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:Here, and are the zonal and meridional flow velocity respectively, is the gravitational acceleration, is the density, and are the components of the bottom drag in the - and -direction respectively, is the average water depth and 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 - and -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. 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 -direction.This implies that zero and is all quantities are homogeneous in the -direction. Therefore, all 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, the advection term, and the frictional term. The latter is nonlinear in two ways. Firstly, because is quadratic in . Secondly, because of 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 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 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. Neglecting friction, the wave speed is given as:Comparing low water to high water levels, the through of a shallow water wave travels slower than the crest. As a result, the crest "catches up" with the trough and a tidal wave becomes asymmetric.
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:
When the tidal range is not small compared to the water depth, i.e. is significant, the flow velocity is not negligible with respect to. Thus, wave propagation speed at the crest is while at the trough, the wave speed is. 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:Here 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, and one originates from advection incorporated in the momentum equation. 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 and are described by a propagating water wave, with a water level amplitude, a radian frequency and a wavenumber. Based on this, the following transformation principles are applied:The non-dimensional variables, denoted by the tildes, are multiplied with an appropriate length, time or velocity scale of the dimensional variable. Plugging in the non-dimensional variables, the governing equations read:The nondimensionalization shows that the nonlinear terms are very small if the average water depth is much larger than the water level variations, i.e. is small. In the case that, a linear perturbation analysis can be used to further analyze this set of equations. This analysis assumes small perturbations around a mean state of :Here.
When inserting this linear series in the nondimensional governing equations, the zero-order terms are governed by:This is a linear wave equation with a simple solution of form:
Collecting the terms and dividing by yields:
Three nonlinear terms remain. However, the nonlinear terms only involve terms of, for which the solutions are known. Hence these can be worked out. Subsequently, taking the -derivative of the upper and subtracting the -derivative of the lower equation yields a single wave equation:
This linear inhomogenous partial differential equation, obeys the following particulate solution:
Returning to the dimensional solution for the sea surface elevation:
This solution is valid for a first order perturbation. The nonlinear terms are responsible for creating a higher harmonic signal with double the frequency of the principal tide. Furthermore, the higher harmonic term scales with, and. Hence, the shape of the wave will deviate more and more from its original shape when propagating in the -direction, for a relatively large tidal range and for shorter wavelengths. When considering a common principal tide, the nonlinear terms in the equation lead to the generation of the harmonic. When considering higher-order terms, one would also find higher harmonics.
Friction
The frictional term in the shallow water equations, is nonlinear in both the velocity and water depth.In order to understand the latter, one can infer from the term that the friction is strongest for lower water levels. Therefore, the crest "catches up" with the trough because it experiences less friction to slow it down. Similar to the nonlinearity induced by the divergence and advection term, this causes an asymmetrical tidal wave.
In order to understand the nonlinear effect of the velocity, one should consider that the bottom stress is often parametrized quadratically:Here is the drag coefficient, which is often assumed to be constant.
Twice per tidal cycle, at peak flood and peak ebb, reaches a maximum,. However, the sign of is opposite for these two moments. Causally, the flow is altered symmetrical around the wave node. This leads to the conclusion that this nonlinearity results in odd higher harmonics, which are symmetric around the node of the principal tide.
Mathematical analysis
Nonlinearity in velocity
The parametrization of contains the product of the velocity vector with its magnitude. At a fixed location, a principal tide is considered with a flow velocity:Here, is the flow velocity amplitude and is the angular frequency. To investigate the effect of bottom friction on the velocity, the friction parameterization can be developed into a Fourier series:
This shows that can be described as a Fourier series containing only odd multiples of the principal tide with frequency. Hence, the frictional force causes an energy dissipation of the principal tide towards higher harmonics. In the two dimensional case, also even harmonics are possible. The above equation for implies that the magnitude of the friction is proportional to the velocity amplitude. Meaning that stronger currents experience more friction and thus more tidal deformation. In shallow waters, higher currents are required to accommodate for sea surface elevation change, causing more energy dissipation to odd higher harmonics of the principal tide.
Nonlinearity in water depth
Although not very accurate, one can use a linear parameterization of the bottom stress:Here is a friction factor which represents the first Fourier component of the more exact quadratical parameterization. Neglecting the advectional term and using the linear parameterization in the frictional term, the nondimensional governing equations read:Despite the linear parameterization of the bottom stress, the frictional term remains nonlinear. This is due to the time dependent water depth in its denominator. Similar to the analysis of the nonlinear advection term, a linear perturbation analysis can be used to analyse the frictional nonlinearity. The equations are given as:Taking the -derivative of the upper equation and subtracting the -derivative of the lower equation, the terms can be eliminated. Calling, this yield a single second order partial differential equation in :In order to solve this, boundary conditions are required. These can be formulated asThe boundary conditions are formulated based on a pure cosine wave entering a domain with length. The boundary of this domain is impermeable to water. To solve the partial differential equation, a separation of variable method can be used. It is assumed that. A solution that obeys the partial differential equation and the boundary conditions, reads:Here,.
In a similar manner, the equations can be determined:Here the friction term was developed into a Taylor series, resulting in two friction terms, one of which is nonlinear. The nonlinear friction term contains a multiplication of two terms, which show wave-like behaviour. The real parts of and are given as:Here the denote a complex conjugate. Inserting these identities into the nonlinear friction term, this becomes:
The above equation suggests that the particulate solution of the first order terms obeys a particulate solution with a time-independent residual flow and a higher harmonic with double the frequency of the principal tide, e.g. if the principal tide has a frequency, the double linearity in the friction will generate an component. The residual flow component represents Stokes drift. Friction causes higher flow velocities in the high water wave than in the low water, hence making the water parcels move in the direction of the wave propagation. When higher order terms in the perturbation analysis are considered, even higher harmonics will also be generated.