imported>John R. Brews |
imported>John R. Brews |
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| {{Image|Earth coordinates.PNG|right|250px|Coordinate system at latitude φ with ''x''-axis east, ''y''-axis north and ''z''-axis upward (that is, radially outward from center of sphere).}}
| | ==Meteorology== |
| ===Rotating sphere===
| | {{Image|Wind deflection.PNG|right|250px|Wind motion in direction of pressure gradient is deflected by the Coriolis force by an amount dependent upon latitude, less at the equator and most at the poles.}} |
| Consider a location with latitude ''φ'' on a sphere that is rotating around the north-south axis.<ref name=Menke>{{Cite book|title=Geophysical Theory |author=William Menke & Dallas Abbott |pages=124–126 |url=http://books.google.com/?id=XP3R_pVnOoEC&pg=PA120&dq=spheres+rotating++Coriolis
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| |isbn=0231067925 |year=1990 |publisher=Columbia University Press}}</ref> 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: | |
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| :<math>\boldsymbol{ \Omega} = \omega \begin{pmatrix} 0 \\ \cos \varphi \\ \sin \varphi \end{pmatrix}\ ,</math> <math>\boldsymbol{ v} = \begin{pmatrix} v_e \\ v_n \\ v_u \end{pmatrix}\ ,</math>
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| :<math>\boldsymbol{ a_{Cor}} =-2\boldsymbol{\Omega \times v}= 2\,\omega\, \begin{pmatrix} v_n \sin \varphi-v_u \cos \varphi \\ -v_e \sin \varphi \\ v_e \cos\varphi\end{pmatrix}\ .</math>
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| 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<sub>u</sub>''=0):
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| :<math> \boldsymbol{ v} = \begin{pmatrix} v_e \\ v_n\end{pmatrix}\ ,</math> <math>\boldsymbol{ a_{Cor}} = \begin{pmatrix} v_n \\ -v_e\end{pmatrix}\ f\ , </math>
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| where ''f'' = {{nowrap|2''ω'' sin''φ''}} is called the ''Coriolis parameter''.<ref name=Norbury>
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| For a discussion of the role of the Coriolis parameter in ocean dynamics see {{cite book |title=Large-scale Atmosphere-ocean Dynamics: Analytical methods and numerical models |author=John Norbury |isbn=052180681X |year=2002 |publisher=Cambridge University Press |url=http://books.google.com/books?id=CRD6VONGtUkC&pg=PA35 |pages=p. 35 ''ff'' |chapter=§6.3 Further geometric and Coriolis approximations}}
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| </ref>
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| 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. In general, observed horizontally, looking along the direction of the movement causing the acceleration, the acceleration always is turned 90° to the right and of the same size regardless of the horizontal orientation. That is:<ref name=Canterbury>{{Cite web|url = http://www.phys.canterbury.ac.nz/newsletter/2005/nl20051202.pdf|author = David Morin, Eric Zaslow, Elizabeth Haley, John Goldne, and Natan Salwen|title = Limerick – May the Force Be With You|work = Weekly Newsletter Volume 22, No 47|publisher = Department of Physics and Astronomy, University of Canterbury|date = 2 December 2005|accessdate = 2009-01-01}}</ref><ref name=Morin>{{Cite book|author=David Morin |url=http://books.google.com/?id=Ni6CD7K2X4MC&pg=PA466&dq=Coriolis+carousel |title=Introduction to classical mechanics: with problems and solutions |page= 466 |isbn= 0521876222 |year=2008 |publisher=Cambridge University Press}}</ref>
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| {{Quote
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| |On a merry-go-round in the night<br>
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| Coriolis was shaken with fright<br>
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| Despite how he walked<br>
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| 'Twas like he was stalked<br>
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| By some fiend always pushing him right |David Morin, Eric Zaslow, E'beth Haley, John Golden, and Nathan Salwen}}
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| As a different case, consider equatorial motion setting φ = 0°. In this case, '''Ω''' is parallel to the North or ''n''-axis, and:
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| :<math>\boldsymbol{ \Omega} = \omega \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\ ,</math> <math>\boldsymbol{ v} = \begin{pmatrix} v_e \\ v_n \\ v_u \end{pmatrix}\ ,</math> <math>\boldsymbol{ a_{Cor}} =-2\boldsymbol{\Omega \times v}= 2\,\omega\, \begin{pmatrix}-v_u \\0 \\ v_e \end{pmatrix}\ .</math>
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| 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]],<ref name=Talwani>
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| The Eötvös effect must be corrected for in determining the effects of gravity aboard ship. See {{cite book |author= Manik Talwani |url=http://books.google.com/books?id=FRrZYTel0DEC&pg=PA46 |pages=p. 46 |chapter=Gravity measurements aboard surface ships: Eötvös effect |title=Gravity anomalies: unsurveyed areas |publisher=American Geophysical Union |year=1906 |editor= Hyman Orlin, ed.}}
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| </ref> and an upward motion produces an acceleration due west.
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| ==Notes== | | ==Notes== |
