Плановые модели волновой гидродинамики с дисперсионным соотношением повышенной точности. III. Линейный анализ в случае неровного дна

Two new fully nonlinear weakly dispersive shallow water models (mSGN and mSGN4) with improved accuracy were developed by Khakimzyanov et al. [1, 2]. The average velocity was used and the bottom mobility was taken into account. Modification of the dispersion parts of the pressure of the well-known Serre – Green – Naghdi (SGN-) model made allows achieving the fourth (for mSGN) and sixth-eighth (for mSGN4) orders of approximation of the dispersion relation of the three-dimensional potential flow model (FNPF-model) in the case of a horizontal stationary bottom. This article addresses a study of the properties for the obtained models in the case of an uneven bottom. The research method is based on the use of the dispersion relation for models, which are linearized to account a slight change of the profile of the bottom [10, 15]. For a hierarchy of long-wave hydrodynamic models using the depth-averaged velocity, relations between the gradients of the amplitude, wavenumber, and bottom are obtained. The dependence between the amplitude and depth is established. A generalization of Green’s law to the case of long-wave models with dispersion is obtained. It is shown that an increase in the order of the long-wave approximation along with an increase in the accuracy of the dispersion relation of shallow water models leads to a more accurate description of both the phase, and the amplitude characteristics of the model for three-dimensional potential flows. At the same time, the mSGN4-model of the fourth order of the long-wave approximation with the eighth order of accuracy of the dispersion relation shows the best approximation of the considered characteristics both in the case of a horizontal and uneven bottoms.