Quantifying Connectivity in the Coastal Ocean With Application to the
Southern California Bight
Mitarai, S., Siegel, D.A., Watson, J.R, Dong, C., McWilliams, J.C.
\begin{ abstract }
The quantification of coastal connectivity is important for a wide range of real-world applications
ranging from assessment of pollutant risk to nearshore fisheries management. For these purposes, coastal connectivity is often defined as the probability that water parcels from one location have advected to another site over a given time interval. Here, we demonstrate how to
quantify coastal connectivity using Lagrangian probability-density functions (PDF) based on
numerical solutions of the coastal circulation of the Southern California Bight (SCB). Ensemble
mean dispersal patterns from a single release site show strong dependencies on particlerelease location, season and year, reflecting annual and interannual circulation patterns in the
SCB. Mean connectivity patterns are heterogeneous for the advection time of around 30 days or
less, due to local circulation patterns, and they become more homogeneous for longer advection times. A given realization of connectivity is stochastic because of eddy-driven transport and
synoptic wind forcing changes. In general, mainland sites are good sources while both Northern
and Southern Channel Islands are poor source sites, although they receive substantial fluxes of
water parcels from the mainland. The predicted connectivity gives useful information to ecological and other applications for the SCB (e.g., designing marine protected areas and predicting
the impact of a pollution event) and provides a path for quantifying connectivity using highresolution numerical solutions of coastal ocean circulations.
\end{ abstract }
1
\section{Introduction}
Solutions for many important problems in the marine environment require the quantification of
the connectivity of one nearshore site with another one via the oceanographic advection of water parcels. For example, the assessment of risk from exposure to a pollutant from a point
source requires the assessment of pollutant concentrations from the source site that have advected to all other sites in the region as a function of time \citep[e.g.,][]{ Fischer:1979 }. Further,
many nearshore fish and invertebrates have a life cycle that includes an obligate pelagic larval
stage that can last from a few days to several months \citep{ Kinlan:2003, Siegel:2003 }. Due to
the small size of marine larvae, advection by coastal circulations is the dominant process driving
larval dispersal which will have an order one influence on their fish stock dynamics \cite[e.g.,][]{
Jackson:1981, Roughgarden:1988; Cowen:2006 }. Hence the quantification of coastal connectivity is important for a wide variety of marine applications rang-ing from the assessment of risk
from pollutant exposure to nearshore fishery management and the evaluation of marine conservation measures.
These real-world applications requires one to rethink how to sample the coastal ocean. It is
natural to consider the dispersal of water parcels in a Lagrangian frame and oceanographers
have developed the observational capability to track surface water parcels using drifting buoys
to characterize near surface trajectories and Lagrangian dispersal statistics \citep[e.g.,][]{ Poulain:1989, Swenson:1996, Dever:1998, LaCasce:2008 }. While these surface-drifter observations have been successfully utilized to characterize regional circulation patterns and the statistics of particle dispersion in the coastal ocean, limitations in their spatio-temporal sampling restricts their ability to address source-to-destination relationships. Finally, only by chance can a
destination be characterized using surface drifter observations.
2
Lagrangian probability-density-function (PDF) modeling approaches, introduced by Taylor
(1921), have been widely used in predicting expected dispersal patterns of materials driven by
turbulent processes \cite[e.g.,][]{ Pope:2000, Mitarai:2003 }. This approach provides the detailed accounting of the probability that water parcels have advected from one location to another over a time interval. This accounting of connectivity can be made over all possible releases,
providing an estimate of the ensemble mean dispersal pattern, or for specific times or time periods, enabling the assessment of seasonal or event scale dispersal patterns. Once estimated,
Lagrangian PDFs can be used to estimate the expected pollutant concentration distribution from
a one-time, point pollutant release or the expectation distribution from a long-term release.
Hence, Lagrangian PDFs provide an appropriate metric for quantifying coastal connectivity.
The limitation in applying these approaches has been the large amount of Lagrangian trajectories required to calculate Lagrangian PDFs from field observations. Here, we overcome this restriction by using Lagrangian trajectories calculated from high spatial resolution coastal numerical model simulations.
The Southern California Bight (SCB) is one of the most-intensively studied coastal regions of
the world’s oceans \citep[e.g.,][]{ Sverdrup:1941, Hickey:1979, Lynn:1987, Hickey:1992, Hickey:1993, Harms:1998, Bograd:2003, Di-Lorenzo:2003, Dong:2007, Hickey:2003, Otero:2004 }.
Many of these studies were motivated by the 1969 Santa Barbara oil spill where more than
80,000 barrels of crude oil were released into the marine environment. This event in environmental history has led in large part to the extensive studies of the circulation of the Santa Barbara (SB) Channel and the consideration of the risks associated with pollutant releases from within
the SBC \cite[e.g.,][]{ Brink:1986, Dever:1998, Harms:1998, Kolpack:1971, Oey:2004,
Winant:1999 }. While these surveys clarified synoptic circulation patterns, their responses to
local wind forcing, water parcel paths, and Lagrangian dispersal scales in the SB Channel,
coastal connectivity throughout the SCB has yet to be precisely defined.
3
The goal of this study is to assess coastal connectivity in the SCB through Lagrangian PDFs
obtained from realistic circulation simulations. We simulate mean dispersal patterns from a single release-site (or Lagrangian PDFs) by implementing Lagrangian particle tracking in the
coastal circulation-simulations of \cite{ Dong:2007 }. We use multiple years of the simulations for
1996--2000, including the strongest El Ni\~no event on record in the equatorial Pacific in 1997-1998 and the rapid transition to strong La Ni\~na conditions in 1998--1999. The release-position
dependence, seasonal variability and interannual variability of Lagrangian PDFs are examined,
by taking advantage of documented surface-drifter and moored observations. We deduce
coastal connectivity of the SCB by evaluating the Lagrangian PDFs for a given source and a
destination. Nearshore waters (within 10 km from the coast) is delineated into 137 sites, and all
possible connectivity among the defined sites are examined by evaluating Lagrangian PDFs for
a given source and destination site. The obtained connectivity patterns are examined as a function of the advection time of Lagrangian particles. The findings of this study will be valuable information for assessing the risks of pollutant exposure and for understanding marine population
dynamics and regulating fish stocks (e.g., designing marine protected areas) in the SCB.
We first present the configuration of the numerical circulation model of the SCB and the formulation of the Lagrangian PDF method used to quantify in Section 2. Lagrangian PDFs and resulting connectivity patterns are then examined in Sections 3 and 4 focusing on the time scales of
connectivity for the SCB. Finally, we discuss the implications of the findings for ecological applications in section \ref{sec:discussion}, along with several other topics.
\section{ Configuration of Numerical Models }
\subsection{ ROMS }
4
We use the three-dimensional numerical hydrodynamic model ROMS (Regional Oceanic Modeling System) of \cite{ Dong:2007 } to determine velocity fields for the SCB. ROMS solves the
rotating primitive equations with a realistic equation of state in a generalized sigma-coordinate
system in the vertical direction and a curvilinear grid in the horizontal plane \citep{
Shchepetkin:2005 }. Three-level nested grids are employed in \citep{ Dong:2007 }. The outer
domain ($L_0$) covers the whole U.S. west coast with about 20-km horizontal grid spacing, and
the second embedded domain ($L_1$) covers a larger SCB area with a 6.7-km grid resolution.
The finest embedded grid $L_2$ with a 1-km horizontal grid resolution covers the SCB region.
Three nested grids share the same 40 vertical-level s-coordinate, with a standard land-mask
algorithm \citep{ Shchepetkin:1996 }. A one-way nesting approach is applied to the momentum
and mass exchange from $L_0$ to $L_1$ and $L_1 to $L_2$ \citep[e.g.,][]{ Penven:2006 }. The
lateral open-boundary conditions for $L_0$ are determined by taking advantage of the monthly
SODA \citep[ Simple Ocean Data Assimilation; ][]{ Carton:2000, Carton:2000a) global oceanic
reanalysis product. In this study, the solution of grid $L_2$ is used to assess coastal connectivity in the SCB (Fig.~\ref{ fig: sites }).
The momentum flux at the surface is calculated from a mesoscale reanalysis wind field, along
with climatological heat and fresh-water flux. A set of nested grids with resolutions 54 km, 18
km, and 6 km were implemented with the regional atmospheric model MM5 (the 5th generation
Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model) by
\citep{ Hughes:2007 }. The highest 6 km resolution contains the SCB region ($L_2$). Model
configuration was forced at its lateral and surface boundaries with 3-hourly data from the NCEP
ETA model reanalysis \citep{ Black:1994 }. The surface heat, freshwater, and short-wave radiation are from the monthly NCEP atmospheric Reanalysis 2. For more detailed descriptions
about the wind model configuration, see \cite{ Conil:2006}. \citep{ Dong:2008 } examined the
5
modeled wind, and found that the simulated seasonal pattern of wind stress and curl is consistent with observational analyses made by \citep{ Koracin:2004, Winant:1997 }.
The simulated flow fields are initialized as the oceanic state on December of 1995 from the SODA product, and the model is integrated using a repeating forcing of a normal year (1996) until
the solution reaches a quasi-steady state. The volume-averaged kinetic energy is monitored as
an indicator of equilibrium state \citep[cf.~][]{ Marchesiello:2003 }. The model was then integrated from 1996 through 2003, forced by the 3-hourly sampled hindcast MM5 wind and SODA
monthly boundary data. \cite{ Dong:2008} analyzed the $L_2$ grid solution with daily-averaged
samples, and found that the model solutions accurately describes mean circulation and interannual, seasonal, and intraseasonal variability (discussed further below).
\subsection{ Lagrangian Particle Tracking }
Estimates of coastal connectivity are made by simulating the trajectories of many water parcels
(or Lagrangian particles) released from nearshore waters \citep[e.g.,][]{ Siegel:2003,
Siegel:2008, Mitarai:2008 }. Lagrangian particles are transported passively with local simulate
currents both in horizontal and vertical directions. We use Lagrangian probability-densityfunction (Lagrangian PDF) methods to quantify site-to-site connectivity patterns in the SCB for a
given time advection time.
Nearshore waters where Lagrangian (or fluid) particles are released are defined as all waters
within 10 km from the coast, and are delineated into 137 sites (Fig.~\ref{ fig: sites } a). The center locations of these sites are located 5-km off the coast, and distributed with an approximately
10-km spacing along the shore. Each site covers a 5-km radius area. These sites cover most
of waters 100 meter deep or shallower where many rockfish larvae are produced (Fig.~\ref{ fig:
6
sites } b). Sites 1 -- 63 are along the mainland. Sites 64 -- 97 are on the shore of the Northern
Channel Islands (San Miguel, Santa Ana, Santa Cruz and Anacapa Islands), and sites 98 -- 137
are on the shore of the Southern Channel Islands (Santa Catalina, San Clemente, San Nicholas
and Santa Barbara). Lagrangian particles are released from each site near the top surface (5
meters below the surface) every 6 hours from January 1, 1996 through December 31, 2002, uniformly distributed with a 1-km resolution. The total number of Lagrangian particles released
from each site is close to 500,000, and sufficient to resolve spatial patterns of Lagrangian PDFs
and resulting connectivity (see below).
Fluid particle properties (e.g., position, velocity, material concentrations, etc) can be expressed
as functions of the initial position a and advection time $\tau$ \citep{ Corrsin:1962, Pope:2000 }.
The position of the n-th (realization of) fluid particle, $ \bm{X}_n (\tau, \bm{a}) $, evolves as
\label{ eq: X }
where $\bm{U}_n (\tau, \bm{a})$ indicates the velocity of the n-th Lagrangian particle. The particle velocity is determined by evaluating the Eulerian flow fields at a given particle location,
\label{ eq: U }
where $ \bm{u} (\bm{x}, t) $ is the Eulerian velocity at a given location $\bm{x}$ and time $t$,
and $t_n$ indicates the re-lease time of the $n$-th Lagrangian particle. Lagrangian particles
are tracked by integrating Eqs.~\label{eq:X} and \label{ eq: U } by using a fourth-order AdamsBashford-Moulton predictor-corrector scheme \citep[e.g.,][]{ Carr:2008, Durran:1999 }, given the
initial location, $\bm{X}_n (0, \bm{a}) = \bm{a}$. The Eulerian velocity at a given particle location is estimated through linear interpolation in time and space of the stored discrete velocity
7
fields. We use 6-hourly averages during ROMS simulations for the stored flow fields. We
tracked Lagrangian particles with a 15-minute time step for 120 days or until they cross the
boundaries, while recording the particle locations every 6 hours.
\subsection{ Estimating Lagrangian PDFs and Quantifying Coastal Connectivity }
A Lagrangian PDF describes the probability density function of particle displacement for a given
advection time $ \tau $. A discrete representation of Lagrangian PDFs, $ f’_{\bm{X}} (\bm{\xi};
\tau, \bm{a}) $, can be determined as
\label{ eq: PDF_discrete }
where $N$ is the total number of Lagrangian particles, $\delta$ is the Dirac delta function, and $
\bm{\xi} $ is the sample space variable for $ X $. We focus only of the evaluation of $
f’_{\bm{X}} (\bm{\xi}; \tau, \bm{a}) $ in the horizontal directions. Discrete Lagrangian PDFs are
evaluated from the center location of the sites defined above, averaged over each site, i.e.,
\label{ eq: PDF_approximation }
where $R$ is the radius of each site (5 km). A Lagrangian PDF is given by spatially-filtering a
discrete PDF in the sample space, i.e.,
\label{ eq: PDF }
8
where $G(\bm{x} - \bm{\xi})$ is a isotropic Gaussian filter with a width of 3 km. Values of a Lagrangian PDF represent the number of particles in km$^2$.
Lagrangian PDFs can be utilized to describe expected dispersal patterns of materials, neglecting molecular diffusion and subsequent chemical reactions, given the initial distributions of material concentrations, $ c(\bm{x}, 0) $. The expected concentration of the materials after a time
interval $\tau$, $ \langle c(\bm{x}, \tau) \rangle $, can be determined as
\label{ eq: PDF_and_production }
This states that the mean concentration at a point is the concentration carried by a particle times
the probability of the particle being there, integrated over all particles that could be there \citep{
Tennekes:1972 }. Different observation times can be assessed by changing the time of release
enabling both the assessment of a single release at different time and over particular time
scales (i.e., a given year or season or ensemble of seasons). The effects of turbulent mixing
and chemical (biological) reactions of materials in altering dispersal patterns will not be investigated. Approaches for accounting for these changes are is summarized in the appendix.
Coastal connectivity via advection of water parcels is quantified here using the calculated Lagrangian PDFs. The degree of connectivity between a given source site $j$ and a given destination site $i$, denoted here as $C_{ji} (\tau)$, is determined by evaluating a Lagrangian PDF
for the source location ($ \bm{x}_j $) to the destination location ($ \bm{x}_i $), i.e.,
\label{ eq: connect }
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where the normalization by $\pi R^2$ converts probability densities to probabilities. Connectivity matrices describe the probability for the event that a water parcel is transported from site $j$
to site $i$. Based upon the site locations shown in Fig.~\ref{ fig: sites }, this results in a 137 x
137 matrix, $ C_{ji}(\tau) $, which we define here as the connectivity matrix. Connectivity matrix
is a function of the advection time $\tau$. We evaluate Lagrangian PDFs and resulting connectivity matrices for $\tau$ = 1, 2, …, 120 days, and use $\tau$ = 30 days as a default case (a typical value for planktonic larval duration for many species; \cite{ Siegel:2003 }). More regional
connectivity can be examined by taking the partial summation of a connectivity matrix over selected source sites.
Values of the destination strength, $ D_i(\tau) $, measure the relative ``attractiveness” of site $i$
for all of the Lagrangian particles released in the domain for a release time of tau. $ D_i(\tau) $
is calculated by summing the connectivity matrix over all source sites within the domain, i.e.,
\label{ eq: destination_strength }
Similarly, the source strength, $ S_j(\tau) $, which measures the relative success that a site’s
particles encounter another site within a time scale $\tau$, can be calculated as
\label{ eq: source_strength }
Distributions of source and destination strength are useful for helping illustrate features of the
calculated connectivity matrices.
\section{ Lagrangian PDFs in the Southern California Bight }
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\subsection{ Turbulent Dispersion and Examples of Expected Dispersal Patterns }
Lagrangian trajectories for particles released off San Nicholas Island (site 130) illustrate the
quasi-chaotic nature of turbulent transport in the coastal ocean (Fig.~\ref{ fig: trajectories }).
The simulated trajectories clearly show sensitivity to the date of release. For example, many
Lagrangian particles released from this site on Jan. 31, 1996 are transported towards the northeast and after 30 days are found relatively close to the release site between the Northern and
Southern Channel Is-lands (Fig.~\ref{ fig: trajectories } c). However, many of those released
just two weeks later (Feb.~15, 1996; Fig.~\ref{ fig: trajectories } d), are transported poleward
along the mainland coast and are found within the SB Channel after 30 days. Differences in the
release times of just two weeks can lead to very different dispersal patterns. This clearly illustrates the characteristics of a turbulent dispersion problem.
Lagrangian length and time scales as well as values of eddy diffusivity are estimated from the
simulated particle-trajectories in order to validate the simulated dispersal patterns. These are
compared with similar statistics obtained from surface-following buoy observations \cite[e.g.,][]{
Swenson:1996 }. We employ the definition of eddy diffusivity proposed by \cite{ Davis:1987 }
following the approach of \cite{ Swenson:1996 }. Lagrangian statistics assessed for a
5$^{\circ}$ by 5$^{\circ}$ box centered at 35$^{\circ}$N 120$^{\circ}$W of \cite{ Swenson:1996 }
can be best compared with the simulation values, because the box overlaps well with our domain (with a broader coverage north of Point Conception). We find Lagrangian time and length
scales of 3.6 $\pm$ 0.8 days and 32.2 $\pm$ 7.1 km, respectively, that correspond to an eddy
diffusivity of 3.6 $\pm$ 1.0 $\times 10^7$ cm$^2$ s$^{-1}$. \cite{ Swenson:1996 } found typical
values for the Lagrangian time and length scales are 1.8 -- 6.8 days and 16 -- 64 km, respectively, while for the eddy diffusivity are 1.7 -- 7.0 $\times 10^7$ cm$^2$ s$^{-1}$. There is good
11
corresponding between Lagrangian statistics obtained from the simulation and the analysis of
field observation, strengthening our confidence for the simulated dispersal patterns of this study.
The Lagrangian PDF statistically describes the expected locations (or displacements) of Lagrangian particles that are released from a single site and transported for a given time interval.
For example, the Lagrangian PDF from a site on San Nicholas Island (site 130) calculated using
4 years of trajectories (1996 -- 1999) is shown in Fig.~\ref{ fig: PDF }. This release site is the
same as the example trajectories shown above (Fig.~\ref{ fig: trajectories }). Determinations of
the Lagrangian PDF from this site are roughly isotropic and they become more homogenous
over the entire domain as the advection time becomes longer, similar to a diffusion process
(Fig.~\ref{ fig: PDF }). This does not mean that materials introduced from this site spread out in
a simple diffusive manner, but rather that their expected distribution is diffusive. Water parcels
from this site can reach most of the locations within the SCB within 20 to 30 days (Figs.~\ref{ fig:
PDF } c -- d). For an advection time of 10 days, strong connectivity patterns are expected to
sites on the south side of Anacapa and Santa Cruz Islands, but not on the mainland or Northern
Islands (Fig.~\ref{ fig: PDF } b). For a longer advection times, the Lagrangian PDF is more uniform and similar levels of connectivity are expected throughout the SCB (Fig.~\ref{ fig: PDF } d).
\subsection{ Release-position dependence of Expected Dispersal Patterns }
The San Nicholas Island source site (\#130; Fig.~\ref{ fig: sites } a) is relatively isolated from the
other Channel Islands and the mainland. Lagrangian particle transport is, therefore, less influenced by the complex coastal topography and distinctive circulation patterns of the SBC. This
results in a roughly isotropic Lagrangian PDF (Fig.~\ref{ fig: PDF }). Lagrangian PDFs calculated from releases from other sites for a 30 day advection time, however, show a strong directionality associated with complex coastline and bathymetry of the SCB (Fig.~\ref{ fig: PDF_sites
12
}). Clearly, Lagrangian PDFs cannot be approximated as simple diffusion for many of these particle release sites (Figs.~\ref{ fig: PDF_sites }).
Simulated mean dispersal patterns can be categorized into four groups, depending on the release-site location. First, Lagrangian particles released on the mainland sites show distinctive
nearly unidirectional mean dispersal patterns (Figs.~\ref{ fig: PDF_sites } a -- d). The mean
poleward transport is observed from sites between the international border and Ventura (site
45), but not from sites in the north of Ventura (Figs.~\ref{ fig: PDF_sites } e -- f). Second, Lagrangian particles released from sites within the SB Channel tend to stay locally within the
Channel (Figs.~\ref{ fig: PDF_sites } e and h). Lagrangian particles released from the outer
edge of the SB Channel do not show this local retention, though (Figs.~\ref{ fig: PDF_sites } f
and \ref{ fig: PDF_sites } i). Third, water parcels originating in the Southern Channel Islands
tend to stay locally, in the southeastern part of the Bight (Figs.~\ref{ fig: PDF_sites } k and l).
This local retention signal is strongest on the eastern shore of San Clemente Island (Fig.~\ref{
fig: PDF_sites } l), and becomes substantially weaker as the site location becomes far from the
location (Fig.~\ref{ fig: PDF_sites } k). The final group includes Lagrangian particles that do not
belong to the first three groups. Lagrangian PDFs from these sites are less affected by complex
topography and distinctive circulation patterns, and do not exhibit strong spatial structures
(Figs.~\ref{ fig: PDF_sites } f, g and i).
The position dependence in Lagrangian PDFs reflects the distinctive circulation-patterns of the
SCB \cite[e.g.,][]{ Sverdrup:1941, Lynn:1987, Hickey:1992, Hickey:1993, Dong:2008 }. Strong
and persistent equatorward winds along the central California coast, which seemingly drive the
strong equatorward current (California Current) throughout the year, separate from the coast in
the vicinity of Point Conception, leaving the winds in the SCB, in general, weak and highly variable in time and space \citep[e.g.,][]{ Winant:1997 }. This wind-sheltering, combined with pole13
ward alongshore pressure gradient, creates a poleward Southern California Countercurrent
(SCC) throughout most of the year \citep[e.g.,][]{ Hickey:1979, Lynn:1987 }. This SCC at times
is found throughout the SBC and can exit at Point Conception \citep[e.g.,][]{ Lynn:1987 }. The
unidirectional Lagrangian PDFs shown in Figs.~\ref{ fig: PDF_sites } a -- d clearly reflect this
SCC. The juxtaposition of the California Current outside of the bight and SCC inside forms a
cyclonic circulation (Southern California Eddy) that is roughly fixed at the shallow off-shore
banks of the SCB \citep{ Bernstein:1977, Lynn:1987 }. The California Current has a large
shoreward component just in the south of the international boarder, and the part of the current
flows northward as the nearshore limb of SCC, creating cyclonic circulation between the Santa
Catalina and San Clement Islands (hereafter the Catalina-Clemente Eddy) \citep[e.g.,][]{
Lynn:1987 }. Similarly, a smaller circulation is observed in in the western part of the SB Channel (hereafter the SB-Channel Eddy), bounded by Northern Channel Islands \citep[e.g.,][]{
Beckenbach:2004, Harms:1998 }. Local retention of Lagrangian particles (Figs.~\ref{ fig:
PDF_sites } e, h and l) coincide with the locations where these cyclonic circulation form. However, this does not necessarily mean that Lagrangian particles are retained by eddies for the
entire advection time, as further explained below.
Within the SB Channel, the Lagrangian PDFs show the highest value at Chinese Harbor (site
84) on the northeastern side of Santa Cruz Island. A string of the Northern Channel Islands
makes local wind and pressure fields in the SBC complicated. Surface-drifters deployed in the
SBC often run aground, most often at Chinese Harbor \cite[e.g.,][]{ Winant:1999 }. This is not
caused by beaching of Lagrangian particles in this simulation as only 0.2 \% of the released water parcels run aground in the simulations and beached particles are not used in evaluating Lagrangian PDFs.
\subsection{ Seasonal and Interannual Variability of Expected Dispersal Patterns }
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Circulation patterns in the SCB show a strong seasonality, driven by variable wind forcing and
alongshore pressure gradient \citep{ Di-Lorenzo:2003, Lynn:1987 }. During winter, the wind
stress within the bight is weaker, spatially homogeneous and temporally variable; during summer, the wind forcing is stronger and more persistent with large spatial gradients \citep{
Winant:1997 }. The core of the California Current is strongest in spring and summer, and tends
to be closer inshore \citep{ Bograd:2000 }. The juxtaposition of the California Current and SCC
(and the Southern California Eddy) becomes much more coherent in summer \citep{ Lynn:1987
}. Although the flow in the SCB is mostly poleward throughout the year, equatorward flow does
occur at the coast near Oceanside (CalCOFI line 90) in the summer and weaker in the spring
\citep[e.g.,][]{ Lynn:1987 }. Seasonality of the circulation in the SBC can be characterized by
different synoptic circulation states responding to differing wind and pressure fields -- upwelling
and eastward flow in spring, a strong cyclonic circulation in summer, poleward relaxation in fall,
and weak and variable circulation in winter \citep[e.g.,][]{ Bray:1997, Winant:1999 }.
Expected 30-day dispersal patterns show strong seasonality reflecting known seasonal changes
in the circulation of the SBC (Fig.~\ref{ fig: PDF_seasonal }). First, poleward transport along the
mainland shows dramatic seasonality (Figs.~\ref{fig: PDF_seasonal } a--d). The mean poleward transport from Oceanside (site 10) is prominent in winter and autumn (Figs.~\ref{ fig:
PDF_seasonal } a and d), while it almost disappears in the spring and summer seasons
(Figs.~\ref{ fig: PDF_seasonal } b and c), corresponding to the appearance of equatorward current in spring and summer. The mean poleward transport is less coherent in winter than in autumn (cf.~Figs.~\ref{ fig: PDF_seasonal } a and d), because of the temporally-variable wind forcing in winter. Second, local retention in the SB Channel is dominant in winter, while this signal
is hardly seen other seasons (Figs.~\ref{ fig: PDF_seasonal } e--h). This feature of local retention is at its maximum in winter when circulation weakens, while comparatively no retention is
15
observed in the spring when upwelling and east-ward flow dominates \citep[e.g.,][]{ Dever:1998,
Winant:1999 }. This implies that Lagrangian particles tend to stay in the Channel simply because the particle transport is limited due to weak and variable flows, and not because cyclonic
eddy retains particles \cite[e.g.,][]{ Nishimoto:2002 }. Finally, retention of Lagrangian particles
off the eastern shore of San Clemente Island shows much less seasonal variability (Figs.~\ref{
fig:PDF_seasonal } i -- l) than is observed for the other retention zones shown here (Figs.~\ref{
fig:PDF_seasonal } a--h). The size of the retention zone off San Clemente Island becomes
larger in winter and autumn (Figs.~\ref{ fig:PDF_seasonal } i and l), and it is smaller in the
spring and summer (Figs.~\ref{ fig:PDF_seasonal } j and k), corresponding to the seasonal coherency of the Catalina-Clemente Eddy (and the Southern California Eddy).
Strong interannual changes in the circulation of the SCB are punctuated by the strong El Ni\~no
event of 1997 to 1998 and the rapid transition to strong La Ni\~na conditions in 1998 and 1999
\citep{ Bograd:2000, Lynn:2002, Schwing:2000 }. The 1997/1998 El Ni\~no is characterized by
a significant increase in dynamic height and a strengthening and broadening of the poleward
nearshore flow, starting the late spring of 1997 and ending the summer of 1998. By early 1999,
the regional circulation features reversed as strong sustained upwelling-favorable winds occurred through late 1998 and much of 1999, resulting in greater than usual southward transport
by the California Current \citep[e.g.,][]{ Schwing:2000 }.
The expected 30-day dispersal patterns represent well the expected inter-annual variability
(Fig.~\ref{ fig: PDF_interannual }). The mean poleward transport of Lagrangian particles along
the mainland is substantially stronger in 1997, reflecting enhanced SCC during the El Ni\~no
event (Fig.~\ref{ fig: PDF_interannual } b). The annual mean poleward transport almost diminishes in 1999, due to an enhanced California Current during the La Ni\~na event (Fig.~\ref{ fig:
PDF_interannual } d). The Lagrangian PDF for 1998 reflects both climatic anomalies as El
16
Ni\~no conditions were still found during the late-winter months while the La Ni\~na started during the fall (Fig.~\ref{ fig: PDF_interannual } c). Local retention in the SB Channel, caused by
weak and variable circulation in the winter, is not during the El Ni\~no event (Fig.~\ref{ fig:
PDF_interannual } f) and is reduced for La Ni\~na periods (Fig.~\ref{ fig: PDF_interannual } h).
The retentive nature of the SB Channel is the greatest during 1996, which is a ``normal'' year
(Fig.~\ref{ fig: PDF_interannual } e). During an El Ni\~no (or La Ni\~na) event, the relaxation (or
upwelling) circulation state is enhanced and poleward (or equatorward) flow becomes stronger.
For either case, more Lagrangian particles are exported from the SB Channel (Figs.~\ref{ fig:
PDF_interannual } f--h). This result is consistent with the analysis remote observations of phytoplankton chlorophyll concentrations in the SB Channel which show the highest concentrations
during ``normal'' years compared with the El Ni\~no or La Ni\~na years \citep{ Otero:2004 }.
Dispersal patterns showing retention off the eastern shore of the San Clemente Island are relatively insensitive to the year of evaluation (Figs.~\ref{ fig: PDF_interannual } i--l) although local
retention appears to be much more substantial in during the El Ni\~no (Figs.~\ref{ fig:
PDF_interannual } j and k).
\section{ Coastal Connectivity of the Southern California Bight }
\subsection{Quantifying Connectivity Matrices}
Coastal connectivity can be quantified by evaluating the simulated Lagrangian PDFs for a given
source and a given destination [Eq.~(\ref{ eq: connect })]. Coastal connectivity is most easily
expressed in matrix form providing what we will call the connectivity matrix. The connectivity
matrix illustrates the degree of connectivity for the possible combinations of nearshore sites given an advection time scale (Fig.~\ref{ fig: connect }). The influence of advection time is apparent on the structure of the coastal connectivity matrix. For a short advection time (10 days),
17
strong connectivity is found mostly amongst nearby sites, i.e., along the diagonal line of the
connectivity matrix (Fig.~\ref{ fig: connect } a). This is because the particle's advection time is
short. The mainland, Northern Channel Islands and Southern Channel Islands are, in general,
not well connected with each other, although there are some exceptions [e.g., transport from
Santa Barbara (site 50) to Chinese Harbor (site 84)]. On the other hand, the connectivity matrix
becomes more homogeneous for a relatively long advection time (60 days; Fig.~\ref{ fig: connect } d), implying that most nearshore sites have a near equal probability of being effected by
any other site (Fig.~\ref{ fig: connect } d). For intermediate advection times (20 to 30 days),
coastal connectivity is not homogeneous, reflecting distinctive circulation patterns of the SCB
(Figs.~\ref{ fig: connect } b and c), as described below. Further for advection time of 20 days
and longer, the calculated connectivity matrices are asymmetric (Figs.~\ref{ fig: connect } b--d).
This implies that the direction of the connection among sites is important beyond just knowing
the distance between sites (as would occur in a purely diffusive environment.).
\subsection{ Source and Destination Strength Distributions }
Spatial distributions of the source and destination strengths [Eqs.~(\ref{ eq: source_strength })
and (\ref{ eq: destination_strength })] provide a regional illustration of much of the information
provided in the connectivity matrix. Source strength distributions for particles that are successfully transported to the mainland sites (Fig.~\ref{ fig: source_strength } a), to the Northern Channel Islands (Fig.~\ref{ fig: source_strength } b), the Southern Channel Islands (Fig.~\ref{ fig:
source_strength } c) or to any nearshore site within the domain (Fig.~\ref{ fig: source_strength }
d) can be constructed by summing the connectivity matrix over the proper destination sites.
Similarly by summing over source sites, the distribution of destination strengths from the mainland, Northern Island, Southern Islands and from all sties can be visualized (Fig.~\ref{
fig:destination_strength }). For both of these summaries, the advection time is set to 30 days.
18
Also note that the color bars for the four panels in Figs.~\ref{ fig: source_strength } and \ref{ fig:
destination_strength } differ.
Source strength distributions show that water parcels with destinations on the mainland, the
Northern Channel Islands or the Southern Channel Islands all have distinct source locations
(Figs.~\ref{ fig: source_strength } a -- c). Water parcels arriving at the Northern Channel Islands
originate mostly from the northern portion of the mainland coast (i.e., between Santa Barbara
and Santa Monica Bay) and the northern shore of the Santa Cruz and Santa Rosa Islands
(Fig.~\ref{ fig: source_strength } b). On the other hand, water particles arriving at the Southern
Channel Islands come mainly from those islands and the mainland centered around Palos
Verdes Peninsula (Fig.~\ref{ fig: source_strength } c). These transports from the mainland to
the Islands are clearly captured by Lagrangian PDFs (Figs.~\ref{ fig: PDF_sites } b -- e).
Water parcels arriving at the mainland sites have come from the mainland and not from the offshore Islands (Fig.~\ref{ fig: source_strength } a). The source strength is greater between
Oceanside and Ventura and peaks near four sites (i.e., Oceanside [10], Long Beach [25], Santa
Monica [32] and Ventura [43]; Fig.~\ref{ fig: source_strength } a) as can also be seen in the
connectivity matrix (Fig.~\ref{ fig: connect } c). Although many water parcels released from
these sites are transported poleward alongshore, a substantial number of water parcels remain
at the release sites (Fig.~\ref{ fig: connect } c) suggesting a self-seeding nature for these sites
for a 30 day advection time. Accounting for water parcels to all sites, sites around Ventura,
Santa Monica and Long Beach (and not Oceanside) are the strongest sources because these
sites export water parcels not only to mainland sites, but also to the Channel Islands (Fig.~\ref{
fig: source_strength } d). Overall, the mainland sites are good sources, while the Island sites
are poor sources, although there is substantial transport of water parcels from the mainland
sites.
19
Destination strength distribution shows regional retention of water parcels for both the Northern
and Southern Channel Islands while the mainland receives many fewer water parcels
(Figs.~\ref{ fig: destination_strength } a -- c). Unlike the source strength distribution (Figs.~\ref{
fig: source_strength } a -- c), destination strengths from the mainland, Northern and Southern
Channel Islands are roughly comparable (Figs.~\ref{ fig: destination_strength } a -- c). Accounting for all water parcels from all sites, Chinese Harbor (northeastern shore of Santa Cruz Island;
site 84) attracts most the water parcels (Fig.~\ref{ fig: destination_strength } d), collecting water
parcels from the mainland due to poleward transport (Figs.~\ref{ fig: PDF_sites } b -- d) and
from the sites within the SB Channel (both on the mainland and Northern Channel Islands) due
to local retention (Figs.~\ref{ fig: PDF_sites } e and h). This is consistent with the location of
beached drifters from deployments in the SB Channel \citep{ Winant:1999 }.
The modeled dispersal patterns differ by season (Figs.~\ref{ fig: PDF_seasonal }) and by year
(Figs.~\ref{ fig: PDF_interannual }) which will obviously have an influence on the source strength
distributions evaluated either by season (Figs.~\ref{ fig: connect_seasonal }) or by year
(Figs.~\ref{ fig: connect_interannual }). Source strength distributions of particles that are advected to the Northern Channel Islands change dramatically from season to season (Figs.\ref{ fig:
connect_seasonal } a--d). In winter, water parcels reach the Northern Channel Islands from a
wide range of source locations including the Southern Channel Islands (Fig.\ref{ fig: connect_seasonal } a). In spring and summer, the strong sources weaken and become more local
to the SB Channel (Figs.~\ref{ fig: connect_seasonal } b -- c). In autumn, strong sources are
found mostly along the central mainland coast (Fig.~\ref{ fig: connect_seasonal } d). Source
distributions to the Southern Channel Islands also change substantially from season to season
(Figs.~\ref{ fig: connect_seasonal } e--h). Summer source strength distributions for particles
advected to the Southern Channel Islands (Fig.~\ref{ fig: connect_seasonal } g) is similar to the
20
mean source distributions (Fig.~\ref{ fig: source_strength } c), although its value is weaker.
However in spring, strong sources are distributed along the northern mainland (Fig.~\ref{ fig:
connect_seasonal } f) while transport from the mainland becomes substantially less in autumn
(Fig.~\ref{ fig: connect_seasonal } h). These source distributions for particles advected to the
Southern Channel Islands corresponds to the seasonal cycle of poleward transport (Figs.~\ref{
fig: PDF_seasonal } a -- d).
Interannual changes in the spatial distribution of particle source strengths are less apparent
(Figs.~\ref{ fig: connect_interannual }). Although variations are found year to year, changes in
the overall regional pattern of the source strength distribution are not found (Figs.~\ref{ fig: connect_interannual }). For example, source strengths (to the Northern Channel Islands or from the
mainland) are only slightly weaker in El Ni\~no year (1997; Figs.~\ref{ fig: connect_interannual }
b and j) than in the normal year (1996; Figs.~\ref{ fig: connect_interannual } a and i). However,
these differences are much smaller than the magnitudes of the observed seasonal changes in
source strength (Figs.~\ref{ fig: connect_seasonal }). For water parcels advected to the Southern Channel Islands, regional connectivity patterns differ for 1997 where San Clemente Island is
a strong and isolated source (Fig.~\ref{ fig: connect_interannual } f) which differs from the other
years which shows a substantial mainland source (Figs.~\ref{ fig: connect_interannual } e, g and
h). Changes in the destination strength to the mainland are observed and in 1999 these patterns become substantially weaker, although the spatial pattern is nearly unchanged (Figs.~\ref{
fig: connect_interannual } i -- l).
\subsection{ Assessing the Time Scales of Coastal Connectivity }
The above analyses of coastal connectivity have shown the temporal dependence of connectivity patterns and strengths as a function of advection time (Figs.~\ref{ fig: connect }), and the
21
season (Figs.~\ref{ fig: connect_seasonal }), or year (Figs.~\ref{ fig: connect_interannual }) of
evaluation. However, individual dispersal patterns include intrinsic variability, reflecting water
parcel transport by coastal eddying motions (Fig.~\ref{ fig: trajectories} ). Clearly, these intraseasonal eddying motions must contribute some to coastal connectivity. Monthly time series of
connectivity strength between six pairs of nearshore sites are shown in Fig.~\ref{ fig: connect_realizations } including examples of ``self connectivity'' of a site with itself. The degree of
connectivity varies substantially on a wide range of time scales between sites showing strong
mean connectivity (Figs.~\ref{ fig: connect_realizations } a and b), sites with weak mean connectivity (Figs.~\ref{ fig: connect_realizations } c and d), and between sites with themselves
(Fig.~\ref{ fig: connect_realizations } e and f). Coastal connectivity is clearly not persistent even
within a season. This intrinsic variability of connectivity is not caused by an insufficient number
of particles released in the simulated flow fields. Connectivity patterns determined with one-half
of the number of particles released resulted in nearly identical dispersal patterns (not shown).
Hence, these intraseasonal peaks in coastal connectivity are created by the transient threads of
connections created by coastal eddying motions \cite[e.g.,][]{ Siegel:2008 }.
In order to understand the role of time scales on connectivity patterns, we first partition the connectivity matrix into a time mean and a standard deviation which measures the time variability
on all sales (Figs.~\ref{ fig: connect_variability } a and b). It is clear that mean connectivity levels are substantially less than those for time variable connections as measured by the standard
deviation distribution. Hence, measuring coastal connectivity by the time mean distribution may
result in the incorrect assessment of connectivity.
The time variable components of connectivity have sources due to seasonal and interannual
variations in circulation patterns as well as eddy-induced water parcel transports. Here, we
quantify the degree of each source of variability by digitally filtering the connectivity time series
22
(Figs.~\ref{ fig: connect_seasonal }) and calculating the root mean variance in the interannual
and seasonal time band (periods of 2 cycles per year and less; Fig.~\ref{ fig: connect_variability
} c) and for the intraseasonal band (periods of 3 cycles per year and greater; Fig.~\ref{ fig: connect_variability } d). The result of this Fourier analysis suggests that intraseasonal band has as
much power as the seasonal and interannual bands combined (Figs.~\ref{ fig: connect_variability } c and d). Further, the seasonal and interannual connectivity (Fig.~\ref{ fig:
connect_variability } c) and the eddy time scale connectivity (Fig.~\ref{ fig: connect_variability }
d) are comparable to the magnitude of the mean connectivity pattern (Fig.~\ref{ fig: connect_variability } a). This result suggests that eddy-induced variability has a power to greatly
control realized connectivity patterns \citep{ Siegel:2008 }.
\section{ Discussion }
We assessed coastal connectivity via advection of water parcels in the SCB by numerically simulating the trajectories of millions of Lagrangian particles. Coastal connectivity was assessed
through the Lagrangian PDF formulation that showed a good qualitative agreement with descriptions based upon available observations. The coastal connectivity is useful information that
can answer some of important questions to ecological and other applications for the SCB, if the
model prediction is accurate. In the following we address potential issues with the modeling,
and demonstrate how to best utilize the model outcomes for real-life applications.
\subsection{ Model Accuracy }
The connectivity patterns shown here are dependent upon the success that the ocean circulation model output correctly models the coastal circulations and that the Lagrangian calculations
done with it are appropriate. Outstanding issues include the degree at which the \cite{
23
Dong:2007 } model simulations represent real ocean flows, the importance of subgrid scale motions on the simulated Lagrangian paths and the effects of boundaries on the assessment of
connectivity. Diffusivity due to subgrid-scale motions (subgrid-scale) may be as substantial as
mesoscale motions. Once particles leave the domain, we cannot keep tracking them, while in
reality they may (or may not) come back to the SCB (boundary effect). Connectivity patterns in
the SCB may be underestimated due to the subgrid-scale diffusivity or the boundary effect. We
address these potential issues or concerns here.
ROMS simulations of \cite{ Dong:2007 } resolve major circulation patterns in the SBC fairly well
\citep{ Dong:2008 }. The model accuracy has been assessed against statistical measures of
extensive observational data, including high-frequency radar data, current meters, acoustic
doppler current profilers data, hydrographic measurements, tide gauges, drifters, altimeters, and
radiometers \citep{ Dong:2008 }. Comparisons with observational data reveal that ROMS reproduces a realistic mean state of the SCB oceanic circulation, as well as its interannual, seasonal and intraseasonal variability. The model has been applied to study the evolution in the
SCB island wakes \citep{ Dong:2007 } and successfully simulate one of the strongest upwelling
events in SCB during March 2002 \citep{ Dong:2007 }. Therefore, the major characteristics of
coastal connectivity, including the mean and seasonal and interannual variability (Fig.~\ref{ fig:
connect_variability } a and c) are expected to be accurate. As it stands it is less likely that the
model can reproduce the exact patterns of realizations (Fig.~\ref{ fig: connect_variability } d),
because modeling of individual eddy-movement is required, although qualitatively it can. The
further model improvements and data assimilation are required.
Subgrid-scale eddy diffusivity is smaller, by orders of magnitude, than resolved eddy diffusivity.
Eddy diffusivity is scaled as $ u’^2 l $, where $ l $ is the scale of motion and $ u’^2 $ is its kinetic energy. Both $ u’^2 $ and $ l $ associated with subgrid-scale motions are much smaller than
24
those of mesoscale eddies. Subgrid-scale diffusivity can be quantified by using available fluiddynamics models. For example, using a conventional Smagorinsky model,
\label{eq:Smagorinsky}
where $ \Delta_G $ is the filter size, $ C_s $ is an empirical constant, and $ \langle S_{ij}
\rangle_l $ is resolved strain rate tensor. The mean strain rate in the SBC is an order of the
Coriolis frequency \citep{ Dong:2007 }. Given $ C_s $ = 0.01, $ \Delta_G $ = 1 km, and $
\langle S_{ji} \rangle_l = f = 7.7 x 10^{-5} s^{-1} $ (corresponding to 32$^{\circ}$ N), $ \gamma_t
$ = 7.7 $ \times $ 10$^3$ cm$^2 $ s$^{-1}$. Notice that resolved eddy diffusivity is 3.6 $ \times
$ 10$^7$ cm$^2$ s$^{-1}$) as mentioned earlier. Subgrid-scale diffusivity is smaller than resolved diffusivity by four orders of magnitude. For example, for the advection time of $ T $ = 30
days, expected displacements due to subgrid-scale motions are only $ (\gamma_t T)^{1/2} $ =
1.44 km. Subgrid-scale diffusivity will be significant in determining each realization of particle
path or two-particle statistics (e.g., separation distance between particles), but not in evaluating
Lagrangian PDFs or coastal connectivity.
The domain covers the SCB eddy, and hence the boundary effects are considered to be less
significant. The SCB will retain particles for a prolonged duration of time, during which coastal
connectivity is established. To illustrate this point, we computed particle residence (e-folding)
times from each site (Fig.~\ref{ fig: residence }). Sites near San Diego show the shortest residence time, i.e., around 10 days. This is mainly due to that these sites are located outside of
the SCB eddy (Fig.~\ref{ fig: residence } b; see also \cite{ Dong:2008 }), not by the boundary
effect. Lagrangian particles are expected to be transported to equatorward due to California
Current attaching back to the mainland around there. Similarly, sites near Point Conception,
San Miguel and Santa Rosa (south shore) Islands have shorter residence time, i.e., around 20
25
days. Sites around Point Conception and San Miguel Island indicate similar residence times
regardless of the difference in distance from the boundaries. This implies that the reduced residence times are mainly not due to the boundary effect, but the strong transport of California
Current. The domain size should be large enough to resolve connectivity in the SCB. We need
a larger domain to assess connectivity with Central California and across the international
boarder. It is however beyond the scope of this study.
In summary, the model is a good first assessment of connectivity patterns in the SCB. The
model outcomes are ready to be utilized for ecological applications. Mean, seasonal and interannual connectivity patterns can be utilized to forecast expected connectivity patterns for a given season under a given climate condition with a reasonable accuracy. The future direction is
data assimilation. We plan to integrate this modeling framework to ongoing data assimilation of
the Southern California Coastal Ocean Observing System (SCCOOS; http://www.sccoos.org/).
Data-driven models will give us further spatio-temporal resolutions in predictability, and can be
used for daily-forecast of applications.
\subsection{ Applications of Lagrangian PDFs }
The advantage of using Lagrangian PDFs is that they can be used to predict future values of
material concentrations over specified time scales. One case where predicted coastalconnectivity can play an important role is in the assessment of risk from exposure to a pollutant
from a point source. The accuracy of the risk assessment will depend on the quantification of
both expected and realized connectivity from the source site to all other sites in the region as a
function of time scale from pollutant release ($ \tau’ $) and of time ($ \tau $) itself. We can evaluate for the expectation or for the exact realization knowing the appropriate $ c(\bm{a}, t) $. For
instance, pollutant spread from a single-point source ($ \bm{a} $) can be expressed as
26
\label{ eq: pollutants }
In the study of ecology, $ f_{\bm{X}} (\bm{x}; \tau, \bm{a}) $, is often referred as to ``dispersal
kernels.’’ The same formalism can be applied in predicting larval dispersal patterns, assuming $
c(\bm{a}, t) $ is larval production. Hence knowing dispersal kernels (Lagrangian PDFs) is key to
successfully predict the future distributions of pollutants and any materials.
Understanding fisheries requires knowledge of recruitment. The number of new recruits at a
given site $ i $ ($ R_i $) can be assessed by knowing larval production at all source sites ($ P_j
$, where $ j = 1, 2, ..., N_j $) and connectivity from site $ j $ to site $ i $. Mathematically,
\label{ eq: recruitment }
where $ m $ is larval mortality, and $ \gamma_i $ is the fraction of larvae that successfully recruit to adults in all arriving (settling) larvae. The summation indicates larval supply to site $ i $
from everywhere. $ C_{ji}(\tau) $ is coastal connectivity via advection of water parcels
(Fig.~\ref{ fig: connect }), whereas $ (1-m) P_j C_{ji}(\tau) $ is connectivity via advection of actual larvae. Different species have different spawning seasons and different planktonic larval
duration ($\tau$). Accordingly, as described above, connectivity patterns can be greatly for different species (cf.~Figs.~\ref{ fig: connect } – \ref{ fig: connect_interannual }). Larval behavior
(e.g., ontogenetic vertical migration) might also alter connectivity patterns as well. Key to successful assessment of recruitment is knowing species-specific connectivity.
\subsection{ Application to MPA siting }
27
Time scales of potential connection is key information for building resilient networks of marine
protected areas (MPAs). The location and magnitude of larval spillover from a given MPA describes its ability to provide new juvenile individuals to other unprotected areas \citep[e.g.,][]{
Botsford:2003 }. Larval-spillover distance depends on planktonic larval durations (PLDs). Given the ability to quantify coastal connectivity described in this paper, we can describe larval
spillover for a range of sedentary or sessile marine species in the SCB. In general, regardless
of the species, the spillover distance should increase as the PLD becomes longer. We assess
‘generic’ larval-spillover patterns through the mean connection time between given sites.
The mean connection time between site $ j $ and site $ i $ can be estimated with the mean advection time weighted by $ C_{ji}(\tau) $, i.e.,
\label{ eq: connect_time }
We estimate the mean connection time by setting the upper limit of the integral in Eq.~(\ref{ eq:
connect_mean }) to 120 days. Sample mean connection times from four different sites (including ongoing MPA sites) are shown in Fig.~(\ref{ fig: connect_time }). Connection times among
the SB Channel sites are relatively short (10 -- 20 days), whereas connection from inner SB
Channel to outer SB Channel takes much longer (40 -- 60 days) (Figs.~\ref{ fig: connect_time }
a and b), reflecting local retention in the SCB (cf.~\ref{ fig: PDF_sites }). Mean connection time
from Oceanside increases along the path of the counter-clockwise SCB eddy, i.e., about 25
days to Palos Verdes Peninsula, 50 days to the SB Channel and Point Conception, and over 2
months to the the outer limb of the SCB eddy (Fig.~\ref{ fig: connect_time } c). Mean connec-
28
tion times from Santa Catalina Island show less spatial variations, less than 50 days to any sites
(Fig.~\ref{ fig: connect_time } d).
PLDs of marine species range from a few days to several months \citep{ Siegel:2003 }. Spillover of species with relatively short PLDs (within a few weeks; e.g., red abalone) will be, in average, limited to local areas, while species with longer PLDs (several months; e.g., urchin, rockfish and lingcod) can spill over to all sites in the SCB (Fig.~\ref{ fig: connect_time }), although at
the same time many of them will be lost to outside of the domain (Fig.~\ref{ fig: residence} d).
Species with moderate PLDs (a month; e.g., kelp bass) will be in between. The mean connection time is not proportional to the Euclidean distance between sites. For example, species with
short PLDs will spillover from Oceanside to Huntington Beach, but not to Southern Islands, despite that they are nearly equally separated from the source (Fig.~\ref{ fig: connect_mean} d).
Fishery managers need to consider circulation patterns in designing MPAs. If more uniform
spillover profits fisheries, Southern Channel Islands are good candidates for MPA sites
(Fig.~\ref{ fig: connect_mean} c).
\section{ Summary and Conclusions }
Assessment of water parcel connectivity in the SCB has been made by using ROMS simulations validated against available observations. The simulated dispersal patterns are consistent
with oceanographic patterns documented in the literature. Expected dispersal patterns (and
coastal connectivity) show strong dependencies on particle-release location, season and year,
reflecting annual and interannual circulation patterns in the SCB. Realizations vary month to
month, reflecting intrinsic variability induced by eddy motions. Intrinsic variability is as much as
the mean and seasonal/interannual variability, and thus sets a strong source of uncertainty in
realized connectivity patterns. Strong connectivity is limited to among local areas for a short
29
advection time (a few weeks), and more sites can be connected for longer time (a month or
longer), with a strong directionality. In general, mainland sites are good sources, and transport
substantial water mass to Islands sites, but not the other way around. The future points to these
techniques holding in a variety of applications. One issue is intrinsic eddy-driven variability
which may be solved by data assimilation models for real-time forecasts, within the framework
of SCCOOS. We will need not only Eulerian validations, but also Lagrangian validations of circulation models by taking advantages of existing and ongoing observations, e.g., drifters, population genetics, microchemistry and fish (invertebrate) recruitment data. Circulation models
have been improved all the time. Real-time forecast is within the reach in the coming decade.
\begin{ acknowledgments }
The authors acknowledge a series of enlightening discussions with Steve Gaines, Bob Warner,
Chris Costello, Bruce Kendall, Ray Hilborn, Steve Polasky, Libe Washburn, Jenn Caselle, Satie
Airame, Dan Rudnick and John Largier, Erik Terrill, Tim Chaffey and James Watson. This work
is a contribution of the ``Flow, Fish and Fishing” biocomplexity project work supported by the
National Science Foundation (NSF grant \# 0308440) and also supported by Coastal Environmental Quality Initiative.
\end{ acknowledgments }
Appendix – PDF methods including diffusion and reactions
The framework of Lagrangian PDF methods used in this study does not account for the changes
of water-parcel composition due to mixing and chemical (biological) reactions of materials. The
mean concentrations of materials can be predicted by utilizing Lagrangian PDFs [Eq.~(\ref{ eq:
PDF_and_production })], if we can assume that water-parcel properties are unchanged during
30
the transport, which may not relevant for some applications (e.g., risk assessment of exposure
to pollutants). In the last decades, PDF methods have been extended to account for these effects \citep{ Pope:2000 }. Here we summarize this modeling approach, which may be utilized
for ecological and other applications.
There exists a technique to theoretically derive the Fokker-Planck equation \citep{ Gardiner:1997 } from an advection-diffusion-reaction equation \cite[e.g.,][]{ Pope:1985 }. Suppose
that species composition of water parcel at position $ x $ at time $ t $ (denoted here as $
\phi_\alpha (\bm{x}, t) $, $ \alpha = 1, 2, ..., N_s $) is determined by an advection-diffusionreaction equation, or
\label{ eq: advection_diffusion_reaction }
where $ u_i( \bm{x}, t ) $ is the Eulerian velocity, $ \kappa $ is the diffusion coefficient, and $
\omega_\alpha ( \bm{\phi} ) $ is the reaction term for species $ \alpha $. In turbulent-flow environment, each realization leads to different species composition at the observation point. The
Fokker-Planck equation describes the expected species composition (or composition PDF), denoted here as $ f_{\bm{\phi}} ( \bm{\psi}; \bm{x}, t ) $. The composition PDF obeys
\label{ eq: Fokker-Planck }
where $ \psi_\alpha $ is the sample space variable for $ \phi_\alpha( \bm{x}, t ) $, and the
bracket $\langle \rangle$ is an ensemble-averaging operator. Notice that the reaction term is
exact and no modeling is required, but diffusion term and advection term are unclosed and re-
31
quire a modeling. Physically-sound models have been developed for the unclosed terms
\citep[e.g.,][]{ Pope:1985, Chen:1989, Colucci:1998, Mitarai:2003 }, e.g.,
\label{ eq: unclosed_advection }, and
\label{ eq: unclosed_diffusion }
where $ \Omega $ is mixing frequency \citep{ Pope:1985 } and $ \kappa_t $ is eddy diffusivity,
which has been discussed in this paper.
The closed (modeled) Fokker-Planck equation [Eqs.~(\ref{ eq: Fokker_Planck }) -- (\ref{ eq: unclosed_diffusion })] can be integrated directly or indirectly through Monte Carlo simulations of
the equivalent stochastic particle system \citep[e.g.,][]{ Gardiner:1997 }. The position and species concentrations of a stochastic particle evolve as
\label{ eq: SDE_position }, and
\label{ eq: SDE_concentration }
where the superscript $ * $ is used to distinguish the stochastic particle system from fluid particle, and $ W_i $ is the Wiener process \citep[e.g.,][]{ Gardiner:1997 }. Species concentrations
change due to the the first term (mixing) and second term (chemical, biological reactions) of the
right hand side of Eq.~(\ref{ eq: SDE_concentration }). Theoretically, Lagrangian PDFs shown
32
in study (Figs.~\ref{ fig: PLD } -- \ref{ fig: PDF_interannual }) can be reproduced by integrating
Eq.~( \ref{ eq: SDE_position }) for a given release site, similarly to Eq.~(\ref{eq: PDF_discrete})
by only using the mean flow statistics, which is less expensive than implementing Lagrangian
particle tracking in ROMS flow fields. The effects of mixing and reactions, if necessarily, can be
assessed by integrating Eq.~( \ref{ eq: SDE_concentration }) at the same time.
The same approach can be applied to describe species concentrations that are spatially-filtered
over a numerical grid-spacing centered at $x$ and at a given $t$. The governing equations are
very similar to Eqs.~(\ref{ eq: Fokker-Planck }) -- (\ref{ eq: SDE_concentration }). For example,
the position of a stochastic particle evolves as
\label{ eq: SDE_position_LES }
where $\gamma_t$ is subgrid-scale eddy-diffusivity (not resolved eddy-diffusivity), and $ \langle
\rangle_l$ is an spatially-filtering operator. This can be considered as more accurate Lagrangian-particle-tracking implementation for ROMS, accounting for numerical resolved velocity ($
\langle u_i \rangle_l $) and subgrid-scale motions (terms including $ \gamma_t $). However, as
mentioned earlier, the subgrid-scale contributions are negligible in evaluating Lagrangian PDFs
($ \gamma_t \ll \kappa_t$).
This Fokker-Planck modeling approaches have been assessed by using direct numerical simulations of Navier-Stokes equations for chemically-reacting turbulent flow problems, and their accuracy has been established \citep[e.g.,][]{ Colucci:1998, Mitarai:2005 }. The accuracy and applicability to marine applications have yet to be investigated.
33