Two-dimensional potential-vorticity inversion ($\psi (\mathbf{x})=\frac{1}{2\pi }\int \zeta ({\mathbf{x}}^{\prime })\log |\mathbf{x}-{\mathbf{x}}^{\prime }|d^2{x}^{\prime }+{\psi }_h$)

In two-dimensional vortex dynamics, solve the Poisson equation for the streamfunction with the prescribed boundary conditions. On the plane its fundamental solution is $(2\pi {)}^{-1}\log |\mathbf{x}|$, so differentiating the displayed inverse gives the planar Biot-Savart kernel. The harmonic function ${\psi }_h$ represents boundary and global-flow data. An additive constant changes no velocity; circulation around holes and a uniform flow are additional degrees of freedom if their boundary or far-field values are not prescribed. The same inverse acts independently on each level in nonrotating layerwise-two-dimensional vortex dynamics.