Date of Award

Spring 2011

Document Type


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Wooyoung Choi

Second Advisor

Michael R. Booty

Third Advisor

Linda Jane Cummings

Fourth Advisor

Richard O. Moore

Fifth Advisor

Philippe Guyenne


The main focus of this dissertation is the asymptotic and numerical modeling of nonlinear ocean surface wave fields. In particular, a development of an accurate numerical model for the evolution of nonlinear ocean waves, including extreme waves known as Rogue/Freak waves. Due to their elusive and destructive nature, the media often portrays Rogue waves as unimaginatively huge and unpredictable monsters of the sea. To address these concerns, derivations of asymptotically reduced models, based on the small wave steepness assumption, are presented and their corresponding numerical simulations via a Fourier pseudo-spectral method are discussed. The simulations are initialized with a well-known JONSWAP wave spectrum and different angular distributions are employed. Both deterministic and Monte-Carlo simulations and the corresponding analysis were carried out. Based on preliminary numerical analysis, certain conclusions are drawn on the validity of the modified nonlinear Schrödinger equation (MNLS) of Dysthe (1979) in relation to realistic ocean surface waves and its ability to predict the occurrence of large amplitude waves known as Rogue or Freak waves.

Furthermore, this dissertation concerns the development of a new computationally efficient numerical model for the short term prediction of evolving weakly nonlinear ocean surface waves. The derivations are originally based on the work of West et al. (1987) and Choi (1995) and since the waves in the ocean tend to travel primarily in one direction, the aforementioned new numerical model is derived based on additional assumption of a weak transverse dependence. In turn, comparisons of the ensemble averaged random initial spectra, as well as deterministic surface-to-surface correlations are presented. The new model is shown to behave quite well in various directional wave fields and can potentially be a candidate for computationally e cient prediction and propagation of large ocean surface waves - Rogue/Freak waves.

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Mathematics Commons