A basing of the diffusion approximation derivation for the four-wavekinetic integral and properties of the approximation amg

 A basing of the diff­usion appr­oxim­ation deri­vat­ion for the Hass­elm­ann kinetic inte­gral desc­rib­ing nonl­inear inte­ract­ions of gravity waves in deep water is disc­ussed. It is shown that the diff­usion appr­oxim­ation cont­ain­ing the second deri­vati­ves of a wave spec­trum in a freq­uency and angle (or in wave vector comp­one­nts) is resu­lting from a step-by-step anal­yti­cal inte­grat­ion of the sixfold Hass­elm­ann inte­gral without invo­lving the quasi-loca­lity hypo­the­sis for nonl­inear inte­ract­ions among waves. A sing­ular­ity anal­ysis of the inte­grand expr­ess­ion gives evid­ence that the appr­oxim­ation ment­ioned above is the small scat­ter­ing angle appr­oxim­ation, in fact, as it was shown for the first time by Hass­elm­ann and Hass­elm­ann (1981). But, in diff­ere­nce to their result, here it is shown that in the course of diff­usion appr­oxim­ation deri­vat­ion one may obtain the final result as a comb­inat­ion of terms with the first, second, and so on deri­vati­ves. Thus, the final kind of appr­oxim­ation can be limited by terms with the second deri­vati­ves only, as it was prop­osed in Zakh­arov and Push­karev (1999). For this version of diff­usion appr­oxim­ation, a nume­rical testing of the appr­oxim­ation prop­ert­ies was carried out. The testing results give a basis to use this appr­oxim­ation in a wave mode­lling prac­tice.