GLY 4734 - Coastal Geomorphology Notes Dr. Peter N. Adams February-March 2010 3 Waves In this section we ll cover waves from their generation to dispersion and travel to shoaling transformation to breaking in the nearshore zone, where they release their energy to do geomorphic work on the coast. Our introduction will cover basic definitions, measurement, analysis, data sources, and wave climate. We can organize our thoughts by loosely categorizing waves into three personas, according to their environment and maturity: 1. Sea: Arising in the localized region of wave generation, where water motions are irregular and appear quite disorganized. 2. Swell: Covering the broad region of wave propagation, wherein the wave water motions transport energy, and the waves themselves organize according to their periodicity. 3. Surf: Occurring in the very narrow band adjacent to a shoreline where waves shoal rapidly, transform, and release energy in breaking and run-up. 3.1 Periodic Waves We can think of waves with respect to a spatial framework, or with respect to a temporal framework. Examine the anatomy of a wave - period, frequency, length, phase velocity, height, displacement, depth, particle orbital diameter, particle orbital velocity. 1
Numerous relationships exist among the wave variables listed above, like for example: There exists a direct relationship between wave length and wave period in deep water, and thus between celerity and wave period. The most basic relationship is: C = L T It is important to pay attention to the details of water motion in waves: 1. Individual particles move in circular patterns with the size of the circles decreasing (exponentially) with water depth. 2. The wave form, however, propagates through the medium. 3. So, a cork floating on the surface of the water makes no net advance in the direction of wave motion - it simply returns back to its starting point after a wave passes. Dispersion: Wave have a tendency to sort themselves out by wave period. So, from a region of wave generation (open ocean storm), the longer period waves travel faster across the ocean, and hence arrive at a point on the coast earlier than short period waves. Superposition: Addition of two or more wave forms leads to constructive and destructive interference as dictated by the heights and periods of the constituent wave forms. See Animation shown in class. Spectral Energy of Water Level Fluctuations - The classification of wave motions is based on the restoring force (e.g. gravity, surface tension, coriolis, etc.). Wind waves occupy the 10 2 10 2 Hz band. (1) 3.2 Measurement of Waves We now understand that a wave field may arise from multiple superimposed wave forms from different sources with different heights, periods, and directions. So accurate measurement of waves, for the purposes of understanding individual constituents, requires more than a meter stick and stopwatch. In addition, the equipment must tolerate the corrosive, oceanic environment. Here we describe some of the techniques employed. 2
3.2.1 In-situ Measurement Devices 1. Surface Piercing. Example: graduated staff with electrodes attached to a the vertical support of a pier or drilling platform. As water surface rises and falls, a specific level of electrodes are recorded as closed circuits providing a time series of water surface levels. 2. Pressure Sensing. Measures time series of height of column of water above the instrument. 3. Surface Following. Example: Waverider buoy with an accelerometer. 3.2.2 Analytical Techniques First, a time series (then a populationdistribution) of waves must be computed from the time series of water level variations. One common technique is the zero upcrossing method (see lecture slide). Statistical Analysis - time domain analysis which analyzes the population distribution of wave heights to determine a series of characteristic heights (H sig, H rms, H 1/10, H max ). For example, the root-mean-square wave height is determined thusly: H rms = 1 N N Hi 2 (2) i=1 Significant wave height (H s ) is commonly reported, and is determined by taking the average of the highest one-third of waves over a specific interval of observation. Spectral Analysis - which is carried out in the frequency domain, and is a fairly standard technique today. This method uses Fourier decomposition. Fourier Analysis is based on the concept that any complex time series can be represented by a combination of various sine and cosine functions. By performing a Fourier Transform of the time domain data, we obtain a function in the frequency domain which describes which frequencies are present in the original function. 3
Several in-class examples of Fourier deconvolution illustrated with MATLAB codes. Sources of wave data are easily accessible (nowadays) online. For example, NOAA maintains a network of buoys (http://www.ndbc.noaa.gov/). Hindcasts are also available (WIS). 3.2.3 Wave Record Analyses Continuous records, if of sufficient length, can be examined for trends in changing wave conditions. Recently, numerous studies have investigated various data sets indicating that changes in ocean wave storminess patterns are indeed occurring (Allan and Komar, 2006; Adams et al., 2008; Komar and Allan, 2008). Example of probability of occurrence analysis. 3.3 Wave Generation The process of wave generation is frequently observed, but the physics and specific mechanisms are still poorly understood. In general, the longer and harder the wind blows, the larger and longer period the waves generated (up to a point). 3.3.1 Processes of Wave Generation Generally accepted theory to account for growth of waves is the Miles-Phillips mechanism, which incorporates 2 processes of energy transfer from wind to waves, covered on pp. 151-153 of Komar: Initial growth stage where there is a linear increase in wave energy with time due to resonance between atmospheric pressure fluctuations and developing, wave-covered, water surface. 2nd mechanism causes exponential growth of already developed waves. Under a logarithmic wind velocity profile, airflow over the sinusoidallyshaped water surface creates a distribution of high pressure over troughs and low pressure over crests, which drives flow separation at the boundary layer, further amplifying wave height. 4
As wave generation continues, increasing energy is pumped toward the long period portion of the spectrum. Long period waves travel more quickly through the medium, and overtake short period waves, sweeping up their momentum and energy. However, short period waves are continuously being generated by the Miles-Phillips mechanism, so they are always present. 3.3.2 Wave Predictions Semi-empirical/semi-theoretical methods have been developed for wave prediction - theory is involved in their formulation, but data are required for the evaluation of various coefficients. Heights and periods of a wave field are dependent on the wind velocity (U), the duration of time that the wind blows (D), and the distance over which the wind blows (F ), also known as the fetch. 1. Significant Wave Approach - developed by Sverdrup, Munk, and Bretschneider, the S-M-B methods relate Hs,Ts as f(u, D, F ), codified in nomograms still used today. 2. Wave Spectrum Approach - in line with the characterization of waves by their spectra, which provide a measure of wave energy at each period/frequency. Produce wave energy spectrum from given wind speed, duration, and fetch. Two ranges - equilibrium range and growth range. Energy first fills the high frequency bins to the equilibrium range, then spills into the longer period portion of the spectrum, which has increasingly greater accommodation for energy in the equilibrium range. Hence, there is a progressive shift in peak period. 3. JONSWAP spectrum - most often used today. 3.4 Wave Theories This section focuses on the motion and energy of waves. Wave movement across the ocean from the source area to coastal locations, where breaking occurs, can be quantitatively explained via several wave theories. In addition, these theories provide information on the motion of water particles as waves pass through the medium. This material is extensively covered on pp. 160-176 of the Komar text. 5
3.4.1 Airy Wave Theory All fluid motions are governed by the mathematical relationships (differential equations) that balance mass and momentum in a control volume. Waves are no different. These relationships boil down to: 1. The equations of motion, which, when applied to an incompressible fluid, are referred to as the Navier-Stokes equations, and when operating under certain assumptions (low viscosity), are known as the Euler equations. 2. Continuity equation, which ensures that mass is conserved. Simultaneous solution of these equations with the right boundary conditions and assumptions provides us with the basis of Airy Wave Theory (AWT), also referred to as Linear Wave Theory (LWT). One of the main assumptions in LWT is that wave height is much smaller than wave length and water depth, so LWT is applicable over much of the open ocean, but breaks down somewhat in shallow, coastal areas. Water Surface Elevation Fluctuation The solution for water surface elevation η as a function of space and time (x and t) is: η(x, t) = H 2 cos(kx σt) (3) where H is wave height, k = 2π 2π is wave number, and σ = is radian frequency, L T with T being wave period, of course. The Airy wave is sinusoidal in profile with respect to time and space. Dispersion Equation Another fundamental equation arising from AWT is the dispersion equation: σ 2 = gk tanh(kh) (4) which can be manipulated with the identities for radian frequency and wave number to yield L = g 2π T 2 tanh( 2πh L ) (5) 6
which is quite useful, as it relates wavelength, period, and water depth, but is problematic because wavelength appears on both sides of the equation and is imprisoned within a hyperbolic trigonometric function for good measure. This will have to be solved iteratively, perhaps by the Newton-Raphson method. Celerity Behavior in Deep and Shallow Water Let s divide both sides of Eqn. 5 by the wave period and see where that gets us. C = gt 2π tanh(2πh L ) (6) Now we have a relationship for wave celerity, C, alternatively referred to as the phase velocity or wave speed. Let s explore this function in deep and shallow water. We must first recall the behavior of the hyperbolic trig functions. Just as the sin, cos, tan, etc. describe ratios between triangle legs on the unit circle, x 2 + y 2 = 1, the hyperbolic trigonometric functions describe ratios between triangle legs on the unit hyperbola, x 2 y 2 = 1. The behavior of these functions is shown in a plot on p. 163 of Komar. Note that the hyperbolic tangent of a large value approaches 1, and the hyperbolic tangent of a small value approaches that small value. So the effect on the general solution for celerity is as follows. In deep water, where tanh(anything) 1: C = gt 2π (7) and in shallow water, where tanh(something) thatsomething: C = gt 2πh 2π L (8) after rearranging and canceling, C 2 = gh, or, C = gh (9) illustrating that in shallow water, all waves should be traveling at the same speed. 7
We can view the behavior of the celerity as a function of water depth, as computed by the deep-water, shallow-water, and general expressions - see figure of Celerity Illustrated, shown in class. As an application, we can examine two examples of shallow water waves: 1. Tsunamis 2. Tow-In, big wave surfing The behavior of wave length, wave celerity, and wave height are nicely summarized in a figure that shows dimensionless variables. Water Particle Movement Water particle orbital path trajectories are also predicted by Airy Wave Theory. The orbital paths divided by wave period provide the wave orbital velocities. The general expressions for horizontal and vertical diameters are d = H cosh[k(z o + h)] sinh(kh) s = H sinh[k(z o + h)] sinh(kh) (10) (11) Deep water: s = d = He kzo = He 2πzo L, circular orbits, whose diameters decrease exponentially with depth from the water surface; at water surface the diameter of particle motion is obviously the wave height, H. Intermediate water: ellipse sizes decrease downward through water column Shallow water: s = 0, d = H ; ellipses flatten to horizontal motions; orbital kh diameter is constant from surface to bottom. Water particle orbital velocities, denoted u for the horizontal and v for the vertical velocity components, respectively, are simply the paths divided by the periods, with a sine or cosine dependence on (kx σt), since they vary spatially at fixed time, and temporally at fixed position. The general forms for horizontal and vertical orbital velocities are: u = πh T cosh[k(z o + h)] sinh(kh) cos(kx σt) (12) 8
w = πh T sinh[k(z o + h)] sinh(kh) sin(kx σt) (13) Wave Energy Density Although no net water movement occurs during the passage of a wave (in Airy Theory), there is still gross movement which constitutes a transfer of energy over the sea surface. Water surface displacement from the flat, stillwater conditions provides a potential energy component, and the movement of the water particles in their orbital paths provides a kinetic energy component. Summing the integrals of both components over one wavelength yields the total energy density (so named because it represents the wave energy over an area of sea surface) of the wavy, sea surface: Wave Energy Flux E = E p + E k = 1 8 ρgh2 (14) E in equation14 is not conserved during shoaling, because it is not truly an energy, but a distribution of energy per area of sea surface. What is conserved is the wave energy flux P = EC g = ECn = 1 8 ρgh2 Cn (15) where C g is wave group speed, and the ratio of group to individual wave speed is given by n = 1 2 [1 + 2kh sinh(2kh) ] (16) Examine how n varies as a function of water depth. In deep water, n = 1 2, implying that the group velocity is 1/2 the individual velocity, whereas in shallow water, n = 1 meaning that the wave group travels at the same speed as an individual wave. Wave Groupiness 9
Radiation Stress This concept is of great power in the study of wave related phenomena. The Radiation Stress is a tensor which represents the excess flow of momentum due to the presence of waves (Longuet-Higgins and Stewart, 1964). Wave advance has an associated momentum flux. The excess momentum is calculated by using the dynamic component of the pressure, i.e. the difference between absolute pressure and the hydrostatic pressure, ensuring that the excess momentum represents the momentum due solely to the presence of waves. Consider that there exists both x-directed and y-directed components of x- and y-oriented momentum, such that [ 2kh S xx = E sinh(2kh) + 1 ] = E(2n 1 2 2 ) (17) [ ] kh S yy = E = E(n 1 sinh(2kh) 2 ) (18) In deep water, when n = 1, S 2 xx = E, and S 2 yy = 0; and in shallow water, when n = 1, S xx = 3E, and S 2 yy = E. We ll return to radiation stresses in our 2 discussion of wave set-up/set-down, and longshore sediment transport. 3.4.2 Stokes Wave Theory Airy wave theory begins to break down in shallow water and, as such, we turn to a wave theory derived by Sir George Stokes in the mid-1800 s. Stokes theory does not make the small wave height assumption. (Show MATLAB code airyequations.m) 3.4.3 Limits of Application of Various Theories Figure 5-29 on p. 175 of Komar text shows appropriate depth-height-wavelength fields for each of the various wave theories. 10
3.5 Wave Transformation This section focuses on the movement and natural alteration of the characteristics of waves as they travel from the source region toward the shore the material is extensively covered on pp. 176-196 of the Komar text. 3.5.1 Propagation At this point, we will redefine the term wave group to mean a bundle of wave energy that travels from source region to the shore. Wave groups travel with speed C g = Cn, a.k.a. the group velocity. Longer period waves outrun and leave behind the shorter period waves. Dispersion produces a narrowing of the energy spectrum (Komar s Fig. 5-30), so the greater the travel distance, the narrower the strong frequency band in the energy density spectrum (PNA Slide). Individual waves arise at the rear of a group, move through the group, and die out at the front. Hence, they cannot be traced across the ocean - but the groups themselves can! Wave Group Rectangle Here, we can introduce the wave group rectangle concept. Imagine a storm area on the open ocean (deep water), whose wind blows for a duration D, and whose fetch (length) is F. This group rectangle will have one period associated with it, and each wave period generated by the storm will have its own conceptual group rectangle. It is truly a continuum, but we ll conceptualize it as a discrete period associated with a discreet rectangle. The width of the group rectangle is the width of the storm front, and the length would be W = DC g. What s C g again? C g = Cn = 1 gt 2 2π (19) The edges of rectangle will not be sharp, as there will be some lateral energy loss along wave crests, and rounded front and back edges due to this pattern of individual wave movement through the group. Importantly, however, there will not be major radial spreading, as experienced by waves generated by a 11
pebble thrown into a pond. Note this clarification, which may conflict a bit with your notion from prior lectures. A wave group rectangle retains most of its original energy during propagation from source to shore. Wave Dispersion The time from when the storm starts to when the waves first reach a shoreline, some distance R away from the storm front is t ob = R C g = 4πR gt (20) Examples of dispersion: 1. Barber and Ursell (1948) - Cornwall coast; 2-hourly spectra for 2.5 days documented a tropical cyclone source of the east coast of the U.S. (Komar Figs. 5-32 and 5-33) 2. Wiegel and Kimberly (1950) - Oceanside, CA coast; southerly swell sourced from (see Google Earth Polygon). 3. Munk and Snodgrass (1957) - Guadalupe Island, Baja CA coast: witnessed swell from Indian Ocean, 15000 km away! These were the studies that verified the group velocity concept - then later, observations were made regarding this through-group movement of individual waves in a group rectangle. Wave Energy Losses Amazingly, very little energy is lost as waves traverse these great distances. See Komar Fig. 5-34 to illustrate spectra of a storm at various distances from the front. Four sources of wave energy loss have been suggested. Viscous Damping - Mathematical relationships have been worked out to compute the amount of viscous damping of waves during propagation, and the result is wave height decays exponentially as a function of time and period to the -4 power. ( ) 32π 4 νt H = H i exp g 2 T 4 (21) 12
This may help explain why short period waves have a tendency to disappear, while long period waves persist. But the result is still negligible, as illustrated by the wave height half life diagram (Komar Fig. 5-35) and equation below. t 1/2 = 0.0088 g 2 T 2 4π 2 ν (22) This shows that it would take a 1 sec period wave about 4.5 hours (12 km) of travel to decrease its height by 1/2, and a 5 sec period wave about 2600 hours (37,000 km) of travel to decrease its height by 1/2! Other Sources of wave energy loss: angular spreading, contrary winds, wave wave interactions/breaking. A nice summary of deep-water wave propagation is provided in a succinct paragraph on p. 183 of the Komar text. 3.5.2 Shoaling Waves moving from deep to intermediate/shallow water change their shape and characteristics significantly. Wave velocity and wavelength decrease, while height increases to conserve wave energy flux. Period remains the same, thank goodness. Just offshore of the breaker zone, the waves have peaked crests and broad troughs; a very different appearance to their deep-water sinusoidal form. Here, we will explore the details of shoaling transformation. Wave Length and Celerity: A commonly accepted relationship, in Airy wave theory, for wavelength in intermediate water ( 1 > h 2 L > 1 ) was provided 20 by Eckart (1952): ( 2πh L = L [tanh L )] 1 2 (23) Since wave period remains constant, and recalling the general and deep-water expressions for wave celerity (Eqns. 6 and 7 in this set of notes), we can safely say that C C = L L = tanh ( ) 2πh = L [ tanh ( 2πh L )] 1 2 (24) 13
which helps explain the behavior of the non-dimensional quantities graphed in Komar s Figure 5-21. Wave length and velocity systematically decrease with decreasing water depth. The shoaling parameter, n, changes smoothly from 1/2 to 1. Since C g /C is the product of n and C/C, the group speed, when entering intermediate water, initially increases, then decreases to coincide with the individual wave speed in shallow water. Wave Height: Figure 5-21 in the Komar text also illustrates the variation in heights of shoaling waves, which can be understood by noting that wave energy flux is constant during the shoaling process: P = ECn = (ECn) = constant (25) A simple algebraic rearrangement of the equated flux relationships, knowing the relationship for wave energy density provided in Eqn. 14, yields ( H 1 = H 2n ) 1/2 C (26) C which explains why orthogonally directed waves tend to increase their height during shoaling it is a direct compensation for the slowing of individual waves and the need to keep a constant wave energy flux. The waves convert a significant fraction of their kinetic energy to potential energy. It is noted, however, that when waves initial enter water of intermediate depth, the height decreases initially, then increases to the break point. This temporary reduction in wave height is associated with the temporary increase in wave group speed, discussed above. As the wave group velocity increases (kinetic energy), the wave energy density necessarily decreases, which is manifested in a decrease in potential energy (wave height). Wave Steepness: This is a fairly straightforward consequence of the combined shoaling behavior of wave height and wave length. Steepness initially decreases upon entry to intermediate water depth, then rapidly increases until the instability condition associated with wave breaking. 14
3.5.3 Refraction At this point it is important to make a distinction between wave crests and wave rays: 1. Wave crests are the line segments that connect the peaks (or troughs) of a wave field. The crests are visible to the observer. 2. Wave rays are the lines orthogonal (perpendicular) to the wave crests, which represent the direction of wave propagation. Waves that approach a coast with their wave crest oriented at an angle to the shoreline orientation will refract when entering intermediate and shallow water. This is visible in the curved appearance of the wave crests and wave rays outlined on the photos shown in class. Straight coasts with shore parallel bathymetric contours will cause obliquely approaching waves to become more shore parallel in shallow water, producing a fan-like appearance in map view. Refraction occurs because of the dependence of shallow water wave speed on water depth: 1. Waves travel more swiftly in deep water. 2. If waves are approaching at an angle to a straight coast, the deeper part of the wave crest will propagate faster than the shallow part of the wave crest. 3. The result is a bending of the wave ray, as the deeper part of the crest sweeps through a greater arc length than the shallow part of the crest, per unit time. The refraction process is analogous to the bending of light rays and governed by the same physical principle Snell s Law sin α 1 C 1 = sin α 2 C 2 = constant (27) In Equation 27, it is obvious that wave speed relates to direction accounting for the refractive behavior visible in the photos shown in class. It should be noted that this process has a profound effect on the wave energy flux distribution on the coast. 15
1. Energy flux per unit length of wave crest is not necessarily conserved. The energy between adjacent wave rays, separated by a distance s, in deep water must is maintained as those rays converge or diverge as a result of refraction to a separation distance of s in shallow water. 2. This can lead to a decrease in wave height during the shoaling and refraction process. 3. This process is sometimes referred to as wave crest stretching. We can summarize the effects of shoaling and refraction with the appropriately named shoaling coefficient, K s, and refraction coefficient, K r, which act to modify the wave height via the following relationship H H = K s K r = [ 1 2n C C ] 1/2 [ s ] 1/2 (28) Simple geometric considerations allow for the simplification of K r below s s = cos α cos α s (29) Examples shown in class demonstrate the effects of shoaling and refraction of a 2m 10s wave over linearly shallowing bathymetry. 3.5.4 Diffraction Lateral translation of energy along a wave crest. Most noticeable where a barrier interrupts a wave train creating a shadow zone. Energy leaks along wave crests into the shadow zone. Also by analogy to light, Huygen s Principle explains the physics of diffraction through a superposition of point sources along the wave crest. 3.5.5 Numerical Models of Wave Propagation, Shoaling, Refraction, and Diffraction Monochromatic wave models Ray tracing models, grid models; e.g. REFDIF (Kirby and Dalrymple, 1986) Spectral wave models e.g. SWAN (Booij et al., 1999) 16
3.6 Breaking The process of wave breaking can be thought of as the release of energy, derived from the wind, along the narrow coastal zone. It leads to geomorphic work done by wind, really, which is translated through medium of water. Wave breaking is responsible for the processes which control beach morphology: (1) nearshore current generation, and (2) sediment transport. 3.6.1 Nearshore Wave Breaking As waves shoal into shallow water, the wave height H increases and the wave length L decreases, dramatically increasing the steepness H/L. This cannot continue indefinitely something has to give. Common misconception: breaking is a result of waves dragging on the bottom, then trip forward due to friction NO! In reality, friction plays a very small role in the dissipation of wave energy. Computer simulations that completely neglect friction can still produce breaking waves. A wave breaks when it becomes overly steep, because the velocity of water particles in the wave crest exceed the velocity of the wave form! Breaker Types The three (or four) main breaker types are: 1. Spilling Breakers display a cascading face of bubbles and foam after peaking and initiating the breaking process. 2. Plunging Breakers abrupt pattern of peaking to a vertical face, overcurling, and plunging downward and forward to unload energy in a very concentrated portion of beach 3. Surging Breakers during the peaking process, the base of the wave destabilizes and surges forward, causing the wave top to implode/collapse 4. Collapsing Breakers not often witnessed and difficult to identify, this wave breaking type is thought to be intermediate between Plunging and Surging. Iribarren Number / Surf Similarity Parameter 17
In reality, breaker types transition from one to the next through a continuum, but in general the type of breaking style correlates well with the ratio of beach slope to wave steepness. This concept was explored by Battjes (1974), and in so doing, he introduced deep-water and shallow-water forms of the Iribarren Number (ξ and ξ b, respectively), which has/have since been referred to as the Surf Similarity Parameter ξ = ξ b = S (H /L ) 1/2 (30) S (H b /L ) 1/2 (31) Spilling breakers tend to occur on gently sloped beaches with waves of high steepness (ξ < 0.5, ξ b < 0.4) Plunging breakers tend to occur on intermediate beaches with waves of intermediate steepness (0.5 < ξ < 3.3, 0.4 < ξ b < 2.0) Surging/Collapsing breakers tend to occur on high gradient beaches with waves of low steepness (3.3 < ξ, 2.0 < ξ b ) Breaking Wave Condition It is convenient to identify a critical condition at which waves break attempts at this have resulted in the following ratio which relates breaking wave height H b to the breaking wave depth h b. γ b = H b h b (32) Laboratory experiments have revealed that this value is not a constant, but varies considerably with wave steepness, H b /gt 2, and beach slope, S. This behavior is illustrated in Komar s Fig. 6-8. For a given wave steepness, higher beach slopes yield higher γ b values. Logically, this observation has led to an attempt to link γ b to the deep-water Iribarren number γ b = 1.2ξ 0.27 (33) 18
Several breaker height prediction relationships have been generated based on deep-water wave conditions including: H b H = H b H = H b H = 1 3.3(H /L ) 1/3 (34) 0.563 (H /L ) 1/5 (35) 0.46 (H /L ) 0.28 (36) These various forms of the breaker height prediction relationship are plotted in Komar s Fig. 6-9. Rearranging Eqn. 35, derived by Komar and Gaughan (1972), we can obtain a relationship for breaking wave height as a function of deep-water height and period H b = 0.39g 1/5 (T H 2 ) 2/5 (37) This relationship is plotted and compared to 3 data sets in Komar s Fig. 6-10. The data span 3 orders of magnitude of breaker heights and are remarkably well-behaved. Plunge Distance As shown in Komar s Fig. 6-11, the ratio of plunge distance to breaking wave height tends to decrease with increasing beach slope. 3.6.2 Surf Zone Wave Decay Within the surf zone, the wave energy dissipation pattern depends on morphology of the beach. General Characterization of Wave Energy Dissipation in Surf Zone On steep, reflective beaches, wave breaking (and energy dissipation) is concentrated through plunging breakers. The broken wave surges up the beach 19
as runup. An example of this setting, provided in class, was the Vilano beach site, just north of St. Augustine Inlet in North Florida. On low-slope, dissipative beaches, there is an extensive, wide surf zone over which spilling breakers dissipate energy. In this setting, at any time, several broken wave bores, and smaller unbroken waves, are visible. An example of this setting, provided in class, was the Anastasia Island site, just south of St. Augustine Inlet in North Florida. In general, we note that where there is a smooth, continuous beach profile, the pattern of wave energy dissipation (breaking) is fairly uniform. In contrast, where there are alongshore bars and troughs, wave dissipation (breaking) is concentrated on the bars and relatively absent over the troughs. Understanding the patterns of wave decay in the surf zone is important for two significant reasons: 1. Wave energy dissipation is inversely related to the alongshore pattern of wave energy delivery so it can help identify relative vulnerability of coastal property. 2. Wave energy expenditure is partially transformed into nearshore currents, which are responsible for sediment transport and beach morphologic modification. In general, natural beaches are subjected to a wide spectrum of wave heights resulting in a wide range of wave breaking conditions larger waves break in deeper water, and smaller waves break in shallower water. This is one of the parameters that is often reported by spectral wave models such as SWAN. Breaking Wave Distributions A now-famous study by Thornton and Guza (1983), used Torrey Pines Beach, a fine sand beach with minimal bars and troughs to study distributions of breaking wave heights within the surf zone on a natural beach. Wave staffs and current meters were emplaced to make measurements from 10 m water depth to inner surf zone. This study documented the following: 1. The histograms of breaking wave height distributions illustrate a greater fraction of broken waves at shallower depths. 2. Histograms of all waves in nearshore (broken and unbroken) show skewed distributions many small waves and few large waves comparable to 20
that of a Rayleigh distribution, which also describe the distribution of deep water wave heights (Longuet-Higgins, 1975). 3. Histograms of broken waves only (cross hatched pattern in Komar s Fig. 6-12) show a more uniform distribution. Energy Saturation Furthermore, in their examination of root-mean-square wave height at various depths in the surf zone, over four different days, Thornton and Guza (1983) found that independent of deep water wave height, waves in the surf zone decay in the same manner, i.e. following the H rms = 0.42h line. Waves are described to be saturated with respect to their energy content within the inner surf zone, where local water depth h controls the wave height. After initial breaking at H b = 0.78h b, surf wave heights appear to decay to the γ = 0.42 ratio, as shown in Komar s Fig.6-14. Models of Wave Height Decay This material is covered on pp. 222-232 of the Komar text. The basis for most predictive models aimed at understanding wave height decay and energy dissipation in the surf zone targets the energy flux relationship (ECn) x = ε(x) (38) This relationship states that the cross shore spatial rate of change of wave energy flux is a function of cross shore position. Regarding the mechanisms of energy dissipation, only a small fraction is expended in frictional losses the bulk of the dissipation occurs as a result of wave breaking and the associated generation of turbulence. Some of the models used in wave height decay/energy dissipation are listed below: 1. Dally et al. (1985) proposed a model where the loss in wave energy per unit area per unit time, ε, is proportional to the difference between the local wave energy flux and the stable wave energy flux. This model has been calibrated with laboratory measurements and since has been incorporated into the USACE model known as RCPWAVE. 21
2. Following the work of Battjes and Janssen (1978), Thornton and Guza (1983) proposed that the energy dissipation term, ε, is proportional to the wave frequency and the cube of the broken wave bore height, while being inversely proportional to the local depth. Undertow 3.6.3 Set-up and Set-down During the breaking process, within the surf zone shoreward of initial breaking, there is a rise in mean water level which slopes up landward. This phenomenon is known as Set-Up. There is a corresponding depression in mean water level oceanward of the break point, known as Set-Down. This process was brought to public attention during the 1938 Hurricane along the Northeast Atlantic coast. Shoreline elevations in exposed regions were 1m higher than in sheltered regions a fact that could not be explained by storm surge, which would elevate shorelines uniformly. Radiation stresses, or the momentum flux of waves, are responsible for this phenomenon. (Longuet-Higgins and Stewart, 1964) In a nutshell, the onshore component of the radiation stress, S xx (the cross-shore component of shoreward directed momentum flux), is balanced by a seaward slope of the water surface providing a pressure gradient or force, in the region of set-up. Recall from Eqn. 40, which, in shallow water, becomes [ 2kh S xx = E sinh(2kh) + 1 ] = E(2n 1 2 2 ) (39) The cross-shore momentum balance is then S xx = 3 2 E = 3 16 ρgh2 (40) S xx x + ρg(η + h) h x = 0 (41) 22
where η is difference in water surface elevation between still water and with waves present, i.e. set-up or set-down. In summary, and paraphrasing from Komar s p. 234, both Set-Down and Set-Up are produced by the shoaling and decay of waves. The momentum flux is conserved by balancing the gradient in the onshore component of the radiation stress, S xx, with the pressure field of the sloping sea surface. This has been verified in laboratory measurements. Notably, the width of the surf zone increases by 20% due to wave set-up. The maximum set-up is at the shoreline and is 29% of the breaking wave height. Set-down at breaker is 5% of breaking wave height. Neglect setup much further than the breaker line. Although there is a lack of dependence on H in set-up, for larger waves, setup begins further seaward and slope is constant so the result is a higher set-up on the beach. 3.6.4 Wave Runup Wave swash represents cutting edge of the ocean s geomorphic buzzsaw, and we want to know how the wave conditions and beach morphology affect the run-up. Methods Several techniques have been employed to measure run-up. 1. Guza and Thornton (1981, 1982) used an 80-m long resistance wire stretched across the beach profile held up at 3 cm height by non-conducting supports same general idea as electronic wave staffs discussed earlier. 2. Holman and Sallenger (1985) used video measurements to document the swash excursion. The run-up edge can be examined frame by frame. 3. Time stack video method by Holland and Holman (1993), on which the slopes of the linear features represent the speed of the run-up bores. Components of Run-Up 1. Wave set-up. 2. Swash of incident waves. 3. Infragravity component (> 20s) of swash oscillations. 23
Previous Estimates of Run-Up Hunt s (1959) estimate: R 2% = 8H s S, from the examination of run-up on rock rubble structures in The Netherlands Battjes s (1971) estimate: R 2% = H s Cξ p, related run-up to Iribarren number, ξ p = S/ H s /L, and an experimentally calibrated coefficient, C, which ranges in value from 1 to 4. This relationship was also used to explore the effect of substrate on run-up, presented in Komar s Fig. 6-31. Run-up was found to be approximately twice as great on a smooth surface, as compared to a rock-covered surface. Guza and Thornton s (1982) estimate of avg. of highest one-third of swashes: R s = 0.7H. This study revealed some unexpected results most of the swash excursion occurred at periods much greater than the incident wave period range. By spectral analysis of the run-up records, they were able to separate into 2 components: 1. An incident wave bore direct swash component, which displayed no dependence on deep-water wave height 2. An infragravity wave component at periods greater than 20 seconds, which DID display a dependence on deep-water wave height 3.6.5 Infragravity Water Motions Infragravity energy is derived from incident wave energy, but how? This material is well covered in Komar pp. 249-269. Evidence from Surf Zone Currents Edge Waves References Adams, P., D. Inman, and N. Graham (2008), Southern california deep-water wave climate: Characterization and application to coastal processes, Journal of Coastal Research, 24 (4), 1022 1035. 24
Allan, J., and P. Komar (2006), Climate controls on us west coast erosion processes, Journal of coastal research, 22 (3), 511 529. Battjes, J. (1974), Surf similarity, Proc 14th International Conference on Coastal.... Battjes, J., and J. Janssen (1978), Energy loss and set-up due to breaking of random waves, Proceedings of the 16th international conference on Coastal Engineering, pp. 569 587. Booij, N., R. Ris, and L. Holthuijsen (1999), A third-generation wave model for coastal regions. i- model description and validation, Journal of Geophysical Research, 104 (C4), 7649 7666. Dally, W., R. Dean, and R. Dalrymple (1985), Wave height variation across beaches of arbitrary profile, Journal of Geophysical Research - Oceans, 90 (C6), 11,917 11,927. Eckart, C. (1952), The propagation of gravity waves from deep to shallow water, Gravity waves, National Bureau of Standards(Circular No. 521), 165 173. Kirby, J., and R. Dalrymple (1986), An approximate model for nonlinear dispersion in monochromatic wave propagation models, Coastal Engineering, 9, 545 561. Komar, P., and J. Allan (2008), Increasing hurricane-generated wave heights along the us east coast and their climate controls, Journal of coastal research, 24 (2), 479 488. Komar, P., and M. Gaughan (1972), Airy wave theory and breaker height prediction, Proceedings of the 13th Coastal Engineering Conference, ASCE, pp. 405 418. Longuet-Higgins, M. (1975), On the joint distribution of the periods and amplitudes of sea waves, Journal of Geophysical Research. Longuet-Higgins, M., and R. Stewart (1964), Radiation stresses in water waves; a physical discussion, with applications, Deep-Sea Research, 11, 529 562. Thornton, E., and R. Guza (1983), Transformation of wave height distribution, J. Geophys. Res, 88 (10), 5925 5938. 25