Applying finite-time lyapunov exponent to study the tidal dispersion on oil spill trajectory in Burrard Inlet
Graphical Abstract
Introduction
Burrard Inlet is a fjord within the city of Vancouver, Canada and is home to Canada’s largest port by tonnage: Port Metro Vancouver (PMV). More than 80% of the inlet’s shoreline is port-industrial, including railyards, terminals for containers and bulk cargo ships, grain elevators, and oil refineries (Haggarty, 2001). Growing environmental concerns for this region include effluent from sewage outfalls, floating debris, and oil spills. (Li and Hodgins, 2004, Stone et al., 2013, Kako et al., 2014, Jambeck et al., 2015, Zhong et al., 2018). However, oil spills have become a dominant issue over the past decades due to the increased transport of petroleum products (Ventikos and Psaraftis, 2004). Moreover, there are plans to increase the capacity for oil transportation, which further increases the risk of oil spills in the region. Should a spill occur, then accurately tracking the spilled oil in PMV would be necessary for a quick and effective response to the spill, mitigating the potential for a severe environmental disaster. Such tracking can be done with computer modelling. However, following the trajectory of an oil spill or predicting the trajectory with a numerical model in coastal areas is not easy. The trajectory is related to the pollutant’s physicochemical properties and depends on tidal currents and dispersion, and transporting the pollutants through estuaries and coastal environments is also associated with tidal dispersion (Geyer and Signell, 1992). Tidal dispersion occurs when the periodic tidal oscillations spread a patch of particles due to the spatial variability of the tidal currents or the background mean flow. This tidal process scatters or dilutes water parcels (U.S. Army Corps of Engineers, 2012). Therefore, to accurately predict the trajectory of an oil spill, it’s necessary to understand the complex relationship between tidal dispersion and an oil spill’s trajectory.
Tidal dispersion is commonly studied using Lagrangian techniques, drifters, and virtual trajectories in a numerical model of ocean flow fields. Over the last two decades, researchers have studied tidal dispersion and its impact on particle transport in estuaries (Rhodes, 1950, Hoitink, 2003, Maity and Maiti, 2016, Ohlmann et al., 2017, Pawlowicz et al., 2019). For instance, researchers tracked surface drifters to estimate residence times, dispersion, and trapping in the Salish Sea, and they concluded that dispersion was crucial to particle transport and is often far more important than mean advection (Pawlowicz et al., 2019). Furthermore, with computer modelling, it has become possible to simulate flow fields with high spatial and temporal resolutions, and, as technology improves and computing power increases, modelling has become a viable, convenient, and economical option for studying dispersion. Several numerical modelling studies on tidal dispersion have been conducted (Geyer and Signell, 1992, Signell and Butman, 1992, Xu and Xue, 2011), with one study noting that the scale of a tidal excursion length relative to the spacing between major bathymetric and shoreline features was the most important effect of the tidal current on tidal dispersion (Geyer and Signell, 1992).
The tidal excursion length is the net horizontal distance a water particle travels from slack low water to slack high water, or vice versa (Ji, 2017). Tidal excursion length is a valuable indicator of hydraulic and mixing characteristics of estuaries (Savenije, 2006), and is usually used to study the movement of pollutants in estuaries over a tidal cycle (Barrett, 1972, Lucas et al., 1999, Schramkowski et al., 2002, Schramkowski et al., 2004, Hibma et al., 2003, Kho et al., 2009, Valle-Levinson, 2013, Ji, 2017). If the tidal excursion were larger than the typical spacing of the topographic features, tidal dispersion could significantly contribute to the flushing of an estuary. Otherwise, the dispersion might only be local without any influence on the overall flushing of the estuary (Geyer and Signell, 1992).
In addition to the length of the tidal excursion, the residual current of the tide is important to tidal dispersion in an estuary. For instance, Xu and Xue (2011) investigated the tidal circulation in Cobscook Bay at different tidal phases and found that the magnitude of the residual currents and the tidal excursion could estimate the effective dispersion coefficient (the degree of separation of particles within the cluster). Xu and Xue found that small residual currents and short tidal excursion limited the tidal dispersion and produced small effective dispersion coefficients, and large residual currents and long tidal excursion enhanced tidal dispersion and produced large effective dispersion coefficients (Xu and Xue, 2011).
Although several advanced methods have been proposed for understanding particle transport using Lagrangian particle trajectories in ocean modelling (De Young and Pond, 1987, Stacey et al., 2002, Stacey and Pond, 2003, Wu et al., 2019), obtaining the Finite-Time Lyapunov Exponent (FTLE) fields and visualising the FTLE ridges is arguably one of the most powerful tools for investigating particle movement in complex flows. Lorenz (1963) first proposed the finite-time method based on the Lyapunov Exponent. After that, Haller, 2001, Haller, 2002 used the ridges of the FTLE field to derive a Lagrangian Coherent System (LCS), which delineates the boundaries between attracting and repelling particles. Scalar FTLE values indicate the convergence and divergence rates between neighbouring particles after a given time interval at a given location in a time-varying flow field (Haller, 2001). High FTLE values indicate ridges when the FTLE field is mapped, and the ridges act as the mathematical definition of an LCS. The ridges represent the most kinematically active material lines of the flow fields (Shadden et al., 2005, Allshouse and Peacock, 2015), so no flux can cross these LCS lines. In other words, LCS lines represent transport boundaries in the flow field. Therefore, mapping the FTLE field not only forecasts the trajectory of particles, but the field predicts the structures that organise the entire flow and determine transport mechanisms in unsteady flows (Brunton and Rowley, 2010, Morel et al., 2014). Compared to traditional modelling approaches, calculating FTLE values does not require having high data accuracy or interpolating the velocity field. Furthermore, FTLE analysis results are not sensitive to data errors.
FTLE and an LCS have been widely used for understanding transport behaviour in unsteady flows (Haller, 2001, Haller, 2002, Orre et al., 2006, Burger et al., 2008, Ferstl et al., 2010, Lipinski and Mohseni, 2010, Sadlo et al., 2011, Fiorentino et al., 2012, Huhn et al., 2012, Suara et al., 2020). For instance, Huhn et al. (2012) studied horizontal Lagrangian surface transport in the Ria de Vigo by comparing real particle trajectories to the LCS. Huhn et al. demonstrated that LCS analysis was helpful for understanding flows when direct velocity field measurements (such as HF Radar) were unavailable. Suara et al. (2020) utilised FTLE to explain the fate of floating material in a coastal tidal embayment. In Suara et al.’s study, FTLE unveiled the pathways for floating transport. Suara et al. found that different tidal phases had different flow structures and locations of the saddle points. The authors also studied the effects of wind on the LCS and stated that near a barrier, the wind increased the rates of contraction and expansion and decreased the mixing strength. Furthermore, the wind changed the particle transport directions. In addition to Suara et al.’s work, FTLE and LCS methods successfully helped researchers to understand how nutrients, pollutants, suspended sediments, and waterborne planktonic biota in coastal oceans would distribute and evolve (Olascoaga and Haller, 2012, Wu et al., 2017, Ku and Hwang, 2018, Ghosh et al., 2021). For example, Ghosh et al. (2021) used FTLE to predict areas of spontaneous material accumulation in Moreton Bay, on the east coast of Australia. Based on the frequency of the FTLE, the authors identified eleven potential areas of spontaneous and persistent accumulation of material in Moreton Bay. They compared the FTLE results with observed areas of debris and found that the identified areas were consistent with their simulated debris accumulation regions (Ghosh et al., 2021).
Most studies of oil spill trajectories in Burrard Inlet, British Columbia, have relied on field observations or simulations from oil spill models (Genwest System Inc, 2015). Although several attempts have been made to track trajectories based on data from Lagrangian observing platforms in the ocean or using simulated Lagrangian particle trajectories from circulation models (Zhong et al., 2018), there are no comprehensive studies using the FTLE method, residual currents, and tidal excursion length to track spill trajectories. Thus, to fill the shortcoming, this study examined how tidal dispersion influences oil spill transport in Burrard Inlet, and FTLE values, residual currents, and tidal excursion length were used as the indicators. This work started by using simulations from a high-resolution numerical model, the Finite-Volume Community Ocean Model (FVCOM), and experiments with surface drifters to explore the influence of tidal dispersion on particle transport. Then, a case study of a simulated oil spill was conducted to verify the particle transport observations. To the best of our knowledge, this is the first attempt at combining FTLE values, residual currents, and tidal excursion length, with a real case study to investigate the influence of tidal dispersion on oil spill trajectories.
Section snippets
Burrard Inlet
The city of Vancouver on the southwest coast of British Columbia, Canada, centres around Burrard Inlet. At its western end, Burrard Inlet connects to the Pacific Ocean via the Strait of Georgia. The inlet is narrow and extends about 30 km inland from the Strait of Georgia to the Indian Arm. In this study, Burrard Inlet (Fig. S1) was defined to include the Outer Harbour, First Narrows (around 880 m wide), the Inner Harbour (about 8.8 km long; also called Vancouver Harbour), Second Narrows
Results
In this study, the FTLE results were normalised with the maximum FTLE (FTLE/FTLEmax) for the forward and backward fields separately (Suara et al., 2020). Then, the forward and backward FTLEs were combined to obtain the hyperbolic FTLE field (see Eq. (4)) (d’Ovidio et al., 2004). This combined FTLE field provided the potential hyperbolic LCS and dominant hyperbolic saddle points, where repelling and attracting material lines meet. A hyperbolic LCS is locally the strongest repelling or attracting
Tidal Dispersion Properties
Several observations and simulations demonstrated that particle trajectories in coastal areas are very sensitive to their initial release locations and dates (Cheng and Casulli, 1982, Signell and Butman, 1992, Brooks et al., 1999, Xu et al., 2006, Beron-Vera and Olascoaga, 2009, Xu and Xue, 2011). FTLE fields, which can be considered sensitivity maps for initial conditions over a finite time interval (Morel et al., 2014), directly address this sensitivity. The FTLE analysis showed that the
Conclusions
This study combined FTLE values, residual currents, and tidal excursion length to explore the influence of tidal dispersion on oil spill transport in Burrard Inlet, BC. The FTLE results showed that the spatial structures of the FTLE fields were sensitive to the differences in the strength of the tidal currents during spring and neap tides, and the FTLE fields were sensitive to the phases of the tide at the time of the particle release. However, there were five consistent features: (1) material
Funding
The Marine Environment Observation Prediction and Response Network (MEOPAR) [Project: 2–02–03–037.3] and Multi-Partner Research Initiative (MPRI) [Project: 6.02] supported this research.
CRediT authorship contribution statement
Conceptualization of this paper was by Xiaomei Zhong, Yongsheng Wu, and Haibo Niu. Xiaomei Zhong analyzed the velocity data and conducted the oil spill simulation, and pre- and post-processed the results under the supervision of Charles Hannah, Haibo Niu, and Yongsheng Wu. Charles Hannah provided drifter observations. Shihan Li helped with the analysis of the FVCOM model results. Writing -original draft was done by Xiaomei Zhong. The review & editing was by Charles Hannah, Yongsheng Wu, and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (74)
- et al.
Tidal circulation and residence Time in a Macrotidal Estuary: Cobscook Bay, Maine
Estuar. Coast. Shelf Sci.
(1999) - et al.
Using Lagrangian Coherent Structures to understand coastal water quality
Cont. Shelf Res.
(2012) - et al.
Tidal-jet and vortex-pair driving of the residual circulation in a tidal estuary
Cont. Shelf Res.
(1994) - et al.
Persistency of debris accumulation in tidal estuaries using Lagrangian coherent structures
Sci. Total Environ.
(2021) Distinguished material surfaces and coherent structures in three-dimensional fluid flows
Phys. Nonlinear Phenom.
(2001)A variational theory of hyperbolic Lagrangian Coherent Structures
Phys. Nonlinear Phenom.
(2011)- et al.
Numerical modelling of shoal pattern formation in well-mixed elongated estuaries
Estuar. Coast. Shelf Sci.
(2003) - et al.
Horizontal Lagrangian transport in a tidal-driven estuary—Transport barriers attached to prominent coastal boundaries
Cont. Shelf Res.
(2012) - et al.
A decadal prediction of the quantity of plastic marine debris littered on beaches of the East Asian marginal seas
Mar. Pollut. Bull.
(2014) - et al.
Characterizing chaotic dispersion in a coastal tidal model
Cont. Shelf Res., Recent Dev. Phys. Oceanogr. Model.: Part III
(2006)
Classical tidal harmonic analysis including error estimates in MATLAB using T_TIDE
Comput. Geosci.
Lagrangian observations of estuarine residence times, dispersion, and trapping in the Salish Sea
Estuar. Coast. Shelf Sci.
The effect of geometry and bottom friction on local bed forms in a tidal embayment
Cont. Shelf Res.
Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows
Phys. Nonlinear Phenom.
Material and debris transport patterns in Moreton Bay, Australia: The influence of Lagrangian coherent structures
Sci. Total Environ.
Spill accident modeling: a critical survey of the event-decision network in the context of IMO’s formal safety assessment
Journal of Hazardous Materials
Representing kelp forests in a tidal circulation model
J. Mar. Syst.
Refining and classifying finite-time Lyapunov exponent ridges
Chaos
Predicting the effect of pollution in estuaries
Proc. R. Soc. Lond. B
An assessment of the importance of chaotic stirring and turbulent mixing on the West Florida Shelf
J. Phys. Oceano
Fast computation of finite-time Lyapunov exponent fields for unsteady flows
Chaos Interdiscip. J. Nonlinear Sci.
A finite volume numerical approach for coastal ocean circulation studies: Comparisons with finite difference models
J. Geophys. Res.: Oceans
An unstructured grid, finite-volume, three-dimensional, primitive equations ocean model: application to coastal ocean and estuaries
J. Atmos. Ocean. Technol.
On Lagrangian residual currents with applications in south San Francisco Bay, California
Water Resour. Res.
Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents
Geophys. Res. Lett.
The internal tide and resonance in Indian Arm, British Columbia
J. Geophys. Res.: Oceans
Interactive separating streak surfaces
IEEE Trans. Vis. Comput. Graph.
A high-resolution assimilating tidal model for the northeast Pacific Ocean
J. Geophys. Res.: Oceans
A reassessment of the role of tidal dispersion in estuaries and bays
Estuaries
Lagrangian coherent structures from approximate velocity data
Phys. Fluids
Cited by (1)
Simulating dispersal in a complex coastal environment: the Eastern Shore Islands archipelago
2024, ICES Journal of Marine Science