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In physics, the Coriolis force is an inertial (or fictitious) 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 day/night cycle, 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

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 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

In Newtonian mechanics, the equation of motion for an object in an inertial reference frame is:

${\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}}$

where ${\displaystyle {\boldsymbol {F}}}$ is the vector sum of the physical forces acting on the object, ${\displaystyle m}$ is the mass of the object, and ${\displaystyle {\boldsymbol {a}}}$ 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 ${\displaystyle {\boldsymbol {\omega }}}$ having variable rotation rate, the equation takes the form:^[8][16]

${\displaystyle {\begin{aligned}{\boldsymbol {F'}}&={\boldsymbol {F}}-m{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r'}}-2m{\boldsymbol {\omega }}\times {\boldsymbol {v'}}-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r'}})\\&=m{\boldsymbol {a'}}\end{aligned}}}$

where

• ${\displaystyle {\boldsymbol {F}}}$ is the vector sum of the physical forces acting on the object
• ${\displaystyle {\boldsymbol {\omega }}}$ is the angular velocity, of the rotating reference frame relative to the inertial frame
• ${\displaystyle {\boldsymbol {r'}}}$ is the position vector of the object relative to the rotating reference frame
• ${\displaystyle {\boldsymbol {v'}}}$ is the velocity of the object relative to the rotating reference frame
• ${\displaystyle {\boldsymbol {a'}}}$ 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, ${\displaystyle -m{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r'}}}$
• Coriolis force, ${\displaystyle -2m({\boldsymbol {\omega }}\times {\boldsymbol {v'}})}$
• centrifugal force, ${\displaystyle -m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r'}})}$

As seen in these formulas the Euler and centrifugal forces depend on the position vector ${\displaystyle {\boldsymbol {r'}}}$ of the object, while the Coriolis force depends on the object's velocity ${\displaystyle {\boldsymbol {v'}}}$ as measured in the rotating reference frame. As expected, for a non-rotating inertial frame of reference ${\displaystyle ({\boldsymbol {\omega }}=0)}$ the Coriolis force and all other fictitious forces disappear.^[21]

Corollaries

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.
• 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.
• 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, to higher latitudes, 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 typical size 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

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, which is the ratio of the velocity, U, of a system to the product of the Coriolis parameter, $$Zdroj:https://en.wikipedia.org?pojem=Coriolis_effect
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