Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 9 5/6 2002 10.5194/npg-9-497-2002 http://www.nonlin-processes-geophys.net/9/497/2002/ http://www.nonlin-processes-geophys.net/9/497/2002/npg-9-497-2002.html http://www.nonlin-processes-geophys.net/9/497/2002/npg-9-497-2002.pdf 497 512 0000-00-00 On the problem of optimal approximation of the four-wave kinetic integral V. G. Polnikov L. Farina Centro de Previsão de Tempo e Estudos Climáticos (CPTEC), Instituto Nacional de Pesquisas Espaciais (INPE), CPTEC/INPE, Cachoeira Paulista, SP, 12630-000, Brazil permanent address: The State Oceanographic Institute, Moscow, Russia The problem of optimization of analytical and numerical approximations of Hasselmann's nonlinear kinetic integral is discussed in general form. Considering the general expression for the kinetic integral, a principle to obtain the optimal approximation is formulated. From this consideration it follows that the most well-accepted approximations, such as Discrete Interaction Approximation (DIA) (Hasselmann et al., 1985), Reduced Integration Approximation (RIA) (Lin and Perry, 1999), and the Diffusion Approximation proposed recently in Zakharov and Pushkarev (1999) (ZPA), have the same roots. The only difference among them is, essentially, the choice of the 4-wave configuration for the interacting waves. To evaluate a quality of any approximation for the 2-D nonlinear energy transfer, a mathematical measure of relative error is constructed and the meaning of approximation efficiency is postulated. By the use of these features it is shown that DIA has better accuracy and efficiency than ZPA. Following to the general idea of optimal approximation and by using the measures introduced, more efficient and faster versions of DIA are proposed.