Theorem: dq/dt = 0 for q = const along characteristic (PV conservation)

Theorem (PV-conservation canonical): along a Lagrangian trajectory (characteristic curve) with dx/dt = u(x, t), the quasi-geostrophic PV q is constant: q(x(t), t) = q(x(0), 0). Hence the total time derivative dq/dt = 0 identically.…