Taylor–Proudman theorem

Taylor–Proudman theorem In fluid mechanics, the Taylor–Proudman theorem (after Geoffrey Ingram Taylor and Joseph Proudman) states that when a solid body[clarification needed] is moved slowly within a fluid that is steadily rotated with a high angular velocity {displaystyle Omega } , the fluid velocity will be uniform along any line parallel to the axis of rotation. {displaystyle Omega } must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.

Contents 1 Derivation 2 Taylor column 3 History 4 References Derivation The Navier–Stokes equations for steady flow, with zero viscosity and a body force corresponding to the Coriolis force, are {displaystyle rho ({mathbf {u} }cdot nabla ){mathbf {u} }={mathbf {F} }-nabla p,} where {displaystyle {mathbf {u} }} is the fluid velocity, {displaystyle rho } is the fluid density, and {displaystyle p} the pressure. If we assume that {displaystyle F=nabla Phi =-2rho mathbf {Omega } times {mathbf {u} }} is a scalar potential and the advective term on the left may be neglected (reasonable if the Rossby number is much less than unity) and that the flow is incompressible (density is constant), the equations become: {displaystyle 2rho mathbf {Omega } times {mathbf {u} }=-nabla p,} where {displaystyle Omega } is the angular velocity vector. If the curl of this equation is taken, the result is the Taylor–Proudman theorem: {displaystyle ({mathbf {Omega } }cdot nabla ){mathbf {u} }={mathbf {0} }.} To derive this, one needs the vector identities {displaystyle nabla times (Atimes B)=A(nabla cdot B)-(Acdot nabla )B+(Bcdot nabla )A-B(nabla cdot A)} and {displaystyle nabla times (nabla p)=0 } and {displaystyle nabla times (nabla Phi )=0 } (because the curl of the gradient is always equal to zero). Note that {displaystyle nabla cdot {mathbf {Omega } }=0} 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: {displaystyle 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.} In coordinates for which {displaystyle Omega _{x}=Omega _{y}=0} , the equations reduce to {displaystyle {frac {partial {mathbf {u} }}{partial z}}=0,} if {displaystyle Omega _{z}neq 0} . 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 {displaystyle {vec {Omega }}} axis, often taken to be the {displaystyle {hat {z}}} 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.[1]: 506 [2] Proudman published another derivation in 1916 and Taylor in 1917, then the effect was demonstrated experimentally by Taylor in 1923.[3]: 648 [4]: 245 [5][6] References ^ Gill, Adrian E. (2016). Atmosphere—Ocean Dynamics. Elsevier. ISBN 9781483281582. ^ Hough, S.S. (January 1, 1897). "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". Phil. Trans. R. Soc. Lond. A. 189: 201–257. Bibcode:1897RSPTA.189..201H. doi:10.1098/rsta.1897.0009. ^ Wu, J.-Z.; Ma, H.-Y.; Zhou, M.-D. (2006). Vorticity and vortex dynamics. Berlin: Springer. ISBN 9783540290285. ^ Longair, Malcolm (2016). Maxwell's Enduring Legacy: A Scientific History of the Cavendish Laboratory. Cambridge University Press. ISBN 9781316033418. ^ Proudman, J. (July 1, 1916). "On the motion of solids in a liquid possessing vorticity". Proc. R. Soc. Lond. A. 92: 408–424. Bibcode:1916RSPSA..92..408P. doi:10.1098/rspa.1916.0026. ^ Taylor, G.I. (March 1, 1917). "Motion of solids in fluids when the flow is not irrotational". Proc. R. Soc. Lond. A. 93: 92–113. Bibcode:1917RSPSA..93...99T. doi:10.1098/rspa.1917.0007. Categories: Fluid dynamicsPhysics theorems

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