  Coherent oscillations have been observed during Type I X-ray bursts from 14
accreting neutron stars in low mass X-ray binaries, providing important
information about their spin frequencies. However, the origin of the brightness
asymmetry on the neutron star surface producing these oscillations is still not
understood. We study the stability of a zonal shearing flow on the neutron star
surface using a shallow water model. We show that differential rotation of >2%
between pole and equator, with the equator spinning faster than the poles, is
unstable to hydrodynamic shear instabilities. The unstable eigenmodes have
properties well-matched to burst oscillations: low azimuthal wavenumber m, wave
speeds 1 or 2% below the equatorial spin rate, and e-folding times close to a
second. Instability is related to low frequency buoyantly driven r-modes that
have a mode frequency within the range of rotation frequencies in the
differentially rotating shell. We discuss the implications for burst
oscillations. Growth of shear instabilities may explain the brightness
asymmetry in the tail of X-ray bursts, although some fine tuning of the level
of differential rotation and a spin frequency near 300 Hz are required in order
for the fastest growing mode to have m=1. If shear instabilities are to operate
during a burst, temperature contrasts of 30% across the star must be created
during ignition and spreading of the flash.
