Force of Coriolis

The force of Coriolis is a inertial Force acting perpendicular to the direction of the movement of a body in displacement in a medium (a reference frame) itself in uniform Rotation, as seen by an observer sharing the same reference frame.

This force is named in the honor of the French engineer Gaspard-Gustave Coriolis.

History

At the end of the 18th century and at the beginning of the 19th century, the Mécanique experienced great theoretical developments. As an engineer, Coriolis was interested to make mechanics theoretical applicable in the comprehension and the development of industrial machines. It is in its article On the equations of the relative movement of the systems of body (1835) that Coriolis mathematically described the force which was to bear its name. In this article, the force of Coriolis seems an additional component with the centrifugal force, felt by a body moving relative with a reference frame in rotation, as that could occur for example in the wheels of a machine.

The argumentation of Coriolis was based on an analysis of the work and potential energy and kinetic in the systems in rotation. Nowadays, the demonstration most used to teach the force of Coriolis uses the tools of the Cinématique.

It is only at the end of the 19th century that this force made its appearance in the weather and oceanographical literature. The term force of Coriolis appeared at the beginning of the 20th century.

Definition

In Newtonian Mécanique, one qualifies the force of Coriolis of fictitious force, or inertia lle , under the terms of the fact that it exists only because the observer is in a reference frame in rotation whereas no force is exerted for an observer at rest or in uniform rectilinear motion (known as reference mark galiléen).

Animation on the right thus shows us the difference between the point of view of a motionless observer and that of an observer which moves with a disc in rotation. For the first, the ball does not make that to move with a constant speed since the center of the disc towards its edge. For him, there is no force concerned and the ball moves in straight line.

For the second (the red point), the ball moves along an arc of circle, towards its left, changing direction constantly. One thus needs a force to explain this displacement. This pseudo-force is the force of Coriolis $\backslash \; vec\; F\_C$. It is perpendicular to the axis of rotation of the reference frame and the vector the speed of the body moving. If the body moves away from the axis of rotation, $\backslash \; vec\; F\_C$ is exerted in the contrary direction of rotation. If the body approaches the axis of rotation, $\backslash \; vec\; F\_C$ is exerted in the same direction as rotation.

Vectorial representation

One can represent $\backslash \; vec\; \{F\_C\}$ like a vector product while using:

$\backslash \; vec\; \{F\_C\}\; =\; -2m\; \backslash \; Omega\; (T)\; (\backslash \; vec\; \{E\}\; \_\; \{axis\}\; \backslash \; wedge\; \backslash \; vec\; \{v\})$

$\backslash \; qquad\; \backslash \; qquad\; \backslash \; begin\; \{boxes\}\; m\; =\; mass\; \backslash \; of\; the\; \backslash \; body\; \backslash \; \backslash \; \backslash \; vec\; \{E\}\; \_\; \{unit\; axis\}\; =\; vector\; \backslash \; \backslash \; parall\; \backslash \; serious\; it\; \backslash \; \backslash \; serious\; has\; \backslash \; the\; axis\; \backslash \; \backslash \; rotation\; \backslash \; \backslash \; \backslash \; Omega\; (T)\; =\; angular\; speed\; \backslash \; \backslash \; instantan\; \backslash \; acute\; \{E\}\; E\; \backslash \; of\; \backslash \; rotation\; \backslash \; \backslash \; \backslash \; vec\; \{v\}\; =\; speed\; \backslash \; of\; the\; \backslash \; body\; \backslash \; end\; \{boxes\}$

However, one can multiply the angular velocity $\backslash \; Omega$ with $\backslash \; vec\; \{E\}\; \_\; \{axis\}$, which produces the vector $\backslash \; vec\; \{\backslash \; Omega\; (T)\}$. This vector instantaneous speed-swivelling $\backslash \; vec\; \{\backslash \; Omega\; (T)\}$ describes thus at the same time the direction and the angular velocity of the reference frame.

$\backslash \; vec\; \{F\_C\}\; =\; -\; 2m\; \backslash \; vec\; \backslash \; Omega\; (T)\; \backslash \; wedge\; \backslash \; vec\; v$

Force of Coriolis and force axifuge

Into the image of the disc and ball seen previously, the latter slips without friction and only the force of Coriolis is present in the reference mark in rotation. In the case of the movement of one body on the surface of the Ground, this last has its movement suitable for the surface of the sphere. It also moves in space, with the rotation of planet, while being attracted by the Gravité. It thus undergoes in more another fictitious force known as Centrifugal force . Both are added:

$\backslash \; vec\; \{F\}\; \_\; \{inertia\}\; =\; \backslash \; vec\; \{F\_C\}\; +\; \backslash \; vec\; \{F\}\; \_\; \{centrifuges\}$

The Centrifugal force and forces it of Coriolis thus appear only in reference frames in swivelling. As one saw previously, the force of Coriolis depends on the vitesse of the body moving. The centrifugal force, actually the force axifuge, definite it like $\backslash \; scriptstyle\; -\; m\; \backslash \; vec\; \{\backslash \; Omega^2\}\; R$ and depends on the position (R) of the body compared to the instantaneous axis of rotation. These two forces can vary if $\backslash \; Omega\; (T)$ varies but for a $\backslash \; Omega\; (T)$ given, one can say that the centrifugal force is the static component of the inertial force appearing in the reference frame in rotation, whereas the force of Coriolis is the kinematic component (cf Inertias)

Simple example

Here another very simple case, which requires the intervention of the force of Coriolis to be interpreted:

That is to say two masses, M and P, describing the same circle with the even angular velocity , in the direct direction and the indirect direction.

• Donc the centrifugal force on each one is identical.
• Each point being balances some in ITS reference frame, there thus exists the same intensity of real force centripetal Fo acting on M and P.
• But reason in the reference frame where P is motionless: M described a circle at the angular velocity doubles there, therefore acceleration has is quadruple. However the real force Fo on M did not change and remains cancelled by the centrifugal force. It is thus necessary well that another force intervenes so that M describes the circle! and it must be worth 4Fo and be centripetal.
• It is well what gives the preceding formula.

Applications

The force of Coriolis allows the interpretation of many phenomena the surface of the Ground; for example the movement of the masses of air and the Cyclone S, the deviation of the trajectory of the projectiles with great range (cf Large Bertha), change of the plan of movement of a pendulum as shown by Foucault in his experiment of the Pendulum of Foucault in 1851 with the the Pantheon of Paris, as well as light the Deviation towards the east at the time of the Freefall.

Coriolis in meteorology and oceanography

The most important application of the pseudo-force of Coriolis is without question in Météorologie and Océanographie. Indeed, the movements with large scales of the terrestrial atmosphere are the result of the difference in pressure between various areas of the atmospheric layer but are rather slow so that displacement due to the rotation of the Ground influence the trajectory of a piece of air. Thus let us consider atmospheric circulation but the same remarks are valid for the movements of water in the Mer S.

Circulation around a depression

The flow of air in a mass of air at rest is naturally between the zones where the pressure is higher towards those where they are minimal. If the Earth were not in rotation, the air pressure would thus be equalized quickly and the atmosphere would quickly become Isotrope without contribution of heat. On the other hand, with the different warming with the pole S and the equator which maintains a difference in pressure, there would be an eternal circulation between these two places. This last circulation exists close to the equator where the effect Coriolis becomes null because $\backslash \; vec\; \{\backslash \; Omega\}\; (T)$ and $\backslash \; vec\; V$ becomes parallel (see Cellules of Hadley).

However, the Earth turns and by using the definition of the force of Coriolis in a reference frame in rotation, one sees that the latter increases as speed obtained by the gradient of pressure increases but in the perpendicular direction. This gives a deviation towards the line in the northern hemisphere (left in that of the south) of a piece of air moving. Thus the air circulation will be anti-clockwise around a depression and schedule around an anticyclone (northern hemisphere). It is the Vent geostrophic.

In the figure on the right, one sees how that occurs by taking the four cardinal points like beginning of the interaction of the forces. The gradient of pressure (blue arrows) starts the air volume displacement but the force of Coriolis (red arrows) the fact of deviating towards the line (black arrows). The gradient of pressure is adjusted in direction with this change as well as the force of Coriolis what makes continuously change the direction of our piece. Quickly, the gradient of pressure and the force of Coriolis are opposed and the air volume displacement is stabilized while following a trajectory perpendicular to the gradient and thus parallel with the lines of équi-pressure (isobar S). In fact, because of the Friction, the centrifugal force and the differences in pressure in an area, balance is never really reached and the direction will remain always slightly towards the basic center pressure (see Spirale of Ekman).

The depressions, also called Cyclone S, cannot be formed close to the equator where the horizontal component of the force of Coriolis is null. The variation of the force of Coriolis thus gives various modes of atmospheric Circulation according to the Latitude.

Inertial ballistics and circles

Another practical use of the force of Coriolis is the calculation of the trajectory of the projectiles in the atmosphere. Once a shell is drawn or that a rocket in suborbital flight exhausted its fuel, its trajectory is controlled only by gravity and the winds (when it is in the atmosphere). Let us suppose now that one removes the deviation due to the wind. In the reference mark in rotation which is the Earth, the ground moves compared to the rectilinear trajectory which a motionless observer in space would see. Thus for a terrestrial observer, it is necessary to add the force of Coriolis to know where the projectile will fall down on the ground.

In the figure of right-hand side, one shows the trajectory which a body would traverse s' it had there only the force of Coriolis which agisse. Let us suppose that the body moves at constant speed of the equator towards the north pole at constant altitude of the ground, it undergoes a displacement towards the line by Coriolis (northern hemisphere). Its speed does not change but its curved direction. In its new trajectory, the force of Coriolis goes back to right angle and the fact of curving even more. Finally, it carries out a complete circle in a given time which depends its speed (v) and on the latitude. The radius of this circle (R) is:

$\backslash ,\; R\; =\; v/f$

Where $\backslash \; begin\; \{boxes\}\; F\; \backslash \; is\; \backslash \; the\; \backslash \; horizontal\; projection\; \backslash \; \backslash \; of\; \backslash \; the\; \backslash \; force\; \backslash \; of\; \backslash \; Coriolis\; \backslash \; \backslash$

\ phi = latitude \ \ F = -2 sin (\ phi) \ Omega (T) \ end {boxes} .

For a latitude around 45 degrees, $\backslash ,\; f$ are about 10⁻⁴ seconde^-1 (giving a rotational frequency 2 p.m.). If a projectile is driven to 800 km/h (approximately 200 m/s), the equation gives a radius of curvature of 2.000 km. It is clearly impossible for a projectile on a ballistic curve to remain in the air 14 hours and it will thus carry out only part of the curved trajectory.

On the other hand, in the case of a piece of air or a volume of water moving in a zone where the pressure is uniform (vast collar of pressure), this trajectory known as inertial is possible. With a typical speed of 10 m/s for the air, the ray is of 100 km whereas with speed of 0,1 m/s for water, one obtains a radius of 1 km. In these two cases one would obtain swirls revolving in sens inverse of that of circulation around a depression. It should be remembered that it is about a case where there is no gradient of pressure. Moreover, like $\backslash ,\; f$ varies with the latitude, the circle would be rather a ellipse.

Coriolis with three dimensions

Until now, we had considered movements according to the horizontal one only. Because the Earth is not punt and that the atmosphere has a certain thickness, the movements generally have a vertical component. The force of Coriolis is thus not exerted only parallel to surface of planet but also according to the vertical. One can think for example of a particle of air on the surface which would move in direction of a star in the direction of rotation of the Earth. As the latter turns, its surface changes direction compared to this oriention and the piece seems to move away to the top from where a pseudo-force attracting it in this direction.

This effect is very weak because the force of Coriolis has little time to be exerted before the piece of air reaches the higher or lower limit of the atmosphere but influence certain objects like the ballistic shootings considering higher. If one looks at the effects according to the direction:

• a piece of air going down will be slightly deflected towards the East.
• Another in rise will be deflected towards the West.
• a movement towards the East will go up slightly.
• a movement towards the West will go down slightly.

Erroneous interpretations

1) Contrary to a popular belief, the force of Coriolis due to the rotation of the terrestrial sphere is too weak to have time to influence the direction of rotation of the flow of the Eau in a Lavabo which is emptied. As it Arsher Shapiro and Lloyd Trefethen showed, to perceive such an influence, it is necessary to observe a water mass stabilized in a very large circular basin (of a diameter about to less several tens of kilometers for an effect in centimetres). In the siphon of a wash-hand basin, the direction of rotation of water is due to the geometry of the wash-hand basin and the water microcourants created during its filling (or during an agitation of water).

to calculate the acceleration of Coriolis, has , one uses this relation:

$a\; =\; 2\; \backslash \; Omega\; \backslash \; cdot\; v\; \backslash \; cdot\; \backslash \; sin\; (\backslash \; phi)$

With:

$\backslash \; Omega\; \backslash ,$: angular velocity of the sidereal swivelling of the Earth. Let us take $\backslash \; Omega\; =\; \backslash \; frac\; \{2\; \backslash \; pi\}\; \{86164\}\; \backslash \; rad\; \backslash \; cdot\; s^\; \{-\; 1\}$

$v\; \backslash ,$: Speed of water moving. Let us take $v\; =\; 1\; \backslash \; m\; \backslash \; cdot\; s^\; \{-\; 1\}$

$\backslash \; phi\; \backslash ,$: Latitude of the place considered. Let us take $\backslash \; phi\; \backslash \; approx\; 60$

numerical Application: $a\; =\; \backslash \; frac\; \{4\; \backslash \; pi\; \backslash \; cdot\; \backslash \; sin\; (60)\}\{86164\}\; \backslash \; approx\; 0,0001\; \backslash \; m\; \backslash \; cdot\; s^\; \{-\; 2\}$

Is 100.000 times less than acceleration due to gravity $(g=9,81\; \backslash \; m\; \backslash \; cdot\; s^\; \{-\; 2\})$. Thus the basin is emptied well before the deviation due to Coriolis is felt.

For the anecdote, George Gamow parodied this generally accepted idea while affirming to have noted at the time of a voyage in Australia that in the southern hemisphere, the cows ruminate while making circulate the grass in opposite direction of the direction in the Northern hemisphere.

2) Rotation in a Tornade is generally anti-clockwise but it is not due to Coriolis. In this case, rotation is initiated by the configuration of the winds in the layer of air close to the ground which gives a horizontal rotation of the air. When the ascending fort running of a storm verticalise this rotation and that it concentrates, the direction is already given. One is still there in a field where the movement of the air is too much fast so that the effect Coriolis has time to have an impact.

3) In the case of a Swirl of dust S, the initiation of rotation is done by a difference of the horizontal winds. There is then a vertical axis of swirl created where the Centrifugal force is counterbalanced by that of pressure. The speed of the particles is too fast and on a too small ray so that the force of Coriolis has time to act. The observations showed that rotation in these vortices are statistically divided also between schedules and anti-clockwise, some is the hemisphere.

4) The force of Coriolis does not depend on the curve of the Earth, only of its rotation and the latitude where one is.

5) The ground being a sphere, the geographical maps in two dimensions are necessarily a projection (see for example the Projection of Mercator) which gives a distortion of terrestrial surface. The trajectory of the ballistic missiles, or the shells, is curved when one traces it on a chart but the curve obtained is a sum of the effect Coriolis, winds and projection which was used to make the chart. However these two last are in general more important than the by-pass Coriolis.

References

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