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 = (8m₀)^1/2, where m₀ 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_m01 and peak T_p wave periods were subsequently calculated based on the first spectral moment of 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_p. Generally, K_s was close to unity throughout the experiment (range 0.90–1.05; mean 0.94).

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^E using T_TIDE (Pawlowicz et al. 2002). A correlation coefficient R_tide for currents is thus defined as the square root of the percentage of total variance in U^E explained by the tidal harmonic analysis.

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 (defined relative to still water) at this site was driven by tides and wave setup, because wind setup was estimated to be only 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 resulting from observed changes in incident wave forcing (not shown). Therefore, in the present study, the specific response of both setup and currents to incident wave forcing was investigated at subtidal frequencies by applying a PL64 filter (Beardsley et al. 1985) with a half-power period of 38 h. Although performing a similar analysis at higher frequencies poses an interesting problem, it is beyond the scope of the present paper and, moreover, not necessary for exploring the fundamental dynamics given in (2). For consistency, when investigating relationships between setup, currents, and wave forcing, each pair of time series were low-pass filtered identically and then decimated to 6-hourly values to form a “subtidal” dataset. In all calculations, significance levels were obtained from the effective degrees of freedom (DOF) estimated from the record length divided by the autocorrelation time scale (Davis 1976); this time scale was typically ∼1.5–2 days.

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 measured at these sites should be at most 5%–10% less than η_r.

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_p ∼ 6 − 13 s), in part generated remotely by several North Pacific storms. Over the longer 10-month deployment (not shown), 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_rh_r. The measurements indicate γ_r ≈ 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_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_r was apparent at these sites, corresponding to the point where the wave-driven circulation roughly shut down (see Figs. 6a,b). For each site H_r was evaluated (based on the rms wave height) and averaged H_r = 0.7 m (range of 0.6–0.8 m; Table 3). Given the typical Kaneohe reef flat depth h_r ≈ 2 m (Falter et al. 2004), the observed ratio H_r/h_r ≈ 0.35 is similar to H_r/h_r ≈ 0.4 proposed by Gourlay (1996).

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 and H²_rms,0T_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^mT_pⁿ − 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 are illustrated in Table 4, where values were calculated for 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 was largest at the shallowest reef flat sites (A2 and A3) near the alongshore center of the bay and slightly smaller at the deeper site (A4). This is not surprising because over this shallower section of reef more wave energy was dissipated by breaking, and also, when comparing equal rates of dissipation, setup gradients will increase in shallower water via (2).

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

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_LL_L ∼ 16 000 m²) is much greater than the cross-sectional area of each reef cell (h_rW_r ∼ 4000 m²). Consequently, an average reef current speed U_r ∼0.1 m s⁻¹ (Table 3) would only induce a lagoon current U_L ∼0.02 m s⁻¹, if it is assumed that mass is conserved across the reef to lagoon (i.e., U_rh_rW_r = U_Lh_LL_L). 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_c; however, we note that in the calculations we assigned L_c to have a large (40%) uncertainty for this very reason. A reduced L_c would ultimately increase our estimates of C_D,c because both q_c and η_L in (12) are fixed by the measurements (Fig. 10). Regardless, any uncertainty in L_c should not alter the dominant momentum balances governing setup and circulation in this system.

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_xy) 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, 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,0²T_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_r ∼ 2 m), compared to many other reefs with crests near mean sea level, the Kaneohe reef also has a much larger threshold breaking height H_r, making setup generation extremely inefficient in this system.

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_r, and fore-reef slope) and incident wave forcing, η_L should increase either when 1) L_c/h_c increases due to an increase in the return flow pathlength and/or a reduction of the lagoon–channel depth, or 2) 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_c, that is, by increasing the lagoon and channel depths (Fig. 13). For this system, the predicted η_r/η_L varies from ∼1 for L_c/h_c ∼ 0.005 to ∼0 for L_c/h_c ∼ 0.01. The variation in η_L between these two limits causes U_r^E to vary by a factor of ∼5.

Finally, bottom roughness and its spatial variability are clearly key factors controlling U_r^E. On the reef flat, estimates of C_D,r ≈ 0.02 were very similar to values measured on other reefs (C_D,r ≈ 0.01 − 0.05; see Lugo-Fernandez et al. 1998; Jones et al. 2008; Hench et al. 2008). Although C_D,r is greater than the 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_D,r = 0.1 assumed by H99 for Kaneohe Bay. Evidently H99 required such an abnormally high C_D,r to produce reasonable currents at this site, given the model’s overprediction of η_r and its neglect of setup inside the lagoon. Estimated lagoon–channel drag coefficients C_D,c ≈ 0.01 − 0.02 were comparable to C_D,r, suggesting that C_D 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 C_Ds to an order of 1 or larger (Reidenbach et al. 2006; Monismith 2007), thus augmenting the spatially integrated C_D,cs 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 U_r^E. 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.