Coriolis effect
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The Coriolis effect is an apparent deflection of a moving object in a rotating frame of reference. This effect is sometimes attributed to the fictitious Coriolis force.
The Coriolis effect caused by the rotation of the Earth is responsible for the precession of a Foucault pendulum and for the direction of rotation of cyclones. In general, the effect deflects objects moving along the surface of the Earth to the right in the Northern hemisphere and to the left in the Southern hemisphere. As a consequence, winds around the center of a cyclone rotate counterclockwise on the northern hemisphere and clockwise on the southern hemisphere. However, contrary to popular belief, the Coriolis effect is not a determining factor in the rotation of water in toilets or bathtubs (see the Draining bathtubs/toilets section below).
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 Laplace in 1778.
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[edit] Formula
The formula for the Coriolis acceleration is
- <math>\mathbf{a_C}=-2\boldsymbol\omega\times\mathbf{v}</math>
where (here and below) <math>\mathbf{v}</math> is the velocity of the particle in the rotating system, and <math>\boldsymbol\omega</math> is the angular velocity vector (which has magnitude equal to the rotation rate and is parallel with 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
- <math>\mathbf{F_C}=-2m(\boldsymbol\omega\times\mathbf{v})</math>.
See Fictitious force for a derivation.
Note that these are cross products. In non-vector terms: at a given rate of rotation of the observer, the magnitude of the Coriolis acceleration of the object will be 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. (Note also that 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
The above formulae use vector notation, and give both the magnitude and direction of the Coriolis effect. For some special cases, a scalar expression might be sufficient, as the direction has already been deduced. For the case of motion restricted to a plane perpendicular to the axis of rotation, such as a rotating turntable, the magnitude of the acceleraton is given by the formula
- <math>a_c = 2 \omega v \ </math>.
When considering atmospheric dynamics, the Coriolis acceleration is only significant in the horizontal equations, due to the short length scale in the vertical direction. However, the horizontal plane is not in general perpendicular to the axis of rotation. The magnitude of the horizontal component of the acceleration is then
- <math>a_c = f v \ </math> ,
where <math>f = 2 \omega \sin(\phi) \ </math> (where <math>\phi \,</math> is the latitude) is called the Coriolis parameter and <math>v \ </math> is the horizontal component of the velocity.
[edit] 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 can be interpreted as being the sum of the effects of two different causes of equal magnitude.
The first cause is the change of velocity 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 (the rate at which the coordinate axes changes direction), and to the velocity. This gives a term <math>-\boldsymbol\omega\times\mathbf{v}</math>. The minus sign stems from the fact that the effect is interpreted from the rotating frame of reference. In the absence of any force, the object will appear to accelerate in the direction opposite of that of rotation.
The second cause is change of velocity in space. Different points in a rotating frame 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 <math>-\boldsymbol\omega\times\mathbf{v}</math>.
[edit] What the Coriolis effect is not
- The Coriolis effect does not depend on the curvature of the Earth, only on its rotation. (However, the value of the Coriolis parameter, <math>f \ </math>, does vary with latitude, and that 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 a great circle is not a straight line on those maps. A ballistic missile fired "straight toward the east (or west)" will appear to turn toward the equator (regardless of hemisphere), because the latitudes, which are projected as straights line on most world maps, are in fact circles around the pole. (Just like the Coriolis effect, this phenomenon is more pronounced near the poles.) The Coriolis effect is of course also present, but its effect on the plotted path is much smaller.
- The Coriolis force should not be confused with the Centrifugal force given by <math>m \boldsymbol\omega\times(\boldsymbol\omega\times\mathbf{r})</math>. 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 and not such that it moves parallel to the rotation axis. Therefore, because the Centrifugal force always exist, it can be easy to confuse the two, making simple explanations of the effect of Coriolis in isolation difficult. In particular, when <math>\mathbf{v}</math> is tangential to the direction of rotation, the Coriolis force will be parallel to the centrifugal force. It is then possible to construct rotating reference frame of a different rotational speed, where <math>\mathbf{v}</math> is zero and there is no Coriolis force. What was considered a Coriolis force in the first frame of reference becomes a part of the centrifugal force in the second.
[edit] Visualisation of the Coriolis effect
To demonstrate the Coriolis effect, a parabolic turntable can be used. On a flat turntable the centrifugal force, which always acts outwards from the rotation axis, would lead to objects being forced out off the edge. But if the surface of the turntable has the correct parabolic bowl shape, and is rotated at the correct rate, then the component of gravity tangential to the bowl surface will exactly balance the centrifugal force. This allows the Coriolis force to be displayed in isolation.
Discs cut from cylinders of dry ice can be used as pucks, moving around almost frictionlessly over the surface of the parabolic turntable, allowing effects of Coriolis on dynamic phenomena to show themselves. To get a view of the motions as seen from a rotating point of view, a video-camera is attached to the turntable in such a way that the camera is co-rotating with the turntable.
When the fluid is rotating on a flat turntable, the surface of the fluid naturally assumes the correct parabolic shape. This fact may be exploited in order to make a parabolic turntable, by using a fluid that sets after several hours, such as a synthetic resin.
In a manner of speaking, the Earth represents such a turntable. The rotation has caused the planet to assume a spheroid shape such that the normal force exactly balances the centrifugal force on a "horizontal" surface. (See equatorial bulge.)
[edit] Draining bathtubs/toilets
A popular misconception is that the Coriolis effect determines the direction in which bathtubs or toilets drain, and whether water always drains in one direction in the Northern Hemisphere, and in the other direction in the Southern Hemisphere. In reality, the Coriolis effect is a few orders of magnitude smaller than other random influences on drain direction, such as the geometry of the sink, toilet, or tub; whether it is flat or tilted; and the direction in which water was initially added to it. Note that toilets typically are designed to only flush in one rotation, by having the flush water enter at an angle.
This is less of a puzzle once one remembers that the earth revolves once per day but that a bathtub takes only minutes to drain. When the water is being drawn towards the drain, the radius with which it is spinning around it decreases, so its rate of rotation increases from the low background level to a noticeable spin in order to conserve its angular momentum (the same effect as ice skaters bringing their arms in to cause them to spin faster).
[edit] Coriolis in Meteorology
Perhaps the most important instance of the Coriolis effect is in the large scale dynamics of the oceans and the atmosphere. In meteorology, it is convenient to use a rotating frame of reference where the Earth is stationary. The fictitious centrifugal and Coriolis forces must then be introduced. The former, however, is cancelled by the non-spherical shape of the earth (see the turn-table analogy above). Hence the Coriolis force is the only fictitious force to have a significant impact on calculations.
[edit] Flow around a low-pressure area
If a low-pressure area forms in the atmosphere, air will tend to flow in towards it, but will be deflected perpendicular to its velocity by the Coriolis acceleration. A system of equilibrium can then establish itself creating circular movement, or a cyclonic flow.
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. Instead of flowing down the gradient, the air tends to flow perpendicular to the air-pressure gradient and forms a cyclonic flow. This is an example of a more general case of geostrophic flow in which air flows along isobars. On a non-rotating planet the air would flow along the straightest possible line, quickly leveling the air pressure. Note that the force balance is thus very different from the case of "inertial circles" (see below) which explains why mid-latitude cycles are larger by an order of magnitude than inertial circle flow would be.
This pattern of deflection, and the direction of movement, is called Buys-Ballot's law. The pattern of flow is called a cyclone. In the Northern Hemisphere the direction of movement around a low-pressure area is counterclockwise. In the Southern Hemisphere, the direction of movement is clockwise because the rotational dynamics is a mirror image there. However, at high altitudes, outward-spreading air rotates in the opposite direction [1] Cyclones cannot form on the equator, and they rarely travel towards the equator, because in the equatorial region the coriolis parameter is small, and exactly zero on the equator.
[edit] Inertial circles
- <math>R=v/f\,</math>.
On the Earth, a typical mid-latitude value for <math>f</math> is 10−4 s−1; hence for a typical atmospheric speed of 10 m/s the radius is 100 km, with a period of about 14 hours. In the ocean, where a typical speed is closer to 10 cm/s, the radius of an inertial circle is 1 km. These inertial circles are clockwise in the northern hemisphere (where trajectories are bent to the right) and anti-clockwise in the southern hemisphere.
If the rotating system is a parabolic turntable, then <math>f</math> is constant and the trajectories are exact circles. On a rotating planet, <math>f</math> varies with latitude and the paths of particles do not form exact circles. Since the parameter <math>f</math> varies as the sine of the latitude, the oscillations associated with a given speed are smallest at the poles (latitude = <math>\pm 90^\circ</math>), and would increase indefinitely at the equator, except the dynamics ceases to apply close to the equator.
The dynamics of inertial circles are different from those of mid-latitude cyclones. In the latter case, the Coriolis force (directed outward) is in an approximate balance with the pressure gradient force (directed inward), a situation known as geostrophic balance. In particular, cyclones rotate in the opposite direction as inertial circles.
[edit] Length scales and the Rossby number
- Further information: Rossby number
The time, space and velocity scales are important in determining the importance of the Coriolis effect. Whether rotation is important in a system can be determined by its Rossby number, which is the ratio of the velocity of a system to the product of the Coriolis parameter, and the lengthscale of the motion:
- <math>Ro = \frac{U}{fL}</math>.
A small Rossby number signifies a system which is strongly affected by rotation, and a large Rossby number signifies a system in which rotation is unimportant. An atmospheric system moving at U = 10m/s occupying a spatial distance of L=1000km, has a Rossby number
- <math>Ro = \frac{10}{10^{-4}\times 1000\times10^3} = 0.1</math>
A man playing catch may throw the ball at U=30m/s in a garden of length L=50m. The Rossby number in this case would be
- <math>Ro = \frac{30}{10^{-4}\times 50} = 6000</math>.
Needless to say, one does not worry about which hemisphere one is in when playing catch in the garden. However, an unguided missile obeys exactly the same physics as a baseball, but may travel far enough and be in the air long enough to notice the effect of Coriolis. Long range shells landed close to, but to the right of where they were aimed until this was noted (or left if they were fired in the southern hemisphere, though most were not).
The Rossby number can also tell us about the bathtub. If the lengthscale of the tub is about L=1m, and the water moves towards the drain at about 60cm/s, then the Rossby number is
- <math>Ro = \frac{0.6}{10^{-4}\times 1} = 6 000</math>.
Thus, the bathtub is, in terms of scales, much like a game of catch, and rotation is likely to be unimportant.
However, if the experiment is very carefully controlled to remove all other forces from the system, rotation can play a role in bathtub dynamics. An article in the British "Journal of Fluid Mechanics" in the 1930's describes this. The key is to put a few drops of ink into the bathtub water, and observing when the ink stops swirling, meaning the viscosity of the water has dissipated its initial vorticity (or curl; i.e. <math>\nabla \times U = 0</math>) then, if the plug is extracted ever so slowly so as not to introduce any additional vorticity, then the tub will empty with a counterclockwise swirl in England.
[edit] Terrestrial effects summarized
The magnitude of the (horizontal) Coriolis effect changes with the latitude and the speed of the air (and water). It is greatest in polar regions where the surface of the Earth is at right angles to the axis of rotation, and it is zero at the equator. It causes air and water masses to turn right in the northern hemisphere and left in the southern hemisphere. This gives rise to geostrophic winds and currents.
The Coriolis effect strongly affects the large-scale atmospheric circulation, leading to the Hadley, Ferrel, and Polar cells. In the oceans, Coriolis is responsible for the propagation of Kelvin waves and the establishment of the Sverdrup balance.
There are also components of the Coriolis effect that are not in the plane tangential to the Earth's surface. Eastward traveling objects will be deflected upwards (feel lighter), while westward traveling objects will be deflected downwards (feel heavier). This is known as the Eötvös effect. In addition objects traveling upwards or downwards will be deflected to the west or east respectively. These effects are greatest near the equator, and require precise instruments to detect since they are quite small.
[edit] Coriolis Elsewhere
[edit] Coriolis flow meter
A practical application of the Coriolis effect is the mass flow meter, an instrument that measures the mass flow rate of a fluid through a tube. The operating principle 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 operating principle essentially involves rotation, though not through a full circle. It works by inducing a vibration of the tube through which the fluid passes, and subsequently monitoring and analysing the inertial effects that occur in response to the combination of the induced vibration and the mass flow.
[edit] Molecular 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. Coriolis effects will therefore be 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.
[edit] Ballistics
The Coriolis effects became important in external ballistics for calculating the trajectories of very long-range artillery shells. The most famous historical example was the Paris gun, used by the Germans during World War I to bombard Paris from a range of about 120 km.
[edit] References
[edit] Physics and meteorology references
- Gill, AE 'Atmospher-Ocean dynamics, Academic Press, 1982.
- Durran, D. R., 1993: Is the Coriolis force really responsible for the inertial oscillation?, Bull. Amer. Meteor. Soc., 74, 2179–2184; Corrigenda. Bulletin of the American Meteorological Society, 75, 261
- Durran, D. R., and S. K. Domonkos, 1996: An apparatus for demonstrating the inertial oscillation, Bulletin of the American Meteorological Society, 77, 557–559.
- Marion, Jerry B. 1970, Classical Dynamics of Particles and Systems, Academic Press.
- Persson, A., 1998 How do we Understand the Coriolis Force? Bulletin of the American Meteorological Society 79, 1373-1385.
- Symon, Keith. 1971, Mechanics, Addison-Wesley
- Norman Ph. A., 2000 An Explication of the Coriolis Effect, Bulletin of the American Meteorological Society: Vol. 81, No. 2, pp. 299–303.
[edit] Historical references
- Grattan-Guinness, I., Ed., 1994: Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Vols. I and II. Routledge, 1840 pp.
1997: The Fontana History of the Mathematical Sciences. Fontana, 817 pp. 710 pp.
- Khrgian, A., 1970: Meteorology—A Historical Survey. Vol. 1. Keter Press, 387 pp.
- Kuhn, T. S., 1977: Energy conservation as an example of simultaneous discovery. The Essential Tension, Selected Studies in Scientific Tradition and Change, University of Chicago Press, 66–104.
- Kutzbach, G., 1979: The Thermal Theory of Cyclones. A History of Meteorological Thought in the Nineteenth Century. Amer. Meteor. Soc., 254 pp.
[edit] Footnotes
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[edit] External links
- The definition of the Coriolis effect from the Glossary of Meteorology
- The Coriolis Effect PDF-file. 17 pages. A general discussion by Anders Persson of various aspects of the coriolis effect, including Foucault's Pendulum and Taylor columns.
- Anders Persson The Coriolis Effect: Four centuries of conflict between common sense and mathematics, Part I: A history to 1885 History of Meteorology 2 (2005)
- Coriolis Force - from ScienceWorld
- The Coriolis Effect: An Introduction. Details of the causes of prevailing wind patterns. Targeted towards ages 5 to 18.
- Coriolis Effect and Drains An article from the NEWTON web site hosted by the Argonne National Laboratory.
- Do bathtubs drain counterclockwise in the Northern Hemisphere? by Cecil Adams.
- Bad Coriolis. An article uncovering misinformation about the Coriolis effect. By Alistair B. Fraser, Emeritus Professor of Meteorology at Pennsylvania State University
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