Mapping exchange and residence time in a model of Willapa Bay

Mapping exchange and residence time in a model of Willapa

Bay, Washington, a branching, macrotidal estuary

N. S. Banas and B. M. HickeySchool of Oceanography, University of Washington, Seattle, Washington, USA

Received 10 March 2005; revised 14 July 2005; accepted 3 August 2005; published 17 November 2005.

[1] The numerical model GETM is used to examine transport pathways and residencetime 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 andthe decrease of the salt intrusion length with increasing river flow with errors of 5–20%.Furthermore, a more stringent test, when the model is run with tidal forcing only, itreproduces the along-channel profile of the effective horizontal diffusivity K, a directmeasure of the strength of subtidal dispersion, which is known from previous empiricalestimates. A Lagrangian, particle-tracking method is used to map subtidal transportpathways at the resolution of the model grid. This analysis reveals an interweaving ofcoherent lateral exchange flows with discontinuous, small-scale dispersion as well as tidalresidual currents that in some locations, sharpen rather than smooth gradients betweenwater masses. Comparison between these Lagrangian results and an Eulerian saltflux decomposition suggests that along-channel complexity (channel junctions andchannel curvature) is at least as important as cross-sectional depth variation in shapingthe subtidal circulation. Finally, a nonconservative tracer method is used to producehigh-resolution, three-dimensional maps of residence time. This analysis shows thatconsistent with previous observational work in Willapa, at all except the highestwinter-storm-level river flows, river- and ocean-density-driven exchanges are discernablebut secondary to tidal stirring. In all seasons, despite the fact that half the volume ofthe bay enters and leaves with every tide, average retention times in the upper third of theestuary are 3–5 weeks.

Citation: Banas, N. S., and B. M. Hickey (2005), Mapping exchange and residence time in a model of Willapa Bay, Washington, a

branching, macrotidal estuary, J. Geophys. Res., 110, C11011, doi:10.1029/2005JC002950.

1. Introduction

[2] This study examines the mechanisms and timescalesof exchange between Willapa Bay, Washington (Figure 1),and the coastal ocean. Willapa is the largest in a series ofshallow, coastal plain estuaries that spans the U.S. PacificNorthwest coast from central Washington to northernCalifornia. Note that the Columbia River, immediatelysouth of Willapa, is so different from its neighbors inmorphology and river input [Hickey and Banas, 2003] thatwe do not treat it as part of this series. These estuaries areforced by strong (2–3 m) tides, and have relatively deepand unchannelized intertidal zones [Emmett et al., 2000]:fully half of Willapa’s area and volume are intertidal, andthis appears to be typical [Hickey and Banas, 2003]. Inaddition, these estuaries are subject to changes in oceanwater properties forced by wind-driven upwelling-downwelling transitions on both seasonal and event (2–10 day) 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 theupwelling-influenced estuaries of Spain and other easternboundary systems [Alvarez-Salgado et al., 2000; Monteiroand Largier, 1999].[3] River input into these estuaries occurs mostly in

winter, outside the growing season [Hickey and Banas,2003]. Accordingly, primary production in Willapaappears to be fueled less by riverine nutrients thanby oceanic nutrients or the direct import of oceanicphytoplankton blooms [Roegner et al., 2002; Ruesinket al., 2003; Newton and Horner, 2003]. The dynamicsof ocean-estuary exchange (its spatial pathways and itsconstancy or variability under changing forcing) arethus essential ecological concerns. These dynamics arethe focus of the modeling work described in thispaper.[4] Recent observational studies have provided a prelim-

inary description of exchange processes in Willapa. Hickeyet al. [2002] found that variations in ocean temperature andsalinity propagated upstream at approximately 10 cm s�1

even in summer, low-river-flow conditions, much fasterthan could be attributed to the river-driven gravitationalcirculation. Banas et al. [2004] found that even duringwinter storms, when river flow is 200 times higher than in

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, C11011, doi:10.1029/2005JC002950, 2005

Copyright 2005 by the American Geophysical Union.0148-0227/05/2005JC002950$09.00

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the late summer dry season, the overall rate of exchangeonly increases by a factor of 3. Thus unlike otherwisesimilar systems like Tomales Bay, California, where astrong seasonal cycle in river flow yields a strong seasonalcycle in flushing rate [Smith et al., 1991], exchangebetween Willapa and the ocean appears to be cushionedby a high ‘‘baseline’’ flushing rate independent of theriver-driven circulation. Banas et al. [2004] suggested thatthe bulk of this baseline exchange is caused by lateral tidalstirring.[5] The observational analyses cited above relied on

multiyear velocity and salinity time series in a few pointlocations, and thus addressed the temporal variability ofWillapa’s circulation more than its spatial complexity. Evenin very simple estuarine geometries, understanding thehorizontal structure of the net circulation is an activeresearch problem [e.g., Valle-Levinson et al., 2003]. Will-apa’s geometry, moreover, is not simple: the bay is anetwork of branching channels formed when a sand spitfrom the Columbia River (the Long Beach Peninsula)

trapped a number of small rivers behind one connectionto the ocean (Figure 1) [Emmett et al., 2000]. The purposeof this modeling study is to provide detailed maps ofexchange and residence time in this complex geometry overa typical seasonal cycle.[6] In many numerical studies of estuaries, flushing rate

and residence time are by necessity pure model predictionsthat can only be validated circumstantially. Here we useindependent, observational estimates of dispersion rates atfour salinity time series locations [Banas et al., 2004](section 3.2) to test modeled dispersion rates and residencetimes directly. The model’s role, then, is principally to fill inspatial detail around these previous empirical conclusions.[7] Section 2 of this paper describes the numerical model.

In section 3 we describe the model’s representation of thetidal circulation and map out the net tidal transport path-ways. In section 4 we add river flow and density effects tothe model, and map the changes in residence time betweenlow-flow (summer) and high-flow (winter) conditions.These results rely on particle-tracking and nonconservative

Figure 1. Map of Willapa Bay. The offshore boundaries of the model domain and samples of the modelgrid are also shown.

C11011 BANAS AND HICKEY: MAPPING EXCHANGE IN WILLAPA BAY

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tracer model methods that are likely to be useful in othercomplex estuaries.

2. Model

2.1. Model Physics

[8] The Willapa circulation model is an implementationof General Estuarine Transport Model (GETM), a finitedifference, primitive equation model designed for shallowwater applications, cases like Willapa where flow overcomplex topography, mixing in strong and changing strat-ification, and flooding and drying of intertidal areas are allimportant. GETM has previously been used to model tidaldynamics in the East Frisian Wadden Sea, which is morethan half intertidal [Stanev et al., 2003]; baroclinic dynamicsand the estuarine turbidity maximum in the Elbe [Burchardet al., 2004]; and seasonal hydrographic patterns in the NorthSea [Stips et al., 2004]. GETM is open source and still underdevelopment: the project home page can be found at http://www.bolding-burchard.com. Below we highlight only somekey features of GETM: For a systematic description ofmodel physics and numerics, see Burchard and Bolding[2002] and Burchard et al. [2004].[9] GETM solves the equations of motion on a curvilin-

ear Arakawa-C grid [Arakawa and Lamb, 1979]. To resolvefast moving surface gravity waves without limiting the timestep for the entire calculation, the solution is split intointernal and external modes, with ten external (barotropic)time steps taken within each internal time step. The modelcontains a simple, stable scheme for handling the floodingand drying of intertidal areas, in which a depth-dependentmultiplier is applied to the equations of motion to smoothlyreduce the role of all terms except tendency, bottom friction,and barotropic pressure gradient as the water depthdecreases to a few cm. In addition, to prevent unphysicalpressure gradients from developing, a correction to sea levelis made where the cell-to-cell change in elevation iscomparable to the water depth [Burchard et al., 2004]. Thisscheme allows the model to accommodate strong tracergradients in intertidal areas.[10] GETM allows a wide choice of high-order advection

schemes for velocity and tracers. High-order schemes ingeneral are far less diffusive than a basic first-order upwindscheme, and therefore better suited to resolving strongvertical and horizontal gradients [Stips et al., 2004] andflows in relatively quiescent areas like shallow banks [Grosset al., 1999]. Both Gross et al. [1999] and Stips et al.[2004], in their evaluations of a number of advectionschemes in coastal and estuarine simulations, ultimatelyrecommend the Superbee [Roe, 1985] and Quickest[Leonard, 1979, 1991] schemes. Both of these are totalvariation diminishing (TVD) and therefore free from nu-merical oscillations and guaranteed to be stable. The twoschemes perform comparably well in both studies [Gross etal., 1999; Stips et al., 2004]; in this study we use Superbee,which is less computationally expensive.[11] GETM also allows a choice of turbulence closures by

coupling to the one-dimensional General Ocean TurbulenceModel (GOTM) [Burchard et al., 1999]. Here we use astandard k-e scheme [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 specialcase of a single two-equation, ‘‘generic length scale’’formulation with variable coefficients. Warner et al.[2005] compared a number of these two-equation specialcases (k-kl, k-w, gen, and the k-e scheme we use) and foundthe differences in their ability to reproduce estuarine salinityfields to be relatively minor, although all performed betterthan the original Mellor-Yamada level 2.5 scheme [Mellorand Yamada, 1982]. For a general review of two-equationclosures, see Burchard [2002].

2.2. Implementation for Willapa Bay

[12] This study uses a model grid with 175 � 82 cells inthe horizontal and 12 sigma levels in the vertical. The bayitself is covered with uniform horizontal squares 255.5 m ona side; beyond the bay mouth lies an idealized rectangular‘‘ocean’’ in which the grid size expands gradually to 6 km(Figure 1). This domain was designed to resolve Willapa’sinternal dynamics only, not the dynamics of the adjacentcoastal ocean. The model ‘‘ocean’’ (Figure 1) is a simplesemienclosed reservoir, open only to the west, with depthlimited to 30 m, since realistic depths would limit the timestep for the whole model. Additionally, the model has noalongshore currents, which on the real Washington coast aretens of cm s�1 and highly variable [Hickey, 1989]. Remov-ing any of these simplifications would provide a false senseof realism unless they were all removed, and even thenmodel validation would be tenuous given the scarcity ofhigh-resolution observations on the adjacent shelf. CouplingWillapa’s internal dynamics to shelf processes thus remainsa future goal.[13] The bathymetric grid within the estuary is interpo-

lated from a finite element model grid developed by theU.S. Army Corps of Engineers Seattle District, who resur-veyed most of the subtidal area of the bay in 1998 [Kraus,2000]. Willapa’s primary channels and many of its second-ary channels are wide enough (500–3000 m) to be well-resolved by our 250 m model grid, although in manylocations real channel edges are steeper than our gridallows. Intertidal bathymetry in our model, like that in thepublished NOAA charts for Willapa, is a composite ofsurveys over many decades. In 2002 the NOAA CoastalServices Center conducted a high-resolution light detectionand ranging (lidar) survey of the shoals of Willapa Bay(http://www.csc.noaa.gov/lidar), but no seamless, georefer-enced bathymetry based on this data set 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 aspossible the analysis below is based on spatially integrativevalidation methods that do not rely on point-by-pointcomparisons.[14] All model scenarios described in this paper are

summarized in Tables 1 and 2. A first set of runs, usedfor model validation (runs A–C), simulates particulartime periods. Another set (E–G) uses idealized tidaland river forcing to represent a typical seasonal cycle,from winter storms (river flow � 1000 m3 s�1) to spring(flow � 100 m3 s�1) to late summer (flow � 0). Run Dis a version of the idealized late summer scenario usedboth for validation and for examining tidal transportpathways in detail (section 3). A final set of runs (H


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Table 1. Overview of Model Runs and Data Sets Used for Model Validation in Sections 3 and 4

Validation Run

A, Tidal Velocity 1a B, Tidal Velocity 2b C, Realistic Yearc D, Mean Tidesd

Simulates 15 Oct to 11 Nov1998

5 May 2000 Jul 1999 to Jun 2000 idealizedsemidiurnal tidee

Tracers — — salt dyeDensity? off off on offModel forcingSea level observations from

Toke Point(NOAA station9440910)

observations fromToke Point

observations fromToke Point

pure M2 tide(±1.2 m)

River input — — observations from USGSgauges 12013500, 12010000,adjusted for watershed area

—

Ocean tracers — — salinity at bay mouth (stationW3 [Banas et al., 2004])

constant dyeconcentration(initial dye fieldin estuary: Figure 3a)

Model Validation

Validation Run

A, Tidal Velocity 1a B, Tidal Velocity 2bC, Realistic Year,c

Criterion 1C, Realistic Year,c

Criterion 2 D, Mean Tidesd

Criterion M2 constituent ofdepth-averagedtidal velocity

instantaneous,cross-sectionallyaveraged tidalvelocity

salt intrusionlength

stratification atBay Center

effective horizontaldiffusivity K

Data ADCP time series atfour stations

small-boat ADCPtransects

salinity at3 stations, Jul1999 to Jun 2000

surface andbottom salinity atBay Center,Oct 2003 toApr 2004

K from salinity timeseries at four stations(method of Banas etal. [2004])

Data source U.S. Army Corps ofEngineersSeattle District[Kraus, 2000]

(previouslyunreported)

Banas et al. [2004] (previouslyunreported;courtesy ofJ. Newton/WashingtonDepartmentof Ecology)

Banas et al. [2004](W3, Bay Center,Oysterville;Shoalwater datapreviously unreported,courtesy of J. Ruesinkand A. Trimble)

aSection 3.1 and Figure 2.bSection 3.1 and Figure 2.cSection 4.1 and Figures 7 and 8.dSection 3.2 and Figure 3.eSame forcing as run E, late summer, Table 2.

Table 2. Overview of Idealized Model Scenarios Used to Examine Residence Time and Exchange Patterns in Section 4

Idealized, Constant Forcing Idealized, Pulsed Ocean Forcing

Run E,Late Summer

Run F,Springa

Run G,Winter

Run H,Pulse

Density Offb

Run I,Pulse

Density Onb

Simulates sustained lowriver flow plusupwelling

sustained moderateriver flow plusupwelling

sustained highriver flow plusdownwelling

ocean variabilityin typical latesummerconditions(�run E)

ocean variabilityin typical springconditions(�run F)

Tracers age age, salt age, salt dye dye, saltDensity? off on on off onModel forcingsea level pure M2 tide

(±1.2 m)pure M2 tide

(±1.2 m)pure M2 tide(±1.2 m)



river input — 100 m3 s�1 1000 m3 s�1 — 100 m3 s�1

ocean tracers age = 0 age = 0;salinity = 30 psu

age = 0;salinity = 20 psu

2 day dyepulsec

2 day dye pulsec;salinity increasestemporarily(28 > 32 > 28 psu)over same 2 days

aSection 4.2 and Figure 9.bSection 4.3 and Figure 10.cAlso repeated with 8 day pulses, section 4.5.


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and I) models the propagation of individual, event-scalepulses of ocean water into the estuary.[15] The tidal, river, and ocean water property forcings

applied in each run are specified in Tables 1 and 2. Thetide in the model was generated by imposing a sea leveltime series on the open ocean boundary (Figure 1). Oceanwater properties, when included, were applied on theopen boundary as well. In runs where river input wasapplied, it was divided among the bay’s three largestrivers according to watershed area: 47% in the NorthRiver, 35% in the Willapa, and 18% in the Naselle(Figure 1). In several runs where river flow was notapplied (A, B, D, E, H: Tables 1 and 2), we eliminatedall density effects, i.e., all density variations in both spaceand 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 hadno effect on the circulation, acting like dye rather thanlike salinity. In addition to ‘‘dye’’ and ‘‘salt,’’ a third typeof tracer, ‘‘age,’’ a nonconservative, passive tracer repre-senting average time spent in the estuary, was included insome runs, as explained in detail in section 4.[16] The tide-only, density-off runs (A, B, D, E, and H)

were spun up from rest over 4 tidal cycles (2.1 days). This isa short adjustment period, but sufficient that when a pureM2 tide was applied, the variation in tidal currents amongsubsequent cycles (which in a fully spun-up model wouldbe zero) was only 0.02%. Runs that included density (C, F,G, and I) were spun up for 100 days, approximately twicethe longest residence times we calculate in section 4, withinitial tracer fields held constant before tracers were releasedand allowed to evolve. These spin-up periods are notincluded in the results below: in all cases the ‘‘initial’’ stateof a run refers to the end of the spin-up period.

3. Tidal Exchange

[17] We begin by evaluating the simplest dynamicalsituation, in which the oceanic tide is the only forcing:river input is zero, as if in late summer, and no densityvariations in either space or time are included. We willbegin by verifying the overall strength of modeled tidalcurrents against data (section 3.1), and then move on to themuch weaker but dynamically more important subtidal (i.e.,tidal cycle average or ‘‘residual’’) circulation.

3.1. Validation of Tidal Currents

[18] For validation of the tidal model (runs A and B,Table 1), we aimed to reproduce the tides during particulartime periods in which velocity data were collected. Notuning of model parameters was done: the bottom roughnessz0, sometimes treated as a free parameter, was set to astandard value of 1 mm. NOAA tidal height observationsfrom Toke Point (station 9440910; Figure 1) were used asthe oceanic boundary condition with no adjustment, exceptfor the shift of time base necessary to match the observedtidal phase at Toke Point. This is a simple and convenientmethod, but a source of error: as this tidal signal propagatesinto the bay it distorts slightly and produces a tidal heightdiscrepancy at Toke Point of 5–10%.[19] Figure 2 shows the depth-averaged amplitude of the

M2 (semidiurnal) tidal constituent for the period 15 October

to 11 November 1998 at four locations, stations at whichthe Army Corps of Engineers Seattle District collectedacoustic Doppler current meter (ADCP) data during thisperiod (run A, Table 1). The M2 constituent was extractedfrom data and model time series using the harmonicanalysis package t_tide [Pawlowicz et al., 2002]. Modelvelocity amplitudes agree with observations to 3–20%, anerror 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 ofM2 currents at the three stations near the mouth: the realflow follows bends in the channel that the modeled flowdoes not, presumably because the model bathymetry issmoother and channel edges less abrupt than in the real bay.[20] 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 largerspatial scale, the model was run for the week ending on5 May 2000, a day on which small-boat transects with a300 kHz ADCP were repeated over a cross section of themain channel for a full flood tide (run B, Table 1).Instantaneous transport through this cross section(expressed as mean velocity) is shown in Figure 2 for boththe observed and modeled flow: each black dot representsan ADCP transect. Instantaneous mean velocities in modeland data agree within 10% for each of the ten transects. Thissuggests that local biases like the directional errors near themouth described above (Figure 2) are indeed local, and thatthe total tidal transport in the model is well represented.

3.2. Validation of Tidal Dispersion: The HorizontalDiffusivity K

[21] The tidal validation above mainly tests the model’sreproduction of the oscillatory part of the tide, which, al-though the largest velocity signal in the estuary (�1 m s�1,Figure 2) by definition does not contribute to net disper-sion or exchange: it simply advects water back and forth.The ‘‘rectified’’ or ‘‘residual’’ tidal currents actuallyresponsible for dispersion in estuaries tend to be only�10% of the tidal amplitude [Zimmerman, 1986], com-parable to model data error in our case as in many cases.Horizontal tidal dispersion in the model must thus beevaluated by other means.[22] Banas et al. [2004] calculated a ‘‘total effective

horizontal diffusivity’’ (K) from salinity time series at eachof four monitoring stations along the main channel and usedthe resulting along-channel profile to characterize the sub-tidal circulation. In this section we will use the sameparameter and compare results directly with those fromobservations. This is a stringent test of a tidal model, onenot often performed for lack of a method for estimating Kfrom data.[23] A brief explanation of K in terms of the estuarine salt

budget follows: A more detailed derivation is given byBanas et al. [2004]. In general, we can express the conser-vation of salt (or some other tracer) on subtidal timescales as

@

@t

Z x

�1ha�sidx ¼ hausi ð1Þ

where x is distance along the main axis of the estuary (x =�1 denotes ‘‘far upstream’’), t time, a cross-sectional area,u velocity perpendicular to the cross section, and s salinity.


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Triangular brackets denote the average over a tidal cycle,and an overbar denotes the cross-sectional average: for anyvariable v, �v a�1

Rv da. Equation (1) simply states that

the rate of change of salt storage upstream of x equals thetotal salt flux across a section at x. The salt flux hausicontains 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 commonapproach [e.g., Ridderinkhof and Zimmerman, 1992;Monismith et al., 2002; Austin, 2004] is to assume thatthese processes combine to form a pattern of advection toocomplicated to describe as anything but diffusion.Following Banas et al. [2004], we can write this as

@

@t

Z x

�1a�sh idxþ Q �sh i ¼ ah iK @�s

@x

� �ð2Þ

where Q is river flow (Q h�si is the seaward flushing of saltby the mean flow) and K is a horizontal diffusivityparameterizing all processes moving salt upstream. If weconfine our attention to low-river-flow conditions, inwhich Q � 0, we can rearrange (2) into a formula forcalculating K from the salinity field:

K ¼ ah i�1 @�s

@x

� ��1 @

@t

Z x

�1a�sh idx ð3Þ

We use salinity here only because it is the most commontracer used to indicate seawater content; to the extent thatK really does represent a large-scale diffusive process, anyconserved tracer can be substituted.[24] To calculate K from the tidal model, we constructed

the following idealized scenario (run D, Table 1). Weimpose a simple, uniform semidiurnal signal whose ampli-tude (1.2 m) was chosen to match the long-term averagetidal height variance at Toke Point. An initial field of ‘‘dye’’with a uniform gradient in the along-channel direction isreleased into the spun-up tidal circulation (Figure 3a). Theinitial dye concentration is high in the ocean and decreasesup estuary as salinity would, although this dye has nodensity and thus does not affect the circulation. The rear-rangement of this dye field by a single tidal cycle is shownin Figures 3a–3c, and its state after 12 cycles (6.2 days) isshown in Figure 3d. The large tongue of high dyeconcentration in Figure 3d indicates water that has enteredthe domain across the open ocean boundary, where aconstant concentration for incoming water was imposed.[25] An average value of K was calculated using (3) for

this tracer field over the 12 tidal cycles shown, at a series ofcross sections along the longest axis of the bay (Figure 3e)from the mouth to Shoalwater Bay in the south. Empiricalvalues of K are given for comparison at four locations.Three are values calculated by Banas et al. [2004] fromlow-river-flow-period salinity data (Figure 3f gives an

Figure 2. Results from runs A and B (Table 1 and section 3.1) compared with observations. Depth-averaged M2 tidal velocities from model and data are shown at four locations (three near the mouth andone near 46�300). Modeled and observed tidal height and the north-south component of instantaneous,cross-sectional mean tidal velocity are shown for one flood tide on 5 May 2000.


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example). The fourth value, previously unreported, wascalculated from a 4 week (August–September 2002)deployment of a YSI temperature-salinity sensor at theedge of the main channel in Shoalwater Bay. The rate ofchange of salt storage at this station was calculated from

the trend in salinity rather than event-scale fluctuations asin Figure 3f but otherwise followed the method describedby Banas et al. [2004].[26] The four empirical K values vary by more than

an order of magnitude from mouth to head. Crucially,

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 fromBanas et al. [2004]).


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the model captures both high and low values accurately(Figure 3e). This spatial gradient in diffusivity can be seenqualitatively in Figures 3a–3d: the dye field in thesouthern part of the bay (low K) changes only gradually,whereas a single tidal cycle (Figures 3a and 3c) is enoughto dramatically displace, stretch, and fold the dye fieldnear the mouth (high K). Note that the only station wheremodel and data values of K differ significantly, Oyster-ville, lies at the junction of two channels (Figure 1), wherethe 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 thisstation.

3.3. In Pursuit of Salt Flux Mechanisms

[27] A map of K as in Figure 3e is a convenient summaryof the subtidal circulation, but obscures the actual mecha-nisms that drive it. What is different about the tidal flow orthe bathymetry in the seaward and landward reaches of theestuary, that the net dispersion rate should vary so muchbetween them? What aspect of the bathymetry (channelwidth, channel curvature, channel junctions, or bank-to-channel depth variations) is most responsible for generating

the tidal asymmetries that in turn generate K? A standardapproach to these questions [Fischer, 1976; Hughes andRattray, 1980; Lewis and Lewis, 1983] is a Reynolds-likedecomposition of the total salt flux hausi (see equation (1)).That is, we can decompose s into mean and varying parts inspace and time:

s ¼ �sh i þ s0h i þ �s1 þ s01 ð4Þ

where triangular brackets and an overbar denote tidal cycleand cross-sectional means as before, a subscript 1 denotesvariations over a tidal cycle, and a tick denotes variationsover a cross section. We can also, following Dronkers andvan de Kreeke [1986], write the mass flux through a smallcross-sectional area element as

dq u da ¼ dqh i þ dq1 ð5Þ

so that the total tidally averaged salt flux through a crosssection becomes

aush i ¼Za

s dqh i ¼ �sh i qh i þZ

s0h i dqh i þ �s1q1h i þZ

s01dq1� �

ð6Þ

If the volume of the estuary is constant and river input Q = 0,hqi = 0, leaving three terms on the right-hand side. Theterm

Rhs0i hdqi (hereinafter referred to as term 1) represents

steady shear dispersion by the tidal residual eddy fieldhu0i: Zimmerman [1986] refers to this mechanism as‘‘Lagrangian chaos,’’ since the velocity field itself issteady and deterministic but randomized by bathymetry tothe point where tracers and particles disperse chaotically.The term h�s1 q1i (hereinafter referred to as term 2), whichdescribes 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 arises whensalt is diverted into side embayments and reenters the mainflow at a different point in the tidal cycle. The cross-termRhs01 dq1i (hereinafter referred to as term 3) involves

variations in both space and time: unsteady sheardispersion, or spatially nonuniform trapping.[28] The relative strength of terms 1, 2, and 3 is shown in

Figure 4 for each of the cross sections where K wascalculated in the previous section. These salt fluxes havebeen converted into diffusivity units (m2 s�1) by combiningequations (3) and (6), i.e., by normalizing by haih@�s/@xi.Total K (dashed line) decreases smoothly from the mouthupstream, but the three constituent diffusivities (bars) riseand fall in a much more complicated pattern. Each of terms1, 2, and 3 is in some location the dominant term; in fact,the cross term 3 (gray bars), the term hardest to interpret, isdominant over much of the upper estuary. This implies thatmuch of the net exchange of ocean water over the sectionsexamined is accomplished not by general, easily parameter-izable features of the flow but by processes localized in bothspace and time. Furthermore, all three terms are in somelocation negative, which could mean that a decompositionbased on cross-sectional averages is not physically mean-ingful in bathymetry this complicated. Alternatively, thesenegative constituent diffusivities could mean that certaintidal residual processes in the bay are actually antidiffusive

Figure 4. Contributions to the total diffusivity K asso-ciated with terms 1 (steady shear), 2 (trapping), and 3 (across term) in the salt flux decomposition (6). Fluxes arecalculated from model run D at the 12 cross sectionsshown in the inset and Figure 3e.


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in a coarse-grained sense, tending to sharpen fronts ratherthan smooth them away. Note that the initially smooth tracerfield shown in Figure 3a does, in fact, become moreinhomogeneous as tidal stirring acts on it (Figure 3d).[29] It appears, then, first, that the cross-sectionally

integrated diffusivities shown in Figures 3e and 4 areaverages over a great deal of small-scale complexity andvariety; and second, that the traditional method of decom-position, based on fixed location tidal cycle and cross-sectional averages, is not an effective way to tease apartthis complexity. In the next section we will describe analternative, flow-following approach that gives clearerresults.

3.4. A Lagrangian Map of Exchange

[30] We begin by releasing an imaginary particle in thecenter of each model grid cell, and track the trajectory ofthese particles as the tidal flow calculated in run D(Figure 3) advects them seaward and landward. Three-dimensional particle tracking is not straightforward insigma coordinates and steep bathymetry; to keep theproblem tractable we follow particles in two dimensionsonly, using depth average currents. This simplificationwould not be dynamically sensible in many scenarios,but in our tide-only case it appears to be fair: when terms1 and 2 in equation (6) are decomposed into vertical meanand varying parts (not shown), horizontal shear dispersiondominates over vertical dispersion by an order of magni-tude. Nevertheless, the two-dimensional particle-trackingapproximation introduces 10–20% volume conservationerrors, which should be taken as a lower limit on theuncertainty in this analysis.[31] The only information we retain from these particle

tracks is the index of the grid cell where each particle islaunched and the cell that contains it one tidal cycle later.The rearrangement done by the subtidal circulation is thusencapsulated in a single two-column table, a function thatmaps the list of grid cell indices back onto itself: in thelanguage of nonlinear dynamics, a ‘‘return map’’ orPoincare map [Beerens et al., 1995]. One application ofthis Lagrangian return map is shown in Figure 5. Here theonly particles drawn are those which cross over a givensection (green lines) within one tidal cycle. Lighter shadesindicate initial position and darker points final positionone tidal cycle later; particles are colored red if they arecrossing up estuary and blue if they are crossing sea-ward, as illustrated in the schematic in Figure 5a. Thisillustration of the crossing regions for each section ofinterest (Figures 5b–5g) highlights not only the totalamount of exchange at each section (which could alsobe inferred from the diffusivity K) but specifically whatwater is being exchanged for what other water in atypical tidal cycle. The details of these results dependon the tidal phase at which particles are released, but thequalitative patterns discussed below are not sensitive toit. Results are shown in Figure 5 for particles released athigh slack, the release time that minimized volumeconservation errors.[32] 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 thatdoes not return on the following tide. The crossing regions

shown in Figure 5b–5g correspond to tidal exchange ratiosof 0.15–0.3, consistent with the range of typical valuesreported by Dyer [1973] and the estimate for Willapa fromdata of Banas et al. [2004].[33] Coherent structures in the net circulation emerge.

Exchange near the mouth (Figures 5b and 5c) consistsmainly of a simple lateral exchange, into the estuary overthe southern part of the mouth and seaward in the deepermain channel to the north. This pattern, in which thesubtidal flow tends to the right both entering and leavingthe estuary is consistent with the expectation for rotation-driven exchange [Valle-Levinson et al., 2003], but it persistseven when the Coriolis acceleration is turned off in themodel (Figure 5h). An alternative explanation may be tidalStokes drift, which tends to produce an up-estuary Lagrang-ian transport through the shallower part of a cross section,balanced by a seaward return flow in the deeper mainchannel [Li and O’Donnell, 1997], as seen here. The lateralexchange pattern persists for 15–20 km upstream from themouth (Figures 5d–5f), but its relationship with the depthof the underlying bathymetry reverses: more than a few kmfrom the mouth (Figures 5d and 5e) flow over the banks isprimarily seaward and flow in the main channel up estuary(bathymetry shown in Figure 1). Analytical studies andobservations of lateral transport gradients in other estuarieshave found both the pattern at Willapa’s mouth (channelflow seaward [Friedrichs and Hamrick, 1996; Li andO’Donnell, 1997]), the pattern in Willapa’s interior (channelflow landward [Valle-Levinson and O’Donnell, 1996;Winant and Gutierrez de Velasco, 2003]), and indeed areversal of the pattern along the estuary [Li, 1996], as weobserve.[34] In the interior of the estuary (Figures 5d–5g) there

are places where the Lagrangian residual circulation appearsmore diffusive than advective, in the sense that holes anddiscontinuities appear in the crossing regions, and in placesthe incoming and outgoing water masses appear to movethrough each other rather than past each other (Figure 5g).Such complexity is especially visible in the exchangesaround channel junctions highlighted in Figure 6. Thesejunction exchanges combine coherent, advective motion(water enters Willapa Channel and then leaves again, ason a counterclockwise conveyor belt, Figure 6b) withdiscontinuities and dispersion on scales from the gridresolution up to 10 km.[35] Thus in contrast to the K formulation (section 3.2)

in which all exchange flows are treated mathematically asdiffusion, and also in contrast to the Eulerian decomposi-tion (section 3.3) in which only simple advective patternsare interpretable, this crossing-region method lets us dis-tinguish in detail between advection-like and diffusion-likeprocesses. It also lets us visualize hybrid, unnameabletransport processes like those around channel junctions.Note that these hybrid junction processes may be impor-tant to exchange overall: the highest diffusivities in thesouthern half of the estuary (Figure 3e) lie at the junctionat the mouth of Shoalwater Bay (Figure 6a).

3.5. Model Visualization Using the LagrangianReturn Map

[36] As noted above, the Lagrangian return map distillsthe net transport accomplished by each tidal cycle into an


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especially concise form. To the extent that one tidal cycleis just like the next (this is definitionally true in theidealized run D), applying the return map repeatedly cangenerate very long subtidal particle trajectories virtuallyinstantaneously compared with the full GETM circulation

model. An interactive demonstration of this method, inwhich a user chooses a launch location for up to a fewthousand drifters and then can watch their flushing anddispersion over 40 tidal cycles, is online at http://coast.ocean.washington.edu/willapa/tidemodel.html. We imposed

Figure 5. (a) Schematic of the method for constructing Lagrangian ‘‘crossing regions’’ from numericalparticle tracking as described in section 3.3. Initial particle positions are shown in light shades, andpositions one tidal cycle later are shown as dark points; for a given cross section (green line), particlescrossing seaward are shown in blue, and particles crossing landward are in red; particles that begin andend on the same side of the cross section are not plotted. (b–g) Crossing regions for six sections underidealized tidal flow (run D, Table 1). (h) Crossing regions for the section shown in Figure 5c when run Dis repeated with the Coriolis acceleration turned off.


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a small random jitter, equivalent to a diffusivity of (gridspacing)2 (tidal period)�1 = 1.5 m2 s�1, to keep particlesfrom being trapped unrealistically at fixed points in thereturn map.[37] Particle trajectories more than a few tidal cycles

long obtained by this method are merely representative,not precise, since the approximations inherent in thereturn map (rounding particle positions to the nearestgrid cell; ignoring vertical shear, unless particles aretracked in three dimensions) quickly add. Nevertheless,this method does illustrate variations in dispersion rateand transport pathways with more immediacy than a mapof diffusivities like Figure 3e. It is also one of the fewways that nonspecialists and casual users can directly

investigate and challenge the details of a complex circu-lation model.

4. Density-Driven Exchange

4.1. Baroclinic Validation: The Salinity Field

[38] So far our analysis has considered low-river-flow,tide-dominated conditions only. In this section we add riversand ocean density to the model, and evaluate their effect onflushing rate relative to the tidal baseline. We must begin,however, by verifying the model’s baroclinic behavior ingeneral: we will do this by validating the model salinityfield against observations across a range of river and oceanforcing.

Figure 6. Crossing regions, like those in Figures 5b–5g, 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 river flow. Results are given for both model run C and observations July1999 to June 2000 (see Table 1).


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[39] For validation, the model was run with time-varyingsea level, river flow, and ocean-salinity forcing for the11-month period July 1999 to June 2000 (run C, Table 1).During this period, as described by Banas et al. [2004], pointsalinity time series were collected at four main-channelstations: W3 near the mouth at 8 m depth, and Bay Center,Oysterville, and Naselle farther up estuary at 1 m depth(Figure 7, insets). For model forcing, sea level from TokePoint was imposed at the ocean boundary as in runs A and B(section 3.1); both tidal and subtidal components wereincluded. River input into the North, Willapa, and NaselleRivers was calculated from USGS gauge data as describedby Banas et al. [2004]. Finally, for lack of a better option forrepresenting changing ocean salinity, tidally averaged salin-ity from W3 was imposed at the ocean boundary as well.[40] This last approximation is the strongest limit on

our ability to simulate particular time periods. Over this11 month run, the distortion as the imposed W3 salinitytime series propagates from the open boundary back to W3is 27% of the signal. As a result, it does not seemmeaningful to compare model results with data instant byinstant, but rather to treat model and data as if theyrepresented similar but distinct time periods. A stricterbaroclinic model validation must wait until better offshoredata or a high-resolution, large-scale coastal model isavailable to provide better salinity boundary conditions.[41] The most integrative test of the model’s baroclinic

circulation is the length of the salt intrusion as a function ofriver flow. A relative measure of salt intrusion length,salinity at Bay Center, Oysterville, and Naselle as a fractionof salinity at the mouth (station W3), is shown for bothobservations and model results in Figure 7. This relativesalinity is displayed as a function of bay total river flow Q:the gap around Q = 30 m3 s�1 corresponds to a 6 weekperiod in autumn 1999 when W3 salinity was unavailable (alinear trend was substituted as the model boundary condi-tion during this period). At Oysterville and Naselle, theslope between relative salinity and river flow is indistin-guishable between model and data at the 95% confidencelevel. Furthermore, the model reproduces, qualitatively, thescatter (the event-to-event variance) in relative salinity at agiven river flow level. At Bay Center, however, thedependence of salinity on river flow in the model is tooweak. This is a sign either that our simplified treatment ofthe coastal ocean distorts the salinity field near the mouth,or that the model is not correctly distributing freshwaterfrom the North and Willapa Rivers around the StanleyChannel–Willapa Channel junction (Figure 1). In eithercase, the bias in the salinity field appears to be a localeffect that does not reach as far south as Oysterville.[42] Furthermore, the model appears to match, qualita-

tively, observations of stratification at Bay Center from amore recent (previously unreported) experiment. TwoSeaBird conductivity-temperature (CT) sensors weredeployed at Bay Center from Oct 2003 to Apr 2004,one at 1 m depth as above, and the other 0.5 m off thebottom. W3 was not occupied during this deployment, andso for lack of a salinity boundary condition we cannotsimulate 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 2003–2004observations (Figures 8a and 8b): at river flow levels

above the long-term median (�100 m3 s�1), the watercolumn stratifies 1–5 psu on each ebb and then destratifiesagain on flood. Similar patterns of strain-induced periodicstratification (SIPS) have been frequently observed inmacrotidal estuaries [Simpson et al., 1990; Stacey et al.,2001], although they have not been documented previouslyin Willapa. Notably for model validation, in both modeland data, the only flow levels at which the flood tide isunable to completely rehomogenize the water column formore than 1–2 cycles at a time are those close to theannual maximum, >1000 m3 s�1 (in observations, earlyFebruary 2004; in model, late December 1999; arrows inFigures 8a and 8b). This model data comparison is shownmore systematically in Figures 8c and 8d. At low-to-moderate river flows, the maximum stratification attainedduring each tidal cycle ranges from 0 to 3 psu in bothdata and model, as the bay responds to various combina-tions of river and ocean water property forcing. At highflows the model underestimates the highest stratificationsat Bay Center, but reproduces the fact that during at leastsome of these high-flow events, tidal cycle minimumstratification does not drop below the observational limitof 0.05 psu, as it does during almost all other conditions(Figures 8c and 8d).[43] Until better boundary condition data are available to

make the baroclinic validation more precise, it would beovertrusting to use the model to map Willapa’s river-drivenexchange circulation in as much detail as we mapped thetidal circulation in section 3.4. Still, we have found that themodeled vertical circulation at midestuary (as indexed byBay Center stratification) is, at least, not grossly wrong and,in fact, that the slope of relative salinity at Oysterville andNaselle versus river flow, i.e., the distance that waterentering the mouth reaches into the southern estuary undercomplex, changing forcing, matches the slope from obser-vations within confidence limits (Figure 7). In the nextsection we describe a numerical method by which river- andocean-density-forced changes in circulation can be translatedinto summary maps of bay-wide residence time.

4.2. A Method for Mapping Residence Time

[44] The method we use to determine residence time,the nonconservative ‘‘age’’ tracer mentioned in section 2,is similar to the ventilation time method (time since awater parcel has been exposed to the atmosphere) used inglobal circulation modeling [Thiele and Sarmiento, 1990;England, 1995]. For a rigorous, general derivation of thisand a variety of other age methods in ocean modeling, seethe thorough review by Deleersnijder et al. [2001] [seealso Monsen et al., 2002, and references therein]. We willexplain this age tracer more briefly here.[45] Consider a passive tracer whose concentration is

initially zero everywhere and that is always zero in theestuary’s river and ocean source waters but that is made togrow at a constant, specified rate in the estuary’s interior.Between the open ocean boundary and the mouth of theestuary, the tracer simply advects and mixes without grow-ing. As time passes, if forcing is held constant, the tracerconcentration will approach an equilibrium in which thesteady creation of tracer within the estuary is balanced byflushing out to sea (which replaces high-concentrationestuarine water with zero-concentration ocean water). At


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this equilibrium, the concentration in each grid cell isdirectly proportional to the average length of time the waterin that grid cell has spent growing, i.e., spent within theestuary. Thus we can think of this tracer as average waterage.[46] As an illustration, consider the case of a riverless,

linear channel: in the cross-sectional average, the conserva-tion equation for age can be written

@c

@t¼ 1

a

@

@xaK

@c

@x

� �þ g ð7Þ

where c is age tracer concentration, a and K are, as before,cross-sectional area and a total effective diffusivity, and g isthe imposed tracer growth rate. We can pick the units of c sothat g = 1. For given profiles of a(x) and K(x), Equation (7)has a steady solution, given by a double integration of

� 1

a

@

@xaK

@c

@x

� �¼ 1 ð8Þ

with the boundary conditions c = 0 at the ocean and (aK)(@c/@x) = 0 (no tracer flux) far upstream. Equation (8)

Figure 8. Time series of stratification at Bay Center and bay total river flow from (a) observations in2003–2004 (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 Figures 8a and 8b,displayed as a function of river flow.


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describes a state in which the along-channel tracer gradient@c/@x has adjusted so that the diffusive loss of tracer (lefthand side) balances the steady, imposed tracer growth rate(right hand side) at every location. If a and K decreaseexponentially toward the head with e-folding lengths la, lK,then at equilibrium, age increases approximately exponen-tially from mouth to head: c(x) / la lK K(x)�1. In this simplecase, as one might expect, age c (an estuarine residence time)is inversely proportional to K (a measure of the estuarineexchange rate). Equation (8) implies that this is likely to betrue in general, at least in a scaling sense, a fact we can use(section 4.4) to check the reasonableness of the residencetimes for Willapa that the age tracer method yields.[47] We implemented this age tracer in GETM by first

removing references to temperature in the equation-of-statecalculation (so that it can be used as a generic passivetracer) and second adding code to the temperature-advectionsubroutine to increment the tracer one unit each time step atgrid points within the estuary. This tracer thus is transportedand mixed using GETM’s full advection and turbulenceschemes, without the need for postprocessing. Cokelet et al.[1991] used a similar method to calculate residence time ina simplified box model of Puget Sound. We have not seenthis method used in a high-resolution estuary model,although it requires only a few lines of code and presum-ably can be added straightforwardly to any primitiveequation model. Note that unlike the simple illustrativecase described in (7) and (8), the full age tracer calculationyields three-dimensional maps of residence time at thesame resolution as the model grid.[48] Tidally averaged maps of age for three model sce-

narios representing an idealized seasonal cycle (runs E–G,Table 2) are shown in Figure 9. The first of these (run E)contains a tidal circulation only, without density effects orriver input, representing the sustained upwelling-favorable,low-river-flow conditions typical in late summer [Hickey etal., 2002; Banas et al., 2004]. This run is identical to run D,discussed in sections 3.2–3.4, except that the age tracerhas been added. The next run (F) adds steady river input of100 m3 s�1 and a constant ocean salinity of 30 psu: theseforcings 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 andocean salinity 20 psu, representing sustained southwardwind, foul weather, downwelling-favorable conditions andintrusion of the Columbia River Plume [Hickey and Banas,2003]. In the following subsections we will examine thetidal residence time pattern (run E) and then compare withthe river-forced cases (F and G).

4.3. Age Distribution Under Tidal Forcing Alone(Run E)

[49] The spatial distribution of ages in Figure 9a encap-sulates much of the information in the dye intrusion(Figure 3d), diffusivity (Figure 3e), and crossing region(Figure 5) views of the tidal circulation discussed above. Atongue of low-age water entering through the southern sideof the mouth corresponds to high, oceanic dye concen-trations in Figure 3d and the up-estuary-directed half ofthe lateral exchange flow shown in Figures 5b–5f. Thecompensating exit flow on the north side of the mouth(Figures 3d, 5b, and 5c) is also visible, above the 3 day

age contour. Where the Lagrangian lateral exchange flowbreaks up in the middle of the estuary (Figure 5g), K has alocal minimum (Figure 3e), and there is a strong gradient,almost a front, in age (Figure 9a). At the other K minimaas well (the tips of the Willapa Channel and ShoalwaterBay) the age gradient is intensified, and wherever K islarge; at the mouth, for example, the age gradient is flatter.The age map thus contains two scales of circulationinformation. First, the age itself indicates the flushing-and-resupply rate for a given region on the whole-estuaryscale: Shoalwater Bay, for example, is �50 days ‘‘distant’’from the mouth in a circulational sense (Figure 9a).Second, while the gradient in age suggests dispersivenessor quiescence on the local scale.[50] There is an intense transverse gradient in age (up to

9 days km�1) across the shallow banks at midestuarywhere the sections in Figures 9d–9f are taken. Thesimplest interpretation is that this gradient, just like thestrong transverse temperature and salinity variations oftenseen in the same location (up to 5�C, 4 psu [Hickey andBanas, 2003]), is caused by differential tidal advection[Huzzey and Brubaker, 1988; O’Donnell, 1993; Hickeyand Banas, 2003]. In this process the channel-to-bankdifference in the strength of flood tide currents strainstracer fields, creating lateral gradients out of along-channelgradients. In other words, just as in summer water onWillapa’s shallowest banks tends to be noticeably warmerand fresher, more like water 5–10 km up estuary, thanwater just 1–2 km away in the main channel, so may thatbank water be 2–3 weeks older, simply through rearrange-ment by the flood tide.

4.4. Age Distribution Under River Forcing(Runs F and G)

[51] As river input increases (Figures 9b and 9c), along-estuary variation in age decreases. (Note that the midestuarybank-to-channel gradient discussed above decreases propor-tionally, consistent with the differential-advection mecha-nism). The northern channel, where 81% of river flowenters (section 2.2), is especially homogenized and quicklyflushed: there is now a source of zero-age water at bothends, and even at moderate river flows (Figure 9b), averageage is <6 days everywhere between the mouth and theWillapa and North rivers. Shoalwater Bay, in contrast, whichhas no river input of its own, is largely bypassed by the river-driven increase in flushing rate. Even at storm-level riverflows (Figure 9c), water at the southern end of ShoalwaterBay is 3 weeks older than water in the central bay.[52] In the absence of river flow (Figure 9d) the age

distribution is vertically uniform, but at high flows(Figure 9f ) water at depth in the main channel is 1–2 daysyounger than surface water. This is a sign that a verticalgravitational circulation has developed, following the clas-sical pattern (inward at depth, outward at the surface) firstdescribed by Pritchard [1956]. Notably, the exchangepattern at flow levels near the long-term mean (Figures 9band 9e) more closely resembles the tide-driven pattern(Figures 9a and 9d) than it does this high-flow, verticalexchange pattern (Figures 9c and 9f). This is consistentwith the results of Banas et al. [2004], who found that atmoderate flows (100 m3 s�1) the total effective diffusiv-ity at Bay Center increases only 40% above its low-river-


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flow, tidal baseline value, while at very high flows(1000 m3 s�1) the total diffusivity is 3.1 times thebaseline value. For comparison, in these model resultsaverage age in the central estuary (from Bay Center toOysterville: see Figure 3e) is 30% higher at moderateflows (100 m3 s�1) than under tidal forcing only and2.4 times higher at high flows (1000 m3 s�1) than undertidal forcing (Figures 9a–9c).

4.5. Intrusion of Ocean Water on the Event Scale(Runs H and I)

[53] The maps of average age discussed above suggestthat at river flow levels from the long-term median down

(i.e., most of spring and summer) river- and ocean-forceddensity effects are secondary to the tides in setting Will-apa Bay’s residence time. It is possible, however, that onthe event (2–10 day) scale, nontidal influences haveintermittent but sometimes much greater dynamical impor-tance. Studies of estuarine adjustment [Kranenburg, 1986;MacCready, 1999] have shown that an estuary may respondto a pulse of river input very differently than to sustainedriver input. In addition, past work on ocean-estuary ex-change on upwelling coasts [Duxbury, 1979; Monteiro andLargier, 1999; Hickey et al., 2002] suggests that the time-dependent response to changing ocean salinity may be animportant modulation of the gravitational circulation.

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). River flow Q and ocean salinity soc are givenfor each case where they are applied. (a–c) Depth-averaged age; arrows mark river inputs. (d–f) Verticalsections of age at midestuary.


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[54] Nevertheless, results from a final set of model runs(H and I, Table 2) suggest that the tide-driven residualcirculation by itself is a good first-order approximation for awide range of spring and summer conditions in Willapa,even on event timescales. Results from these runs, whichsimulate the propagation of an individual 2 day pulse ofocean water into the bay, are shown in Figure 10. The firstof these runs (H) is identical to the idealized tide run Ddescribed in section 3, except that now the initial dyeconcentration is zero everywhere, and after the tidal flowis spun up dye is injected along the open ocean boundaryfor 2 days and then shut off. The resulting dye patch, whichmight be thought of as a marker for a mass of nutrient- andbiomass-rich or nutrient- and biomass-poor water producedby an individual upwelling or downwelling event, propa-gates into the estuary as shown in Figure 10 (top). In thesecond of these runs (I, Figure 10, bottom), a constantbackground river flow of 100 m3 s�1 has been added as inrun F (Figure 9b), and in addition, ocean salinity smoothlyincreases from 28 to 32 psu and back again during the2 days that dye is injected. This combination of forcingsrepresents a rapid spring upwelling event [Hickey andBanas, 2003]. With river and ocean density forcing added,28% more dye (i.e., new ocean water) enters initially(Figure 10, bottom, day 4) and the half-life of the dye inthe estuary is 10% longer. These are measurable effects, butstill only fractional changes. When the duration of the dyerelease/upwelling event is extended to 8 days (not shown),

the river- and ocean-forced modulation of the intrusion iseven weaker: 10% more dye enters, and has a half-life only5% longer, than in the tide-only case.

5. Discussion

[55] In the pulse propagation experiment just described,not only the horizontal spreading of an isolated oceanwater mass, but also its steady migration up estuary, isprimarily controlled by tidal stirring. This may seemcounterintuitive, given that tidal stirring is so often treatedmathematically, as the label ‘‘stirring’’ implies, as a simpleFickian diffusion process [Hansen and Rattray, 1965;MacCready, 2004] (section 3.2). Indeed, throughout ouranalysis we have been, inevitably, shuttling between thelanguage of ‘‘diffusion,’’ ‘‘dispersion,’’ and ‘‘stirring’’ onthe one hand and ‘‘advection,’’ ‘‘transport pathways,’’ and‘‘conveyor belts’’ on the other. In this section, we will tryto reconcile these ideas, and show what we can infer aboutthe actual mechanisms of tidal exchange in Willapa whenall the components of our analysis are taken together.

5.1. Tidal Flushing and Tidal Retention

[56] A schematic of tidal exchange is shown in Figure 11.Arrows indicate the Lagrangian transport pathways mostvisible in the crossing-region analysis from section 3.4.The most coherent feature is the strong lateral exchangeflow in the seaward reach. The inflow curves around the

Figure 10. Evolution of pulses of ocean water marked by a 2 day dye release in the model. Each frameshows depth-averaged dye concentration relative to the initial source concentration. Two scenarios areshown: propagation under idealized tides only, without density effects (run H), and propagation whenriver flow of 100 m3 s�1 is added and ocean salinity temporarily increases from 28 to 32 psu during thedye release (run I; see Table 2). The first case represents typical late summer conditions, and the secondrepresents a typical spring upwelling event.


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southern and western sides of the entrance, crossing in andout of the curving main channel as shown (Figures 5b–5gand 11). The compensating outflow, through the deepchannel to the north of the entrance, draws water from abroad area on both sides of the Stanley-Willapa junction,although some water from the central estuary dispersesinto the Willapa Channel (Figure 6b) before entering thisoutflow, in a tangled, ‘‘diffusive’’ process. A similarinterleaving of incoming and outgoing water massesoccurs at the junction at the Shoalwater Bay entrance(Figures 6a and 11).[57] Thus as we commented above (section 3.4), the

along-channel profile of effective diffusivity K (Figure 3e)describes not just diffusion but also coherent, advective

flow structures. Furthermore, on large scales these flowssometimes actually appear antidiffusive, creating fronts anddiscontinuities instead of smoothing them away (Figure 3d).The key to this phenomenon, in mathematical terms, is thata gradient in diffusivity acts on a tracer as a kind of‘‘psuedoadvection.’’ That is, the general tracer diffusionequation

@s

@t¼ @

@xK@s

@x

� �ð9Þ

(where s is tracer concentration) can be rewritten

@s

@tþ � dK

dx

� �@s

@x

� �¼ K

@2s

@x2ð10Þ

as if K were locally constant and a background velocity�dK/dx were advecting the tracer in the direction ofdecreasing diffusivity. This idea helps explain why anyparticular water mass tends to collect or stop advancingwhere K is low: southern Shoalwater Bay, the head of thenorthern channel (Figures 9a and 10, top), and themidestuary minimum in K (Figures 3e and 5g and dashedline in Figure 11). A local minimum in K, i.e., a regionwhere the interaction of tides and bathymetry happens togenerate few residual currents, acts as a semipermeablebarrier to tidal exchange. A minimum in K is a gap insubtidal transport, a valley that can be crossed only slowly,and thus a place where relatively sharp breaks in waterproperties may form (Figures 3d and 9a).[58] The potential significance of such a ‘‘dispersion

gap’’ (and additional support for our model results) can beseen in a fact well known in the Willapa Bay oyster culturecommunity, although not well documented in the scientificliterature [Hedgpeth and Obreski, 1981; J. Ruesink andA. Trimble, personal communication, 2004]. Natural settle-ment of oyster larvae, which requires retention in the watercolumn for three weeks after spawning, occurs in thesouthern reaches of Willapa Bay but not in the northern.The dividing line lies north of the northern tip of LongIsland, close to where we find the K minimum and sharpgradient in age discussed above (Figure 11). Just as newocean water advances quickly up to this point in the bay butcontinues past it only weakly (Figure 3d), so can water upestuary of this point (carrying recently spawned oysterlarvae, say) be trapped behind it. The online drifter-disper-sion visualization described in section 3.5 shows thisdirectly. Our model thus reproduces a striking feature ofWillapa’s circulation: despite the fact that half the volume ofthe bay enters and leaves with every tide (section 1), atypical water parcel in the upper third of the estuary isretained for several weeks or even months.

5.2. Importance of Along-Channel Complexity

[59] In the Lagrangian view, the high diffusivities andrapid exchange seaward of the midestuary dispersion gapappear to be the result of a single, not especially complexlateral shear flow structure (Figures 5 and 11). The Eulerianview of the same area, however, is decidedly lacking insimple shear structures (Figure 4): in fact, near the mouth thesalt flux constituent that dominates in equation (6) is term 2,the only term that does not involve lateral shear at all.

Figure 11. Schematic of the tide-driven residual circula-tion based on Figures 3, 5, and 9a. Arrows indicateLagrangian transport pathways. A dashed line marksthe water-mass-isolating ‘‘dispersion gap’’ discussed insection 5.1.


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[60] Dronkers and van de Kreeke [1986] present aninterpretation of the salt flux constituents in equation (6)that helps explain this discrepancy. They refer to term 1 asthe ‘‘local salt flux,’’ since it involves only the structure of uand s within a cross section. They refer to terms 2 and 3together as the ‘‘nonlocal salt flux,’’ since the temporalvariations in these terms (s1, dq1) are really signs of spatialvariation outside the cross section being analyzed: i.e.,variations over a tidal excursion in the along-channeldirection. They then demonstrate that the nonlocal saltflux is approximately equal to the difference between thelocal flux through a stationary section and the local fluxthrough a section moving with the flow. In other words,where terms 2 and 3 are large (in Willapa Bay, nearlyeverywhere, Figure 4), it is definitional that the Euleriancirculation pattern does not reflect the underlying Lagrang-ian circulation, as we have found.[61] Furthermore, the fact that the nonlocal terms are

nonnegligible or dominant over most of the estuary(Figure 4) can be taken to mean that the Lagrangiantransport pathways we have described are driven at leastas much by along-channel (nonlocal) bathymetric complex-ity as by transverse complexity. It is not surprising, then,that as discussed in section 3.4, Willapa’s exchange circu-lation is not readily schematized as a lateral, channel flowversus bank flow pattern, despite the ubiquity of suchpatterns in the estuarine literature [Li and O’Donnell,1997]. There can be no typical lateral pattern if the localsalt flux (term 1) is weak and changeable; in Willapa term 1changes sign and weakens over the 20 km in from themouth (Figure 4), and indeed the landward and seawardhalves of the Lagrangian exchange flow migrate betweenbank and channel and dissipate over the same distance(Figures 5 and 11). If the circulation in estuaries likeWillapa is to be schematized, we propose that one startnot with lateral depth variation but along-estuary complex-ity: channel curvature, branching and junctions, and sets ofebb-dominant/flood-dominant channel pairs. To date, thesecommon features of coastal plain estuaries have receivedmore attention in the morphodynamics literature [e.g., vanVeen, 2002; Hibma et al., 2003] than in the hydrodynamicsliterature.

6. Conclusions

[62] This modeling study and a previous observationalanalysis [Banas et al., 2004] paint a consistent picture ofthe processes that control flushing in Willapa Bay. Underhigh-flow, winter conditions the density-driven and tide-driven circulations are both first-order contributors: neithercan be neglected (Figure 9c). During spring and summer,however, when river input is generally below the long-term average, tidal effects dominate, and baroclinic effectsare discernable (Figure 9e) but a second-order correction.This appears to be true on both seasonal timescales(Figure 9) and the event scale (Figure 10).[63] 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 river flowand drops in ocean salinity (a set of forcings that tends tooccur in concert on the Washington coast in winter and earlyspring [Hickey and Banas, 2003; Banas et al., 2004]).

Examining a range of such scenarios in more detail maybe important to understanding phenomena that unfold notover days or weeks but over individual tidal cycles: e.g.,patterns of larval settlement in the seaward reach of theestuary [Eggleston et al., 1998; Roegner et al., 2003].Before such processes near the mouth can be examinedreliably in detail, however, better offshore data is needed formodel forcing and validation.[64] 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 onlydynamically central but extremely intricate and variegated,diffusion-like in some places and conveyor-belt-like inothers (section 5). Our description of tidal dispersion inWillapa relies on two Lagrangian methods: the residualcirculation return map, built through particle tracking(section 3.4), and the steady state age distribution, calculatedusing a nonconservative tracer (section 4.2). The agedistribution, in particular, provides the most comprehensivepicture of the flushing of the estuary of all the methods usedin this study. It does not seem coincidental that it is also theonly method that does not impose an artificial geometry (apreferred ‘‘along-channel’’ direction, straight line crosssections) on the flow. We recommend particle and tracermethods along these lines, and attention to exactly thosechannel bends and flares and junctions that have beenhardest to treat analytically, in the analysis of flushing andresidence time in other complex systems.

[65] Acknowledgments. This work was supported by grants NA76RG0119 Project R/ES-9, NA76 RG0119 AM08 Project R/ES-33, NA16RG1044 Project R/ES-42, NA76 RG0485 Project R/R-1 and NA96OP0238 Project R/R-1 from the National Oceanic and AtmosphericAdministration to Washington Sea Grant Program, University of Wash-ington. N.S.B. was also supported in part by a National Defense Scienceand Engineering Graduate Fellowship. Sue Geier was responsible for initialdata processing. Thanks are given to Jan Newton, Judah Goldberg, andthe Washington Department of Ecology, who generously provided theBay Center stratification data. Thanks go to Brett Dumbauld and theWashington Department of Fish and Wildlife, who provided field supportand boat time, Jennifer Ruesink and Alan Trimble at the University ofWashington, who provided the Shoalwater Bay salinity time series, andAdele Militello and the U. S. Army Corps of Engineers Seattle District,who made their model grid available to us. Finally, many thanks aregiven to Karsten Bolding and Manuel Ruiz Villareal for their unstintinghelp setting up GETM on our computer and to Parker MacCready andRocky Geyer for their insights.

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��N. S. Banas and B. M. Hickey, School of Oceanography, Box

355351, University of Washington, Seattle, WA 98195, USA.([email protected])


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Mapping exchange and residence time in a model of Willapa Bay

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