Jupiter's atmosphere, as observed in the 1979 Voyager space craft images, is characterized by 12 zonal jet streams and about 80 vortices, the largest of which are the Great Red Spot and three White Ovals that had formed in the 1930s. The Great Red Spot has been observed continuously since 1665 and, given the dynamical similarities between the Great Red Spot and the White Ovals, the disappearance of two White Ovals in 1997-2000 was unexpected. Their longevity and sudden demise has been explained however, by the trapping of anticyclonic vortices in the troughs of Rossby waves, forcing them to merge. Here I propose that the disappearance of the White Ovals was not an isolated event, but part of a recurring climate cycle which will cause most of Jupiter's vortices to disappear within the next decade. In my numerical simulations, the loss of the vortices results in a global temperature change of about 10K, which destabilizes the atmosphere and thereby leads to the formation of new vortices. After formation, the large vortices are eroded by turbulence over a time of ~60 years-consistent with observations of the White Ovals-until they disappear and the cycle begins again.