Numerical modeling the development and transport of
lobster larvae (Homarus americanus) in the Coastal Gulf of Maine
Danya Xu, Huijie Xue, Steve Cousins
1.Introduction^{}
Pelagic marine
organisms may be transported considerable distances away from their original
spawning or hatching locations by the ocean currents. In the Gulf of Maine,
zooplankton and larvae may experience a long journey before they settle down
due to strong surface currents. Incze and Naimie (2000) used a coupled physical biological,
individualbased model (IBM) to calculate the trajectories of larval and
postlarval lobsters in the Gulf of Maine. Their model results show that the
local population is supplied by upstream sources.
Lagrangian
stochastic modeling of the advection and dispersion of clusters of particles
has been a common technique both in atmospheric boundary layer study and in
oceanography. Few studies (e.g.,
Visser,1997, Brickman et al,2002, Proehl et al., 2004) have been done to
simulate the distribution of planktonic cells and larvae development. Most of
these studies focus on dispersion in the vertical instead of horizontal
direction in the ocean. In this study, a robust Lagrangian particletracking
model is embedded within a 3D coastal ocean model to study the transport and
development of pelagic lobster larvae in the coastal region of the Gulf of
Maine. The ocean model calculates the circulation in the Gulf of Maine and its
variability in response to wind, surface heat flux, river discharge, and tides.
Particles can be tracked in the 3D, timedependent Eulerian flow field to
examine their Lagrangian trajectories. The particle trajectory model has been
extended to include the random walk mechanism that simulates transport
processes by subgrid scale turbulence. Multiple tracers can be released at any
given position, specific time and be tracked in different trajectories. After
some time, the tracers of the same group may end at totally different places
due to random walk associated with turbulence diffusion. This process is very
hard to be conducted in drifter experiment in reality but could be simulated in
the model.
The model is used
to simulate the life history of lobster larvae released at the potential
spawning sites near the coast of the Gulf of Maine. In the model, larvae are
restricted at the depth of 5m and develop to the postlarvae stage depending on
the ambient temperature. Model results showed that larvae hatched at different
times around the Gulf result in a great deal of spatial and temporal
differences in life history since both the flow field and the surface
temperature change seasonally and spatially.
2. Method^{}
2.1 Ocean
circulation model
First of all, we have been successfully
simulating the seasonal circulation of the Gulf of Maine using the Princeton Ocean Model (POM). Our study region covered the entire Gulf of Maine
area. The model has 180 x 120 horizontal grid points and 22 vertical levels to map the Gulf of Maine,
Georges Bank, Scotian Shelf and the adjacent slope region with realistic topography to 3000 meter isobath. The model resolution varies
from 35 km. At the surface, The gulf of Maine nowcast/forecast system is
driven at the surface with the heat, moisture, and momentum fluxes from the National
Center for Environmental Prediction (NECP) ’s Eta mesoscale
atmospheric forecast model with a
spatial resolution of 32 km and a temporal resolution of 3h. Boundary forcing
includes daily river outflows from St. John, Penobscot, Kennebec, Androscoggin,
Saco, and Merrimack, tidal (M2,S2,N2,K1,O1,P1) and subtidal forcing from the
open ocean, which is interpolated from the daily nowcast of the NCEP regional Ocean Forecast System
(ROFS). This version
also includes SST
assimilation. The model results are on the web as a part of the Gulf of Maine
Ocean Observing System (GoMOOS)
effort.
Detailed discussions of the Gulf of Maine
circulation model can be found in Xue et al, 2005: The GoMOOS Nowcast/forecast
System (Continental Shelf Research, 25(2005), 21222146).
2.2
Particletracking algorithm
The
tracer model is based on Jarle Berntsen (1994). The original model only had the advection term,
and particles could move freely in all 6 directions. A random walk term has
been added to this tracer algorithm. As we do not know details of the
biological behavior and vertical migration of lobster larvae, in our model,
larvae move at a fixed depth of 5m.
Random walk
formulations are based on the same assumptions as the advectiondispersion
equation in which the overall transport of particles during a time interval Dt results from an advection component (model
resolved flows) and a dispersive component that accounts for the unresolved
flow processes. The advection component is calculated for each particle by
interpolating the 3D current velocity, V(t), to the particle¡¯s position and
transporting it to a new position at the next time step. Furthermore, particles
tend to disperse with time in turbulent flows due to smallscale fluctuations
in the field. Since the gridded mean current velocity cannot resolve these
smaller scale turbulence, a numerical technique must be employed to simulate
particle dispersion. In a Lagrangian framework, this is idealized by
superimposing a random walk for each particle, equivalent to an approximate
solution to the diffusion term in an Eulerian frame. The equation that
describes the particle position can thus be written as
where X_{i}
(t) is the position vector of the i^{th} particle at time t, V(t)
is the local current velocity, Z(t) is a vector representing turbulent
dispersion (random walk term), which contains three component Gaussian random
numbers with zero mean and a standard deviation related to the rate of
diffusion. It could be explained as: the position of a particular particle at
next time step is calculated based on the position of the current time step
plus the advection distance by the local currents and the diffusion effect.
In order to keep
all of the terms’ units in the above formula uniform, we do the dimensional
analysis. The dimension of the random walk term is kinematic viscosity (m^{2}s^{1}) divided by spatial
interval of our model grid.
In the model, the random
walk term is calculated from kinematic viscosity (m^{2}s^{1}) divided by the spatial interval of our model grid instead of the length scale
of eddy diffusion because this term represents an averaged effect of subgrid
scale processes.
Another widely used parameterization of the random walk term is (A_{m}^{.}Dt)^{1/2}. More
sophisticated methods of including the gradient of the kinematic viscosity
(Visser,1997) have also been tested and compared with observed drifter trajectories,
which will soon be incorporated in this study in the near future. The kinematic
viscosity in
the horizontal is
calculated by using the Smagorinski (1963) mixing scheme. Using this algorithm, a series of
random walk numerical simulation tests have been finished. The model tracer
trajectories are consistent with drifter observations.
2.3 Lobster
larvae development formula
The development of Homarus
americanus larvae can be divided into 5 stages. Spawning takes place
offshore and planktonic stages migrate inshore towards the end of larval
development (Incze and Naimie, 2000). Larvae are considered to be neutrally
buoyant. The biological component of our model consists of
temperaturedependent development rates modified from the polynomial equations
of MacKenzie (1988) for different larval stages. The lobster larvae duration
time is given in table 1. When the lobster larvae become postlarvae, they are
ready to settle.
Table 1. Lobster larvae stages development eq.
(MacKenzie,1988)
Stage I 
Dev I=(851(T0.84)^{1.91})/2.5 
Stage II 
Dev II=(200(T4.88)^{1.47})/2.5 
Stage III 
Dev III=(252(T5.3)^{1.54})/2.5 
Stage IV 
Dev IV=(0.358833T^{2}14.316T+156.895)/5.^{} 
where T is the
temperature in ^{o}C , Numerical lobster larvae develop depending on
the local temperature.
In the process of
the transportation of the lobster larvae, when larvae hit land, we think the larvae
are dead and just remove from the ocean.
3. Results^{}
The
lobster eggs are released in the regions along the coast where water depth less
than 100m, include part of South of Scotian shelf, German Bank area and Browns
Banks. There are 10 larvae released in each model grid (a total of 2,174 x10
larvae) in the entire spawing area. All larvae begin with a same initial stage
equal to 1. Larvae are released every 10 days from June 1 to September 21 and
are carried through time in each case for 2 months. We have 12 cases. The initial larvae are released at 15m (near the upper
thermocline) and transported at this fixed depth till stage IV. After
developing to stage IV, the larvae rise up to 1m beneath the surface.
From the
circulation and temperature distribution, it is easy to find that Maine Coastal
Currents (MCC) has different pattern and this annual variability of the circulation
of GOM is thought due to many different factors such as atmospheric forcing,
slop/shelf water intrusion. The annual variability of MCC would affect the
larval development and transportation. For example: In 2003, maybe because the
former colder winter, the Scotia Shelf water intruded into GOM is colder than
normal year. The MCC is very strong, it extent to west coast of Maine and New
Hampshire. But in 2002, the main branch of MCC turned back into inner Gulf at
southwest of Penobscot Bay, only a weak branch of MCC goes to west. In 2004,
the situation is in the middle of these two different patterns. We run
2002,2003,2003 total 3 years to examine the variation of the lobster larval
development and transportation.
Till
now we already finished experiments of 2002,2003,2004. The lobster larvae
development and distribution show annual variability due to different
circulation pattern.
1) Plots of lobster larval
trajectories
Model results of lobster larval development (2002)
Model results of lobster larval development (2003)
Model results of lobster larval development (2004)
2) Time series of larvae number
variability (chart 17 represent zone AG)
Time series of
number of larvae in different stages for Zone AG are shown in this table.
2002 

2003 

2004 
3) Stage 4.5
When larvae reach
stage IV, they are float up to surface layer (1m). We count the number of stage4.5 larvae in different LMZ.
2002 

2003 

2004 
4) Connection Matrix
The entire coastal
region of the Gulf of Maine is divided into 16 different zones (including Browns Bank, German
Bank, West Nova Scotia Shelf, Bay of Fundy, Grand Manan Island, 7 Maine lobster
manage zone AG, New Hampshire, Massachusetts Bay, outer Cape Cod)
2002 

2003 

2003 
5) Average
duration time
2002 

2003 

2003 
4. Discussion^{}
Development of lobster larvae
depends on the initial release position and ambient temperature. Model results
show that larvae hatched at different times around the Gulf result in a great
deal of spatial and temporal differences in life history since both the flow
field and the surface temperature change seasonally and spatially.
In future work,
vertical immigration, grazing, prey and mortality will be considered in the
lobster larval transport model.
Acknowledgements
We thank Dr. Lewis S. Incze
for lobster larvae development formula and helpful suggestions. This research
was funded by NOAA.
References
Berntsen, J.,
Skagen, D.W. and Svendsen, E., 1994, Modeling the drift of particles in the
North Sea with reference to sandeel larvae, Fisheries Oceanography, 3,
8191.
Brickman D. and Smith P.C.,
2002, Lagrangian Stochastic model in coastal oceanography, Journal of
Atmospheric And Ocean Technology, 19, 8399.
Incze L.S. and C. Naimie,
2000, Modelling the transport of lobster (homarus americanus) larvae and postlarvae
in the Gulf of Maine, Fisheries Oceanography, 9, 99113.
Proehl, J.A., D.R.Lynch,
D.J.McGillicuddy Jr., 2004, Modeling turbulent dispersion on the North flank of
Georges Bank using Lagrangian Particle Methods, Continental Shelf Research.
Visser A. W., 1997, Using
random walk models to simulate the vertical distribution of particles in a
turbulent water column. Mar. Ecol. Prog. Ser., 158: 275281.
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