Department of Oceanography and Coastal Sciences,College of the Coast and Environment,Louisiana State University,Baton Rouge 70803,USA

Effectof stratification on currenthydrodynam icsover Louisiana shelf during Hurricane Katrina

Mohammad Nabi Allahdadi*,Chunyan Li

Department of Oceanography and Coastal Sciences,College of the Coast and Environment,Louisiana State University,Baton Rouge 70803,USA

Abstract

Numericalexperimentswere conducted using the finite volume community oceanmodel(FVCOM)to study the impactof the initialdensity stratification on simulated currents over the Louisiana shelf during Hurricane Katrina.Model results for two simulation scenarios,including an initially stratified shelf and an initially non-stratified shelf,were exam ined.Comparison of two simulations for two-dimensional(2D)currents, the time series of current speed,and variations of cross-shore currents across different sections showed that the smallest differences between simulated currents for these two scenarios occurred over highlymixed regionsw ithin 1 radius ofmaximum wind(RMW)under the hurricane. For areas farther from them ixed zone,differences increased,reaching themaximum values off Terrebonne Bay.These large discrepancies correspond to significantdifferencesbetween calculated verticaleddy viscosities for the two scenarios.The differenceswere addressed based on the contradictory behavior of turbulence in a stratified fluid,as compared to a non-stratified fluid.Incorporation of thisbehavior in theMellor-Yamada turbulent closuremodel established a Richardson number-based stability function thatwas used for estimation of the vertical eddy viscosity from the turbulentenergy andmacroscale.The resultsof this study demonstrate the necessity for inclusion of shelf stratificationwhen circulationmodeling isconducted using three-dimensional(3D)baroclinicmodels.To achievehigh-accuracy currents,the parametersassociated w ith the turbulence closures should be calibrated w ith field measurements of currents at different depths.

?2017 Hohai University.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Hurricane Katrina；Louisiana shelf；Hydrodynamics；Baroclinic and barotropicmodels；Stratification

1.Introduction

Water column stratification is a prom inent feature ofmost oceanic,shelf,and estuarine waters.A lthough m ixing forces, including w inds,waves,and tides,occasionally or continuously m ix the water column,density gradients re-stratify thewater column partially or fully,even in the vicinity of a hurricane's track(Keen and Glenn,1999).Hence,these water bodies are always affected by some degree of stratification. Stratification highly modulates bio-geochem ical processes across the water column and affects the concentration of different chem ical substances(Katsev et al.,2010).Strong summertime stratification over the Louisiana-Texas continental shelf is the main physical contributor to the seasonal hypoxia,as it prevents re-oxygenation of bottom water (W iseman etal.,1997；Allahdadietal.,2013；Chaichitehrani, 2012).Asa resultof confinementby the stratification,colored dissolved organic matter(CDOM)is exposed to the sunlight and loses its properties through the photo-bleaching process (Tehrani et al.,2013；Chaichitehrani et al.,2014).

A relevant aspect of the stratification/de-stratification impact is the contribution in terms of modulating thecirculation.Several studies have demonstrated the substantial effect of stratification on current hydrodynam ics in different water bodies(Csanady,1972；Park and Kuo,1996；Saenko, 2006；Zhang and Steele,2007；A llahdadi et al.,2011,2017). Park and Kuo(1996)pointed out that,in an estuary,m ixing across the water column can weaken the circulation by enhancing verticalmomentum flux,while circulationmay be strengthened by increasing salinity gradientsalong theestuary. For a large stratified lake,the interaction between w ind stress and Coriolis force can produce complicated circulation structures(Csanady,1972).Csanady(1972)used an analytical model to study theeffectofw ind stresson largestratified lakes and concluded that baroclinic Kelvin waves that cause substantial transport along the shore are major features of the response to spatially uniform w ind.Baroclinicity significantly contributes to the formation of a cyclonic summertime circulation pattern in Lake M ichigan(Schwab and Beletsky,2003). As an example for shelf waters,a study for the Rine ROFI demonstrated that the progression of tidal waves in stratified waters could result in significantly different current patterns, as compared to cases w ith pre-m ixed waters.Current observation data were used along w ith an analytical model to conclude that tidal ellipses were substantially different for stratified and m ixed conditions(Visser et al.,1994).During the well-mixed periods,tidal currents were essentially rectilinear,w ith the major component directed along the coast, while the stratification enhanced the cross-shore componentup to about 40%of the alongshore component.Over the Gorgeous Bank,a remarkable summertime stratification effect on amplification of current speed and transport,as well as a strengthening of the tidalmixing front,was reported(Naim ie, 1996；Chen etal.,2003).

In the open ocean or large water bodies like the Gulf of Mexico,stratification contributes to the amplification of currents through inertial oscillations.In this case,stratification controls the rate of w ind momentum exchange from the surface across thewater column by controlling the vertical eddy viscosity(Davies,1985).Studiesusing both observationaldata and numerical models have demonstrated the intensifi cation effect of summertime stratification on shelf currents in the northern Gulf ofMexico(Chen and Xie,1997；DiMarco etal., 2000).

Hurricane w inds m ixing the water column produce an extreme case of stratification/m ixing effects on circulation.In this case,a rather complicated spatial and temporal pattern of m ixing across the water column is produced(Elsberry et al., 1976；Price,1981).Cooper and Thompson(1989)reported thatskipping the stratification could havebeen oneof themain reasons for discrepancies between measured and simulated current speeds in the northern Gulf of Mexico during Hurricane Eloise.Consideration of an initially stratified water column and the effect of turbulent m ixing across the water column is essential to some 3D numerical models used for studying hurricane-induced currents(for example,Ly,1994；Keen and Glenn,1999；Ly and Kantha,1993).Keen and Glenn(1999)simulated the hydrodynam ics induced by Hurricane Andrew,which passed over the Louisiana shelf in 1992, and compared the results w ith available measurements at several stations.They exam ined both stratified and nonstratified models,and concluded that for moorings that are located w ithin 1 RMW of the hurricane center,the results obtained from both stratified and non-stratifiedmodelsagreed w ith themeasurements.For stations farther than 1RMW from the hurricane center,and especially farther than 2 RMW,results of the non-stratified model were significantly different from themeasurements.This indicates that simulated current magnitudes are affected by the rate of turbulent energy exchange and dissipation w ithin the water column(Allahdadi et al.,2011,2017).The same approach was applied by Keen and Glenn(1998)to calibrating the simulated surface currents during Hurricane Andrew.They found that the overestimation/underestimation of surface/bottom currents was due to the small rateof turbulentenergy dissipation.Hence,by changing the empirical parameter representing the dissipation term in the turbulent closure model,current speeds were modified to agree w ith the measurements.However,these studies did not address the detailed variations of currents in relation to stratification and turbulent m ixing.This study attempted to examine the temporal and spatial variations of simulated currents in a stratified and non-stratified water column during ahurricane and address the role of verticalm ixing in causing the differences between the simulated currents for different scenarios of the initial shelf stratification.

2.M aterials and methods

2.1.Model formulation

The finite volume community ocean model(FVCOM),a 3D prim itive equation ocean model,was used to implement the numerical tests and study the effect of stratification on hurricane-induced circulation in the northern Gulf of Mexico. The horizontal equations ofmotion in a 3D case(as included in FVCOM)consider the local acceleration term,nonlinear acceleration terms,Coriolis effect,pressure gradient,and verticaland horizontal internal friction terms.Eqs.(1)and(2) include all the mentioned terms forxandydirections, respectively:

wherex,y,andzare east-west,north-south,and vertical Cartesian coordinate axes,respectively；u,v,andware the currentvelocity components in thexdirection,ydirection,andzdirection,respectively；tis time；fis the Coriolis parameter；Pis the pressure；ρ0is the referencewater density；Kmis thevertical eddy viscosity；andFuandFvare the horizontalmomentum diffusion terms inxandydirections,respectively.

The second term on the right-hand side of each equation accounts for theeffectofverticalmomentum flux on currentsinxandydirections.This term is a function of the vertical eddy viscosity,which is controlled by stratification across thewater column(Chen etal.,2006).Determining theappropriatevalues for thisparameterhasalwaysbeenachallenge in3D circulation models.FVCOM uses the Mellor-Yamada level 2.5 turbulent closuremodel to parametrize theverticaleddy viscosity.In this approach,Kmisdeterm ined from the follow ing equation:

where 0.5q2is the turbulent kinetic energy,lis the turbulent macroscale,andSmis a stability function.Parametersqandlare calculated by solving a set of differential equations, incorporated in FVCOM.ParameterSmis a function ofq,l, and the GradientRichardson number ateach depth(discussed further in Section 4).FVCOM also solves the equations for temperature and salinity variations.Hence,vertical variations of density and stratification and,thereby,the stratification strength are determ ined for further application in the turbulent closure model.More about the FVCOM model formulation and its numerical scheme can be found in Chen et al.(2006).

Fig.1.Modeling area.

2.2.Model setup

Themodeling area extended from Mobile,Alabama to the Sabine Bank,Texas,and comprised Louisiana shallow and deep waters(Fig.1(a)).A computationalmesh,composed of triangular elements,which was refined over the shelf,was used(Fig.1(b)).Mesh resolution varied from 10 km along the offshore boundary to about500m over the inner shelf.Fig.2 shows the track of Hurricane Katrina in the northern Gulf of Mexico.In the vertical direction,25 sigma layersw ith higher resolution at the surfacewere considered(Fig.3),where layers and bed profi le are plotted along section 1,and the Southwest Pass is located at an along-shelf distance of 440 km.

Themodelwas forced with a Hurricane Katrinaw ind field (see Fig.2 for the track in the northern Gulf of Mexico)obtained from the combination of H-W ind and w ind fields from the NationalCenters for EnvironmentalPrediction(NCEP).HWind is a high-resolution hurricane w ind product from the National Oceanic and Atmospheric Adm inistration(NOAA)'s Hurricane Research Division(Powell etal.,1996,1998).This w ind field is produced using available surface weather observations including ships,buoys,coastal platforms,surface aviation reports,and reconnaissance aircraft data adjusted to the surface.The final w ind field is represented at a 10-m height on a 1000 km×1000 km movingboxw ith a spatial resolution of about 6 km centered at the hurricane's central position(Dietrich et al.,2011；Wang and Oey,2008).In this study,a hurricane w ind field was blended w ith the NCEP/ NARR(North American Regional Reanalysis)w ind at a 36-km resolution(Allahdadi et al.,2011)to include the background w inds away from the hurricane center.Themain objective of this study was to demonstrate the effect of stratification on the current pattern generated only by a hurricane force.In this case,the pre-stratified water column wasm ixed under the pure effect of the hurricane,and all changes in stratification and current pattern could be attributed to the hurricane.Therefore,tidal forceswere not included.Also,the Louisiana shelf is am icrotidal region where the average tidal domain is 0.4m(A llahdadietal.,2013).The current inducedby this small tidal force is negligible inmost cases(DiMarco and Reid,1998).

Fig.2.Track of Hurricane Katrina in northern Gulf of Mexico.

Fig.3.Water depth distribution and sigma layers used for model discretization in z direction.

To avoid instabilities caused by the reflected waves from the boundary,the explicit Orlanski radiation(ORE)was used as the boundary condition along w ith the appropriate number of sponge layers.Further details about the model setup and verification w ith field data can be found in Allahdadi(2014) and Allahdadiand Li(2017).

2.3.Simulation scenarios

To study the effect of stratification on the Louisiana shelf currents during Hurricane Katrina,two simulation scenarios, stratified and non-stratified shelves,were considered.Model setup and input parameters for both scenarioswere the same, w ith the only difference being the initial temperature and salinity variations across the water column.For the stratified scenarios,vertical variations of temperature and salinity were introduced into the model from the climatological database, prepared by NOAA(http://www.nodc.noaa.gov/access/ allproducts.htm l)for August,the month in which Hurricane Katrinapassed over the Louisianashelf.Fig.4 shows the initial temperature stratification at the beginning of the simulation across thewater column for the east-west section 1 shown in Fig.2.The patternsof initialsalinity stratification and thereby density stratificationaresim ilar.Forthenon-stratified scenarios, constant values of 25°C forwater temperature and 35 psu for salinity wereused at thebeginning of the simulation.

Fig.4.Initial temperature stratification over shelf for section 1 in stratified scenario.

Verticaleddy viscosity forboth scenarioswas resolvedusing the Mellor-Yamada level 2.5 turbulent closure model,as described in Section 2.1.Two parameters,thebackground eddy viscosity and the constant for calculating turbulent energy dissipation(B1),were specified for both stratified and nonstratified simulation scenarios.Note that the background eddy viscosity isa partof the verticaleddy viscosity accounting for the initial stratification conditions at the beginning of simulations,and the rest of the vertical eddy viscosity is calculated based on a turbulentclosuremodel.The background eddy viscosity was considered 10-4based on Kantha and Clayson (1994).This small value of 10-4for verticaleddy viscosity at thebeginningofmodelingwasused toaccountforinternalwave effects at the start time.The constant for calculating turbulent energy dissipation wasused to calculate the rate of kinetic energy dissipation asa tuning constant.The rateof kinetic energy dissipation in theMellor-Yamadaclosuremodeliscontrolledby the follow ing relationship:

where3is the rate of kinetic energy dissipation；andB1is a coefficient holding a range between 12 and 25,which was considered as its intermediatevalueof16.6 in thisstudy(Keen and Glenn,1998).

3.Sim ulation results

Using the model described in the previous section,simulation was completed for a 15-day period.This simulation period included aweek before August29(the day thatKatrina passed over the Louisiana shelf)and aweek after the landfall in the northern Gulf of Mexico.Simulated currents for stratified and non-stratified scenarios were compared at different times,and some significant differenceswere identified,especially in terms of the velocity values.Fig.5 shows the shelfw ide simulated surface currents for each scenario at two different times on August 29,2005,where the blue solid line represents the hurricane's track and the red circular points on the hurricane's track show the location of the eye corresponding to the simulation at7:00 and 10:00 UTC.

At7:00UTC on August29,2005,when the hurricane'seye was located about 50 km southwest of the Bird's Foot Delta, just west of the M ississippi Southwest Pass,northeasterly w indson the leftsideof the hurricane produced southward-tosouthwestward currents in the near-shore and offshore regions of Terrebonne Bay.The overall current pattern for both simulation scenarios was sim ilar.However,the pattern of simulated currents for the non-stratified scenario was spatially smoother and more uniform,exhibiting larger velocities over the inner shelf,especially off Terrebonne Bay.The cyclonic pattern of the hurricane-induced currents ismore pronounced for the stratified scenario,due to the substantial decline of simulated currents over the offshore areas under the non-stratified scenario.Over the offshore area of Terrebonne Bay, speeds of simulated currents under the assumption of a nonstratified water column reached 1 m/s,while,for the stratified scenario,the current speed over this areawas 0.2-0.3m/ s.At10:00 UTC,the eyewas located about20 km southwest of the Southwest Pass.The southward and southwestward currents,flowing from thewestof Terrebonne Bay toward the area in the south of Barataria Bay,were still smoother and larger in magnitude for the non-stratified scenario.Currents around the hurricane center at 10:00 UTC followed patterns for both simulation scenarios sim ilar to those at 7:00 UTC.

Fig.5.Simulated currents over Louisiana shelf at two different times in stratified and non-stratified scenarios.

The difference in currents resulting from stratified and non-stratified simulation scenarios was also exam ined by comparing time series of current speeds at three different points(P1,P2,and P3)over the Louisiana shelf(Fig.6). A lthough the adopted approaches for calculating the vertical eddy viscosity and theassociated parametersare the same for both simulation scenarios,the dependency of eddy viscosity on the stability of the water column(stratification strength) can result in completely different values of eddy viscosity at a particular location.In order to explore the effectof vertical eddy viscosity on currents,for each point,the values of the vertical eddy viscosity in stratified and non-stratified scenarios(as calculated by the Mellor-Yamada level 2.5 turbulent closure model)are compared(Fig.7).Time series of current speed and vertical eddy viscosity are presented for about40 h,including 20 h before the time the eye reached the Southwest Pass(represented as negative values)and 20 h after that(represented as positive values).The time that the eye reached the Southwest Pass is considered the reference time(t=0).

It can be seen from Fig.6 that,for the non-stratified scenario,the current speed peak(almost corresponding tot=0) atpointP1,located about50 km westof Katrina's track(about 1.4 RMW,when the eyewas located justwest of the Southwest Pass),was almost twice that of the stratified scenario. Thenon-stratified scenario results in almostzero currentspeed att＜-10 h andt＞10 h,while the stratified scenario results in current speeds between 0.15 and 0.4 m/s att＜-10 h andt＞10 h.Asshown in Fig.7,time seriesof calculated vertical eddy viscosity atpoint P1 are completely different for the two simulation scenarios.For the non-stratified scenario,the parameter has a maximum of 20 m2/s,while the stratified scenario results in very small values(less than 0.01m2/s).

Fig.6.Comparison of time series of simulated surface current velocities in stratified and non-stratified scenarios for points P1,P2,and P3 over Louisiana shelf.

ForpointP2,located 25 km(about0.7RMW)from the left side of the eye,differencesbetween currentspeeds of the two simulation scenariosare smaller,as compared to point P1,but are stillsignificant.At this location,simulated peaksof current for the stratified and non-stratified scenariosare about0.5 and 0.7 m/s,respectively(Fig.6(b)).The corresponding time series of the vertical eddy viscosity at a depth of 10 m depict sim ilar variations for both scenarios.The vertical eddy viscosity increases from small values to the peak(corresponding to the approximate timewhen the hurricane's eye is located in the west part of the Southwest Pass)and decreases after the landfall.However,for the stratified scenario,the values are substantially smaller than for the non-stratified scenario before the peak.The peak value of vertical eddy viscosity for the stratified scenario is almost half of what it is for the nonstratified scenario and occurs about three hours after that of the eddy viscosity peak for the non-stratified scenario.After the peak of verticaleddy viscosity for the stratified scenario is reached,eddy viscosities for the two simulation scenarios matchwell,presumably because thehurricane-inducedm ixing produces identicalstability valuesacross thewater column for both scenarios.Simulated currents in the two scenarios at point P3 agree very well.This point is located about 10 km (about0.28 RMW)from the rightside of the hurricane's track. This point is located in the area that is confined by the Bird's Foot Delta and is affected by the rightward bias of the hurricane'sw ind.Hence,it isexpected thathigh currentspeeds are produced over thisarea,and theassociatedmixing ismuch stronger than it is at points P1 and P2.With the substantial m ixing in this area,the calculated vertical eddy viscosities for both stratified and non-stratified scenarios should be sim ilar. As Fig.7(c)illustrates,time series of estimated vertical eddy viscosities are almost the same for both scenarios.Similar values of vertical eddy viscosity maintained sim ilar amounts of turbulentenergy across thewater column,leading to similar surface currents(Fig.6(c)).

Fig.7.Comparison of time series of vertical eddy viscosities at a depth of 10m from surface in stratified and non-stratified scenarios for points P1,P2,and P3 over Louisiana shelf.

The difference between the shelf currents for the stratified and non-stratified scenarios was also investigated by examining current variations for three cross-sections,including the north-south sections A and B and the east-westsection 1(see Fig.2 for locations).During the severalhours thatKatrinawas approaching the shelf and passing over it,currentshad a strong cross-shelf component(forexample,Fig.5),so only the northsouth current component is shown in Fig.8,where the arrowsrepresent the overall current directions in upper and lower layersacross thewater column.Locationsof sections A and B are selected to present the patterns of stratified and nonstratified shelf areas during the hurricane,respectively.

Fig.8.Contours of simulated cross-shelf currents at10:00 UTC on August25,2005 across north-south and east-west sections for stratified and non-stratified scenarios.

For section A,located south of Terrebonne Bay,the distance to Katrina's track is about 100 km(2.8 RMW).At this location,hurricane-induced m ixing is weak,and the water column is significantly stratified(Allahdadi,2014).The pattern of the north-south component of currents across this section for the two simulation scenarios is compared at the timewhen Katrina's eyewas located at the Southwest Pass,in Fig.8(a)and(b).A lthough over the offshore part of sectionsthere are some sim ilarities,some significant differences are noticeable over the inner-shelf area.A well-developed twolayer current w ith the surface layer flow ing away from the shore and the lower layer flow ing shoreward is depicted in the non-stratified scenario,while there ismore variability of currents across the water column for the stratified scenario.An approximately uniform current,especially for the lower layer, is produced in the non-stratified scenario.For the stratified scenario,the simulated current pattern across the lower layer, particularly over the m iddle part of section A(a cross-shore distance of 20-40 km in Fig.8(a))is more complex.Over them iddle cross-section region and beneath thewater depth of 10m,a rectangular zonew ith a relative high-velocity core is formed.Forsection B,located about10 km from the rightside of the Katrina's track(about 0.28 RMW),simulated current patterns in the two scenarios aremore sim ilar(Fig.8(c)and (d)).The results for both scenarios show well-developed twolayer circulations.The upper layer flows shoreward under the effect of southerly hurricane w inds on the right side of Katrina's track.At the same time,compensatory currents running away from the shore are produced in the lower layer (lower than them id-depth).Currentpatternsacrossboth layers are sim ilar for the two scenarios.However,more variability occurs in the stratified scenario.The vertical structure of the simulated current was also investigated for an east-west section(section 1 in Fig.2),extending westward from the Southwest Pass.Like north-south sections,variations of crossshelf current for section 1 at the timewhen theeyewas located at the Southwest Pass are presented in Fig.8(e)and(f),where the dashed line shows the location of thehurricane'seyeat this time.Current patterns demonstrate similar two-layer flow systems,as illustrated for north-south sections.For the upper layer,the current direction is mainly away from the shore (negative velocities)at the left side of the hurricane's eye. Current patterns for the two simulation scenarios are sim ilar for the areas closer to the hurricane's track.Significant differences are identified for areas further to the west.These remote areas exhibit more oscillatory current patterns and more variability across thewater column.

4.Discussion

Simulation results of Louisiana shelf currents for stratified and non-stratified scenariosduring Hurricane Katrinashow that current patterns and values can be either sim ilar or highly different,depending on the simulated location w ith respect to thehurricane'seyeataspecific time.Itisinferred that,forareas closeto theeye,especiallyon therightsideofKatrina'strack,the results from the two simulation scenarioswereconsistent,while differencesweremore pronounced as the distance from theeye (in this case to thewest)increased.This is consistentw ith the modeling study of Keen and Glenn(1999)forcurrents induced by HurricaneAndrew over theLouisianashelf.In theirstudy,for areaswithin1RMW of thehurricanecenter,where the turbulent m ixingwasstrong,w ith significantm ixing occurring across the water column,sim ilar currentswere simulated in the stratified and non-stratified scenarios,and the simulated results were consistent w ith the measurements.For areas farther than 2.5 RMW from the hurricane center,turbulentm ixing wasweak. Hence,differences in simulated currents between the two scenarioswere large,and themeasured currents cannotbe reproduced in the non-stratified scenario.This conclusion is illustrated in Fig.9,where the solid line shows the hurricane's track.Fig.9(a)shows the difference(Dc)between average simulated surface currents for20 h(from when hurricanew ind started to affecttheshelfuntilseveralhoursafter the landfall)in the two simulation scenariosacross the study area.For the sake of convenience,adashed circledepicting thearea1RMW from the hurricane center(1RMW zone),when theeyewas located westof theSouthwestPass,issketched.Asexpected,forthearea thatwasaffectedbyhurricanewindw ithin the1RMW zone,the difference in simulated currents between the two scenarios is generally small.Themaximum differencesare observed in the area located in thesouthof TerrebonneBay.Theclosestdistance of thisarea to theaverageeye location isabout1.8RMW,while the farthest point is about 3 RMW from the hurricane center. Differentresponsesofsimulated currentsregarding thedistance from the hurricane center and the intensity of turbulentm ixing can be justified based on them ixing characteristicsof the shelf during Katrina.Fig.9(b)shows the averagem ixed layer depth (MLD)over theshelf during the time Katrinaaffected theshelf(the averaging period is consistentw ith Fig.9(a)).Over the shelf,the greatest values of the MLD were found within the 1RMW zone.Thisarea corresponds to the smallvaluesof the difference between the simulated currents for two simulated scenarios.Thegeneraldeclineof them ixing strength in thewest and the areaoff Terrebonne Bay(shown as lesserm ixing depth in Fig.9(b))coincideswith the increase of difference between simulated currents for two scenarios.The higher valuesof differences right at the northwest of the Southwest Pass are consistentw ith the decrease in them ixed layer depth in the vicinity of the delta as a result of Southwest Pass geometry blocking northward currents.

Fig.9.Difference between average simulated surface currents in two simulation scenarios over Louisiana shelf and corresponding average m ixed layer depths inmeters.

The effect of stratification on themomentum balance and the role of vertical eddy viscosity in controlling stratification is described in Eqs.(1)and(2).According to these equations, the conclusion above implies that,over the areas that show larger differences between simulated currents in the stratified and non-stratified water columns,corresponding values for the vertical eddy viscosity should be substantially different(also illustrated in Fig.7 for point P1).This is basically due to the different behaviors of turbulence in stratified and nonstratified flows(Dickey and Mellor,1980).Based on laboratory experiments,Dickey and Mellor(1980)found that the decay of turbulent energy is sim ilar in stratified and nonstratified waters until the Richardson number decreases to a critical value.At this time,turbulent decay ceases abruptly and then continuesw ith amuch lower rate,as compared to the time before the critical stage.At this new stage,internal gravity waves are dom inant.Including the effect of stratification in turbulent closure algorithms like Mellor-Yamada level 2.5,which calculates vertical eddy viscosity in the model,has always been a challenge(Galperin et al.,1988；Kantha and Clayson,1994).In the original Mellor-Yamada turbulent closure model(Mellor and Yamada,1974,1982), one of the main problems was insufficient turbulentmixing for Richardson numbers larger than a typical value like 0.21 (Kantha and Clayson,1994；Burchard and Baumert,1995). Durski et al.(2004)used two different closure models, including Mellor-Yamada level 2.5 and the enhanced version of K-profi le parameterization(KPP)to study turbulentm ixing and circulation,over an idealized continental shelf area.They concluded that the selected closure model and the initial stratification could significantly alter the coastal circulation.

The inconsistency was highly reduced by parameterization of the stability functionSm,as suggested by Kantha and Clayson(1994)and Galperin et al.(1988).As illustrated in Eq.(3),the stability function is used to calculate the vertical eddy viscosity based on turbulent energy and the turbulent macroscale.Themodified equation used in FVCOM for estimating the stability function is(Chen et al.,2006)

whereGhis the gradient Richardson number calculated as follows:

wheregis the acceleration due to gravity,ρ0is the reference water density,andρzis thewater density atvertical coordinatez.Galperin et al.(1988)assigned an upper value of 0.023 forGh,representing the case of unstable stratification(ρz＞0)and a lower bond value of-0.28 for the stable stratification case (ρz＜0).

Time series of the gradient Richardson number at m iddepths for points P1,P2,and P3 are shown in Fig.10.At points P1 and P2,the gradientRichardson number approaches zero only for several hours before and after the time the eye reaches the SouthwestPass(t=0).AtpointP3,it reacheszero att=-10 h and remainsclose to zero forat least thenext60h.

The calculated results of shelf-w ide vertical eddy viscosity for both the stratified and non-stratified scenarios across an east-west section(section 1 in Fig.2),along w ith the variations of water temperature(temperature resulting from the simulation for a stratified water column)are presented in Fig.11.Note that variations ofwater salinity across thewater column for each scenario followed patterns sim ilar to the temperature and therefore were not shown here.Exam ining simulated vertical eddy viscosity and temperature across this section demonstrates the effect of stratification/Richardson number on estimating the vertical eddy viscosity.Profi les are presented for 10:00UTC on August29,2005 in Fig.11,while the eyewas located justwestof the Southwest Pass.A lthough there are sim ilaritiesbetween the calculated eddy viscosities in the stratified and non-stratified scenarios in terms of area and depths of influences,some remarkable differences are also observed.As illustrated in Fig.11(a)and(b),over themixedarea located on the right side of Katrina's track(very right corner of the shown profi le),the calculated eddy viscosities are almost identical,both in values and pattern.Over the other parts of the shelf,the differences are more noticeable.A smoothed pattern is simulated for the non-stratified scenario, whilemany oscillationsand irregularities are observed for the stratified scenario.

Fig.10.Time seriesofgradientRichardson number for three different locations.

Fig.11.Variationsof calculated verticaleddy viscosity acrosssection 1 at 10:00 UTC on August29,2005 in stratified and non-stratified scenariosand corresponding variations of simulated water temperature in stratified scenario.

The oscillation is presumably produced by the internal waves,as a result of the interaction between the turbulence and the stratified flow as mentioned by Dickey and Mellor (1980).Development of amixed layer across the water column over the areas w ithin 1 RMW of the hurricane center (Fig.11(c))is consistentw ith the zone associated w ith smaller differencesbetween the simulated currents in two scenarios as illustrated in Fig.9(a).Asexpected,the layer of high turbulent mixing was interrupted by temperature oscillationsat the base of themixed layer.This behavior is consistentwithmodeling results of Elsberry et al.(1976).

5.Summ ary and conclusions

In this study,the effect of stratification on the current hydrodynamicswas examined through numerical simulations of the Louisiana shelf circulation under Hurricane Katrina.Two simulation scenarios,including an initially stratified shelf and an initially non-stratified shelf,were exam ined using FVCOM. The differentbehaviors of vertical eddy viscosity for the two simulation scenarioswere addressed based on the differences of turbulent decay in stratified and non-stratified flows.The Mellor-Yamada level 2.5 turbulent closure model and the associatedmodifications applied by Galperin etal.(1988)and Kantha and Clayson(1994)were incorporated in FVCOM,as an appropriate approach to parameterizing vertical eddy viscosity.The parameterization considers the effect of the gradientRichardson numberon the stability function,which is applied to the estimation of vertical eddy viscosity.The dependency of verticaleddy viscosity on thegradientRichardson number was demonstrated by exam ining values of the parameter for locationsw ith differentstratification conditions. The follow ing conclusions are drawn:

(1)Model results showed that circulation patterns under the two scenariosweregenerally sim ilar.However,somenoticeable differenceswere identified.Over them ixed area,especially on the right side of Katrina's track,the differences were small, while over the stratified area on the left side(the area located outside of the 1 RMW zone,which was far from the intense hurricane-induced turbulence),the differences were conspicuously larger.Exam ining the simulated cross-shelf velocities across north-south and east-west sections confi rmed this conclusion.Furthermore,comparison of the calculated vertical viscosities for thestratified and non-stratified scenarios for three pointsof different locationsw ith respect to the hurricane center strongly supported this conclusion.

(2)Stratificationmodulates the vertical eddy viscosity that directly affects the current through a term in themomentum equations.Hence,treatment of vertical eddy viscosity is of great importance in circulation modeling.The parameterization of vertical eddy viscosity is based on the particular behavior of turbulence in stratified flow,which is sim ilar to that of non-stratified flow until it reaches a critical gradient Richardson number.

(3)The results of the present study demonstrate that the effect of stratification on the 3D baroclinicmodeling of shelf waters may not be neglected and,to achieve high-accuracy results for circulation,stratification and its associated parameters should be treated appropriately.In this regard,there are two key aspects:First,the initial fields of temperature and salinity that specify the stratification condition at the beginning of simulation should be provided based on reliable data. Lack of appropriate data indicating the initial stratification can degrade the accuracy,even when compared to the more unrealistic non-stratified scenario(Keen and Glenn,1999).Second,suitable turbulent closuremodels should be applied for resolving the vertical eddy viscosity.Although the Mellor-Yamada level 2.5 is a w idely used model,the performance of other models,including K-profi le parameterization(KPP)and thek-3model,can be examined in order to determ ine the best approach(Li et al.,2005).However,some parameters, including the background eddy viscosity andB1(for Mellor-Yamada level 2.5),should be tuned using the available hydrodynam ics data across thewater column.

Acknow ledgem ents

The authors would like to thank Dr.Chang-sheng Chen (University of Massachusetts-Dartmouth)for his kindness in sharing the FVCOM code.

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Received 11 September 2016；accepted 24March 2017 Available online 7 June 2017

This work was supported by grants from Louisiana's Coastal Protection and Restoration Authority(CPRA)and the Stennis Space Center,the Lake Pontchartrain Basin Foundation,the National Science Foundation(Grants No. OCE-0554674,DEB-0833225,OCE-1140268,and OCE-1140307),the Hypoxia Project of NOAA(Grant No.NA06NPS4780197),the Shanghai Universities First-Class DisciplinesProject,and the ShanghaiOcean University International Center for Marine Studies.

*Corresponding author.

E-mail address:nabiallahdadi@gmail.com(Mohammad Nabi A llahdadi).

Peer review under responsibility of Hohai University.

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