The potential vorticity equation

eq:aniso:q is the equation satisfied by the potential vorticity Ω = ω_θ / r of an axisymmetric classical solution. It is eq:aniso:theta divided by r: the stretching term (u_r / r) ω_θ of the azimuthal vorticity equation is exactly the term produced by the radial transport of the quotient, and the cylindrical Laplacian obeys the operator identity

r⁻¹ (∂_r² + r⁻¹ ∂_r - r⁻²) (r Ω) = (∂_r² + 3 r⁻¹ ∂_r) Ω,

so the first-order radial term acquires the coefficient 3.

Away from the axis Ω agrees with the quotient ω_θ / r on a full neighbourhood, so its classical derivatives are those of the quotient; the quotient rules below are obtained from the product rule against the radial weight r⁻¹, whose only nonvanishing classical derivative in the meridional plane is the radial one.

Derivatives of a function of the radial coordinate

Classical derivatives see only the off-axis branch

The quotient rules for the radial weight

The potential vorticity equation

theorem CIV.potential_vorticity_pde (u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (p : CKN.Foundation.Parabolic.ParabolicPoint → ℝ) (f : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (hsol : IsClassicalSolutionOn u p f unitCylinder) (haxi : IsAxisymmetricOn u unitCylinder) (z : CKN.Foundation.Parabolic.ParabolicPoint) (hz : z ∈ unitCylinder) (hplane : z.1 1 = 0) (hr : z.1 0 ≠ 0) : CKN.timePartial (potentialVorticity u) z + u z 0 * CKN.spatialPartial (potentialVorticity u) 0 z + u z 2 * CKN.spatialPartial (potentialVorticity u) 2 z = CKN.spatialSecondPartial (potentialVorticity u) 0 0 z + 3 / z.1 0 * CKN.spatialPartial (potentialVorticity u) 0 z + CKN.spatialSecondPartial (potentialVorticity u) 2 2 z + CKN.spatialPartial (fun (w : CKN.Foundation.Parabolic.ParabolicPoint) => u w 1 ^ 2 / w.1 0 ^ 2) 2 z + potentialVorticity f z

The potential vorticity equation eq:aniso:q: away from the axis, the material derivative of the potential vorticity Ω = ω_θ / r of a classical, forced, axisymmetric solution equals its rescaled cylindrical Laplacian ∂_r² Ω + 3 r⁻¹ ∂_r Ω + ∂_z² Ω, the source ∂_z (u_θ² / r²), and the potential vorticity of the force.