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 $Tapprox 0.5$ 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.
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