The rescaled potential vorticity equation

eq:aniso:zoom:finite:equation is the equation satisfied by the rescaled potential vorticity Ω_n = zoomOmega = lam³ δ Ω of eq:aniso:zoom:fields in the zoomed variables ((R, Z), τ) of eq:aniso:zoom:finite:variables, with δ = lam ^ (2h) and μ = lam ^ (1 - 2h), so that δ μ = lam.

It is lam⁵ δ times the potential vorticity equation eq:aniso:q of CIV.potential_vorticity_pde, evaluated at the zoom point. Term by term, using the chain rules ∂_R = lam ∂_r, ∂_Z = μ ∂_z, ∂_τ = lam² ∂_t of CIV.Zoom.ScalarChainRule:

• ∂_τ Ω_n = lam² · lam³δ ∂_tΩ = lam⁵δ ∂_tΩ;
• V_n ∂_R Ω_n = (lam u_r)(lam · lam³δ ∂_rΩ) = lam⁵δ u_r ∂_rΩ;
• W_n ∂_Z Ω_n = (lam δ u_z)(μ lam³δ ∂_zΩ) = lam⁴δ²μ u_z ∂_zΩ = lam⁵δ u_z ∂_zΩ;
• ∂_RR Ω_n = lam² lam³δ ∂_rrΩ = lam⁵δ ∂_rrΩ and (3/R) ∂_R Ω_n = lam⁵δ (3/r) ∂_rΩ, the second because the zoomed radius satisfies r = lam R;
• δ² ∂_ZZ Ω_n = δ²μ² lam³δ ∂_zzΩ = lam⁵δ ∂_zzΩ;
• ∂_Z (S_n²/R²) = lam⁵δ ∂_z (u_θ²/r²), since S_n²/R² = lam⁴δ² (u_θ²/r²) ∘ zoomPoint;
• the force term is lam⁵δ times the potential vorticity of f.

Away from the axis Ω = ω_θ / r is the smooth quotient of the smooth azimuthal vorticity by the radius, so every chain rule is applied on the open off-axis part D = {z ∈ unitCylinder | z₁ ≠ 0} of the cylinder rather than on unitCylinder itself: Ω is merely continuous across the axis.

theorem CIV.zoomOmega_pde (lam h zc : ℝ) (hlam : 0 < lam) (u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (pres : CKN.Foundation.Parabolic.ParabolicPoint → ℝ) (f : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (hsol : IsClassicalSolutionOn u pres f unitCylinder) (haxi : IsAxisymmetricOn u unitCylinder) (p : (ℝ × ℝ) × ℝ) (hp : zoomPoint lam h zc p ∈ unitCylinder) (hr : p.1.1 ≠ 0) : dtPast (zoomOmega lam h zc u) p + zoomV lam h zc u p * dr (zoomOmega lam h zc u) p + zoomW lam h zc u p * dz (zoomOmega lam h zc u) p = dr (dr (zoomOmega lam h zc u)) p + 3 / p.1.1 * dr (zoomOmega lam h zc u) p + (lam ^ (2 * h)) ^ 2 * dz (dz (zoomOmega lam h zc u)) p + dz (fun (q : (ℝ × ℝ) × ℝ) => zoomS lam h zc u q ^ 2 / q.1.1 ^ 2) p + lam ^ 5 * lam ^ (2 * h) * potentialVorticity f (zoomPoint lam h zc p)

The rescaled potential vorticity equation eq:aniso:zoom:finite:equation: at a zoomed point off the axis whose image lies in the unit cylinder, the rescaled potential vorticity Ω_n of a classical, forced, axisymmetric solution is transported by (V_n, W_n) against the anisotropic diffusion ∂_RR + 3R⁻¹∂_R + δ²∂_ZZ, the rescaled swirl source ∂_Z (S_n²/R²) and lam⁵δ times the potential vorticity of the force. It is lam⁵δ times CIV.potential_vorticity_pde at the zoom point.