Atoll Lagoon Flushing Forced by Waves ⁎ David P

Home , Atoll, Coastal engineering, Manihiki, Rakahanga, Swash

Coastal Engineering 53 (2006) 691–704 www.elsevier.com/locate/coastaleng

Atoll lagoon flushing forced by waves ⁎ David P. Callaghan a, , Peter Nielsen a, Nick Cartwright b, Michael R. Gourlay a, Tom E. Baldock a

a Division of Civil Engineering, The University of Queensland, Australia b School of Engineering, Griffith University, Australia

Received 22 August 2005; received in revised form 22 December 2005; accepted 17 February 2006 Available online 19 April 2006

Abstract

Water level and current measurements from two virtually enclosed South Pacific atolls, Manihiki and Rakahanga, support a new lagoon flushing mechanism which is driven by waves and modulated by the ocean tide for virtually enclosed atolls. This is evident because the lagoon water level remains above the ocean at all tidal phases (i.e., ruling out tidal flushing) and because the average lagoon water level rises significantly during periods with large waves. Hence, we develop a model by which the lagoons are flushed by waves pumping of ocean water into the lagoon and gravity draining water from the lagoon over the reef rim. That is, the waves on the exposed side push water into the lagoon during most of the tidal cycle while water leaves the lagoon on the protected side for most of the tidal cycle. This wave-driven through flow flushing is shown to be more efficient than alternating tidal flushing with respect to water renewal. Improved water quality should therefore be sought through enhancement of the natural wave pumping rather than by blasting deep channels which would change the system to an alternating tide-driven one. © 2006 Elsevier B.V. All rights reserved.

Keywords: Atoll lagoon flushing; Hydrodynamics; Water quality; Wave pumping; Modelling; Wave set-up; Pearl farming

1. Introduction several wide reef flats which are elevated above MSL (mean sea level). Consequently, the lagoons are virtually disconnected There are two natural sources of power for flushing atoll from the ocean (Solomon, 1997). This configuration is quite lagoons: tides and waves. Their relative importance depends on different from the other atolls, also located in the northern the topography of the atoll rim as well as on the local wave and group, which have several deep reef passes that allow the ocean tide climates. If the atoll has wide reef passes, which are deep tide to drive the lagoon flushing. However, similar to Penrhyn, compared with the wave height and the tidal range, the flushing Cook Islands, the lagoon is used to grow black lipped oysters will be generated mainly by the tide and the lagoon water level for their black pearls, which is the primary source of income for will oscillate within the range of the ocean tide as shown in Fig. the Manihiki populace (McKenzie, 2004). The water quality 1b. If, on the other hand, the atoll has an almost unbroken rim of within the lagoon is therefore of great importance to the living coral growing to a few decimetres above mean sea level profitability of this industry. To increase the pearl yield from the (MSL), the flushing will be driven by the waves as shown in lagoon, the number of oysters has been increased leading to Fig. 1a. That is, the side facing the largest waves will have large several episodic large scale oyster deaths from disease directly amounts of water pushed over the reef rim while water will linked to poor water quality (Sharma et al., 2001; McKenzie, drain to the ocean on the leeward side. The tide will modulate 2004). This reduction in water quality has also increased the this process to a degree that depends on Atide /H, i.e., the ratio number of shells rejecting the seed (nucleus for pearl between the tidal amplitude and the wave height, Fig. 1a. development) or dying after being seeded. One method to The Manihiki and Rakahanga atolls (Fig. 2) are located in the overcome these problems is improving water quality by northern group of the Cook Islands. Both atolls consist of increasing lagoon flushing. This should however be done in a way which is in harmony with the natural system. This paper ⁎ Corresponding author. Tel.: +61 7 3365 3914; fax: +61 7 3365 4599. aims to provide understanding of the natural system before E-mail address: [email protected] (D.P. Callaghan). proposing methods to enhance the flushing.

Fig. 1. Two different lagoon flushing processes possible in atoll lagoons exposed to waves (with larger waves on the exposed coast compared to the leeward coast) and ocean tide. a. The wave-driven lagoon flushing process present at Manihiki and Rakahanga, illustrated by the elevated lagoon water surface level and the persistent one-way flow across the atoll. b. The tidal flushing process that occurs when deep and wide channels connect the ocean and the lagoon. Tidal flushing typically generates alternating flow patterns in the lagoon with the lagoon water surface variations contained within the ocean tide variations. The alternating flow patterns associated with tidal flushing give less water renewal than the wave-driven through flows. Panels c. and d. show the tidal ranges and relative position for the ocean and lagoon under wave-driven and tidal-driven flushing respectively.

This paper demonstrates the mechanism of wave-driven models have been formulated which compare well with the field lagoon flushing occurring at Manihiki and Rakahanga atolls measurements. These models and the field measurements using observed lagoon dynamics. The processes observed were; demonstrate that as wave energy flux increases, the lagoon inflow driven by waves via the wave pump concept where the water level height above the ocean increases. This dynamic wave breaking lifts water onto the reef flats, well above the process is different to tidal flushing, where the lagoon ocean water level (Bruun and Viggoson, 1977; Nielsen et al., fluctuations are always contained within the ocean tide 1999, 2001), outflow controlled by critical flow conditions at variations. the leeward reef edge and gravity driving the flows across the This paper is arranged as follows. Section 2 describes the lagoon from the exposed to leeward coasts. Combining these field sites and measurements obtained during two field processes with the conservation of water volume, two new experiments. Sections 3 and 4 derive respectively an analytical

Fig. 2. a. Location of the Cook Islands in the South Pacific Ocean; b. Manihiki and c. Rakahanga atolls of the Cook Islands. The grey shading indicates; land ( ), reef flats ( ) and lagoon ( ). Under normal wave conditions, water enters the lagoon via the reef flats only. However, during Tropical Cyclone Martin (1997), very large waves from the west overtopped the land at Tauhunu Village (western atoll coastline). The predominant wave and swell directions for both atolls are from easterly directions (Thompson, 1986). D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704 693 and a numerical model. The wave flushing process is discussed and 40% of infragravity wave motion (periods between 25 and in Section 5 as it influences Manihiki and similar atolls. Final 250 s is recorded). To avoid aliasing when measuring at conclusions are communicated in Section 6. locations exposed to infragravity motion, five readings at 1-min intervals were taken and averaged to represent the mean water 2. Field data level at that time. During all field experiments, manual readings were taken throughout the collection period at between 10- and The ocean and lagoon water surface levels were obtained 30-min intervals between 7 am and 10 pm. This ensured that using co-located damped stilling wells similar to those water level measurements were obtained throughout the field described by Nielsen (1999) with time constants (τ)ofca100 experiment and avoided data loss due to instrument failure. s and self logging pressure transducers (in situ MiniTROLLs The stilling wells were installed to measure both ocean and advance). From Nielsen (1999) and laboratory testing, the lagoon water surface levels. To compare stilling well readings, stilling wells used act in a manner analogously to a linear filter the well tops were surveyed to a common datum established where the relation between actual and measured signals can be during the field investigation using a Topcon AT-G7 Auto written as Level. Permanent concrete structures were used to establish the arbitrary level datum. This survey was undertaken several times Bm þ ¼ ð Þ during the measurement period with survey closure errors s B m a 1 t (Muskett, 1995) ranging from 0 to 2 mm at Manihiki and from 1 where a and m are the actual and measured signals respectively. to 2 mm at Rakahanga to the common datum, which confirms The amplitude response function (FLF) for Eq. (1) is that the stilling wells did not move vertically during the measurement period and that the correct well top levels were −1 FLFðxsÞ¼ð1 þ ixssÞ ð2Þ measured with an accuracy of ±2 mm. The measuring scale is pffiffiffiffiffiffi graduated in 2 mm increments. Consequently, the accuracy of where i ¼ −1 and ω is the angular frequency. To measure the s manually read water levels is estimated at ±4 mm (i.e., response time constant, a half height test is preformed where the centimetre accuracy), excluding human errors. time ðt = Þ is measured for the water within the stilling well to 1 2 The self logging pressure transducers (PT) were logged at fall from its initial height above the filter to half that height. 20 s intervals, corrected for atmospheric variations (using an in Under this scenario, Eq. (1) simplifies to situ BaroTROLL). This water depth signal was converted to Bm water surface level using the following procedure. First, the s þ m ¼ 0 ð3Þ Bt measured PT depth was centrally averaged over ten minute intervals to remove incident and infragravity motions from the and solving yields signal; Second, the average depth was vertically fitted to the − = manual stilling well level measurements by minimising the mðtÞ¼mðt ¼ 0Þe t s ð4Þ difference squared between PT and manual stilling well and rearranging yields measurements. Detrending was not included as the PTs were tested and found to be stable in time and measurements fitted t1 2 well to manual stilling well readings (see Figs. 5 and 6). The s ¼ c1:44t1 ð5Þ ln 2 2 purpose of deploying PTs was to obtain higher resolution measurements during the day and to provide night readings Laboratory testing indicates that τ∈[90; 110 s] with a mean when manual readings were stopped for practical reasons. The near 100 s. Wells outside this range had their filters modified to stilling wells' purpose was to establish water surface levels at ensure τ ∼100 s. centimetre accuracy. Fig. 3 shows the range of response expected in the measured The stilling wells (both atolls) and co-located pressure signal and demonstrates that the stilling wells remove 98% of transducers (only at Manihiki) were installed to measure the incident wave motion. However, it also shows that between 7% ocean and lagoon water surface levels during two field

Fig. 3. The amplitude response function for the lower limit (▬▪▬, τ=90 s), average (▬▬, τ=100 s) and higher limit (▪▪▪▪, τ=110 s) time constants characterising the stilling wells deployed at Manihiki and Rakahanga, determined from Eq. (1) using the measured response time range. 694 D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704

Fig. 4. a. Tauhunu (western) Harbour panorama photograph, taken from the concrete slab damaged during Tropical Cyclone Martin in 1997. The left extent looks south-west and the right extent looks north-west. Two stilling wells are located on the reef flat and adjacent to the wharf. The harbour benchmark (labelled BM) is also shown, which was used to relate all stilling well measurements. (This harbour is not connected to the lagoon.) b. Sketch plan (not to scale) showing the harbour, Tauhunu Village and the lagoon which shows the circulation pattern observed at both Manihiki and Rakahanga where water typically discharges from the reef flats into the harbour (the location of Tauhunu Village is shown on Fig. 2b). campaigns at Manihiki and one short field visit to Rakahanga. to the ocean and an internal basin through the reef flats (see Fig. Survey equipment failure resulted in the discarding of the 4). Dunn et al. (2000) measured wave set-up in river entrances, measurements from the first Manihiki field experiment (an which has a similar arrangement to these harbours. They accurate level datum was not established). The field measure- showed that while significant wave set-up occurred on the ments presented in this paper are from the second Manihiki (24 surrounding beaches, there was no measurable wave set-up October through to 8 November 2004) and the short Rakahanga within the river entrance. This absence of wave set-up within (13 February 2004) field experiments, where centimetre the harbour was visually observed at Manihiki, where the accurate water surface measurements from stilling wells were surrounding reef flats (equivalent to the beaches) persistently obtained. discharged into the harbour basin resulting in a persistent The ocean tides were measured in harbours at both atolls. offshore flow direction in the channel connecting the basin to Both harbours are located on the external reef flats near land and the ocean (c.f., flow patterns indicated in Fig. 4b). Consequent- are not hydraulically connected to the atoll lagoon by channels, ly, the water level measured within the harbour was adopted as reef passes or reef flats. Both harbours consist of a deep channel approximately the ocean water level.

Fig. 5. Water surface levels measured at Manihiki during October 2004. Manual stilling well measurements are shown by × for the ocean and by + for the lagoon. The converted pressure transducer measurements are shown using continuous lines with ▬ for the ocean and — for the lagoon. D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704 695

Fig. 6. Measured ocean tide (•) and lagoon water surface level (+) at Rakahanga on the 13 February 2004.

The lagoon measurements for Manihiki were recorded across level. This eliminates the ocean tide as a driver of the lagoon from the harbour (Fig. 4b). Similarly, the Rakahanga lagoon flushing for these atolls (c.f. Fig. 1). Further, the inflow occurs stilling well was located in the lagoon across from the harbour despite the elevated lagoon water surface level (indicated by with continuous land separating them (Fig. 2c). The lagoon rising lagoon levels in Figs. 5 and 6). wells were located as close as possible to the ocean wells, thus Fig. 7 shows the wave energy flux derived from hindcast allowing regular manual measurements using limited field wave height, period and direction data from the NOAA's Wave workers. Watch III model (Tolman, 1997, 1999) together with the The field experiment on Manihiki was conducted over a averaged (over one tidal cycle) height that the lagoon water period of three weeks. The self logging pressure transducers level above the ocean water level (the lagoon overheight) at were deployed twice for 6.5 days, with instrument fouling Manihiki. The average overheight clearly increases with towards the end of the first deployment resulting in five increasing wave energy flux, as previously observed by Sharma continuous days of reliable pressure data when compared to et al. (2001). the manual stilling well readings. No reliable pressure data The very similar shapes of ▬▬▬and ▬▬ in Fig. 7 show was obtained during the second deployment due to the that the inflows are driven by wave pumping similar to that original fouling of the pressure transducers. Nevertheless, Fig. reported by Nielsen et al. (1999, 2001) for rip currents. In this 2 5 presents the manual stilling well and converted pressure model, the water flow rate qi [m /s] crossing the exposed reef transducer water level measurements for the period where the rim times the equivalent pressure increase ρgΔ(ρ is water den- pressure transducers were operating reliably, which extends sity, g is gravitational acceleration and Δ is change in elevation from the 24th through to the 29th of October 2004. The of the water surface) is assumed to be proportional to E (wave f,c D standard deviation of the differences between the manual energy flux at the coastline), i.e., pump efficiency ¼ qg qi. E ; stilling well and converted pressure transducer water level That is, the incident wave energy usefully pumps water ontof c measurements was 2.4 and 2.6 cm for the lagoon and harbour the reef flats at an elevated level compared to the ocean. Fig. respectively. This standard deviation is much less than the 8 shows the wave pump in action on the exposed side of amplitude of the two signals and the difference between the Manihiki, during mild wave conditions (deepwater wave two signals. height, H0 ∼1.5 m). The waves pump water onto the reef flat, Fig. 6 shows the field measurements obtained at Rakahanga, lifting the water surface ca 0.3 m above the ocean. The blocks which are exclusively manual stilling wells readings. of dead coral on the reef flat and the little waterfall represent The measurements shown in Figs. 5 and 6 reveal that, at both removable obstructions to the inflow; this is discussed later in atolls, the lagoon water level is at all times above the ocean Section 5 in terms of improving flushing efficiency.

Fig. 7. Wave energy flux derived from hindcast wave height, period and direction data from the WWIII model (▬▬▬) with the averaged (12.25-h box car filter) lagoon overheight (——) and instantaneous lagoon overheight ( ) at Manihiki during October 2004. 696 D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704

Fig. 8. Typical example of the wave breaking process pumping water onto the reef flat. The little waterfall illustrates that the water level on this section of the reef flat is ca 30 cm above the lagoon level, which at 15:04, the time of the photograph, was estimated from measurements to be 0.5 cm below the ocean level (which was nearing high tide, see insert).

Fig. 9 shows the outflow on the leeward side of Manihiki. made origin, but in the case of Manihiki and Rakahanga no The rim exerts hydraulic control via critical flow conditions. major channels exist (Solomon, 1997). The lagoon water level (left) is clearly above the ocean (right), Conservation of water volume for the system when assuming as measured (c.f. Fig. 5). the lagoon fluctuates uniformly over its surface area with Level surveys of the reef flats indicate that the reef flats are vertical side walls (over the lagoon water level variation) is above MSL and the reef edge is typically above the highest high expressed by tide level in the ocean during the field campaign. The reef flats and edges also have coral growing on them. It could perhaps be dg A ¼ Q −Lq ð6Þ argued that the water level measurements showing the lagoon dt i o water level well above that of the ocean is an unusual scenario. However, the healthy coral growth on the reef flats and edges where A is the lagoon surface area, η is the lagoon water level, t implies that they are being consistently wet (Schlager, 1998) is time, Qi is the total rate of inflow, L the length of crest where and so this observation dismisses this argument. outflow can occur and qo is the unit length outflow rate The following two sections provide new analytical and (subscripts ‘i’ and ‘o’ refer to in- and outflow respectively numerical models of lagoon flushing based on wave-driven throughout this paper). inflow and gravity-driven outflow. 3.1. Simple harmonic inflow 3. Analytical model for constant wave conditions The essence of the dynamics of the system can be gleaned Consider an atoll with cross section as shown in Fig. 10. from the analytical solution for the case where Qi is simple The rim of the lagoon consists of sub-aerial vegetated harmonic, i.e., ‘motus’ and lower lying ‘hoas’ (reef flats) where either wave- ¼ P þ ̂ ð Þ driven inflow or gravity-driven outflow can occur (c.f., Figs. 2a, Qi Qi Qicosxt 7 8, and 9). The crest of the ‘hoas’ is above MSL and the edge is ¯ ˆ slightly higher (zedge −zcrest ∼10–30 cm) due to the corals and where Qi and Qi are the mean and oscillating inflow vegetation being kept wet by the wave splashing at the edge. components and ω is the tidal angular frequency (see Eqs. The rim may also be broken by deep channels of natural or man- (23) and (24)). The outflow (qo) corresponds to critical flow

Fig. 9. Typical example of gravity-driven outflow on the leeward side of the atoll. The ocean is to the right and the lagoon to the left and the flow direction is left to right. The steep water surface gradients at the reef rim indicate hydraulic control by critical flow conditions. D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704 697

Fig. 10. Definition sketch, MSL = mean sea level; SWL = still water level; and MWS = mean water surface.

That is, the mean lagoon water level increases with with a driving head equal to the lagoon water surface height P above the reef edge, y (c.f. Fig. 10), i.e., increasing wave height (which increases Qi and y¯) and Eq. rffiffiffi (13) indicates that the system responds faster when the waves 2 2 pffiffiffiffiffi q ¼ y gy ð8Þ are large. Conversely, we expect a greater time lag for the o 3 3 lagoon water surface when the waves and hence y¯ are small. This behaviour was observed at Manihiki with the measured where g is the gravitational acceleration. In this scenario, the lagoon lag increasing from 2.1 to 4 h as the deepwater wave conservation of volume may be written as, rffiffiffi height decreased from 2 to 1.1 m. dy P 2 2 pffiffiffiffiffi A ¼ Q þ Q̂cosxt− Ly gy: ð9Þ dt i i 3 3 3.2. Inflow by wave pumping Eq. (9) is non-linear and would require numerical integration Consider next a few details of the relation between wave in a dynamic scenario. However, it becomes linear and easily conditions and inflow rate. The local per unit length inflow rate, solvable if the outflow rate is approximated by qi, will for a given cross section of the lagoon rim, depend on the pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi deepwater wave height (H ), the wave period (T), and the height ¼ ¼ þ = ð − Þþ ½ð − Þ2ðÞ 0 qo y gy ¯y g¯y 3 2 g¯y y ¯y O y ¯y 10 that the waves on the inflow side have to lift the water (ηmax − η where y¯ is the time–mean depth at the outflow reef edge. For tide), i.e., the simple harmonic inflow Eq. (7) we then get a simple ¼ ð ; ; − ÞðÞ qi qi H0 T gmax gtide 16 harmonic lagoon water surface oscillation: where η is the maximum water level on the inflow edge, c.f. ð Þ¼ þ ̂ ð − ÞðÞ max y t ¯y ycos xt uy 11 Fig. 10. One inflow model is the wave pump model by which a where rffiffiffi certain fraction (Cpump) of the wave energy flux per unit length ̂ along the coastline is converted to useful pumping power ̂¼ 3 p1ffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQi ; y u ()rffiffiffi (Bruun and Viggoson, 1977; Nielsen et al., 1999, 2001): 2 L g¯y u 2 t þ 3 pAffiffiffiffiffiffi 1 x qgðg −g Þq ¼ C E ; ð17Þ 2 L g¯y max tide i pump f c rffiffiffi where E is the wave energy flux, per unit length along the ¼ −1 3 pxAffiffiffiffiffi ð Þ f,c uy tan 12 2 L g¯y coastline available to drive the pump. The left hand side of Eq. (17) is the amount of power required to pump water at the flow i.e., the linearised system has the time constant rate of qi against a water head of ηmax −ηtide and the right hand rffiffiffi side is the power available from the waves to pump water onto 3 A the reef flat. Tlagoon ¼ pffiffiffiffiffi ð13Þ 2 L g¯y Fig. 11 contains a large amount of field and laboratory data on the pump efficiency Cpump, plotted as function of (ηmax − which is the time lag of the lagoon water level relative to the η tide)/R. R is thep runupffiffiffiffiffiffiffiffiffi heightp estimatedffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from Hunt's (1959) ocean tide. 2 formula R ¼ tanb HLo ¼ tanb HðgT =2kÞ. The data indi- The time averaged outflow depth is cate a strong dependence on the ramp slope β with the “steepest” β P 2=3 ( =45°) of Gourlay (1996) reaching efficiencies above 0.4 3 Q ¯y ¼ piffiffiffi ð14Þ while field data (Nielsen et al., 2001) from rip systems with 2 L g much milder slopes (β≈1− 2°) give efficiencies that are typically ten times smaller (0.035±0.01). The improvement of η and hence, the mean lagoon water level, ¯,is pump efficiencies with slope is consistent with higher swash P 2=3 overtopping flow rates at larger beach slopes (Peregrine and 3 Qi g¯ ¼ zedge;o þ pffiffiffi ð15Þ Williams, 2001; Baldock et al., 2005). The ramp slope on 2 L g Manihiki Atoll is estimated to be in the range 1:10 to 1:5. 698 D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704

Fig. 11. Experimental data on the wave pump efficiency as function of dimensionless freeboard from tide level. Bruun and Viggoson (1977) ramps 1:4 to 1:6, Gourlay (1996) 1:1 slope, , , 1:5 concrete slope, 1:10 plywood slope with 0, 1.24 and 2.4 m plate behind the slope crest, rips in the field, mild slopes.

Slope roughness is likely to diminish Cpump and the reef where H0 is the deepwater wave height. The total inflow rate is slopes on Manihiki are very rough compared to the laboratory obtained by integrating this equation around the part of the atoll slopes of wood or smooth concrete. perimeter which is open to inflow Solving the wave pump equation for the inflow rate yields Qiðh0Þ¼qi;maxlcos½hðsÞ−h0ds ¼ qi;maxLiðh0Þ: ð21Þ ¼ Ef ;c ¼ Ef ;c ð Þ qi Cpump ð − Þ Cpump ð þ − Þ 18 Values of the effective inflow crest length L (θ ) estimated qg gmax gtide qg g Dhi gtide i 0 from an aerial photograph of Manihiki Atoll are shown in Fig. where Δ is the inflow headloss. The inflow rate (18) will vary hi 12 as a function of offshore wave angle θ0. around the atoll because the Ef,c available to drive the wave The presence of the lagoon water level η in the denominator pump changes with the shoreline orientation. Assuming a of Eq. (19) prevents simple analytical solutions of (6) based on straight shoreline and using Snell's law, the inflow becomes, this form of qi. However, an analytical solution is possible if the – E cos½hðsÞ−h water level difference in the wave pump Eqs. (18) (20) is q ðsÞ¼C f 0 ð19Þ replaced by the reef edge height above the ocean tide level. Eq. i pump qgðg þ D −g Þ hi tide (20) then becomes where s is the curvilinear axis running along the coastline, θ(s) 2 is the bearing of the shoreline normal at s, θ is the deepwater gH0 T 0 qiðsÞcCpump cos½hðsÞ−h0: ð22Þ 32kðz ; −g Þ wave direction and Ef is the wave energy flux per unit length of edge i tide wave crest. Replacing Ef with the linear wave theory expression and ignoring inflow headloss (i.e., assuming Δhi =0) yields Writing the ocean tide level as

2 g ¼ Atidecosxt ð23Þ ð Þ¼ gH0 T ½ ð Þ− tide qi s Cpump kð − Þ cos h s h0 32 g gtide where A is the tidal amplitude. We can obtain an inflow ¼ ½ ð Þ− ðÞ tide qi;maxcos h s h0 20 expression in the form (7), under the assumption that the tidal

Fig. 12. The effective inflow crest length Li(θ0) as function of wave direction for Manihiki Atoll. D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704 699

amplitude is small compared with zedge,i,orAtide ≪zedge,i.We water level, (η¯) above the ocean tide level. Eq. (20) then get becomes gH 2T gH 2T A ð Þc 0 ½ ð Þ− ðÞ ð Þc 0 þ tide ð ÞðÞ qi s Cpump kð − Þ cos h s h0 25 Qi t Cpump 1 cosxt Li h0 24 32 g¯ gtide 32kzedge;i zedge;i We can obtain an inflow expression in the form (7), under the While Eq. (24) will work, Eq. (22) becomes meaningless for assumption that the tidal amplitude is small compared with η¯,or η tide =zedge,i. Atide ≪η¯. We get Fig. 13 shows the analytical model (continuous thick black 2 line) formed from Eqs. (11) and (24) and field measurements gH T Atide Q ðtÞcC 0 1 þ cosxt L ðh Þð26Þ from Manihiki previously presented in Section 2. The ocean i pump 32kg¯ g¯ i 0 tide amplitude (Atide) and radian frequency (ω) were estimated from the ocean water level measurements and used in Eq. and from the form of Eq. (7), we get (24) to predict the wave pumping inflow as modulated by the 2 P gH T ocean tide (i.e., the vertical lift that the waves pump against Q cC 0 L ðh Þð27Þ i pump 32kg¯ i 0 varies with the tide). The wave energyP flux was calculated using the average wave height ðH ¼ 2:2mÞ and period P 0 and substituting this expression for Q and η¯=z +y¯ into Eq. (T¯=13 s) for the comparison period. The predictions exhibit i edge,o (14), yields the following implicit equation for y¯. an overestimation of mean lagoon overheight and water level fluctuation amplitude. That is, the model is either; under- 2 ð Þ 2=3 estimating outflow or overestimating inflow. The three main 2=3 3 CpumpgH0 TLi h0 ¯yðzedge;o þ ¯yÞ ¼ pffiffiffi ð28Þ simplifications employed to yield the solution were; linearis- 2 32kL g ing the outflow; excluding friction and setting the pumping head to the reef edge level less the ocean tide level. Fig. 13 also shows this adjustment to the analytical model Numerically integrating Eq. (6) shows that linearising the (continuous thick grey line). Increasing the head which the outflow expression (8) and including outflow friction does not waves were pumping against to the mean lagoon level has significantly change predictions (see discussion in Section 4). corrected the lagoon water level fluctuation amplitude towards Further, including outflow friction is expected to worsen the field measurements. predictions. The assumption of uniform water level fluctuations in the The approximation that zedge,i ≈η, while allowing an explicit lagoon and vertical side walls required for the application (6) expression for y¯, does not appear to be an acceptable with constant A is considered acceptable for Manihiki. That is, approximation, given that, from field measurements, the lagoon surface area is 4.3×107 m2 compared to the active 6 2 zedge,i ≈0.2 m and η∈[0.2;0.4 m], i.e., factor two variation. reef top surface area of 2.7×10 m (i.e., an order of magnitude To improve this approximation, the water level difference in the less). Hence, the inclusion of water level variations across the wave pump Eqs. (18)–(20) is now replaced by the mean lagoon reef flats due to friction or changes in the overall lagoon area

Fig. 13. Comparison between measured (two tidal cycles starting at 26 October 2004 13:10) and the analytical model (Eqs. (11) and (24)) where the solid square symbol (▪) and thin continuous line (—) are the measured and fitted ocean tide (Atide and ω estimated from the measured ocean water levels and used in Eq. (24)), the cross symbol (×) is the measured lagoon water levels and the thick black ( ) and grey ( ) continuous lines are the predicted lagoon water surface levels using Eqs.

(24) and (27) respectively for the inflow expression (Cpump =0.035). 700 D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704 due to non-vertical side walls at Manihiki for mass conservation Colleter (2005). The effect of the water level behind or on top of are second order adjustments. the reef flats or breakwater is incorporated in our wave pump The wave pump efficiency (Cpump) was obtained from efficiency formulation. average parameters for the atoll. That is, the offshore reef edge Another group of overtopping expressions follow the slope was visually estimated at between 1:5 and 1:10 and (ηmax analytical work of Peregrine and Williams (2001) and extended − ηtide)/R ∈[0.1;0.5] during the period compared against, by Baldock et al. (2005) or the empirical work of Hedges and which yields Cpump ∼0.035 (c.f. Fig. 11). Hence, the Reis (1998, 2004), where the overtopping rate is determined by assumptions and simplifications made to achieve the analytical considering a swash truncation process. These expressions also solution are not impacting significantly on the prediction generally ignore the influence on qi of the water level behind the accuracy for the field measurements presented. This may imply crest, which is important in the present application. that the wave pump process is strong compared to other The fully and partially oscillatory overtopping processes, processes in operation at Manihiki during the field data addressed by Caceres et al. (2005), occur when the structure collection period. crest is located below ca MSL+H/2, where H is the wave The model has predicted the average overheight reasonably height at the structure. In these processes, the water mass is well, with less accurate prediction of lagoon water level timing transferred in an oscillatory manner (albeit an oscillatory motion and shape. This may be explained by assuming constant wave modified by the structure). This is not the scenario at either height and period or wave energy flux over the two tidal cycles. Manihiki or Rakahanga, where the waves break on the reef edge This is particularly relevant given the analytical solution resulting in unsteady channel flow across the reef flat into the predicts a lagoon response time of the order of 2 h compared lagoon. to the tidal period of 12.5 h. That is, wave variations occurring We thus adopted the approach of Nielsen et al. (1999, 2001) over the 2-h period are impacting on the lagoon dynamics. in the analytical solution (see Eq. (17)) as it is based upon physical arguments and deals with the effect of the water level 3.3. Applicability of other wave overtopping formulations behind the crest. A valid alternative is the more detailed approach of Gourlay and Colleter (2005). However this There are numerous wave overtopping formulations relating approach probably would not lead to a simple analytical solution discharge to wave properties (see for example, Owen, 1980; of Eq. (6). Allsop et al., 1995; van der Meer and Janssen, 1995; Hedges The following section provides a modified model where the and Reis, 1998; Besley, 1999; Caceres et al., 2005; van der assumptions of constant wave height, period and direction, Meer et al., 2005). Many of these expressions employ the form constant tidal amplitude and approximate pumping head are relaxed. However, these additional refinements result in a q R i ¼ f pcffiffiffiffiffiffiffiffi ð29Þ numerical model since no analytical solutions exist. gTmHs Tm gHs 4. Numerical model for variable wave conditions or The analytical model provides quasi-steady predictions by pqffiffiffiffiffiffiffiffiffii ¼ Rc ð Þ f 30 assuming a constant wave energy flux. We now develop a gH 3 Hs s simple numerical model, easily implemented in a spread sheet, where Rc is the structure freeboard, Tm is the mean wave period to predict lagoon dynamics under variable wave conditions. at the structure toe. These two forms were originally developed Further, as it is convenient, the linearisation approximation by Weggel (1976), Owen (1980), and van der Meer and Janssen required by the analytical model will be removed and outflow (1995). Recently, Steendam et al. (2004) plotted relative crest friction will be included. R qi height, c against dimensionless flow rate, pffiffiffiffiffiffiffiffiffi for more than For inflow (ignoring headloss, Δ , for simplicity) the wave- H gH 3 hi ten thousands test results, their Fig. 3, which showss significant driven inflow is (see Fig. 14) scatter when geometry and other parameters are ignored (five orders of magnitude in dimensionless flow rate for relative 2 gH0 T qiðs; g; tÞcCpump cos½hðsÞ−h0 height (Rc /Hs) in the range 0.3 up to 3.5). Besley (1999), kð − Þ 32 g gtide however, confirmed for particular geometry, this empirical ¼ ð ; ; Þ ½ ð Þ− ðÞ R qi h0 g t cos h s h0 31 approach was applicable for 0:05 < pcffiffiffiffiffiffiffiffi < 0:30 and Le Fur et Tm gHs al. (2004) extended this with additional experimental data to Rc An outflow model is composed of a more or less uniform, 0:05 < pffiffiffiffiffiffiffiffi < 0:60. However, as these formulations are Tm gHs quasi-steady flow with gradually varying flow over the flat strongly dependent on slope geometry, which varies along the crest, followed by critical flow over the reef edge. If the friction edge of an atoll (and is often unknown), application of these over the flat part of the reef top generates the headloss Δ (see formulations is impractical for atolls. Further, most of the ho Fig. 15) the outflow velocity at the edge (u ) is thus overtopping data/formulae from the breakwater literature apply edge to situations where the water level behind the breakwater is rffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi below the crest and thus does not influence qi. That influence is 2 uedge ¼ gðg−Dho−zedge;oÞ: ð32Þ important in the reef flat scenario, see for example, Gourlay and 3 D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704 701

Fig. 14. Inflow definition sketch.

Assuming that the flow velocity at the centre of the reef flat conditions strengthened, as visually observed and confirmed (ucrest) is proportional to uedge, we get from hindcast wave energy flux, from the Wave Watch III 0 1 model (see Fig. 7). The outflow friction factor of 0.1 was 2 2 u2 − − estimated from reef top current and water level gradient f ucrest fW 2 edge Bg zedge;o DhoC fW Dho ¼ @ A measurements undertaken at Manihiki. 2g dcrest 3 2g Dho Dho g−zcrest− g−zcrest− The model, however, has poorly predicted the dynamic 0 12 2 details, particularly the phasing where the model persistently 3 reacts quicker than the field measurements. However, the model 2 Bg−z ; −D C fW ¼ @ edge o hoA ð33Þ does capture the essential dynamic behaviour of this lagoon. 3 Dho 2 g−zcrest− This numerical model does perform better than the analytical 2 model for variable wave conditions, as expected given the where d is the flow depth halfway across the reef crest. Eq. relaxation in the constant wave energy flux assumption required crest D D (33) can be simplified, using ho ≪1, ho ≪1,(1−ε)3 ≈1 in the analytical model. That is, the analytical model currently g−zedge;o g−zcrest −3ε and (1−ε)− 3 ≈1+3ε for ε≪1, to provides the quasi-steady dynamics whereas the numerical model provides the instantaneous dynamics. Fig. 17 compares 2 fW the performance between the two models for approximately c 3 2ð Þ constant wave conditions and shows that both models predicts Dho 3 34 g−zcrest 2 fW 1 1=2 the lagoon overheight to similar accuracy. That is, including þ3 − g−zedge;o 3 2 g−zedge;o g−zcrest outflow friction and not linearising the outflow in the numerical model has not dramatically changed predictions. Consequently, which yields an outflow rate of the choice of linearising approaches does not appear significant 0 1 3=2 for lagoon dynamics at Manihiki and other physical processes rffiffiffi B 2 fW C not included in these two models are required to improve 2 2pffiffiffiB 3 2 C qoc gBg−zedge;o− C 3 3 @ g−z 3 2 fW 1 1=2 A predictions further. crest þ3 − g−zedge;o 3 2 g−zedge;o g−zcrest ð35Þ 5. Discussion Fig. 16 shows that the numerical model formed from Eqs. The field data collected on both Manihiki and Rakahanga (6), (31), and (35) reproduces the general overheight pattern shows that the lagoon water level is consistently elevated above when compared against field measurements. The model predicts the MSL, indicating that ocean tides are not driving the lagoon the additional overheight that occurred when the wave flushing. The super-elevation was established to depend

Fig. 15. Outflow definition sketch. 702 D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704

Fig. 16. Comparison between measured (in 2004) and the numerical model (Eqs. (6), (31), and (35)), where the thin continuous line (—) is the measured ocean tide, the cross symbol (×) and the thick continuous line (▬▬) are the measured and predicted lagoon water surface level respectively and the thick dashed line (▬▬▬)is the hindcast wave energy flux from the Wave Watch III model. Adopted parameters used were Cpump =0.035 and f=0.1. principally on the ocean wave conditions as it was also Fig. 18 shows the monthly averaged wave energy flux for observed that the lagoon quickly responded to changing wave Manihiki from late 1986 through to late 1989 using GEOSAT conditions. Further, the field measurements and hindcast satellite wave measurements (Barstow and Haug, 1994), with an wave statistics confirm that wave energy flux and lagoon overall average of 26 kW/m. Using this average wave energy overheight are strongly correlated (i.e., as wave energy flux flux and the analytical model, the total mean outflow is ca 3 increases, lagoon water level increases) (c.f. Fig. 7). This 470 m /s (using Eq. (27) with Ef ∼26 kW/m). The average behaviour was reproduced by both of the new models lagoon depth, determined from survey of the lagoon provided presented here, which are based upon wave pumping and by Smith (2004, pers. comm.), is 47 m. Consequently, wave gravity draining describing inflow and outflow respectively. pumping turns over the entire water volume of the lagoon every This modelling also demonstrated the role of the ocean tide in 50 days (assuming full vertical mixing). The most efficient tidal modulating the wave pumping process. Both models are flushing possible involves installing deep channels between the formulated for the continuous rim scenario, which is lagoon and the ocean. Under this arrangement, the largest consistent with site surveys for Manhiki and Rakahanga. amount of water that can be exchanged between the ocean and For atolls with strongly noncontinuous rims, inflows and lagoon during one tidal cycle is 2AAtide. Hence, tidal flushing outflows will primarily be driven by water level gradients replaces the lagoon water every 62 days under spring tidal between the ocean and the lagoon, balanced by bed friction in conditions (i.e., a 2Atide ≈0.4 m average spring tidal range). the usual manner. These two scenarios are the limiting cases Under the predominant easterly wave direction (Thompson, for microtidal atoll lagoon flushing, with intermediate cases 1986) and using the analytical model, wave pumping driven by being a mixture of both flushing mechanisms. Consequently, Ef ∼12.5 kW/m flushes the lagoon in the same time as the very improved lagoon flushing, can be achieved by either best tidal flushing scenario. Hence, from the wave energy fluxes improving the wave flushing or the tidal flushing efficiencies. shown in Fig. 18, wave pumping provides more efficient For management purposes it is important to know which flushing for 32 out of the 34 months where wave measurements option is the more efficient. are available (i.e., 95% of the time).

Fig. 17. Comparison between measured (in 2004) and the analytical and numerical models, where the thin continuous line (—) is the measured ocean tide, the cross symbol (×) is measured lagoon water surface level, the thick grey and black continuous lines ( and ) are the analytical and numerical predictions of lagoon water surface level respectively and the thick dashed line (▬▬▬) is the hindcast wave energy flux from the Wave Watch III model. D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704 703

Fig. 18. Estimated monthly averaged wave energy flux from the GEOSATsatellite wave height measurements over the region covering Manihiki (source, Barstow and Haug, 1994). The average wave energy flux for this period is ca 26 kW/m.

Fig. 2b shows the Manihiki Atoll together with the extents of Acknowledgements land and reef tops. The southern region is typically formed of ancient coral extending ca 1 to 1.5 m above MSL, blocking an This work is financially supported by CRC for Sustainable estimated 80% of inflow crest length available (c.f., Fig. 8). The Tourism Project #52001. The assistance from Cook Islands eastern (inflow) and the north western (outflow) reef tops are Ministry of Marine Resources personnel at Rarotonga, Manihiki typically open (c.f., Fig. 9) and do not impede the flow. The and Rakahanga is also gratefully acknowledged. The field work southern reef top length is 6 km and increasing the percentage assistance from, in alphabetical order, Ms Ellen Milnes, Prof. available to transfer water into the lagoon by removing the Pierre Perrochet, Dr Frederic Saint-Cast and Ms Christie ancient coral represents a factor five increase in the inflow crest Schacht is also gratefully acknowledged. length (c.f., Fig. 12) and consequently more rapid lagoon Robert Smith from SOPAC has kindly provided the survey flushing. Hence, minor clearing (ca 1 km) of ancient coral on data of Manihiki lagoon. the southern reef top is expected to generate a two fold increase We gratefully acknowledge the useful, critical and construc- in lagoon flushing. tive comments and suggestions on the manuscript by the anonymous reviewers. 6. Conclusion References Field measurements from the South Pacific atolls of Manihiki and Rakahanga show that the flushing of such full- rimmed atolls in microtidal mid ocean locations is driven by Allsop, N.W.H., Besley, P., Madurini, L., 1995. Overtopping performance of wave pumping and not by tidal forcing. This is evident from the vertical and composite breakwaters, seawalls and low reflection alternatives. Final Proceedings of MCS Project, Hannover, Germany. fact that the lagoon water level is at all times above the Baldock, T.E., Hughes, M.G., Day, K., Louys, J., 2005. Swash overtopping and surrounding ocean and that the overheight increases with sediment overwash on a truncated beach. Coastal Engineering 52 (7), increasing wave height. This is confirmed by living corals that 633–645. are located above the measured high tide (and maintained wet Barstow, S., Haug, O., 1994. The Wave Climate of the Cook Islands. South by the wave pumping process). Pacific Applied Geoscience Commission (SOPAC) Technical Report, vol. 200. Oceanographic Company of Norway. For steady/constant wave conditions, a simple analytical Besley, P., 1999. Overtopping of Seawalls: Design and Assessment Manual. model gives a good understanding of how the ocean tide Environment Agency, Bristol. 37 pp. modulates the wave pumping process and good predictions are Bruun, P.F., Viggoson, G., 1977. The wave pump: conversion of wave energy to obtained for the lagoon tidal amplitude and time lag. current energy. Journal of the Waterway, Port, Coastal and Ocean Division – For variable wave conditions and tidal amplitude a numerical 103 (WW4), 449 469. Caceres, I., Sanchez-Arcilla, A., Zanuttigh, B., Lamberti, A., Franco, L., 2005. model is required. Under a constant wave height and tidal Wave overtopping and induced currents at emergent low crested structures. amplitude the numerical model performs as well as the Coastal Engineering 52 (10–11), 931–947. analytical model, but no better. Dunn, S.L., Nielsen, P., Madsen, P.A., Evans, P., 2000. Wave Setup in River The through flow system driven by the waves gives more Entrances. In: Edge, B.L. (Ed.), 27th International Conference on Coastal – efficient water renewal than an alternating tide-driven system. Engineering. ASCE, New York, pp. 3432 3445. Gourlay, M.R., 1996. Wave set-up on coral reefs: 1. Set-up and wave generated Enhanced lagoon flushing could be achieved by increasing the flows on an idealised two dimensional horizontal reef. Coastal Engineering present mode of inflow by removing some of the dead coral 27 (3–4), 161–193. obstructions and flow constrictions, see Fig. 8. A twofold Gourlay, M.R., Colleter, G., 2005. Wave-generated flow on coral reefs—an increase in water renewal rate seems easily achievable in this analysis for two-dimensional horizontal reef-tops with steep faces. Coastal – way. Blasting of deep channels which will change the flushing Engineering 52 (4), 353 387. Hedges, T.S., Reis, M.T., 1998. Random wave overtopping of simple sea walls: regime to an alternating tidal system would be less efficient and a new regression model. Water and Maritime 130. would of course also be an unacceptable change to the natural Hedges, T.S., Reis, M.T., 2004. Accounting for random wave run-up in system. overtopping predictions. Maritime Engineering 157 (MA3), 113–121. 704 D.P. Callaghan et al. / Coastal Engineering 53 (2006) 691–704

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