The potential vorticity inversion concept is extended to obtain two-dimensional symmetrically unstable basic states. The goal is to invert a lens-like patch of negative potential vorticity embedded within a large area of uniform positive potential vorticity to create the basic state. In order to circumvent the non-ellipticity in areas of negative potential vorticity, a variational formulation of the invertibility principle is proposed. The functional incorporates the thermal wind balance as a constraint between the mass and the wind field. This condition is necessary and sufficient to define a steady solution to the inversion problem that is suitable to serve as a basic state. The uniqueness of this solution, however, is not addressed. Inversions with different potential vorticity distributions are performed, enabling us to highlight the dependence of the behaviour of the symmetric rolls (initial linear growth rate, wavelength) on the shape of the negative potential vorticity area (aspect ratio, amplitude and surface). Particular attention is paid to defining the domain of parameters (frontal environment and anomalies) that enables symmetric instability in a convectively-and inertially-stable flow. Results from the stability analysis for embedded zones are compared to those obtained for uniform potential vorticity in the linear regime. Results are generally confirmed and new ones are found. It is shown, for example, that the linear growth rate increases when the unstable surface decreases. The inversion technique is finally extended to the moist (saturated) case. The variational approach is, therefore, shown to be able to handle areas of negative potential vorticity, opening new areas for a more systematic use of the concept of inversion in synoptic and dynamic meteorology.
