The linear stability of two-layer primitive equation ocean rings is co
nsidered in the case when the rings are wide compared with a deformati
on radius, as is usually observed. Asymptotic theory is developed to s
how the existence of solutions for arbitrarily wide rings, and these s
olutions can be followed as the rings are made successively narrower.
An exponential cubic radial dependence is used for the mean flow, rath
er than the more usual Gaussian structure. There are two reasons: a Ga
ussian shape was fully discussed in a previous paper, and a Gaussian h
as exceptional properties, unlike other power laws. The specific cases
of warm and cold Gulf Stream rings are considered in detail. The theo
ry provides an accurate prediction of phase velocity and growth rate f
or cold rings and a reasonable prediction for warm rings. Solutions in
the asymptotic regime have a larger growth rate than other (nonasympt
otic) solutions for cold rings, but not for warm rings. Attention is g
iven to the role of the mean barotropic circulation, which had been fo
und in earlier work to have a strong effect on ring stability. There i
s still evidence for stabilization when the mean flow in the lower lay
er vanishes, but other features are also involved. In particular, the
linear stability of a ring appears to be as sensitive to subtle shape
details as it is to the sense of the deep flow. The authors generally
find warm co-rotating rings with a cubic exponential form to be unstab
le, although somewhat less so than counterrotating rings.