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Inversion of potential vorticity density P-eta* = (omega(a)center dot del eta)/(partial derivative eta/partial derivative Z) with absolute vorticity omega(a) and function eta is explored in eta coordinates. This density is shown to be the component of absolute vorticity associated with the vertical vector of the covariant basis of eta coordinates. This implies that inversion of P-eta* in eta coordinates is a two-dimensional problem in hydrostatic flow.

Examples of inversions are presented for eta = theta (theta is potential temperature) and eta = p (p is pressure) with satisfactory results for domains covering the North Pole. The role of the boundary conditions is investigated and piecewise inversions are performed as well. The results shed new light on the interpretation of potential vorticity inversions.