Wave transformation and wave-driven circulation at two different sites, on the Island of O'ahu, Hawaii

Jérôme Aucan
June 14, 2001

Problem overview

The purpose of this study is to examine wave transformations and wave-driven currents at two very different nearshore settings on an insular coastline. Two study areas emphasizing 1) strong bottom friction dissipation over a coral reef and 2) energetic wave forcing on a steep coral sand beach, are chosen to improve particular aspects of prediction models and to improve the understanding of the underlying physics.
The first study site, Kailua Bay on the windward side of O'ahu, is a long sandy beach, protected behind a wide living and fossil reef platform. In situ observations and numerical simulations using a phase-averaged wave model will be used to validate the parametrisation of bottom friction in a rough, non-erodible, reef environment. Results from the wave model will then be used to force a wave-driven circulation model, and additional data will be used to validate the parametrisation of the forcing term through the concept of radiation stress.
The second study site, Waimea Bay, on the north shore of O'ahu, is a steep-slope pocket beach subject to large northwesterly swells during the winter. During these very energetic wave events, the beach often experiences rapid erosion, followed by unexpectedly fast recovery during relatively weaker wave conditions. The main forcing mechanisms for this across shore sediment transport are believed to be incident and reflected swell, infragravity and wave-driven currents. Waimea also posseses an intermittent breaker line offshore, which produces drastically different surf zone morphologies. This study will use in situ and video imagery data in conjunction with wave and circulation models to study the relative importance of each process in the establishment of the nearshore circulation, illustrated by a noticeable rip current, beach cusps and rapid changes in the beach morphology. The study will also carry out a careful examination of the different infragravity waveforms and their dependence on the incoming deep water swell in an attempt to describe the dissipation of bound long waves and the generation of free long wave in the surf zone.

Introduction

Wave transformation from deep water to finite-depth

Surface gravity waves or swell are usually formed during mid to high latitudes open ocean storms. Momentum is transfered from the wind to the surface waves, until the newly generated swell propagates out of the storm area. Once propagating freely, the swell energy spectrum is modified due to its dispersive nature in deep water, weak nonlinear interactions, and weak dissipation due to whitecapping. When entering shallow water, the wave spectrum is strongly affected by the bottom through shoaling, refraction, diffraction, reflection and bottom friction. In finite-depth water, nonlinear effects such as depth-induced wave breaking or wave-wave interactions arise. The nature of the bottom and the topogaphy have a strong impact on the nearshore wave field. A rough or irregular bottom will induce larger wave dissipation than a smooth, flat bottom. A reef environment typically presents an extremely rough bottom, to the extent that wave dissipation due to bottom friction approaches that due to wave breaking. The process of wave energy dissipation at the seabed can be modeled by the action of a stress :

$\begin{displaymath} \tau =-C_f \rho_f {\bf u}_b \vert{\bf u}_b\vert \end{displaymath}$

(1)

on the water moving at a velocity ${\bf u}_b$ just above the boundary layer (Putnam & johnson, 1949).

is the friction coefficient which is expected to vary with flow conditions. Collins (1972) found that if equation (1) is used, the dissipation rate due to the bottom friction is :

$\begin{displaymath} C_{bf}(\sigma,\theta)=-C_{f} \frac{\sigma^2}{g^2\sinh^2(kd)}E( \sigma,\theta) \end{displaymath}$

(2)

has been shown to be a function of the bottom roughness to semiexcursion ratio

(jonsson, 1980). The determination of the bed roughness

depends on the properties of the bed material, and in the case of erodible beds, on the presence or not of bed ripples. In the case of coral reefs, the bed roughness is likely to be correlated to the type of coral cover (Gerritsen, 1981).

Wave transformation models

Numerical models for the prediction of open ocean surface gravity waves spectrum are in common use today. These models have been implemented for deep water where only the atmospheric boundary layer has an interaction with the wave spectrum (SWAMP Group, 1985). When entering shallow water, other processes need to be taken into account. Deep water wave models are then modified to include refraction, shoaling and dissipation due to bottom friction (WAMDI Group, 1988). In coastal finite-depth areas, nonlinear effects such as depth-induced wave breaking and wave-wave interactions also need to be included in the model. Even in those areas, it is still possible to use the previous wave models, based on the linear wave propagation theory, by adding nonlinear correction terms (SWAN Model, Booij et al. 1999). Different options for the parametrisation of the bottom friction in equation (2) are available. Hasselman & Collins, (1973) used a constant

, chosen to best fit the data of the Joint North Sea Wave Project (JONSWAP) experiment. Collins (1972) used a drag law model with $C_f=C_{bottom} g U_{rms}$ . It's apparent from previous studies that the friction coefficient

needs to be an order of magnitude higher in a typically rough reef environment than over open shelves (Lugo-fernandez et al., 1998). The incorporation of a realistic friction parametrisation into a regional wave model, based on an in situ estimation of the coral reef roughness elements, has not yet been well tested with field measurements.

Wave induced circulation

The knowledge of wave-driven currents over reefs is an essential step for the study of coral reefs, as it determines the advection of nutrients and various water properties (Roberts et al., 1992), and possibly determines the community distribution and production rates (Hearn, 1999). Attempts to couple a numerical wave model to a circulation model on a regional grid have been limited due to the prohibitive calculation time required for both models. Recent advances in computer performance now allow such coupled models to run efficiently. The wave model produces spatial distributions of the radiation stress in the bay. This stress can be incorporated with a surface wind-stress to force a barotropic circulation model (Longuet-Higgins & Stewart, 1964).

Radiation stress in the nearshore

The radiation stress can be seen as a flux of excess momentum carried by the wave (Longuet-Higgins & Stewart, 1964). The effect of a gradient in the radiation stress across the nearshore zone is partitioned between wave-induced currents and wave-induced setup (Symonds et al., 1995). The radiation stress depends on the local wave properties and can therefore be computed by the wave model. The variations of the radiation stress with the heights of the incoming sea and swell waves also accounts for some of the longer periods oscillations observed in the surf zone, the infragravity motions.

Infragravity Motions

Long waves of period between 1 and 5 minutes can occur along beaches during energetic swell events and are found to be correlated to the swell envelope (Munk, 1949). This nonlinear interaction between waves was explained theoretically through the concept of radiation stress (Longuet-Higgins & Stewart, 1962). It is related to the groupiness of the incident short waves, the long wave trough corresponding to the group of high short waves. This long wave has been called a forced or bound wave and travels at the group velocity of the short wave. When swell waves steepen and break in the surf zone, the infragravity waveform is somehow released (it Herbers et al., 1994). Another type of long wave can coexist in the nearshore, called the free wave or edge wave, and it travels at the long wave speed of $\sqrt{gh}$ . The fate of forced long waves and the generation of free long waves in the surf zone are still poorly understood (Herbers et al., 1995). Another possible source for infragravity generation is the temporal variations in the wave setup (Symonds et al., 1982).
Free and forced infragravity levels have been measured on the very narrow shelves of the hawaiian archipelago (Okihiro et al., 1993), and the relative amount of forced infragravity to total infragravity was shown to increase with increasing swell energy. The presence of long, bound or free, infragrivity waves also modulates the nearshore currents. Visual observations suggest that the position and characteristics of the surf zone seem to have a strong influence on both the circulation pattern and the infragravity field.

Study plan

Kailua Bay

Site description

Kailua Bay, O'ahu, is comprised of a large living and fossil reef platform that separates a coral sand beach in the nearshore from the deep ocean. Cutting across the reef is a sand filled channel that provides a potential conduit for sand exchange between the nearshore and offshore zones. The bay is exposed to year long easterly trade wind swells, with typical periods and heights of 8-12 sec and 2-3 meters. During the winter season, northerly swell produced by storms in the North Pacific also impact Kailua Bay, with typical periods and heights of 14-18 sec and up to 4-5 meters. There is a significant decrease in wave energy as the waves propagate across the reef platform as a result of both depth-induced breaking and bottom dissipation.
The goal of the study in Kailua Bay is to validate the formulation of the bottom friction over a highly frictional reef environement. Several formulations for the bottom dissipation term are available in SWAN, based on different physical assumption. The specification of a spatially varying friction coefficient is also possible. An attempt will be made to relate the nature and type of the coral cover to a roughness length, that can then be incorporated in the expression of the bottom friction coefficient.
The SWAN model can also compute the radiation stress field. We will try to investigate the partitioning of the radiation stress gradient between generating wave-driven flow and wave-induced set-up over the reef. The coupling of the wave model with the circulation model through the simple concept of radiation stress will then be tested, and the effects of tides and surface gravity waves on the circulation in the bay will be examined.

Measurements

To study the wave dissipation, spectral energy measurements need to be performed at several locations spanning the outer bay, where the swell is still unaffected by the presence of the bottom, and close to the beach where most of the energy has been dissipated across the reef. Two different types of instruments will be used for these measurements : A directional waverider buoy, moored in 120 meters of water 2 nautical miles outside the bay, provides the outer boundary input spectrum for the wave model. Displacements are recorded at 1Hz sampling rate, and every half an hour the complete frequency directional energy spectrum is calculated. An additional waverider buoy is moored inside the bay to monitor to effects of refraction and diffraction on swell direction when waves travel into shallower water. Wave sensors are also used at depths between 30 and 2.5 meters. They are bottom mounted pressure sensors, sampling at 1 hz and producing non-directional wave energy spectra.

Study content

The study will consist of several field measurements, to obtain wave and current informations at several locations in the bay, covering the widest range of offshore wave conditions. Those measurements will allow us to estimate energy loss due to bottom friction or depth-induced wave-breaking. Efforts will be made to focus on water depths where friction is important but wave breaking has not yet occured. A similar study was made by Gerritsen, 1981, on the Ala Moana Beach Park fringing reef on O'ahu south shore. Our study differs in that we will focus on wave dissipation prior to breaking, while Gerritsen focused on dissipation under broken waves bores. The use of the numerical wave model also will help to differentiate between the different dissipation mechanisms. The correlation between bottom type and bottom dissipation is illustrated in the various parametrisations of

in eq. 2. There is no previous field data evidence to give preference to a particular friction model (Booij et al., 1999). From the available data set, a few scenarios will be chosen to run the model with various parametrisations and compare it with in-situ data. The most satisfactory results will be used to force the circulation model.

Waimea Bay

Description

This study will attempt to document the evolution of the spectral composition of the wave field inside the bay as a function of the incoming swell, and its effects on the nearshore circulation. Inside of Waimea Bay, there is a pocket beach with steep slopes (.07-.20), coarse grained coral sand, that experiences energetic wave and wave driven currents. Large excursions in shoreline position occur associated with beach wide erosion and accretion, beach cusps and strong wave driven flow (Dail et al. 1999). During the most energetic swell events (period $\geq 17$ sec , height $\geq 3$ meters), waves break well offshore in the bay, then propagate as a bore before reforming and breaking again in the shorebreak.

The study in Waimea will also investigate the respective effects of infragravity, incident and reflected short gravity waves on the circulation. Various nearshore motions have been documented over the infragravity range (1-5 minutes) and the far infragravity range (15-25 minutes). The physical origin of these motions is of different types. It is a partition of bound infragravity wave, leaky or trapped edge waves, or shear waves. The effects of set-up on those waves, and its temporal modulation by the the incoming swell also has an effect on these motions but hasn't been extensively documented yet. Those process are not equally related to the incoming swell forcing (Holland & Holman, 1999). Generation of free long wave or the transformation of forced long waves during breaking is still poorly understood (Herbers et al., 1995). The different infragravity levels are likely to be sensitive to the presence or not of the offshore breaker line.

Measurements

An Argus station overlooking the bay, and comprised of 4 color cameras, will be used to document the wave transformation from deep water to the swash zone, and the extent and strength of the wave driven current. An Argus station is a computer controled image processing station that can either take snapshot images or time exposure images. The Waimea Bay station is one of the numerous Argus sites installed worlwide by the Coastal Imaging Laboratory (http://cil-www.oce.orst.edu:8080). High density sampling in the nearshore is used for the capture of infragravity motions, and the swash variability along the beach.
Simultaneously, several in situ wave measuring instruments will be used : A directional waverider buoy moored in 200 meters of water 1.5 nautical miles in the north-west of Waimea Bay will provide the characteristics of almost unaltered deep ocean incident swell, and possibly some reflected swell. In addition, several bottom mounted pressure sensors inside the bay will provide wave energy spectra, and point validation for the measurement of wave characteristics obtained with the Argus system. Additionaly, a bottom mounted acoustic doppler profiler (ADP) will be used in the surf zone.

Study content

The simultaneous measurements of the wave field in deep and shallow water and the current in the surf zone will be used to investigate their correlation. The rip current in the northern part of the bay is a prominent feature of the surf zone during energetic swell events. With the help of in situ measurements of currents in the surf zone, we will attempt to document the structure of this feature, and relate the variations in strength and extent of the rip to the gravity and infragravity nearshore wave climate. We will use the Argus station in the swash zone as a mean of investigating the presence and relative importance of the different waveforms illustrated by their frequency-wavenumber characteristics. In deeper water, the bound long wave is coupled with the groups of the incoming short wave, so a bispectral analysis is used to estimate the relative amounts of free and forced infragravity. Forced or free motions seem to not have the same relation to the incoming short wave.

Expected outcomes

A primary goal of this study is to improve the understanding of wave dissipation over coral reefs and test its parametrisation into a regional numerical wave model.
This study will also examine the results of a circulation model over a realistic bathymetry, forced by the wave field computed by the regional wave model. The degree of accuracy of such a coupled model is important for further studies of reef-dominated nearshore environments. Finally, this study will document the relative importance of various forcing processes in establishing circulation on a steep, high energy beach.

Bibliography

1

Booij, N., R. C. Ris, and L. H. Holthuijsen, A third-generation wave model for coastal regions : 1. Model description and validation, J. Geophys. Res., 104, 7649-7666, 1999.

2

Collins, J. I., Prediction of shallow water spectra, J. Geophys. Res., 77, 2693-2707, 1972.

3

Hasselman, K. and J. I. Collins, Spectral dissipation of finite-depth gravity waves due to turbulent bottom friction, J. Mar. Res., 26, 1-12, 1968.

4

Dail, H. J., M. A. Merrifield, and M. Bevis, Steep beach morphology changes due to energetic wave forcing, Mar. Geol. 162 443-458, 2000.

5

Gerritsen, F. 1981. Wave attenuation and wave setup on a coastal reef. Look Lab. Tech. Report No 48, University of Hawaii Sea Grant Program, Universisty of Hawaii.

6

Hearn, C. J., Wave-breaking hydrodynamics within coral reef systems and the effect of changing sea level, J. Geophys. Res. 104, 30007-30019, 1999.

7

Herbers, T. H. C., S. Elgar, R. T. Guza, Generation and propagation of infragravity waves, J. Geophys. Res., 100, 24863-24872, 1995.

8

Herbers, T. H. C., S. Elgar, R. T. Guza, Infragravity-Frequency (0.005-0.05 Hz) Motions on the shelf. Part I : Forced Waves, J. Phys. Ocean., 24, 917-927, 1994.

9

Herbers, T. H. C., S. Elgar, R. T. Guza, Infragravity-Frequency (0.005-0.05 Hz) Motions on the shelf. Part II : Free Waves, J. Phys. Ocean., 25, 1063-1079, 1995.

10

Holland, K. T., and R. A. Holman, Wavenumber-frequency structure of infragravity swash motions, J. Geophys. Res. 104, 13479-13488, 1999.

11

Jonsson, I.J., a new approach to oscillatory rough turbulent boundary layers, Ocean Eng., 7,109-152,1980.

12

Longuet-Higgins, M. S. and R. W. Stewart, Radiation stress and mass transport in surface gravity waves with application to surf beats, J. Fluid Mech., 13, 481-504, 1962.

13

Longuet-Higgins, M. S. and R. W. Stewart, Radiation stress in water waves : A physical discussion with applications, Deep Sea Res., 11, 529-562, 1964.

14

Lugo-Fernandez, A., Roberts H. H., Wiseman W.J. and B.L. Carter, Water Level and currents of tidal and infragravity periods at Tague Reef, St Croix (USVI), Coral Reefs, 17, 343-349, 1998.

15

Munk, W. H., Surf beats, EOS Trans. AGU,30, 849-854, 1949.

16

Okihiro, M., R. T. Guza, and R. J. Seymour, Excitation of seiche observed in a small harbor J. Geophys. Res., 98, 18201-18211, 1993.

17

Putnam, J. A. and J. W. Johnson, The dissipation of wave energy by bottom friction, EOS Trans. AGU, 30, 67-74, 1949.

18

Roberts, H. H., P. A. Wilson, and A. Lugo-Fernandez, Biologic and geologic responses to physical processes : Exemples from modern reef systems of the Caribean-Atlantic region, Cont. Shelf Res., 12, 809-834, 1992.

19

Sverdrup, H. U., and W. H. Munk, Wind, sea, and swell : Theory of relations for forecasting, U. S. Hydrographic office, Technical Report Number 1, 1947.

20

Symonds, G., D. A. Huntley, and A. J. Bowen, Two-dimensional surf beat : Long wave generation by a time varying breakpoint J. Geophys. Res., 87, 492-498, 1982.

21

Symonds, G., K. P. Black and I. R. Young, Wave-driven flow over shallow reefs J. Geophys. Res., 100, 2639-2648, 1995.

22

WAMDI Group, The WAM model-A third generation ocean wave prediction model, J. Phys. Oceanogr., 18, 1775-1810, 1988.

About this document ...

This document was generated using the LaTeX2HTML translator Version 2002-2-1 (1.70)

The command line arguments were:
latex2html draft.tex -split 1

The translation was initiated by Jerome Aucan on 2004-05-18

Jerome Aucan 2004-05-18