In physics, the **Coriolis force** is an inertial or fictitious force^{[1]} that seems to act on objects that are 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 force and centrifugal force appear. Both forces are proportional to the mass of the object. The Coriolis force is proportional to the rotation rate and the centrifugal force is proportional to the square of the rotation rate. 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 (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*.^{[2]} They allow the application of Newton's laws to a rotating system. They are correction factors that do not exist in a non-accelerating or inertial reference frame.

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 per day, so for motions of everyday objects the Coriolis force is usually quite small compared to other forces; 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 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.^{[3]} 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 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).

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 you go, the smaller the "horizontal diameter" of the Earth (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).^{[4]}^{[5]} For objects moving east-west in the northern hemisphere, the Coriolis deflection can be intuitively explained as follows. The speed of an object moving east at a given latitude is faster than the rotation of the earth at that latitude. This means the object will take a path of increasing radius due to centripetal acceleration. This deflects the object to the south. Similarly, an object moving west in the northern hemisphere is moving too slow relative to the rotation of the earth and takes a path of decreasing radius. This deflects the object to the north.

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.^{[6]} 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.^{[7]} The Coriolis acceleration equation was derived by Euler in 1749^{[8]}^{[9]} and the effect was described in the tidal equations of Pierre-Simon Laplace in 1778.^{[10]}

Gaspard-Gustave Coriolis published a paper in 1835 on the energy yield of machines with rotating parts, such as waterwheels.^{[11]} 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.^{[12]}^{[13]} The effect was known in the early 20th century as the "acceleration of Coriolis",^{[14]} and by 1920 as "Coriolis force".^{[15]}

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.^{[16]}

The understanding of the kinematics of how exactly the rotation of the Earth affects airflow was partial at first.^{[17]} 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.^{[18]}

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 non-inertial reference frame with a rotation rate about the origin, the equation takes the form

where

- is the rotation vector, with magnitude , of the rotating reference frame relative to the inertial frame
- is the velocity relative to the rotating reference frame
- is the position vector of the object
- is the acceleration relative to the rotating reference frame

The additional terms on the force side of the equation are fictitious forces as they are perceived in the rotating frame as additional forces that contribute to the apparent acceleration just like the real external forces.^{[19]}^{[20]} The ficitious force terms on the force side of the equation are, reading from left to right, the Euler force , the Coriolis force , and the centrifugal force , respectively.^{[21]} Unlike the Euler and centrifugal forces, which depend on the position vector of the object, 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.^{[22]}

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 on the equator moving north or south relative to 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 on 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 on 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.
^{[23]}) - if the velocity is against the direction of rotation, the Coriolis force is inward to the axis. (On Earth, this situation occurs for a body on the equator moving west, which would deflect downward as seen by an observer.)

The Coriolis force 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, the Coriolis force is a consequence of inertia,^{[24]} 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, the 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 (Merry-go-round ), who are throwing a ball to each other. From the children's point of view, this ball's path is curved sideways by the Coriolis force. Suppose the roundabout spins anticlockwise when viewed from above. From the thrower's perspective, the deflection is to the right.^{[25]} From the non-thrower's perspective, deflection is to the left. *For a mathematical formulation see Mathematical derivation of fictitious forces.* In meteorology, a rotating frame (the Earth) with its Coriolis force provides a more natural framework for explanation of air movements than a non-rotating, inertial frame without Coriolis forces.^{[26]} In long-range gunnery, sight corrections for the Earth's rotation are based on the Coriolis force.^{[27]} 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) is 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). For an object to move in a straight line, it must accelerate so that its velocity changes from point to point by the same amount as the velocities of the frame of reference. The force 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 .

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,, and the length scale, *L*, of the motion:

The Rossby number 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, in low-pressure systems it is low, and in oceanic systems it is around 1. 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.^{[28]}

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.

A baseball pitcher may throw the ball at U = 45 m/s (100 mph) for a distance of L = 18.3 m (60 ft). The Rossby number in this case would be 32,000.

Baseball players don't care about which hemisphere they're playing in. However, an unguided missile obeys exactly the same physics as a baseball, but 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 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.^{[29]}^{[30]}^{[31]}

The animation at the top of this article is a classic illustration of Coriolis force. Another visualization of the Coriolis and centrifugal forces is this animation clip.

Given the radius *R* of the turntable in that animation, the rate of angular rotation ω, and the speed of the cannonball (assumed constant) *v*, the correct angle θ to aim so as to hit the target at the edge of the turntable can be calculated.

The inertial frame of reference provides one way to handle the question: calculate the time to interception, which is *t _{f}* =

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 the figure.

To determine the components of acceleration, a general expression is used from the article fictitious force:

in which the term in **Ω** × **v**_{B} is the Coriolis acceleration and the term in **Ω** × (**Ω** × **r**_{B}) is the centrifugal acceleration. The results are (let α = θ − ω*t*):

Producing a centrifugal acceleration:

Also:

producing a Coriolis acceleration:

These accelerations are shown in the diagrams 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 center of rotation):^{[32]}

and an additional component of acceleration perpendicular to **r**_{B}(*t*):

The "centripetal" component of acceleration resembles that for circular motion at radius *r*_{B}, 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 2ω*v*. 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 **r**_{B}(*t*). Agreement of the two approaches demonstrates that one could start from the general expression for fictitious acceleration above and derive the trajectories shown here. However, working from the acceleration to the trajectory is more complicated than the reverse procedure used here, which 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 focus on displaying the Coriolis effect particularly, that the Coriolis force is the only factor that must be considered,^{[33]} but that is not so.^{[34]} 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 are countered by gravitational attraction.^{[35]}^{[36]}

The figure illustrates a ball tossed from 12:00 o'clock toward the center of a counter-clockwise 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 center, while the ball-thrower rotates counter-clockwise 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 center of the carousel (providing the ball-thrower's line of sight), and the other points from the center of the carousel to the ball. (This arrow gets shorter as the ball approaches the center.) 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 center of the carousel is aligned with 12:00 o'clock. The other arrow of the pair locates the ball relative to the center 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 a *counter-clockwise* 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.

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).

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.

The concept "Coriolis force" is specially suitable for the description of motion of atmosphere (i.e. winds) over the surface of the Earth. The Earth (like all rotating celestial bodies) has taken an oblate shape such that the gravitational force is slightly off-set towards the Earth axis as illustrated in the figure.

For a mass point at rest on the Earth surface the horizontal component of the gravitation counteracts the "centrifugal force" preventing it to slide away towards the equator. This means that the vector term of the "equation of motion" above

is directed straight down, orthogonal to the surface of the Earth. The force affecting the motion of air "sliding" over the Earth surface is therefore (only) the horizontal component of the Coriolis term

This component is orthogonal to the velocity over the Earth surface and is given by the expression

where

- is the spin rate of the Earth
- is the latitude, positive in northern hemisphere and negative in the southern hemisphere

In the northern hemisphere where the sign is positive this force/acceleration is to the right of the direction of motion, in the southern hemisphere where the sign is negative this force/acceleration is to the left of the direction of motion

As the Earth turns around its axis, everything attached to it, including the atmosphere, turns with it (imperceptibly to our senses). An object that is moving without being dragged along with the surface rotation or atmosphere such as an object in ballistic flight or an independent air mass within the atmosphere, travels in a straight motion over the turning Earth. From our rotating perspective on the planet, the direction of motion of an object in ballistic flight changes as it moves, bending in the opposite direction to our actual motion. When viewed from a stationary point in space directly above the north pole, any land feature in the Northern Hemisphere turns anticlockwise—and, fixing our gaze on that location, any other location in that hemisphere rotates around it the same way. The traced ground path of a freely moving body in ballistic flight traveling from one point to another therefore bends 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.

Consider a location with latitude *φ* on a sphere that is rotating around the north-south axis.^{[37]} 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 *v _{u}* = 0):

where is called the Coriolis parameter.

By setting *v _{n}* = 0, it can be seen immediately that (for positive φ and ω) a movement due east results in an acceleration due south. Similarly, setting

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, and an upward motion produces an acceleration due west.

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.^{[38]}

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 spiraling pattern in these gyres. The spiraling 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.^{[39]}

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.^{[40]}

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.^{[41]} 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 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.^{[42]} At high altitudes, outward-spreading air rotates in the opposite direction.^{[43]} Cyclones rarely form along the equator due to the weak Coriolis effect present in this region.^{[44]}

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 = ±90°), and increase toward the equator.^{[45]}

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.

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-travelling objects are deflected downwards (feel heavier), while Eastward-travelling objects are deflected upwards (feel lighter).^{[46]} This is known as the Eötvös effect. 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 travelling 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. However, 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.

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).^{[47]} 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 are 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%)
^{[47]}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.

- 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%)
- 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.

- From the fixed inertial frame of reference above Earth, the train travelling 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.^{[48]}

The above example can be used to explain why the Eötvös effect starts diminishing when an object is travelling 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 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 experiences an upward force instead. In the figure, the Eötvös effect is illustrated for a 10 kilogram 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.

Contrary to popular misconception, water rotation in home bathrooms under *normal* circumstances is not related to the Coriolis effect or to the rotation of the Earth, and no consistent difference in rotation direction between toilet drainage in the Northern and Southern Hemispheres can be observed.^{[49]}^{[50]}^{[51]}^{[52]} The formation of a vortex over the plug hole may be explained by the conservation of angular momentum: 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 they pull their arms in. Any rotation around the plug hole that is initially present accelerates as water moves inward.

The Coriolis force still affects the direction of the flow of water, but only minutely. 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 is likely to 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]} Despite this, the idea that toilets and bathtubs drain differently in the Northern and Southern Hemispheres has been popularized by several television programs and films, including *Escape Plan*, *Wedding Crashers*, *The Simpsons* episode "Bart vs. Australia", *Pole to Pole*,^{[56]}^{[57]} and *The X-Files* episode "Die Hand Die Verletzt".^{[58]} Several science broadcasts and publications, including at least one college-level physics textbook, have also stated this.^{[59]}^{[60]}

In 1908, the Austrian physicist Ottokar Tumlirz described careful and effective experiments that demonstrated the effect of the rotation of the Earth on the outflow of water through a central aperture.^{[61]} The subject was later popularized in a famous 1962 article in the journal *Nature*, which described an experiment in which all other forces to the system were removed by filling a 6 ft (1.8 m) tank with 300 U.S. gal (1,100 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 an anticlockwise 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 conditions.^{[62]}^{[63]}

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 a 1000-yard shot would be deflected 3 inches 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.^{[27]}^{[64]}

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.

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.^{[65]}^{[66]}

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 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.^{[67]} 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.

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.^{[68]}

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.^{[69]}

When an external torque is applied to a spinning gyroscope along an axis that is at right angles to the spin axis, the rim velocity that is associated with the spin becomes radially directed in relation to the external torque axis. This causes a Coriolis force to act on the rim in such a way as to tilt the gyroscope at right angles to the direction that the external torque would have tilted it. This tendency has the effect of keeping spinning bodies stably aligned in space.

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".^{[70]}

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.^{[71]}

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.^{[72]} 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.^{[73]}^{[72]}

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}, L_{2}, L_{3}) lie along the line connecting the two large bodies, while the last two points (L_{4} and L_{5}) each form an equilateral triangle with the two large bodies. The L_{4} and L_{5} 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.^{[74]} The stability can result in orbits around just L_{4} or L_{5}, known as tadpole orbits, where trojans can be found. It can also result in orbits that encircle L_{3}, L_{4}, and L_{5}, known as horseshoe orbits.

- Analytical mechanics
- Applied mechanics
- Classical mechanics
- Dynamics (physics)
- Earth's rotation
- Equatorial Rossby wave
- Frenet–Serret formulas
- Gyroscope
- Kinetics (physics)
- Mechanics of planar particle motion
- Reactive centrifugal force
- Secondary flow
- Statics
- Uniform circular motion
- Whirlpool

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**^**Graney, Christopher M. (2015).*Setting Aside All Authority: Giovanni Battista Riccioli and the Science Against Copernicus in the Age of Galileo*. Notre Dame, Indiana: University of Notre Dame Press. pp. 115–125. ISBN 9780268029883.**^**Schneider, Stephen H.; Root, Terry L.; Mastrandrea, Michael, eds. (2011).*Encyclopedia of Climate and Weather*.**3**. Oxford University Press. p. 310.**^**John M. Wallace; Peter V. Hobbs (1977).*Atmospheric Science: An Introductory Survey*. Academic Press, Inc. pp. 368–371. ISBN 978-0-12-732950-5.**^**Roger Graham Barry; Richard J. Chorley (2003).*Atmosphere, Weather and Climate*. Routledge. p. 113. ISBN 978-0-415-27171-4.- ^
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**^**Tumlirz, Ottokar (1908). "Ein neuer physikalischer Beweis für die Achsendrehung der Erde".*Sitzungsberichte der Math.-nat. Klasse der Kaiserlichen Akademie der Wissenschaften IIa*.**117**: 819–841.**^**Shapiro, Ascher H. (1962). "Bath-Tub Vortex".*Nature*.**196**(4859): 1080–1081. Bibcode:1962Natur.196.1080S. doi:10.1038/1961080b0.**^**(Vorticity, Part 1). Web.mit.edu. Retrieved 8 November 2011.**^**"Do Snipers Compensate for the Earth's Rotation?".*Washington City Paper*. 25 June 2010. Retrieved 16 July 2018.**^**When a container of fluid is rotating on a turntable, the surface of the fluid naturally assumes the correct parabolic shape. This fact may be exploited to make a parabolic turntable by using a fluid that sets after several hours, such as a synthetic resin. For a video of the Coriolis effect on such a parabolic surface, see Geophysical fluid dynamics lab demonstration Archived 20 November 2005 at the Wayback Machine John Marshall, Massachusetts Institute of Technology.**^**For a java applet of the Coriolis effect on such a parabolic surface, see Brian Fiedler Archived 21 March 2006 at the Wayback Machine School of Meteorology at the University of Oklahoma.**^**John Marshall; R. Alan Plumb (2007).*Atmosphere, Ocean, and Climate Dynamics: An Introductory Text*. Academic Press. p. 101. ISBN 978-0-12-558691-7.**^**Omega Engineering. "Mass Flowmeters".**^**califano, S (1976).*Vibrational states*. Wiley. pp. 226–227. ISBN 978-0471129967.**^**Fraenkel, G.; Pringle, W.S. (21 May 1938). "Halteres of Flies as Gyroscopic Organs of Equilibrium".*Nature*.**141**(3577): 919–920. Bibcode:1938Natur.141..919F. doi:10.1038/141919a0.**^**Dickinson, M. (1999). "Haltere-mediated equilibrium reflexes of the fruit fly, Drosophila melanogaster".*Phil. Trans. R. Soc. Lond*.**354**(1385): 903–916. doi:10.1098/rstb.1999.0442. PMC 1692594. PMID 10382224.- ^
^{a}^{b}Sane S., Dieudonné, A., Willis, M., Daniel, T. (February 2007). "Antennal mechanosensors mediate flight control in moths" (PDF).*Science*.**315**(5813): 863–866. Bibcode:2007Sci...315..863S. CiteSeerX 10.1.1.205.7318. doi:10.1126/science.1133598. PMID 17290001.CS1 maint: Multiple names: authors list (link) **^**Fox, J; Daniel, T (2008). "A neural basis for gyroscopic force measurement in the halteres of Holorusia".*Journal of Comparative Physiology*.**194**(10): 887–897. doi:10.1007/s00359-008-0361-z. PMID 18751714.**^**Spohn, Tilman; Breuer, Doris; Johnson, Torrence (2014).*Encyclopedia of the Solar System*. Elsevier. p. 60. ISBN 978-0124160347.

- 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, pp. 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, pp. 142–154

(Original article [in French] PDF file, 400 KB, scanned images of complete pages.) - Gill, A. E.
*Atmosphere-Ocean dynamics*, Academic Press, 1982. - Robert Ehrlich (1990).
*Turning the World Inside Out and 174 Other Simple Physics Demonstrations*. Princeton University Press. p.*Rolling a ball on a rotating turntable*; p. 80 ff. ISBN 978-0-691-02395-3. - Durran, D. R., 1993:
*Is the Coriolis force really responsible for the inertial oscillation?*, Bull. Amer. Meteor. Soc., 74, pp. 2179–2184; Corrigenda. Bulletin of the American Meteorological Society, 75, p. 261 - Durran, D. R., and S. K. Domonkos, 1996:
*An apparatus for demonstrating the inertial oscillation*, Bulletin of the American Meteorological Society, 77, pp. 557–559. - Marion, Jerry B. 1970,
*Classical Dynamics of Particles and Systems*, Academic Press. - Persson, A., 1998
*[1] How do we Understand the Coriolis Force?*Bulletin of the American Meteorological Society 79, pp. 1373–1385. - Symon, Keith. 1971,
*Mechanics*, Addison–Wesley - Akira Kageyama & Mamoru Hyodo:
*Eulerian derivation of the Coriolis force* - James F. Price:
*A Coriolis tutorial*Woods Hole Oceanographic Institute (2003) - McDonald, James E. (May 1952). "The Coriolis Effect" (PDF).
*Scientific American*: 72–78. Retrieved 2016-01-04.Everything that moves over the surface of the Earth – water, air, animals, machines and projectiles – sidles to the right in the Northern Hemisphere and to the left in the Southern

. Elementary, non-mathematical; but well written.

- Grattan-Guinness, I., Ed., 1994:
*Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences*. Vols. I and II. Routledge, 1840 pp.

1997:*The Fontana History of the Mathematical Sciences*. Fontana, 817 pp. 710 pp. - Khrgian, A., 1970:
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- The definition of the Coriolis effect from the Glossary of Meteorology
- The Coriolis Effect — a conflict between common sense and mathematics PDF-file. 20 pages. A general discussion by Anders Persson of various aspects of the coriolis effect, including Foucault's Pendulum and Taylor columns.
- The coriolis effect in meterology PDF-file. 5 pages. A detailed explanation by Mats Rosengren of how the gravitational force and the rotation of the Earth affect the atmospheric motion over the Earth surface. 2 figures
- 10 Coriolis Effect Videos and Games- from the About.com Weather Page
- Coriolis Force – from ScienceWorld
*Coriolis Effect and Drains*An article from the NEWTON web site hosted by the Argonne National Laboratory.- Catalog of Coriolis videos
*Coriolis Effect: A graphical animation*, a visual Earth animation with precise explanation*An introduction to fluid dynamics*SPINLab Educational Film explains the Coriolis effect with the aid of lab experiments*Do bathtubs drain counterclockwise in the Northern Hemisphere?*by Cecil Adams.*Bad Coriolis.*An article uncovering misinformation about the Coriolis effect. By Alistair B. Fraser, Emeritus Professor of Meteorology at Pennsylvania State University*The Coriolis Effect: A (Fairly) Simple Explanation*, an explanation for the layperson- Observe an animation of the Coriolis effect over Earth's surface
- Animation clip showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.
- Vincent Mallette
*The Coriolis Force*@ INWIT - NASA notes
- Interactive Coriolis Fountain lets you control rotation speed, droplet speed and frame of reference to explore the Coriolis effect.

Ageostrophy is the real condition that works against geostrophic wind or geostrophic currents in the ocean, and works against an exact balance between the Coriolis force and the pressure gradient force. While geostrophic currents or winds come from an equilibrium of a particular system, ageostrophy is more often observed because of other forces such as friction or the centrifugal force from curved fluid flow.

AnticycloneAn anticyclone (that is, opposite to a cyclone) is a weather phenomenon defined by the United States National Weather Service's glossary as "a large-scale circulation of winds around a central region of high atmospheric pressure, clockwise in the Northern Hemisphere, counterclockwise in the Southern Hemisphere". Effects of surface-based anticyclones include clearing skies as well as cooler, drier air. Fog can also form overnight within a region of higher pressure. Mid-tropospheric systems, such as the subtropical ridge, deflect tropical cyclones around their periphery and cause a temperature inversion inhibiting free convection near their center, building up surface-based haze under their base. Anticyclones aloft can form within warm core lows such as tropical cyclones, due to descending cool air from the backside of upper troughs such as polar highs, or from large scale sinking such as the subtropical ridge.

The evolution of an anticyclone depends on a few variables such as its size, intensity, moist-convection, Coriolis force etc .

Baer–Babinet lawIn geography, the Baer–Babinet law, sometimes called Baer's law, identifies a way in which the process of formation of rivers is influenced by the rotation of the earth. According to the hypothesis, because of the rotation of the earth, erosion occurs mostly on the right banks of rivers in the Northern Hemisphere, and in the Southern Hemisphere on the left banks.The concept was originally introduced by a French physicist Jacques Babinet in 1859 using mathematical deduction and Coriolis force. A more definitive explanation was given by an Estonian scientist Karl Ernst von Baer in 1860.Although it is possible that an aggregate measurement of all rivers would lead to a correlation with the Baer–Babinet law, the Coriolis force is orders of magnitude weaker than the local forces on the river channel from its flow. Therefore, this is unlikely to be important in any given river. Albert Einstein wrote a paper in 1926 explaining the true causes of the phenomenon (see tea leaf paradox).

Balanced flowIn atmospheric science, balanced flow is an idealisation of atmospheric motion. The idealisation consists in considering the behaviour of one isolated parcel of air having constant density, its motion on a horizontal plane subject to selected forces acting on it and, finally, steady-state conditions.

Balanced flow is often an accurate approximation of the actual flow, and is useful in improving the qualitative understanding and interpretation of atmospheric motion.

In particular, the balanced-flow speeds can be used as estimates of the wind speed for particular arrangements of the atmospheric pressure on Earth’s surface.

Coriolis effect (perception)In psychophysical perception, the Coriolis effect (also referred to as the Coriolis illusion) is the misperception of body orientation and induced nausea due to the Coriolis force.

The Coriolis effect is a concern for pilots, where it can cause extreme disorientation.

Coriolis frequencyThe **Coriolis frequency** *ƒ*, also called the **Coriolis parameter** or **Coriolis coefficient**, is equal to twice the rotation rate *Ω* of the Earth multiplied by the sine of the latitude *φ*.

The rotation rate of the Earth (*Ω* = 7.2921 × 10^{−5} rad/s) can be calculated as 2*π* / *T* radians per second, where *T* is the rotation period of the Earth which is one sidereal day (23 hr 56 m 4.1 s). In the midlatitudes, the typical value for is about 10^{−4} rad/s. Inertial oscillations on the surface of the earth have this frequency. These oscillations are the result of the Coriolis effect.

Consider a body (for example a fixed volume of atmosphere) at latitude moving at velocity in the earth's rotating reference frame. In the local reference frame of the body, the vertical direction is parallel to the radial vector pointing from the center of the earth to the location of the body and the horizontal direction is perpendicular to this vertical direction (and hence in the meridional direction). The Coriolis force (proportional to ), however, is perpendicular to the plane containing both the earth's angular velocity vector (where ) and the body's own velocity in the rotating reference frame . Thus, the Coriolis force is always at an angle with the local vertical direction. The local horizontal direction of the Coriolis force is thus . This force acts to move the body along longitudes or in the meridional directions.

Suppose the body is moving with a velocity such that the centripetal and Coriolis (due to ) forces on it are balanced. We then have

where is the radius of curvature of the path of object (defined by ). Replacing we obtain

Thus the Coriolis parameter, , is the angular velocity or frequency required to maintain a body at a fixed circle of latitude or zonal region. If the Coriolis parameter is large, the effect of the earth's rotation on the body is significant since it will need a larger angular frequency to stay in equilibrium with the Coriolis forces. Alternatively, if the Coriolis parameter is small, the effect of the earth's rotation is small since only a small fraction of the centripetal force on the body is canceled by the Coriolis force. Thus the magnitude of strongly affects the relevant dynamics contributing to the body's motion. These considerations are captured in the nondimensionalized Rossby number.

In stability calculations, the rate of change of along the meridional direction becomes significant. This is called the Rossby parameter and is usually denoted

where is the in the local direction of increasing meridian. This parameter becomes important, for example, in calculations involving Rossby waves.

Fictitious forceA fictitious force (also called a pseudo force, d'Alembert force, or inertial force) is an apparent force that acts on all masses whose motion is described using a non-inertial frame of reference, such as a rotating reference frame. Examples are the forces that act on passengers in an accelerating or braking automobile, and the force that pushes objects toward the rim of a centrifuge.

The fictitious force F is due to an object's inertia when the reference frame does not move inertially, and thus begins to accelerate relative to the free object. The fictitious force thus does not arise from any physical interaction between two objects (that is, it is not a "contact force"), but rather from the acceleration a of the non-inertial reference frame itself, which from the viewpoint of the frame now appears to be an acceleration of the object instead, requiring a "force" to make this happen. As stated by Iro:

Such an additional force due to nonuniform relative motion of two reference frames is called a pseudo-force.

Assuming Newton's second law in the form F = ma, fictitious forces are always proportional to the mass m.

The fictitious force on an object arises as an imaginary influence, when the frame of reference used to describe the object's motion is accelerating compared to a non-accelerating frame. The fictitious force "explains," using Newton's mechanics, why an object does not follow Newton's laws and "floats freely" as if weightless. As a frame can accelerate in any arbitrary way, so can fictitious forces be as arbitrary (but only in direct response to the acceleration of the frame). However, four fictitious forces are defined for frames accelerated in commonly occurring ways: one caused by any relative acceleration of the origin in a straight line (rectilinear acceleration); two involving rotation: centrifugal force and Coriolis force; and a fourth, called the Euler force, caused by a variable rate of rotation, should that occur.

Gravitational force would also be a fictitious force based upon a field model in which particles distort spacetime due to their mass, such as General Relativity.

Geostrophic currentA geostrophic current is an oceanic current in which the pressure gradient force is balanced by the Coriolis effect. 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 Southern Hemisphere. This concept is familiar from weather maps, whose isobars show the direction of geostrophic flow in the atmosphere. Geostrophic flow may be either barotropic or baroclinic. A geostrophic current may also be thought of as a rotating shallow water wave with a frequency of zero. The principle of geostrophy is useful to oceanographers because it allows them to infer ocean currents from measurements of the sea surface height (by combined satellite altimetry and gravimetry) or from vertical profiles of seawater density taken by ships or autonomous buoys. The major currents of the world's oceans, such as the Gulf Stream, the Kuroshio Current, the Agulhas Current, and the Antarctic Circumpolar Current, are all approximately in geostrophic balance and are examples of geostrophic currents.

Geostrophic windThe geostrophic wind () is the theoretical wind that would result from an exact balance between the Coriolis force and the pressure gradient force. This condition is called geostrophic balance. The geostrophic wind is directed parallel to isobars (lines of constant pressure at a given height). This balance seldom holds exactly in nature. The true wind almost always differs from the geostrophic wind due to other forces such as friction from the ground. Thus, the actual wind would equal the geostrophic wind only if there were no friction and the isobars were perfectly straight. Despite this, much of the atmosphere outside the tropics is close to geostrophic flow much of the time and it is a valuable first approximation. Geostrophic flow in air or water is a zero-frequency inertial wave.

Inertial waveInertial waves, also known as inertial oscillations, are a type of mechanical wave possible in rotating fluids. Unlike surface gravity waves commonly seen at the beach or in the bathtub, inertial waves flow through the interior of the fluid, not at the surface. Like any other kind of wave, an inertial wave is caused by a restoring force and characterized by its wavelength and frequency. Because the restoring force for inertial waves is the Coriolis force, their wavelengths and frequencies are related in a peculiar way. Inertial waves are transverse. Most commonly they are observed in atmospheres, oceans, lakes, and laboratory experiments. Rossby waves, geostrophic currents, and geostrophic winds are examples of inertial waves. Inertial waves are also likely to exist in the molten core of the rotating Earth.

Jet streamJet streams are fast flowing, narrow, meandering air currents in the atmospheres of some planets, including Earth. On Earth, the main jet streams are located near the altitude of the tropopause and are westerly winds (flowing west to east). Their paths typically have a meandering shape. Jet streams may start, stop, split into two or more parts, combine into one stream, or flow in various directions including opposite to the direction of the remainder of the jet.

The strongest jet streams are the polar jets, at nine–twelve km (30,000–39,000 ft) above sea level, and the higher altitude and somewhat weaker subtropical jets at 10–16 km (33,000–52,000 ft). The Northern Hemisphere and the Southern Hemisphere each have a polar jet and a subtropical jet. The northern hemisphere polar jet flows over the middle to northern latitudes of North America, Europe, and Asia and their intervening oceans, while the southern hemisphere polar jet mostly circles Antarctica all year round.

Jet streams are the product of two factors: the atmospheric heating by solar radiation that produces the large-scale Polar, Ferrel, and Hadley circulation cells, and the action of the Coriolis force acting on those moving masses. The Coriolis force is caused by the planet's rotation on its axis. On other planets, internal heat rather than solar heating drives their jet streams. The Polar jet stream forms near the interface of the Polar and Ferrel circulation cells; the subtropical jet forms near the boundary of the Ferrel and Hadley circulation cells.Other jet streams also exist. During the Northern Hemisphere summer, easterly jets can form in tropical regions, typically where dry air encounters more humid air at high altitudes. Low-level jets also are typical of various regions such as the central United States. There are also jetstreams in the thermosphere.

Meteorologists use the location of some of the jet streams as an aid in weather forecasting. The main commercial relevance of the jet streams is in air travel, as flight time can be dramatically affected by either flying with the flow or against, which results in significant fuel and time cost savings for airlines. Often, the airlines work to fly 'with' the jet stream for this reason. Dynamic North Atlantic Tracks are one example of how airlines and air traffic control work together to accommodate the jet stream and winds aloft that results in the maximum benefit for airlines and other users. Clear-air turbulence, a potential hazard to aircraft passenger safety, is often found in a jet stream's vicinity, but it does not create a substantial alteration on flight times. These are narrow belts.

Kelvin waveA 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 group speed of the wave energy for all frequencies. This means that it retains its shape as it moves in the alongshore direction over time.

A Kelvin wave (fluid dynamics) is also a long scale perturbation mode of a vortex in superfluid dynamics; in terms of the meteorological or oceanographical derivation, one may assume that the meridional velocity component vanishes (i.e. there is no flow in the north–south direction, thus making the momentum and continuity equations much simpler). This wave is named after the discoverer, Lord Kelvin (1879).

Non-inertial reference frameA non-inertial reference frame is a frame of reference that is undergoing acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will in general detect a non-zero acceleration. In a curved spacetime all frames are non-inertial. The laws of motion in non-inertial frames do not take the simple form they do in inertial frames, and the laws vary from frame to frame depending on the acceleration. To explain the motion of bodies entirely within the viewpoint of non-inertial reference frames, fictitious forces (also called inertial forces, pseudo-forces and d'Alembert forces) must be introduced to account for the observed motion, such as the Coriolis force or the centrifugal force, as derived from the acceleration of the non-inertial frame.

As stated by Goodman and Warner, "One might say that F = ma holds in any coordinate system provided the term 'force' is redefined to include the so-called 'reversed effective forces' or 'inertia forces'."

Rossby numberThe **Rossby number** (**Ro**) named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of inertial force to Coriolis force, terms and in the Navier–Stokes equations, respectively. It is commonly used in geophysical phenomena in the oceans and atmosphere, where it characterizes the importance of Coriolis accelerations arising from planetary rotation. It is also known as the **Kibel number**.

The Rossby number (Ro and not R_{o}) is defined as:

where *U* and *L* are, respectively, characteristic velocity and length scales of the phenomenon and *f* = 2 Ω sin φ is the Coriolis frequency, where Ω is the angular frequency of planetary rotation and φ the latitude.

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 and centrifugal forces dominate. For example, in tornadoes, the Rossby number is large (≈ 10^{3}), in low-pressure systems it is low (≈ 0.1 – 1) and in oceanic systems it is of the order of unity, but depending on the phenomena can range over several orders of magnitude (≈ 10^{−2} – 10^{2}). As a result, in tornadoes the Coriolis force is negligible, and balance is between pressure and centrifugal forces (called *cyclostrophic balance*). Cyclostrophic balance also commonly occurs in the inner core of a tropical cyclone. In low-pressure systems, centrifugal force is negligible and balance is between Coriolis and pressure forces (called *geostrophic balance*). In the oceans all three forces are comparable (called *cyclogeostrophic balance*). For a figure showing spatial and temporal scales of motions in the atmosphere and oceans, see Kantha and Clayson.

When the Rossby number is large (either because *f* is small, such as in the tropics and at lower latitudes; or because *L* is small, that is, for small-scale motions such as flow in a bathtub; or for large speeds), the effects of planetary rotation are unimportant and can be neglected. When the Rossby number is small, then the effects of planetary rotation are large and the net acceleration is comparably small allowing the use of the geostrophic approximation.

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. (This article considers only frames rotating about a fixed axis. For more general rotations, see Euler angles.)

Rotating spheresIsaac Newton's rotating spheres argument attempts to demonstrate that true rotational motion can be defined by observing the tension in the string joining two identical spheres. The basis of the argument is that all observers make two observations: the tension in the string joining the bodies (which is the same for all observers) and the rate of rotation of the spheres (which is different for observers with differing rates of rotation). Only for the truly non-rotating observer will the tension in the string be explained using only the observed rate of rotation. For all other observers a "correction" is required (a centrifugal force) that accounts for the tension calculated being different from the one expected using the observed rate of rotation. It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space. Alternatively, these experiments provide an operational definition of what is meant by "absolute rotation", and do not pretend to address the question of "rotation relative to what?" General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.

Secondary circulationA secondary circulation is a circulation induced in a rotating system. For example, the primary circulation of Earth's atmosphere is zonal. If however a parcel of air, that moves in a purely zonal direction, is accelerated or decelerated zonally, the Coriolis force will add a meridional component to its velocity. This meridional circulation is then the secondary circulation.

Taylor–Proudman theoremIn fluid mechanics, the **Taylor–Proudman theorem** (after Geoffrey Ingram Taylor and Joseph Proudman) states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity , the fluid velocity will be uniform along any line parallel to the axis of rotation. must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.

Tropical cyclogenesis is the development and strengthening of a tropical cyclone in the atmosphere. The mechanisms through which tropical cyclogenesis occurs are distinctly different from those through which temperate cyclogenesis occurs. Tropical cyclogenesis involves the development of a warm-core cyclone, due to significant convection in a favorable atmospheric environment.Tropical cyclogenesis requires six main factors: sufficiently warm sea surface temperatures (at least 26.5 °C (79.7 °F)), atmospheric instability, high humidity in the lower to middle levels of the troposphere, enough Coriolis force to develop a low-pressure center, a pre-existing low-level focus or disturbance, and low vertical wind shear.Tropical cyclones tend to develop during the summer, but have been noted in nearly every month in most basins. Climate cycles such as ENSO and the Madden–Julian oscillation modulate the timing and frequency of tropical cyclone development. There is a limit on tropical cyclone intensity which is strongly related to the water temperatures along its path.An average of 86 tropical cyclones of tropical storm intensity form annually worldwide. Of those, 47 reach hurricane/typhoon strength, and 20 become intense tropical cyclones (at least Category 3 intensity on the Saffir–Simpson Hurricane Scale).

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