The one-dimensional Green–Naghdi equations with a time dependent bottom topography and their conservation laws

Abstract

This paper deals with the one-dimensional Green–Naghdi equations describing the behavior of fluid flow over an uneven bottom topography depending on time. Using Matsuno’s approach, the corresponding equations are derived in Eulerian coordinates. Further study is performed in Lagrangian coordinates. This study allowed us to find the general form of the Lagrangian corresponding to the analyzed equations. Then, Noether’s theorem is used to derive conservation laws. As some of the tools in the application of Noether’s theorem are admitted generators, a complete group classification of the Green–Naghdi equations with respect to the bottom depending on time is performed. Using Noether’s theorem, the found Lagrangians, and the group classification, conservation laws of the one-dimensional Green–Naghdi equations with uneven bottom topography depending on time are obtained.

Group classification of the two-dimensional Green-Naghdi equations with a time dependent bottom topography

“…The solution of these determining equations gives the general form of elements of the equivalence group of the class (2.1). Because of the cumbersome calculations we extend the equivalence transformations, found in [12] for one-dimensional case, to the two-dimensional case…”

“…In the classical Green-Naghdi equations, there are no equivalence transformations defined by Galilean invariance. To overcome this obstacle, the Green-Naghdi equations with bottom topography depending on time were derived in [12]. For deriving these equations the authors used Matsuno's approach [9].…”

“…Group properties of the one-dimensional Green-Naghdi equations with uneven bottom depending on time were studied in [12]. One of the main focuses of [12] was the representation of one-dimensional equations in a variational form. For this purpose, the equations were rewritten in mass Lagrangian coordinates.…”

Balance in non-hydrostatic rotating shallow-water flows

“…Miles & Salmon 53 in 1985 established its variational foundation, ensuring conservation, and the model has since seen widespread application to non-rotating, often unidirectional flows. [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67] . Given the many people who could be credited for this model, it might be fairer to call it the 'non-hydrostatic shallow-water model', especially since the only difference from the SW model is the relaxation of the hydrostatic approximation.…”