Coastal Engineering Prof. K. A. Rakha Cairo University Faculty of Engineering 2013

Transcription

1 Coastal Engineering Prof. K. A. Rakha Cairo University Faculty of Engineering 2013

2 Table of Contents 1 Waves Description of Waves Wind and Waves Sea and Swell Small Amplitude Wave Theory Solving the Dispersion Equation Reflected Waves Short Term Wave Analysis Time Domain Analysis Short-Term Wave Height Distribution Frequency Domain Analysis Wave Generation Wave Hindcasting Wave Transformation Refraction and Shoaling Wave Diffraction Wave Breaking Wave Models Water Level Variations Astronomic Tides Equilibrium Tide (Moon) Daily Inequality Spring/Neap Tides Other Effects Tide Analysis and Prediction

3 2.1.6 Datums Storm Surge Barometric Surge Seiche Tsunami Eustatic (Sea) Level Change Isostatic (Land) Rebound and Subsidence Global Climate Change Currents in the Marine Environment Tidal Currents Wind Generated Currents Stratification and Density Currents Wave Induced Currents Shore-normal currents Shore-parallel currents Two-dimensional Currents Hydrodynamic Models Nearshore Sediment Transport Longshore Sediment Transport Predicting Potential Littoral Drift Littoral Drift Budget On/Offshore Sediment Transport Coastal Sediment Cells Sediment Transport Models Morphology and Shoreline Change Models One-line Models K.A.Rakha 2

4 4.5.1 Analytical Solution Model Classification according to Time and Space Reducing Uncertainty References... 1 K.A.Rakha 3

5 1 Waves Knowledge of waves and he forces they generate are essential for the design of coastal projects since they are the major factor that determines the geometr of beaches, the planning and design of marinas, waterways, shore protection measures, hydraulic structures, and other coastal works. 1.1 Description of Waves The subject of water waves covers phenomena ranging from capillary waves that have very short wave periods (order 0.1 seconds) to tides, tsunamis (earthquake generated waves) and seiches (basin oscillations), where wave periods are expressed in minutes or hours (Kamphuis, 2000). Wave heights also vary in height from a few millimeters for capillary waves to 10 s of meters for long waves. A classification by wave frequency of the various types of waves is given in Fig In the middle of the range of frequencies are the waves that are the focus of this chapter. They are normally known as gravity waves or windgenerated waves. Their periods range from 1 to 20 (to 30) seconds and their wave heights are seldom greater than 10 m. Yet, because of their prevalence, these waves account for most of the total available wave energy. Mangor (2004) divides waves into short waves and long waves with short waves of periods less than 20 second. Long waves are defined as the waves with periods ranging from 20 sec to 40 min and are divided into surf-beats, harbour resonance, seiche and tsunamis. Water level oscillations with periods or recurrence intervals larger than an hour such as astronomical tides and storm surge are referred to as water-level variations. The shape of a water surface subjected to wind is so complex that it almost defies description. Even when the first puffs of wind impact an otherwise flat water surface the resulting distortions present non-linearities that make rigorous analysis impossible. When the first ripples generated by these puffs are subsequently strengthened by the wind and interact with each other, the stage has been set for what is known as a confused sea. The waves will continue to grow ever more complex through processes known only to the sea itself. It is necessary to simplify the confusion and to use these simplified concepts in design. This chapter will establish a bridge from the confusing and complex sea state to theoretical expressions that are simple and can be used for most design purposes.

6 Waves Period 24 h 12 h 5 min 30 s 1 s 0.1 s Wave Transtidal Infragravity Ultragravity Long-period Capillary band Gravity Primary Storm systems, tsunamis disturbing Sun, Moon Wind force Primary Coriolis force Surface tension restoring Gravity force Energy (L 2 ) 24 h 12 h 5 min 30 s 1 s 0.1 s Time (s) Figure 1.1: Wave Classification by Frequency (after Kinsman, 1965). 1.2 Wind and Waves For theoretical analysis of wave generation, the reader is referred to more extensive references on this subject such as Dean and Dalrymple (1991), Dingemans (1997), Horikawa (1988), Ippen (1966), Kinsman (1965), Sarpkaya and Isaacson (1981), who discuss various theoretical models at length. In general, it may be said that wind speed and wave activity are closely related. There are other important variables to consider such as depth of water, duration of the storm and fetch (the distance over which the wind blows over the water and generates waves). At this stage only wind is considered and water depth, wind duration and fetch are assumed to be unlimited. The resulting waves are often called Fully Developed Sea and these conditions are approximated only in the deep, open sea. The relationship between wind and waves in the open sea is so predictable that sailors have for centuries drawn a close parallel between wind and waves. The Beaufort Scale in Table 1.1 is a formalized relationship between sea state and wind speed that can be used to obtain an estimate of waves in the open sea when wind speed is known. K.A.Rakha 1-2

7 Waves Table 1.1: Beaufort Scale of Wind And Sea State 1) Beaufort Wind Force Wind Speed (knots) 2) Description of wind Description of Sea Approx H s (m) ApproxT (sec) Calm Sea like a mirror Light airs Ripples are formed Light breeze Gentle breeze Moderate breeze Fresh breeze Strong breeze Moderate gale Fresh gale Strong gale Whole gale 3) Storm 3) Small wavelets, still short but more pronounced; crests have a glassy appearance, but do not break Large wavelets, crests begin to break. Perhaps scattered white caps. Small waves, becoming larger; fairly frequent white capping. Moderate waves, taking a more pronounced long form; many white caps are formed (chance of some spray). Large waves begin to form; the white foam crests are more extensive everywhere (probably some spray). Sea heaps up and white foam from breaking waves begins to be blown in streaks along the direction of the wind (spindrift). Moderately high waves of greater length; edges of crests break into spindrift. The foam is blown in well-marked streaks along the direction of the wind. Spray affects visibility. High waves. Dense streaks of foam along the direction of the wind. Sea begins to roll. Visibility affected. Very high waves with long overhanging crests. The resulting foam is in great patches and is blown in dense white streaks along the direction of the wind. On the whole, the surface of the sea takes a white appearance. The rolling of the sea becomes heavy and shocklike. Visibility is affected. Exceptionally high waves (small and medium sized ships might for a long time be lost to view behind the waves). The sea is completely covered with long white patches of foam lying along the direction of the wind. Visibility affected Hurricane 3) completely white with driving spray; Air filled with foam and spray. Sea visibility very seriously affected ) 20 4) 40 4) 22 4) 1) 2) 3) 4) Fully developed sea - unlimited fetch and duration. 1 knot 1.8 km/hr 0.5 m/s Required durations and fetches are seldom attained to generate fully developed sea. Really only a m deep interface between sea and air. K.A.Rakha 1-3

8 Waves 1.3 Sea and Swell Waves generated locally by wind are generally known as sea, which consists of waves of many different wave heights and periods as shown in the time series in Fig. 1.2 (irregular waves). The waves in Fig. 1.2 form what is called a wave train. The waves, on average propagate more or less in the wind direction. WL (m) Time (sec) Fig. 1.2: Record of Locally Generated Sea On large bodies of water, the waves will travel beyond the area in which they are generated. For example, waves generated by a storm off the French coast may travel southward and eventually arrive to Tunisia. While the waves travel such long distances, the energy of the individual waves is dissipated by internal friction and wave energy is transferred from the higher frequencies to lower frequencies. The resulting waves arriving in Tunisia will be more orderly than the initial sea, generated off France, with longer wave periods (10-20 sec) and smaller wave heights. Such waves, which are generated some distance away and travel into an area, are called swell (see Fig. 1.3). On most coasts, sea and swell occur simultaneously. The exceptions are enclosed bodies of water such as lakes, reservoirs and inland seas, where swell cannot arrive from long distances away. K.A.Rakha 1-4

9 Waves Fig. 1.3: Sea and Swell 1.4 Small Amplitude Wave Theory In this section a simplified method of representing wave motion will be introduced. It is called Small Amplitude Wave Theory. It may appear to be almost impossible to adequately represent locally generated, confused sea as in Fig It might also be expected that any simple theory would be more applicable to the more regular swell conditions. Yet over the years, it was found that for most problems there is no need to differentiate between sea and swell or to use a more complicated wave theory. Small Amplitude Wave Theory can be confidently applied to both sea and swell (Kamphuis, 2000). More complex wave theories have been developed, but they are normally used only for research and complex designs. For most straightforward designs small amplitude wave theory has been found sufficient. A wave is periodic if its motion and surface profile recur in equal intervals of time termed the wave period. A wave form that moves horizontally relative to a fixed point is called a progressive wave and the direction in which it moves is termed the direction of wave propagation. A progressive wave is called wave of permanent form if it propagates without experiencing any change in shape. The basis for small amplitude wave theory is the sinusoidal wave, shown in Fig Furthermore it is assumed that the ocean waves are two dimensional, small in amplitude, and progressively definable by their wave height and period for a given water depth. A right hand system of coordinates is used with its origin at still water level (SWL). The SWL is defined as the water surface that would exist in the absence of any wave action. The x axis is horizontal and parallel to the direction of wave propagation. The y axis is horizontal and perpendicular to the x axis. The z-axis is vertically up and therefore the K.A.Rakha 1-5

10 Waves position of the bottom is at z = -d. The highest point of the wave is the crest and the lowest point is the trough. The sinusoidal water surface η may be described by, 2x 2t = a cos (kx- t) a cos (1.1) L T where a is the amplitude of the wave, x is distance in the direction of wave propagation, t is time, k is the wave number (the angular frequency at which the wave pattern repeats itself in space), is the angular wave frequency (the angular frequency of repetition in time), L is the wave length and T is the wave period. The values of k and are calculated from, 2 2 k = ; = (1.2) L T The maximum vertical distance between crest and trough of the wave is called the wave height, H(=2a). Since in an actual wave train, such as in Fig. 1.2, the wave heights and lengths are not all the same, statistical representative values are used. The ratio of wave height to wave length (H/L) is called wave steepness. The wave form moves forward and the velocity of propagation of the wave (or phase speed) is calculated from, L C = (1. 3) T Mean water level (MWL) is defined as the level midway between wave crest and trough. In small amplitude wave theory, MWL is the same as SWL, but for higher order wave theories MWL will be above SWL Further, waves are differentiated as Long-Crested or Short-Crested which refers to the length of the wave crest perpendicular to the wave shape and its velocity of propagation. Swell is normally long crested (the wave is recognizable as a single crest over a hundred meters or so) and Sea is normally short crested. Waves are considered to be in deep water when d/l > 0.5 and in shallow water when d/l >.0.0. Between these limiting conditions, the water depth is called transitional. The Small Amplitude Wave Theory expressions are summarized in Table 1.2. Equation [1] (equation numbers in square brackets refer to those in Table 1.2) describes the water surface fluctuation as shown in Fig Equation [2] calculates the velocity of propagation, C, assuming the wave retains a constant form. The 'tanh' term has two asymptotic values. For large depths, kd (or d/l) is large resulting in, For small depths, 2d tanh kd= tanh 1 (1.4) L 2 d 2 d tanh kd= tanh ( ) ( ) (1. 5) L L Thus, it is possible to give deep and shallow water asymptotic values for C as in Table 1.2. It has been customary to define deep water as d/l>0.5 (tanh kd = 0.996) and shallow water is usually defined as d/l<0.05 (kd = 0.592, while tanh kd = 0.531). K.A.Rakha 1-6

11 Waves c L Crest z x H a=h/2 SWL d Trough z = -d Fig. 1.4: Sinusoidal Wave and Wave Parameters Waves propagate at velocity C, but the individual water particles do not propagate; they move in particle orbits as shown in Fig For small amplitude wave theory, such particle orbits are elliptical and if the water is 'deep', they become circular. Their size decreases with depth. Horizontal and vertical orbital velocity components, u and w, and orbit semi-axes, A and B, are given in Eqs. [4] to [7]. K.A.Rakha 1-7

12 Waves Table 1.2: Common Expressions for Linear Progressive Waves Parameter 1. Water Surface General H = cos 2 (kx- t) w Deep (d/l > 0.5) Shallow (d/l < 0.05) 2. Velocity of Propagation (Dispersion Equation) L gt C = = = tanh kd T k 2 gl = tanh kd 2 gt C o = C = gd 2 gt 3. Wave Length L = CT = tanh kd 2 H coshk(z+ d) 4. Horizontal u = cos Orbital Velocity T sinhkd 2 w u o 2 gt Lo = L CT 2 H = T o e k o z cos w H u = 2 g d cos w 5. Vertical Orbital Velocity H w = T sinhk(z+ d) sinhkd sin w w o H = T o e k o z sin w w = H 1 T z d sin w 6. Horizontal Semi- Axis 7. Vertical Semi- Axis A = B = H coshk(z+d) 2 sinh kd H sinh k(z+d) 2 sinh kd p 8. Pressure = - z+kp g 9. Pressure Response Factor K p coshk(z+d) = cosh kd A o H = 2 B K o = A e k o z HT A = 4 B = 1 o o p = e g d H z 2 d k o z Kp = Energy Density 1 E gh Wave Power P = EC o ECG P = o 2 P = EC 12. Group Velocity CG = n C C G o C = 2 o CG = C 13. Group Velocity Parameter 1 2 kd n = 1+ 2 sinh2 kd 1 n o = n = 1 2 K.A.Rakha 1-8

13 Waves A A B B SWL Elliptical Orbits A > B w = 0 u > 0 Bottom z = -d Fig. 1.5: Orbital Motion of Particles. The pressure fluctuations at any point below the water surface are related to the water level fluctuations at the surface. If the wave were infinitely long, the water level would be horizontal at any time and the pressure fluctuations would be hydrostatic. The pressure fluctuation would be (gh), where is the fluid density and g is the gravitational acceleration. For waves of limited length the pressure fluctuations are smaller than (gh). The ratio of the actual pressure fluctuations to (gh), is called the pressure response factor, K p, and it is a function of wave length (or wave period) and depth below the surface. For longer waves or for locations close to the water surface, the pressure response factor approaches 1. For shorter waves or for locations far below the water surface, the pressure response factor approaches zero. Eqs. [8] and [9] quantify the pressure response. K.A.Rakha 1-9

14 Waves Wave Energy is expressed per unit surface area as Energy Density, E, in joules/m 2 as in Eq. [10]. It is made up of half Potential Energy and half Kinetic Energy. Eq. [11] gives Wave Power, P, arriving at any location. Its units are watts/m of wave crest. Eq. [2] indicates that longer period waves travel faster than shorter period waves. A real wave train, as in Fig. 1.2, contains many different wave periods and therefore it would stretch out (disperse) as it traveled. The longest waves would lead and run further and further ahead with time and distance, while the shortest waves would lag further behind. Hence Eq. [2] is called the Dispersion Equation. Equation [2] also means that waves of roughly the same period tend to travel together. Waves of almost the same period interfere to form beats or wave groups, resulting in two wave speeds involved: the speed of the individual waves given by Eq. [2] and the speed of the wave group, which is C multiplied by the factor n, given in Eq. [13]. In deep water n approaches ½ and in shallow water n approaches 1. Thus C G <C, but in very shallow water C G approaches C Solving the Dispersion Equation To solve Eq. [2] and all the other equations in Table 1.2, it is necessary to know the wave length, L, which may be calculated using Eq. [3]. However, Eq. [3] is implicit and can only be solved numerically. Tables of solutions have been prepared that yield L as well as other important wave characteristics (see Table 1.3). Such tables are known as Wave Tables and have been published in Shore Protection Manual (l984) and Wiegel (l964). To use the wave tables, the deep water approximation of wave length is first calculated as given by Eq. [3]. Then using the depth of water, d, it is possible to enter the wave tables with d/l o to evaluate all the remaining wave parameters. The use of the wave table is suitable for only a few calculations. For a large number of calculations, L or C may be calculated using a root finding technique such as Newton-Raphson, but such a technique requires iteration. To speed up such computations, approximations may be used such as the one proposed by Hunt (1979), = gd C y +( y y y y ) (1. 6) where 2 d y = Lo (1. 7) K.A.Rakha 1-10

15 Waves Table 1.3: Wave Table d L 0 tanh kd d L kd sinh kd cosh kd 2kd sinh2 kd K s K.A.Rakha 1-11

16 Waves d L 0 tanh kd d L kd sinh kd cosh kd 2kd sinh2 kd K s K.A.Rakha 1-12

17 Waves Example 1.1 In this example the small amplitude wave parameters given in Table 1.2 are calculated for a wave of period, T = 10 sec, with a wave height, H = 1.5 m in a depth of water, d = 9.4 m. First, it is necessary to calculate the deep water wave length and relative depth: L o 2 gt 1.56T (100) 156m; The wave table (Table 1.3) yields the following: d L o d = ; tanhkd = ; L sinhkd = ; coshkd 1.22 ; n = 0.5*( ) From the value of L d, the wave length in 9.4 m of water and wave number, k, may now be calculated: d L 90.4m; k L From these, the following parameters may be computed; is assumed to be 1035 kg/m 3 for sea water. At the bottom: L C = = 9.04 m/s; T 2 gh E C G j/m; nc 0.881(9.04) 7.96 m/s; P EC G 22,730 w/m of wavecrest z d; k( z d) 0; sinh k( z d) 0; coshk ( z d) 1.0 and the horizontal component of orbital velocity is: u B H T 1 sinh (1.5) cos( kxt) kd 10 1 cos( kxt) 0.67cos( kxt).703 Thus, at the bottom, u B has a maximum value u B = 0.67 m/s and the vertical velocity component of orbital motion at the bottom is zero. The amplitude of the orbital motion at the bottom is K.A.Rakha 1-13

18 Waves H AB= 2 sinh 1.5 = =1.07 m. kd 2(0.703) and the orbital diameter is 2A B = 2.14 m. The pressure response factor K p at the bottom is: ( K ) p B = 1 1 = cosh kd 1.22 =0.82 which means that the maximum pressure fluctuation caused by the wave height H = 1.5 m is: K p H =0.82 ( 1.5) =1.23 (m of water) K.A.Rakha 1-14

19 Waves 1.5 Reflected Waves When a wave reaches a rigid, impermeable vertical wall the wave is completely reflected. After some time, under well controlled conditions, the reflected waves and the incident waves together form a system of waves whose form no longer moves forward in space, commonly known as a standing wave. A theoretical expression for such a standing wave, as shown in Fig. 1.6, may be obtained by superposition of the equations for an incident and a reflected wave. It may be seen that the pattern repeats itself every half wave length and that the first location of the maximum wave height (antinode) is at the structure, while the first location of zero wave height (node) is located L/4 from the wall. The maximum wave height is twice the height of the original incident wave. At t = 0, T,... L At t = T/2, 3T/2,... Node Node Anti-Node Anti-Node Fig. 1.6: Standing Waves K.A.Rakha 1-15

20 Waves Partial wave reflection will result if the reflecting surface is sloping, flexible or porous and yields a variation in wave height. The partial antinodes (H max ) are less than twice the incident wave height, while the partial nodes (H min ) are greater than zero. 1.6 Short Term Wave Analysis In the first part of this chapter, waves on the sea surface were assumed to be nearly sinusoidal with constant height, period and direction (i.e., monochromatic waves). Visual observation of the sea surface and measurements indicate that the sea surface is composed of waves of varying heights and periods moving in differing directions. In the first part of this chapter, wave height, period, and direction could be treated as deterministic quantities. Once we recognize the fundamental variability of the sea surface, it becomes necessary to treat the characteristics of the sea surface in statistical terms. This complicates the analysis but more realistically describes the sea surface. The term irregular waves will be used to denote natural sea states in which the wave characteristics are expected to have a statistical variability in contrast to monochromatic waves, where the properties may be assumed constant. Monochromatic waves may be generated in the laboratory but are rare in nature. In analysis of wave data, it is important to distinguish between Short-Term and Long-Term wave analysis. Short-Term analysis refers to analysis of waves that occur within one wave train or within one storm; Long-Term analysis refers to the derivation of distributions that cover many years. This section deals with short term wave analysis. Two approaches exist for short term analysis of irregular waves: spectral methods and wave-by-wave (wave train) analysis. Spectral approaches are based on the Fourier Transform of the sea surface. This analysis is usually called frequency domain analysis since the wave spectra is used rather than a time series. Indeed this is currently the most mathematically appropriate approach for analyzing a time-dependent, three-dimensional sea surface record. Unfortunately, it is exceedingly complex and at present few measurements are available that could fully tap the potential of this method. However, simplified forms of this approach have been proven to be very useful. The other approach used is wave-by-wave analysis. In this analysis method, a time-history of the sea surface at a point is used, the undulations are identified as waves, and statistics of the record are developed. This method is used called the time domain analysis since it deals with a time series of the water surface. The primary drawback to the wave-by-wave analysis is that it cannot tell anything about the direction of the waves. Indeed, what appears to be a single wave at a point may actually be the local superposition of two smaller waves from different directions that happen to be intersecting at that time. Disadvantages of the spectral approach are the fact that it is linear and can distort the representation of nonlinear waves Time Domain Analysis In the time-domain analysis of irregular or random seas, wave height and period, wavelength, wave crest, and trough have to be carefully defined for the analysis to be performed. The definitions provided earlier in the regular wave section of this chapter assumed that the crest of a wave is any maximum in the wave record, while the trough can be any minimum. However, these definitions may fail when two crests occur within an K.A.Rakha 1-16

21 Waves intervening trough lying below the mean water line. Also, there is not a unique definition for wave period, since it can be taken as the time interval between either two neighbouring wave troughs or two crests. Other more common definitions of wave period are the time interval between successive crossings of the mean water level by the water surface in a downward direction called zero down-crossing period or zero up-crossing period for the period deduced from successive up-crossings. Using these definitions of wave parameters for an irregular sea state, the periods and heights of irregular waves are not constant with time, changing from wave to wave. Waveby-wave analysis determines wave properties by finding average statistical quantities (i.e., heights and periods) of the individual wave components present in the wave record. Wave records must be of sufficient length to contain several hundred waves for the calculated statistics to be reliable. Average statistical representations for an irregular sea state may be defined in several ways. These include the mean height H, the root-mean-square height, and the mean height of the highest one-third of all waves known as the significant height. Among these, the most commonly used is the significant height, denoted as H s or H 1/3. Significant wave height has been found to be very similar to the estimated visual height by an experienced observer (Kinsman, 1965). The average of the highest 10% (H 0.1 ) or the highest 1% (H 0.01 ) is also sometimes used for design purposes. The average statistical period could be the mean period, or average zero-crossing period, etc Short-Term Wave Height Distribution The heights of individual waves may be regarded as a stochastic variable represented by a Probability Distribution Function (PDF). From an observed wave record, such a function can be obtained from a histogram of wave heights normalized with the mean heights in several wave records measured at a point. The Rayleigh distribution was found to be the most suitable distribution for representing wave heights within a storm (short term). Figure 1.7 shows the Rayleigh distribution (p curve) together with the cumulative Rayleigh distribution (P curve). Equation (1.8) provides the equation for the Rayleigh PDF, H p H H H exp 2 H 4 H 2 (1. 8) The Cumulative Distribution Function (CDF) of wave heights based on the Rayleigh distribution (the probability that any individual wave of height H' is not higher than a specified wave height H) can be written as, H P H H exp 4 H 2 (1. 9) The Rayleigh distribution is generally adequate, except in shallow water in which it may overestimate the number of large waves. Investigations of shallow-water wave records from numerous studies indicate that the distribution deviates from the Rayleigh, and other distributions have been shown to fit individual observations better (SPM, 1984). The K.A.Rakha 1-17

22 Probability Waves primary cause for the deviation is that the large waves suggested in the Rayleigh distribution break in shallow water. Using the Rayleigh distribution the following relationships can be derived: H H 0.1 =1.27 H =0.63 H s s H H 0.01 rms =1.67 H =0.707 H s s (1. 10) H / H p P Fig. 1.7: Rayleigh distribution Frequency Domain Analysis Considering a single-point time-history of surface elevation, spectral analysis proceeds from viewing the record as the variation of the surface from the mean and recognizes that this variation consists of several periodicities. In contrast to the wave-by-wave approach, which seeks to define individual waves, the spectral analysis seeks to describe the distribution of the variance with respect to the frequency of the signal. By convention, the distribution of the variance with frequency is written as S(f) with the underlying assumption that the function is continuous in frequency space (see Fig. 1.8). The reason for this assumption is that all observations are discretely sampled in time, and thus, the analysis should produce estimates as discrete frequencies which are then statistically smoothed to estimate a continuum. S(f) is known as the Wave Variance Spectral Density Function or Wave Spectrum. Variance is a statistical term and it is preferable to develop a physical explanation for the wave spectrum. This is attained by using the frequency energy spectrum E(f). Assuming linear wave theory valid, the energy of the wave field may be estimated by multiplying S(f) by ρg. K.A.Rakha 1-18

23 Waves The surface can be envisioned not as individual waves but as a three-dimensional surface, which represents a displacement from the mean and the variance to be periodic in time and space. The simplest spectral representation is to consider E(f,θ), which represents how the variance is distributed in frequency f and direction θ. E(f,θ) is called the 2-D or directional energy spectrum because it can be multiplied by ρg to obtain wave energy. The advantage of this representation is that it tells the engineer about the direction in which the wave energy is moving. The different wave height statistics (e.g. significant wave height) can be determined from the moments of the wave spectra. The moments of the wave spectrum are defined as, m n = f f 0 f n S(f) df The zero moment is therefore the area under the spectrum (1. 11) m o = f f 0 S(f) df = 2 f (1. 12) From the area under the wave spectrum, assuming the wave height distribution to be Rayleigh, the various wave heights may be estimated. To distinguish between significant wave height (derived from time domain analysis) and its counterpart, derived from frequency analysis, the latter is called the Characteristic Wave Height or Zero Moment Wave Height. H ch= H mo= 4 f (1. 13) The representation of the wave energy distribution with frequency is a large improvement over the time-domain analysis methods discussed earlier. With this information it is possible to study resonant systems such as the response of drilling rigs, ships' moorings, etc. to wave action, since it is now known in which frequency bands the forcing energy is concentrated. It is also possible to separate sea (shorter period waves) and swell (longer period waves) via the wave spectrum, when both occur simultaneously. Since there are many wave frequencies (or wave periods) represented in the spectrum it is usual to characterize the wave spectrum by its peak frequency f p, the frequency at which the spectrum displays its largest variance (or energy). The peak period may be defined as, 1 Tp = (1. 14) f p Since the measured spectra show considerable similarity, a number of attempts have been made to formulate parametric expressions. One commonly used spectrum in wave hindcasting and forecasting projects is the single-parameter spectrum of Pierson- Moskowitz PM (Pierson and Moskowitz 1964). An extension of the PM spectrum is the JONSWAP spectrum (Hasselmann et al. 1973, 1976). K.A.Rakha 1-19

24 Waves S(f) (m 2 /Hz) Area = 2 f f p f (Hz) 1.7 Wave Generation Fig. 1.8: Wave spectra. When a gentle breeze blows over water, the turbulent eddies in the wind field will periodically touch down on the water, causing local disturbances of the water surface. The wind speed must be in excess of 0.23 m/s to overcome the surface tension in the water. Theory (Phillips, 1957) shows wind energy is transferred to waves most efficiently when they both travel at the same speed. But wind speed is normally greater than the wave speed. For this reason the generated waves will form as an angle to the wind direction so that the component of wind speed in the direction of wave propagation approaches the wave speed. The generated wave crests are short crested, irregular waves. Once the initial wavelets have been formed and the wind continues to blow, energy is transferred from the wind to the waves. Much of the wind energy is transferred to the higher frequency waves, i.e., the wind causes more ripples to form on top of existing waves, rather than increasing the size of the larger waves directly by shear and pressure differences. This pool of high frequency energy is then transferred to lower frequencies by the interaction of the high frequency movement with the adjacent slower moving water particles. This wave-wave interaction transfers wave energy to the lower frequencies of the wave spectrum. Earlier we stated that wave height and wave period is closely related to wind speed. It should therefore be possible to derive wave conditions from known wind conditions. In fact, it should be possible to reconstruct a wave climate at a site from historical, measured wind records. Such a computation is known as Wave Hindcasting. K.A.Rakha 1-20

25 Waves Wave Hindcasting For most locations, it is difficult to find long term wave data that is essential for the design of any coastal project. Hindcasted wave data is usually used for such purposes Parametric Methods The theory of wave generation has had a long and rich history. Beginning with some of the classic works of Kelvin (1887) and Helmholtz (1888) in the 1800's, many scientists, engineers, and mathematicians have addressed various forms of water wave motions and interactions with the wind. In the early 1900's, the work of Jeffreys (1924, 1925) hypothesized that waves created a "sheltering effect" and hence created a positive feedback mechanism for transfer of momentum into the wave field from the wind. However, it was not until World War II that organized wave predictions began in earnest. During the 1940's, large bodies of wave observations were collated and the bases for empirical wave predictions were formulated. Sverdrup and Munk (1947) presented the first documented relationships among various wave-generation parameters and resulting wave conditions. The method was later extended by Bretschneider (e.g., Bretschneider, 1958) to form the empirical method, now known as the SMB Method. The method is described fully in Shore Protection Manual (1977). In Shore Protection Manual (1984) this method was replaced by the Jonswap Method, based on research on wave spectra in growing seas by Hasselmann et al (1973). The Jonswap, SMB and similar methods are called parametric methods because they use wind parameters to produce wave parameters, rather than develop a detailed description of the physics of the processes. Although, these methods produce only Significant Wave Height (Hs) and Significant Wave Period (Ts), they may be extended to provide estimates of the parametric wave spectra. Waves are not only a response to wind speed (U). Wind direction (θ) determines the general direction of wave travel (wind and wave directions are defined as the directions from where they come). Fetch (F), the distance over which the wind blows over the water to generate the waves, is important. Storm duration (t) is important and finally the depth of water in the generating area (d) influences the wave conditions through bottom friction. Parametric wave hindcasting derives H and T from U, F, t and d. The wave direction is usually assumed to be the wind direction. This assumption can be a source of substantial errors in wave direction, that will result in large errors in the computation of responses such as alongshore sediment transport rate. If F, t and d are all infinite, the result is a Fully Developed Sea. The waves are fully developed so that any added wind energy is balanced by wave energy dissipation rate resulting from internal friction and turbulence. In that case, the resulting wave conditions are a function of wind speed only, as described by the Beaufort Scale (Table 1.1). When F, t or d are limited, the resulting waves will be smaller. The Jonswap method of wave hindcasting uses the following dimensionless expressions. gt * gf * gh mo * p * gt * gd =, H mo =, T p =, t =, d = (1. 15) 2 2 U U U U U F 2 K.A.Rakha 1-21

26 Waves These are dimensionless versions of fetch length, characteristic (zero moment) wave height, peak period of the spectrum, storm duration and depth of water. Note that F, H, and d are in metres, t and T are in seconds and U is in m/sec. The Jonswap relationships are: 1 * * 2 H mo (F ) (1. 16) and 1 * * 3 T p = (F ) (1. 17) 2 * * 3 t 68.8 ( F ) (1. 18) Three different conditions must be distinguished for waves generated in deep water. They can be Fetch Limited, Duration Limited or Fully Developed Sea. On a small water body, the waves would be limited by a short fetch and H mo and T p can be calculated directly from Eqs and On a larger body of water, the same equations apply, but wind duration may limit the size of waves. Eq is then used to calculate an effective fetch (the fetch needed to produce the same wave height if the duration had been infinite) F * eff * t = /2 (1. 19) When F * < F eff *, the waves are fetch limited and Eqs and 1.15 are used with F * ; when F eff * < F * the waves are duration limited and Eqs and 1.15 are used with F eff *. Finally, for a large body of water and a large duration a fully developed sea exists, which is calculated using the following upper limits: * * * H mo = ; T p = ; t =71,500 (1. 20) The procedure of computing H mo and T p by Jonswap has been published as a nomogram in the Shore Protection Manual (1984) Numerical Models For many applications, the above simplistic hindcast methods are good enough for first estimates especially of maximum conditions. However, for many applications, it is necessary to have a long-term hindcast wave climate relating waves to wind at hindcast intervals which usually are 1 hour, 3 hours or 6 hours. For this purpose, numerical models are used. These models can be one dimensional 1D as explained in Kamphuis (2000) or two dimensional 2D. K.A.Rakha 1-22

27 Waves Two dimensional models calculate the spectral wave fields over large areas. The WAM model (WAMDI, 1988), and the Wavewatch (Tolman, 1991) are examples of such models. Figure 1.9 provides a sample of the wave field calculated over the Mediterranean Sea at a certain instant using the Wavewatch model (Eldeberky et al., 2002). Fig. 1.9: Sample of Wavewatch results over the Mediterranean (Eldeberky et al. 2002). These models can be run in forecast mode using wind forecasted over the water body. Many centers world wide sell hindcasted data obtained from advanced offshore wave models (e.g. British Met Office BTO). Many other centers provide wave forecasts based on advanced offshore wave models (e.g. These forecasts range from global forecasts to local forecasts. 1.8 Wave Transformation Coastal engineering considers problems near the shoreline normally in water depths of less than 20 m. The study of shoreline change and beach protection frequently requires analysis of coastal processes over entire littoral cells, which may span over tens of kilometres. K.A.Rakha 1-23

28 Waves Wave data are generally not available at the site or depths required. Often a coastal engineer will find that data have been collected or hindcast at sites offshore in deeper water or nearby in similar water depths. Thus it is essential in such case to transform the waves from offshore or nearby locations to nearshore locations. Waves propagating through shallow water are strongly influenced by the underlying bathymetry and currents. A sloping or undulating bottom, or a bottom characterized by shoals or underwater canyons, can cause large changes in wave height and direction of travel. Shoals can focus waves, causing an increase in wave height behind the shoal. Other bathymetric features can reduce wave heights. The magnitude of these changes is particularly sensitive to wave period and direction and how the wave energy is spread in frequency and direction. In addition, wave interaction with the bottom can cause wave attenuation. Wave height is often the most significant factor influencing a project. Designing with a wave height that is overly conservative can greatly increase the cost of a project and may make it uneconomical. Conversely, underestimating wave height could result in catastrophic failure of a project or significant maintenance costs. Approaches for transforming waves are numerous and differ in complexity and accuracy. Processes that can affect a wave as it propagates from deep into shallow water include: Refraction. Shoaling. Diffraction. Dissipation due to friction. Dissipation due to percolation. Breaking. Additional growth due to the wind. Wave-current interaction. Wave-wave interactions The first three processes are propagation effects because they result from convergence or divergence of waves caused by the shape of the bottom topography, which influences the direction of wave travel and causes wave energy to be concentrated or spread out. Diffraction also occurs due to structures that interrupt wave propagation. The dissipation and breaking processes are sink mechanisms because they remove energy from the wave field through dissipation. The wind is a source mechanism because it represents the addition of wave energy if wind is present. The presence of a large-scale current field can affect wave propagation and dissipation. Wave-wave interactions result from nonlinear coupling of wave components and result in transfer of energy from some waves to others Refraction and Shoaling Wave shoaling is the change in wave height due to the change in water depth. Refraction is the turning of the direction of wave propagation when the wave front travels at an angle with the depth contours in shallow water. The refraction is caused by the fact that the K.A.Rakha 1-24

29 Waves waves propagate more slowly in shallow water than in deep water. A consequence of this is that the wave fronts tend to become aligned with the depth contours. The wave-propagation problem can often be readily visualized by construction of wave rays. If a point on a wave crest is selected and a wave crest orthogonal is drawn, the path traced out by the orthogonal as the wave crest propagates onshore is called a ray (Fig.1.10). Wave Ray Wave Crest b Shoreline Fig. 1.10: Wave Rays for straight and parallel contours. Wave Refraction causes the waves to be focused on headlands or over shoals (Fig. 1.11). In bays or submarine canyons the wave energy is reduced due to refraction. K.A.Rakha 1-25

30 Waves Bay Head land Bay Contours Orthogonals Fig. 1.11: Wave refraction at headlands and in bays. K.A.Rakha 1-26

31 Waves Assuming energy flux is conserved between the wave rays, nceb constant (1. 21) This equation can be reduced to the following (see Dean and Dalrymple, 1991), H K K H (1. 22) s r o Where the shoaling coefficient K s can be calculated from, K s noco 1 (1. 23) nc 2ntanh kd The refraction coefficient Kr is calculated from, bo Kr (1. 24) b Straight and Parallel Contours For straight and parallel contours Snell s law can be used to determine the wave direction α at any depth based on the deep water wave direction, sin sin C o (1. 25) C o Where the subscript o denotes deep water conditions. Equation (1.24) can also be simplified to be, cos K o r (1. 26) cos Wave Diffraction. Wave diffraction is a process of wave propagation that can be as important as refraction and shoaling. The classical introduction to diffraction treats a wave propagating past the tip of a breakwater (see Fig. 1.12). In Fig Region I would not include any waves if diffraction did not occur. The spilling of energy across the wave rays into the shadow zone is termed as diffraction. Any process that produces an abrupt or very large gradient in wave height along a wave crest also produces diffracted waves that tend to move energy away from higher waves to the area of lower waves. Thus initial wave energy is reduced as diffracted waves are produced. Refraction and diffraction of course take place simultaneously in most cases and therefore the above distinction is an academic separation of two closely related processes. K.A.Rakha 1-27

32 Waves Wave crest L Region I (Perfect calm) Region II Breakwater Region III No diffraction Breakwater With Diffraction Effects Fig. 1.12: Wave diffraction at the tip of a breakwater. K.A.Rakha 1-28

33 Waves Figure 1.13 shows the diffraction of irregular waves in a port obtained using a numerical model. Such models are important tools for the design of new ports. Fig. 1.13: Wave diffraction in a port using a short wave model (Mangor, 2004). For simple harbours with small changes in depth it is possible to use diffraction templates (SPM, 1984). For more complex situations numerical models that include refraction and diffraction need to be used as discussed later Wave Breaking Wave shoaling causes wave height to increase to infinity in very shallow water. There is, however, a physical limit to the height of the waves: the ratio of wave height to wave length or the wave steepness (H/L). When this physical limit is exceeded, the wave breaks and dissipates its energy. At this point Eq. (1.21) is no longer valid. Wave shoaling, refraction and diffraction transforms waves from deep water to the point where they break and then their wave height begins to decrease markedly, because of energy dissipation. The sudden decrease in the maximum value of wave height defines the breaking point and determines the breaking parameters (H b, and d b ). The breaker type is a function of the beach slope (m) and the wave steepness (H/L). Spilling breakers, occur on flat beach slopes as shown in Fig In spilling breakers (Fig. 1.15), the wave crest becomes unstable and cascades down the shoreward face of the wave producing a foamy water surface. Several wave crests may be breaking simultaneously, giving the appearance of several rows of breaking waves throughout the breaking zone. K.A.Rakha 1-29

34 Waves Plunging breakers occur on steeper beaches. In plunging breakers, the crest curls over the shoreward face of the wave and falls into the base of the wave, resulting in a high splash. They are, for example, predominant when swell breaks on flat sandy beaches. They are also the most common breaker type in hydraulic model studies, in which the beach steepness is often exaggerated. Collapsing breakers occur on steep beaches. In collapsing breakers the crest remains unbroken while the lower part of the shoreward face steepens and then falls, producing an irregular turbulent water surface. Surging breakers occur on very steep beaches. The waves simply surge up and down the beach and there is very little or no breaking. Many studies have been performed to develop relationships to predict the wave height at incipient breaking H b. Several of these formulas are available in Kamphuis (1991) including criterion for irregular waves. The simplest of these formulas is the solitory wave criterion, Hb b 0.78 (1. 27) d b Where γ b is the breaker index. K.A.Rakha 1-30

35 Waves Spilling breaker Air entrainment Very flat beach slope Plunging breaker Steep beach slope Surging breaker Very steep beach slope Fig. 1.14: Breaker types. K.A.Rakha 1-31

36 Waves Spilling Plunging Fig. 1.15: Photos of spilling and plunging breakers. K.A.Rakha 1-32

37 Waves 1.9 Wave Models Several types of short wave models exist and are applied to different applications. These models include some of the processes discussed earlier. These processes will not all dominate or exist at a certain location as shown in Fig Thus, different models exist that include the relevant physical processes for certain applications. It is essential to select the most suitable model for a certain application by determining the important physical processes involved. Then the suitable model is selected to perform the calculations accurately and efficiently. Battjes (1994) classified wave models into phase-averaged and phase-resolving models. Figure 1.17 provides a chart of the different types of models used in the coastal environment. The Boussinesq type of models are usually used for harbour agitation studies (see Fig.1.13). Such models include refraction, diffraction and wave-wave interaction in shallow water. The FUNWAVE Model is an example of a 2D Bousinesq model available at the University of Delaware The Mild Slope Equations MSE include refraction and diffraction (the elliptic form includes reflection also) and are thus commonly used for modeling areas where both refraction and diffraction are important. The REFDIF Model is an example of a parabolic MSE model available at the University of Delaware. Spectral wave models are used for the transformation of wave spectra from deep water to the shallow area. Such models do not include diffraction. The SWAN model, STWAVE part of CEDAS package), and NSW (part of the MIKE21 package) models are examples of such models. K.A.Rakha 1-33

38 Waves Process Oceans Shelf Seas Nearshore Harbours Diffraction Refraction and Shoaling Wind Input White capping Depth-Breaking Bottom friction Reflection Wave-wave interaction Blank dominant significant but not dominant of minor Importance negligible Fig. 1.16: Different wave processes relevant for different marine and coastal applications. K.A.Rakha 1-34

39 Waves Phase Averaged Phase Resolving Snell's Law Ray Tracing Domain Refraction Spectral Mild Slope Equation Boussinesq Others Rcpwave Approximation Parabolic Approximation Hyperbolic Approximation Fig. 1.17: Different types of wave models used for different applications. K.A.Rakha 1-35

40 2 Water Level Variations Although the design of structures is normally considered to be a function of wave conditions, water levels are also very important. A structure close to shore that is subject to waves will be exposed to larger waves for higher water levels because the water depth determines where waves break. This results in increased forces on the structure and overtopping of water that will damage the structure and areas behind it. Conversely, when the water level drops, the same structure may not be exposed to waves at all. Thus most damage to structures occurs when the water levels are high. Similarly, high water levels cause retreat of sandy shores, even if they are backed by substantial dunes. The higher water levels allow larger waves to come closer to the shore. These waves will erode the dunes and upper beach and deposit the sand offshore. If the water level rise is temporary, most of this loss will be regained at the next low water. Permanent water level rise, however, will result in permanent loss of sand. Shorelines consisting of bluffs or cliffs of erodable material are continuously eroded by wave action. High water levels, however, will allow larger waves to attack the bluffs directly, causing a temporary rapid rate of shoreline recession. According to Kamphuis (2000), there are several types of water level fluctuations and they can be classified according to their return period as: Short Term Astronomic Tides Storm Surge Seiche Long Term Eustatic (Sea) Level Rise Isostatic (Land) Emergence and Subsidence Climate Change Other short term water level changes such as wave setup will be discussed in the next chapter. 2.1 Astronomic Tides Astronomic tides are observed as the periodic falling and rising of the water surface for major water bodies on the earth. Astronomic tides are the result of a combination of forces acting on individual water particles. The main forces are: Gravitational attraction of the earth,

41 Water Level Variations Centrifugal force generated by the rotation of the earth moon combination, Gravitational attraction of the moon, Gravitational attraction of the sun. Because of its relative closeness, the moon induces the greatest effect on the tides Equilibrium Tide (Moon) Kamphuis (2000) considered only the first three forces (neglecting the force of the sun) and assumed that the whole earth is covered with water to describe the tidal movement. The resultant force on the water particles can be shown to be a small horizontal force that moves the water particle A in Fig. 2.1 toward the moon and particle B away from the moon, resulting in two bulges of high water, (Defant, 1961; Ippen, 1966). As we turn with the earth s angular velocity ω E around the earth's axis at C E in the direction of the arrow, we turn through this deformed sphere of water and experience two high water levels and two low water levels per day. The resulting tidal period would be 12 hrs. However, the moon-earth system also rotates around C ME with velocity ω ME in the same direction as the earth's rotation. The bulges move with the moon and hence the tidal period is hrs (12 hr & 25 min). The tide in Fig. 2.1 is called Equilibrium Tide since it results from the assumption that the tidal forces act on the water for a long time so that equilibrium is achieved between the tide generating force and the slope of the water surface. The sun's gravity forms a second, smaller set of bulges toward the sun and away from the sun. Since our day is measured with respect to the sun, the period of the tide generated by the sun is 12 hrs. Both these equilibrium tides occur at the same time and they will add up when the moon and sun are aligned (at new moon and full moon). At those times, the tides are higher than average. At quarter moon, the forces of the sun and moon are 90 out of phase and the equilibrium tides subtract from each other. At such a time, the tides will be lower than average. The higher tides are called Spring Tides and the lower ones Neap Tides. Fig. 2.2 demonstrates this. The phases of the moon are shown at the bottom of the figure and it is seen that, except for some phase lag, the maximum tides (spring tides) in Fig. 2.2 correspond to new and full moon, while the neap tides correspond to the quarter moon. K.A.Rakha 2-2

42 Water Level Variations B E A ME C C ME Moon E C E Earth Equilibrium Tide Fig. 2.1: Equilibrium Tide. K.A.Rakha 2-3

43 Level (m) Level (m) Level (m) Water Level Variations Khasab, Hormuz (Oman) Time (Hr) Al-Ahmadi, Kuwait Time (Hr) Bushehr, Iran Time (Hr) Fig. 2.2: Tide Predictions for Stations in the Arabian/Persian Gulf. K.A.Rakha 2-4

44 Water Level Variations Daily Inequality Fig. 2.1 was drawn looking down on the earth s axis. Since the equilibrium tide is three dimensional in shape (it forms a distorted sphere), the picture is the same when the earth is viewed from the side, as shown in Fig An observer, C, travelling along a constant latitude would experience two tides of equal height per day. However, the moon or sun is seldom in the plane of the equator. When the moon or sun has a North or South Declination with respect to the equator, as shown in Fig. 2.4, one bulge of the equilibrium tide will lie above the equator and one below the equator. An observer moving along constant latitude would now experience two tides per day of unequal height. This is called Daily Inequality. The daily inequality is most pronounced when the moon or sun is furthest North or South of the equator. It generally increases with latitude and there is no daily inequality at the equator. Daily inequality is demonstrated in Fig The daily inequality cycle generated by the moon repeats itself every 29.3 days. For the tide generated by the sun, the daily inequality is greatest shortly after mid-summer and mid-winter, causing higher tides in early January and early July. K.A.Rakha 2-5

45 Water Level Variations N Latitude E C Equator Earth Moon Equilibrium Tide Fig. 2.3: Equilibrium Tide (from side) K.A.Rakha 2-6

46 Water Level Variations N Latitude E C Equator Earth Declination Moon Fig. 2.4: Daily Inequality. K.A.Rakha 2-7

47 Water Level Variations Spring/Neap Tides The semidiurnal rise and fall of tide can be described as nearly sinusoidal in shape, reaching a peak value every 12 hr and 25 min. This period represents one-half of the lunar day. Two tides are generally experienced per lunar day because tides represent a response to the increased gravitational attraction from the (primarily) moon on one side of the earth, balanced by a centrifugal force on the opposite side of the earth. These forces create a "bulge" or outward deflection in the water surface on the two opposing sides of the earth. The magnitude of tidal deflection is partially a function of the distance between the moon and earth. When the moon is in perigee, i.e., closest to the earth, the tide range is greater than when it is furthest from the earth, in apogee. Conversely, when the moon is in apogee, the potential term is at a minimum value. This difference may be as large as 20 percent. The tidal force envelope produced by the moon's gravitational attraction is accompanied by a tidal force envelope of considerably smaller amplitude produced by the sun. The tidal force exerted by the sun is a composite of the sun's gravitational attraction and a centrifugal force component created by the revolution of the earth's center-of-mass around the centerof-mass of the earth-sun system, in an exactly analogous manner to the earth-moon relationship. The position of this force envelope shifts with the relative orbital position of the earth in respect to the sun. Because of the great differences between the average distances of the moon (238,855 miles) and sun (92,900,000 miles) from the earth, the tide producing force of the moon is approximately 2.5 times that of the sun. Spring tides occur when the sun and moon are in alignment. This occurs at either a new moon, when the sun and moon are on the same side of the earth, or at full moon, when they are on opposite sides of the earth. Neap tides occur at the intermediate points, the moon's first and third quarters. Figure 2.6 is a schematic representation of these predominant tidal phases. Lunar quarters are indicated in the tidal time series shown in Fig When the moon is at new phase and full phase, the gravitational attractions of the moon and sun act to reinforce each other. Since the resultant or combined tidal force is also increased, the observed high tides are higher and low tides are lower than average. This means that the tidal range is greater at all locations which display a consecutive high and low water. Such greater-than-average tides results are known as spring tides - a term which merely implies a "welling up" of the water and bears no relationship to the season of the year. At first- and third-quarter phases (quadrature) of the moon, the gravitational attractions of the moon and sun upon the waters of the earth are exerted at right angles to each other. Each force tends in part to counteract the other. In the tidal force envelope representing these combined forces, both maximum and minimum forces are reduced. High tides are lower and low tides are higher than average. Such tides of diminished range are called neap tides, from a Greek word meaning "scanty". K.A.Rakha 2-8

48 Water Level Variations Fig. 2.6: Spring and Neap Tides. K.A.Rakha 2-9

49 Water Level Variations Other Effects So far we have explained the characteristics of tides, based on four influences, the gravitational attraction of the sun and moon, and the declination of the sun and moon. There are many other, secondary effects. For example, we have assumed that the sun and the moon travel in circular orbits relative to the earth. These orbits are actually elliptical and therefore the distances between the earth and the sun and moon change in a periodic fashion. This effect (and many others) can be viewed as a separate tide generator (like the moon in Fig. 2.1). Each such tide generator has its own strength, frequency and phase angle with respect to the others. The resulting tide is, therefore, a complex addition of effects of the moon, the sun and many secondary causes. Each component is called a tide constituent (Dronkers, 1964). Until now we have assumed that the earth is completely covered with water and that the same forces act everywhere continuously. It was seen that the tide moves relatively slowly, while the earth turns more rapidly through the tide. In reality, the earth s large land masses will not turn through the tide, but will move the water masses along with them, disrupting our picture. The only place where an equilibrium tide can possibly develop is in the Southern Hemisphere, where the earth is circled by one uninterrupted band of water. An equilibrium tide can form there and it will progress into the various oceans. It takes time to travel along those oceans and hence the actual tide constituent (water level fluctuation) lags behind its related theoretical tide constituent (from equilibrium theory), causing high water to occur after the moon crosses the local meridian and causing spring tide some time after full (or new) moon. The earth s geography not only confines the water and moves it along with the surface of the earth, it also causes certain tidal constituents to resonate locally in the various oceans, seas, bays and estuaries. Thus some constituents are magnified in certain locations, while others simply disappear, making the tide at each location quite unique. One aspect that is often magnified by the land mass is the daily inequality, increasing the difference between the larger and smaller daily tides so that the small tides become virtually non-existent. The Semi-Diurnal (twice per day) tides then become Diurnal (once per day). An example of this is shown in Fig Tide Analysis and Prediction The equilibrium theory of tides is a hypothesis that the waters of the earth respond instantaneously to the tide-producing forces of the sun and moon. For example, high water occurs directly beneath the moon and sun, i.e., at the sublunar and subsolar points. This tide is referred to as an equilibrium tide. The tide-producing forces can be written in a polynomial expansion approximation. These expansion terms involve astronomical arguments describing the location of the sun and moon as well as the location of the observer on the earth. Although several variational forms of the series expansion have been published, the development presented in Schureman (1924) is given below. Alternate forms of expansion are discussed in Dronkers (1964). According to equilibrium theory, the theoretical tide can be predicted at any location on the earth as a sum of a number of harmonic terms contained in the polynomial expansion K.A.Rakha 2-10

50 Water Level Variations representation of the tide-producing forces. However, the actual tide does not conform to this theoretical value because of friction and inertia as well as differences in the depth and distribution of land masses of the earth. Because of the above complexities, it is impossible to exactly predict the tide at any place on the earth based on a purely theoretical approach. However, the tide-producing forces (and their expansion component terms) are harmonic; i.e., they can be expressed as a cosine function whose argument increases linearly with time according to known speed criteria. If the expansion terms of the tide-producing forces are combined according to terms of identical period (speed), then the tide can be represented as a sum of a relatively small number of harmonic constituents. Each set of constituents of common period are in the form of a product of an amplitude coefficient and the cosine of an argument of known period with phase adjustments based on time of observation and location. Observational data at a specific time and location are then used to determine the coefficient multipliers and phase arguments for each constituent, the sum of which are used to reconstruct the tide at that location for any time. This concept represents the basis of the harmonic analysis, i.e., to use observational data to develop site-specific coefficients that can be used to reconstruct a tidal series as a linear sum of individual terms of known speed. Tide Analysis consists of separating a measured tide into as many of its constituents as can be identified from the length of record available. The tide is assumed to be represented by the harmonic summation, H( t) H o f n H n n cos a t V o u n n ( 2.1 ) where H(t) = Water level at time t (t is measured from start of the year) H o = Mean water level above some defined datum H n = Mean amplitude of tidal constituent n f n = Factor for adjusting mean amplitude (for each year) a n = speed of constituent n ( 2π / T where T is the tidal period ) (V o +u) n = Equilibrium argument (for each year) κ n = Phase shift of tidal constituent n K.A.Rakha 2-11

51 Water Level Variations For any location the tide can be calculated provided that the values of H o, H n and κ n are known. These values are computed from observed tidal time series data, usually from a least squares analysis. The time-specific arguments (f n and V o + u) are determined from formulas or tables. Most of the constituents listed in Table 2.1 are associated with a subscript indicating the approximate number of cycles per solar day (24 hr). Constituents with subscripts of 2 are semidiurnal constituents and produce a tidal contribution of approximately two high tides per day. Diurnal constituents occur approximately once a day and have a subscript of 1. Symbols with no subscript are termed long-period constituents and have periods greater than a day; for example, the Solar Annual constituent S a has a period of approximately 1 year. There are also constituents that describe interactions between other constituents. One year's record will comfortably provide the amplitudes and phase angles of 60 such tide constituents. One important tidal constituent has a period of 18.6 years. It cannot be calculated from a reasonable record length and is therefore introduced by formulas. Factors are computed (f n and V o + u) that adjust the amplitude and phase shift as function of time relative to this 18.6 year cycle. For many construction projects, local tidal information will not be available and tides need to be measured and analysed specifically for a project. In that case, it is usual to collect rather short tidal records. For record lengths of a month or so, tide analysis can only yield the lunar and solar, semi-diurnal tides, daily inequality, lunar declinational tides and at most five or six other constituents that can readily be separated. But that is often sufficient for approximate predictions. K.A.Rakha 2-12

52 Water Level Variations Datums Water level and its change with respect to time have to be measured relative to some specified elevation or datum in order to have a physical significance. In the fields of coastal engineering and oceanography this datum represents a critical design parameter because reported water levels provide an indication of minimum navigational depths or maximum surface elevations at which protective levees or berms are overtopped. It is therefore necessary that coastal datums represent some reference point which is universally understood and meaningful, both onshore and offshore. The following are some of the commonly used datums, HAT MHWS MHWN MSL MLWN MLWS LAT Highest Astronomical Tide Mean High Water Springs Mean High Water Neaps Mean Sea Level Mean Low Water Neaps Mean Low Water Springs Lowest Astronomical Tide Mean sea level (MSL) was widely adopted as a primary datum on the assumption that it could be accurately computed from tidal elevation records measured at any well-exposed tide gauge. MSL determinations are based on the arithmetic average of hourly water surface elevations observed over a long period of time. The ideal length of record is approximately 19 years, a period that accounts for the 18- to 19-year long-term cycle in tides and is sufficient to remove most meteorological effects. When estimates of MSL are required, but less than 19 years of data are available, computations should be based on an integral number of tidal cycles, for example, an integral number of years or 29-day spring/neap cycles. For gauges where hourly data are not available, or their use is impractical, MSL can be approximated as the tidal datum midway between MHW and MLW. This datum, referred to as Mean Tide Level (MTL), may differ from MSL depending on the local relative importance of the diurnal components of the tide. Table 2.1: Tidal period and speed for some important Constituents. Constituent Tidal Period (hr) Speed (deg/hr) M S K O Sa Ssa K.A.Rakha 2-13

53 Water Level Variations Mm Msf Mf S Q P N υ 2 (NU 2 ) K L (2N) μ 2 (MU 2 ) T M (MS) (2MS) Storm Surge Storms are atmospheric disturbances characterized by low pressures and high winds. A storm surge represents the water surface response to wind-induced surface shear stress and pressure fields. Storm-induced surges can produce short-term increases in water level that rise to an elevation considerably above mean water levels. The water level fluctuation due to storm surge is an increase in water level resulting from shear stress by onshore wind over the water surface (Fig. 2.7). This temporary water level increase occurs at the same time as major wave action and it is the cause of most of the world's disastrous flooding and coastal damage. Parts of Bangladesh are flooded regularly by storm surge resulting from passing cyclones with the loss of thousands of lives. In a 1990 cyclone, the water levels rose by 5-10 m and it was estimated that more than 100,000 lives were lost. The shorelines along the southern borders of the North Sea, particularly the Netherlands, were flooded in 1953, because storm surge caused dike breaches. During storm surge the water level at a downwind shore will be raised until the slope of the water surface counteracts the shear stress from the wind. Computations of storm surge are carried out using the same depth-averaged two dimensional equations of motion and continuity that are used for tidal computations. In this case wind-generated shear stress is the main driving force. For simple problems, the equations can be reduced to a onedimensional computation, K.A.Rakha 2-14

54 Water Level Variations ds dx U cos gd 2 ( 2.2 ) where S is the storm surge (the setup of the water level by the wind), x is the direction over which the storm surge is calculated, ζ is a constant (=3.2x10-6 ), U is the wind speed, φ is the angle between the wind direction and the x-axis and D is the new depth of water (=d+s). Equation 2.2 clearly shows that storm surge is greatest in shallow water. 2.3 Barometric Surge Since strong winds are the result of large pressure fluctuations, a barometric surge will accompany storm surge. Suppose there is a difference in barometric pressure Δp between the sea and the shore, then an additional water level Δh rise will be generated: p h g ( 2.3 ) where ρ is the density of water. Equation 2.3 results in a water level rise of about 0.1 m for each kpa of pressure difference. A major depression can easily generate a pressure difference of 5 kpa, resulting in a potential barometric surge of 0.5 m. K.A.Rakha 2-15

55 Water Level Variations W s W Closed Basin D s d Open Sea Fig. 2.7: Storm surge in closed and open seas. K.A.Rakha 2-16

56 Water Level Variations 2.4 Seiche Seiches are standing waves or oscillations of the free surface of a body of water in a closed or semiclosed basin. These oscillations are of relatively long period, extending from minutes in harbors and bays to over 10 hr in the Great Lakes. Any external perturbation to the lake or embayment can force an oscillation. In harbors, the forcing can be the result of short waves and wave groups at the harbor entrance. The oscillations will continue for some time because friction forces are quite small. The wave length of the fundamental mode of the oscillation (a standing wave) for a closed basin (Fig. 2.8) is twice the effective basin length (B). In general, the wave length is 2B/(1+nh) for the nh harmonic. For an open ended basin (open coast), the fundamental wave length is 4 times the effective length of the shelf (B) over which the storm surge was initially set up. In general, for the nh harmonic it is 4B/(1+2nh). The period of oscillation (T=L/C) for a closed basin may be calculated as: L/4 L/2 B B 3L 1 /4 2L 1 /2 Open Basin Closed Basin Fig. 2.8: Seiche wavelengths. K.A.Rakha 2-17

57 Water Level Variations 2.5 Tsunami Tsunami is a single wave generated by sub-sea earthquakes and typically has a period of 5 to 60 minutes. Tsunami waves can travel long distances and is normally not very high in deep water. In shallow water the wave shoaling can reach a height more than 10 m. Tsunamis are rare and coastal structures seldom take them into account. 2.6 Eustatic (Sea) Level Change The term Eustatic refers to a global change in ocean water levels; the result of melting or freezing of the polar ice caps and the thermal expansion of the water mass with temperature change. The water levels 25,000 years ago were 150 m below the present level (Kamphuis, 2000). Between then and 3,000 years ago, water level rose at a more-or-less steady rate of about 7 mm/yr to almost the present water level. The present average rate of eustatic rise is small and therefore difficult to measure. Estimate range from 1 to 3 mm/yr. This relatively small rate of rise, nevertheless, causes the ocean shores to be submerging and is at least partly responsible for the fact that most beaches around the world are eroding over the long term. 2.7 Isostatic (Land) Rebound and Subsidence The common natural cause for isostatic (land) elevation change is a result of the adjustment of the earth's crust to the release of pressure exerted by the 1 to 2 km thick ice sheet that covered it during the last glaciation. Typically, the earth s crust was severely depressed by the ice and a rise (forebulge) was formed in the earth s crust ahead of the glaciers. When the ice retreated, the earth's surface rebounded (upward) where the glaciers had been and lowered where the forebulge had occurred. This process still takes place today, but at a much reduced rate. Most areas in the higher latitudes experience isostatic rebound and areas at more intermediate latitudes experience some subsidence. Although subsidence does occur naturally, often it is man-made. Pumping groundwater, petroleum and natural gas are common causes. Subsidence exacerbates the effects of eustatic sea level rise since the relative sea level rise with respect to the land will now be greater. K.A.Rakha 2-18

58 Water Level Variations Fig. 2.9: Tsunami generation and propagation. K.A.Rakha 2-19

59 Water Level Variations 2.8 Global Climate Change The final and potentially most dangerous water level change results from trends in global climate. In the discussion of eustatic sealevel rise, we have already seen that global warming after the last glaciation has resulted in a sealevel rise of 100 to 150 m through melting of the polar ice caps and thermal expansion of the water in the ocean. The present rate has slowed down to 1 to 1.5 mm/yr, but any additional warming would increase this rate of sealevel rise. Concern is centered around the production of the so-called greenhouse gases. These combustion products are thought to act as an insulating blanket over the earth, decreasing the net longwave radiation from the earth into space and thus trapping the sun's heat to cause global warming. It is a controversial subject and indeed there is a contingent of respected scientists that disputes the whole idea. According to Kamphuis (2000), predicted rise in water level for the year 2025 varies from 0.1 to 0.2 m. For 2050, the estimates vary from 0.2 to 1.3 m and for 2100 the estimates are 0.5 to 2 m. K.A.Rakha 2-20

60 3 Currents in the Marine Environment Various types of currents exist in the marine environment. These currents may exist in the open sea or in the nearshore area. In the open sea the currents are mainly tidal and wind driven currents. In the nearshore area wave induced currents can also exist. 3.1 Tidal Currents Tidal currents are induced by the gravity forces of the sun, the moon, and the planets. Tidal currents are oscillatory currents with typical periods of about 12 or 24 hours (semi-diurnal and diurnal). Tidal currents are influenced by the sea bottom contours and by coastal morphology. They are strongest at large water depths and in estuaries or straits where the current is forced into a narrow area. The most important tidal currents for coastal morphology are the currents generated at tidal inlets. In the deep, open ocean, the fluid velocity (tidal current or horizontal tide) is in phase with the tidal water level fluctuations (vertical tide). At high water there is a maximum current velocity in the direction of tide propagation. When the tide approaches land, however, the phase relationship between horizontal and vertical tide changes. In the case of a tidal inlet or bay, the water level fluctuations in the bay are driven by the tidal water level in the sea. Rising water levels in the sea cause a current to flow into the bay, raising its water level. This inflow of water is called flood and the outflow current during the other half of the tidal cycle is called ebb. 3.2 Wind Generated Currents Wind generated currents are caused by the wind shear stress along the sea surface. These currents are normally located in the upper layer of the water body and are thus not very important from a morphological point of view. Wind currents can have an important role however in the movement of pollutants and oil spills. In shallow waters and in lagoons, wind generated currents can be important. Wind generated currents are typically less than 5 per cent of the wind speed. 3.3 Stratification and Density Currents An estuary is defined as a tidal area where a river meets the sea. It has salt water on its downstream limit (sea) and fresh water on the upstream limit (river). The salt sea water normally has a salinity in the vicinity of 35 parts per thousand (ppt) and a density of 1035 kg/m3. The fresh water has a density of 1000 kg/m 3. The way the transition from salt to fresh water takes place depends on the amount of mixing that takes place in the estuary. In a well-mixed estuary (an estuary with much turbulence), salt and fresh water are thoroughly mixed at any location. Salinity simply varies along the estuary from 35 ppt in the sea to zero ppt in the river and at any specific location, salinity and density will vary with the tide. If there is little mixing in the estuary, the lighter fresh water will lie over the heavier salt water, resulting in a stratified estuary. The density differences will induce currents. In deep

61 Currents in the Marine Environment oceans, density currents also exist due to stratification caused by temperature and salinity differences over the water depth. 3.4 Wave Induced Currents Besides the wave-induced oscillatory currents the breaking of waves induces other nonoscillatory currents. These currents are directed in the on/offshore and alongshore directions Shore-normal currents The breaking phenomena and the asymmetric wave form in the nearshore area results in a mass transport of water with in the upper layers. This results in a water surplus in the surf zone (wave setup). This surplus water returns to the sea via rip and undertow currents Rip Currents At certain intervals along the coastline, the longshore current will form a rip current. The rip currents are directed in the offshore direction. The rip opening in the bars will often form the lowest section of the coastal profile with a local setback in the shoreline opposite the rip opening (see Fig. 3.1). Field observations showed that the rip current velocity might exceed 1.0 m/sec and occasionally extend more than 500 m from the breaker line. Presently there is no proven method to predict rip current generation and the spacing between rips. Figure 3.2 shows the typical nearshore currents for different wave incidence (Harris, 1969) Undertow The undertow current is a return flow concentrated near the bed (see Fig. 3.3). This current is important in the formation of bars. The mass transport carried toward the beach due to waves is concentrated between the wave trough and crest elevations. Because there is no net mass flux through the beach, the wave-induced mass transport above the trough is largely balanced by a reverse flow or undertow below the trough. The undertow current at the bottom may be relatively strong, generally 8-10 percent of the wave celerity. The wave setup at the still water line is about 0.15d b Shore-parallel currents Longshore currents are the dominating current in the nearshore zone, generated by obliquely approaching breaking waves. This current has its maximum close to the breaker line (see Fig. 3.2). During storms the longshore current can reach values of 2.5 m/s. The longshore current carries sediment along the shoreline (littoral drift) as explained later. K.A.Rakha 3-2

62 Currents in the Marine Environment Fig. 3.1: Photo of rip currents observed along a beach. K.A.Rakha 3-3

63 Currents in the Marine Environment b Typical current distribution V l A. Oblique ( b large) Breaker Rip Current B. Normal ( b ~ 0) b C. Slightly Oblique ( b small) Fig. 3.2: Nearshore Circulation systems (Harris, 1969) K.A.Rakha 3-4

64 Currents in the Marine Environment Wave Setup MWL SWL Undertow Fig. 3.3: Schematic of the undertow current. K.A.Rakha 3-5

65 Currents in the Marine Environment Two-dimensional Currents Along a straight coastline the above mentioned shore parallel and shore normal currents exist. When, combined they are three-dimensional in nature. For complex bathymetries two dimensional currents exist. These currents occur due to irregular bathymetries or due to the existence of structures in the nearshore zone (such as groins or breakwaters). Coastal structures influence the current pattern in two ways: by obstructing the shoreparallel currents and by setting up secondary circulation currents. The nature of the obstruction to the shore-parallel currents depends on the extent and geometry of the coastal structure. If the structure is located within the breaker zone, the obstruction leads to offshore directed currents that will cause loss of beach material. If the structure is a harbour, the current will follow the upstream breakwater and reach the entrance area (see Fig. 3.4). These currents will cause sedimentation and will affect the navigation. It is thus important to provide smooth currents that will be acceptable for the navigation and will reduce the sedimentation. A smooth layout of the main and secondary breakwaters with a narrow entrance is the best alignment, rather than the alignment provided in Fig At the leeward side of coastal structures, special current patterns can develop caused by the sheltering effect of the structure in the diffraction area. The wave setup in the sheltered areas will be lower than that in the adjacent exposed areas generated a gradient in water level that will drive currents (e.g. see Fig. 3.5). These circulation currents can be dangerous to swimmers who might swim in the sheltered areas during storms. If the structure extends beyond the breaker zone, the shoe parallel current will be directed along the structure where the increasing depth will cause the currents to be reduced. This will cause the deposition of sediment forming a shoal off the breaker zone. In the lee of a major coastal structure the effect of return currents towards the sheltered area will also be pronounced. In this case however, the current patterns will be smoother and less dangerous for swimmers. K.A.Rakha 3-6

66 Currents in the Marine Environment Fig. 3.4: Schematic of the currents at the SUMED Harbour at Sedi Kerir, Egypt (Rakha and Abul-Azm, 2003) K.A.Rakha 3-7