Continental Shelf Research 58 (2013) 96–106

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Research papers

Please check cross mark logo is notplacedLagrangian model of a surface-advectedriver plume

Please check cross mark logo is not placedAlexanderA. Osadchiev, Peter O. Zavialov n

Please check cross mark logo is not placedP.P. ShirshovOceanology Institute, Moscow, Russia

a r t i c l e i n f o

Article history:Received 3 February 2012Received in revised form6 March 2013Accepted 20 March 2013Available online 30 March 2013

Keywords:River plumeLagrangian modelMomentum budgetWind stress

43/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.csr.2013.03.010

espondence to: P.P. Shirshov Institute of Oces, 36, Nakhimovsky Prospect Avenue, Moscow499 124 5994; fax: +7 499 124 5983.ail address: peter@ocean.ru (P.O. Zavialov).

a b s t r a c t

A Lagrangian model of a surface-advected river plume is developed. The model combines the deterministicapproach resulting from the momentum budget with a random-walk scheme. We validated and appliedthe model to the conditions of the region adjacent to the Mzymta River estuary in the Black Sea. Further,using the model, we investigated the dependencies between the spatial extent of the plume and the windstress τ. The character of these dependencies proved to depend critically on the wind directionwith respectto the geometry of the coast and the river mouth. With the offshore and “downwelling favorable”alongshore winds, the plume area initially grows with the increase of τ, and then starts to decrease when τexceeds a certain critical value. With the onshore wind, the area decreases steadily (though not linearly) asτ increases. The most complex behavior is observed for the “upwelling favorable” alongshore wind: theplume area initially decreases until the wind stress exceeds a certain critical value, then starts to increasewith τ and attains a maximum, and then eventually drops to zero as τ continues to increase. As expected,the increase of the inflow velocity, i.e., discharge rate, resulted in the growth of the plume area, while thepatterns of the dependencies between the plume area and the wind stress configurations were robust inthe broad range of intermediate discharge values. We also investigated the dependence of the plume areaon the latitude, with all other conditions kept fixed. This dependence exhibited a distinct “M-shaped”pattern, with minima corresponding to the equatorial and polar regions, and maxima in the tropics.However, even moderate wind stress largely offsets the effect of the changing Coriolis parameter andstrongly distorts the shape of the curve.

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1. Introduction

Freshwater input from fluvial runoffs plays an important role inphysical, chemical and biological processes in the ocean, especiallyin the shelf regions. The volume of the total river inflow to theworld ocean is estimated as 39,000 km3 a year (Lvovich, 1986),which is about a mere one-tenth of the input part of the waterbudget of the ocean. However, for the inland seas and certain shelfareas, fluvial discharges can play much more significant role. Onegood example is the Black Sea, where the long-term averageannual volume of the continental runoff is about 338 km3, whichexceeds the atmospheric precipitation (238 km3 a year) and isclose to the evaporation (396 km3 a year) (Simonov and Altman,1991).

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anology, Russian Academy of117997, Russia.

Once the river water inflows into the sea, it forms distinctstructures on the ocean surface adjacent to the river mouth,commonly called “plumes”. Salinity within such a structure isgenerally lower than that of the surrounding ocean waters, andmany other properties also differ. The horizontal scale of a plumecan be of tens and even hundreds of kilometers, and yet theinterface between the plume and the sea water often remains asdistinct as a few centimeters. River plumes can significantlymodify the physical structure of the shelf waters. In the sea areasadjacent to river estuaries, the buoyancy flux from fluvial dis-charges can be comparable or even exceed the buoyancy inputassociated with seasonal heating (Simpson, 1997).

An important consequence of continental runoff consists inaltering the along- and cross-shelf transport pathways (Gan et al.,2008; Banas et al., 2009; Xia et al., 2011). River discharges withhigh nutrient content modulate the primary productivity, signifi-cantly affect fisheries and the food webs in general (Mallin et al.,2005), and may cause eutrophication and hypoxia over broad shelfareas (Wiseman and Kelly, 1994; Rabailais et al., 1996). Continental

Fig. 1. Study area in the eastern part of the Black Sea shelf: Mzymta River mouthregion and bottom topography.

A.A. Osadchiev, P.O. Zavialov / Continental Shelf Research 58 (2013) 96–106 97

runoff is an important source of momentum and vorticity in thecoastal regions (Fong and Geyer, 2002) and one of the mainmechanisms that determine transport and accumulation of sedi-ments in the shelf zone (Geyer et al., 2000). Furthermore, themotion of river plumes has been shown to generate large-amplitude internal waves, which may have important implicationsfor the turbulent mixing (Nash and Moum, 2005).

River plumes have been a subject of a significant number ofobservational, numerical, and theoretical studies. The free motionof the plume is perhaps understood best. In the absence ofexternal forcing such as wind, oceanic currents, or tides, plumebehavior is fully determined by the following parameters: widthand depth of the mouth, river inflow velocity, bathymetry andvertical stratification on the adjacent shelf, and the Coriolisparameter (Chao and Boicort, 1986; Garvine, 1987; O'Donnell,1990; Avicola and Huq, 2003; Zavialov et al., 2003; Horner-Devine et al., 2006).

The behavior of plumes under external forcing such as the windstress or the interaction with an ambient coastal circulation isunderstood much worse. The principal forcing mechanismsinclude local winds (e.g., Bowman, 1978; Bowman and Iverson,1978; Chao, 1988; Fennel and Mutzke, 1997; Pullen and Allen,2000; Janzen and Wong, 2002; Jonson et al., 2003; Hetland, 2005;Choi and Wilkin, 2006; Hunter et al., 2006; Whitney and Garvine,2006), background coastal circulations (e.g., Garvine, 1987; Fongand Geyer, 2002), and tides (Simpson, 1997; Blumberg et al., 1999;Peters, 1999; Warner et al., 2002; Cheng and Casulli, 2004). Thesefactors can significantly change the plume area, shape, andstructure, but the degree of the influence vary significantly(Zhang et al., 1987; Kourafalou et al., 1996; Fong and Geyer,2002; Xia et al., 2007).

A number of models simulating the formation and evolution ofa river plume have been developed. Most of them (e.g., Chao andBoicort, 1986; Weaver and Hsieh, 1987; Chao, 1988; Johnes et al.,1992; Pinazo et al., 1996; Yankovsky and Chapman, 1997; Pullenand Allen, 2000; Garvine, 2001; Fong and Geyer, 2002; GarciaBerdeal et al., 2002; Choi and Wilkin, 2006; Xia et al., 2007) werebased on three-dimensional momentum equations and use var-ious turbulence closure schemes for parameterizing the turbulentmixing (e.g., Mellor and Yamada, 1982; Galperin et al., 1988). Someof these models use also Lagrangian technique as a post-processing procedure to simulate the spreading of pollutants andother admixtures to the ocean water (e.g., Elliot et al., 1986;Hunter, 1987; Clarke and Elliot, 1998; Black and Parry, 1999;Roberts, 1999; Kim et al., 2000; Black and Vincent, 2001; Hillet al., 2003; Korotenko et al., 2004; Nassehi and Passone, 2005) orthe dynamics of buoyant discharges (Whitney and Garvine, 2006;Banas et al., 2009; Xia et al., 2011). Models of this type, however,encounter difficulties in simulating processes in the areas withsharp gradients like those at the plume borders. The fixedhorizontal and vertical grid resolution is generally insufficient toreproduce processes adequately at the frontal discontinuitiesbounding the plume (Estournel et al., 1997).

Another group of models (Garvine, 1987; O'Donnell, 1990) havecircumvented this problem by representing a plume as water massof homogeneous density with fronts treated as border disconti-nuities. Models of this type, however, encounter difficulties takinginto account the external forcing such as winds, ambient seacurrents, and tides (Estournel et al., 1997).

An alternative ideology can be based on using Lagrangianformalism for simulating motion and dissipation of a plume. Inthis approach, the plume is represented by a set of particles whosetrajectories and properties are determined by integrating themomentum and diffusion equations for every individual particle.One advantage of the models of this type is their relativecomputational ease compared with Eulerian models. Lagrangian

models are suitable to simulate processes with sharp gradientsand do not have limitations connected with inappropriate gridresolution. They can be used for running series of plume simula-tion experiments with a variety of forcing factors, such as wind,ambient sea currents, river inflow velocity, Coriolis parameter, etc.,for obtaining a better insight into the influence of the externalforcing on river plume dynamics. The intrinsic deficiency ofLagrangian models lies in the fact that they cannot take intoaccount any feedbacks from the plume onto the ambient ocean.While this limitation is generally very serious, it may be lesssignificant for the plumes of rivers of small to medium size whosereciprocal impact on the larger scale coastal circulation is negli-gible or small.

We are aware of only one previous work where the fullyLagrangian approach was used to simulate the plume dynamics(Mestres et al., 2003). In this paper, we present a Lagrangian modelwhich is based on the same general principles, but use different,perhaps, more physically based, parameterizations for the termsand coefficients involved in the momentum and diffusion equa-tions, and also combines the deterministic approach with anappropriately tuned random walk scheme.

Because the model is essentially two-dimensional and restri-cted to consideration of surface-advected plumes we dubbed it asSurface-Trapped River Plume Evolution (STRiPE). The model isintended to simulate a plume of a small to medium river. To thisend, we apply the model and tune it for the plume of the MzymtaRiver in the Black Sea. We use our recent in situ measurements tovalidate the model. After that we use STRiPE to perform numericalexperiments of the river plume behavior under various externalforcing conditions.

2. Study area and data used

2.1. Study area

Although the Mzymta is the biggest river in Russia feeding theBlack Sea (Fig. 1), it is relatively small with its catchment area of885 km2, discharge rate of 49.5 m3 s−1on the long-term average,and mean annual runoff of about 1.562 km3. Its catchment basinlies in an elevated mountainous region (average altitude of basin is1309 m) and is mainly fed by snow melting in springtime, and rain

A.A. Osadchiev, P.O. Zavialov / Continental Shelf Research 58 (2013) 96–10698

events throughout the year (Jaoshvili, 2002). During floods, theriver discharge can be more than 15 times larger than its averagevalue (the maximum value on the record was 764 m3 s−1)(Prokhorov and Waxman, 1973). The region is non-tidal, as theastronomical tide at the Black Sea is negligible. The ambientsurface salinity of the Black Sea in the study area normally spansbetween 17 and 19 psu (e.g., Zatsepin et al., 2003). The shelf in theregion is quite narrow, and the steep continental slope dominatedby the northwestward flowing Rim Current is located at thedistance of only about 3 km from the coast.

The Mzymta merges into the sea in an urban area within thedensely populated city of Sochi, perhaps the most important resortof Russia. Yet, the waters at this part of the shelf have poorecological conditions (e.g., Amirkhanov et al., 1997). The rivercarries a large volume of suspended sediments, especially after thecommencement of the ongoing construction works in the newport of Sochi next to the mouth, as well as other pollutants andnutrients. Hence, better understanding of the Mzymta plumedynamics is of significant economic and social importance, parti-cularly, in view of the forthcoming winter Olympic Games to beheld in Sochi in 2014. The other reason for selecting the areaadjacent to the Mzymta River mouth as the study site was theavailability of the necessary observational data (see below).

2.2. Data used

The in situ hydrographic data used in this study were obtainedduring the recent monitoring surveys of the Shirshov Institute ofOceanology to the Mzymta mouth region of the Black Sea coast.The measurements were conducted during the flood periods onMay 25–30, 2010 and May 26–30, 2011. Continuous surfacesalinity and temperature measurements were performed everyday along the ship track (Fig. 2) using a shipboard pump-throughCTD system. In addition to these continuously recorded data, topto bottom CTD profiles at 5–7 stations situated within the plumeand outside it were collected every day during the field study.

A mechanical current meter was deployed at the depth 9 m, atthe distance of about 2 km from the river mouth (Fig. 2). Thisinstrument measured the velocity beneath the plume with theintervals of 10 min. This flow, generally directed along the shore,was interpreted as the ambient sea current in the study region andused by the model. The mean velocity of the ambient along-shoresea current was 4.7 cm s−1 in 2010 and 4.4 cm s−1 in 2011. Wind

Fig. 2. Map of the measurements conducted in the study area.

conditions were moderate during both periods of observations,with the maximumwind speed 4.3 m s−1 in the survey of 2010 and6.2 m s−1 in the survey of 2011. The detailed account of thesemeasurements can be found in our paper (Korotkina et al., 2011).

Another current meter was deployed in the river itself, some100 m upstream from the mouth, and it registered the dischargerate at 2 min intervals. Also, a complete set of meteorological datawere registered as 10 min averages at an automatic weatherstation installed in the immediate proximity of the river mouth.The measured Mzymta discharge rate spanned between 60 and185 m3 s−1, while the river inflow velocity varied from 0.5 to1.6 m s−1.

Maps of surface salinity were composed for every day ofobservations as a result of the measurements described above.The position and area of the plume for each day can be clearlyidentified from these maps. In addition, every day the plume wascontoured directly by recording the GPS coordinates while theship was sailing along the easily visible boundary of the plume.Furthermore, the resulting maps of the plume are in goodaccordance with MERIS-Envisat satellite images showing thedistribution of the total suspended matter (see also Section 4).Generally, different methods for tracing the plume yielded similarresults.

Overall, during the measurements, the actual extent of the riverplume along the sea shore varied between 2.8 and 8.2 km, whilethe plume area ranged from 9.7 to 14.7 km2. The maximum depthof the plume layer was about 5 m, which is small compared withthe characteristic sea depth D420 m. Hence, we can assume thatthe influence of the bottom friction on the plume is negligible(Huq, 2009).

3. Model

We develop here a Lagrangian approach based on tracking themotion and dissipation of imaginary individual “particles” of theriver water discharged into the sea. Hereinafter, we omit thequotes and use the term particle for simplicity, even though itmay not be very precise. Such a particle is considered to be ahomogeneous elementary water column extending from the sur-face down to the boundary between the plume and the underlyingsea water. The particles are released from the river mouth,separated by regular time intervals, and assigned with the initialvelocity w0 depending on the discharge rate (see Appendix fordetails).

The initial coordinates, density and height of every newelementary column are set equal to the coordinates of the rivermouth, the river water density, and the river depth at the mouth,respectively, while the initial accelerations of these particles arenull. The number of the particles appearing during a time step ofthe integration is proportional to the river discharge rate at thegiven step.

The subsequent motion of the particle is determined by themomentum equation applied to this specific particle. The overallset of particles represents the river plume, and hence the temporalevolution of the plume structure is obtained. We presume that thebuoyant plume remains confined to the surface layer, therefore,the model describes the 2D-motion of the particles, although thewater belonging to the particle is allowed to be vertically mixedwith the underlying sea water, so that the salinity and densitychange in time until the particle eventually dissipates, i.e., dis-appears. The details of the equations and parameterizationsinvolved will be given below. At every step of the computation,the model reads the corresponding values from the input timeseries of river discharge rate, wind stress, and ambient seacurrents velocity data. The following forces are applied to the

Fig. 3. Schematic of the forces applied to an individual particle of the plume.

A.A. Osadchiev, P.O. Zavialov / Continental Shelf Research 58 (2013) 96–106 99

individual particle: the Coriolis force, the pressure gradient force,the wind stress force, the friction at the lower boundary of theplume, and the lateral friction (see the schematic in Fig. 3). Themodel then calculates the acceleration components (ax, ay) andthen the final velocity components (u, v).

Because we are tracking individual particles treated as materialpoints, we do not need to use any spatial grid for solving theequations of motion. However, we do introduce an auxiliaryhorizontal grid with the respective increments Δx and Δy inzonal and meridional directions in order to calculate the spatialderivatives that take part in the parameterizations of some of theforces applied to the particle, as specified below. To this end, thevalues of the velocity, plume height, and water density areassigned to the required grid nodes by interpolating the respectivevalues from all the particles.

The momentum equations for an individual particle read:

aiþ1x ¼ f vi þ τix

ρihi−μiv

hiui−ui

sea

hi

þ μh

hi

uixþΔx;y þ ui

x−Δx;y−2ui

Δxþ ui

x;yþΔy þ uix;y−Δy−2u

i

Δy

!−gϰ

hixþΔx;y−h

ix−Δx;y

Δx

aiþ1y ¼ −f ui þ τiy

ρihi−μiv

hivi−visea

hi

þ μh

hi

vixþΔx;y þ vix−Δx;y−2vi

Δxþ vix;yþΔy þ vix;y−Δy−2v

i

Δy

!−gϰ

hix;yþΔy−hix;y−Δy

Δy

ð1Þwhere ðux;y; vx;yÞ are interpolated velocity components at the (x, y)grid node, f is the Coriolis parameter, ðτx; τyÞ are the wind stresscomponents, ρ is the density of water in the particle, h is theheight of the particle (once again, interpreted as the water columnwithin the plume extending from the surface to the underlying seawater), hx;y is the interpolated height at the (x, y) grid node, μh; μvare the horizontal and vertical eddy viscosity coefficients whoseparameterizations will be specified below, ðusea; vseaÞ are theambient sea currents velocity components, g is the gravity accel-eration, and ϰ is a dimensionless pressure gradient scaling coeffi-cient. The superscripts stand for the number of a discrete timestep. In the equations above, the first term in the right hand side

stands for the acceleration of the Coriolis force, the second one forthat of the force applied from the wind, the third and the fourthfor those from the friction with the underlayer and lateral friction,respectively, and the last term symbolizes the acceleration due tothe pressure gradient force.

The scaling coefficient ϰ was selected based on the assumptionthat plume dynamics without external forcing is governed byquasi-geostrophic balance (Kao, 1981; Munchow and Garvine,1993; Fong and Geyer, 2002; Choi and Wilkin, 2006). For theunforced river plume, the numerical value ϰ ¼ 25� 10−6 wasfound to provide for approximate balance between the pressuregradient, the Coriolis force, and boundary friction, as well as thebest agreement with the observational results.

After the acceleration components ðax; ayÞ are obtained fromthe momentum equations, the final velocities (u, v) for the period(t; t þ Δt) are calculated from kinematic formulas:

uiþ1 ¼ ui þ aiþ1x Δt; viþ1 ¼ vi þ aiþ1

x Δt ð2ÞIn order to simulate the small-scale horizontal turbulent mix-

ing, the deterministic approach described above was complemen-ted by the random-walk Monte Carlo method (e.g., Hunter, 1987)tuned based on diffusivity coefficients as specified in Appendix.

This approach that combines deterministic description of themotion with a stochastic random-walk scheme was used in anumber of previous works, mainly for simulating the transport ofpollutants (Korotenko, 1994; Mestres et al., 2003; Lardner et al.,2006).

The simulation of the vertical dissipation of a plume particle isbased on the salinity diffusion equation:

∂S∂t

¼Dv∂S∂z

ð3Þ

where S is the salinity, and Dv is the vertical diffusion coefficient.We assume that density depends linearly on salinity, and the latterequation takes the following discrete form for salinity and densityof a particle:

Siþ1 ¼ Si þ DivðSsea−SiÞ

hiΔt

ρiþ1 ¼ ρi þ Divðρsea−ρiÞ

hiΔt ð4Þ

A.A. Osadchiev, P.O. Zavialov / Continental Shelf Research 58 (2013) 96–106100

where ρsea is the ambient sea water density, and Ssea is theambient sea water salinity.

Hence, as the salty water from below is entrained into theplume gradually replacing the freshwater, the density of water inthe elementary column increases according to Eq. (4), while theheight of the elementary column decreases according to thefollowing linear equation:

∂h∂t

¼−Dv ð5Þ

or, in a discrete form,

hiþ1 ¼ hi−DivΔt ð6Þ

It can be shown that the latter equation is actually equivalent toEq. (4).

The horizontal and vertical diffusion coefficients used in Eqs.(3)–(6) were calculated through Smagorinsky formula and para-meterizations based on Richardson number, respectively, as spe-cified in Appendix.

The parameterization of the vertical diffusion through theRichardson number seems to be physically adequate in this case.Indeed, the vertical mixing beneath the plume should be mainlygoverned by two competing processes: on the one hand, the shearbetween the plume easily moving under the wind forcing and theunderlying sea water tends to increase mixing. On the other hand,the enhanced vertical stratification tends to damp it. The ratiobetween the two mechanisms is quantified by the Richardsonnumber.

Finally, the boundary conditions at the coastline were formu-lated as the no-normal flow condition, i.e., once the particle hitsthe shore, its cross-shore velocity component vanishes, while thealong-shore component is retained without change.

Fig. 4. The modeled unforced plume at t¼200 h: the modeled particles and thecontour of the interpolated plume.

4. Validation of the model

The ability of the model to produce realistic results was testedin two ways. Firstly, we performed a series of simulations undersimplified forcing conditions and compared their results with theidealized solutions known from previous studies. Secondly, thedistributions of sea surface salinity, density, and plume thicknessobtained from the model forced by the “real” winds and dischargewere verified against the in situ measurements and satelliteimages of the Mzymta River plume.

The first series of the validating and calibrating experimentswas held with the unforced river plume. The model domain wasthe area adjacent to the Mzymta mouth with the realistic shore-line (Fig. 4). The freshwater particles were released from theMzymta mouth with the mean inflow azimuth αr ¼ 2251 (countingfrom the North clockwise) and the inflow velocity was set towr¼0.5 m s–1. The depth of the river at the mouth h0 was 3 m, theriver and sea waters density were set to the respective 1000 kg m-3

and 1017 kg m-3. The Coriolis parameter f was selected as1.028�10−4 s−1corresponding to the latitude 43.411 of theMzymta River mouth. The model integration time step was equalto 10 min.

In the absence of winds and sea currents, the simulatedMzymta River plume attained a characteristic shape of a bulgeadjacent to the river mouth and a compensating northwestwardalongshore coastal current (Fig. 4), as it has been well establishedfor a freely moving surface-advected plume in the previous studies(e.g., Garvine, 1987; Chao, 1988; O'Donnell, 1990; Oey and Mellor,1993). The bullets in Fig. 4 depict the positions of 1009 modeledparticles released from the mouth during 200 h of the simulation,with the grayscale indicating the salinity (darker tones correspondto fresher water), while the contour indicates the limit of theinterpolated plume, as represented by 16 psu isoline. Fig. 4, there-fore, illustrates the conversion from the discrete particles to thegridded surface salinity field. Another known feature of the bulgereproduced well by the model is the so-called “ballooning”, i.e., itspermanent growth during the entire modeling period (Fig. 5),associated with the fact that the compensating coastal currentflow is always smaller than the river inflow, see also (Nof, 1988;Pichevin and Nof, 1997; Fong, 1998; Horner-Devine et al., 2000;Garvine, 2001; Nof and Pichevin, 2001; Fong and Geyer, 2002).

We repeated the simulations of the free plume with 3 differentriver inflow velocities, namely, 0.25, 0.75, and 1.25 m s−1 and allthese experiments yielded similar results. However, model simu-lations of the plume in the presence of an along-shore north-westward wind showed significant change in the plume behavior,even if the wind was light. The unforced river plume wassimulated for 50 h, and then a constant along-shore northwest-ward wind (τ¼10−3 N m-2) was applied. In this case, the bulge isstrongly modified and advected in the direction of the ambient seacurrent (Fig. 6). Similar conclusions were drawn in previousmodeling studies which investigated the impact of wind (GarciaBerdeal et al., 2002) and a slow along-shore ambient sea current(Yankovsky and Chapman, 1997; Fong and Geyer, 2002; GarciaBerdeal et al., 2002).

The other set of the model validation experiments consisted insimulations of the Mzymta plume under the “real” conditionsduring the two periods, namely, May 26–30, 2010, and May 27–31,2011. The input data used were the series of the wind stress, riverdischarge rate, and sea current velocity, as obtained from the fieldmeasurements at 10 min intervals (see Section 2 for details). Theconfigurations of the plume observed on the first days of thesimulation experiments, i.e., May 26, 2010, and 27, 2011, weregiven as the initial conditions.

At the end of May, 2010, the average measured wind speed inthe Mzymta mouth region was only about 0.8 m s−1, while the

Fig. 5. Behavior of the unforced plume. Surface salinity structure (psu) for the model run without external forcing at t¼50, 100, 150, and 200 h with river discharge velocitywr¼0.5 m/s, and Coriolis parameter f¼1.028�10−4 l/s. The thin isolines are 4, 8, and 12 psu. The bold isoline representing the plume limit is 16 psu.

Fig. 6. Behavior of the plume under wind forcing conditions. Surface salinity structure (psu) under light SE wind forcing with τ¼10−3 N/m2 at t¼50, 60, 70 and 80 h. The thinisolines are 4, 8, and 12 psu. The bold isoline representing the plume limit is 16 psu.

A.A. Osadchiev, P.O. Zavialov / Continental Shelf Research 58 (2013) 96–106 101

ambient sea current velocity was below 0.1 m s−1, so it can be saidthat the plume was under conditions of moderate external forcing.Mild eastern and southeastern winds prevailed throughout theperiod, and, as it can be seen from the plume maps, they causedthe plume spread in the western and northwestern directions (seeFig. 7a).

At the end of May, 2011, the wind forcing was stronger thanthat observed one year earlier. The average wind stress was about3 times higher, whilst the ambient sea current velocity was aboutthe same. The wind direction was subject to strong variability. As aresult, the plume area, shape, and orientation changed signifi-cantly from one day to another. On May 27 and 29, when thewinds were mainly onshore, the plume was arrested near theshore on both sides of the mouth, while on May 28 and 31 itpropagated in the southern direction under moderate northernand northwestern winds (Fig. 7b).

In order to validate the model, the “snapshots” of the simulatedplume corresponding to the noon of every day during the simula-tion period were compared to those obtained from the in situ andsatellite observations. As illustrated by Fig. 7a and b, the position,pattern, and area of the modeled plume for the simulation periodsin May, 2010 and 2011 generally agreed well with those observedon the respective days.

Hence, the series of validation experiments described in thissection, demonstrated that the model is capable of reproducingthe plume rather realistically, at least, under the conditions of lowto moderate forcing (wind speed below 7 m s−1). There is no in situdata on Mzymta River plume dynamics for higher wind speed tovalidate the model against, so we cannot judge on the efficiency ofthe model under strong wind forcing conditions. However, themodel does reproduce decay of the plume under strong windconditions due to enhanced mixing, which is what should beexpected.

5. Diagnostic experiments

We now proceed to a series of diagnostic simulations of theMzymta river plume under various forcing scenarios. Our objectiveis to investigate the dependencies between the parameters of theplume, particularly its spatial extent, on the one hand, and thewind stress, the river discharge rate, and the Coriolis parameter onthe other. Although the model experiments were conducted withthe shoreline and other geometric settings of the area adjacent tothe Mzymta River mouth, we believe that the insights obtainedthereby hold promise to be of a broader applicability in the generalcontext of the dynamics of small and medium river plumes.

The model configuration used in the diagnostic experimentswas as described above in Section 4. The initial condition was “no-plume”, and then the freshwater inflow from the mouth waspermitted to generate the plume. The river inflow velocity wr,wind stress field τ! and Coriolis parameter f were prescribed andkept constant during each simulation run, but varied from oneexperiment to another. The ambient sea current velocity was setequal to zero. The plume evolution was simulated for the period of1000 min, and then the parameters of the resulting plume wererecorded.

5.1. Dependence on wind forcing

Wind forcing is the most important external factor that affectsthe area, shape and orientation of the plume. As mentioned above,a number of previous observational and modeling investigationsaddressed this issue however, most of these studies dealt witheither the specific measured winds or some illustrative idealizedcases. No generalized understanding of the plume dependence onthe wind forcing applied in different directions (with respect tothe shoreline and river mouth) has been achieved. In this part ofthe study, we investigate the dependence of the plume on thewind forcing applied in different directions with respect to the

Fig. 7. Plume contours modeled under “real” conditions (“m”), in situ observedplume contours (“o”), and satellite images of the plume (“s”) during the periodsMay 28–30, 2010 (a) and May 28–30, 2011 (b). The modeled and observed plumecontours are defined by 15.5 psu isoline (a), and 13.5 psu isoline (b). The dashedlines do not correspond to the real plume limits, and only indicate the southern-most part of the ship track.

Fig. 8. Modeled plume area as a function of wind stress magnitude for four“archetypal” wind forcings regimes at t¼1000 min, river discharge velocitywr¼1.05 m/s, and Coriolis parameter f¼1.028�10−4 l/s.

Fig. 9. Modeled area of the unforced plume as a function of latitude as determinedby Coriolis parameter at t¼1000 min under various river discharge rates.

A.A. Osadchiev, P.O. Zavialov / Continental Shelf Research 58 (2013) 96–106102

shoreline and river mouth. Low computational cost of the Lagran-gian model enabled us to perform the total of 35� 72� 4¼ 10080simulation runs, changing the wind speed (35 different windspeed values, from 0 up to 6 m s−1 with the increment0.175 m s−1) 72 wind direction with the increment of 51, and fourriver inflow velocity values from 0.35 to 1.4 m s−1 with theincrement of 0.35 m s−1.

The discussion below focuses on plume spreading under four“archetypal” wind forcing regimes, namely, the onshore, offshore,and the two alongshore wind directions. Hereinafter, we refer tothe alongshore wind blowing to the right of the river mouth as“downwelling favorable”, and the alongshore wind blowing to theleft of the mouth as “upwelling favorable”. The river discharge rateand Coriolis parameter were kept constant at wr¼1.05 m s−1 andf¼1.028�10−4 s−1 during each run. The four resulting plotsexpressing the dependence between the plume and the windstress are shown in Fig. 8. Two of these graphs, namely, thosecorresponding to the downwelling favorable and the offshorewinds, are very similar. The plume area initially increases withthe wind stress and attains certain maximum value. Further growth ofthe wind stress reduces the plume area, and eventually destroys theplume completely. On the contrary, the dependencies for the twoothers “archetypal” winds are different from this pattern, as well asfrom each other. The growth of the onshore wind stress causescontinuous, albeit not uniform, decrease of the plume area, whilethe increase of the “upwelling favorable” wind results in initial

decrease of the plume area, followed by an intermittent increaseand then eventual decay of the plume. The onshore wind forcingdefinitely is the most diminishing for the plume area. Under weakforcing conditions the plume area is greater for the “downwellingfavorable” winds than for the “upwelling favorable” and offshore,while under stronger winds we observe the opposite situation. Somesimilar findings were previously reported by Xia et al. (2011) who useda 3D hydrodynamic model.

In the present study, the low computational cost of theLagrangian approach enabled us to perform similar experimentsat much smaller increment of the input data, and hence obtainmore detailed dependencies. In addition to these 4 basic winddirections, we also performed the simulations for all other direc-tions with the increment of 51. The intermediate regimes arecharacterized by the curves SðτÞ representing the passagesbetween the basic types as described above. The possible under-lying mechanisms are discussed in Section 6.

5.2. Dependence on latitude

In order to compare the behaviors of river plumes at differentlatitudes, we performed the series of plume simulations withoutapplying any external forcing, but changing the Coriolis parameter

A.A. Osadchiev, P.O. Zavialov / Continental Shelf Research 58 (2013) 96–106 103

within the entire range of latitudes from 901 S to 901 N. Eachsimulation was run with 4 different river inflow velocities, namely,0.35, 0.7, 1.05, and 1.4 m s−1, kept constant during each simulation.

All of the obtained curves expressing the dependence of theresulting plume area on the Coriolis parameter have characteristic“M-shaped” pattern, approximately symmetric with respect tof¼0, and exhibited maxima at the tropical latitudes (Fig. 9). Thesymmetry is only approximate because the geometry of thecoastline and the mouth used in the simulations is non-symmetric and implies slightly different outcome for differentsigns of Coriolis force. As the river discharge rate grows, thelatitude where the maximum area is achieved increases fromabout 201 (for inflow velocity 0.35 m s−1) to about 301 (for1.4 m s−1).

We also repeated these experiments adding southeasterly windforcing. It was found that even moderate wind stress at1.5�10−2 N m-2 largely offsets the effect of the changing f andstrongly distorts the symmetric shape of the curve (Fig. 10). Thus,the “M-shaped” dependence of the plume area on latitude in realconditions is largely marked by other dynamical factors. However,we believe that this dependence is illustrative as far as the generalmechanisms of the plume formations are concerned.

Fig. 10. Modeled plume area as a function of latitude as determined by Coriolisparameter at t¼1000 min with river discharge velocity wr¼1.05 m/s, withoutexternal forcing and under light and moderate NW wind forcing.

Fig. 11. Modeled non-dimensional plume area as a function of normalized windstress for NW wind forcing at t¼1000 min under various river discharge rates andCoriolis parameter f¼1.028�10−4 l/s.

5.3. Dependence on river discharge rate

To investigate the influence of the river discharge rate on theplume dynamics, we repeated the simulations of the plume withthe forcing specified above with 4 different river inflow velocities,namely, 0.35, 0.7, 1.05 and 1.4 m s−1. The Coriolis parameter waskept constant at f¼1.028�10−4 s−1 during each run. As expected,the increase of the inflow velocity resulted in the growth of theplume area for most of the configurations. The above discussedfeatures of the dependence between the plume area and forcingconfigurations were rather robust in the range of intermediatedischarge values (Figs. 9 and 11). The four plots expressing thedependence between the dimensionless plume area (normalizedwith the square of barotropic Rossby radius) and the dimension-less wind stress are shown in Fig. 11. In this figure, the wind stresswas normalized by f ρH

ffiffiffiffiffiffiffiffig0H

p, which can be interpreted as a

Coriolis force applied to a particle of the plume spreading underthe gravity force.

However, very low and very high inflow velocities (whosesimulations are not presented in this article) changed the plumedynamics significantly. For example, under extremely small runoffconditions, the bulge does not form for the free plume because ofits quick dissipation.

6. Discussion

The notably different shapes of the curves expressing thedependencies of the area of small to medium river plume on thewind stress for different wind directions (Fig. 8) and Coriolisparameter (Fig. 9) raise the question of their physical meaning.First of all, we note that, regardless of their direction, sufficientlystrong winds should destroy the plume due to enhancement ofvertical and horizontal mixing. Therefore, all the curves in Fig. 8should tend to zero in the limit of high wind stress values, which isindeed the case. However, the behavior is less obvious in the lowand intermediate wind stress ranges. Geometrically, the fourarchetypal winds can be identified for any estuary: two blowingperpendicular to the shore (onshore and offshore), and the othertwo blowing along the shore (one to the left of the initialmomentum of the river water, which we call “upwelling favor-able”, and the “downwelling favorable” one in the oppositedirection to it). In the region of this study, these four basic winddirections are represented by the respective SW, NE, NW, and SEwinds. We now speculate that the water particles in a freelypropagating, unforced plume naturally tend to move offshorebecause of their inertia, and along the shore to the right/left ofthe mouth (for the Northern/Southern hemisphere) because of theCoriolis force. Further, if the wind forcing is applied, the wind dragcan act either in favor or against these “preferred” directions.

For example, both the offshore and the “downwelling favor-able” alongshore winds produce drags that are codirectional withthe free motion of the plume. Therefore, as long as the wind stressis not strong enough to destroy the plume, it facilitates itsspreading, and the plume area initially increases with τ (Fig. 8,triangles and circles). As the wind stress grows further, thedissipation increases, so the plume area attains its maximumvalue at some critical τcr (estimated in this particular case asabout 4.5�10−3 N m−2 from Fig. 8), and then starts to decrease.

In contrast with the winds of the directions mentioned in thepreceding paragraph, the onshore wind is always against thenatural propagation of the plume. It presses the plume towardsthe coast and tends to arrest it within a narrow belt adjacent to theshore. The wind-induced dissipation adds to the decay of theplume. Therefore, under the conditions of the onshore wind stress,

A.A. Osadchiev, P.O. Zavialov / Continental Shelf Research 58 (2013) 96–106104

the area of the plume decreases steadily, though not linearly(Fig. 8, squares).

The most complex dependence on the wind stress module isevident for the “upwelling-type” winds (Fig. 8, diamonds) whosedrag is also against the free motion of the plume. Once such a windis applied to the initial “bulge-shaped” plume, its action ismanifested in a retreat of the “downstream”, far edge of theplume towards the river mouth. At the same time, the waterparticles forming the “upstream” edge of the plume near themouth are less sensible to the wind due to their higher initialmomentum, stronger Coriolis force, and larger vertical extent ofthe plume layer. Therefore, as long as τ is relatively small, the“upstream” edge remains almost intact, while the “downstream”

one displaces towards the mouth, and the plume initially narrowsand attains a minimum area, as reflected in the left portion of thecorresponding curve in Fig. 8. However, if the wind stress growsfurther, the entire plume is eventually forced to migrate to the leftof the river mouth and stretches with the wind, so its spatialextent starts to increase with τ. Finally, as the growth of the windstress continues, the dissipation prevails and the plume vanishes.

The “M-shaped” pattern of the dependence of the plume areaon the latitude can also be explained in a fairly simple manner(Fig. 9). In the absence of the surface forcing and ambient seacurrents, the Coriolis force represents the only mechanism thatreturns the water parcels of the plume to the coast and therebyhelps to maintain the plume bounded. Hence, if the Coriolis forceis small (i.e., in the equatorial regions), the plume is subjected tomore rapid horizontal dissipation while spreading unlimitedly intothe open sea, and the residual plume area becomes relativelysmall. On the other hand, if the Coriolis parameter is large (i.e., thepolar regions), the Coriolis force traps the river discharge leavingthe mouth in a narrow stripe immediately adjacent to the shore,and the plume area is relatively small, too. Therefore, betweenthese two minima, there must be a range of “plume favorable”latitudes, where the area attains its maximum, and this is exactlywhat is seen in Fig. 9 for the tropics (20–301). It follows from theseexperiments that under other equal conditions, small to mediumriver plumes in the tropical and subtropical latitudes tend tooccupy larger areas than those in the equatorial or sub-polarregions. However, the dependence on the Coriolis parameter isrelatively weak compared with that on the wind stress, and islargely offset even by moderate wind (Fig. 10).

7. Conclusions

A Lagrangianmodel of awind-forced, surface-advected river plumeis developed. The model simulates the plume by tracking individualparticles. This allows avoiding problems of insufficient grid resolutionin the areas of sharp gradients, typical for Eulerian approach tomodeling of river plumes. Another advantage of this model is its highspeed at relatively low computational cost, as compared with Eulerianmodels. On the other hand, a significant drawback of the model lies inthe fact that it ignores any feedback from the plume onto the ambientocean. Therefore, the model is intended to simulate plumes of small tomedium rivers whose reciprocal impact on the ambient coastalcirculations is considered negligible.

The model was used to investigate the dependencies betweenthe spatial extent of the plume and the wind stress. The characterof these dependencies proved to depend critically on the winddirection. For the offshore and “downwelling favorable” along-shore winds, the plume area initially grows with the increase of τ,and then starts to decrease when τ exceeds a certain critical value.For the onshore wind, the area decreases steadily (though notlinearly) as τ increases. The most complex behavior is observed forthe “upwelling favorable” alongshore wind: the plume area

initially decreases until the wind stress exceeds a certain criticalvalue, then starts to increase with τ and attains a maximum, andthen eventually drops to zero as τ continues to increase.

We also investigated the dependence of the plume area on thelatitude, keeping all other conditions fixed. It was found that if allother conditions were equal, the most spatially extensive plumesformed in the tropical regions. The exact latitude of the maximumplume area depended on the river inflow velocity.

The model was also applied to the real wind stress, ambient seacurrents and latitude conditions of the region adjacent to theMzymta River mouth in the Black Sea. The model simulatedvariability of the Mzymta River plume during two selected trialperiod of 5 days each close to that observed through in situmeasurements and satellite imagery.

Overall, the proposed model appears to be a useful and handytool for simulating the synoptic variability of small and mediumriver plumes under real forcing conditions, as well as the generalaspects of the plume dynamics. The findings discussed above mayhave practical implications in the context of marine pollution, aswell as sediment transport and nutrient cycling in thecoastal zone.

Acknowledgments

This work was supported by the Russian Ministry of Educationand Science, and the EU Framework Program 7 (CLIMSEAS Project).The authors are grateful to many colleagues who participated in thefield campaigns, whose data were used in this study. They also wishto thank Dmitry Soloviev (MHI, Ukraine) for assistance with satelliteimagery. Valuable comments from 3 anonymous reviewers and theEditor helped to significantly improve the manuscript.

Appendix. Parameterizations used by the model

The mean river inflow velocity wr and the mean river inflowangle αr determined by the geometry of the estuary implies theuniform probability distribution law for the initial velocities andinflow angles of the newly appearing particles from the rivermouth at every time step:

w0 � U 0; wr½ � ; α0 � U αr−π

12; αr þ

π

12

h iðA2:1Þ

which imitates the continuous turbulent flow of river water intothe plume.

In order to simulate the small-scale horizontal turbulent mix-ing, the deterministic approach described in the main text (Eq. (1))was complemented by the following random-walk Monte Carlomethod (Hunter, 1987):

xiþ1 ¼ xi þ uiþ1Δt− aiþ1x Δt22 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Di

hΔtq

ηx

yiþ1 ¼ yi þ viþ1Δt− aiþ1y Δt2

2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Di

hΔtq

ηyðA2:2Þ

where (x, y) are the coordinates of an individual particle, Δt is thetime step, Dh is the horizontal diffusion coefficient depending onthe velocity field as specified below, and ηx; ηy are the independentrandom variables with standard normal distribution, which wereproduced by a random number generator.

The horizontal and vertical diffusion coefficients used inEqs. (3)–(6) at the Section 3 were calculated from the following

A.A. Osadchiev, P.O. Zavialov / Continental Shelf Research 58 (2013) 96–106 105

formulas:

Dih ¼ ζh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuixþΔx;y−u

ix−Δx;y

Δx

� �2

þ 12

vixþΔx;y−vix−Δx;y

Δx þ uix;yþΔy−u

ix;y−Δy

Δy

� �2

þ vix;yþΔy−vix;y−Δy

Δy

� �2vuut

ðA2:3Þ

Div ¼ ζvð1−minð1; RiiÞ2Þ3 ðA2:4Þ

where ζh; ζv are the horizontal and vertical scaling coefficie-

nts, Rii ¼Ni2=Si2 is the Richardson number, Ni ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðg=ρiÞðρsea−ρi=hi

qÞ

is the buoyancy frequency, Si ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðui −ui

seaÞ2 þ ðvi −viseaÞ2=hiq

is thevertical shear. The first equation is the well-known Smagorinskyformula (Smagorinsky, 1963) used in plume simulations in a numberof previous studies (e.g., Xia et al., 2011), while formula (A2.4) forvertical diffusion was adopted from Large et al. (1994).

According to Linden (1979), density stratification inhibitsturbulence and therefore reduces vertical viscosity and diffusitiv-ity as Ri−0:5. The horizontal and vertical diffusitivity and viscositycoefficients used in the model were assigned with the followingparameterizations (Yankovsky and Chapman, 1997):

μh ¼ 4� 10−4; ζh ¼ 1 ðA2:5Þ

μiv ¼ 10−5 þ 9� 10−5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 0:3Rii

p ; ζv ¼ 2� 10−5 ðA2:6Þ

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