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In [[fluid mechanics]], the '''Taylor–Proudman theorem'''<ref>The Taylor-Proudman theorem was first derived by Sydney Samuel Hough (1870-1923), a mathematician at Cambridge University.  See:  {{cite journal |author=Hough, S.S. |title=On the application of harmonic analysis to the dynamical theory of the tides. Part I. On Laplace’s "oscillations of the first species," and on the dynamics of ocean currents |journal=Phil. Trans. R. Soc. Lond. A |volume=189 |pages=201–257 |date=January 1, 1897 |doi=10.1098/rsta.1897.0009 |bibcode = 1897RSPTA.189..201H }}</ref> (after [[G. I. Taylor]]<ref>{{cite journal |author=Taylor, G.I. |title=Motion of solids in fluids when the flow is not irrotational |journal=Proc. R. Soc. Lond. A |volume=93 |issue= |pages=92–113 |date=March 1, 1917 |doi=10.1098/rspa.1917.0007 |bibcode = 1917RSPSA..93...99T }}</ref> and [[Joseph Proudman]]<ref>{{cite journal |author=Proudman, J. |title=On the motion of solids in a liquid possessing vorticity |journal=Proc. R. Soc. Lond. A |volume=92 |pages=408–424 |date=July 1, 1916 |doi=10.1098/rspa.1916.0026 |bibcode = 1916RSPSA..92..408P }}</ref>) states that when a solid body is moved slowly within a fluid that is steadily rotated with a high [[angular velocity]] <math>\Omega</math>,  the fluid [[velocity]] will be uniform along any line parallel to the axis of rotation. <math>\Omega</math> must be large compared to the movement of the solid body in order to make the [[coriolis force]] large compared to the acceleration terms.


That this is so may be seen by  considering the [[Navier–Stokes equations]] for steady flow, with zero viscosity and a body force corresponding to the [[Coriolis force]], which are:
:<math>
\rho({\mathbf u}\cdot\nabla){\mathbf u}={\mathbf F}-\nabla p,</math>
where <math>{\mathbf u}</math> is the fluid velocity, <math>\rho</math> is the fluid density, and <math>p</math> the pressure.  If we now make the assumption that <math>F=\nabla\Phi</math> is [[scalar potential]] and the advective term may be neglected (reasonable if the [[Rossby number]] is much less than unity) and that the flow is incompressible (density is constant) then the equations become:


:<math>
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2\rho\Omega\times{\mathbf u}=\nabla \Phi -\nabla p,</math>
where <math>\Omega</math> is the [[angular velocity]] vector. If the [[Curl (mathematics)|curl]] of this equation is taken, the result is the Taylor–Proudman theorem:
:<math>
({\mathbf\Omega}\cdot\nabla){\mathbf u}={\mathbf  0}.
</math>


To derive this, one needs the vector identities
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:<math>\nabla\times(A\times B)=A(\nabla\cdot B)-(A\cdot\nabla)B+(B\cdot\nabla)A-B(\nabla\cdot A)</math>
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:<math>\nabla\times(\nabla p)=0\ </math>
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:<math>\nabla\times(\nabla \Phi)=0\ </math>
 
(because the [[Curl (mathematics)|curl]] of the gradient is always equal to zero).
Note that <math>\nabla\cdot{\mathbf\Omega}=0</math> is also needed (angular velocity is divergence-free).
 
The vector form of the Taylor–Proudman theorem is perhaps better understood by expanding the dot product:
 
:<math>
\Omega_x\frac{\partial {\mathbf u}}{\partial x} + \Omega_y\frac{\partial {\mathbf u}}{\partial y} + \Omega_z\frac{\partial {\mathbf u}}{\partial z}=0.
</math>
 
Now choose coordinates in which <math>\Omega_x=\Omega_y=0</math> and then the equations reduce to
 
:<math>
\frac{\partial{\mathbf u}}{\partial z}=0,</math>
if <math>\Omega_z\neq 0</math>. Note that the implication is that ''all three'' components of the velocity vector are uniform along any line parallel to the z-axis.
 
==Taylor Column==
 
The Taylor column is an imaginary cylinder projected above and below a real cylinder that has been placed parallel to the rotation axis (anywhere in the flow, not necessarily in the center). The flow will curve around the imaginary cylinders just like the real due to the Taylor–Proudman theorem, which states that the flow in a rotating, homogenous, inviscid fluid are 2-dimensional in the plane orthogonal to the rotation axis and thus there is no variation in the flow along the <math>\vec{\Omega}</math>axis, often taken to be the <math>\hat{z}</math> axis.  
 
The Taylor column is a simplified, experimentally observed effect of what transpires in the Earth's atmospheres and oceans.
 
== References ==
{{Reflist}}
 
{{DEFAULTSORT:Taylor-Proudman theorem}}
[[Category:Fluid dynamics]]
[[Category:Physics theorems]]

Latest revision as of 11:12, 27 December 2014


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