Talk:Coriolis force/proposed

This page was inserted as the new version on 2005/08/22. William M. Connolley 11:08:02, 2005-08-22 (UTC).

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Hurricane Isabel east of the Bahamas on September 15, 2003.

In physics, the Coriolis effect is the apparent deflection of a freely moving object as observed from a rotating frame of reference. The effect can be accounted for in the rotating frame by the introduction of the fictitious Coriolis force which then balances the the equations of the apparent motion. The effect is named after Gaspard-Gustave Coriolis, a French scientist, who discussed it in 1835, though the mathematics appeared in the tidal equations of Laplace in 1778.

One example of the Coriolis effect is mid-latitude cyclones.

Formula

The formula for the Coriolis acceleration is

$-2{\boldsymbol {\omega }}\times \mathbf {v}$

where v is the velocity in the rotating system, omega is the angular velocity (effectively, the rotation rate). The equation may be multiplied by the mass of the relevant object to produce the Coriolis force.

At a given rotation speed, the acceleration will be proportional to the velocity and the sine of the angle between the velocity vector and the axis.

The Coriolis effect is the behavior added by the Coriolis force. The formula implies that the Coriolis force is perpendicular both to the velocity of the moving mass and to the rotation axis. So in particular:

• if the velocity is zero, the Coriolis force is zero
• if the velocity is parallel to the rotation axis, the Coriolis force is zero
• if the velocity is straight (perpendicularly) inward to the axis, the force will follow the direction of rotation

When considering atmospheric dynamics, the Coriolis acceleration (strictly a 3-d vector in the formula above) appears only in the horizontal equations due to the neglect of products of small quantities and other approximations. The term that appears is then

$-f\mathbf {k} \times (u,v)$

where $\mathbf {k}$ is a unit local vertical, $f=2\omega \sin({\text{latitude}})$ is called the Coriolis parameter and (u,v) are the horizontal components of the velocity.

Flow around a low pressure area

Schematic representation of flow around a low-pressure area. The pressure gradient force is represented by blue arrows, the Coriolis force (always perpendicular to the velocity) by red arrows

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 force. 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 force 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. 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.

Draining bathtubs/toilets

People often ask whether 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. The answer is almost always no. 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. If one takes great care to create a flat circular pool of water with a small, smooth drain; to wait for eddies caused by filling it to die down; and to remove the drain from below (or otherwise remove it without introducing new eddies into the water) – then it is possible to observe the influence of the Coriolis effect in the direction of the resulting vortex. There is a good deal of misunderstanding on this point, as most people (including many scientists) do not realize how small the Coriolis effect is on small systems.¹ 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. The increase in rotational speed around the plug hole is because water is being drawn towards the plughole and so its radius of its mass to the point it is spinning around 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 bringing your arms in on a swivel chair making it spin faster)

The time and space scales are important in determining the importance of the coriolis effect. Weather systems are large enough to feel the curvature of the earth and generally rotate less than once a day so a similar timescale to the earth's rotation so the coriolis effect is dominant. An unguided missile, if fired far enough, will travel far enough and be in the air long enough to notice the effect but the dominant effect is the direction it was fired in. 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 weren't). You don't worry about which hemisphere you're in when playing catch in the garden though this is exactly the same physics at a smaller scale. A bathtub is best approximated (in terms of scale) by a game of catch.

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.

Ballistics

In firing projectiles over a significant distance, the rotation of the Earth must be taken into account. During its flight, the projectile moves in a straight line (not counting gravitation and air resistance for now). The target, co-rotating with the Earth, is a moving target, so the gun must be aimed not directly at the target, but at a point where the projectile and the target will arrive simultaneously.

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. A 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.

Visualisation of the Coriolis effect

Turntable used to study Coriolis dynamics

To demonstrate the Coriolis effect, a turntable can be used. If the turntable is flat, then the centrifugal force, which always acts outwards from the rotation axis, would lead to objects being forced out off the edge. If the turntable has a bowl shape, then the component of gravity tangenital to the bowl surface will tend to counteract the centrifugal force. If the bowl is parabolic, and spun at the appropriate rate, then gravity exactly counteracts the centrifugal force and the only net force (bar friction, which can be minimised) acting is then the Coriolis force. If the turntable is a dish with a rim and filled with liquid, then when the liquid is rotating it naturally assumes a parabolic shape (for the same reasons). If a liquid that sets after several hours is used, such as a synthetic resin, a permanent shape is obtained.

Disks cut from cylinders of dry ice can be used as pucks, moving around almost frictionlessly over the surface of the parabolic turntable, allowing dynamic phenomena to show themselves. To also 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. This type of setup, with a parabolic turntable, at the center about a centimeter deeper than the rim, is used at Massachusetts Institute of Technology (MIT) for teaching purposes. ²

Inertial circles

Schematic representation. Inertial circles of air masses in the absence of other forces.

If an object moves in a rotating system subject only to the Coriolis force, it will move in a circular trajectory called an 'inertial circle'.

In an inertial circle, the force balance is between the centrifugal force (directed outwards) and the Coriolis force (directed inwards). The dynamics is thus quite different to mid-latitude cyclones or hurricanes, in which cases the force balance is between the pressure gradient force (directed inwards) and the Coriolis force (directed out). In particular, this means that the direction of orbit is oppositite to that of mid-latitude cyclones.

The frequency of these oscillations is given by f, the coriolis parameter; and their radius by [1]:

$v/f$,

where v is the velocity of the air mass. On the Earth, a typical mid-latitude value for f is 10^-4; hence for a typical atmospheric speed of 10 m/s the radius is 100 km, with a period of about 14 hours. For a turntable rotating about once every 6 seconds, f is one, hence the radius of the circles, in cm, is numerically the same as the speed, in cm/s.

The centrigufal force is $v^{2}/r$ and the Coriolis force $vf$, hence the forces balance when $v^{2}/r=vf$, i.e. $v/f=r$, giving the expression above for the radius of the circles.

If the rotating system is a turntable, then $f$ is constant and the trajectories are exact circles. On a rotating planet, $f$ varies with latitude and the circles do not exactly close.

Closer to the equator the component of the velocity towards or away from the Earth's axis is smaller, this component varies as $sin(latitude)$, and this is taken into account in the parameter f. For a given velocity the oscillations are smallest at the poles as shown by the picture and would increase indefinitely at the equator, except the dynamics ceases to apply close to the equator. On a rotating planet the oscillations are only approximately circular [2] and do not form closed loops as indicated in the drastically simplified picture.

References

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
  1200 KB PDF-file of the above article

• Durran, D. R., and S. K. Domonkos, 1996: An apparatus for demonstrating the inertial oscillation. Bulletin of the American Meteorological Society, 77, 557–559.
  1300 KB PDF-file of the above article

• 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.
  374 KB PDF-file of the above article

• 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.
  200 KB PDF-file of the above article

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.

External links

• http://amsglossary.allenpress.com/glossary/search?id=coriolis-force1

• ^Note 1 : Coriolis effect misconceptions

• ^Note 2 : Rotating turntable setup The concave turntable used at MIT for educational purposes.

• ^Note 3 : Taylor columns The counterintuitive behavior of a rotating fluid. Demonstration at MIT for educational purposes

• 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.

Category:Classical mechanics Category:Dynamical systems Category:Oceanography Category:Meteorology Category:Celestial mechanics Category:Force

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