G Winter

B Castelle

R J Lowe

J E Hansen

R Mccall

When is flow re-entrainment important for the flushing time in coastal reef systems?

Keywords: Reef, wave-driven circulation, flushing, Lagrangian, drifter

The rates of water exchange between coastal reef systems and the surrounding ocean are key physical drivers of water quality and reef ecosystems. It is generally assumed that water exiting a reef system through reef channels is predominantly replaced by 'new' water from offshore. However, exiting water may also recirculate back into the reef system reducing the rate of exchange between the reef and the ocean, which has implications for reef water temperatures, nutrient fluxes and population connectivity. To quantify flow re-entrainment at a rocky reef site in southwestern Australia, flow patterns were measured with GPS-tracked drifters during a two-week field experiment. The field observations were extended via a set of idealized numerical experiments to determine the effect of variable oceanic forcing and reef geometry on flow re-entrainment. The observations demonstrate that re-entrainment can vary significantly and the numerical results support the hypothesis that re-entrainment increases with increasing offshore wave height, increasing alongshore currents outside of the reef, and decreasing reef channel spacing but is largely not impacted by reef roughness. Reentrainment was correlated with a predictor variable R, which is a measure of wave forcing versus the total offshore flow cross-section, and alongshore currents outside the reef. For large values of R and strong alongshore currents, flow re-entrainment increases the effective flushing time by a factor of three or more. The results suggest that flow re-entrainment may be particularly important in small-scale reef systems or reefs exposed to an energetic wave climate and/or strong alongshore currents.

Introduction

The circulation of water within reef systems controls the exchange of dissolved and particulate material between reefs and the surrounding deeper ocean, which substantially influences ecosystems of both tropical coral reefs [see Lowe and Falter, 2015 for a review] and temperate rocky reefs [e.g. [START_REF] Morgan | Surfzone hydrodynamics as a key determinant of spatial variation in rocky intertidal communities[END_REF]. For example, the influx of water to reefs from the surrounding ocean often has higher nutrient concentrations and is cooler [START_REF] Lowe | Oceanic Forcing of Coral Reefs[END_REF]. As the water flows over shallow reefs, nutrients in various forms are taken up by reef communities at rates influenced by local water motion [START_REF] Falter | Mass-transfer limitation of nutrient uptake by a wave-dominated reef flat community[END_REF].

Furthermore, this influx of water contributes to the regulation of reef water temperatures, which due to the relatively shallow water depths, can either heat or cool relative to the surrounding ocean [START_REF] Zhang | Atmospheric forcing intensifies the effects of regional ocean warming on reef-scale temperature anomalies during a coral bleaching event[END_REF]. Water that exits through the reef channel can also remove suspended sediments [e.g. [START_REF] Storlazzi | Waveand tidally-driven flow and sediment flux across a fringing coral reef: Southern Molokai, Hawaii[END_REF], larvae [e.g. [START_REF] Lugo-Fernández | Inferring probable dispersal of Flower Garden Banks Coral Larvae (Gulf of Mexico) using observed and simulated drifter trajectories[END_REF] as well as pollutants from the reef system (e.g., associated with terrestrial discharge).

Reef circulation can be driven by a combination of different processes including tides [START_REF] Black | Residence times of neutrally-buoyant matter such as larvae, sewage or nutrients on coral reefs[END_REF], buoyancy differences [START_REF] Herdman | Heat balances and thermally driven lagoon-ocean exchanges on a tropical coral reef system (Moorea, French Polynesia)[END_REF], wind stresses [START_REF] Tartinville | The water residence time in the Mururoa atoll lagoon: sensitivity analysis of a three-dimensional model[END_REF] and breaking waves [START_REF] Hoeke | Hydrodynamics of a bathymetrically complex fringing coral reef embayment: Wave climate, in situ observations, and wave prediction[END_REF][START_REF] Lowe | Wave-Driven Circulation of a Coastal Reef-Lagoon System[END_REF][START_REF] Taebi | Nearshore circulation in a tropical fringing reef system[END_REF].

However, many reef systems that fringe coastlines are exposed to wind (sea-swell) wave energy that provides the main forcing to reef circulation. For these wave-exposed reefs, radiation stress gradients [Longuet-Higgins, 1962] that originate from wave dissipation on the forereef or reef crest generate variations in the mean water level over the reef flat and crossreef currents. In the absence of reef channels (i.e., a one-dimensional, cross-shore system) the setup over a coastal reef is maximum near the shoreline and the net depth-averaged cross-reef mass flux is zero [START_REF] Buckley | Wave Setup over a Fringing Reef with Large Bottom Roughness[END_REF][START_REF] Vetter | Wave setup over a Pacific Island fringing reef[END_REF]. When the reef flat is intersected by reef channels, the net forcing, which is the residual difference between radiation stress and setup gradients, drives a net cross-reef onshore flow that is countered by bed friction [START_REF] Symonds | Wave-driven flow over shallow reefs[END_REF]. The return flow in the channel may also be fed directly by alongshore flows entering the channel laterally from the reef platform. The relative contribution of water entering the channel from the lagoon or directly from the reef depends on the relative importance of flow resistance across the reef and along the lagoon, which is governed by the roughness properties as well as geometry of the reef system [START_REF] Monismith | Flow through a rough, shallow reef[END_REF].

In many reef studies, water that exits through a reef channel is assumed to leave the reef system and to be replaced by 'new' offshore water [START_REF] Herdman | Heat balances and thermally driven lagoon-ocean exchanges on a tropical coral reef system (Moorea, French Polynesia)[END_REF][START_REF] Lowe | Morphological constraints to wave-driven circulation in coastal reef-lagoon systems: A numerical study[END_REF][START_REF] Taebi | Nearshore circulation in a tropical fringing reef system[END_REF]. While this approach allows ocean-reef exchange rates to be more easily quantified from fixed (Eulerian) measurements, it does not account for water that has initially flowed out through the channel and then returns back into the reef-lagoon system, which will effectively increase the flushing time of the system. While water exchange and flow re-entrainment in wave-dominated reefs have been previously estimated from numerical simulations, these have typically been done for either a specific reef system [e.g. [START_REF] Zhang | Atmospheric forcing intensifies the effects of regional ocean warming on reef-scale temperature anomalies during a coral bleaching event[END_REF] or for cases where the wave forcing was not modeled explicitly [e.g. [START_REF] Herdman | Circulation, Residence Time and Retention in a Tropical Coral Reef[END_REF].

Thus, a systematic approach to flow re-entrainment in wave-dominated reef environments that incorporates the impact of different oceanic forcing and reef geometry is currently lacking and motivates the present study.

The aim of this manuscript is to identify the parameters that govern flow reentrainment in wave-dominated coastal reef systems. To assess the hydrodynamic response to different parameters, we validate the numerical model XBeach [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF] with field observations from moored instruments and Lagrangian drifters in a coastal reef system.

The model is then extended to test the effects of wave forcing and along-shelf currents as well as geometric parameters on flow re-entrainment in an idealized wave-dominated reef system. The specific objectives of this study are to: (i) investigate the variability of flow reentrainment, (ii) identify the governing parameters that drive this variability and (iii) to assess the implications of variable flow re-entrainment on flushing time estimates for coastal reef systems, more generally. The results can thus help to classify coastal reef systems as being predominantly open or closed systems [START_REF] Cowen | Connectivity of Marine Populations: Open or Closed?[END_REF] based on their specific geometrical characteristics and predominant forcing.

Section 2 provides background on water exchange processes between the nearshore and the open ocean and explains different methods to quantify these exchange processes.

Section 3 describes the field experiment and observations. In section 4 we present a numerical model of the field site that has been used to test the sensitivity of flow reentrainment to variable hydrodynamic forcing and reef geometry in a simplified reef system.

Section 5 discusses the physical processes that change with the tested parameters and the impact of the re-entrainment rate on flushing time estimates.

Background: Quantifying water exchange rates

Water exchange between a coastal basin (e.g., an estuary or lagoon) and the surrounding open ocean is typically quantified in terms of a flushing time. This time scale is often estimated as the average time (Tf) it takes for the entire volume (V) of water inside the basin to be replaced by 'new' water and can be estimated as [START_REF] Fischer | Mixing in Inland and Coastal Waters[END_REF]:

𝑇 𝑓 = 𝑉 𝑄 ( 1
)
where Q is the flux of water into the basin. Importantly, Eq. ( 1) assumes that the outflow from an enclosed water body is not re-entrained once it exits the basin through defined boundaries [START_REF] Monsen | A comment on the use of flushing time, residence time, and age as transport time scales[END_REF].

For tidally-flushed systems, Eq. ( 1) has been extended to include a re-entrainment fraction b that accounts for water that exits the basin and subsequently re-enters the system at a later time [START_REF] Sanford | Model for Estimating Tidal Flushing of Small Embayments[END_REF]:

𝑇 𝑓 = 𝑉 (1 -𝑏)𝑄 (2)
In prior studies of reef systems, re-entrainment is usually assumed to be zero (b = 0 in Eq.

( 2)) [START_REF] Coronado | On the circulation in the Puerto Morelos fringing reef lagoon[END_REF][START_REF] Kench | Physical processes in an Indian Ocean atoll[END_REF][START_REF] Kraines | Rapid water exchange between the lagoon and the open ocean at Majuro Atoll due to wind, waves and tide[END_REF][START_REF] Lowe | Morphological constraints to wave-driven circulation in coastal reef-lagoon systems: A numerical study[END_REF][START_REF] Taebi | Nearshore circulation in a tropical fringing reef system[END_REF]. Such a flushing time will therefore represent the minimum time for flushing to occur. However, [START_REF] Herdman | Circulation, Residence Time and Retention in a Tropical Coral Reef[END_REF] found re-entrainment rates in a large-scale tropical coral reef system in Moorea (Polynesia), where circulation is driven by buoyancy and waves, to be 23-50% suggesting that re-entrainment would substantially increase the true flushing time.

Accurate estimates of b in Eq. ( 2) can be difficult to obtain in the field because flow re-entrainment can only be directly measured in a Lagrangian reference frame where flow pathways are tracked. To overcome this, it is common to evaluate re-entrainment using

Lagrangian approaches (such as GPS-tracked drifters or dye), which has been frequently applied in the study of rip currents along beaches [START_REF] Austin | Temporal observations of rip current circulation on a macro-tidal beach[END_REF][START_REF] Brown | Field Observations of Surf Zone-Inner Shelf Exchange on a Rip-Channeled Beach[END_REF][START_REF] Hally-Rosendahl | Cross-shore tracer exchange between the surfzone and inner-shelf[END_REF]MacMahan et al., 2010b;R Jak McCarroll et al., 2018;[START_REF] Reniers | Surf zone surface retention on a ripchanneled beach[END_REF][START_REF] Spydell | Observing surf-zone dispersion with drifters[END_REF]. These sandy beach environments are often characterized by having shore-parallel sand bars, over which water moves shoreward, and defined channels, through which water returns seaward [START_REF] Castelle | Rip current types, circulation and hazard[END_REF][START_REF] Dalrymple | Rip Currents[END_REF]. In these studies, the cross-shore exchange has typically been expressed in terms of an exit rate (or its opposite property, the retention rate). This exit (or retention) rate is defined by the number of drifters leaving (or remaining within) the surf zone indefinitely, relative to the total number of drifters initially seeded [START_REF] Reniers | Surf zone surface retention on a ripchanneled beach[END_REF]. Thus, drifters that recirculate and ultimately remain inside the surf zone are accounted for in this definition. In beach environments, the exit rate has been found to correlate with the so-called 'exit parameter', which is a function of the surf zone width and the incident wave forcing [START_REF] Reniers | Surf zone surface retention on a ripchanneled beach[END_REF]. This exit rate can be enhanced by geomorphic features such as headlands [START_REF] Castelle | Surf zone flushing on embayed beaches[END_REF]R. Jak McCarroll et al., 2014] and by intermittendly spaced rip channels and rip head shoals [START_REF] Castelle | Bathymetric control of surf zone retention on a rip-channelled beach[END_REF], but can be reduced due to the presence of alongshore currents and obliquely incident waves [START_REF] Spydell | The Suppression of Surfzone Cross-shore Mixing by Alongshore Currents[END_REF][START_REF] Winter | Rip currents under obliquely incident wind waves and tidal longshore currents[END_REF].

While the nearshore circulation patterns of barred beaches have some analogies to the flows in fringing reef systems, there can be a number of key differences, for example: (1) reef systems can have much larger spatial scales [see [START_REF] Falter | Physical and Biological Controls on the Carbonate Chemistry of Coral Reef Waters: Effects of Metabolism, Wave Forcing, Sea Level, and Geomorphology[END_REF], so that the impact of parameters that describe the reef geometry, such as channel spacing, may differ; (2) bed roughness over reefs [START_REF] Reidenbach | Boundary layer turbulence and flow structure over a fringing coral reef[END_REF] as well as bathymetry gradients are often much greater than over sandy bottoms; and (3) incident waves break on a forereef, which has typically a steeper slope than a sandy beach, so that the width of the surf zone can be much narrower and less variable [START_REF] Symonds | Two-dimensional surf beat: Long wave generation by a time-varying breakpoint[END_REF]. It remains unclear how these differences will affect ocean-reef water exchange rates.

3 Field Experiment

Site description

A two-week field study was conducted during the austral winter (May-June) of 2014 along a 600 m stretch of Garden Island in southwestern Australia (Figure 1a). The beach is fronted by several ~1 m deep limestone reefs, which is typical for much of the southwest coast of Australia. A shore-attached reef platform is located near the center of the site and is flanked to the south by another reef and by a deeper (2-3 m) lagoon (Figure 1c). To the north this group of reef platforms is bounded by a channel and to the south by a more pronounced and deeper channel, which is orientated in a southwesterly direction. The bathymetry of the region was surveyed in 2009 using aerial bathymetric LiDAR with 5 m horizontal resolution and ±0.45 m uncertainty in the vertical [Department of Transport Western Australia, 2009].

In addition, a detailed bathymetric survey was conducted during the experiment using a single beam echosounder and RTK-GPS system by small boat and a backpack mounted RTK-GPS system near the shoreline with an estimated uncertainty in the vertical of 0.1 m for surveys by boat [START_REF] Macmahan | Hydrographic surveying from a personal watercraft[END_REF] and 0.05 m by foot, respectively [START_REF] Barnard | Synthesis study of an erosion hot spot, Ocean Beach, California[END_REF].

The dimensions of the reef platforms and the lagoon are typical of many rocky coastlines as well as many nearshore fringing coral reefs, but smaller than those of some previously studied coral reef systems [e.g. [START_REF] Herdman | Circulation, Residence Time and Retention in a Tropical Coral Reef[END_REF][START_REF] Zhang | The combined influence of hydrodynamic forcing and calcification on the spatial distribution of alkalinity in a coral reef system[END_REF]. The effect of reef scale on flow re-entrainment is discussed in section 5.1.3. The southwest coast of Australia is a micro-tidal environment and during the field experiment the tidal range was between 0.3 and 0.6 m so that tidally induced currents are negligible. Alongshore currents outside the reef on the inner continental shelf (depth up to 20 m) are predominantly driven by a balance between local wind stresses and regional alongshore pressure gradients [START_REF] Zaker | Dynamics of the Coastal Boundary Layer off Perth, Western Australia[END_REF]. The currents can become particularly strong (up to 0.3 m s -1 ) during the summer months due to persistent southwesterly winds [START_REF] Gersbach | Upwelling on the south-west coast of Australia-source of the Capes Current?[END_REF]. Episodic fluctuations in the shelf-scale pressure gradients due to coastally trapped waves and eddies can enhance the variability, and episodically reverse the along-and cross-shore currents on the inner shelf [START_REF] Ruiz-Montoya | Summer circulation dynamics within the Perth coastal waters of southwestern Australia[END_REF]. Section 5.1.2 discusses the impact of variable alongshore currents on flow re-entrainment. The southwest coast of Australia receives the highest wave energy events in the austral winter months from May to September [START_REF] Bosserelle | Inter-annual variability and longer-term changes in the wave climate of Western Australia between 1970 and 2009[END_REF] in the period that the experiment took place.

Instrument Layout

Surf zone exit and re-entrainment rates were quantified using Lagrangian drifters, similar to the design by [START_REF] Schmidt | A GPS-tracked surf zone drifter[END_REF], over 8 days during the experiment. When floating freely, all but the uppermost ~5 cm of the drifter bodies were submerged. Based on a similar drifter design, wind drift was estimated at 1% of the wind speed measured 0.5 m above the water surface [START_REF] Schmidt | A GPS-tracked surf zone drifter[END_REF] and therefore is not expected to affect the drifters during this experiment (expected drift due to windage < 0.05 ms -1 ). Drifter positions were recorded at 10 Hz by a Qstarz BT-Q1000eX GPS logger placed in the top of each drifter. For each of the 8 deployment days, 12 to 15 drifters were released for two to four hours at a time (Table 1). Drifters were deployed in clusters, mostly inside the southern lagoon (8 of the 11 deployments), with the remaining deployments inshore of the northern channel (yellow stars in Figure 1c). This manuscript focuses primarily on the deployments in the southern lagoon because the lagoon connects to the better defined southern channel.

During each deployment, any drifters that beached or exited the reef-system were retrieved and redeployed at the initial release location. Data recorded by the GPS loggers were initially averaged in 1 Hz blocks to remove noise. Short wave motions were then removed by applying a moving average filter of 25 s so that only infragravity and slowly varying mean current motions were preserved in the recorded drifter tracks. The low-pass filtered positions were numerically-differentiated to determine the velocities of the drifters. The velocity and vorticity of all drifter observations was averaged over cells of 20 m by 20 m. In addition to the drifters, acoustic Doppler velocimeters and profilers measured velocities and wave conditions throughout the experiment at six sites within the reef-lagoon system (Table 2 andFigure 1c). For instruments in the shallow reef-lagoon system (depths <5 m), the velocities were low-pass filtered to remove fluctuations shorter than 15 minutes, and in the case of the velocity profilers, the velocities were depth-averaged. On the inner shelf (site A1 in 11 m water depth, Figure 1b), an Acoustic Wave and Current meter (AWAC) recorded offshore waves and currents. Drifter deployments were conducted at similar tidal water levels of 0.25 m above Australian Height Datum (AHD, approximately mean sea level)

during falling tide (Figure 2a). During the deployments the incident significant wave heights at A1 ranged from Hm0 = 1.0-1.8 m, peak periods from 12-18 s and directions from 260-270º

(Figure 2b-d). At sites A1 and A2 the hourly mean velocities were averaged over the uppermost 1.5 m of the water column, where they were usually strongest. At site A1, the surface currents varied between -0.1 m s -1 and 0.05 m s -1 during the deployments (Figure 2e).

W1

Young Ultrasonic Anemometer 85106 10 s average

Wind was measured on the dune near the salient with an ultrasonic anemometer (Young 85106) that sampled average wind speeds at 10 s intervals. This anemometer did not sample continuously due to power interruptions and missing data were filled in by wind speeds and directions sampled half-hourly 700 m inland from the field site [Bureau of Meterology Commonwealth of Australia, 2014]. In the overlapping periods, the locally measured wind data correlated well with the inland measurements (R 2 = 0.90 and R 2 = 0.94 for cross-shore and alongshore wind speeds, respectively). During the deployments, wind speeds varied between 2-5 m s -1 and came from a range of directions (Figure 2f). releases and the numbers from 1 to 8 denote the day of drifter deployment. See Figure 1 for the locations of sites A1 and W1.

Quantifying re-entrainment

To provide a quantitative measure of flow re-entrainment, we evaluated the exchange rate between lagoon and ocean waters using two different definitions of the exit rate, based on either excluding or including the effect of re-entrainment. Both these exit rates were calculated for the drifters deployed in the field as well as those simulated in the numerical model XBeach (Section 4). The first exit rate E1 is based on the commonly used minimum flushing time described by Eq. ( 1) and is defined as the ratio of the number of drifters that leave the reef-lagoon system in seaward direction (L) and the total number of drifters initially seeded (N), i.e.

𝐸 1 = 𝐿 𝑁 (3)
The offshore boundary of the reef-lagoon system is defined as the cross-shore position where the alongshore-averaged roller energy (as calculated in XBeach) exceeded 10% of the maximum roller dissipation in accordance with studies on rip-channeled beaches [e.g. [START_REF] Reniers | Surf zone surface retention on a ripchanneled beach[END_REF].

The second exit rate definition (E2) accounts for re-entrainment and thus corresponds to a more realistic estimate of the amount of flushing. The exit rate E2 is defined as the ratio of drifters retrieved outside the reef-lagoon system at the end of the deployment over the total number of drifters released within the reef system (N). The total number of drifters outside the reef-lagoon system is the difference between the number of drifters that left the lagoon seaward through the channel (L) and the number of drifters that return back to the reef-lagoon system (B):

𝐸 2 = 𝐿 -𝐵 𝑁 (4)
This definition is similar to the definition used in surf zone exchange studies on beaches [MacMahan et al., 2010a]. When the drifters were released, the number of drifters outside the reef-lagoon system (E2) typically increased and then fluctuated around a constant value (see, for example, Figure 4, Day 1 and 6). These fluctuations were related to groups of drifters simultaneously exiting and re-entering the lagoon. We note that this exit rate definition E2

does not distinguish between re-entrained drifters and drifters that remained inside the reeflagoon system, i.e. there is no difference between low exchange rates due to stagnant flow and due to energetic recirculating flow [similar to the residence time approach in [START_REF] Zhang | The combined influence of hydrodynamic forcing and calcification on the spatial distribution of alkalinity in a coral reef system[END_REF]. E2 is thus not a suitable measure to quantify the return flow regime of water that exits the channel.

A parameter that specifically describes the return flow regime is the flow reentrainment b (Eq. ( 2)), which is defined as the number of drifters re-entering the lagoon (B)

divided by the number of drifters that have exited the channel (𝐿), and was quantified by combining Eqs. ( 3) and ( 4):

𝑏 = 𝐵 𝐿 = 𝐸 1 -𝐸 2 𝐸 1 (5)
We note that water exiting the channel may be mixed with the surrounding ocean water through lateral dispersion and that b may also be smaller than estimated from Lagrangian drifter measurements or simulations. Numerical modelling indicates that sub-grid mixing using the [START_REF] Smagorinsky | General circulation experiments with the primitive equations[END_REF] model accounted for less than 5% in the momentum balance equation and was negligible. Hence, advection is the dominant process to transport material offshore or to re-entrain it back into the reef-lagoon system.

Field Observations

The drifter pathways displayed two main patterns over the course of the study: (i) complete drifter re-entrainment (Figure 3a typically left the reef system within an hour or less after their deployment inside the southern lagoon (see colors in Figure 3c andd). In contrast, re-entrained drifters remained within the reef lagoon system for two hours or more (see colors in Figure 3a andb). We classified the eight drifter deployments into predominantly exit or re-entrainment flow patterns. Day 1, 2, 3, 6 and 8 were characterized by (partial) drifter re-entrainment. Initially, all drifters were seeded inside the reef-lagoon system (green areas, Figure 4). Drifters then exited the reef system (yellow) and re-entered so that at any point in time a proportion of drifters was inside and the remaining drifters were outside the reef-lagoon system. On Day 1, drifters circulated in a wide eddy so that at any given time E2 ~ 50% of all drifters were offshore of the breaker zone (Figure 4). All of these drifters re-entered the lagoon and the re-entrainment rate b was 100%. On Day 2, drifters remained mostly onshore of the breaker line and thus E2 was low (10%). Drifters were retrieved gradually inside the lagoon after circulating numerous times, hence the gradual decrease of total drifters. On Day 3, a large number of drifters (E2 = 76%) exited the reef-lagoon system indefinitely and only a small number of drifters (b = 26%) recirculated. Similarly, on Day 6 only 46% of the drifters were re-entrained. Exiting drifters were retrieved and redeployed again inside the lagoon so that the total number of drifters floating freely fluctuated. On Day 4, 5 and 7, all drifters exited the reef system, were retrieved offshore and re-deployed all at once within the lagoon. On Day 8, part of the drifters left the reef system, mostly re-entered the lagoon and were retrieved both outside and inside the reef-lagoon system. In general, drifters were more likely to re-enter the lagoon when they floated alongside the channel edge where they were caught in an eddy (marked by the high vorticity in Figure 5a) than when they floated along the center of the channel in a south-westerly direction (Figure 5b). The re-entrainment pattern was dominant when wave heights were large, water levels were low and alongshore velocities were directed northward and thus favoring drifters returning over the reef to the north of the southern channel (Table 1).

Numerical Model

Model description

The general influence of reef geometry parameters and offshore hydrodynamic conditions on water exchange rates were further investigated using the numerical model XBeach configured in a short wave-group (surf beat) mode (refer to [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF] for details of the model). This model couples a module describing the wave-group varying wave energy with a non-linear shallow water wave module to describe wave-averaged flow, including wave setup, infragravity waves and wave-current interactions, and has been successfully used in other two-dimensional reef studies [START_REF] Van Dongeren | Numerical modeling of low-frequency wave dynamics over a fringing coral reef[END_REF] as well as numerical experiments on drifter retention on rip channeled beaches [START_REF] Castelle | Bathymetric control of surf zone retention on a rip-channelled beach[END_REF].

The analysis of the model output was conducted in two stages. Firstly, to gain confidence that the model is capable of accurately reproducing drifter exits and retention, we performed hindcast simulations of the drifter observations based on the bathymetry of the field site at Garden Island. The model was forced by JONSWAP spectra on the offshore boundary (~11 m depth) that represented the significant wave height, peak period, directional and frequency spreading of the frequency-direction variance density spectra measured at the offshore AWAC A1 (Figure 1b). Bathymetry within the model domain was derived from the aerial LiDAR and single beam echosounder surveys. Wave breaking was simulated using the dissipation formulation proposed by [START_REF] Roelvink | Dissipation in random wave groups incident on a beach[END_REF] with γ = 0.7, which is similar to previous reef hydrodynamic models [START_REF] Lowe | Morphological constraints to wave-driven circulation in coastal reef-lagoon systems: A numerical study[END_REF] and provided better results than other values of γ tested in this study within the typical range from 0.5 to 0.7 [START_REF] Van Dongeren | Numerical modeling of low-frequency wave dynamics over a fringing coral reef[END_REF]. The wave forcing that generates wave setup and currents within the flow module included radiation stresses due to short waves and wave rollers. Sub-grid turbulence was modelled using the [START_REF] Smagorinsky | General circulation experiments with the primitive equations[END_REF] turbulence closure model. The domain consisted of 190 grid cells in the alongshore by 103 cells in the cross-shore with a resolution of 6 x 6 m in the areas of interest over the reef and in the lagoon. Each simulation was allowed one-hour spin-up time, which was adequate to establish a steady state wave-driven reef circulation, and the subsequent hourly output was used to calculate time-averaged flow and wave quantities.

For computational efficiency, the hindcasts were performed for specific one-hour periods each day coinciding to daily maximum and minimum water levels (diurnal high and low tide)

as well as intermediate tide stages. This resulted in 64 hindcast simulations (four per day), which were used to assess the model performance for variable wave and water level conditions.

Bottom friction was modeled as [START_REF] Feddersen | Velocity moments in alongshore bottom stress parameterizations[END_REF][START_REF] Ruessink | Modeling the alongshore current on barred beaches[END_REF]:

𝜏 𝑏 = 𝑐 𝑓 𝜌√(1.16𝑢 𝑟𝑚𝑠 ) 2 + (𝑢 𝐸 + 𝑣 𝐸 ) 2 (6)
where 𝜏 𝑏 is the bottom friction, cf is the depth-independent friction coefficient, ρ the water density, 𝑢 𝑟𝑚𝑠 the root-mean-squared orbital wave velocity and 𝑢 𝐸 and 𝑣 𝐸 are the depthaveraged cross-shore and alongshore velocities. Within the sandy areas (i.e. lagoon, channels and offshore), we assumed that bed friction was comparable to typical values reported at beaches using an equivalent friction formulation [START_REF] Feddersen | Velocity moments in alongshore bottom stress parameterizations[END_REF] and set the bed friction coefficient to cf = 0.003 [e.g. [START_REF] Feddersen | Alongshore momentum balances in the nearshore[END_REF]. Based on initial model testing we calibrated the bed friction over the reef areas (darker shades in Figure 3) to cf = 0.01, which is lower than values reported for coral reefs using an equivalent friction parameterization (values have typically been found to range from 0.009-0.027 over reefs; see Table 1 in [START_REF] Rosman | A framework for understanding drag parameterizations for coral reefs[END_REF]). This lower friction coefficient is conceivable for a weathered limestone reef with algae cover. Wind stresses were applied in the model based on the measured wind velocities and a quadratic wind drag coefficient of CD = 0.002 [START_REF] Large | Radiation stress and mass transport in gravity waves, with application to 'surf beats[END_REF]. However, including wind forcing terms did not affect the model skill even during the storm event, when the wind speed exceeded 12 m s -1 because the wind stress terms were an order of magnitude smaller than the dominant wave and pressure forcing terms.

Along-shelf currents offshore of the reef were simulated by imposing an alongshore pressure gradient to reproduce the weak, moderate or strong north-or southward directed currents observed in the surface layer at A1. In the second stage of the analysis, we simplified the bathymetry to isolate the impact of various hydrodynamic parameters and reef geometries on flow re-entrainment. For this analysis, the bathymetry at Garden Island was schematized with rectangular reefs that were intersected by regularly spaced channels (Figure 6). This geometry was similar to the reeffringed lagoon to the south of the platform, where most of the drifters were deployed. In a series of simulations, parameters of the idealized reef system (channel spacing and reef friction) and hydrodynamic forcing (offshore wave height and along-shelf current) were covaried, while other parameters were kept constant. The model reef was 150 m wide in the cross-shore and was located 150 m from the shoreline. The lagoon and channel were 3 m deep and connected to a beach face with a concave profile 𝑧 = 𝐴𝑥 2/3 [START_REF] Dean | Equilibrium beach profiles: US Atlantic and Gulf coasts[END_REF][START_REF] González | Equilibrium beach profile model for perched beaches[END_REF], where x is the cross-shore coordinate, and A was set to 0.15. All simulations were forced at the offshore boundary by waves comprised of a JONSWAP spectrum of normally incident waves with peak period Tp = 14 s (the average observed during the experiment at Garden Island), a peak enhancement factor of γjonswap = 3.3 and a directional spreading of σ = 25º. Simulations with the same wave height were forced by identical time series. After a one-hour spin-up time, 2121 uniformly spaced drifters were seeded as passive particles inside the reef-lagoon system and their tracks were simulated for three hours using Lagrangian particle tracking.

The reef parameters (channel spacing and reef friction) and hydrodynamic forcing (offshore wave height and along-shelf current) were varied independently according to Table 3, resulting in a total of 320 simulations. The choice of hydrodynamic parameters was motivated by the field observations (see section 3.4) and the reef parameters were chosen based on initial parameter tests and also varied to cover a range of typical fringing reef dimensions with channel spacing from 300 m to 700 m [START_REF] Falter | Physical and Biological Controls on the Carbonate Chemistry of Coral Reef Waters: Effects of Metabolism, Wave Forcing, Sea Level, and Geomorphology[END_REF] and roughness values, which have been previously reported to range from cf = 0.009 to 0.027 [START_REF] Rosman | A framework for understanding drag parameterizations for coral reefs[END_REF]. To determine the exit and re-entrainment rates the modelled drifters were first allowed to propagate for an hour through the reef-lagoon system. As drifters exited and re-entered the lagoon in groups, the number of drifters offshore from the reef fluctuated. To eliminate the effect of these fluctuations, the exit rate E2 was averaged over the last two hours of each simulation. This time was sufficient for drifters to finish at least one circulation cycle even in large lagoons. This approach is similar to the time averaged values used by Castelle et al.

[2014], [START_REF] Castelle | Surf zone flushing on embayed beaches[END_REF] and [START_REF] Reniers | Surf zone surface retention on a ripchanneled beach[END_REF].

Hindcast Model Results

The overall model performance was quantified in terms of the root mean square error (RMSE), bias and the Willmott skill (WS) [START_REF] Willmott | Statistics for the Evaluation and Comparison of Models[END_REF], which are defined as:

𝑅𝑀𝑆𝐸 = ∑ √|𝑋 𝑚𝑜𝑑 -𝑋 𝑜𝑏𝑠 | 2 𝑁 𝑖=1 (7) 𝑏𝑖𝑎𝑠 = ∑ 𝑋 𝑚𝑜𝑑 -𝑋 𝑜𝑏𝑠 𝑁 𝑖=1 (8) 𝑊𝑆 = 1 - ∑ |𝑋 𝑚𝑜𝑑 -𝑋 𝑜𝑏𝑠 | 2 𝑁 𝑖=1 ∑ (|𝑋 𝑚𝑜𝑑 -𝑋 ̅ 𝑜𝑏𝑠 | + |𝑋 𝑜𝑏𝑠 -𝑋 ̅ 𝑜𝑏𝑠 |) 2 𝑁 𝑖=1 (9)
where Xmod and Xobs are the modelled and observed values (here sea-swell wave height Hm0, umean and vmean), the overbar indicates time averaging of these values and N is the number of samples (N = 64, four simulations per day).

At sites inside the lagoon (S7), offshore from the reef (R7) and on the reef platform (R2), the wave heights were accurately predicted over the duration of the experiment (RMSE = 0.08 -0.13 m, bias = -0.07 -0.12 m and WS = 0.82 -0.92, Table 2). However, in the channel at CS2 and CS4 the model overestimated wave heights consistently by 20%. This may be due to visually observed bathymetric features inside the channel, which are smaller than the resolution of the LiDAR bathymetry (~5 m), and thus not resolved in the model.

Further, wave diffraction may play a role at this site, which is not in the model. The alongshore velocities in the channel at CS2 and CS4 were consistently overestimated by ~0.1 m s -1 but followed the trends in the observations well. This resulted in a flow that was directed slightly more alongshore in the model, despite the total velocity magnitude being predicted well (not shown). Overall, the RMSE for the modelled velocities ranged between 0.05 and 0.13 m s -1 , the bias between -0.11 and 0.12 m s -1 and the WS was between 0.78 and 0.95 (Table 2).

The model results were predominantly compared spatially to the observed drifter tracks as for the objectives of this study the spatial patterns were most important. The model produced a circulation cell to the north of the channel, which is in agreement with the observed drifter re-entrainment (Figure 3a,b). However, the modelled re-entrainment cell had a somewhat wider radius than in the field observations, which is most likely due to the combination of bathymetry and model resolution not being able to capture the very steep bed level gradient from the channel to the reef resulting in a more alongshore directed flow. The model also reproduced the southwesterly trajectory of flow exiting the southern channel (Figure 3c,d). Overall, the model was able to replicate the flow magnitudes, directions and patterns with reasonable accuracy and was able to qualitatively reproduce the observed flow patterns.

Effect of hydrodynamic and reef geometry parameters

The primary goal of the numerical model was to identify the parameters and mechanisms that control flow re-entrainment. For each simulation with varying reef geometry and hydrodynamic forcing the exit rates E1 and E2 (Eq. ( 3) and (4), respectively) and reentrainment b (Eq. ( 5)) were calculated. The mean offshore and onshore velocity across all reef channels and reef structures, respectively, were determined at the cross-shore location where the offshore current in the channel was maximum. This location was typically just offshore from the channel exit. The drifter re-entrainment b increased (exit rate E2 decreased) with increasing wave height (Figure 7a) and increasing alongshore current outside the reef (Figure 7b). Large waves caused the maximum offshore flow velocity in the channel and the onshore flow velocity over the reef to increase (Figure 7a) while the alongshore velocity outside the reef had no impact on the cross-shore velocities (Figure 7b). With large channel spacing reentrainment b decreased (exit rate E2 increased) while the onshore velocities decreased and the offshore velocities increased (Figure 7c). Large bed roughness is a characteristic feature of many (or most) reefs that distinguishes these systems from analogous rip-channeled beach environments. Large reef friction equally reduced the onshore velocities over the reef and the offshore velocities in the channel. Drifters were thus neither more likely to exit offshore nor to return onshore over the reef and the re-entrainment of the drifters remained unchanged (Figure 7d).

Discussion

The dynamics of wave-driven flows in reef environments and ocean-reef exchange have been either studied in a Eulerian reference frame that quantifies the incoming water flux across the reef platform and the offshore flux exiting the reef channels [e.g. [START_REF] Hench | Episodic circulation and exchange in a wave-driven coral reef and lagoon system[END_REF][START_REF] Hoeke | Hydrodynamics of a bathymetrically complex fringing coral reef embayment: Wave climate, in situ observations, and wave prediction[END_REF][START_REF] Lowe | Wave-Driven Circulation of a Coastal Reef-Lagoon System[END_REF][START_REF] Taebi | Nearshore circulation in a tropical fringing reef system[END_REF] or by tracking simulated particles [START_REF] Zhang | The combined influence of hydrodynamic forcing and calcification on the spatial distribution of alkalinity in a coral reef system[END_REF]. These approaches have not allowed to identify the governing drivers of reef exchange processes. For estuarine exchange processes, the drivers of recirculation have been identified to be tidal residual currents and baroclinically driven currents [see Geyer and MacCready, 2014, for a review]. This study provides insight into the processes related to reef geometry and forcing conditions that control the proportion of water that recirculated in a reef system.

Field observations suggest that re-entrainment can vary substantially due to variations in hydrodynamic forcing conditions alone (incident wave heights and along-shelf currents). This is consistent with the numerical model that assessed the effect of a range of reef geometries along with a larger range of variable hydrodynamic conditions, so that reentrainment varied between 7% and 73%. Importantly, these results demonstrate that the definition of the exit rate E1 (Eq. ( 3)), which ignores the effect of re-entrainment and counts all drifters that flow seaward through the channel as exiting, was consistently higher over the range of tested parameters. This definition is analogous to flushing time estimates commonly used with Eulerian velocity measurements obtained using fixed (moored) instruments [e.g. [START_REF] Taebi | Nearshore circulation in a tropical fringing reef system[END_REF]. Our results suggest that flushing time estimates purely based on the offshore velocity are not always representative because they discount the effect of flow reentrainment. In the following, we investigate the effect of a number of hydrodynamic and geometric parameters on reef re-entrainment to determine the conditions and environments in which flow re-entrainment is an important process to consider.

Influence of hydrodynamic and reef geometry parameters on flow dynamics

To further investigate how the physical mechanisms responsible for re-entrainment are controlled by reef geometry parameters and hydrodynamic forcing conditions, we first assessed the momentum balances (Appendix A) across the reef-lagoon system for the default simulation that was forced with 1.5 m waves. The channels were spaced 400 m apart and no alongshore current was imposed outside the reef (Table 3). In this case, 71% (E1) of all seeded drifters floated offshore through the channel and 43% (E2) remained outside of the reef-lagoon system. Of the drifters that floated offshore 40% (b) were re-entrained back into the reef system. The wave forcing induced by wave breaking on the forereef and the reef platform (Figure 8a) was balanced mostly by cross-shore pressure gradients (Figure 8b, see also, e.g. [START_REF] Symonds | Wave-driven flow over shallow reefs[END_REF][START_REF] Taebi | A numerical study of the dynamics of the wave-driven circulation within a fringing reef system[END_REF]. The net forcing, which we refer to as the sum of the offshore directed pressure gradient and onshore directed wave forcing, is directed onshore over the reef platform (Figure 8c), where it drives the cross-reef current that contributes to a relevant bottom shear stress over shallow and rough reefs (Figure 8e). Near the channels, the net forcing is balanced by advection (Figure 8d) and generates counterrotating eddies (Figure 8f) that, together with the onshore flow over the reef, were responsible for re-entrainment (see also Figure 5). The following sections assess how this momentum balance changes, particularly the strength of the net forcing term, when hydrodynamic forcing or reef geometry are altered.

Wave height

In both the field observations and the numerical model, the re-entrainment increased with larger offshore wave heights. Larger waves strengthen the net forcing responsible for the wave-driven flows (i.e., the difference between the wave forcing and pressure gradient) just offshore from the reef (x = 1050m, Figure 9a) and on the parts of the reef near the channel from where the offshore flow is primarily fed [START_REF] Svendsen | Analysis of Rip Current Systems[END_REF]. This net forcing drives a stronger onshore flow just outside the reef and a stronger offshore flow in the channel (Figure 7a), which also enhances the strength of the channel vortices (Figure 9b) that ultimately re-entrain water back into the reef-lagoon system. Therefore, although larger waves drive a stronger seaward flow out the channel (Figure 7a), which was also observed at the field site (r 2 = 0.89 between offshore waves and seaward flow velocities) and other reef sites [e.g. [START_REF] Lowe | Wave-Driven Circulation of a Coastal Reef-Lagoon System[END_REF], larger waves also enhance flow re-entrainment. The increase of drifter re-entrainment with increasing wave height causes the exit rate E2 to decrease, which is consistent with field observations and modeling studies on ripchanneled beaches [START_REF] Castelle | Bathymetric control of surf zone retention on a rip-channelled beach[END_REF][START_REF] Scott | Controls on macrotidal rip current circulation and hazard[END_REF]. These studies have linked the decrease in exit rate to: (1) increased onshore flow over the sand bars, which transports drifters back towards shore [START_REF] Scott | Controls on macrotidal rip current circulation and hazard[END_REF], and to (2) wider surf zones [START_REF] Castelle | Bathymetric control of surf zone retention on a rip-channelled beach[END_REF][START_REF] Reniers | Surf zone surface retention on a ripchanneled beach[END_REF]. However, the surf zone width depends largely on the forereef slope and varies less on steep forereefs. Here, we attribute the increased flow-re-entrainment to the stronger onshore mass flux over the reef. An increase of re-entrainment with larger waves implies that large storm waves may be increasingly less efficient in flushing costal reef systems.

Alongshore current

In the presence of an inner-shelf alongshore current, the offshore current in the channel is redirected towards the downdrift direction (Figure 10a). The cross-shore current locally blocks the alongshore current resulting in a local increase in the water level updrift of the channel and conversely a local decrease in the water level downdrift of the channel (Figure 10). The modified pressure gradient, and thus the net forcing, are redirected towards the downstream reef (Figure 10). This allows water to be transported towards the reef and then back shoreward by the wave-driven cross-reef flow such that the re-entrainment b increases and the exit rate E2 decreases. This is consistent with the field observations at Garden Island where drifters returned via the reef to the north of the channel when the alongshore current outside the reef was directed northward. [START_REF] Herdman | Circulation, Residence Time and Retention in a Tropical Coral Reef[END_REF] observed similar dynamics at a large-scale coral reef where it was found that more drifters returned over a downdrift reef in the presence of an increasing alongshore current. Similar observations have been made on rip-channeled beaches where drifter re-entrainment increased in the presence of alongshore currents associated with tidal flows [START_REF] Winter | Rip currents under obliquely incident wind waves and tidal longshore currents[END_REF].

Channel spacing

As the channel spacing increases, the ratio of channel width to alongshore reef width decreases. To balance the volume of water flowing across the reef into the lagoon with the volume of water flowing out through the channel, the seaward velocity in the channel increases for larger channel spacing while the maximum onshore velocity over the reef decreases (Figure 7c). Hence, water is advected further offshore rather than back towards the reef. Drifter retention also decreases with increasing channel spacing on rip-channeled beaches [START_REF] Castelle | Bathymetric control of surf zone retention on a rip-channelled beach[END_REF]. However, this study shows that this effect weakens with larger channel spacing until a limit is reached where re-entrainment becomes approximately constant (i.e. in larger-scale reef systems, re-entrainment becomes insensitive to the channel spacing). Within the parameter space that we tested, this limit was ~500 m. Our results are consistent with observations of less re-entrainment in large reef systems at Moorea, where reef channels are 4 to 5 km apart and re-entrainment was less than 50% [START_REF] Herdman | Circulation, Residence Time and Retention in a Tropical Coral Reef[END_REF].

Thus, small-scale reef systems appear to promote large re-entrainment rates.

5.2 A predictor for re-entrainment and its implications for flushing times

Based on the sensitivity of re-entrainment b to the tested reef geometry and hydrodynamic parameters, we define a re-entrainment predictor variable R as:

𝑅 = 𝐻 𝑚0 ℎ 𝑟𝑒𝑒𝑓 𝑊 𝑐 𝜆 (10)
Here the ratio of the offshore significant wave height to the water depth over the reef (𝐻 𝑚0 /ℎ 𝑟𝑒𝑒𝑓 ) provides an indication of the intensity of wave dissipation on the reef; whereas the ratio of the channel width to the channel spacing (𝑊 𝑐 /𝜆) provides a measure for the relative cross-sectional areas available for onshore and offshore mass transport. Both of these ratios affect the balance between onshore flow velocity over the reef and offshore flow velocity inside the channel. Stronger onshore flow over the reef favors drifter-re-entrainment and stronger offshore flow transports drifters further offshore from where they are less likely to be re-entrained. The alongshore current is considered separately because the reentrainment mechanism is fundamentally different (see section 5.1). An alongshore current does not affect the cross-shore flow velocities but enhances flow re-entrainment because it transports drifters towards an area of onshore flow.

We found that the exit rate E2 decreases with R (r 2 = 0.70 in the absence of alongshore To demonstrate the implications of the large variability in re-entrainment (b) for reef flushing times, we calculated the flushing time with re-entrainment (Eq. ( 2)) and without reentrainment (Eq. ( 1)). In many simulations, the inclusion of b in the estimate substantially increased the flushing time compared to an estimate without re-entrainment: for 58% of the simulations the flushing time increased by more than a factor of 1.5 and in 17% of the simulations by more than two. The largest increase (greater than three times) was observed for the simulations with large R (a combination of short channel spacing λ = 300 m and wave heights of Hm0 > 2 m) and strong alongshore currents. In contrast, there was little difference between the two flushing times in simulations with small R (channel spacing λ > 500 m and Hm0 = 1 m) and no alongshore current.

Conclusions

This study examined the processes responsible for re-entrainment in wave-dominated The present study demonstrates that re-entrainment rates can be large and highly variable among reefs, and thus should be included when assessing flushing times and material exchange of reef systems with the surrounding ocean. High re-entrainment rates may limit the ability of a reef to exchange material with both the surrounding ocean and other reef systems, and thus further emphasizes the need to accurately quantify flow re-entrainment to make robust estimates of, for example, larval dispersal, reef connectivity [START_REF] Cowen | Connectivity of Marine Populations: Open or Closed?[END_REF][START_REF] Teske | Oceanography promotes self-recruitment in a planktonic larval disperser[END_REF] and reef water quality [START_REF] Falter | Physical and Biological Controls on the Carbonate Chemistry of Coral Reef Waters: Effects of Metabolism, Wave Forcing, Sea Level, and Geomorphology[END_REF][START_REF] Lowe | Oceanic Forcing of Coral Reefs[END_REF]. Flow reentrainment is generally important for reefs that are frequently intercepted by channels and reefs that are exposed to large waves and strong alongshore currents outside the reef.

Figure 1 .

 1 Figure 1. a) Location of the Garden Island study site within southwestern Australia, b) aerial image (Google Earth) of the field site and the locations of the Acoustic Wave and Current meters (AWAC) A1 and A2, c) locations of velocity profilers (red dots), velocimeters (blue

Figure 2 .

 2 Figure 2. Site A1: a) Tidal water level relative to AHD, b) significant wave height at, c) peak

  and b), (ii) complete drifter ejection offshore (Figure3c and d), and a combination of pattern (i) and (ii).

Figure 3 .

 3 Figure 3. Examples of observed drifter trajectories that become re-entrained on a) Day 1 and

Figure 4 .

 4 Figure 4. Number of drifters inside (green) and outside (yellow) the reef-lagoon system as a function of time for each day. Day 1, 2, 3, 6 and 8 were characterized by (partial) drifter reentrainment and Day 4, 5 and 7 by drifter exits.

Figure 5 .

 5 Figure 5. Drifter velocity (black arrows) and vorticity (colours) averaged over all drifter deployments, in which retention dominated (left) and in which exits dominated (right) interpolated on a 20 m x 20 m grid. Red (blue) circles in the left panel indicate eddies rotating in clockwise (anti-clockwise) direction. In the right panel, information on drifter velocity is mostly absent along the channel edges because drifters generally floated along the centre of the channel in exit dominated flow regimes.

Figure 6 .

 6 Figure 6. Schematized reef dimensions: a) top view and b) cross-sectional view. The

Figure 7 .

 7 Figure 7. Re-entrainment b, exit rate E2, average offshore velocity in the channel and average onshore velocity at the reef edge as functions of variations in a) wave height, b) alongshore current outside the reef, c) channel spacing and d) lagoon length. The circle marks the mean value of all simulations and the black bar indicates the range in which 50% of the simulation

Figure 8 .

 8 Figure 8. Simulated magnitude and direction of the momentum terms (cross-and alongshore, see Eq. A1-2) for the default case. From top left to bottom right, the contributions from: a) wave forcing, b) pressure gradient, c) sum of wave forcing and pressure gradient, d) nonlinear advection, e) bottom shear stress and f) mean velocity with Stokes drift (vectors) and vorticity (colors). Viscosity terms were negligible and are not shown. Black contour lines indicate the -2.5 m and -1 m isobaths. For clarity only every third vector is shown in cross-

Figure 9

 9 Figure 9. a) Modeled net forcing as the sum of the pressure gradient and wave forcing for a

Figure 10

 10 Figure 10. a) Mean velocity field for a simulation with a strong alongshore current (v0 = 0.15 ms -1 ). b) Net forcing as the sum of the pressure gradient and wave forcing for a simulation with strong alongshore current (v0 = 0.15 ms -1 , b = 62%, red arrows) compared to

  currents) following a negative reciprocal function while re-entrainment (b) increases with increasing values of R and plateaus with high values of R following a positive reciprocal function. In the absence of an alongshore current, b and the reciprocal of parameter R are correlated (r 2 = 0.73). With increasing alongshore current outside the reef the correlationbetween R and b reduces, i.e. when the current outside the reef is 0.15 m s -1 , the correlation coefficient drops to r 2 = 0.25 and re-entrainment is consistently higher across all values of R (Figure11). The alongshore current outside the reef then dominates the re-entrainment process. For large values of R (>0.4), re-entrainment is less sensitive to the alongshore current outside the reef and the re-entrainment rates converge for all tested alongshore current magnitudes. To summarize, R and the alongshore current both increase re-entrainment, but when the alongshore current is strong it reduces the sensitivity to variations of R and vice versa.

Figure 11 .

 11 Figure 11. Exit rate E2 (a) and re-entrainment b (b) versus the re-entrainment prediction parameter R for all simulations. The different marker colors denote variable alongshore currents outside the reef.

  reef systems. Drifter observations in a wave-dominated rocky limestone reef in southwestern Australia illustrated two distinct flow patterns of either complete drifter ejection or drifter reentrainment. These observations motivated an idealized numerical study that investigated the effect of reef channel spacing and reef roughness along with offshore wave height and alongshore currents on flow re-entrainment. The model demonstrated that large waves and strong alongshore currents outside the reef enhance flow re-entrainment. The model results further showed that large reef channel spacing reduces flow re-entrainment, while the reef roughness had no effect. A single re-entrainment prediction parameter R is proposed, which incorporates the effect of wave forcing and reef geometry. For large values of R (i.e. large offshore wave height and small channel spacing) and strong alongshore currents outside the reef flow re-entrainment is important to consider in reef flushing time estimates. Large wave heights can dominate the re-entrainment mechanism and reduce the positive effect of an alongshore current outside the reef and vice versa.

Table 1

 1

	Day	Number of drifters	Duration (hours)	Hm0 [m]	Tp [s]	θp [ºN]	Water depth on the reef [m]	Along-shelf current [m s -1 ]	Return flow regime in channel the southern	Deployment location
	1	12	3:40 1.66 14 266	1.00	0.04	Re-entrainment	Southern lagoon
	2	15	4:24 1.49 13 265	0.99	0.05	Re-entrainment	Southern lagoon
										Southern
	3	15	2:28 1.48 14 262	0.96	-0.02 Combination	lagoon and northern
										channel
	4	15	1:46 1.24 14 268	1.21	-0.06 Exiting	Southern lagoon
										Southern
	5	15	2:56 1.31 13 261	1.21	-0.1 Exiting	lagoon and northern
										channel
	6	15	3:58 1.13 12 261	1.11	-0.05	Re-entrainment	Southern lagoon
	7	15	3:22 1.24 18 267	1.09	-0.05 Exiting	Southern lagoon
	8	14	2:47 1.40 15 266	1.06	-0.02	Re-entrainment	Southern lagoon and

. Drifter deployments: Number of drifters deployed on each day, total duration of each drifter deployment, offshore wave conditions, observed return flow regime (only in the southern channel) and deployment location.

Table 2

 2

					Hm0	u	v
	Site	Instrumen t	Depth	Sampling configurations	(RMSE, bias,	(RMSE, bias,	(RMSE, bias,
					WS)	WS)	WS)
	Offshore
	A1	Nortek AWAC	10.5 m	Pressure and surface velocities in hourly bursts of 2048 s at 2 Hz, velocity profile hourly in 0.5 m bins
	Within the reef-lagoon system
		Nortek		Pressure and	0.42 m,	0.02 m s -1 ,	0.11 m s -1 ,
	CS2	ADP High Resolutio	3.1 m	velocity profile at 1 Hz continuous in	0.42 m, 0.58	0.00 m s -1 , 0.95	-0.11 m s -1 , 0.86
		n (HR)		0.05 m bins
	CS4	RDI ADCP	5.4 m	Pressure and velocity profile at 1	0.35 m, 0.35 m, 0.64	0.06 m s -1 , 0.05 m s -1 , 0.90	0.11 m s -1 , -0.11 m s -1 , 0.88

. Instruments and sampling configuration. "A" denotes offshore instruments, "CS" instruments in the southern respectively, "R" instrument on the reef platform and "S" south of it.

Table 3 .

 3 Parameters (see also Figure6) and parameter values used in the sensitivity test.

	Parameter values for the default case are highlighted in bold.
	Parameter	Values
	Variable
	Channel spacing λ	300, 400, 500, 600, 700 m
	Reef friction cf,R	0.003, 0.01, 0.03, 0.1
	Significant offshore wave height Hm0	1.0, 1.5, 2.0, 2.5 m
	Alongshelf current	0, 0.05, 0.10, 0.15 m s -1

  𝜌 is the water density, 𝜂 is setup, h the still water depth, u and v are the cross-and alongshore velocities, 𝜐 𝐻 the horizontal viscosity and 𝜏 𝑥 𝑏 and 𝜏 𝑦 𝑏 are the bottom shear stresses in cross-and alongshore direction. The wave forcing is calculated from the radiation stress gradients 𝑆 𝑖𝑗 as follows:

	𝐹 𝑥 𝜌	-𝑔(𝜂 + ℎ)	𝜕𝜂 𝜕𝑥	-(𝜂 + ℎ) (𝑢	𝜕𝑢 𝜕𝑥	+ 𝑣	𝜕𝑢 𝜕𝑦	) -𝜐 𝐻 (𝜂 + ℎ) (	𝜕 2 𝑢 𝜕𝑥 2 +	𝜕 2 𝑢 𝜕𝑦 2 )
											(A1)
			-	𝑏 𝜏 𝑥 𝜌	= 0
	𝐹 𝑦 𝜌	-𝑔(𝜂 + ℎ)	𝜕𝜂 𝜕𝑦	-(𝜂 + ℎ) (𝑢	𝜕𝑣 𝜕𝑥	+ 𝑣	𝜕𝑣 𝜕𝑦	) -𝜐 𝐻 (𝜂 + ℎ) ( 𝜕 2 𝑣 𝜕𝑥 2 +	𝜕 2 𝑣 𝜕𝑦 2 )
											(A2)
			-	𝑏 𝜏 𝑦 𝜌	= 0
	where 𝐹 𝑥 = -( 𝜕(𝑆 𝑥𝑥 + 𝑆 𝑥𝑥,𝑟𝑜𝑙𝑙𝑒𝑟 ) 𝜕𝑥	+	𝜕(𝑆 𝑥𝑦 + 𝑆 𝑥𝑦,𝑟𝑜𝑙𝑙𝑒𝑟 ) 𝜕𝑦	)	(A3)
		𝐹 𝑦 = -(	𝜕(𝑆 𝑦𝑥 + 𝑆 𝑦𝑥,𝑟𝑜𝑙𝑙𝑒𝑟 ) 𝜕𝑥	+	𝜕(𝑆 𝑦𝑦 + 𝑆 𝑦𝑦,𝑟𝑜𝑙𝑙𝑒𝑟 ) 𝜕𝑦	)	(A4)

Acknowledgements

We are grateful to Andrew Pomeroy, Mike Cuttler, Anton Kuret, Laura Segura and Mark Buckley for assistance with the instrument deployment and recovery. Funding for this project was provided by an ARC Future Fellowship (FT110100201) and ARC Discovery Project (DP140102026) to RJL. This work was undertaken by GW as part of a PhD at The University of Western Australia and was funded by an International Postgraduate Research Scholarship, an Australian Postgraduate Award and an EA and CH Jenkins scholarship. GW also acknowledges a Sue Baker Convocation Travel Award, which allowed her to visit the University of Bordeaux, where a large part of this manuscript was developed. BC was financially supported by projects DECA (INSU/EC2CO-DRIL) and CHIPO (grant number ANR-14-ASTR-0004-01) supported by the Agence Nationale de la Recherche (ANR). Funding for RM was provided by the "Hydroand morphodynamics during extreme events" research program of Deltares. The numerical modeling work was supported by resources provided by The Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. The data are available under http://doi.org/10.5281/zenodo.1292064.

Appendix A. Momentum equations

Output from the idealized XBeach simulations was used to assess the relative importance of the terms of the steady (d/dt = 0) depth and time (wave) averaged momentum balances in cross (x)-and alongshore (y) direction including the wave (radiation stress) forcing, 𝐹 𝑥 and 𝐹 𝑦 , as source term and pressure gradients, advection, turbulent mixing, and bottom shear stress, as sink terms: