  An initial zonally symmetric quasigeostrophic potential-vorticity (PV)
distribution q_i(y) is subjected to complete or partial mixing within some
finite zone |y| < L, where y is latitude. The change in M, the total absolute
angular momentum, between the initial and any later time is considered. For
standard quasigeostrophic shallow-water beta-channel dynamics it is proved
that, for any q_i(y) such that dq_i/dy > 0 throughout |y| < L, the change in M
is always negative. This theorem holds even when "mixing" is understood in the
most general possible sense. Arbitrary stirring or advective rearrangement is
included, combined to an arbitrary extent with spatially inhomogeneous
diffusion. The theorem holds whether or not the PV distribution is zonally
symmetric at the later time. The same theorem governs Boussinesq
potential-energy changes due to buoyancy mixing in the vertical. For the
standard quasigeostrophic beta-channel dynamics to be valid the Rossby
deformation length L_D >> \epsilon L where \epsilon is the Rossby number; when
L_D = \infty the theorem applies not only to the beta-channel, but also to a
single barotropic layer on the full sphere, as considered in the recent work of
Dunkerton and Scott on "PV staircases". It follows that the M-conserving PV
reconfigurations studied by those authors must involve processes describable as
PV unmixing, or anti-diffusion, in the sense of time-reversed diffusion.
Ordinary jet self-sharpening and jet-core acceleration do not, by contrast,
require unmixing, as is shown here by detailed analysis. Mixing in the jet
flanks suffices. The theorem extends to multiple layers and continuous
stratification. A corollary is a new nonlinear stability theorem for shear
flows.
