Taylor–Proudman balance in a homogeneous fluid
For steady inviscid incompressible homogeneous fluid in the leading small-Rossby rotating balance, taking the curl of 2Ω×u=−∇(p/ρ+Φ_eff) gives (Ω·∇)u=0.
Key knowledge
For steady inviscid incompressible homogeneous fluid in the leading small-Rossby rotating balance, taking the curl of 2Ω×u=−∇(p/ρ+Φ_eff) gives (Ω·∇)u=0. With uniformΩ along z,the balanced velocity is independent of z. This ideal rotational constraint helps explain Taylor columns and the contrast with baroclinic thermal-wind shear.
Values and equations
2Ω×u=−∇(p/ρ+Φ_eff); (Ω·∇)u=0
- Ω
- uniform rotation vector s⁻¹
- u
- balanced velocity field m s⁻¹
- p
- pressure Pa
- ρ
- constant fluid density kg m⁻³
- Φ_eff
- conservative gravitational/centrifugal potential per mass m² s⁻²
Conditions and scope
- Uniform constant nonzero rotation,constant positive density,smooth divergence-free velocity and conservative effective body potential. Inertial and frictional terms are negligible in the displayed leading balance.
- Small Rossby number alone does not justify arbitrary time-dependent acceleration neglect. Stratification/baroclinicity,boundary friction and significant unsteadiness can break the constraint; ideal columns are not rigid solid structures or a universal law for every atmospheric flow.
Sources · 2
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Massachusetts Institute of Technology2 sources
- Taylor Columns: Theory, actual curl derivation equations1–3MIT Weather in a Tank — John Marshall and collaboratorsReference guide
- Lecture18 §18.2, slow frictionless barotropic scaling and Taylor–Proudman resultMIT OpenCourseWare — Fluid Dynamics of the Atmosphere and Ocean (2004)Reference guide · PDF
Evidence and review
Read actual Weather-in-a-Tank balance/curl equations1–3 and lecture18.2 scaling/result. Rechecked curl(Ω×u)=Ωdivu−(Ω·∇)u for constantΩ. Narrowed to constant-density incompressible steady balance; excluded the lecture's missing-gradient typo in18.2 and indiscriminate small-Rossby/time-acceleration equivalence.
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