Tides in shallow seas and estuaries

Although the tide-generating forces are all-pervasive and no water particle can evade its influence, they are of no consequence to the mean large-scale oceanic circulation. The reason is that the tides are a form of low amplitude periodic motion and do not contribute to the balance of forces for the steady state. A more formal way of expressing this is that tidal motion and the mean water movement of the steady state circulation can be studied independently, and their joint effect on the movement of water particles and on the distribution of water properties can be found by adding the results from the two independent studies. (In the framework of mathematics this is known as a linear system. Linear systems have the useful property that the sum of any two solutions, such as the solution to the mean steady state equations and the solution to the tidal equations, is again a solution to the system. This allows us to build the full description of the water's movement through a succession of solutions to simple problems.) Our review of deep ocean dynamics in Chapter 2 therefore did not include a review of the tides.

The situation is different when it comes to the balance of forces in shallow seas, even if we consider only the steady state. In many shallow seas tidal movement, though still periodic, is no longer weak, and can result in mean water movement known as the residual flow. More importantly, tidal currents cause mixing strong enough to determine the stratification of some shallow seas. A description of shallow water dynamics therefore has to include the effect of tidal movement.

Estuarine dynamics depend entirely on the tides. The fact that in estuaries even the mean circulation cannot be understood without consideration of a strictly periodic phenomenon makes the study of estuarine dynamics intrinsically more difficult than the study of the dynamics of the deep ocean. (Mathematically, estuaries represent a non-linear system. Solutions to non-linear systems cannot be found by adding solutions of simpler sub-systems.) The balance of forces in estuaries inevitably includes a representation of tidal effects. This is not always evident from the equations used for the study of estuarine dynamics, which rarely if ever include the tide-generating forces in their original periodic form. Most models of estuaries include the effect of tidal mixing by choosing appropriate formulations for the terms which represent the effects of friction and diffusion. The presence of tidal motion is not obvious from the resulting equations. As we shall see in Part 2, it can also take on different forms depending on the type of estuary. For a correct assessment of the circulation in an estuary it is important to understand how the tides act in the estuary and how they can be represented in the balance of forces.

This chapter gives a very brief review of the tide-generating forces and their impact on the world ocean. Emphasis is on those aspects which are of consequence for the coastal ocean and estuaries. Many important aspects of the tides (for example details about constituents, inequalities etc.) are not included; these are left to dedicated texts on tides. The approach here is similar to the procedure adopted with earlier topics: We begin with a review of the situation in the deep ocean and proceed to a discussion of the modifications produced by shallow water.

The tide-generating forces

Tides are the result of gravitational attraction between stellar bodies. A discussion of the tide-generating forces therefore begins with a look at the forces which act on the earth as it travels through space.

Newton's Law of Gravitation states that the force of gravity between two stellar bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. Since the distances from our earth to nearly all other large objects in space are so enormous, the only stellar bodies of tidal significance are the moon and the sun. The effect of the two can be studied separately, and most properties of the tides as they are observed can be understood by assuming that earth, moon and sun are all perfect spheres of homogeneous mass composition. This allows us to apply Kepler's First Law which describes the movement of stellar bodies around each other and states that the earth and the moon, or the sun and the earth, follow elliptical paths around one of the two focal points of an ellipse. Since the sun's mass is about 332,000 times larger than the mass of the earth, the focal point of the sun-earth system is located inside the sun. Likewise, the focal point of the earth-moon system is located inside the earth, since the moon's mass is only 1.2% of the mass of the earth.

The movement of the earth on its elliptical path is illustrated in Figure 5.1. To understand what generates the tides it is convenient to forget for the moment that the earth rotates around its axis once every day and consider its movement with respect to sun or moon without its daily rotation. The earth revolves around the focal point without changing its orientation in space. This type of movement is known as revolution without rotation. We can see an example of revolution with rotation when looking at the moon: It does not maintain its orientation in space but always faces the earth with the same side; the moon's "backside" was not known to us until we could send spacecraft up to look at the moon from behind. The moon's movement around the earth is therefore similar to the movement of a stone twirled around at the end of a string. In contrast, to model the earth's rotation at the end of a stick would require a mechanism which keeps the orientation of the earth's axis fixed in space.

Tides are the result of the balance between the force of gravity and the centrifugal force; they cannot exist under revolution with rotation. Figure 5.1 shows the balance of forces under revolution without rotation. The gravitational force varies with distance from the attracting body; it is larger at points on the earth's surface closer to the sun (or moon) and smaller at points on the opposite side. The centrifugal force, on the other hand, is determined by the angular velocity of the earth's movement, which under revolution without rotation is the same for all points inside and on the surface of the earth. The two forces balance each other exactly at the earth's centre and when integrated over the mass of the earth. On the earth's surface the balance is not exact, and the remaining force varies in strength and direction. It is directed outward, acting in the opposite direction to gravity (ie in the vertical, indicated by the open arrows in the Figure), at the point directly under the sun (or moon) and at the point directly opposite. This produces a minuscule variation of gravity, not enough to be noticeable without extremely sensitive instruments and certainly not enough to produce oceanic tides of the observed magnitude. More importantly, over most of the earth's surface the remaining force acts horizontally, ie in a direction which in an ocean at rest does not experience any other forcing. It is this horizontal component (indicated by the full brown arrows) which is responsible for tides in the ocean and is therefore known as the tide-generating force.

The tide-generating force is periodic around the earth and produces a series of convergences and divergences (see Figure 5.1). If we now allow for the daily rotation around the earth's axis, we find that the sequence of divergences and convergences of the tide-generating force sweeps around the earth once every day. As a result, water is moved and accumulates in one region and is drained away in another region, and these bulges and depressions travel across the ocean. In other words, oceanic tides are waves of very long wavelength driven by currents which are produced by the horizontally acting tide-generating force. To put it even more succinctly, the rise and fall of the sea level which we generally associate with the word "tides" is a result of horizontal water movement; it is not the primary response to the tide-generating force.

The equilibrium tide, the dynamic theory of tides

It is tempting to think that on an earth without continents the tidal wave follows the distribution of the tide-generating force exactly, so that the arrows of Figure 5.1 can be interpreted directly as a depiction of the tidal currents: Water accumulates under the sun or moon and at the opposite point on the earth's surface, while tidal currents are largest along a line half-way between these points; the entire pattern is tied to the sun's or moon's position and follows its apparent path around the earth. This theoretical idea is known as the equilibrium tide. It was first proposed by Bernoulli in 1740 who used it to explain many tidal features such as periodicity, inequalities between successive high waters and low waters, and the occurrence of spring tides near full and new moon. The equilibrium tide theory was a great step forward in explaining the tides, being based on a correct understanding of the tide-generating force. Before Bernoulli's theory the tides were explained as being generated by the ocean breathing in and out and other unrealistic ideas.

Nevertheless, it is clear that the tides cannot react to the tide-generating force in the way assumed by the equilibrium tide. To move the tidal wave around the earth within one day would require the movement of enormous amounts of water with the speed of modern aircraft and is physically simply not possible. This was recognized by Laplace in 1775 when he developed the dynamic theory of tides. Laplace accepted Bernoulli's idea of the tides as long waves but pointed out that every finite volume of fluid has its own preferred wave frequencies. If some force tries to excite periodic motion, the reaction of the fluid will be much stronger if the forcing occurs at one of these resonance frequencies than if it occurs at other frequencies. As an example, imagine you hold a bowl or a small tank filled with water and move it gently back and forth. At frequencies other than a resonance frequency the water in the vessel will only follow the movement of the vessel. If the vessel movement occurs at a resonance frequency the water level will undergo strong oscillation, possibly causing some water to spill over. It is not difficult to identify the resonance frequencies of the vessel by slowly varying the frequency of the forcing. Figure 5.2 shows how the response to the forcing varies as you come close to resonance. The amplitude of the response grows rapidly as the resonance frequency is approached, while the phase indicates a change from direct to inverse response.

The resonance frequencies depend on the dimensions of the vessel; the larger (wider or deeper) the vessel, the longer the resonance periods. It is possible to determine the resonance frequencies for a water body of a given size from its dimensions. Alternatively, it is possible to determine the size a water body must have to resonate at given tidal frequencies. Laplace calculated the depth the ocean must have to be at resonance with the major periods of the lunar and solar tide, assuming again an earth without continents. He found that the ocean is at resonance with the semidiurnal lunar tide if its depth equals 1965 m and again at 7937 m; resonance with the semidiurnal solar tide occurs at 2248 m and at 8894 m. At other depths the ocean would not be at resonance but would show an inverse response for all depths between the two resonance depths, while a shallower or deeper ocean would show a direct response. (Thus, an ocean of 2000 m depth would have a direct semidiurnal solar tide but an inverse semidiurnal lunar tide.)

Later refinements of the dynamic theory repeated Laplace's calculations for ocean basins limited by coastlines of various shapes. This modifies the resonance frequencies of the basins, but the principle remains: The closeness of the frequency of the tide-generating force to one of the resonance frequencies determines the amplitude of the tidal wave which is generated in the basin. It also determines the phase, ie the occurrence of low and high water relative to the passage of the moon and sun. Newton's dynamic theory was the first theory capable of explaining why tidal amplitudes and phases vary widely throughout the world ocean.

In propagating waves (such as wind waves seen to travel across the sea surface) all points on the sea surface undergo periodic uplift and sinking and experience horizontal movement. The experiment with the bowl or small tank demonstrates that tides of ocean basins are standing waves. Particles moving in a standing wave do not all experience the same type of motion. Some particles - in our bowl or tank experiment the particles near the walls of the vessel - experience only periodic uplift and sinking. Halfway between the ends of the vessel the particles experience only horizontal motion. These particles are located on a line known as a nodal line or node (Fig. 5.3a).

Not many ocean basins contain tidal nodes. The reason is that on a rotating earth the Coriolis force deflects particle movement from a straight path, causing water to accumulate on the right in the northern hemisphere and on the left in the southern hemisphere. As a result, the standing plain wave observed in the non-rotating tank is changed into a system in which the wave moves along the vessel walls around a central point (Fig. 5.3b). This produces clockwise wave propagation in the southern hemisphere and anti-clockwise wave propagation in the northern hemisphere. The system is easily modelled with a circular bowl by moving it in a circular motion at one of its resonance frequencies.

The central point around which the wave propagates is called an amphidromic point. It is the rotating system's equivalent of the non-rotating system's nodes, sharing with them the property that all vertical movement disappears in it.

Tides in shallow seas

This very brief summary of tidal dynamics in deep water is still quite incomplete, but it is sufficient to understand the action of the tides on shelves and in estuaries. The first and fundamental observation with respect to tides in shallow water is that the tide-generating force is of global scale. Only the largest water bodies such as the major oceans can therefore experience tidal forcing in the way described by Laplace. Smaller water bodies such as marginal seas or estuaries cannot produce a response to astronomical tidal forcing. If there is tidal movement in these regions, it is forced by the tidal currents of the deep ocean which enter and leave the region periodically at the connection to the ocean. Tides generated in this way are known as co-oscillation tides. Marginal seas have their own resonance frequencies, determined again by their dimensions. As a consequence, the amplitudes and phases of co-oscillation tides depend on the closeness of a resonance frequency to one of the tidal frequencies and on the amplitude of the tidal currents in the deep ocean at the connecting line with the marginal sea. This explains, for example, why mediterranean seas are virtually tide-free; their connection with the open ocean is so restricted that the oceanic tides cannot produce co-oscillation.

Because co-oscillation tides are a resonance phenomenon they usually display the largest tidal range near the coast of the marginal sea or at the inner end of a bay. (You can see this demonstrated in an animation.) This can give rise to extreme tidal ranges if the co-oscillation occurs at resonance. The largest tidal range occurs in the Bay of Funday on the Canadian Atlantic coast. This bay is 151 km long and 31 km wide and at spring tide experiences a tidal range of 21 m. The North West Shelf of Australia is another region with large tidal resonance; the tidal range on the North West Shelf reaches 8 m and more.

A large tidal range is of course always associated with strong tidal currents, and tidal currents on the shelf are always larger than tidal currents in the open ocean. In some locations tidal currents can become unusually strong even under a moderate or small tidal range. This occurs where constrictions prevent the free flow of the tidal wave and force it to rush through narrow openings.

The most spectacular tidal current of this type is the famous "maelstrom" in the Saltfjord of northern Norway. This 500 m deep fjord is connected with the North Atlantic Ocean by a 3 km long channel of only 150 m width and 31 m depth. The channel is much too small to allow the fjord to follow the oceanic tide, and the difference in water level between the two ends of the channel can reach up to 1 m. This produces a periodic current through the channel of speeds in excess of 20 knots (up to 40 km/h) which produces intense whirlpools (maelstroms) of 10 - 15 m diameter. Calm conditions every 6 hours allow ships to pass through the channel, before the current starts again. For centuries it has been said that the Saltstraumen, as the current is known, runs strongest on Good Friday (the Friday before Easter). This is easily understood from tidal theory if we recall that the Christian church sets the date of Easter as the first Sunday after the full moon following the vernal equinox: By definition the tide generating potential of the sun and moon act in concert at that time.

A coastal inlet in the Kimberleys of Western Australia shows even stronger tidal currents. Its connection to the North West Shelf is only a few hundred meters long and barely 50 m wide. The difference in water level on either side of the connection is clearly visible from the top of the cliff, as a tidal waterfall rushes through the gap, changing direction every six hours.

Shallow seas which are close to resonance with one of the tidal periods are of great importance for the world's fishing industry. The flow of strong tidal currents over a shallow ocean floor produces turbulence of sufficient intensity to keep the entire water column well mixed throughout most of the year. Nutrients which usually accumulate in the sediment and are no longer available to support marine life, are continuously kept in suspension under such conditions. These coastal seas are therefore among the most productive fishing regions of the world ocean, rivalling the great coastal upwelling regions and the fertile Southern Ocean. The North Sea or the Newfoundland Banks are two examples of regions where tidal mixing keeps nutrient concentrations in the water column at a high level.

Tides in shallow water are generally a mixture of propagating waves and standing waves. One major difference between these two types of waves is the phase relationship between elevation and tidal current. As could be seen from the example of the water bowl or tank, currents and water level are 90° (or a quarter period) out of phase: Currents are strongest when the water surface is flat and vanish when the water level is at its highest and lowest (high and low tide). In propagating waves, on the other hand, currents are strongest at high and low tide, ie they are in phase with the elevation. For a given coastal location the time of strongest tidal current relative to high tide therefore depends on the type of tidal wave in the region.

Sudden changes in water depth can lead to a change of the tide from a standing wave to a propagating wave. This occurs because the propagation speed of shallow water waves depends on the water depth. If such a wave encounters a sudden change of depth, its propagation speed is slower over the shallower region than over the deeper region; its propagation speed on either side of the sudden depth change is mismatched, and the wave cannot continue unchanged across the changing topography. This leads to partial reflection of the wave. If a wave approaches a steep rise of the sea floor (Fig. 5.4), part of the wave continues as a propagating wave in the shallow water, while part of it is reflected back into the deeper water and combines with the incoming wave to form a partially standing wave. Tidal currents and elevation are thus in phase in the shallow part but out of phase, by a degree determined by the wave's reflection coefficient (Fig. 5.4), in the deeper part. This explains the wide range of observed phase relationships between tidal currents and high or low tide in the world ocean's shelf seas.

Tides in estuaries

The dominant influence on the tides in estuaries is the change of water depth and estuary width as the tide propagates up the estuary. Effects produced by sudden changes in depth, as discussed above in the context of shelf seas, occur in estuaries as well, so the phase relationship between tidal currents and high or low water can vary strongly along estuaries. More importantly, the shoaling and narrowing of the estuary slows the progress of the tidal wave, increasing its amplitude. The wave becomes distorted and displays a shorter rising tide than falling tide. Mathematically this change in wave shape can be described by the development of higher harmonics, tidal periods which do not have their origin in a corresponding forcing period but are simple fractions (1/2, 1/3 etc.) of the forcing period (Fig. 5.5). As just described, these higher harmonics occur as a result of shallowing water; they do not exist in the deep ocean and are therefore known as shallow water tides. They are ubiquitous in estuaries but are also found in extremely shallow shelf regions, particularly those bordered by large intertidal flats.

In all situations where shallow water tides are observed it is useful to remember that shallow water tides are a convenient way of describing the shape of the tidal wave through a combination of harmonic waves. Physically, the tidal wave is still experienced with the same basic period but appears deformed, with a shorter rising tide than falling tide. When this process is driven to extremes (as a result of particular topographic circumstances), the rising tide can take on the form of a wall of water which travels up the estuary, causing a nearly instantaneous rise of water level as the water wall passes. This phenomenon is known as a bore. Some bores are spectacular natural phenomena; look up some photos of bores if you wish.

Tidal bores can cause great inconvenience to boats. Where a bore occurs in an otherwise navigable estuary it brings ship traffic to a halt for significant parts of the day. Port authorities then try and eliminate the bore by changing the shape and depth of the estuary.

© 1996 - 2000 M. Tomczak

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