Math 2.0

Fluid mechanics Theorem

Taylor's constraint

Builds on Coriolis forceBoussinesq approximation for buoyancyMagnetohydrodynamicsGeodynamoTaylor–Proudman theoremRossby number

Statement

Let a sphere of radius r0r_0 be filled with an incompressible, electrically conducting fluid of constant density ρ\rho, rotating with constant angular velocity 𝛀=Ω𝐳^\boldsymbol{\Omega} = \Omega \hat{\mathbf{z}}. Use cylindrical coordinates (s,ϕ,z)(s, \phi, z) with axis along 𝛀\boldsymbol{\Omega}. Assume the flow 𝐮\mathbf{u} satisfies ∇⋅𝐮=0\nabla \cdot \mathbf{u} = 0 in the sphere and 𝐮⋅𝐧=0\mathbf{u} \cdot \mathbf{n} = 0 on its boundary. Let pp be the pressure, 𝐉×𝐁\mathbf{J} \times \mathbf{B} the Lorentz force density, ϑ\vartheta a scalar buoyancy field and 𝐠\mathbf{g} a radial gravity field. Neglect inertia and viscosity, and assume the magnetostrophic balance 2ρ 𝛀×𝐮=−∇p+𝐉×𝐁+ρ ϑ 𝐠.2 \rho \, \boldsymbol{\Omega} \times \mathbf{u} = -\nabla p + \mathbf{J} \times \mathbf{B} + \rho \, \vartheta \, \mathbf{g}. For 0<s<r00 < s < r_0, let C(s)C(s) be the part of the cylinder of radius ss about the axis that lies inside the sphere. Then for every ss, ∫C(s)(𝐉×𝐁)ϕ dS=0.\int_{C(s)} (\mathbf{J} \times \mathbf{B})_\phi \, dS = 0.

Remarks

Proof idea: take the ϕ\phi-component of the balance and integrate over C(s)C(s). The pressure term integrates to zero because pp is single-valued around each circle. The buoyancy term vanishes because a radial vector has no ϕ\phi-component. The Coriolis term gives 2ρΩ∫C(s)us dS2\rho\Omega \int_{C(s)} u_s \, dS, which is the net mass flux through the cylinder, and it is zero by incompressibility and the boundary condition. Taylor proved this in 1963 for the Earth's core, where the Rossby number and the Ekman number are very small. A field that satisfies the constraint is called a Taylor state. Without the magnetic field, the same balance leads to the Taylor–Proudman theorem.

Going deeper

For a student

In the geodynamo, Taylor's constraint is the starting point of the magnetostrophic theory of the core. When a field departs from a Taylor state, the cylinders are free to rotate about the axis, and small inertia lets them oscillate. These torsional oscillations are used to explain decadal changes in the length of day. Numerical geodynamo models, which keep viscosity and inertia much larger than in the real core, satisfy the constraint only approximately. The balance uses the Boussinesq approximation of magnetohydrodynamics.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first.

Show the path as a graph
Boussinesq approximation for buoyancy, concept, no evidence levelBoussinesq approximat…conceptCross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptGeodynamo, concept, no evidence levelGeodynamoconceptGeophysical fluid dynamics, concept, no evidence levelGeophysical fluid dyn…conceptIdentity matrix, object, no evidence levelIdentity matrixobjectIncompressible flow, concept, no evidence levelIncompressible flowconceptInviscid flow, concept, no evidence levelInviscid flowconceptMagnetohydrodynamics, concept, no evidence levelMagnetohydrodynamicsconceptMatrix, definition, no evidence levelMatrixdefinitionOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptRossby number, concept, no evidence levelRossby numberconceptVector space, definition, no evidence levelVector spacedefinitionVelocity, concept, no evidence levelVelocityconceptClassical mechanics, concept, no evidence levelClassical mechanicsconceptTaylor–Proudman theorem, statement, no evidence levelTaylor–Proudman theor…statementSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionMomentum, concept, no evidence levelMomentumconceptInertial frame of reference, concept, no evidence levelInertial frame of ref…conceptMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionNewton's laws of motion, statement, no evidence levelNewton's laws of moti…statementAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectAngular velocity, concept, no evidence levelAngular velocityconceptRotating reference frame, concept, no evidence levelRotating reference fr…conceptFictitious force, concept, no evidence levelFictitious forceconceptCoriolis force, concept, no evidence levelCoriolis forceconceptTaylor's constraint, statement, no evidence levelTaylor's constraintstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Boussinesq approximation for buoyancy concept
  2. Cross product concept
  3. Derivative concept
  4. Euclidean space object
  5. Euclidean vector concept
  6. Geodynamo concept
  7. Geophysical fluid dynamics concept
  8. Identity matrix object
  9. Incompressible flow concept
  10. Inviscid flow concept
  11. Magnetohydrodynamics concept
  12. Matrix definition
  13. Ordinary differential equation concept
  14. Rossby number concept
  15. Vector space definition
  16. Velocity concept
  17. Classical mechanics concept
  18. Taylor–Proudman theorem statement
  19. Square matrix definition
  20. Transpose definition
  21. Linear map definition
  22. Momentum concept
  23. Inertial frame of reference concept
  24. Multilinear map definition
  25. Orthogonal matrix definition
  26. Newton's laws of motion statement
  27. Alternating multilinear map definition
  28. Orthogonal group object
  29. Determinant definition
  30. Special orthogonal group object
  31. Angular velocity concept
  32. Rotating reference frame concept
  33. Fictitious force concept
  34. Coriolis force concept
  35. Taylor's constraint statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dynamo_theory
About this page
TagCGQT9BB4Q9
URIm2://CGQT9BB4Q9
Kindstatement, theorem
WritingW0 stub
MSC76U05, 76W05, 86A25
StatusThis copy of the site has no build data. Run m2 status CGQT9BB4Q9.
AboutGeodynamo
UsesCoriolis force, Boussinesq approximation for buoyancy, Magnetohydrodynamics, Geodynamo, Taylor–Proudman theorem, Rossby number
Changed3 November 2026

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