Wave turbulence describes the statistical behavior of a large ensemble of waves. The wave turbulence theory (WTT) predicts a power-law spectrum as a result of nonlinear wave-wave interactions, which has applications to many ocean wave systems. While the theory is expected to be valid at some “kinetic limit” of a small nonlinearity and a large domain, the wave turbulence behavior in a finite domain (say a computational domain or experimental facility) is still an open question. We study this problem in the context of a generic wave model Majda-McLaughlin-Tabak (MMT) equation in a two-dimensional periodic domain. At low nonlinearity level, we find that the turbulence spectrum critically depends on the aspect ratio of the 2D domain (in particular as a rational or irrational number). This phenomenon is explained through the discrete resonant manifold which varies with both the wave dispersion relation and domain geometry. This particular result questions the validity of using periodic-domain simulations to represent homogeneous wave turbulence. At high nonlinearity level, we study the WTT closure model for high-order correlators. Our results show that the traditional closure model has to be modified to describe wave turbulence in a finite domain. Finally, we show that, for a certain parameter regime of the MMT equation, a new type of rogue wave is discovered that only occurs at low nonlinearity level. Speaker(s): Yulin Pan (University of Michigan, Naval Architecture and Marine Engineering)