Nonlinear tides explained

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:^[1] $$\backslash frac+\backslash frac[(D\_0\; +\; \backslash eta)u]+\backslash frac[(D\_0\; +\; \backslash eta)v]=0,$$$$\backslash frac+u\backslash frac+v\backslash frac=-g\backslash frac-\backslash frac,$$$$\backslash frac+u\backslash frac+v\backslash frac=-g\backslash frac-\backslash frac.$$Here,

u and v are the zonal (x ) and meridional (y ) flow velocity respectively, g is the gravitational acceleration, \rho is the density, \tau _{b,x} and \tau _{b,y} are the components of the bottom drag in the x - and y -direction respectively, D _{0} is the average water depth and η is the water surface elevation with respect to the mean water level. The former of the three equations is referred to as the continuity equation while the others represent the momentum balance in the x - and y -direction respectively.

These equations follow from the assumptions that water is incompressible, that water does not cross the bottom or surface and that pressure variations above the surface are negligible. The latter allows the pressure gradient terms in the standard Navier-Stokes equations to be replaced by gradients in

η . Furthermore, the coriolis and molecular mixing terms are omitted in the equations above since they are relatively small at the temporal and spatial scale of tides in shallow waters.

For didactic purposes, the remainder of this article only considers a one-dimensional flow with a propagating tidal wave in the positive

x -direction.This implies that v =0 zero and is all quatities are homogeneous in the y -direction. Therefore, all \partial / \partial y terms equal zero and the latter of the above equations is arbitrary.

Nonlinear contributions

In this one dimensional case, the nonlinear tides are induced by three nonlinear terms. That is, the divergence term

\partial( η u)/\partial x , the advection term u \partial u/\partial x , and the frictional term \tau _{b/(D} _{0+η)} . The latter is nonlinear in two ways. Firstly, because \tau _{b} is (nearly) quadratic in u . Secondly, because of η in the denominator. The effect of the advection and divergence term, and the frictional term are analysed separately. Additionally, nonlinear effects of basin topography, such as intertidal area and flow curvature can induce specific kinds of nonlinearity. Furthermore, mean flow, e.g. by river discharge, may alter the effects of tidal deformation processes.

Harmonic analysis

A tidal wave can often be described as a sum of harmonic waves. The principal tide (1st harmonic) refers to the wave which is induced by a tidal force, for example the diurnal or semi-diurnal tide. The latter is often referred to as the

M _{2} tide and will be used throughout the remainder of this article as the principal tide. The higher harmonics in a tidal signal are generated by nonlinear effects. Thus, harmonic analysis is used as a tool to understand the effect the nonlinear deformation. One could say that the deformation dissipates energy from the principal tide to its higher harmonics. For the sake of consistency, higher harmonics having a frequency that is an even or odd multiple of the principle tide may be referred to as the even or odd higher harmonics respectively.

Divergence and advection

In order to understand the nonlinearity induced by the divergence term, one could consider the propagation speed of a shallow water wave.^[2] Neglecting friction, the wave speed is given as:^[3]

$$c\_0\; \backslash approx\; \backslash sqrt$$

Comparing low water (LW) to high water (HW) levels (

η _{LW} < η _{HW} ), the through (LW) of a shallow water wave travels slower than the crest (HW). As a result, the crest "catches up" with the trough and a tidal wave becomes asymmetric.^[4]

In order to understand the nonlinearity induced by the advection term, one could consider the amplitude of the tidal current. Neglecting friction, the tidal current amplitude is given as:

$$U\_0\; \backslash approx\; c\_0\; \backslash frac$$

When the tidal range is not small compared to the water depth, i.e.

η /D _{0} is significant, the flow velocity u is not negligible with respect to c _{0} . Thus, wave propagation speed at the crest is c _{0} + u while at the trough, the wave speed is c _{0} - u . Similar to the deformation induced by the divergence term, this results in a crest "catching up" with the trough such that the tidal wave becomes asymmetric.

For both the nonlinear divergence and advection term, the deformation is asymmetric. This implies that even higher harmonics are generated, which are asymmetric around the node of the principal tide.

Mathematical analysis

The linearized shallow water equations are based on the assumption that the amplitude of the sea level variations are much smaller than the overall depth. This assumption does not necessarily hold in shallow water regions. When neglecting the friction, the nonlinear one-dimensional shallow water equations read:$$\backslash frac+\backslash underbrace\_i+(D\_+\backslash underbrace\_i=0,$$$$\backslash frac+\backslash underbrace\_=-g\backslash frac.$$Here

D _{0} is the undisturbed water depth, which is assumed to be constant. These equations contain three nonlinear terms, of which two originate from the mass flux in the continuity equation (denoted with subscript i ), and one originates from advection incorporated in the momentum equation (denoted with subscript ii ). To analyze this set of nonlinear partial differential equations, the governing equations can be transformed in a nondimensional form. This is done based on the assumption that u and η are described by a propagating water wave, with a water level amplitude H _{0} , a radian frequency \omega and a wavenumber k . Based on this, the following transformation principles are applied:$$\backslash left\backslash$$

Notes and References

1. Book: Cushman-Roisin . Benoit . Beckers . Jean-Marie . Introduction to geophysical fluid dynamics: physical and numerical aspects . 2011 . Academic Press . 978-0-12-088759-0 . 2nd . Waltham, MA . 760173075.
2. Book: B., Parker, Bruce . Tidal hydrodynamics . 1991 . Wiley . 0-471-51498-5 . 231330044.
3. Book: Pond, Stephen . Introductory dynamical oceanography . 1991 . George L. Pickard . 978-0-08-057054-9 . Second . Oxford . 886407149.
4. Dronkers . J. . 1986-08-01 . Tidal asymmetry and estuarine morphology . Netherlands Journal of Sea Research . en . 20 . 2 . 117–131 . 10.1016/0077-7579(86)90036-0 . 1986NJSR...20..117D . 0077-7579.