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## Material Information- Title:
- Nearshore Infragravity Wave Generation: A Numerical Model and Parametric Study
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- CZARNECKI, EILEEN M (
*Author, Primary*) - Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
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Bathymetry ( jstor ) Boundary conditions ( jstor ) Deep water ( jstor ) Low frequencies ( jstor ) Modeling ( jstor ) Shorelines ( jstor ) Stress waves ( jstor ) Water depth ( jstor ) Waves ( jstor )
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- Copyright Eileen M Czarnecki. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 8/31/2006
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- 649814524 ( OCLC )
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NEARSHORE INFRAGRAVITY WAVE GENERATION: A NUMERICAL MODEL AND PARAMETRIC STUDY By EILEEN M. CZARNECKI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Eileen M. Czarnecki This document is dedicated to my father, William John Czarnecki. ACKNOWLEDGMENTS First and foremost, the author wishes to thank her advisor and supervisory committee chairman, Dr. Andrew Kennedy, for his support, patience, guidance, instruction, and assistance throughout this study. Gratitude is extended to the other members of the committee, Dr. Robert Dean and Dr. Robert Thieke, for their assistance. The author also wishes to thank the entire faculty of the Civil and Coastal Engineering Department, who helped to shape her intellectual development as an engineer, both during undergraduate and graduate studies. Finally, the author would like to thank her family and friends for their love and support. Special thanks go to Elaine Czarnecki, Billy Czarnecki, Barbara Lewis, and Xiaoyan Zheng. TABLE OF CONTENTS page ACKNOW LEDGM ENTS ........................................ iv LIST OF TABLES .................. .......................................... ...... .. .............. viii LIST OF FIGURES ..................................... ix ABSTRACT.................................... xii CHAPTER 1 INTRODUCTION ................... .................. .............. .... ......... ....... 1.1. Definition and Importance of Infragravity W aves.................................................1 1.2 Objectives ................... ...................................... ............ ........ 1 1.3 Literature Review ................................................ ........ 1 1.4 Thesis O outline ...................................... ......................... .... ....... .4 2 NUMERICAL MODEL .................6............. .......... 6 2.1 C onceptual O verview .................................................. ...............6 2.2 Bathymetry ........................................ ........6 2.3 W ave Spectrum ................... ...... .... .... ........ ............... .......... 7 2.4 Wave Transformation ........................ ............... ...............9 2.5 Long W ave G eneration................................................. 12 2.5.1 Governing Equations .................................... ...... ............ ........ 12 2.5.2 Infragravity W ave Surface Elevation.............. .................................... 13 2.5.3 Boundary Conditions...................................................... 17 2.5.3.1 Shoreline boundary condition ................. ............................17 2.5.3.2 Offshore boundary condition .................................. ....18 2.5.4 Derivation of Radiation Stress..................................23 2.5.4.1 Steady radiation stress ........................ ...... ................... 26 2.5.4.2 Low frequency radiation stress ................ ................. ...........27 2.6 Root Mean Square Long Wave Surface Elevation................ .................28 2.7 M ean W after Level ...................... ........ .... ..................28 2.8 Model Validation ......... ......... ......................30 v 3 NUMERICAL SIMULATION OF LOW FREQUENCY WAVE CLIMATE..........34 3.1 Use of the M odel ......................................... ...... ..... ...34 3.2 Parameters.............................. ...... ......... 35 3.3 Results...................... .... .. ............ ......... 35 3.3.1 B ase C ase....................................35 3.3.2 Offshore Significant Wave Height and Peak Period ................................37 3.3.3 Jonswap Peak Enhancement Factor............ ....................49 3.3.4 Deep Water Directional Width...............................50 3.3.5 Peak Wave Direction ............... .... ...................51 3.3.6 B ottom Friction .............................. ............................. 53 3.3.7 Bar Amplitude ............................................. .... .....54 3.3.8 Bar Width .......................................... .........56 3.3.9 Distance of Bar from Shore........................................... 57 3.3.10 W ater Depth at Offshore Boundary............ ....................58 3.3.11 Domain Length ...........___ ............ ......... .60 4 CONCLUSIONS ...................................................62 4.1 Summary...................... .. .......................62 4.2 D discussion and Conclusions ........................................... ............... 63 4.3 Recommendations for Further Work................................67 APPENDIX A DERIVATION OF EQUATION FOR INFRAGRAVITY WAVE SURFACE ELEVATION ...................................... ................................. ........ 69 A. 1 Second Order Partial Differential Equation..............................................69 A.2 Second Order Ordinary Differential Equation in the Frequency Domain...........70 B DERIVATION OF EQUATIONS FOR BOUNDARY CONDITIONS....................72 B 1 Shoreline B boundary Condition.............................................. 72 B.2 Offshore Boundary Condition............................... ................... 73 B.2.1 Characteristic Equations ............................... ............... 73 B.2.1.1 Incom ing bound wave ........................... ..... ............... 73 B.2.1.2 Outgoing free wave ............................. ............... 74 B.2.1.3 Combined Characteristic Equation................. .................74 B.2.2 Incoming Bound W ave Amplitude.................................. ...... ........75 B.2.3 Offshore Boundary Condition Equation............................................... 77 C VALUES OF ROOT MEAN SQUARE LONG WAVE SURFACE ELEVATION FOR EACH TEST CA SE .............................................................................79 LIST O F R EFER EN CE S ..................................... ....................................................... ....... 83 SUPPLEM ENTARY REFERENCES ....................................................... 86 BIOGRAPHICAL SKETCH .................................................. ............... 90 LIST OF TABLES Table page 3-1 List of test cases with corresponding results section and equations ........................34 C-i Root mean square wave surface elevation for each test case, at the offshore boundary and at 25 m seaward of the shoreline cutoff depth............... ...............79 LIST OF FIGURES Figure page 2-1 Definition sketch of coordinate axes............. ....... .................... 13 2-2 Definition sketch of the incoming short wave direction 0 ........................15 2-3 Definition sketch of infragravity wave direction: incoming bound wave angle Oin, outgoing free wave angle ,,ut, and actual outgoing free wave angle Oout,actual = ot + 7 ................................. .................. ..............19 2-4 Overlay of analytic and numerical solutions................................................ ...... 32 3-1 Model results for the base case: (a) root mean square long wave surface elevation (r-hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. 36 3-2 Effect of variation of peak wave period (Tp) (for offshore Hs=0.4 m) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (H). ......................................................38 3-3 Effect of variation of peak wave period (Tp) (for offshore Hs =0.7 m) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (H). ......................................................39 3-4 Effect of variation of peak wave period (Tp) (for offshore Hs =1.0 m) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (H). ......................................................40 3-5 Effect of variation of peak wave period (Tp) (for offshore Hs =2.0 m) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (H). ......................................................42 3-6 Effect of variation of peak wave period (Tp) (for offshore Hs =3.0 m) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (H). ......................................................43 3-7 Effect of variation of offshore significant wave height (Hs) (for T,=4 s) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (H). ......................................................44 3-8 Effect of variation of offshore significant wave height (Hs) (for T,=6 s) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (H). ......................................................45 3-9 Effect of variation of offshore significant wave height (Hs) (for T,=8 s) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (H). ......................................................46 3-10 Effect of variation of offshore significant wave height (Hs) (for T,=10 s) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (H). ......................................................47 3-11 Effect of variation of offshore significant wave height (Hs) (for T,=12 s) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (H). ......................................................48 3-12 Effect of variation of Jonswap peak enhancement factor (y) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (H ). ............................................................... 49 3-13 Effect of variation of deep-water directional width (dir-widtho) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (H ).. ............................. .................. ........ 51 3-14 Effect of variation of peak wave direction (0,) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs).................52 3-15 Effect of variation of bottom friction coefficient (ff) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs)...........53 3-16 Effect of variation of bar amplitude (a2) on (a) root mean square long wave surface elevation (r-hatms), (b) significant wave height (Hs), and (c) bathymetric profile. ............................. .................. ......... 55 3-17 Effect of variation of bar width on (a) root mean square long wave surface elevation (r-hatms), (b) significant wave height (Hs), and (c) bathymetric profile. ......................................... ........................................ . 5 6 3-18 Effect of variation of distance of bar from shore (x,) on (a) root mean square long wave surface elevation (r-hatms), (b) significant wave height (Hs), and (c) bathym etric profile.. ............................... ............................... 58 3-19 Effect of variation of water depth at offshore boundary offshorer) on (a) root mean square long wave surface elevation (r-hatrms), (b) significant wave height (Hs), and (c) bathym etric profile.. .............................................. 59 3-20 Effect of variation of domain length (ld) on (a) root mean square long wave surface elevation (r-hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. .........................................................61 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science NEARSHORE INFRAGRAVITY WAVE GENERATION: A NUMERICAL MODEL AND PARAMETRIC STUDY By Eileen M. Czarnecki August 2006 Chair: Andrew Kennedy Major Department: Civil and Coastal Engineering The purpose of this study was to develop and present a linear frequency-domain numerical model of nearshore infragravity wave generation on an alongshore-uniform beach, and then to utilize this model to investigate the importance of various parameters in the generation of these infragravity waves. The numerical model was composed of two parts: wave transformation and infragravity wave generation. Wave transformation was modeled using linear wave theory. Infragravity wave generation was based on the theory that these waves are forced by spatial changes in the radiation stresses associated with the incoming short wave groups, then released to become free waves when these short waves break near the shoreline, and subsequently reflected at the shoreline to become outgoing free infragravity waves. The total infragravity wave surface elevation was determined by summing all of the infragravity contributions due to difference interactions between pairs of incoming short wave spectral components with frequency differences in the infragravity range ( Af <: 0.05Hz). The results of the numerical simulation of the low frequency wave climate indicated the relative importance of various parameters in the generation of nearshore infragravity waves. These parameters affect the infragravity wave response in the following manner: a higher peak wave period leads to a higher magnitude of response; a higher offshore significant wave height leads to a higher magnitude of response; a narrower directional width leads to higher magnitude of response; increased bottom friction leads to a decreased magnitude of response; a shallower bathymetric profile leads to an increased magnitude of response; a higher bar amplitude leads to decreased magnitude of response, especially near the shoreline; a wider bar leads to a slightly higher magnitude of response; the bar being closer to the shore results in a narrower response pattern; obliquely incident waves lead to a decreased magnitude of response at the offshore boundary, although peak wave direction has little effect on response near the shoreline; a narrower frequency spectrum leads to a slightly decreased magnitude of response. CHAPTER 1 INTRODUCTION 1.1. Definition and Importance of Infragravity Waves Infragravity waves, also known as long waves or surf beat, are defined as low- frequency waves, with periods typically between 20 and 200 seconds (Henderson and Bowen, 2002). In contrast to short-period waves, infragravity, or long-period, waves are not obvious from visual observation because they cause a very slow variation of sea surface elevation. Nevertheless, they are important in shallow water and contribute to the majority of water surface elevation variance at the shoreline (Holman and Bowen, 1984). Infragravity waves are thought to be important in the generation of nearshore currents (Bryan and Bowen, 1998), harbor resonance (Janssen et al., 2003), nearshore sediment transport, and bar formation (Bowen and Inman, 1971; Holman and Bowen, 1982). 1.2 Objectives The objectives of the current study are two-fold. The first objective is to develop and present a linear frequency-domain numerical model of nearshore infragravity wave generation on an alongshore-uniform beach. The second objective is to utilize the model to investigate the effects of various parameters on nearshore infragravity wave generation. 1.3 Literature Review The slow temporal variation of sea surface elevation in the nearshore zone was first observed by Munk (1949) and Tucker (1950). Their observations showed that the low- frequency infragravity motion was correlated with the incoming short-wave groups. Longuet-Higgins and Stewart (1962, 1964) later hypothesized a mechanism for the generation of infragravity waves: that these waves are forced, or bound, by spatial changes in the radiation stresses associated with the incoming short wave groups, then released to become free waves when these short waves break near the shoreline, and subsequently reflected at the shoreline to become outgoing free infragravity waves. Since that time, this theory has been explored extensively in analytical, numerical, field and laboratory studies. Recently, an extensive cross-correlational study of laboratory data conducted by Janssen et al. (2003) confirmed that the incoming bound infragravity waves travel at the group velocity Cg of the incoming short wave groups and that the outgoing free infragravity waves travel at the shallow water wave velocity gh The numerical model presented in this thesis is based upon this radiation stress theory. The other main theory of long wave generation by incoming short wave groups involves a time-varying breakpoint (Symonds et al., 1982). The variation in wave amplitude due to the wave groups would cause these incoming short waves to break at different locations in a periodic manner. Theoretically, the time varying breakpoint could generate free waves radiating both in the onshore and offshore directions. However, research has shown that the long waves generated by a time-varying breakpoint are only a small contribution to the total infragravity wave energy, compared to those generated by spatial variation in radiation stresses associated with short-wave groups (List, 1992a, 1992b). In their cross-correlational study, Janssen et al. (2003) found no indications of infragravity waves being generated at the breaker bar. Therefore, the model presented in this thesis does not include a time-varying breakpoint. As mentioned previously, the incoming infragravity waves are bound by the incoming short wave groups. The outgoing free infragravity waves, depending on their frequency and alongshore wave number, may refract repeatedly and be trapped near the shoreline, traveling in the alongshore direction as edge waves, or they may continue to travel in the offshore direction as leaky waves (Schaffer, 1994). Free waves are more energetic than bound waves, both near the shore and in deeper water (Okihiro et al., 1992; Herbers et al., 1994). The cross-shore structure of infragravity waves shows an outgoing progressive wave at the offshore boundary and partial standing waves near the shoreline, where edge waves often predominate (Symonds et al., 1982). The model developed in this thesis produced the same cross-shore pattern of infragravity waves. Lippmann et al. (1999) found that shear waves (instabilities in the alongshore current) contribute to infragravity velocity variance; however, since shear waves do not significantly affect the infragravity wave surface elevation, they are not considered as a factor in the present model. Edge waves of the same frequency can resonate with each other, and thus, it is important to include friction in numerical models in order to prevent unbounded growth of infragravity waves (Reniers et al. 2002, Schaffer, 1993). Bathymetry is another important factor in infragravity waves. Herbers and colleagues (1995b) found that shoreline morphology affects the generation and reflection of free infragravity waves and that the shelf-wide topography affects the propagation and trapping of free infragravity waves. On barred beaches, edge waves tend to be trapped at the location of the bar (Bryan and Bowen, 1998). Sallenger and Holman (1987) found that the dominant peak of the infragravity wave surface elevation occurs at the bar crest. Edge waves, which propagate in the alongshore direction, may be either progressive or standing. Standing edge waves may be generated by topographic changes or obstructions in the seabed which can act as reflectors (Huntley et al., 1981). Obviously, when modeling infragravity waves, it is important to take into account the local bathymetry. The numerical model presented in this thesis is based upon the two-dimensional numerical model of infragravity wave generation developed by Reniers et al. (2002). Reniers and colleagues (2002) assumed an alongshore uniform bathymetry in their model and found that results of their numerical simulation compared well with field data gathered at Duck, North Carolina. Later, in 2003, Van Dongeren and colleagues explored whether modeling the bathymetry in two or three dimensions significantly affects the numerical simulation of low frequency wave climate at Duck, North Carolina. They found that long-shore variability in bathymetry had little effect on their results and concluded that, for bathymetries that do not have much long shore variability, it is valid to model infragravity wave generation in two dimensions. 1.4 Thesis Outline The current Chapter 1 begins with a brief description of infragravity waves and their importance. Next, the objective of the study is stated, followed by a literature review of research on infragravity waves. Chapter 2 presents a detailed description of the numerical model. Chapter 3 presents results of the model, used to investigate the low frequency wave climate under varying conditions, in order to determine the importance of various parameters and their effects on infragravity wave generation. Chapter 4 includes a summary, conclusions, a discussion of results, and recommendations for further work. Appendix A and Appendix B give detailed derivations of the main equations that constitute the numerical model. Appendix C tabulates the results of the numerical model for each test case described in Chapter 3. The list of references cited in 5 this thesis is followed by supplementary references pertinent to the topic of infragravity waves. CHAPTER 2 NUMERICAL MODEL 2.1 Conceptual Overview The numerical model developed in this chapter is composed of two main parts: (1) wave transformation, and (2) infragravity wave generation. The two main inputs for the wave transformation model were the bathymetry and the wave spectrum (which was composed of many frequencies and directions). Equations for wave transformation were based on linear wave theory. The numerical model of infragravity wave surface elevation was based upon the model presented by Reniers et al. (2002), with some modifications. This model mathematically describes the generation of infragravity waves, which are composed of incoming bound waves generated by pairs of incoming short wave components and outgoing free waves that are generated by reflection of the incoming waves at the shoreline. 2.2 Bathymetry The bathymetric profile was assumed to be uniform in the alongshore (y) direction and was modeled in the cross-shore (x) direction as an equilibrium beach profile with an alongshore bar superimposed. The equation for the equilibrium beach profile (Dean and Dalrymple, 2002) is h(x)= Axy (2.2.1) where h represents water depth, x represents the cross-shore coordinate, and A is the profile scale factor. The equation for the bar, given by Yu and Slinn (2003) to approximate the alongshore bar measured at Duck, North Carolina, is h(x) = -a, exp 5 x x (2.2.2) where a2 is the bar amplitude and xc is the distance of the center of the bar from the shoreline. In equations (2.2.1) and (2.2.2), x = 0 at the onshore boundary. However, since the numerical model requires that x = 0 at the offshore boundary, the order of the x vector was reversed according to x = Xshore -xe (2.2.3) in order to satisfy this condition. Equations (2.2.1) through (2.2.3) were combined to form the equation for the bathymetric profile h(x') = Ax' -a exp (2.2.4) with an onshore water depth of 0.25 m and a maximum onshore slope of 1/20. In equation (2.2.4), c, is the bar width coefficient. 2.3 Wave Spectrum The wave spectrum describing the incoming short waves was composed of a frequency spectrum and a directional spectrum. The frequency spectrum was modeled as a Jonswap frequency spectrum. The expression for the Jonswap spectrum is given by Kamphius (2000) as -4 ag y2 exp(a) 5 f SA, f )= aexp -2 (2.3.1) (2;r)45 4 fp In the above equation,fis the frequency,f, is the peak frequency, g is the gravitational constant, y is the peak enhancement factor (with a higher value of y implying a narrower, more peaked frequency spectrum), a is a coefficient related to frequency, and aj is a coefficient related to the wave-generating conditions. The coefficient a is given by a 22 (2.3.2) where 3 = 0.07 for f < fp, and 3 = 0.09 for f > fp. (2.3.3) The coefficient aj is given by a = 0.076 gFJ 022 (2.3.4) where F is the fetch length and U is the wind velocity. In this model, fetch length was assigned the value of F = 1000 m, and wind velocity was assigned the value of U= 5 m/s. The offshore significant wave height Hs was specified as an input value, which was then converted to the root mean square wave height (Hrm,). H,,,,, = H,2 (2.3.5) The frequency spectrum was then proportioned to the root mean square wave height (Hrm,) according to the equation H,,,,, = (2.3.6) where p is the density of seawater and E is the total spectral energy, equal to the sum of the energies of each individual frequency component. The spectral frequency resolution was set at 0.00083 Hz, to give a total of 602 components. The directional spectrum was calculated as D(O)= cos"(0-0) (2.3.7) In the above equation, m determines the directional width of the spectrum (with a higher value of m corresponding to a narrower spectrum), 0 is the incoming short wave direction, and O, is the peak incoming short wave direction. The peak wave direction (0p) and the deep water directional width (dir-widtho) were specified as input for the directional spectrum. The deep water directional width (dir-widtho) was related to the directional width at the offshore boundary (dir-widthi) by the following equation: ko sin(dir 1i /,h.,) = k, sin(dir 1i //h,) (2.3.8) where ko and k1 are the wave numbers in deep water and at the offshore boundary, respectively. The value of m in equation (2.3.7) was chosen to match the directional width at the offshore boundary. The directional spectrum was then normalized such that the area under the spectrum was equal to one. Finally, the frequency and directional spectrums were combined into one spectrum, with one direction per frequency, according to the following equation: S(f,0)= S,(f)D(O) (2.3.9) 2.4 Wave Transformation The two main inputs for the wave transformation program are the bathymetry and the wave frequency-directional spectrum. For each component of the frequency- directional spectrum, the wave length, celerity, and wave number, wave angle, and energy dissipation were calculated. From linear wave theory (Dean and Dalrymple, 1991), deep water wave length Lo and deep water wave group velocity cgo were L= gT(2.4.1) Cgo = L1T (2.4.2) where Tis the wave period (s), calculated as the inverse of the wave frequency (Hz). The wave length L, the wave celerity c, the wave number k, and the wave group velocity cg were calculated (from linear wave theory) for each component frequency, at each cross-shore location: L = L, tanh 2z (2.4.3) c= LT (2.4.4) k =2 (2.4.5) c = c I+-kh (2.4.6) g 2 sinh(2kh) (2.4.6) The wave angle 0 was calculated for each wave component, at each cross-shore location, from Snell's law, as i = sin- C *sin (2.4.7) where j represents the cross-shore location. Energy dissipation was based upon the model presented by Reniers and Battjes (1997). The dissipation S is given by the differential equation d (Ecg cos)= S (2.4.8) dx where S = aH 2Qb (2.4.9) 4T max P In the above equation, p is the density of water, g is the gravitational constant, Tp is the peak period, a is a coefficient (equal to one in this model), Qb is the fraction of breaking waves, and Hmax is the maximum wave height. The fraction of breaking waves Qb (Battjes and Janssen, 1978) is given by the implicit relationship Qb = exp b )2 (2.4.10) Hrms1 max The maximum wave height is given as H 0.88 tanhi2 (2.4.11) k k 0.88) with the wave breaking parameter yb given as b =0.5 + 0.4tanh 3Hrso"" (2.4.12) In the above equation, Lo and Hrmso (root mean square wave height in deep water) were defined from the peak values in the spectrum, with rms0 rms,offshore Cg peak (2.4.13) cg0, peak The Newton-Raphson method of root-finding (Hornbeck, 1975) was used to solve equation (2.4.10). The energy dissipation and thus the wave height at each cross-shore location was calculated for each wave component by using forward differences to estimate the differential equation (2.4.8). 2.5 Long Wave Generation 2.5.1 Governing Equations The continuity equation and the cross-shore and alongshore momentum equations form the basis of the numerical model. The equation for infragravity wave surface elevation was derived from these three basic linearized long wave equations, given below. These equations assume that q, u, and v are small and therefore, that the nonlinear terms can be neglected. Long wave continuity equation: aq 0(hu) 0(hv) + + = 0 (2.5.1) at x 9y Cross-shore momentum equation: au a7n aSxx aSx ph -+pgh (2.5.2) at ax ax ay Alongshore momentum equation: Qv Qy 9Sxy 9Syx ph +pgh- as aS- (2.5.3) at 9y y ax In the above equations, r represents the infragravity wave surface elevation, u represents the cross-shore velocity component, v represents the along-shore velocity component, t represents time, and x and y represent the coordinate axes. In this model, the x-axis is oriented in the cross-shore direction, with x=0 at the offshore boundary. The y-axis is oriented in the along-shore direction. A definition sketch of coordinate axes is given in Figure 2-1. S Sy, and Syy represent the radiation stresses, which force the infragravity waves. Radiation stress symbolized as Smn represents the flux of the m-component of momentum in the n-direction. Thus, for example, Sy represents the flux of the x- component of momentum in the y-direction. y 10 O c Figure 2-1. Definition sketch of coordinate axes 2.5.2 Infragravity Wave Surface Elevation Equations (2.5.1)-(2.5.2) were combined into one equation. This resulted in the second order partial differential equation for infragravity wave surface elevation: 1 a2u dh7 a8 ch 1 aS 2a2S 5$2S( +h + +h + + (2.5.4) g t2 g at X2 dx ax ay pg dx 8x8y ay 2 with the linear damping term a7 added in order to prevent unbounded growth in the g at case of infragravity wave resonance. The term [t is a resistance factor, given by p(x) = f,/V(x) (2.5.5) and ff is an empirical coefficient of bottom friction. The derivation of equation (2.5.4) is given in Appendix A. Equation (2.5.4) was transformed into a series of second order ordinary differential equations (2.5.6) in the frequency domain so that it could later be solved using finite difference representations of derivatives with respect to x. h d dh d + 4;-2Af2 iu27rAf hAk2 2 h-+- + --- hAk 7 dx2 dx dx) g g Y drA d2S +2iAk y +k 2S (2.5.6) dx2 y dx y The derivation of equation (2.5.6) is given in Appendix A. This transformation necessitates defining functional representations of infragravity wave surface elevation r, and radiation stresses Sx, Sx, and Syy: )(x, y, t, f,, f,, k,,, k,2) = I r(x, f,, f2, k,,, k,,) exp[i(2rAft Aky)+ (2.5.7) Sx(x,y,t, fl, f2k,k,ky2) l=SI(x, f,ky,ky2)exp[i(2;Aft -Ak yY)]+* (2.5.8) S,(x,y,t, f,, f,1,f,,k,2)= I (x, f,, f,k,k 2)exp[i(2Aft-Ak y )]* (2.5.9) 2 S,(x, y, t,,, f,, k, k2) = (x, f,, f, k, k) exp[i2Aft Ak y)]+* (2.5.10) 2 where is the complex conjugate, and frequency and wave number are from a combination of two incoming short wave spectral components (indicated by the subscripts 1 and 2). The infragravity wave frequency is defined as Af = f, f2 (2.5.11) The radial frequency is defined as cr = 2.Af (2.5.12) The alongshore wave number is defined as Aky = ky = ky2 = k sin -k sin (2.5.13) where 0 represents the incoming short wave direction. The cross-shore wave number (which will later be used to calculate the amplitude of infragravity wave surface elevation 77 and the amplitudes of the radiation stresses S, Sx,, and S,) is defined as Ak, = k, k = k, cosO, -k, cosO, (2.5.14) A definition sketch of the incoming short wave direction 0 is given in Figure 2-2. The wave direction of each incoming short wave component was between -7T/2 and 7T/2 radians. Wave angles were measured with respect to the positive x-axis, according to the right-hand-rule (counterclockwise positive and clockwise negative). 7c/2 x 7/2 Figure 2-2. Definition sketch of the incoming short wave direction 0 In order to numerically solve equation (2.5.7), derivatives of i (and similarly of S, and S,) were estimated using second order central differences. = -(2 .5.15) 9x 2 Ax (2.5.16) 2x Ax2 where the subscript j represents the cross-shore (x) location and Ax represents the grid resolution, which was set to 5 m in this model. The central difference equations were substituted into equation (2.5.7), in order to obtain the solution for (7 at a typical cross-shore locationj. h, dh 1 4iA2 f2 i2rAf 2 2h, h, dh 1 Ax2 dx 2Ax 1 g g x 2 x2 dx 2AxJ+ 1 I -S +2S -S S -S I -(+1) + 2xj) XX-1) +iAk xy+1) xy1-)/ Ak 2 Pg A2 A y (2.5.17) In equation (2.5.17), r, is unknown, the right hand side consists of known quantities which may be calculated numerically, and the coefficients of Q^, 77, and ,,,1 may also be calculated numerically. Equation (2.5.17) is valid at all interior points in the cross-shore grid. It was solved in matrix form and was represented as a _1 + bj + c 1 = Rj (2.5.18) with additional coefficients at the onshore and offshore boundaries. The coefficients of equation (2.5.18) are h dh 1 a _2 dx 212 (2.5.19) bi = f- h, Ak _2 (2.5.20) [ g g J y Ax 2 = h dh\ 1 cJ 2 d+ 2AX- (2.5.21) and the right-hand-side is R =1 SJ+)+2J) + 2-) + iAk xy(J+) xy(J-1)+ 2 (2.5.22) pg 2 A Ax Note that j goes from 1 to n, with j = n at the shoreline and j = 1 at the offshore boundary. Forj = 1 to n, if n = 6, the matrix would be set up as such: bi c, d, 0 0 0 I R,~ a2 b2 c2 0 0 0 72 R2 0 a b3 3 0 0 3 R 3 3 3 3 3 (2.5.23) 0 0 a4 b4 4 0 4 R4 0 0 0 a5 b5, c,5 R, O 0 0 z6 a6 b,6 R The coefficients b c1, and d6 and z,, an, and b, as well as right-hand-sides R1 and Rn will be defined in the following section on boundary conditions. 2.5.3 Boundary Conditions Boundary conditions were needed because: (1) at the shoreline, equation (2.5.17) would have terms with the subscript (j+1), and atj = n, (j+1) does not exist; and (2) at the offshore boundary, equation (2.5.17) would have terms with the subscript (j-1), and atj =1, (j-1) does not exist. Thus, boundary condition equations were formed by mathematically describing the conditions at both the shoreline and at the offshore boundary. Second order backward differences were used to estimate the derivatives at the shoreline boundary; second order forward differences were used to estimate the derivatives at the offshore boundary. 2.5.3.1 Shoreline boundary condition At the shoreline boundary, where j = n, it was assumed that there exists a "wall" at a very small depth. Then, obviously, un = 0, since there could be no x-directed velocity through that wall. This assumption was applied to equation (2.5.2) in the frequency domain, using the second order backward difference representation for derivatives with respect to x, to obtain (1 2 3 1 3,(n 4S + 2) Sn) 2 + -- + 1 "=i -QA -- --;M. Sln) K2Axj 2 Ax I 2Axf pghn 2Ax (2.5.24) The complete derivation of this equation is given in Appendix B. Equation (2.5.24) forms the shoreline boundary condition for the matrix (2.5.23) and may be represented as zA-2 + ank-I +b,,A = Rn (2.5.25) with coefficients ( =(2.5.27) -22 2.5.3.2 Offshore boundary condition At the offshore boundary, it was assumed that the overall long wave surface elevation was formed by the superposition of the incoming bound and outgoing free waves, such that r = -r +routS (2.5.30) )7 = )7b +)out (2.5.30) where wave surface elevation for the incoming bound wave subscriptt b) and the outgoing free wave subscriptt out) were defined as 7b = 7 exp(i[ot K,, (x cos O,, + y sin 8, }+ (2.5.31) 2ou = 2 q,,, exp{i[ot + K0o, (x cos 6,,, + y sin )] + *(2.5.32) Wave angles 0,, and O,,t were measured with respect to the positive x-axis, according to the right-hand-rule (counterclockwise positive and clockwise negative). A definition sketch of infragravity wave direction is given in Figure 2-3. 7c/2 Oout,actual in X out a -7t/2 Figure 2-3. Definition sketch of infragravity wave direction: incoming bound wave angle Oin, outgoing free wave angle Oout, and actual outgoing free wave angle Oout,actual = 0out + 71 The value of the incoming bound wave angle 0,n was between -7T/2 and 7T/2 radians. The outgoing free wave angle O0ut was defined such that its value was also between -7T/2 and 7T/2 radians. This was done because, for inverse trigonometric functions, Matlab returns a value between -7T/2 and 7T/2 radians, and But would have to be calculated using the inverse sine function (according to equation (2.5.39)). However, the actual value of 0out was outactual = out + (2.5.33) The wave number of the incoming bound wave, Kn, was defined as K,, = Ak+ Ak2 (2.5.34) where Ak = K,,, cosO,, (2.5.35) Ak, = K,, sin 8,, (2.5.36) The wave number of the outgoing free wave, Kout, was defined as Kot = k +kk. (2.5.37) where kx,,, = Ko,, cos0B,, (2.5.38) Ak5 = -Ko,, sin 6,,, (2.5.39) Equivalently, since the speed of the outgoing free wave is Vgh , Ko,, = 2TAf/ gh (2.5.40) Given the above expressions, equations (2.5.31) and (2.5.32) may be expressed equivalently in terms of wave numbers: 7b = 1 exp(- iAkx)exp[i(2;TAft Ak,y)]* (2.5.41) 2 7out 17b exp(kxoux)exp[i(2'Aft Ak,y) +* (2.5.42) The offshore boundary condition was formed by starting with characteristic equations for both the incoming bound wave and the outgoing free wave. The characteristic equation for the surface elevation of the incoming bound wave is a77b 2.'zAf a77b 2.'zAf a77b + cos + sinO = 0 (2.5.43) at K rn Ox K y The characteristic equation for the surface elevation of the outgoing free wave is 077o t k amut Aut Ak out - gh k+ gh k =0 (2.5.44) at Kout ax K0 Qy Equations (2.5.43) and (2.5.44) were combined with 7 = b, + out to obtain 7 gh kx j7+ V-Ak^au ig h ~ a7 + gh K 0 1a at Kou ') Kout C St 2Af cs Ak 2Af O 17b (2.5.45) + cosrn + sgh K -0 sin J Kou8 Kaot K I y Equation (2.5.45) was the starting point for forming the offshore boundary condition. The derivation of this equation as well as proof that equations (2.5.43) and (2.5.44) are characteristic equations for rb and fout are given in Appendix B. The amplitude ?b of the incoming bound wave was obtained analytically by applying equation (2.5.4) to the incoming bound wave at the offshore boundary. Akxt) 21 2 ) + 2Akx1)AkS + Ak2,, (1) 4p;r2Af2 2ipu,Af pghAkx()2 pgh, Ak2 In the above equation, the subscript 1 represents the cross-shore location at the offshore boundary. The radiation stress amplitudes Sx, S, and S, were also found analytically and will be discussed in section 2.5.4. The derivation of equation (2.5.46) is given in Appendix B. Finally, equation (2.5.45) was transformed into the frequency domain. Then, using the second order backward difference representation for the derivative of 7 with respect to x, the offshore boundary condition equation (2.5.47) was obtained. S 3 k A 2 Vgh\ r Vgh kx0, t ( g i;rAf +- gh k + + 77 4Ax Kout 2 Ko) Ax K) y 4Ax Ko) 2ld 2 2ld Old = iAkx + cosV + iAk + sin 8, 2 Kout Kin 2 1 K out Kin) (2.5.47) The complete derivation of this equation is given in Appendix B. Equation (2.5.47) forms the shoreline boundary condition for the matrix (2.5.23) and may be represented as b,7, +c,, +dI, =3 RI (2.5.48) with coefficients b Af xough Ak1 2 gh (2.5.49) 4Ax K 2 Ko) c, = h (2.5.50) d= 4 Kagh (2.5.51) and the right-hand-side 2 K in 2 (out n 2.5.52) (2.5.52) 2.5.4 Derivation of Radiation Stress Radiation stresses were derived from the wave energies of the incoming short waves. The free surface elevation C of a component incoming short wave is 1H = exp(iyV)+* (2.5.53) 22 where the component wave height H is a real number, and where the short wave phase function x is defined by its derivatives, as follows: = -kcosO (2.5.54) = -k sin 0 (2.5.55) ay S=a (2.5.56) Recall that the wave number k and the wave angle 0 of the incoming short wave are functions of x. Also, the long wave phase q is 0 = (2TAf Akyy) (2.5.57) where AkA = K,,, sin 0,,, and neither K,, nor 0,r are functions of x. The numerical model employs a large number of component incoming short waves, such that the free surface elevation is S= Y,. (2.5.58) m=l where m is the individual component and Nis the total number of components. The square of the free surface elevation C2 can be represented in summation notation. The following formula shows that C2 can be reduced to two-component interactions of incoming short waves. N N-1 N =2 ;- 2 + 2V Y, ;, (2.5.59) m=l m=1 n=m+l where 2i H f2 2 r2n = m exp(2iVy)+ m exp(- 2i V,) + H (2.5.60) 16 16 8 2-m(,-n HmHn exp[i(ym + V, )+ H n exp[i(y, -V n)] 8 8 (2.5.61) + HmHn exp[- i (ym + l, )+ H exp[i(V, m )] 8 8 Terms involving summing the phases of the two component incoming short waves, such as Vf, + n, or 2 Vf, are high frequency terms. Terms involving the difference of the phases of the two component incoming short waves, such as Vm f, or Vf, Vm, are low frequency terms. Terms with no exponential are steady terms. High frequency terms will be discarded, since this numerical model involves radiation stresses that force the long waves, which are, by definition, low frequency waves. From equations (2.5.60) and (2.5.61), keeping only the low frequency and steady terms, 2 ow = mn exp[i(Vy Vn)]+* (2.5.62) 8 fi 2 msteady =- (2.5.63) Next, the radiation stresses S, S, and S, were derived in terms of the component wave heights. The general equations for radiation stress, valid only for steady or slowly varying waves, (Dean and Dalrymple, 1991) are S, = En(cos20+1) -1 (2.5.64) S, =E n(sin20+1)-1 (2.5.65) S = Sy = nsin20 (2.5.66) "2 where n = c c (2.5.67) As mentioned previously, S., is the wave momentum flux in the x-direction of the x- component of momentum, Syis the wave momentum flux in the y-direction of the y- component of momentum, and Sy is the wave momentum flux in the x-direction of the y- component of momentum. Since the general equation for wave energy is E = pgH2 (2.5.68) 8 then the radiation stresses are proportional to the wave heights squared. The component wave heights H were obtained from the component energies of the input wave energy spectrum according to equation (2.5.68). The radiation stresses S^,S S, and S, were calculated at each cross-shore location, for each arbitrary combination of two incoming short wave components. Low frequency radiation stresses were derived from the interaction between two different components, for difference frequencies within the infragravity range ( Af: < 0.05Hz). Steady radiation stresses were derived from component self-self interactions. For the low frequency components of radiation stresses, the average incoming wave angle 0a, and the average ratio nv were used, where o,, = (, +O)/2 (2.5.69) n2= [cg +fcg /2 (2.5.70) The low frequency wave energy is Eow =Pg;lw (2.5.71) In terms of wave heights, this may be written as EHow = pg ,, exp[i(,m -V ,)]+* (2.5.72) The steady wave energy is 1 Eteady = PgCseady =1 2 m (2.5.73) 8 Equations (2.5.8)-(2.5.10) give functional representations of the low frequency components of radiation stress. However, the total radiation stress at any cross-shore location is composed of both the steady and low frequency (unsteady) components, which can be represented in summation notation as Stota = ZS steady + S,,ow exp[i(2Aft Ak)] + (2.5.74) Syotal = Soysteady + IS ,,ow exp[i(2rAft Aky)]+* (2.5.75) 2 S0,tot0 =Sns.Ptearp +- nZSo exp[i(2'Aft -Aky)]+* (2.5.76) 2.5.4.1 Steady radiation stress The steady component radiation stresses are Smstea = Esteady nm(cos2 m +1)- ] (2.5.77) Symea = Estea [n (sin2 Om +1)- (2.5.78) om,steady sey 21 .nymsrtea = -- (n1 sin 20,) (2.5.79) 2.5.4.2 Low frequency radiation stress The low frequency (unsteady) component radiation stress amplitudes are Sxx low = lw no,(cos Ol, + 1)- (2.5.80) w = ow na(sin 2a, +1)- (2.5.81) E, = w (n sin 2O a) (2.5.82) It can be seen from equation (2.5.72) that the low frequency wave energy E1ow is a function of the incoming short wave phase Vy(equations (2.5.54)- (2.5.56)), which is a function of both x, y, and t. In order to obtain the low frequency components of radiation stress at each cross-shore or x-location, it was necessary to know the phase at each cross- shore location. In the numerical model, it was assumed that y = 0 and t = 0 for all x and that Vf= 0 at the offshore boundary. Thus, with this simplifying assumption, the short wave phase ywas considered to be a function of x only. Then, from equation (2.5.54) for y'/8x and from forward differences, Vwas calculated at each location in the cross- shore, beginning at the offshore boundary and continuing towards the shoreline, using the following formula (where j represents the cross-shore location) y,1 = Ax(- k, cosO )+ y, (2.5.83) 2.6 Root Mean Square Long Wave Surface Elevation As stated previously, this numerical model combines two component incoming short waves from the input frequency-directional spectrum. Every possible combination of components was calculated, such that each component was combined with every other component, including itself. When two different components, with difference frequencies within the infragravity range ( Af < 0.05Hz), were combined, the result was the forcing of a long wave. When a component was combined with itself, the result was setup or setdown, as will be described in the following section on mean water level. Since the long wave surface elevation 7 is a periodic function, it is appropriate to represent its amplitude as a root mean square. Thus, the root mean square wave surface elevation, is given by ms(j) ( m jO) 2 (2.6.1) where m,n indicates the combination of two different components andj represents the cross-shore location. Note that the term r,,,,( in equation (2.6.1) is equivalent to r, from equation (2.5.17). 2.7 Mean Water Level As stated in the previous section, when a component was combined with itself, the result was setup or setdown. At the offshore boundary, the mean water level is given by linear wave theory (Dean & Dalrymple, 1991) as a2k + C() (2.7.1) 2sinh2kh g where a = H/2 and C(t) = 0 if the mean water level is zero in deep water. Thus, the mean water level for each wave component at the offshore boundary is given by i= k (2.7.2) 8 sinh 2kh The mean water level at each location in the cross-shore was derived from the cross-shore momentum equation (2.5.2), where r is the water surface elevation, and S, and Sx are the steady components of radiation stress. Taking the time average, 9u/at = 0, and thus the equation reduces to a1steady 1 Sx,steady aSxy,steady |, - h 1(2.7.3) ax pg x y where the long wave phase 0 = 0 for a steady wave and thus o7steady 1 steady (2.7.4) ax 2 ax sxx, steady 1 x,steadyv 7 ax 2 ax dS 1 asxy, steady 1 I xystea = --iykS (2.7.6) ay 2 xysteady( Then, using forward differences to estimate derivatives with respect to x, one obtains h hj 1 xxsteady, xx,steady . steady+=1 steady Ak Sxy,steady j (2.7.7) Ax A Ax 9x Ax where j represents the cross-shore location. Since AkA = 0 when the two incoming short wave components are the same, the above equation reduces to steady ~ = steady xx,stead vj+ xx,steady) (2.7.8) Thus, the mean water level or setup is calculated from steady, for each self-self interaction of incoming wave components, at each cross-shore location, beginning with steady = r at the offshore boundary. Then, the sum of seady for all components gives the mean water level at each cross-shore location. IWL(J) I'tea-(j) (2.7.9) m where m represents the self-self combination of an incoming wave component,andj represents the cross-shore location. The accuracy of this calculation was checked by assuring that the mean water level at the shoreline was approximately 15% of the breaking wave height and also approximately equal to the magnitude of root mean square infragravity wave surface elevation multiplied by F2 . 2.8 Model Validation The model was validated by analytically solving the second order partial differential equation for infragravity wave surface elevation (equation (2.5.4)) for a simple test case: a simple input spectrum of two incoming wave components each with wave direction 0 = 0O, no bottom friction ([t = 0), a flat bed with a constant water depth of 3 m, and a reflecting boundary condition at the shoreline. The simplified version of equation (2.5.4) for these given conditions is given by equation (2.8.1). Additional criteria for the simple test case were fi = 0.19 Hz, f2 = 0.20 Hz, and domain length = 7000 m. The governing second order partial differential equation for infragravity wave surface elevation was 1 02 7 21 1 02S. g +h pg (2.8.1) g Ot2 a2 2g a2 which is satisfied everywhere in the domain by 1 = 7b + 1f (2.8.2) where 1ib (bound wave) is the particular solution to equation (2.8.1), given by 7b = Ib exp[i(2;Aft Akx)]+ (2.8.3) 2 and where ?if (free wave) is the homogeneous solution to equation (2.8.1), given by r7f = j exp 2 2fAft 2, f x] + 2- 2f exp i 2rAft + 2f x +* (2.8.4) 2 2 _gh The amplitude ?b of the incoming bound wave was already solved analytically (equation (2.5.46)), which simplifies for this test case to Ak 2 b = S (2.8.5) 2+ 2. 2Af2 2+ g where Sxx is equivalent to S^ at x = 0 and is constant across the domain because of the constant depth. The shoreline boundary condition was taken from the cross-shore momentum equation (2.5.2), for u = 0 (for perfect reflection at the shoreline) and Aky = 0 (from all 0 = 00), reduced to 07 The offshore boundary condition was taken from equation (2.5.45), reduced to d- gh r a2AfBb (2.8.7) with K,, = Akx since Aky = 0. By substituting equations (2.8.2) (2.8.4) into the boundary conditions, one obtains ,jf = 0 and 2f() exp -i KAkx + 2fx 1 (2.8.8) 2;TAf pgh [h where the subscript j represents the cross-shore location. Then, making substitutions into equation (2.8.2), the analytic solution for the infragravity wave surface elevation is = Iexp(i2zAf) k exp(- iAkx) + ) exp .2,Af (2.8.9) 2 L I gh J_ The root mean square wave surface elevation for the analytic solution was then calculated as 77rs(J) = -exp(- iAkx- )+ 2f exp igh x (2.8.10) The results of the analytic solution (7r,,,,) were then compared to the results of the numerical solution used by the model (7,,, ), equation (2.6.1). Figure 2-4 shows an overlay of the analytic and numerical solutions for the simple test case discussed in this section. Both solutions show an almost complete standing wave across the entire cross- 0.2 E 5 I I I I I I fI 0.1 I I I I I I I I I I I I 005 I I I 0 1000 2000 3000 4000 5000 6000 7000 x(m) Figure 2-4. Overlay of analytic (77,r ) and numerical (7,, ) solutions. Legend: -- 7,,,, Irmp shore domain. A test of errors in the numerical solution showed that errors in the root mean square infragravity wave surface elevation converged to the exact solution with second-order accuracy. CHAPTER 3 NUMERICAL SIMULATION OF LOW FREQUENCY WAVE CLIMATE 3.1 Use of the Model The model described in Chapter 2 was used to investigate the low frequency wave climate under varying conditions. Conditions were varied according to eleven basic parameters: offshore significant wave height, offshore peak period, the Jonswap peak enhancement factor, deep water directional width, peak wave direction, bottom friction, bar amplitude, bar width, distance of the bar from the shore, water depth at the offshore boundary, and domain length. Table 3-1 lists the test cases with the corresponding results section and relevant equations for each case. The base case was the basis against which all other test cases were compared. The parameters for the other test cases are reviewed in section 3.2. Table 3-1. List of test cases with corresponding results section and equations Test Case Results Section Relevant Equations Base Case 3.3.1 Offshore Significant Wave Height (Hs) 3.3.2 (2.3.5) Offshore Peak Period (Tp) 3.3.2 (2.3.1) Jonswap Peak Enhancement Factor (y) 3.3.3 (2.3.1) Deep Water Directional Width (dir-widtho) 3.3.4 (2.3.8) Peak Wave Direction (Op) 3.3.5 (2.3.7) Bottom Friction (ff) 3.3.6 (2.5.4)-(2.5.5) Bar Amplitude (a2) 3.3.7 (2.2.4) Bar Width (c,) 3.3.8 (2.2.4) Distance of Bar From Shore (xc) 3.3.9 (2.2.4) Water Depth at Offshor Boundary offshorer) 3.3.10 (2.2.4) Domain Length (ld) 3.3.11 (2.2.4) 3.2 Parameters Significant wave height (Hs) at the offshore boundary, peak period (Tp), and the Jonswap peak enhancement factor (y) were specified as input for the Jonswap frequency spectrum. The offshore significant wave height (Hs) is given by equation (2.3.5). The peak period (Tp) was calculated as the inverse of the peak frequency (f,). Both the peak frequency (fp) and the Jonswap peak enhancement factor (y) are given in equation (2.3.1), which defines the Jonswap frequency spectrum. The deep water directional width (dir- widtho) and peak wave direction (0p) were specified as input for the directional spectrum (equations (2.3.7)-(2.3.8)). The coefficient of bottom friction ff influenced the infragravity wave response (see equations (2.5.4)-(2.5.5)) and served to prevent unbounded growth in the case of infragravity wave resonance. The remaining parameters were specified as input for the bathymetric profile (equation (2.2.4)): bar amplitude (a2), bar width coefficient (c,), distance of the bar from the shore (xc), domain length, and water depth at the offshore boundary, which was varied by varying the profile scale factor (A). 3.3 Results 3.3.1 Base Case The base case was the basis against which all other test cases were compared. The parameters for the base case were as follows: offshore significant wave height Hs = 1 m, offshore peak period Tp = 8 s, Jonswap peak enhancement factor y = 3.3, deep water directional width dir-widtho = 150, peak wave direction O, = 0O, coefficient of bottom friction ff= 1/200 s-1, bar amplitude a2 = 1.5 m, bar width coefficient c, = 5, distance of the bar from the shore xc = 120 m, water depth at the offshore boundary offshore = 10.09 m (from the profile scale factor A = 0.1 m), and domain length Id = 1000 m. Figure 3-1 shows the model results for the base case. Part (c) of this figure displays the bathymetric profile. Unless otherwise noted, this is the bathymetric profile used in the subsequent test cases. Part (b) of this figure shows the transformation of significant wave height (Hs); as the incoming short waves approach the shore, they shoal and break. 0.2 0.1 100 200 300 400 500 600 700 x(m) 800 900 1000 100 200 300 400 500 600 700 800 900 x(m) 1000 -5- -10 -15 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-1. Model results for the base case: (a) root mean square long wave surface elevation (r-hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. (a)II I - As expected, wave breaking is pronounced in the region of the bar. Part (a) of Figure 3-1 displays the root mean square long wave surface elevation (r-hatms), calculated from equation (2.6.1)*. As expected, at the offshore boundary, the infragravity wave has a small magnitude, and near the wave breaking region, the infragravity wave has a larger magnitude; a time animation of model results showed the pattern of a small outgoing infragravity wave at the offshore boundary and a larger magnitude partial standing wave near the wave breaking region. The vertical line near the shoreline indicates the cutoff depth. The cutoff depth was calculated, for each case, as the shallowest depth greater than 0.5 m; shoreward of this cutoff depth, the model results are considered to be invalid. For the base case, the cutoff depth was 0.59 m. At the offshore boundary, the root mean square long wave surface elevation was 0.042 m; 25 m seaward of the shoreline cutoff depth, the root mean square wave surface elevation was 0.140 m. Appendix C tabulates the values of the root mean square long wave surface elevation at the offshore boundary and at 25 m seaward of the shoreline cutoff depth, for each test case. 3.3.2 Offshore Significant Wave Height and Peak Period Figures 3-2 3-6 show the effect of variation of peak wave period (Tp) on model results, for offshore significant wave heights (Hs) of 0.4 m, 0.7 m, 1.0 m, 2.0 m, and 3.0 m, respectively. These figures clearly show that, for all cases, a higher peak wave period leads to a higher magnitude of long wave response (r-hatms), across the entire cross- shore (x) domain. This seems to be consistent with results of a numerical study by Battjes et al. (2004), in which it was found that incoming bound infragravity waves *In this chapter, the root mean square infragravity wave surface elevation is denoted as r-hatms in order to be consistent with axis labels in the figures. This is equivalent to r,,, in equation (2.6.1). generated by higher frequency incoming wave components experience significantly more dissipation than the incoming bound waves generated by lower frequency incoming wave components. Figures 3-7 3-11 show the effect of variation of offshore significant wave height (Hs) on model results, for peak wave periods (Tp) of 4 s, 6 s, 8 s, 10 s and 12 s, respectively. These figures clearly show that, for all cases, a higher offshore significant wave height leads to a higher magnitude of long wave response (r-hatms), across the entire cross-shore (x) domain. This makes sense, because the radiation stresses that force 0.1 0.05 0 100 200 300 400 500 x(m) 600 700 800 900 1000 0.8 0.6 E 0.4 0.2 0 O 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-2. Effect of variation of peak wave period (Tp) (for offshore H,=0.4 m) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs). Legend: -T,=4 s, Tp=6 s, -- Tp=8 s, Tp=10 s, ..... T,=12 s. (I) the infragravity waves are proportional to the wave heights squared; thus, more energetic (higher) incoming short waves lead to greater forcing and thus higher amplitudes of the infragravity waves. Figure 3-2 shows the effect of the variation of peak wave period for an offshore significant wave height of 0.4 m. In Figure 3-2 (a), the offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.006 m for Tp = 4 s to 0.025 m for Tp = 12 s. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.024 m for Tp = 4 s to 0.073 m for 0.2 (a) E 0.15 - m 0.1 0.05 0 100 200 300 400 500 600 700 800 900 1000 x(m) E0.5 0 100 200 300 400 500 600 700 800 900 1000 X(m) Figure 3-3. Effect of variation of peak wave period (Tp) (for offshore Hs =0.7 m) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs). Legend: -T,=4 s, Tp=6 s, -- Tp=8 s, Tp=10 s, ..... T,=12 s. T, = 12 s. Figure 3-2 (b) clearly shows that, for an offshore significant wave height of 0.4 m, a higher peak wave period leads to a higher significant wave height, particularly in the wave shoaling region. Figure 3-3 shows the effect of the variation of peak wave period for an offshore significant wave height of 0.7 m. In Figure 3-3 (a), the offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.010 m for Tp = 4 s to 0.042 m for Tp = 12 s. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.040 m for Tp = 4 s to 0.128 m for 0.3 (a) f 0.2 0 100 200 300 400 500 600 700 800 900 1000 1.5 (b) x 0.5 - 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-4. Effect of variation of peak wave period (Tp) (for offshore Hs =1.0 m) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs). Legend: -Tp=4 s, Tp=6 s, Tp=8 s, Tp=10 s, ..... T,=12 s. T, = 12 s. Figure 3-3 (b) clearly shows that, for an offshore significant wave height of 0.7 m, a higher peak wave period leads to a higher significant wave height, particularly in the wave shoaling region. Figure 3-4 shows the effect of the variation of peak wave period for an offshore significant wave height of 1.0 m. The base case (Tp = 8 s) is shown as a bold black line. In Figure 3-4 (a), the offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.012 m for Tp = 4 s to 0.065 m for Tp = 12 s. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.053 m for Tp = 4 s to 0.197 m for Tp = 12 s. Figure 3-4 (b) clearly shows that, for an offshore significant wave height of 1.0 m, a higher peak wave period leads to a higher significant wave height, particularly in the wave shoaling region. Figure 3-5 shows the effect of the variation of peak wave period for an offshore significant wave height of 2.0 m. In Figure 3-5 (a), the offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.018 m for Tp = 4 s to 0.130 m for Tp = 12 s. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.072 m for Tp = 4 s to 0.357 m for Tp = 12 s. Figure 3-5 (b) shows that, for an offshore significant wave height of 2.0 m, values of Hs through the cross-shore domain for Tp = 6 to 12 s are quite similar, although there is still a general trend for higher peak wave period leading to a higher significant wave height. However, when Tp = 4 s, the model causes the waves to break near the offshore boundary, limiting the wave height to Hmax, which is dependent upon wave period (equation (2.4.11)). 42 0.6 I 0.4 S0.2 - 0 100 200 300 400 500 600 700 800 900 1000 x(m) 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-5. Effect of variation of peak wave period (Tp) (for offshore Hs =2.0 m) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs). Legend: -T,=4 s, Tp=6 s, -- Tp=8 s, Tp=10 s, ..... T,=12 s. Figure 3-6 shows the effect of the variation of peak wave period for an offshore significant wave height of 3.0 m. In Figure 3-6 (a), the offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.047 m for Tp = 4 s to 0.193 m for Tp = 12 s. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.105 m for Tp = 4 s to 0.443 m for Tp = 12 s. Figure 3-6 (b) shows that, for an offshore significant wave height of 3.0 m, values of Hs through the cross-shore domain for Tp = 8 to 12 s are quite similar. When Tp = 6 s, significant wave height is less than that for the higher wave periods through most of the cross-shore domain. When Tp = 4 s, the model causes significant wave breaking near the offshore boundary, limiting the wave height to Hmax (equation (2.4.11)). 0.8 0 6. (a) S0.4 --------^ -- ^ .**-\ 0 100 200 300 400 500 600 700 800 900 1000 x(m) (b) 0.. 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-6. Effect of variation of peak wave period (Tp) (for offshore Hs =3.0 m) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs). Legend: -T,=4 s, Tp=6 s, -- Tp=8 s, Tp=10 s, ..... T,=12 s. Figure 3-7 shows the effect of the variation of offshore significant wave height for a peak wave period of 4 s. In Figure 3-7 (a), the offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.006 m for Hs = 0.4 m to 0.047 m for Hs = 3 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.024 m for Hs = 0.4 m to 0.105 m for Hs = 3 m. Figure 3-7 (b) shows that, in general, a higher offshore significant wave height 44 0 .2 II IIiII O 100 200 300 400 500 600 700 800 900 1000 x(m) 0.15 - x(m) Figure 3-7. Effect of variation of offshore significant wave height (Hs) (for T,=4 s) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (Hs). Legend: Hs=0.4 m, Hs=0.7 m, -- Hs=l m, Hs=2 m, ..... Hs 3 m. corresponds to a higher Hs in the entire cross-shore domain, although wave heights tend to converge in the nearshore after the waves have broken. It is notable that, when the offshore H1 = 3 m, the model causes significant wave breaking near the offshore boundary, limiting the root mean square wave height Hons (equation (2.3.5)) to Hmax (equation (2.4.11)); this causes convergence of Hs with the case of offshore Hs = 2.0 m. However, although wave heights (Hs) are identical throughout most of the cross-shore region for these two cases (offshore Hs = 2 m and 3m), it can be seen from Figure 3-7 (a), that a higher offshore significant wave height leads to a higher magnitude of long wave response (-hato a highe), across the entire cross-shore domain. Mathematically, this can be response (q-hatrm,), across the entire cross-shore domain. Mathematically, this can be explained by the fact that the offshore boundary condition is affected by the offshore significant wave height, due to the amplitude of the incoming bound wave rb (equation (2.5.46)) increasing with increasing radiation stress at the offshore boundary. However, since radiation stresses are proportional to wave heights squared, and since, for cases where the offshore Hrms is greater than Hmax (such as in the case where Hs, = 3 m and Tp = 4 s), the radiation stresses at the offshore boundary are most likely overestimated and thus the results for the long wave response (in such cases) are most likely invalid. 0.3 - U0.2 0 100 200 300 400 500 600 700 800 900 1000 x(m) (b) 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-8. Effect of variation of offshore significant wave height (Hs) (for T,=6 s) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (Hs). Legend: Hs=0.4 m, Hs=0.7 m, -- Hs=l m, Hs,=2 m, ..... H 3 m. Figure 3-8 shows the effect of the variation of offshore significant wave height for a peak wave period of 6 s. In Figure 3-8 (a), the offshore value of the root mean square 46 long wave surface elevation (r-hatms) ranges from 0.011 m for Hs = 0.4 m to 0.060 m for Hs, = 3 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.045 m for Hs, = 0.4 m to 0.193 m for Hs, = 3 m. Figure 3-8 (b) shows that, for a peak wave period of 6 s, a higher offshore significant wave height corresponds to a higher Hs, in the entire cross-shore domain. 0.4 - 0.2 0 100 200 300 400 500 600 700 800 900 1000 x(m) 4 3 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-9. Effect of variation of offshore significant wave height (Hs) (for T,=8 s) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs). Legend: Hs=0.4 m, Hs=0.7 m, H,_ Hs= m, Hs,=2 m, ..... H 3 m. Figure 3-9 shows the effect of the variation of offshore significant wave height for a peak wave period of 8 s. The base case (Hs = 1 m) is shown as a bold black line. In Figure 3-9 (a), the offshore value of the root mean square long wave surface elevation (r- hatms) ranges from 0.017 m for H, = 0.4 m to 0.106 m for H, = 3 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.059 m for Hs = 0.4 m to 0.282 m for Hs, = 3 m. Figure 3-9 (b) shows that, for a peak wave period of 8 s, a higher offshore significant wave height corresponds to a higher Hs in the entire cross-shore domain. 0.6 (a) 0.4 0.2 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-10. Effect of variation of offshore significant wave height (Hs) (for T,=10 s) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs). Legend: -Hs=0.4 m, Hs=0.7 m, -- Hs=l m, Hs=2 m, .... Hs 3 m. Figure 3-10 shows the effect of the variation of offshore significant wave height for a peak wave period of 10 s. In Figure 3-10 (a), the offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.021 m for Hs = 0.4 m to 0.149 m for Hs = 3 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.067 m for Hs = 0.4 m to 0.356 48 m for Hs = 3 m. Figure 3-10 (b) shows that, for a peak wave period of 10 s, a higher offshore significant wave height corresponds to a higher Hs in the entire cross-shore domain. 0.8 (a) 0.6 0.4 0 100 200 300 400 500 600 700 800 900 1000 x(m) Hs=2 m, .... Hs 3 m. Figure 3-11 shows the effect of the variation of offshore significant wave height for a peak wave period of 12 s. In Figure 3-11 (a), the offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.025 m for Hs = 0.4 m to 0.193 m for Hs = 3 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.073 m for Hs = 0.4 m to 0.443 m for Hs = 3 m. Figure 3-11 (b) shows that, for a peak wave period of 12 s, a higher 49 offshore significant wave height corresponds to a higher Hs in the entire cross-shore domain. 3.3.3 Jonswap Peak Enhancement Factor Figure 3-12 shows the effect of the variation of the Jonswap peak enhancement factor y (equation (2.3.1)) on model results. According to Kamphius (2000), y has an average value of 3.3 and typically ranges between 1.0 and 7.0. A higher value of y implies a narrower, more peaked frequency spectrum. The base case (y = 3.3) is shown as a bold black line. Figure 3-12 (b) shows that the variation of y has little effect on E 0.2 O 0 0 100 200 300 400 500 600 700 800 x(m) 900 1000 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-12. Effect of variation of Jonswap peak enhancement factor (y) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (Hs). Legend: -y=1.0, y=3.4, y=5.0, -- y=7.0. significant wave height (Hs). This is expected, since the total wave energy for each case was the same. Figure 3-12 (a) shows that a narrower frequency spectrum (increased y) (. . leads to a slightly decreased root mean square long wave surface elevation (r-hatrms). This is unexpected, and further investigation would have to be done to explain this pattern of results. The offshore value of the root mean square long wave surface elevation (r-hatrms) ranges from 0.043 m for y = 1.0 to 0.036 m for y = 7.0. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.147 m for y = 1.0 to 0.120 m for y = 7.0. 3.3.4 Deep Water Directional Width Figure 3-13 shows the effect of the variation of deep water directional width, dir- widtho, (equation (2.3.8)) on model results. The deep water directional width was varied from 5' to 300. The base case, with dir-widtho = 15', is shown as a bold black line. Figure 3-13 (a) shows that a narrower directional width leads to a greater magnitude of long wave response. This finding is consistent with those of Van Dongeren et al. (2003); in employing a numerical model (SHORECIRC), it was found that eliminating directional spreading from the incoming wave spectrum caused significant amplification of infragravity wave heights. In analyzing field data, Herbers et al. (1995a, 1995b) found a similar relationship between directional spreading and infragravity wave response. The offshore value of the root mean square long wave surface elevation (r-hatrms) ranges from 0.051 m for dir-widtho = 5' to 0.030 m for dir-widtho = 300. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.159 m for dir-widtho = 50 to 0.110 for dir-widtho = 300. Figure 3-13 (b) shows that deep water directional width has little effect on significant wave height throughout the cross-shore domain. 51 (a) 0.2 - 0.1 0 100 200 300 400 500 600 700 800 900 1000 x(m) 1.5 0.5 O 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-13. Effect of variation of deep-water directional width (dir-widtho) on (a) root mean square long wave surface elevation (r-hatrms) and (b) significant wave height (Hs). Legend: -dir-widtho=50, dir-widtho=150, dir-widtho=300. 3.3.5 Peak Wave Direction Figure 3-14 shows the effect of the variation of peak wave direction, O,, (equation (2.3.7)) on model results. Peak wave direction was varied from 0O (shore-normal, base case) to 600 (oblique incidence). Figure 3-14(a) shows that peak wave direction has little effect on long wave response near the shoreline. However, at the offshore boundary, more oblique waves (higher Op) lead to a decreased magnitude of long wave response. One possible explanation for this is that more obliquely incident short waves lead to more 52 0O.2 J_ 0.1 0 100 200 300 400 500 600 700 800 900 1000 x(m) 1.5 1 O 100 200 300 400 500 600 700 800 900 1000 Figure 3-14. Effect of variation of peak wave direction (,p) on (a) root mean square long wave surface elevation (r-hatms) and (b) significant wave height (Hs). Legend: O,=0o, p=200, O,=400, -- =600. trapping of the outgoing free infragravity waves and thus less leaky waves reaching the offshore boundary. The offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.042 m for O, = 00 to 0.029 m for O, = 600. Figure 3- 14(b) shows that the obliquely incident waves (O, = 600) have a lower significant wave height; this is due to greater refraction and less shoaling than the non-obliquely incident waves. It is important to note that, although the significant wave height (Hs) and thus the wave energy of the obliquely incident waves (O, = 600) is lower than the Hs for all of the other cases as the waves approach the bar, the magnitude of the infragravity wave response in the nearshore region is similar in all cases. This is consistent with the explanation that more obliquely incident waves generate more edge waves (trapped infragravity free waves), and thus the case of O, = 600 shows similar nearshore infragravity wave energy as the other cases, although it shows less incoming short wave energy than the other cases. 3.3.6 Bottom Friction Figure 3-15 shows the effect of the variation of the bottom friction coefficient,ff, (equations (2.5.4)-(2.5.5)) on model results. The bottom friction coefficient was varied from ff= 1/400 s-1 to 1/50 s-1. The base case, with ff= 1/200 s-1, is shown as a bold black line. Figure 3-15 (a) shows that increased bottom friction decreased the magnitude of long wave response across the entire cross-shore domain. This is expected, because E 0.2 - 0 100 200 300 400 500 600 700 800 900 1000 x(m) 1.5- 0.5 - 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-15. Effect of variation of bottom friction coefficient (ff) on (a) root mean square long wave surface elevation (r-hat,,s) and (b) significant wave height (Hs). Legend: ff=/50 s-1, ff=l/100 s1, ffl1/200 s1, -- ff=/300 s1, ffl1/400 s1. bottom friction serves to damp the long wave response (equations (2.5.4)- (2.5.5)). The offshore value of the root mean square long wave surface elevation (r-hatms) ranges from 0.048 m for ff = 1/400 s-1 to 0.016 m for ff = 1/50 s-1. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.150 m for ff = 1/400 s-1 to 0.098 m for ff = 1/50 s-1. Figure 3-15 (b) shows that the bottom friction coefficient has no effect on significant wave height throughout the cross- shore domain. This is because bottom friction was not a factor in the wave-breaking portion of the model. 3.3.7 Bar Amplitude Figure 3-16 shows the effect of the variation of bar amplitude, a2, (equation 2.2.4) on model results. The bar amplitude was varied from a2 = 0 m to 2.14 m. The base case, with a2 = 1.5 m, is shown as a bold black line. The variation of the bathymetric profile due to bar amplitude is shown in Figure 3-16 (c). The maximum bar amplitude of 2.14 m was chosen in order to limit the minimum water depth over the bar to 0.3 m. Figure 3-16 (b) shows that an increased bar amplitude leads to stronger wave breaking and a decreased significant wave height from the bar to the shoreline. Figure 3-16 (a) shows a more complex pattern of results for the long wave response. In general, a higher bar amplitude leads to a decreased magnitude of long wave response, especially near the shoreline; however, this pattern seems to be reversed at the peak of the bar for the maximum bar amplitude of 2.14 m. One explanation for this is the trapping of edge waves due to the bar. Also, it is notable that, for the case of no bar (a2 = 0 m), r-hatms shows a sharp increase at a location approximately 40 m seaward of the shoreline cutoff depth. This may be due to the predominance of infragravity wave reflection at the shoreline but an absence of trapping due to a bar. E 0.2 E 0.1 0 1.5 0 100 200 300 400 500 600 700 800 900 1000 x(m) 0 100 200 300 400 500 600 700 800 900 1000 x (m) 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-16. Effect of variation of bar amplitude (a2) on (a) root mean square long wave surface elevation (r-hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Legend: a2=0 m, a2=0.5 m, -- a2=1.0 m, az=2.0 m, ***** az=2.14 m. a2=1.5 m, 56 3.3.8 Bar Width Figure 3-17 shows the effect of the variation of bar width on model results. Equation (2.2.4) shows how the bar width coefficient c, affects the bathymetric profile. The bar width coefficient was varied from c, = 2 to 8. A lower c, implies greater width. 0.2 0.1 100 200 300 400 500 600 700 x(m) 100 200 300 400 500 x(m) 800 900 1000 600 700 800 900 1000 -5 -10 -15 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-17. Effect of variation of bar width on (a) root mean square long wave surface elevation (r-hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Width coefficient is c,. Legend: c,=2, c,=5, w=8. IIIa) I I I I I III The base case, with c, = 5, is shown as a bold black line. The variation of the bathymetric profile due to bar width is shown in Figure 3-17 (c). Figure 3-17 (b) shows that a wider bar leads to a more gradual pattern of wave breaking. Figure 3-17 (a) shows that a wider bar leads to a slightly increased magnitude of long wave response, especially near the shoreline. One possible explanation for this is that bars tend to trap edge waves; therefore, a wider bar might lead to more trapping of these edge waves near the shoreline. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.150 m for c, = 2 to 0.124 m for c, = 8. 3.3.9 Distance of Bar from Shore Figure 3-18 shows the effect of the variation of the distance of the bar from shore, xc, (equation (2.2.4)) on model results. The distance xc was varied from 80 m to 150 m. The base case, with xc = 120 m, is shown as a bold black line. The variation of the bathymetric profile due to the distance of the bar from shore is shown in Figure 3-18 (c). Figure 3-18 (b) shows that the bar being closer to shore leads to waves breaking closer to shore. Figure 3-18 (a) shows that the bar being closer to shore results in a narrower long wave response pattern near the shoreline. This makes sense, because the bar trapped edge waves would be nearer to shore if the bar is nearer to the shore; as the bar is located further from the shoreline, the long wave response pattern becomes more spread out in the near-shore region. 0.2 0.1 100 200 300 400 500 600 700 800 x(m) 0 1 I 1 1 1 1 0 100 200 300 400 500 x (m) 600 700 800 900 1000 -156 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-18. Effect of variation of distance of bar from shore (x,) on (a) root mean square long wave surface elevation (r-hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Legend: xc=80 m, xc=100 m, xc=120m, --xc=150m. 3.3.10 Water Depth at Offshore Boundary Figure 3-19 shows the effect of the variation of water depth at the offshore boundary offshorer) on model results. The water depth at the offshore boundary was (III)I 900 1000 59 varied by varying the profile scale factor A (equation (2.2.4). The bar amplitude was scaled to the offshore water depth by multiplying the bar amplitude a2 (in equation (2.4.3)) by the ratio A/Abase where A is the profile scale factor and Abase is the profile E 0.2 E - 0.1 100 200 300 400 500 600 700 800 900 1000 x(m) 0 100 200 300 400 500 600 700 800 900 1000 x(m) 0 100 200 300 400 500 600 700 800 900 1000 x(m) Figure 3-19. Effect of variation of water depth at offshore boundary offshorer) on (a) root mean square long wave surface elevation (r-hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Legend: -hoffshore=7.19 m, hoffshore=8.16 m,-- hoffshore=9.13 m, hoffshore=10.09 m, hoffshore=l 11.97 m, ***** offshore= 12.91 m. scale factor for the base case. The offshore water depth was varied from offshore = 7.19 m (for A = 0.07) to offshore = 12.91 m (for A = 0.13). The base case, with offshore = 10.09 m (for A = 0.1), is shown as a bold black line. The variation of the bathymetric profile is shown in Figure 3-19 (c). Figure 3-19 (b) shows waves breaking farther from shore for the shallower bathymetric profiles. Figure 3-19 (a) shows that a shallower bathymetric profile (smaller offshore and bar amplitude) results in a higher amplitude of long wave response in the entire cross-shore domain. This makes sense conceptually because infragravity waves tend to be predominant in shallow water. The offshore value of the root mean square long wave surface elevation (r-hatrms) ranges from 0.047 m for offshore = 7.19 m to 0.037 m for offshore = 12.91 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.151 m for offshore = 7.19 m to 0.131 m for offshore = 12.91 m. 3.3.11 Domain Length Figure 3-20 shows the effect of the variation of the domain length (ld) on model results. The distance Id was varied from 750 m to 1625 m. The base case, with Id = 1000 m, is shown as a bold black line. Figures 3-20 (a) (c) were plotted such that for all cases, x = 1625 m at the onshore boundary. These figures show convergence of the results for all cases. This indicates that the chosen cross-shore domain of 1000 m was adequately long to give accurate results. (a) S0.2 0 0 200 400 600 800 1000 1200 1400 160C x(m) 1.5 (b) 0.5- 0. 0 200 400 600 800 1000 1200 1400 160OC x(m) 0 -5 0P -o10r I I5 I I" " 0 200 400 600 800 x (m) 1000 1200 1400 1600 Figure 3-20. Effect of variation of domain length (ld) on (a) root mean square long wave surface elevation (r-hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Legend: ld=1625 m ld=750 m, d=1000 m, -- 1d=1375 m, CHAPTER 4 CONCLUSIONS 4.1 Summary The purpose of this study was to develop and present a linear frequency-domain numerical model of nearshore infragravity wave generation on an alongshore-uniform beach, and then to utilize this model to investigate the importance of various parameters in the generation of these infragravity waves. The model of infragravity wave generation was based upon the model presented by Reniers et al. (2002), which was found by the same researchers to agree well with field data from Duck, North Carolina. Chapter 2 of this thesis presented detailed derivations of this numerical model, as well as a validation of the numerical model against an analytical solution for a simple test case. The results presented in Chapter 3 of this thesis demonstrate the relative importance of various parameters in the generation of nearshore infragravity waves. The results of the numerical simulation of the low frequency wave climate indicated that the most important parameters in the generation of nearshore infragravity waves are: peak wave period (Tp), offshore significant wave height (Hs), bottom friction (ff), deep water directional width (dir-widtho), and water depth at the offshore boundary offshorere. These parameters affect the infragravity wave response in the following manner: * A higher Tp leads to a higher magnitude of response; * A higher Hs leads to a higher magnitude of response; * A narrower directional width leads to higher magnitude of response; * Increased bottom friction leads to a decreased magnitude of response; * A shallower bathymetric profile leads to an increased magnitude of response. The remaining parameters investigated in the numerical simulation were found to be moderately important in the generation of nearshore infragravity waves. These parameters are: bar amplitude (a2), the bar width coefficient (cw), the distance of the bar from the shoreline (xc), peak wave direction (,p), and the Jonswap peak enhancement factor (y). These parameters affect the infragravity wave response in the following manner: * A higher bar amplitude leads to decreased magnitude of response, especially near the shoreline; * A wider bar leads to a slightly higher magnitude of response; * The bar being closer to the shore results in a narrower response pattern; * Obliquely incident waves (higher Op) lead to a decreased magnitude of response at the offshore boundary, although peak wave direction has little effect on response near the shoreline; * A narrower frequency spectrum (increased y) leads to a slightly decreased magnitude of response. 4.2 Discussion and Conclusions One of the main findings in this study is that a larger peak wave period (Tp) of the incoming short waves leads to a greater magnitude of infragravity wave surface elevation. It is important to note that this pattern is evident for small incident waves (i.e. Hs=0.4m), where Hs is higher across the domain for higher peak periods (i.e. greater dissipation of short wave energy for lower Tp), and also for large incident waves (i.e. Hs=2.0 m) where Hs is similar across the domain for all peak periods. This indicates that, regardless of the dissipation of the primary (short) waves, there is still greater dissipation of infragravity wave energy when the peak period of the incident short waves is lower. This seems to be consistent with results of a numerical study by Battjes et al. (2004), in which it was found that incoming bound infragravity waves generated by higher frequency incoming wave components experience significantly more dissipation than the incoming bound waves generated by lower frequency incoming wave components. Another parameter that has an important impact on infragravity wave surface elevation is offshore significant wave height (Hs). Results of the numerical simulation of the low frequency wave climate show that higher offshore significant wave height leads to higher magnitude of infragravity wave response. This is consistent with the findings of Ruessink (1998a) that the infragravity wave energy is significantly positively correlated with the wave energy of the incoming short waves. This also makes sense mathematically, because the radiation stresses that force the infragravity waves are proportional to the wave heights squared; thus, more energetic (higher) incoming short waves lead to greater forcing and thus higher amplitudes of the infragravity waves. However, as seen in Figure 3.3.2.6, it is important to specify a realistic wave height at the offshore boundary. If the offshore Hrms exceeds Hmax at the offshore boundary, then the value of the radiation stresses at the offshore boundary will be unrealistically high, and the model results for infragravity wave surface elevation will be inaccurate. Results of the numerical simulation of the low frequency wave climate also show that a narrower directional width leads to a greater magnitude of long wave response. This finding is consistent with that of Van Dongeren et al. (2003); in employing a numerical model (SHORECIRC), it was found that eliminating directional spreading from the incoming wave spectrum caused significant amplification of infragravity wave heights. In analyzing field data, Herbers et al. (1995a, 1995b) found a similar relationship between directional spreading and infragravity wave response. As expected, increased bottom friction decreased the magnitude of long wave response across the entire cross-shore domain. Bottom friction serves to damp the long wave response (equations (2.5.4)- (2.5.5)) and is necessary to prevent unbounded growth in the case of infragravity wave resonance. The bathymetric parameter that seems to most significantly affect the long wave response is the offshore water depth offshorere, which was varied in the numerical simulation by varying the profile scale factor (A, equation (2.2.4)). A shallower bathymetric profile (smaller offshore and bar amplitude) results in higher amplitude of long wave response in the entire cross-shore domain. This makes sense conceptually because infragravity waves tend to be predominant in shallow water. Results of the numerical simulation of the low frequency wave climate show that, in general, a higher bar amplitude leads to a decreased magnitude of long wave response, especially near the shoreline; however, this pattern seems to be reversed at the peak of the bar for the maximum bar amplitude. One explanation for this is the trapping of edge waves due to the bar. Also, it is notable that, for the case of no bar, the root mean square infragravity wave surface elevation shows a sharp increase at a location approximately 40 m seaward of the shoreline cutoff depth. This may be due to the predominance of infragravity wave reflection at the shoreline but an absence of trapping due to a bar. The variation of bar width also affects the low frequency wave climate. Results of the numerical simulation show that a wider bar leads to a slightly increased magnitude of long wave response, especially near the shoreline. One possible explanation for this is that bars tend to trap edge waves; therefore, a wider bar might lead to more trapping of these edge waves near the shoreline. Results of the numerical simulation of the low frequency wave climate also show that the bar being closer to shore results in a narrower long wave response pattern near the shoreline. This makes sense, because the bar trapped edge waves would be nearer to shore if the bar is nearer to the shore; as the bar is located further from the shoreline, the long wave response pattern becomes more spread out in the near-shore region. Findings also indicate that peak wave direction has little effect on long wave response near the shoreline. However, at the offshore boundary, more oblique waves (higher Op) lead to a decreased magnitude of long wave response. One possible explanation for this is that more obliquely incident short waves lead to more trapping of the outgoing free infragravity waves and thus less leaky waves reaching the offshore boundary. It is important to note that, although the significant wave height (Hs) and thus the wave energy of the obliquely incident waves (Op = 600) is lower than the Hs for all of the other cases as the waves approach the bar, the magnitude of the infragravity wave response in the nearshore region is similar in all cases. This is consistent with the explanation that more obliquely incident waves generate more edge waves (trapped infragravity free waves), and thus the case of O, = 600 shows similar nearshore infragravity wave energy as the other cases, although it shows less incoming short wave energy than the other cases. The one result of this numerical simulation of low frequency wave climate that is not consistent with expectations is that a narrower frequency spectrum (increased y) leads to a slightly decreased root mean square long wave surface elevation (r-hatrms). Further investigation would have to be done to explain this pattern of results. 4.3 Recommendations for Further Work It is possible to expand this numerical model to include solutions for the velocities u and v, in the cross-shore and alongshore directions, respectively. In order to do this, rather than combining equations (2.5.1)- (2.5.3) into a single equation for infragravity wave surface elevation fl (2.5.4), one would transform equations (2.5.1)- (2.5.3) into the frequency domain and solve for the three unknowns 7, ui, and i. This would necessitate formulating more complex boundary conditions and would also entail more computational time. The benefit of this expansion would be the ability to analyze the relationships between infragravity waves and nearshore currents or vorticity. Another possibility is to expand the model to include an alongshore-varying bathymetry. Thus, the model would be valid for cases other than those with an alongshore uniform bathymetry. Perturbation expansions could be used to generate this alongshore-varying bathymetry. This would provide results showing the alongshore variation of infragravity wave surface elevation and would thus allow for the depiction of edge waves. With the model in its current state, one could analyze the output in order to separate the incoming (bound) and outgoing (free) long waves. It would also be possible to further analyze the model output to separate the outgoing long waves into edge waves and leaky waves by forming a frequency alongshore wave number spectrum. This would allow one to analyze which factors or parameters are most important in the 68 generation of the long waves traveling in each direction, and in trapped versus leaky waves. APPENDIX A DERIVATION OF EQUATION FOR INFRAGRAVITY WAVE SURFACE ELEVATION This appendix gives the derivations of the two forms of the equation for infragravity wave surface elevation: the second order partial differential equation, equation (2.5.4), and the second order ordinary differential equation in the frequency domain, equation (2.5.6). A.1 Second Order Partial Differential Equation Equation (2.5.4), the second order partial differential equation for infragravity wave surface elevation, 1 8a2 Pa a2 7 dh a a2 1 SX 22S, 2S +h + +h + + g at2 g t x2 dx& x y pg dx2 9xy 2 j was obtained by combining equations (2.5.1)-(2.5.3), as follows. 1. Differentiate equation (2.5.1) with respect to time: +- h- +- h = 0 (A. 1) 9t2 ax at) 9 y at) 2. Differentiate equation (2.5.2) with respect to x: a 2, SB^ 92SX p- h -+pg --ah (A.2) ax at ax ax axay 3. Differentiate equation (2.5.3) with respect to y: a ( av a99 8 7 2S 2S( p- h -+pg h-- (A.3) S \y at yq ( y Ay .x y 4. Rearrange equation (A. 1): -h- -- --7 ha (A.4) ax at 9t2 9y at 5. Substitute equation (A.4) into equation (A.2) to obtain: 0 2 7 9 2h7 0 92S. 02S" S 9y ajt t ax ax x a2 2 x 9xy 6. Substitute equation (A.5) into equation (A.3) to obtain: 1 2r 9h 9r 1 9 h 9r 92r 12Sy 2S -+ +h + --+h- -+2- (A.6) g at Ox ax a2x 9hy y 2y pg da2x 9xy 2 y 2 7. Assuming h varies only with x, not with y: 1 a2h7 a7+ dha 17 +h, +2 +a) (A.7) g at2 9x2 dxax 9 y pgK dx2 9xay ay 2 This is the same as equation (2.5.4), except for the added term on the left hand g at side. A.2 Second Order Ordinary Differential Equation in the Frequency Domain Equation (2.5.6), the second order ordinary differential equation for infragravity wave surface elevation, d + dh du + 4.2Af2 iu2zAf hAk2 2 dx dx dx g g 1 dXS dSd 2 d + 2iAk + Ak 2 pg 2 dx dx was obtained by transforming equation (2.5.4) into the frequency domain, as follows: 1. Symbolize the infragravity wave phase q as S= (2zAft Aky) (A.8) 2. From equation (2.5.7) for q: S= iTzAfiexp[i0]+* (A.9) at at -2;r2A 2exp[i]+* (A.10) 8t a7 1 ^ expli]+* (A.11) cx 2 cx 2 exp[iO]+* (A.12) ax2 2 ax2 a27 1 -- Ak 2exp[li]+* (A.13) ay 2 3. From equation (2.5.10) for Syy: 1 Ak 2 exp[iq ]+* (A.14) y 2 2 4. From equation (2.5.9) for Sxy: 02S 1 dS i i =iAk, exp[iq]+* 8x-y 2 dx (A.15) 5. From equation (2.5.8) for S,,xx: a2SX 1 d2S x S 2 exp[i*]+* (A.16) x 2 2 dxC2 6. Equations (A.9)-(A. 16) were substituted into equation (2.5.4) to obtain equation (2.5.6). APPENDIX B DERIVATION OF EQUATIONS FOR BOUNDARY CONDITIONS B.1 Shoreline Boundary Condition The cross-shore momentum equation (2.5.2) was the starting point in deriving the reflecting boundary condition at the shoreline. au a as8xx as, ph-+ pgh at ax ax ay The following steps were followed in order to obtain the shoreline boundary condition equation (2.5.24). 1. Assume that there is perfect reflection at the shoreline (x=n). Therefore, let Un = 0. Equation (2.5.2) then becomes 7 1 as as (B.1) dx pgh d x Sy ) 2. Transform equation (B. 1) to the frequency domain. Recall that the infragravity wave phase is represented as = (27rAft -Aky) (B.2) From equations (2.5.7)-(2.5.9) for 7, Sxx, and Sxy: a7 1 a^ exp(i)+* (B.3) ax 2 ax as lasexp(io)+* (B.4) ax 2 ax as -iAk = Y S, exp(i)+* (B.5) Sy 2 Substitute equations (B.3)-(B.5) into equation (B.1) to obtain a- iAk, S pgh Ox " (B.6) 3. Use second order backward differences to estimate the derivatives with respect to x: 897 9,_ 4^-i + 39 Na n2 -4 +n (B.7) ax 2Ax a ., ":(n-2) ~4xx(n-l) +3S(n) (B.8) Ax 2Ax 4. Substitute equations (B.7) and (B.8) into equation (B.6) to obtain equation (2.5.24): 1 -( 2 3 1 r3 -(n) 4S,(nl) + Sxn iAk Sy(n) 2Ax A x 2Axj pghn 2Ax B.2 Offshore Boundary Condition B.2.1 Characteristic Equations B.2.1.1 Incoming bound wave Equation (2.5.43) can be proven to be a characteristic equation for the surface elevation of the incoming bound wave. 1. Given equation (2.5.43): a77b 2zAf a7b 2'zAf a 8qb 7, + cosO 0 + 2 sino 7 = 0 at K x K y 2. Substitute c=27zAf and take the analytic derivatives of rb from equation (2.5.31) to obtain iyCb + cos 8O (- iK,,, cos ,,, )7b + sin 8O, (- iK, sin 8,)17b = 0 3. Cancel terms to obtain the trigonometric identity cos2 O,, + sin2 O,, = 1 (B.9) (B.10) which is true, by definition. B.2.1.2 Outgoing free wave Equation (2.5.44) can be proven to be a characteristic equation for the surface elevation of the outgoing free wave. 1. Given equation (2.5.44): d07,,, kxout dr0,, Aky 07o 0 gh + gh 0 St Kot ax Kout y 2. Take the analytic derivatives of r,,ut from equation (2.5.42) to obtain iuuou kgxoh (ik-xo0u)u0i + gh -ihk( ik =) 0 (B.l) Kout Kout 3. Substitute Ko,, = then cancel and rearrange terms to obtain Vgh K = Ak+kxo (B.12) which is true by definition ofKout (equation (2.5.37)). B.2.1.3 Combined Characteristic Equation The combined characteristic equation (2.5.45) was the starting point for forming the offshore boundary condition. It was obtained by the following steps: 1. Substitute 77o = 1r rb (from equation 2.5.30) into equation (2.5.44) to obtain d- r boa Sbb- 0 (B.13) at at Ko,, x x ) K aout 2. Add together equations (B.13) and (2.5.43) and rearrange terms to obtain equation (2.5.45): a gh k, a. + gh 7k ja7 at Ko,,, x Ko0t gh + os,, O + gh Kout- f sin 8,, B.2.2 Incoming Bound Wave Amplitude Equation (2.5.46) gives the analytic solution of the incoming bound wave amplitude rb It was obtained by the following steps: 1. Apply equation (2.5.4) to the incoming bound wave at the offshore boundary. 1 82 b 8b a2 1b dh b 2b 1a0Sxx 22x + 2S +h + +h +a + g t2 g at a2 dxax y2 pg2 dx2 axay ay2 (B.14) Note that the radiation stresses force the incoming bound wave and not the outgoing free wave, and thus it was not necessary to apply the subscript b to the radiation stresses in the above equation. 2. In order to take the analytic derivatives of radiation stresses with respect to x, radiation stresses were defined at the offshore boundary as Sxx = S x exp(-iAkxx)expi(2Aft Aky) +* (B.15) S, = exp(- iAkxx)exp [i (2Aft Aky)]+ (B.16) SA = 1 exp(- iAkx)exp[i(2Aft Aky)] +* (B.17) 2 Equations (B.15)-(B.17) are similar in form to equation (2.5.41), which defines the surface elevation of the incoming bound wave at the offshore boundary. 3. By comparing the above three equations with equations (2.5.8)-(2.5.10), it can be seen that S, = S exp(- iAkx) (B.18) S, = S, exp(- iAk,x) (B.19) Sy = S,, exp(- iAkx) (B.20) 4. Take the analytic derivatives of r7b from equation (2.5.41) and of radiation stresses from equations (B. 15)- (B. 17). Recall that = (27TAft-Akyy). a-7b = izAf exp(- iAkx)exp(i) +* (B.21) at = -2zi f Ar2b exp(- iAkxx)exp(i) +* (B.22) at2 ax2 AkX2 7b exp(- iAkx)exp(io) + (B.23) X2 2 a -7b Ak 2 exp(- iAkxx)exp(io) + (B.24) oy 2 12S -Ak1 exp(- iAkxx)exp(i) + (B.25) na2 2 SAkxAk, S exp(- iAkx)exp(i) + (B.26) axay 2 a2 -1 Ak, 2S exp(- iAkxx)exp(ij) +* (B.27) Cy 2 dh 5. Assume a flat bed at the offshore boundary; therefore, the term = 0 in equation dx (2.5.4). Substitute into equation (2.5.4) the above analytic derivatives from equations (B.21)-(B.27). Note that since x = 0 at the offshore boundary, the term exp(-iAkx) = 1. Rearrange terms to obtain Akx S + 2Ak AkSA + Ak S 17b = x) xx -x(l ) 2 (B.28) 4p'r Af2 2ippnf' pghAk ) pghAk 6. As can be seen from equations (B.18)- (B.20), since x = 0 at the offshore boundary, S, S,y, and S., are equivalent to S, S,,, and SY, respectively, at the offshore boundary. Therefore, the above equation becomes equation (2.5.46): Akxct) + 2AkrM)+AkS,( + 2Ak .2 , 4 pr;2Af2 2ip1n, Af pgh, Ak (2 pgh, AkY B.2.3 Offshore Boundary Condition Equation The offshore boundary condition equation (2.5.47) was obtained by taking the following steps: 1. Begin with equation (2.5.45). gh 0a7 + gV- a7k at K0ut ax K0ut jy g- + cos O,, + gh K, ] ,,L Ksin O y 2. Take the analytic derivates of r (from equation 2.5.7). =7 izAf exp(i )+* at ar 1 a;7 exp(if)+* ax 2 ax = -- iAk, exp(i) + * cy 2 (B.29) (B.30) (B.31) 3. Equation (2.5.45) was transformed into the frequency domain by substitution of the above equations (B.29)-(B.31). Af k xo 2 KOut Cx 1 Ak , 2gh K iAky 2 Kout K 2'Af (1 gh- + cos Akxb exp(-iAkxx) + gh 2Af sin*,, 1 iAkyb exp(-iAkxx) Kout Kin 2 (B.32) 4. Use the second order forward differences representation of at the offshore boundary. S 3-i1+472(B.373) Ox 2Ax 5. Substitute equation (B.33) into equation (B.32) and rearrange terms to obtain the offshore boundary condition equation (2.5.47). 3 r k 1 2 gh ( ghkxout kxout g irrAf +- -gh Ak + + 3 y 4Ax K4A 2 K ) Ax K 4Ax Kou lOut ~Ofrt 2OA/ l ~2 = iAkx + g cos + iAkn sin 8,n 2 out Kin 2 Iout Kin APPENDIX C VALUES OF ROOT MEAN SQUARE LONG WAVE SURFACE ELEVATION FOR EACH TEST CASE Table C-1. Root mean square wave surface elevation ( 7ms) for each test case, at the offshore boundary and at 25 m seaward of the shoreline cutoff depth. Root mean square wave surface elevation (rms) (m) At offshore boundary At 25 m seaward of shoreline cutoff depth 3.3.1 Base Case 0.0417 0.1400 3.3.2 Offshore Significant Wave Height and Peak Period HI(m), Tn(s) 0.4, 4 0.0056 0.0239 0.4,6 0.0110 0.0448 0.4, 8 0.0167 0.0587 0.4, 10 0.0212 0.0669 0.4, 12 0.0249 0.0734 0.7, 4 0.0096 0.0398 0.7, 5 0.0189 0.0736 0.7, 6 0.0279 0.0975 0.7,7 0.0358 0.1126 0.7, 8 0.0419 0.1277 1,4 0.0122 0.0526 1,6 0.0284 0.1050 1,8 0.0417 0.1400 1, 10 0.0524 0.1646 1, 12 0.0651 0.1966 2,4 0.0176 0.0717 2,6 0.0442 0.1581 2, 8 0.0716 0.2220 2, 10 0.1036 0.2945 2, 12 0.1295 0.3569 Table C-1. Continued Root mean square wave surface elevation (r,,) (m) At offshore boundary At 25 m seaward of shoreline cutoff depth 3.3.2 Offshore Significant Wave Height and Peak Period (continued) H, (m), T,(s) 3,4 0.0465 0.1053 3,6 0.0598 0.1927 3, 8 0.1063 0.2821 3, 10 0.1492 0.3562 3, 12 0.1925 0.4428 3.3.3 Jonswap Peak Enhancement Factor 1 0.0433 0.1469 2 0.0412 0.1405 3 0.0417 0.1405 3.3 0.0417 0.1400 4 0.0381 0.1290 5 0.0374 0.1225 6 10.0358 10.1195 7 10.0358 10.1200 3.3.4 Deep Water Directional Width dir-width_0 (deg) 5 0.0510 0.1587 10 0.0451 0.1461 15 0.0417 0.1400 20 0.0343 0.1208 25 0.0340 0.1212 30 0.0302 0.1099 3.3.5 Peak Wave Direction O, (deg) 0 0.0417 0.1400 10 0.0378 0.1322 20 0.0393 0.1359 30 0.0379 0.1361 40 0.0384 0.1393 50 0.0331 0.1417 60 0.0288 0.1316 Table C-1. Continued Root mean square wave surface elevation ( mrm) (m) At offshore boundary At 25 m seaward of shoreline cutoff depth 3.3.6 Bottom Friction ff 1/50 0.0156 0.0982 1/100 0.0278 0.1149 1/150 0.0345 0.1267 1/200 0.0417 0.1400 1/250 0.0432 0.1418 1/300 0.0460 0.1424 1/350 0.0434 0.1419 1/400 0.0484 0.1504 3.3.7 Bar Amplitude a2 0 0.0420 0.1336 0.5 0.0441 0.1394 1.0 0.0442 0.1435 1.5 0.0417 0.1400 2.0 0.0333 0.1200 2.14 0.0277 0.1093 3.3.8 Bar Width Cw 2 0.0426 0.1500 3 0.0435 0.1489 4 0.0407 0.1411 5 0.0417 0.1400 6 0.0410 0.1352 7 0.0381 0.1265 8 0.0381 0.1241 Table C-1. Continued Root mean square wave surface elevation (r,,) (m) At offshore boundary At 25 m seaward of shoreline cutoff depth 3.3.9 Distance of Bar from Shore 80 0.0373 0.1371 90 0.0358 0.1277 100 0.0397 0.1325 110 0.0397 0.1339 120 0.0417 0.1400 130 0.0399 0.1343 140 0.0422 0.1400 150 0.0434 0.1432 3.3.10 Water Depth at Offshore Boundary 7.19 0.0474 0.1506 8.16 0.0437 0.1413 9.13 0.0395 0.1343 10.09 0.0417 0.1400 11.97 0.0367 0.1284 12.91 0.0368 0.1308 3.3.11 Domain Length Id (m)__ _ 750 0.0455 0.1379 1000 0.0417 0.1400 1375 0.0377 0.1389 1625 0.0359 0.1411 LIST OF REFERENCES Battjes, J. A., and Janssen, J. P. F. M., Energy loss and set-up due to breaking of random waves, Proceedings of the 16th International Coastal Engineering Conference, Hamburg, pp. 569-587, American Society of Civil Engineers, New York, 1978. Battjes, J. A., Bakkenes, H. J., Janssen, T. T., and van Dongeren, A. R., Shoaling of subharmonic gravity waves, Journal of Geophysical Research, 109, C02009, 2004. Bowen, A. J., and Inman, D. L., Edge waves and crescentic bars, Journal of Geophysical Research, 76(36), 8662-8671, 1971. Bryan, K. R., and Bowen, A. J., Bar-trapped edge waves and longshore currents, Journal of Geophysical Research, 103(C 12), 27,867-27,884, 1998. Dean, R. G., and Dalrymple, R. A., Water Wave Mechanics for Engineers and Scientists, World Scientific Publishing Co., River Edge, NJ, 1991. Dean, R. G., and Dalrymple, R. A., Coastal Processes n i/l Engineering Applications, Cambridge University Press, New York, NY, 2002. Henderson, S. M., and Bowen, A. J., Observations of surf beat forcing and dissipation, Journal of Geophysical Research, 10 7(C 11), 14-1-14-10, 2002. Herbers, T. H. C., Elgar, S., and Guza, R. T., Infragravity-frequency (0.005-0.05 Hz) motions on the shelf, part I, Forced waves, Journal of Physical Oceanography, 24, 917-927, 1994. Herbers, T. H. C., Elgar, S., and Guza, R. T., Generation and propagation of infragravity waves, Journal of Geophysical Research, 100(C12), 24,863-24,872, 1995a. Herbers, T. H. C., Elgar, S., Guza, R. T., and O'Reilly, W.C., Infragravity-frequency (0.005-0.05 Hz) motions on the shelf, part II, Free waves, Journal of Physical Oceanography, 25, 1063-1079, 1995b. Holman, R. A., and Bowen, A. J., Bars, bumps, and holes: models for the generation of complex beach topography, Journal of Geophysical Research, 87(C 1), 457-468, 1982. Holman, R. A., and Bowen, A. J., Longshore structure of infragravity wave motions, Journal of Geophysical Research, 89, 6446-6452, 1984. 84 Hornbeck, R. W., Numerical Methods, Quantum Publishers, Inc., New York, NY, 1975. Huntley, D. A., Guza, R. T., and Thornton, E. B., Field observations of surf beat 1. Progressive edge waves, Journal of Geophysical Research, 86(C7), 6451-6466, 1981. Janssen, T. T., Battjes, J. A., and van Dongeren, A. R., Long waves induced by short- wave groups over a sloping bottom, Journal of Geophysical Research, 108(C8), 2003. Kamphius, J. W., Introduction to Coastal Engineering and Management, World Scientific Publishing Co., River Edge, NJ, 2000. Lippmann, T. C., Herbers, T. H. C., and Thornton, E. B., Gravity and shear wave contributions to nearshore infragravity motions, Journal of Physical Oceanography, 29, 231-239, 1999. List, J. H., Breakpoint-forced and bound long waves in the nearshore: a model comparison, Proceedings of the 23rd International Coastal Engineering Conference, Venice, Italy, pp. 860-873, American Society of Civil Engineers, New York, 1992a. List, J. H., A model for the generation of two-dimensional surf beat, Journal of Geophysical Research, 97(C4), 5623-5635, 1992b. Longuet-Higgins, M. S., and Stewart, R. W., Radiation stresses and mass transport in gravity waves, with application to "surf beats," Journal ofFluidMechanics, 13, 481-504, 1962. Longuet-Higgins, M. S., and Stewart, R. W., Radiation stresses in water waves: a physical discussion with applications, Deep Sea Research, 11, 529-562, 1964. Munk, W.H., Surf beats, EOS Transactions, American Geophysical Union, 30, 849-854, 1949. Okihiro, M., Guza, R. T., and Seymour, R. J., Bound infragravity waves, Journal of Geophysical Research, 97(C7), 11,453-11,469, 1992. Reniers, A. J. H. M., and Battjes, J. A., A laboratory study of longshore currents over barred and non-barred beaches, Coastal Engineering, 30, 1-22, 1997. Reniers, A. J. H. M., van Dongeren, A. R., Battjes, J. A., and Thornton E. B., Linear modeling of infragravity waves during Delilah, Journal of Geophysical Research, 107(C10), 3137, 2002. Ruessink, B. G., The temporal and spatial variability of infragravity energy in a barred nearshore zone, Continental ShelfResearch, 18, 585-605, 1998a. 85 Sallenger, A. H., and Holman, R. A., Infragravity waves over a natural barred profile, Journal of Geophysical Research, 92(C9), 9531-9540, 1987. Schaffer, H. A., Infragravity waves induced by short-wave groups, Journal of Fluid Mechanics, 247, 551-588, 1993. Schaffer, H. A., Edge waves forced by short-wave groups, Journal ofFluid Mechanics, 259, 125-148, 1994. Symonds, G., Huntley, D.A., and Bowen, A. J., Two-dimensional surfbeat: Long wave generation by a time varying breakpoint, Journal of Geophysical Research, 87, 492-498, 1982. Tucker, M. J., Surf beats: sea waves of 1 to 5 min. period, Proceedings of the Royal Society ofLondon, Series A, Mathematical and Physical Sciences, 202, 565-573, 1950. Van Dongeren, A., Reniers, A., and Battjes, J., Numerical modeling of infragravity wave response during DELILAH, Journal of Geophysical Research, 108(C9), 3228, 2003. Yu, J., and Slinn, D. N., Effects of wave-current interaction on rip currents, Journal of Geophysical Research, 108(C3), 33-1-33-19, 2003. SUPPLEMENTARY REFERENCES Baldock, T. E., and Simmonds, D. J., Separation of incident and reflected waves over sloping bathymetry, Coastal Engineering, 38, 167-176, 1999. Battjes, J. A., and Stive, M. J. F., Calibration and verification of a dissipation model for random breaking waves, Journal of Geophysical Research, 90(C5), 9159-9167, 1985. Elgar, S., Herbers, T. H. C., Okihiro, M., Oltman-Shay, J., and Guza, R. T., Observations of Infragravity Waves, Journal of Geophysical Research, 9 7(C 10), 15,573-15,577, 1992. Foda, M. A., and Mei, C. C., Nonlinear excitation of long-trapped waves by a group of short swells, Journal ofFluid Mechanics, I]], 319-345, 1981. Gallagher, B., Generation of surf beat by non-linear wave interactions, Journal of Fluid Mechanics, 49, 1971. Guza, R. T., and Davis, R. E., Excitation of edge waves by waves incident on a beach, Journal of Geophysical Research, 79(9), 1285-1291, 1974. Guza, R. T., and Bowen, A. J., Finite amplitude edge waves, Journal ofMarine Research, 34(2), 269-293, 1976. Guza, R. T., and Thornton, E. B., Observations of surf beat, Journal of Geophysical Research, 90(C2), 3161-3172, 1985. Henderson, S. M., Elgar, S., and Bowen, A. J., Observations of surf beat propagation and energetic, Proceedings of the 27th International Coastal Engineering Conference, Sydney, pp. 1412-1421, American Society of Civil Engineers, New York, 2000. Herbers, T. H. C., Elgar, S., Guza, R. T., and O'Reilly, W.C., Infragravity-frequency (0.005-0.05 Hz) motions on the shelf, Proceedings of the 23rd International Coastal Engineering Conference, Venice, Italy, pp. 846-859, American Society of Civil Engineers, New York, 1992. Holman, R. A., Infragravity energy in the surf zone, Journal of Geophysical Research, 86(C7), 6442-6450, 1981. Holman, R. A., and Sallenger, A. H., Setup and swash on a natural beach, Journal of Geophysical Research, 90(C1), 945-953, 1985. Howd, P. A., Oltman-Shay, J., and Holman, R. A., Wave variance partitioning in the trough of a barred beach, Journal of Geophysical Research, 96(C7), 12,781-12,795, 1991. Huntley, D. A., Long-period waves on a natural beach, Journal of Geophysical Research, 81(36), 6441-6449, 1976. Janssen, T. T., Kamphius, J. W., Van Dongeren, A. R., and Battjes, J. A., Observations of long waves on a uniform slope, Proceedings of the 27th International Coastal Engineering Conference, Sydney, pp. 2192-2205, American Society of Civil Engineers, New York, 2000. Kostense, J. K., Measurements of surf beat and set-down beneath wave groups, Proceedings of the 19th International Coastal Engineering Conference, Houston, pp. 724-740, American Society of Civil Engineers, New York, 1984. Lippmann, T. C., Holman, R. A., and Bowen, A. J., Generation of edge waves in shallow water, Journal of Geophysical Research, 102(C4), 8663-8679, 1997. List, J. H., Wave groupiness variations in the nearshore, Coastal Engineering, 15, 475- 496, 1991. Liu, P. L.-F., A note on long waves induced by short-wave groups over a shelf, Journal ofFluid Mechanics, 205, 163-170, 1989. Longuet-Higgins, M. S., and Stewart, R. W., Changes in the form of short gravity waves on long waves and tidal currents, Journal ofFluidMechanics, 8, 565-583, 1960. Madsen, 0. S., On the generation of long waves, Journal of Geophysical Research, 76(36), 8672-8683, 1971. Madsen, P. A., Sorensen, 0. R., And Schaffer, H. A., Surf zone dynamics simulated by a Boussinesq type model. Part II: surf beat and swash oscillations for wave groups and irregular waves, Coastal Engineering, 32, 289-319, 1997. Masselink, G., Group bound long waves as a source of infragravity energy in the surf zone, Continental ShelfResearch, 15(13), 1525-1547, 1995. Mei, C. C., The Applied Dynamics of Ocean Surface Waves, World Scientific Publishing Co., River Edge, NJ, 1992. Mei, C. C., and Benmoussa, C., Long waves induced by short-wave groups over an uneven bottom, Journal ofFluidMechanics, 139, 219-235, 1984. Nakamura, S., and Katoh, K., Generation of infragravity waves in breaking process of wave groups, Proceedings of the 23rd International Coastal Engineering Conference, Venice, Italy, pp. 990-1003, American Society of Civil Engineers, New York, 1992. |

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PAGE 1 NEARSHORE INFRAGRAVITY WAVE GE NERATION: A NUMERICAL MODEL AND PARAMETRIC STUDY By EILEEN M. CZARNECKI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2006 PAGE 2 Copyright 2006 by Eileen M. Czarnecki PAGE 3 This document is dedicated to my father, William John Czarnecki. PAGE 4 iv ACKNOWLEDGMENTS First and foremost, the author wishes to thank her advisor and supervisory committee chairman, Dr. Andrew Kennedy, for his support, patience, guidance, instruction, and assistance thr oughout this study. Gratitude is extended to the other members of the committee, Dr. Robert Dean a nd Dr. Robert Thieke, for their assistance. The author also wishes to thank the en tire faculty of the Civil and Coastal Engineering Department, who helped to sh ape her intellectual development as an engineer, both during undergra duate and graduate studies. Finally, the author would lik e to thank her family and friends for their love and support. Special thanks go to Elaine Czarn ecki, Billy Czarnecki, Barbara Lewis, and Xiaoyan Zheng. PAGE 5 v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT......................................................................................................................xii CHAPTER 1 INTRODUCTION........................................................................................................1 1.1. Definition and Importance of Infragravity Waves.................................................1 1.2 Objectives...............................................................................................................1 1.3 Literature Review...................................................................................................1 1.4 Thesis Outline.........................................................................................................4 2 NUMERICAL MODEL...............................................................................................6 2.1 Conceptual Overview.............................................................................................6 2.2 Bathymetry.............................................................................................................6 2.3 Wave Spectrum.......................................................................................................7 2.4 Wave Transformation.............................................................................................9 2.5 Long Wave Generation.........................................................................................12 2.5.1 Governing Equations..................................................................................12 2.5.2 Infragravity Wave Surface Elevation.........................................................13 2.5.3 Boundary Conditions..................................................................................17 2.5.3.1 Shoreline boundary condition..........................................................17 2.5.3.2 Offshore boundary condition...........................................................18 2.5.4 Derivation of Radiation Stress....................................................................23 2.5.4.1 Steady radiation stress......................................................................26 2.5.4.2 Low frequency radiation stress........................................................27 2.6 Root Mean Square Long Wave Surface Elevation...............................................28 2.7 Mean Water Level................................................................................................28 2.8 Model Validation..................................................................................................30 PAGE 6 vi 3 NUMERICAL SIMULATION OF LOW FREQUENCY WAVE CLIMATE..........34 3.1 Use of the Model..................................................................................................34 3.2 Parameters.............................................................................................................35 3.3 Results...................................................................................................................35 3.3.1 Base Case....................................................................................................35 3.3.2 Offshore Significant Wave Height and Peak Period..................................37 3.3.3 Jonswap Peak Enhancement Factor............................................................49 3.3.4 Deep Water Directional Width...................................................................50 3.3.5 Peak Wave Direction..................................................................................51 3.3.6 Bottom Friction..........................................................................................53 3.3.7 Bar Amplitude............................................................................................54 3.3.8 Bar Width...................................................................................................56 3.3.9 Distance of Bar from Shore........................................................................57 3.3.10 Water Depth at Offshore Boundary..........................................................58 3.3.11 Domain Length.........................................................................................60 4 CONCLUSIONS........................................................................................................62 4.1 Summary...............................................................................................................62 4.2 Discussion and Conclusions.................................................................................63 4.3 Recommendations for Further Work....................................................................67 APPENDIX A DERIVATION OF EQUATION FOR IN FRAGRAVITY WAVE SURFACE ELEVATION..............................................................................................................69 A.1 Second Order Partial Differential Equation.........................................................69 A.2 Second Order Ordinary Differentia l Equation in the Frequency Domain...........70 B DERIVATION OF EQUATIONS FOR BOUNDARY CONDITIONS....................72 B.1 Shoreline Boundary Condition.............................................................................72 B.2 Offshore Boundary Condition..............................................................................73 B.2.1 Characteristic Equations............................................................................73 B.2.1.1 Incoming bound wave.....................................................................73 B.2.1.2 Outgoing free wave.........................................................................74 B.2.1.3 Combined Characteristic Equation.........................................................74 B.2.2 Incoming Bound Wave Amplitude............................................................75 B.2.3 Offshore Boundary Condition Equation....................................................77 C VALUES OF ROOT MEAN SQUARE LONG WAVE SURFACE ELEVATION FOR EACH TEST CASE...........................................................................................79 LIST OF REFERENCES...................................................................................................83 PAGE 7 vii SUPPLEMENTARY REFERENCES...............................................................................86 BIOGRAPHICAL SKETCH.............................................................................................90 PAGE 8 viii LIST OF TABLES Table page 3-1 List of test cases with corres ponding results section and equations........................34 C-1 Root mean square wave surface eleva tion for each test case, at the offshore boundary and at 25 m seaward of the shoreline cutoff depth...................................79 PAGE 9 ix LIST OF FIGURES Figure page 2-1 Definition sketch of coordinate axes........................................................................13 2-2 Definition sketch of the incoming short wave direction .......................................15 2-3 Definition sketch of infragravity wa ve direction: incoming bound wave angle in, outgoing free wave angle out, and actual outgoing free wave angle out,actual = out + ...................................................................................................................19 2-4 Overlay of analytic and numerical solutions............................................................32 3-1 Model results for the base case: (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile.36 3-2 Effect of variation of peak wave period (Tp) (for offshore Hs=0.4 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................38 3-3 Effect of variation of peak wave period (Tp) (for offshore Hs =0.7 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................39 3-4 Effect of variation of peak wave period (Tp) (for offshore Hs =1.0 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................40 3-5 Effect of variation of peak wave period (Tp) (for offshore Hs =2.0 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................42 3-6 Effect of variation of peak wave period (Tp) (for offshore Hs =3.0 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................43 3-7 Effect of variation of offs hore significant wave height (Hs) (for Tp=4 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................44 PAGE 10 x 3-8 Effect of variation of offs hore significant wave height (Hs) (for Tp=6 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................45 3-9 Effect of variation of offs hore significant wave height (Hs) (for Tp=8 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................46 3-10 Effect of variation of offs hore significant wave height (Hs) (for Tp=10 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................47 3-11 Effect of variation of offs hore significant wave height (Hs) (for Tp=12 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)................................................................................................................48 3-12 Effect of variation of Jons wap peak enhancement factor ( ) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)...........................................................................................................................49 3-13 Effect of variation of deep-w ater directional width (dir-width0) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)...........................................................................................................................51 3-14 Effect of variation of peak wave direction ( p) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)...................52 3-15 Effect of variation of bottom friction coefficient (ff) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs)...........53 3-16 Effect of variation of bar amplitude ( a2) on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile.......................................................................................................................55 3-17 Effect of variation of bar width on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. ............................................................................................................................... ...56 3-18 Effect of variation of distance of bar from shore ( xc) on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile...................................................................................................58 PAGE 11 xi 3-19 Effect of variation of wa ter depth at offshore boundary (hoffshore) on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile..............................................................................59 3-20 Effect of variation of domain length (ld) on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile.......................................................................................................................61 PAGE 12 xii Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science NEARSHORE INFRAGRAVITY WAVE GE NERATION: A NUMERICAL MODEL AND PARAMETRIC STUDY By Eileen M. Czarnecki August 2006 Chair: Andrew Kennedy Major Department: Civil and Coastal Engineering The purpose of this study was to devel op and present a linear frequency-domain numerical model of nearshore infragravity wave generation on an alongshore-uniform beach, and then to utilize this model to inve stigate the importance of various parameters in the generation of these infragravity wave s. The numerical model was composed of two parts: wave transformation and infragra vity wave generation. Wave transformation was modeled using linear wave theory. In fragravity wave generation was based on the theory that these waves are forced by spatia l changes in the radiation stresses associated with the incoming short wave groups, then re leased to become free waves when these short waves break near the shoreline, and s ubsequently reflected at the shoreline to become outgoing free infragravity waves. Th e total infragravity wave surface elevation was determined by summing all of the infr agravity contributions due to difference interactions between pairs of incoming shor t wave spectral components with frequency differences in the infragravity range ( Hz f 05 . 0 ). The results of the numerical PAGE 13 xiii simulation of the low frequency wave climate indicated the relative importance of various parameters in the generation of nearshore infr agravity waves. These parameters affect the infragravity wave response in the followi ng manner: a higher peak wave period leads to a higher magnitude of response; a higher o ffshore significant wave height leads to a higher magnitude of response; a narrower dire ctional width leads to higher magnitude of response; increased bottom friction leads to a decreased magnitude of response; a shallower bathymetric profile leads to an in creased magnitude of re sponse; a higher bar amplitude leads to decreased magnitude of re sponse, especially near the shoreline; a wider bar leads to a s lightly higher magnitude of respons e; the bar being closer to the shore results in a narrower re sponse pattern; obliquely incide nt waves lead to a decreased magnitude of response at the offshore boundar y, although peak wave direction has little effect on response near the shoreline; a narr ower frequency spectrum leads to a slightly decreased magnitude of response. PAGE 14 1 CHAPTER 1 INTRODUCTION 1.1. Definition and Importance of Infragravity Waves Infragravity waves, also known as long wa ves or surf beat, are defined as lowfrequency waves, with periods typically between 20 and 200 seconds (Henderson and Bowen, 2002). In contrast to short-period wa ves, infragravity, or long-period, waves are not obvious from visual observation because th ey cause a very slow variation of sea surface elevation. Nevertheless, they are important in shallow water and contribute to the majority of water surface elevation variance at the shoreline (Hol man and Bowen, 1984). Infragravity waves are thought to be important in the gene ration of nearshore currents (Bryan and Bowen, 1998), harbor resonance (Janssen et al., 2003), nearshore sediment transport, and bar formation (Bowen and Inman, 1971; Holman and Bowen, 1982). 1.2 Objectives The objectives of the current study are twofold. The first objective is to develop and present a linear frequency-domain numerical model of nearshore infragravity wave generation on an alongshore-uniform beach. The second objective is to utilize the model to investigate the effects of various para meters on nearshore infragravity wave generation. 1.3 Literature Review The slow temporal variation of sea surface elevation in the nearshore zone was first observed by Munk (1949) and Tucker (1950). Their observations showed that the lowfrequency infragravity motion was correlate d with the incoming short-wave groups. PAGE 15 2 Longuet-Higgins and Stewart (1962, 1964) later hypothesized a mechanism for the generation of infragravity waves: that these waves are forced, or bound, by spatial changes in the radiation stre sses associated with the in coming short wave groups, then released to become free waves when these short waves break near the shoreline, and subsequently reflected at the shoreline to become outgoing free infragravity waves. Since that time, this theory has been explor ed extensively in analytical, numerical, field and laboratory studies. Recently, an extens ive cross-correlational study of laboratory data conducted by Janssen et al. (2003) conf irmed that the incoming bound infragravity waves travel at the group velocity cg of the incoming short wa ve groups and that the outgoing free infragravity waves travel at the shallow water wave velocity gh. The numerical model presented in this thesis is based upon this radiation stress theory. The other main theory of long wave generation by incoming short wave groups involves a time-varying breakpoint (Symonds et al., 1982). The variation in wave amplitude due to the wave groups would cause these incoming short waves to break at different locations in a periodic manner. Theoretically, the time varying breakpoint could generate free waves radiating both in th e onshore and offshore directions. However, research has shown that the long waves gene rated by a time-varying breakpoint are only a small contribution to the total infragravity wave energy, compared to those generated by spatial variation in radiati on stresses associated with short-wave groups (List, 1992a, 1992b). In their cross-correlational study, Jans sen et al. (2003) found no indications of infragravity waves being generated at the brea ker bar. Therefore, the model presented in this thesis does not includ e a time-varying breakpoint. PAGE 16 3 As mentioned previously, the incomi ng infragravity waves are bound by the incoming short wave groups. The outgoing fr ee infragravity waves, depending on their frequency and alongshore wave number, may re fract repeatedly and be trapped near the shoreline, traveling in the al ongshore direction as edge waves, or they may continue to travel in the offshore direc tion as leaky waves (Schaffer, 1994). Free waves are more energetic than bound waves, both near the shor e and in deeper water (Okihiro et al., 1992; Herbers et al., 1994). The cross-shore structure of infragravity waves shows an outgoing progressive wave at the offshore boundary and partia l standing waves near the shoreline, where edge waves often pre dominate (Symonds et al., 1982). The model developed in this thesis produced the same cr oss-shore pattern of infragravity waves. Lippmann et al. (1999) found that shear waves (instabilities in the alongshore current) contribute to infragravity velocity variance; however, since shear waves do not significantly affect the infragravity wave su rface elevation, they are not considered as a factor in the present model. Edge waves of the same frequency can resonate with each other, and thus, it is important to include fric tion in numerical models in order to prevent unbounded growth of infragravity waves (Reniers et al. 2002, Schaffer, 1993). Bathymetry is another important factor in infragravity waves. Herbers and colleagues (1995b) found that shoreline mor phology affects the generation and reflection of free infragravity waves and that the shel f-wide topography affect s the propagation and trapping of free infragravity waves. On barre d beaches, edge waves tend to be trapped at the location of the bar (Bryan and Bowe n, 1998). Sallenger and Holman (1987) found that the dominant peak of the infragravity wa ve surface elevation occurs at the bar crest. Edge waves, which propagate in the alongshor e direction, may be ei ther progressive or PAGE 17 4 standing. Standing edge waves may be gene rated by topographic cha nges or obstructions in the seabed which can act as reflectors (Huntley et al., 1981). Obviously, when modeling infragravity waves, it is important to take into account the local bathymetry. The numerical model presented in this thesis is based upon the two-dimensional numerical model of infragravity wave gene ration developed by Re niers et al. (2002). Reniers and colleages (2002) assumed an alongs hore uniform bathymetry in their model and found that results of their numerical simulation compared well with field data gathered at Duck, North Carolina. La ter, in 2003, Van Dongeren and colleagues explored whether modeling the bathymetry in two or three dimensions significantly affects the numerical simulation of low freque ncy wave climate at Duck, North Carolina. They found that long-shore variab ility in bathymetry had littl e effect on their results and concluded that, for bathymetries that do not have much long shore variability, it is valid to model infragravity wave ge neration in two dimensions. 1.4 Thesis Outline The current Chapter 1 begins with a brie f description of infragravity waves and their importance. Next, the objective of th e study is stated, followed by a literature review of research on infragravi ty waves. Chapter 2 presents a detailed description of the numerical model. Chapter 3 presents results of the model, used to investigate the low frequency wave climate under varying conditi ons, in order to dete rmine the importance of various parameters and their effects on infragravity wave generation. Chapter 4 includes a summary, conclusi ons, a discussion of results , and recommendations for further work. Appendix A and Appendix B gi ve detailed derivations of the main equations that constitute the numerical model. Appendix C tabulates the results of the numerical model for each test case described in Chapter 3. The list of references cited in PAGE 18 5 this thesis is followed by supplementary refere nces pertinent to the topic of infragravity waves. PAGE 19 6 CHAPTER 2 NUMERICAL MODEL 2.1 Conceptual Overview The numerical model developed in this chapte r is composed of two main parts: (1) wave transformation, and (2) infragravity wave generation. The two main inputs for the wave transf ormation model were the bathymetry and the wave spectrum (which was composed of many frequencies and directions). Equations for wave transformation we re based on linear wave theory. The numerical model of infragravity wa ve surface elevation was based upon the model presented by Reniers et al. (2002), with some modifications. This model mathematically describes the generation of in fragravity waves, which are composed of incoming bound waves generated by pairs of incoming short wave components and outgoing free waves that are generated by reflection of the incoming waves at the shoreline. 2.2 Bathymetry The bathymetric profile was assumed to be uniform in the alo ngshore (y) direction and was modeled in the cross-s hore (x) direction as an equili brium beach profile with an alongshore bar superimposed. The equation fo r the equilibrium beach profile (Dean and Dalrymple, 2002) is 3 2) ( Ax x h (2.2.1) PAGE 20 7 where h represents water depth, x represents the crossshore coordinate, and A is the profile scale factor. The equation for the bar, given by Yu and Slinn (2003) to approximate the alongshore bar measured at Duck, North Carolina, is 2 25 exp ) (c cx x x a x h (2.2.2) where a2 is the bar amplitude and xc is the distance of the center of the bar from the shoreline. In equations (2.2.1) and (2.2.2), x = 0 at the onshore boundary. However, since the numerical model requires that x = 0 at the offshore boundary, the order of the x vector was reversed according to x x xshoreline ' (2.2.3) in order to satisfy this condition. Equations (2.2.1) through (2.2.3) were combined to form the equation for the bathymetric profile 2 2 3 2' exp ' ) ' (c c wx x x c a Ax x h (2.2.4) with an onshore water depth of 0.25 m a nd a maximum onshore slope of 1/20. In equation (2.2.4), cw is the bar width coefficient. 2.3 Wave Spectrum The wave spectrum describing the incoming short waves was composed of a frequency spectrum and a directional spectrum . The frequency spectrum was modeled as a Jonswap frequency spectrum. The expres sion for the Jonswap spectrum is given by Kamphius (2000) as 4 5 4 exp 24 5 exp 2p a J Jf f f g f S (2.3.1) PAGE 21 8 In the above equation, f is the frequency, fp is the peak frequency, g is the gravitational constant, is the peak enhancement fact or (with a higher value of implying a narrower, more peaked frequency spectrum), a is a coefficient related to frequency, and J is a coefficient related to the wave-generating conditions. The coefficient a is given by 2 2 22p pf f f a (2.3.2) where 07 . 0 for pf f , and 09 . 0 for pf f . (2.3.3) The coefficient J is given by 22 . 0 2076 . 0 U gFJ (2.3.4) where F is the fetch length and U is the wind velocity. In this model, fetch length was assigned the value of F = 1000 m, and wind velo city was assigned the value of U = 5 m/s. The offshore significant wave height Hs was specified as an input value, which was then converted to the root mean square wave height ( Hrms). 2s rmsH H (2.3.5) The frequency spectrum was then proportioned to the root mean square wave height ( Hrms) according to the equation g E Hrms* 8 (2.3.6) where is the density of seawater and E is the total spectral energy, equal to the sum of the energies of each individual frequency co mponent. The spectral frequency resolution was set at 0.00083 Hz, to give a total of 602 components. PAGE 22 9 The directional spectrum was calculated as ) ( cosp mD (2.3.7) In the above equation, m determines the directional widt h of the spectrum (with a higher value of m corresponding to a narrower spectrum), is the incoming short wave direction, and p is the peak incoming short wave dire ction. The peak wave direction ( p) and the deep water directional width (dir-width0) were specified as input for the directional spectrum. The deep water directional width (dir-width0) was related to the directional width at the offshore boundary (dir-width1) by the following equation: ) _ sin( ) _ sin(1 1 0 0width dir k width dir k (2.3.8) where k0 and k1 are the wave numbers in deep water and at the offshore boundary, respectively. The value of m in equation (2.3.7) was chos en to match the directional width at the offshore boundary. The directional spectrum was then normalized such that the area under the spectrum was equal to one. Finally, the frequency and directional spec trums were combined into one spectrum, with one direction per frequency, ac cording to the following equation: ) ( ) ( ) , ( D f S f SJ (2.3.9) 2.4 Wave Transformation The two main inputs for the wave transf ormation program are the bathymetry and the wave frequency-directional spectrum. For each component of the frequencydirectional spectrum, the wave length, cel erity, and wave number, wave angle, and energy dissipation were calculated. From linear wave theory (Dean and Da lrymple, 1991), deep water wave length L0 and deep water wave group velocity cg0 were PAGE 23 10 22 0gT L (2.4.1) T L cg20 0 (2.4.2) where T is the wave period (s), calculated as the inverse of the wave frequency f (Hz). The wave length L , the wave celerity c , the wave number k , and the wave group velocity cg were calculated (from linea r wave theory) for each comp onent frequency, at each cross-shore location: L h L L2 tanh0 (2.4.3) T L c (2.4.4) L k 2 (2.4.5) kh kh c cg2 sinh 2 1 (2.4.6) The wave angle was calculated for each wave component, at each cross-shore location, from Snellâ€™s law, as 1 1 1sin * sinj j j jc c (2.4.7) where j represents the cross-shore location. Energy dissipation was based upon the m odel presented by Reniers and Battjes (1997). The dissipation S is given by the differential equation S Ec dx dgcos (2.4.8) where PAGE 24 11 b pQ H T g S2 max4 (2.4.9) In the above equation, is the density of water, g is the gravitational constant, Tp is the peak period, is a coefficient (equal to one in this model), Qb is the fraction of breaking waves, and Hmax is the maximum wave height. The fraction of breaking waves Qb (Battjes and Janssen, 1978) is gi ven by the implicit relationship 2 max1 exp H H Q Qrms b b (2.4.10) The maximum wave height is given as 88 . 0 tanh 88 . 0maxkh k Hb (2.4.11) with the wave breaking parameter b given as peak rms bL H, 0 033 tanh 4 . 0 5 . 0 (2.4.12) In the above equation, L0 and Hrms0 (root mean square wave he ight in deep water) were defined from the peak values in the spectrum, with peak g peak g offshore rms rmsc c H H, 0 , , 0 (2.4.13) The Newton-Raphson method of root-finding (Hornbeck, 1975) was used to solve equation (2.4.10). The energy dissipation and thus the wave height at each cross-shore location was calculated for each wave component by using forward differences to estimate the differential equation (2.4.8). PAGE 25 12 2.5 Long Wave Generation 2.5.1 Governing Equations The continuity equation and the crossshore and alongshore momentum equations form the basis of the numerical model. The equation for infragravity wave surface elevation was derived from these three ba sic linearized long wave equations, given below . These equations assume that , u , and v are small and therefore, that the nonlinear terms can be neglected. Long wave continuity equation: 0 ) ( ) ( y hv x hu t (2.5.1) Cross-shore momentum equation: y S x S x gh t u hxy xx (2.5.2) Alongshore momentum equation: x S y S y gh t v hyx xy (2.5.3) In the above equations, represents the infragravity wave surface elevation, u represents the cross-shore velocity com ponent, v represents the along-s hore velocity component, t represents time, and x and y represent the coordi nate axes. In this model, the x-axis is oriented in the cross-shore di rection, with x=0 at the o ffshore boundary. The y-axis is oriented in the along-shore direction. A defin ition sketch of coordinate axes is given in Figure 2-1. Sxx, Sxy, and Syy represent the radiation stresses, which force the infragravity waves. Radiation stress symbolized as Smn represents the flux of the m-component of PAGE 26 13 momentum in the n-direction. Thus, for example, Sxy represents the flux of the xcomponent of momentum in the y-direction. offshore boundary shoreline0 x y Figure 2-1. Definition sket ch of coordinate axes 2.5.2 Infragravity Wave Surface Elevation Equations (2.5.1)-(2.5.2) were combined into one equation. This resulted in the second order partial differential equation for infragravity wave surface elevation: 2 2 2 2 2 2 2 2 2 2 22 1 1 y S y x S x S g y h x dx dh x h t g t gyy yx xx (2.5.4) with the linear damping term t g added in order to prevent unbounded growth in the case of infragravity wave resonance. The term is a resistance factor, given by ) ( ) ( x h f xf (2.5.5) and ff is an empirical coefficient of bottom fric tion. The derivation of equation (2.5.4) is given in Appendix A. Equation (2.5.4) was transformed into a seri es of second order ordinary differential equations (2.5.6) in the frequency domain so that it could later be solved using finite difference representations of derivatives with respect to x. PAGE 27 14 yy y yx y xx yS k dx S d k i dx S d g k h g f i g f dx d dx dh dx d h 2 1 2 4 2 2 2 2 2 2 2 2 (2.5.6) The derivation of equation (2.5.6) is given in Appendix A. This transformation necessitates defining f unctional representations of infragravity wave surface elevation , and radiation stresses Sxx, Sxy, and Syy: * 2 exp ) , , , , ( 2 1 ) , , , , , , (2 1 2 1 2 1 2 1 y k ft i k k f f x k k f f t y xy y y y y (2.5.7) * 2 exp ) , , , , ( 2 1 ) , , , , , , (2 1 2 1 2 1 2 1 y k ft i k k f f x S k k f f t y x Sy y y xx y y xx (2.5.8) * 2 exp ) , , , , ( 2 1 ) , , , , , , (2 1 2 1 2 1 2 1 y k ft i k k f f x S k k f f t y x Sy y y xy y y xy (2.5.9) * 2 exp ) , , , , ( 2 1 ) , , , , , , (2 1 2 1 2 1 2 1 y k ft i k k f f x S k k f f t y x Sy y y yy y y yy (2.5.10) where * is the complex conjugate, and fr equency and wave number are from a combination of two incoming short wave spectral components (indicated by the subscripts 1 and 2). The infragravity wave frequency is defined as 2 1f f f (2.5.11) The radial frequency is defined as f 2 (2.5.12) The alongshore wave number is defined as 2 2 1 1 2 1sin sin k k k k ky y y (2.5.13) PAGE 28 15 shoreline x where represents the incoming short wave di rection. The cross-shore wave number (which will later be used to calculate the am plitude of infragravity wave surface elevation and the amplitudes of the radiation stresses , , xy xxS Sand yyS ) is defined as 2 2 1 1 2 1cos cos k k k k kx x x (2.5.14) A definition sketch of the incoming short wave direction is given in Figure 2-2. The wave direction of each incoming short wave component was between -/2 and /2 radians. Wave angles were measured with respect to the positive x-axis, according to the right-hand-rule (counterclockwise positive and clockwise negative). /2 /2 Figure 2-2. Definition sketch of the incoming short wave direction In order to numerically solve e quation (2.5.7), derivatives of (and similarly of xyS and xxS) were estimated using sec ond order central differences. x xj j 2 1 1 (2.5.15) 2 1 1 2 2 2 x x j j j (2.5.16) where the subscript j represents the cross-shore (x) location and x represents the grid resolution, which was set to 5 m in this model. The central difference equations were PAGE 29 16 substituted into equation (2.5.7), in order to obtain the solution for j ( at a typical cross-shore location j). yy y j xy j xy y j xx j xx j xx j j j j j y j j j jS k x S S k i x S S S g x dx dh x h x h k h g f i g f x dx dh x h 2 1 2 1 2 2 4 2 12 ) 1 ( ) 1 ( 2 ) 1 ( ) ( ) 1 ( 1 2 2 2 2 2 1 2 (2.5.17) In equation (2.5.17), j is unknown, the right hand side consists of known quantities which may be calculated numerica lly, and the coefficients of 1j , j , and 1j may also be calculated numerically. Equation (2.5.17) is valid at al l interior points in the cross-sh ore grid. It was solved in matrix form and was represented as j j j j j j jR c b a 1 1 (2.5.18) with additional coefficients at the onshore and offshore boundaries. The coefficients of equation (2.5.18) are x dx dh x h aj j j2 12 (2.5.19) 2 2 2 22 2 4x h k h g f i g f bj y j j (2.5.20) x dx dh x h cj j j2 12 (2.5.21) and the right-hand-side is PAGE 30 17 yy y j xy j xy y j xx j xx j xx jS k x S S k i x S S S g R 2 12 ) 1 ( ) 1 ( 2 ) 1 ( ) ( ) 1 ( (2.5.22) Note that j goes from 1 to n, with j = n at the shoreline and j = 1 at the offshore boundary. For j = 1 to n, if n = 6, the ma trix would be set up as such: 6 5 4 3 2 1 6 5 4 3 2 1 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 R R R R R R b a z c b a c b a c b a c b a d c b (2.5.23) The coefficients b1, c1, and d1 and zn, an, and bn as well as right-hand-sides R1 and Rn will be defined in the following section on boundary conditions. 2.5.3 Boundary Conditions Boundary conditions were needed because: (1) at the shoreline, equation (2.5.17) would have terms with the subscr ipt (j+1), and at j = n, (j+1 ) does not exist; and (2) at the offshore boundary, equation (2.5.17) would have terms with the subscript (j-1), and at j =1, (j-1) does not exist. Thus, boundary condition equations were formed by mathematically describing the conditions at both the shoreline and at the offshore boundary. Second order backward differences we re used to estimate the derivatives at the shoreline boundary; second or der forward differences were used to estimate the derivatives at the offshore boundary. 2.5.3.1 Shoreline boundary condition At the shoreline boundary, where j = n, it was assumed that there ex ists a â€œwallâ€ at a very small depth. Then, obviously, un = 0, since there could be no x-directed velocity through that wall. This assumption was appl ied to equation (2.5.2) in the frequency PAGE 31 18 domain, using the second order backward di fference representation for derivatives with respect to x, to obtain ) ( ) 2 ( ) 1 ( ) ( 1 2 2 4 3 1 2 3 2 2 1n xy y n xx n xx n xx n n n nS k i x S S S gh x x x (2.5.24) The complete derivation of this equation is given in Appendix B. Equation (2.5.24) forms the shoreline bounda ry condition for the matrix (2.5.23) and may be represented as n n n n n n nR b a z 1 2 (2.5.25) with coefficients x zn2 1 (2.5.26) x an2 (2.5.27) x bn2 3 (2.5.28) and the right-hand-side ) ( ) 2 ( ) 1 ( ) ( 2 4 3 1n xy y n xx n xx n xx n nS k i x S S S gh R (2.5.29) 2.5.3.2 Offshore boundary condition At the offshore boundary, it was assumed that the overall long wave surface elevation was formed by the superpositi on of the incoming bound and outgoing free waves, such that out b (2.5.30) PAGE 32 19 shoreline x where wave surface elevation for the incoming bound wave (subscript b ) and the outgoing free wave (subscript out ) were defined as * sin cos exp ~ 2 1 in in in b by x K t i (2.5.31) * sin cos exp ~ 2 1 out out out out outy x K t i (2.5.32) Wave angles in and out were measured with respect to the positive x-axis, according to the right-hand-rule (countercloc kwise positive and clockwise negative). A definition sketch of infragravity wave direction is given in Figure 2-3. /2 out,actual in out /2 Figure 2-3. Definition sketch of infragravity wave direction: incoming bound wave angle in, outgoing free wave angle out, and actual outgoing free wave angle out,actual = out + The value of the incoming bound wave angle in was between /2 and /2 radians. The outgoing free wave angle out was defined such that its value was also between /2 and /2 radians. This was done because, for inve rse trigonometric functi ons, Matlab returns a value between /2 and /2 radians, and out would have to be calculated using the PAGE 33 20 inverse sine function (accordi ng to equation (2.5.39)). Ho wever, the actual value of out was out actual out, (2.5.33) The wave number of the incoming bound wave, Kin, was defined as 2 2 x y ink k K (2.5.34) where in in xK k cos (2.5.35) in in yK k sin (2.5.36) The wave number of the outgoing free wave, Kou t, was defined as 2 2 xout y outk k K (2.5.37) where out out xoutK k cos (2.5.38) out out yK k sin (2.5.39) Equivalently, since the speed of the outgoing free wave is gh, gh f Kout 2 (2.5.40) Given the above expressions, equations (2 .5.31) and (2.5.32) may be expressed equivalently in terms of wave numbers: * 2 exp exp ~ 2 1 y k ft i x k iy x b b (2.5.41) * 2 exp exp ~ 2 1 y k ft i x iky xout b out (2.5.42) PAGE 34 21 The offshore boundary condition was formed by starting with characteristic equations for both the incoming bound wa ve and the outgoing free wave. The characteristic equation for the surface elevation of the incoming bound wave is 0 sin 2 cos 2 y K f x K f tb in in b in in b (2.5.43) The characteristic equation for the surface elevation of the outgoing free wave is 0 y K k gh x K k gh tout out y out out xout out (2.5.44) Equations (2.5.43) and (2.5 .44) were combined with out b to obtain y K f K k gh x K f K k gh y K k gh x K k gh tb in in out y b in in out xout out y out xout sin 2 cos 2 (2.5.45) Equation (2.5.45) was the starting point fo r forming the offshore boundary condition. The derivation of this equation as well as proof that equations (2.5.43) and (2.5.44) are characteristic equations for b and out are given in Appendix B. The amplitude b ~ of the incoming bound wave was obtained analytically by applying equation (2.5.4) to the incoming bound wave at the offshore boundary. 2 1 2 ) 1 ( 1 1 2 2 ) 1 ( 2 ) 1 ( ) 1 ( ) 1 ( 2 ) 1 (2 4 2 ~y x yy y xy y x xx x bk gh k gh f i f S k S k k S k (2.5.46) In the above equation, the subscript 1 repres ents the cross-shore location at the offshore boundary. The radiation stress amplitudes , , xy xxS Sand yyS were also found analytically and will be discussed in section 2.5.4. The derivation of equation (2.5.46) is given in Appendix B. PAGE 35 22 Finally, equation (2.5.45) was transformed into the frequency domain. Then, using the second order backward difference representation for the derivative of with respect to x, the offshore boundary condition equation (2.5.47) was obtained. in in out y b y in in out xout b x out xout out xout out y out xoutK f K k gh k i K f K k gh k i K k x gh K k x gh K gh k i K k gh x f i sin 2 ~ 2 1 cos 2 ~ 2 1 4 2 1 4 33 2 1 2 (2.5.47) The complete derivation of this equation is given in Appendix B. Equation (2.5.47) forms the shoreline bounda ry condition for the matrix (2.5.23) and may be represented as 1 3 1 2 1 1 1 R d c b (2.5.48) with coefficients out y out xoutK gh k i K k gh x f i b2 12 1 4 3 (2.5.49) out xoutK k x gh c1 (2.5.50) out xoutK k x gh d 41 (2.5.51) and the right-hand-side in in out y b y in in out xout b xK f K k gh k i K f K k gh k i R sin 2 ~ 2 1 cos 2 ~ 2 11 (2.5.52) PAGE 36 23 2.5.4 Derivation of Radiation Stress Radiation stresses were de rived from the wave energies of the incoming short waves. The free surface elevation of a component incoming short wave is * exp 2 2 1 i H (2.5.53) where the component wave height H is a real number, and where the short wave phase function is defined by its derivatives, as follows: cos k x (2.5.54) sin k y (2.5.55) t (2.5.56) Recall that the wave number k and the wave angle of the incoming short wave are functions of x. Also, the long wave phase is y k fy 2 (2.5.57) where in in yK k sin , and neither Kin nor in are functions of x. The numerical model employs a large numb er of component incoming short waves, such that the free surface elevation is N m m 1 (2.5.58) where m is the individual component and N is the total number of components. The square of the free surface elevation 2 can be represented in summation notation. The PAGE 37 24 following formula shows that 2 can be reduced to two-co mponent interactions of incoming short waves. 1 11 1 2 22N m N m n n m N m m (2.5.59) where 8 2 exp 16 2 exp 16 2 2 2 2m m m m m mH i H i H (2.5.60) m n n m n m n m n m n m n m n m n mi H H i H H i H H i H H exp 8 exp 8 exp 8 exp 8 2 (2.5.61) Terms involving summing the phases of the two component incoming short waves, such as m + n or 2m are high frequency terms. Terms involving the difference of the phases of the two component incoming short waves, such as m n or n m, are low frequency terms. Terms with no exponential are steady terms. High frequency terms will be discarded, since this numerical model i nvolves radiation stresse s that force the long waves, which are, by definition, low freque ncy waves. From equations (2.5.60) and (2.5.61), keeping only the low frequency and steady terms, * exp 8 2 n m n m lowi H H (2.5.62) 8 2 2m steadyH (2.5.63) Next, the radiation stresses , , yy xxS S and xyS were derived in terms of the component wave heights. The general equati ons for radiation stress, valid only for steady or slowly varying waves, (D ean and Dalrymple, 1991) are PAGE 38 25 2 1 1 cos2n E Sxx (2.5.64) 2 1 1 sin2n E Syy (2.5.65) 2 sin 2 n E S Syx xy (2.5.66) where c c ng (2.5.67) As mentioned previously, Sxx is the wave momentum flux in the x-direction of the xcomponent of momentum, Syy is the wave momentum flux in the y-direction of the ycomponent of momentum, and Sxy is the wave momentum flux in the x-direction of the ycomponent of momentum. Since the general equation for wave energy is 28 1 gH E (2.5.68) then the radiation stresses are proportional to the wave hei ghts squared. The component wave heights H were obtained from the component en ergies of the input wave energy spectrum according to equation (2.5.68). The radiation stresses , , yy xxS S and xyS were calculated at each cross-shore location, for each arbitrary combination of two incomi ng short wave components. Low frequency radiation stresses were derived from the in teraction between two different components, for difference frequencies within the infragravity range ( Hz f05 . 0 ). Steady radiation stresses were derived from co mponent self-self interactions. For the low frequency components of ra diation stresses, the average incoming wave angle av and the average ratio nav were used, where PAGE 39 26 2n m av (2.5.69) 2 n g m g avc c c c n (2.5.70) The low frequency wave energy is 2low lowg E (2.5.71) In terms of wave heights, this may be written as * exp 4 1 n m n m lowi H H g E (2.5.72) The steady wave energy is 2 2 8 1 m steady steadyH g g E (2.5.73) Equations (2.5.8)-(2.5.10) give functiona l representations of the low frequency components of radiation stress. However, th e total radiation stress at any cross-shore location is composed of both the steady a nd low frequency (uns teady) components, which can be represented in summation notation as * )] 2 ( exp[ 2 1 , , ,y k ft i S S Sy low xx steady xx total xx (2.5.74) * )] 2 ( exp[ 2 1 , , ,y k ft i S S Sy low xy steady xy total xy (2.5.75) * )] 2 ( exp[ 2 1 , , ,y k ft i S S Sy low yy steady yy total yy (2.5.76) 2.5.4.1 Steady radiation stress The steady component radiation stresses are 2 1 1 cos 2 , m m steady steady xxmn E S (2.5.77) PAGE 40 27 2 1 1 sin 2 , m m steady steady yymn E S (2.5.78) 1 1 ,2 sin 2 n E Ssteady steady xym (2.5.79) 2.5.4.2 Low frequency radiation stress The low frequency (unsteady) component radiation stress amplitudes are 2 1 1 cos 2 , av av low low xxn E S (2.5.80) 2 1 1 sin 2 , av av low low yyn E S (2.5.81) av av low low xyn E S2 sin 2 , (2.5.82) It can be seen from equation (2.5.72) that the low frequency wave energy lowE is a function of the incoming short wave phase (equations (2.5.54)(2.5.56)), which is a function of both x, y, and t. In order to obt ain the low frequency components of radiation stress at each cross-shore or x-location, it was necessary to know the phase at each crossshore location. In the numerical model, it was assumed that y = 0 and t = 0 for all x and that = 0 at the offshore boundary. Thus, with this simplifying assumption, the short wave phase was considered to be a function of x only. Then, from equation (2.5.54) for x and from forward differences, was calculated at each location in the crossshore, beginning at the offshore boundary and continuing towards the shoreline, using the following formula (where j represents the cross-shore location) j j j jk x cos1 (2.5.83) PAGE 41 28 2.6 Root Mean Square Long Wave Surface Elevation As stated previously, this numerical model combines two component incoming short waves from the input frequency-direct ional spectrum. Every possible combination of components was calculated, such that each component was combined with every other component, including itself . When two different components, with difference frequencies within the infragravity range ( Hz f05 . 0 ), were combined, the result was the forcing of a long wave. When a component was combined with itself, the result was setup or setdown, as will be described in the following section on mean water level. Since the long wave surface elevation is a periodic function, it is appropriate to represent its amplitude as a root mean square . Thus, the root mean square wave surface elevation, is given by n m j n m j rms , 2 ) ( , ) ( 2 1 (2.6.1) where m,n indicates the combination of two different components and j represents the cross-shore location. Note that the term ) ( ,j n m in equation (2.6.1) is equivalent to j from equation (2.5.17). 2.7 Mean Water Level As stated in the previous section, when a component was combined with itself, the result was setup or setdown. At the offs hore boundary, the mean water level is given by linear wave theory (Dean & Dalrymple, 1991) as g t C kh k a) ( 2 sinh 22 (2.7.1) where 2 H a and 0 ) ( t C if the mean water level is zero in deep water. Thus, the mean water level for each wave compone nt at the offshore boundary is given by PAGE 42 29 kh k H 2 sinh 8 2 . (2.7.2) The mean water level at each location in the cross-shore was derived from the cross-shore momentum equation (2.5.2), where is the water surface elevation, and Sxx and Sxy are the steady components of radiation stress. Taking the time average, 0 t u, and thus the equation reduces to y S x S g x hsteady xy steady xx steady, ,1 (2.7.3) where the long wave phase = 0 for a steady wave and thus x xsteady steady 2 1 (2.7.4) x S x Ssteady xx steady xx , , 2 1 (2.7.5) steady xy y steady xyS k i y S, , 2 1 (2.7.6) Then, using forward differences to estimate derivatives with respect to x, one obtains j steady xy y j steady xx j steady xx j steady j j steady jS k i x S S g x h x h, , 1 , 1 1 (2.7.7) where j represents the cro ss-shore location. Since 0 yk when the two incoming short wave components are the same, the above equation reduces to j steady xx j steady xx j j steady j steadyS S gh, 1 , 1 1 (2.7.8) Thus, the mean water level or setup is calculated from steady , for each self-self interaction of incoming wave components, at each cross-shore location, beginning with PAGE 43 30 steady at the offshore boundary. Then, the sum of steady for all components gives the mean water level at each cross-shore location. m j m steady jMWL) ( , ) ( (2.7.9) where m represents the self-self combination of an incoming wave component,and j represents the cross-shore location. The accuracy of this calculation was checked by assuring that the mean water level at the shoreline was approximately 15% of the breaking wave height and also approximately equal to the magnitude of root mean square infragravity wave surface elevation multiplied by 2. 2.8 Model Validation The model was validated by analytically solving the sec ond order partial differential equation for infragravity wave surface elevation (equation (2.5.4)) for a simple test case: a simple input spectrum of two incoming wave components each with wave direction = 0o, no bottom friction ( = 0), a flat bed with a constant water depth of 3 m, and a reflecting boundary condition at the shoreline. The simplified version of equation (2.5.4) for these given conditions is given by equation (2.8.1). Additional criteria for the simple test case were f1 = 0.19 Hz, f2 = 0.20 Hz, and domain length = 7000 m. The governing second order partial differe ntial equation for infragravity wave surface elevation was 2 2 2 2 2 21 1 x S g x h t gxx (2.8.1) which is satisfied everywhere in the domain by PAGE 44 31 f b (2.8.2) where b (bound wave) is the particular so lution to equation (2.8.1), given by * 2 exp ~ 2 1 x k ft ix b b (2.8.3) and where f (free wave) is the homogeneous solu tion to equation (2.8.1), given by * 2 2 exp ~ 2 1 2 2 exp ~ 2 12 1 x gh f ft i x gh f ft if f f (2.8.4) The amplitude b ~ of the incoming bound wave was alrea dy solved analytically (equation (2.5.46)), which simplifies for this test case to xx x x bS k gh f k ~ 2 2 2 ~2 2 2 2 (2.8.5) where xxS ~ is equivalent to xxS at x = 0 and is constant acr oss the domain because of the constant depth. The shoreline boundary condition was take n from the cross-shore momentum equation (2.5.2), for u = 0 (for perf ect reflection at the shoreline) and ky = 0 (from all = 0o), reduced to x S gh xxx 1 (2.8.6) The offshore boundary condition was taken fr om equation (2.5.45), reduced to x K f gh x gh tb in 2 (2.8.7) with Kin = kx since ky = 0. By substituting equations (2.8.2) (2.8.4) into the boundary conditions, one obtains 0 ~1 f and PAGE 45 32 j x b xx x j fx gh f k i gh S f gh k 2 exp ~ ~ 2 ~) ( 2 (2.8.8) where the subscript j represents the cross-shor e location. Then, making substitutions into equation (2.8.2), the analytic solution for the infragravity wave surface elevation is j f j x b jx gh f i x k i ft i 2 exp ~ exp ~ 2 exp 2 12 (2.8.9) The root mean square wave surface elevation fo r the analytic solution was then calculated as 2 2 ) (2 exp ~ exp ~ 2 1 j f j x b j rmsx gh f i x k i (2.8.10) The results of the analytic solution (rms ) were then compared to the results of the numerical solution used by the model (rms ), equation (2.6.1). Figure 2-4 shows an overlay of the analytic and numerical solutions for the simple test case discussed in this section. Both solutions show an almost co mplete standing wave ac ross the entire crossFigure 2-4. Overlay of analytic (rms ) and numerical (rms ) solutions. Legend: __.__.rms , .. rms . PAGE 46 33 shore domain. A test of errors in the numeri cal solution showed that errors in the root mean square infragravity wave surface elev ation converged to th e exact solution with second-order accuracy. PAGE 47 34 CHAPTER 3 NUMERICAL SIMULATION OF LO W FREQUENCY WAVE CLIMATE 3.1 Use of the Model The model described in Chapter 2 was used to investigate the low frequency wave climate under varying conditions. Conditions were varied according to eleven basic parameters: offshore significant wave height , offshore peak period, the Jonswap peak enhancement factor, deep water directional wi dth, peak wave direction, bottom friction, bar amplitude, bar width, distance of the bar from the shore, water depth at the offshore boundary, and domain length. Table 3-1 lists the test cases with the corresponding results section and relevant equations for each case. The base case was the basis against which all other test cases were compared. The para meters for the other test cases are reviewed in section 3.2. Table 3-1. List of test cases with corresponding results se ction and equations Test Case Results Section Relevant Equations Base Case 3.3.1 Offshore Significant Wave Height (Hs) 3.3.2 (2.3.5) Offshore Peak Period (Tp) 3.3.2 (2.3.1) Jonswap Peak Enhancement Factor ( ) 3.3.3 (2.3.1) Deep Water Directional Width (dir-width0) 3.3.4 (2.3.8) Peak Wave Direction ( p) 3.3.5 (2.3.7) Bottom Friction (ff) 3.3.6 (2.5.4)-(2.5.5) Bar Amplitude ( a2) 3.3.7 (2.2.4) Bar Width ( cw) 3.3.8 (2.2.4) Distance of Bar From Shore ( xc) 3.3.9 (2.2.4) Water Depth at Offshor Boundary (hoffshore) 3.3.10 (2.2.4) Domain Length (ld) 3.3.11 (2.2.4) PAGE 48 35 3.2 Parameters Significant wave height (Hs) at the offshore boundary, peak period (Tp), and the Jonswap peak enhancement factor ( ) were specified as input for the Jonswap frequency spectrum. The offshore si gnificant wave height (Hs) is given by equation (2.3.5). The peak period (Tp) was calculated as the inverse of the peak frequency (fp). Both the peak frequency (fp) and the Jonswap peak enhancement factor ( ) are given in equation (2.3.1), which defines the Jonswap frequency spectrum. The deep water directional width (dirwidth0) and peak wave direction ( p) were specified as input for the directional spectrum (equations (2.3.7)-(2.3.8)). The coefficient of bottom friction ff influenced the infragravity wave response (see equations (2.5.4)-(2.5.5)) and served to prevent unbounded growth in the case of infragravity wa ve resonance. The remaining parameters were specified as input for the bathymetric profile (equation (2.2.4 )): bar amplitude ( a2), bar width coefficient ( cw), distance of the bar from the shore ( xc), domain length, and water depth at the offshore boundary, which was varied by varying the profile scale factor (A). 3.3 Results 3.3.1 Base Case The base case was the basis against which al l other test cases were compared. The parameters for the base case were as fo llows: offshore significant wave height Hs = 1 m, offshore peak period Tp = 8 s, Jonswap peak enhancement factor = 3.3, deep water directional width dir-width0 = 15o, peak wave direction p = 0o, coefficient of bottom friction ff = 1/200 s-1, bar amplitude a2 = 1.5 m, bar width coefficient cw = 5, distance of the bar from the shore xc = 120 m, water depth at the offshore boundary hoffshore = 10.09 m (from the profile scale factor A = 0.1 m), and domain length ld = 1000 m. PAGE 49 36 Figure 3-1 shows the model results for the base case. Part (c) of this figure displays the bathymetric profile. Unless ot herwise noted, this is the bathymetric profile used in the subsequent test cases. Part (b) of this figur e shows the transformati on of significant wave height (Hs); as the incoming short waves approach the shore, they shoal and break. Figure 3-1. Model results for the base case: (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. PAGE 50 37 As expected, wave breaking is pronounced in th e region of the bar. Part (a) of Figure 3-1 displays the root mean square long wave surface elevation ( -hatrms), calculated from equation (2.6.1) * . As expected, at the offshore boundary, the infragravity wave has a small magnitude, and near the wave breaking re gion, the infragravity wave has a larger magnitude; a time animation of model result s showed the pattern of a small outgoing infragravity wave at the offshore boundary a nd a larger magnitude partial standing wave near the wave breaking region. The vertical line near the sh oreline indicates the cutoff depth. The cutoff depth was calculated, for each case, as the shallowest depth greater than 0.5 m; shoreward of this cutoff depth, th e model results are considered to be invalid. For the base case, the cutoff depth was 0.59 m. At the offshore boundary, the root mean square long wave surface elevation was 0.042 m; 25 m seaward of the shoreline cutoff depth, the root mean square wave surface elevation was 0.140 m. Appendix C tabulates the values of the root mean square long wave surface elevation at the offshore boundary and at 25 m seaward of the shoreline cutoff depth, for each test case. 3.3.2 Offshore Significant Wave Height and Peak Period Figures 3-2 â€“ 3-6 show the effect of variation of peak wave period (Tp) on model results, for offshore si gnificant wave heights (Hs) of 0.4 m, 0.7 m, 1.0 m, 2.0 m, and 3.0 m, respectively. These figures clearly show th at, for all cases, a hi gher peak wave period leads to a higher magnitude of long wave response ( -hatrms), across the entire crossshore (x) domain. This seems to be consistent with results of a numerical study by Battjes et al. (2004), in which it was found that incoming bound infragravity waves * In this chapter, the root mean square infr agravity wave surface elevation is denoted as -hatrms in order to be consistent with axis labels in the figures. This is equivalent to rms in equation (2.6.1). PAGE 51 38 generated by higher frequency incoming wave components experience significantly more dissipation than the incoming bound waves gene rated by lower frequency incoming wave components. Figures 3-7 â€“ 3-11 show the effect of vari ation of offshore significant wave height (Hs) on model results, for peak wave periods (Tp) of 4 s, 6 s, 8 s, 10 s and 12 s, respectively. These figures clearly show that , for all cases, a higher offshore significant wave height leads to a higher ma gnitude of long wave response ( -hatrms), across the entire cross-shore (x) domain. This makes sense, because the ra diation stresses that force Figure 3-2. Effect of vari ation of peak wave period (Tp) (for offshore Hs=0.4 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Tp=4 s, .. Tp=6 s, __.__. Tp=8 s, __ __ Tp=10 s, ***** Tp=12 s. PAGE 52 39 the infragravity waves are proportional to the wave heights squared; thus, more energetic (higher) incoming short waves lead to greater forcing and thus higher amplitudes of the infragravity waves. Figure 3-2 shows the effect of the variat ion of peak wave period for an offshore significant wave height of 0.4 m. In Figure 3-2 (a), the offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.006 m for Tp = 4 s to 0.025 m for Tp = 12 s. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.024 m for Tp = 4 s to 0.073 m for Figure 3-3. Effect of vari ation of peak wave period (Tp) (for offshore Hs =0.7 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Tp=4 s, .. Tp=6 s, __.__. Tp=8 s, __ __ Tp=10 s, ***** Tp=12 s. PAGE 53 40 Tp = 12 s. Figure 3-2 (b) clearly shows that, for an offshore significa nt wave height of 0.4 m, a higher peak wave period leads to a high er significant wave height, particularly in the wave shoaling region. Figure 3-3 shows the effect of the variat ion of peak wave period for an offshore significant wave height of 0.7 m. In Figure 3-3 (a), the offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.010 m for Tp = 4 s to 0.042 m for Tp = 12 s. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.040 m for Tp = 4 s to 0.128 m for Figure 3-4. Effect of vari ation of peak wave period (Tp) (for offshore Hs =1.0 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Tp=4 s, .. Tp=6 s, __ Tp=8 s, __ __ Tp=10 s, ***** Tp=12 s. PAGE 54 41 Tp = 12 s. Figure 3-3 (b) clearly shows that, for an offshore significa nt wave height of 0.7 m, a higher peak wave period leads to a high er significant wave height, particularly in the wave shoaling region. Figure 3-4 shows the effect of the variat ion of peak wave period for an offshore significant wave height of 1.0 m. The base case (Tp = 8 s) is shown as a bold black line. In Figure 3-4 (a), the offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.012 m for Tp = 4 s to 0.065 m for Tp = 12 s. The value of the root mean square long wave surface elevati on at 25 m seaward of the shoreline cutoff depth ranges from 0.053 m for Tp = 4 s to 0.197 m for Tp = 12 s. Figure 3-4 (b) clearly shows that, for an offshore significant wave he ight of 1.0 m, a higher peak wave period leads to a higher signif icant wave height, particularly in the wave shoaling region. Figure 3-5 shows the effect of the variat ion of peak wave period for an offshore significant wave height of 2.0 m. In Figure 3-5 (a), the offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.018 m for Tp = 4 s to 0.130 m for Tp = 12 s. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.072 m for Tp = 4 s to 0.357 m for Tp = 12 s. Figure 3-5 (b) shows that, for an offshore significant wave height of 2.0 m, values of Hs through the cross-shore domain for Tp = 6 to 12 s are quite similar, although there is still a general trend for higher peak wave period lead ing to a higher significant wave height. However, when Tp = 4 s, the model causes the waves to break near the offshore boundary, limiting th e wave height to Hmax, which is dependent upon wave period (equation (2.4.11)). PAGE 55 42 Figure 3-5. Effect of vari ation of peak wave period (Tp) (for offshore Hs =2.0 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Tp=4 s, .. Tp=6 s, __.__. Tp=8 s, __ __ Tp=10 s, ***** Tp=12 s. Figure 3-6 shows the effect of the variat ion of peak wave period for an offshore significant wave height of 3.0 m. In Figure 3-6 (a), the offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.047 m for Tp = 4 s to 0.193 m for Tp = 12 s. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.105 m for Tp = 4 s to 0.443 m for Tp = 12 s. Figure 3-6 (b) shows that, for an offshore significant wave height of 3.0 m, values of Hs through the cross-shore domain for Tp = 8 to 12 s are quite similar. When Tp = 6 s, significant wave height is less than that for the higher wave periods through most PAGE 56 43 of the cross-shore domain. When Tp = 4 s, the model causes significant wave breaking near the offshore boundary, limiting the wave height to Hmax (equation (2.4.11)). Figure 3-6. Effect of vari ation of peak wave period (Tp) (for offshore Hs =3.0 m) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Tp=4 s, .. Tp=6 s, __.__. Tp=8 s, __ __ Tp=10 s, ***** Tp=12 s. Figure 3-7 shows the effect of the variati on of offshore significant wave height for a peak wave period of 4 s. In Figure 3-7 (a), the offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.006 m for Hs = 0.4 m to 0.047 m for Hs = 3 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.024 m for Hs = 0.4 m to 0.105 m for Hs = 3 m. Figure 3-7 (b) shows that, in genera l, a higher offshore significant wave height PAGE 57 44 Figure 3-7. Effect of variation of offshore significant wave height (Hs) (for Tp=4 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Hs=0.4 m, .. Hs=0.7 m, __.__. Hs=1 m, __ __ Hs=2 m, ***** Hs 3 m. corresponds to a higher Hs in the entire cross-shore domain, although wave heights tend to converge in the near shore after the waves have broken. It is notable that, when the offshore Hs = 3 m, the model causes significan t wave breaking near the offshore boundary, limiting the root mean square wave height Hrms (equation (2.3.5)) to Hmax (equation (2.4.11)); this causes convergence of Hs with the case of offshore Hs = 2.0 m. However, although wave heights (Hs) are identical th roughout most of the cross-shore region for these two cases (offshore Hs = 2 m and 3m), it can be seen from Figure 3-7 (a), that a higher offshore significan t wave height leads to a hi gher magnitude of long wave response ( -hatrms), across the entire cross-shore doma in. Mathematically, this can be PAGE 58 45 explained by the fact that the offshore boundary condition is aff ected by the offshore significant wave height, due to the amplitude of the incoming bound wave b ~ (equation (2.5.46)) increasing with increasing radiati on stress at the offshore boundary. However, since radiation stresses are proportional to wa ve heights squared, and since, for cases where the offshore Hrms is greater than Hmax (such as in the case where Hs = 3 m and Tp = 4 s), the radiation stresses at the offshore boundary are most likely overestimated and thus the results for the long wave response (in such cases) are most likely invalid. Figure 3-8. Effect of variation of offshore significant wave height (Hs) (for Tp=6 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Hs=0.4 m, .. Hs=0.7 m, __.__. Hs=1 m, __ __ Hs=2 m, ***** Hs 3 m. Figure 3-8 shows the effect of the variati on of offshore significant wave height for a peak wave period of 6 s. In Figure 3-8 (a), the offshore value of the root mean square PAGE 59 46 long wave surface elevation ( -hatrms) ranges from 0.011 m for Hs = 0.4 m to 0.060 m for Hs = 3 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.045 m for Hs = 0.4 m to 0.193 m for Hs = 3 m. Figure 3-8 (b) shows that, for a peak wave period of 6 s, a higher offshore significant wave height corresponds to a higher Hs in the entire cross-shore domain. Figure 3-9. Effect of variation of offshore significant wave height (Hs) (for Tp=8 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Hs=0.4 m, .. Hs=0.7 m, __ Hs=1 m, __ __ Hs=2 m, ***** Hs 3 m. Figure 3-9 shows the effect of the variati on of offshore significant wave height for a peak wave period of 8 s. The base case (Hs = 1 m) is shown as a bold black line. In Figure 3-9 (a), the offshore value of the root mean square long wave surface elevation ( hatrms) ranges from 0.017 m for Hs = 0.4 m to 0.106 m for Hs = 3 m. The value of the PAGE 60 47 root mean square long wave surface elevati on at 25 m seaward of the shoreline cutoff depth ranges from 0.059 m for Hs = 0.4 m to 0.282 m for Hs = 3 m. Figure 3-9 (b) shows that, for a peak wave period of 8 s, a higher offshore significant wave height corresponds to a higher Hs in the entire cross-shore domain. Figure 3-10. Effect of variation of offshore significant wave height (Hs) (for Tp=10 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Hs=0.4 m, .. Hs=0.7 m, __.__. Hs=1 m, __ __ Hs=2 m, ***** Hs 3 m. Figure 3-10 shows the effect of the variati on of offshore significant wave height for a peak wave period of 10 s. In Figure 310 (a), the offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.021 m for Hs = 0.4 m to 0.149 m for Hs = 3 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cuto ff depth ranges from 0.067 m for Hs = 0.4 m to 0.356 PAGE 61 48 m for Hs = 3 m. Figure 3-10 (b) shows that, for a peak wave period of 10 s, a higher offshore significant wave hei ght corresponds to a higher Hs in the entire cross-shore domain. Figure 3-11. Effect of variation of offshore significant wave height (Hs) (for Tp=12 s) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ Hs=0.4 m, .. Hs=0.7 m, __.__. Hs=1 m, __ __ Hs=2 m, ***** Hs 3 m. Figure 3-11 shows the effect of the variati on of offshore significant wave height for a peak wave period of 12 s. In Figure 311 (a), the offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.025 m for Hs = 0.4 m to 0.193 m for Hs = 3 m. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cuto ff depth ranges from 0.073 m for Hs = 0.4 m to 0.443 m for Hs = 3 m. Figure 3-11 (b) shows that, for a peak wave period of 12 s, a higher PAGE 62 49 offshore significant wave hei ght corresponds to a higher Hs in the entire cross-shore domain. 3.3.3 Jonswap Peak Enhancement Factor Figure 3-12 shows the effect of the vari ation of the Jonswap peak enhancement factor (equation (2.3.1)) on model results . According to Kamphius (2000), has an average value of 3.3 and typically ranges between 1.0 and 7.0. A higher value of implies a narrower, more peaked frequency spectrum. The base case ( = 3.3) is shown as a bold black line. Figure 3-12 (b) shows that th e variation of has little effect on Figure 3-12. Effect of variation of Jonswap peak enhancement factor ( ) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ =1.0, __ =3.4, .. =5.0, __.__. =7.0. significant wave height (Hs). This is expected, since the total wave energy for each case was the same. Figure 3-12 (a) shows that a narrower frequency spectrum (increased ) PAGE 63 50 leads to a slightly decreased root mean square long wave surface elevation ( -hatrms). This is unexpected, and further investigati on would have to be done to explain this pattern of results. The offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.043 m for = 1.0 to 0.036 m for = 7.0. The value of the root mean square long wave surface elev ation at 25 m seaward of the shoreline cutoff depth ranges from 0.147 m for = 1.0 to 0.120 m for = 7.0. 3.3.4 Deep Water Directional Width Figure 3-13 shows the effect of the variati on of deep water directional width, dirwidth0, (equation (2.3.8)) on model results. Th e deep water directional width was varied from 5o to 30o. The base case, with dir-width0 = 15o, is shown as a bold black line. Figure 3-13 (a) shows that a na rrower directional wi dth leads to a greater magnitude of long wave response. This findi ng is consistent with those of Van Dongeren et al. (2003); in employing a numerical model (SHOREC IRC), it was found that eliminating directional spreading from the incoming wa ve spectrum caused significant amplification of infragravity wave heights. In analyz ing field data, Herbers et al. (1995a, 1995b) found a similar relationship between di rectional spreading and infrag ravity wave response. The offshore value of the root mean s quare long wave surface elevation ( -hatrms) ranges from 0.051 m for dir-width0 = 5o to 0.030 m for dir-width0 = 30o. The value of the root mean square long wave surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.159 m for dir-width0 = 5o to 0.110 for dir-width0 = 30o. Figure 3-13 (b) shows that deep water directional width ha s little effect on significant wave height throughout the cross-shore domain. PAGE 64 51 Figure 3-13. Effect of variation of deep-water directi onal width (dir-width0) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ dir-width0=5o, __ dir-width0=15o, .. dir-width0=30o. 3.3.5 Peak Wave Direction Figure 3-14 shows the effect of the va riation of peak wave direction, p, (equation (2.3.7)) on model results. Peak wa ve direction was varied from 0o (shore-normal, base case) to 60o (oblique incidence). Figu re 3-14(a) shows that peak wave direction has little effect on long wave response near the shor eline. However, at the offshore boundary, more oblique waves (higher p) lead to a decreased magnit ude of long wave response. One possible explanation for this is that more obliquely incident short waves lead to more PAGE 65 52 Figure 3-14. Effect of varia tion of peak wave direction ( p) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: __ p=0o, ___ p=20o, .. p=40o, __.__. p=60o. trapping of the outgoing free infragravity wave s and thus less leaky waves reaching the offshore boundary. The offshore value of th e root mean square long wave surface elevation ( -hatrms) ranges from 0.042 m for p = 0o to 0.029 m for p = 60o. Figure 314(b) shows that the obliquely incident waves ( p = 60o) have a lower significant wave height; this is due to greater refraction a nd less shoaling than the non-obliquely incident waves. It is important to note that, although th e significant wave heig ht (Hs) and thus the wave energy of the obliquely incident waves ( p = 60o) is lower than the Hs for all of the other cases as the waves approach the bar, the magnitude of the infragravity wave response in the nearshore region is similar in all cases. This is consistent with the explanation that more obliquely incident waves generate more edge waves (trapped PAGE 66 53 infragravity free waves) , and thus the case of p = 60o shows similar nearshore infragravity wave energy as the other cas es, although it shows less incoming short wave energy than the other cases. 3.3.6 Bottom Friction Figure 3-15 shows the effect of the variat ion of the bottom friction coefficient,ff, (equations (2.5.4)-(2.5.5)) on model results. The bottom friction coefficient was varied from ff = 1/400 s-1 to 1/50 s-1. The base case, with ff = 1/200 s-1, is shown as a bold black line. Figure 3-15 (a) shows that increased bottom friction decreased the magnitude of long wave response across the entire cross-s hore domain. This is expected, because Figure 3-15. Effect of variation of bottom friction coefficient (ff) on (a) root mean square long wave surface elevation ( -hatrms) and (b) significant wave height (Hs). Legend: ___ ff=1/50 s-1, .. ff=1/100 s-1, __ ff=1/200 s-1, __.__. ff=1/300 s-1, __ __ ff=1/400 s-1. PAGE 67 54 bottom friction serves to damp the long wave response (equations (2.5.4)(2.5.5)). The offshore value of the root mean s quare long wave surface elevation ( -hatrms) ranges from 0.048 m for ff = 1/400 s-1 to 0.016 m for ff = 1/50 s-1. The value of the root mean square long wave surface elevation at 25 m seaw ard of the shoreline cutoff depth ranges from 0.150 m for ff = 1/400 s-1 to 0.098 m for ff = 1/50 s-1. Figure 3-15 (b) shows that the bottom friction coefficient has no effect on si gnificant wave height throughout the crossshore domain. This is because bottom friction was not a factor in the wave-breaking portion of the model. 3.3.7 Bar Amplitude Figure 3-16 shows the effect of the variation of bar amplitude, a2, (equation 2.2.4) on model results. The bar amplitude was varied from a2 = 0 m to 2.14 m. The base case, with a2 = 1.5 m, is shown as a bold black line. The variation of the bathymetric profile due to bar amplitude is shown in Figure 3-16 (c). The maximum bar amplitude of 2.14 m was chosen in order to limit the minimum wa ter depth over the bar to 0.3 m. Figure 3-16 (b) shows that an increased bar amplitude leads to stronger wave breaking and a decreased significant wave height from the ba r to the shoreline. Figure 3-16 (a) shows a more complex pattern of results for the long wave response. In general, a higher bar amplitude leads to a decreased magnitude of long wave response, especially near the shoreline; however, this pattern seems to be reversed at the peak of the bar for the maximum bar amplitude of 2.14 m. One expl anation for this is the trapping of edge waves due to the bar. Also, it is notable that, for th e case of no bar ( a2 = 0 m), -hatrms shows a sharp increase at a location approxima tely 40 m seaward of the shoreline cutoff PAGE 68 55 depth. This may be due to the predominan ce of infragravity wave reflection at the shoreline but an absence of trapping due to a bar. Figure 3-16. Effect of va riation of bar amplitude ( a2) on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Legend: ___ a2=0 m, .. a2=0.5 m, __.__. a2=1.0 m, __ a2=1.5 m, __ __ a2=2.0 m, ***** a2=2.14 m. PAGE 69 56 3.3.8 Bar Width Figure 3-17 shows the effect of the vari ation of bar width on model results. Equation (2.2.4) shows how the bar width coefficient cw affects the bathymetric profile. The bar width coefficient was varied from cw = 2 to 8. A lower cw implies greater width. Figure 3-17. Effect of variation of bar widt h on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Width coefficient is cw. Legend: ___ cw=2, .. cw=5, __ cw=8. PAGE 70 57 The base case, with cw = 5, is shown as a bold black line. The variation of the bathymetric profile due to ba r width is shown in Figure 317 (c). Figure 3-17 (b) shows that a wider bar leads to a more gradual patt ern of wave breaking. Figure 3-17 (a) shows that a wider bar leads to a slig htly increased magnitude of lo ng wave response, especially near the shoreline. One possible explanation for this is that bars tend to trap edge waves; therefore, a wider bar might lead to more trap ping of these edge waves near the shoreline. The value of the root mean square long wa ve surface elevation at 25 m seaward of the shoreline cutoff depth ranges from 0.150 m for cw = 2 to 0.124 m for cw = 8. 3.3.9 Distance of Bar from Shore Figure 3-18 shows the effect of the variati on of the distance of the bar from shore, xc, (equation (2.2.4)) on model results. The distance xc was varied from 80 m to 150 m. The base case, with xc = 120 m, is shown as a bold black line. The variation of the bathymetric profile due to the di stance of the bar from shore is shown in Figure 3-18 (c). Figure 3-18 (b) shows that the ba r being closer to shore leads to waves breaking closer to shore. Figure 3-18 (a) shows th at the bar being closer to sh ore results in a narrower long wave response pattern near the shoreline. This makes sense, because the bar trapped edge waves would be nearer to shore if the bar is nearer to the shore; as the bar is located further from the shoreline, the long wave re sponse pattern becomes more spread out in the near-shore region. PAGE 71 58 Figure 3-18. Effect of variation of distance of bar from shore ( xc) on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Legend: ___ xc=80 m, .. xc=100 m, __ xc=120 m, __.__. xc=150 m. 3.3.10 Water Depth at Offshore Boundary Figure 3-19 shows the effect of the vari ation of water depth at the offshore boundary (hoffshore) on model results. The water depth at the offshore boundary was PAGE 72 59 varied by varying the profile scale factor A (equation (2.2.4). The bar amplitude was scaled to the offshore water dept h by multiplying the bar amplitude a2 (in equation (2.4.3)) by the ratio A/Abase where A is the profile scale factor and Abase is the profile Figure 3-19. Effect of variation of water depth at offshore boundary (hoffshore) on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Legend: ___ hoffshore=7.19 m, .. hoffshore=8.16 m, __.__. hoffshore=9.13 m, __ hoffshore=10.09 m, __ __ hoffshore=11.97 m, ***** hoffshore=12.91 m. PAGE 73 60 scale factor for the base case. The offshore water depth was varied from hoffshore = 7.19 m (for A = 0.07) to hoffshore = 12.91 m (for A = 0.13). The base case, with hoffshore = 10.09 m (for A = 0.1), is shown as a bold black line. The variation of the bathymetric profile is shown in Figure 3-19 (c). Figure 3-19 (b) s hows waves breaking farther from shore for the shallower bathymetric profiles. Figure 3-19 (a) shows that a shallower bathymetric profile (smaller hoffshore and bar amplitude) results in a higher amplitude of long wave response in the entire cross-shore domain. This makes sense conceptually because infragravity waves tend to be predominant in shallow water. The offshore value of the root mean square long wave surface elevation ( -hatrms) ranges from 0.047 m for hoffshore = 7.19 m to 0.037 m for hoffshore = 12.91 m. The value of the root mean square long wave surface elevation at 25 m seaward of the s horeline cutoff depth ranges from 0.151 m for hoffshore = 7.19 m to 0.131 m for hoffshore = 12.91 m. 3.3.11 Domain Length Figure 3-20 shows the effect of th e variation of the domain length (ld) on model results. The distance ld was varied from 750 m to 1625 m. The base case, with ld = 1000 m, is shown as a bold black line. Figures 320 (a) â€“ (c) were plotted such that for all cases, x = 1625 m at the onshore boundary. These figures show convergence of the results for all cases. This indicates that the chosen cross-shore domain of 1000 m was adequately long to give accurate results. PAGE 74 61 Figure 3-20. Effect of va riation of domain length (ld) on (a) root mean square long wave surface elevation ( -hatrms), (b) significant wave height (Hs), and (c) bathymetric profile. Legend: ___ ld=750 m, __ ld=1000 m, __.__. ld=1375 m, __ __ ld=1625 m PAGE 75 62 CHAPTER 4 CONCLUSIONS 4.1 Summary The purpose of this study was to develop and present a linear frequency-domain numerical model of nearshore infragravity wave generation on an alongshore-uniform beach, and then to utilize this model to inve stigate the importance of various parameters in the generation of these infragravity waves. The model of infragravity wave generation was based upon the model presented by Reni ers et al. (2002), which was found by the same researchers to agree well with field data from Duck, No rth Carolina. Chapter 2 of this thesis presented detailed derivations of th is numerical model, as well as a validation of the numerical model against an analytical solution for a simple test case. The results presented in Chapter 3 of this thesis de monstrate the relative importance of various parameters in the generation of nearshore infragravity waves. The results of the numerical simulati on of the low frequency wave climate indicated that the most important parameters in the generation of n earshore infragravity waves are: peak wave period (Tp), offshore significan t wave height (Hs), bottom friction (ff), deep water directional width (dir-width0), and water depth at the offshore boundary (hoffshore). These parameters affect the infragravity wave response in the following manner: A higher Tp leads to a higher magnitude of response; A higher Hs leads to a higher magnitude of response; A narrower directional width leads to higher magnitude of response; PAGE 76 63 Increased bottom friction leads to a decreased magnitude of response; A shallower bathymetric profile leads to an increased magnitude of response. The remaining parameters investigated in the numerical simulation were found to be moderately important in the generation of nearshore infragravity waves. These parameters are: bar amplitude (a2), the bar width coefficient (cw), the distance of the bar from the shoreline (xc), peak wave direction ( p), and the Jonswap peak enhancement factor ( ). These parameters affect the infragravity wave response in the following manner: A higher bar amplitude leads to decreased magnitude of response, especially near the shoreline; A wider bar leads to a slightly higher magnitude of response; The bar being closer to the shore resu lts in a narrower response pattern; Obliquely incident waves (higher p) lead to a decreased ma gnitude of response at the offshore boundary, although peak wave di rection has little effect on response near the shoreline; A narrower frequency spectrum (increased ) leads to a slightly decreased magnitude of response. 4.2 Discussion and Conclusions One of the main findings in this study is that a larger peak wave period (Tp) of the incoming short waves leads to a greater ma gnitude of infragravity wave surface elevation. It is important to note that this pattern is evident for small incident waves (i.e. Hs=0.4m), where Hs is higher across the domain for higher peak periods (i.e. greater dissipation of short wave energy for lower Tp), and also for large incident waves (i.e. Hs=2.0 m) where Hs is similar across the domain for all peak periods. This indicates that, regardless of the dissipation of the primary (shor t) waves, there is still greater dissipation of infragravity wave energy when the peak peri od of the incident short waves is lower. PAGE 77 64 This seems to be consistent with results of a numerical study by Battjes et al. (2004), in which it was found that incoming bound in fragravity waves generated by higher frequency incoming wave components experience significantly more dissipation than the incoming bound waves generated by lower frequency incoming wave components. Another parameter that has an important impact on infragravity wave surface elevation is offshore signi ficant wave height (Hs). Results of the numerical simulation of the low frequency wave climate show that higher offshore significan t wave height leads to higher magnitude of infragravity wave res ponse. This is consis tent with the findings of Ruessink (1998a) that the infragravity wave energy is significantly positively correlated with the wave energy of the inco ming short waves. This also makes sense mathematically, because the radiation stress es that force the infragravity waves are proportional to the wave height s squared; thus, more ener getic (higher) incoming short waves lead to greater forcing and thus highe r amplitudes of the infragravity waves. However, as seen in Figure 3.3.2.6, it is impor tant to specify a realistic wave height at the offshore boundary. If the offshore Hrms exceeds Hmax at the offshore boundary, then the value of the radiation stresse s at the offshore boundary will be unrealistically high, and the model results for infragravity wave surface elevation will be inaccurate. Results of the numerical simulation of th e low frequency wave climate also show that a narrower directional wi dth leads to a greater magnitu de of long wave response. This finding is consistent with that of Van Dongeren et al . (2003); in employing a numerical model (SHORECIRC), it was found that eliminating directional spreading from the incoming wave spectrum caused signi ficant amplification of infragravity wave PAGE 78 65 heights. In analyzing field data, Herbers et al. (1995a, 1995b) found a similar relationship between direc tional spreading and infrag ravity wave response. As expected, increased bottom friction decreased the magnitude of long wave response across the entire cross-shore domain. Bottom friction serves to damp the long wave response (equations (2.5.4)(2.5.5)) a nd is necessary to pr event unbounded growth in the case of infragravity wave resonance. The bathymetric parameter that seems to most significantly affect the long wave response is the offshore water depth (hoffshore), which was varied in the numerical simulation by varying the profile scale f actor (A, equation (2.2.4)). A shallower bathymetric profile (smaller hoffshore and bar amplitude) results in higher amplitude of long wave response in the entire cross-shor e domain. This makes sense conceptually because infragravity waves tend to be predominant in shallow water. Results of the numerical simulation of th e low frequency wave climate show that, in general, a higher bar amplitude leads to a decreased magnitude of long wave response, especially near the shoreline; however, this patt ern seems to be reversed at the peak of the bar for the maximum bar amplitude. One expl anation for this is the trapping of edge waves due to the bar. Also, it is notable that , for the case of no bar, the root mean square infragravity wave surface elevation shows a sh arp increase at a location approximately 40 m seaward of the shoreline cutoff depth. This may be due to the predominance of infragravity wave reflection at the shorelin e but an absence of trapping due to a bar. The variation of bar width also affects th e low frequency wave climate. Results of the numerical simulation show that a wider bar leads to a slightly increased magnitude of long wave response, especially near the shoreline. One possi ble explanation for this is PAGE 79 66 that bars tend to trap edge waves; therefor e, a wider bar might lead to more trapping of these edge waves near the shoreline. Results of the numerical simulation of th e low frequency wave climate also show that the bar being closer to shore results in a narrower l ong wave response pattern near the shoreline. This makes sense, because th e bar trapped edge waves would be nearer to shore if the bar is nearer to the shore; as th e bar is located further from the shoreline, the long wave response pattern becomes more spread out in the near-shore region. Findings also indicate that peak wave direction has little effect on long wave response near the shoreline. However, at the offshore boundary, more oblique waves (higher p) lead to a decreased magnitude of long wave response. One possible explanation for this is that mo re obliquely incident short wa ves lead to more trapping of the outgoing free infragravity waves and t hus less leaky waves reaching the offshore boundary. It is important to note that, although the significant wave height (Hs) and thus the wave energy of the obliquely incident waves ( p = 60o) is lower than the Hs for all of the other cases as the waves approach the ba r, the magnitude of the infragravity wave response in the nearshore region is similar in all cases. This is consistent with the explanation that more obliquely incident waves generate more edge waves (trapped infragravity free waves) , and thus the case of p = 60o shows similar nearshore infragravity wave energy as the other cas es, although it shows less incoming short wave energy than the other cases. The one result of this numerical simulation of low frequency wave climate that is not consistent with expectat ions is that a narrower fr equency spectrum (increased ) leads PAGE 80 67 to a slightly decreased root mean square long wave surface elevation ( -hatrms). Further investigation would have to be done to explain this pattern of results. 4.3 Recommendations for Further Work It is possible to expand this numerical mode l to include solutions for the velocities u and v , in the cross-shore and alongshore directi ons, respectively. In order to do this, rather than combining equations (2.5.1)(2.5 .3) into a single equa tion for infragravity wave surface elevation (2.5.4), one would transform e quations (2.5.1)(2.5.3) into the frequency domain and solve for the three unknowns , u , and v . This would necessitate formulating more complex boundary conditions and would also entail more computational time. The benefit of this e xpansion would be the ab ility to analyze the relationships between infragravity wave s and nearshore currents or vorticity. Another possibility is to expand the m odel to include an alongshore-varying bathymetry. Thus, the model would be va lid for cases other than those with an alongshore uniform bathymetry. Perturbation expansions could be used to generate this alongshore-varying bathymetry. This woul d provide results showing the alongshore variation of infragravity wave surface elevation and would t hus allow for the depiction of edge waves. With the model in its current state, one c ould analyze the output in order to separate the incoming (bound) and outgoing (free) long wa ves. It would also be possible to further analyze the model outpu t to separate the outgoing lo ng waves into edge waves and leaky waves by forming a frequency al ongshore wave number spectrum. This would allow one to analyze which factors or parameters are most important in the PAGE 81 68 generation of the long waves traveling in eac h direction, and in trapped versus leaky waves. PAGE 82 69 APPENDIX A DERIVATION OF EQUATION FOR IN FRAGRAVITY WAVE SURFACE ELEVATION This appendix gives the derivations of the two forms of the equation for infragravity wave surface elevation: the second order partial differential equation, equation (2.5.4), and the second order ordina ry differential equation in the frequency domain, equation (2.5.6). A.1 Second Order Partial Differential Equation Equation (2.5.4), the second order partial di fferential equation for infragravity wave surface elevation, 2 2 2 2 2 2 2 2 2 2 22 1 1 y S y x S x S g y h x dx dh x h t g t gyy yx xx was obtained by combining equations (2.5.1)-(2.5.3), as follows. 1. Differentiate equation (2.5.1) with respect to time: 02 2 t v h y t u h x t (A.1) 2. Differentiate equation (2.5.2) with respect to x: y x S x S x h x g t u h xxy xx 2 2 2 (A.2) 3. Differentiate equation (2.5.3) with respect to y: y x S y S y h y g t v h yyx yy 2 2 2 (A.3) 4. Rearrange equation (A.1): PAGE 83 70 t v h y t t u h x2 2 (A.4) 5. Substitute equation (A.4) into equation (A.2) to obtain: y x S x S x h x x h g t t v h yxy xx 2 2 2 2 2 2 2 (A.5) 6. Substitute equation (A.5) into equation (A.3) to obtain: 2 2 2 2 2 2 2 2 2 2 22 1 1 y S y x S x S g y h y y h x h x x h t gyy xy xx (A.6) 7. Assuming h varies only with x, not with y: 2 2 2 2 2 2 2 2 2 2 22 1 1 y S y x S x S g y h x dx dh x h t gyy xy xx (A.7) This is the same as equation (2.5.4), except for the added term t g on the left hand side. A.2 Second Order Ordinary Differential Equation in the Frequency Domain Equation (2.5.6), the second order ordinary differential equation for infragravity wave surface elevation, yy y yx y xx yS k dx S d k i dx S d g k h g f i g f dx d dx dh dx d h 2 1 2 4 2 2 2 2 2 2 2 2 was obtained by transforming equation (2.5.4) into the frequency domain, as follows: 1. Symbolize the infragravity wave phase as ) 2 (y k fty (A.8) 2. From equation (2.5.7) for : PAGE 84 71 * exp i f i t (A.9) * exp 22 2 2 2 i f t (A.10) * exp 2 1 i x x (A.11) * exp 2 12 2 2 2 i x x (A.12) * exp 2 12 2 2 i k yy (A.13) 3. From equation (2.5.10) for Syy: * exp 2 12 2 2 i S k y Syy y yy (A.14) 4. From equation (2.5.9) for Sxy: * exp 2 12 i k i dx S d y x Sy xy xy (A.15) 5. From equation (2.5.8) for Sxx: * exp 2 12 2 2 2 i dx S d x Sxx xx (A.16) 6. Equations (A.9)-(A.16) were substituted into equation (2.5.4) to obtain equation (2.5.6). PAGE 85 72 APPENDIX B DERIVATION OF EQUATIONS FOR BOUNDARY CONDITIONS B.1 Shoreline Boundary Condition The cross-shore momentum equation (2.5.2 ) was the starting point in deriving the reflecting boundary conditi on at the shoreline. y S x S x gh t u hxy xx The following steps were followed in order to obtain the shoreline boundary condition equation (2.5.24). 1. Assume that there is perfect reflection at the shoreline (x=n). Therefore, let un = 0. Equation (2.5.2) then becomes y S x S gh xxy xx 1 (B.1) 2. Transform equation (B.1) to the frequenc y domain. Recall that the infragravity wave phase is represented as ) 2 (y k fty (B.2) From equations (2.5.7)-(2.5.9) for , Sxx, and Sxy: * exp 2 1 i x x (B.3) * exp 2 1 i x S x Sxx xx (B.4) * exp 2 i S k i y Sxy y xy (B.5) Substitute equations (B.3)-(B.5) into equation (B.1) to obtain PAGE 86 73 xy y xxS k i x S gh x 1 (B.6) 3. Use second order backward differences to estimate the derivatives with respect to x: x xn n n 2 3 4 1 2 (B.7) x S S S x Sn xx n xx n xx xx 2 3 4 ) ( ) 1 ( ) 2 ( (B.8) 4. Substitute equations (B.7) and (B.8) into e quation (B.6) to obtain equation (2.5.24): ) ( ) 2 ( ) 1 ( ) ( 1 2 2 4 3 1 2 3 2 2 1n xy y n xx n xx n xx n n n nS k i x S S S gh x x x B.2 Offshore Boundary Condition B.2.1 Characteristic Equations B.2.1.1 Incoming bound wave Equation (2.5.43) can be proven to be a characteristic equation for the surface elevation of the incoming bound wave. 1. Given equation (2.5.43): 0 sin 2 cos 2 y K f x K f tb in in b in in b 2. Substitute =2f and take the analyt ic derivatives of b from equation (2.5.31) to obtain 0 sin sin cos cos b in in in in b in in in in biK K iK K i (B.9) 3. Cancel terms to obtain th e trigonometric identity 1 sin cos2 2 in in (B.10) PAGE 87 74 which is true, by definition. B.2.1.2 Outgoing free wave Equation (2.5.44) can be proven to be a characteristic equation for the surface elevation of the outgoing free wave. 1. Given equation (2.5.44): 0 y K k gh x K k gh tout out y out out xout out 2. Take the analytic derivatives of out from equation (2.5.42) to obtain 0 out y out y out xout out xout outk i K k gh ik K k gh i (B.11) 3. Substitute gh Kout , then cancel and rearrange terms to obtain 2 2 2 xout y outk k K (B.12) which is true by definition of Kout (equation (2.5.37)). B.2.1.3 Combined Characteristic Equation The combined characteristic equation (2.5 .45) was the starting point for forming the offshore boundary condition. It wa s obtained by the following steps: 1. Substitute b out (from equation 2.5.30) into equation (2.5.44) to obtain 0 y y K k gh x x K k gh t tb out y b out xout b (B.13) 2. Add together equations (B.13) and (2.5. 43) and rearrange term s to obtain equation (2.5.45): PAGE 88 75 y K f K k gh x K f K k gh y K k gh x K k gh tb in in out y b in in out xout out y out xout sin 2 cos 2 B.2.2 Incoming Bound Wave Amplitude Equation (2.5.46) gives the analytic solution of the incoming bound wave amplitude b ~ . It was obtained by the following steps: 1. Apply equation (2.5.4) to the incomi ng bound wave at the offshore boundary. 2 2 2 2 2 2 2 2 2 2 22 1 1 y S y x S x S g y h x dx dh x h t g t gyy yx xx b b b b b (B.14) Note that the radiation stresses fo rce the incoming bound wave and not the outgoing free wave, and thus it was not neces sary to apply the subscript b to the radiation stresses in the above equation. 2. In order to take the analy tic derivatives of radiation stresses with respect to x, radiation stresses were define d at the offshore boundary as * 2 exp exp ~ 2 1 y k ft i x k i S Sy x xx xx (B.15) * 2 exp exp ~ 2 1 y k ft i x k i S Sy x xy xy (B.16) * 2 exp exp ~ 2 1 y k ft i x k i S Sy x yy yy (B.17) Equations (B.15)-(B.17) are similar in form to equation (2.5.41), which defines the surface elevation of the incoming bound wave at the offshore boundary. 3. By comparing the above three equations with equations (2.5.8)-(2.5.10), it can be seen that PAGE 89 76 x k i S Sx xx xx exp ~ (B.18) x k i S Sx xy xy exp ~ (B.19) x k i S Sx yy yy exp ~ (B.20) 4. Take the analytic derivatives of b from equation (2.5.41) and of radiation stresses from equations (B.15)(B.17). Recall that = (2 ftkyy). * ) exp( exp ~ i x k i f i tx b b (B.21) * ) exp( exp ~ 22 2 2 2 i x k i f tx b b (B.22) * ) exp( exp ~ 2 12 2 2 i x k i k x x b x b (B.23) * ) exp( exp ~ 2 12 2 2 i x k i k yx b y b (B.24) * ) exp( exp ~ 2 12 2 2 i x k i S k x Sx xx x xx (B.25) * ) exp( exp ~ 2 12 i x k i S k k y x Sx xy y x xy (B.26) * ) exp( exp ~ 2 12 2 2 i x k i S k y Sx yy y yy (B.27) 5. Assume a flat bed at the offs hore boundary; therefore, the term 0 dx dh in equation (2.5.4). Substitute into equation (2.5.4) the above analytic derivatives from equations (B.21)-(B.27). Note that sin ce x = 0 at the offshore boundary, the term exp(-ikxx) = 1. Rearrange terms to obtain 2 1 2 ) 1 ( 1 1 2 2 2 ) 1 ( 2 ) 1 (2 4 ~ ~ 2 ~ ~y x yy y xy y x xx x bk gh k gh f i f S k S k k S k (B.28) PAGE 90 77 6. As can be seen from equations (B.18)(B.20), since x = 0 at the offshore boundary, , ~ , ~ xy xxS S and yyS ~ are equivalent to , , xy xxS S and yyS , respectively, at the offshore boundary. Therefore, the above equation becomes equation (2.5.46): 2 1 2 ) 1 ( 1 1 2 2 ) 1 ( 2 ) 1 ( ) 1 ( ) 1 ( 2 ) 1 (2 4 2 ~y x yy y xy y x xx x bk gh k gh f i f S k S k k S k B.2.3 Offshore Boundary Condition Equation The offshore boundary condition equation (2.5.47) was obtained by taking the following steps: 1. Begin with equation (2.5.45). y K f K k gh x K f K k gh y K k gh x K k gh tb in in out y b in in out xout out y out xout sin 2 cos 2 2. Take the analytic derivates of (from equation 2.5.7). * ) exp( i f i t (B.29) * ) exp( 2 1 i x x (B.30) * ) exp( 2 1 i k i yy (B.31) 3. Equation (2.5.45) was transformed into th e frequency domain by substitution of the above equations (B.29)-(B.31). ) exp( ~ 2 1 * sin 2 ) exp( ~ 2 1 cos 2 2 1 2 1 x k i k i K f K k gh x k i k i K f K k gh k i K k gh x K k gh f ix b y in in out y x b x in in out xout y out y out xout (B.32) PAGE 91 78 4. Use the second order forward differences representation of x at the offshore boundary. x x 2 4 3 3 2 1 (B.33) 5. Substitute equation (B.33) into equation (B .32) and rearrange terms to obtain the offshore boundary condition equation (2.5.47). in in out y b y in in out xout b x out xout out xout out y out xoutK f K k gh k i K f K k gh k i K k x gh K k x gh K gh k i K k gh x f i sin 2 ~ 2 1 cos 2 ~ 2 1 4 2 1 4 33 2 1 2 PAGE 92 79 APPENDIX C VALUES OF ROOT MEAN SQUARE LO NG WAVE SURFACE ELEVATION FOR EACH TEST CASE Table C-1. Root mean squa re wave surface elevation (rms ) for each test case, at the offshore boundary and at 25 m seaward of the shoreline cutoff depth. Root mean square wave surface elevation (rms ) (m) At offshore boundary At 25 m seaw ard of shoreline cutoff depth 3.3.1 Base_Case 0.0417 0.1400 3.3.2 Offshore Significant Wave Height and Peak Period Hs(m), Tp(s) 0.4, 4 0.0056 0.0239 0.4, 6 0.0110 0.0448 0.4, 8 0.0167 0.0587 0.4, 10 0.0212 0.0669 0.4, 12 0.0249 0.0734 0.7, 4 0.0096 0.0398 0.7, 5 0.0189 0.0736 0.7, 6 0.0279 0.0975 0.7, 7 0.0358 0.1126 0.7, 8 0.0419 0.1277 1, 4 0.0122 0.0526 1, 6 0.0284 0.1050 1, 8 0.0417 0.1400 1, 10 0.0524 0.1646 1, 12 0.0651 0.1966 2, 4 0.0176 0.0717 2, 6 0.0442 0.1581 2, 8 0.0716 0.2220 2, 10 0.1036 0.2945 2, 12 0.1295 0.3569 PAGE 93 80 Table C-1. Continued Root mean square wave surface elevation (rms ) (m) At offshore boundary At 25 m seaw ard of shoreline cutoff depth 3.3.2 Offshore Significant Wave Height and Peak Period (continued) Hs(m), Tp(s) 3, 4 0.0465 0.1053 3, 6 0.0598 0.1927 3, 8 0.1063 0.2821 3, 10 0.1492 0.3562 3, 12 0.1925 0.4428 3.3.3 Jonswap Peak Enhancement Factor 1 0.0433 0.1469 2 0.0412 0.1405 3 0.0417 0.1405 3.3 0.0417 0.1400 4 0.0381 0.1290 5 0.0374 0.1225 6 0.0358 0.1195 7 0.0358 0.1200 3.3.4 Deep Water Directional Width dir-width_0 (deg) 5 0.0510 0.1587 10 0.0451 0.1461 15 0.0417 0.1400 20 0.0343 0.1208 25 0.0340 0.1212 30 0.0302 0.1099 3.3.5 Peak Wave Direction p (deg) 0 0.0417 0.1400 10 0.0378 0.1322 20 0.0393 0.1359 30 0.0379 0.1361 40 0.0384 0.1393 50 0.0331 0.1417 60 0.0288 0.1316 PAGE 94 81 Table C-1. Continued Root mean square wave surface elevation (rms ) (m) At offshore boundary At 25 m seaw ard of shoreline cutoff depth 3.3.6 Bottom Friction f f 1/50 0.0156 0.0982 1/100 0.0278 0.1149 1/150 0.0345 0.1267 1/200 0.0417 0.1400 1/250 0.0432 0.1418 1/300 0.0460 0.1424 1/350 0.0434 0.1419 1/400 0.0484 0.1504 3.3.7 Bar Amplitude a2 0 0.0420 0.1336 0.5 0.0441 0.1394 1.0 0.0442 0.1435 1.5 0.0417 0.1400 2.0 0.0333 0.1200 2.14 0.0277 0.1093 3.3.8 Bar Width cw 2 0.0426 0.1500 3 0.0435 0.1489 4 0.0407 0.1411 5 0.0417 0.1400 6 0.0410 0.1352 7 0.0381 0.1265 8 0.0381 0.1241 PAGE 95 82 Table C-1. Continued Root mean square wave surface elevation (rms ) (m) At offshore boundary At 25 m seaw ard of shoreline cutoff depth 3.3.9 Distance of Bar from Shore xc (m) 80 0.0373 0.1371 90 0.0358 0.1277 100 0.0397 0.1325 110 0.0397 0.1339 120 0.0417 0.1400 130 0.0399 0.1343 140 0.0422 0.1400 150 0.0434 0.1432 3.3.10 Water Depth at Offshore Boundary hoffshore 7.19 0.0474 0.1506 8.16 0.0437 0.1413 9.13 0.0395 0.1343 10.09 0.0417 0.1400 11.97 0.0367 0.1284 12.91 0.0368 0.1308 3.3.11 Domain Length ld (m) 750 0.0455 0.1379 1000 0.0417 0.1400 1375 0.0377 0.1389 1625 0.0359 0.1411 PAGE 96 83 LIST OF REFERENCES Battjes, J. A., and Janssen, J. P. F. M., En ergy loss and set-up due to breaking of random waves, Proceedings of the 16th International Coastal Engineering Conference, Hamburg , pp. 569-587, American Society of Civil Engineers, New York, 1978. Battjes, J. A., Bakkenes, H. J., Janssen, T. T., and van Dongeren, A. 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J., Finite amplitude edge waves, Journal of Marine Research, 34 (2), 269-293, 1976. Guza, R. T., and Thornton, E. B., Observations of surf beat, Journal of Geophysical Research, 90 (C2), 3161-3172, 1985. Henderson, S. M., Elgar, S., and Bowen, A. J ., Observations of surf beat propagation and energetics, Proceedings of the 27th Internati onal Coastal Engineering Conference, Sydney , pp. 1412-1421, American Society of Civil Engineers, New York, 2000. Herbers, T. H. C., Elgar, S., Guza, R. T ., and Oâ€™Reilly, W.C., Infragravity-frequency (0.005-0.05 Hz) motions on the shelf, Proceedings of the 23rd International Coastal Engineering Conference, Venice, Italy , pp. 846-859, American Society of Civil Engineers, New York, 1992. Holman, R. A., Infragravity energy in the surf zone, Journal of Geophysical Research, 86 (C7), 6442-6450, 1981. Holman, R. A., and Sallenger, A. H., Setup and swash on a natural beach, Journal of Geophysical Research, 90 (C1), 945-953, 1985. PAGE 100 87 87 Howd, P. A., Oltman-Shay, J., and Holman, R. A., Wave variance partitioning in the trough of a barred beach, Journal of Geophysical Research, 96 (C7), 12,781-12,795, 1991. Huntley, D. A., Long-period waves on a natural beach, Journal of Geophysical Research, 81 (36), 6441-6449, 1976. Janssen, T. T., Kamphius, J. W., Van Dongeren, A. R., and Battjes, J. A., Observations of long waves on a uniform slope, Proceedings of the 27th International Coastal Engineering Conference, Sydney , pp. 2192-2205, American Society of Civil Engineers, New York, 2000. Kostense, J. K., Measurements of surf beat and set-down beneath wave groups, Proceedings of the 19th International Coastal Engineering Conference, Houston, pp. 724-740, American Society of Civil Engineers, New York, 1984. Lippmann, T. C., Holman, R. A., and Bowen, A. J., Generation of edge waves in shallow water, Journal of Geophysical Research, 102 (C4), 8663-8679, 1997. List, J. H., Wave groupiness va riations in the nearshore, Coastal Engineering, 15 , 475496, 1991. Liu, P. L.-F., A note on long waves indu ced by short-wave groups over a shelf, Journal of Fluid Mechanics, 205 , 163-170, 1989. Longuet-Higgins, M. S., and Stew art, R. W., Changes in the form of short gravity waves on long waves and tidal currents, Journal of Fluid Mechanics, 8 , 565-583, 1960. Madsen, O. S., On the generation of long waves, Journal of Geophysical Research, 76 (36), 8672-8683, 1971. Madsen, P. A., Sorensen, O. R., And Schaffer, H. A., Surf zone dynamics simulated by a Boussinesq type model. Part II: surf beat and swash oscillations for wave groups and irregular waves, Coastal Engineering, 32 , 289-319, 1997. Masselink, G., Group bound long waves as a sour ce of infragravity energy in the surf zone, Continental Shelf Research, 15 (13), 1525-1547, 1995. Mei, C. C., The Applied Dynamics of Ocean Surface Waves , World Scientific Publishing Co., River Edge, NJ, 1992. Mei, C. C., and Benmoussa, C., Long waves induced by short-wave groups over an uneven bottom, Journal of Fluid Mechanics, 139 , 219-235, 1984. Nakamura, S., and Katoh, K., Generation of infragravity waves in breaking process of wave groups, Proceedings of the 23rd International Coastal Engineering Conference, Venice, Italy , pp. 990-1003, American Society of Civil Engineers, New York, 1992. PAGE 101 88 88 Oltman-Shay, J., and Guza, R. T., Infragrav ity edge wave observations on two California beaches, Journal of Physical Oceanography, 17 , 644-663, 1987. Roelvink, J. A., Petit, H. A. H., and Kostense, J. K., Verification of a one-dimensional surfbeat model against laboratory data, Proceedings of the 23rd International Coastal Engineering Conference, Venice, Italy , pp. 960-973, American Society of Civil Engineers, New York, 1992. Ruessink, B. G., Bound and free infragravity waves in the nearshore zone under breaking and nonbreaking conditions, Journal of Geophysical Research, 103 (C6), 12,79512,805, 1998b. Sand, S. E., Long waves in directional seas, Coastal Engineering, 6 , 195-208, 1982. Schaffer, H. A., and Svendsen, I. A., Su rf beat generation on a mild-slope beach, Proceedings of the 21st International Coastal Engineering Conference, Malaga , pp. 1058-1072, American Society of Civil Engineers, New York, 1988. Schaffer, H. A., and Jonsson, I. G., Theory versus experiments in two-dimensional surf beats, Proceedings of the 22nd International Coastal Engineering Conference, Delft , pp. 1131-1143, American Society of Civil Engineers, New York, 1990. Sheremet, A., Guza, R. T., Elgar, S., and Herb ers, T. H. C., Observations of nearshore infragravity waves: Seaward and shoreward propagating components, Journal of Geophysical Research, 107 (C8), 10-1-10-10, 2002. Thornton, E. B., and Guza, R. T., Energy saturation and phase speeds measured on a natural beach, Journal of Geophysical Research, 87 (C12), 9499-9508, 1982. Van Dongeren, A., Bakkenes, H. J., and Jan ssen, T., Generation of long waves by short wave groups, Proceedings of the 28th International Coastal Engineering Conference, Wales, pp. 1093-1105, American Societ y of Civil Engineers, New York, 2002. Van Dongeren, A. R., and Svendsen, I. A ., Absorbing-generating boundary condition for shallow water models, Journal of Waterway, Port, Coastal, and Ocean Engineering, 123 (6), 303-312, 1997. Van Dongeren, A. R., and Svendsen, I. A., Non linear and 3D effects in leaky infragravity waves, Coastal Engineering, 41 , 467-496, 2000. Van Leeuwen, P. J., and Battjes, J. A., A model for surf beat, Proceedings of the 22nd International Coastal Engi neering Conference, Delft , pp. 32-40, American Society of Civil Engineers, New York, 1990. PAGE 102 89 89 Watson, G., and Peregrine, D. H., Lo w frequency waves in the surf zone, Proceedings of the 23rd International Coastal Engi neering Conference, Venice, Italy , pp. 818-831, American Society of Civil Engineers, New York, 1992. PAGE 103 90 BIOGRAPHICAL SKETCH Eileen Czarnecki was born in 1971 as the first child of William and Elaine Czarnecki. She and her younger brother Billy were raised in Parsippany, New Jersey. Eileen attended Rutgers University in New Brunswick, New Je rsey, and obtained a Bachelor of Arts degree in psychology in 1993. She work ed for several years as a research assistant at the Univer sity of Miami in Miami, Florida. In 2000, she decided to study engineering full time and was admitted to the University of Florida in Gainesville, Florida, where she obtained her Bachelor of Sc ience degree in civil engineering in 2003. In January of 2004, she began graduate st udy in coastal engin eering at the same university. Upon graduation, she plans to return to the New Jersey area to work as an engineer, combining her knowledge of and in terest in both coas tal and structural engineering. |