Eddy-Induced Variability in Larval Settlement of Nearshore Marine Species Mitarai, S., Siegel, D.A., Warner, R.R., Winters, K.B, Kendall, B.E., Gaines, S.D. Will be submitted to MEPS Abstract Quantitative descriptions of larval dispersal are critical for the predictive understanding of many nearshore marine populations. For sessile species with a planktonic life stage, larval dispersal is the predominant means that connects spatially segregated populations. Larval dispersal is driven by mean currents, wind-driven Ekman circulation and coastal eddy motions as modified by the larval development time course and larval movements. Eddy motions predominate in the variability of ocean circulation, and have been considered to be a major source of stochasticity in settlement and recruitment events. In contemporary marine ecology, however, the vast majority of models oversimplify the processes of larval dispersal by simply neglecting eddy motions and often describe it as a simple diffusion process. In this study, we clarify intrinsic stochasticity arising from coastal eddy motions in larval dispersal by introducing simple scaling theory that counts the number of arriving eddies to habitats for a single spawning season. This scaling theory suggests that coastal eddy motions set a strong source of uncertainties in settlement patterns and population connectivity of coastal marine species for a single spawning season even when extreme abundance of larval production is available, and provides a means to quantify eddy-induced stochasticity in larval settlement patterns. We demonstrate that the scaling theory can accurately account for eddy-induced stochasticity in simulated dispersal patterns obtained from idealized coastal circulation simulations of the California Current. These results give new insights into the nature of larval dispersal and its potential for regulating population dynamics of nearshore marine species. 1 Introduction 1 Quantitative descriptions of larval dispersal are critical for the predictive understanding of many nearshore marine populations. Many of the marine species that comprise nearshore populations are relatively sessile as adults, with a dispersive planktonic larval stage. The planktonic larval stage is often the predominant means that connect spatially separated habitats for sessile species. Hence a predictive knowledge of larval dispersal, including source and destination locations and the degree of their connectivity, is key information for the study of nearshore marine population dynamics (Roughgarden et al., 1988; Kinlan and Gaines, 2003; Largier, 2003; Sale et al., 2005; Warner and Cowen, 2002). Due to the small size of marine larvae, larval transport by coastal circulation processes is likely the dominant process driving larval dispersal (e.g., Jackson and Strathmann, 1981; Siegel et al., 2003), although larval behavior (such as vertical migratory behavior and late-developmental period swimming) can also have some influence on integrated larval transport (Leis, 2006; Leis et al., 2007; Gerlach et al., 2007; Paris and Cowen, 2004; Siegel et al., in press). Larval settlement and recruitment observations provide many hints of the stochastic nature of larval dispersal for many fish and invertebrate populations on intra-seasonal to inter-annual time scales (Caffey, 1985; Caselle and Warner, 1996; Farrell et al., 1991; Hamilton et al., 2006; Swearer et al., 1999; Myers, 2001). One major source of stochasticity is changes in the circulation of the coastal ocean driven by mean currents, wind-driven Ekman circulation and coastal eddy motions. Among these physical drivers, coastal eddy motions predominate in the variability of coastal systems, creating seemingly chaotic flow patterns (e.g., Poulain and Niiler, 1989; Swenson and Niiler, 1996; Signell and Geyer, 1991; Strub et al., 1991; Haidvogel et al., 1991; Richardson, 1993; Bernstein et al., 1977; Dong and McWilliams, 2007). These eddy motions have recently been considered to be a major source of stochasticity in larval settlement patterns, and this can have an influence on stock recruitment dynamics (e.g. Roughgarden et al., 1998; Siegel et al., 2003, in press; Cowen et al., 2006; PfeifferHerbert et al., 2007; Mitarai et al., 2008). 2 The vast majority of models in contemporary marine ecology fails to account for the chaotic nature of coastal eddy motions. For example, a conventional diffusion modeling approach (e.g, Largier, 2003; Jackson and Strathmann, 1981) describes larval dispersal as a smooth and homogeneous diffusive process (often as a Gaussian process), and eddy-induced stochasticity in larval dispersal is simply ignored. By definition, a diffusion modeling approach describes time-averaged dispersal patterns of water parcels, smoothing out turbulent eddies (e.g., Pope, 2000; Tennekes and Lumley, 1972). In another example, an unstructured larval pool has been often assumed in many ecological studies, where larvae are produced in the nearshore, enter a common pool offshore, and are then returned to nearshore habitats often by downwelling favorable winds (Bakun and Parrish, 1982; Farrell et al., 1991; Parrish et al., 1981; Roughgarden et al., 1991). While the larval pool modeling approach can account for pulses of settlement, homogeneous mixing makes all sites potential sources of larvae for all other sites, and thus may connect populations that are in reality unconnected. A larval pool has to be spatially structured by ocean circulation. Numerical simulations of coastal circulation processes are a powerful tool to assess population connectivity via advection of water parcels (e.g., James et al., 2002; Siegel et al., 2003, in press; Cowen et al., 2006; Pfeiffer-Herbert et al., 2007; Aiken et al., 2007; Mitarai et al., 2008). However, very few of these models have been utilized to assess the underlying mechanism of eddy-induced stochasticity in population connectivity and larval settlement patterns. Do coastal eddy motions set spatio-temporal variations in larval settlement and recruitment patterns? Under what circumstances can conventional diffusion-based models be approximately valid? More importantly, how does eddy-induced stochasticity in larval dispersal affect the understanding and predicting of ecological processes and the design of fishery management? Many of these questions are yet to be answered because we still lack a clear picture of the intrinsic stochasticity arising from coastal eddy motions. The goal of this study is to clarify intrinsic stochasticity in larval settlement patterns and population connectivity arising from chaotic coastal eddy motions. We introduce simple 3 scaling theory that accounts for this eddy-induced stochasticity in larval dispersal by counting the number of coastal eddies arriving to coastal habitats. Predictions by the proposed scaling theory suggest that coastal eddy motions set a strong source of uncertainties in larval settlement patterns and population connectivity when viewed on annual time scales, even when extreme abundance of larval production is available. We test this scaling theory by using simulated larval dispersal patterns obtained from idealized coastal circulation simulations of the California Current along the central California coastline (Mitarai et al. 2008). We show that the proposed scaling theory successfully accounts for eddy-induced stochasticity in the simulated dispersal patterns. Based on the results, we discuss important consequences of scaled eddy-induced stochasticity in stock dynamics and community structure of nearshore marine populations. We illustrate fundamental sources of stochasticity in larval dispersal and marine population dynamics by using simple scaling theory. We first present scaling theory for eddy-induced stochasticity in larval dispersal in Section 2. The proposed scaling theory is then examined by using simulated larval dispersal patterns obtained from the idealized coastal circulation simulations of Mitarai et al (2008) in Section 3. Finally, we discuss the importance of coastal eddy motions in marine population dynamics and a spatial fishery management in section 4. 2 Scaling Theory for Eddy-induced Stochasticity 2.1 Eddy-induced stochasticity in larval dispersal Larval dispersal of coastal marine species takes place in the turbulent coastal ocean, where coastal eddies are ubiquitous and drive the stirring and mixing of water parcels (e.g., Fig. 1 a). Coastal eddy motions evolve rapidly in response to winds, while being advected with the mean currents and wind-driven Ekman circulation. Studies show that ocean surface drifters are advected by these turbulent currents (Winant et al., 1999; Dever et al., 1998; Swenson and Niiler, 1996; Poulain and Niiler, 1989; Ohlmann et al., 2001). Surface drifters are advected by the ocean currents, mostly along coastal eddies 4 due to geostrophic currents, while transported by mean currents and wind-driven Ekman circulation, as illustrated in Fig. 1 b (e.g., see also Ohlmann et al., 2001). Hence eddies act to sweep larvae together into streaks, and transport them as coherent packets (see also Roughgarden et al., 1988, Mitarai et al., 2008, Siegel et al., in press). Population connectivity among nearshore habitats, therefore, has to be strongly controlled by the departure and arrival of these “larval packets.” This is very different from a diffusion modeling approach, where larval dispersal is described as a homogeneous and small-scale diffusion process. While a diffusion modeling approach always results in smooth and homogeneous connectivity among coastal populations, larval packet transport can lead to sparse and heterogeneous population connectivity (Mitarai et al., 2008; Siegel et al., in press). An unstructured larval pool has been often assumed in many ecological studies, where larvae are produced in the nearshore, enter a common pool offshore. Coastal eddies, however, will make a larval pool heterogeneous in space. Larval production occurs within a few kilometer of the shoreline, at least in the California Current (Refs; need help from Bob here), while the size of coastal eddies is typically much larger and the center of eddies is located 10’s to 100’s km offshore (as explained below). Larvae are then seeded and transported in the peripheral of eddies due to Geostrophic currents, leaving the center of eddies rather unoccupied (Fig. 1 b). This makes a larval pool spatially heterogeneous structured by coastal eddy motions. Settlement pulses observation driven by downwelling favorable winds (Bakun and Parrish, 1982; Farrell et al., 1991; Parrish et al., 1981; Roughgarden et al., 1991) may be caused by, among other things, this heterogeneous larval pool formed by coastal eddies. Settlements pulses can occur when downwelling favorable winds bring coherent larval packets formed by eddies from an offshore larval pool towards a shore. Coastal eddy motions can be, therefore, a strong source of uncertainties in determining larval settlement patterns. The exact patterns in larval dispersal processes are expected to be different for each independent spawning season. This occurs because the statistics of coastal circulations are 1) actually different in different years (what we will call real inter-annual variability) 5 or 2) the flows are statistically identical but the spawning seasons samples a different collection of eddy motions resulting in different dispersal patterns (here defined as apparent inter-annual variability). We examine apparent inter-annual variations in dispersal patterns, due to chaotic coastal eddy motions, by introducing simple scaling theory to describe them below. 2.2 Scaling eddy-induced stochasticity in settlement events The scaling of the number of settlement pulses will be key to understanding eddyinduced stochasticity in larval dispersal processes. The fluid dynamics of ocean circulation systems suggests that larvae are swept into packets by coastal eddies, while transported by mean currents and wind-driven Ekman circulation, as described above. Larval settlement patterns along a shore can be then modeled as a superposition of arriving pulses of settlements (packets) formed by coastal eddies. The resulting settlement patterns and degree of eddy-induced stochasticity will depend on the total number of settlement pulses arriving for the domain, N_{ev}, and the spatial extent for each settlement pulse, δ_{ev}, which is normalized by the domain size. A large number of events, each providing occurring over relatively large spatial scales, will result in a smooth settlement pattern; whereas fewer, smaller sized events will result in a patchy pattern of settlement (e.g. Siegel et al., in press). The number of coastal eddies arriving in the domain can be estimated as the ratios of the domain size, L, to the eddy size, l (the number of eddies that occupy the domain) and the duration of the spawning season, T, to the eddy residence time, τ (the number of eddies arriving at a site per season) and the fraction of arriving eddies that contain settling larvae, f_{sv} , i.e., 6 The survivability fraction, f_{sv}, should be f_{sv} = 1 if all coastal eddies contain particles (larvae), and f_{sv} < 1 otherwise. Since the goal of this study is to assess stochasticity induced by coastal eddies, and not due to biological variability, we assume f_{sv} = 1 for the rest of this paper. The relative spatial scale, δ_{ev}, is set solely by the eddies in the flow field, and estimated as δ_{ev} = l/L. The time and length scales of coastal eddy motions can be estimated by using available oceanographic information. The first-mode baroclinic Rossby radius of deformation is a natural scale in the ocean associated with boundary phenomena such as boundary currents, fronts, and eddies (Gill, 1982). As an example, let us estimate the number of arriving eddies to the central California coast using the baroclinic Rossby radius of deformation. Along the US west coast, the first-mode baroclinic Rossby radius of deformation is approximately 20 to 30 km (Chelton et al., 1998). Hence the portion of the coastline covered by each coastal eddy can be estimated as l = 2r = 40 – 60 km, where r is the first-mode baroclinic Rossby radius of deformation. The mean current speed of California Current is about u ≈ 4 cm/s (e.g., Swenson and Niiler, 1996; Poulain and Niiler, 1989; Siegel et al., 2003). The eddy residence time can then be scaled by τ = l/u = 12 – 17 days. The central California coast extends approximately 250 km (approximately between Pt. Conception and Monterey). Hence each eddy covers 20% of the coastline, i.e., δ_{ev} = 50 / 250 = 0.2. For a single spawning season (e.g., 90 days), the total number of arriving packets can be scaled by N_{ev} = 30, given the parameters L = 250 km, l = 50 km, T = 90 d, τ = 15 d and f_{sv} = 1. One way to assess the spatial variability of the number of settling larvae among sites is to use a coefficient of variation statistic (CV_s). If the probability that a given larval packet lands on a particular site is δ_{ev} and each event is independent, then the expected number of packets arriving at a site is δ_{ev} N_{ev} and the expected value of the CV_s can be approximated by using binomial sampling theory as 7 If N_{ev} or δ_{ev} increases, the settlement pattern will become less stochastic. Given the parameters L = 250 km, l = 50 km, T = 90 d, τ = 15 d and f_{sv} = 1, for example, Eqs. (1) and (2) yield <CV_s> ≈ 0.37. Thus, eddy-induced stochasticity causes about 40% of variation in settlement events among sites, even without biological variability in larval production (f_{sv} = 1). Adding biological source of variability makes spatial settlement patterns for a single spawning season even more stochastic. The scales of coastal eddies are regional. While the first-mode baroclinic Rossby radius of deformation is approximately 20 to 30 km along the US west coast, the radius quickly increases towards the equator, e.g., south of Baja California (Chelton et al., 1998). The presence of topographic features such as irregular coastlines will have a strong influence on the surrounding circulation. Small topographic features, for example, may create smaller coastal eddies. Given a larger (or smaller) eddy size, the eddy-induced stochasticity may vary substantially. However, scaling theory, Eqs. (1) and (2), suggests that the eddy-induced stochasticity is rather insensitive to the choice of eddy size. For a case with a larger eddy, for example, the number of arriving eddies is reduced [Eq. (1)], which will inflate eddy-induced stochasticity [Eq. (2)]. However, the domain covered by an eddy is increased, which reduces spatial heterogeneity. These two counter-acting effects nearly cancel out, leaving eddy-induced stochasticity rather unchanged. For example, the scaling theory yields <CV_s> ≈ 0.44, given the parameters L = 250 km, l = 100 km, T = 90 d, τ = l / 0.04 ≈ 30 d and f_{sv} = 1. Stochastic settlement due to coastal eddies are expected, not only in the California Current, but also other coastal oceans. 2.3 Eddy-induced stochasticity in population connectivity Connectivity between given nearshore sites is often quantified using a “connectivity matrix” (e.g., James et al., 2002; Largier, 2003; Cowen, 2002; Cowen et al., 2006; Siegel et al., in press). Definitions for the connectivity matrix differ widely. A connectivity matrix can be defined by 1) accounting for water-parcel transport from a source site j to a destination site i within spawning and larval development time-course windows as 8 modified by larval behavior characteristics; 2) the spawned larvae from site j for the season that successfully settle at site i; or 3) those spawned larvae from site j that successfully recruit to the adult stage at site i. The first definition excludes any biotic sources of stochasticity other than larval life history characteristics. The second and third superimpose the biotic spatial processes of production and post-settlement interactions onto the first abiotic definition. Cowen et al. (2006), for example, used the third definition in their illustration of a connectivity matrix. In this study, we refer to the first, abiotic, definition as the connectivity matrix, so that we can separate the effect of eddy-induced stochasticity from other biological sources of variability. The scaling relationship can be used to develop a “packet model” for population connectivity to aid in spatial population dynamics modeling. The idea is to portray population connectivity as the departure and arrival of N_{ev} independent, equallysized, settlement packets. The source and destination locations of each packet are determined by random sampling of the long-term averaged Gaussian dispersal kernels (e.g., see Siegel et al, 2003; Largier, 2003). Destination locations for the N_{ev} packets (X) are selected randomly from within the domain and their source locations (Y) are also determined randomly selecting from the Gaussian distributions [Eq. (3)], which accounts for the downcoast displacement, i.e., where x and y are sample space variables for X and Y, respectively; f_x and f_y are the probability density functions for X and Y, respectively; and µ and σ, respectively, represent the mean and the standard deviation of the dispersal distance of successfully settling larvae. Here values of Gaussian parameters µ and σ can be estimated from oceanographic information for given larval development time courses (see Siegel et al., 2003). Connectivity matrices are then modeled based upon the number of packets between a given source and destination. Mathematically, 9 where X_n and Y_n indicate the source and destination locations for the n-th larval packet; Θ_x (a, b) is the boxcar function that is equal to 1 for a < x < b and 0 otherwise, representing a destination (source) area covered by each eddy. As he number of packets increases (approaches infinity, precisely), modeled connectivity matrices become identical to Gaussian dispersal kernels, i.e., which gives a smooth and homogeneous connectivity among populations (diffusion models). We numerically demonstrate when modeled connectivity [Eq. (4)] can be reasonably approximated by smooth Gaussian connectivity [Eq. (5)] below. The packet model suggests that coastal eddy motions set a strong source of stochasticity in population connectivity when viewed on annual time scales. In other words, the number of larval packets formed by eddies is not large enough to achieve smooth and homogeneous population connectivity, even when extreme abundance of larval production is available. Figures 2 a − c show the connectivity matrices obtained from three different realizations of the packet model predictions (predicted using a different random seed in determining the source and destination locations of packets mimicking chaotic coastal eddy motions), given the parameters L = 250 km, l = 50 km, T = 30 d, τ = 15 d, f_{sv} = 1, µ = -68 km and σ = 71 km. Here the values of µ and σ are determined by using the regression formula by Siegel et al. (2003), assuming 20 days of pelagic larval durations in the California Current. The realized connectivity matrices show that some sites receive less settling larvae, while others receive pulses of settlements from a wide array of source locations. There are “hot spots” that indicate strong connections between particular source and destination locations. The connectivity matrices are not only heterogeneous in space, but also intermittent in time. Different realizations produce distinctly different patterns in settlement pulses and 10 different connections among nearshore sites; these are still spatially heterogeneous, but the locations and intensities of the hot spots change (cf. Figs. 2 a - c). These realizations illustrate that coastal eddy motions alter larval settlement patterns from one spawning season to the next, due to the chaotic nature of turbulence. The realized connectivity matrices are very different from a diffusion model prediction (cf. Figs. 2 a – c and d). Graphically, the packet model for population connectivity can be considered as randomly placing N_{ev} of eddy-size “patches” in the smooth, Gaussian connectivity matrix. When averaged over many independent spawning seasons, the domain is filled with many successful settlement events and the signature of spatial heterogeneity in population connectivity becomes smoother (Siegel et al., 2003, in press). Figure 3 shows the packet model prediction for 1 and then averaged over 5, 10 and 100 independent spawning seasons. Comparison of Fig. 2 d and Fig 3. d shows that the resulting connectivity patterns become smoother approaching the Gaussian patterns of connectivity predicted by a diffusion model. “Hot spots” in the connectivity matrices are still discernible after 10 spawning seasons (Fig. 3 c), although connectivity patterns are smoother. The connectivity matrices become very similar to the diffusion model prediction when averaged over 100 spawning seasons (Fig. 3 d). The scaling analysis, Eqs. (1) and (2), clearly describes this change in connectivity. By substituting Eq. (1) to Eq. (2), we obtain Larval settlement (and connectivity) patterns will become smoother as T , δ_{ev} , or f_{sv} increase or as τ decreases. Given the parameters L = 250 km, l = 50 km, τ = 15 d, f_{sv} = 1 and T = 90, 450, 900 and 9000 d, Eq. (6) yields <CV_s> ≈ 0.37, 0.21, 0.12 and 0.04. These values show a good agreement with CV_s values computed from the realized connectivity matrices in Fig. 2 (see the caption of Fig. 2). 11 Note that the connectivity described here is independent of adult population demographics. Adding variability in larval production will only make the connectivity diagrams presented here more stochastic, and requires even longer time until spatial heterogeneity in the simulated population connectivity is smoothed out. These results suggest that population connectivity should be stochastic even when extreme abundance of larval production is available, and chaotic coastal eddy motions, thus, set avoidable uncertainties in population connectivity for a single spawning season. Also note that the packet model assumes that the probability that a given larval packet lands on a particular site is equal [Eq. (3)]. Complex coastal topography, however, will modify the probability distributions for the packet landing location, and may create preferable destination sites over seasons if persistent topographic eddies are formed (e.g., Graham and Largier, 1997). This topographic effect may make population connectivity even more heterogeneous in space, but probably less intermittent in time. While it would be possible to include topographic effects in scaling theory (with much more complexity), it is beyond the scope of this study. Our primary focus is to clarify eddyinduced stochasticity in larval dispersal processes, separated from other sources of stochasticity. The proposed scaling theory shows that ocean stirring makes larval connections among nearshore sites a stochastic process that is both spatially heterogeneous and temporally intermittent. 3 A Test of Scaling Theory In the previous section, we introduced scaling theory that accounts for stochasticity in population connectivity and settlement events driven by coastal eddy motions. The natural next step would be to examine the proposed scaling theory (the packet model). It would be challenging to test scaling theory by using realistic circulation simulations (e.g., James et al., 2002; Cowen et al., 2006; Pfeiffer-Herbert et al., 2007; Aiken et al., 2007) and in-situ settlement and recruitment observation data (Caffey, 1985; Caselle and Warner, 1996; Farrell et al., 1991; Hamilton et al., 2006; Swearer et al., 1999; Myers, 2001) because of the difficulty to separate eddy-induced stochasticity from other sources of biological and topographic variability. In this study we test the scaling theory 12 against idealized simulations of coastal circulation processes modeled after the California Current (Mitarai et al., 2008). The coastal circulation-simulations of Mitarai et al. (2008) assume homogeneous environment in the along-shore direction and separate eddy-induced stochasticity from other sources of variability. This numerical configuration, thus, allows us to test the packet model effectively. 3.1 Idealized coastal circulation-simulations Model forcing and domain configuration of Mitarai et al. (2008) are patterned after typical flow conditions of the California Current at line 70 (off shore of Pt. Sur, California) of the California Oceanic Cooperative Fisheries Investigations (CalCOFI) (Lynn and Simpson, 1987; Chelton et al., 1998). The domain is modeled to be homogeneous in the along-shore direction with a domain size of 256 km in the along-shore direction and 288 km in the cross-shore direction. The domain is discretized horizontally by a 2-kmresolution grid (128 grid points in the along-shore direction and 144 grid locations in the cross-shore direction). Twenty vertical (depth) levels are considered, with enhanced resolution near the top and bottom boundaries. The bathymetry has a steep continental slope modeled after CalCOFI line 70. No bathymetric variations are considered in the along-shore direction, and periodic boundary conditions are used at the northern and southern boundaries. This scenario reasonably represents the US west coast while still providing a numerical system capable of addressing the fundamental processes of larval dispersal driven by eddy motions. The simulated flow fields are mainly driven by stochastic wind stress applied to the top surface. The wind field is assumed to vary on spatial scales much larger than the alongshore scale of the simulated domain while its magnitude decreases towards the shore (Pickett and Paduan, 2003; Capet et al., 2004). Each component of the wind vector is modeled as a statistically-stationary Gaussian random process, given the statistics estimated from hourly buoy wind data of the National Data Buoy Center (stations 46028, 46012 and 46042) for summer and winter and the spatial wind observations of Pickett and Paduan (2003). We simulate two distinctive flow regimes, i.e., a strong upwelling 13 condition in summer (July) and a weak upwelling condition in winter (January). Here we simulate 28 different realizations of flow fields for each condition by changing the random number seed in wind forcing, while the other parameters and configuration are unchanged. Using these realizations of simulated flow fields, we examine apparent inter-annual variations in dispersal patterns. 3.2 Modeling of larval dispersal in the simulated flow fields A large number of Lagrangian particles are released and tracked in the simulated flow fields to simulate dispersal of larvae. Lagrangian particles are passively advected by coastal circulation processes in horizontal directions while they are capable of changing their vertical locations, mimicking ontogenetic development of vertically migrating behaviors. Values of planktonic larval duration (PLD) can range from a month to several months for typical reef fish (Victor, 1986; Wellington and Victor, 1989). Only larvae transported to nearshore suitable habitats during their settlement competency time windows (at the end of the PLD) are counted as successful settlers. Nearshore habitats where particles are released and settle are defined here as all waters shallower than 100 m in depth (within 10 km from the coast). The exaggerated offshore extent of suitable habitat (at least for the California coast) was selected to account for active swimming towards suitable habitat in the last stages of larval development. After the simulation flow fields reach statistically stationary conditions, 1,000 Lagrangian particles are released daily for 90 days in the upper 10 m of the water column, uniformly distributed in nearshore waters. The total number of released particles within the domain is 90,000 for each realization. We examine two different scenarios for vertical positioning. For the first scenario, Lagrangian particles are released near the sea surface (within the upper 10 m) and stay at this depth while they are passively transported horizontally. For the second scenario, particles are released near the top surface, and shift their vertical locations to 30-m deeper after 5 days from their release, keeping their depth after that. Most fish larvae are found within the upper 50 m or so (Leis, 1991; Cowen, 2002). These two behaviors are expected to give a 14 general idea of the importance of larval vertical positioning, coupled with coastal eddy motions, in determining population connectivity. Successful settlement is defined as an event that occurs when the Lagrangian particles are found within the nearshore habitat region within competency time windows. We examine four sets of competency time windows, i.e., 10 – 20, 20 – 40, 30 – 60 and 40 – 80 days. As stated before, the pelagic larval durations (PLDs) of many benthic reef fish are about a month or two and the duration of the competency time window tends to be greater in fishes with larger PLD (Victor, 1991). Lagrangian particles are transported by coastal circulation until the end of the competency time windows. During the time period, Lagrangian particles are allowed to record successful settlement more than once, in order to account for individual differences in development within a larval cohort. Particles settling to (and from) outside of the domain are also accounted for by taking advantage of the along-shore periodicity counting the number of periodic domains the particle has traversed. 3.3 Larval packet transport in coastal circulation-simulations Coastal circulation processes and eddy water parcel motions can be estimated using the temporal evolution of the sea level distribution because contours of sea level provide geostrophic streamlines for flow at the sea surface. Figure 4 shows model depictions of sea level contours and the trajectories of Lagrangian particles under the strong upwelling conditions characteristic of summer. In the figure, low sea level features (blue regions) correspond to cyclones and support counter-clockwise geostrophic currents. Anti-cyclones, high sea level features (yellow region), create clockwise circulations. The sea level contours represent coastal eddy motions well. Coastal eddy motions evolve rapidly in response to statistically forced winds, while being advected with the mean equatorward currents. Comparison of each panel of Fig. 4 shows that eddy patterns change greatly over the period. Lagrangian particles are advected by the simulated currents, mostly along lines of constant sea level, and these patterns evolve rapidly as the sea level patterns change (Fig. 4). Although Lagrangian 15 particles are uniformly released from nearshore waters, their density quickly becomes non-uniform as the particles are advected by the coastal eddy motions. Eddies act to sweep larvae together into streaks, and transport them as coherent packets, as illustrated in the previous section (cf. Figs. 1 and 4). Eddy-induced stochasticity in larval settlements can be seen by examining a time series showing the temporal and spatial patterns of successful settlements (Fig. 5). Figures 5 a – d show time series of the density and along-shore destination locations of particles that successfully settle in the domain for the default case. Four different realizations of the simulations are shown, using different number seeds in the wind forcing, which results in the circulation patterns being different for each of the realizations, corresponding to four independent spawning seasons or years. Successful settlements occur in infrequent pulses, because settling larvae are accumulated by coastal eddies and delivered to nearshore habitats as a coherent group (Fig. 4). Different realizations produce different patterns in settlement pulses that are still spatially heterogeneous, but the locations and intensities of the hot spots change (cf. Figs. 5 a – d). Simulated settlement patterns are stochastic regardless of the spawning season, PLDs and vertically migrating behavior. In the winter off California, upwelling-favorable winds are diminished along with the strength of the currents (Parrish et al., 1981; Pickett and Paduan, 2003), and reduced mean offshore surface currents keep more released larvae within the region of suitable habitat, which may diminish eddy-induced stochasticity. The simulated dispersal patterns, however, show that the same larval release schedule applied to a typical winter flow field still shows a high level of stochasticity (Fig. 5 e). Ontogenetic descents move larvae from higher speed surface flows to deeper, slower flows, which is expected to reduce eddy-induced stochasticity. When we model vertical migrating ontogenetic behavior as a descent from the surface to 30 m after 5 days from release, the arrival locations and times of settlements pulses are modified at certain times, but the highly heterogeneous nature of the settlement remains (cf. Figs. 5 a and h). Dispersal patterns can change greatly depending on the PLD because larvae are transported further in time and distance as their PLD increases. Figures 5 f and g show 16 arrival time series obtained with two different competency time windows, i.e., 10 – 20 and 30 – 60 days, in exactly the same flow fields as in Fig. 5 a. While simulated dispersal patterns are distinctly different under these differing competency time windows, they all remain stochastic. 3.4 Spatio-temporal scales emerging from the simulated dispersal We can measure the temporal and spatial scales of arriving settlement pulses using a variogram (Rossi et al., 1992). Time scales are calculated using the variogram of the arrival density with arrival location held constant, while length scales are calculated using the variogram range of the arrival density with arrival location held constant. Arrivals before the first settlement is observed (e.g., day 20 in Fig. 5 a) and after the last settlement occurs (e.g., day 130 in Fig. 5 a) are not used in the computation. The variogram range is defined as the minimum spatial lag at which the variogram values reach the variance of the arrival densities. The obtained time and length scales are summarized in Table 1. The temporal and spatial scales of settlement pulses were rather consistent regardless of the seasonal upwelling conditions, PLDs and vertically migrating behavior, although some variation occurs. The arrival time scales of settlement pulses range from 11 to 16 days. The arrival length scales of settlement pulses measure 35 to 57 km. These values are similar to the eddy time (τ = 12 – 17 d) and length scales (l = 40 – 60 km) estimated by the scaling theory for California Current. Nearshore eddies appear to set the spatio-temporal scales of successful larval settlement events in the simulated flow fields from this analysis, although we cannot really prove it from this analysis. 3.5 Scaling theory vs the circulation simulations In order to assess connectivity via advection of water parcels in the simulated flow fields, nearshore waters are delineated into multiple equal-area sites with 4-km spacing in the along-shore direction (64 sites in the domain). Sites are identified by their alongshore locations as i (or j) = 1, 2, ..., 64. Connectivity matrices are obtained based upon 17 the number of settling particles between a given source and destination for each realization of the simulations, i.e., where S^n_{ji} indicates the number of successfully setting particles that are transported from site j to site i during the n-th spawning season (here, realization of the simulations), and N_j is the number of particles released from site j. This matrix defines coastal connectivity independent of adult population demographics. The connectivity matrices obtained from the simulated flow fields are neither smooth nor homogeneous for a single spawning season, similarly to the predictions by the packet model (cf. Figs. 2 and 6). Figure 6 shows the resulting connectivity obtained from four different realizations of the simulations (simulated using a different random initial seed in the wind forcing), corresponding to the arrival time series shown in Figs. 5 a – d. Connectivity matrices obtained from the coastal circulation simulations show a good qualitative agreement with the packet model (cf. Figs. 2 and 6), that is, spatial heterogeneity in connectivity predicted by the packet model shows similar patterns to the results obtained from the coastal circulation simulations. Coastal eddy motions alter connectivity patterns from one time period (spawning season) to the next. As averaged over many independent spawning seasons, the spatial heterogeneity in connectivity decreases both in the coastal circulation simulations and the packet model predictions (cf. Figs. 3 and 7). Equally, both connectivity matrices are smoothed out, and can be approximated by a diffusion model. Thus, the packet model shows a good qualitative agreement with the circulation simulations. The packet model also accurately quantifies eddy-induced stochasticity in the simulated settlement events. Figure 8 compares estimates of <CV_s> obtained from the circulation simulations and with those predicted by scaling theory, as a function of the number of spawning seasons. The prediction from scaling theory [Eq. (6)] shows good 18 agreement with the results obtained from circulation simulations. For the case with shorter larval release duration, there are less settlement pulses for a single spawning season (smaller N_{ev}). Accordingly, <CV_s> is higher, and it decays more slowly than in the default case when averaged over many independent spawning seasons. The packet model reasonably approximates the decay of <CV_s> for each case. The good agreement between the scaling theory and the circulation simulations suggests that scales of spatial settlement variations, induced by ocean circulation, can be estimated by counting the number of coastal eddies and their spatial extent in the target area. Averaged over many independent spawning seasons, more eddies bring more larvae to nearshore habitats, and spatial settlement variations are smoothed out. Coastal eddy motions set the spatial and temporal scales of larval settlement patterns at least in the simulated flow fields. 3.7 Caveats of scaling theory Stochasticity of settlement events is reduced when settling larvae stay within nearshore waters during their planktonic duration. For short PLDs or under the weak upwelling conditions (as in winter), a considerable number of settling particles may stay within nearshore waters during their entire PLD. For example, for the case with the short PLD (around two weeks) under the weak upwelling condition of winter, approximately 25% of the settling particles stay in nearshore waters throughout their PLDs (Fig. 9 a), and their cross-shore dispersal range (within which 99% of the settling particles are contained) is limited to 50 km from the coast at most (Fig. 9 b). Dispersal patterns of settling larvae that are not advected away by currents or eddies from nearshore waters will not be much affected by coastal eddy motions. Reflecting that these settlers that stay within nearshore waters, eddy-induced stochasticity is reduced for shorter PLDs and under the weak upwelling condition of winter (Fig. 8). For many of the cases considered in the dispersal simulations (and for many benthic species), however, not many larvae remain nearshore, and coastal eddy motions are important. Typically, more than 90% of settlers are advected out of nearshore waters before settling (Fig. 9 a). Under the strong 19 upwelling condition in summer, in particular, almost all settlers do not stay within nearshore waters, and dispersal range exceed 100 km. 5 Discussion The proposed scaling theory (the packet model), counting the number of arriving eddies to nearshore habitats, suggests that coastal eddy motions should set a strong source of uncertainties in larval dispersal patterns, even when extreme abundance of larval production is available. Larvae are swept into streaks (or packets) by coastal eddies, while transported by mean currents and wind-driven Ekman circulation (Fig. 1). The number of larval packets formed by coastal eddies is not large enough to achieve smooth and homogeneous larval settlement patterns when viewed on annual time scales (Fig. 3). Therefore, even without any biological or topographic sources of variability, population connectivity patterns are expected to be heterogeneous in space, creating only a few strong connections among sites (Fig. 2). This scaling analysis is good news and bad news for the study of marine ecology. While the packet model provides a means to quantify this eddy-induced stochasticity in larval settlement patterns (Fig. 8), it also implies that coastal eddy motions may make larval settlement and population connectivity patterns almost unpredictable, due to the chaotic nature of turbulence (Figs. 2 a – c). However, this eddy-induced stochasticity in larval dispersal is important only if it alters the predictions and/or understanding of nearshore marine population dynamics. We discuss the importance of these findings below. One case where coastal eddy motions can play an important role in marine population dynamics is in the case of interspecific interactions where competition among larvae may be important for determining post-settlement recruitment rates. Larvae from different spawning periods can “catch” different eddies, resulting in different dispersal patterns on a year-to-year or generation-to-generation basis. Hence, there is the possibility that larvae from an inferior competitor will occasionally land in locations that are free of the superior competitor’s larvae – if it happens often enough, the two species can coexist. This can be considered as a spatial variant of the storage effect where rare 20 recruitment events can lead to coexistence of interacting species (e.g., Warner and Chesson, 1985, Berkley et al. in prep). Another important effect of coastal turbulence could be in the condition to be satisfied for retention of local production and upstream invasion. Byers and Pringle (2006) derived the condition for the upstream-transport/retention condition in an advective environment, in terms of the Gaussian parameters µ and σ in Eq. (3), assuming a smooth, Gaussian larval dispersal and no intra-specific and interspecific competition at recruitment. The packet model, however, suggests that larval should be transported as coherent packets, not as a diffusion process, and dispersal patterns can be approximated by Gaussian distributions only when averaged over many independent spawning seasons (Fig. 4). Hence the upstream-transport/retention condition may be altered by accounting stochastic larval transport and also intra- and inter-specific interactions at recruitment. Chaotic eddy transport of larvae may be an important mechanism that makes species coexistence in an advective environment possible. Imagine two identical species that disperse in an advective environment with initial conditions in which one species is distributed upstream from the other (cf. Byers et al, 2006). If the two species are demographically identical (e.g., the same mortality, same fecundity, same competency time windows and post-settlement density dependence factors), and competing for limited resources (or space), the upstream species has a great advantage because it can send more larvae to settling sites downstream due to the mean advection in the system. Downstream species may be able to invade upstream because their larvae can be occasionally transported upstream as a coherent packet, not as a weak diffusion processes, into areas where upstream species happen to be rare. The number of successful settlements at a site will show substantial variation year to year, even without any other sources of uncertainties (Fig. 2). The “packet” transport of larvae by coastal eddy motions, coupled with life history, may structure nearshore marine populations because it adds strong inter-annual variation in settlements. 21 The stochastic nature of population connectivity influences many other ecological processes. For example, the pulsed nature of larval transport, even at long distances from a source, can act to alleviate the Allee effects that limit the success of longdistance colonization (Gascoigne and Lipcius, 2004). Further, occasional large, isolated pulses of larval settlement predicted here should enhance the contribution of the storage effect on species persistence and coexistence (Warner and Chesson, 1985). Similarly, the long lives and high fecundity that are characteristic of most marine organisms may in fact be evolutionary responses to the inherent uncertainty of recruitment. Different life histories (e.g., various vertical positioning behavior, spawning timing and PLDs) may be employed in order to take advantage of chaotic coastal eddy motions. While the present study identifies important features of complex nearshore systems, more work is required to understand the interactions among variable coastal circulation, organism life cycles, and the management of these important ecosystems and the fisheries they support. The cohort transport of fish larvae will also make the management of nearshore fisheries difficult. Fishery managers must assess stocks on annual time scales, and here we show that at that population connectivity is inherently stochastic. Fisheries scientists and marine ecologists have long tried to explain the sources of recruitment variability so that recruitment could be predicted knowing only environmental factors (Hjort, 1914; Chavez et al., 2003; Cowen et al., 2006; Walters and Martell, 2004; Myers, 2001). But, because larval dispersal is not a deterministic diffusion process, local rates of larval settlement will be largely decoupled from local stocks; thus stochastic population connectivity may provide an unexplored source of noise in observed stockrecruitment relationships (Walters and Martell, 2004; Myers, 2001). The proper design of a network of marine protected areas (MPAs) requires accounting for the spatial distribution of reproductive adults and the connectivity among populations as well as the spatial distribution of fishing effort. In most MPA modeling studies, larval dispersal is described as a smooth and homogeneous diffusion process, given certain regional oceanographic information (e.g., Largier, 2003; Siegel et al., 2003). Our work indicates that the connectivity among MPAs will not be persistent year to year, reflecting chaotic 22 coastal eddy motions. The connectivity among MPAs may be very different depending on species characteristics as well (e.g., spawning timing and ontogenetic behaviors). Fishery managers must cope with this unavoidable unpredictability in nearshore populations and the fisheries they support. Acknowledgments The authors acknowledge a series of enlightening discussions with Chris Costello, Ray Hilborn, Steve Polasky, Libe Washburn, Sally McIntyre, Jenn Caselle, Brian Kinlan, Tim Chaffey, James Watson, Heather Berkley, Charles Dong, Jim McWilliams and Jamey Pringle. This work is a contribution of the “Flow, Fish and Fishing” biocomplexity project work and is supported by the National Science Foundation (NSF grant # 0308440). Figure 1. (a) Sea-surface temperature adjacent to the central California coastal revealed by the Advanced Very High Resolution Radiometer (AVHRR). Images are colored from dark blue (~ 5 C), though light blue to green to yellow, ending in red (~ 15 c) to indicate a progression from cold to warm water. Dates of photographs are May 18, 1982, April 6, 1983, and May 21, 1984. Notice that the California Current consists of many coastal eddies that predominates in variability in the circulation system. (b) Schematic diagram of larval transport in California Current system adjacent to central California. Coastal eddy motions indicated with orange circles (clock-wise eddy motions) and light blue circles (counter-clockwise eddy motions) evolve rapidly in response to winds, while being advected with the mean currents (blue arrow) and wind-driven Ekman circulation (red arrow). These eddies act to sweep larvae (green circles) together into streaks, and transport them as coherent packets. [This diagram is Fig. 7 of 23 Roughgarden et al. (1988) with additional cartoons on top of it. I will make a new diagram getting some pictures from Erik Fields for the panel a and using Adobe illustrator for the panel b.] 24 Figure 2. a) - c) Connectivity matrices obtained from three different realizations (using different random number seed in determining source and destinations of packets) of the packet model [Eqs. (1), (3) and (4)], given the parameters L = 250 km, l = 50 km, T = 30 d, τ = 15 d, f_{sv} = 1, µ = -68 km and σ = 71 km. d) Connectivity matrix predicted by a diffusion model [Eqs. (3) and (5)], given the parameters µ = -68 km and σ = 71 km. The source and destination locations are given by along-shore location. Smaller values indicate locations that are downstream. Values less than 0 or greater than the domain size (250 km) indicate source locations that are outside of the domain. The dashed line in the connectivity matrices represent self settlement; i.e., the source and destination locations of larval packets are identical. Points above the self-settlement line indicate larval transport from upstream to downstream. Color scales for the diffusion model show the probability densities for the source (or destination) locations of successfully settling larvae to (from) a given site. Each row (or column) of the connectivity predicted by the diffusion model has a Gaussian profile given by Eq. (3). Color scales for the packet model are adjusted so that predicted connectivity becomes identical to the diffusion model as averaged over infinite number of spawning seasons (realizations), as given by Eqs. (3) and (5). Figure 3. Connectivity matrices obtained from the packet model [Eqs. (1), (3) and (4)] for 1 season and averaged over 5, 10 and 100 independent spawning seasons. Here, the model parameters are set to L = 250 km, l = 50 km, τ = 15 d and T = 90, 450, 900 and 9000 d. Mean dispersal scales are set to µ = -68 km and σ = 71 km, corresponding to the condition used in Fig. 2. The coefficients of variation for the number of the settlement among sites (variation for summation of each column of the matrix) are 0.38, 0.18, 0.10 and 0.02 (from left to right) for this particular realization. For the detailed descriptions of connectivity matrices, see the caption of Fig. 2. Figure 4. Model depictions of sea level (color contours in cm) and the trajectories of Lagrangian particles for days 45, 60, 75 and 90. Here, only 2000 randomly chosen particles are shown. The circles show the location of the particles while the white trails behind each show their previous 3-day trajectories. Red circles indicate successfully settling particles (i.e., found within 10 km from the coast during the competency time window.) The competency time window is set to 20 to 40 days. The vertical dashed red line indicates the boundary for the nearshore habitat from which particles (larvae) are released and where settlement can occur. Low sea level features correspond to cyclones and support counter-clockwise geostrophic currents. Anti-cyclones, high sea level features, create clockwise circulations. The flow field is modeled to represent conditions found in the central coast of California (CalCOFI line 70) during a typical July (high upwelling conditions). The simulated domain is 256 km in the along-shore direction and 288 km in the cross-shore direction (only the inner 200 km are shown in the cross-shore direction). Particles are released near the top surface, and keep their depth while they are passively transported by coastal circulation in horizontal directions. There is a mean southward flow in the domain. Figure 5. Arrival time series showing the density (color) and arrival to along-shore locations (vertical axis) of successfully settling particles from all source locations. Left panels: arrival time series obtained from four different realizations of the simulations (using different random number seeds in the wind forcing) for the base case (strong upwelling condition of summer). In this case, Lagrangian particles remain near the top surface while being passively transported in horizontal directions, and the competency time window is set from 20 to 40 days. Settling particles from outside of the domain are simulated by taking advantage of the along-shore periodic flow fields. Right panels: arrival time series e) under the weak upwelling condition in winter, f) when the competency time window is set from 10 to 20 days and g) from 30 to 60, and h) for vertically-migrating larvae (migrating to 30 m after 5 days, mimicking an ontogenetic vertical migration) days. The flow fields are identical for the panels a, f – h. The three vertical lines in the time series indicate the day when the first larvae are able to settle; when larval releases stop; and when settlement ends, respectively. The densities (colors) are given by the number of settlement events per 4 km per day normalized by the total number of settlement events for each case. Table 1. Length (km) and time (days) of the arriving pulses of settlements (see Fig. 5) calculated using a variogram range (Rossi et al., 1992) under different conditions depending on seasonal upwelling conditions, planktonic larval durations (PLDs) and vertically migrating behavior. Two scenarios are tested for vertically migrating behaviors: i) staying near the top surface and ii) vertically migrating to 30 m after 5 days. Values are shown for a strong upwelling condition (summer) and a weak upwelling condition (winter). The time scales are calculated using the variogram range of the arrival density with arrival location held constant, while the length scales are calculated using the variogram range of the arrival density with arrival location held constant. Figure 6. Connectivity matrices obtained from four different realizations of the simulations (using different random number seeds in the wind forcing) under the strong upwelling condition in summer. Lagrangian particles are transported passively in horizontal directions while they stay near the top surface. The competency time window is set to 20 to 40 days. Nearshore waters are delineated into multiple equalarea sites with 4-km spacing in the along-shore direction (64 sites in the domain). Connectivity (color) is given by the number of successful settlers advected from a source site j to a destination site i divided by the total number of particles released from site j [Eq. (7)]. The source and destination site locations are shown with along-shore location here. Smaller values indicate site locations that are downstream. Values less than 0 or greater than the domain size (256 km) indicate source site locations that are outside of the domain. Settling particles from outside of the domain are simulated by taking advantage of the alongshore periodic flow fields. The dashed line in the connectivity matrices represent self settlement; i.e., the source and destination locations of settling particles are identical. Figure 7. Connectivity matrices obtained from the coastal circulation-simulations for a) a single spawning season, b) averaged over 5 spawning seasons, c) averaged over 10 spawning seasons and d) averaged over 20 spawning seasons under a strong upwelling conditions of summer. Connectivity matrices for a number of seasons are generated by combining the same number of randomly-chosen realizations of the simulations. The coefficients of variation for the number of the settlement among sites (variation for summation of each column of the matrix) decreases as 0.58, 0.23, 0.13 and 0.08. For detailed descriptions of connectivity matrices, see the caption of Fig. 6. Figure 8. Comparison of the coefficient of variation for the number of settlements among sites (CV_s) obtained from the circulation simulations (symbols) and predicted by scaling theory (packet model), Eqs. (6) as a function of the number of spawning seasons (or realizations). The expected CV_s value, <CV_s>, is plotted as a function of the number of spawning seasons. As averaged over many independent spawning seasons, spatial heterogeneity in settlement patterns decrease (see Figs. 3 and 7), and accordingly <CV_s> decreases. We examine a) the default case with 90-day larval release duration for a single spawning season and b) the reduced 30-day release case. Accordingly, T is set to a) T = 90, 180, ..., 2250 d and b) T = 30, 60, ..., 750 d for the scaling theory version. The other parameters are the same between the two cases: L = 256 km, l = 50 km, τ = 15 d and f_{sv} = 1. For the circulation simulations, the competency time window is set to 20 to 40 days. Two seasonal upwelling conditions are tested: a strong upwelling condition of summer (black circles) and a weak upwelling condition of winter (open circles). Two different scenarios are examined for ontogenetic vertical migration: staying near the top surface (circles) and migrate 30 m after 5 days from release (triangles). Simulated settlement patterns for a number of spawning seasons are generated by combining the same number of randomly-chosen realizations of the simulations. The expected value is computed using 20 different combinations. The horizontal line indicates CV_s = 0.13, corresponding to the rather smooth connectivity matrix shown in Fig. 3 panel c. Figure 9. Top panels: fraction of the total number settling particles that remained in nearshore waters (i.e., within 10 km from the coast) in the total number of settling particles obtained from the circulation simulations as a function of the competency time window under two seasonal upwelling conditions: a) a strong upwelling condition of summer and b) a weak upwelling condition of winter. Two scenarios are tested for vertical migrating behavior: staying near the top surface (black bars) and vertically migrating to 30 m after 5 days (white bars). Bottom panels: cross-shore dispersal range of the settlers (the distance from the coast within which 99% of the settling particles are contained) as a function of the competency time window under c) a strong upwelling condition of summer and d) a weak upwelling condition of winter. References Aiken, C. M., Navarrete, S. A., Castillo, M. I., Castilla, J. C., 2007. 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