Coriolis effect

# Coriolis effect

Overview
In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame
Rotating reference frame
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. A rotating frame of reference is a special case of a non-inertial reference...

. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right. The mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave Coriolis
Gaspard-Gustave Coriolis
Gaspard-Gustave de Coriolis or Gustave Coriolis was a French mathematician, mechanical engineer and scientist. He is best known for his work on the supplementary forces that are detected in a rotating frame of reference. See the Coriolis Effect...

, in connection with the theory of water wheels, and also in the tidal equations
Theory of tides
The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans, under the gravitational loading of another astronomical body or bodies...

of Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

in 1778.
Discussion

Recent Discussions
Encyclopedia
In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame
Rotating reference frame
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. A rotating frame of reference is a special case of a non-inertial reference...

. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right. The mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave Coriolis
Gaspard-Gustave Coriolis
Gaspard-Gustave de Coriolis or Gustave Coriolis was a French mathematician, mechanical engineer and scientist. He is best known for his work on the supplementary forces that are detected in a rotating frame of reference. See the Coriolis Effect...

, in connection with the theory of water wheels, and also in the tidal equations
Theory of tides
The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans, under the gravitational loading of another astronomical body or bodies...

of Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

in 1778. And even earlier, Italian scientists Giovanni Battista Riccioli
Giovanni Battista Riccioli
Giovanni Battista Riccioli was an Italian astronomer and a Catholic priest in the Jesuit order...

and his assistant Francesco Maria Grimaldi
Francesco Maria Grimaldi
Francesco Maria Grimaldi was an Italian Jesuit priest, mathematician and physicist who taught at the Jesuit college in Bologna....

described the effect in connection with artillery in the 1651 Almagestum Novum, writing that rotation of the Earth should cause a cannon ball fired to the north to deflect to the east. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology
Meteorology
Meteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...

.

The Coriolis effect is caused by the rotation of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

and the inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

of the mass experiencing the effect. Newton's laws of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

govern the motion of an object in a (non-accelerating) inertial frame of reference
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

. When Newton's laws are transformed to a rotating frame of reference, the Coriolis and centrifugal forces appear. Both forces are proportional to the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

of the object. The Coriolis force is proportional to the rotation rate and the centrifugal force is proportional to its square. The Coriolis force acts in a direction perpendicular to the rotation axis and to the velocity of the body in the rotating frame and is proportional to the object's speed in the rotating frame. The centrifugal force acts outwards in the radial direction and is proportional to the distance of the body from the axis of the rotating frame. These additional forces are termed either inertial forces, fictitious force
Fictitious force
A fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame....

s or pseudo forces. They allow the application of simple Newtonian laws to a rotating system. They are correction factors that do not exist in a true non-accelerating "inertial" system.

Perhaps the most commonly encountered rotating reference frame is the Earth. Because the Earth completes only one rotation per day, the Coriolis force is quite small, and its effects generally become noticeable only for motions occurring over large distances and long periods of time, such as large-scale movement of air in the atmosphere or water in the ocean. Such motions are constrained by the 2-dimensional surface of the earth, so only the horizontal component of the Coriolis force is generally important. This force causes moving objects on the surface of the Earth to appear to veer to the right in the northern hemisphere
Northern Hemisphere
The Northern Hemisphere is the half of a planet that is north of its equator—the word hemisphere literally means “half sphere”. It is also that half of the celestial sphere north of the celestial equator...

, and to the left in the southern
Southern Hemisphere
The Southern Hemisphere is the part of Earth that lies south of the equator. The word hemisphere literally means 'half ball' or "half sphere"...

. Rather than flowing directly from areas of high pressure to low pressure, as they would on a non-rotating planet, winds and currents tend to flow to the right of this direction north of the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

, and to the left of this direction south of it. This effect is responsible for the rotation of large cyclones (see Coriolis effects in meteorology).

## History

Gaspard-Gustave Coriolis
Gaspard-Gustave Coriolis
Gaspard-Gustave de Coriolis or Gustave Coriolis was a French mathematician, mechanical engineer and scientist. He is best known for his work on the supplementary forces that are detected in a rotating frame of reference. See the Coriolis Effect...

published a paper in 1835 on the energy yield of machines with rotating parts, such as waterwheels. That paper considered the supplementary forces that are detected in a rotating frame of reference. Coriolis divided these supplementary forces into two categories. The second category contained a force that arises from the cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

of the angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

of a coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

and the projection of a particle's velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

into a plane perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to the system's axis of rotation. Coriolis referred to this force as the "compound centrifugal force" due to its analogies with the centrifugal force
Centrifugal force
Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...

already considered in category one. By the early 20th century the effect was known as the "acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...

of Coriolis". By 1919 it was referred to as "Coriolis' force" and by 1920 as "Coriolis force".

In 1856, William Ferrel
William Ferrel
William Ferrel , an American meteorologist, developed theories which explained the mid-latitude atmospheric circulation cell in detail, and it is after him that the Ferrel cell is named. He was born in southern Pennsylvania. His family moved to what would become West Virginia in 1829...

proposed the existence of a circulation cell in the mid-latitudes with air being deflected by the Coriolis force to create the prevailing westerly winds
Westerlies
The Westerlies, anti-trades, or Prevailing Westerlies, are the prevailing winds in the middle latitudes between 30 and 60 degrees latitude, blowing from the high pressure area in the horse latitudes towards the poles. These prevailing winds blow from the west to the east, and steer extratropical...

.

Understanding the kinematics of how exactly the rotation of the Earth affects airflow was partial at first. Late in the 19th century, the full extent of the large scale interaction of pressure gradient force
The pressure gradient force is not actually a 'force' but the acceleration of air due to pressure difference . It is usually responsible for accelerating a parcel of air from a high atmospheric pressure region to a low pressure region, resulting in wind...

and deflecting force that in the end causes air masses to move 'along' isobars was understood.

## 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 the magnitude and direction of the Coriolis acceleration is

where (here and below) is the acceleration of the particle in the rotating system, is the velocity of the particle in the rotating system, and Ω is the angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

vector which has magnitude equal to the rotation rate ω and is directed along the axis of rotation of the rotating reference frame, and the × symbol represents the cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

operator.

The equation may be multiplied by the mass of the relevant object to produce the Coriolis force:
.

See fictitious force
Fictitious force
A fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame....

for a derivation.

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 frame's 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 vector cross product can be evaluated as the determinant of a matrix:

where the vectors i, j, k are unit vectors in the x, y and z directions.

## Causes

The Coriolis effect exists only when one uses a rotating reference frame. In the rotating frame it behaves exactly like a real force (that is to say, it causes acceleration and has real effects). However, Coriolis force is a consequence of inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

, and is not attributable to an identifiable originating body, as is the case for electromagnetic or nuclear forces, for example. From an analytical viewpoint, to use Newton's second law in a rotating system, Coriolis force is mathematically necessary, but it disappears in a non-accelerating, inertial frame of reference. For example, consider two children on opposite sides of a spinning roundabout (carousel
Carousel
A carousel , or merry-go-round, is an amusement ride consisting of a rotating circular platform with seats for riders...

), who are throwing a ball to each other (see Figure 1). From the children's point of view, this ball's path is curved sideways by the Coriolis effect. Suppose the roundabout spins counter-clockwise when viewed from above. From the thrower's perspective, the deflection is to the right. From the non-thrower's perspective, deflection is to left. For a mathematical formulation see Mathematical derivation of fictitious forces.

A denizen of a rotating frame, such as an astronaut in a rotating space station, very probably will find the interpretation of everyday life in terms of the Coriolis force accords more simply with intuition and experience than a cerebral reinterpretation of events from an inertial standpoint. For example, nausea due to an experienced push may be more instinctively explained by Coriolis force than by the law of inertia. See also Coriolis effect (perception)
Coriolis effect (perception)
In psychophysical perception, the Coriolis effect is the misperception of body orientation and induced nausea due to the Coriolis force ....

. In meteorology, a rotating frame (the Earth) with its Coriolis force proves a more natural framework for explanation of air movements than a hypothetical, non-rotating, inertial frame without Coriolis forces. In long-range gunnery, sight corrections for the Earth's rotation are based upon Coriolis force. These examples are described in more detail below.

The acceleration entering the Coriolis force arises from two sources of change in velocity that result from rotation: the first 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 change direction), and to the component of velocity of the object in a plane perpendicular to the axis of rotation. This gives a term .
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 is the change of velocity in space. Different positions 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 to the component of the velocity of the object in a plane perpendicular to the axis of rotation (which determines how quickly it moves between those points). This also gives a term .

## Length scales and the 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
Rossby number
The Rossby number, named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of inertial to Coriolis force, terms v\cdot\nabla v\sim U^2 / L and \Omega\times v\sim U\Omega in the Navier–Stokes equations, respectively...

, which is the ratio of the velocity, U, of a system to the product of the Coriolis parameter,, and the length scale, L, of the motion:.
The Rossby number is the ratio of inertial to Coriolis forces. A small Rossby number signifies a system which is strongly affected by Coriolis forces, and a large Rossby number signifies a system in which inertial forces dominate. For example, in tornadoes, the Rossby number is large, in low-pressure systems it is low and in oceanic systems it is of the order of unity. As a result, in tornadoes the Coriolis force is negligible, and balance is between pressure and centrifugal forces. In low-pressure systems, centrifugal force is negligible and balance is between Coriolis and pressure forces. In the oceans all three forces are comparable.

An atmospheric system moving at U = 10 m/s occupying a spatial distance of L = 1000 km (621 mi), has a Rossby number of approximately 0.1.
A man playing catch may throw the ball at U = 30 m/s in a garden of length L = 50 m. The Rossby number in this case would be about = 6000.
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 in the Northern Hemisphere landed close to, but to the right of, where they were aimed until this was noted. (Those fired in the southern hemisphere landed to the left.) In fact, it was this effect that first got the attention of Coriolis himself.

### Intuitive explanation

As the Earth turns around its axis, everything attached to it turns with it (imperceptibly to our senses). An object that is moving without being dragged along with this rotation travels in a straight motion over the turning Earth, seeming (from our rotating perspective upon the planet) to change its direction of motion as it moves, thus appearing to travel along a curved path that bends in the opposite direction to our actual motion and tracing out a path on the ground below that curves the same way. When viewed from a stationary point in space above, any land feature in the Northern Hemisphere turns counter-clockwise, and, fixing our gaze on that location, any other location in that hemisphere will rotate around it the same way. The traced ground-path of a freely moving body traveling from one point to another will therefore bend the opposite way, clockwise, which is conventionally labeled as "right," where it will be if the direction of motion is considered "ahead" and "down" is defined naturally.

### Rotating sphere

Consider a location with latitude φ on a 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 (listing components in the order East (e), North (n) and Upward (u)) are:

When considering atmospheric or oceanic dynamics, the vertical velocity is small and the vertical component of the Coriolis acceleration is small compared to gravity. For such cases, only the horizontal (East and North) components matter. The restriction of the above to the horizontal plane is (setting vu=0):

where is called the Coriolis parameter.

By setting vn = 0, it can be seen immediately that (for positive φ and ω) a movement due east results in an acceleration due south. Similarly, setting ve = 0, it is seen that a movement due north results in an acceleration due east. In general, observed horizontally, looking along the direction of the movement causing the acceleration, the acceleration always is turned 90° to the right and of the same size regardless of the horizontal orientation. That is:
As a different case, consider equatorial motion setting φ = 0°. In this case, Ω is parallel to the North or n-axis, and:

Accordingly, an eastward motion (that is, in the same direction as the rotation of the sphere) provides an upward acceleration known as the Eötvös effect
Eötvös effect
The Eötvös effect is the change in perceived gravitational force caused by the change in centrifugal acceleration resulting from eastbound or westbound velocity...

, and an upward motion produces an acceleration due west.

### Distant stars

The apparent motion of a distant star as seen from Earth is dominated by the Coriolis and centrifugal forces. Consider such a star (with mass m) located at position r, with declination
Declination
In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...

δ, so Ω · r = |r| Ω sin(δ), where Ω is the Earth's rotation vector. The star is observed to rotate about the Earth's axis with a period of one sidereal day in the opposite direction to that of the Earth's rotation, making its velocity v = –Ω × r. The fictitious force, consisting of Coriolis and centrifugal forces, is:
,

where uΩ = Ω−1Ω is a unit vector in the direction of Ω. The fictitious force Ff is thus a vector of magnitude m Ω2|r| cos(δ), perpendicular to Ω, and directed towards the center of the star's rotation on the Earth's axis, and therefore recognisable as the centripetal force that will keep the star in a circular movement around that axis.

### Meteorology

Perhaps the most important instance of the Coriolis effect is in the large-scale dynamics of the oceans and the atmosphere. In meteorology and oceanography
Oceanography
Oceanography , also called oceanology or marine science, is the branch of Earth science that studies the ocean...

, it is convenient to postulate a rotating frame of reference wherein the Earth is stationary. In accommodation of that provisional postulation, the otherwise fictitious centrifugal and Coriolis forces are introduced. Their relative importance is determined by the applicable Rossby numbers. Tornado
A tornado is a violent, dangerous, rotating column of air that is in contact with both the surface of the earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. They are often referred to as a twister or a cyclone, although the word cyclone is used in meteorology in a wider...

es have high Rossby numbers, so, while tornado-associated centrifugal forces are quite substantial, Coriolis forces associated with tornados are for practical purposes negligible.

High pressure systems rotate in a direction such that the Coriolis force will be directed radially inwards, and nearly balanced by the outwardly radial pressure gradient. This direction is clockwise in the northern hemisphere and counter-clockwise in the southern hemisphere. Low pressure systems rotate in the opposite direction, so that the Coriolis force is directed radially outward and nearly balances an inwardly radial pressure gradient. In each case a slight imbalance between the Coriolis force and the pressure gradient accounts for the radially inward acceleration of the system's circular motion
Circular motion
In physics, circular motion is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation , or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of...

.

#### 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 force. A system of equilibrium can then establish itself creating circular movement, or a cyclonic flow. Because the Rossby number is low, the force balance is largely between the pressure gradient force
The pressure gradient force is not actually a 'force' but the acceleration of air due to pressure difference . It is usually responsible for accelerating a parcel of air from a high atmospheric pressure region to a low pressure region, resulting in wind...

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, large scale motions in the atmosphere and ocean tend to occur perpendicular to the pressure gradient. This is known as geostrophic flow. On a non-rotating planet fluid would flow along the straightest possible line, quickly eliminating pressure gradients. Note that the geostrophic balance is thus very different from the case of "inertial motions" (see below) which explains why mid-latitude cyclones 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
In meteorology, Buys Ballot's law may be expressed as follows: In the Northern Hemisphere, if a person stands with his back to the wind, the low pressure area will be on his left. This is because wind travels counterclockwise around low pressure zones in the Northern Hemisphere...

. In the atmosphere, the pattern of flow is called a cyclone
Cyclone
In meteorology, a cyclone is an area of closed, circular fluid motion rotating in the same direction as the Earth. This is usually characterized by inward spiraling winds that rotate anticlockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere of the Earth. Most large-scale...

. In the Northern Hemisphere the direction of movement around a low-pressure area is anticlockwise. In the Southern Hemisphere, the direction of movement is clockwise because the rotational dynamics is a mirror image there. At high altitudes, outward-spreading air rotates in the opposite direction. Cyclones rarely form along the equator due to the weak Coriolis effect present in this region.

#### Inertial circles

An air or water mass moving with speed subject only to the Coriolis force travels in a circular trajectory called an 'inertial circle'. Since the force is directed at right angles to the motion of the particle, it will move with a constant speed, and perform a complete circle with frequency . The magnitude of the Coriolis force also determines the radius of this circle:

On the Earth, a typical mid-latitude value for is 10−4 s−1; hence for a typical atmospheric speed of 10 m/s the radius is 100 km (62 mi), with a period of about 14 hours. For an ocean current with a typical speed of 10 cm/s, the radius of an inertial circle is 1 km (0.621372736649807 mi). 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 is constant and the trajectories are exact circles. On a rotating planet, varies with latitude and the paths of particles do not form exact circles. Since the parameter varies as the sine of the latitude, the radius of the oscillations associated with a given speed are smallest at the poles (latitude = ±90°), and increase toward the equator.

#### Other terrestrial effects

The Coriolis effect strongly affects the large-scale oceanic and atmospheric circulation
Atmospheric circulation
Atmospheric circulation is the large-scale movement of air, and the means by which thermal energy is distributed on the surface of the Earth....

, leading to the formation of robust features like jet stream
Jet stream
Jet streams are fast flowing, narrow air currents found in the atmospheres of some planets, including Earth. The main jet streams are located near the tropopause, the transition between the troposphere and the stratosphere . The major jet streams on Earth are westerly winds...

s and western boundary currents. Such features are in geostrophic
Geostrophic
A geostrophic current is an oceanic flow in which the pressure gradient force is balanced by the Coriolis force. The direction of geostrophic flow is parallel to the isobars, with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the...

balance, meaning that the Coriolis and pressure gradient forces balance each other. Coriolis acceleration is also responsible for the propagation of many types of waves in the ocean and atmosphere, including Rossby wave
Rossby wave
Atmospheric Rossby waves are giant meanders in high-altitude winds that are a major influence on weather.They are not to be confused with oceanic Rossby waves, which move along the thermocline: that is, the boundary between the warm upper layer of the ocean and the cold deeper part of the...

s and Kelvin wave
Kelvin wave
A Kelvin wave is a wave in the ocean or atmosphere that balances the Earth's Coriolis force against a topographic boundary such as a coastline, or a waveguide such as the equator. A feature of a Kelvin wave is that it is non-dispersive, i.e., the phase speed of the wave crests is equal to the...

s. It is also instrumental in the so-called Ekman dynamics in the ocean, and in the establishment of the large-scale ocean flow pattern called the Sverdrup balance
Sverdrup balance
The Sverdrup balance, or Sverdrup relation, is a theoretical relationship between the wind stress exerted on the surface of the open ocean and the vertically integrated meridional transport of ocean water.- History :...

.

### Eötvös effect

The practical impact of the Coriolis effect is mostly caused by the horizontal acceleration component produced by horizontal motion.

There are other components of the Coriolis effect. 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
Eötvös effect
The Eötvös effect is the change in perceived gravitational force caused by the change in centrifugal acceleration resulting from eastbound or westbound velocity...

. This aspect of the Coriolis effect is greatest near the equator. The force produced by this effect is similar to the horizontal component, but the much larger vertical forces due to gravity and pressure mean that it is generally unimportant dynamically.

In addition, objects traveling upwards or downwards will be deflected to the west or east respectively. This effect is also the greatest near the equator. Since vertical movement is usually of limited extent and duration, the size of the effect is smaller and requires precise instruments to detect.

### Draining in bathtubs and toilets

In 1908, the Austrian physicist Otto Tumlirz
Otto Tumlirz
Otto Tumlirz, or Ota Tumlíř was a Czech-Austrian psychologist, researcher for pedagogy...

described careful and effective experiments which demonstrated the effect of the rotation of the Earth on the outflow of water through a central aperture. The subject was later popularized in a famous article in the journal Nature
Nature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...

, which described an experiment in which all other forces to the system were removed by filling a 6 feet (1.8 m) tank with 300 gallons (1,135.6 l) of water and allowing it to settle for 24 hours (to allow any movement due to filling the tank to die away), in a room where the temperature had stabilized. The drain plug was then very slowly removed, and tiny pieces of floating wood were used to observe rotation. During the first 12 to 15 minutes, no rotation was observed. Then, a vortex appeared and consistently began to rotate in a counter-clockwise direction (the experiment was performed in Boston, Massachusetts, in the Northern hemisphere). This was repeated and the results averaged to make sure the effect was real. The report noted that the vortex rotated, "about 30,000 times faster than the effective rotation of the earth in 42° North (the experiment's location)". This shows that the small initial rotation due to the earth is amplified by gravitational draining and conservation of angular momentum to become a rapid vortex and may be observed under carefully controlled laboratory
Laboratory
A laboratory is a facility that provides controlled conditions in which scientific research, experiments, and measurement may be performed. The title of laboratory is also used for certain other facilities where the processes or equipment used are similar to those in scientific laboratories...

conditions.

The formation of a vortex over the plug hole may be explained by the conservation of angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

. The radius of rotation decreases as water approaches the plug hole so the rate of rotation increases, for the same reason that an ice skater's rate of spin increases as she pulls her arms in. Any rotation around the plug hole that is initially present accelerates as water moves inward. If the water is so still that the effective rotation rate of the earth (once per day at the poles, once every 2 days at 30 degrees of latitude) is faster than that of the water relative to its container, and if externally applied torques (such as might be caused by flow over an uneven bottom surface) are small enough, the Coriolis effect may determine the direction of the vortex. Without such careful preparation, the Coriolis effect may be much smaller than various other influences on drain direction, such as any residual rotation of the water
and the geometry of the container.
Despite this, the idea that toilets and bathtubs drain differently in the Northern and Southern Hemispheres has been popularized by several television programs, including The Simpsons
The Simpsons
The Simpsons is an American animated sitcom created by Matt Groening for the Fox Broadcasting Company. The series is a satirical parody of a middle class American lifestyle epitomized by its family of the same name, which consists of Homer, Marge, Bart, Lisa and Maggie...

episode "Bart vs. Australia
Bart vs. Australia
"Bart vs. Australia" is the sixteenth episode of the sixth season of The Simpsons. It originally aired on the Fox network in the United States on February 19, 1995. In the episode, Bart is indicted for fraud in Australia, and the family travels to the country so Bart can apologize...

" and The X-Files
The X-Files
The X-Files is an American science fiction television series and a part of The X-Files franchise, created by screenwriter Chris Carter. The program originally aired from to . The show was a hit for the Fox network, and its characters and slogans became popular culture touchstones in the 1990s...

episode "Die Hand Die Verletzt
Die Hand Die Verletzt
"Die Hand Die Verletzt" is the fourteenth episode of the second season of the science fiction television series The X-Files. It premiered on the Fox network on January 27, 1995. It was written by Glen Morgan and James Wong, directed by Kim Manners, and featured guest appearances by Susan Blommaert,...

". Several science broadcasts and publications, including at least one college-level physics textbook, have also stated this.

### Ballistic missiles and satellites

Ballistic missiles and satellites appear to follow curved paths when plotted on common world maps mainly because the Earth is spherical and the shortest distance between two points on the Earth's surface (called a great circle
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

) is usually not a straight line on those maps. Every two-dimensional (flat) map necessarily distorts the Earth's curved (three-dimensional) surface. Typically (as in the commonly used Mercator projection
Mercator projection
The Mercator projection is a cylindrical map projection presented by the Belgian geographer and cartographer Gerardus Mercator, in 1569. It became the standard map projection for nautical purposes because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as...

, 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 effects became important in external ballistics for calculating the trajectories of very long-range artillery
Artillery
Originally applied to any group of infantry primarily armed with projectile weapons, artillery has over time become limited in meaning to refer only to those engines of war that operate by projection of munitions far beyond the range of effect of personal weapons...

shells. The most famous historical example was the Paris gun
Paris Gun
The Paris Gun was a German long-range siege gun used to bombard Paris during World War I. It was in service from March-August 1918. When it was first employed, Parisians believed they'd been bombed by a new type of high-altitude zeppelin, as neither the sound of an airplane nor a gun could be heard...

, used by the Germans during World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...

to bombard Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

from a range of about 120 km (74.6 mi).

### Cannon on turntable

Figure 1 is an animation of the classic illustration of Coriolis force. Another visualization of the Coriolis and centrifugal forces is this animation clip. Figure 3 is a graphical version.

Here is a question: given the radius of the turntable R, the rate of angular rotation ω, and the speed of the cannonball (assumed constant) v, what is the correct angle θ to aim so as to hit the target at the edge of the turntable?

The inertial frame of reference provides one way to handle the question: calculate the time to interception, which is tf = R / v . Then, the turntable revolves an angle ω tf in this time. If the cannon is pointed an angle θ = ω tf = ω R / v, then the cannonball arrives at the periphery at position number 3 at the same time as the target.

No discussion of Coriolis force can arrive at this solution as simply, so the reason to treat this problem is to demonstrate Coriolis formalism in an easily visualized situation.

The trajectory in the inertial frame (denoted A) is a straight line radial path at angle θ. The position of the cannonball in (x, y) coordinates at time t is:
In the turntable frame (denoted B), the x- y axes rotate at angular rate ω, so the trajectory becomes:
and three examples of this result are plotted in Figure 4.

To determine the components of acceleration, a general expression is used from the article fictitious force
Fictitious force
A fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame....

:
in which the term in Ω × vB is the Coriolis acceleration and the term in Ω × ( Ω × rB) is the centrifugal acceleration. The results are (let α = θ − ωt):
producing a centrifugal acceleration:
Also:
producing a Coriolis acceleration:

Figure 5 and Figure 6 show these accelerations for a particular example.

It is seen that the Coriolis acceleration not only cancels the centrifugal acceleration, but together they provide a net "centripetal", radially inward component of acceleration (that is, directed toward the centre of rotation):

and an additional component of acceleration perpendicular to rB (t):
The "centripetal" component of acceleration resembles that for circular motion
Circular motion
In physics, circular motion is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation , or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of...

at radius rB, while the perpendicular component is velocity dependent, increasing with the radial velocity v and directed to the right of the velocity. The situation could be described as a circular motion combined with an "apparent Coriolis acceleration" of 2ωv. However, this is a rough labelling: a careful designation of the true centripetal force refers to a local reference frame that employs the directions normal and tangential to the path, not coordinates referred to the axis of rotation.

These results also can be obtained directly by two time differentiations of rB (t). Agreement of the two approaches demonstrates that one could start from the general expression for fictitious acceleration above and derive the trajectories of Figure 4. However, working from the acceleration to the trajectory is more complicated than the reverse procedure used here, which, of course, is made possible in this example by knowing the answer in advance.

As a result of this analysis an important point appears: all the fictitious accelerations must be included to obtain the correct trajectory. In particular, besides the Coriolis acceleration, the centrifugal force plays an essential role. It is easy to get the impression from verbal discussions of the cannonball problem, which are focussed on displaying the Coriolis effect particularly, that the Coriolis force is the only factor that must be considered; emphatically, that is not so. A turntable for which the Coriolis force is the only factor is the parabolic turntable. A somewhat more complex situation is the idealized example of flight routes over long distances, where the centrifugal force of the path and aeronautical lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

are countered by gravitational attraction
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

.

### Tossed ball on a rotating carousel

Figure 7 illustrates a ball tossed from 12:00 o'clock toward the centre of an anticlockwise rotating carousel. On the left, the ball is seen by a stationary observer above the carousel, and the ball travels in a straight line to the centre, while the ball-thrower rotates anticlockwise with the carousel. On the right the ball is seen by an observer rotating with the carousel, so the ball-thrower appears to stay at 12:00 o'clock. The figure shows how the trajectory of the ball as seen by the rotating observer can be constructed.

On the left, two arrows locate the ball relative to the ball-thrower. One of these arrows is from the thrower to the centre of the carousel (providing the ball-thrower's line of sight), and the other points from the centre of the carousel to the ball.(This arrow gets shorter as the ball approaches the centre.) A shifted version of the two arrows is shown dotted.

On the right is shown this same dotted pair of arrows, but now the pair are rigidly rotated so the arrow corresponding to the line of sight of the ball-thrower toward the centre of the carousel is aligned with 12:00 o'clock. The other arrow of the pair locates the ball relative to the centre of the carousel, providing the position of the ball as seen by the rotating observer. By following this procedure for several positions, the trajectory in the rotating frame of reference is established as shown by the curved path in the right-hand panel.

The ball travels in the air, and there is no net force upon it. To the stationary observer the ball follows a straight-line path, so there is no problem squaring this trajectory with zero net force. However, the rotating observer sees a curved path. Kinematics insists that a force (pushing to the right of the instantaneous direction of travel for an anticlockwise rotation) must be present to cause this curvature, so the rotating observer is forced to invoke a combination of centrifugal and Coriolis forces to provide the net force required to cause the curved trajectory.

### Bounced ball

Figure 8 describes a more complex situation where the tossed ball on a turntable bounces off the edge of the carousel and then returns to the tosser, who catches the ball. The effect of Coriolis force on its trajectory is shown again as seen by two observers: an observer (referred to as the "camera") that rotates with the carousel, and an inertial observer. Figure 8 shows a bird's-eye view based upon the same ball speed on forward and return paths. Within each circle, plotted dots show the same time points. In the left panel, from the camera's viewpoint at the center of rotation, the tosser (smiley face) and the rail both are at fixed locations, and the ball makes a very considerable arc on its travel toward the rail, and takes a more direct route on the way back. From the ball tosser's viewpoint, the ball seems to return more quickly than it went (because the tosser is rotating toward the ball on the return flight).

On the carousel, instead of tossing the ball straight at a rail to bounce back, the tosser must throw the ball toward the right of the target and the ball then seems to the camera to bear continuously to the left of its direction of travel to hit the rail (left because the carousel is turning clockwise). The ball appears to bear to the left from direction of travel on both inward and return trajectories. The curved path demands this observer to recognize a leftward net force on the ball. (This force is "fictitious" because it disappears for a stationary observer, as is discussed shortly.) For some angles of launch, a path has portions where the trajectory is approximately radial, and Coriolis force is primarily responsible for the apparent deflection of the ball (centrifugal force is radial from the center of rotation, and causes little deflection on these segments). When a path curves away from radial, however, centrifugal force contributes significantly to deflection.

The ball's path through the air is straight when viewed by observers standing on the ground (right panel). In the right panel (stationary observer), the ball tosser (smiley face) is at 12 o'clock and the rail the ball bounces from is at position one (1). From the inertial viewer's standpoint, positions one (1), two (2), three (3) are occupied in sequence. At position 2 the ball strikes the rail, and at position 3 the ball returns to the tosser. Straight-line paths are followed because the ball is in free flight, so this observer requires that no net force is applied.

## Visualization of the Coriolis effect

To demonstrate the Coriolis effect, a parabolic turntable can be used. On a flat turntable, the inertia of a co-rotating object would force it off the edge. But if the surface of the turntable has the correct parabolic bowl shape (see Figure 9) and is rotated at the correct rate, the force components shown in Figure 10 are arranged so the component of gravity tangential to the bowl surface will exactly equal the centripetal force necessary to keep the object rotating at its velocity and radius of curvature (assuming no friction). (See banked turn.) This carefully contoured surface allows the Coriolis force to be displayed in isolation.

Discs cut from cylinders of dry ice
Dry ice
Dry ice, sometimes referred to as "Cardice" or as "card ice" , is the solid form of carbon dioxide. It is used primarily as a cooling agent. Its advantages include lower temperature than that of water ice and not leaving any residue...

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 the reference frame rotating with the turntable, a video camera is attached to the turntable so as to co-rotate with the turntable, with results as shown in Figure 11. In the left panel of Figure 11, which is the viewpoint of a stationary observer, the gravitational force in the inertial frame pulling the object toward the center (bottom ) of the dish is proportional to the distance of the object from the center. A centripetal force of this form causes the elliptical motion. In the right panel, which shows the viewpoint of the rotating frame, the inward gravitational force in the rotating frame (the same force as in the inertial frame) is balanced by the outward centrifugal force (present only in the rotating frame). With these two forces balanced, in the rotating frame the only unbalanced force is Coriolis (also present only in the rotating frame), and the motion is an inertial circle. Analysis and observation of circular motion in the rotating frame is a simplification compared to analysis or observation of elliptical motion in the inertial frame.

Because this reference frame rotates several times a minute rather than only once a day like the Earth, the Coriolis acceleration produced is many times larger and so easier to observe on small time and spatial scales than is the Coriolis acceleration caused by the rotation of the Earth.

In a manner of speaking, the Earth is analogous to such a turntable. The rotation has caused the planet to settle on a spheroid shape, such that the normal force, the gravitational force and the centrifugal force exactly balance each other on a "horizontal" surface. (See equatorial bulge
Equatorial bulge
An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force of its rotation. A rotating body tends to form an oblate spheroid rather than a sphere...

.)

The Coriolis effect caused by the rotation of the Earth can be seen indirectly through the motion of a Foucault pendulum
Foucault pendulum
The Foucault pendulum , or Foucault's pendulum, named after the French physicist Léon Foucault, is a simple device conceived as an experiment to demonstrate the rotation of the Earth. While it had long been known that the Earth rotated, the introduction of the Foucault pendulum in 1851 was the...

.

### Coriolis flow meter

A practical application of the Coriolis effect is the mass flow meter
Mass flow meter
A mass flow meter, also known as an inertial flow meter is a device that measures mass flow rate of a fluid traveling through a tube. The mass flow rate is the mass of the fluid traveling past a fixed point per unit time....

, an instrument that measures the mass flow rate
Mass flow rate
Mass flow rate is the mass of substance which passes through a given surface per unit time. Its unit is mass divided by time, so kilogram per second in SI units, and slug per second or pound per second in US customary units...

and density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

of a fluid flowing through a tube. The operating principle involves inducing a vibration of the tube through which the fluid passes. The vibration, though it is not completely circular, provides the rotating reference frame which gives rise to the Coriolis effect. While specific methods vary according to the design of the flow meter, sensors monitor and analyze changes in frequency, phase shift, and amplitude of the vibrating flow tubes. The changes observed represent the mass flow rate and density of the fluid.

### 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
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...

.

### Insect flight

Flies (Diptera
Diptera
Diptera , or true flies, is the order of insects possessing only a single pair of wings on the mesothorax; the metathorax bears a pair of drumstick like structures called the halteres, the remnants of the hind wings. It is a large order, containing an estimated 240,000 species, although under half...

) and moths (Lepidoptera
Lepidoptera
Lepidoptera is a large order of insects that includes moths and butterflies . It is one of the most widespread and widely recognizable insect orders in the world, encompassing moths and the three superfamilies of butterflies, skipper butterflies, and moth-butterflies...

) utilize the Coriolis effect when flying: their halteres
Halteres
Halteres are small knobbed structures modified from the hind wings in some two-winged insects. They are flapped rapidly and function as gyroscopes, informing the insect about rotation of the body during flight....

, or antennae in the case of moths, oscillate rapidly and are used as vibrational gyroscopes. See Coriolis effect in insect stability. In this context, the Coriolis effect has nothing to do with the rotation of the Earth.

• Analytical mechanics
Analytical mechanics
Analytical mechanics is a term used for a refined, mathematical form of classical mechanics, constructed from the 18th century onwards as a formulation of the subject as founded by Isaac Newton. Often the term vectorial mechanics is applied to the form based on Newton's work, to contrast it with...

• Applied mechanics
Applied mechanics
Applied mechanics is a branch of the physical sciences and the practical application of mechanics. Applied mechanics examines the response of bodies or systems of bodies to external forces...

• Centrifugal force
Centrifugal force
Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...

• Centrifugal force (rotating reference frame)
• Centripetal force
Centripetal force
Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...

• Classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

• Dynamics (physics)
• Earth's rotation
• Equatorial Rossby wave
• Frenet-Serret formulas
Frenet-Serret formulas
In vector calculus, the Frenet–Serret formulas describe the kinematic properties of a particle which moves along a continuous, differentiable curve in three-dimensional Euclidean space R3...

• Geostrophic wind
Geostrophic wind
The geostrophic wind is the theoretical wind that would result from an exact balance between the Coriolis effect and the pressure gradient force. This condition is called geostrophic balance. The geostrophic wind is directed parallel to isobars . This balance seldom holds exactly in nature...

• Gyroscope
Gyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...

• Kinetics (physics)
Kinetics (physics)
In physics and engineering, kinetics is a term for the branch of classical mechanics that is concerned with the relationship between the motion of bodies and its causes, namely forces and torques...

• Mass flow meter
Mass flow meter
A mass flow meter, also known as an inertial flow meter is a device that measures mass flow rate of a fluid traveling through a tube. The mass flow rate is the mass of the fluid traveling past a fixed point per unit time....

• Mechanics of planar particle motion
Mechanics of planar particle motion
This article describes a particle in planar motion when observed from non-inertial reference frames. The most famous examples of planar motion are related to the motion of two spheres that are gravitationally attracted to one another, and the generalization of this problem to planetary motion....

• Reactive centrifugal force
Reactive centrifugal force
In classical mechanics, reactive centrifugal force is the reaction paired with centripetal force. A mass undergoing circular motion constantly accelerates toward the axis of rotation. This centripetal acceleration is caused by a force exerted on the mass by some other object. In accordance with...

• Secondary flow
Secondary flow
In fluid dynamics, a secondary flow is a relatively minor flow superimposed on the primary flow, where the primary flow usually matches very closely the flow pattern predicted using simple analytical techniques and assuming the fluid is inviscid...

• Statics
Statics
Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...

• Uniform circular motion
Uniform circular motion
In physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. The distance of the body from the axis of rotation remains constant at all times. Though the body's speed is constant, its velocity is not constant: velocity, a vector quantity, depends...

### Further reading: physics and meteorology

• Riccioli, G.B., 1651: Almagestum Novum, Bologna, pp. 425–427
(Original book [in Latin], scanned images of complete pages.)
• Coriolis, G.G., 1832: Mémoire sur le principe des forces vives dans les mouvements relatifs des machines. Journal de l'école Polytechnique, Vol 13, 268–302.
(Original article [in French], PDF-file, 1.6 MB, scanned images of complete pages.)
• Coriolis, G.G., 1835: Mémoire sur les équations du mouvement relatif des systèmes de corps. Journal de l'école Polytechnique, Vol 15, 142–154
(Original article [in French] PDF-file, 400 KB, scanned images of complete pages.)
• Gill, AE Atmosphere-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