# TR-08-13:Chapter1

## Contents

## Introduction

### Overview

The U.S. Army Corps of Engineers (USACE) maintains a large number of navigation structures in support of federal navigation projects nationwide. These structures constrain currents to promote scouring of the navigation channel, stabilize the location of the inlet channel and entrance, and provide wave protection to vessels transiting the navigation channel. Such structures are subject to degradation from the continual impact of currents and waves impinging upon them. Questions arise about the necessity and consequences of engineering actions taken to rehabilitate or modify the structures. A long-range maintenance and rehabilitation plan to manage navigation structures and support the federal navigation projects requires a life-cycle forecast of waves and currents in District projects along with a quantification of potential evolutionary changes in wave climates decadally with impacts to analyses and decisions.

The Coastal Inlets Research Program (CIRP) of the U.S. Army Engineer Research and Development Center (ERDC) operates a Coastal Modeling System (CMS) that has established and maintained multidimensional numerical models integrated to simulate waves, currents, water level, sediment transport, and morphology change in the coastal zone. Emphasis is on navigation channel performance and sediment management for inlets, adjacent beaches, and estuaries. The CMS is verified with field and laboratory data and provided within a user-friendly interface running in the Surface-Water Modeling System (SMS).

CMS-Wave (Lin et al. 2006b, Demirbilek et al. 2007), previously called WABED (Wave-Action Balance Equation Diffraction), is a two-dimensional (2D) spectral wave model formulated from a parabolic approximation equation (Mase et al. 2005a) with energy dissipation and diffraction terms. It simulates a steady-state spectral transformation of directional random waves co-existing with ambient currents in the coastal zone. The model operates on a coastal half-plane, implying waves can propagate only from the seaward boundary toward shore. It includes features such as wave generation, wave reflection, and bottom frictional dissipation.

CMS-Wave validation and examples shown in this report indicate that the model is applicable for propagation of random waves over complicated bathymetry and nearshore where wave refraction, diffraction, reflection, shoaling, and breaking simultaneously act at inlets. This report presents general features, formulation, and capabilities of CMS-Wave Version 1.9. It identifies basic components of the model, model input and output, and provides application guidelines.

### Review of wave processes at coastal inlets

As waves approach coastal inlets and their navigation channels, their height and direction can change as a result of shoaling, refraction, diffraction, reflection, and breaking. Waves interact with the bathymetry and geometric features, and also with the currents and coastal structures. Accurate numerical predictions of waves are required in engineering studies for coastal inlets, shore protection, nearshore morphology evolution, harbor design and modification, navigation channel maintenance, and navigation reliability.

Nearshore wave modeling has been a subject of considerable interest, resulting in a number of significant computational advances in the last two decades. Advanced wave theories and solution methods for linear and nonlinear waves have led to development of different types of wave transformation models for monochromatic and irregular or random waves, from deep to shallow waters, over a wide range of geometrically different bathymetry approaches (Nwogu and Demirbilek 2001; Demirbilek and Panchang 1998). In general, each wave theory and associated numerical model has certain advantages and limitations, and appropriateness of the models depends on the relative importance of various physical processes and the particular requirements of a project.

Natural sea waves are random, and their characteristics are different from those of monochromatic waves. A spectrum of natural sea waves is considered as the sum of a large number of harmonic waves, each with constant amplitude and phase, randomly chosen for each observance of a true record (Holthuijsen et al. 2004). Wave transformation models for practical applications must represent irregular wave forms and provide estimates of wave parameters demanded in engineering studies. Mase and Kitano (2000) classified random wave transformation models into two categories. The first category consists of simplified models that include parametric models and probabilistic models. The second category comprises more refined models including time-domain (phase-resolving) and frequency-domain (phase-averaged) models. Wave spectral models based on the wave-energy or wave-action balance equation fall within the frequency-domain class. These types of spectral wave models are appropriate for applications of directional random wave transformation over large areas, and they have in recent years become increasingly popular in the estimation of nearshore waves.

Phase-averaged wave models did not account for wave diffraction and reflection until recently, and these processes are necessary for accurate wave prediction at coastal inlets and their navigation channels, particularly if coastal structures are present. To accommodate the diffraction effect, Mase (2001) introduced a term derived from a parabolic wave equation and incorporated it into the wave-action balance equation. The equation is solved by a first-order upwind difference scheme and is numerically stable. A Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme is employed in the model to reduce numerical diffusion from the effect of wave diffraction (Mase et al. 2005a).

The wave-current interaction is another common concern in navigation and inlet projects. The effect of tidal currents on wave propagation through an inlet determines wave height and steepness, channel maintenance, jetty design, and so on. Waves shortened and steepened by ebb currents can lead to considerable breaking in the inlet. Wave blocking becomes a navigation if encountering strong currents. Wave-induced nearshore and alongshore currents can develop in areas where convergence (divergence) of wave energy occurs over complicated bathymetry. Therefore, the effect of ambient currents on wave propagation through inlets cannot be neglected in numerical wave models.

Longuet-Higgins and Stewart (1961) introduced the concept of radiation stresses and showed the existence of energy transformation between waves and currents. Bretherton and Garrett (1968) demonstrated that wave action, and not wave energy, was conserved for coexisting current and waves and in the absence of wave generation and dissipation. This led to the use of the wave-action balance equation in wave models. Interactions of currents with random waves are more complicated than the interaction with regular waves. Huang et al. (1972) derived equations to investigate the change of spectral shape caused by currents. Tayfun et al. (1976) considered the effect on a directional wave spectrum resulting from to a combination of varying water depth and current.

Depth-limited wave breaking must be considered in numerical models simulating coastal waves because it normally dominates wave motion in the nearshore. In the absence of ambient currents, Zhao et al. (2001) examined five different parameterizations for wave breaking in a 2D elliptic wave model. They determined, using laboratory and field data, that the formulations of Battjes and Janssen (1978) and Dally et al. (1985) were the most robust for use with the mild-slope wave equation.

Smith (2001a) evaluated five breaking parameterizations with a spectral wave model by comparing them to the Duck94 field data (http://www.frf. usace.army.mil/ duck94/DUCK94.stm), and concluded that the Battjes and Janssen (1978) parameterization yielded the smallest error if applied with the full Rayleigh distribution to estimate percentage of wave breaking. Subsequently, Zubier et al. (2003) indicated that the formulation of Battjes and Janssen (1978) could provide a better fit to field data than the Dally et al. (1985) formula. More recently, Goda (2006) demonstrated that different random wave breaking models can yield different estimates of wave height.

In the presence of an ambient current, however, no such comprehensive evaluation of different wave breaking formulations has been conducted, although many studies have been carried out to improve the representation of currents on wave breaking. Yu (1950) described the condition of wave breaking caused by an opposing current using the ratio of the wave celerity to current speed. Iwagaki et al. (1980) verified that Miche’s (1951) breaking criterion holds if the wavelength considering the current effect was included. Hedges et al. (1985) expressed a limiting spectral shape for wave breaking on a current in deep water and tested it with four different spectra in a wave-current flume.

Lai et al. (1989) performed a flume experiment of wave breaking and wave-current interaction kinematics. They concluded that the linear theory predicted kinematics well if the Doppler shift was included. A downshifting of the peak frequency was observed for wave breaking on a strong current. The study further confirmed the occurrence of wave blocking in deep water if the ratio of the ebb current velocity to wave celerity exceeded 0.25. Suh et al. (1994) extended the formula of Hedges et al. (1985) to finite water depth by testing nine spectral conditions in a flume study. They developed an equation for the equilibrium range spectrum for waves propagating on an opposing current. Comparison agreed reasonably well with laboratory data that showed the change in the high-frequency range of the wave spectrum.

Sakai et al. (1989) measured the effect of opposing currents on wave height over three sloping bottoms for a range of wave periods and steepness. They extended the breaker criterion of Goda (1970) by formulating a coefficient accounting for the combined effect of the flow, bottom slope, incident wavelength, and local water depth. Takayama et al. (1991) and Li and Dong (1993) noted that breaker indices for regular and irregular waves in the presence of opposing currents could be classified into geometric, kinematic, and dynamic criteria. The study suggested that the wave breaking empirical parameter in Miche’s (1951) formula should be 0.129 and that in Goda’s (1970) formula should be 0.15 for irregular spilling breakers caused by opposing currents.

Briggs and Liu (1993) conducted laboratory experiments to investigate the interaction of ebb currents with regular waves on a 1:30 bottom slope at an entrance channel. They found little effect on wave period, but a significant increase in wave height and nonlinearity with increasing current strength. Raichlen (1993) carried out a laboratory study on the propagation of regular waves on an adverse three-dimensional jet. He found increases in incident wave height by a factor of two or more for ebb current to wave celerity ratios as small as 10 percent. Briggs and Demirbilek (1996) performed experiments using monochromatic waves to study the wave-current interaction at inlets, and found similar results.

Ris and Holtuijsen (1996) analyzed the field data of Lai et al. (1989) to evaluate breaking criteria and indicated that the white-capping formulation of Komen et al. (1984) underestimated wave dissipation. The Battjes and Janssen (1978) breaking algorithm gave significantly better agreement with the data if it was supplemented with an adjustment for white-capping.

Smith et al. (1998) examined wave breaking on a steady ebbing current through laboratory experiments in an idealized inlet. They applied a one-dimensional wave action balance equation to evaluate wave dissipation formulas in shallow to intermediate water depths. The study showed that white-capping formulas, which are strongly dependent on wave steepness, generally underpredicted dissipation, whereas the Battjes and Janssen (1978) breaking algorithm predicted wave height well in the idealized inlet. They concluded that the breaking criterion applied at a coastal inlet must include relative depth and wave steepness, as well as the wave-current interaction. The depth factor was more influential for longer period waves whereas the steepness was more influential for shorter period waves.

Chawla and Kirby (2002) proposed an empirical bulk dissipation formula using a wave slope criterion, instead of the standard wave height to water depth ratio proposed originally by Thornton and Guza (1983). They also modified the wave dissipation formula of Battjes and Janssen (1978) using a wave slope criterion. Calculations made with a one-dimensional spectral model were compared to laboratory measurements, showing that both modified bore-based dissipation models worked well. They concluded that depth-limited breaking differed from current-limited breaking and suggested formulating the breaking as a function of water depth, wave characteristics, and current properties.

There are a number of wave breaking formulas in the literature that consider effects of both current and depth. However, only a few have been tested in 2D numerical models for random waves over changing bathymetries and with strong ambient currents. Recent studies have revealed the necessity of evaluating wave breaking and dissipation in wave models. For instance, Lin and Demirbilek (2005) used a set of wave data collected around an ideal tidal inlet in the laboratory (Seabergh et al. 2002) as a benchmark to examine the performance of two spectral wave models, GHOST (Rivero et al. 1997) and STWAVE (Smith et al. 1999), for random wave prediction around an inlet. Both models tended to underestimate the wave height seaward and bayside of the inlet. Further enhancement of wave breaking and wave-current interaction near an inlet would be necessary for improving spectral wave models.

Mase et al. (2005a) applied CMS-Wave and SWAN ver.40.41 (Booij et al. 1999) to simulate the wave transformation over a sloping beach to simulate a rip current. The two models behaved differently, caused mainly by different formulations of current-related wave breaking and energy dissipation in addition to different treatment of wave diffraction. CMS-Wave is designed for inlet and navigation channel applications over a complicated bathymetry. It is capable of predicting inlet wave processes including wave refraction, reflection, diffraction, shoaling, and coupled current and depth-limited wave breaking. Additional features such as wind-wave generation, bottom friction, and spatially varied cell sizes have recently been incorporated into CMS-Wave to make it suitable for more general use in the coastal region.

Concurrently, the CIRP is developing a Boussinesq-type wave model BOUSS-2D (Nwogu and Demirbilek 2001), which is a phase-resolving nonlinear model, capable of dealing with complex wave-structure interaction problems in inlets and navigation projects. CIRP also made improvements to the harbor wave model CGWAVE (Demirbilek and Panchang 1998) and coupled it to the BOUSS-2D for its special needs. The combination of these three categories of wave models provides the USACE the most appropriate wave modeling capability.

### New features added to CMS-Wave

Specific improvements were made to CMS-Wave in four areas: wave breaking and dissipation, wave diffraction and reflection, wave-current interaction, and wave generation and growth. Wave diffraction terms are included in the governing equations following the method of Mase et al. (2005a). Four different depth-limiting wave breaking formulas can be selected as options including the interaction with a current. The wave-current interaction is calculated based on the dispersion relationship including wave blocking by an opposing current (Larson and Kraus 2002). Wave generation and whitecapping dissipation are based on the parameterization source term and calibration using field data (Lin and Lin 2004a and b, 2006b). Bottom friction loss is estimated based on the classical drag law formula (Collins 1972).

Other useful features in CMS-Wave include grid nesting capability, variable rectangular cells, wave overtopping, wave runup on beach face, and assimilation for full-plane wave generation. More features such as the nonlinear wave-wave interaction and an unstructured grid are presently under investigation.

CMS-Wave prediction capability has been examined by comparison to comprehensive laboratory data (Lin et al. 2006b). More evaluation of the model performance is presented in this report for two additional laboratory data sets. The first laboratory data set is from experiments representing random wave shoaling and breaking with steady ebb current around an idealized inlet (Smith et al. 1998), covering a range of wave and current parameters. This data set is examined here in evaluation of wave dissipation formulations for current-induced wave breaking. The second laboratory data set is from experiments for random wave transformation accompanied with breaking over a coast with complicated bathymetry and strong wave-induced nearshore currents. Comparisons of measurements and calculations are used to (a) validate the predictive accuracy of CMS Wave, (b) investigate the behavior of different current and depth-limited wave breaking formulas, and (c) select formulas best suitable for spectral models in nearshore applications. The diffraction calculations by CMS-Wave are tested for a gap between two breakwaters and behind a breakwater.

The content of this report is as follows: model formulation and improvements to CMS-Wave are described in Chapter 2; the model interface in the SMS is summarized in Chapter 3; validation studies using experimental data and model applications are presented in Chapter 4; and practical applications are given in Chapter 5.

Table of Contents | Chapter 2 - Model Description |