Hurricane eyewalls are often observed to be nearly circular structures, but they are occasionally observed to take on distinctly polygonal shapes. The shapes range from triangles to hexagons and, while they are often incomplete, straight line segments can be identified. Other observations implicate the existence of intense mesovortices within or near the eye region. Is there a relation between polygonal eyewalls and hurricane mesovortices? Are these phenomena just curiosities of the hurricane's inner-core circulation, or are they snapshots of an intrinsic mixing process within or near the eye that serves to determine the circulation and thermal structure of the eye?As a first step toward understanding the asymmetric vorticity dynamics of the hurricane's eye and eyewall region, these issues are examined within the framework of an unforced barotropic nondivergent model. Polygonal eyewalls are shown to form as a result of barotropic instability near the radius of maximum winds. After reviewing linear theory, simulations with a high-resolution pseudospectral numerical model are presented to follow the instabilities into their nonlinear regime. When the instabilities grow to finite amplitude, the vorticity of the eyewall region pools into discrete areas, creating the appearance of polygonal eyewalls. The circulations associated with these pools of vorticity suggest a connection to hurricane mesovortices. At later times the vorticity is ultimately rearranged into a nearly monopolar circular vortex. While the evolution of the finescale vorticity field is sensitive to the initial condition, the macroscopic end-states are found to be similar. In fact, the gross characteristics of the numerically simulated end-states are predicted analytically using a generalization of the minimum enstrophy hypothesis. In an effort to remove some of the weaknesses of the minimum enstrophy approach, a maximum entropy argument developed previously for rectilinear shear flows is extended to the vortex problem, and end-state solutions in the limiting case of tertiary mixing are obtained.Implications of these ideas for real hurricanes are discussed.