Nonlinear tides

Nonlinear tides are generated by hydrodynamic distortions of tides. A tidal wave is said to be nonlinear when its shape deviates from a pure sinusoidal wave. In mathematical terms, the wave owes its nonlinearity due to the nonlinear advection and frictional terms in the governing equations. These become more important in shallow-water regions such as in estuaries. Nonlinear tides are studied in the fields of coastal morphodynamics, coastal engineering and physical oceanography. The nonlinearity of tides has important implications for the transport of sediment.

Framework From a mathematical perspective, the nonlinearity of tides originates from the nonlinear terms present in the Navier-Stokes equations. In order to analyse tides, it is more practical to consider the depth-averaged shallow water equations: ∂ η ∂ t + ∂ ∂ x [ ( D 0 + η ) u ] + ∂ ∂ y [ ( D 0 + η ) v ] = 0 , {\displaystyle {\frac {\partial \eta }{\partial t}}+{\frac {\partial }{\partial x}}[(D_{0}+\eta )u]+{\frac {\partial }{\partial y}}[(D_{0}+\eta )v]=0,}

∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = − g ∂ η ∂ x − τ b , x ρ ( D 0 + η ) , {\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=-g{\frac {\partial \eta }{\partial x}}-{\frac {\tau _{b,x}}{\rho (D_{0}+\eta )}},}

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