  The coherence properties of mechanical resonators are often limited by
multiple unavoidable forms of loss -- including phonon-phonon and phonon-defect
scattering -- which result in the scattering of sound into other resonant modes
and into the phonon bath. Dynamic suppression of this scattering loss can lift
constraints on device structure and can improve tolerance to defects in the
material, even after fabrication. Inspired by recent experiments, here we
introduce a model of phonon losses resulting from disorder in a whispering
gallery mode resonator with acousto-optical coupling between optical and
mechanical modes. We show that a typical elastic scattering mechanism of high
quality factor (Q) mechanical modes flips the direction of phonon propagation
via high-angle scattering, leading to damping into modes with the opposite
parity. When the optical mode overlaps co-propagating high-Q and bulk
mechanical modes, the addition of laser cooling via sideband-resolved damping
of the mechanical mode of a chosen parity also damps and modifies the response
of the bulk modes of the same parity. This, in turn, simultaneously improves
the quality factor and reduces the thermal load of the counter-propagating
high-Q modes, leading to the dynamical creation of a cold phononic shield. We
compare our theoretical results to the recent experiments of Kim et al., and
find quantitative agreement with our theory.
