  Floquet theory is used to describe the unstable spectrum at large scales of
the beta-plane equation linearized about Rossby waves. Base flows consisting of
one to three Rossby wave are considered analytically using continued fractions
and the method of multiple scales, while base flow with more than three Rossby
waves are studied numerically. It is demonstrated that the mechanism for
instability changes from inflectional to triad resonance at an O(1) transition
Rhines number Rh, independent of the Reynolds number. For a single Rossby wave
base flow, the critical Reynolds number Re^c for instability is found in
various limits. In the limits Rh --> infinity and k --> 0, the classical value
Re^c = sqrt(2) is recovered. For Rh --> 0 and all orientations of the Rossby
wave except zonal and meridional, the base flow is unstable for all Reynolds
numbers; a zonal Rossby wave is stable, while a meridional Rossby wave has
critical Reynolds number Re^c = sqrt(2). For more isotropic base flows
consisting of many Rossby waves (up to forty), the most unstable mode is purely
zonal for 2 <= Rh < infinity and is nearly zonal for Rh = 1/2, where the
transition Rhines number is again O(1), independent of the Reynolds number and
consistent with a change in the mechanism for instability from inflectional to
triad resonance.
