Isolated spheroidal geophysical vortices

28 pages, 13 figures, 3 appendixThe steady flow of three-dimensional isolated axisymmetric baroclinic geophysical vortices in cyclo-geostrophic and hydrostatic balance with spheroidal pressure anomaly gradients is investigated through the analysis of three-parameter vortex families. The vortices have an inner core in near solid-body rotation and zero, or negligibly small, pressure anomaly gradient at the vortex boundary. Thus, these isolated vortices have zero amount of potential vorticity anomaly. As the vortex amplitude parameter a1 is increased in cyclones, static instability at some depth along the vortex vertical axis is reached before centrifugal instability occurs at the minimum of negative vertical vorticity at the surface vortex outer core. In anticyclones, as |a1| increases, centrifugal instability occurs before static instability at the vortex centre. The first vortex family analyzed, denoted V(p,q), is characterized by spheroidal pressure anomaly gradients F′(p,q;x) such that the first ⌊p⌋ (integer part of p) derivatives of F′′(p,q;x) at the vortex centre x=0, and the first ⌊q⌋ derivatives of F(p, q; x) at the vortex boundary x=1 are zero. For comparison purposes two other three-parameter vortex families are introduced, namely the exponential E and the hyperbolic tangent H vortex families. The relation between the parameter values in each vortex family leading to statically stable vortices is provided. The main static instability properties of the three vortex families are similar, but each vortex family has their own qualities which make them more or less appropriate to a particular applicationPartial support for this research has come from the Spanish Ministerio de Economía y Competividad (grant numbers CTM2011-2886 and CTM2014-56987-P)Peer Reviewe