Ocean and shelf tides

Introduction

Tidal motion is the oscillation of ocean waters under influence of the attractive gravitational forces of the moon and the sun. The response of the ocean to the gravitational forces follows a pattern of rotating ('amphidromic') systems, as a consequence of the earth's rotation. The frequencies are determined by the relative periodic motions of moon, sun and earth surface. The amplitude of the tidal oscillation is very small compared to ocean depths. The tidal oscillation in each point can therefore be represented by a linear superposition of sinusoidal tidal components. The most important tidal components have a periodicity which is close to semidiurnal or diurnal, due to earth's rotation.

This article is an adaptation of the sections on ocean tides from the book Dynamics of Coastal Systems .

Tide generation

Moon and sun

The regular daily upward and downward motion of the water surface along the coastline is the most visible expression of tidal forcing. In ancient Greece it was recognized that tides are in some way related to sun and moon, but this relationship could not be explained . The explanation of tidal motion as a consequence of gravitational forces was given by Newton. The lunar and solar tide generating forces acting on the oceans are extremely small; they are a factor [math]\, \approx 5.10^{-7} \,[/math] smaller than the gravitational force of the earth. The tide-generating force of the moon is about twice as large as that of the sun.

Tidal wave and ocean basin resonance

Ocean tides owe their strength for a large part to resonance; tidal motion is amplified for tidal components with a frequency close to the frequency of free oscillations in the ocean basins. This is the case in particular for the semi-diurnal tide in the North Atlantic and Pacific Oceans , and also for the diurnal tide in the Pacific ocean . Tidal waves in wide ocean basins (width typically larger than a few thousand km) rotate around points of zero amplitude, the amphidromic points; the rotation is counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. This due to earth's rotation, which induces Coriolis acceleration of tidal currents. Fig.1 shows the semi-diurnal tidal wave in the world's oceans; it comprises a large number of amphidromic systems, where the tide rotates in resonance with the semi-diurnal component of the gravitational force. This ocean tidal wave pattern can be described in first approximation by a combination of Kelvin and Sverdrup waves (also called Poincaré waves, see Tidal motion in shelf seas).

Fig.1. System of semidiurnal lunar tidal waves (M2) in the oceans represented by lines of equal tidal phase (solid, intervals of 30[math]^{\circ}[/math]) and lines of equal tidal amplitude (dashed, intervals of 0.25 m). Three ranges are indicated for the tidal amplitude [math]a[/math] on the continental shelf: microtidal (white fringe, [math]a \lt [/math] 1 m), mesotidal (light grey fringe, 1 [math] \lt a \lt [/math] 2 m) and macrotidal (dark grey fringe, [math]a \gt [/math] 2 m). Redrawn after Bearman (1991)

Tidal components

Semidiurnal periodicity

In order to explain tidal motion, it is not sufficient to consider only the gravitational forces exerted by moon and sun on the earth's water masses. Tide generation results from the local imbalance at the earth's surface of two opposing factors: the gravitational forces acting between the earth and the moon (and between the earth and the sun), and the outward centrifugal acceleration related to the orbital motions of these celestial bodies. In fact, only the tangential component of the resulting force is relevant, which points towards the equator. Because the gravitational force and the centrifugal force cancel at any location on the earth's surface twice during each diurnal rotation, the major periodicity is approximately semidiurnal (a little more than 12 hours because of moon's orbital motion).

Diurnal tide and spring-neap cycle

Because the solar and lunar orbits do not coincide with the equatorial plane (the angle between orbital plane and equatorial plane is called declination), a daily inequality arises in the semidiurnal cycle. The daily inequality is strongest in shelf seas that resonate at diurnal frequency and which are situated close to amphodromic points of the semidiurnal tide. In such regions the tide is mainly diurnal.

Due to the [math]\approx[/math] 30-day orbital motion of the moon, the moon-earth and sun-earth axes approximately coincide every 15 days (syzygy). This causes a 15-day cycle of neap tide and spring tide; spring tide follows full moon and new moon and neap tide follows half-moon (with a delay of one or two days due to frictional dissipation ). In fact, the 15-day period corresponds to the frequency difference of the semidiurnal lunar component ([math]M_2[/math]) and the semidiurnal solar component ([math]S_2[/math]); the interference of these components produces the neap-spring variation of the tidal amplitude.

Other tidal components

Fig.2. Coastal zones with dominant semi-diurnal tide.

The different cycles in the relative motions of moon, sun and earth surface generate tidal waves with corresponding periods. The lunar semidiurnal tidal component ([math]M_2[/math], period [math]\approx[/math] 12 h 25 min) is generally the largest tidal constituent (see Fig.2), followed by the semidiurnal solar tide [math]S_2[/math]. Other important tidal constituents are:

• the diurnal component [math]K_1[/math], which is related to the declination of the lunar and solar orbits relative to the equatorial plane (period [math]\approx[/math] 24 h),
• the diurnal lunar component [math]O_1[/math] (period [math]\approx[/math] 26 h) and
• the diurnal solar component [math]P_1[/math] (period [math]\approx[/math] 24 h).

There are many other tidal components of smaller magnitude and lower frequency related to periodicity in the lunar and terrestrial orbits. A modulation of the mean tidal amplitude of the order of 5% is related to the 18.6 year oscillation in the declination of the lunar orbit.

Asymmetric ocean tides

Because of the minor role of friction in the propagation of ocean tides, individual astronomic tidal components are well described by sinusoidal functions. This implies that for each component the rising and falling branches are symmetric. This also holds on average for a superposition of tidal components, if the frequencies [math]\omega_i[/math] of the major tidal components are not linearly related with integer coefficients ([math]\omega_i[/math] is different from any combination of sums and differences of [math]\omega_j, \; j \neq i[/math]). Taken over a sufficiently long time interval the average period of rising tide equals the average period of falling tide. Hence, at the continental shelf boundary there is symmetry between the astronomical forcing of flood currents and ebb currents.

However, certain combinations of astronomic components yield asymmetric tides, irrespective of the averaging period. The reason is that most tidal constituents result from a superposition of a limited number of basic cycles in the relative motions of earth, moon and sun. The frequencies of these tidal constituents correspond to sums and differences of the basic frequencies . This implies that different tidal constituents may interfere in such a way that flood and ebb are modulated in a systematic, asymmetric way. Such asymmetries may become significant in the case of strong diurnal tides; for dominant semidiurnal tides this effect is small .

A particular example is the combination of the [math]M_2, K_1[/math] and [math]O_1[/math] tides. The sum of the frequencies of the [math]K_1[/math] and [math]O_1[/math] tidal constituents equals the frequency of the [math]M_2[/math] tidal constituent. In regions where the amplitudes of the [math]K_1[/math] and [math]O_1[/math] tides are not small compared to the amplitude of the [math]M_2[/math] tide, the sum of these three tidal constituents yields an asymmetric tide, with an asymmetry depending on the respective phases of these tidal constituents. In this case the astronomic forcing of flood and ebb currents at the continental shelf boundary is nor symmetric, with consequences for residual sediment transport.

Tides on the continental shelf

Tidal amplification

The tides generated in the ocean propagate to the continental shelf, where the tidal range may increase further. Different phenomena contribute to this amplification: resonant dimensions of the shelf sea, slowing down of wave-energy propagation ('shoaling') or concentration of the tidal energy flux in areas of reduced width ('funneling'). The maximum spring tidal range can exceed 14 m in some funnel-shaped bays, such as Cobequid Bay (Canada), Bristol Channel (UK) and Baie du Mont Saint Michel (France).

Tidal amplification in shelf seas is counteracted by frictional momentum dissipation. The strong tidal motion occurring in many shelf seas is thus not locally produced by tide generating forces, but results from co-oscillation with ocean tides and from local topographic amplification. Numerical tidal studies for the northwest European shelf and for the East China shelf show that the local tide generating force influences the co-oscillating semidiurnal tide in these shelf seas by no more than about 1%. This is the main reason why no substantial tidal motion is generated in shallow enclosed seas or lakes. Energy dissipation of ocean tidal energy is mainly due to frictional dissipation in shelf seas ; however, non-negligible dissipation takes place also in the ocean . Tidal resonance in shelf seas generally has a damping effect on ocean tides .

Higher harmonic components

Fig.3. Coastal zones with important quarter-diurnal tide.

Tides on the continental shelf are due to co-oscillation with the ocean tides at the continental shelf boundaries, as noted earlier. However, the non-linearity of tidal propagation in shallow environments with strong topography generates additional tidal components, corresponding to multiples of the dominant ocean tidal frequencies . These locally generated higher harmonic components are associated with tidal-wave distortion and tidal asymmetry. Tidal motion in continental waters has an asymmetric character: the periods of tidal rise and tidal fall are not equal. Flood currents and ebb currents have different strength, related in particular to the quarter-diurnal tidal component. In Fig. 3 coastal regions are indicated with an important M4 tidal component; they coincide largely with coastal zones where the semidiurnal tide is strong (Fig. 2). Tidal asymmetry in shallow coastal waters, estuaries and tidal rivers, plays a crucial role in tide-topography interaction , because of the highly non-linear dependence of sediment transport on tidal current strength, see Morphology of estuaries.

Further reading

Cartwright, D.E. 1999. Tides, a scientific history. Cambridge Univ.Press, UK, 292 pp.

Hendershot, M.C. 2007. Long Waves and Ocean Tides. In: Evolution of Physical Oceanography (Eds. B. Warren, and C. Wunsch) MIT OpenCourseWare, https://ocw.mit.edu

References

The main author of this article is Job Dronkers Please note that others may also have edited the contents of this article. Citation: Job Dronkers (2018): Ocean and shelf tides. Available from http://www.coastalwiki.org/wiki/Ocean_and_shelf_tides [accessed on 22-11-2024] For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Dronkers J