Rip current pulses tied to Lagrangian coherent structures

Transcription

1 Click Here for Full Article GEOPHYSICAL RESEARCH LETTERS, VOL. 37,, doi: /2009gl041443, 2010 Rip current pulses tied to Lagrangian coherent structures A. J. H. M. Reniers, 1,2 J. H. MacMahan, 3 F. J. Beron Vera, 1 and M. J. Olascoaga 1 Received 20 October 2009; revised 21 January 2010; accepted 28 January 2010; published 13 March [1] The trapping and ejection of surfzone floating material is examined by unveiling Lagrangian Coherent Structures (LCSs) hidden in the pulsating rip current surface velocity field produced by a three dimensional numerical model resolving wave group induced Very Low Frequency motions (VLFs). LCSs explain the typically observed patchiness of flotsam within the surf zone and the streaky distribution outside of the surf zone. The ejection of surfzone material occurs when filament like LCSs separate form the main ripcurrent circulation corresponding to a situation where eddies temporarily extend the rip current beyond the surf zone and subsequently detach. The LCSs support the idea that VLFs form the dominant exchange mechanism of surfzone floating material with the inner shelf. Citation: Reniers, A. J. H.M.,J.H.MacMahan,F.J.Beron Vera, and M. J. Olascoaga (2010), Rip current pulses tied to Lagrangian coherent structures, Geophys. Res. Lett., 37,, doi: /2009gl Introduction [2] The flow circulation on a rip channeled beach is far from steady [MacMahan et al., 2006]. Temporal variations induced by the changes in the incident wave conditions, tidal modulation, very low frequency (VLF) motions with periods of the order of 10 minutes, infragravity motions with periods of order 1 minute, as well as the incident swell with periods of order 0.1 minute all contribute to the dynamic signature of the rip current circulations. As a result it is not clear when it is likely for a swimmer, or any floating matter present within the surf zone, to be transported out of the surf zone. Although rip currents are generally viewed as a conduit for transport of surfzone material onto the inner shelf [Shepard et al., 1941; Inman and Brush, 1973], recent field experiments with GPS equipped drifters on a rip channeled beach in Sand City, Monterey Bay, showed a surprisingly high hourly retention rate of the drifters of about 80% [MacMahan et al., 2010]. This high retention rate was also observed at two other rip channeled beaches, Truc Vert, France and Perranporth, United Kingdom, respectively, and corroborated by numerical modeling results [Reniers et al., 2009]. During these experiments drifters within the surf zone tended to collect within rip current circulation cells 1 Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida, USA. 2 Department of Hydraulic Engineering, Delft University of Technology, Delft, Netherlands. 3 Department of Oceanography, Naval Postgraduate School, Monterey, California, USA. Copyright 2010 by the American Geophysical Union /10/2009GL thereby traveling in a circular fashion. In contrast, during the 2009 Rip Exchange (REX) experiment at Sand City, drifters released just outside of the surf zone often formed narrow streaks. This behaviour is consistent with the distribution of persistent foam generated by breaking waves (Figure 1) which tends to be patchy within the surf zone and streak like outside of the surf zone. The narrow streaks of flotsam originating from the turbulent surf zone, indicative of exchange between surf zone and inner shelf, are also observed on days without wind and therefore not related to the collection of surface floating material as a result of Langmuir circulations [e.g., Kukulka et al., 2009]. [3] In the following we examine what physical mechanisms are responsible for the surfzone trapping and occasional exit of surface drifters and the ensuing disjointed distribution of surface floating material in and outside the surf zone. Understanding the mechanisms behind surf zone trapping and ejection is clearly of importance to swimmer safety, but also for surfzone pollution, sediment transport, water quality and biology. However, such an understanding cannot be attained without the use of appropriate tools. In this work we consider Lagrangian Coherent Structures (LCSs), a novel dynamical systems notion [Haller, 2000; Shadden et al., 2005], which has recently been applied to shelf scale circulation systems [Lekien et al., 2005; Olascoaga et al., 2006; Lekien and Coulliette, 2007; Olascoaga et al., 2008; Beron Vera and Olascoaga, 2009; Shadden et al., 2009] to explain the pathways of surface floating material and potential trapping of pollutants and harmful algae. In layman s terms, LCSs are (almost) material curves (i.e., transport barriers) that, hidden in the flow, wholly control the motion of passively advected traces in unsteady flows. Consequently, by computing LCSs one can unambiguously identify the transport by rip current pulses, which is difficult if not impossible to be revealed by simple inspection of surfzone velocity snapshots [Reniers and MacMahan, 2008; Geiman et al., 2008, 2009]. LCSs are here computed based on model predicted surface velocities to study the effects rip current pulses associated with flow motions at the VLF time scale on the transport and fate of floating surfzone material. 2. Methods [4] The wave and flow modeling used to predict the unsteady surface velocity field is explained in detail by Reniers et al. [2009, and references therein]. The flow model is driven by wave momentum and pressure gradients varying on the wave group scale, thus resolving the mean and wave group induced vortical and infragravity motions, and calculates the Generalized Lagrangian Mean (GLM) flow velocity thus accounting for the Stokes drift [Andrews and McIntyre, 1978; Groeneweg and Klopman, 1998; 1of5

2 MANGEN software package ( by evaluating equation (1) on a regular grid with Dx = 2 m and Dy = 2 m. Trajectories are computed using a fourth order Runge Kutta Fehlberg integrator with a fixed time step Dt = 0.1 s. The required interpolations are carried out using a third order scheme. The time integration interval is set to t = 10 min representing the VLF flow dynamics observed in the rip current circulations. By considering t < 0 we are restricting attention to LCSs of attracting type, which are suitable to determine passive tracer pathways. Choosing a larger t would introduce long term diffusion and dispersion associated with the larger scale nearshore flow circulations, thereby obscuring the VLF contribution [Brown et al., 2009]. Figure 1. Picture of a rip current at the Sand City field site with narrow streaks of surface floating material (indicated by the sequence of green arrows) present outside the surf zone and more patchy distribution within the surf zone. Drifters used in the field experiment described by MacMahan et al. [2010] are shown on the beach. Walstra et al., 2000]. The three dimensional (3D) model utilized here has been verified with drifter inferred mean velocities and in situ measurements of both the mean and VLF flow obtained during the 2007 RCEX field experiment at Sand City [MacMahan et al., 2010], showing good agreement [Reniers et al., 2009]. Model predicted hourly retention rates of about 80% are in good correspondence with the observations provided that both Stokes drift and VLF motions were included in the prediction of the fate of surface floating material. Excluding VLF motions in the drifter trajectory calculations resulted in near complete retention indicating the relevance of VLFs in ejecting surfzone material. [5] However, even though the processes responsible for the drifter trajectories have been identified the question remains what mechanisms are responsible for the trapping and occasional surfzone exits at this beach. To that end the fate of surface floating material is examined by extracting LCSs from the nearshore flow field. Given the fact that synoptic unsteady flow fields are not available from observations, the verified 3D wave and flow model is used to hindcast the unsteady surface velocity flow field during the Sand City field drifter deployment on yearday 124 [MacMahan et al., 2010]. [6] LCSs can be identified with maximizing curves or ridges in Finite Time Lyapunov Exponent (FTLE) fields. The FTLE gives information on the maximum expansion or contraction rate for pairs of passively advected particles (e.g., representative for surface drifters) and is defined by t ðxþ ¼ jj 1 ln kr tþ ðxþk : Here kkstands for spectral norm and t t+t : x(t) x(t + t) where x(t) denotes the position at time t of a particle on the ocean s surface. The FTLEs are calculated with the t ð1þ 3. Results [7] Model predictions of surface floating material are based on the fate of virtual surface drifters propagated by the GLM velocity field (Figure 2). Positions of rip channels and shoals can be inferred from the narrow respectively wide separation between the 0 m and 1.5 m bottom contours. Drifters are initially uniformly seeded at 2 m intervals in the inner surf zone. The subsequent drifter response is strongly influenced by its initial position. Drifters that are initially present in the rip channels get advected in the offshore direction by rip currents. At locations of flow contraction, associated with the pairing of counter clock wise (CCW) and clock wise (CW) eddies in the rip channels, the drifters cluster together while traveling offshore (Figures 2a and 2b). Many of the outgoing drifters travel obliquely with respect to the rip channel orientation and remain entrained in the surfzone circulation. However, the presence of the VLFs changes the rip current orientation over time occasionally creating conditions that promote the rip current to extend beyond the surf zone (e.g., around Y = 0 m in Figure 2b). As time progresses, these offshore traveling drifters start forming streaks even though the underlying rip current flow field diverges as can be inferred from the vorticity patterns (e.g., around Y = 0 m). The drifter streaks are maintained as the drifters pass through the outer surf zone and become even more pronounced on the inner shelf (compare Figures 2c, 2d and 2e). In contrast, drifters that are initially located between rip channels and the shoals tend to become trapped within the surf zone forming patches of drifters propagating in a circular motion adjacent to the rip channels (e.g., around Y = 150 m and Y = 100 m). Drifters initially present near the center of the shoals travel toward the shore line followed by a lateral movement by means of the feeder flows ending up in the rip currents or the adjacent circulation resulting in very low drifter density at the shoal centers. The fact that some drifters make it out of the surf zone and others do not as function of their initial position suggests the presence of organized flow structures which will be examined by analyzing the computed LCSs. [8] The computed LCSs in the nearshore display many thin layers centered loosely around a core (Figure 3 and Animation 1 1 ). Each layer constitutes a transport barrier and as a result surface floating material, represented by the 1 Animation is available in the HTML. 2of5

3 Figure 2. Snapshots of GLM vorticity field in s 1 indicated by the colorbar in Figure 2a where warm (cold) colors correspond to CW (CCW) rotation. Corresponding computed drifter positions (a) 1, (b) 3, (c) 5, (d) 9 and (e) 17 minutes (black dots) after initial virtual drifter seeding for yearday 124 drifter field deployment. Bottom contours at 1.5 m depth and shore line (solid white lines) given as a reference. Approximate surfzone edge is indicated by the dashed white line. virtual drifters, is trapped between the layers moving in a circular fashion. Note that these structures are not apparent in the instantaneous GLM vorticity field snapshots shown in Figure 2. The space between the adjacent LCSs is often very narrow, resulting in the collection of surface floating material in thin streaks. This is amply demonstrated in the rapid transition just after the deployment of the virtual drifters (compare Figures 3a and 3b), where the initially widely distributed drifters quickly converge along the LCSs. Only in the inner core the drifters can move freely resulting in patch like distributions (e.g., around Y = 150 m). [9] At locations of rip currents the LCSs are elongated in the cross shore allowing the transport of drifters offshore. Once the drifters reach the offshore extent of the LCS they can only move in the alongshore direction, thereby generally re entering the surf zone. Only occasionally the filamentlike LCSs peel off and become detached from the inner layers. If this happens at the outer surf zone, material Figure 3. Snapshots of backward time FTLE field in s 1 indicated by the colorbar in Figure 3a and computed drifter positions (a) 1, (b) 3, (c) 5, (d) 9 and (e) 17 minutes (black dots) after virtual drifter seeding for yearday 124 drifter field deployment displaying the time evolution of attracting LCSs (FTLE ridges roughly corresponding to most intense green tones) associated with VLF dynamics. Initially uniformly distributed drifters quickly converge along the LCSs forming narrow streaks with occasional exits from the surf zone (indicated by the dashed white line). Bottom contours at 1.5 m depth and shore line (solid white lines) given as a reference. See Animation 1. 3of5

4 Figure 4. Detailed snapshot of virtual trajectories in the vicinity of the detaching VLF eddies calculated from the velocity field slice 10 minutes after initial virtual drifter seeding for yearday 124 drifter field deployment with an integration interval of 200 s. Underlying GLM vorticity field in s 1 indicated by the colorbar in the upper right corner where warm (cold) colors correspond to CW (CCW) rotation. Width of arrows corresponds to velocity magnitude and red tip indicates direction. Corresponding computed drifter positions indicated by the black dots. Approximate surfzone edge is indicated by the dashed white line. trapped by the detaching filament(s) is transported offshore outside of the surf zone (around Y = 0 m and Y = 250 m in Figure 3) with the drifters converging on the LCS forming a thin line (Figure 3e). [10] Computation (not shown) of repelling LCSs (i.e., ridges in forward time FTLE fields) as in the work by Shadden et al. [2006] reveals that the aforementioned (attracting) LCS corresponds to the outer edge of a VLF motion made up of an eddy pair that is slowly detaching from the rip current. The effect of the eddies on the transport of drifters can be unmasked by integrating from t = t 0 to t = t 0 + dt the vector field defined by the t = t 0 velocity field slice. The resulting virtual particle trajectories are shown in Figure 4. With an integration period of 200 s the corresponding trajectory lengths show a decrease toward the offshore edge of the VLF eddies resulting in a convergence of particles. Given the fact that at the same time the flow diverges along the edge, the particles are also transported along the interface resulting in an increasing streakiness and elongation of the drifter distribution (compare the drifter distribution in the corresponding area in Figures 3d and 3e). 4. Conclusions [11] The effect of VLFs on the ejection of surfzone floating material on a rip channeled beach has been assessed by calculating LCSs within the nearshore surface velocity field obtained with a verified three dimensional wave and flow model resolving the wave group dynamics. The LCSs explain the occasional exit of surface drifters from the surf zone due to VLF eddy motions as the outer filament like LCSs detach from the nearshore rip circulation. This occurs when the presence of VLFs extends the rip current flow beyond the surf zone and the corresponding VLF eddies subsequently detach from the main rip current. These results support the idea that this is the dominant exchange mechanism of surfzone floating material with the inner shelf for rip current flows. In addition, the frequently observed narrow streaks of remnant surface floating material outside of the surf zone on rip channeled beaches (Figure 1) is explained by the attracting LCSs associated with offshore eddies detached from the rip current flow. In contrast, the distribution of surface floating material within the surf zone can be quite patchy where drifters collect at the cores of the LCSs. This result is consistent with Talbot and Bate [1987] and MacMahan et al. [2010] who found largest concentrations of diatoms and drifters, respectively, in the center of rip current circulation. This difference in distribution is well explained by the LCSs underlying nearshore flow dynamics. [12] Acknowledgments. Reniers was supported by ONR contract N and the National Science Foundation OCE MacMahan was supported by ONR contract N , N , N WR20226, N WR20006, and the National Science Foundation OCE Beron Vera and Olascoaga were supported by NSF grants CMG and CMG We thank DELTARES for the use of their Delft3D software. References Andrews, D. G., and M. E. McIntyre (1978), An exact theory of non linear waves on a Lagrangian mean flow, J. Fluid Mech., 89(4), , doi: /s Beron Vera, F. J., and M. J. Olascoaga (2009), An assessment of the importance of chaotic stirring and turbulent mixing on the West Florida Shelf, J. Phys. Oceanogr., doi: /2009jpo Brown, J., J. MacMahan, A. Reniers, and E. Thornton (2009), Surf zone diffusivity on a rip channeled beach, J. Geophys. Res., 114, C11015, doi: /2008jc Geiman, J. D., J. T. Kirby, A. J. H. M. Reniers, J. H. MacMahan, J. W. Brown, J. A. Brown, and T. P. 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