effects of bathymetry, friction, and rotation on estuary...

NOVEMBER 2003 2375V A L L E - L E V I N S O N E T A L .

q 2003 American Meteorological Society

Effects of Bathymetry, Friction, and Rotation on Estuary–Ocean Exchange

ARNOLDO VALLE-LEVINSON, CRISTOBAL REYES,* AND ROSARIO SANAY

Center for Coastal Physical Oceanography, Ocean, Earth and Atmospheric Sciences Department, Old Dominion University,Norfolk, Virginia

(Manuscript received 9 September 2002, in final form 23 April 2003)

ABSTRACT

An analytical model that includes pressure gradient, friction, and the earth’s rotation in both components ofthe flow is used to study the transverse structure of estuarine exchange flows and the nature of transversecirculation in estuaries of arbitrary bathymetry. Analytical results are obtained for generic bathymetry and alsoover real depth distributions and are compared with observations. This study extends previous efforts on thetopic of transverse structure of density-induced exchange flows in three main aspects: 1) the analytical modelexplores any arbitrary bathymetry; 2) the results reflect transverse asymmetries, relative to a midchannel cen-terline, associated with the effects of the earth’s rotation; and 3) the transverse circulation produced by theanalytical model is examined in detail. Analytical results over generic bathymetry show, in addition to the alreadyreported dependence of exchange flow structure on the Ekman number, two new features. First, the transversestructure of along-estuary flows shows the earth’s rotation effects, even in relatively narrow systems, thusproducing transverse asymmetries in these flows. The asymmetries disappear under strongly frictional (highEkman number) conditions, thus illustrating the previously documented pattern of inflow in channels and outflowsover shoals for typical estuaries. Second, transverse flows resemble a ‘‘sideways gravitational circulation’’ whenfrictional effects are apparent (Ekman number greater than ;0.1) responding to a transverse balance betweenpressure gradient and friction. These transverse flows reverse direction under very weak friction and reflectCoriolis deflection of along-estuary flows, that is, geostrophic dynamics. All examples of observed flows aresatisfactorily explained by the dynamics included in the analytical model.

1. Introduction

Several studies that have explored the transverse orlateral structure of along-estuary flows driven by den-sity gradients under bathymetric influences (Wong1994; Friedrichs and Hamrick 1996; and Kasai et al.2000) have shown that net inflows tend to be con-centrated in channels while outflows appear overshoals. In an illuminating paper, Kasai et al. (2000)characterized the estuary–ocean exchange in terms ofthe competition between friction and the earth’s ro-tation as captured by the vertical Ekman number E.Under large frictional influences (E . 1) the exchangetook place as proposed by Wong (1994) and Friedrichsand Hamrick (1996): inflow from surface to bottomin the channel and outflow entirely occupying theflanks. As frictional effects decreased, net inflow was

* Current affiliation: Universidad del Mar, Puerto Angel, Oaxaca,Mexico.

Corresponding author address: Dr. Arnoldo Valle-Levinson, Cen-ter for Coastal Physical Oceanography, Ocean, Earth and Atmo-spheric Sciences Dept., Old Dominion University, Norfolk, VA23529.E-mail: [email protected]

still found in the channel but now restricted to a bot-tom Ekman layer, whereas outflow occupied the entirenear-surface layer (Fig. 1). Therefore, estuary–oceanexchange shifted from large transverse variability un-der strong friction to large vertical variability underweak friction. This friction/rotation competitionhelped explain the nature of several observations indifferent systems. However, the analytical resultsshowed symmetric distributions about the deepest partof the channel (thalweg ) unlike expected asymmetriesthat would arise from the earth’s rotation effects.These asymmetries consist of the core of maximumoutflow leaning toward the left (looking into the es-tuary in the Northern Hemisphere) and the core ofmaximum inflow tilting toward the right. Indeed, thereare observations over bathymetry consisting of achannel flanked by shoals (e.g. in the James River;Valle-Levinson et al. 2000) and numerical results(Valle-Levinson and O’Donnell 1996) that show theasymmetric distributions related to the earth’s rotationeffects.

Our objective is to extend Kasai et al.’s work in orderto (i) explain the asymmetric nature of exchange flowsand (ii) characterize the nature of mean transverse orcross-estuary flows over various bathymetries. We ex-

2376 VOLUME 33J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

FIG. 1. Contours of along-estuary flow obtained with Kasai et al. (2000) solution over a triangular crosssection for different Ekman numbers E. Corresponding values of Az are (a) 0.1, (b) 0.01, and (c) 0.001 m2

s21. Contours are nondimensional and have been scaled to the density-induced flow expected at the surfaceand in the channel’s centerline. Representations in this and subsequent figures are looking into the estuary.Also, negative, shaded values denote net inflow.

plore an analytical solution of the transverse variabilityof the flow in an estuary with arbitrary cross-estuarybathymetry. Kasai et al’s work is extended in three as-pects: 1) the analytical solution may be examined overany prescribed arbitrary bathymetry, 2) the qualitativeeffects of the earth’s rotation are evident in the solution,and 3) the resulting transverse flows are scrutinized. Theanalytical results derived with the extended solution are

compared with observations that have been obtained indifferent systems in order to show the robustness of thesolution.

2. Analytical solution

The objective of the solution is to describe the trans-verse structure of the nontidal or mean along-estuary


u and transverse y flows produced by pressure gradi-ents and modified by Coriolis and frictional influences.In a right-handed coordinate system (x, y, z) in whichx points seaward, y across the estuary and z upward(Fig. 2), the nontidal (or steady) momentum balanceto be solved becomes a set of two differential equa-tions:

2]h g ]r ] u2 f y 5 2g 1 z 1 A andz 2]x r ]x ]z0

2]h g ]r ] yfu 5 2g 1 z 1 A , (1)z 2]y r ]y ]z0

where f , g, r 0 , r, h, Az are the Coriolis parameter (13 1024 s21), the gravity acceleration (9.8 m s22), ref-erence water density (1025 kg m23), variable waterdensity (kg m23), surface elevation (m), and verticaleddy viscosity (m2 s21), respectively. An importantaspect of the analytical solution is the restrictive as-sumption that the vertical eddy viscosity is homoge-neous in z and y. Despite of this assumption, the resultsobtained with the analytical model are quite revealing.The set of equations (1) may be represented in termsof a complex velocity w 5 u 1 iy , where i 2 5 21 isthe imaginary number,

2] wgN 2 Dz 5 A 2 ifw. (2)z 2]z

In (2), N and D represent the barotropic and baroclinicpressure gradients, respectively:

]h ]hN 5 1 i and

]x ]y

g ]r ]rD 5 1 i .1 2r ]x ]y0

Then, following Kasai et al. (2000), the flow w hascontributions from the sea level slope and from the hor-izontal density gradients; that is,

w(z, y) 5 gNF (z, y) 1 F (z, y),1 2 (3)

and F1 and F 2 represent functions that depict the ver-tical and transverse structures of the barotropic (fromsea level slope) and baroclinic (from density gradient)

contributions to the flow. Using (3), (2) can be re-written as

2] F if 11 2 F 5 and12]z A Az z

2] F if Dz2 2 F 5 2 . (4)22]z A Az z

Assuming, as boundary conditions, no stress at thesurface (]F1 /]z 5 ]F 2 /]z 5 0 at z 5 0) and no slipat the bottom (F1 and F 2 5 0 at z 5 2H ), and thatthe horizontal density gradient is independent ofdepth, the solution of (4) as obtained by Kasai et al.(2000) is

i cosh(az)F 5 1 2 and1 [ ]f cosh(aH )

iD cosh(az)az 2aHF 5 (e 2 az) 2 (e 1 aH ) . (5)2 [ ]f a cosh(aH )

In this solution, H is any arbitrary depth distributionas a function of y. The parameter a equals (1 1 i )/DE ,where DE is the Ekman layer depth (2Az / f )1/2 . Noticethat solution (5) requires prescription of the densitygradient D and the eddy viscosity Az . Solution (3) re-quires a surface slope N that is dynamically consistentwith D. In order to derive the value of N we use aboundary condition that assumes no net volume fluxacross the channel, that is,

B 0

w dz dy 5 0, (6)E E0 2H

where B is the estuary’s width. Condition (6) impliesthat the river flux is much smaller than the flux producedby the gravitational circulation, which is restrictive butmay be the case in estuaries (at least by one order ofmagnitude: e.g., Wong and Valle-Levinson 2002). Al-ternatively, solutions (5) and (3) may allow the pre-scription of N with subsequent estimation of D that sat-isfies (6). We first adopt the approach of Kasai et al.(2000) of prescribing D and reproduce their results fora bottom of triangular shape.

The value of the surface slope that satisfies condition(6) with a prescribed D is

B

2aH 2aH 2 2D [(e 1 aH ) tanh(aH ) 2 (1 2 e 1 a H /2)] dyE0DI2N 5 2 5 2 . (7)

BagI1ag [tanh(aH ) 2 aH ] dyE

0


FIG. 2. Schematic representation of the estuary’s reference frame.

Prescribing D and Az and with H as any function of y,the solution to (3) is obtained with (5) and (7). Theintegrals in (7) are solved numerically.

a. Solution over triangular bathymetryThe solutions may be cast in terms of the vertical

Ekman number E 5 Az/( f ). Figure 1 presents the2Hmax

solutions of along-estuary flows over a triangular crosssection with the origin in the thalweg. These are in-tended to reproduce the results of Kasai et al. (2000)for f and Hmax of 1024 s21 and 30 m, respectively, andthree different values of Az (1021, 1022 and 1023 m2

s21), corresponding to E of 1, 0.1, and 0.01, respec-tively. The density gradient is taken as D 5 1024 1 i0kg m24; that is, the density field only changes in thealong-estuary direction.

The solutions clearly show different effects of fric-tion. For large frictional effects relative to Coriolis ac-celerations (E 5 1), the estuary–ocean exchange showstransverse reversals in sign: inflow in the deepest partof the channel, reaching the surface, and outflow overthe shallow sides. This is the same as the Wong (1994)solution that disregards Coriolis effects. The main fea-ture of the solution of Fig. 1 is that, as rotation effectsbecome more prevalent or friction is less prominent (E, 1), outflow is restricted to a surface layer that extendsacross the estuary, and inflow appears restricted to abottom layer; that is, exchange now shows vertical re-versals in sign. The puzzling aspect of these results isthat the qualitative effects of rotation, that is, flow de-flection to the right in the Northern Hemisphere, are notapparent. The solution is symmetric about the thalwegwhereas reality reflects asymmetries.

In contrast to the reference frame of Kasai et al.(2000), we take the origin (y 5 0) at one of the coastsinstead of at the thalweg (or centerline) of the estuary(Fig. 2) and prescribe arbitrary bathymetries. Under thisreference frame, the solution remains symmetric for anyarbitrary bathymetry (Fig. 3). Different values of E re-flect the same symmetry explored over the triangularsection. The analytical solution is correct but its rep-resentation seems to be physically unrealistic, so thefirst step to attack this puzzle was to try explain thesymmetry of the solution. In order to determine thereasons for symmetry around the thalweg, we decidedto go back to examine simpler dynamics focusing onthe geostrophic balance over a flat bottom in the trans-verse direction.

b. Geostrophic balance over f lat bottom

The momentum balance in the transverse directionthat includes both barotropic and baroclinic contribu-tions to the pressure gradient is

]h g ]rfu 5 2g 1 z. (8)

]y r ]y

The solution should show an exchange flow that leanstoward the coast as, for example, in the numerical so-lution of Valle-Levinson et al. (1996). It also shouldbe symmetric about a tilting plane that slopes accordingto the Margules equation (Gill 1982). Following thesame approach as before, we prescribe the transversedensity gradient ]r/]y and take the boundary condition

B 0

u dz dy 5 0 (9)E E0 2H

to obtain a dynamically consistent sea surface slope ]h/]y. Because ]r/]y and ]h/]y are constant across theestuary, the geostrophic solution

g ]r Hu 5 z 1 (10)1 2r f ]y 20

is also independent of the transverse coordinate y andonly depends on z (Fig. 4a). Therefore, the prescribedpressure gradient should depend on y to obtain a morerealistic approximation of the exchange flows.

First we prescribe a density gradient with exponentialdecay across the estuary:

]r ]r2ky 2ky5 e 5 ge , (11)1 2]y ]y

max

where g is positive and represents the value of the trans-verse density gradient at the origin (y 5 0). The param-eter k is the rate of exponential decay that could berelated to the internal radius of deformation R1; that is,k 5 1/R1. Thus, using (8) and (11) and applying con-dition (9) to derive a dynamically consistent surfaceslope, we obtain


gg H2ky 2kBu 5 e z 1 (1 2 e ) . (12)[ ]r f 2kB0

The solution to (12) with g of 1 3 1024 kg m24 and k5 2/B clearly yields an asymmetric result that now de-pends on y and z. This result is, however, physicallyuntenable (Fig. 4b). The strongest inflow appears on theleft (looking into the estuary) and the strongest outflowappears on the right, contrary to the correct distribution.In addition, the isotachs slope in the opposite directionto the expected proper direction.

The next approach was to prescribe the sea surfaceslope instead of the density gradient, also with an ex-ponential decay:

]h ]h2ky 2ky5 e 5 ze , (13)1 2]y ]y

max

where z is negative and represents the value of the trans-verse sea level slope at the origin (y 5 0). Once againusing (8) and (13) and applying condition (10) to derivea dynamically consistent density gradient, we obtain

gz 2 z2kB 2kyu 5 (e 2 1) 2 e . (14)[ ]f kB H

This solution, with z 5 21 3 1026, is physically morerealistic than the previous two (Fig. 4c). Outflows andinflows appear on the correct side of the channel. The onlyunexpected detail of the solution is perhaps the downwardconcavity of the outflow isotachs. The condition shownin (13) yields the desired asymmetry relative to the cen-terline of the channel. In order to refine solution (14) abit more, the following condition was used:

]h 2 22k y5 ze . (15)]y

This condition yields the solution

gz 2I z 2 22k yu 5 2 1 e , where1 2f B HB

2 22k yI 5 e dy, (16)E0

which is consistent with theoretical expectations (Fig.4d). The lateral distributions of the sea surface slopeexpressed in (13) and (15) were inspired by the distri-bution obtained with a numerical experiment of a den-sity gradient adjusting under the influence of rotation[experiment 4 in Valle-Levinson and O’Donnell(1996)]. The numerical distribution is in general agree-ment with (13) (Fig. 4e) but much more similar to (15)in terms of the concavity of the curves.

Further justification of the use of (15) is provided bythe vertically averaged dynamics expressed in (1). Ex-amining only the transverse component of the verticallyaveraged momentum, the transverse slope may be givenby

]h f H ]r C y VD B B5 2 u 2 2 , (17)]y g r ]y gH

where the overbar denotes vertical average, CD is thenondimensional bottom drag coefficient (0.0025), Vrepresents the current speed, and the B subscript isassociated with near-bottom variables. This expression(17) indicates that the transverse slope in an estuaryhas contributions from Coriolis, baroclinic, and fric-tional effects, all of which can be evaluated with ad-equate transverse distributions of water density andvelocity in an estuary. Data from the James River es-tuary (Valle-Levinson et al. 2000) are used to evaluateeach contribution to the lateral slope as portrayed inFig. 4f. The main contributions arise from Coriolis andbaroclinic effects, with bed stress effects playing a sec-ondary role. Baroclinic effects are nearly constantacross the estuary, consistently with the analytical pre-scription of Kasai et al. (2000), but Coriolis effectsshow significant transverse variability, proportional tothe vertical average of u. The shape of the lateral slopecalculated with observations is indeed very similar tothat prescribed by (15) and to that derived from nu-merical results.

The distributions given by (15) and obtained numer-ically (Fig. 4e) and from observations (Fig. 4f) reflecta buoyant current of width R1, affected obviously byCoriolis accelerations. Therefore, on the basis of ob-servations and numerical results, the transverse distri-bution of sea surface slope represented by (15) is ex-pected to be dynamically consistent for density-inducedflows influenced by the earth’s rotation. The salient fea-ture of this exercise is that the problem of representingsolution (3) in an estuary requires (i) prescription of thebarotropic pressure gradient and elucidation of a dy-namically consistent baroclinic pressure gradient and(ii) the barotropic pressure gradient to have an expo-nential decay across the estuary as portrayed in (15).These two requirements should yield solutions that areasymmetric around the thalweg or centerline of the es-tuary. Such an approach is explored next to describeexchange flows resulting from the dynamics representedby (1) or, equivalent, by (2).

c. Solution with arbitrary bathymetry

The same solutions (3) and (5) apply but now con-dition (7) is modified, as learned from the previous ex-ercise of geostrophic exchange, to reflect prescriptionof the surface slope N and estimation of a dynamicallyconsistent density gradient D. This approach is essentialto allow physically consistent asymmetries in exchangeflows and is given by

agI3D 5 2 , (18)I2

where I3 is represented by


FIG. 3. Kasai’s solution over arbitrary bathymetry for the same E as in Fig. 1 and for two different bathymetries.Transverse flows are now presented also (arrows). Both velocity components are nondimensionally scaled by thedensity-induced along-estuary flow at z 5 0 and y/B 5 0.5. The value of this along-estuary flow is indicated bythe numbers next to the arrows.

B

I 5 N(y)[tanh(aH ) 2 aH ] dy3 E0

and I2 is the same as in (7). The solution is applied todifferent values of the eddy viscosity coefficient over atypically shaped channel with maximum depth of 30 m

and the surface slope only varying in the transversedirection

2 226 2k yN 5 21 3 10 (1 1 ie ). (19)

This condition indicates that sea level decreases seawardalong the system and from the left coast to the right


FIG. 3. (Continued )

coast (looking into the estuary). Other parameters in thesolution remain the same as in Figs. 1 and 3.

3. Results over arbitrary bathymetry

Solution (3) and (5) with condition (19) is appliedover a channel with shape:

2 2H 5 30 exp[2(y/B 2 0.5) /0.3 ],

as illustrated in Fig. 5 for different Ekman numbers.The most striking feature of the along-estuary flow, rel-ative to Fig. 3, is the effect of the earth’s rotation. Theflow remains asymmetric about the thalweg (y/B 5 0.5)for E , 1 and the inflow remains detached from thesurface at E , 0.2. This particular value depends onthe shape of the channel. Note that the outflow tends tobranch off in the shallow areas as E increases. Thisbranching caused by frictional effects may explain the


FIG. 4. Along-channel geostrophic flows (m s21), (a) prescribing a constant density gradient (1 3 1024 kg m24, appearingon the right panel) and calculating the corresponding sea level slope; (b) prescribing an exponentially decaying density gradient(right panel) and calculating the sea level slope, solution (12); (c) prescribing an exponentially decaying sea level slope (rightpanel) and calculating the corresponding density gradient, solution (14); and (d) same approach as (c) but with solution (16).(e) A comparison between the sea level slope obtained with a numerical experiment (dark line) and (13) (dotted line) and (15)(gray line) with z 5 21 3 1026. (f ) Different contributions to the transverse slope (continuous, thickest line) calculated with(17) from observations in the James River estuary.


FIG. 4. (Continued)

relatively low salinity water usually observed in thelower Chesapeake Bay toward the right-hand coast(looking into the estuary; Valle-Levinson and Lwiza1997). Under strongly frictional conditions (E . 1) thesolution appears like that of Wong (1994) as the ex-change flows become symmetric about the thalweg.

Another interesting aspect of the solution is thetransverse flow that develops under different E. LowE indicates that the dynamics is geostrophic in bothalong-estuary and transverse directions and reflects thetransverse flow adjusting to the along-estuary pressuregradient: flow to the left at the surface and to the rightat depth. As frictional effects enter the solution (in-creased E ), the transverse flow tends to emulate a‘‘sideways estuary’’: surface flow to the right and bot-tom flow to the left, in contrast to the E K 1 solution.This is because the dynamics tends to resemble thatresponsible for the gravitational circulation in estuaries(pressure gradient balanced by friction). Note also thatthe transverse flows show net convergences over theedge of the channels. These convergences move towardthe thalweg as friction increases. This is consistent withobservations in the moderately frictional James Riverestuary (Valle-Levinson et al. 2000), where conver-gence lines tend to be observed over the channelslopes, and the strongly frictional Conwy estuary (Nu-nes and Simpson 1984), where convergence lines tendto be aligned with the thalweg.

Other channel configurations portray additional re-vealing features of the solution. By prescribing the fol-lowing bathymetry,

2 2H 5 30 exp[2(y/B 2 0.15) /0.5 ],

the deepest channel is shifted toward the left-hand coast-line (y/B 5 0.15). Under no friction (E K 1) the deepestchannel reflects more of an exchange flow (Fig. 6a) than

the situation where the deepest point is in the middle(Fig. 5a). This is because the strongest inflow, whichfollows the strongest baroclinic pressure gradient at thelocation of the deepest point, is found where the geo-strophic outflow should be expected. The strongest out-flow appears toward the left-hand coast, but, as E in-creases, the outflow tends to become symmetric as be-fore. The inflow does not reach the surface, even underthe influence of E k 1 (not shown). The transversecirculation is consistent with the previous channel con-figuration although the surface flow to the right andbottom flow to the left, characteristic of gravitationalcirculation, appears better developed at lower E (cf.Figs. 5b and 6b).

By shifting the channel to the right-hand coast (y/B5 0.9), where geostrophic inflow is expected, that is,

2 2H 5 30 exp[2(y/B 2 0.9) /0.5 ],

some differences appear relative to the previous con-figurations (Fig. 7). For the frictionless case, inflow oc-cupies most of the deep channel. For all cases, outflownow appears concentrated in one branch as E increasesinstead of the two branches that develop with the pre-vious configurations. At high E, inflows almost reachthe surface but do not quite touch it, even at E k 1(not shown). The transverse circulation develops itsgravitational character at higher E than before. This isbecause the shift of the deepest part to the right allowsthe exchange flow to resemble the geostrophic ex-change.

4. Comparison with observations

The revealing results shed by the analytical solutionare now applied to real bathymetry distributions acrossvarious systems over which observations of mean flowsare readily available. In every application, flow obser-vations have been obtained with a towed acoustic Dopp-ler current profiler (ADCP) over one full diurnal tidalcycle (;25 h). Model–observation comparisons are car-ried out in three estuaries and two other systems thatexhibit inverse estuarine conditions. The comparisonsconcentrate on the general distributions of the along-estuary and transverse flows. Best match between modeland observations for both flow components was soughtthrough prescription of different values of Az. We reportonly the values of the eddy viscosity that reflect obser-vations best.

a. Observations in estuaries: James River

We first use observations carried out across the JamesRiver estuary in November 1996, same data used forFig. 4f, as reported by Valle-Levinson et al. (2000). Theanalytical results over the James River bathymetry com-pare very favorably to observations (Fig. 8). This isachieved with a value of Az 5 0.002 m2 s21, which isclose to the values of the eddy viscosity obtained in that


FIG. 5. Analytical solution as in Fig. 3 but under different initial condition—condition (15). Contours and arrowsdenote along-estuary and transverse flows, respectively.

paper with a simple turbulence closure. The eddy vis-cosity corresponds to an E (estimated with the deepestpoint) of 0.1. This value of E may indicate that frictionplays a lesser role but most of the section is rather

shallow so that frictional influences are indeed mani-fested by the mean inflows occupying the whole watercolumn in a portion of the channel.

The observed and analytical along-estuary flows have


FIG. 6. As in Fig. 5 but with channel off-center to the left.

the same general structure that features inflow in thechannel and outflow concentrated over the left shoal(looking into the estuary). The modeled outflow regionthat appears over the right shoal (Fig. 8b) decreases,

and it resembles the observations even more if thealong-estuary sea level slope is also prescribed with anexponential decay across the estuary (from left to right),analogous to the prescription of the transverse sea level


FIG. 7. As in Fig. 5 but with channel off-center to the right.

slope (Fig. 8c). Furthermore, decreases in Az restrict theinflow only to the deep region of the channel, and out-flow extends throughout the section (not shown).

Transverse flows are also comparable in the sense thatboth observations and model show near-surface flowfrom left to right and near-bottom flow toward the left.

As discussed in the solutions over simplified bathymetry(Fig. 5), this is an indication of frictional effects enteringinto the transverse dynamics and tending to balance thepressure gradient thus emulating a ‘‘sideways gravita-tional circulation.’’

In general the similarity of both components of the


FIG. 8. Comparison between observations (cm s21) and analytical solution (Az 5 0.002 m2 s21) for realbathymetry over the James River estuary. Both components of the analytical results (maximum arrow length fortransverse component) are scaled by 5 cm s21.

flow is remarkable. Discrepancies appear only as a fewdetails, like the outflow not observed over the rightshoal. Such discrepancies are attributed mainly to fourfactors, which also apply to subsequent examples.First, the dynamic factor: the analytical solution doesnot include advective accelerations and also assumesuniform Az . Geometric considerations related to cur-vature effects and nonuniformities of bathymetry inthe along-estuary direction enter into this factor. Sec-ond, the continuity constraint [(6)] factor: the solution

does not allow net volume transports into or out of theestuary, which is usually the case in reality. Third, thefactor related to the sectional area covered by the ob-servations: observations do not reach the surface orbottom (owing to ADCP sidelobe effects) or the endsof the section. Fourth, the factor related to the lack ofdynamic consistency between the two components ofthe barotropic pressure gradient (sea level slope),which are prescribed independently of each other andin reality should be linked. The main message is that


FIG. 9. Same as Fig. 8 but for Chesapeake Bay mouth with (b) Az 5 0.0015 m2 s21 and (c) 0.005 m2 s21.Scaling velocity for both flow components of analytical results is 11 cm s21.

the dominant dynamics in the James River example arecaptured by the balance portrayed in (1) or (2).

b. Observations in estuaries: Chesapeake Bayentrance

Next example includes observations over a morecomplicated bathymetry at the entrance to Chesapeake

Bay on 12–13 May 1997 (Fig. 9) reported in Valle-Levinson et al. (1998). Two channels of different widthsand depths are located near the ends of the cross sectionand are separated by a shoal. The along-estuary com-ponent of the flow is marginally similar between ob-servations and analytical solution with Az 5 0.0015 m2

s21 (Figs. 9a,b). The similarity of the shape of this com-ponent improves if the eddy viscosity increases to Az 5


0.005 m2 s21 (Fig. 9c). In particular, the outflow in thechannels resembles more the observations than the caseof low viscosity.

Transverse flows with Az 5 0.0015 m2 s21 (Fig. 9b)are consistent with the generic results of low E number(Fig. 5a). This means that transverse flows during theperiod of observations were mostly linked to the along-estuary dynamics, rather than the transverse dynamics.The same increase in viscosity that makes along-estuaryflows be more similar to analytical predictions alters thetransverse flows and produces a departure from obser-vations. Discrepancies between observations and the-oretical results can be attributed, in addition to the fourfactors mentioned above, to wind forcing conditionsduring the observation period. The eddy viscosities usedfor the analytic results indicate that, during the obser-vation period, friction was important only in the alongestuary dynamics and played a lesser role in the trans-verse direction.

c. Observations in estuaries: Lower Chesapeake Bay

Another example, the last one of estuaries, refers toobservations obtained over even more complicated ba-thymetry in lower Chesapeake Bay on 5–6 October1993 (Fig. 10) reported in Valle-Levinson and Lwiza(1995). Three channels of different characteristics ap-pear over the cross section portrayed. This is perhapsthe most challenging test for the model and is not a fullyreliable test because the observations lack the qualityof those at other sites. The observations covered thewidest cross section and were collected over two day-time semidiurnal tidal cycles. Nonetheless, the along-estuary component of the flow shows rough resemblancebetween observations and analytical results. The generallocation of inflows/outflows is reproduced reasonablywell by the model results with an Az 5 0.006 m2 s21.The transverse component of the flow also shows rea-sonably good agreement except in the region delineatedby the 10–20-km distances and surface to 5-m depths.In general, the areas of mean convergence at the edgeof the channels appear in both observations and model.The distribution of along-channel and transverse flows,as well as the value of the eddy viscosity coefficient Az,strongly suggests that frictional effects are relevant forboth components of the flow. This indicates that thedynamics are appropriately represented by (1) or (2).The discrepancies between observations and theory canbe attributed to the factors mentioned in the previoustwo cases.

d. Observations in other systems: Guaymas Bay

We now test the performance of the analytical solu-tion for a Mexican coastal lagoon that shows inverseestuarine behavior, at least during part of the year (Valle-Levinson et al. 2001). The application of the solution

to an inverse estuary requires that condition (19) berewritten as

2 226 2k (y2B)N 5 1 3 10 [1 1 ie ]. (20)

This condition indicates an along-estuary seaward in-crease of sea level and a transverse increase in sea levelfrom left to right (looking into the system). These sealevel slopes are in opposite direction to those of a typicalestuary. Using condition (20) at the entrance to the Bayof Guaymas ( f 5 6.4 3 1025 s21) yields analyticalresults that are very similar to observations obtained on11–12 June 1999 (Fig. 11). Outflows are now restrictedto the channel, as expected for an inverse estuary, andinflows appear over shallow areas. Transverse flows fea-ture near-surface flow from right to left and bottom flowin the opposite direction. Such distribution of transverseflows, together with the value of Az 5 0.0015 m2 s21

used, indicates frictional influences on the flows. Notethat the observations encompass a subset of the entirecross section depicted in the analytical results. Withinthat subset is where both along-estuary and transverseflows are quite similar and consistent. This is anotherexample in which the dynamics are dominated by Cor-iolis, pressure gradient, and friction in both componentsof the flow.

e. Observations in other systems: Gulf of Fonseca

The final example of the applicability of the analyticalmodel features observations obtained in a tropical es-tuary on the Pacific side of Central America, the Gulfof Fonseca ( f 5 3.3 3 1025 s21). The data were ob-tained during the dry season on 16–17 March 2001 (Val-le-Levinson and Bosley 2003). During the dry season,it is reasonable to expect net circulation resembling in-verse estuaries or salt-plug estuaries. At the mouth ofthe system, whether it behaves as inverse or salt plug,outflow is expected at depth and inflow near the surface,as observed (Fig. 12). This is the example with deepestcross section and minimal frictional effects. This is sub-stantiated by the structure of the along- and transverseflows in both model and observations. The outflow mayreach the surface but outside the deepest channel, incontrast to it reaching the surface within the channelunder strong frictional effects. The transverse flows ap-pear to the right of the inflow/outflow as expected fromdominant influences of Coriolis accelerations. The lowvalue of Az 5 0.0002 m2 s21 also implies that the dom-inant dynamics during this neap tides period was qua-sigesotrophic. Observations during spring tides (notshown) revealed increased frictional influences.

5. Conclusions

The analytical model featured in this work is an ex-tension of Kasai et al. (2000) in three aspects: 1) pre-scription of an initial condition that yields asymmetricflow, as in reality, across the estuary; 2) application to


FIG. 10. As in Fig. 8 but for lower Chesapeake Bay (approx 20 km landward from section at bay entrance) and Az

5 0.006 m2 s21. The scaling velocity of analytical results is 5 cm s21.

any arbitrary bathymetric structure across the estuary;and 3) examination of transverse flows over differentbathymetries and different Ekman numbers E. In con-trast to Kasai et al.’s work, we found that the sea levelslope needs to be prescribed and then a dynamicallyconsistent density gradient derived, rather than vice ver-sa. Only this approach warrants physically tenable

asymmetries in the exchange flows. The sea level slopeprescribed in the analytical model featured a transverseexponential decay, consistent with numerical results andestimates from observations, which would be worth-while exploring with direct observations as a challeng-ing hypothesis to test.

The analytical model yields reliable distributions of


FIG. 11. As in Fig. 8 but for Guaymas Bay, Mexico, with Az 5 0.0015 m2 s21. Scaling for both flow components(maximum arrow length for transverse component) is 5 cm s21.

along-estuary and transverse flows in systems wherethe mean flow is mostly driven by density gradientsand their associated sea level slopes. The model is quiteuseful in the interpretation of the dominant dynamicsthat drive observed mean flows, as in the examplespresented here. The analytical model should also helpverify results of numerical models in cases with com-plex bathymetry. The limitations of the analytical mod-el, which produce discrepancies relative to observa-tions only in few details of the solution, are 1) omission

of advective accelerations and assumption of uniformAz , 2) assumption of net zero volume flux, and 3) lackof dynamic consistency between the two componentsof the prescribed barotropic pressure gradient. Theselimitations, however, are minor in comparison with theinsights gained on the lateral distribution of along-estuary flows, on transverse flows, and on the dynamicsassociated with these flows over any arbitrarybathymetry.

Most important, in addition to documenting the tran-


FIG. 12. As in Fig. 8 but for the Gulf of Fonseca, Central America, with Az 5 0.0002 m2 s21. Scaling for bothflow components (maximum arrow length for transverse component) is 20 cm s21.

sition of the exchange flow structure from low to highE as already presented in Kasai et al. (2000), the ana-lytical results have provided the following physical in-sights in terms of the lateral structure of the along-estuary flow and of the transverse flows: (i) The lateralstructure of the along-estuary flow shows asymmetriesrelated to Coriolis accelerations, even in relatively nar-row systems like Guaymas Bay. These asymmetrieswould disappear and give way to symmetric distribu-tions about the estuary’s thalweg under strongly fric-

tional environments (E . 1). As pointed out by Kasaiet al. (2000), the length scale that determines whetherrotation is relevant or not is the depth of the Ekmanlayer relative to the water column depth, not the widthof the system. (ii) Transverse flows are linked to along-estuary pressure gradients and flows (to the right ofinflows/outflows) under frictionless conditions in sucha way that near surface flow is to the left (looking intothe estuary) and near-bottom is to the right. This trans-verse circulation tends to reverse direction as frictional


effects become important (typically E . 0.01) thus pro-ducing a ‘‘sideways’’ estuarine circulation that respondsto the transverse pressure gradient. These conclusionsshould allow us to establish the dynamic state of anysystem by elucidating the general shape of exchangeflows.

Acknowledgments. This study was inspired by thework of A. Kasai and remarks by D. Farmer. Also, J.H.Simpson suggested evaluation of (17) to justify theshape of the transverse slope. This work is funded byNSF Project 9983685. The data used for model com-parisons required the effort of many people whose ded-ication is greatly appreciated. The James River datawere obtained with NSF Project 9529806. ChesapeakeBay entrance data were obtained with MMS Project 14-35-0001-30807 and NOAA Office of Sea GrantNA56RG0489. Lower Chesapeake Bay data were ob-tained with CCPO funding. Data from the Bay of Guay-mas were collected thanks to the coordination of J. Del-gado. Gulf of Fonseca data were obtained with NOAAfunding through NOS. This paper was completed whileAVL was on research assignment at the Centro de In-vestigacion Cientıfica y Educacion Superior de Ensen-ada, Estacion La Paz Baja California Sur. Thanks aregiven to J. H. Simpson, G. Gutierrez de Velasco, A.Kasai, and C. Winant for their comments.

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