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The nonlinear Schr\"odinger equation is widely used as an approximate modelfor the evolution in time of the water wave envelope. In the context ofsimulating ocean waves, initial conditions are typically generated from ameasured power spectrum using the random phase approximation, and periodized onan interval of length $L$. It is known that most realistic ocean waves powerspectra do not exhibit modulation instability, but the most severe ones do; itis thus a natural question to ask whether the periodized random phaseapproximation has the correct stability properties. In this work we specify arandom phase approximation scaling so that, in the limit of $L\to\infty,$ thestability properties of the periodized problem are identical to those of thecontinuous power spectrum on the infinite line. Moreover, it is seen throughconcrete examples that using a too short computational domain can completelysuppress the modulation instability.

Modulation instability and convergence of the random phase approximation for stochastic sea states