## 1. Introduction

*η*

*x*is the cross-shore distance,

*h*is the water depth,

*τ*is the bed stress, and

_{b}*q*=

*U*(

*η*

*h*) is the volume flux per unit width based on the Lagrangian mean velocity

*U*=

*U*+

^{E}*U*(i.e., the sum of an Eulerian velocity

^{S}*U*and Stokes drift

^{E}*U*). The radiation stress

^{S}*S*for normally incident waves is generally computed from linear wave theory as

_{xx}*k*is the wavenumber and

*E*= 1/8

_{w}*ρgH*

^{2}

_{rms}is the wave energy density inferred from the root-mean-square wave height

*H*

_{rms}.

*H*

_{0}and period

*T*, and assumes the wave height

*H*through the surf zone is depth limited by

*γ*is an empirical breaking parameter (note that for natural reefs with random wave forcing

*H*

_{rms}is typically assumed for

*H*). For the case where

*γ*is spatially uniform, H99 neglects convective accelerations and friction in (2), and using (4) predicts the maximum setup on the reef

*η*

_{r}will be

*H*=

_{r}*γh*is the depth-limited wave height over the reef flat. An alternative approach for predicting

_{r}*η*

_{r}was developed by GC05, who empirically related

*η*

_{r}to the incident wave energy flux (proportional to

*H*

_{0}

^{2}

*T*) and the overall energy dissipated within the surf zone. Consistent with the H99 model result [(5)], the GC05 model also predicts that setup will increase above some breaking wave threshold height

*H*; that is, setup will be absent when

_{r}*H*

_{0}<

*H*.

_{r}*η*

_{r}and the lagoon

*η*

_{L}can drive an Eulerian flow

*U*across the reef (Fig. 1). For

_{r}^{E}*η*

_{r}≪

*h*, and assuming negligible wave dissipation on the reef flat (i.e.,

_{r}*dS*/

_{xx}*dx*≈ 0, where

*dh*/

*dx*≈ 0), (2) predicts that a balance will exist between the mean water level gradient and the bed stress. In current-dominated systems (i.e.,

*U*≫

_{r}^{E}*U*, where

_{w}*U*is a representative near-bed wave velocity), bed stresses are often well-described using a quadratic drag law

_{w}*τ*=

_{b}*ρC*|

_{D}*U*|

^{E}*U*with drag coefficient

^{E}*C*, such that

_{D}*d*

*η*

*dx*can be estimated as (

*η*

_{r}−

*η*

_{L})/

*L*given a cross-shore reef flat length

_{r}*L*(Fig. 1a). To obtain a solution to (6), H99 and GC05 assumed in their models that

_{r}*η*

_{L}was approximately zero, such that

*d*

*η*

*dx*could be estimated as ∼

*η*

_{r}/

*L*. This may be a reasonable assumption for those reefs that do not bound landmasses or enclosed bodies of water. Examples of such reefs can be found in the “open” Indo-Pacific atolls that form incomplete rings, or in many of the smaller individual reefs forming the Great Barrier Reef, which lack a true lagoon. However, many reefs that form adjacent to coastal landmasses, for example, “fringing reefs,” have enclosed lagoons in which exchange with the surrounding ocean is often restricted by friction through narrow, rough, and sometimes tortuous channels. Assuming

_{r}*η*

_{L}∼ 0 for these reef systems may no longer be valid, given that

*η*

_{L}may become a substantial fraction of

*η*

_{r}(Jago et al. 2007). Therefore, while

*η*

_{r}may still be primarily controlled by incident wave forcing under these circumstances, the magnitude of

*U*should also depend strongly on the momentum dynamics governing

_{r}^{E}*η*

_{L}.

The overall dynamics controlling wave-driven flows on coastally bounded reefs may be similar to those, for example, governing rip currents that form over submerged bars on sandy beaches. In this case, setup gradients can drive a flow over a shallow bar crest that returns to the ocean through narrow rip channels (e.g., see MacMahan 2006 for a review). However, while the beach literature may serve as a foundation for understanding and predicting wave-driven flows on coastally bounded reefs, these reefs may possess a number of key differences that could significantly distinguish their dynamics from beaches. First, the bottoms of coral reefs are often very rough compared to sandy beaches because of the presence of canopy-forming benthic organisms and the irregular morphology of the platforms on which they grow. The resulting generated bottom friction can play an important or even dominant role in the overall wave-energy dissipation on the fore reef (Roberts et al. 1975; Lowe et al. 2005), which could significantly reduce the wave setup (Longuet-Higgins 2005; Dean and Bender 2006). Over the reef flat, this enhanced friction may also significantly reduce wave-driven mean flows in comparison to hydraulically smoother beaches. Second, reefs display a much wider range of lagoon and channel geometries than beaches, for example, ranging from the very deep (>50 m) lagoons of atolls to the much shallower (<5 m) lagoons of fringing reefs. Finally, reefs exhibit a wide range of fore-reef bottom slopes that can vary from an order of 1:100 to an order of 1:1. Given that setup on reefs and beaches is strongly slope dependent (Stockdon et al. 2006; Callaghan et al. 2006), the relationship between incident wave forcing and wave setup on reefs may differ appreciably from beaches.

The main objective of this paper is to investigate how the fundamental momentum balances governing wave setup and circulation on coastal reefs behave under different incident wave conditions, and how these balances are affected by reef morphology. To accomplish this, a detailed investigation into the dynamics of wave-driven flows in Kaneohe Bay, Hawaii, was conducted (Fig. 2) using observations from a 10-month field program. The influence of winds, tides, and buoyancy on the lagoon circulation of Kaneohe will be presented elsewhere. Previously, the most detailed physical oceanographic study of this system was conducted by Bathen (1968), who did not consider wave forcing. In contrast, H99 proposed that Kaneohe Bay’s reef and lagoon are dominantly flushed by wave-driven flows; this conclusion was based largely on the model discussed above, resulting from the lack of any detailed field observations. To aid in the interpretation of the wave setup and circulation observations, as well as to broaden the scope of the study, a simple model that considers the mass and momentum balances governing setup and circulation in the cross-reef direction (both over the reef flat and within the lagoon–channel region) is developed. Through application of this model, we are able to investigate how key properties (e.g., frictional characteristics and reef morphological parameters) associated with coastal reefs, govern setup and circulation within these systems in a more general way.

This paper is organized as follows. In section 2 we formulate a simple model for estimating setup and circulation on coastally bounded reefs that extends the 1D mass and momentum balances traditionally considered in the reef literature for the fore-reef and reef flat regions, to also incorporate the coupled dynamics of the lagoon–channel return flows. Descriptions of the study site, field program, and data analysis are presented in section 3. Results from the field studies are described in section 4 and compared with the numerical model predictions and existing reef models in section 5. The results are summarized and discussed in section 6.

## 2. Wave-driven reef circulation in the presence of lagoon setup

*h*and

_{r}*h*are representative depths of the reef and channel regions, respectively;

_{c}*W*and

_{r}*W*are the alongshore reef and channel widths; and

_{c}*U*and

_{r}*U*are the depth-averaged mass transport velocities (i.e., the sum of the Eulerian and Stokes drift velocities). In terms of the total circulation cell width

_{c}*W*

_{total}=

*W*+

_{r}*W*, (7) can be rewritten as

_{c}*η*

*h*≪ 1 (valid for this study site, but not always for reefs), the cross-shore momentum balance [(2)] simplifies to

*η*

*h*, the full momentum balance given by (2) must be considered. Although the wave-driven flow pattern in Fig. 1b is two-dimensional (2D), that is, some alongshore flow must be present from the lagoon to the channel, analogous 1D models developed for predicting rip currents on barred beaches have often found (9) can still provide accurate estimates of cross-shore variations in

*η*

*η*

*S*(

_{xx}*x*), (9) can be numerically integrated across the reef (along transect

*A*–

*B*, see Fig. 1b) for a given cross-reef discharge

*q*, from offshore (

_{r}*η*

*η*

*η*

_{L}),

*C*is a representative drag coefficient for the reef flat. Note that in wave-dominated coastal regions (i.e.,

_{D,r}*U*≫

_{w}*U*), a linear drag law may be more appropriate yet can easily be substituted into (10), that is,

_{r}^{E}*τ*=

_{b}*ρC′*, with a linear drag coefficient

_{D}U_{w}U_{E}*C′*(Mei 1989).

_{D}*q*to be constant along

_{r}*A–B*; however, it otherwise remains unconstrained by the offshore boundary condition (

*η*

*η*

_{L}in (10) can be obtained for any

*q*. Thus, to evaluate

_{r}*q*, the additional momentum balance on the lagoon–channel return flow is also considered. If it is assumed that 1) wave forcing is negligible in this deep channel region (i.e.,

_{r}*dS*/

_{xx}*dx*≈ 0), 2)

*h*is spatially uniform in this region (i.e.,

_{c}*dh*/

*dx*≈ 0), and 3) Stokes drift is negligible (i.e.,

*q*≈

_{c}^{E}*q*), then the simplified momentum balance [(2)] becomes

_{c}*C*is a representative drag coefficient for this lagoon–channel region (possibly different from the reef value

_{D,c}*C*). Given that

_{D,r}*η*

*d*

*η*

*dx*≈

*η*

_{L}/

*L*, where

_{c}*L*can be taken as the total return flow path (transect

_{c}*B–C–D*; see Fig. 1b). This gives

*q*has been related to the flow across the reef

_{c}*q*using (7). Both (10) and (12) represent a closed set of equations that can be numerically solved by first guessing a value of

_{r}*q*, evaluating

_{r}*η*

_{L}using (10), comparing this value with

*η*

_{L}predicted from (12), and then iterating to find a

*q*that satisfies both (10) and (12).

_{r}*S*for (10). A number of 1D models have been developed to predict wave transformation on beaches (e.g., Thornton and Guza 1983, hereafter TG83; Lippmann et al. 1996; Baldock et al. 1998); these models should also be appropriate for reefs with relatively mild slopes (e.g., <1:10). It is not the goal of the present study to conduct a detailed evaluation of such wave models (for such a review, see Apotsos et al. 2008a). For this application, the model by TG83 is applied, in part because it was found to accurately predict wave transformation on the Kaneohe reef in prior work (Lowe et al. 2005). Like many parametric wave models, TG83 assume that cross-shore gradients in wave energy flux are balanced by the local rate of energy dissipation by both breaking ε

_{xx}*and bottom friction ε*

_{b}*,*

_{f}*and ε*

_{b}*are parameterized via semiempirical relationships. Given*

_{f}*E*,

_{w}*S*is evaluated with (3). While the enhancement of

_{xx}*S*by rollers is sometimes added to (3), this is not included in the present study. We note, however, that some studies of beaches have found that the inclusion of a roller has only a minimal effect on the total setup generated shoreward of the surf zone (e.g., Apotsos et al. 2007).

_{xx}## 3. Field experiment

### a. Site description

Kaneohe Bay (Fig. 2), located on the northeastern coast of Oahu, Hawaii (21°29′N, 157°48′W), is approximately 13 km long by 4 km wide. A 5-km-long, ∼2-km-wide, shallow (<3 m) reef extends over much of the area, and is covered by coral, algae, coral rubble, and sand. An ∼1-km-wide lagoon (12–16 m deep) separates the reef from shore, with a bottom comprised mostly of sand and mud, although there are a number of coral patch reefs that rise to within ∼1 m of the surface distributed throughout. The lagoon exchanges with the ocean through the northern Ship Channel (mean depth ∼12 m) and the shallower Sampan Channel in the south (mean depth ∼5 m).

### b. Measurement of waves, currents, and wave setup

A series of instrument deployments were conducted between June 2005 and March 2006 (Table 1). Waves were measured by an offshore directional wave buoy (WO), two Seabird Electronics (SBE) pressure sensors (model 26) located on the fore reef (W1 and W2) and two on the back reef (W3 and W4), as well as the pressure sensors on three RD Instruments acoustic Doppler current profilers (ADCPs) located on the reef flat (A2, A3, and A4; Table 1). The offshore wave buoy, operated and maintained by the University of Hawaii Sea Level Center, was located ∼8 km southeast of Mokapu Point. SBE 26s at sites W1 and W2 each collected 2048 pressure samples at 2 Hz every 30 min, while at W3 and W4 they collected 1024 samples at 1 Hz every hour. The ADCPs at sites A2, A3, and A4 were burst sampled such that 900 pressure samples were obtained at 1 Hz every hour. An additional SBE 26 was operated as a water level gauge (no waves) and was positioned on a patch reef in the lagoon (site W5).

Pressure data were analyzed by dividing each burst into 16 sections of equal length, each with 50% overlap, applying a Hanning window to the segments, and computing spectra. These were converted to one-dimensional wave spectra *S* using linear wave theory, from which rms wave heights were calculated as *H*_{rms} = (8*m*_{0})^{1/2}, where *m*_{0} is the zeroth moment of *S* based on the energy between 2 and 30 s (this generally contained >99% of the total wave energy). Mean *T _{m}*

_{01}and peak

*T*wave periods were subsequently calculated based on the first spectral moment of

_{p}*S*and the spectral peak, respectively. At deeper sites (W1 and W2), the spectra between 2 and 4 s was modeled by assuming a frequency to the −4 power relationship (Jones and Monismith 2007), in order to account for energy that could not be resolved in this high-frequency region because of the attenuation of the pressure signal with depth. This correction generally only contributed a small amount (<2%) to the total energy.

Theories developed to predict *η*_{r} on reefs are often cast as being dependent on the deep-water wave height *H*_{rms,0}. However, because of large-scale wave refraction, wave heights measured by the offshore wave buoy (site WO) may not accurately reflect the deep-water wave height directly offshore of the study site. Therefore, *H*_{rms,0} was calculated from the rms wave height measured on the fore-reef *H*_{FR}, corrected to a deep-water value using a shoaling coefficient *K _{s}* =

*H*

_{FR}/

*H*

_{rms,0}(Dean and Dalrymple 1991) based on the peak period

*T*. Generally,

_{p}*K*was close to unity throughout the experiment (range 0.90–1.05; mean 0.94).

_{s}Current profiles were measured using seven ADCPs deployed at sites A1–A7, with bin sizes ranging from 0.05 to 0.5 m (Table 1). All of the ADCPs were burst sampled at 1 Hz for 15 min each hour. Hourly current profiles were obtained for each ADCP by averaging all samples in a burst and were depth averaged to produce time series of *U ^{E}*. These were rotated into the principal component axes of the velocity variance (Emery and Thompson 2001), such that positive flow along the major axis represented flow into the bay. Tidal current amplitudes and phases were calculated from

*U*using T_TIDE (Pawlowicz et al. 2002). A correlation coefficient

^{E}*R*

_{tide}for currents is thus defined as the square root of the percentage of total variance in

*U*explained by the tidal harmonic analysis.

^{E}No sites were located inside the surf zone, so the wave-induced transport was estimated from the depth-averaged Stokes drift velocities *U ^{S}* computed from the wave spectra following Kenyon (1969). Given that most wave measurements did not include directional information, the dominant Stokes drift directions were inferred from directions output from a numerical wave model [Simulating Waves Nearshore (SWAN); Booij et al. 1999] simulation of the winter experiment. Wave directions in Kaneohe Bay were mostly insensitive to large variations in offshore wave direction, because they are mainly controlled by refraction patterns over the shallow reef (Lowe et al. 2005).

Mean water level variability *η**O*(1 mm) based on a typical 5 m s^{−1} wind speed (Lugo-Fernandez et al. 1998), and thus could be neglected from the momentum balance. Although wave setup on both reefs and beaches can be modulated at tidal frequencies (e.g., Thornton and Kim 1993; MacMahan et al. 2006), discriminating these relatively small modulations in setup from much larger variations in the tides occurring at the same frequencies can be difficult. Through application of the model from section 2, we estimated these tidal modulations in setup to be relatively insignificant in comparison to the much greater variability in *η*

To compute wave setup, a modified version of the approach by Raubenheimer et al. (2001) was used, which is detailed in the appendix. Setup was calculated only for sites where instruments were deployed on a hard reef platform (i.e., A2–A4, W3–W5). Thus, data collected by instruments in the channels (A1 and A5) were precluded from the analysis, because they gradually sank into the sand by up to 5 cm during the experiment, making it difficult to maintain an accurate reference level. Note that data from sites A2 and A3 was used to estimate the maximum reef setup *η*_{r}, because they were located only ∼100–200 m shoreward of a consistent surf zone (Fig. 1). This is equivalent to only 5%–10% of the overall cross-reef length *L _{r}*, so given the momentum balance in (6),

*η*

*η*

_{r}.

## 4. Observations

### a. Offshore forcing conditions

The winter deployment captured four spring–neap tidal cycles (Fig. 3a) and offshore rms wave heights *H*_{rms,0} averaged 1.6 m (Fig. 3b). During the first 2 weeks (18–30 January), trade wind wave conditions dominated, with persistent waves coming from the east with *H*_{rms,0} ∼ 2 m and *T _{p}* ∼ 7 − 9 s. For the remainder of the experiment, waves were variable (

*H*

_{rms,0}∼1–3 m;

*T*∼ 6 − 13 s), in part generated remotely by several North Pacific storms. Over the longer 10-month deployment (not shown),

_{p}*H*

_{rms,0}averaged ∼1.4 m (range ∼0.5–3.5 m), and propagated mostly from 50° to 80°.

### b. Wave energy distribution

Figure 4a shows time series of measured *H*_{rms} at representative fore-reef and reef flat sites. On average, *H*_{rms} on the fore reef (sites W1 and W2) was 25% lower than offshore (Table 2), with only ∼6% of this discrepancy explained by wave shoaling (see section 3b). As discussed by Lowe et al. (2005), this wave attenuation is likely caused by bottom friction across the rough and relatively gently sloping fore-reef offshore of sites W1 and W2.

In general, wave heights in Kaneohe Bay can be controlled by changes to either offshore wave conditions or tidal elevation (or a combination of both). To quantify these effects, correlation coefficients between the local wave height and both the offshore wave height and local water depth were computed (Table 2). Wave heights on the fore reef (W1 and W2), measured prior to breaking, were only correlated with offshore wave heights (*R*_{wave}∼0.9), while on the shallow reef flat (A2 and A3) wave heights were instead primarily controlled by the tidal elevation (*R*_{tide}∼0.8). Thus, given that A2 and A3 sit shoreward of a saturated surf zone, local wave heights were mostly insensitive to changes in incident wave forcing (Fig. 4a) because their height was depth limited according to *H*_{rms,max} = *γ _{r}h_{r}*. The measurements indicate

*γ*≈ 0.3 at sites A2 and A3 (Fig. 4c), which is comparable to values found on other reefs and also on some beaches (e.g., TG83; Hardy and Young 1996). At A4, W3, and W4, all located on the reef but in deeper water (∼3–5 m), wave heights were only weakly correlated with tides (

_{r}*R*

_{tide}< 0.5) and were more correlated with the offshore wave height (

*R*

_{wave}∼0.6–0.8).

### c. Dominant circulation patterns

On average, two persistent circulation cells dominated the north and south regions of the bay (Fig. 5a), with onshore flow generated over the shallow reef sections (A2 and A3) followed by return flows out of the channels (A1 and A5); the division of these cells occurred at Kapapa Island, roughly at the alongshore center of the bay. The major axes of the principal component ellipses at most sites were aligned with time-averaged current vectors (Fig. 5b). Mean flow inside the lagoon (A6 and A7) was typically much weaker than over the reef, averaging ∼1–2 cm s^{−1}; however, it instantaneously attained values up to ∼10 cm s^{−1} (Fig. 6c).

Variability in *U _{E}* occurring at tidal frequencies may result from a combination of the ebbing and flooding of the bay, as well as tidal modulations of the wave-driven currents. On the reef (A2–A4), these tidal flows were oriented across the reef; note that at A4 this implies that the flow was roughly orthogonal to the dominant flow direction at this site (Fig. 5d). Tidal currents were much stronger in the channels and lagoon than over the reef (by factors ranging from 2 to 4), resulting from preferential flow through these deeper regions (Table 3). In general, flow variability inside the lagoon (A6 and A7) was dominantly driven by tides, with

*R*

_{tide}ranging between 0.7 and 0.8. Flow variability in the channels (A1 and A5) was only partially correlated with tides (

*R*

_{tide}= 0.5–0.6), with very weak correlation

*R*

_{tide}< 0.3 on the reef flat (A2 and A3).

### d. Wave-driven transport

To evaluate the role of wave forcing, the response of the subtidal circulation to the magnitude of *H*_{rms,0} was investigated. Subtidal flows across the reef (A2, A3, and A4) and in the channels (A1 and A5) were dominantly wave driven (Figs. 6a,b), with *R*_{wave}∼ 0.8–0.9. However, for the lagoon sites (A6 and A7), subtidal currents were only weakly correlated with *H*_{rms,0} (*R*_{wave} < 0.3; Fig. 6c).

For the reef and channel sites, where subtidal flows were dominantly wave driven (Fig. 6a,b), *U ^{E}* increased roughly linearly with

*H*

_{rms,0}. Moreover, a threshold wave height

*H*was apparent at these sites, corresponding to the point where the wave-driven circulation roughly shut down (see Figs. 6a,b). For each site

_{r}*H*was evaluated (based on the rms wave height) and averaged

_{r}*H*= 0.7 m (range of 0.6–0.8 m; Table 3). Given the typical Kaneohe reef flat depth

_{r}*h*≈ 2 m (Falter et al. 2004), the observed ratio

_{r}*H*/

_{r}*h*≈ 0.35 is similar to

_{r}*H*/

_{r}*h*≈ 0.4 proposed by Gourlay (1996).

_{r}Mean Stokes drift vectors during the winter experiment were directed across the reef (Fig. 5c). Computed *U ^{S}* was most variable on the fore reef (range of ∼1–16 cm s

^{−1}; mean of ∼3 cm s

^{−1}), but on average it was comparable to values on the reef at A2 and A3 (range of ∼1–5 cm s

^{−1}; mean of ∼3 cm s

^{−1}). On the reef, Stokes drift on average contributed ∼25% of the total transport, but its importance varied as offshore wave conditions changed (Fig. 7). Given that waves on the reef (sites A2 and A3) were mostly independent of offshore wave conditions, the relative importance of Stokes drift increased as the incident wave forcing decreased to a point where it sometimes dominated over the Eulerian flow.

### e. Wave setup

A strong correlation was present between *η**H*^{2}_{rms,0}*T _{p}* for representative sites on the reef flat and lagoon (Fig. 8b,c), indicating that setup increased roughly proportionally to the incident (deep water) wave energy flux. Results of a nonlinear least squares fitting of

*η*

*aH*

_{rms,0}

*−*

^{m}T_{p}^{n}*b*[(15)] to the data indicated that

*m*and

*n*ranged between 1.8–2.9 and 0.7–1.0, respectively, averaging

*m*= 2.3 and

*n*= 0.8 (Table 4). This scaling is very different from the

*m*= 1 and

*n*= 0 scaling from H99 [(5)], but it is consistent with the

*m*= 2 and

*n*= 1 scaling of GC05.

Spatial variations in *η**H*_{rms,0} = 2 m and *T _{p}* = 7 s using the nonlinear regression coefficients computed for each dataset. For this wave condition,

*η*

_{r}∼ 6 − 8 cm on the reef flat (A2 and A3) and ∼4 cm on both the back reef (W3 and W4) and lagoon (W5). Thus,

*η*

_{L}was not zero inside the lagoon (Fig. 8a) as assumed in existing reef models, instead representing ∼60%–80% of the maximum reef flat values

*η*

_{r}. Wave setup also varied somewhat alongshore (Table 4). For a given wave height,

*η*

## 5. Model application

To apply the model in section 2 to Kaneohe, we first defined representative reef geometry parameters (i.e., *L _{r}*,

*W*,

_{r}*h*,

_{r}*W*,

_{c}*h*and fore-reef slope). Given that the water depth is not uniform over the reef and channel regions, we assigned uncertainties to these depths based on the typical range within each region, and propagated this uncertainty through the calculations via Monte Carlo simulations. Therefore, a representative reef flat depth would be

_{c}*h*= 2 ± 1 m, while the depths of the southern and northern channels were assigned

_{r}*h*= 5 ± 2 m and 12 ± 3 m, respectively. A fore-reef slope (∼1:60) was taken from Lowe et al. (2005, see their Fig. 15). The overall widths of the two circulation cells

_{c}*W*

_{total}were defined as the distances from Kapapa Island to the northern and southern extent of the channels (Table 5). Apportioning

*W*

_{total}between the channels and reef can be done by calculating the slope from a linear regression of

*U*versus

_{c}*U*for each cell (Fig. 9), to estimate the fraction of the total cell

_{r}*W*/

_{r}*W*

_{total}required to account for the observed flow exiting the channels via (8). This gave

*W*/

_{r}*W*

_{total}= 0.71 ± 0.16 and 0.73 ± 0.13 for the southern and northern cells, respectively (Table 5).

Both (10) and (12) assume a quadratic drag law holds; however, for cases where *U _{w}* ≫

*U*, a linear drag model may be more suitable (e.g., Mei 1989). Choosing an appropriate drag law is important, because this will directly control how the reef circulation scales with incident wave forcing, that is,

^{E}*U*will scale according to

_{r}^{E}*H*

_{rms,0}(or

*η*

_{r}

^{1/2}) if drag is quadratic, and to

*H*

_{rms,0}

^{2}(or

*η*

_{r}) if drag is linear. On the reef flat (A2–A3) where wave heights were mostly time invariant (Fig. 4a), a typical rms near-bed wave velocity was

*U*∼ 0.4 m s

_{w}^{−1}, making

*U*comparable to

_{w}*U*, and thus ambiguous as to which drag form should be applied. Therefore, a quadratic drag law was considered; however, for the sake of completeness the influence of a linear formulation on the model results was investigated. A linear regression between the setup and current data was then used to compute quadratic drag coefficients assuming an estimated cross-reef length

_{r}^{E}*L*= 1500±200 m and lagoon–channel return path

_{r}*L*= 5000±2000 m for both cells (Fig. 10). Reef

_{c}*C*and lagoon–channel

_{D, r}*C*drag coefficients [estimated from (6) and (11), respectively] were generally comparable and also similar for both the southern and northern cells, ranging between 0.01 and 0.03 (Table 5). A similar analysis was also conducted to calculate linear drag coefficients (not shown); however, this required estimating

_{D, c}*U*, which spatially decayed across the reef flat. Given that

_{w}*U*typically ranged from 0.4 m s

_{w}^{−1}on the leading edge of the reef flat (A2–A3) to zero on the back reef, for simplicity a spatially averaged 0.2 m s

^{−1}was assumed for the system. This gave values of

*C*′ and

_{D,r}*C*′ for the reef and lagoon–channel regions, respectively, ranging between 0.01 and 0.02 (Table 5), which are thus similar to the quadratic coefficients.

_{D,c}To illustrate the predicted cross-shore variation in *H*_{rms}, *η**U ^{E}* over the reef for the southern cell, we first considered a “typical” wave condition as an example (

*H*

_{rms,0}=2 m,

*T*= 7). The TG83 wave model was applied both with friction present [i.e., using the energy dissipation factor

_{p}*f*= 0.24 measured on the Kaneohe reef by Lowe et al. (2005)] and with friction turned off. We assumed a spatially uniform

_{e}*C*∼ 0.02, given the similarity between measured

_{D}*C*and

_{D,r}*C*(see above). In addition, we assumed an initial offshore wave height that reproduced the same

_{D,c}*H*

_{rms}on the fore reef (i.e., as directly measured at site W1)

*after*the effects of shoaling and friction had been assessed. Thus, including friction required a slightly larger initial

*H*

_{rms,0}= 2.4 m to produce the same fore-reef

*H*

_{rms}(Fig. 11b), implying that frictional damping offshore of W1 reduced the wave height by ∼20%, which is comparable to the difference observed between the fore-reef and offshore wave buoy (Table 2). When compared to the no-friction case, the inclusion of friction also significantly reduced

*H*

_{rms}on the reef flat, which gradually decayed shoreward.

Although the model slightly overpredicted observed *η*_{r} and *η*_{L} by ∼20%, it did predict that *η*_{L} will be ∼60% of *η*_{r}, a ratio consistent with observations (Fig. 11c). Furthermore, the model also matched observed *U _{r}^{E}* when setup in the lagoon was accounted for (Fig. 11d). However, because the height of waves incident to the surf zone was fixed for comparison purposes, there were no appreciable differences in both

*η*

*U*between the runs with and without wave friction. In simulations of the winter experiment period, the model based on the default quadratic drag law reproduced

^{E}*U*well, but tended to slightly underestimate the variability in current fluctuations that result from variation in

_{r}^{E}*H*

_{rms,0}(Fig. 12). The model with linear drag tended to better reproduce this wave-driven variability in

*U*, but overpredicted the amplitude of the current fluctuations some of the time. Finally, the model also indicated that Stokes drift should be the dominant transport mechanism on the fore reef and also in the vicinity of the reef crest (Fig. 11e).

_{r}^{E}In contrast, it is clear that H99 [via (5)] significantly overpredicts *η*_{r} (by a factor ∼3) (Fig. 11c). While the *H*_{rms} distribution from H99 [via (4)] correctly predicted the total reduction in *H _{r}* by breaking (not surprising because the observed

*γ*= 0.3 was used), the decrease in

_{r}*H*

_{rms}through the surf zone occurs much more rapidly than predicted by TG83 (Fig. 11b), which led to much stronger radiation stress gradients that enhanced

*η*

_{r}. Combined with the fact that H99 neglects lagoon setup, it overpredicted

*U*by a factor of ∼2. GC05 do not attempt to predict the particular

_{r}^{E}*H*

_{rms}distribution and instead relate

*η*

_{r}to the total dissipation in the surf zone, which increases proportionally to an empirical “reef profile shape factor”

*K*that they find increases with reef slope from ∼0.2 to 0.8 [see (17) in their paper for details]. While GC05 with

_{p}*K*= 0.8 overpredicted

_{p}*η*

_{r}by a factor of ∼4 (not shown), predictions using

*K*= 0.2 were in good agreement with observations (Fig. 11c). Nonetheless, GC05 still overpredicted

_{p}*U*by ∼20%, because of the model’s neglect of

_{r}^{E}*η*

_{L}.

## 6. Discussion and conclusions

### a. Wave-driven circulation of Kaneohe Bay

Results from the study revealed that wave breaking was the dominant mechanism driving flow over the reef flat and out the reef channels. Comparison of the reef flat (A2 and A3) and channel (A1 and A5) current speeds showed that the effective reef width *W _{r}* of each circulation cell was ∼3 times larger than the corresponding channel width

*W*, indicating that the bulk of the total alongshore reef length supported an onshore wave-driven flow.

_{c}In contrast, the weaker subtidal currents in the lagoon (A6 and A7) suggested that wave forcing may play a much smaller role in the overall lagoon circulation. To some degree this is expected, because the cross-sectional area of the deep lagoon (*h _{L}L_{L}* ∼ 16 000 m

^{2}) is much greater than the cross-sectional area of each reef cell (

*h*∼ 4000 m

_{r}W_{r}^{2}). Consequently, an average reef current speed

*U*∼0.1 m s

_{r}^{−1}(Table 3) would only induce a lagoon current

*U*∼0.02 m s

_{L}^{−1}, if it is assumed that mass is conserved across the reef to lagoon (i.e.,

*U*=

_{r}h_{r}W_{r}*U*). As a result, any wave-driven current signal measured in the lagoon should indeed be very weak, and may be comparable or dominated by subtidal currents induced by winds and buoyancy (Bathen 1968). The conceptual model did assume that flow originating near the reef crest (A2 and A3) crosses the entire reef flat before entering the channels; however, it is impossible to precisely quantify how much of the total flow crossing the reef directly penetrated into the lagoon from limited point measurements of these very weak lagoon currents. It is possible that only some fraction of the volume of water initially crossing the reef (A2 and A3) actually entered the lagoon and, for example, fed the channels along the back edge of the reef. This would serve to reduce the return-flow pathlength

_{L}h_{L}L_{L}*L*; however, we note that in the calculations we assigned

_{c}*L*to have a large (40%) uncertainty for this very reason. A reduced

_{c}*L*would ultimately increase our estimates of

_{c}*C*because both

_{D,c}*q*and

_{c}*η*

_{L}in (12) are fixed by the measurements (Fig. 10). Regardless, any uncertainty in

*L*should not alter the dominant momentum balances governing setup and circulation in this system.

_{c}Further support for the robustness of using 1D cross-shore mass and momentum equations in this inherently 2D system is that it appears to provide accurate estimates of both setup and wave-driven transport. However, this is not entirely surprising given that many beach studies have also found that the application of these same 1D equations along a sequence of transects alongshore can often be sufficient for estimating flows that may be ultimately 2D in terms of their spatial structure. For example, for topographic rip currents, Bellotti (2004) used data from Haller et al. (2002) to show that consideration of two 1D cross-shore momentum transects across a bar and through a rip channel could accurately predict the alongshore setup variability responsible for driving these flows when *W _{c}*/

*W*

_{total}was small, a reasonable assumption for Kaneohe Bay (Table 5). Other more general studies of topographically driven alongshore flows have shown that when the ratio of cross-shore to alongshore length scales of the bottom topography (∼

*L*

_{r}/

*W*

_{total}) is small, a cross-shore momentum balance governed by (2) that neglects gradients in the alongshore component of the radiation stresses (

*S*) should accurately predict the alongshore setup gradients that dominate the momentum balances of these flows (e.g., Putrevu et al. 1995; Apotsos et al. 2008b). For the circulation cells in Kaneohe Bay,

_{xy}*L*

_{r}/

*W*

_{total}∼ 0.4–0.7, which is not very small but is still less than unity. In general, the detailed spatial structure of wave-driven flows on reefs can certainly be most accurately captured through direct application of sophisticated 2D coupled wave–circulation numerical models. Nevertheless, this simple model based on a pair of 1D cross-shore momentum balances appears to be a particularly useful tool for illustrating how the morphology and bottom roughness of reefs fundamentally controls setup and circulation in coastally bounded reef systems.

### b. Role of reef morphology and roughness on setup and circulation of reefs

Maximum setup *η*_{r} on the Kaneohe reef increased proportional to the incident wave energy flux (i.e., *η*_{r} ∝ *H*_{rms,0}^{2}*T _{p}*), consistent with the scaling proposed by GC05 and also predictions from the TG83 model (Fig. 12). However, for the range of incident wave heights encountered during the experiment, the magnitude of

*η*

_{r}itself (<10 cm) was much smaller than previous observations on steeper atolls and barrier reefs (>50 cm; see Gourlay 1996 for a review). Apparently, this is due to the mild slope of the Kaneohe reef (∼1:60), which serves to reduce

*η*

_{r}in at least the following two ways: 1) it allows for significant frictional wave dissipation on the fore reef prior to breaking, and 2) it alters the dynamics of breaking and distributes the overall rate of dissipation over a wider surf zone. The former effect was apparent in Fig. 11b, where it was necessary to initialize the wave model with friction using a ∼20% larger

*H*

_{rms,0}(or ∼40% larger offshore wave energy flux) to generate the same

*η*

_{r}as the case without friction. The latter effect has been studied in detail on reefs by, for example, Gourlay (1996) and Nielsen et al. (2008), who showed that for a given incident wave condition,

*η*

_{r}can increase by a factor of ∼4 as fore-reef bottom slopes increase from an order of ∼1:100 to ∼1:1. Combined with the fact that the Kaneohe reef is deep (

*h*∼ 2 m), compared to many other reefs with crests near mean sea level, the Kaneohe reef also has a much larger threshold breaking height

_{r}*H*, making setup generation extremely inefficient in this system.

_{r}Once *η*_{r} is generated, the strength of the wave-driven flow will ultimately be controlled by the morphology and bottom roughness of the reef–lagoon region shoreward of the surf zone. In a classic view of coral reefs, the mean water level in the lagoon is generally assumed to be approximately equal to the open ocean; that is, it is assumed that the lagoon and channels are sufficiently deep and/or wide such that friction on the return flow is negligible. Conversely, on beaches setup is traditionally assumed to progressively increase past the break point as wave energy is dissipated gradually shoreward (Bowen 1968). Even in topographic rip systems, where a shoreward flow is present over a bar, frictional resistance in the rip current cell is strong enough, in many cases, to maintain a dominant momentum balance across the bar between setup and radiation stress gradients alone (i.e., Reniers and Battjes 1997; Haller et al. 2002). Reef–lagoon systems such as Kaneohe Bay generally sit somewhere between these two extremes, because *η*_{L} is not negligible yet is still much less than *η*_{r}. With the flow across the reef *U _{r}^{E}* specifically controlled by

*η*

_{r}−

*η*

_{L}, the momentum balance on the return flow that controls

*η*

_{L}must be carefully considered, with particular attention being given to the lagoon–channel geometry. For a given reef geometry (fixed

*W*,

_{r}*L*,

_{r}*h*, and fore-reef slope) and incident wave forcing,

_{r}*η*

_{L}should increase either when 1)

*L*/

_{c}*h*increases due to an increase in the return flow pathlength and/or a reduction of the lagoon–channel depth, or 2)

_{c}*W*/

_{c}*W*

_{total}decreases due to a relative reduction in channel width. As an example, the former effect can be investigated using the simple model of Kaneohe Bay by holding the reef geometry and channel width constant, and hypothetically varying

*L*/

_{c}*h*, that is, by increasing the lagoon and channel depths (Fig. 13). For this system, the predicted

_{c}*η*

_{r}/

*η*

_{L}varies from ∼1 for

*L*/

_{c}*h*∼ 0.005 to ∼0 for

_{c}*L*/

_{c}*h*∼ 0.01. The variation in

_{c}*η*

_{L}between these two limits causes

*U*to vary by a factor of ∼5.

_{r}^{E}Finally, bottom roughness and its spatial variability are clearly key factors controlling *U _{r}^{E}*. On the reef flat, estimates of

*C*≈ 0.02 were very similar to values measured on other reefs (

_{D,r}*C*≈ 0.01 − 0.05; see Lugo-Fernandez et al. 1998; Jones et al. 2008; Hench et al. 2008). Although

_{D,r}*C*is greater than the

_{D,r}*C*∼

_{D}*O*(0.001) typically observed on relatively smooth sandy beaches (e.g., Feddersen et al. 2003), this value is still much smaller than the

*C*= 0.1 assumed by H99 for Kaneohe Bay. Evidently H99 required such an abnormally high

_{D,r}*C*to produce reasonable currents at this site, given the model’s overprediction of

_{D,r}*η*

_{r}and its neglect of setup inside the lagoon. Estimated lagoon–channel drag coefficients

*C*≈ 0.01 − 0.02 were comparable to

_{D,c}*C*, suggesting that

_{D,r}*C*may be surprisingly uniform across this system. This may be due to the fact that the lagoon and channels are interspersed by numerous “patch reefs” (biogenic carbonate structures formed by coral that extend over most of the water column; see Fig. 1). These patch reefs may locally serve to increase effective

_{D}*C*s to an order of 1 or larger (Reidenbach et al. 2006; Monismith 2007), thus augmenting the spatially integrated

_{D}*C*s that were measured. Finally, both the measurements and model predictions indicate that wave–current interactions on coral reef flats can potentially have a major influence on the response of the system-wide circulation to incident wave forcing, by controlling the scaling of bed stresses with

_{D,c}*U*. For systems such as Kaneohe, the importance of these wave–current interactions may vary considerably across the system, because of the high degree of spatial variability in wave energy, current speeds, and physical bed roughness.

_{r}^{E}## Acknowledgments

The authors are grateful to Dan Schar, Pete Newkirk, Nicole Jones, Cameron McDonald, Sarah Giddings, and Kristen Davis for assisting with the field experiments, and the constructive comments from two anonymous reviewers and Jerry Smith that helped to greatly improve the original manuscript. SGM thanks Jim Hench for discussions into reef dynamics. We especially thank Jerome Aucan, Mark Merrifield, and the Coastal Data Information Program for providing the offshore wave buoy data. This work was supported by the National Science Foundation through Grants OCE-0452800, OCE-0453117, OCE-0622967, and OISE- 0601787, and the Singapore Stanford Partnership. RJL acknowledges support from an Australian Research Council Discovery Project grant DP0770094.

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## APPENDIX

### Wave Setup Analysis

Subtidal mean water level variability *η**P = P*_{atm} + *η* + *h*, where *P*_{atm} is atmospheric pressure, *h* is the still-water depth, and *η* is the water level associated with tides and wave setup. We assume that *η* on the fore-reef sites (W1 and W2) only varied with tides, that is, set-down was negligible here (this was justified a posteriori; see Fig. 11c). This defines a reference pressure *P*_{0}, such that for each site, we obtain Δ*P* = *η* + *h* − *h*_{0}, where Δ*P* ≡ *P* − *P*_{0}. We note that possible flow noise effects on Δ*P*, resulting from the dynamic pressure field of the waves (see Raubenheimer et al. 2001), were estimated to be negligible based on observed and modeled (via section 5) near-bed wave velocities evaluated between each pair of sites.

*η*

*η*

*a*and

*b*are empirical constants and

*m*and

*n*are coefficients describing the scaling of

*η*

*aH*

_{rms,0}

*−*

^{m}T_{p}^{n}*c*, where

*c*≡

*b*−

*h*+

*h*

_{0}is a constant for each site. Through a nonlinear least squares regression of

*H*

_{rms,0}and

*T*, the coefficients (

_{p}*a*,

*c*,

*m*, and

*n*) and their uncertainty were evaluated. However, to evaluate the magnitude of

*η*

*b*must first be calculated given that the pressure sensors were not leveled. This coefficient can be obtained from the current measurements by evaluating the threshold wave height

*H*below which the wave-driven circulation turns off (see Fig. 6). At this threshold, where

_{r}*H*

_{0}=

*H*, setup

_{r}*η*

*b*=

*aH*.

_{r}^{m}T_{p}^{n}Kaneohe Bay, Oahu, HI. (a) Quickbird aerial photograph (Digitalglobe) with the instrument locations superimposed. The offshore wave buoy (not shown) is located ∼8 km southeast of Mokapu Peninsula bounding the southern end of the bay. (b) The bathymetry highlights its dominant morphological features.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Kaneohe Bay, Oahu, HI. (a) Quickbird aerial photograph (Digitalglobe) with the instrument locations superimposed. The offshore wave buoy (not shown) is located ∼8 km southeast of Mokapu Peninsula bounding the southern end of the bay. (b) The bathymetry highlights its dominant morphological features.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Kaneohe Bay, Oahu, HI. (a) Quickbird aerial photograph (Digitalglobe) with the instrument locations superimposed. The offshore wave buoy (not shown) is located ∼8 km southeast of Mokapu Peninsula bounding the southern end of the bay. (b) The bathymetry highlights its dominant morphological features.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Physical forcing conditions during the winter experiment. (a) Tidal elevation. (b) Offshore rms wave height *H*_{rms,0} measured by the offshore wave buoy (WO). (c) Offshore peak wave period *T _{p}*. (d) Offshore mean wave direction

*θ*(note that 45° is the shore-normal direction).

_{m}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Physical forcing conditions during the winter experiment. (a) Tidal elevation. (b) Offshore rms wave height *H*_{rms,0} measured by the offshore wave buoy (WO). (c) Offshore peak wave period *T _{p}*. (d) Offshore mean wave direction

*θ*(note that 45° is the shore-normal direction).

_{m}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Physical forcing conditions during the winter experiment. (a) Tidal elevation. (b) Offshore rms wave height *H*_{rms,0} measured by the offshore wave buoy (WO). (c) Offshore peak wave period *T _{p}*. (d) Offshore mean wave direction

*θ*(note that 45° is the shore-normal direction).

_{m}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

(a) Time series of rms wave height *H*_{rms} measured on the fore reef (site W1) and reef flat (site A2) during the winter experiment, based on raw (nonsubtidally filtered) data. (b) Local wave height vs local water depth on the fore reef at site W1. (c) Local wave height vs local water depth on the reef flat at site A2. The dashed line represents the depth-limited maximum wave height *H*_{rms,max} = *γ _{r}h_{r}* with

*γ*= 0.3, above which waves were not observed.

_{r}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

(a) Time series of rms wave height *H*_{rms} measured on the fore reef (site W1) and reef flat (site A2) during the winter experiment, based on raw (nonsubtidally filtered) data. (b) Local wave height vs local water depth on the fore reef at site W1. (c) Local wave height vs local water depth on the reef flat at site A2. The dashed line represents the depth-limited maximum wave height *H*_{rms,max} = *γ _{r}h_{r}* with

*γ*= 0.3, above which waves were not observed.

_{r}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

(a) Time series of rms wave height *H*_{rms} measured on the fore reef (site W1) and reef flat (site A2) during the winter experiment, based on raw (nonsubtidally filtered) data. (b) Local wave height vs local water depth on the fore reef at site W1. (c) Local wave height vs local water depth on the reef flat at site A2. The dashed line represents the depth-limited maximum wave height *H*_{rms,max} = *γ _{r}h_{r}* with

*γ*= 0.3, above which waves were not observed.

_{r}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

(a) Time-averaged Eulerian current vectors, (b) principal component ellipses (radii represent one current velocity standard deviation), (c) time-averaged Stokes drift vectors, and (d) *M*_{2} tidal ellipses. All data are from the winter (∼2 month) deployment and are based on the total flow variability (i.e., nonsubtidally filtered). Note the differences in the velocity scales between the figures.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

(a) Time-averaged Eulerian current vectors, (b) principal component ellipses (radii represent one current velocity standard deviation), (c) time-averaged Stokes drift vectors, and (d) *M*_{2} tidal ellipses. All data are from the winter (∼2 month) deployment and are based on the total flow variability (i.e., nonsubtidally filtered). Note the differences in the velocity scales between the figures.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

(a) Time-averaged Eulerian current vectors, (b) principal component ellipses (radii represent one current velocity standard deviation), (c) time-averaged Stokes drift vectors, and (d) *M*_{2} tidal ellipses. All data are from the winter (∼2 month) deployment and are based on the total flow variability (i.e., nonsubtidally filtered). Note the differences in the velocity scales between the figures.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Subtidal Eulerian current speeds *U ^{E}* along the principal major axis, measured at the (a) reef flat (site A2), (b) channel (site A1), and (c) lagoon (site A7) as a function of the rms offshore wave height

*H*

_{rms,0}.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Subtidal Eulerian current speeds *U ^{E}* along the principal major axis, measured at the (a) reef flat (site A2), (b) channel (site A1), and (c) lagoon (site A7) as a function of the rms offshore wave height

*H*

_{rms,0}.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Subtidal Eulerian current speeds *U ^{E}* along the principal major axis, measured at the (a) reef flat (site A2), (b) channel (site A1), and (c) lagoon (site A7) as a function of the rms offshore wave height

*H*

_{rms,0}.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Time series of subtidal Eulerian velocity *U ^{E}* and subtidal Stokes drift velocity

*U*, along the principal major axis at A2.

^{S}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Time series of subtidal Eulerian velocity *U ^{E}* and subtidal Stokes drift velocity

*U*, along the principal major axis at A2.

^{S}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Time series of subtidal Eulerian velocity *U ^{E}* and subtidal Stokes drift velocity

*U*, along the principal major axis at A2.

^{S}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

(a) Time series of subtidal variability in wave setup *η**H*^{2}_{rms,0} *T _{p}* plotted for the entire 10-month experiment, as measured on the reef flat (A2) and inside the lagoon (W5).

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

(a) Time series of subtidal variability in wave setup *η**H*^{2}_{rms,0} *T _{p}* plotted for the entire 10-month experiment, as measured on the reef flat (A2) and inside the lagoon (W5).

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

(a) Time series of subtidal variability in wave setup *η**H*^{2}_{rms,0} *T _{p}* plotted for the entire 10-month experiment, as measured on the reef flat (A2) and inside the lagoon (W5).

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Linear regression (*y* = *Ax*) used to estimate an effective reef width *W _{r}* for the two circulation cells. Note that a zero

*y*intercept was assumed in order to satisfy (7). (a) South (Sampan Channel) cell using the depth-averaged transport (Eulerian plus Stokes drift) at reef A2 and channel A1 sites (

*R*

^{2}= 0.78). (b) North (Ship Channel) cell using the depth-averaged transport at reef A3 and channel A5 sites (

*R*

^{2}= 0.56). Solid gray lines represent the best linear fit to the data; dashed lines bound the 95% confidence limits of the line of best fit.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Linear regression (*y* = *Ax*) used to estimate an effective reef width *W _{r}* for the two circulation cells. Note that a zero

*y*intercept was assumed in order to satisfy (7). (a) South (Sampan Channel) cell using the depth-averaged transport (Eulerian plus Stokes drift) at reef A2 and channel A1 sites (

*R*

^{2}= 0.78). (b) North (Ship Channel) cell using the depth-averaged transport at reef A3 and channel A5 sites (

*R*

^{2}= 0.56). Solid gray lines represent the best linear fit to the data; dashed lines bound the 95% confidence limits of the line of best fit.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Linear regression (*y* = *Ax*) used to estimate an effective reef width *W _{r}* for the two circulation cells. Note that a zero

*y*intercept was assumed in order to satisfy (7). (a) South (Sampan Channel) cell using the depth-averaged transport (Eulerian plus Stokes drift) at reef A2 and channel A1 sites (

*R*

^{2}= 0.78). (b) North (Ship Channel) cell using the depth-averaged transport at reef A3 and channel A5 sites (

*R*

^{2}= 0.56). Solid gray lines represent the best linear fit to the data; dashed lines bound the 95% confidence limits of the line of best fit.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Linear regression (*y* = *Ax*) of subtidal water level differences vs Eulerian current speeds squared used to infer quadratic drag coefficients. (a) Water level difference between A2 and W5 vs the reef current squared at A2 (*R*^{2} = 0.57). (b) Water level difference between A3 and W5 vs the reef current squared at A3 (*R*^{2} = 0.34). (c) Water level difference between W5 and the channel exit (where *η**R*^{2} = 0.70). (d) Water level difference between W5 and the channel exit vs the channel current squared at A5 (*R*^{2} = 0.63). Solid gray lines represent the best linear fits to the data; dashed lines bound the 95% confidence limits of the lines of best fit.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Linear regression (*y* = *Ax*) of subtidal water level differences vs Eulerian current speeds squared used to infer quadratic drag coefficients. (a) Water level difference between A2 and W5 vs the reef current squared at A2 (*R*^{2} = 0.57). (b) Water level difference between A3 and W5 vs the reef current squared at A3 (*R*^{2} = 0.34). (c) Water level difference between W5 and the channel exit (where *η**R*^{2} = 0.70). (d) Water level difference between W5 and the channel exit vs the channel current squared at A5 (*R*^{2} = 0.63). Solid gray lines represent the best linear fits to the data; dashed lines bound the 95% confidence limits of the lines of best fit.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Linear regression (*y* = *Ax*) of subtidal water level differences vs Eulerian current speeds squared used to infer quadratic drag coefficients. (a) Water level difference between A2 and W5 vs the reef current squared at A2 (*R*^{2} = 0.57). (b) Water level difference between A3 and W5 vs the reef current squared at A3 (*R*^{2} = 0.34). (c) Water level difference between W5 and the channel exit (where *η**R*^{2} = 0.70). (d) Water level difference between W5 and the channel exit vs the channel current squared at A5 (*R*^{2} = 0.63). Solid gray lines represent the best linear fits to the data; dashed lines bound the 95% confidence limits of the lines of best fit.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Model results across the reef obtained using TG83 (with and without wave friction), applied to the southern circulation cell for *H*_{rms,0} = 2 m, *T _{p}* = 7 s (note the lagoon–channel region is not shown). Squares are derived from field data, based on representative values when

*H*

_{rms,0}= 2 m (i.e., based on the regression coefficient in Table 4 for setup). Error bars are derived from uncertainties in the regression coefficients. For comparison, predictions from H99 and GC05 (

*K*= 0.2) are also included. (a) Assumed cross-shore depth profile, (b) wave height distribution

_{p}*H*, (c) setup distribution

_{rms}*η*

*U*, and (e) depth-averaged Stokes drift speed

^{E}*U*. Note that the H99 model is applied using the same parameters recommended in that paper (i.e., using different

^{S}*γ*’s on the fore reef and reef flat), except we use a slightly smaller

*γ*= 0.3 to reflect the observed reef flat wave heights (see Fig. 4). The TG83 model is applied with the identical parameter values used in Lowe et al. (2005).

_{r}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Model results across the reef obtained using TG83 (with and without wave friction), applied to the southern circulation cell for *H*_{rms,0} = 2 m, *T _{p}* = 7 s (note the lagoon–channel region is not shown). Squares are derived from field data, based on representative values when

*H*

_{rms,0}= 2 m (i.e., based on the regression coefficient in Table 4 for setup). Error bars are derived from uncertainties in the regression coefficients. For comparison, predictions from H99 and GC05 (

*K*= 0.2) are also included. (a) Assumed cross-shore depth profile, (b) wave height distribution

_{p}*H*, (c) setup distribution

_{rms}*η*

*U*, and (e) depth-averaged Stokes drift speed

^{E}*U*. Note that the H99 model is applied using the same parameters recommended in that paper (i.e., using different

^{S}*γ*’s on the fore reef and reef flat), except we use a slightly smaller

*γ*= 0.3 to reflect the observed reef flat wave heights (see Fig. 4). The TG83 model is applied with the identical parameter values used in Lowe et al. (2005).

_{r}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Model results across the reef obtained using TG83 (with and without wave friction), applied to the southern circulation cell for *H*_{rms,0} = 2 m, *T _{p}* = 7 s (note the lagoon–channel region is not shown). Squares are derived from field data, based on representative values when

*H*

_{rms,0}= 2 m (i.e., based on the regression coefficient in Table 4 for setup). Error bars are derived from uncertainties in the regression coefficients. For comparison, predictions from H99 and GC05 (

*K*= 0.2) are also included. (a) Assumed cross-shore depth profile, (b) wave height distribution

_{p}*H*, (c) setup distribution

_{rms}*η*

*U*, and (e) depth-averaged Stokes drift speed

^{E}*U*. Note that the H99 model is applied using the same parameters recommended in that paper (i.e., using different

^{S}*γ*’s on the fore reef and reef flat), except we use a slightly smaller

*γ*= 0.3 to reflect the observed reef flat wave heights (see Fig. 4). The TG83 model is applied with the identical parameter values used in Lowe et al. (2005).

_{r}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Time se**r**ies of field observations and model predictions using TG83 (including wave friction), during the winter experiment period, of (a) reef-top setup *η*_{r} (at A2), (b) lagoon setup *η*_{L} (at W5), and (c) Eulerian current speed *U _{r}^{E}* (at A2). Model predictions are shown for both quadratic friction (

*C*= 0.02) and linear friction (

_{D}*C*= 0.02), based on a representative near-bed wave orbital velocity

_{D}^{L}*U*= 0.2 m s

_{w}^{−1}. Model skill values (i.e., calculated via Warner et al. 2005) were 0.94 (quadratic) and 0.94 (linear) for

*η*

_{r}, 0.90 (quadratic) and 0.86 (linear) for

*η*

_{L}, and 0.84 (quadratic) and 0.90 (linear) for

*U*. Perfect agreement will yield a skill of one, whereas complete disagreement will yield a skill of zero. Corresponding

_{r}^{E}*R*

^{2}values were 0.94 (quadratic) and 0.94 (linear) for

*η*

_{r}, 0.82 (quadratic) and 0.81 (linear) for

*η*

_{L}, and 0.80 (quadratic) and 0.78 (linear) for

*U*.

_{r}^{E}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Time se**r**ies of field observations and model predictions using TG83 (including wave friction), during the winter experiment period, of (a) reef-top setup *η*_{r} (at A2), (b) lagoon setup *η*_{L} (at W5), and (c) Eulerian current speed *U _{r}^{E}* (at A2). Model predictions are shown for both quadratic friction (

*C*= 0.02) and linear friction (

_{D}*C*= 0.02), based on a representative near-bed wave orbital velocity

_{D}^{L}*U*= 0.2 m s

_{w}^{−1}. Model skill values (i.e., calculated via Warner et al. 2005) were 0.94 (quadratic) and 0.94 (linear) for

*η*

_{r}, 0.90 (quadratic) and 0.86 (linear) for

*η*

_{L}, and 0.84 (quadratic) and 0.90 (linear) for

*U*. Perfect agreement will yield a skill of one, whereas complete disagreement will yield a skill of zero. Corresponding

_{r}^{E}*R*

^{2}values were 0.94 (quadratic) and 0.94 (linear) for

*η*

_{r}, 0.82 (quadratic) and 0.81 (linear) for

*η*

_{L}, and 0.80 (quadratic) and 0.78 (linear) for

*U*.

_{r}^{E}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Time se**r**ies of field observations and model predictions using TG83 (including wave friction), during the winter experiment period, of (a) reef-top setup *η*_{r} (at A2), (b) lagoon setup *η*_{L} (at W5), and (c) Eulerian current speed *U _{r}^{E}* (at A2). Model predictions are shown for both quadratic friction (

*C*= 0.02) and linear friction (

_{D}*C*= 0.02), based on a representative near-bed wave orbital velocity

_{D}^{L}*U*= 0.2 m s

_{w}^{−1}. Model skill values (i.e., calculated via Warner et al. 2005) were 0.94 (quadratic) and 0.94 (linear) for

*η*

_{r}, 0.90 (quadratic) and 0.86 (linear) for

*η*

_{L}, and 0.84 (quadratic) and 0.90 (linear) for

*U*. Perfect agreement will yield a skill of one, whereas complete disagreement will yield a skill of zero. Corresponding

_{r}^{E}*R*

^{2}values were 0.94 (quadratic) and 0.94 (linear) for

*η*

_{r}, 0.82 (quadratic) and 0.81 (linear) for

*η*

_{L}, and 0.80 (quadratic) and 0.78 (linear) for

*U*.

_{r}^{E}Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Effect of varying lagoon–channel dimensions on (a) setup distribution *η**U ^{E}* speed for

*H*

_{rms,0}= 2 m,

*T*= 7 s. The model is applied using the same reef dimensions for Kaneohe Bay used in Fig. 11 (

*L*,

_{r}*h*,

_{r}*W*, fore-reef slope) and a fixed channel width

_{r}*W*with

_{c}*C*= 0.02. The return pathlength

_{D}*L*is also held fixed and the lagoon–channel depth

_{c}*h*is varied. Note that for small

_{c}*h*/

_{c}*L*,

_{c}*U*on the fore reef (

^{E}*x*< 0) is directed offshore to offset the relatively strong Stokes drift in this region, that is, to conserve the total mass flux

*q*across the reef.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Effect of varying lagoon–channel dimensions on (a) setup distribution *η**U ^{E}* speed for

*H*

_{rms,0}= 2 m,

*T*= 7 s. The model is applied using the same reef dimensions for Kaneohe Bay used in Fig. 11 (

*L*,

_{r}*h*,

_{r}*W*, fore-reef slope) and a fixed channel width

_{r}*W*with

_{c}*C*= 0.02. The return pathlength

_{D}*L*is also held fixed and the lagoon–channel depth

_{c}*h*is varied. Note that for small

_{c}*h*/

_{c}*L*,

_{c}*U*on the fore reef (

^{E}*x*< 0) is directed offshore to offset the relatively strong Stokes drift in this region, that is, to conserve the total mass flux

*q*across the reef.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Effect of varying lagoon–channel dimensions on (a) setup distribution *η**U ^{E}* speed for

*H*

_{rms,0}= 2 m,

*T*= 7 s. The model is applied using the same reef dimensions for Kaneohe Bay used in Fig. 11 (

*L*,

_{r}*h*,

_{r}*W*, fore-reef slope) and a fixed channel width

_{r}*W*with

_{c}*C*= 0.02. The return pathlength

_{D}*L*is also held fixed and the lagoon–channel depth

_{c}*h*is varied. Note that for small

_{c}*h*/

_{c}*L*,

_{c}*U*on the fore reef (

^{E}*x*< 0) is directed offshore to offset the relatively strong Stokes drift in this region, that is, to conserve the total mass flux

*q*across the reef.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3958.1

Instrument locations, configurations, and deployment information. Two sampling strategies were employed: 1) a long-term (∼10 month) deployment at representative sites on the fore reef, reef flat, and in the lagoon; and 2) a shorter (∼2 month) deployment in winter 2006 where several additional instruments were added to the base array.

Root-mean-square wave heights (*H*_{rms}, mean, and standard deviation), correlation coefficients *R*_{tide} between *H*_{rms} and the local tidal depth, as well as *R*_{wave} between *H*_{rms} and the offshore wave height *H*_{rms,0}, Stokes drift estimates (*U ^{S}*, mean, and standard deviation), and the median fraction of the local mass flux resulting from Stokes drift. Italicized correlations are significant to 95%; bold correlations to 99%. Asterisks denote results based on the entire 10-month record.

Depth-averaged Eulerian current statistics (cm s^{−1}). Time-averaged current vectors and current ellipse axes, with the axes values representing one standard deviation of the variability. Magnitude of the tidal current amplitude for the *M*_{2} component inferred via harmonic analysis. *R*_{tide} is defined as the square root of the total current variability explained by the harmonic analysis. Correlation coefficient *R*_{wave} of the subtidal (38 h, low-pass filtered) current (positive into the bay) with the offshore rms wave height *H*_{rms,0}. Threshold rms wave height *H _{r}* inferred from subtidal currents. Italicized correlations are significant to 95%; bold correlations to 99%. Asterisks denote results based on the entire 10-month record.

Wave setup scaling. Nonlinear regression coefficients *m* and *n* based on a fit governed by *η**aH*_{rms,0}* ^{m}T_{p}^{n}* −

*b*[(15)], and the coefficient of determination for the best fit (

*R*

^{2}). Setup estimated from the regression coefficients for

*H*

_{rms,0}= 2 m with

*T*= 7 s. Uncertainties represent the 95% confidence intervals for the nonlinear least squares parameter estimates. Asterisks denote results based on the entire 10-month record.

_{p}Reef geometry parameters, and estimated quadratic (*C _{D,r}*,

*C*) and linear (

_{D,c}*C*′,

_{D,r}*C*′) drag coefficients derived for the reef and lagoon–channel regions, respectively. Values are shown for each of the two dominant circulation cells (southern and northern). Uncertainties denote the 95% confidence intervals of the parameter estimates.

_{D,c}