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Arnoldo Valle-LevinsonAU2

Civil and Coastal Engineering DepartmentUniversity of FloridaGainesvilleFLUSAE-mail: arnoldo@coastal.ufl.edu

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Biographical Sketch

Arnoldo is a Professor at the Civil and Coastal Engineering Department at the University of Florida. He earned a PhD incoastal oceanography from the State University of New York, Stony Brook in 1992. He was an Assistant and AssociateProfessor at the Oceanography Department, Old Dominion University in Virginia, USA, from 1996 to 2005. He thenmoved to the University of Florida, where he became Professor in 2008. His work deals mainly with the study of estuarineand coastal hydrodynamics. Arnoldo received in 2000 an early career award from the US National Science Foundation,which is the highest honor awarded by the foundation to a young scientist. He has received a Fulbright SpecialistFellowship, which took him to Chile, a Gledden Fellowship, which allowed him to visit Australia, and a fellowshipfrom the Mexican Academy of Sciences. Arnoldo has worked extensively in several Latin-American countries, where he alsoteaches courses on estuarine and coastal hydrodynamics. He has more than 90 peer-reviewed publications and is serving asan Associate Editor of two professional journals.

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1.06a0005 Classification of Estuarine CirculationA Valle-Levinson, University of Florida, Gainesville, FL, USA

© 2012 Elsevier Inc. All rights reserved.

1.06.1 Classification of Gravitational Circulation According to Estuarine Origin/Geomorphology 41.06.1.1 Coastal Plain/Drowned River Valley Estuaries 41.06.1.2 Tectonic Estuaries 41.06.1.3 Fjords 61.06.1.4 Bar-Built Estuaries 71.06.2 Classification of Gravitational Circulation According to Water Balance 71.06.3 Classification of Gravitational Circulation According to the Competition between Tidal Flow

and River Discharge 81.06.4 Estuarine Circulation 9References 12

Abstract

In the absence of the Earth’s rotation effects, estuarine circulation represents the interaction among the contributions fromgravitational circulation, tidal residual circulation, and circulation driven by tidally asymmetric vertical mixing. In turn,gravitational circulation is driven by river discharge and density gradients. Gravitational circulation tends to be dominant inmany estuaries and can be classified according to the basin’s morphology or origin, to its water balance, or to the competitionbetween tidal forcing and river discharge. Such classification as well as the circumstances under which the residual circulationinduced by tidal flows and tidally asymmetric mixing are discussed in this chapter.

p0005 Throughout the years, several schemes have been proposed toclassify estuaries on the basis of their definition as semi-enclosedcoastal bodies of water where ocean water is diluted by land-derived freshwater discharges. Traditional approaches to categor-ize estuaries have dealt with (1) their origin or geomorphology,(2) their water balance, (3) the competition between tidal flowand river discharge, and (4) the stratification and water-circula-tion characteristics of the system. An overview of suchclassification, of estuaries as basin entities, may be found else-where (e.g., Valle-Levinson, 2010). This chapter seeks to describethe circulation in estuaries, not necessarily as basin entities but asdynamic entities, but still, following similar schemes that havebeen used in the past.

p0010 It is necessary to point out, however, that fitting a givenestuary or coastal basin into any type of classification is a riskyundertaking. This is because most estuaries will show charac-teristics of different categories according to intra-tidal (ebb orflood) and sub-tidal (springs or neaps) phases, buoyancy con-ditions (dry or wet season), and atmospheric forcing (cooling/heating and wind direction). According to the phasing of theforcing, a given estuary will be stratified or destratified and willexhibit different circulation patterns. Therefore, classificationsof estuaries should be used cautiously. In any case, a discussionof several approaches is presented next.

p0015 In order to classify the along-estuary (in the x direction)nontidal, or sub-tidal, circulation ue, it is essential to acknowl-edge the contribution from four basic flows: (1) density-drivenflow ud; (2) tidally rectified flow ut; (3) river-driven flow ur; and(4) flow induced by tidally asymmetric mixing ua (Cheng andValle-Levinson, 2010):

ue ¼ ud þ ut þ ur þ ua ½1�

p0020We begin with, and concentrate mostly in this chapter on, adescription of the typical gravitational circulation, as pro-posed by Pritchard (1956). Herein, we refer to gravitationalcirculation ug as the combined flow produced by densitygradients ud and by the water-level slope associated withriver discharge ur. In any basin where fresh/riverine andsalty/oceanic waters interact, the long-term circulation shouldconsist of lighter, riverine water moving at the surface towardthe ocean and heavier, oceanic water flowing near the bottomtoward the river source (Figure 1). This presumed two-layercirculation is vertically sheared, bidirectional, and also knownas baroclinic circulation. The various appellations describeflow resulting from the adjustment of the density gradientunder the influence of gravity. The density-driven flowreverses direction with depth and the river-driven flow isunidirectional with depth.

p0025It is important to note that the direct observation of gravita-tional circulation in nature may or may not be possible becauseof typical masking of instantaneous flow by tides. Whenpossible, the observation of estuarine circulation requires per-sistent measurements throughout one or several tidal cycles.This is because coastal basins are normally affected by forcingfrom tides, the atmosphere (winds, barometric pressure, cool-ing/heating, and evaporation/precipitation), and the adjacentocean (coastally trapped waves and planetary waves with per-iods greater than the tidal period). Therefore, at any given time,the circulation profile may show unidirectional flow through-out the water column. The gravitational circulation should thenbe extracted from data of flow profiles after averaging over one,or several, or many tidal cycles (Figure 2). The result of aver-aging flow profiles represents the interaction between watersof different densities. Even then, the average flow may have

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influence from tidally asymmetric mixing (e.g., Stacey et al.,2008) causing ua, which will be discussed in Section 1.06.4. Inmany cases, however, the gravitational circulation may beobserved instantaneously during periods of slack tidal currents(e.g., around hour 7 in Figure 2), when only density gradientsdrive the circulation observed.

p0030 The gravitational circulation ug often results from a dynamicbalance between pressure gradient and stress divergence(frictional effects):

1ρ∂p∂x

¼ g∂η∂x

þ g

ρ∫0

−H

∂p∂x

dz ¼ ∂∂z

Az∂ug∂z

� �½2�

where Az is an eddy viscosity coefficient with units m2 s−2, ρis a reference density of seawater, and z is the verticalcoordinate, positive upward. The pressure gradient ∂p/∂xhas a contribution from the density difference ∂p/∂xbetween riverine and oceanic waters, and a contributionfrom the water-level slope ∂η/∂x that develops between theriver and the ocean. Riverine waters, being less dense thanoceanic waters, stand taller and are forced to flow seaward.Heavier oceanic waters flow near the bottom toward thefreshwater source. The stress divergence that balances thepressure gradient arises mainly from the interactionbetween flows and the bottom of the basin (bottom

f0010 Figure 2 Time series of principal-axis flow profiles during one tidal cycle at the entrance to the Chesapeake Bay. The tidally averaged flow profile appearson the right panel (seaward flow is positive).

–1.0–1.0 –0.5 0.0

Nondimensional flow

Non

dim

ensi

onal

dep

th

0.5 1.0

–0.8

–0.6

–0.4

–0.2

0.0

River

ug

Ocean

udur

f0005 Figure 1 Representation of gravitational circulation ug (in blue), illustrating seaward flow at the surface and landward flow at the bottom. The red line isthe density-driven circulation ud and the orange line is the river-induced circulation ur.

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friction). This interaction causes vertical shears that allowtransfer of horizontal momentum in the vertical direction,the stress divergence. The analytical solution to eqn [2]for ug can be obtained by integrating it twice and applyingboundary conditions of no stress at the surface, no flowat the bottom, and net transport provided by riverdischarge per unit width R (in m2 s−1), As in Officer(1976):

ugðzÞ ¼ g ∂ρ=∂x H3

48 ρ Az9 1−

z2

H2

� �−8 1þ z3

H3

� �� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ud

þ 32R

H1−

z2

H2

� �

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}ur

½3�

p0035 This velocity profile is presented in Figure 1 with both ud and urcontributions. Notice that ud is a third-degree polynomial, ahyperbola with two inflection points, which has a zero verticalaverage. Its amplitude is proportional to the density gradient(∂p/∂x) and the cube of the water-column depth (H3), andinversely proportional to mixing (as prescribed by Az). In turn,ur is a second-degree polynomial with a parabolic profile thatrepresents seaward flow throughout the entire water column. Itsmagnitude is proportional to R and inversely proportional toH.Other forms of this solution can be obtained with a differentboundary condition at the bottom (e.g., no stress instead of noslip) and express it as a function of ∂p/∂x:

udðzÞ ¼ g ∂ρ=∂x H3

24 ρ Az1−4

z3

H3−6

z2

H2

� �� �½4�

or of ∂η/∂x:

udðzÞ ¼ g ∂η=∂x H2

12 Az1−4

z3

H3 −6z2

H2

� �� �: ½5�

p0040These two profiles ([4] and [5]) show symmetric distributionsaround mid-depth with maxima at surface and bottom.

p0045Moreover, a classification of gravitational circulationrequires a conceptual picture of typical water-column stratifica-tion. The interaction between riverine and oceanic watersproduces a profile of water density that increases from thesurface to the bottom (Figure 3). The density increase canadopt different shapes that may be linked to the nature ofgravitational circulation, as described in detail in other chaptersof this treatise. AU3On one end of the possibilities, the densityincrease with depth can be very slow or virtually absent (prac-tically uniform density with depth), which indicates thatmixing from tides or atmospheric forcing overpowers the buoy-ancy input to the water column (Figure 4). On the other end ofpossibilities, the density may increase monotonically withdepth, which means that there is uniform vertical exchangethroughout the water column (Figure 5). Between the twoends of possibilities, the density may (1) increase monotoni-cally at a surface layer (forming a pycnocline that extendsthroughout a surface layer) and then become uniform or nearlyuniform in a bottom layer (Figure 6), or (2) be uniform at asurface layer and be segregated by a pycnocline from a uniformbottom layer (Figure 7), or (3) increase slightly with depthwithin a surface layer and be separated by a pycnocline froma bottom layer where density also increases slightly (Figure 8),or (4) even be uniform in a surface layer that is separated froma monotonically increasing bottom layer (Figure 9). It is the

12

15

16

14

12

10

8

6

4

2

20

Sal

25 30 2.0

16

14

12

10

8

6

4

2

2.5

Tem (°C)

3.0 3.5 4.0

16

14

12

10

8

6

4

2

14 16 18

Den (kg m–3)

20 22

f0015 Figure 3 Tidally averaged profiles of salinity, temperature, and density anomaly at the entrance to Hampton Roads (linking the James River andChesapeake Bay) on 26 January 2005.

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vertically sheared gravitational circulation, as well as the verti-cal density distribution, that will be described for differenttypes of basins in the discussions of this chapter.

s00051.06.1 Classification of Gravitational CirculationAccording to Estuarine Origin/Geomorphology

s00101.06.1.1 Coastal Plain/Drowned River Valley Estuaries

p0050These are found mostly in temperate climates and are themost studied of all estuaries. Extensive studies have beenreported in coastal plain, temperate estuaries in Japan,Australia, Europe, and North America. The gravitational cir-culation is typically well developed and vertically sheared,although it may be influenced by bathymetry (see 00208). Inthese systems, the stress divergence that balances the pressuregradient arises mostly from tidal flows interacting with thebottom. Such interaction represents the main agent for verti-cal mixing of momentum and mass. In some cases, it ispossible that mixing at the pycnocline, from internal stresses(or stresses within the water column around the pycnoclineregion), also contributes to balance the pressure gradient.Moreover, advective accelerations could influence thedynamics of the exchange flow. The density profile mayacquire any of the configurations described in the previousparagraph (Figures 3–9).

s00151.06.1.2 Tectonic Estuaries

p0055These estuaries operate dynamically in many ways that aresimilar to coastal plain estuaries. For instance, the processesand dynamics described in coastal plain estuaries apply, forthe most part, to San Francisco Bay, a tectonic estuary. Thereare tectonic estuaries, however, like some rias in the north-west coast of Spain (e.g., Gilcoto et al., 2007), where thegravitational circulation is not as well developed as in coastalplain estuaries. This is because the circulation forced fromthe coast, typically during upwelling conditions, masks thegravitational circulation. Moreover, rias are deeper thancoastal plain estuaries and therefore tidal currents tend tobe weaker. This means that the bottom stress divergence isless effective in balancing the pressure gradient and the windstress, together with Coriolis and advective accelerations,enter the dynamic picture. Depending on the season, thedensity profile may exhibit (1) a monotonically increasing

f0020 Figure 4 Tidally averaged profiles of salinity, temperature, and densityanomaly at St. Augustine Inlet, Florida, on 2 February 2006. Profilesillustrate essentially vertically mixed conditions.

f0025 Figure 5 Generic tidally averaged density profile illustrating uniformly stratified water column.

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f0030 Figure 6 Salinity, temperature, and density profiles at Reloncavi Fjord, Chile, on 10 March 2002, illustrating a surface stratified layer on top of ahomogeneous water column (profiles extended to >100m but density did not change appreciably below 20m).

f0035 Figure 7 Generic tidally averaged density profile illustrating weakly stratified surface and bottom layers, separated by a pycnocline.

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density within a surface layer that is separated from a homo-geneous bottom layer, or (2) a weakly varying distribution.This pattern that develops in rias is similar to what happensin fjords.

s0020 1.06.1.3 Fjords

p0060 These estuaries are the deepest and among the most stratifiedduring the season of largest buoyancy input. The residual (orlong-term) circulation may be interpreted as a gravitationalcirculation consisting of a thin (typically <0.3 of the water-column depth) surface layer flowing seaward and a sluggish,thick (often >0.7 of the water-column depth) bottom layerflowing landward. On the other hand, the residual circula-tion may also be interpreted as the result of a thin lens ofseaward-flowing freshwater that overlays a rather sluggishbasin. In the sluggish basin, underneath the freshwater lens,

the residual circulation arises from the rectification (non-linear modification) of the tidal wave reflecting at the endof the fjord (e.g., Ianniello, 1977). This tidal distortion yieldsa two-layer or three-layer (depending on the specific geome-try of the basin) residual flow, as observed in several fjordsand channels of southern Chile (e.g., Valle-Levinson et al.,2007). The alternative interpretations of the residual circula-tion in fjords require further studies to resolve whether theresidual circulation is driven by density gradients throughoutthe water column. Regardless of whether the residual circula-tion is density driven (gravitational) throughout the watercolumn or not, it is likely that in fjords the pressure gradientis balanced mainly by advective accelerations with somecontribution from Coriolis accelerations. It is also likelythat internal friction becomes dynamically important inregions of geometry variations, where vertical excursionsof the pycnocline will cause increased drag and energy

f0040 Figure 8 Generic tidally averaged density profile illustrating mixed surface and bottom layers, separated by a pycnocline.

f0045 Figure 9 Generic tidally averaged density profile illustrating a mixed surface layer separated by a continuously stratified water column underneath.

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dissipation. The density profile will typically exhibit a mono-tonically increasing density within a thin surface layer,separated from a homogeneous bottom layer that occupiesmost of the water column.

s0025 1.06.1.4 Bar-Built Estuaries

p0065 These estuaries tend to be shallow (a few meters deep) withweak river discharge and rapid dissipation of tidal energy asthe tidal wave enters the estuary. They are mostly found insubtropical and temperate low-lying areas with little area fora well-developed river basin, that is, with small watershedareas. The gravitational circulation may be restricted to thearea proximate to the source of freshwater but may not bevertically sheared in the rest of the system. Tidal currentsand mixing are typically relevant at the transition with theocean but become weak inside the basin. These systems aremostly driven by wind forcing but the dynamics remainsbetween pressure gradient and friction, except that frictionacts both at the surface, through wind stress, and at thebottom, through stresses produced by wind-driven currents.The density profile tends to be homogeneous or with weakvertical structure, when compared to the other types ofestuaries.

s0030 1.06.2 Classification of Gravitational CirculationAccording to Water Balance

p0070 In the most generic way, estuarine circulation may be classifiedaccording to whether the net volume outflow, associated withthe gravitational circulation, is larger or smaller than the netvolume inflow. In positive or normal estuaries, there is a net

volume export to the coastal ocean. The gravitational circula-tion will exhibit this characteristic with stronger outflow at thesurface than inflow near the bottom. All of the estuaries dis-cussed above for coastal plain, tectonic, fjord, and bar built fitinto this category.

p0075Of great interest are the little-studied estuaries where there isno freshwater input or where it is rather low. These basins aregenerically known as low-inflow estuaries (e.g., Largier, 2010).Within this group, the following systems may develop: inverseestuaries, thermal estuaries, salt-plug estuaries, and thermal-plug (or thermal bar) estuaries (Figure 10).

p0080Inverse estuaries are those whose salinity and density aregreater than the ocean. It is said that these basins are hypersa-line and hyperpycnal. The circulation resulting from suchcondition consists of net outflow near the bottom and netinflow at the surface, that is, there is a net volume inflow intothe estuary that is lost by net evaporation. These estuariesappear preferentially in low-latitude, arid regions where eva-poration exceeds precipitation, such as Guaymas Bay in theGulf of California (Valle-Levinson et al., 2001), Shark Bay inWestern Australia, and Spencer Gulf in South Australia.

p0085Thermal estuaries also appear in arid or semiarid cli-mates. They may exhibit slight hypersaline conditions butremain hypopycnal, that is, the density of the basin is lowerthan oceanic density. This can occur when the temperatureof the basin is warmer than that of the ocean because ofcoastal upwelling. In this case, there will be a gravitationalcirculation resembling a coastal-plain estuary where thevolume outflow is larger than the inflow. These systemsare peculiar because in some cases when hypersaline condi-tions develop, the inflow may be larger than the outflow,like in San Diego Bay in the southwestern United States(Largier, 2010).

Land

Inverse

Ocean

Dep

th

ρ

Land

Thermal

Ocean

Dep

th

ρ

Land

Salt plug

Ocean

Dep

th

ρ

Land

Thermal plug

Ocean

Dep

th

ρ

f0050 Figure 10 Types of estuaries according to their water balance (low-inflow estuaries).

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p0090 Salt-plug estuaries are a combination of inverse and positiveestuaries. These develop in relatively shallow basins of semiaridor strongly seasonal climates where a river reaches the coast.During the dry season, the river discharge is low but a positiveestuary is still present around the area where the river enters theestuary. In the rest of the basin, hypersaline and inverse condi-tions develop, creating a region of maximum salinity within theestuary. This region of maximum salinity acts as a salt plug thatseparates a zone of positive gravitational circulation near theriver/estuary area and a zone of inverse gravitational circulationbetween the salt plug and the coastal ocean. The salt plugshould have important water-quality conditions as it effectivelyacts as a barrier for the flushing of the estuarine region.Examples of systems similar to this one are found in AlligatorRiver, Northern Territory of Australia, and the Gulf of Fonsecain the Pacific side of Central America (Valle-Levinson andBosley, 2003).

p0095 Thermal-plug estuaries are analogous to salt-plug estuaries.They may develop transitionally from a thermal estuary thatbecomes hypersaline and inverse near its landward end. Thus,the landward portion of the estuary will exhibit inflow at sur-face and outflow near the bottom, whereas the seaward portionwill exhibit outflow at the surface and inflow at depth. Theconvergence of bottom flows toward the landward end of theestuary will represent a thermal plug that also acts as a barrierfor flushing. These systems are much less documented, but SanDiego Bay is thought to show these properties (Largier, 2010).

s0035 1.06.3 Classification of Gravitational CirculationAccording to the Competition between Tidal Flowand River Discharge

p0100 Traditionally, estuaries may be classified as vertically homoge-neous, weakly stratified or partially mixed, and highlystratified. This classification has been proposed on the basisof the competition between tidal forcing and buoyancy forcing.The key of this classifying idea is how to characterize tidal andbuoyancy forcing. One way of assessing tidal forcing is throughthe tidal prism volume, and a way of evaluating buoyancyforcing in estuaries dominated by freshwater input is by thevolume of the river discharged. Tidal prism is the volume ofwater that enters the estuary with every tidal cycle. It equals thetidal range multiplied by the surface area of the basin.Therefore, much larger tidal prisms than freshwater volumes,more than one order of magnitude larger, will result in weaklystratified or vertically homogeneous estuaries. In these estu-aries, the gravitational circulation may be vertically sheared orhorizontally sheared, depending on the geometry of the basin(see 00208).

p0105 When the tidal prism is of the order of the freshwatervolume or slightly larger, the estuary will be partially mixed.These estuaries support the strongest longitudinal density gra-dients because of the active mixing, albeit not sufficientlyenergetic to homogenize the basin. In this case, the gravita-tional circulation is most robust because of the strength of thehorizontal density gradients. Many temperate estuaries may fitinto this category.

p0110 Finally, when the freshwater volume overwhelms the tidalprism, by one or more orders of magnitude, strong stratifica-tion will ensue. This will cause well-developed gravitational

circulation but not as robust as in partially mixed estuariesbecause the longitudinal density gradients will be weaker.Moreover, the vertical exchange of properties is hindered bythe presence of a pycnocline. These estuaries are found in wetclimates.

p0115Once again, it is pertinent to raise caution about this type ofclassification. Any given estuary may show every characteristicor type as seasonal forcing changes. It may also change fromspring to neap tides or even from flood to ebb phases of thetidal cycle.

p0120In the widely used classification of estuaries based on acirculation/stratification diagram (Hansen and Rattray, 1966),the nature of both the gravitational circulation and the waterstratification are used to classify an estuary. This classificationdoes not describe the gravitational circulation but uses it tocharacterize the estuary. A detailed explanation of this classifica-tion is presented in most texts that deal with estuaries and thereader is referred to those explanations (e.g., Valle-Levinson,2010). In essence, the gravitational circulation is described as acirculation parameter uf, which is the ratio between the net sur-face outflow speed us and the cross-sectional average of the netflow uc. When the gravitational circulation is well developed, thenet outflow will be similar to the net inflow in such a way that uctends to zero. In this case of well-developed gravitational circula-tion, the circulation parameter uf will tend to infinity, that is, itbecomes a large number. When the gravitational circulation isweak, the net surface outflow is nearly representative of theentire cross-sectional average flow in which case uf tends to 1.

p0125The stratification of the estuary is characterized by the stra-tification parameter Sp, which is the ratio between the top-to-bottom salinity difference ΔS and the cross-sectional mean ofsalinity So. The values of Sp range from nearly zero for a mixedestuary to near 1, for a highly stratified estuary. On the basis ofits values of uf and Sp, an estuary may be strongly stratified,partially mixed, mixed, or salt wedge. However, in order tocharacterize the estuary, these diagnostic parameters areneeded. In other words, by the time the values of uf and Spare determined, the type of estuary is already known.

p0130Arguably, a more useful approach is a prognostic scheme inwhich simple estuarine properties such as the tidal currentamplitude and the river discharge velocity are used to character-ize the estuary and the nature of the circulation. This approachhas been proposed independently by Prandle (2009) and Geyer(2010). In essence, Geyer’s approach predicts the stratification ofthe estuary ΔS, normalized by a reference salinity S0, as a func-tion of tidal current amplitude UT and river velocity Ur:

ΔSSo

¼ 10U

4=5r

U2=5T ðβgSohÞ1=5

½6�

p0135Where β is the coefficient of haline contraction (7.7 × 10−4), g isthe acceleration caused by gravity, and h is the water-columndepth. Contours of normalized stratification are shown inFigure 11 for a range of commonly observable Ur and UT. Theupper right corner of the parameter space characterizes estu-aries that are short and flush quickly, such as salt-wedgeestuaries with strong tidal forcing (e.g., Columbia River inWashington/Oregon and Merrimack River in Massachusetts,both in the United States). The lower left corner of Figure 11includes long- and slow-flushing estuaries. Moreover, theupper part of the parameter space in the ordinate, that of

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large river velocities, characterizes highly stratified estuariessuch as the Mississippi River for weak tidal currents. Thelower part of the ordinate illustrates vertically homogeneousestuaries, such as the Delaware River, and the middle portion ofthe ordinate includes partially mixed estuaries such as theJames River and the Hudson River. This approach provides arough characterization of processes in estuaries but does nottake into account temporal or spatial variations in the particu-lar systems. Further research is needed to determine othervariables that should enter this scheme.

s0040 1.06.4 Estuarine Circulation

p0140 The introductory part of this chapter mentions that estuarinecirculation ue can be regarded as the sum of the gravitationalcirculation ug, the tidal residual circulation ut, and the

circulation induced by asymmetric tidal mixing ua. Up tothis point, discussions have concentrated on ug but ut and uawill now be presented for idealized vertically mixed, weaklystratified, and strongly stratified estuaries, following theresults of Cheng and Valle-Levinson (2010). Tidal residualflows develop from distortions on the tidal signal as it enterssemi-enclosed systems. The distortions cause stronger floodsthan ebbs, or vice versa, which result in residual flows. Inturn, tidal asymmetries in vertical mixing are illustrated inFigure 12. Asymmetric distributions from flood to ebb ofeddy viscosity and stress (therefore stress divergence) pro-files show larger values in flood than in ebb. The strongermixing in flood than ebb is also related to asymmetricvelocity profiles at both phases of the tidal cycle. Theseasymmetries produce a residual flow that can either enhanceor compete against gravitational circulation. In a verticallywell-mixed estuary with a horizontal density gradient, ug is

Longslow flushing

Partially mixed

Highly stratified1.000

0.100

0.010

0.001

Shortrapid flushing

Well mixed

1

UT (m s−1)

Ur (

m s

−1)

f0055 Figure 11 Stratification contours obtained from the relation that includes the influence of tidal flows (abscissa) and river flows (ordinate). Only thecontours that approximately separate stratified from mixed estuaries are shown. Modified from Geyer, R.W., 2010. Estuarine salinity structure andcirculation. In: Valle-Levinson, A. (Ed.), Contemporary Issues in Estuarine Physics. Cambridge University Press, Cambridge, pp. 12–26.

Dep

th (

m)

0

−5

−10−1 −0.5 0 0.5

u (m s−1) 10−3 (m2 s−1)Az

1 1.5 0 10 20

712 1 8

11 910

2

6 35

4

34 2 5 1 6 7

12 8 11 910

Shear stress, 10 −4 Pa

1050−5−10

(a) (b) (c)5 64 3

72 8 1

10 12

9 11

f0060 Figure 12 Numerical results from an idealized flat bottom estuary for which lateral variations are neglected. Numbers indicate hours after the end of ebb.Continuous lines are for flood phases. (a) Evolution of tidal velocity profiles throughout the tidal cycle. Positive is seaward. (b) Evolution of the eddyviscosity profiles through the tidal cycle. (c) Evolution of the stress profiles throughout the tidal cycle. From Cheng, P., Valle-Levinson, A., 2010. Estuarineresidual flows produced by tidal asymmetries in mixing, submitted to J. Geophys. Res.

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as expected, with surface outflow and bottom inflow(Figure 13). The strongest gravitational circulation occursat the region of the estuary with largest horizontal densitygradient, which is near the estuary mouth. Tidal residualflows indicate net outflow throughout the water columnwith a similar distribution to ur, indicating that tidal residualflows enhance the gravitational outflow but oppose thegravitational inflow. By contrast, tidal asymmetries in mix-ing produce a residual circulation that reinforcesgravitational circulation. It even has a larger magnitudethan ug and is strongest at the mouth, where the asymme-tries in mixing and tidal currents are largest in the estuary.The sum of all contributions to ue, as in eqn [1], is essen-tially the same as the straight tidal average of tidal flowsalong the idealized estuary. Therefore, in a vertically mixedestuary, the estuarine circulation can be reliably representedas the linear combination of the four contributors discussed.

p0145 In a weakly stratified estuary, the main driver of estuarinecirculation that is markedly modified is the circulation pro-duced by tidal asymmetries in mixing (Figure 14). In thiscase, ua exhibits a three-layered structure in the estuarine region(where the density driven flow is most robust), representinginflow at surface and bottom. Therefore, ua competes with

gravitational circulation in portions of the water column butreinforces ug in others. The magnitude of ua is similar to that ofud and therefore it is still relevant. Once again, in a weaklystratified estuary, the estuarine circulation can be dependablyrepresented as the linear superposition of differentmechanisms.

p0150In a strongly stratified estuary (Figure 15), ut and ua aredifferent from the other two types of estuaries. The tidalresidual flow shows a two-layered pattern that acts in con-cert with the density-driven flow, but the flow driven bytidal asymmetries in mixing opposes it. The magnitude ofua is smaller than ud so its influence, albeit important, isless than in the other types of estuaries. It can be seen thatthe linear superposition of the four mechanisms is a goodrepresentation of the estuarine circulation. There is nodoubt that the mechanism that drives ua should be impor-tant in estuaries and its influence requires more detailedobservational scrutiny. A relevant concept advanced inthis chapter is that estuarine circulation ue is not necessarilythe same as gravitational circulation ug. As discussed above,ue and ug will be the same only if the flows produced bytidal rectification and tidally asymmetric mixing arenegligible.

Dep

th (

m)

Dep

th (

m)

Dep

th (

m)

0

0.8

1.2 1.6

0.8

ur ud

uaut

ur + ud + ut + ua

−4

−8

0

−4

−8

−4

−8

240 260 280x (km) x (km)

260 280240

0

0.4

0

0

−2

−2

−2

4

8

8

35

15

10

25

5

0

−5−515

5

0

20

30

12 12 12

4

0

2

646

88

4 8

4

0

2016

12

−4

8−

−8

−4

35

2015

10

0−5

−5

0

−10

3025

5

105

u

f0065 Figure 13 Along-estuary distributions, derived from a numerical model, of the different mechanisms that produce estuarine circulation and of tidallyaveraged flow in an idealized vertically mixed estuary. Shaded contours indicate negative flows (inflows). The bottom right panel is the straight tidalaverage, whereas the bottom left panel is the sum of each individual contribution. From Cheng, P., Valle-Levinson, A., 2010. Estuarine residual flowsproduced by tidal asymmetries in mixing, submitted to J. Geophys. Res.

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epth

(m

)D

epth

(m

)D

epth

(m

)

0

ur ud

uaut

ur + ud + ut + ua

−4

−8

0

−4

−8

−4

−8

240 260 280x (km) x (km)

260 280240

0

0

u

1

2

4 8 1616

12

4

4

3026

10

5

20

15

25

20

20

3

5

0

−5

−5−5

0

−5

−5−10

−5

05

5

10 1520

2530

35

−50

5 55

0

5101

5

20−15

−3

−3

−3

−50−5−10

0

036

8

0

1518 1215

f0070 Figure 14 Same as Figure 13 but for a weakly stratified estuary. From Cheng, P., Valle-Levinson, A., 2010. Estuarine residual flows produced by tidalasymmetries in mixing, submitted to J. Geophys. Res.

Dep

th (

m)

Dep

th (

m)

Dep

th (

m)

0

ur ud

uaut

ur + ud + ut + ua

−4

−8

0

−4

−8

−4

−8

220 240 260 280 220 240 260 280x (km) x (km)

0

u

5

1020

0

5101520 25

0

102050 50

0

0 0 0

20305060 70 70

10

0

0

10

010

10

304060 70

50

0

0

0

20

0

10

010

f0075 Figure 15AU1 Same as Figure 13 but for a strongly stratified estuary. From Cheng, P., Valle-Levinson, A., 2010. Estuarine residual flows produced by tidalasymmetries in mixing, submitted to J. Geophys. Res.

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References

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bib0010 Geyer, R.W., 2010. Estuarine salinity structure and circulation. In: Valle-Levinson, A.(Ed.), Contemporary Issues in Estuarine Physics. Cambridge University Press,Cambridge, pp. 12–26.

bib0015 Gilcoto, M., Pardo, P.C., Álvarez-Salgado, X.A., Pérez, F.F., 2007. Exchange fluxesbetween the Rı´a de Vigo and the shelf: a bidirectional flow forced by remote wind.Journal of Geophysical Research 112, C06001. doi: 10.1029/2005JC003140.

bib0020 Hansen, D.V., Rattray, M., Jr., 1966. New dimensions in estuary classification.Limnology and Oceanography 11, 319–325.

bib0025 Ianniello, J.P., 1977. Tidally induced residual currents in estuaries of constant breadthand depth. Journal of Marine Research 35, 755–786.

bib0030 Largier, J., 2010. Low-inflow estuaries: hypersaline, inverse and thermal scenarios. In:Valle-Levinson, A. (Ed.), Contemporary Issues in Estuarine Physics. CambridgeUniversity Press, Cambridge, pp. 247–272.

bib0035 Officer, C.B., 1976. Physical Oceanography of Estuaries (and Associated CoastalWaters). Wiley, New York, NY, 465 pp.

bib0040Prandle, D., 2009. Estuaries: Dynamics, Mixing, Sedimentation and Morphology.Cambridge University Press, Cambridge, 246 pp.

bib0045Pritchard, D.W., 1956. The dynamic structure of a coastal plain estuary. Journal ofMarine Research 15, 33–42.

bib0050Stacey, M.T., Fram, J.P., Chow, F.K., 2008. Role of tidally periodic density stratificationin the creation of estuarine subtidal circulation. Journal of Geophysical Research113, C08016. doi: 10.1029/2007JC004581.

bib0055Valle-Levinson, A. (Ed.), 2010. Definition and classification of estuaries. In:Contemporary Issues in Estuarine Physics. Cambridge University Press, Cambridge,pp. 1–11.

bib0060Valle-Levinson, A., Bosley, K.T., 2003. Reversing circulation patterns in a tropicalestuary. Journal of Geophysical Research 108 (C10), 3331. doi: 10.1029/2003JC001786.

bib0065Valle-Levinson, A., Delgado, J.A., Atkinson, L.P., 2001. Reversing water exchangepatterns at the entrance to a Semiarid Coastal Lagoon. Estuarine, Coastal and ShelfScience 53, 825–838.

bib0070Valle-Levinson, A., Sarkar, N., Sanay, R., Soto, D., León, J., 2007. Spatial structure ofhydrography and flow in a Chilean Fjord, Estuario Reloncaví. Estuaries and Coasts30 (1), 113–126.

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