MODELING WAVES IN SURFZONES AND AROUND ... - CiteSeerX

MODELING W A V E S IN SURFZONES AND AROUND ISLANDS By James T. Kirby, 1 A. M. ASCE and Robert A. Dalrymple, 2 M. ASCE ABSTRACT: A semi-empirical model for surf zone wave height decay is adapted to the parabolic equation method in order to include the effect of depth-limited wave breaking in combined refraction/diffraction calculations. Several examples for plane beaches are presented in order to show correspondence between the empirical model, its numerical formulation, and previous laboratory data. The model is then applied to the study of wave breaking and diffraction around offshore islands of various planforms. An analytic method for constant depth, which can be used to extend computed solutions to the farfield downwave of the island, is provided, and several simple examples treating the island as a finite width breakwater are examined.

INTRODUCTION

In recent years, the parabolic equation method for wave propagation has seen a rapid development in the context of predicting surface water waves over areas of variable bathymetry, including both the effects of refraction and diffraction. The original development of the method for water waves was based either on a splitting of the elliptic mild slope equation of Berkhoff (2) into coupled equations for forward- and backscattered wave motion, as in Ref. 22, or on direct perturbation expansion of the governing equations using the Wentzel-Kramers-Brillouin formalism, as in Ref. 14. Reference may be made to the article by Tappert (24) for a discussion of the splitting method and earlier applications in other fields. Recent extensions of the capabilities of the parabolic method include the modeling of wave-current interaction (3,10), iterative calculation of the reflected wave field (15), and the inclusion of lowest order nonlinear effects in the Stokes wave formulation (11,12,16). An advantage of the parabolic method over solution techniques for elliptic and hyperbolic equations in two space dimensions is that no downwave boundary condition is needed for the solution of the initial boundary value problem. However, in applications of wave models to coastal areas, accurate modeling of the behavior of waves in the vicinity of a physical downwave boundary consisting of an actual coastline or an offshore island is of primary importance to the prediction of known physical effects such as wave-induced runup, longshore currents, and scour. Wave breaking in the surf zone is a complex, highly nonlinear phenomenon. It is obvious that the parabolic equation method, which is limited to the representation of linear or weakly nonlinear wave fields, 'Asst. Prof., Coastal and Oceanographic Engrg. Dept., Univ. of Florida, Gainesville, FL 32611. 2 Prof., Dept. of Civ. Engrg., Univ. of Delaware, Newark, DE 19716. Note.—Discussion open until June 1, 1986. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 17, 1984. This paper is part of the Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 112, No. 1, January, 1986. ©ASCE, ISSN 0733-950X/ 86/0001-0078/$01.00. Paper No. 20322. 78

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H/H b

_

0

0.2

0.4 0.6 h/h K

0.8

1.0

FIG. 1.—Measured Wave Heights in Laboratory Surfzone; Beach Slope s = 1:65 (from Ref. 8)

is basically incapable at present of representing the underlying physics of the breaking process. However, some progress can be made by shifting our view of the model from its physical basis to its use as a predictive tool. The forces leading to the generation and maintenance of setup and wave-induced currents depend on a physical balance between gradients of excess momentum fluxes, pressure forces due to changes in mean surface elevation, and bottom shear stresses. The role of a wave model in determining the balance consists of predicting the local wave energy density and direction of propagation of the wave field. Thus, as a lowest approximation of the overall physics, it suffices that the wave model be able to predict the local wave amplitude in the breaking zone with some degree of reliability. The simplest model of wave decay in the surf zone, the spilling breaker model, is based on the assumption that the ratio of wave height to local water depth has the same value everywhere in the surf zone as at the breaker line. This assumption has been used extensively in the literature, from predictions of setup (18) and lohgshore currents (17) up to the latest applications of numerical refraction schemes to the study of wave-induced circulation over arbitrary bottoms (7). However, it has long been known that breaking waves, especially of the plunging type, do not follow so simple a rule. Extensive model tests of normally incident wave trains breaking on laboratory beaches have shown that the pattern of wave height decay across the surf zone_ is strongly a function of the beach slope. A representative measurement by Horikawa and Kuo (8) is shown for example in Fig. 1 in comparison to their dissipation model. 79

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The purpose of the present study is to relate an empirical model of surf zone wave energy decay to the dissipation coefficient, w, of the dissipative wave model of Dalrymple et al. (5), and to detail the application of the model to the prediction of wave height in the surf zone. Here, the model of Dally et al. (4) is used, although any of the related models for dissipation in bores could be applied just as easily (Refs. 1, 6, 8, e.g.). ENERGY DECAY MODEL

Dally et al. (4) have proposed that the decay of energy flux with distance in the surf zone is proportional to the excess of energy flux over a stable value, given for waves propagating shoreward in the x direction by the relation f (ECg) = ~ [EC, - ( E Q J (1) ox h where h = the local water depth; and K = a constant to be determined, which is related to the rate of energy decay. The quantity (ECg)s = the "stable" energy flux for a broken wave in water of depth h. Here, E = 1/8 pgH2, H = the local wave height; and Cg = (1 + 2/c/i/sinh 2kh)/2 where p = the fluid density; g = the acceleration of gravity; and k and h are related by the dispersion relationship, co2 = gk tanh kh. Here co = 2ir/T, where T = the wave period. Dally et al. show that this model of wave energy decay is analogous to the energy loss in a hydraulic jump. The stable energy flux may be related to the height obtained asymptotically by a wave propagating over a flat bottom or a plane slope. Measurements by Horikawa and Kuo (8) indicate that a value of Hs = 0.4/i is approached for waves breaking on a plane slope. In the following derivation, we denote (H/h)s = 7, where 7 = an empirical constant. Eq. 1 may be related to a wave energy equation for dissipative motion after assuming a time-steady wave field. For normally incident waves, the energy equation becomes ^ ( E Q = -WE

(2)

Noting that Cgs = Cg, W may be written as K !A = ^ (1 _ ^' W =^i V(l _ I"? =V lH- £ h \ E/ h \

where H = 2\A\; and A = a generally complex measure of wave amplitude and phase. For a plane beach with slope s, and neglecting the effect of setup (which is not calculated by the wave model directly), a simple analytic solution of Eq. 1 was obtained by Dally et al. Their comparison to laboratory data of breaking waves shows that this model very successfully represents wave height decay across the surf zone. This analytical model will therefore be used for comparison with the numerical model. Defining a = K/s, the wave height in dimensionless form is related to the local water depth by 80

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FIG. 2.—Surfzone Wave Height, Plane Beach ( — a = 1-10; — a = 5/2): (1) Constant Height/Depth, H = K/J; (2) H = yh

, (1 4)

©'-(s) [ - (s)" where

a

A= a

[HJ

=

M+A

5 2

fy\ Hb I - I ; K = — (at breaking) 5 \K/ hb 2

\hj L1 ~ 2 w

ln

w

(4)

(5)

(6)

Based on comparisons to the laboratory data of Horikawa and Kuo (8), Dally et al. chose the value K = 0.15. The special case a = 5/2 then corresponds to a beach slope s = 0.06. Results for a range of a values of 1 < a < 10, corresponding to the range of beach slopes 0.15 > s > 0.015, are given in Fig. 2 for 7 = 0.4, K = 0.78. The lines labeled 1 and 2 correspond to the constant decay H = K!I = 0.78 h and to the stable wave height Hs = yh = 0.4 h, respectively. The effect of beach slope, s, is clearly apparent. For mild slopes, a. is large and the wave heights across the surf zone are much less than given by the spilling breaker model, while for steep beaches, a is small and the wave heights exceed the spilling breaker model (H = KH) heights. The paper by Dally et al. contains an extensive comparison between these results and the data of Horikawa and Kuo (8). 81


APPLICATION OF MODEL IN PARABOLIC EQUATION METHOD

The results (Eqs. 4-6) provide a check for determining the accuracy of iterative schemes using the wave damping term (Eq. 3). Noting that Hs = yh and H = 2\A\, Eq. 3 may be written as W='

1 - fit 4\Af

h

(7)

Assuming that the reflected wave (in the minus x direction) is small, we modify the linear parabolic equation given by Dalrymple et al. (5) to obtain Ax -i(k-ko)A v

1 + — C„A 2C„ 8

'

i 2Cco

(CQA,),, + wA = 0 s y,y

(8)

where subscripts x and y denote differentiation. Here, A{x,y) is the complex amplitude of a steady wave train T)(x,y,t) = Re {A{x,y)ei{k"x^t]}

(9)

in which k0 = a reference value of the w a v e n u m b e r based on a constant water depth that is characteristic of the domain, and w =

W 2C„

K ( y2h2^ = — 11 -

2h

(10)

4A

Here we choose a real value of w in contrast to the results for b o u n d a r y layer damping (13), since the wave breaking process would not be expected to distort the wavelength in the same m a n n e r as a small dissipation due to viscous effects. Eq. 8 is written in finite difference form according to the Crank-Nicolson method: A'+1 L - A'

i +1

.^

-—Ax -^w

/c0) A)+l + (k) - k0)A}] +

/-•I+l _

pi s->i+1

4Ax

/-"i

i

+

c;+i

4co

+ vijA)) = 0

C'Si

(11)

where y-derivative terms are given by

tec A v = (cq;+1

l2{h)+lf

K 1

2/i'+1

4|Af

= •

0

..

;

M K —>-

or already breaking

\A y —< h i

or not yet broken

(15) y

If waves are unbroken in all / grid blocks, w'j+x is set equal to zero and the scheme proceeds based on Eq. 14 with no second iteration. NORMALLY INCIDENT WAVES ON PLANE SLOPE

Several cases were run for waves starting in a depth of 2 m and propagating directly towards shore over a plane slope. The program checks at each step that the wave height has not exceeded the breaking criterion. When H becomes greater than K/Z, the program begins calculating values of the damping coefficient, Eq. 15. Breaking continues until w falls to a value of zero, which does not occur on the plane beaches studied here but would be expected to occur readily for waves propagating over uneven topography. For each case examined, the wave height was

000

Q20

0.40

QSO

Q80

1.00

FIG. 3.—Numerical Results for Wave Height Decay; Normal Incidence (a = 3, s 0.05; — Analytic Solution; @, Ax = 1.0 m; O, Ax = 2.0 m; • , Ax = 5.0 m)

H Hb

000

Q20

040

0.60

0.80

I.00

FIG. 4.—Numerical Results for Wave Height Decay; Normal Incidence (a = 10, s = 0.015; — Analytic Solution; • , Ax = 2.0 m; O, Ax = 5.0 m; • , Ax = 10.0 m) 83


assumed to be 1.0 m at a depth of 2 m, and wave period was assumed to be 5 sec. Tests using values of a = 3 and 10 are presented here using various computational grid spaces Ax. Results for a. = 3 and Ax = 1.0, 2.0, and 5.0 m are shown in Fig. 3 and a = 10 and Ax = 2.0, 5.0 and 10.0 m in Fig. 4. The exact solution, Eq. 4, is included in each figure for comparison, with hb being taken as the average of the depth at the last grid point before breaking and at the first grid point after breaking. In each case, the numerical results provide an adequate representation of the exact solution. APPLICATION TO OFFSHORE ISLANDS

Due to a complex combination of wave breaking, refraction, and diffraction, the wave field in the vicinity of an offshore island, either natural or man-made, is extremely complicated. For pure refraction Pocinki (20) has provided a solution, and for nonbreaking waves Jonsson, et al. (9) and Smith and Sprinks (23) have developed solutions that include refraction and diffraction. However, an island will generally possess a surf zone and the shadow zone created behind the island will persist for longer distances than for the nonbreaking wave case due to loss of energy near the island boundary. Several numerical (two-dimensional) examples of waves in the vicinity of islands have been computed here using Eqs. 11 and 14 and including wave breaking. The inclusion of dry shoreline boundaries or vertical walls in the computational domain of a wave model generally requires a modification of the domain, which can include the specification of stair-step boundaries in the case of complicated obstacles. However, if the surface-piercing obstacle consists of a sloping structure that is expected to be surrounded by a surfzone, a major simplification may be introduced. By replacing the surface-piercing island by a shoal with a flat top and covered by a thin layer with a depth on the order of a centimeter, the entire island area may be included in the unmodified computational grid. In this "thin film" model, wave breaking then reduces wave height across the sufzone to a small value at the "real" shoreline, after which further breaking reduces the height of a wave propagating over the island to approximately 7 times the film depth. This wave is transmitted beyond the island and plays no role in the downwave region due to its small height. This approach alleviates the need for internal grid boundaries unless reflective structures are to be included. The islands in the present computations have an elliptical planform, and are located in water of constant depth h0 = 10 m. The depth contours of the islands are given by

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