Tides in marginal seas

The transition from the deep ocean to the continental shelf, known as the continental slope, is characterized by a sudden decrease in water depth. In order to apply to the conservation of energy, the tidal wave has to deform as a result of the decrease in water depth. The total energy of a linear progressive wave per wavelength is the sum of the potential energy (PE) and the kinetic energy (KE). The potential and kinetic energy integrated over a complete wavelength are the same, under the assumption that the water level variations are small compared to the water depth (${\displaystyle \eta <<H}$).[4]

${\displaystyle \int _{0}^{\lambda }PE=\int _{0}^{\lambda }KE={\frac {1}{2}}\rho g\int _{0}^{\lambda }\eta ^{2}dx}$

Figure 1: Tidal wave transmission on the continental shelf. At the transition from the open ocean to the continental shelf, the water depth decreases abruptly. As a result the tidal wave speed decreases as its phase and group speed are dependent on depth. In order to conserve the energy flux, the amplitude of the tidal wave has to increase on the continental shelf.

where ${\displaystyle \rho }$ is the density, ${\displaystyle g}$ the gravitation acceleration and ${\displaystyle \eta }$ the vertical tidal elevation. The total wave energy becomes:

${\displaystyle E=\rho g\int _{0}^{\lambda }\eta ^{2}dx}$

If we now solve for a harmonic wave ${\displaystyle \eta (x)=Acos(kx)}$, where ${\displaystyle k}$ is the wave number and ${\displaystyle A}$ the amplitude, the total energy per unit area of surface becomes:[5]

${\displaystyle E_{s}={\frac {1}{2}}\rho gA^{2}}$

A tidal wave has a wavelength that is much larger than the water depth. And thus according to the dispersion of gravity waves, they travel with the phase and group velocity of a shallow water wave: ${\displaystyle c_{p}=c_{g}={\sqrt {gh}}}$. The wave energy is transmitted by the group velocity of a wave[6] and thus the energy flux (${\displaystyle F_{E}}$) is given by:

${\displaystyle F_{E}={\frac {1}{2}}\rho gA^{2}{\sqrt {gh}}}$

The energy flux needs to be conserved and with ${\displaystyle \rho }$ and ${\displaystyle g}$ constant, this leads to:

${\displaystyle F_{E,1}=F_{E,2}\Longrightarrow {A_{1}}^{2}{\sqrt {gh_{1}}}={A_{2}}^{2}{\sqrt {gh_{2}}}}$

where ${\displaystyle h2<h1}$ and thus ${\displaystyle A_{2}>A_{1}}$.

When the tidal wave propagates onto the continental shelf, the water depth ${\displaystyle (h)}$ decreases. In order to conserve the energy flux, the amplitude of the wave needs to increase (see figure 1).

After entering the continental shelf, a tidal wave quickly faces a boundary in the form of a landmass. When the tidal wave reaches a continental margin, it continues as a boundary trapped Kelvin wave. Along the coast, a boundary trapped Kelvin is also known as a coastal Kelvin wave or Edge wave. A Kelvin wave is a special type of gravity wave that can exist when there is (1) gravity and stable stratification, (2) sufficient Coriolis force and (3) the presence of a vertical boundary.[12] Kelvin waves are important in the ocean and shelf seas, they form a balance between inertia, the Coriolis force and the pressure gradient force. The simplest equations that describe the dynamics of Kelvin waves are the linearized shallow water equations for homogeneous, in-viscid flows. These equations can be linearized for a small Rossby number, no frictional forces and under the assumption that the wave height is small compared to the water depth (${\displaystyle \eta <<h}$). The linearized depth-averaged shallow water equations become:

u momentum equation:

• ${\displaystyle {\frac {\partial u}{\partial t}}-fv=-g{\frac {\partial \eta }{\partial x}}}$

v momentum equation:

• ${\displaystyle {\frac {\partial v}{\partial t}}+fu=-g{\frac {\partial \eta }{\partial y}}}$

the continuity equation:

• ${\displaystyle {\frac {\partial \eta }{\partial t}}+h({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}})=0}$

where ${\displaystyle u}$ is the zonal velocity (${\displaystyle x}$ direction), ${\displaystyle v}$ the meridional velocity (${\displaystyle y}$ direction), ${\displaystyle t}$ is time and ${\displaystyle f}$ is the Coriolis frequency.

Kelvin waves are named after Lord Kelvin, who first described them after finding solutions to the linearized shallow water equations with the boundary condition ${\displaystyle u(x,y,t)=0}$.[13] When this assumption is made the linearized depth-averaged shallow water equations that can describe a Kelvin wave become:

u momentum equation:

• ${\displaystyle v={\frac {-g}{f}}{\frac {\partial \eta }{\partial x}}}$

v momentum equation:

• ${\displaystyle {\frac {\partial v}{\partial t}}=-g{\frac {\partial \eta }{\partial y}}}$

the continuity equation:

• ${\displaystyle {\frac {\partial \eta }{\partial t}}+h{\frac {\partial v}{\partial y}}=0}$

Now it is possible to get an expression for ${\displaystyle \eta }$, by taking the time derivative of the continuity equation and substituting the momentum equation:

• ${\displaystyle {\frac {\partial ^{2}\eta }{\partial t^{2}}}-gh{\frac {\partial ^{2}\eta }{\partial y^{2}}}=0}$

The same can be done for ${\displaystyle v}$, by taking the time derivative of the v momentum equation and substituting the continuity equation

• ${\displaystyle {\frac {\partial ^{2}v}{\partial t^{2}}}-gh{\frac {\partial ^{2}v}{\partial y^{2}}}=0}$

Both of these equations take the form of the classical wave equation, where ${\displaystyle c={\sqrt {gh}}}$. Which is the same velocity as the tidal wave and thus of a shallow water wave. These preceding equations govern the dynamics of a one-dimensional non-dispersive wave, for which the following general solution exist:

• ${\displaystyle \eta =-h\ F(y+ct)\ e^{\frac {-x}{R}}}$
• ${\displaystyle v={\sqrt {gh}}\ F(y+ct)\ e^{\frac {-x}{R}}}$

where length ${\displaystyle R={\frac {\sqrt {gh}}{f}}}$ is the Rossby radius of deformation and ${\displaystyle F(y+ct)}$ is an arbitrary function describing the wave motion. In the most simple form ${\displaystyle F}$ is a cosine or sine function which describes a wave motion in the positive and negative direction. The Rossby radius of deformation is a typical length scale in the ocean and atmosphere that indicates when rotational effects become important. The Rossby radius of deformation is a measure for the trapping distance of a coastal Kelvin wave.[14] The exponential term results in an amplitude that decays away from the coast.

The expression of tides as a bounded Kelvin wave is well observable in enclosed shelf seas around the world (e.g. the English channel, the North Sea or the Yellow sea). Animation 1 shows the behaviour of a simplified case of a Kelvin wave in an enclosed shelf sea for the case with (lower panel) and without friction (upper panel). The shape of an enclosed shelf sea is represented as a simple rectangular domain in the Northern Hemisphere which is open on the left hand side and closed on the right hand side. The tidal wave, a Kelvin wave, enters the domain in the lower left corner and travels to the right with the coast on its right. The sea surface height (SSH, left panels of animation 1), the tidal elevation, is maximum at the coast and decreases towards the centre of the domain. The tidal currents (right panels of animation 1) are in the direction of wave propagation under the crest and in the opposite direction under the through. They are both maximum under the crest and the trough of the waves and decrease towards the centre. This was expected as the equations for ${\displaystyle \eta }$ and ${\displaystyle v}$ are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term. Therefore this set of equations describes a wave that travels along the coast with a maximum amplitude at the coast which declines towards the ocean. These solutions also indicate that a Kelvin wave always travels with the coast on their right hand side in the Northern Hemisphere and with the coast at their left hand side in the Southern Hemisphere. In the limit of no rotation where ${\displaystyle f\rightarrow 0}$, the exponential term increase without a bound and the wave will become a simple gravity wave orientated perpendicular to the coast.[14] In the next section, it will be shown how these Kelvin waves behaves when traveling along a coast, in an enclosed shelf seas or in estuaries and basins.

Tides in enclosed shelf seas

Animation 1: A traveling Kelvin wave is shown for the Northern hemisphere with (lower panel) and without (upper panel) friction, the coast is always at the right of the direction of motion. The domain is closed on the right hand side, to mimic an enclosed shelf sea. The wave enters the domain on the lower left hand side and travels towards the right. On the right hand side the wave is reflected and travels back towards the left. On the closed side the reflection happens through the creation of Poincare waves which are not modelled here. The panels on the left shows the sea surface height and the panels on the right the velocity. The colours are indicated in the colour bar, the arrows are scaled velocity vectors

Figure 2: The time averaged sea surface height is shown in red shading and the dashed lines are cotidal lines at intervals of roughly an hour. The Amphidromic points are located at the intersect of all lines. The upper panel shows the case for without friction and the lower panel for the case with friction

The expression of tides as a bounded Kelvin wave is well observable in enclosed shelf seas around the world (e.g. the English channel, the North Sea or the Yellow sea). Animation 1 shows the behaviour of a simplified case of a Kelvin wave in an enclosed shelf sea for the case with (lower panel) and without friction (upper panel). The shape of an enclosed shelf sea is represented as a simple rectangular domain in the Northern Hemisphere which is open on the left hand side and closed on the right hand side. The tidal wave, a Kelvin wave, enters the domain in the lower left corner and travels to the right with the coast on its right. The sea surface height (SSH, left panels of animation 1), the tidal elevation, is maximum at the coast and decreases towards the centre of the domain. The tidal currents (right panels of animation 1) are in the direction of wave propagation under the crest and in the opposite direction under the through. They are both maximum under the crest and the trough of the waves and decrease towards the centre. This was expected as the equations for ${\displaystyle \eta }$ and ${\displaystyle v}$ are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term.

On the enclosed right hand side, the Kelvin wave is reflected and because it always travels with the coast on its right, it will now travel in the opposite direction. The energy of the incoming Kelvin wave is transferred through Poincare waves along the enclosed side of the domain to the outgoing Kelvin wave. The final pattern of the SSH and the tidal currents is made up of the sum of the two Kelvin waves. These two can amplify each other and this amplification is maximum when the length of the shelf sea is a quarter wavelength of the tidal wave.[2] Next to that, the sum of the two Kelvin waves result in several static minima's in the centre of the domain which hardly experience any tidal motion, these are called Amphidromic points. In the upper panel of figure 2, the absolute time averaged SSH is shown in red shading and the dotted lines show the zero tidal elevation level at roughly hourly intervals, also known as cotidal lines. Where these lines intersect the tidal elevation is zero during a full tidal period and thus this is the location of the Amphidromic points.

In the real world, the reflected Kelvin wave has a lower amplitude due to energy loss as a result of friction and through the transfer via Poincare waves (lower left panel of animation 1). The tidal currents are proportional to the wave amplitude and therefore also decrease on the side of the reflected wave (lower right panel of animation 1). Finally, the static minima's are no longer in the centre of the domain as wave amplitude is no longer symmetric. Therefore, the Amphidromic points shift towards the side of the reflected wave (lower panel figure 2).

The dynamics of a tidal Kelvin wave in enclosed shelf sea is well manifested and studied in the North Sea.[15]