GreenNaghdi

GreenNaghdi, often written Green–Naghdi, refers to a class of depth-averaged models for shallow water waves named after Green and Naghdi. Developed in the 1970s, these equations extend the classical shallow water (Saint-Venant) framework by incorporating nonlinear and dispersive effects that arise from the vertical structure of the flow. The model describes the evolution of the free surface and the depth-averaged horizontal velocity in one or two horizontal dimensions and is widely used in coastal and ocean engineering.

In one common formulation, the dependent variables are the total water depth h(x,t) and the depth-averaged velocity u(x,t), with h = H0 + eta where eta is the free-surface displacement. The governing equations consist of a continuity equation, h_t + (h u)_x = 0, and a momentum equation, (h u)_t + (h u^2 + 1/2 g h^2)_x + dispersive terms = 0. The dispersive terms incorporate higher-order spatial derivatives of h and u and are of higher order in the shallowness parameter. The Green–Naghdi model is fully nonlinear and weakly dispersive, reducing to the shallow water equations when dispersion is neglected.

Applications include modeling coastal wave propagation, shoaling, runup, and tsunami dynamics, as well as interactions of waves with complex bottom topography. The equations exist in one and two horizontal dimensions and are solved with various numerical schemes, such as finite differences, finite volumes, or finite elements. They provide a balance between nonlinear accuracy and dispersive effects, bridging fully nonlinear shallow-water models and fully dispersive deep-water theories.