tidal hydraulics - cwcold.cwc.gov.in/cpdac-website/training/12_tidal_hydraulics.pdf · training...

TIDAL HYDRAULICS

A.V.MAHALINGAIAH Senior Research Officer

Central Water and Power Research Station, Pune

1.0 INTRODUCTION In the field of coastal hydraulics, tides, waves, currents, winds are the important parameters, which are to be considered for any development in this region. The basic knowledge on tides, short waves, density currents and coastal processes is very much essential for effective planning and implementation of projects related to coastal, estuarine and harbour. This will not only aid in understanding the whole system with all the relevant aspects, but also in the analysis of various alternatives and making a viable choice and practical implementation of the chosen alternative. To illustrate this further, let us have an idea of the hydraulic phenomena that play a role, through an example. Consider the planning of a harbour in an estuary flanked by sandy coastline (see Fig.1).

We consider an estuary with a straight sandy coastline with tide, short waves, long shore sediment transport and river with fresh water discharge. The proposed harbour area consists of harbour basins located inside an estuary and breakwaters at the entrance with approach channel (depth -15m). The phenomenon, which play a role are: • Breakwater stops the long shore sediment transport, so accretion and erosion

will occur on the up drift and down drift sides respectively. • Possible siltation of the approach channel. • Wave penetration in to the harbour area. • Intrusion of the salt water from the sea because of the deeper and wider

entrance of the river. • Cross currents and circulation at the entrance of the harbour basins due to the

tide. Thus from the above example, it is evident that in order to examine all the relevant phenomenon that play a role in the planning and design of a harbour or to use the correct tools for the study of the phenomenon, one should have the knowledge of the following:

Tides Short waves Salinity intrusion Coastal process (long shore sediment transport)

The present lecture is, however, intended to give an overview of the tidal

hydraulics and the various aspects related to tides will only be discussed.

Training Course on Coastal Engineering & Coastal Zone Management

Tidal Hydraulics by A.V. Mahalingaiah, SRO, CWPRS, Pune 2

1.1 Tides

The periodic rise and fall of the water level in the ocean is known as the astronomical tide or pulse of the earth. The period of the vertical movement is about 12 hours 25 minutes, known as the tidal period and is denoted a T. The highest level is called high water (HW) and the lowest level is called the low water (LW). The difference between HW and LW is called the tidal range.

If we measure the vertical movement of the water level for about one day

(24 hours) we would observe that the second HW and LW would differ form the first HW and LW. This difference in consecutive high water and low waters is called the daily inequality. The seaward tidal flow from high tide to low tide is called ebb tide and landward tidal flow from low to high tide is called floodtide. Time of no flow between ebb and flood tide is called slack water (period). If we observe the tide for a longer period (say about 1 month), then we observe that the tidal range varies with time. There are periods with large tidal ranges and periods with small tidal ranges. The period with large tidal range is called the spring tide and the period with smaller tidal ranges is called the neap tide. The time between two periods of spring tides is about 15 days (half a month). The above observations are in relation to a fixed location. The tidal wave however is a long wave of wavelength, which is of the order of several hundreds of kilometers. Associated with the vertical movement of the water surface, there is also horizontal movement of the water particles known as tidal currents.

HIGH WATER ( HW)

LOW WATER ( LW)

TIDAL PERIOD = 12 h 25

TIDAL RANGE

Water Level

Time

Tidal inequality

T

Fig.2



A more detailed discussion on tides will be covered under the following topics:

Generation of tides. Tidal analysis Tidal prediction Tidal waves. Some examples of tide dominated areas

1.2 Generation of tides : Tides are generated by the force of attraction between the earth, moon and the sun. The Newton’s Law of gravity gives the force of attraction between two bodies. The attractive force is counter acted by the centrifugal force due to the rotation of the earth and moon around their common centre of gravity. The common centre of gravity is inside the earth and the earth moon system rotates around the common centre of gravity in 27.32 days.

For the earth moon system there is a balance between the attractive force and the centrifugal force. It can be shown that the centrifugal force on every point P on the surface of the earth is directed parallel to the line that connects the centers of the earth and moon, and the centrifugal force is equal for all the points on the earth’s surface.

On the other hand, the attractive force on the earth’s surface varies with

the distance from the moon. Tides are caused by forces acting on the water particles on the surface of the earth. The resultant of the centrifugal force and the attractive force is the tide generating force that acts like a tractive force over the surface of the earth. The sun also causes similar tractive force on the surface of the earth. The ratio of the tractive forces caused by the moon and the sun is about 2:1. Thus the effect of the sun on the tide is also important. The hydrostatic pressure force balances the tractive force on the fluid, over the surface of the earth. Equilibrium is reached as a result of the slope in the water level. The resultant water level can be sketched as shown in the Fig.3.

Moon

Earth

Slope = max

Slope = 0

P

Fig.3



As it can be seen from the figure the fluid on the surface of the earth is

distributed in the form of an ellipsoid. Increase in water level occurs at the sides of the earth facing the moon and in the opposite side. A similar ellipsoid results from the attraction of the sun. The daily inequality of the tide occurs due to the inclination in the orbit of the moon.

When the sun, earth and the moon are in one line the solar bulge and the moon bulge are working together (i.e. in phase). This occurs on the new moon and full moon causing spring tides. During this time the high waters are extra high and the low waters are extra low.

When sun, earth and the moon are aligned at right angle, the bulges of the moon and the sun are out of phase. This occurs during the first quarter and last quarter of the moon causing the neap tides. The high waters during this time are low and low waters are less low.

As mentioned earlier, if we observe the tide for more than a month and observe the time history, one can observe the periodic variation in the tidal ranges distinguishing the spring tides from the neap tides. The period `T’ of this phenomenon can be derived from the angular speeds of the moon and the sun (relative to the earth). The period `T’ is around 14.8 days and thus we will have spring/neap tides about two times per month.

1.3 Tidal Analysis

Due to the constant changes in the position of the moon and the sun, changes in the respective distances from the earth and the changes in the elliptical orbits and the common centre of gravity with the earth, there is a periodical fluctuation in the tide generating force. Such a continuous cycle can be represented as a sum of simple harmonic oscillations. A simple harmonic oscillation is determined once we know its period, its amplitude and its phase. The constituent of the tide due to the moon and the sun are referred with the abbreviation M and S respectively. The subscripts 1,2, and 4 indicate whether it is a diurnal, semi diurnal or quarter diurnal component respectively. According to the dynamic theory the amplitude of the actual tide is the sum of these tidal constituents. Table I shows the important constituents of the tides. By means of long terms observations at a given point it would be possible to obtain the amplitude and phase of each tidal constituent for that particular point. Such a procedure is called harmonic analysis of tides. While only the periods of each tidal constituent can be obtained from theory, the amplitude and phase can be obtained only from the analysis of the actual observations.



Table-1 Name of Component

Symbol Period (solar hours)

Amplitude ratio

(M = 100)

Principal lunar M2 12.42 100 Principal lunar S2 12.00 46.60 semi

diurnal Larger lunar elliptic N2 12.66 19.20 Lunar-solar semi diurnal

K2 11.97 12.70

Lunar-solar diurnal K1 23.93 58.40

Principal lunar diurnal

O1 25.82 41.50 diurnal

Principal solar diurnal

P1 24.07 19.40

The astronomical analysis of the tide gives the frequencies and relative

magnitudes of each component. The relative strength or amplitude ratio of a component is given in relation to the strength of the M component that has been set to 100%. If we measure the tide at a certain location on the earth, the relative magnitude of the components can differ considerably from the astronomical components. This is due to the irregularities in the oceans and the seas. In the shallow coastal shelf region, bottom friction, and the variable propagation speed of the tidal wave influence the tide. In the deep oceans, however, the tide can be completely described by the sum of the astronomical components.

In practice the observed tide at any given location can be used to find various harmonic components. The harmonic components are used to predict the tidal signal for the future. The characteristics of the components are derived from the observed tide. The methods that can be used to find the constituents for an observed tidal signal are as follows:

Method of least squares Fourier analysis

Fig.4 shows the amplitude Hi in meters as a function of the angular speed ω

obtained as a result of tidal analysis at a certain location on the earth. As it can be seen, the different groups of components that consist are:

- diurnal components - semi-diurnal components (M2 is the most important) - terra-diurnal components (not important) - quarter-diurnal components. - six-diurnal components.



It can also be observed that within a group the difference in angular speed (frequency) is very small. Two things are important while measuring the tide for carrying out the tidal analysis of the observed signal.

- Sample interval

- Length of the observation time.

Consider the observed water level at certain location as shown in Fig.5. The

entire signal cannot be used for the tidal analysis. The values at discrete points are to be taken which are known as the samples. The time between two samples is known as the sample interval. This is taken as constant. As it can be seen the tidal curve follows the sine function and it can be proved that any sine function must be sampled at least two times per period. For example, the tidal components M that has a period of 3 hours, the sample interval should be less that 1-1/2 hours. A common sample interval, in practice, is 1 hour for tidal analysis.

The second element is the length of the observation time required for the

tidal analysis. Referring to Fig.4, it can be observed from the results of the tidal analysis that with in the same group of components the angular speed are very close to each other. For instance the M2 and S2 components have angular speeds of 29o /hour and 30o/hour respectively. In order to separate these two components in the tidal analysis certain length of the observation is required depends upon the angular speeds of the components. In the case of M2 and S2 components the tidal measurement should be at least 15 days. A period of 30 days is more or less accepted as the standard observation period for minimum tidal analysis. In order to separate the components S2 and K2 or P1 and K1 we need an observation period of half a year. A period of 369 days is considered as the standard length of observation for the tidal analysis.

Hi t

Sample Interval

Fig.



1.3.1 Tidal Prediction :

The prediction of the tide is the inverse operation of the analysis of the tide. Having found the harmonic constituents (amplitude and phases) for a given location we can predict the tide for any time in the future. It should however be noted that the physical condition of the area should not change. If there are any major civil engineering works in the particular area then the harmonic constituents of the tide will get affected. In such a situation the tidal analysis may have to be carried out for the changed situation.

The prediction of the tide can be done using the formula :

)cos()(1

0 iiiii

n

ii guvtHfhth −+++= ∑

=

ω

Where, ho = mean sea level Hi = amplitude of component i fi = node factor w = angular speed of component i (v +u1) = astronomic argument gi = phase at the location

The following example illustrates the tidal prediction for certain location on

the surface of the earth. In this example, it is required to estimate the water level at a location X on January 5th 1800 hrs. of the year 1991. The components M2 and S2 are only to be used.

For the restricted number of components i.e. M2 and S2 we can write the

formula for prediction of tide is as follows:

))(cos())(cos()(2)2222222220 SSSSSMMMMM guvtHfguvtHfhth −+++−+++= ωω

The various components for the location ‘X’ different sources is as follows:

ho = 0.03m fm2 = 0.988 fs2 = 1.0 hm2 = 0.76 m hs2 = 0.19 m ωm2 = 28.9841 ωm2 = 30.0 (v+u)M2= 0.5 (v+u)M2= 0.0 gm2 = 90o gm2 = 151o

Substituting the above values in the equation we find that the magnitude of water level at the location X on January 5th 18.00 hrs. 1991 is +0.82m. 1.3.2 Types of tides :

The tides are classified based on form number denoted by `F’, which is the ratio of the sum of the amplitudes of the two main diurnal components K1` and O1 to the sum of the two main semi-diurnal components M2 and S2 . A high value of F (say above 3.0) implies a diurnal tidal cycle i.e., only one high tide occurs in a day. Low values of F (say less than 0.25) implies a semi-diurnal tide, i.e. two high waters will occur in a day.



1.4 Tidal waves :

In the analysis and prediction of the tide we have so far looked at the tide as it occurs at one location. Now we shall investigate the tidal motion in a larger area. In order to describe the tidal motion in a certain area we must derive the equations that describe the propagation of tidal wave. With these equations we are able to make the tidal computations. One of the main reasons for making the tidal computations is to see how water levels and velocities will change when civil engineering works are carried out in tidal regions. The changes in water motion may also affect the morphology, mixing process, salt intrusion in the area of interest.

The flow motion is characterized by the water level r and the velocity

components u, v, w. These variables are the function of time `t’ and the space co-ordinates x, y and z. The velocities u. v and w are directed in the x, y and z directions respectively. In the derivation of the equations one can consider either one-dimensional or two-dimensional flow depending upon the nature of the problem. Here, we will consider only the one-dimensional equation. The following assumptions are made in deriving the equations :

1. Vertical velocities are small = by this assumption the flow is treated to be nearly horizontal i.e., the pressure in the water is proportional to the depth (hydrostatic pressure distribution).

2. Width of the channel is small – this implies that the water level in the cross direction is horizontal. To give an idea, the width should be less than 10 km.

3. Density of water is constant. The equations that describes the water motion in tidal waves are :

- equation of continuity and - equation of motion

0=∂∂

+∂∂

tb

xQ η

(Continuity equation )

0122 =+

∂∂

+∂∂

RACQQ

gx

gtQ

Aη

( Equation of motion )

Inertia gravitation friction Where, Q = Discharge n = Water Level B = width of the cross section A = Cross sectional area G = Acceleration due to gravity C= Chezy’s constant (bed roughness) R = Hydraulic mean radius t = time X = Space co-ordinate (one dimension)

In the equation of motion we distinguish three terms, namely, the inertia term (acceleration of the flow), the gravitational term (slope of the water level) and



the friction term (bottom friction). For the analysis of the tidal waves the above three terms are important. It is not possible to solve the above equations analytically and the equations are solved numerically on a computer. The equations, however, can be simplified and solved analytically. The simplified equations that are derived from the complete equations describe what we call as harmonic waves. The harmonic waves are periodic in time, sinusoidal in shape and have amplitudes, which are much less than the water depth. The harmonic waves also propagate without any loss in their original shape. This is due to the fact that the equations neglect the friction. In nature, however, we will never find pure harmonic waves. In the shallow sea and estuaries we have to deal with friction, damping reflection etc.

The dimensions of the bay or an estuary have a certain relation to the tidal

wave length that will give rise to resonance of the tidal wave. The mode of oscillation in a bay or estuary that is closed at one end is governed by the ratio of the length of the tidal wave and the length of the bay. This will give rise to some times what is known as a standing wave. When standing wave is formed the tidal amplitude increases considerable between the open end and the closed end of the bay. The high waters and low waters will occur in the whole bay simultaneously. Resonance does not happen only due to the tidal wave that has periods of several hours (12h 25min). There are also periodic long waves of relative short periods ranging from 5 to 30 min and are called Seiches. These are caused by meteorological phenomenon like moving depressions. The amplitudes of Seiches are small compared to the water depth. This means that the friction can be neglected and the waves can be described in terms of the harmonic waves. It is important to note that in the design of harbour basins one should be aware of the resonance that can occur due to Seiches. The practical consequences of this are high velocities at the mouth of the harbour basin and presence of cross currents in the river and excessive vertical movement for moored ships.

The other practical application of tidal wave theory is the tidal motion in relatively short bays or estuaries or short harbour basins. The terms short basin is related to the parameter I/L, where l-length of the basin and L–length of the tidal wave. If I/L is less than 0.02 then the basin will be filled in horizontal layers. A small increase in the water level at the mouth can be seen as a disturbance and a small translation wave will run through the basin. The time required to propagate up to the end of the basin and back again is very small compared to the period of the tidal wave.

In many rivers/estuaries the incoming tidal wave is so much modified that a bore may develop during which the water surges up the river with a steep front. The phenomenon is associated with shallow depths and sudden increase in the gradient of the river. The tidal wave traveling up the shallow estuary breaks when its height above the surface of water is equal to the depth below the surface. If the speed of the first wave is reduced due to bottom friction, the succeeding wave over takes it resulting in the wave traveling with almost vertical crest. This phenomenon is known as bore.



So far we have considered the tidal motion in one horizontal dimension. In

large estuaries, seas and oceans however we have to deal with two-dimensional tidal motion. The tidal motion here is different from the cases we have considered until now.

- the flow is 2-dimensional - the rotation of the earth has effect on the flow.

The following three equations are required in describing the flow in this case:

- the equation of continuity - the equation of motion in x-direction - the equation of motion in y-direction

In the case of two-dimensional flow as compared to one-dimensional flow

extra terms would arise due to fluid friction along the walls, wind force and Coriolis force. The Coriolis force is caused by the rotation of the earth and is significant in oceans, seas and very wide estuaries. The rotation of the earth introduces additional acceleration force on the fluid and is directed to the right in the north hemisphere and to the left in the south hemisphere of the earth. The Coriolis force on the surface of the earth is a function of the latitude. The force is maximum near the poles and zero near the equator. With out going into the details of the two dimensional equations we will only examine the effect of the Coriolis force on the tidal system in seas and oceans.

In very wide rivers the cross slope of the water level in the rivers balances

the Coriolis force. For instance, if we consider a river of 500m width having velocity of 1m/sec and located at 50o latitude on the surface of the globe, the Coriolis force will cause a cross slope of 1 in 100000. This will give rise to a level difference of 0.5 cm in the cross direction and this will not have any influence on the main flow of the river. The situation, however, will change if the width is increased to several kilometers. For large widths the Coriolis force can cause cross velocities. If we consider large areas like the North Sea the gravity term and Coriolis term in the equations of motion can be of the same order of magnitude. In this case we have to deal with Kelvin waves (see Fig.6). These are progressive wave propagating along the walls (boundaries) and the amplitude of these waves is a function of the width (y). Near the wall the amplitude is highest and it decreases rapidly away from the wall. In practice the high waters and low waters are turning around in the basin. The rotation is anti-clock wise in the north-hemisphere and clockwise in the south-hemisphere. The nodal line is reduced to a nodal point and the water level at the nodal point is constant. This point is known as the amphidromic point. The wave system is called the amphidromic system. The lines of equal phases around the amphidromic point are known as the co-tidal lines. The lines of equal tidal ranges are known as the co-range lines. In nature we find many amphidromic systems. 1.5 Some examples of tide dominated areas :



India has a large coastline having tides of different ranges at various locations along the coast. The Fig.7, shows the Cotidal lines and tidal ranges at various locations along the Indian coastline. A brief description of the tidal phenomenon in the Gulf of Kutch, the Hooghly estuary and the Bay of Fundy (Canada) is given below: 1.5.1 Tidal Phenomenon in the Gulf of Kutch :

The Gulf of Kutch is situated in Gujarat coast about 600 km north of Mumbai. In connection with tidal power project enormous data were collected in respect of tide and tide induced current, both in the Gulf and in the offshore region. The positions of tide and current observations are shown in Fig.8. The offshore position is about 100 km away from the coast. The offshore tide as observed at locations 1 and 7 are shown in Fig.9(a) and 9(b). From the figures it may be noted that tides in this region are semidiurnal in nature with diurnal inequality giving rise to prominent difference between spring and neap tidal ranges. Simultaneous tide at location 1 and 7 are shown in Fig.10. Tides are symmetric during flood phase as well as ebb phase. Comparing tide at location 1 and 7 it may be seen that there is phase difference as well as amplification, when tide is progressing from offshore to inshore. Propagation characteristics of the tide at different position inside the Gulf are shown in the Fig.12. Due to funneling effect the tide gets amplified. The tidal range, which is about 3 m at Okha, is almost 6 m at Navlakhi. A typical simultaneous tide and velocity observation is shown in Fig.11. The peak velocity is not occurring exactly at the mid tide but about 1.1/2 hour after mid tide. 1.5.2 Tides in Hoogly estuary :

The Hooghly river shown in Fig.13 has tidal effect for a length of about 300 km. The river has a moderate tidal range and a considerable tidal length, with the result that when high water occurs at the mouth, the previous high water just reached the tidal limit. Tidal behavior in an estuary can be observed by recording the rise and fall of water simultaneously at several stations spread along its length. Some typical observations are reproduced in Fig.14.

The Hooghly, with a tidal range of about 5.5m during springs, has channel depths below low water that vary considerably, and averages about 6m over long distances. The variation of depth is between 6 and 12m during spring tides and causes considerable distortion of the tidal wave as shown in Fig.14. During neap tides, the tidal range can be about 2m, and the neap tide low water level is more than 1m above the spring tide low water. Depths during the neap tide vary between 7m and 9m, and as a result of this relatively small variation in depth, the tidal wave undergo only slight distortion as shown in Fig.15. The effect of distortion can also be seen in Fig.16, in which the tidal observations shown in Fig.14 have been also plotted to show simultaneous water surface profile along the estuary.

A further effect of tidal propagation deserves comment. The mean tide level at the sea-face at Saugor is shown in Fig.14 and 15 is almost same in each case,



but at stations further inland, the mean level during spring tides is higher than that observed during neaps. This effect can be so great that at some land ward stations, low water during neap tides can be lower than low water during springs, which is not the usual case. For example, at 204 km from Saugor, low water spring tide (LWS) is 2.8 m above datum. Were as the low water neap tide (LWN) is only 2.2m above datum. Evidently a large volume of water can accumulate in the upper part of an estuary during a spring tide cycle. This causes an increase of salinity during spring tides and a decrease during neaps. It also contributes to net land ward movement of sediment during spring tides and net seaward movement during neaps.

1.5.3 Tides in the Bay of Fundy (Canada):

As mentioned earlier the mode of oscillations in a bay or estuary, which is closed at one end, is governed by the ratio of the length of the tidal wave and the length of the bay. Resonance will take place if the length of the bay is 1/4th or 3/4th or 5/4th of the length of the tidal wave. The resonance of the wave some times results in the formation of a standing wave, which results in considerable increase in the tidal amplitude between the open and closed ends of the bay. An example of standing wave in which the length of the bay is about 1/4th of the wavelength is found in the Bay of Fundy in the east coast of Canada.

The Bay of Fundy is shown in Fig.17. The bay is connected to open sea and has a closed end. The tidal amplitude increases considerably between the open and closed end of the bay. The amplitude at the open end is about 1.5 m and that at the closed end is about 6m. The high waters and low waters occur in the whole bay almost simultaneously. Both the above features indicate a standing wave in resonance. The following calculations further confirm this aspect. Length of the (1) = 300 km. Average depth of the bay (h) = 75m The principle tide is M2 tide with a period of 12hrs 25min. (T) = 44,700 sec. The wave length (L) is given by the equation:

ThgcTL .,== This gives L = 1200 km REFERENCE : 1. Cmd.Macmillan, D.H.(1966) ‘TIDES’ CR Books Limited, London 2. Brown, Son & Ferguson, Ltd.(1958), ‘Manual of Tidal Prediction’, Nautical

Publishers, Glasgow. 3. Ippen, A.T. (1975) ‘Tidal Dynamics in Estuaries’, MIT 4. U.S.A CERC (1984) ` Shore Protection Manual’ 5. CWPRS Sp. Note No.2494, (1984) ` Feasibility Studies for the Tidal Power

Project in the Gulf of Katchch (Vol.I).



Fig.1 Example of a Harbour



Fig.4 Amplitude of tidal constitutes as function of frequency

Fig.6 Kelvin waves in the Northern hemisphere



Fig.7 Co-tidal lines and tidal ranges 9 feet) along the Indian coast



Fig.8 Location of field observation in the Gulf of Kuchch

Fig.9(a) Tide at station -1

Fig.9(b) Tide at station -7



Fig.10 Plot of simulation observations at station 1 & 7

Fig.11 Typical tide and current observations at S2 in the Gulf of Kuchch

Fig.12 Co-tidal range and Co-tidal lines for the Gulf of Kuchch



Fig.13 Plan of Hooghly river

Fig.14 Spring tide in Hooghly river



Fig.15 Neap tide in Hooghly river

Fig.16 Instantaneous water profiles along Hooghly Estuary

Fig.17 Bay of Fundy ( Canada)

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