3.10 A COMBINATION OF EMPIRICAL ORTHOGONAL FUNCTION AND NEURAL NETWORK APPROACHES FOR PARAMETERIZING NONLINEAR INTERACTIONS IN WIND WAVE MODELS. Vladimir M. Krasnopolsky *, Hendrik L. Tolman, and Dmitry V. Chalikov Science Applications International Corp. at the National Centers for Environmental Predictions 1. INTRODUCTION Ocean wind wave modeling for hindcast and forecast purposes has been at the center of interest for many decades. Numerical prediction models are generally based on a form of the spectral energy or action balance equation sw ds nl in S S S S Dt DF + + + = (1) Interaction Approximation (DIA, Hasselman et al 1985). The development of the DIA allowed for the successful development of the first third- generation wave model WAM (WAMDI Group 1988). More than a decade of experience with the WAM model and its derivatives has identified shortcomings of the DIA. The DIA tends to unrealistically increase the directional width of spectra, has a systematic spurious impact on the shape of the spectrum near the spectral peak frequency, and has a much too strong signature at high frequencies. In present third generation wave models, these deficiencies can be countered at least in part by the dissipation source term S ds , which is generally used for tuning the energy balance in the equation (1). Although this approach gives good results, it is counterproductive, because it prohibits development of dissipation source terms based on solid physical considerations. With our increased understanding in the physics of wave generation and dissipation, this becomes an even bigger obstacle impeding further development of third-generation wave models. where F is the spectrum, S in is the input source term, S nl is the nonlinear interaction source term, S ds is the dissipation or 'whitecapping' source term, and S sw represents additional shallow water source terms. Several studies (Hasselman et al 1973) identified the active role of the nonlinear interactions in wave growth and the need for explicit modeling of S nl in wave models. State-of-the-art or so-called third generation wave models therefore explicitly model this source term. In its full form (e.g., Hasselmann and Hasselmann 1985), the calculation of the interactions S nl requires the integration of a six- dimensional Bolzmann integral: 2. NN PARAMETERIZATION OF S nl Considering the above, it is of crucial importance for the development of third generation wave models to develop an economical yet accurate approximation for S nl . Here, we explore a Neural Network Interaction Approximation (NNIA) to achieve this goal (see also Krasnopolsky et al 2002). NNs can be applied here because the nonlinear interaction (2) is essentially a nonlinear mapping (symbolically represented in eq. (2) by T) which relates two vectors (2-D fields in this case). Thus, the nonlinear interaction source term can be considered as a nonlinear mapping between a spectrum F and a source term S nl S k T Fk Gk k k k k k k k n n n n n n n n dk dk dk nk Fk gk kh nl ( ) () ( , , , ) ( ) ( ) [ ( ) ( )] () () ; tanh( ) r r r r r r r r r r r r r r r 4 4 1 2 3 4 1 2 3 4 1 2 3 4 1 3 4 2 2 4 3 1 1 2 3 2 = ⊗ = = ⋅ + − − ⋅ + − × ⋅ ⋅ − + ⋅ ⋅ − = = ⋅ ⋅ ∫ ω δ δω ω ω ω (2) − ω ω where the complicated coupling coefficient G contains moving singularities (K. Hasselmann 1973). This integration requires roughly 10 3 to 10 4 times more computational effort than all other aspects of the wave model combined. Present operational constraints require that the computational effort for the estimation of S nl should be of the same order of magnitude as the remainder of the wave model. This requirement was met with the development of the Discrete S nl = T(F) , (3) * Corresponding author address: Vladimir Krasnopolsky, SAIC at EMC/NCEP/NOAA, 5200 Auth Rd., Camp Spring, MD 20906, e-mail: Vladimir.Krasnopolsky@noaa.gov where T is the exact nonlinear operator given by the full Bolzmann interaction integral (2) (Hasselmann and Hasselmann 1985). Discretization of S and F (as is necessary in any MMAB Contribution No. 224