Editing Taylor–Proudman theorem

In [[fluid mechanics]], the '''Taylor–Proudman theorem''' (after [[G. I. Taylor|Geoffrey Ingram Taylor]] and [[Joseph Proudman]]) states that when a solid body{{clarify|reason=The derivation doesn't mention a solid body anywhere, while this statement seems to imply that a solid body is necessary for this theorem to apply.|date=August 2020}} 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.

==Derivation==
The [[Navier–Stokes equations]] for steady flow, with zero [[viscosity]] and a body force corresponding to the Coriolis force, 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 assume that <math>F=\nabla\Phi=-2\rho\mathbf\Omega\times{\mathbf u}</math> is a [[scalar potential]] and the [[Advection|advective]] term on the left may be neglected (reasonable if the [[Rossby number]] is much less than unity) and that the [[Incompressible flow|flow is incompressible]] (density is constant), the equations become:

:<math>
2\rho\mathbf\Omega\times{\mathbf u}=-\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 calculus identities|vector identities]]

:<math>\nabla\times(A\times B)=A(\nabla\cdot B)-(A\cdot\nabla)B+(B\cdot\nabla)A-B(\nabla\cdot A)</math> 

and 

:<math>\nabla\times(\nabla p)=0\ </math>

and 
:<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>

In coordinates for which <math>\Omega_x=\Omega_y=0</math>, the equations reduce to
:<math>
\frac{\partial{\mathbf u}}{\partial z}=0,</math>
if <math>\Omega_z\neq 0</math>.  Thus, ''all three'' components of the velocity vector are uniform along any line parallel to the z-axis.

==Taylor column==
{{Main article|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, homogeneous, 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.

==History==

The result known as the Taylor-Proudman theorem was first derived by Sydney Samuel Hough (1870-1923), a mathematician at Cambridge University, in 1897.<ref>{{cite book|last1=Gill|first1=Adrian E.|author-link=Adrian Gill (meteorologist)|title=Atmosphere—Ocean Dynamics|date=2016|publisher=Elsevier|isbn=9781483281582}}</ref>{{rp|506}}<ref>{{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 |doi-access=free }}</ref> Proudman published another derivation in 1916 and Taylor in 1917, then the effect was demonstrated experimentally by Taylor in 1923.<ref>{{cite book|last1=Wu|first1=J.-Z.|last2=Ma|first2=H.-Y.|last3=Zhou|first3=M.-D.|title=Vorticity and vortex dynamics|date=2006|publisher=Springer|location=Berlin|isbn=9783540290285}}</ref>{{rp|648}}<ref>{{cite book|last1=Longair|first1=Malcolm|author-link=Malcolm Longair|title=Maxwell's Enduring Legacy: A Scientific History of the Cavendish Laboratory|date=2016|publisher=Cambridge University Press|isbn=9781316033418|url=https://books.google.com/books?id=vAWpDAAAQBAJ}}</ref>{{rp|245}}<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 |issue=642 |doi=10.1098/rspa.1916.0026 |bibcode = 1916RSPSA..92..408P |doi-access=free }}</ref><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 |pages=92–113 |date=March 1, 1917 |issue=648 |doi=10.1098/rspa.1917.0007 |bibcode = 1917RSPSA..93...99T |doi-access=free }}</ref>

== References ==
{{Reflist}}

{{DEFAULTSORT:Taylor-Proudman theorem}}
[[Category:Eponymous theorems of physics]]
[[Category:Fluid dynamics]]