Editing Coriolis force (section)

==Applied to the Earth==
The acceleration affecting the motion of air "sliding" over the Earth's surface is the horizontal component of the Coriolis term
<math display="block"> -2 \, \boldsymbol{\omega} \times \mathbf{v}</math>
This component is orthogonal to the velocity over the Earth surface and is given by the expression
<math display="block"> \omega \, v\  2 \, \sin \phi </math>
where
* <math> \omega </math> is the spin rate of the Earth
* <math> \phi </math> is the latitude, positive in the [[northern hemisphere]] and negative in the [[southern hemisphere]]

In the northern hemisphere, where the latitude is positive, this acceleration, as viewed from above, is to the right of the direction of motion. Conversely, it is to the left in the southern hemisphere.

===Rotating sphere===
[[File:Earth coordinates.svg|thumb|Coordinate system at latitude φ with ''x''-axis east, ''y''-axis north, and ''z''-axis upward (i.e. radially outward from center of sphere)]]

Consider a location with latitude ''φ'' on a sphere that is rotating around the north–south axis. A local coordinate system is set up with the ''x'' axis horizontally due east, the ''y'' axis horizontally due north and the ''z'' axis vertically upwards. The rotation vector, velocity of movement and Coriolis acceleration expressed in this local coordinate system [listing components in the order east (e), north (n) and upward (u)] are:<ref name=Menke>{{Cite book| title=Geophysical Theory |author = Menke, WIlliam & Abbott, Dallas |pages=124–126 |url=https://books.google.com/books?id=XP3R_pVnOoEC&pg=PA120 |isbn=9780231067928 |year=1990 |location = New York, NY | publisher=Columbia University Press}}</ref>
<math display="block">\boldsymbol{ \Omega} = \omega \begin{pmatrix} 0 \\ \cos \varphi \\ \sin \varphi \end{pmatrix}\ , \mathbf{ v} = \begin{pmatrix} v_{\mathrm e} \\ v_{\mathrm n} \\ v_{\mathrm u} \end{pmatrix}\ ,</math>
<math display="block">\mathbf{a}_{\mathrm C } =-2\boldsymbol{\Omega} \times\mathbf{v}= 2\,\omega\, \begin{pmatrix} v_{\mathrm n} \sin \varphi-v_{\mathrm u} \cos \varphi \\ -v_{\mathrm e} \sin \varphi \\ v_{\mathrm e} \cos\varphi\end{pmatrix}\ .</math>
When considering atmospheric or oceanic dynamics, the vertical velocity is small, and the vertical component of the Coriolis acceleration (<math>v_e \cos\varphi</math>) is small compared with the acceleration due to gravity (g, approximately {{convert|9.81|m/s2|abbr=on}} near Earth's surface). For such cases, only the horizontal (east and north) components matter.{{citation needed|date = June 2023}} The restriction of the above to the horizontal plane is (setting ''v<sub>u</sub>''&nbsp;=&nbsp;0):{{citation needed|date = June 2023}}
<math display="block"> \mathbf{ v} = \begin{pmatrix} v_{\mathrm e} \\ v_{\mathrm n}\end{pmatrix}\ , \mathbf{ a}_{\mathrm C} = \begin{pmatrix} v_{\mathrm n} \\ -v_{\mathrm e}\end{pmatrix}\ f\ , </math>
where <math>f = 2 \omega \sin \varphi \,</math> is called the Coriolis parameter.

By setting ''v''<sub>n</sub> = 0, it can be seen immediately that (for positive ''φ'' and ''ω'') a movement due east results in an acceleration due south; similarly, setting ''v''<sub>e</sub> = 0, it is seen that a movement due north results in an acceleration due east.{{citation needed|date = June 2023}} In general, observed horizontally, looking along the direction of the movement causing the acceleration, the acceleration always is turned 90° to the right (for positive ''φ'') and of the same size regardless of the horizontal orientation.{{citation needed|date = June 2023}}

In the case of equatorial motion, setting ''φ'' = 0° yields:
<math display="block">
  \boldsymbol{ \Omega} = \omega \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\ ,
  \mathbf{ v} = \begin{pmatrix} v_{\mathrm e} \\ v_{\mathrm n} \\ v_{\mathrm u} \end{pmatrix}\ ,
</math>
<math display="block">
  \mathbf{ a}_{\mathrm C}  = -2\boldsymbol{\Omega} \times\mathbf{v} = 2\,\omega\, \begin{pmatrix} -v_{\mathrm u } \\0 \\ v_{\mathrm e} \end{pmatrix}\ .
</math>
'''Ω''' in this case is parallel to the north-south axis.

Accordingly, an eastward motion (that is, in the same direction as the rotation of the sphere) provides an upward acceleration known as the [[Eötvös effect]], and an upward motion produces an acceleration due west.{{citation needed|date = June 2023}}<ref>{{Cite journal |last=Persson |first=Anders. O |date=2005 |title=The Coriolis Effect: Four centuries of conflict between common sense and mathematics, Part I: A history to 1885 |journal=History of Meteorology |issue=2 |via=University of Exeter}}</ref>

{{Self-reference|For additional examples in other articles, see [[rotating spheres]], [[Centrifugal force (rotating reference frame)#Apparent motion of stationary objects|apparent motion of stationary objects]], and [[Fictitious force#Crossing a carousel|carousel]].}}

===Meteorology and oceanography===
[[File:Northern vs Southern hemisphere tropical cyclones.jpg|thumb|Due to the Coriolis force, low-pressure systems in the Northern hemisphere, like [[Typhoon Nanmadol (2022)|Typhoon Nanmadol]] (left), rotate counterclockwise, and in the Southern hemisphere, low-pressure systems like [[Cyclone Darian]] (right) rotate clockwise.]]
[[File:Coriolis effect10.svg|thumbnail|Schematic representation of flow around a '''low'''-pressure area in the Northern Hemisphere. The Rossby number is low, so the centrifugal force is virtually negligible. The pressure-gradient force is represented by blue arrows, the Coriolis acceleration (always perpendicular to the velocity) by red arrows]]
[[File:Coriolis effect14.png|thumb|Schematic representation of inertial circles of air masses in the absence of other forces, calculated for a wind speed of approximately {{convert|50|to|70|m/s|mph|sp=us|abbr=on}}.]]
[[File:The Earth seen from Apollo 17.jpg|right|thumbnail|Cloud formations in a famous image of Earth from Apollo 17, makes similar circulation directly visible]]

Perhaps the most important impact of the Coriolis effect is in the large-scale dynamics of the oceans and the atmosphere. In meteorology and [[oceanography]], it is convenient to postulate a rotating frame of reference wherein the Earth is stationary. In accommodation of that provisional postulation, the [[Centrifugal force|centrifugal]] and Coriolis forces are introduced. Their relative importance is determined by the applicable [[Coriolis effect#Length scales and the Rossby number|Rossby numbers]]. [[Tornado]]es have high Rossby numbers, so, while tornado-associated centrifugal forces are quite substantial, Coriolis forces associated with tornadoes are for practical purposes negligible.<ref name=Holton2>{{cite book| title=An Introduction to Dynamic Meteorology |year=2004 |author=Holton, James R.  |url=https://books.google.com/books?id=fhW5oDv3EPsC&pg=PA64|page=64 |isbn=9780123540157 |publisher=Elsevier Academic Press |location=Burlington, MA}}</ref>

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 [[hurricane|cyclones]] as well. Many of the ocean's largest currents circulate around warm, high-pressure areas called [[gyre]]s. Though the circulation is not as significant as that in the air, the deflection caused by the Coriolis effect is what creates the spiralling pattern in these gyres. The spiralling wind pattern helps the hurricane form. The stronger the force from the Coriolis effect, the faster the wind spins and picks up additional energy, increasing the strength of the hurricane.<ref name="Coriolis effect">{{cite web|last=Brinney|first=Amanda |title= What Is the Coriolis Effect? |url=https://www.thoughtco.com/what-is-the-coriolis-effect-1435315 | work=ThoughtCo.com}}</ref>{{better source needed|date = June 2023}}

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]].<ref>{{Cite news| url=https://www.nationalgeographic.org/encyclopedia/coriolis-effect/ |title=The Coriolis Effect: Earth's Rotation and Its Effect on Weather | format = grades 3-12 teaching resource | author = Evers, Jeannie (Ed.) | date=May 2, 2023| location = Washington, DC | publisher =National Geographic Society| access-date=2018-01-17|language=en}}</ref>{{better source needed|date = June 2023}}

====Flow around a low-pressure area====
{{Main|Low-pressure area}}
If a low-pressure area forms in the atmosphere, air tends to flow in towards it, but is deflected perpendicular to its velocity by the Coriolis force. A system of equilibrium can then establish itself creating circular movement, or a cyclonic flow. Because the Rossby number is low, the force balance is largely between the [[pressure-gradient force]] acting towards the low-pressure area and the Coriolis force acting away from the center of the low pressure.

Instead of flowing down the gradient, large scale motions in the atmosphere and ocean tend to occur perpendicular to the pressure gradient. This is known as [[geostrophic flow]].<ref name=Barry>{{cite book| title=Atmosphere, Weather and Climate |author = Barry, Roger Graham & Chorley, Richard J. | url=https://books.google.com/books?id=MUQOAAAAQAAJ&pg=PA115|page=115 |isbn=9780415271714 |year=2003 | location = Abingdon-on-Thames, Oxfordshire, England | publisher=Routledge-Taylor & Francis}}</ref> On a non-rotating planet, fluid would flow along the straightest possible line, quickly eliminating pressure gradients. The geostrophic balance is thus very different from the case of "inertial motions" (see below), which explains why mid-latitude cyclones are larger by an order of magnitude than inertial circle flow would be.{{citation needed|date = June 2023}}

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.<ref>{{cite web | last = Nelson | first = Stephen | title = Tropical Cyclones (Hurricanes) | work = Wind Systems: Low Pressure Centers | location = New Orleans, LA | publisher = [[Tulane University]] | date = Fall 2014 | url = http://www.tulane.edu/~sanelson/New_Orleans_and_Hurricanes/tropical_cyclones.htm | access-date = 2016-12-24 }}</ref> At high altitudes, outward-spreading air rotates in the opposite direction.{{citation needed|date = June 2023}}<ref>For instance, see the image appearing in this source: {{cite web | author = NASA Staff | date =  | title = Cloud Spirals and Outflow in Tropical Storm Katrina from Earth Observatory | work = JPL.NASA.gov | url = https://photojournal.jpl.nasa.gov/catalog/PIA04384 | access-date =  | location =  | publisher = [[NASA]]  }}{{full citation needed|date = June 2023}}</ref>{{full citation needed|date = June 2023}} Cyclones rarely form along the equator due to the weak Coriolis effect present in this region.<ref>{{Cite book| url=https://books.google.com/books?id=XRxzAwAAQBAJ&pg=PA326 |title=Encyclopedia of Disaster Relief |last1=Penuel|first1=K. Bradley |last2=Statler|first2=Matt |date=2010-12-29 |publisher=SAGE Publications |isbn=9781452266398 |page=326|language=en}}</ref>

====Inertial circles====
An air or water mass moving with speed <math>v\,</math> 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 <math>R</math> is given by:
<math display="block">R = \frac{v}{f}</math>
where <math>f</math> is the Coriolis parameter <math>2 \Omega \sin \varphi</math>, introduced above (where <math>\varphi</math> is the latitude). The time taken for the mass to complete a full circle is therefore <math>2\pi/f</math>. The Coriolis parameter typically has a mid-latitude value of about 10<sup>−4</sup>&nbsp;s<sup>−1</sup>; hence for a typical atmospheric speed of {{convert|10|m/s|mph|sp=us|abbr=on}}, the radius is {{convert|100|km|0|sp=us|abbr=on}} with a period of about 17&nbsp;hours. For an ocean current with a typical speed of {{convert|10|cm/s|mph|sp=us|abbr=on}}, the radius of an inertial circle is {{convert|1|km|1|sp=us|abbr=on}}. 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 <math>f</math> is constant and the trajectories are exact circles. On a rotating planet, <math>f</math> varies with latitude and the paths of particles do not form exact circles. Since the parameter <math>f</math> varies as the sine of the latitude, the radius of the oscillations associated with a given speed are smallest at the poles (latitude of ±90°), and increase toward the equator.<ref name=Marshall2>{{Cite book|title= Atmosphere, Ocean and Climate Dynamics: An Introductory Text | page = 98 |author1=Marshall, John|author2=Plumb, R. Alan |url=https://books.google.com/books?id=aTGYbmVaA_gC&pg=PA98 |isbn=9780125586917 |year=2007 |publisher=Elsevier Academic Press |location=Amsterdam}}</ref>

====Other terrestrial effects====
The Coriolis effect strongly affects the large-scale oceanic and [[atmospheric circulation]], leading to the formation of robust features like [[jet stream]]s and [[western boundary current]]s. 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 wave]]s and [[Kelvin wave]]s. It is also instrumental in the so-called [[Ekman layer|Ekman dynamics]] in the ocean, and in the establishment of the large-scale ocean flow pattern called the [[Sverdrup balance]].

===Eötvös effect===
{{Main|Eötvös effect}}
The practical impact of the "Coriolis effect" is mostly caused by the horizontal acceleration component produced by horizontal motion.

There are other components of the Coriolis effect. Westward-traveling objects are deflected downwards, while eastward-traveling objects are deflected upwards.<ref>{{cite book |title=Fundamentals of Geophysics |edition=illustrated |first1=William |last1=Lowrie |publisher=Cambridge University Press |year=1997 |isbn=978-0-521-46728-5 |page=45 |url=https://books.google.com/books?id=7vR2RJSIGVoC}} [https://books.google.com/books?id=7vR2RJSIGVoC&pg=PA45 Extract of page 45]</ref> This is known as the [[Eötvös effect]]. This aspect of the Coriolis effect is greatest near the equator. The force produced by the Eötvös effect is similar to the horizontal component, but the much larger vertical forces due to gravity and pressure suggest that it is unimportant in the [[hydrostatic equilibrium]]. However, in the atmosphere, winds are associated with small deviations of pressure from the hydrostatic equilibrium. In the tropical atmosphere, the order of magnitude of the pressure deviations is so small that the contribution of the Eötvös effect to the pressure deviations is considerable.<ref>{{cite journal |last1=Ong |first1=H. |last2=Roundy |first2=P.E. |title=Nontraditional hypsometric equation |journal=Q. J. R. Meteorol. Soc. |date=2020 |volume=146 |issue=727 |pages=700–706 |doi=10.1002/qj.3703|arxiv=2011.09576|bibcode=2020QJRMS.146..700O |s2cid=214368409 |doi-access=free }}</ref>

In addition, objects traveling upwards (i.e. ''out'') or downwards (i.e. ''in'') are deflected to the west or east respectively. This effect is also the greatest near the equator. Since vertical movement is usually of limited extent and duration, the size of the effect is smaller and requires precise instruments to detect. For example, idealized numerical modeling studies suggest that this effect can directly affect tropical large-scale wind field by roughly 10% given long-duration (2 weeks or more) heating or cooling in the atmosphere.<ref>{{cite journal |last1=Hayashi |first1=M. |last2=Itoh |first2=H. |title=The Importance of the Nontraditional Coriolis Terms in Large-Scale Motions in the Tropics Forced by Prescribed Cumulus Heating |journal=J. Atmos. Sci. |date=2012 |volume=69 |issue=9 |pages=2699–2716 |doi=10.1175/JAS-D-11-0334.1|bibcode=2012JAtS...69.2699H |doi-access=free }}</ref><ref>{{cite journal |last1=Ong |first1=H. |last2=Roundy |first2=P.E. |title=Linear effects of nontraditional Coriolis terms on intertropical convergence zone forced large-scale flow |journal=Q. J. R. Meteorol. Soc. |date=2019 |volume=145 |issue=723 |pages=2445–2453 |doi=10.1002/qj.3572|arxiv=2005.12946 |bibcode=2019QJRMS.145.2445O |s2cid=191167018 }}</ref> Moreover, in the case of large changes of momentum, such as a spacecraft being launched into orbit, the effect becomes significant. The fastest and most fuel-efficient path to orbit is a launch from the equator that curves to a directly eastward heading.

====Intuitive example====
Imagine a train that travels through a [[friction]]less 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&nbsp;m/s).<ref name=Persson>{{cite journal |last1=Persson |first1=Anders |title=The Coriolis Effect – a conflict between common sense and mathematics |page=8 |url=http://met.no/english/topics/nomek_2005/coriolis.pdf |access-date=6 September 2015 |publisher=The Swedish Meteorological and Hydrological Institute |location=[[Norrköping]], [[Sweden]] |url-status=dead |archive-url=https://web.archive.org/web/20050906101226/http://met.no/english/topics/nomek_2005/coriolis.pdf |archive-date=6 September 2005}}</ref> 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 reference frame|rotating frame of reference]] on [[Earth]] first, and then checked against a fixed [[Inertial frame of reference|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 [[rotation around a fixed axis|axis of rotation]]; the train is denoted by a few red pixels, fixed at the left side in the leftmost picture, moving in the others <math>\left(1\text{ day} \mathrel\overset{\land}{=} 8\text{ s}\right):</math>

[[File:Earth and train 2FPS.gif|center|Earth and train]]

# 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.{{paragraph}}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 of Earth|gravity]] and the reaction from the track. This force is greater (by 0.34%)<ref name=Persson /> 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.
# 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.{{paragraph}}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.
# 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.{{paragraph}}[[File:Eotvos efect on 10Kg.png|thumb|350 px|alt=|Graph of the force experienced by a {{Convert|10|kg|adj=on}} object as a function of its speed moving along Earth's equator (as measured within the rotating frame). (Positive force in the graph is directed upward. Positive speed is directed eastward and negative speed is directed westward).]] From the fixed inertial frame of reference above Earth, the train traveling east now rotates at twice the rate as when it was at rest—so the amount of centripetal force needed to cause that circular path increases leaving less force from gravity to act on the track. This is what the Coriolis term accounts for on the previous paragraph.{{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]].<ref>{{cite book |last1=Lowrie |first1=William |title=A Student's Guide to Geophysical Equations |date=2011 |publisher=[[Cambridge University Press]] |isbn=978-1-139-49924-8 |page=141 |url=https://books.google.com/books?id=HPE1C9vtWZ0C&pg=PA141 |access-date=25 February 2020 |language=en}}</ref>

The above example can be used to explain why the Eötvös effect starts diminishing when an object is traveling westward as its [[tangential speed]] increases above Earth's rotation (465&nbsp;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 {{Convert|930|m/s|mph|abbr=on}} 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 {{Convert|930|m/s|mph|abbr=on|sp=us}} experiences an upward force instead. In the figure, the Eötvös effect is illustrated for a {{Convert|10|kg|adj=on}} object on the train at different speeds. The parabolic shape is because the [[centripetal force]] is proportional to the square of the tangential speed. On the inertial frame, the bottom of the parabola is centered at the origin. The offset is because this argument uses the Earth's rotating frame of reference. The graph shows that the Eötvös effect is not symmetrical, and that the resulting downward force experienced by an object that travels west at high velocity is less than the resulting upward force when it travels east at the same speed.

===Draining in bathtubs and toilets===
Contrary to popular misconception, bathtubs, toilets, and other water receptacles do not drain in opposite directions in the Northern and Southern Hemispheres. This is because the magnitude of the Coriolis force is negligible at this scale.<ref name = SciAmer2>{{cite web| author = Scientific American Staff, and Hanson, Brad; Decker, Fred W.; Ehrlich, Robert & Humphrey, Thomas | date = January 28, 2001 | title = Can Somebody Finally Settle This Question: Does Water Flowing Down a Drain Spin in Different Directions Depending on Which Hemisphere You're In? And If So, Why? | work=ScientificAmerican.com | format = serial expert interviews | location = Berlin | publisher = Scientific American-Springer Nature | url=https://www.scientificamerican.com/article/can-somebody-finally-sett/ |access-date=June 28, 2023}}</ref><ref>{{cite web| author = Fraser, Alistair B. | date = | title = Bad Coriolis... Bad Meteorology | work = EMS.PSU.edu | format = teacher's resource | url=http://www.ems.psu.edu/~fraser/Bad/BadCoriolis.html|access-date=June 28, 2023}}</ref><ref>{{cite web|url=http://www.snopes.com/science/coriolis.asp|title=Flush Bosh |access-date=2016-12-21 |work=www.snopes.com|date=28 April 2003 }}</ref><ref>{{cite web |url=http://science.howstuffworks.com/science-vs-myth/everyday-myths/rotation-earth-toilet-baseball2.htm|title=Does the rotation of the Earth affect toilets and baseball games?|date=2009-07-20|access-date=2016-12-21}}</ref> Forces determined by the initial conditions of the water (e.g. the geometry of the drain, the geometry of the receptacle, preexisting momentum of the water, etc.) are likely to be orders of magnitude greater than the Coriolis force and hence will determine the direction of water rotation, if any. For example, identical toilets flushed in both hemispheres drain in the same direction, and this direction is determined mostly by the shape of the toilet bowl.

Under real-world conditions, the Coriolis force does not influence the direction of water flow perceptibly. Only if the water is so still that the effective rotation rate of the Earth is faster than that of the water relative to its container, and if externally applied torques (such as might be caused by flow over an uneven bottom surface) are small enough, the Coriolis effect may indeed determine the direction of the vortex. Without such careful preparation, the Coriolis effect will be much smaller than various other influences on drain direction<ref>{{cite book| url=https://books.google.com/books?id=8hOLs-bmiYYC&pg=PA168| pages=168–9| title=Physics: A World View |author1=Kirkpatrick, Larry D.  |author2=Francis, Gregory E.  |year=2006 |publisher=Cengage Learning |isbn=978-0-495-01088-3}}</ref> such as any residual rotation of the water<ref>{{cite journal |journal=Journal of Fluid Mechanics |title=Stationary bathtub vortices and a critical regime of liquid discharge |author1=Y. A. Stepanyants |author2=G. H. Yeoh |volume=604 |issue=1 |pages=77–98 |year=2008 |bibcode=2008JFM...604...77S |doi=10.1017/S0022112008001080 |s2cid=53071268 |url=http://eprints.usq.edu.au/5726/2/Stepanyants_Yeoh_2008_PV.pdf |access-date=13 July 2019 |archive-date=23 July 2022 |archive-url=https://web.archive.org/web/20220723091453/https://eprints.usq.edu.au/5726/2/Stepanyants_Yeoh_2008_PV.pdf |url-status=dead }}</ref> and the geometry of the container.<ref>{{cite book|title=A Student's Guide to Earth Science: Words and terms |author=Creative Media Applications|publisher=Greenwood Publishing Group|year=2004 |isbn=978-0-313-32902-9 |page=22 |url=https://books.google.com/books?id=fF0TTZVQuZoC&pg=PA22}}</ref>

====Laboratory testing of draining water under atypical conditions====
In 1962, [[Ascher H. Shapiro|Ascher Shapiro]] performed an experiment at [[Massachusetts Institute of Technology|MIT]] to test the Coriolis force on a large basin of water, {{Convert|2|m|sp=us}} across, with a small wooden cross above the plug hole to display the direction of rotation, covering it and waiting for at least 24 hours for the water to settle. Under these precise laboratory conditions, he demonstrated the effect and consistent counterclockwise rotation. The experiment required extreme precision, since the acceleration due to Coriolis effect is only <math>3\times 10^{-7}</math> that of gravity. The vortex was measured by a cross made of two slivers of wood pinned above the draining hole. It takes 20 minutes to drain, and the cross starts turning only around 15 minutes. At the end it is turning at 1 rotation every 3 to 4 seconds.

He reported that,<ref>{{cite journal |last1=Shapiro |first1=Ascher H. |title=Bath-Tub Vortex |journal=Nature |date=December 1962 |volume=196 |issue=4859 |pages=1080–1081 |doi=10.1038/1961080b0|bibcode=1962Natur.196.1080S |s2cid=26568380 }}</ref>
{{Blockquote|text= Both schools of thought are in some sense correct. For the everyday observations of the kitchen sink and bath-tub variety, the direction of the vortex seems to vary in an unpredictable manner with the date, the time of day, and the particular household of the experimenter. But under well-controlled conditions of experimentation, the observer looking downward at a drain in the northern hemisphere will always see a counter-clockwise vortex, while one in the southern hemisphere will always see a clockwise vortex. In a properly designed experiment, the vortex is produced by Coriolis forces, which are counter-clockwise in the northern hemisphere.}}

Lloyd Trefethen reported clockwise rotation in the [[Southern Hemisphere]] at the University of Sydney in five tests with settling times of 18 h or more.<ref>{{cite journal |last1=Trefethen |first1=Lloyd M. |last2=Bilger |first2=R. W. |last3=Fink |first3=P. T. |last4=Luxton |first4=R. E. |last5=Tanner |first5=R. I. |title=The Bath-Tub Vortex in the Southern Hemisphere |journal=Nature |date=September 1965 |volume=207 |issue=5001 |pages=1084–1085 |doi=10.1038/2071084a0|bibcode=1965Natur.207.1084T |s2cid=4249876 }}</ref>

===Ballistic trajectories===
The Coriolis force is important in [[External ballistics#Coriolis drift|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 {{convert|120|km|sp=us|abbr=on}}. The Coriolis force minutely changes the trajectory of a bullet, affecting accuracy at extremely long distances. It is adjusted for by accurate long-distance shooters, such as snipers. At the latitude of [[Sacramento]], California, a {{convert|1000|yd|abbr=on}} northward shot would be deflected {{convert|2.8|in|abbr=on}} 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.<ref name=Taylor0>The claim is made that in the Falklands in WW I, the British failed to correct their sights for the southern hemisphere, and so missed their targets. {{Cite book|title=A Mathematician's Miscellany |author=John Edensor Littlewood |page=[https://archive.org/details/mathematiciansmi033496mbp/page/n62 51] |url=https://archive.org/details/mathematiciansmi033496mbp |year=1953 |publisher=Methuen And Company Limited}} {{Cite book|title=Classical Mechanics |author=John Robert Taylor |page=364; Problem 9.28 |url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PA364 |isbn=978-1-891389-22-1 |year=2005 |publisher=University Science Books}} For set up of the calculations, see Carlucci & Jacobson (2007), p. 225</ref><ref>{{cite news |title=Do Snipers Compensate for the Earth's Rotation? |url=https://www.washingtoncitypaper.com/columns/straight-dope/article/13039128/do-snipers-compensate-for-the-earthrsquos-rotation-what-the-coriolis |access-date=16 July 2018 |work=Washington City Paper |date=25 June 2010 |language=en}}</ref>

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.<ref>{{cite book |first1=Barry A. |last1=Klinger |first2=Thomas W. N. |last2=Haine |date=2019 |chapter-url=https://books.google.com/books?id=Kr2GDwAAQBAJ&pg=PA291 |title=Ocean Circulation in Three Dimensions |chapter=Deep Meridional Overturning |department=Thermohaline Overturning |isbn=978-0521768436 |publisher=Cambridge University Press |access-date=2019-08-19 }}</ref>

[[File:Trajectory-groundtrack-drift.png|thumb|Trajectory, ground track, and drift of a typical projectile. The axes are not to scale.]]

The Coriolis force on a moving [[projectile]] depends on velocity components in all three directions, [[latitude]], and [[azimuth]]. The directions are typically downrange (the direction that the gun is initially pointing), vertical, and cross-range.<ref>{{Citation |last=McCoy|first= Robert L.|year= 1999|title=Modern Exterior Ballistics   |publisher=Schiffer Military History  |isbn=0-7643-0720-7 }}</ref>{{rp|178}}

<math display="block"> A_\mathrm{X} = -2 \omega ( V_\mathrm{Y} \cos \theta_\mathrm{lat} \sin \phi_\mathrm{az} + V_\mathrm{Z}  \sin \theta_\mathrm{lat} )  </math>
<math display="block"> A_\mathrm{Y} = 2 \omega ( V_\mathrm{X} \cos \theta_\mathrm{lat} \sin \phi_\mathrm{az} + V_\mathrm{Z}  \cos \theta_\mathrm{lat} \cos \phi_\mathrm{az})  </math>
<math display="block"> A_\mathrm{Z} = 2 \omega ( V_\mathrm{X} \sin \theta_\mathrm{lat}  - V_\mathrm{Y}  \cos \theta_\mathrm{lat} \cos \phi_\mathrm{az})  </math>
where
* <math> A_\mathrm{X} </math>, down-range acceleration.
* <math> A_\mathrm{Y} </math>, vertical acceleration with positive indicating acceleration upward.
* <math> A_\mathrm{Z} </math>, cross-range acceleration with positive indicating acceleration to the right.
* <math> V_\mathrm{X} </math>, down-range velocity.
* <math> V_\mathrm{Y} </math>, vertical velocity with positive indicating upward.
* <math> V_\mathrm{Z} </math>, cross-range velocity with positive indicating velocity to the right.
* <math> \omega </math> = 0.00007292&nbsp;rad/sec, angular velocity of the earth (based on a [[sidereal day]]).
* <math> \theta_\mathrm{lat} </math>, latitude with positive indicating Northern hemisphere.
* <math> \phi_\mathrm{az} </math>, [[azimuth]] measured clockwise from due North.