Mapping exchange and residence time in a model of Willapa Bay, Washington,
a branching, macrotidal estuary
N. S. Banas
B. M. Hickey
School of Oceanography, Box 355351
University of Washington
Seattle, WA 98195
J. Geophys. Res., in press
1
Abstract
The numerical model GETM is used to examine transport pathways and residence time in Willapa
Bay, Washington, a macrotidal estuary with a complex channel geometry. When the model is run
with realistic forcing, it reproduces both tidal velocities and the decrease of the salt-intrusion length
with increasing riverflow with errors of 5-20%. Furthermore—a more stringent test—when the
model is run with tidal forcing only, it reproduces the along-channel profile of the effective
horizontal diffusivity K, a direct measure of the strength of subtidal dispersion, which is known
from previous empirical estimates. A Lagrangian, particle-tracking method is used to map subtidal
transport pathways at the resolution of the model grid. This analysis reveals an interweaving of
coherent lateral exchange flows with discontinuous, small-scale dispersion, as well as tidal-residual
currents that in some locations sharpen rather than smooth gradients between water masses.
Comparison between these Lagrangian results and an Eulerian salt-flux decomposition suggests
that along-channel complexity—channel junctions, channel curvature—is at least as important as
cross-sectional depth variation in shaping the subtidal circulation. Finally, a nonconservative-tracer
method is used to produce high-resolution, three-dimensional maps of residence time. This analysis
shows that, consistent with previous observational work in Willapa, at all except the highest, winterstorm-level riverflows, river- and ocean-density-driven exchanges are discernable but secondary to
tidal stirring. In all seasons, despite the fact that half the volume of the bay enters and leaves with
every tide, average retention times in the upper third of the estuary are 3-5 weeks.
2
1. Introduction
This study examines the mechanisms and timescales of exchange between Willapa Bay,
Washington, USA (Fig. 1) and the coastal ocean. Willapa is the largest in a series of shallow,
coastal-plain estuaries that spans the U.S. Pacific Northwest coast from central Washington to
northern California. Note that The Columbia River, immediately south of Willapa, is so different
from its neighbors in morphology and river input (Hickey and Banas 2003) that we do not treat it
as part of this series. These estuaries are forced by strong (2–3 m) tides, and have relatively deep
and unchannelized intertidal zones (Emmett 2000): fully half of Willapa's area and volume are
intertidal, and this appears to be typical (Hickey and Banas 2003). In addition, these estuaries are
subject to changes in ocean water properties forced by wind-driven upwelling-downwelling
transitions on both seasonal and event (2-10 d) timescales. The best analogies are the shallow,
macrotidal estuaries of Britain and Northern Europe (Bowden and Gilligan 1971, Zimmerman
1976, Dronkers and van de Kreeke 1986, Simpson et al. 2001) and the upwelling-influenced
estuaries of Spain and other eastern boundary systems (Álvarez-Salgado et al. 2000, Monteiro
and Largier 1999).
River input into these estuaries occurs mostly in winter, outside the growing season
(Hickey and Banas 2003). Accordingly, primary production in Willapa appears to be fueled less
by riverine nutrients than by oceanic nutrients or the direct import of oceanic phytoplankton
blooms (Roegner et al. 2002, Ruesink et al. 2003, Newton and Horner 2003). The dynamics of
ocean-estuary exchange—its spatial pathways, its constancy or variability under changing
forcing—are thus essential ecological concerns. These dynamics are the focus of the modeling
work described in this paper.
3
Recent observational studies have provided a preliminary description of exchange
processes in Willapa. Hickey et al. (2002) found that variations in ocean temperature and salinity
propagated upstream at approximately 10 cm s-1 even in summer, low-riverflow conditions,
much faster than could be attributed to the river-driven gravitational circulation. Banas et al.
(2004) found that even during winter storms, when riverflow is 200 times higher than in the latesummer dry season, the overall rate of exchange only increases by a factor of 3. Thus unlike
otherwise similar systems like Tomales Bay, California, where a strong seasonal cycle in
riverflow yields a strong seasonal cycle in flushing rate (Smith et al. 1991), exchange between
Willapa and the ocean appears to be cushioned by a high "baseline" flushing rate independent of
the river-driven circulation. Banas et al. (2004) suggested that the bulk of this baseline exchange
is caused by lateral tidal stirring.
The observational analyses cited above relied on multi-year velocity and salinity time
series in a few point locations, and thus addressed the temporal variability of Willapa's
circulation more than its spatial complexity. Even in very simple estuarine geometries,
understanding the horizontal structure of the net circulation is an active research problem (e.g.,
Valle-Levinson et al. 2003). Willapa's geometry, moreover, is not simple: the bay is a network of
branching channels formed when a sand spit from the Columbia River (the Long Beach
Peninsula) trapped a number of small rivers behind one connection to the ocean (Fig. 1; Emmett
et al. 2000). The purpose of this modeling study is to provide detailed maps of exchange and
residence time in this complex geometry over a typical seasonal cycle.
In many numerical studies of estuaries, flushing rate and residence time are by necessity
pure model predictions that can only be validated circumstantially. Here we use independent,
4
observational estimates of dispersion rates at four salinity time-series locations (Banas et al.
2004; Sec. 3b below) to test modeled dispersion rates and residence times directly. The model's
role, then, is principally to fill in spatial detail around these previous empirical conclusions.
Section 2 of this paper describes the numerical model. In Sec. 3 we describe the model's
representation of the tidal circulation and map out the net tidal transport pathways. In Sec. 4 we
add riverflow and density effects to the model, and map the changes in residence time between
low-flow (summer) and high-flow (winter) conditions. These results rely on particle-tracking and
nonconservative-tracer model methods that are likely to be useful in other complex estuaries.
2. The model
a. Model physics
The Willapa circulation model is an implementation of GETM (General Estuarine
Transport Model), a finite-difference, primitive-equation model designed for shallow-water
applications, cases like Willapa where flow over complex topography, mixing in strong and
changing stratification, and flooding and drying of intertidal areas are all important. GETM has
previously been used to model tidal dynamics in the East Frisian Wadden Sea, which is more
than half intertidal (Stanev et al. 2003); baroclinic dynamics and the estuarine turbidity
maximum in the Elbe (Burchard et al. 2004); and seasonal hydrographic patterns in the North
Sea (Stips et al. 2004). GETM is open-source and still under development: the project home page
can be found at http://www.bolding-burchard.com. Below we highlight only some key features
of GETM: for a systematic description of model physics and numerics, see Burchard and
5
Bolding (2002) and Burchard et al. (2003).
GETM solves the equations of motion on a curvilinear Arakawa-C grid (Arakawa and
Lamb 1977). To resolve fast-moving surface gravity waves without limiting the timestep for the
entire calculation, the solution is split into internal and external modes, with ten external
(barotropic) timesteps taken within each internal timestep. The model contains a simple, stable
scheme for handling the flooding and drying of intertidal areas, in which a depth-dependent
multiplier is applied to the equations of motion to smoothly reduce the role of all terms except
tendency, bottom friction, and barotropic pressure gradient as the water depth decreases to a few
cm. In addition, to prevent unphysical pressure gradients from developing, a correction to sea
level is made where the cell-to-cell change in elevation is comparable to the water depth
(Burchard et al. 2004). This scheme allows the model to accommodate strong tracer gradients in
intertidal areas.
GETM allows a wide choice of high-order advection schemes for velocity and tracers.
High-order schemes in general are far less diffusive than a basic first-order upwind scheme, and
therefore better suited to resolving strong vertical and horizontal gradients (Stips et al. 2004) and
flows in relatively quiescent areas like shallow banks (Gross et al. 1999). Both Gross et al.
(1999) and Stips et al. (2004), in their evaluations of a number of advection schemes in coastal
and estuarine simulations, ultimately recommend the Superbee (Roe 1985) and Quickest
(Leonard 1979, 1991) schemes. Both of these are total-variation-diminishing (TVD) and
therefore free from numerical oscillations and guaranteed to be stable. The two schemes perform
comparably well in both studies (Gross et al. 1999, Stips et al. 2004); in this study we use
Superbee, which is less computationally expensive.
6
GETM also allows a choice of turbulence closures by coupling to the one-dimensional
model GOTM (General Ocean Turbulence Model: Burchard et al. 1999). Here we use a standard
k-
cheme (e.g., Burchard and Bolding 2001) with the stability functions suggested by Canuto
et al. (2001). This scheme is among those that Umlauf and Burchard (2003) have shown can be
represented as a special case of a single two-equation, "generic length scale" formulation with
variable coefficients. Warner et al. (2005) compared a number of these two-equation special
cases (k-kl, k-, gen, and the k- scheme we use) and found the differences in their ability to
reproduce estuarine salinity fields to be relatively minor, although all performed better than the
original Mellor-Yamada level 2.5 scheme (Mellor and Yamada 1982). For a general review of
two-equation closures, see Burchard (2002).
b. Implementation for Willapa Bay
This study uses a model grid with 175-by-82 cells in the horizontal and 12 sigma levels in
the vertical. The bay itself is covered with uniform horizontal squares 255.5 m on a side; beyond
the bay mouth lies an idealized rectangular "ocean" in which the grid size expands gradually to 6
km (Fig. 1). This domain was designed to resolve Willapa's internal dynamics only, not the
dynamics of the adjacent coastal ocean. The model "ocean" (Fig. 1) is a simple semienclosed
reservoir, open only to the west, with depth limited to 30 m, since realistic depths would limit the
timestep for the whole model. Additionally, the model has no alongshore currents, which on the
real Washington coast are tens of cm s-1 and highly variable (Hickey 1989). Removing any of
these simplifications would provide a false sense of realism unless they were all removed, and
even then model validation would be tenuous given the scarcity of high-resolution observations
7
on the adjacent shelf. Coupling Willapa's internal dynamics to shelf processes thus remains a
future goal.
The bathymetric grid within the estuary is interpolated from a finite-element model grid
developed by the U. S. Army Corps of Engineers Seattle District, who resurveyed most of the
subtidal area of the bay in 1998 (Kraus 2000). Willapa's primary channels and many of its
secondary channels are wide enough (500–3000 m) to be well-resolved by our 250 m model grid,
although in many locations real channel edges are steeper than our grid allows. Intertidal
bathymetry in our model, like that in the published NOAA charts for Willapa, is a composite of
surveys over many decades. In 2002 the NOAA Coastal Services Center conducted a highresolution LIDAR (Light Detection and Ranging) survey of the shoals of Willapa Bay
(http://www.csc.noaa.gov/lidar), but no seamless, georeferenced bathymetry based on this
dataset has yet appeared. Willapa's morphology is, in any case, a moving target—channels can
migrate 100 m or more in a single year (e.g., Hands and Shepsis 1999)—and therefore as much
as possible the analysis below is based on spatially integrative validation methods that do not
rely on point-by-point comparisons.
All model scenarios described in this paper are summarized in Tables 1 and 2. A first set
of runs, used for model validation (runs A-C), simulates particular time periods. Another set (EG) uses idealized tidal and river forcing to represent a typical seasonal cycle, from winter storms
(riverflow ~ 1000 m3 s-1) to spring (flow ~ 100 m3 s-1) to late summer (flow ~ 0). Run D is a
version of the idealized late-summer scenario used both for validation and for examining tidal
transport pathways in detail (Sec. 3). A final set of runs (H-I) models the propagation of
individual, event-scale pulses of ocean water into the estuary.
8
The tidal, river, and ocean-water-property forcings applied in each run are specified in
Tables 1 and 2. The tide in the model was generated by imposing a sea-level time series on the
open ocean boundary (Fig. 1). Ocean water properties, when included, were applied on the open
boundary as well. In runs where river input was applied, it was divided among the bay’s three
largest rivers according to watershed area: 47% in the North River, 35% in the Willapa, and 18%
in the Naselle (Fig. 1). In several runs where riverflow was not applied (A, B, D, E, H: Tables 1
and 2), we eliminated all density effects—i.e., all density variations in both space and time—in
order to isolate the role of tidal dynamics. This was done by switching off the call to the
equation-of-state calculation in the code, so that model tracers had no effect on the circulation,
acting like dye rather than like salinity. In addition to "dye" and "salt," a third type of tracer—
"age," a nonconservative, passive tracer representing average time spent in the estuary—was
included in some runs, as explained in detail in Sec. 4.
The tide-only, density-off runs (A, B, D, E, H) were spun up from rest over 4 tidal cycles
(2.1 days). This is a short adjustment period, but sufficient that when a pure M2 tide was applied,
the variation in tidal currents among subsequent cycles (which in a fully spun-up model would
be be zero) was only 0.02%. Runs that included density (C, F, G, I) were spun up for 100 days—
approximately twice the longest residence times we calculate in Sec. 4—with initial tracer fields
held constant before tracers were released and allowed to evolve. These spin-up periods are not
included in the results below: in all cases the "initial" state of a run refers to the end of the spinup period.
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3. Tidal exchange
We begin by evaluating the simplest dynamical situation, in which the oceanic tide is the
only forcing: river input is zero, as if in late summer, and no density variations in either space or
time are included. We will begin by verifying the overall strength of modeled tidal currents
against data (Sec. 3a), and then move on to the much weaker but dynamically more important
subtidal (i.e., tidal-cycle-average or "residual") circulation.
a. Validation of tidal currents
For validation of the tidal model (runs A and B: Table 1), we aimed to reproduce the tides
during particular time periods in which velocity data were collected. No tuning of model
parameters was done: the bottom roughness z0, sometimes treated as a free parameter, was set to
a standard value of 1 mm. NOAA tidal-height observations from Toke Point (station 9440910;
Fig. 1) were used as the oceanic boundary condition with no adjustment, except for the shift of
timebase necessary to match the observed tidal phase at Toke Point. This is a simple and
convenient method, but a source of error: as this tidal signal propagates into the bay it distorts
slightly and produces a tidal-height discrepancy at Toke Point of 5-10%.
Figure 2 shows the depth-averaged amplitude of the M2 (semidiurnal) tidal constituent
for the period Oct 15 – Nov 11, 1998 at four locations, stations at which the Army Corps of
Engineers Seattle District collected acoustic doppler current meter (ADCP) data during this
period (run A: Table 1). The M2 constituent was extracted from data and model time series using
the harmonic-analysis package t_tide (Pawlowicz et al. 2002). Model velocity amplitudes agree
10
with observations to 3-20%, an error not much larger, encouragingly, than the 5-10% error
caused by the imprecision of the boundary forcing. The model does not, however, reproduce the
direction of M2 currents at the three stations near the mouth: the real flow follows bends in the
channel that the modeled flow does not, presumably because the model bathymetry is smoother
and channel edges less abrupt than in the real bay.
This sort of point-by-point bathymetry-driven bias may or may not translate into bias in
overall tidal transport, however. To test the tidal model more integrally, on a larger spatial scale,
the model was run for the week ending on May 5, 2000, a day on which small-boat transects with
a 300 kHz ADCP were repeated over a cross-section of the main channel for a full flood tide (run
B: Table 1). Instantaneous transport through this cross-section (expressed as mean velocity) is
shown in Fig. 2 for both the observed and modeled flow: each black dot represents an ADCP
transect. Instantaneous mean velocities in model and data agree within 10% for each of the ten
transects. This suggests that local biases like the directional errors near the mouth described
above (Fig. 2) are indeed local, and that the total tidal transport in the model is well-represented.
b. Validation of tidal dispersion: the horizontal diffusivity K
The tidal validation above mainly tests the model's reproduction of the oscillatory part of
the tide, which, although the largest velocity signal in the estuary (~ 1 m s-1: Fig. 2) by definition
does not contribute to net dispersion or exchange: it simply advects water back and forth. The
“rectified” or “residual” tidal currents actually responsible for dispersion in estuaries tend to be
only ~10% of the tidal amplitude (Zimmerman 1986), comparable to model-data error in our
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case as in many cases. Horizontal tidal dispersion in the model must thus be evaluated by other
means.
Banas et al. (2004) calculated a “total effective horizontal diffusivity” (K) from salinity
time series at each of four monitoring stations along the main channel, and used the resulting
along-channel profile to characterize the subtidal circulation. In this section we will use the same
parameter and compare results directly with those from observations. This is a stringent test of a
tidal model, one not often performed for lack of a method for estimating K from data.
A brief explanation of K in terms of the estuarine salt budget follows: a more detailed
derivation can be found in Banas et al. (2004). In general, we can express the conservation of salt
(or some other tracer) on subtidal timescales as
t
as dx =
x
aus
(1)
where x is distance along the main axis of the estuary (x = – denotes "far upstream"), t time, a
cross-sectional area, u velocity perpendicular to the cross section, and s salinity. Triangular
brackets denote the average over a tidal cycle, and an overbar denotes the cross-sectional
average: for any variable v, v a-1 v da. Equation (1) simply states that the rate of change of
salt storage upstream of x equals the total salt flux across a section at x. The salt flux < aus >
contains all processes moving salt in and out of the estuary, including the river-driven
circulation, tidal stirring, wind-driven exchange, and interactions among them. A common
approach (e.g., Ridderinkhof and Zimmerman 1990, Monismith et al. 2002, Austin 2004) is to
assume that these processes combine to form a pattern of advection too complicated to describe
as anything but diffusion. Following Banas et al. (2004) we can write this as
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t
x
s
as dx + Q s = a K
x
(2)
where Q is riverflow—Q < s> is the seaward flushing of salt by the mean flow—and K is a
horizontal diffusivity parameterizing all processes moving salt upstream. If we confine our
attention to low-riverflow conditions, in which Q 0, we can rearrange (2) into a formula for
calculating K from the salinity field:
K= a
1
s
x
1
t
as dx
x
(3)
We use salinity here only because it is the most common tracer used to indicate seawater content;
to the extent that K really does represent a large-scale diffusive process, any conserved tracer can
be substituted.
To calculate K from the tidal model, we constructed the following idealized scenario (run
D: Table 1). We impose a simple, uniform semidiurnal signal whose amplitude (1.2 m) was
chosen to match the long-term-average tidal-height variance at Toke Point. An initial field of
"dye" with a uniform gradient in the along-channel direction is released into the spun-up tidal
circulation (Fig. 3a). The initial dye concentration is high in the ocean and decreases up-estuary
as salinity would, although this dye has no density and thus does not affect the circulation. The
rearrangement of this dye field by a single tidal cycle is shown in Figs. 3a-c, and its state after 12
cycles (6.2 days) is shown in Fig. 3d. The large tongue of high dye concentration in Fig. 3d
indicates water that has entered the domain across the open ocean boundary, where a constant
concentration for incoming water was imposed.
An average value of K was calculated using (3) for this tracer field over the 12 tidal
cycles shown, at a series of cross-sections along the longest axis of the bay (Fig. 3e) from the
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mouth to Shoalwater Bay in the south. Empirical values of K are given for comparison at four
locations. Three are values calculated by Banas et al. (2004) from low-riverflow-period salinity
data (Fig. 3f gives an example). The fourth value, previously unreported, was calculated from a
four-week (Aug - Sep 2002) deployment of a YSI temperature-salinity sensor at the edge of the
main channel in Shoalwater Bay. The rate of change of salt storage at this station was calculated
from the trend in salinity rather than event-scale fluctuations as in Fig. 3f, but otherwise followed
the method described by Banas et al. (2004).
The four empirical K values vary by more than an order of magnitude from mouth to
head. Crucially, the model captures both high and low values accurately (Fig. 3e). This spatial
gradient in diffusivity can be seen qualitatively in Figs. 3a-d: the dye field in the southern part of
the bay (low K) changes only gradually, whereas a single tidal cycle (Figs. 3a,c) is enough to
dramatically displace, stretch, and fold the dye field near the mouth (high K). Note that the only
station where model and data values of K differ significantly, Oysterville, lies at the junction of
two channels (Fig. 1), where the geometric assumptions behind the Banas et al. (2004)
calculation may be violated: either the model or the "empirical" estimate may be the source of
error at this station.
c. In pursuit of salt-flux mechanisms
A map of K as in Fig. 3e is a convenient summary of the subtidal circulation, but
obscures the actual mechanisms that drive it. What is different about the tidal flow or the
bathymetry in the seaward and landward reaches of the estuary, that the net dispersion rate
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should vary so much between them? What aspect of the bathymetry—channel width, channel
curvature, channel junctions, bank-to-channel depth variations—is most responsible for
generating the tidal asymmetries that in turn generate K? A standard approach to these questions
(Fischer 1976, Hughes and Rattray 1980, Lewis and Lewis 1983) is a Reynolds-like
decomposition of the total salt flux < aus > (see equation (1)). That is, we can decompose s into
mean and varying parts in space and time:
s = < s> + <s'> + s1 + s'1
(4)
where triangular brackets and an overbar denote tidal-cycle and cross-sectional means as before,
a subscript 1 denotes variations over a tidal cycle, and a tick denotes variations over a crosssection. We can also, following Dronkers and van de Kreeke (1986), write the mass flux through
a small cross-sectional area element as
dq u da = <dq> + dq1
(5)
so that the total tidally averaged salt flux through a cross-section becomes
< aus > = a <s dq>
= < s><q> + <s'><dq> + < s1 q1> + <s'1 dq1>
(i)
(ii)
(6)
(iii)
If the volume of the estuary is constant and river input Q = 0, <q> = 0, leaving three terms on the
right-hand side. Term (i) represents steady shear dispersion by the tidal-residual eddy field <u'>:
Zimmerman (1986) refers to this mechanism as "Lagrangian chaos," since the velocity field itself
is steady and deterministic but randomized by bathymetry to the point where tracers and particles
disperse chaotically. Term (ii), which describes correlations over the tidal cycle between s and q,
can be thought of as "tidal trapping" (Okubo 1973, Fischer 1976), the result of a phase lag that
15
arises when salt is diverted into side embayments and reenters the main flow at a different point
in the tidal cycle. The cross-term (iii) involves variations in both space and time: unsteady shear
dispersion, or spatially non-uniform trapping.
The relative strength of (i), (ii), and (iii) are shown in Fig. 4 for each of the cross-sections
where K was calculated in the previous section. These salt fluxes have been converted into
diffusivity units (m2 s-1) by combining equations (3) and (6), i.e., by normalizing by <a>< s/x>.
Total K (dashed line) decreases smoothly from the mouth upstream, but the three constituent
diffusivities (bars) rise and fall in a much more complicated pattern. Each of (i), (ii), and (iii) is
in some location the dominant term; in fact, the cross-term (iii) (gray bars), the term hardest to
interpret, is dominant over much of the upper estuary. This implies that much of the net
exchange of ocean water over the sections examined is accomplished not by general, easily
parameterizable features of the flow but by processes localized in both space and time.
Furthermore, all three terms are in some location negative, which could mean that a
decomposition based on cross-sectional averages is not physically meaningful in bathymetry this
complicated. Alternatively, these negative constituent diffusivities could mean that certain tidalresidual processes in the bay are actually anti-diffusive in a coarse-grained sense, tending to
sharpen fronts rather than smooth them away. Note that the initally smooth tracer field shown in
Fig. 3a does in fact become more inhomogeneous as tidal stirring acts on it (Fig 3d).
It appears, then, first, that the cross-sectionally integrated diffusivities shown in Figs. 3e
and 4 are averages over a great deal of small-scale complexity and variety; and second, that the
traditional method of decomposition, based on fixed-location tidal-cycle and cross-sectional
averages, is not an effective way to tease apart this complexity. In the next section we will
16
describe an alternative, flow-following approach that gives clearer results.
d. A Lagrangian map of exchange
We begin by releasing an imaginary particle in the center of each model grid cell, and
track the trajectory of these particles as the tidal flow calculated in run D (Fig. 3) advects them
seaward and landward. Three-dimensional particle tracking is not straightforward in sigma
coordinates and steep bathymetry; to keep the problem tractable we follow particles in two
dimensions only, using depth-average currents. This simplification would not be dynamically
sensible in many scenarios, but in our tide-only case it appears to be fair: when terms (i) and (iii)
in equation (6) are decomposed into vertical-mean and -varying parts (not shown), horizontal
shear dispersion dominates over vertical dispersion by an order of magnitude. Nevertheless, the
two-dimensional particle-tracking approximation introduces 10-20% volume-conservation
errors, which should be taken as a lower limit on the uncertainty in this analysis.
The only information we retain from these particle tracks is the index of the grid cell
where each particle is launched and the cell that contains it one tidal cycle later. The
rearrangement done by the subtidal circulation is thus encapsulated in a single two-column table,
a function that maps the list of grid-cell indices back onto itself: in the language of nonlinear
dynamics, a "return map" or Poincaré map (Beerens et al. 1994). One application of this
Lagrangian return map is shown in Fig. 5. Here the only particles drawn are those which cross
over a given section (green lines) within one tidal cycle. Lighter shades indicate initial position
and darker points final position one tidal cycle later; particles are colored red if they are crossing
up-estuary and blue if they are crossing seaward, as illustrated in the schematic in Fig. 5a. This
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illustration of the crossing regions for each section of interest (Fig. 5b-g) highlights not only the
total amount of exchange at each section (which could also be inferred from the diffusivity K)
but specifically what water is being exchanged for what other water in a typical tidal cycle. The
details of these results depend on the tidal phase at which particles are released, but the
qualitative patterns discussed below are not sensitive to it. Results are shown in Fig. 5 for
particles released at high slack, the release time that minimized volume-conservation errors.
Note that dividing the volume of a given section's crossing regions by the intertidal
volume upstream gives the "tidal exchange ratio," or fraction of each tidal prism that does not
return on the following tide. The crossing regions shown in Fig. 5b-g correspond to tidal
exchange ratios of 0.15 - 0.3, consistent with the range of typical values reported by Dyer (1973)
and the estimate for Willapa from data in Banas et al. (2004).
Coherent structures in the net circulation emerge. Exchange near the mouth (Figs. 5b,c)
consists mainly of a simple lateral exchange, into the estuary over the southern part of the mouth
and seaward in the deeper main channel to the north. This pattern, in which the subtidal flow
tends to the right both entering and leaving the estuary is consistent with the expectation for
rotation-driven exchange (Valle-Levinson et al. 2003), but it persists even when the Coriolis
acceleration is turned off in the model (Fig. 5h). An alternative explanation may be tidal Stokes
drift, which tends to produce an up-estuary Lagrangian transport through the shallower part of a
cross-section, balanced by a seaward return flow in the deeper main channel (Li and O'Donnell
1997), as seen here. The lateral exchange pattern persists for 15-20 km upstream from the mouth
(Fig. 5d-f), but its relationship with the depth of the underlying bathymetry reverses: more than a
few km from the mouth (Fig. 5d,e) flow over the banks is primarily seaward and flow in the
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main channel up-estuary (bathymetry shown in Fig. 1). Analytical studies and observations of
lateral transport gradients in other estuaries have found both the pattern at Willapa's mouth
(channel flow seaward: Friedrichs and Hamrick 1996, Li and O'Donnell 1997), the pattern in
Willapa's interior (channel flow landward: Valle-Levinson and O'Donnell 1996, Winant and
Gutiérrez de Velasco 2003), and indeed a reversal of the pattern along the estuary (Li 1996), as
we observe.
In the interior of the estuary (Fig. 5d-g) there are places where the Lagrangian residual
circulation appears more diffusive than advective, in the sense that holes and discontinuities
appear in the crossing regions, and in places the incoming and outgoing water masses appear to
move through each other rather than past each other (Fig. 5g). Such complexity is especially
visible in the exchanges around channel junctions highlighted in Fig. 6. These junction
exchanges combine coherent, advective motion (water enters Willapa Channel and then leaves
again, as on a counterclockwise conveyor belt: Fig. 6b) with discontinuities and dispersion on
scales from the grid resolution up to 10 km.
Thus in contrast to the K formulation (Sec. 3b) in which all exchange flows are treated
mathematically as diffusion, and also in contrast to the Eulerian decomposition (Sec. 3c) in
which only simple advective patterns are interpretable, this crossing-region method lets us
distinguish in detail between advection-like and diffusion-like processes. It also lets us visualize
hybrid, unnameable transport processes like those around channel junctions. Note that these
hybrid junction processes may be important to exchange overall: the highest diffusivities in the
southern half of the estuary (Fig. 3e) lie at the junction at the mouth of Shoalwater Bay (Fig. 6a).
19
e. Model visualization using the Lagrangian return map
As noted above, the Lagrangian return map distills the net transport accomplished by
each tidal cycle into an especially concise form. To the extent that one tidal cycle is just like the
next (this is definitionally true in the idealized run D), applying the return map repeatedly can
generate very long subtidal particle trajectories virtually instantaneously compared with the full
GETM circulation model. An interactive demonstration of this method, in which a user chooses a
launch location for up to a few thousand drifters and then can watch their flushing and dispersion
over 40 tidal cycles, is online at http://coast.ocean.washington.edu/willapa/tidemodel.html. We
imposed a small random jitter, equivalent to a diffusivity of (grid spacing)2 (tidal period)-1 = 1.5
m2 s-1, to keep particles from being trapped unrealistically at fixed points in the return map.
Particle trajectories more than a few tidal cycles long obtained by this method are merely
representative, not precise, since the approximations inherent in the return map (rounding
particle positions to the nearest grid cell; ignoring vertical shear, unless particles are tracked in
three dimensions) quickly add. Nevertheless, this method does illustrate variations in dispersion
rate and transport pathways with more immediacy than a map of diffusivities like Fig. 3e. It is
also one of the few ways that nonspecialists and casual users can directly investigate and
challenge the details of a complex circulation model.
4. Density-driven exchange
a. Baroclinic validation: the salinity field
So far our analysis has considered low-riverflow, tide-dominated conditions only. In this
20
section we add rivers and ocean density to the model, and evaluate their effect on flushing rate
relative to the tidal baseline. We must begin, however, by verifying the model's baroclinic
behavior in general: we will do this by validating the model salinity field against observations
across a range of river and ocean forcing.
For validation, the model was run with time-varying sea-level, riverflow, and oceansalinity forcing for the 11-month period July 1999 – June 2000 (run C: Table 1). During this
period, as described by Banas et al. (2004), point salinity time series were collected at four mainchannel stations: W3 near the mouth at 8 m depth, and Bay Center, Oysterville, and Naselle
farther up-estuary at 1 m depth (Fig. 7, insets). For model forcing, sea level from Toke Point was
imposed at the ocean boundary as in runs A and B (Sec. 3a); both tidal and subtidal components
were included. River input into the North, Willapa, and Naselle Rivers was calculated from
USGS gauge data as described in Banas et al. (2004). Finally, for lack of a better option for
representing changing ocean salinity, tidally-averaged salinity from W3 was imposed at the
ocean boundary as well.
This last approximation is the strongest limit on our ability to simulate particular time
periods. Over this 11-month run, the distortion as the imposed W3 salinity time series propagates
from the open boundary back to W3 is 27% of the signal. As a result, it does not seem
meaningful to compare model results with data instant by instant, but rather to treat model and
data as if they represented similar but distinct time periods. A stricter baroclinic model validation
must wait until better offshore data or a high-resolution, large-scale coastal model is available to
provide better salinity boundary conditions.
The most integrative test of the model's baroclinic circulation is the length of the salt
21
intrusion as a function of riverflow. A relative measure of salt intrusion length—salinity at Bay
Center, Oysterville, and Naselle as a fraction of salinity at the mouth (station W3)—is shown for
both observations and model results in Fig. 7. This relative salinity is displayed as a function of
bay-total riverflow Q: the gap around Q = 30 m3 s-1 corresponds to a six-week period in autumn
1999 when W3 salinity was unavailable (a linear trend was substituted as the model boundary
condition during this period). At Oysterville and Naselle, the slope between relative salinity and
riverflow is indistinguishable between model and data at the 95% confidence level. Furthermore,
the model reproduces, qualitatively, the scatter—the event-to-event variance—in relative salinity
at a given riverflow level. At Bay Center, however, the dependence of salinity on riverflow in the
model is too weak. This is a sign either that our simplified treatment of the coastal ocean distorts
the salinity field near the mouth, or that the model is not correctly distributing freshwater from
the North and Willapa Rivers around the Stanley Channel-Willapa Channel junction (Fig. 1). In
either case, the bias in the salinity field appears to be a local effect that does not reach as far
south as Oysterville.
Furthermore, the model appears to match, qualitatively, observations of stratification at
Bay Center from a more recent (previously unreported) experiment. Two SeaBird conductivitytemperature (CT) sensors were deployed at Bay Center from Oct 2003 to Apr 2004, one at 1 m
depth as above, and the other 0.5 m off the bottom. W3 was not occupied during this
deployment, and so for lack of a salinity boundary condition we cannot simulate this time period
in the model directly. The 1999-2000 model run described above (run C) does, however,
reproduce the most striking feature of the 2003-04 observations (Fig. 8a,b): at riverflow levels
above the long-term median (~ 100 m3 s-1), the water column stratifies 1-5 psu on each ebb and
22
then destratifies again on flood. Similar patterns of strain-induced periodic stratification (SIPS)
have been frequently observed in macrotidal estuaries (Simpson et al. 1990, Stacey et al. 2001),
although they have not been documented previously in Willapa. Notably for model validation, in
both model and data, the only flow levels at which the flood tide is unable to completely rehomogenize the water column for more than 1-2 cycles at a time are those close to the annual
maximum, > 1000 m3 s-1 (in observations, early Feb 2004; in model, late Dec 1999; arrows, Fig.
8a,b). This model-data comparison is shown more systematically in Fig. 8c,d. At low-tomoderate riverflows, the maximum stratification attained during each tidal cycle ranges from 0-3
psu in both data and model, as the bay responds to various combinations of river and ocean
water-property forcing. At high flows, the model underestimates the highest stratifications at Bay
Center, but reproduces the fact that during at least some of these high-flow events, tidal-cycleminimum stratification does not drop below the observational limit of 0.05 psu, as it does during
almost all other conditions (Fig. 8c,d).
Until better boundary-condition data are available to make the baroclinic validation more
precise, it would be over-trusting to use the model to map Willapa's river-driven exchange
circulation in as much detail as we mapped the tidal circulation in Sec. 3d. Still, we have found
that the modeled vertical circulation at mid-estuary (as indexed by Bay Center stratification) is,
at least, not grossly wrong; and in fact that the slope of relative salinity at Oysterville and Naselle
vs. riverflow—i.e., the distance that water entering the mouth reaches into the southern estuary
under complex, changing forcing—matches the slope from observations within confidence limits
(Fig. 7). In the next section we describe a numerical method by which river- and ocean-densityforced changes in circulation can be translated into summary maps of bay-wide residence time.
23
b. A method for mapping residence time
The method we use to determine residence time—the nonconservative "age" tracer
mentioned in Sec. 2—is similar to the ventilation-time method (time since a water parcel has
been exposed to the atmosphere) used in global circulation modeling (Thiele and Sarmiento
1990, England 1995). For a rigorous, general derivation of this and a variety of other age
methods in ocean modeling, see the thorough review by Deleersnijder et al. (2001; see also
references in Monsen et al. 2002). We will explain this age-tracer more briefly here.
Consider a passive tracer whose concentration is initially zero everywhere, and that is
always zero in the estuary's river and ocean source waters, but that is made to grow at a constant,
specified rate in the estuary's interior. Between the open ocean boundary and the mouth of the
estuary, the tracer simply advects and mixes without growing. As time passes, if forcing is held
constant, the tracer concentration will approach an equilibrium in which the steady creation of
tracer within the estuary is balanced by flushing out to sea (which replaces high-concentration
estuarine water with zero-concentration ocean water). At this equilibrium, the concentration in
each grid cell is directly proportional to the average length of time the water in that grid cell has
spent growing, i.e., spent within the estuary. Thus we can think of this tracer as average water
age.
As an illustration, consider the case of a riverless, linear channel: in the cross-sectional
average, the conservation equation for age can be written
c 1 c =
aK + t a x x 24
(7)
where c is age-tracer concentration, a and K are, as before, cross-sectional area and a total
effective diffusivity, and is the imposed tracer growth rate. We can pick the units of c so that = 1. For given profiles of a(x) and K(x), Equation (7) has a steady solution, given by a double
integration of
1 c aK = 1
a x x (8)
with the boundary conditions c = 0 at the ocean and (aK)(c/x) = 0 (no tracer flux) far upstream.
Equation (8) describes a state in which the along-channel tracer gradient c/x has adjusted so
that the diffusive loss of tracer (left hand side) balances the steady, imposed tracer growth rate
(right hand side) at every location. If a and K decrease exponentially toward the head with efolding lengths la, lK, then at equilibrium, age increases approximately exponentially from mouth
to head: c(x) la lK K(x)-1. In this simple case, as one might expect, age c (an estuarine residence
time) is inversely proportional to K (a measure of the estuarine exchange rate). Equation (8)
implies that this is likely to be true in general, at least in a scaling sense, a fact we can use (Sec.
4d below) to check the reasonableness of the residence times for Willapa that the age-tracer
method yields.
We implemented this age-tracer in GETM by first removing references to temperature in
the equation-of-state calculation (so that it can be used as a generic passive tracer) and second
adding code to the temperature-advection subroutine to increment the tracer one unit each
timestep at grid points within the estuary. This tracer thus is transported and mixed using
25
GETM's full advection and turbulence schemes, without the need for postprocessing. Cokelet
and Ebbesmeyer (1991) used a similar method to calculate residence time in a simplified box
model of Puget Sound. We have not seen this method used in a high-resolution estuary model,
although it requires only a few lines of code and presumably can be added straightforwardly to
any primitive-equation model. Note that unlike the simple illustrative case described in (7) and
(8), the full age-tracer calculation yields three-dimensional maps of residence time at the same
resolution as the model grid.
Tidally-averaged maps of age for three model scenarios representing an idealized
seasonal cycle (runs E-G: Table 2) are shown in Fig. 9. The first of these (run E) contains a tidal
circulation only, without density effects or river input, representing the sustained upwellingfavorable, low-riverflow conditions typical in late summer (Hickey et al. 2002, Banas et al.
2004). This run is identical to run D, discussed in Sec. 3b-d, except that the age tracer has been
added. The next run (F) adds steady river input of 100 m3 s-1 and a constant ocean salinity of 30
psu: these forcings represent the fair-weather (upwelling-favorable) conditions typical in spring
and early summer (Banas et al. 2004). In the third run (G), river input is 1000 m3 s-1 and ocean
salinity 20 psu, representing sustained southward-wind, foul-weather, downwelling-favorable
conditions and intrusion of the Columbia River Plume (Hickey and Banas 2003). In the
following subsections we will examine the tidal residence-time pattern (run E) and then compare
with the river-forced cases (F,G).
c. Age distribution under tidal forcing alone (run E)
The spatial distribution of ages in Fig. 9a encapsulates much of the information in the
26
dye-intrusion (Fig. 3d), diffusivity (Fig. 3e), and crossing-region (Fig. 5) views of the tidal
circulation discussed above. A tongue of low-age water entering through the southern side of the
mouth corresponds to high, oceanic dye concentrations in Fig. 3d and the up-estuary-directed
half of the lateral exchange flow shown in Fig. 5b-f. The compensating exit flow on the north
side of the mouth (Fig. 3d, 5b,c) is also visible, above the 3-day age contour. Where the
Lagrangian lateral-exchange flow breaks up in the middle of the estuary (Fig. 5g), K has a local
minimum (Fig. 3e), and there is a strong gradient, almost a front, in age (Fig. 9a). At the other K
minima as well—the tips of the Willapa Channel and Shoalwater Bay—the age gradient is
intensified, and wherever K is large—at the mouth, for example, the age gradient is flatter. The
age map thus contains two scales of circulation information. The age itself indicates the flushingand-resupply rate for a given region on the whole-estuary scale—Shoalwater Bay, for example,
is ~ 50 days "distant" from the mouth in a circulational sense (Fig. 9a)—while the gradient in age
suggests dispersiveness or quiescence on the local scale.
There is an intense transverse gradient in age (up to 9 d km-1) across the shallow banks at
mid-estuary where the sections in Fig. 9d-f are taken. The simplest interpretation is that this
gradient, just like the strong transverse temperature and salinity variations often seen in the same
location (up to 5 °C, 4 psu: Hickey and Banas 2003), is caused by differential tidal advection
(Huzzey and Brubaker 1988, O'Donnell 1993, Hickey and Banas 2003). In this process the
channel-to-bank difference in the strength of flood-tide currents strains tracer fields, creating
lateral gradients out of along-channel gradients. In other words, just as in summer water on
Willapa's shallowest banks tends to be noticeably warmer and fresher—more like water 5-10 km
up-estuary—than water just 1-2 km away in the main channel, so may that bank water be 2-3
27
weeks older, simply through rearrangement by the flood tide.
d. Age distribution under river forcing (runs F, G)
As river input increases (Fig. 9b,c), along-estuary variation in age decreases. (Note that
the mid-estuary bank-to-channel gradient discussed above decreases proportionally, consistent
with the differential-advection mechanism). The northern channel, where 81% of riverflow
enters (Sec. 2b), is especially homogenized and quickly flushed: there is now a source of zeroage water at both ends, and even at moderate riverflows (Fig. 9b), average age is < 6 days
everywhere between the mouth and the Willapa and North rivers. Shoalwater Bay, in contrast,
which has no river input of its own, is largely bypassed by the river-driven increase in flushing
rate. Even at storm-level riverflows (Fig. 9c), water at the southern end of Shoalwater Bay is 3
weeks older than water in the central bay.
In the absence of riverflow (Fig. 9d) the age distribution is vertically uniform, but at high
flows (Fig. 9f) water at depth in the main channel is 1-2 d younger than surface water. This is a
sign that a vertical gravitational circulation has developed, following the classical pattern
(inward at depth, outward at the surface) first described by Pritchard (1952). Notably, the
exchange pattern at flow levels near the long-term mean (Fig. 9b,e) more closely resembles the
tide-driven pattern (Fig. 9a,d) than it does this high-flow, vertical exchange pattern (Fig. 9c,f).
This is consistent with the results of Banas et al. (2004), who found that at moderate flows (100
m3 s-1) the total effective diffusivity at Bay Center increases only 40% above its low-riverflow,
tidal baseline value, while at very high flows (1000 m3 s-1) the total diffusivity is 3.1 times the
baseline value. For comparison, in these model results average age in the central estuary (from
28
Bay Center to Oysterville: see Fig. 3e) is 30% higher at moderate flows (100 m3 s-1) than under
tidal forcing only and 2.4 times higher at high flows (1000 m3 s-1) than under tidal forcing (Fig.
9a-c).
e. Intrusion of ocean water on the event scale (runs H, I)
The maps of average age discussed above suggest that at riverflow levels from the longterm median down—i.e., most of spring and summer—river- and ocean-forced density effects
are secondary to the tides in setting Willapa Bay's residence time. It is possible, however, that on
the event (2-10 d) scale, non-tidal influences have intermittent but sometimes much greater
dynamical importance. Studies of estuarine adjustment (Kranenburg 1986, MacCready 1999)
have shown that an estuary may respond to a pulse of river input very differently than to
sustained river input. In addition, past work on ocean-estuary exchange on upwelling coasts
(Duxbury 1979, Monteiro and Largier 1999, Hickey et al. 2002) suggests that the time-dependent
response to changing ocean salinity may be an important modulation of the gravitational
circulation.
Nevertheless, results from a final set of model runs (H-I: Table 2) suggest that the tidedriven residual circulation by itself is a good first-order approximation for a wide range of spring
and summer conditions in Willapa, even on event timescales. Results from these runs, which
simulate the propagation of an individual 2 d pulse of ocean water into the bay, are shown in Fig.
10. The first of these runs (H) is identical to the idealized-tide run D described in Sec. 3, except
that now the initial dye concentration is zero everywhere, and after the tidal flow is spun up dye
is injected along the open ocean boundary for 2 d and then shut off. The resulting dye patch—
29
which might be thought of as a marker for a mass of nutrient- and biomass-rich or -poor water
produced by an individual upwelling or downwelling event—propagates into the estuary as
shown in Fig. 10a. In the second of these runs (I: Fig. 10b), a constant background riverflow of
100 m3 s-1 has been added as in run F (Fig. 9b), and in addition, ocean salinity smoothly increases
from 28 to 32 psu and back again during the 2 d that dye is injected. This combination of
forcings represents a rapid spring upwelling event (Hickey and Banas 2003). With river and
ocean-density forcing added, 28% more dye (i.e., new ocean water) enters initially (Fig. 10b, day
4) and the half-life of the dye in the estuary is 10% longer. These are measurable effects, but still
only fractional changes. When the duration of the dye release/upwelling event is extended to 8 d
(not shown), the river- and ocean-forced modulation of the intrusion is even weaker: 10% more
dye enters, and has a half-life only 5% longer, than in the tide-only case.
5. Discussion
In the pulse-propagation experiment just described, not only the horizontal spreading of
an isolated ocean water mass, but also its steady migration up-estuary, is primarily controlled by
tidal stirring. This may seem counterintuitive, given that tidal stirring is so often treated
mathematically, as the label "stirring" implies, as a simple Fickian diffusion process (Hansen and
Rattray 1965, MacCready 2004, Sec. 3b above). Indeed, throughout our analysis we have been,
inevitably, shuttling between the language of "diffusion," "dispersion," and "stirring" on the one
hand and "advection," "transport pathways," and "conveyor belts" on the other. In this section,
we will try to reconcile these ideas, and show what we can infer about the actual mechanisms of
tidal exchange in Willapa when all the components of our analysis are taken together.
30
a. Tidal flushing and tidal retention
A schematic of tidal exchange is shown in Fig 11. Arrows indicate the Lagrangian
transport pathways most visible in the crossing-region analysis from Sec. 3d. The most coherent
feature is the strong lateral exchange flow in the seaward reach. The inflow curves around the
southern and western sides of the entrance, crossing in and out of the curving main channel as
shown (Figs. 5b-g, 11). The compensating outflow, through the deep channel to the north of the
entrance, draws water from a broad area on both sides of the Stanley-Willapa junction, although
some water from the central estuary disperses into the Willapa Channel (Fig 6b) before entering
this outflow, in a tangled, "diffusive" process. A similar interleaving of incoming and outgoing
water masses occurs at the junction at the Shoalwater Bay entrance (Figs. 6a, 11).
Thus as we commented above (Sec. 3d), the along-channel profile of effective diffusivity
K (Fig. 3e) describes not just diffusion but also coherent, advective flow structures. Furthermore,
on large scales these flows sometimes actually appear anti-diffusive, creating fronts and
discontinuities instead of smoothing them away (Fig. 3d). The key to this phenomenon, in
mathematical terms, is that a gradient in diffusivity acts on a tracer as a kind of
"psuedoadvection." That is, the general tracer-diffusion equation
s s = K t x x (10)
(where s is tracer concentration) can be rewritten
s dK s 2s
=
K
+ t dx x x 2
(11)
31
as if K were locally constant and a background velocity –dK/dx were advecting the tracer in the
direction of decreasing diffusivity. This idea helps explain why any particular water mass tends
to collect or stop advancing where K is low: southern Shoalwater Bay, the head of the northern
channel (Figs. 9a, 10a), and the mid-estuary minimum in K (Figs. 3e, 5g; dotted line in Fig. 11).
A local minimum in K—i.e. a region where the interaction of tides and bathymetry happens to
generate few residual currents—acts as a semipermeable barrier to tidal exchange. A minimum
in K is a gap in subtidal transport, a valley that can be crossed only slowly, and thus a place
where relatively sharp breaks in water properties may form (Figs. 3d, 9a).
The potential significance of such a "dispersion gap" (and additional support for our
model results) can be seen in a fact well-known in the Willapa Bay oyster-culture community,
although not well documented in the scientific literature (Hedgpeth and Obreski 1981; J. Ruesink
and A. Trimble, pers. comm.) Natural settlement of oyster larvae, which requires retention in the
water column for three weeks after spawning, occurs in the southern reaches of Willapa Bay but
not in the northern. The dividing line lies north of the northern tip of Long Island, close to where
we find the K minimum and sharp gradient in age discussed above (Fig. 11). Just as new ocean
water advances quickly up to this point in the bay but continues past it only weakly (Fig. 3d), so
can water up-estuary of this point (carrying recently spawned oyster larvae, say) be trapped
behind it. The online drifter-dispersion visualization described in Sec. 3e shows this directly. Our
model thus reproduces a striking feature of Willapa's circulation: despite the fact that half the
volume of the bay enters and leaves with every tide (Sec. 1), a typical water parcel in the upper
third of the estuary is retained for several weeks or even months.
32
b. The importance of along-channel complexity
In the Lagrangian view, the high diffusivities and rapid exchange seaward of the midestuary dispersion gap appear to be the result of a single, not especially complex lateral-shear
flow structure (Figs. 5, 11). The Eulerian view of the same area, however, is decidedly lacking in
simple shear structures (Fig. 4): in fact, near the mouth the salt-flux constituent that dominates in
equation (6) is term (ii), the only term that does not involve lateral shear at all.
Dronkers and van de Kreeke (1986) present an interpretation of the salt-flux constituents
in equation (6) that helps explain this discrepancy. They refer to term (i) as the "local salt flux,"
since it involves only the structure of u and s within a cross-section. They refer to (ii) and (iii)
together as the "nonlocal salt flux," since the temporal variations in these terms (s1, dq1) are
really signs of spatial variation outside the cross-section being analyzed: i.e., variations over a
tidal excursion in the along-channel direction. They then demonstrate that the nonlocal salt flux
is approximately equal to the difference between the local flux through a stationary section and
the local flux through a section moving with the flow. In other words, where terms (ii) and (iii)
are large (in Willapa Bay, nearly everywhere: Fig. 4), it is definitional that the Eulerian
circulation pattern does not reflect the underlying Lagrangian circulation, as we have found.
Furthermore, the fact that the nonlocal terms are non-negligible or dominant over most
of the estuary (Fig. 4) can be taken to mean that the Lagrangian transport pathways we have
described are driven at least as much by along-channel (nonlocal) bathymetric complexity as by
transverse complexity. It is not surprising, then, that as discussed in Sec. 3d, Willapa's exchange
circulation is not readily schematized as a lateral, channel-flow-versus-bank-flow pattern, despite
33
the ubiquity of such patterns in the estuarine literature (Li and O'Donnell 1997, other references
above). There can be no typical lateral pattern if the local salt flux (i) is weak and changeable; in
Willapa term (i) changes sign and weakens over the 20 km in from the mouth (Fig. 4), and
indeed the landward and seaward halves of the Lagrangian exchange flow migrate between bank
and channel and dissipate over the same distance (Figs. 5, 11). If the circulation in estuaries like
Willapa is to be schematized, we propose that one start not with lateral depth variation but alongestuary complexity: channel curvature, branching and junctions, and sets of ebb-dominant/flooddominant channel pairs. To date, these common features of coastal-plain estuaries have received
more attention in the morphodynamics literature (e.g., van Veen 1950, Hibma et al. 2003) than in
the hydrodynamics literature.
6. Conclusions
This modeling study and a previous observational analysis (Banas et al. 2004) paint a
consistent picture of the processes that control flushing in Willapa Bay. Under high-flow, winter
conditions, the density-driven and tide-driven circulations are both first-order contributors:
neither can be neglected (Fig. 9c). During spring and summer, however, when river input is
generally below the long-term average, tidal effects dominate, and baroclinic effects are
discernable (Fig. 9e) but a second-order correction. This appears to be true on both seasonal
timescales (Fig. 9) and the event scale (Fig. 10).
We did not, however, test a full suite of event-scale scenarios involving, for example,
bursts of storm-level, northward winds accompanied by sharp pulses of riverflow and drops in
34
ocean salinity (a set of forcings that tends to occur in concert on the Washington coast in winter
and early spring: Hickey and Banas 2003, Banas et al. 2004). Examining a range of such
scenarios in more detail may be important to understanding phenomena that unfold not over days
or weeks but over individual tidal cycles: patterns of larval settlement in the seaward reach of the
estuary, for example (Eggleston et al. 1998, Roegner et al. 2003). Before such processes near the
mouth can be examined reliably in detail, however, better offshore data is needed for model
forcing and validation.
Tidal dispersion has often been represented in the theory of estuarine circulation in
particularly stripped-down, generic terms. In Willapa the tidal circulation is not only dynamically
central but extremely intricate and variegated, diffusion-like in some places and conveyor-beltlike in others (Sec. 5). Our description of tidal dispersion in Willapa relies on two Lagrangian
methods: the residual-circulation return map, built through particle tracking (Sec. 3d), and the
steady-state age distribution, calculated using a nonconservative tracer (Sec. 4b). The age
distribution, in particular, provides the most comprehensive picture of the flushing of the estuary
of all the methods used in this study. It does not seem coincidental that it is also the only method
that does not impose an artificial geometry (a preferred "along-channel" direction, straight-line
cross-sections) on the flow. We recommend particle and tracer methods along these lines, and
attention to along-estuary geometric intricacies—exactly those channel bends and flares and
junctions that have been hardest to treat analytically—in the analysis of flushing and residence
time in other complex systems.
35
Acknowledgements
This work was supported by grants NA76 RG0119 Project R/ES-9, NA76 RG0119 AM08
Project R/ES-33, NA16 RG1044 Project R/ES-42, NA76 RG0485 Project R/R-1 and NA96
OP0238 Project R/R-1 from the National Oceanic and Atmospheric Administration to
Washington Sea Grant Program, University of Washington. NSB was also supported in part by a
National Defense Science and Engineering Graduate Fellowship. Sue Geier was responsible for
initial data processing. Thanks are given to Jan Newton, Judah Goldberg, and the Washington
Department of Ecology, who generously provided the Bay Center stratification data; Brett
Dumbauld and the Washington Department of Fish and Wildlife, who provided field support and
boat time; Jennifer Ruesink and Alan Trimble at the University of Washington, who provided the
Shoalwater Bay salinity time series; and Adele Militello and the U. S. Army Corps of Engineers
Seattle District, who made their model grid available to us. Finally, many thanks are given to
Karsten Bolding and Manuel Ruiz Villareal for their unstinting help setting up GETM on our
computer, and to Parker MacCready and Rocky Geyer for their insights.
36
Alvarez-Salgado, X. A., J. Gago, B. M. Miguez, M. Gilcoto, and F. F. Pérez (2000), Surface
waters of the NW Iberian margin: upwelling on the shelf versus outwelling of upwelled waters
from the Rías Baixas, Estuarine Coastal Shelf Sci, 51, 821-837.
Arakawa, A., and V. R. Lamb (1979), Computational design of the basic dynamical processes of
the UCLA general circulation model, Meth. Comput. Phys, 17, 173-265.
Austin, J. A. (2004), Estimating effective longitudinal dispersion in the Chesapeake Bay,
Estuarine Coastal Shelf Sci, 60, 359-368.
Banas, N. S., B. M. Hickey, P. MacCready, and J. A. Newton (2004), Dynamics of Willapa Bay,
Washington: A highly unsteady, partially mixed estuary, J. Phys. Oceanogr, 34, 2413-2427.
Beerens, S. P., H. Ridderinkhof, and J. T. F. Zimmerman (1994), An analytic study of chaotic
stirring in tidal areas, in Chaos Applied to Fluid Mixing, edited by Aref, H., and M. S. E. Naschie,
Pergamon, Oxford.
Bowden, K. F., and R. M. Gilligan (1971), Characterisation of features of estuarine circulation as
represented in the Mersey Estuary, Limnol. Oceanogr, 16, 490-502.
Burchard, H. (2002), Applied turbulence modelling in marine waters, Lecture Notes in Earth
Sciences, Vol. 100, Springer, Berlin.
Burchard, H., and K. Bolding (2001), Comparative analysis of four second-moment turbulence
closure models for the oceanic mixed layer, J. Phys. Oceanogr, 31, 1943-1968.
Burchard, H., and K. Bolding (2002), GETM, a general estuarine transport model: Scientific
documentation. Technical report, European Commission, Ispra.
Burchard, H., K. Bolding, and M. R. Villareal (1999), GOTM, A general ocean turbulence model.
Technical report EUR 18745 EN, European Commission.
Burchard, H., K. Bolding, and M. R. Villareal (2004), Three-dimensional modelling of estuarine
turbidity maxima in a tidal estuary, Ocean Dynamics, 54, 250-265.
Canuto, V. M., A. Howard, Y. Cheng, and M. S. Dubovikov (2001), Ocean turbulence I: onepoint closure model. Momentum and heat vertical diffusivities with and without rotation, J.
Phys. Oceanogr, 31, 1413-1426.
37
Cokelet, E. D., R. J. Stewart, and C. C. Ebbesmeyer (1991), Concentrations and ages of
conservative pollutants in Puget Sound, in Puget Sound Research '91, Vol. 1, 99-108,
Puget Sound Water Quality Authority.
Deleersnijder, E., J.-M. Campin, and E. J. M. Delhez (2001), The concept of age in marine
modelling: I. Theory and preliminary model results, J. Mar. Sys, 28, 229-267.
Dronkers, J., and J. v. d. Kreeke (1986), Experimental determination of salt intrusion mechanisms
in the Volkerak estuary, Neth. J. Sea Res, 20, 1-19.
Dyer, K. R. (1973), Estuaries: A physical introduction, 140pp, Wiley.
Eggleston, D., D. Armstrong, W. Elis, and W. Patton (1998), Estuarine fronts as conduits for
larval transport: hydrodynamics and spatial distribution of megalopae, Mar Ecol Prog Ser, 164,
73-82.
Emmett, R., R. Llanso, J. Newton, R. Thom, M. Hornberger, C. Morgan, C. Levings, A.
Copping, and P. Fishman (2000), Geographic signatures of North American West Coast
estuaries, Estuaries, 23, 765-792.
England, M. H. (1995), The age of water and ventilation timescales in a global ocean model, J.
Phys. Oceanogr, 25, 2756-2777.
Fischer, H. B. (1976), Mixing and dispersion in estuaries, Annual Review of Fluid Mechanics, 8,
107-133.
Friedrichs, C. T., and J. H. Hamrick (1996), Effects of channel geometry on cross sectional
variation in along channel velocity in partially stratified estuaries, in Buoyancy Effects on Coastal
and Estuarine Dynamics, edited by Aubrey, D. G., and C. T. Friedrichs, pp. 283-300, American
Geophysical Union, Washington, D.C.
Gross, E. S., J. R. Koseff, and S. G. Monismith (1999), Evaluation of advective schemes for
estuarine salinity simulations, J. Hydr. Engin, 125, 32-46.
Hands, E. B., and V. Shepsis (1999), Cyclic channel movement at the entrance to Willapa Bay,
Washington, USA. Proceedings, Coastal Sediments '99, 1522, Amer. Soc. Civil Engin.
Hansen, D. V., and M. Rattray Jr (1965), Gravitational circulation in straits and estuaries, J.
Mar. Res, 23, 104-122.
38
Hedgpeth, J. W., and S. Obreski (1981), Willapa Bay: A historical perspective and a rationale for
research. FWS/OBS-81/03, 52pp, Office of Biological Services, U. S. Fish and Wildlife Service,
Washington, DC.
Hibma, A., H. J. d. Vriend, and M. J. F. Stive (2003), Numerical modelling of shoal pattern
formation in well-mixed elongated estuaries, Estuarine Coastal and Shelf Sci, 57, 981-991.
Hickey, B. M. (1989), Patterns and processes of circulation over the shelf and slope, in Coastal
oceanography of Washington and Oregon, edited by Hickey, B. M., and M. R. Landry, pp. 41115, Elsevier.
Hickey, B. M., and N. S. Banas (2003), Oceanography of the U. S. Pacific Northwest coastal
ocean and estuaries with application to coastal ecology, Estuaries, 26, 1010-1031.
Hickey, B. M., X. Zhang, and N. Banas (2002), Coupling between the California Current System
and a coastal plain estuary in low riverflow conditions, J. Geophys. Res, 107, 3166,
10.1029/1999JC000160.
Hughes, F. W., and M. R. Jr (1980), Salt flux and mixing in the Columbia River Estuary,
Estuarine Coastal Mar. Sci, 10, 479-494.
Huzzey, L. M., and J. M. Brubaker (1988), The formation of longitudinal fronts in a coastal
plain estuary, J. Geophys. Res, 93, 1329-1334.
Kranenburg, C. (1986), A timescale for long-term salt intrusion in well-mixed estuaries, J. Phys.
Oceanogr, 16, 1329-1331.
Kraus, N. C. (2000), Study of navigation channel feasibility, Willapa Bay, Washington. Final
Report, 440pp, U. S. Army Corps of Engineers Seattle District.
Leonard, B. P. (1979), A stable and accurate convective modeling procedure based on quadratic
upstream interpolation, Comput. Meth. Appl. Mech. Engin, 19, 59-98.
Leonard, B. P. (1991), The ULTIMATE conservative difference scheme applied to unsteady
one-dimensional advection, Comput. Meth. Appl. Mech. Engin, 88, 17-74.
Lewis, R. E., and J. O. Lewis (1983), The principal factors contributing to the flux of salt in a
narrow, partially stratified estuary, Est. Coast. Shelf Sci, 33, 599-626.
39
Li, C., and J. O'Donnell (1997), Tidally induced residual circulation in shallow estuaries with
lateral depth variation, J. Geophys. Res, 102, 27915-27929.
MacCready, P. (1999), Estuarine adjustment to changes in river flow and tidal mixing, J. Phys.
Oceanogr, 29, 708-726.
MacCready, P. (2004), Toward a unified theory of tidally-averaged estuarine salinity structure,
Estuaries, 27, 561-570.
Mellor, G. L., and T. Yamada (1982), Development of a turbulence closure model for geophysical
fluid problems, Rev. Geophys, 20, 851-875.
Monismith, S. G., W. Kimmerer, J. R. Burau, and M. T. Stacey (2002), Structure and flowinduced variability of the subtidal salinity field in northern San Francisco Bay, J. Phys.
Oceanogr, 32, 3003-3019.
Monsen, N. E., J. E. Cloern, L. V. Lucas, and S. G. Monismith (2002), A comment on the use of
flushing time, residence time, and age on transport time scales. Limnol. Oceanogr., 47, 15451553.
Monteiro, P. M. S., and J. L. Largier (1999), Thermal stratification in Saldhana Bay (South
Africa) and subtidal, density-driven exchange with the coastal waters of the Benguela upwelling
system, Estuarine and Coastal Shelf Science, 49, 877-890.
Newton, J. A., and R. A. Horner (2003), Use of phytoplankton species indicators to track the
origin of phytoplaknton blooms in Willapa Bay, Washington, Estuaries, 26, 1071-1078.
O'Donnell, J. (1993), Surface fronts in estuaries: A review, Estuaries, 16, 12-39.
Okubo, A. (1973), Effect of shoreline irregularities on streamwise dispersion in estuaries and
other embayments, Neth. J. Sea Res, 6, 213-224.
Pawlowicz, R., B. Beardsley, and S. Lentz (2002), Classical tidal harmonic analysis including
error estimates in MATLAB using T_TIDE, Computers and Geosciences, 28, 929-937.
Pritchard, D. W. (1956), The dynamic structure of a coastal plain estuary, J. Mar. Res, 15, 33-42.
Ridderinkhof, H., and J. T. F. Zimmerman (1992), Chaotic stirring in a tidal system, Science, 258,
1107-1111.
40
Roe, P. L. (1985), Some contributions to the modeling of discontinuous flows, Lecture Notes in
Applied Mathematics, 22, 163-193.
Roegner, G. C., B. M. Hickey, J. A. Newton, A. L. Shanks, and D. A. Armstrong (2002), Windinduced plume and bloom intrusions into Willapa Bay, Washington, Limnol. Oceanogr, 47, 10331042.
Roegner, G. C., D. A. Armstrong, B. M. Hickey, and A. L. Shanks (2003), Ocean Distribution of
Dungeness Crab Megalopae and Recruitment Patterns to Estuaries in Southern Washington State,
Estuaries, 26, 1058-1070.
Ruesink, J. L., G. C. Roegner, B. R. Dumbauld, J. A. Newton, and D. A. Armstrong (2003),
Contributions of coastal and watershed energy sources to secondary production in a
Northeastern Pacific estuary, Estuaries, 26, 1079-1093.
Simpson, J. H., J. Brown, J. Matthews, and G. Allen (1990), Tidal straining, density currents,
and stirring in the control of estuarine stratification, Estuaries, 26, 1579-1590.
Simpson, J. H., R. Vennell, and A. J. Souza (2001), The salt fluxes in a tidally-energetic estuary,
Est. Coast. Shelf Sci, 52, 131-142.
Smith, S. V., J. T. Hollibaugh, S. J. Dollar, and S. Vink (1991), Tomales Bay metabolism: CNP
stoichiometry and ecosystem heterotrophy at the land-sea interface, Estuarine Coastal Shelf Sci,
33, 223-257.
Stacey, M. T., J. R. Burau, and S. G. Monismith (2001), Creation of residual flows in a partially
stratified estuary, J. Geophys. Res, 106, 17013-17037.
Stanev, E., J.-O. Wolff, H. Burchard, K. Bolding, and G. Flöser (2003), On the circulation in the
East Frisian Wadden Sea: numerical modeling and data analysis, Ocean Dynamics, 53, 27-51.
Stips, A., K. Bolding, T. Pohlmann, and H. Burchard (2004), Simulating the temporal and spatial
dynamics of the North Sea using the new model GETM (general estuarine transport model),
Ocean Dynamics, 54, 266-283.
Thiele, G., and J. L. Sarmiento (1990), Tracer dating and ocean ventilation, J. Geophys. Res, 95,
9377-9391.
Umlauf, L., and H. Burchard (2003), A generic length-scale equation for geophysical turbulence
models, J. Mar. Res, 61, 235-265.
41
Valle-Levinson, A., C. Reyes, and R. Sanay (2003), Effects of bathymetry, friction, and Earth's
rotation on estuary/ocean exchange, J. Phys. Oceanogr, 33, 2375-2393.
Valle-Levinson, A., and J. O'Donnell (1996), Tidal interaction with buoyancy driven flow in a
coastal-plain estuary, in Buoyancy effects on coastal and estuarine dynamics, edited by D. G.
Aubrey, and C. T. Friedrichs., pp. 265-281, Amer. Geophys. Union.
van Veen, J. (2002), Ebb and flood channel systems in the Netherlands tidal waters. Orig. publ. J.
Royal Dutch Geogr. Soc., 67, 303-325, 1950. Introd., annot., and transl. H. Bonekamp, E. Elias,
A. Hibma, C. van de Kreeke, M. van Ledden, D. Roelvink, H. Schuttelaars, H. de Vriend, Z.-B.
Wang, A. van der Spek, M. Stive, and T. Zitman, Delft Univ Press.
Warner, J. C., C. R. Sherwood, H. G. Arango, and R. P. Signell (2005), Performance of four
turbulence closure models implemented using a generic length scale method, Ocean Modelling, 8,
81-113.
Winant, C. D., and G. Gutierrez de Velasco (2003), Tidal dynamics and residual circulation in a
well-mixed inverse estuary, J. Phys. Oceanogr, 33, 1365-1379.
Zimmerman, J. T. F. (1976), Mixing and flushing of tidal embayments in the western Dutch
Wadden Sea, Part I: Distribution of salinity and calculation of mixing time scales, Neth. J. Sea
Res, 10, 149-191.
Zimmerman, J. T. F. (1986), The tidal whirlpool: A review of horizontal dispersion by tidal and
residual currents, Neth. J. Sea Res, 20, 133-154.
42
Figure 1. Map of Willapa Bay. The offshore boundaries of the model domain and samples of the
model grid are also shown.
Figure 2. Results from runs A and B (Table 1, Sec. 3a) compared with observations. Depthaveraged M2 tidal velocities from model and data are shown at four locations (three near the
mouth, one near 46° 30'). Modeled and observed tidal height and the north-south component of
instantaneous, cross-sectional-mean tidal velocity are shown for one flood tide on May 5, 2000.
Figure 3. (a-d) Evolution of an initially uniform tracer gradient under idealized tidal flow (run
D). (e) Total effective horizontal diffusivity K, calculated for this run from tracer fluxes across
12 sections (green lines); also, empirical values of K calculated from salinity time series at four
stations (blue dots) using the method of Banas et al. (2004). (f) An example of the empirical K
calculation (adapted from Banas et al. 2004).
Figure 4. Contributions to the total diffusivity K associated with terms (i) (steady shear), (ii)
(trapping), and (iii) (a cross-term) in the salt-flux decomposition (6). Fluxes are calculated from
model run D at the 12 cross-sections shown in the inset and Fig. 3e.
Figure 5. (a) Schematic of the method for constructing Lagrangian "crossing regions" from
numerical particle tracking as described in Sec. 3c. Initial particle positions are shown in light
shades, positions one tidal cycle later as dark points; for a given cross-section (green line),
particles crossing seaward are shown in blue, particles crossing landward in red; particles that
43
begin and end on the same side of the cross-section are not plotted. (b-g) Crossing regions for six
sections under idealized tidal flow (run D: Table 1). (h) Crossing regions for the section shown
in (c) when run D is repeated with the Coriolis acceleration turned off.
Figure 6. Crossing regions, like those in Fig. 5b-g, for two sections at channel junctions.
Figure 7. Salinity at three stations (Bay Center, Oysterville, Naselle) relative to salinity at the
mouth (W3), as a function of bay-total riverflow. Results are given for both model run C and
observations Jul 1999 – Jun 2000 (see Table 1).
Figure 8. Time series of stratification at Bay Center and bay-total riverflow, from (a)
observations in 2003-04 (see Table 1) and (b) model run C. Sustained stratification events are
marked by arrows. (c,d) Maximum and minimum stratification during each tidal cycle shown in
(a) and (b), displayed as a function of riverflow.
Figure 9. Maps of water age (average time that the water in each grid cell has spent within the
estuary) under three constant-forcing scenarios (runs E-G: Table 2). Riverflow Q and ocean
salinity soc are given for each case where they are applied. (a-c) Depth-averaged age; arrows
mark river inputs. (d-f) Vertical sections of age at mid-estuary.
Figure 10. Evolution of pulses of ocean water marked by a two-day dye release in the model.
Each frame shows depth-averaged dye concentration relative to the initial source concentration.
44
Two scenarios are shown: propagation under idealized tides only, without density effects (run
H), and propagation when riverflow of 100 m3 s-1 is added and ocean salinity temporarily
increases from 28 to 32 psu during the dye release (run I: see Table 2). The first case represents
typical late-summer conditions, the second a typical spring upwelling event.
Figure 11. Schematic of the tide-driven residual circulation, based on Figs. 3, 5, and 9a. Arrows
indicate Lagrangian transport pathways. A dotted line marks the water-mass-isolating
"dispersion gap" discussed in Sec. 5a.
45
Table 1. Overview of model runs and datasets used for model validation in Secs. 3 and 4.
validation runs
run ID
A
B
C
D
TIDAL VELOCITY 1
TIDAL VELOCITY 2
REALISTIC YEAR
MEAN TIDES
(Sec. 3a, Fig. 2)
(Sec. 3a, Fig. 2)
(Sec. 4a, Figs. 7-8)
(Sec. 3b, Fig. 3)
simulates
Oct 15 - Nov 11, 1998 May 5, 2000
Jul 1999 - Jun 2000
tracers
—
—
salt
idealized semidiurunal tide
(same forcing as run E, LATE
SUMMER: Table 2)
dye
density?
off
model forcing
off
on
off
sea level
observations from
Toke Point
observations from
Toke Point
pure M2 tide (± 1.2 m)
river input
observations from
Toke Point (NOAA
station 9440910)
—
—
observations from USGS gauges
12013500, 12010000,
adjusted for watershed area
—
ocean tracers
—
—
salinity at bay mouth (station W3: constant dye concentration
Banas et al. 2004)
(initial dye field in estuary:
Fig. 3a)
model validation
criterion
M2 constituent of
depth-averaged tidal
velocity
instantaneous, crosssectionally-averaged
tidal velocity
salt intrusion
length
stratification at effective horizontal
Bay Center
diffusivity K
data
ADCP time series at
four stations
small-boat ADCP
transects
salinity at 3
surface and
K from salinity time series at
stations, Jul 1999 bottom salinity four stations (method of
- Jun 2000
at Bay Center, Banas et al. 2004)
Oct 2003 - Apr
2004
data source
US Army Corps of
Engineers Seattle
District: Kraus 2000
(previously
unreported)
Banas et al. 2004 (previously
unreported;
courtesy of J.
Newton/ Wash.
Dept. of
Ecology)
Banas et al. 2004
(W3, Bay Center, Oysterville;
Shoalwater data previously
unreported, courtesy of J.
Ruesink and A. Trimble)
46
Table 2. Overview of idealized model scenarios used to examine residence time and exchange
patterns in Sec. 4.
E
idealized, constant forcing
F
G
LATE SUMMER
SPRING
WINTER
run ID
idealized, pulsed ocean forcing
H
I
PULSE -
PULSE -
DENSITY OFF
DENSITY ON
simulates
sustained low
riverflow +
upwelling
(Sec. 4b, Fig. 9)
sustained moderate
riverflow +
upwelling
sustained high
riverflow +
downwelling
tracers
age
age, salt
age, salt
(Sec. 4c, Fig. 10)
ocean variability in ocean variability in
typical late-summer typical spring
conditions (~ run E)
conditions
(~ run F)
dye
dye, salt
density ?
off
on
on
off
on
model forcing
—— all runs: pure M2 tide (± 1.2 m) ——
sea level
river input
—
ocean tracers age = 0
3 -1
100 m s
1000 m3 s-1
—
age = 0
salinity = 30 psu
age = 0
salinity = 20 psu
2 d dye pulse
100 m3 s-1
2 d dye pulse;
salinity increases
temporarily
(28 > 32 > 28 psu)
over same 2 d
(also repeated with 8 d pulses: Sec. 4e)
model grid
expands offshore
TOKE PT.
Willapa Bay
Columbia R.
closed boundary
50 km
model grid
in estuary
8–30 m
above
MLLW
10 km
MLLW –
8m
124°20'
124°10'
124°0'
123°50'
Figure 1. Map of Willapa Bay. The offshore boundaries of the model domain and samples of the model grid
are also shown.
Depth-averaged
M2 tidal velocity
amplitude
Toke Pt.
DATA
RUN A
1 m s–1
46°40'
2
Tidal height at Toke Pt. (m from MSL)
0
model
data
-2
1
Mean N-S velocity over thalweg (m s–1)
model
46°30'
0
data
RUN B
-1
06:00
12:00
18:00
MAY 5, 2000
Figure 2. Results from runs A and B (Table 1, Sec. 3a) compared with observations. Depth-averaged M2 tidal
velocities from model and data are shown at four locations (three near the mouth, one near 46° 30'). Modeled
and observed tidal height and the north-south component of instantaneous, cross-sectional-mean tidal velocity
are shown for one flood tide on May 5, 2000.
Concentration of a passive tracer
0
50
(a)
100%
(c)
(b)
RUN D
initial field
(early ebb)
after 5 h
(low water)
(d)
(e)
after 1
tidal cycle
Effective horizontal diffusivity K (m2 s–1)
710±150
W3
DATA
MODEL
270±40
Bay Center
(f)
210±80
Oysterville
20
Low-riverflow-period
observations 1998–2001
at Bay Center
0
after 12
cycles
60±30
Shoalwater
-20
-0.2
0
0.2
0.4
Axial salinity gradient
(psu km-1)
Figure 3. (a-d) Evolution of an initially uniform tracer gradient under idealized tidal flow (run D). (e) Total
effective horizontal diffusivity K, calculated for this run from tracer fluxes across 12 sections (green lines);
also, empirical values of K calculated from salinity time series at four stations (blue dots) using the method
of Banas et al. (2004). (f) An example of the empirical K calculation (adapted from Banas et al. 2004).
steady shear
up-gradient
–200
RUN D
0
trapping
cross-term
down-gradient (landward)
200
800 m2 s–1
K
20
40
Figure 4. Contributions to the total diffusivity K associated with terms (i) (steady shear), (ii) (trapping), and
(iii) (a cross-term) in the salt-flux decomposition (6). Fluxes are calculated from model run D at the 12 crosssections shown in the inset and Fig. 3e.
step 2
Lagrangian
tidal residuals
step 3 CROSSING REGIONS
step 1
(a)
particle-tracking
in full tidal flow
after
1 cycle
model
grid
particles crossing IN
and OUT
across a given section
(b)
(c)
RUN D
(d)
(e)
without
Coriolis
(h)
(f)
(g)
Figure 5. (a) Schematic of the method for constructing Lagrangian "crossing regions" from numerical particle
tracking as described in Sec. 3c. Initial particle positions are shown in light shades, positions one tidal cycle
later as dark points; for a given cross-section (green line), particles crossing seaward are shown in blue,
particles crossing landward in red; particles that begin and end on the same side of the cross-section are not
plotted. (b-g) Crossing regions for six sections under idealized tidal flow (run D: Table 1). (h) Crossing regions
for the section shown in (c) when run D is repeated with the Coriolis acceleration turned off.
RUN D
Stanley–Willapa
Channel junction
Shoalwater Bay
entrance
(a)
(b)
Figure 6. Crossing regions, like those in Fig. 5b-g, for two sections at channel junctions.
RUN C
Data
salinity / salinity at mouth
Model
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
Bay Center
0.2
0.2
0
10
100
1000
Bay-total riverflow (m3 s–1)
0
Oysterville
10
100
1000
Bay-total riverflow (m3 s–1)
0.2
0
Naselle
10
100
1000
Bay-total riverflow (m3 s–1)
Figure 7. Salinity at three stations (Bay Center, Oysterville, Naselle) relative to salinity at the mouth (W3),
as a function of bay-total riverflow. Results are given for both model run C and observations Jul 1999 – Jun
2000 (see Table 1).
Surface-bottom salinity difference
at Bay Center (psu)
DATA (2003–04)
4
2
0
1500
Riverflow (m3 s–1)
1000
500
0
Nov 03
Dec 03
Jan 04
Feb 04
(a)
MODEL (1999–2000)
RUN C
4 Surface-bottom salinity difference
at Bay Center (psu)
2
0
1500
Riverflow (m3 s–1)
1000
500
0
Nov 99
Dec 99
Jan 00
Feb 00
Mar 00
(b)
8
1.5
MAX stratification
over each tidal cycle (psu)
Data
Model
MIN stratification
over each tidal cycle (psu)
6
1
4
0.5
2
obs. limit
0
(c)
100
Riverflow (m3 s–1)
1000
0
(d)
100
Riverflow (m3 s–1)
1000
Figure 8. Time series of stratification at Bay Center and bay-total riverflow, from (a) observations in 200304 (see Table 1) and (b) model run C. Sustained stratification events are marked by arrows. (c,d) Maximum
and minimum stratification during each tidal cycle shown in (a) and (b), displayed as a function of riverflow.
Age (average time spent within estuary: days)
0
12
24
36
(b)
(a)
48
60
(c)
6
6
12
30
42
RUN E LATE SUMMER
RUN F SPRING
RUN G WINTER
Q=0
density off (tides only)
Q = 100 m3 s–1
soc = 30 psu
Q = 1000 m3 s–1
soc = 20 psu
2
4
6
Distance from western shore (km)
(d)
(e)
(f)
0
10
20
0
Figure 9. Maps of water age (average time that the water in each grid cell has spent within the estuary) under
three constant-forcing scenarios (runs E-G: Table 2). Riverflow Q and ocean salinity soc are given for each
case where they are applied. (a-c) Depth-averaged age; arrows mark river inputs. (d-f) Vertical sections of
age at mid-estuary.
ocean dye
injected for 2 d
DYE CONCENTRATION
1
4.0
6.6
9.2
RUN H
tides only
RUN I
tides + rivers + ocean density
100%
TIME (days)
10 tidal cycles
1.4
10
11.7
19.5
27.3
Figure 10. Evolution of pulses of ocean water marked by a two-day dye release in the model. Each frame
shows depth-averaged dye concentration relative to the initial source concentration. Two scenarios are shown:
propagation under idealized tides only, without density effects (run H), and propagation when riverflow of
100 m3 s-1 is added and ocean salinity temporarily increases from 28 to 32 psu during the dye release (run I:
see Table 2). The first case represents typical late-summer conditions, the second a typical spring upwelling
event.
Figure 11. Schematic of the tide-driven residual circulation, based on Figs. 3, 5, and 9a. Arrows indicate
Lagrangian transport pathways: note that point-by-point, Eulerian residual velocities are generally substantially
different. A dotted line marks the water-mass-isolating "dispersion gap" discussed in Sec. 5a.