Editing Coriolis force (section)

===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]].}}