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Notice: ©1996 American Geophysical Union. An edited version of this paper was published by AGU. This publication may be cited as: Smith, N. P. (1996). Tidal and Low Frequency Flushing of a Coastal Lagoon Using a Flexible Grid Model. In Pattiaratchi, C. (Ed.), Mixing in Estuaries and Coastal Seas, Coastal Estuarine Studies, 50(171‐183), doi:10.1029/CE050 11

Tidal and Low Frequency Flushing of a Coastal Lagoon Using a Flexible Grid Model

N. P. Smith

Abstract

Tidal and low frequency nontidal exchanges of water between a coastal lagoon and adjacent continental shelf are investigated within the context of flushing, using a computer model based on the continuity equation. Flushing is quantified by the 50% renewal time. The arrival of new ocean water occurs as a result of longitudinal diffusion and a degree of advective transport governed by the flexibility of segment boundaries to move with the ebb and Oood of Ule tide. The model is applied to the Indian River lagoon system, lying along the Atlantic coast of Florida, USA. Flushing rates are quantified for three sub-basins of Indian River lagoon, and for Banana River lagoon, connected to the northern sub-basin uf Indian River lagoon. The lagoons are microtidal and depend upon low frequency exchanges to maintain water quality. One-year simulations for the Indian River lagoon system as a whole show Olat the 50% renewal time is approximately 140 days when transport by advection. When the renewal of lagoon water is by longitudinal diffusion alone, Ole 50% renewal level is not reached after 365 days. A second series of simulations compares flushing rates for the Olree sub-basins of Indian River lagoon and for Banana River lagoon, assuming a completely flexible grid. The soulliern and central sub-basins of Indian River lagoon receive a 50% renewal of new ocean water in about 5 and 12 days, respectively; Ole northern sub-basin of Indian River lagoon and Banana River lagoon reach approximately 30% renewal by Ole end of Ole one-year simulation.

MiXing in Estuaries and Coastal Seas Coastal and Estuarine Studies Volume 50, Pages 171-183 Copyright 1996 by the American Geophysical Union

171 172 Flexible Grid Model

Introduction

Mixing processes in estuaries dictate the rates at which salt water and fresh water enter and leave, respectively, and thus are of crucial importance in determining temporal and spatial variations of salinity, and the flushing characteristics of the estuary in general. Energetic mixing within an estuary, in combination with active estuarine-shelf exchanges, reduces the residence time of fresh water, and at the same time encourages the incursion of salt water. Thus, whether flushing is quantified by the flushing time (Officer, 1976) or, for example, the 50% renewal time of new ocean water (Pritchard, 1960), mixing within an estuary determines the estuary's natural ability to maintain or reestablish water quality. Early flushing models (Ketchum, 1951; Stommel and Arons, 1951) used time steps of one semidiurnal tidal cycle and assumed complete mixing within segments partitioned according to the local intertidal volume. While simple flushing models continue to be used (Dyer and Taylor, 1973; Robinson, 1983; Merino et al., 1990; Miller and McPherson, 1991) and serve a useful purpose ~ quicklook tools (Wood, 1979), three basic features compromise their ability to provide realistic results. First is the assumption of complete mixing, which was challenged at an early date (Austin, 1954), although Wood (1979) has suggested a modification to deal with this issue. Second is the need to work with a single tidal constituent, which eliminates the opportunity to investigate variations in flushing rates over a synodic month, for example. Third, although freshwater outflow can be incorporated into tidal prism models, low frequency nontidal exchanges between estuarine and continental shelf waters cannot. Nontidal exchanges often provide an important supplement to tidal flushing. This paper constitutes one of a series of studies that has been designed to quantify flushing rates for Indian River lagoon, lying along the Atlantic coast of Florida (Figure 1). The lagoon is, 196 km long and generally 2-4 km wide. Water depths are characteristically between 1 and 3 m, though the Atlantic Intracoastal Waterway, forming the longitudinal axis of the lagoon, has a depth of 3.5 m. The lagoon is divided into three sub-basins, defined by three inlets, all of which are in the southern half of the lagoon. Banana River lagoon is connected to the northern sub-basin. The northern end of Indian River is connected to the southern end of Mosquito Lagoon by Haulover Canal (Smith, 1993a). Indian River lagoon is microtidal, with the semidiurnal M2 tide serving as the principal constituent (Smith, 1987). M2 amplitudes in the northern sub-basin are generally 0-5 cm. Amplitudes in the central and southern sub-basins are 5-10 cm and 10-15 cm, respectively. Banana River lagoon is virtually tideless, with amplitudes of all tidal constituents less than 0.5 em. The first paper in the series (Smith, 1993b) quantified the intertidal volume by using harmonic constants of the principal semidiurnal and diurnal tidal constituents recorded at 28 study sites. Amplitudes were multiplied by the surface areas they represent, and phase angles were incorporated to account for the movement of tidal waves through the lagoon. A precise measure of the intertidal volume was obtained Smith 173

for spring and neap conditions, including effects of diurnal inequalities. The second paper (Smith, 1993c) applied these results to a flushing model that incorporated both advective and diffusive transport of fresh and salt water, and that compared flushing rates with and without ancillary nontidal exchanges. An important distinction between this approach and hydrodynamic models is that the rise and fall of the tide is specified within the model, using the predicted tide (Schureman, 1958), rather than simulated by adjusting friction within the model. As a result, tidal exchanges are modeled more precisely. Mixing within the lagoon must be specified, however, just as with a hydrodynamic model, and model verification will involve matching hydrographic measurements.

ATLANTIC OCEAN

Inlet

Ft. Pierce Inlet

o 50Km

St. Lucie Inlet

Figure 1. The Indian River lagoon system on Florida's Atlantic coast, including Indian River lagoon and Banana River lagoon. Dots show locations of water level recorders; lateral lines define segments 1-16 in Indian River lagoon and segments Ib-3b in Banana River lagoon. 174 Flexible Grid Model

Neither of the above studies incorporated Banana River due to a lack of information on tidal conditions. Also, the flushing model assumed a complete mixing of water moving from one segment to the next, as have most of the earlier tidal prism models. This third study expands the geographic scope of earlier work on Indian River lagoon by incorporating the exchange of water between the northern sub-basin and Banana River lagoon. Also, the model used here has a flexible grid that controls the extent to which segment boundaries move along the longitudinal axis of the lagoon with the ebb and flood of the tide. When the grid flexes freely with the longitudinal flow, no advective transport from one segment to the next results, and transport is by diffusion only. When the grid is fixed and inflexible, advective transport between adjacent segments is similar to that specified in the tidal prism models. In that case, diffusion is not considered. Indian River lagoon is well understood in terms of tidal dynamics (Smith, 1987, 1990), but a dearth of hydrographic data has prevented the verification of results of flushing studies (Sheng and others, 1990, Smith, 1993c). Models can be useful in the diagnostic mode for gaining insight, however, even while the data base necessary for verification is being assembled. The model used in this study is intended to quantify flushing rates, not elucidate the dynamics of the lagoon. For that, one must tum to hydrodynamic models. The scope of this study is intentionally restricted to flushing in response to two-way lagoon-shelf exchanges. Fresh water effects are considered in other studies (see Smith, 1993c). The specific aims of the study are to obtain flushing estimates of flushing rates for the three sub-basins of Indian River "lagoon and Banana River lagoon, and to compare the assumption of complete mixing. with the assumption that mixing occurs by diffusion only.

Observations

Water level records from the 31 locations shown in Figure 1 were obtained between, 1969 and 1992 using Stevens Model A and Model F analog recorders, and a Stevens Model 7031 digital recorder. Water levels were read to the nearest 0.1 cm or 0.01 foot (0.3 cm). Time series varied in length, but most were between two and three months long. All records provided information on the nontidal rise and fall in lagoon water level (Smith, 1986); longer time series, in excess of one year, were used to characterize seasonal cycles. The surface areas of Indian River and Banana River lagoons represented by each of the 31 study sites were determined using a compensating polar planimeter and navigational charts. Study sites were not distributed uniformly, and individual water level time series represented surface areas of from 1.4 to 98.0 km2. Some of the smaller segments were combined, especially near the inlets, where small volumes could be completely flushed within the one-hour time steps used in the simulations. In its final form, the model included three segments in Banana River lagoon and 16 Smith 175

segments in Indian River lagoon, as shown in Figure 1. The model does not require data from a dense network of tide gauges. The necessary number is that required to reproduce the spatial variability of amplitudes and phase angles for the principal tidal constituents. Historical water level records were also used to quantify low frequency nontidal variations in lagoon water level. Eight years of data from a water level recorder located in segment II indicated that in the interior of the lagoon 95% of the high and low tide levels are within 20 em of median values. Histograms constructed from tidal predictions (not shown) indicate that diurnal inequalities, combined with spring-neap tide conditions, explain about half of this variability. The rest can be attributed to nontidal processes. Shorter time series from throughout Indian River were used to characterize both the time scales of the low frequency rise and fall in lagoon water level and the magnitude of the local non tidal fluctuations. Current measurements from the southern end of Banana River lagoon, made with a General Oceanics Model 2010 film recording inclinometer, were used to conflI1l1 that tidal and nontidal exchanges between Indian River segment 4 and Banana River segment 3b were being simulated properly. Figure 2 is a composite of total (top), tidal (middle) and low-pass filtered nontidal currents (bottom) based on measurements made over a 52-day period in early, 1983. Amplitudes of the principal tidal constituents were compared to the flood and ebb current speeds generated by the model in the absence of nontidal exchanges. Nontidal currents, shown at the bottom of Figure 2, are generally within ±20 em S-I. The restricted cross-sectional area at the southern end of Banana River forced a ±17 cm s-I current speed when a ±20 em variation in sea level and an extinction coefficient of 0.003 h m- I (see equation 3) were used in the model.

Methods

Harmonic analysis (Dennis and Long, 1971) provided the amplitudes and local phase i~ angles needed to identify six principal tidal constituents (M2, S2, N2. Kl> 0 1 and f'!" PI)· These harmonic constants were used in tum to predict water levels needed to quantify the hour-by-hour tide-induced changes in volume for each segment (Schureman, 1958). When time series from individual study sites were longer than about 35 days, analyses of overlapping 29-day time periods provided a sample of amplitudes and phase angles which could be vector-averaged, as suggested by Haurwitz and Cowley (1975), to obtain a more representative measure of the constituent. Low frequency variations in coastal sea level over time scales on the order of one week are frequent enough and of sufficient magnitude to augment tidal flushing in a meaningful way, especially in the northern sub-basin, where tidal motions are significantly damped. Low frequency water level variations penetrate further into the

> .. '" ..

. -~ ~....,.. 176 Flexible Grid Model

40

0

~ III "-- -40 E U ~

(/) -80 0 w w TOTAL 0- f (/) -120 14 11 28 4 11 \8 25 4 >-- UARCH Z JANUAR'f fEBRUARY W a:: a:: :::J U --' w z 'OJo . z <{ TIDAL I U -40 i i i I i I \4 21 28 4 11• 18 25 4 L? JANUARY FEBRUARY MARCH Z 0 --' ':1- ....

-40 i i i I i 14 21 28 4 11 4 JANUARY F[BRUARY MARCH

Figure 2. Composite of current meter data recorded at the southern end of segment 3b in Banana River lagoon, January 15 to March 8, 1983. Total along-channel component flow is shown at the top of the composite, the predicted tide is given by the middle plot and the low­ pass filtered nontidal flow appears at the boltom. Positive along-channel speeds indicate flow into Banana River lagoon.

lagoon and thus can impact areas beyond the reach of tidal exchanges. Time scales of nontidal water level variations were specified using plots (not shown) of low-pass filtered time series of water level recorded throughout the lagoon. In some cases, low frequency variations in lagoon water level are of lower amplitude in summer months than in fall and winter months, but a distinct seasonality does not appear consistently in time series from different locations and different years. In general, the nontidal rise and fall in lagoon water level relative to the local seasonal mean is 5-10 em, with higher values occurring close to inlets (Smith, 1986). In some cases, the range in low frequency water level in the interior of the lagoon can reach 15-20 em, but low frequency variations greater than ±20 em are rare. Low frequency water level variations were modeled using

(1) Smith 177

where Am is the amplitude of the nontidal water level in the m1h segment, t is time, t is the time lag of high water in the mth segment relative to that at the nearest inlet, and T is the period of the nontidal water level variation. The double prime notation is used to denote nontidal processes. The time lag in the mth segment was modeled by assuming that low frequency water level fluctuations move through the lagoon as shallow-water waves:

m Xj = L (2) j=1 (gHj )1/2 '

where g is gravity, H is the mean depth of the segment and x is the length of the segment along the longitudinal axis of the lagoon. An unpublished 157-day study of non tidal water level variations in the northern sub-basin of Indian River showed an average time delay of just under 5 hours for water levels recorded in segment 4 relative to segment 8. This is equivalent to a propagation speed of about 8 kIn h-l. While this implies a mean depth of only 0.2 m, this speed is within the 5-10 kIn h- l range of M2 tidal wave speeds found throughout the lagoon (Smith, 1987). The spatial decrease in amplitude of low frequency water level fluctuations was modeled as an exponential extinction: m IXj j=1 -k- ), (3) T'

where Ao is the amplitude of the low frequency water level variation at the inlet. A value of 0.003 h m- l was used for the extinction coefficient, k. By 'combining the extinction coefficient with the ratio of the distance to the period of the low frequency fluctuation, the decrease in amplitude is relatively small at locations near inlets, and for sufficiently long periodicities.

The intertidal volume of the mth segment, Vm, is obtained from the product of the :: surface area, S, and the sum of the tidal and nontidal water levels, assuming that the surface area of the segment does not vary with water level: ::; (4)

where the prime notation refers to tidal fluctuations.

To quantify the flushing of Indian River lagoon, the simulations begin with the assumption that the lagoon is filled with "lagoon water"- an undefined mixture of fresh and salt water. The time required for tidal and low frequency nontidal exchanges to replace half of the lagoon water with new ocean water is referred to as the "50% renewal time," Rso. This time interval can be determined for the lagoon as a whole, for specific sub-basins, or for any individual segment. With both tidal and L 178 Flexible Grid Model nontidal exchanges, the fraction of new ocean water at a given location can fluctuate significantly over time scales on the order of several hours to several days, as new ocean water slowly replaces lagoon water. Thus, Rso is defined to be the midpoint between the first time the new ocean water fraction reaches 50% and the last time a fraction below 50% is recorded.

The Model

The one-dimensional numerical model used to quantify flushing incorporates both advective and diffusive transport terms. In differential form, the time rate of change in an ocean water tracer, T, is given by

(5) where U' is the cross-sectionally averaged current speed in the x-direction relative to the movement of a flexible boundary separating two segments, and Kx is a longitudinal diffusion coefficient, given by

(6)

(McDowell and O'Connor, 1977), where Cr is Manning's roughness coefficient and R is the hydraulic radius. The ocean water tracer can be expressed in parts per thousand and treated exactly like salinity. The diffusion coefficient varies significantly in both space and time. The mean for the lagoon was about 40 m2 s-l, but mean values near inlets were nearly double this value. In the northern part of the northern sub-basin, the mean was 10-15 m2 s-l. The cross-sectionally-averaged current speed at segment boundaries is calculated using a finite difference form of the continuity equation:

(7) where the summation gives the hourly change of the volume of the lagoon landward of the lateral cross-section, and Am is the cross-sectional area of the seaward side of the mth segment. The cross-sectionally-averaged current speed is related to the relative speed given in Equation (5) by

U' = nUm, (8)

where n can vary from 0.0 to 1.0. A value of 0.0 produces a segment boundary that expands and contracts freely with the ebb and flood of the total current (tidal plus Smith 179

nontidal). This will restrict the accumulation of new ocean water to that resulting from longitudinal diffusion. A value of 1.0 produces inflexible (stationary) segment boundaries, and the accumulation of new ocean water will occur as a result of advection only. Both the central and southern sub-basins of Indian River lagoon are served by two .,.:_,.1-;,0_. inlets. In these cases, phase angles of the M2 tidal constituent (Smith, 1987) were used to determine the segments with maximum phase lags relative to tbe rise and fall of the tide at the nearest inlet, and Urn was calculated with the boundary condition that no long-term net transport occurred through the interior segment with tbe greatest phase lag.

Results

Results are sub-divided into two parts. In tbe first part, simulations trace the accumulation of new ocean water in tbe Indian River lagoon system, including Banana River lagoon, which is treated as a fourth sub-basin. Two simulations compare the assumptions of no intersegment advective transport (n = 0 in equation (8», then complete advective mixing (n = I). These assumptions correspond to completely flexible and completely inflexible segment boundaries, respectively. In the second part of this section, a series of simulations, based on the assumption of no advective transport, compares the rates of accumulation of ocean water individually for thl:: three sub-basins of Indian River lagoon and for Banana River lagoon. Figure" 3 shows the accumulation of new ocean water in the Indian River lagoon system under the assumption of completely inflexible segment boundaries (curve a) and completely flexible boundaries (curve b). Simulations were based on observed tidal conditions, and coastal sea level varied between +20 and -20 em with a period of 168 hours. Results indicate that at the end of a one-year simulation the renewal of new ocean water resulting from unrestricted advective transport (curve a) is 167% of the renewal occurring in response to transport associated with longitudinal diffusion only. The 50% renewal time indicated by curve (a) is approximately 140 days, while Rso for curve (b) has not been determined at the end of tbe simulation. Combining Banana River lagoon with tbe three sub-basins of Indian River lagoon to quantify flushing by lagoon-shelf exchanges for the Indian River lagoon system as a whole masks significant differences in flushing rates of individual sub-basins. A final series of simulations considers flushing individually for Banana River lagoon and for the three sub-basins of Indian River lagoon, as defined by the three inlets that connect the lagoon with shelf waters of tbe Atlantic Ocean. In these simulations, the three segments in direct contact with the three inlets were not included. They flush almost completely on each tidal cycle, and with the exception of segment 16 it was not clear how a given segment should be sub-divided and assigned to the two adjacent sub-basins. •

180 Flexible Grid Model

100 c o o CJl 80 o -..J C

L 60 Q) 4-' o ~ o (1) 40 (J)

c (1) () 20 L Q) tl-

O-+-----r---~--.__-__.--__._--____r--__r_--...... o 100 200 300 400 Simulation Length, Days

Figure 3. Composite showing the accumulation of new ocean water in the Indian River lagoon system (three sub-basins of Indian River lagoon plus Banana River lagoon), assuming complete mixing of water advected into a new segment (curve a) and no advection due to flexible segment boundaries (curve b). Simulations include tidal exchariges and assume coastal sea level variations of ±20 ern with a period of seven days.

Figure 4 summarizes results for the four sub-basins individually and reveals distinct differences in flushing rates. The southern and central sub-basins of Indian River lagoon accumulate new sea water at relatively rapid rates. Curves (a) and (b) indicate Rso values of approximately 5 and 12 days, respectively. In the lower part of the plot, curves (c) and (d), representing the northern sub-basin of Indian River . '-. lagoon and Banana River lagoon, respectively, indicate decidedly slower flushing rates. By the end of the one-year simulation, both sub-basins contain less than 30% new ocean water. It is of interest to note that initially the accumulation of new ocean water in Banana River lagoon is much slower than the accumulation of new ocean water in the northern sub-basin of Indian River lagoon. This is because Banana River lagoon gets its ocean water through segment 4 of Indian River lagoon (see Figure 1), which in turn is well removed from Sebastian Inlet. Once new ocean water has begun to accumulate in segment 4, however, Banana River lagoon accumulates ocean water faster than does the northern sub-basin of Indian River lagoon. This is because the total volume of segments 1-3 in Indian River lagoon is one-third greater than the total volume of Banana River lagoon. The crossing of the curves after about 340 days is probably not physically meaningful, however, given the very slow rates at which both sub-basins accumulate new ocean water through Sebastian Inlet. Smith 181

(0) 100 c -- (b) if) 0 D I 80 D :J VI c -- 60 "- Q) ->--' 0 S -10 0 Q) VI (c) _~ c 20 (]) u ------.------"------Q) ~ ~(d) ll- // 0 -- 0 100 200 300 400 Days

Figure 4. Same as Figure 3, but for the southern (curve a), central (curve b) and northern (curve c) sub-basins of Indian River lagoon; and for Banana River lagoon (curve d)_ Simulations assumed a completely flexible grid and no net advcctive transport of sea water.

Discussion

Results from this study quantify the expected differences between flushing rates calculated with unrestricted advective transport, and flushing rates calculated with no advective transport of water between segments. While the flushing rates themselves, whether for the lagoon system as a whole or for individual sub-basins, are undoubtedly subject to revision in view of the roles played by freshwater outflow (Smith, 1993c), wind stress (Sheng et al., 1990) and density currents (Smith, 1990), the divergence of the curves in Figure 3 suggests that the specification of transport is an important feature of any flushing model. While it is tempting to follow Wood (1979) and use partial advective transport (i.e., intermediate values of n) to represent intersegment transfers associated with freshwater outflow or with wind-driven or density-driven transport, such processes are more appropriately investigated explicitly with a hydrodynamic model than i implicitly with a model based on the continuity equation. Similarly, it is better to I L

- ~ < {~-, 182 Flexible Grid Model

use a model that sub-divides lateral cross-sections to account explicitly for flood­ dominant and ebb-dominant layers and channels than to use values for n (equation 8) lying between zero and unity. Thus, as hydrographic data become available to verify simulations, it will be preferable to pursue the investigation of lagoonal flushing with a hydrodynamic model than to search for a best-fit value of n. lbree limitations should be kept in mind when using a model based on the continuity equation. First, while one can reproduce tidal flushing with considerable accuracy, one learns more about effects of tidal flushing than about its causes. The dynamics of flushing are not addressed. A second limitation stems from the model's inability to incorporate freshwater effects, whether associated with rainfall, surface runoff or groundwater seepage. The boundary condition specifying the speed of the longitudinal flow relative to the speed of the segment boundary (equation 7) determines what percent of the longitudinal transport will be retained in a given segment. Thus, when a segment contains a quasi-steady freshwater source, even a partial retention results in the quasi-steady growth of the segment. Segments in the interior of the lagoon will eventually expand to the point where they force other segments out of the lagoon through the inlets. Only with tidal and/or low frequency co-oscillating flow patterns will segments alternately expand and contract, and thereby retain manageable sizes. Finally, this approach does not quantify the long­ term net transport through the interior of a lagoon between inlets of unequal size (van de Kreeke and Cotter, 1974). In the case of Indian River lagoon, however, this will occur only in the central and southern segments of the lagoon, where flushing rates are already relatively high. Even with a model that is restricted to flushing by lagoon-shelf exchanges, however, distinctly different rates at which the four sub-basins receive new ocean water (Figure 4) suggest that the regular renewal of ocean water in the central and southern sub-basins occurs substantially faster than in the northern sub-basins, including Banana River lagoon. Indeed, incorporating freshwater sources and sinks has the effect of establishing a net outflow that opposes the importation of new "'. ocean water and reduces the 50% renewal time still further. Smith (1993c) found 50% renewal times of less than one week for the southern sub-basin of Indian River lagoon, but still well in excess of one year for the northern sub-basin under high rainfall conditions.

Acknowledgement. Water level records from 17 of the 31 study sites were collected and provided by the National Ocean Service (NOS); additional water level records were provided by the Florida Medical Entomology Laboratory in Vero Beach, Florida. Ms. Elizabeth Smith assembled and edited the NOS portion of the data base. Partial support for this work was provided by the Florida Department of Environmental Regulation through Contract No. CM-llS, and by the Florida Department of Natural Resources through Grant Agreement No. 6598. Harbour Branch Oceanographic Institution, Contribution Number 1012. Smith 183

'.- ".- -.' ~ References

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