Tides in marginal seas

Tides are water level variations caused by the gravitational interaction between the Moon, the Sun and the Earth.

However, due to global and local ocean responses different tidal patterns are generated.

The complicated ocean responses are the result of the continental barriers, resonance due to the shape of the ocean basin, the tidal waves impossibility to keep up with the Moons tracking, the Coriolis acceleration and the elastic response of the solid earth.

In order to apply to the conservation of energy, the tidal wave has to deform as a result of the decrease in water depth.

When the tidal wave propagates onto the continental shelf, the water depth

In order to conserve the energy flux, the amplitude of the wave needs to increase (see figure 1).

The above explanation is a simplification as not all tidal wave energy is transmitted, but it is partly reflected at the continental slope.

At the continental shelf the reflection and transmission of the tidal wave can lead to the generation of internal tides on the pycnocline.

The shoaling of the internal tide drives mixing across the pycnocline, high levels carbon sequestration and sediment resuspension.

[9][10] Furthermore, through nutrient mixing the shoaling of the internal tide has a fundamental control on the functioning of ecosystems on the continental margin.

[11] After entering the continental shelf, a tidal wave quickly faces a boundary in the form of a landmass.

These preceding equations govern the dynamics of a one-dimensional non-dispersive wave, for which the following general solution exist: where length

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.

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 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.

are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term.

[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.

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 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.

are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term.

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.

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).

Therefore, the Amphidromic points shift towards the side of the reflected wave (lower panel figure 2).

The deformation of the tide is largely controlled by the competition between bottom friction and channel convergence.

[16] Channel convergence increases the tidal amplitude and phase speed as the energy of the tidal wave is traveling through a smaller area while bottom friction decrease the amplitude through energy loss.

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.
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