Ocean Dynamics DOI 10.1007/s10236-010-0371-6
A numerical study of horizontal dispersion in a macro tidal basin Danya Xu & Huijie Xue
Received: 12 August 2010 / Accepted: 8 December 2010 # Springer-Verlag 2011
Abstract Tidal circulation in Cobscook Bay, a macro tidal basin, is simulated using the three-dimensional, nonlinear, finite element ocean model, QUODDY_dry. Numerical particles are released from various transects in the bay at different tidal phases and tracked for several tidal cycles. Initially, nearby particles in the main tidal channel experience a great deal of spreading and straining, and after a few tidal cycles, they are separated in different parts of the bay. The fundamental mechanism for particle dispersion is the chaotic advection that arises from long tidal excursions passing through many residual eddies. A loosely correlated, inverse relationship between the two dimensionless parameters, ν (the ratio of the residual current to the tidal current) and l (the ratio of the tidal excursion to the main topographic scale), can be constructed for large values of ν. Several Lagrangian statistical measures are used to quantify and distinguish dispersion regimes in different parts of Cobscook Bay. It is found that the effective Lagrangian dispersion coefficient can be estimated using the product of the magnitude of residual currents and the tidal excursion. Keywords Tidal dispersion . Langrangian chaotic advection . Langrangian dispersion coefficients . Cobscook Bay Responsible Editor: Tal Ezer This article is part of the Topical Collection on 2nd International Workshop on Modelling the Ocean 2010 D. Xu : H. Xue (*) School of Marine Sciences, University of Maine, Orono, ME 04469-5706, USA e-mail:
[email protected] D. Xu Center for Environmental Sensing and Modeling (CENSAM) Singapore-MIT Alliance for Research and Technology (SMART), S16-05-08, 3 Science Drive 2, Singapore 117543, Singapore
1 Introduction Mixing, stirring, and transporting of particles in tidal basins and estuaries are not only interesting scientific problems but also important processes that control the distribution and fate of nutrients, suspended sediments, pollutions, and other waterborne planktonic biota in coastal oceans. Two basic mechanisms of horizontal dispersion, shear dispersion and chaotic advection, have traditionally been proposed to interpret particle mixing and dispersion in fluid environments. Shear dispersion results from a combined effect of shear velocity and turbulence mixing, which leads to the horizontal dispersion of a patch of particles (Taylor 1954). On the other hand, chaotic advection suggests that significant horizontal dispersion and mixing can be induced in oscillatory flows (Aref 1984; Ottino 1989). However, most of these idealized models cannot be applied directly to complex tidal basins or estuaries in the real world where the velocity field is organized in many different time and length scales (Zimmerman 1986). In coastal oceans and estuaries, dye experiments showed that the relationship between the horizontal diffusivity and the scale of tidal motions follows the Richardson’s “4/3 law”, i.e., the horizontal effective diffusivity (1–100 m2 s−1) is proportional to the 4/3 power of the scale of the motion (Okubo 1971). Several in situ investigations have been conducted over the past decades. For example, dye experiments were carried out to estimate the near-bottom diffusivity across the foot of the shelf-slope front (Houghton 1997) and across the Oregon shelf (Dale et al. 2006) and in the bottom boundary layer near the tidal mixing front on Georges Bank (Houghton and Ho 2001) and in Hudson River (Chant et al. 2007). During the coastal mixing and optics experiment, both fluorescein and rhodamine WT dyes were released in the southern New England shelf to examine the diapycnal diffusion (Ledwell et al. 2004) and the lateral diffusion
Ocean Dynamics
(Sundermeyer and Ledwell 2001). In particular, Sundermeyer and Ledwell (2001) concluded that the shear dispersion and the dispersion due to interleaving water masses cannot account for the observed dye distribution, and they speculated that dispersion by episodic vortex motions might be responsible in some cases. The use of numerical models to study tidal dispersion can be traced back to the early 1980s. Awaji et al. (1980) and Awaji (1982) modeled a large number of particles in a tidally dominated strait and found that even without considering the subgrid scale turbulence, spatial variations in magnitude and phase of the oscillatory current can still give rise to large particle displacements and the mixing coefficient can reach as high as 800 m2 s−1. Stochastic motion associated with turbulence may enhance the horizontal dispersion. Signell and Butman (1992) calculated tidal exchange and dispersion in Boston Harbor and demonstrated that the rapid exchange and mixing occurred near the inlets of the harbor due to the asymmetry in the ebb and flood response. The tidal residual circulation was not sufficient to explain the net transport of water in regions where the tidal flow is strongly nonlinear. Geyer and Signell (1992) reviewed various tidal dispersion theories and suggested that the effectiveness of tidal dispersion depends on the relative scale of the tidal excursion to the spacing between major bathymetric and shoreline features. Flow separation and transient eddies depend highly on position and tidal phase, which makes parameterization of the eddy diffusion coefficient almost impossible. Ridderinkhof and Loder (1994) analyzed the Lagrangian circulation in the outer Gulf of Maine and found chaotic regimes in both tidal and residual flows when small-scale topography features on Georges Bank were resolved. Proehl et al. (2005) estimated the cross- and along-bank dispersion rate by simulating an ensemble of particles in the three-dimensional advective and diffusive field on Georges Bank. They found that the ensemble is essentially non-diffusive off the bank, whereas the dispersion rate is enhanced by vertical diffusion as the ensemble encounters the front. Bilgili et al. (2005) found that the mixing and exchange in and out of the Great Bay Estuary can be increased considerably by adding a diffusivity as low as 1 m2 s−1, but further increase of diffusivity to 10 m2 s−1 has little effect on the overall exchange characteristics. Orre et al. (2006) used different Lagrangian measures to depict spatial variations of mixing in the western shelf of Norway and found that particles initiated in nearby locations may experience very distinct evolutions and that chaotic mixing properties are highly inhomogeneous over the whole shelf. Aref (1984) found that Lagrangian trajectory might be chaotic in particular circumstances, which can lead to a large dispersion even in completely deterministic, periodic
flows without any turbulent motion. Chaotic advection (also known as Lagrangian chaos) was thought of as the principal mechanism that influences the Lagrangian transport due to interaction of tidal currents and complex topographic features (Zimmerman 1986; Beernes et al. 1994). Zimmerman (1986) suggested that the shear dispersion alone might underestimate the dispersion process. In a macro tidal basin like Cobscook Bay, particle trajectories show a great deal of sensitivity to release location and time, and they tend to be unpredictable even under pure advection (Brooks et al. 1999; Xu et al. 2006). We hypothesize that chaotic advection plays an important role in particle dispersion in Cobscook Bay. Cobscook Bay (Fig. 1) is located in the easternmost part of Maine, USA, bordering Canada’s maritime province New Brunswick. It connects with Passamaquoddy Bay and the offshore water at the mouth of the Bay of Fundy. Generally, Cobscook Bay, Passamaquoddy Bay, and the adjacent water and land areas are called the Quoddy region. From west to east, the bay is commonly divided into the inner, central, and outer bay, which are connected via a narrow ( ¼ uðxi ; yi ; tÞ < dt > : dyi ¼ vðx ; y ; tÞ i i dt
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Here the subscript i denotes the ith particle and (xi, yi) is the position of the particle. In this study, dispersion due to subgrid scale mixing is excluded. (u, v) thus represents the instantaneous velocity at the location of the particle (xi, yi), which can be interpolated from the ocean model, QUODDY_dry. A fourth-order Runge–Kutta scheme is used for tracking particles numerically (Blanton 1995). The time step used in the particle-tracking algorithm is 24 s and the particle positions are recorded every 6 min. Once a particle hits the shore or the sea floor, its last active position in the water is recorded and the particle is then eliminated from further computation. 2.3 Hourly release experiments In theory, particles need to be released at every node and every time step, and then followed for some tidal cycles. This is unrealistic for field observations and is also computationally overwhelming in high-resolution models. A possible compromise is that we divide the whole domain into several distinct subregions and release sufficient particles incrementally in these subregions to depict tidal dispersion characteristics. The advantage of this approach is that it dramatically reduces the computational expense. The
Ocean Dynamics Fig. 1 Bathymetry and mesh of the Cobscook–Passamaquoddy Bay model (a). The insert shows the Gulf of Maine with the Quoddy region highlighted by the circle. The small area enclosed by the rectangle is Cobscook Bay, shown in b with marked geographic features. Also shown in b are locations of 16 transects where particles are released every hour throughout a tidal cycle to depict dispersion behaviors in subregions of Cobscook Bay
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key to the selection of subregions and release times, however, is that they must neither sacrifice the generality nor alter statistical measures of dispersion characteristics. Following this principal, 16 transect lines are selected to represent both dynamically active and sluggish areas in Cobscook Bay. The lines are shown in Fig. 1b. The main tidal channel is separated by 7 transects (lines 5, 6, 7, 8, 14, 15, and 16) into the inner, central, and outer bay. In the outstretching arms of the inner and central bay, sluggish and active regions are also distinguished using the, more or less, east–west-oriented lines (lines 1, 2, 3, 4, 9, 10, 11, 12, and 13). Previous studies show that particles can end at different places even when they are released at the same place but different tidal phases (e.g., Cheng et al. 1982; Signell and Butman 1992; Brooks et al. 1999; Xu et al.
2006). In this study, particles are released on these transect lines in the model every hour from hour 48 to hour 60 to cover the whole tidal cycle for a total of 13 experiments.
3 Model results 3.1 Simulations of drifter experiments To validate the dynamic model and the Lagrangian tracking algorithm, we simulated and compared trajectories with two drifter experiments conducted on 7 October 2003 and 5 August 2004 (Fig. 2). The modeled current were compared very well with the observed current at the Gulf of Maine Ocean Observing System buoy J at 66°59′49″ W, 44°53′12″
Fig. 2 The observed (a, c) and the modeled (b, d) drifter trajectories. The upper panels show the experiment conducted on 7 October 2003 and the lower panels on 5 August 2004. Black dots represent the initial positions of drifters
Ocean Dynamics
N when a total of ten tidal constituents (M2, N2, S2, K1, O1, L 2, M 4, NU 2, 2N 2 , and K 2) were included in the simulations (Xu et al. 2006). Particles were also released in the model from locations and time corresponding to those of the drifter deployments. Both experiments lasted for about 6 h. The wind observed at Eastport (National Oceanic and Atmospheric Administration, National Ocean Service (NOAA/NOS), http://www.co-ops.nos.noaa.gov) was applied as the surface meteorological forcing in the drifter simulations. The simulated drifter trajectories (Fig. 2b, d) showed the split of drifters similar to the observed trajectories. However, in both experiments, the observed trajectories had crossovers not long after deployment (Fig. 2a, c), which the model did not reproduce, perhaps due to inadequate resolution of the model in Western and Head Harbor passages. During the second drifter experiment, the sole drifter in Western Passage was trapped in a small cyclonic eddy, but in the model the drifter only showed a curved path. The present model simulates the main tendency of the drifter trajectories. Moreover, the ending positions between the modeled and the observed drifters are close, from which the tidal excursion can be estimated for the intended dispersion study. 3.2 Tidal residual currents The semidiurnal lunar tide, M2, is the predominant tidal constituent in the Gulf of Maine and the large tidal range in the Quoddy region results from the near resonance of the Gulf of Maine and the Bay of Fundy to M2 (Garrett 1972; Greenberg 1979). In order to examine the tidally induced dispersion rather than to simulate real time tidal currents, analyses from this point on are focused on the experiments in which the surface wind forcing is ignored and only the M2 tidal wave is introduced at the open boundary because the primary objective of this study is to demonstrate that a simple oscillatory tidal current can lead to extraordinary horizontal mixing and dispersion. As the water depth in Cobscook Bay is generally shallow (less than 25 m), the model with M2 only reaches the equilibrium after being integrated for one to two tidal cycles. All simulations are run for 14 days. The model result over the fifth tidal cycle is averaged to depict the residual circulation. Figure 3 shows the vertically averaged residual currents in Cobscook Bay. Tidal residual flows in a large part of the bay are very small, less than 5 cm s−1, except in the main tidal channel especially those around Falls Island and in Lubec Narrows. Long arms stretch from the main tidal channel of Cobscook Bay northward and southward in an almost geometric symmetry. They appear to induce several pairs of counter rotating eddies in the residual current field on both sides of the main channel, especially near the entrance of narrow
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necks. In addition, many small eddies form in the inner and outer bay with diameters ranging from a few hundred meters to several kilometers. In the central bay, four eddies meet to form a butterfly configuration: a pair of small eddies (diameters less than 1 km) on the western side and a pair of larger eddies (diameters about 2–3 km) on the eastern side. The formation of the eddy dipoles has been attributed to the asymmetry of flooding and ebbing tide (Brooks et al. 1999) and the convergence/divergence of tidal currents in the main channel due to phase lags between the east and the west end of the bay (Xu et al. 2006). Together the pair of eddy dipoles results in convergence in the east–west direction and divergence in the north–south direction. The center where the pair of eddy dipoles meets is nearly stagnant, corresponding to a point of large uncertainty in the Lagrangian point of view (Samelson and Wiggins 2006; Wiggins 2005). This further complicates the transport process in the central bay. 3.3 Ratio of the residual current to the tidal current (ν) Ridderinkhof and Zimmerman’s (1992) kinematic model suggested that in an idealized tidal flow system with antisymmetric blinking vortices, particle dispersions are governed by two dimensionless parameters l and ν, where l is the ratio of the tidal excursion to the residual eddy diameter and ν is the ratio of the residual eddy velocity to the tidal velocity amplitude. Straining increases as ν increases, but chaotic advection emerges for certain values of l. The authors also showed signs of chaotic stirring in the Dutch Wadden Sea with l between 2 and 4 and ν between 0.05 and 0.3. We calculate the distribution of ν in the Quoddy region. As M2 is the only forcing, it is straightforward to calculate
Ocean Dynamics
the tidal velocity amplitude (i.e., the peak speed over a tidal cycle) and the residual velocity (Fig. 3) at a given location. ν is then calculated as the ratio of the latter to the former, which is smaller than 0.3 almost everywhere in the study area except for a small pocket in Western Passage where ν is ∼0.4 (Fig. 4). Compared with the residual currents shown in Fig. 3, it is obvious that high values of ν occur from the center to the shoreward side of the residual eddies. In Cobscook Bay, ν varies from 0 to 0.3, which is consistent with the range of values in the western Dutch Wadden Sea (Ridderinkhof and Zimmerman 1992) and the analytical range given by Beernes et al. (1994). 3.4 Tidal excursion In order to calculate the dimensionless parameter l, we need first to estimate the tidal excursion. Particles released in the outstretching arms of the inner bay tend to move coherently in the sluggish water without separating from the group over many tidal cycles (Xu 2008). In these areas, most of the particles have similar Lagrangian behavior and therefore a small number of particles released on each line might produce consistent statistics. But in energetic areas like the main tidal channel of Cobscook Bay, particles can be transported to more than 10 km away over half a tidal cycle. Meanwhile, strong tidal currents interact with spatially varying residual currents to result in significantly different trajectories even for particles that are released nearby (Xu et al. 2006). To determine the number of particles needed to represent the extent of spread over a tidal cycle, the first 250 particles are released equally distanced along the transect line 8. Distances traveled by the particles in 6-min intervals are calculated and are shown in Fig. 5a for a complete tidal cycle. The release is around early ebb. Due to the sheared
structure of the tidal jet, particles in the center of the channel are transported farther compared to those near both ends of the line and the displacements reach their maximum around the mid ebb. The incremental displacements gradually decrease to zero when slack occurs. All particles appear to drift systematically from their starting locations without much divergence during the first ebb. After that, the tide turns around during flood and particles are washed back into the bay. In this period, particles show a great deal more spread. There are obvious phase lags in reaching maximum displacement among particles. Few particles hit the shore (those with zero displacement in the middle of flood and remaining to be zero afterwards). Figure 5b shows a similar plot with only 25 equally distanced particles released along line 8 at the same time as in Fig. 5a. The initial line length is about 1.5 km and the initial separation between 25 particles on the line is about 60 m. This is roughly the model resolution at this location. Time evolution of the displacements between 25 particles and 250 particles are quite similar, suggesting that 25 particles released even in highly dynamical areas are likely to result in robust statistical measures. The tidal excursion is the sum of the displacements over a tidal cycle. Results from the 13 hourly releases are used to form the averaged tidal excursions. Although not shown, the averaged tidal excursions are often greater for particles near the middle of transects because of stronger tidal velocities there. Such a pattern, however, can be altered by the residual eddies so that the particles trapped in eddies have smaller averaged tidal excursions. The mean of the averaged tidal excursions from the 25 particles for each of the 16 transects is listed in Table 1. In order to compare different transect lines, hereafter we divide the 16 transects into 2 groups: the transects in the outstretching arms of the inner and central bay (i.e., lines 1, 2, 3, 4, 9, 10, 12, and 13 in Fig. 1b) as group 1 (G1) and the transects in the main tidal channel (i.e., lines 5, 6, 7, 8, 11, 14, 15, and 16 in Fig. 1b) as group 2 (G2). The mean tidal excursions vary from transect to transect with a range from ∼5 to ∼20 km. Generally, transects in G1 have smaller mean tidal excursions than those from G2. 3.5 Relationship between ν and l
Fig. 4 Distribution of ν (the ratio of the residual velocity to the tidal current amplitude) in Cobscook Bay
From the vorticity equation, Beernes et al. (1994) concluded that the parameter ν tends to decrease as l increases. Our estimates of ν in Cobscook Bay are consistent with their analytic range of 0–0.3. To calculate l, we need not only the tidal excursion (see Section 3.4 above) but also the size of the residual eddies. Unlike the idealized studies of Ridderinkhof and Zimmerman (1992) and Beerens et al. (1994), the diameter of the residual eddies cannot be readily deduced because most eddies are not circular as they are stretched or
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squeezed by the complex topography and coastline. On the other hand, bottom topography and coastline are known to determine the scales of residual eddies (Zimmerman 1978, 1980; Robinson 1981; Geyer and Signell 1992; Ridderinkhof and Loder 1994). The two-dimensional spectral analysis (e.g., FFT2 in MATLAB) can readily be used to determine topographic scales as in Salvador et al. (1999). The spectral peak occurs at wavenumbers ∼0.25 cycle per km (cpkm) and ∼0.19 cpkm in the W–E direction and N–S direction, respectively, which correspond to a wavelength of 3.3 km in the two-dimensional plane. This value can be thought of as the main topographic scale in Cobscook Bay. There are several other wavenumbers (0.5, 0.7, and 0.9 cpkm) with minor spectral energy concentrations corresponding to a series of smaller scale (2, 1.4, and 1.1 km) topographic features. When the averaged tidal excursions are compared with topographic scales, the dimensionless parameter l in Cobscook Bay ranges from 0.1 to 9 (Fig. 6). Most transects in G1 have smaller values of ν and l except for transects 10 and 12. Although the scatter-plots of the two estimated Table 1 Averaged residual speed, dimensionless parameter ν, tidal excursion, and the initial effective dispersion coefficient K for each of the 16 transect lines
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parameters do not show a tight correlation, it appears that ν is somewhat inversely proportional to l as suggested by Beerens et al. (1994), especially for larger values of ν and l associated with both G2 and G1. For a given transect, the range of ν is relatively small, whereas l can vary greatly. Take the transect mean values of ν and tidal excursion listed in Table 1 and exclude those transects with the averaged ν less than 0.07 (approximately the lower limit shown in Beernes et al. 1994), there exists an inverse relationship between the transect mean ν and tidal excursion with a correlation coefficient of −0.78 for the rest of the transects.
4 Statistical measures of dispersion Particle trajectories are useful in observing Lagrangian behaviors in a qualitative way, but to quantify dispersion processes in a tidal regime, a large number of trajectories from repeated releases are needed. In this study, particles are released in the model every hour from hour 48 to hour
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Ocean Dynamics Fig. 6 Scatter plots of the estimated dimensionless parameters ν and l for all particles released on the 16 transects. a For the transects in G1 and b for G2
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significant differences in line segment growth among releases initiated at different tidal phases. The envelope of growth is defined by the maximum (the thin solid line) and the minimum (the thin dashed line). The thick solid line between the limits shows the mean growth of all 13 releases from a particular transect. Figure 7a shows the line growth for transect 1, which represents the southern arm in the inner Cobscook Bay near Freds Islands (see Fig. 1b). Both tidal currents and residual flows in this part of the bay are weak. There is not much spatial variation of tidal currents either. Particles move back and forth following the rhythm of the semidiurnal tide. The net displacements of particles over a tidal cycle are very
60 at all 16 transects (see Fig. 1b) for a total of 13 experiments. Statistical measures are introduced to quantify the dispersion and transport of particles. 4.1 Line stretching Ridderinkhof and Zimmerman (1992) used the growth rate of line length to depict chaotic regimes. When chaotic advection occurs, the growth of a line segment is exponential rather than linear. Figure 7a–d are examples of line segment growth over the first 10 tidal cycles after their releases, from which one can distinguish different tidal mixing regimes and Lagrangian behaviors. There are
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Fig. 7 The mean, maximum, and minimum growth of line segments 1 (a), 8 (b), 9 (c), and 15 (d) normalized to their initial lengths, respectively. The thin solid curve corresponds to the release with the maximum growth of the line segment among the 13 releases, the thin dash curve is the minimum growth, and the thick solid curve is the averaged growth from the 13 releases
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counterparts in the central bay (Fig. 7b). After the period of fast growth, some particles are trapped in eddies (similar to segment 9 in Fig. 7c, see below); some are transported to sluggish waters or stranded; some may even move out of the domain (in which case the last position inside the domain is used). All these contribute to the slower growth at later times. Between the sluggish water in the outstretching arms of the inner bay and the actively stirred water in the mid and outer main tidal channel, there are areas in the arms of the central bay (lines 9, 10, 11, 12, and 13) where the line segment growth is slow initially, followed by a period of relatively fast growth before stabilizing (Fig. 7c). Apparently, particles originated in these areas are highly influenced by the pairs of residual eddy dipoles in the central bay (see Fig. 3) and the strong tidal jet in the main channel. Particles are trapped within the eddy dipoles in the beginning, which results in a slow growth of the line segment initially. A fraction of particles may escape from these eddies and be entrained in the tidal channel. Once there, the tidal jet can transport particles over large distances and more active dispersion arises, similar to those originated in the main tidal channel as shown in Fig. 7b, d. One measure of dispersion can be deduced from the mean growth of the line segments, which is shown in Fig. 8 for all 16 line clusters. In general, the lengths increase initially and gradually asymptote to different constants after about three to four tidal cycles. Note that the asymptotic values and the time taken to reach the asymptotic are affected by the choice of number of particles. However, the focus here is how fast the lengths grow in the first one to two tidal cycles because after that the particles, especially those in the main tidal channel, move away from their initial locations, and the dispersion and straining the particles experience are no longer representative of the original transect lines. It is clear from Fig. 8 that tidal stirring and dispersion are inhomogeneous across different parts of Cobscook Bay. In the main channel of the central and outer bay, the early growth is nearly exponential with an e-folding time of about one tidal cycle (clusters 8 and
limited. Water parcels tend to systematically drift away from their starting points during the first half of a tidal cycle and then move back to near their initial positions by the end of the second half of the cycle. Model simulations show that even after 10 tidal cycles, most particles are still locked in the inner bay without significant exchanges with other places. At the end of 10 cycles, the line element grows to ∼10 times its original length but still shows obvious periodic collapses at the end of each cycle. This suggests the ordinary advection driven by the simple rhythmic tide. The initial growth rate of the line segment is linear and there is no sign of chaotic stirring. The dispersion is greater for ebb releases than for flood releases in this region, implying the preferred flushing condition during local ebb in the southern arm of the inner Cobscook Bay. Of course, it is possible for fluctuations induced by wind or other causes to modify the dispersion, which will be a topic for future studies. In this region, l ranges from 0.1 to 2.5, ν is smaller than 0.05 (see Fig. 6a), from which one can conclude that tidal stirring and dispersion in the southern arm of the inner Cobscook Bay are weak and the numerical simulation illustrates that the residence time of this area is more than 5 days, consistent with the result of Brooks et al. (1999). In contrast, transect 8 is located in the energetic section of the main tidal channel along with transect 15. Particles spread very quickly all over the basin. Panels b and d of Fig. 7 depict the line growth with exponential rates in the first few tidal cycles. Tidal stirring occurs in ways that are far less predictable in the main tidal channel, which results in rapid exchanges with different parts of the bay (Xu et al. 2006). Particles on these transects have tidal excursions ranging from 20 to 30 km. Many clockwise and counterclockwise rotating residual eddies exist in the main channel with diameters from several hundred meters to a couple of kilometers. l varies from 5 to 9, while ν varies from 0.05 to 0.2 (see Fig. 6b). Due to large tidal excursions, a particle has more chances to modify its trajectory as it travels through a large number of residual eddies. The length growth rates suggest that the tidal stirring and dispersion in the outer bay (Fig. 7d) are even stronger than their Fig. 8 Growth of the line segments for G1 (a) and G2 (b)
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the transect line 12. This is because of not only the smaller tidal excursions but also less spatial variation of the residual currents to perturb the pathways of the particles of G1, which make the curves of G1 less convoluted than those of G2.
15). The exponential growth can last from the first to the third tidal cycle. Signell (1989) calculated the exponential growth of the perimeter of a rectangular particle array over 12 h and found that the e-folding scale is approximately half a tidal cycle. The difference between the e-folding scales can be attributed to different study areas that could naturally have different growth rates. Another reason for the differences is that Signell (1989) calculated the perimeter growth and we are calculating the line segment growth. Nevertheless, the exponential growth of line length is a clear sign of chaotic advection that can lead to significant horizontal dispersion according to Ridderinkhof and Zimmerman (1992).
4.3 Dispersion coefficient Besides the line stretching and line folding, we also calculated the absolute dispersion as defined by Provenzale (1999) and the relative dispersion as defined by LaCasce and Bower (2000) for the individual line clusters of particles (not shown). As functions of time, the absolute dispersion describes how a cluster of particles systematically drifts away from their starting locations, whereas the relative dispersion indicates how particles within the cluster separate from each other. Discussed in detail in Xu (2008), the absolute and relative dispersions can also be used to distinguish the different dispersion regimes between the outstretching arms and the main tidal channel of Cobscook Bay. For the clusters released in the outstretching arms, both the absolute and the relative dispersion collapse periodically at the end of each tidal cycle, suggesting regular advection in these sluggish waters. However, for the clusters released in the main tidal channel, both variances asymptotically reach much greater values in three to four tidal cycles, suggesting more disordered spreading. In short, all the statistical measures illustrate inhomogeneous dispersion and mixing in Cobscook Bay. In order to quantify spreading in the oscillatory flow and reveal the difference in dispersion for different parts of Cobscook Bay, we follow Signell and Geyer (1990) and define the effective Lagrangian dispersion coefficient as follows:
4.2 Line folding All transects are straight lines initially. They are deformed and distorted to become convoluted curves as time goes. The length growth alone cannot reflect how complex these curves become. On the other hand, the folding of a curve can be measured by how big an area the curve passes through. The box counting technique (a proxy for fractal dimension) is straightforward enough to be implemented (Dubuc et al. 1989). First, the whole domain is subdivided into N small square boxes. The number of boxes on land (L) is subtracted from the total number of boxes (N) so that there are N–L boxes as the water area. The boxes that a particle line passes through are shaded and counted (M) at the end of each tidal cycle and the percentage of the area the curve fills can be calculated simply as M/(N−L). Hence, the mixing areas can be easily compared between different particle lines. Three different box sizes, 160, 80, and 40 m, have been used with consistent results. Thus only the 160-m size box counting result is shown in Fig. 9. The growth of mixing areas illustrates a similar picture to the length growth shown in Fig. 8, such that the areas of potential mixing (the complexity of the curves) grow as the tidal cycle evolves. The proportionally mixed areas of G1 (Fig. 9a) are about 1/3 of those of G2 (Fig. 9b) except for
s 2x ðt þ T Þ s 2x ðtÞ s 2y ðt þ T Þ s 2y ðtÞ þ T T
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Fig. 9 Growth of the mixing area for the line clusters in G1 (a) and G2 (b). Percentage of the mixing area is defined as the ratio between the number of boxes that the particle line passes through and the total number of water boxes
1 KðtÞ ¼ 4
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Ocean Dynamics
where σx and σy are the variances of particle displacements in two directions, defined by ( ) N N n o X 1 X 2 2 2 2 s x ðtÞ; s y ðtÞ ¼ ðxi ðtÞ xðtÞÞ ; ðyi ðtÞ yðtÞÞ N i¼1 i¼1 ð3Þ T is the tidal period. (xi, yi), i=1, 2... N, are particle positions at time t, and is the position of the center of mass at time t. Figure 10 shows the effective dispersion coefficient, K, evolving with time for the 16 line clusters. Generally, the effective dispersion coefficient increases in the first one to two tidal cycles then declines. By definition, the effective Lagrangian dispersion coefficient is the rate of change in variance of two consecutive tidal cycles. The rate is high in the first one to two tidal cycles because the particles were close to each other initially. The particles continue to spread for many tidal cycles, but the rate of spread decreases despite that there might be mix-up of particles. Exactly when the effective dispersion coefficient starts to decrease may be affected by the number of particles. On the other hand, the definition is valid only in the first tidal cycle. After that, the particles, especially those in the main tidal channel, move away from their initial locations, and the dispersion and straining the particles experience are no longer representative of the original transect lines. For clusters in the outstretching arms of the inner and central bay (i.e., G1), values of the effective dispersion coefficient are usually low (less than 20 m2 s−1). The highest value can reach 25–40 m2 s−1 for clusters 10 and 12. This is similar to the value reported by Geyer and Signell (1992) for an idealized tidal headland. However, the maximum tidal currents and residual currents in Geyer and Signell (1992) are ∼1.5 and 0.3 m s−1, respectively. These values are similar to the maximum tidal currents and residual currents in the main tidal channel in Cobscook Bay, where values of the effective dispersion coefficient are much larger than those of G1. The clusters in the outer bay have higher values reaching 120 m2 s−1 (Fig. 10b). This
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Fig. 10 Similar to Fig. 8 but for the effective Lagrangian dispersion coefficient. a For the line clusters in G1 and b for G2
value of dispersion coefficient is larger than that found in Geyer and Signell (1992), which is probably due to much more regular topography and Eulerian flow fields in their idealized study hence less dispersal. The effective dispersion coefficient in the first tidal cycle is used to quantify the strength of horizontal dispersion in various parts of the bay (Table 1; Fig. 11a). As expected, clusters in G1 have much lower values (1–20 m2 s−1) than those in G2 (20–120 m2 s−1). The latter is still much smaller than the one Awaji (1982) reported (i.e., 800 m2 s−1), but it is approximately the same as the dispersion coefficient Ridderinkhof and Zimmerman (1992) reported for the western Dutch Wadden Sea (∼100 m2 s−1). The unusually large value of dispersion coefficient that Awaji (1982) obtained resulted likely from much larger tidal currents (maximum ∼2.5 m s−1) and residual currents (maximum ∼0.6 m s−1) in his model. Brooks et al. (1999) used the diameter of the central eddy dipoles (∼3 km) and the speed of the eddy dipoles (∼0.1 m s−1) and estimated the effective horizontal mixing coefficient (product of eddy diameter and eddy velocity) in the main channel of the center Cobscook Bay to be about 300–400 m2 s−1. It is also 2–3 times greater than our estimation. Zimmerman (1986) concluded that the dispersion coefficient as the product of eddy velocity and length scale could overestimate its actual value. He further suggested that there should be a constant of proportionality applied to this product, which is at most on the order of 0.1. We calculate the eddy dispersion (Ke =burL) as the product of the speed of residual currents (ur) and tidal excursion (L) for each transect line (see Table 1) with the constant b=0.1. The spatial variation of dispersion can be predicted reasonably well using the product of the residual current and the tidal excursion, which is shown as the dashed line with stars in Fig. 11a. Compared with the effective dispersion coefficient obtained using the variance of Lagrangian particles (solid line with circles in Fig. 11a), the similarities between the predictions are obvious and the correlation coefficient between these two curves is about 0.8 (Fig. 11b). We use the tidal excursion instead of the eddy diameter to estimate
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Ocean Dynamics 140
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Fig. 11 The effective Lagrangian dispersion coefficient in the first tidal cycle (solid line with circles) and the dispersion coefficient predicted using the product of residual currents and tidal excursion (dashed line with stars) for the 16 line clusters (a). Linear regression between these two estimations is shown in b
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the effective Lagrangian dispersion coefficient because the tidal excursion is a Lagrangian variable. In Fig. 11a, the tendency reflecting the dispersal property is dominated by tidal excursion such that small/large tidal excursions may limit/promote the horizontal dispersion in a tidal basin.
5 Summary Horizontal dispersion in a macro tidal basin, Cobscook Bay, is studied using the unstructured grid, finite element coastal ocean model, QUODDY_dry. When considering multiple tidal constituents and surface wind, the modeled drifter trajectories agree with observations (Fig. 2). Forced by the dominant tidal constituent M2, without considering the stochastic turbulence motion and wind forcing, particles are released on transects in different parts of the tidal basin at different tidal phases and tracked for several tidal cycles. The overall horizontal dispersion properties are obtained from the averaged results of a total of 13 hourly release experiments. Residual currents in Cobscook Bay show considerable spatial complexity with many small eddies. A pair of counter rotating eddy dipoles occur in the central bay and numerous small-scale eddies form around the headlands, convex and concave coastline regions from the inner bay to the outer bay (Fig. 3). The formation of these eddies is mainly due to asymmetric tidal currents interacting with the complex coastline and topography in the narrow channels of Cobscook Bay. Whether or not an eddy can affect nearby particles’ trajectories is determined by the size and strength of the eddy. The spatial distribution of the dimensionless parameter ν in Cobscook Bay (Fig. 4) suggests that the largest ν is biased on the shoreward side of the residual eddies. Larger ν implies that it is easier for eddies to pull particles away from the main tidal channel, which strengthens the mixing. On the other hand, particles entrained in these eddies may be trapped, which leads to a decrease in tidal excursion and an increase in residence time. For example, clusters 7 and 8 are both initially located in the main channel
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of the central bay, but line 7 cuts across the two counter rotating eddies on the western side. The averaged value of ν is 0.13 on line 7 and 0.08 on line 8. As a few particles of line 7 are trapped in the eddy dipole, the mean tidal excursion of line 7 is smaller than that of line 8 (see Table 1). The particle-tracking experiments show highly inhomogeneous mixing in Cobscook Bay. Two different mixing regimes (regular advection and chaotic advection) can be distinguished by examining Lagrangian statistical measures. In the outstretching arms of Cobscook Bay, tidal currents are smaller (∼0.5 m s −1 ), so are the residual currents (∼0.01 m s−1), which makes the dimensionless parameter ν very small (∼0.02). These regions can be thought of as sluggish waters where mixing is weak. Particle arrays released in these areas experience very little strain and distortion. In contrast, the vigorous tidal jet in the main channel (∼1.5 m s−1) can lead to large tidal excursions (∼20 km). During the journey over a tidal cycle, a particle traveling along a “would be deterministic” trajectory is perturbed frequently by residual eddies, which steer the particle away from the “would be deterministic” pathway. Current separation often occurs after passing the tip of peninsulas or around islands. Thus, initially, nearby particles may separate and enter different flows (e.g., trapped in different residual eddies) and lose connections thereafter. Similar processes repeat as the tidal cycle repeats, and the overall effect is that chaotic advection becomes the dominant mixing mechanism in areas with large tidal excursions. Unlike the idealized kinematic model of Ridderinkhof and Zimmerman (1992), in which chaotic advection can be determined by two dimensionless parameters ν and l, in a dynamic model, it is difficult to estimate the diameter of residual eddies due to their irregular distributions and shapes. As noted by Zimmerman (1978), Geyer and Signell (1992), and Ridderinkhof and Loder (1994), the horizontal dispersion is related to tidal excursions and scales of the major bathymetric and shoreline features. A twodimensional spectral analysis is applied in this study and the major topographic scale in the main channel of
Ocean Dynamics
Cobscook Bay is determined to be about 3 km. To characterize the chaotic advection, we adopt the dimensionless parameter ν and calculate l as the ratio of the tidal excursion to the topographic scale of the main tidal channel. Table 1 lists the averaged residual velocity, dimensionless parameter ν, tidal excursion, and the initial effective dispersion coefficient for each of the transect lines. An inverse relationship between the parameter ν and the tidal excursion (or l) agrees qualitatively with the analytic result of Beerens et al. (1994), but it can only be defined when ν is greater than 0.07 because in the real world too many topographic wavenumbers tend to muddle the relationship (see Fig. 6). Chaotic advection promotes particle dispersion and water exchange dramatically. Theoretical studies (see Ottino 1989) suggest that Lagrangian statistical measures like length growth and relative dispersion can be used to identify chaotic advection when the growth rates of these measures increase exponentially instead of linearly or in power laws. Numerical experiments conducted in this study depict exponential growths with an e-folding time scale of about one tidal cycle. The rapid growth of relative dispersion is usually a clear sign of chaotic advection. This generally occurs in areas with large tidal excursion and strong residual currents as indicated by the product of ur (speed of the residual current) and L (tidal excursion) shown in Fig. 11a. For particles that originate in the regular advection regime and subsequently drift into these areas, they might be affected by chaotic advection and an exponential growth of relative dispersion may occur in later tidal cycles. The chaotic advection leads to large horizontal dispersion (∼120 m2 s−1) in areas of large tidal excursion and strong residual eddies. In areas of small tidal excursion and weak residual currents, regular advection dominates and particles drift in a systematic way rather than being strained. Stochastic turbulence motion may enhance horizontal mixing and dispersion in a tidal basin. Bilgili et al. (2005) noted in their study of Great Bay that the addition of a small diffusion coefficient of 1 m2 s−1 can increase particle dispersion significantly, while a further increase of diffusion coefficient to 10 m2 s−1 does not enhance the overall exchange. Tidal currents are on the order of 50 cm s−1 in the Great Bay, much weaker compared to those in Cobscook Bay. How significantly the turbulence dispersion can modify the chaotic advection and how these two different mechanisms interact and compete with each other in different tidal regimes are interesting topics for a future study. Secondly, including the vertical dimension can be important in certain areas (e.g., near a front as suggested by Proehl et al. (2005)). Lastly, surface wind can interact with topography to result in residual currents in addition to enhanced mixing. It would be another interesting topic in future research to investigate wind-induced
dispersion in the absence or presence of tidally induced chaotic advection. Regardless, the message is clear that the Lagrangian effective dispersion coefficient is high enough in the absence of subgrid scale turbulence to represent an important mixing mechanism. The chaotic advection is expected to produce rapid water exchanges and large horizontal dispersion along the main tidal channel in many shallow tidal basins. Acknowledgments The authors thank Dr. David Greenberg (Bedford Institute of Oceanography) greatly for his generosity in sharing the ocean model QUODDY_dry and for his advice in using the model. Mr. Stephen Cousins (the University of Maine) and Mr. Randy Losier (St. Andrews Biological Station) provided frequent technical assistance to enable computations in this study. We thank Dr. David Brooks (Texas A&M University) for insightful discussions on the circulation in Cobscook Bay and Dr. Emmanuel Boss (the University of Maine) for valuable suggestions on Lagrangian statistics. The digitized topography of Cobscook Bay was provided by Dr. David Brooks. The drifter data were processed by Heidi Leighton and Will Hopkins at the Cobscook Bay Resource Center. We also thank two anonymous reviewers for constructive comments on the manuscript. This study was supported by the Maine Department of Environmental Protection grant (MOSAC06-02) to the University of Maine.
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