Coriolis force

From Wikipedia, the free encyclopedia

(Redirected from Coriolis effect)
"Coriolis effect" redirects here. For the effect in psychophysical perception,
see Coriolis effect (perception). For the film, see The Coriolis Effect.

In the inertial frame of reference (upper part of the

picture), the black ball moves in a straight line. However, the observer (red dot) who
is standing in the rotating/non-inertial frame of reference (lower part of the picture)
sees the object as following a curved path due to the Coriolis and centrifugal forces
present in this frame.[1]

Part of a series on

Classical mechanics

Second law of motion

 History
 Timeline
 Textbooks
Branches









Fundamentals





o
o



















Formulations



o
o
o
o
o
o

Core topics














o
o



Rotation




o





Scientists















 









 Physics portal
 Category

 v
 t
 e

In physics, the Coriolis force is a pseudo force that acts on objects in motion within
a frame of reference that rotates with respect to an inertial frame. In a reference
frame with clockwise rotation, the force acts to the left of the motion of the object. In
one with anticlockwise (or counterclockwise) rotation, the force acts to the
right. Deflection of an object due to the Coriolis force is called the Coriolis effect.
Though recognized previously by others, the mathematical expression for the
Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de
Coriolis, in connection with the theory of water wheels. Early in the 20th century, the
term Coriolis force began to be used in connection with meteorology.

Newton's laws of motion describe the motion of an object in an inertial (non-

accelerating) frame of reference. When Newton's laws are transformed to a rotating
frame of reference, the Coriolis and centrifugal accelerations appear. When applied
to objects with masses, the respective forces are proportional to their masses. The
magnitude of the Coriolis force is proportional to the rotation rate, and the magnitude
of the centrifugal force is proportional to the square of the rotation rate. The Coriolis
force acts in a direction perpendicular to two quantities: the angular velocity of the
rotating frame relative to the inertial frame and the velocity of the body relative to the
rotating frame, and its magnitude is proportional to the object's speed in the rotating
frame (more precisely, to the component of its velocity that is perpendicular to the
axis of rotation). 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 inertial forces, fictitious forces, or pseudo forces. By
introducing these fictitious forces to a rotating frame of reference, Newton's laws of
motion can be applied to the rotating system as though it were an inertial system;
these forces are correction factors that are not required in a non-rotating system.

In popular (non-technical) usage of the term "Coriolis effect", the rotating reference
frame implied is almost always the Earth. Because the Earth spins, Earth-bound
observers need to account for the Coriolis force to correctly analyze the motion of
objects. The Earth completes one rotation for each sidereal day, so for motions of
everyday objects the Coriolis force is imperceptible; its effects 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, or where high
precision is important, such as artillery or missile trajectories. Such motions are
constrained by the 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 be deflected to the right (with respect to the direction of travel) in
the Northern Hemisphere and to the left in the Southern Hemisphere. The horizontal
deflection effect is greater near the poles, since the effective rotation rate about a
local vertical axis is largest there, and decreases to zero at the equator. Rather than
flowing directly from areas of high pressure to low pressure, as they would in a non-
rotating system, winds and currents tend to flow to the right of this direction north of
the equator ("clockwise") and to the left of this direction south of it ("anticlockwise").
This effect is responsible for the rotation and thus formation
of cyclones (see: Coriolis effects in meteorology).

History

Image from Cursus seu Mundus Mathematicus (1674) of

C.F.M. Dechales, showing how a cannonball should deflect to the right of its target
on a rotating Earth, because the rightward motion of the ball is faster than that of the

tower. Image from Cursus seu Mundus

Mathematicus (1674) of C.F.M. Dechales, showing how a ball should fall from a
tower on a rotating Earth. The ball is released from F. The top of the tower moves
faster than its base, so while the ball falls, the base of the tower moves to I, but the
ball, which has the eastward speed of the tower's top, outruns the tower's base and
lands further to the east at L.
Italian scientist Giovanni Battista Riccioli and his assistant Francesco Maria
Grimaldi described the effect in connection with artillery in the 1651 Almagestum
Novum, writing that rotation of the Earth should cause a cannonball fired to the north
to deflect to the east.[2] In 1674, Claude François Milliet Dechales described in
his Cursus seu Mundus Mathematicus how the rotation of the Earth should cause a
deflection in the trajectories of both falling bodies and projectiles aimed toward one
of the planet's poles. Riccioli, Grimaldi, and Dechales all described the effect as part
of an argument against the heliocentric system of Copernicus. In other words, they
argued that the Earth's rotation should create the effect, and so failure to detect the
effect was evidence for an immobile Earth.[3] The Coriolis acceleration equation was
derived by Euler in 1749,[4][5] and the effect was described in the tidal
equations of Pierre-Simon Laplace in 1778.[6]

Gaspard-Gustave de Coriolis published a paper in 1835 on the energy yield of

machines with rotating parts, such as waterwheels.[7][8] 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 of the angular velocity of
a coordinate system and the projection of a particle's velocity into a
plane perpendicular 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 already considered in category one.[9][10] The effect was known in the early 20th
century as the "acceleration of Coriolis",[11] and by 1920 as "Coriolis force".[12]

In 1856, William Ferrel 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.[13]

The understanding of the kinematics of how exactly the rotation of the Earth affects
airflow was partial at first.[14] Late in the 19th century, the full extent of the large scale
interaction of pressure-gradient force and deflecting force that in the end causes air
masses to move along isobars was understood.[15]

Formula
See also: Fictitious force
In Newtonian mechanics, the equation of motion for an object in an inertial reference
frame is:where is the vector sum of the physical forces acting on the object, is the
mass of the object, and is the acceleration of the object relative to the inertial
reference frame.
Transforming this equation to a reference frame rotating about a fixed axis through
the origin with angular velocity having variable rotation rate, the equation takes the
form:[8][16]where the prime (') variables denote coordinates of the rotating reference
frame (not a derivative) and:
 is the vector sum of the physical forces acting on the object
 is the angular velocity of the rotating reference frame relative to the inertial frame
 is the position vector of the object relative to the rotating reference frame
 is the velocity of the object relative to the rotating reference frame
 is the acceleration of the object relative to the rotating reference frame
The fictitious forces as they are perceived in the rotating frame act as additional
forces that contribute to the apparent acceleration just like the real external forces. [17]
[18][19]
The fictitious force terms of the equation are, reading from left to right: [20]

 Euler force,
 Coriolis force,
 centrifugal force,
As seen in these formulas the Euler and centrifugal forces depend on the position
vector of the object, while the Coriolis force depends on the object's velocity as
measured in the rotating reference frame. As expected, for a non-rotating inertial
frame of reference the Coriolis force and all other fictitious forces disappear.[21]

Direction of Coriolis force for simple cases

As the Coriolis force is proportional to a cross product of two vectors, it is
perpendicular to both vectors, in this case the object's velocity and the frame's
rotation vector. It therefore follows that:

 if the velocity is parallel to the rotation axis, the Coriolis force is zero. For
example, on Earth, this situation occurs for a body at the equator moving north or
south relative to the Earth's surface. (At any latitude other than the equator,
however, the north–south motion would have a component perpendicular to the
rotation axis and a force specified by the inward or outward cases mentioned
below).
 if the velocity is straight inward to the axis, the Coriolis force is in the direction of
local rotation. For example, on Earth, this situation occurs for a body at the
equator falling downward, as in the Dechales illustration above, where the falling
ball travels further to the east than does the tower. Note also that heading north
in the Northern Hemisphere would have a velocity component toward the rotation
axis, resulting in a Coriolis force to the east (more pronounced the further north
one is).
 if the velocity is straight outward from the axis, the Coriolis force is against the
direction of local rotation. In the tower example, a ball launched upward would
move toward the west.
 if the velocity is in the direction of rotation, the Coriolis force is outward from the
axis. For example, on Earth, this situation occurs for a body at the equator
moving east relative to Earth's surface. It would move upward as seen by an
observer on the surface. This effect (see Eötvös effect below) was discussed by
Galileo Galilei in 1632 and by Riccioli in 1651.[22]
 if the velocity is against the direction of rotation, the Coriolis force is inward to the
axis. For example, on Earth, this situation occurs for a body at the equator
moving west, which would deflect downward as seen by an observer.
Intuitive explanation
For an intuitive explanation of the origin of the Coriolis force, consider an object,
constrained to follow the Earth's surface and moving northward in the Northern
Hemisphere. Viewed from outer space, the object does not appear to go due north,
but has an eastward motion (it rotates around toward the right along with the surface
of the Earth). The further north it travels, the smaller the "radius of its parallel
(latitude)" (the minimum distance from the surface point to the axis of rotation, which
is in a plane orthogonal to the axis), and so the slower the eastward motion of its
surface. As the object moves north it has a tendency to maintain the eastward speed
it started with (rather than slowing down to match the reduced eastward speed of
local objects on the Earth's surface), so it veers east (i.e. to the right of its initial
motion).[23][24]
Though not obvious from this example, which considers northward motion, the
horizontal deflection occurs equally for objects moving eastward or westward (or in
any other direction).[25] However, the theory that the effect determines the rotation of
draining water in a household bathtub, sink or toilet has been repeatedly disproven
by modern-day scientists; the force is negligibly small compared to the many other
influences on the rotation.[26][27][28]

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 force. Whether rotation is important in a system can be determined by
its Rossby number (Ro), 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:Hence, it is
the ratio of inertial to Coriolis forces; a small Rossby number indicates a system is
strongly affected by Coriolis forces, and a large Rossby number indicates a system
in which inertial forces dominate. For example, in tornadoes, the Rossby number is
large, so in them the Coriolis force is negligible, and balance is between pressure
and centrifugal forces. In low-pressure systems the Rossby number is low, as the
centrifugal force is negligible; there, the balance is between Coriolis and pressure
forces. In oceanic systems the Rossby number is often around 1, with all three
forces comparable.[29]
An atmospheric system moving at U = 10 m/s (22 mph) occupying a spatial distance
of L = 1,000 km (621 mi), has a Rossby number of approximately 0.1.[30]

An unguided missile can travel far enough and be in the air long enough to
experience the effect of Coriolis force. Long-range shells in the Northern
Hemisphere can land to the right of where they were aimed until the effect was noted
(those fired in the Southern Hemisphere landed to the left.) It was this effect that first
drew the attention of Coriolis himself.[31][32][33]

Simple cases
Tossed ball on a rotating carousel

Left Figure: The trajectory of a ball

thrown from the edge of a rotating disc, as seen by an external observer. Because of
the rotation, the ball has both an initial tangential velocity and a radial velocity given
by the thrower. These velocities bring it to the right of the center. Right Figure: The
trajectory of a ball thrown from the edge of a rotating disc, as seen by the thrower,
the rotating observer. It is deviating from the straight line.
The figures illustrate a ball tossed from 12:00 o'clock toward the center of a counter-
clockwise rotating carousel. In the first figure, the ball is seen by a stationary
observer above the carousel, and the ball travels in a straight line slightly to the right
of the center, because it had an initial tangential velocity given by the rotation (blue
arrow) and a radial velocity given by the thrower (green arrow). The resulting
combined velocity is shown as a solid red line, and the trajectory is shown as a
dotted red line. In the second figure, the ball is seen by an observer rotating with the
carousel, so the ball-thrower appears to stay at 12:00 o'clock, and the ball trajectory
has a slight curve.

Bounced ball

Bird's-eye view of carousel. The carousel

rotates clockwise. Two viewpoints are illustrated: that of the camera at the center of
rotation rotating with the carousel (left panel) and that of the inertial (stationary)
observer (right panel). Both observers agree at any given time just how far the ball is
from the center of the carousel, but not on its orientation. Time intervals are 1/10 of
time from launch to bounce.
The figure 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. The figure 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).[citation needed]

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.[citation needed]

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 1. From the
inertial viewer's standpoint, positions 1, 2, and 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.

Applied to the Earth

The acceleration affecting the motion of air "sliding" over the Earth's surface is the
horizontal component of the Coriolis termThis component is orthogonal to the
velocity over the Earth surface and is given by the expressionwhere
 is the spin rate of the Earth
 is the latitude, positive in the Northern Hemisphere and negative in the Southern
Hemisphere
In the Northern Hemisphere, where the latitude is positive, this acceleration, as
viewed from above, is to the right of the direction of motion. Conversely, it is to the
left in the southern hemisphere.

Rotating sphere

Coordinate system at latitude φ with x-axis

east, y-axis north, and z-axis upward (i.e. radially outward from center of 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:[34]When considering atmospheric or oceanic dynamics, the vertical velocity is
small, and the vertical component of the Coriolis acceleration () is small compared
with the acceleration due to gravity (g, approximately 9.81 m/s2 (32.2 ft/s2) near
Earth's surface). For such cases, only the horizontal (east and north) components
matter.[citation needed] The restriction of the above to the horizontal plane is (setting vu = 0):
[citation needed]
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.[citation needed] In general,
observed horizontally, looking along the direction of the movement causing the
acceleration, the acceleration always is turned 90° to the right (for positive φ) and of
the same size regardless of the horizontal orientation.[citation needed]

In the case of equatorial motion, setting φ = 0° yields:Ω in this case is parallel to the
north–south axis.
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, and an upward
motion produces an acceleration due west.[citation needed][35]

For additional examples in other articles, see rotating spheres, apparent motion of
stationary objects, and carousel.
Meteorology and oceanography

Due to the Coriolis force, low-pressure systems

in the Northern Hemisphere, like Typhoon Nanmadol (left), rotate counterclockwise,
and in the Southern Hemisphere, low-pressure systems like Cyclone Darian (right)

rotate clockwise. Schematic representation of flow

around a low-pressure area in the Northern Hemisphere. The Rossby number is low,
so the centrifugal force is virtually negligible. The pressure-gradient force is
represented by blue arrows, the Coriolis acceleration (always perpendicular to the

velocity) by red arrows Schematic representation

of inertial circles of air masses in the absence of other forces, calculated for a wind
speed of approximately 50 to 70 m/s (110 to 160 mph).
 Cloud formations in a famous image of Earth
from Apollo 17, makes similar circulation directly visible
Perhaps the most important impact of the Coriolis effect is in the large-scale
dynamics of the oceans and the atmosphere. In meteorology and oceanography, it is
convenient to postulate a rotating frame of reference wherein the Earth is stationary.
In accommodation of that provisional postulation, the centrifugal and Coriolis forces
are introduced. Their relative importance is determined by the applicable Rossby
numbers. Tornadoes have high Rossby numbers, so, while tornado-associated
centrifugal forces are quite substantial, Coriolis forces associated with tornadoes are
for practical purposes negligible.[36]

Because surface ocean currents are driven by the movement of wind over the
water's surface, the Coriolis force also affects the movement of ocean currents
and cyclones as well. Many of the ocean's largest currents circulate around warm,
high-pressure areas called gyres. Though the circulation is not as significant as that
in the air, the deflection caused by the Coriolis effect is what creates the spiralling
pattern in these gyres. The spiralling wind pattern helps the hurricane form. The
stronger the force from the Coriolis effect, the faster the wind spins and picks up
additional energy, increasing the strength of the hurricane.[37][better source needed]

Air within high-pressure systems rotates in a direction such that the Coriolis force is
directed radially inwards, and nearly balanced by the outwardly radial pressure
gradient. As a result, air travels clockwise around high pressure in the Northern
Hemisphere and anticlockwise in the Southern Hemisphere. Air around low-pressure
rotates in the opposite direction, so that the Coriolis force is directed radially outward
and nearly balances an inwardly radial pressure gradient.[38][better source needed]

Flow around a low-pressure area

Main article: Low-pressure area
If a low-pressure area forms in the atmosphere, air tends to flow in towards it, but is
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 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.[39] On a non-rotating planet, fluid would flow along the straightest
possible line, quickly eliminating pressure gradients. 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.[citation needed]

This pattern of deflection, and the direction of movement, is called Buys-Ballot's law.
In the atmosphere, the pattern of flow is called a cyclone. 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.[40] At high altitudes, outward-spreading air
rotates in the opposite direction.[citation needed][41][full citation needed] Cyclones rarely form along the
equator due to the weak Coriolis effect present in this region.[42]

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 moves with a constant speed around a circle whose
radius is given by:where is the Coriolis parameter , introduced above (where is the
latitude). The time taken for the mass to complete a full circle is therefore . The
Coriolis parameter typically has a mid-latitude value of about 10−4 s−1; hence for a
typical atmospheric speed of 10 m/s (22 mph), the radius is 100 km (62 mi) with a
period of about 17 hours. For an ocean current with a typical speed of 10 cm/s
(0.22 mph), the radius of an inertial circle is 1 km (0.6 mi). These inertial circles are
clockwise in the Northern Hemisphere (where trajectories are bent to the right) and
anticlockwise 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 of ±90°), and increase toward the equator.[43]

Other terrestrial effects

The Coriolis effect strongly affects the large-scale oceanic and atmospheric
circulation, leading to the formation of robust features like jet streams and western
boundary currents. Such features are in geostrophic 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 waves and Kelvin waves. 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.

Eötvös effect
Main article: 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. Westward-traveling objects are
deflected downwards, while eastward-traveling objects are deflected upwards.[44] This
is known as the Eötvös effect. This aspect of the Coriolis effect is greatest near the
equator. The force produced by the Eötvös effect is similar to the horizontal
component, but the much larger vertical forces due to gravity and pressure suggest
that it is unimportant in the hydrostatic equilibrium. However, in the atmosphere,
winds are associated with small deviations of pressure from the hydrostatic
equilibrium. In the tropical atmosphere, the order of magnitude of the pressure
deviations is so small that the contribution of the Eötvös effect to the pressure
deviations is considerable.[45]

In addition, objects traveling upwards (i.e. out) or downwards (i.e. in) are 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. For example, idealized numerical
modeling studies suggest that this effect can directly affect tropical large-scale wind
field by roughly 10% given long-duration (2 weeks or more) heating or cooling in the
atmosphere.[46][47] Moreover, in the case of large changes of momentum, such as a
spacecraft being launched into orbit, the effect becomes significant. The fastest and
most fuel-efficient path to orbit is a launch from the equator that curves to a directly
eastward heading.

Intuitive example
Imagine a train that travels through a frictionless railway line along the equator.
Assume that, when in motion, it moves at the necessary speed to complete a trip
around the world in one day (465 m/s).[48] The Coriolis effect can be considered in
three cases: when the train travels west, when it is at rest, and when it travels east.
In each case, the Coriolis effect can be calculated from the rotating frame of
reference on Earth first, and then checked against a fixed inertial frame. The image
below illustrates the three cases as viewed by an observer at rest in a (near) inertial
frame from a fixed point above the North Pole along the Earth's axis of rotation; the
train is denoted by a few red pixels, fixed at the left side in the leftmost picture,
moving in the others

1. The train travels toward the west: In that case, it moves against the direction
of rotation. Therefore, on the Earth's rotating frame the Coriolis term is
pointed inwards towards the axis of rotation (down). This additional force
downwards should cause the train to be heavier while moving in that
direction.

If one looks at this train from the fixed non-rotating frame on top of the center
of the Earth, at that speed it remains stationary as the Earth spins beneath it.
Hence, the only force acting on it is gravity and the reaction from the track.
 This force is greater (by 0.34%)[48] than the force that the passengers and the
train experience when at rest (rotating along with Earth). This difference is
what the Coriolis effect accounts for in the rotating frame of reference.

2. The train comes to a stop: From the point of view on the Earth's rotating
frame, the velocity of the train is zero, thus the Coriolis force is also zero and
the train and its passengers recuperate their usual weight.

From the fixed inertial frame of reference above Earth, the train now rotates
along with the rest of the Earth. 0.34% of the force of gravity provides
the centripetal force needed to achieve the circular motion on that frame of
reference. The remaining force, as measured by a scale, makes the train and
passengers "lighter" than in the previous case.

3. The train travels east. In this case, because it moves in the direction of Earth's
rotating frame, the Coriolis term is directed outward from the axis of rotation
(up). This upward force makes the train seem lighter still than when at rest.

Graph of the force

experienced by a 10-kilogram (22 lb) object as a function of its speed moving
along Earth's equator (as measured within the rotating frame). (Positive force
in the graph is directed upward. Positive speed is directed eastward and
negative speed is directed westward).From the fixed inertial frame of
reference above Earth, the train traveling east now rotates at twice the rate
as when it was at rest—so the amount of centripetal force needed to cause
that circular path increases leaving less force from gravity to act on the track.
This is what the Coriolis term accounts for on the previous paragraph.

As a final check one can imagine a frame of reference rotating along with the
train. Such frame would be rotating at twice the angular velocity as Earth's
rotating frame. The resulting centrifugal force component for that imaginary
frame would be greater. Since the train and its passengers are at rest, that
would be the only component in that frame explaining again why the train and
the passengers are lighter than in the previous two cases.
This also explains why high-speed projectiles that travel west are deflected down,
and those that travel east are deflected up. This vertical component of the Coriolis
effect is called the Eötvös effect.[49]
The above example can be used to explain why the Eötvös effect starts diminishing
when an object is traveling westward as its tangential speed increases above Earth's
rotation (465 m/s). If the westward train in the above example increases speed, part
of the force of gravity that pushes against the track accounts for the centripetal force
needed to keep it in circular motion on the inertial frame. Once the train doubles its
westward speed at 930 m/s (2,100 mph) that centripetal force becomes equal to the
force the train experiences when it stops. From the inertial frame, in both cases it
rotates at the same speed but in the opposite directions. Thus, the force is the same
cancelling completely the Eötvös effect. Any object that moves westward at a speed
above 930 m/s (2,100 mph) experiences an upward force instead. In the figure, the
Eötvös effect is illustrated for a 10-kilogram (22 lb) object on the train at different
speeds. The parabolic shape is because the centripetal force is proportional to the
square of the tangential speed. On the inertial frame, the bottom of the parabola is
centered at the origin. The offset is because this argument uses the Earth's rotating
frame of reference. The graph shows that the Eötvös effect is not symmetrical, and
that the resulting downward force experienced by an object that travels west at high
velocity is less than the resulting upward force when it travels east at the same
speed.

Draining in bathtubs and toilets

Contrary to popular misconception, bathtubs, toilets, and other water receptacles do
not drain in opposite directions in the Northern and Southern Hemispheres. This is
because the magnitude of the Coriolis force is negligible at this scale.[27][50][51][52] Forces
determined by the initial conditions of the water (e.g. the geometry of the drain, the
geometry of the receptacle, preexisting momentum of the water, etc.) are likely to be
orders of magnitude greater than the Coriolis force and hence will determine the
direction of water rotation, if any. For example, identical toilets flushed in both
hemispheres drain in the same direction, and this direction is determined mostly by
the shape of the toilet bowl.

Under real-world conditions, the Coriolis force does not influence the direction of
water flow perceptibly. Only if the water is so still that the effective rotation rate of the
Earth 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 indeed determine the direction of the vortex. Without
such careful preparation, the Coriolis effect will be much smaller than various other
influences on drain direction[53] such as any residual rotation of the water[54] and the
geometry of the container.[55]

Laboratory testing of draining water under atypical conditions

In 1962, Ascher Shapiro performed an experiment at MIT to test the Coriolis force on
a large basin of water, 2 meters (6 ft 7 in) across, with a small wooden cross above
the plug hole to display the direction of rotation, covering it and waiting for at least 24
hours for the water to settle. Under these precise laboratory conditions, he
demonstrated the effect and consistent counterclockwise rotation. The experiment
required extreme precision, since the acceleration due to Coriolis effect is only that
of gravity. The vortex was measured by a cross made of two slivers of wood pinned
above the draining hole. It takes 20 minutes to drain, and the cross starts turning
only around 15 minutes. At the end it is turning at 1 rotation every 3 to 4 seconds.

He reported that,[56]
Both schools of thought are in some sense correct. For the everyday observations of
the kitchen sink and bath-tub variety, the direction of the vortex seems to vary in an
unpredictable manner with the date, the time of day, and the particular household of
the experimenter. But under well-controlled conditions of experimentation, the
observer looking downward at a drain in the northern hemisphere will always see a
counter-clockwise vortex, while one in the southern hemisphere will always see a
clockwise vortex. In a properly designed experiment, the vortex is produced by
Coriolis forces, which are counter-clockwise in the northern hemisphere.
Lloyd Trefethen reported clockwise rotation in the Southern Hemisphere at the
University of Sydney in five tests with settling times of 18 h or more.[57]

Ballistic trajectories
The Coriolis force is 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 (75 mi). The Coriolis force minutely changes the trajectory of a bullet,
affecting accuracy at extremely long distances. It is adjusted for by accurate long-
distance shooters, such as snipers. At the latitude of Sacramento, California, a
1,000 yd (910 m) northward shot would be deflected 2.8 in (71 mm) to the right.
There is also a vertical component, explained in the Eötvös effect section above,
which causes westward shots to hit low, and eastward shots to hit high.[58][59]

The effects of the Coriolis force on ballistic trajectories should not be confused with
the curvature of the paths of missiles, satellites, and similar objects when the paths
are plotted on two-dimensional (flat) maps, such as the Mercator projection. The
projections of the three-dimensional curved surface of the Earth to a two-dimensional
surface (the map) necessarily results in distorted features. The apparent curvature of
the path is a consequence of the sphericity of the Earth and would occur even in a
non-rotating frame.[60]

Trajectory, ground track, and drift of a typical

projectile. The axes are not to scale.
The Coriolis force on a moving projectile depends on velocity components in all three
directions, latitude, and azimuth. The directions are typically downrange (the
direction that the gun is initially pointing), vertical, and cross-range.[61]: 178

where
 , down-range acceleration.
 , vertical acceleration with positive indicating acceleration upward.
 , cross-range acceleration with positive indicating acceleration to the right.
 , down-range velocity.
 , vertical velocity with positive indicating upward.
 , cross-range velocity with positive indicating velocity to the right.
 = 0.00007292 rad/sec, angular velocity of the Earth (based on a sidereal day).
 , latitude with positive indicating Northern Hemisphere.
 , azimuth measured clockwise from due North.
Visualization

Fluid assuming a parabolic shape as it is rotating

Object moving frictionlessly over the surface of a

very shallow parabolic dish. The object has been released in such a way that it
follows an elliptical trajectory.
Left: The inertial point of view.

Right: The co-rotating point of view. The forces at

play in the case of a curved surface.
Red: gravity
Green: the normal force
Blue: the net resultant centripetal force.
To demonstrate the Coriolis effect, a parabolic turntable can be used. On a flat
turntable, the inertia of a co-rotating object forces it off the edge. However, if the
turntable surface has the correct paraboloid (parabolic bowl) shape (see the figure)
and rotates at the corresponding rate, the force components shown in the figure
make the component of gravity tangential to the bowl surface exactly equal to 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.[62][63]
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 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 the figure.
In the left panel of the figure, 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 with analysis and 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.[64] 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.)

The Coriolis effect caused by the rotation of the Earth can be seen indirectly through
the motion of a Foucault pendulum.

In other areas
Coriolis flow meter
A practical application of the Coriolis effect is the mass flow meter, an instrument
that measures the mass flow rate and density 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 not completely circular, provides the rotating reference
frame that 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.[65]

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 are therefore present, and make the atoms
move in a direction perpendicular to the original oscillations. This leads to a mixing in
molecular spectra between the rotational and vibrational levels, from which Coriolis
coupling constants can be determined.[66]
Insect flight
Flies (Diptera) and some moths (Lepidoptera) exploit the Coriolis effect in flight with
specialized appendages and organs that relay information about the angular
velocity of their bodies. Coriolis forces resulting from linear motion of these
appendages are detected within the rotating frame of reference of the insects'
bodies. In the case of flies, their specialized appendages are dumbbell shaped
organs located just behind their wings called "halteres".[67]

The fly's halteres oscillate in a plane at the same beat frequency as the main wings
so that any body rotation results in lateral deviation of the halteres from their plane of
motion.[68]

In moths, their antennae are known to be responsible for the sensing of Coriolis
forces in the similar manner as with the halteres in flies.[69] In both flies and moths, a
collection of mechanosensors at the base of the appendage are sensitive to
deviations at the beat frequency, correlating to rotation in the pitch and roll planes,
and at twice the beat frequency, correlating to rotation in the yaw plane.[70][69]

Lagrangian point stability

In astronomy, Lagrangian points are five positions in the orbital plane of two large
orbiting bodies where a small object affected only by gravity can maintain a stable
position relative to the two large bodies. The first three Lagrangian points (L 1, L2, L3)
lie along the line connecting the two large bodies, while the last two points (L 4 and L5)
each form an equilateral triangle with the two large bodies. The L4 and L5 points,
although they correspond to maxima of the effective potential in the coordinate frame
that rotates with the two large bodies, are stable due to the Coriolis effect. [71] The
stability can result in orbits around just L4 or L5, known as tadpole orbits,
where trojans can be found. It can also result in orbits that encircle L3, L4, and L5,
known as horseshoe orbits.