Relating River Plume Structure to Vertical Mixingpong.tamu.edu/~rob/pubs/hetland_jpo_2005.pdf ·...

Relating River Plume Structure to Vertical Mixing

ROBERT D. HETLAND

Department of Oceanography, Texas A&M University, College Station, Texas

(Manuscript received 30 March 2004, in final form 24 February 2005)

ABSTRACT

The structure of a river plume is related to the vertical mixing using an isohaline-based coordinate system.Salinity coordinates offer the advantage of translating with the plume as it moves or expanding as the plumegrows. This coordinate system is used to compare the relative importance of different dynamical processesacting within the plume and to describe the effect each process has on the structure of the plume. Verticalmixing due to inertial shear in the outflow of a narrow estuary and wind mixing are examined using anumerical model of a wind-forced river plume. Vertical mixing, and the corresponding entrainment ofbackground waters, is greatest near the estuary mouth where inertial shear mixing is large. This region isdefined as the near field, with the more saline, far-field plume beyond. Wind mixing increases the mixingthroughout the plume but has the greatest effect on plume structure at salinity ranges just beyond the nearfield. Wind mixing is weaker at high salinity classes that have already been mixed to a critical thickness, apoint where turbulent mixing of the upper layer by the wind is reduced, protecting these portions of theplume from further wind mixing. The work done by mixing on the plume is of similar magnitude in both thenear and far fields.

1. Introduction

River plumes are central to a number of importantsocietal oceanographic problems. For example, a toxicdinoflagellate, Alexandrium spp., is associated with thethe Kennebec–Penobscot River plume in the Gulf ofMaine (Franks and Anderson 1992). Stratificationcaused by Mississippi–Atchafalaya outflow preventsventilation of lower-layer waters, allowing hypoxic con-ditions to develop on the continental shelf (Rabalais etal. 1999). Nearly one-half of all oceanic carbon burialoccurs in large river deltas (Hedges and Keil 1995).

Many papers have reported on the various featuresof river plumes, particularly a recirculating bulge thatforms in the vicinity of the outflow (e.g., Garvine 1987;O’Donnell 1990; Yankovsky and Chapman 1997; Fong1998; Nof and Pichevin 2001; Garvine 2001; Yankovskyet al. 2001) and the on- and offshore motion of theplume in response to upwelling and downwelling windstresses (Fong et al. 1997; Pullen and Allen 2000; Fong

and Geyer 2001; García Berdeal et al. 2002; Hetlandand Signell 2005). However, analysis of the plume isdifficult, particularly interpreting observations, becauseof the changing position of the plume. In the case ofwind forcing, the plume may change position so muchthat, at many points, the plume may be only occasion-ally present. Also, even when the plume is present, dif-ferent regimes of the plume may be measured, for ex-ample, frontal regions versus the core of the plume.Many of these difficulties stem from a Cartesian, orEulerian, view of the plume.

This paper examines the plume in salinity coordi-nates, a natural coordinate system for the plume. Al-though this approach is not Lagrangian, in that theplume may be steady in salinity space even while waterflows through it, salinity coordinates offer the advan-tage of translating with the plume as it moves or ex-panding as the plume grows. For example, the additionof wind causes the plume to mix as well as to changehorizontal position. Salinity coordinates are used tohere to examine changes in vertical mixing in isolationby following the plume as it is shifted by currents.

The focus in this paper is buoyancy-forced flow fromnarrow estuaries, where the local internal deformationradius is larger than the width of estuary mouth. Forthe narrow-estuary case, as water leaves the estuary, it

Corresponding author address: Robert D. Hetland, 3146TAMU, Department of Oceanography, College Station, TX77843-3146.E-mail: [email protected]

SEPTEMBER 2005 H E T L A N D 1667

© 2005 American Meteorological Society

JPO2774

spreads and shoals, becoming supercritical. Here, theestuary mouth acts as a hydraulic constriction for theupper layer (Armi and Farmer 1986). The acceleratingflow soon becomes unstable, and strong shear mixingoccurs in the near field (Wright and Coleman 1971;MacDonald 2003). Beyond this region, mixing is pri-marily caused by wind stress through a mechanism de-scribed by Fong and Geyer (2001). Ekman transport inthe upper layer may become large enough that shearinstability is induced. At this point, the plume will mixand thicken until the local Richardson number is againabove the critical value. This model of wind mixingconsiders only the local wind stress and stratification. Itis not yet clear how this balance affects, and is affectedby, the horizontal plume structure. A cartoon of thevarious dynamical regions within the plume shown inFig. 1 demonstrates the mixing history of a water parcelas it leaves the river/estuary and eventually becomespart of the background waters.

The goal of this paper is to relate vertical mixing indifferent dynamical regions of the plume to changes inplume structure. In particular, this paper will comparethe relative importance of different dynamical pro-cesses acting within different parts of the plume, thestructure of the plume, and the role of wind mixing indetermining that structure. Garvine (1999) notes thatthe steady-state alongshore scale of a river plume with-out wind forcing depends on, among other things, thevalue of background mixing, the minimum value fordiffusivity, and viscosity in a turbulence closure scheme.Garvine’s results show that increasing the backgroundmixing decreases the alongshore scale of the plume(roughly related to the total area of the plume). Thispaper expands Garvine’s basic result by including wind

stress and by relating the size of the plume to differentvertical mixing processes.

2. Salinity coordinates

Interpreting measurements of a river plume in Car-tesian space may be difficult—for instance, the plumemay be only occasionally present at certain locations.The analysis methods presented below are less sensitiveto the motions of the plume, because these methodsconsider the water mass structure of the plume as awhole using a coordinate system based on salinity. Thissalinity-based coordinate system follows the plume as itmoves and allows the freshwater introduced into thedomain to be followed as it is mixed with the back-ground waters. The analysis presented below is basedon the approach of MacCready et al. (2002), who ex-amine long-term estuarine salt balances by calculatingsalt fluxes across isohalines. The derivation below ex-tends MacCready et al.’s analysis by demonstratinghow changes in isohaline surface area can be used toestimate salt flux at particular salinity classes within theplume, rather than across the entire isohaline surface.

Here, we will consider a volume V bounded by thesea surface and ocean floor, a face within the riverwhere s � 0, and on the seaward edge by an isohaline,sA (the shaded area in Fig. 2). A portion of the bound-ing area A defined by the isohaline surface sA com-pletely divides fresher plume water (s � sA) from therest of the ocean. There is a net freshwater flux of QR

across the face of the volume within the river.The three-dimensional salt balance equation,

�s

�t� � · �su� � �� · f, �1�

FIG. 1. This conceptual model of river plume anatomy shows the major regions andindicates the dominant mixing mechanisms.

1668 J 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 VOLUME 35

integrated over volume V is

�

�t �V

s dV � sA �A

uA · dA � sA �A

u · dA

� ��A

f · dA, �2�

where u is the three-dimensional flow vector, uA is thenormal velocity of the surface A itself (such that u � uA

is the flow through A), and f is the diffusive salt flux.The generalized Leibnitz theorem (Kundu 1990, p. 75)is used to take the time derivative outside the integralin the first term. The advective and diffusive salt fluxesthrough the faces of V are nonzero only on the isoha-line surface A. Because sA is defined to be constantalong A, sA may be taken outside the area integrals. Astatement of mass balance within the volume, againderived by integrating over V, is

�V

�t� �

A

uA · dA � �A

u · dA � QR. �3�

The mass and salt balance equations are combined toform

sA

�

�tVfA � sAQR � �

A

f · dA, �4�

where

VfA�sA� � �V

sA � s

sAdV �5�

is the freshwater content within V, relative to sA. If Eq.(4) is divided by sA, an intuitive freshwater balance isformed—the freshwater from the river, QR, must eitherincrease VfA in time or be compensated by a freshwaterflux, f/sA, across A.

By knowing QR and the change in freshwater contentover time, the average salt flux, f across A may be es-timated. However, it is expected that within the riverplume the flux will change at different points within theplume, so that an area average of the salt flux over alarge isohaline may be difficult to interpret: are changesin average flux due to intense localized mixing orbroad-scale changes?

Assuming a thin pycnocline, the salt flux across twoisohalines within the pycnocline will be similar. Thismay be used to derive an estimate of the salt flux as ata particular salinity class, instead of as an averageacross an entire isohaline surface. For example, assumethat the plume is approximated as a single, active layerwith horizontally varying salinity overlying a quiescentlayer with a uniform, background salinity of s0. Thearea A may now be related exactly to upper-layer sa-linity, sl, alone:

A�sl� � ��s�sl

dA�z��

. �6�

The integral over A may be converted to an integralover sl by converting dA to �A/�sl dsl. DifferentiatingEq. (4) with respect to sA, noting for the case of a thinpycnocline that sA � sl in the area integral, gives

FIG. 2. The volume V, bounded on the seaward edge by area A, is shaded gray; the area Ais defined by isohaline s � sA. A freshwater flux, QR, is input into the volume on the oppositeface of V. The second isohaline, sA � s � sA, is used in an example in the text. The differencein isohaline area between sA � s and sA is A.


�

�t

�

�sA�sAVfA� � QR � fA

�A

�sl, �7�

where fA is the average vertical salt flux associated withsalinity class sA. If Eq. (7) is differentiated with respectto sA, it becomes a local statement of the freshwaterbalance at salinity class sA, just as Eq. (5) becomes aglobal statement of the freshwater flux after dividingby sA.

A steady-state version of Eq. (7) may be derivedheuristically by taking the difference of Eq. (4) evalu-ated at two neighboring isohalines, sA and sA � s. Be-cause the pycnocline is thin, the salt flux across theoverlapping area is identical. Here, fA is the average saltflux over A, the difference in the two areas considered(see Fig. 2). In the limit where s → 0, the steady-stateform of Eq. (7) is recovered.

3. Numerical setup

Model configuration

The archetype for this configuration used in this pa-per is the Kennebec River plume in the Gulf of Maine,which has been the subject of a number of observa-tional and numerical studies (Fong et al. 1997; Hetlandand Signell 2005; Geyer et al. 2004). The simulationsemploy version 2.1 of the Regional Ocean ModelingSystem (ROMS; Haidvogel et al. 2000). ROMS is afree-surface, hydrostatic, primitive equation oceanmodel that uses stretched, terrain-following coordi-nates in the vertical and orthogonal curvilinear coordi-nates in the horizontal. The code design is modular, sothat different choices for advection and mixing, for ex-ample, may be applied by simply modifying preproces-sor flags. ROMS is open source and freely available.

The numerical domain is a narrow estuary attachedto a uniformly sloping shelf with a straight coastline(Fig. 3). The oceanic part of the domain is approxi-mately 250 km long and 80 km wide, with variable reso-lution concentrated near the estuarine outflow regionand along the coast. Resolution is decreased near theedges to inhibit small-scale alongshore variability at thenorthern and southern boundaries. The model has 20vertical s layers, with resolution focused near the sur-face (s-coordinate parameters used are hc � 10 m, s �5.0, and b � 0.01). This is equivalent to better than 1-mvertical resolution in the upper 5 m of the water columnover the entire domain. Conservative splines are usedto estimate vertical gradients.

The estuary is 10 m deep, approximately 20 km long,and 1.5 km wide. Freshwater is introduced as a bound-ary condition on the westward end. Tides are not ex-plicitly modeled. Tidal mixing within the estuary is pa-

rameterized by specifying a constant vertical diffusivity/viscosity of 1 � 10�4 m2 s�1, so that the length scale ofthe salt intrusion is not significantly greater than thechannel length, and the time scale of estuarine adjust-ment is rapid enough to come into a steady state withinapproximately one upwelling/downwelling period (Het-land and Geyer 2004). The coastal ocean has a 10-mwall along the coast, and the bottom has a uniformslope of 1/1500, resulting in a maximum depth at theeastern boundary of about 70 m. Horizontal resolutionof 500-m resolution in the immediate vicinity of theoutflow results in three grid points across the estuarymouth.

Austin and Lentz (2002) show how the Ekman layermay be shut down on a stratified shelf under wind forc-ing when the near-shore water becomes vertically wellmixed, trapping water in the shallow region near thecoast. In other simulations in which the topography waskept very shallow along the coast with a very weakbottom slope (not shown here), this caused a portion ofthe plume to be trapped near the coast, even when theplume was upwelled. Regions with stronger near-shorebottom slopes, such as in the vicinity of the KennebecRiver plume, are not affected by this process. Also, inthis idealized study, maintaining simply connected iso-haline surfaces is important in these first attempts atisohaline analysis. Lentz and Helfrich (2002) show howthe propagation speed of a coastally trapped buoyantjet depends on the bottom slope, and it is plausible thatthis parameter is important in wind-driven plumes aswell. The present study considers only the steep slopelimit, as topographic dependence is not the focus of thisstudy.

Garvine (2001) suggests that model configuration isimportant in influencing some of the aspects of simu-lated river plumes. In particular, he notes that main-taining a very shallow depth at the coast prevents theformation of a backward-propagating (against theKelvin wave propagation direction) bulge at the estuarymouth. Practically, the shallowest possible coastal walldepths are a few tens of centimeters in models likeROMS that do not support wetting and drying. In thisstudy, even in the cases with no wind and no back-ground flow (not shown here), a backward-propagatingbulge was not a problem.

The model is initiated with no flow and a flat seasurface. The initial tracer distribution is uniform back-ground salinity of 32 psu, with vertical temperaturestratification typical for an east coast continental shelfin summer (a 10-m homogeneous mixed layer aboveexponential stratification with a 20-m decay scale, rang-ing from 20°C at the surface to 5°C at depth). Theplume never interacts directly with the thermocline.


The estuary was initialized with a vertically uniformalong-channel salt gradient, linearly transitioning fromoceanic salinity to freshwater with a 50-km length scale,in order to decrease the estuarine adjustment time. Themodel is forced with freshwater at the river end of theestuary and spatially uniform but temporally oscillatingnorth–south wind stress; both are ramped over 1 day.The depth-integrated flow at the eastern boundary isset equal to the Ekman transport.

The northern boundary (the upstream boundary inthe Kelvin wave sense) depth-integrated flow is relaxedto results from a two-dimensional experiment to pre-vent drift in the alongshore transport through the do-main. In experiments without this boundary condition,the transport averaged over an upwelling/downwellingcycle tended to drift O(0.10 m s�1). The two-dimen-

sional experiment has no mass flow into the domainfrom the river, but identical wind stress. The domain isidentical, except that it is periodic in the north–southdirection, with the same cross-shore topography as thethree-dimensional simulations. The flow is initially atrest, with no sea surface height anomalies. The easternboundary transport is again set to the Ekman transport.A mean alongshore background flow is added to thetwo-dimensional results before they are applied to thethree-dimensional model.

In the three-dimensional simulations, mass is con-served except for the gain and loss of mass due to theEkman transport through the eastern open boundaryby requiring the northern and southern boundariescarry the same alongshore transport. This is accom-plished by integrating the flow along these two bound-

FIG. 3. The model domain includes a flat-bottomed, prismatic estuary attached to a uni-formly sloping coast. The shallowest depths are 10 m in the estuary and at the coast, and 70m along the seaward edge. (upper right) A detail of the grid shows the model grid focused nearthe estuary mouth, with 500-m resolution in this region. Moderate resolution increases gradu-ally away from this point to 3 km, until very near the edges where it is more telescoped tomuch coarser resolution at the boundaries. This was done to increase the domain size andreduce grid-scale noise at the boundaries.


aries, then applying a small transport along all openboundaries to correct any mass transport imbalances.

All of the simulations have a freshwater flux of 1000m3 s�1 applied at the river end of the estuary, rampedup over a day to prevent numerical shocks from form-ing. For the cases with wind, a spatially uniform, oscil-latory in time, alongshore wind stress with an amplitudeof 0.5 � 10�4 m2 s�2 and a period of 4 days was used. Amean background current, flowing in the Kelvin wavepropagation direction, of �0.05 m s�1 is specified at thenorthern boundary (in addition to the two-dimensionalwind-driven transport). The main effect of the back-ground current is to increase the alongshore freshwaterflux in the wind-driven simulations.

The numerical configuration uses fourth-order hori-zontal advection for tracers with a grid-scaled horizon-tal diffusivity equivalent to 10.0 m2 s�1 for a 1 km2 gridcell (ranging from 5.0 to 50.0 m2 s�1 in the resolvedportion of the domain). Many numerical models arerun with a much smaller horizontal diffusivity, underthe assumption that horizontal mixing should be kept aslow as possible. However, in earlier simulations therewas considerable numerical noise near the estuarymouth in salinity that considerably affected the watermass structure of the plume, in some cases causing spu-rious numerical mixing comparable to the vertical mix-ing calculated within the turbulent closure scheme. Itshould be noted that the Mellor–Yamada scheme hadvery little noise even when using a horizontal diffusionnearly two orders of magnitude smaller. A third-order,upwind scheme is used for horizontal momentum ad-vection, with no explicit horizontal viscosity applied.

Plume dimensions, such as bulge diameter, are sur-prisingly insensitive to advection scheme, providing thedomain has adequate resolution near the estuary mouth(approximately three to five grid points across themouth). The qualitative structure of the solution is al-ways the same, even using poor resolution and a low-order advection scheme. All of the simulations withoutwind have a bulge forming directly downstream of theestuary outflow. The largest difference between advec-tion schemes is in the formation of spurious fresh anddense water. Numerical over and undershoots in thevicinity of the front would create slightly saltier andfresher water on both sides of the front. The freshwateris lighter than the surrounding water and remains at thesurface. However, the saltier water is denser than thesurrounding water and sinks. This artificial unmixingcreates a pool of salty water along the seafloor, as wellas a spurious source of freshwater near the surface. Inthe simulations presented in this paper, since back-ground salinity is constant, the magnitude of the spuri-

ous freshwater source may be calculated by equating itto the spurious, salty bottom water.

In simulations (not shown) with poor resolution orusing low-order advection schemes, the spurious fresh-water source may reach 10% of the real, specified fresh-water source. The spurious freshwater source can bereduced by applying a moderate horizontal mixing. Us-ing higher resolution and the Mellor–Yamada scheme(which produces the smoothest fields), even when usingvery weak horizontal mixing, reduces the spuriousfreshwater flux to essentially nil. The spurious freshwa-ter flux is 0.5% of the specified freshwater flux over thecontinental shelf region with no wind forcing and0.05% with wind forcing.

Two common turbulence closure schemes are used tocalculate vertical mixing: Mellor–Yamada and k–�(Mellor and Yamada 1974; Umlauf and Burchard2003). The differences in plume water mass structurebetween different grid resolutions and advectionschemes (assuming at least three grid points across theestuary mouth, and at least third-order advection) aredwarfed by the differences in water mass structure us-ing different vertical closure schemes. Because thestructure of vertical mixing is an important theme inthis paper, numerical results both schemes have beenincluded to gain a basic appreciation of the sensitivityof the results presented to the choice of closure scheme.The background, or minimum, mixing used was identi-cal for both closures: 5 � 10�6 m2 s�1 for both momen-tum and tracers. These and other parameters used bythe closure schemes were the default parameters forROMS version 2.1. Both shear and stratification wereaveraged horizontally before mixing rates were calcu-lated. The Kantha–Clayson stability function formula-tion was used for the Mellor–Yamada scheme (Kanthaand Clayson 1994), the Galperin stability function for-mulation was used for the k–� scheme (Galperin et al.1988).

The configuration used in this paper was chosen tobalance numerical accuracy with computational speed.This type of configuration could be applied to a realisticriver plume simulation without significant modifica-tions to the standard code, and with reasonable inte-gration time on modern computers. It takes about 9 h ofwall clock time to integrate the simulation for a monthon a single 2.3-GHz Intel Pentium-4 processor.

4. Results

a. Plume structure in physical space

1) WIND-FORCED CASE

The plume moves off- and onshore in response toupwelling and downwelling wind stresses, respectively.


Results for the case with wind forcing is shown in Fig.4. For this figure, the Mellor–Yamada mixing scheme isused for this particular simulation; the results from thek–� closure are qualitatively similar. The upper panelsshow sea surface salinity with surface currents overlaid,and lower panels show the freshwater thickness for thefinal four days of the simulation. Freshwater thickness,relative to a reference salinity s0, is defined as the ver-tical integral of the salinity anomaly, (s0 � s)/s0. Theplume changes position considerably over one cycle ofthe wind stress forcing. During upwelling, the plumenearly loses contact with the coast, a trait that becomesmore pronounced as the duration of upwelling in-creases. During downwelling, the plume is pressedagainst the coast, developing a strong coastal current.The plume’s response to upwelling and downwelling isnot symmetric: during upwelling the plume is blownoffshore, during downwelling the plume is pushedalongshore. This causes most of the alongshore fresh-water flux to occur during downwelling conditions;alongshore freshwater flux is essentially halted duringupwelling. Wind also affects the thickness of the plumeby stretching the plume out as it moves away from theshore during upwelling, and pressing plume to the coastduring downwelling. This affect is seen clearly in thesnapshots of freshwater thickness (Fig. 4), where thefreshwater thickness is least after upwelling, and great-est after downwelling.

2) NO-WIND CASE

Figure 5 shows the properties of the plume with nowind forcing on day 16 of the simulation. A bulge hasformed near the outflow, with a recirculating gyre. Thebulge grows in time; as noted by Fong (1998), only aportion of the freshwater introduced continues down-coast as a coastal current. In the case presented here,about half of the freshwater input into the domain iscarried away by the coastal current. Because of this,freshwater accumulates within this bulge and the bulgeexpands and thickens. The vertical salt flux along thesurface defining the upper layer ( �s/�z|z��h, discussedin more detail below) is plotted in the third panel of Fig.5 and is used to estimate the regions where verticalmixing is strong. There is a region of strong verticalmixing near the estuary outflow about two orders ofmagnitude larger than the vertical mixing found in therest of the plume. When the distribution of maximumvertical salt flux is compared with the Froude number(the fourth panel in Fig. 5), it is apparent that the regionof high mixing is associated with supercritical flow. TheFroude number in the upper layer is defined operation-ally here as

Fr �u

�g�h, �8�

where

u � ��h

�

|u| dz, g� � g��

�0, and

�� 0 � ��h

�

�|u| dz��h

�

|u| dz, �9�

and the upper-layer thickness h is defined as the pointwhere � � 1⁄2(�0 � �min). Note that this definition of ��is used only for the Froude number calculations. Thedepth-dependent flow speed in the upper layer is |u|;the depth-dependent density is �. The integrations areover the upper layer, between the free surface and theinterface defined by h. The upper-layer density isweighted by the flow speed to be consistent with a layermodel.

The numerically simulated near-field outflow regionhas a similar structure to that described by Wright andColeman (1971), where the flow from South Pass isshown to rapidly shoal at the mouth of the pass, withFroude numbers over 2 just past the point where thepycnocline shallows during ebb tide. Beyond this point,the flow decelerates and becomes saltier due to entrain-ment of denser, sluggish background waters. In ap-proximately 8–10 channel widths (depending on thephase of the tide) the South Pass outflow has entrainedenough background water so that the Froude number isbelow 1. Note that it is the decrease in momentum,rather than the increase in density, that is responsiblefor the decrease in the Froude number due to entrain-ment.

In the simulations presented here, the estuary mouthacts as a constriction. The results of Armi and Farmer(1986) show how flow through a constriction must besupercritical, even when the estuarine exchange is notmaximal (Stommel and Farmer 1953). The simulatedestuarine exchange is not maximal. Hetland and Geyer(2004) argue this is not expected for a prismatic estuarychannel), but the simulated upper-layer outflow is su-percritical near the mouth of the estuary. Thus, thesimulated estuarine outflow is similar to flow through aconstriction in the case where only the upper-layer flowbecomes supercritical (e.g., the bottom three panels inFig. 2 of Armi and Farmer 1986). One notable differ-ence is that Armi and Farmer (1986) show a relativelygradual transition to higher Froude numbers, with arapid transition back to subcritical flow in the form of ahydraulic jump. In the numerical simulations presentedhere, as well as in the results of Wright and Coleman(1971), the transition to supercritical flow by shoaling


FIG. 4. (top) Sea surface salinity and (bottom) freshwater thickness are plotted on a color scale for the final 4 days of the simulationwith wind using the Mellor–Yamada closure. Instantaneous surface current vectors are overlaid in the upper panels. Contours in thelower panels are sea surface salinity. The yellow arrow indicates the direction of the wind during maximum upwelling and downwelling;the yellow dot indicates zero wind stress at that instant.


Fig 4 live 4/C

of the pycnocline is relatively rapid (occurring over lessthan 1 km) in relation to the gradual transition back tosubcritical flow through mixing (occurring over ap-proximately 5 km). This difference is due to the pres-ence of mixing within the model and actual plumes thatis absent in the inviscid solutions of Armi and Farmer.

3) PLUME REGIONS

Approximate salinity ranges may be estimated forthe boundaries between the estuary, near-field, and far-field regions. The salinity ranges of the three plumeregions are slightly different, depending on if the plumeis considered in three-dimensional space, or as a singleactive layer (see the appendix for more exact defini-tions of each view of the plume). These slight differ-ences are to be expected, given the different definitionsof plume salinity. However, the three regions are al-ways discernible and the salinity ranges are similar forboth views of the plume. The estuarine outflow surfacesalinity (s|z � �) ranges from 15 to 18 psu, with highersalinity outflow during downwelling. The upper-layersalinity (sl) leaving the estuary ranges from 18 to 20 psu,also with higher salinity during downwelling. The salin-ity range of the near field is seen most clearly in thecases without wind. Water leaving the near field has anupper-layer salinity of approximately sl � 25–26 psu.This water recirculates within the bulge that forms justdownstream, in the Kelvin wave sense, of the estuarine

outflow, creating a thick, homogeneous mass of waterwithin the bulge.

b. Plume structure in salinity space

1) FRESHWATER BUDGET

Freshwater volume within different salinity classes isused in order to examine changes in whole plume struc-ture. The freshwater volume, relative to the referencesalinity, s0, is defined as the integral of the freshwaterfraction

Vf�sA� � ��s�sA

s0 � s

s0dV, �10�

where the volume integral is bounded by the isohalinesA, such that all of the water fresher than sA is containedin the integral. To determine the distribution of fresh-water as a function of salinity class, �Vf /�sA is plottedfor the two turbulent closure schemes. Integrating �Vf /�sA over a range of salinities will give the total fresh-water contained within those salinity classes. The inte-gral over the entire range is identical for all cases, sincethe each case has the same freshwater input.

As mixing increases, freshwater will generally bemoved toward higher salinity classes. Freshwater distri-butions are shown for the two turbulent mixing closuresused in this paper (Fig. 6, top). Although the qualitativestructure is similar between the different closureschemes, there are significant differences in the actual

FIG. 5. Four properties of the plume are shown on day 16 of the simulation without wind: sea surface salinity (with surface currentvectors overlaid), freshwater thickness, a weighted average of the vertical salt flux [log( �s/�z)|z��h], and the Froude number. Contoursin the three right panels are sea surface salinity. Property definitions are given in the text.


Fig 5 live 4/C

amount of freshwater found at different salinity classes.Notably, the k–� closure seems to mix more than theMellor–Yamada closure for the case with wind. Be-cause of the large differences between these twoschemes, results that depend on the turbulence closureare presented using both schemes. The local maximumin �Vf /�sA for the no-wind case shows the buildup ofwater within the recirculating bulge at approximately24 psu.

For the wind-driven cases, the largest changes infreshwater enclosed by an isohaline, sA, are at theboundary between the near and far fields (sA � 26 psu).The lower panel of Fig. 6 shows the time-dependentanomalies of Vf as a percent of the mean. Freshwater islost from the region of high variability (24 psu � sA �28 psu) during upwelling, and the freshwater is replen-ished during downwelling. This variability is due en-tirely to changes in mixing within the plume and thecorresponding flux of freshwater across isohaline sur-faces, since sA otherwise follows the plume as it ismoved adiabatically.

2) HORIZONTAL PLUME STRUCTURE

The area Al, enclosed by the upper-layer salinity con-tour sl, is calculated as a function of sl for the fourmodel runs. Both turbulence closure schemes are

shown for cases with wind and no wind. Values of Al,averaged over one upwelling/downwelling cycle (theperiod of time between days 16 and 20), are shown inupper panel of Fig. 7. The results show that, betweenthe two turbulent closure schemes, the largest differ-ence in Al is within the near field, with surface salinitiesof approximately 20–26 psu, both with and withoutwind. Within the estuary (sl � 20 psu) and in the farfield (sl � 26 psu) the both schemes produce similarvalues for Al.

Wind affects both the near field and far field, with thelargest difference in Al at the interface between thenear and far fields. Wind has very little affect on Al

within the estuary and at very high salinity values. Anexample of time-dependent changes in Al due to windforcing is shown in the lower panel of Fig. 7, wherepercent changes in Al are are plotted as a function of sl

and time. These results are based on the Mellor–Yamada run with wind; results for the k–� scheme werequalitatively similar. The largest anomalies in Al format salinity ranges between the near and far fields justafter downwelling, similar to the wind-driven anomaliesin Vf. This high anomaly propagates toward higher sa-linity values during upwelling, again suggesting watermoves from the near field to the far field in a coherentpulse during upwelling.

FIG. 6. (top) The distribution of freshwater in salinity space, �Vf /�sA, compared for the twodifferent mixing schemes with and without wind forcing. (bottom) Time-dependent anomaliesin Vf, plotted as percent deviation from mean values with a contour interval of 10%, for theMellor–Yamada case with wind.


The variability in plume area contains both adiabaticplume motions, increases and decreases in area dueonly to the plume being stretched by upwelling andcompressed by downwelling, and nonadiabatic changesto plume structure due to mixing within the plume.Based on the freshwater anomalies shown in Fig. 6(bottom), which are entirely nonadiabatic, approxi-mately one-half of the variability in wind-forced plumearea is due to adiabatic motions, the other half to nona-diabatic mixing processes.

3) VERTICAL PLUME STRUCTURE

An average salinity profile was calculated as a func-tion of sea surface salinity by finding the average depthof each isohaline underneath a particular range of seasurface salinity. The area covered by each sea surfacesalinity range is integrated, and an along-plume coor-dinate with units of distance is calculated by taking thesquare root of this quantity. The isohaline depth andalong-plume distance calculated from each of the fourstandard cases were averaged over one upwelling/downwelling period (from days 16 to 20). The result isan idealized average cross section of the plume in sa-linity space. The results are shown as a salinity spaceprofile (black lines in Fig. 8). A similar calculation was

done with the layer model where upper-layer thicknesswas averaged over salinity classes (red lines in Fig. 8).Again, the square root of the area covered by eachsalinity range was integrated and summed along salinityclass to estimate an along-plume coordinate.

The two cases without wind are very similar. Thehalocline remains at a relatively constant depth with aslight decrease in depth at the end of the near-fieldregion. Both the Mellor–Yamada and k–� closure pro-duce a similar vertical structure at all portions of theplume. The horizontal changes discussed above are ap-parent, with the k–� scheme producing a larger near-field region.

In salinity space, the simulations with wind are, onaverage, thinner in the near field and thicker in the farfield when compared with the no-wind cases. The k–�simulations were 20% thinner in the near field and 20%thicker in the far field after the inclusion of wind. Windcaused the Mellor–Yamada simulations to be 10% thin-ner in the near field and 60% thicker in the far field.The wind-driven surface layer is less stratified in theMellor–Yamada case, and the halocline for the k–�case is shallower than the Mellor–Yamada case every-where. Both closure schemes show that, beyond theestuary, wind causes the plume to thicken; as the plumegets saltier, the decrease in plume thickness at the end

FIG. 7. (top) The area, Al, averaged over one upwelling/downwelling wind stress cycle(16–20 day) enclosed by the upper-layer salinity contour, sl. (bottom) Time-dependent anoma-lies, plotted as percent deviation from mean values with a contour interval of 25%, for theMellor–Yamada case with wind.


FIG. 8. Ideal cross sections of the salinity structure. Thin lines show isohalines with an interval of 1 psu. Thick lines are drawnevery 5 psu (e.g., 20, 25, and 30 psu). The rightmost thick line shows the 30-psu isohaline in all four panels.


Fig 8 live 4/C

of the near-field region seen in the no-wind case is ab-sent.

The largest time-dependent changes in wind-forcedplume thickness are at the interface between the nearand far fields, at the same point in salinity space as themaximum percent variability in plume area and fresh-water volume. The plume thins during upwelling, andthickens during downwelling, as expected. Time-dependent changes in plume thickness due to windstress (not shown) indicate that the plume thicknesschanges by approximately 60% at the interface be-tween the near and far fields, but only about 15% atboth higher salinity classes (sl � 30) and lower salinityclasses (20 � sl � 22). The Mellor–Yamada and k–�cases are similar, with the principal differences beingthe same as discussed above for the average profiles.

c. Far-field wind mixing

Away from the mouth of the estuary, mixing in theplume is due primarily to surface wind stress. The basicmechanism was described by Fong and Geyer (2001),who describe a one-dimensional model in which abuoyant layer is mixed by shear mixing. The shear be-tween the upper and lower layers is created by the Ek-man transport of the upper layer. If the Ekman trans-port is large enough to induce shear instability in theupper layer, it will mix, entraining lower-layer wateruntil the Richardson number rises above the criticalvalue. Fong and Geyer calculated the depth at whichthis criteria would be reached, based on a density dif-ference between the upper and lower layers and aspecified wind stress. In their calculation, the upper-layer density was approximated as constant.

This theory can be extended to include densitychanges in the upper layer due to entrainment of lower-layer water. Consider a layer of purely freshwater, withdensity �f and hf thick, overlaying denser (�0) oceanwater, infinitely deep. A wind stress, �, blows over thewater causing and Ekman transport in the upper layerof uh � �/�0 f, where u is the velocity in the upper layerand h is the upper-layer thickness. The upper layer mayundergo shear mixing, and entrain water from the un-derlying ocean when the bulk Richardson number, Ri� ��gh/�0�u2 (where �� is the density difference, and�u is the velocity difference between the two layers),becomes lower than a critical value Ric. Now, allow thedensity of the upper layer to decrease as deeper wateris entrained, so that hf ��f � h��, where ��f � (�f � �0)is the density difference between freshwater and thereference state, and

hf � ��H

� s0 � s

s0dz �11�

is the freshwater thickness. Solving for h � hc, the criti-cal upper-layer thickness when Ri � Ric, gives

hc �2�

�0 f� Richf g��f ��0

� 2hf �RicFrd � 2hf

Frd

Frc,

�12�

stating that the final upper-layer thickness dependsonly on the initial conditions, the value of Ric, and themagnitude of the (maximum) wind stress. This equa-tion may be cast in terms of an upper-layer, freshwaterFroude number, Fd � uf /cf, where uf � �/f�0hf and cf ��hf g��f /�0. This upper-layer, freshwater Froudenumber is similar to the densimetric Froude numberoften used in studies of estuarine circulation (e.g.,Hansen and Rattray 1966; MacCready 1999). If thecritical Richardson number is written as a criticalFroude number, Ric � 1/Fr2

c, the critical thickness maybe simply calculated as 2 times the freshwater thicknesstimes a ratio of the freshwater Froude number to thecritical Froude number.

Equation (12) may be converted to a critical salinityby conserving the total freshwater in the water column(hf � hc�s/s0) to get

sc � s0�1 �hf

hc� � s0�1 �

Frc

2Frd�. �13�

If the local salinity in the plume is less than sc, mixingwill occur, if the local salinity is greater than sc, thewater column is stable.

Equation (13) was derived assuming a slab-like upperlayer, with constant velocity and density. If the layer isconsidered to have uniform gradients, as in Fong andGeyer (2001), there will be an additional 2�1/2 factor onthe right-hand side. However, changes such as this areequivalent to changes in the critical bulk Richardsonnumber, and do not change the dynamical meaning ofthe equation.

After the plume leaves the near-field region, mixingdue to the inertia of the estuarine outflow is suppressedand wind mixing dominates. The critical thickness ofthe plume as a function of freshwater thickness a windstress [Eq. (12)] suggests that wind mixing may act tostabilize the water mass structure of the plume. Even ifthe upper layer is mixed by some other mechanism be-fore the maximum in wind stress occurs, wind mixingwill still mix the upper layer to the same thickness, solong as hc is larger than the thickness of the upper layerafter the initial mixing. However, this assumes that thehorizontal distribution of freshwater is given. It is likelythat this distribution is in some ways related to thewater mass structure itself. For instance, vertical thick-ness of the mixed layer will determine the speed of the


Ekman transport in the upper layer, and densityanomalies between the plume and the background flowwill determine the propagation speed of the coastal jet.If, on the other hand, the plume has already been mixedto or beyond the critical thickness, the plume will notmix further. This mixing could be caused by a previouslarge wind event, for example. In this case, the theorysuggests that the plume will be protected from furtherwind mixing; the water mass structure of the plume willbe determined by the largest previous wind event.

To examine how well this theory can predict the sa-linity structure of a river plume, the plume sea surfacesalinity was compared to the estimated critical salinityin the upper layer. In Fig. 9, the simulated sea surfacesalinity is compared to the critical upper-layer salinitycalculated from Eq. (12) (noting that hf S0 � h�S) usingthe local freshwater thickness and Ric � 3.0. Duringmaximum upwelling, there is a band of sea surface sa-linity associated with high mixing that close to the pre-dicted critical upper-layer salinity. These parts of theplume are being mixed by the wind, such that the sa-linity is not greater than the critical salinity. Locationsin the plume that have a salinity greater than the criticalsalinity are associated with lower mixing. These parts ofthe plume are not as affected by wind mixing becausethe salinity there is greater than the critical salinity.This suggests that these portions of the plume with sa-linity higher than the critical salinity are protected fromfurther wind mixing. During downwelling, nearly all ofthe plume has subcritical values of surface salinity, andstrong vertical mixing is confined to fresher waters inthe near-field region, where mixing always occurs re-gardless of the phase of the wind stress. This is in agree-ment with the freshwater volume analysis above, whichsuggests that freshwater is moved from lower to highersalinity classes during upwelling.

The value of Ric used is only an effective value, sinceactual mixing in the plume is controlled not only byshear mixing of the upper-layer Ekman flow, but alsoshear caused by geostrophic or inertial flow in additionto the Ekman flow. The enhanced shear in the upperlayer requires that the effective critical Richardsonnumber used in the theory here be larger than typicallyused, since the theory of critical thickness only includesEkman induced shear.

d. Average salt flux over one upwelling/downwelling cycle

After the plume has reached a quasi-steady state, theaverage entrainment into the plume over an upwelling/downwelling cycle can be calculated from the steadystate form of Eq. (7),

fA � QR� �A

�sA��1

, �14�

where A is the average area contained within isohalinesA over one upwelling/downwelling cycle.

A direct estimate of the salt flux is calculated byinterpolating the salt flux to the depth of the upper-layer thickness

f � ��s

�z�z��h, �15�

again averaged over one upwelling/downwelling cycle.Here, is the turbulent diffusivity calculated within themodel. A comparison between fA and f for the twoturbulence closure schemes is shown in Fig. 10. The twoestimates of the salt flux show the same structure foreach turbulence closure, with high salt flux in the near-field where shear mixing is strong, decreasing by nearlyan order of magnitude at higher salinities, where windmixing is the dominant entrainment process. The twoclosures, however, are distinct. The Mellor–Yamadascheme has a much higher salt flux in the near field ascompared with the k–� scheme.

The assumption of a steady state appears to be validfor salinity classes that have a time scale smaller thanthe period of forcing, here about 4 days. The upperpanel of Fig. 11 shows the plume time scales at differentsalinity classes by integrating the freshwater volumecontained within an isohaline and dividing this quantityby the freshwater flux. This provides a filling time, thetime it would take the freshwater flux, Qf, to replace thefreshwater volume, Vf, for each salinity class.

Different portions of the plume have different timescales with respect to both wind and freshwater forcingdue to the differences in the volume of water at differ-ent salinity classes. In the above estimate of salt flux, asteady state is assumed. The validity of this assumptionmay be estimated by taking a ratio of the time-depen-dent term to the freshwater forcing term in equation.Integrating over one upwelling/downwelling cycle ofperiod T gives

Change in fresh water contentFresh water flux input

� � �

�sA�sAVfA��

t

t�T

� �TQf��1. �16�

The ratio of these terms is plotted in the lower panel ofFig. 11. The steady-state assumption is valid for lower-salinity classes lower than a certain value. This valuechanges, depending on whether wind is included, but itseems to correspond to salinity classes that have a fill-ing times, Vf Q�1

f , of about 3–5 days.


FIG. 9. (top) The maximum vertical salt flux for extreme upwelling and downwelling winds in the upper two panels, and thecorresponding critical salinity vs (bottom) the actual plume surface salinity. In all, the logarithmic color scale represents a weightedaverage of the vertical salt flux ( �s/�z|z��h) for a particular horizontal point in the domain. The lower panels are based on Eq. (13).The solid line shows the relation sc � smodel; the dashed line shows a 5-psu offset, defined by sc � smodel � 5.


Fig 9 live 4/C

e. Work done by vertical mixing

Turbulent mixing does work against buoyancy byraising the center of mass of the water column. The rateof work, dW/dt, done by vertical mixing on the densitystructure of the plume, equal to the rate of nonadiabaticchanges in potential energy due to vertical mixing, iscalculated as g times a volume integral of the verticalsalt flux,

dW

dt� �

V

g��

�zdV, �17�

referred to simply as the rate of mixing work. In thecalculations here, only density changes due to salt fluxwere considered; this was accomplished by holdingtemperature constant (the mean temperature of the

surface mixed layer) when calculating the density. Po-tential energy changes due to changes in sea surfaceheight, adiabatic changes in plume structure, andchanges due to mixing of the thermocline were all largecompared to potential energy changes due to salt flux,and have been excluded from this calculation.

Wind increases the total rate of mixing work. Figure12 (top) shows time series of the rate of mixing work forthe four standard cases. Work done by mixing is highestwhen the wind stress is greatest. Wind increases thework done by mixing for both closure schemes, with thetotal rate of mixing work using the Mellor–Yamadascheme greater than the k–� scheme. The asymmetry inrate of mixing work due to the wind is also greater inthe Mellor–Yamada scheme, with most work done dur-ing periods of upwelling wind. For the case without

FIG. 10. Two estimates of fA, the vertical salt flux at salinity class sA, are shown. Red lines show �s/�z|z��h, the vertical salt fluxcalculated from the model at the base of the upper layer (z � �h). The blue lines are calculated from Eq. (14), assuming a steady state.Both estimates have been filtered so that the resolution in salinity space is approximately 1 psu. Thin gray lines in the upper panelrepresent the colored lines in the bottom panel, and vice versa..


Fig 10 live 4/C

wind, the rate of mixing work calculated using the k–�closure is 60% larger than that calculated using theMellor–Yamada scheme, excluding mixing in the estu-ary. Mixing within the estuary is due only changes in thedensity of water entering the estuary, since mixing isheld constant within estuary.

The combined rate of mixing work done in the estu-ary and near field is comparable to the rate of mixingwork done in the far field, and the rates of work donein the estuary and near field are similar. The additionalrate of mixing work added by the wind, calculated asthe rate of work in the case with no wind subtractedfrom the case with wind, is about 10% less than the rateof work in the no wind case (again, excluding mixing inthe estuary) for the k–� closure. However, the addi-tional rate of work from including wind is almost 3times larger, excluding the estuary, than the no windcase for the Mellor–Yamada closure.

The two closure schemes mix different salinity classeswithin the plume at different rates. The rate of workdone by mixing as a function of upper-layer salinity, sl,was calculated as

�

�sl�dW

dt � ��

�sl��

Al��

�H

�

g��

�zdz� dA, �18�

where Al is the area defined by upper-layer salinitiesless than sl. The rate of work done by mixing as a func-tion of salinity class is averaged over one upwelling/downwelling cycle (from days 16 to 20), and the resultsare plotted in the middle panel of Fig. 12. The rate ofmixing work within the estuary (sl � 20 psu) is nearlyidentical at all times and for both closure schemes, asexpected. For the no-wind case, the rate of work doneby mixing is higher both in the near field (at sl � 25 psu)as well as at higher salinity classes (at sl � 28 psu) forthe k–� scheme. Wind significantly increases the rate ofwork done by mixing at higher salinity classes. For thek–� scheme, wind decreases the rate of work done bymixing in the near field for the k–� scheme, and shiftsthe mixing to lower salinity classes. Wind increases therate of work done by mixing at all salinity classes for theMellor–Yamada scheme.

Mixing rates (e.g., Fig. 10) are related to work perunit area rather than total work in a given salinity class.The lower panel of Fig. 12 shows the rate of work doneby mixing per unit area as a function of salinity class.The total increase rate of work done by the wind mixingis very large within the far field; however, because thearea of the plume is large at these salinity classes, thework per unit area in this region remains small. Al-though wind forcing raises the rate of work per unit

FIG. 11. (top) Flushing time scales (volume of freshwater over the freshwater flux) of plumewaters within isohaline surfaces sA. The freshwater volume relative to sA is averaged for days16–20, over one upwelling/downwelling cycle. Plume time scales range from just over 1 day at thestart of the near field to 18 days at the highest salinity classes (the average total time of integrationfor this calculation). (bottom) The relative importance of the time-dependent term in Eq. (7).


area done in the far field, rate of work per unit areadone in the near field remains about an order of mag-nitude larger. Also, although the total rate of workdone in the near field is the same for both schemes, the

rate of work done per unit area in the near filed issmaller for the k–� scheme. This is consistent with thek–� simulations having weaker mixing and larger valuesof Al in the near field.

FIG. 12. (top) A time series of total work done by vertical mixing for different simulations. (middle) The workdone by mixing averaged over days 1–20, one upwelling/downwelling cycle, plotted as a function of upper-layersalinity. (bottom) The average work done by mixing per unit area as a function of upper-layer salinity.


Fig 12 live 4/C

5. Discussion

The various regions of the plume are most apparentin the cases without wind forcing. In this case, the nearfield is apparent as a local maximum in the work done,a local high in �Al/�sl, as well as feeding a local maxi-mum in �Vf /�s�. It is interesting to note that, althoughthe mixing rates are lower for the k–� closure (Fig. 10),the amount of mixing is greater (Fig. 12) because thearea over which the mixing occurs is larger (Fig. 7).

The largest changes between the cases with wind andthe cases without wind are in the range of salinitiesbetween the near and far field. At these salinity classes,the wind takes water leaving the near field and mixesthis water toward higher salinity classes, primarily dur-ing upwelling (see Fig. 7). In general, the interface be-tween the near and far field is apparent in the anomalyfields of the cases that include wind, but the near andfar field are not apparent when simply looking at time-averaged area enclosed by upper-layer salinity con-tours; the wind forcing tends to erase this boundary inthe mean.

Wind moves water from near-field to far-field salinityclasses when the wind stress is the highest. Because ofthe asymmetries in mixing with regard to the phase ofthe wind stress, the calculations using the Mellor–Yamada scheme tend to move water toward higher sa-linity classes more during upwelling than downwelling.The percent anomalies in freshwater volume (Fig. 6)and plume area (Fig. 7) tend to be highest at the inter-face between the near and far fields and become lowerat higher salinity classes. The percent anomaly de-creases because the freshwater volume and plume areaat higher salinity classes is larger, resulting in a rela-tively smaller percent anomaly. In addition to this, how-ever, water will be mixed quickly up to its local criticalsalinity [Eq. (13)], after which mixing will be sup-pressed, further reducing wind-forced anomalies infreshwater volume and plume area at higher salinityclasses.

Wind stress may act to stabilize the structure of theplume in salinity space by actively mixing the plumeonly to a particular point. Equation (12) and the cor-responding analysis presented in Fig. 9 suggest thatwind mixing will be strong until the plume has reacheda particular critical thickness, after which mixing willdecrease. Portions of the plume at or above the criticalsalinity will be protected from further turbulent mixingby the wind. Given the same horizontal freshwaterthickness distribution, the plume will mix to the sameend member independent of mixing history, so if poornumerical resolution causes more or less mixing in thenear-field region, wind mixing may compensate for this

error as long as the critical thickness (or correspondingupper-layer salinity) has not yet been reached.

The fact that wind mixing is reduced after the plumereaches its critical thickness may be the reason that thetotal area of the plume at very high salinity classes (sl �30) changes very little with the inclusion of wind. Sincethe average mixing rate in the far field is still orders ofmagnitude smaller than in the near field and that ratedoes not change substantially with the inclusion of windwithin the far field, the area of the plume at very highsalinity classes may remain the same, since the area isinversely related to the mixing across that isohaline andthat mixing is near to the background mixing regardlessof the presence of the wind.

Clearly the wind will also change the geographicalposition of the plume, stretching it out during upwelling(increasing the area) and pressing it against the coastduring downwelling (decreasing the area). However, onaverage, the area of the plume at very high salinityclasses seemed to be relatively insensitive to the pres-ence of wind. This result would change if the mean ofthe wind stress were nonzero. The plume areas were allsimilar (within 10%) with and without wind, except thatthe k–� scheme was about 20–30% larger at the highestsalinity classes than the other three cases (not shown).Changes in the plume area at very high salinity classes(sl � 31) are dominated by the increasing freshwaterintroduced to the system, since changes in the freshwa-ter volume contained within the highest salinity classare exactly related to the total freshwater introducedinto the system.

The freshwater volume of the plume is the truestrepresentation of the plume in salinity space, since it iscompletely independent of plume position. However,because water mass modification is related to both themixing rate and the local changes plume area [see Eq.(7)], changes in plume structure do depend indirectlyon plume position through the plume area. In this re-spect, the salinity space view of the plume is incompletewithout some understanding of the Cartesian view ofthe plume.

6. Conclusions

The water mass structure of an idealized river plumewas examined in the context of changing wind stressamplitude for two common turbulence closureschemes, Mellor–Yamada and k–�. Wind stress changesthe position of the plume and increases the mixingwithin the plume. The focus of this paper is the rela-tionship between plume horizontal dimensions and ver-tical mixing, so the changing position of the plume isdeemphasized through the use of salinity coordinates.


This coordinate system allows the size of the plume tobe investigated independently from the position of theplume.

The plume can be divided into two dynamically dis-tinct regions: the near field, characterized by both in-ertial shear mixing and wind mixing, and the far field,characterized by only wind mixing when present. Thenear field mixing is localized at lower salinity values inthe cases presented here between approximately 20 and26 psu. Wind mixing caused the largest changes in sa-linity space plume structure at salinity ranges boundingthe near and far field: 24–29 psu in the cases presentedhere. Wind mixing did not affect higher salinity classesbecause the plume had reached its critical thickness (orequivalently, its critical salinity). Turbulent mixingcaused by the wind is suppressed after this point, andthe mixing approaches background levels. The criticalthickness depends on the magnitude of the wind stressand the equivalent freshwater thickness. The mixing atthe highest salinity classes is therefore largely con-trolled by background mixing.

Mixing within the plume, caused by advective shearmixing and wind stress, is related to the surface area ofisohalines within the plume. Given the same freshwaterflux, strong mixing requires only a small isohaline area,whereas weaker mixing requires a larger isohaline areato maintain the same total freshwater flux across theisohaline. Strong mixing due to inertial shear is con-fined to the near field. Wind mixing is strongest duringupwelling; however, the average rate of mixing in thefar field is much smaller than in the near field. Thus, thefar-field plume has a much larger surface area andlonger time scales of water mass modification. Time-scale analysis shows that the near-field plume is in asteady balance by the end of the integration time (20days), but the far-field plume is still changing. Thehigher rates of mixing and correspondingly lower arealextent in the near field are responsible for the fasteradjustment times within the near field.

A reasonable approximation to the plume structureis a single active layer in which salinity may vary hori-zontally within the layer, but the pycnocline is very thineverywhere. A single layer approximation may be usedto obtain an estimate of the salt flux across the pycno-cline that has the same characteristics as a direct esti-mate using a weighted average of the vertical salt flux.

One of the difficulties in comparing numerical simu-lations of river plumes to observations is that riverplumes change position, so at any given point, theplume may be present only some of the time. Becausethe analysis presented in this paper focuses on distri-butions of salinity, the analysis methods may be appliedto both numerical output and hydrographic observa-

tions (providing the measurements resolve the thin sur-face plume). Salinity coordinates offer an integrativeview of the plume that does not depend on smallchanges in frontal position, may therefore be used tosupplement more conventional model/data compari-sons based in geographic space. Many of the resultsdiscussed in this paper would be difficult to show usinga Cartesian view of the plume. Salinity coordinates areextremely useful in examining mixing processes withinthe plume, since adiabatic changes in position are ig-nored by the calculation.

Garvine’s (1999) results can now be interpreted usingisohaline coordinates. In a steady state, all of the fresh-water input into the system through the river must passthrough each isohaline in the plume. In order of isoha-lines beyond the near-field mixing region to pass thisfreshwater through at background values of diffusivity,the surface area of that isohaline must be large. If thebackground diffusivity is reduced by one-half, the sur-face area must double to compensate, so that the fresh-water flux through the surface remains constant. Windforcing modifies the structure of the regions that aresusceptible to wind mixing, that are fresher than thecritical salinity, by increasing the mixing and reducingthe area proportionally. However, at higher salinityclasses the plume is typically protected from the effectsof further wind mixing, the local salinity is above thecritical salinity, preventing strong mixing and increasingthe plume area. Thus, although differing in details, thisstudy confirms Garvine’s basic result that plume struc-ture depends fundamentally on mixing and the mixingparameterization.

Acknowledgments. I thank Parker MacCready,Rocky Geyer, Steve Lentz, and Rich Signell for manyhelpful comments and suggestions. This project wassupported by ONR Grant N00014-03-1-0398.

APPENDIX

Definition of Terms

In this paper, there are two complementary views ofthe plume within salinity space. The first is a three-dimensional view, using an isohaline surface to bounddifferent regions of the plume. The critical variables inthis view of the plume are the isohaline sA, the areadefined by this isohaline A, and the volume enclosed bythis isohaline V. Freshwater contained within the iso-haline surface may be defined in one of two ways. Thefreshwater relative to a constant reference salinity, s0, isVf. The freshwater relative to the salinity bounding iso-haline is VfA. Derivatives with respect to salinity aredefined using sA.


The second view of the plume assumes the plumeacts as a singe. The upper-layer salinity is defined as theaverage salinity above h, the upper-layer thickness. Thelocation of h is defined as the point where s � (smin �smax)/2. The area of the plume enclosed by a contour ofupper-layer salinity, sl, is Al. In this case, derivativeswith respect to salinity are defined using sl.

A complete list of terms used is below.

A(sA) isohaline surface s � sA (m2)Al(sl) surface area enclose by upper-layer salinity

contour sl (m2)cf freshwater phase speed, �hf g��f /�0 (m s�1)f salt flux vector (psu m s�1)fA(sl) vertical salt flux into upper layer (psu m s�1)f salt flux at z � �h (psu m s�1)Fr upper-layer Froude number, u/g�hFrc critical upper-layer Froude number, Ri�1/2

c

Frd upper-layer freshwater Froude number, uf /cf

g gravitational acceleration (m s�2)g� reduced gravity, g��/�0 (m s�2)sA salinity defining isohaline surface A (psu)h upper-layer thickness (m)hc critical thickness (m)hf freshwater thickness (m)H bottom depth (m)� sea surface height (m) vertical diffusivity of salt (m2 s�1)QR freshwater flux (m3 s�1)� density (kg m�3)Ric critical bulk Richardson number�� density difference between upper and lower

layer (kg m�3)�0 reference density (kg m�3)�f density of freshwater (kg m�3)s salinity (psu)sA salinity of bounding isohaline surface (psu)sc critical salinity (psu)sl upper-layer salinity (psu)s0 reference salinity (psu)t time (s)� wind stress (m2 s�2)uA motion vector of isohaline surface A (m s�1)u current vector (m s�1)uf Ekman flow speed of freshwater layer, �/f�0hf

(m s�1)V(sA) volume enclosed by isohaline surface s � sA

(m3)Vf(sA) freshwater volume, referenced to s0 (m3)VfA(sA) freshwater volume, referenced to sA (m3)dW/dt rate of work done on the density structure by

mixing (W)x, y, z spatial dimensions (m)

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