  The frustrated magnet SrDy$_2$O$_4$ exhibits a field-induced phase with a
magnetization plateau at $1/3$ of the saturation value for magnetic fields
applied along the $b$-axis. We report here a neutron scattering study of the
nature and symmetry of the magnetic order in this field-induced phase. Below
$T\approx 0.5$ K, there are strong hysteretic effects, and the order is short
or long ranged for zero-field and field cooling, respectively. We find that the
long-range ordered magnetic structure within the zig-zag chains is identical to
that expected for the one-dimensional axial next-nearest neighbour Ising
(ANNNI) model in longitudinal field. The long-range ordered structure in field
contrasts with the short-range order found at zero field, and is probably
reached through enhanced quantum fluctuations with increasing fields.
