotation period of Earth: 23 hours, 56 minutes, 4 seconds.

According to general relativity, space-time has intrinsic orientation (but no preferred orientation!). The spinning gyroscopes lock on to the intrinsic orientation of space-time, showing the observer which frame isn't rotating.


Note 2: EDN Access 2003-06-30 (http://www.e-insite.net/ednmag/index.asp?layout=article&stt=000&articleid=CA305490&pubdate=6%2F26%2F2003)

  Results from FactBites:
Coriolis effect - Wikipedia, the free encyclopedia (2017 words)
The Coriolis effect is an apparent deflection of a freely moving object as observed from a rotating frame of reference.
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.
The force balance is largely between the pressure gradient force acting towards the low-pressure area and the Coriolis acceleration acting away from the center of the low pressure.
  More results at FactBites »



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