Journal of Marine Science and Engineering

Article Lateral Circulation in a Partially Stratified Tidal Inlet

Linlin Cui *, Haosheng Huang , Chunyan Li and Dubravko Justic

Department of Oceanography and Coastal Sciences, College of the Coast and Environment, Louisiana State University, Baton Rouge, LA 70803, USA; [email protected] (H.H.); [email protected] (C.L.); [email protected] (D.J.) * Correspondence: [email protected]; Tel.: +1-225-578-5117



Received: 30 September 2018; Accepted: 3 December 2018; Published: 19 December 2018


Abstract: Using a three-dimensional, hydrostatic, primitive-equation ocean model, this study investigates the dynamics of lateral circulation in a partially stratified tidal inlet, the Barataria Pass in the Gulf of Mexico, over a 25.6 h diurnal tidal cycle. Model performance is assessed against observational data. During flood tide, the lateral circulation exhibits the characteristics similar to those induced by differential advection, i.e., lateral flow consists of two counter-rotating cells and is convergent at the surface. The analysis of momentum balance indicates that, in addition to the pressure gradient and vertical stress divergence, nonlinear advection and horizontal stress divergence are also important contributors. During ebb phase, the lateral circulation is mostly toward the right shoal (when looking into the estuary) for the whole water column and persisting for almost the whole period. The surface divergence suggested by the differential advection mechanism lasts for a very short period, if it ever exists. The main momentum balance across most of the transect during ebb is between the along-channel advection of cross-channel momentum and pressure gradient. The sectional averaged lateral velocity magnitude during ebb is comparable to that during flood, which is different from the idealized numerical experiment result.

Keywords: estuarine modeling; lateral circulation; tidal currents; momentum balance

1. Introduction The lateral circulation in tidally dominant estuaries can be driven by various mechanisms. Nunes and Simpson [1] identified the effect of differential advection on the generation of secondary circulation. They pointed out that due to frictional retardation, the along-channel velocity is stronger in the channel than over the shoals. When acting upon an along-channel density gradient, it results in greater (smaller) density at the thalweg than at the shoals during flood (ebb) tides. This produces a cross-channel pressure gradient toward the channel (shoals) on the surface and a pressure gradient toward the shoals (channel) at the bottom during flood (ebb) tides. Thus, the lateral flows are convergent (divergent) at the surface over the deep channel and divergent (convergent) at the bottom during flood (ebb) tides. However, the surface axial convergence was only observed during flood tides in Nunes and Simpson’s work. In an idealized, narrow straight channel with weak stratification, Lerczak and Geyer [2] confirmed that secondary circulation was driven by differential advection. Differential rotation of tidal ellipse was also identified as a mechanism for axial convergence fronts [3]. Interactions between barotropic pressure gradient and bathymetry can generate convergence of lateral flow, producing flows rotating toward the channel from the shoals [4,5]. In curved estuaries, an alternative driving mechanism for lateral circulation is the centrifugal acceleration [6–8] and advection [9]. Winds can enhance or degrade the local-curvature-induced, two-layer flow and can drive three-layer flow [10]. Ekman-forced lateral circulation varies with the Ekman number. When the boundary layer is comparable to the channel

J. Mar. Sci. Eng. 2018, 6, 159; doi:10.3390/jmse6040159 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2018, 6, 159 2 of 23 depth (large Ekman number), lateral flow is a single circulation cell; while for thin tidal boundary layer (small Ekman number), lateral flow is complex and varies over the tidal cycle [2]. Boundary mixing on a no-flux boundary layer was confirmed to be one of the driving mechanisms of lateral circulation [2,11]. Cheng et al. [12] investigated the lateral circulation during stratified ebb tides due to the lateral baroclinic pressure gradient, which is generated by differential diffusion caused by a lateral asymmetry in vertical mixing. In the same idealized numerical experiment, Lerczak and Geyer [2] found that lateral flow is about four times stronger during flood tides than during ebb tides. They attributed it to the interaction between the along-channel tidal currents and nonlinear advective processes over a tidal cycle. This flood–ebb asymmetry in the lateral circulation strength was also observed by Scully et al. [13] in the Hudson River estuary, where stronger lateral flows were observed during flood tides while lateral flows were suppressed during ebb tides. However, in the numerical modeling of James River Estuary, Li et al. [14] found flood–ebb asymmetry during neap tides with stronger lateral circulation on ebb, while flood–ebb asymmetry was reduced during spring tides. Lateral circulation plays an important role in estuarine dynamics. Many observational and numerical simulation results [2,7,12,15,16] have demonstrated the existence of secondary currents, and discussed its dependence and feedback on density stratification [7,12,17,18], streamwise momentum budget [2], estuarine circulation [12,13], as well as its impacts on sediment transport and geomorphology [19–21]. In Barataria Pass, the area of focus for this study, observations showed that there existed a distinct asymmetry in stratification within the diurnal cycle [22]. However, in that analysis no consideration was given to the contribution of the lateral circulation to density stratification. In this study, we use a three-dimensional (3-D), high-resolution hydrodynamic model to examine the lateral circulation structure in the partially stratified tidal inlet, Barataria Pass, which connects Barataria Bay with the continental shelf in the southeastern Louisiana. The objectives of this study are to elucidate the tidal evolution of lateral circulation and determine its driving mechanisms. The remainder of this paper is organized as follows: Section2 describes the study area, and the configuration of the 3-D finite volume numerical ocean model. Section3 presents the validation of the numerical model and the temporal evolution of lateral circulation over a 25.6 h diurnal tidal cycle. In Section4, we quantify the 2-D and 3-D momentum balance and examine the driving forcing for lateral circulation. Flood–ebb variations in lateral circulation pattern are also discussed in Section4. Finally, conclusions are given in Section5.

2. Materials and Methods

2.1. Study Area Barataria Bay (Figure1) is located at the southeastern Louisiana, on the western side of the Mississippi Birdfoot Delta. It is connected to the Gulf of Mexico through several tidal inlets including Barataria Pass, which is between two barrier islands, Grand Isle Island and Grand Terre Island (Figure1b). The most significant freshwater source inside the Barataria Bay is the Davis Pond freshwater diversion. The maximum diversion discharge is ~300 m3 s−1 [23]. Barataria Pass is an 800 m wide narrow channel. It is one of the four main tidal passes of Barataria Bay, accounting for ~66% of total water exchange [24]. Tidal currents account for ~85% of the total flow variance in the inlet, with equal contributions from the O1 and K1 constituents. Tidal amplitudes for both O1 and K1 constituents are about 0.5 m s−1 [25]. Maximum tidal currents reach as high as 2 m s−1 during tropic tides. Barataria Bay is composed of broad shallow waters (average depth of 2 m), islands and a 5 m deep main shipping channel, the Barataria Waterway. The shipping channel ends at Barataria Pass. Around Barataria Pass the main channel has an average depth of ~20 m and is being periodically dredged, causing a depression of ~50 m deep close to the inlet (Figure2). A cross-sectional view of the Barataria Pass, along which all analyses are performed, is shown in Figure1c. Bathymetrically this transect has one 20 m deep channel in the center bordered by extensive shoal regions. The shoal at the left-hand side (looking upstream) is ~2 m deep and 250 m wide, while that at the right-hand side is ~3 m deep and 100 m wide. J. Mar. Sci. Eng. 2018, 6, x FOR PEER REVIEW 2 of 23 cycle [2]. Boundary mixing on a no-flux boundary layer was confirmed to be one of the driving mechanisms of lateral circulation [2,11]. Cheng et al. [12] investigated the lateral circulation during stratified ebb tides due to the lateral baroclinic pressure gradient, which is generated by differential diffusion caused by a lateral asymmetry in vertical mixing. In the same idealized numerical experiment, Lerczak and Geyer [2] found that lateral flow is about four times stronger during flood tides than during ebb tides. They attributed it to the interaction between the along-channel tidal currents and nonlinear advective processes over a tidal cycle. This flood–ebb asymmetry in the lateral circulation strength was also observed by Scully et al. [13] in the Hudson River estuary, where stronger lateral flows were observed during flood tides while lateral flows were suppressed during ebb tides. However, in the numerical modeling of James River Estuary, Li et al. [14] found flood–ebb asymmetry during neap tides with stronger lateral circulation on ebb, while flood–ebb asymmetry was reduced during spring tides. Lateral circulation plays an important role in estuarine dynamics. Many observational and numerical simulation results [2,7,12,15,16] have demonstrated the existence of secondary currents, and discussed its dependence and feedback on density stratification [7,12,17,18], streamwise momentum budget [2], estuarine circulation [12,13], as well as its impacts on sediment transport and geomorphology [19–21]. In Barataria Pass, the area of focus for this study, observations showed that there existed a distinct asymmetry in stratification within the diurnal cycle [22]. However, in that analysis no consideration was given to the contribution of the lateral circulation to density stratification. In this study, we use a three-dimensional (3-D), high-resolution hydrodynamic model to examine the lateral circulation structure in the partially stratified tidal inlet, Barataria Pass, which connects Barataria Bay with the continental shelf in the southeastern Louisiana. The objectives of this study are to elucidate the tidal evolution of lateral circulation and determine its driving mechanisms. The remainder of this paper is organized as follows: Section 2 describes the study area, and the configuration of the 3-D finite volume numerical ocean model. Section 3 presents the validation of the numerical model and the temporal evolution of lateral circulation over a 25.6 h diurnal tidal cycle. In Section 4, we quantify the 2-D and 3-D momentum balance and examine the driving forcing forJ. Mar. lateral Sci. Eng. circulation.2018, 6, 159 Flood–ebb variations in lateral circulation pattern are also discussed in Section3 of 23 4. Finally, conclusions are given in Section 5.

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2. Materials and Methods

2.1. Study Area Barataria Bay (Figure 1) is located at the southeastern Louisiana, on the western side of the Mississippi Birdfoot Delta. It is connected to the Gulf of Mexico through several tidal inlets including Barataria Pass, which is between two barrier islands, Grand Isle Island and Grand Terre Island (Figure 1b). The most significant freshwater source inside the Barataria Bay is the Davis Pond freshwater diversion. The maximum diversion discharge is ~300 m3 s−1 [23]. Barataria Pass is an 800 m wide narrow channel. It is one of the four main tidal passes of Barataria Bay, accounting for ~66% of total water exchange [24]. Tidal currents account for ~85% of the total flow variance in the inlet, with equal contributions from the O1 and K1 constituents. Tidal amplitudes for both O1 and K1 constituents are about 0.5 m s−1 [25]. Maximum tidal currents reach as high as 2 m s−1 during tropic tides. Barataria Bay is composed of broad shallow waters (average depth of 2 m), islands and a 5 m deep main shipping channel, the Barataria Waterway. The shipping channel ends at Barataria Pass.

Around Barataria Pass the main channel has an average depth of ~20 m and is being periodically Figuredredged, 1. (causinga) Geographic a depression location of of ~50 the m BaratariaBaratarideep closea Estuary. to the inlet S1–S6 (Figure are USGS 2). A stations. cross-sectional ( b) Location view of of Baratariathe Barataria Pass. Pass, The The coordinatealong coordinate which is is definedall defined analyses as as positive positive are pe rformed,xx toto the the eastern easternis shown bank, bank, in Figurepositive positive 1c. yy toBathymetricallyto the the upstream. upstream. Thethis redtransect line indicateshas one 20 the m cross-sectiondeep channel shown in the centin (cer) andbordered is usedused by inin extensive laterlater analysis.analysis. shoal (regions.c) Cross-sectional The shoal viewat the of left-hand Barataria side Pass. (looking The black upstream) lines indicate is ~2 mCTD deep measurements. and 250 m wide, C1–C7 while are thatlocations at the used right-hand for 2-D momentumside is ~3 m equation deep and analysis. 100 m wide.

FigureFigure 2. Unstructured 2. Unstructured grid grid configured configured forthe forFinite the Finite Volume Volume Coastal Coastal Ocean Ocean Model Model (FVCOM) (FVCOM) Barataria Pass model:Barataria ( aPass) whole model: computational (a) whole computational domain; (b )domain; local domain (b) local of domain Barataria of Estuary;Barataria (Estuary;c) local domain(c) of Baratarialocal domain Pass, of with Barataria horizontal Pass, with resolution horizontal ~50 resolution m in the ~50cross-channel m in the cross-channel direction direction and 30 and m 30 in the m in the along-channel direction. Contours are interpolated bathymetry. White dash line indicates along-channel direction. Contours are interpolated bathymetry. White dash line indicates cross section cross section in this study. in this study.

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2.2. Model Description and Configuration The Finite Volume Coastal Ocean Model (FVCOM) is used in this study to simulate the hydrodynamics of the Barataria Basin and adjacent continental shelf. FVCOM is a 3-D, hydrostatic, free surface, primitive-equation ocean model [26,27]. In the finite volume method, the computational domain is discretized using a mesh of non-overlapping triangles in the horizontal and sigma-coordinate (σ-coordinate) in the vertical. The governing equations are solved in their integral forms in each individual control volume. The triangular grid in the horizontal can resolve complex coastal and bathymetric geometries. It uses a cell-vertex-centered (similar to the finite-difference C-grid) method, which facilitates the enforcement of mass conservation in tracer advection and tracer open boundary conditions. Vertical mixing uses modified Mellor and Yamada level 2.5 turbulence model [28,29] and horizontal diffusion uses Smagorinsky eddy parameterization [30]. The model employs mode split approaches (barotropic 2D (external) mode and baroclinic 3D (internal) mode) to solve the momentum equations with second-order accuracy. The bottom boundary conditions apply an exact form of the no flux boundary conditions. A flooding–drying scheme is implemented in FVCOM to simulate motions in intertidal zones and wetlands. If vertical water column thickness at the cell center is less than a criterion value (typically 5 cm), then the cell is designated as a dry cell and its velocity is set to zero. Whenever the vertical water column thickness exceeds the criterion value, the cell becomes wet and water level and velocity are computed from control equations. The advantage of triangular mesh to accurately represent complex bathymetry and coastlines makes FVCOM ideally suited for Barataria Pass study. A high-resolution FVCOM Barataria Pass model was developed by configuring FVCOM version 2.6 to the Northwestern Gulf of Mexico continental shelf region with inclusion of the intertidal zones inside the Barataria Bay. The computational domain extends longitudinally from Mobile Bay (Alabama) to west of Galveston Bay (Texas) and offshore to about 27◦ N (Figure2). It consists of 146,266 triangular nodes and 283,721 triangular cells. The horizontal grid resolution varies from about 10 m in the upper estuary to about 8 km near the open boundary. Near Barataria Pass, grid cells are fine enough to ensure that the 800 m wide inlet cross section is resolved by ~20 triangles (Figure2c). Vertically FVCOM employs 19 uniform sigma layers, which is ~0.1 m over the shoal and ~1 m in the central depression of the tidal inlet. Computational time steps are 0.2 s and 2.0 s for the external and internal modes, respectively. Model results are saved every 10 min for further diagnostic analysis. Model bathymetry was obtained from various sources. Using an inverse distance weighted interpolation method, a 5 m by 5 m resolution digital elevation model constructed from Light Detection and Ranging (LIDAR) measurement was interpolated into model wetland region. The water depth in channels, bayous, and lakes was interpolated from Coastal Louisiana Ecosystem Assessment and Restoration Report (CLEAR), US Army Corp Survey, and NOAA nautical charts. Shelf and open ocean water depth was interpolated from a coarse resolution ADCIRC model bathymetry. The water depth values in the inlet were obtained by vessel-based surveys [22,31]. Since temperature difference is minute across much of our modeling region during summer time [32], salinity is considered to be the most important factor that influences water density and vertical stratification in Barataria Bay and adjacent coastal oceans in this study. Thus it is the only prognostic tracer variable in FVCOM simulations. Temperature is kept as a spatial and temporal constant (20 ◦C). The coefficients for horizontal viscosity and diffusivity are both set to be 0.4 m2 s−1. The conventional quadratic bottom friction formulation is applied, with drag coefficient Cd determined by matching a logarithmic bottom boundary layer velocity to that of the numerical model at the lowest sigma-layer height. However, bottom drag coefficient over the wetlands is defined as five times greater than that in the estuarine channels, mimicking the vegetation damping effect [33].

2.3. Model Forcing, Initial, and Boundary Conditions FVCOM is driven by winds at the surface, sea level elevation at the open boundary, and freshwater inflows from various Mississippi River and Atchafalaya River passes, and the Davis Pond Diversion. J. Mar. Sci. Eng. 2018, 6, 159 5 of 23

It is initialized on 1 October 2007 and run until 31 December 2008. We use 3-hourly wind data from NOAA National Centers for Environmental Prediction (NCEP) North American Regional Reanalysis (NARR) products and interpolate it onto the entire computational domain. The initial values of sea level elevation and velocity are specified as zero throughout the computational domain. The initial salinity field over the continental shelf is interpolated from HYCOM Gulf of Mexico 1/25◦ reanalysis product, while salinity inside the Barataria Bay is interpolated from observations at the United States Geological Survey (USGS) stations. Experiments show that this technique gives more accurate salinity simulation result than that with linear interpolation from estuarine head to the mouth. At the open boundary, the 6-min interval sea level time series at four stations are downloaded from NOAA tides and currents website. Time series of Dauphin Island, Southwest Pass, Freeport, and Galveston Pier 21 are directly used to prescribe sea surface elevations at the easternmost node, the southeastern node, the southwestern node, and the westernmost node, respectively. Sea level elevation at other open boundary nodes are piecewise linearly interpolated from these four nodes. Observed 15-min freshwater discharge at four locations, Mississippi River at Belle Chasse, David Pond diversion, Atchafalaya River at Morgan City, and Wax Lake Outlet, are injected into the computational domain with flux boundary conditions of zero salinity and specified volume and momentum. All model forcing functions are ramped up from zero over a period of 10 days.

2.4. Observations Simulated water elevation, velocity and salinity are compared with in situ observations. Water elevation data is obtained from National Water Information System of USGS shown in Figure1, including six stations, Barataria Bay near Grand Terre Island (S1), Barataria Bay North of Grand Isle (S2), Hackberry Bay near NW of Grand Isle (S3), Barataria Waterway (S4), Little Lake near Bay Dosgris (S5), and Little Lake near Bay Cutoff (S6). The unit of water elevation is converted to meter and vertical datum is adjusted to mean sea level in order to be compared with model results. Velocity and salinity data are from the field observations conducted at Barataria Pass from 11:30 31 July to 11:10 1 August 2008 UTC (see Li et al. [31] for details about field observation and data processing).

2.5. Analysis Methods The vertically averaged cross- and along-channel momentum equations are written as:

0  0  0 2 ! Z Z Z 1 ∂uD 1 ∂u D ∂u vD ∂ζ g ∂ 0 ∂D = − + + f v −g − [ D ρdσ dσ + σρdσ] D ∂t D ∂x ∂y |{z} ∂x ρ0 ∂x ∂x | {z } | {z } −1 σ −1 | {z } | {z } (1) τsx τbx 1 + − + Fex + Gx Dρ0 Dρ0 |{z} D |{z} | {z } | {z }

! 0 0 0 1 ∂vD 1 ∂u vD ∂v2D ∂ζ g Z ∂ Z ∂D Z = − + − f u−g − [ (D ρdσ0)dσ + σρdσ] D ∂t D ∂x ∂y |{z} ∂y ρ0 ∂y ∂y | {z } COR | {z } −1 σ −1 DDT | {z } DPBP | {z } ADV DPBC (2) τsy τby 1 + − + Fey + Gy Dρ0 Dρ0 |{z} D |{z} | {z } HDIF | {z } WIND FRIC AV2D where (u, v) are the vertical integrated cross- and along-axis velocity components. The positive u is pointed to the eastern bank, the positive v to the upstream. Terms from the left to the right are local acceleration (DDT), nonlinear advection (ADV), Coriolis force (COR), barotropic pressure gradient (DPBP), baroclinic pressure gradient (DPBC), wind stress (WIND), bottom friction (FRIC), horizontal diffusion (HDIF), and difference between nonlinear terms of vertically-averaged 2-D variables and vertical integration of 3-D variables (AV2D). The expressions for HDIF and AV2D can be referred to [26,27]. Consistent with 3-D currents converted to cross- and along-channel directions, J. Mar. Sci. Eng. 2018, 6, 159 6 of 23 all terms in Equations (1) and (2) are rotated from the original FVCOM computation x-y coordinates (see AppendixA for details). AJ. Mar. 3-D Sci. FVCOM Eng. 2018, momentum6, x FOR PEER REVIEW equation analysis was used to identify the mechanisms 6that of 23 drive lateral circulation. The equation is written as: all terms in Equations (1) and (2) are rotated from the original FVCOM computation x-y coordinates

(see Appendix A for details).   0  2 Z A 3-D1 ∂uD FVCOM1 momentum∂u D 1 ∂ equationuvD 1 analysis∂uω was used∂ζ to gidentify∂ the mechanisms0 ∂D that drive = − − − + f v −g −  D dσ + σρ  lateral circulation.D ∂t TheD equation∂x D is∂ writteny D as:∂σ |{z} ∂x ρ0 ∂x ∂x | {z } | {z } | {z } | {z } | {z } σ (3) | {z } 1 𝜕𝑢𝐷 1 𝜕𝑢 𝐷 1 𝜕𝑢𝑣𝐷 11 𝜕𝑢𝜔∂  ∂u  𝜕𝜁 𝑔 𝜕 𝜕𝐷 =− − + − + 𝑓𝑣 −𝑔+ − 𝐷 𝑑𝜎 +𝜎𝜌 2Km Fx 𝐷 𝜕𝑡 𝐷 𝜕𝑥 𝐷𝜕𝑦 D𝐷∂σ𝜕𝜎 ∂σ 𝜕𝑥|{z}𝜌𝜕𝑥𝜕𝑥 1 𝜕 |𝜕𝑢 {z } (3) + 𝐾 +𝐹⏟ 𝐷𝜕𝜎𝜕𝜎  0  2 Z 1 ∂vD 1 ∂uvD 1 ∂v D 1 ∂vω ∂ζ g ∂ 0 ∂D = − − − − f u −g − [ D dσ + σρ ] D 1∂𝜕𝑣𝐷t D1 𝜕𝑢𝑣𝐷∂x D1 𝜕𝑣∂y𝐷 D1 𝜕𝑣𝜔∂σ |{z} 𝜕𝜁∂y 𝑔ρ0 ∂𝜕y 𝜕𝐷∂y | {z } =−| {z }|− {z }|− {z }− 𝑓COR𝑢 −𝑔| {z } − [ 𝐷σ 𝑑𝜎 +𝜎𝜌 ] DDT𝐷𝜕𝑡 ADVU𝐷𝜕𝑥 ADVV𝐷𝜕𝑦 ADVW𝐷𝜕𝜎 DPBP𝜕𝑦 | 𝜌 𝜕𝑦 {z 𝜕𝑦 } DPBC (4) 1∂  ∂v  (4) 1 𝜕 + 𝜕𝑣 Km + Fy + 𝐾 D2 ∂σ+𝐹 ∂σ 𝐷𝜕𝜎𝜕𝜎 ⏟ |{z} | {z } HDIF VDIF wherewhere(u, v )(𝑢,are 𝑣) cross- are cross- and and along-channel along-channel velocity velocity components.components. The The advection advection terms terms are aremoved moved to to the samethe same side side of the of the pressure pressure gradient, gradient, and and each each termterm in in Equations Equations (3) (3) and and (4) (4)is calculated is calculated with with its its correspondingcorresponding sign sign (e.g., (e.g.,−∂ (−𝜕(𝑢u2D)𝐷)/D⁄∂x 𝐷𝜕𝑥).).

3. Results3. Results

3.1. Two-Month3.1. Two-Month Water Water Elevation Elevation Comparisons Comparisons We use the Pearson correlation coefficient [34] as evaluation metrics. Two-month (1 July to 31 We use the Pearson correlation coefficient [34] as evaluation metrics. Two-month (1 July to 31 August 2008) time series of water elevation for both observation and model simulation are shown in August 2008) time series of water elevation for both observation and model simulation are shown in Figure 3. The model reproduces the observed tidal variations, including tropic-equatorial Figuremodulation,3. The model as well reproduces as wind-driven the observed water level tidal set-up variations, and set-down including with tropic-equatorial all correlation coefficients modulation, as wellabove as wind-driven 0.9. water level set-up and set-down with all correlation coefficients above 0.9.

Figure 3. Water elevation comparison between USGS observations (black) and model simulation (red) Figure 3. Water elevation comparison between USGS observations (black) and model simulation from 1 July to 31 August 2008. Grayed areas represent missing data. (red) from 1 July to 31 August 2008. Grayed areas represent missing data.

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3.2.3.2. Velocity Velocity SimilarSimilar to theto the treatment treatment in Li etin al.Li [22 et], theal. x-[22], and ythe-velocity x- and components y-velocity were components counterclockwise were ◦ rotatedcounterclockwise by 52.7 from rotated the eastby and52.7° north from directionthe east toand obtain north the direction cross- andto obtain along-channel the cross- velocity and componentsalong-channel (Figure velocity1b), whichcomponents are shown (Figure in Figure1b), which4. Positive are shown along-channel in Figure 4. velocity Positive is along-channel flood current. Asvelocity shown is in flood Figure current.4a, the observedAs shown along-channel in Figure 4a, the velocity observed has aalong-channel stronger tidal velocity signal, withhas a maximum stronger −1 magnitudetidal signal, of with ~1.5 maximum m s , than magnitude the model of simulated~1.5 m s−1, than velocity, the model which simulated has a maximum velocity, magnitude which has a of − ~1.0maximum m s 1. magnitude The tidal phaseof ~1.0 is m in s agreement−1. The tidal with phase the is observations. in agreement Bothwith observedthe observations. and modeled Both cross-channelobserved and velocities modeled (Figurecross-channel4b) are velocities much smaller (Figure and 4b) noisier, are much and smaller the tidal and signal noisier, is notand clear the comparedtidal signal to is the not along clear compared channel velocity to the along component. channel velocity The discrepancy component. between The discrepancy the observed between and modeledthe observed velocity andis, modeled in part, velocity due to theis, in fact part, that due observed to the fact velocity that observed data points velocity were data chosen points along were a 530chosen m long along transect a 530 within m long a 90transect m band within (45 m a on90 eachm band side (45 [22 ]),m whileon each modeled side [22]), velocity while data modeled points arevelocity exactly data along points the chosenare exactly transect. along the chosen transect.

FigureFigure 4. 4.(a ()a Along-channel) Along-channel velocity velocity atat 1.321.32 mm belowbelow the surface for for observed observed (red (red dot) dot) and and modeled modeled (blue(blue dot) dot) data; data; ( b(b)) Cross-channel Cross-channel velocityvelocity atat 1.321.32 m below the surface surface for for observed observed (red (red dot) dot) and and modeledmodeled (blue (blue dot) dot) data. data.

3.3.3.3. Vertical Vertical Salinity Salinity Profile Profile AA seabird seabird Electronics Electronics Conductivity, Conductivity, Temperature, Temperature, and and Depth Depth Sensor Sensor (CTD) (CTD) was usedwas forused vertical for profilesvertical of profiles salinity. of A salinity. total of 28A total CTD of casts 28 wereCTD madecasts were between made 31 between July and 31 1 AugustJuly and 2008 1 August over a 2008 25.6 h periodover a (see 25.6 Table h period 1 in (see Li et Table al. [31 1] in for Li details). et al. [31] Vertical for details). salinity Vertical profiles salinity from profiles FVCOM from simulation FVCOM at threesimulation locations, at three eastern, locations, middle, eastern, and western middle, side an ofd western the channel side (see of the Figure channel1c for (see station Figure locations), 1c for arestation compared locations), with theare CTDcompared measurements with the inCTD Figure measurements5. The magnitude in Figure of observed 5. The salinitymagnitude ranges of betweenobserved 19 salinity and 28.5. ranges The between maximum 19 and vertical 28.5. salinityThe maximum difference vertical is about salinity 5.5. difference The magnitude is about of 5.5. the simulatedThe magnitude salinity variesof the betweensimulated 15 andsalinity 27. Generally,varies between the model 15 and underestimates 27. Generally, salinity, the perhapsmodel partiallyunderestimates due to the salinity, neglect perhaps of evaporation. partially due However, to the theneglect model of successfullyevaporation. capturesHowever, characteristic the model featuressuccessfully in salinity captures vertical characteristic profile. For features example, in castsalinity 6 (Figure vertical5a), profile. which was For madeexample, 4 h aftercast maximum6 (Figure flood,5a), which has a weakwas made stratification 4 h after at maximum the top of flood, the water has a columnweak stratification and well-mixed at the state top atof thethe bottom.water Ourcolumn model and captures well-mixed thisfeature. state at Otherthe bottom. casts, suchOur model as casts captures 19, 26, 18,this 8, feature. and 16, Other have similarcasts, such vertical as profilescasts 19, with 26, the18, 8, simulation. and 16, have The similar temporal vertical evolution profiles of with modeled the simulation. salinity is The also temporal consistent evolution with the observations.of modeled Forsalinity example, is also the consistent sequence with of cast the numbers observations. (from lowFor toexample, high) for the both sequence the observation of cast andnumbers the model (from show low thatto high) the salinityfor both tends the observation to decrease and in Figure the model5f (ebb show tide, that west the station). salinity This tends gives to usdecrease confidence in Figure to apply 5f the(ebb model tide, towest do station). further qualitativeThis gives dynamicus confidence analysis. to apply the model to do further qualitative dynamic analysis.

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Figure 5.5. Vertical profilesprofiles ofof salinitysalinity comparisoncomparison betweenbetween inin situsitu observationobservation (red)(red) andand simulationsimulation (black).(black). The numbersnumbers in thethe plotplot representrepresent thethe sequencesequence ofof CTDCTD casts.casts. Left panels are forfor floodflood tidestides ((aa,,c ,ce, ),and while e), rightwhile panels right panels for ebb for tides ebb ( btides,d,f). (bTop, d, middleand f). ,Top and, middlebottom, rowand arebottom at the row eastern are at (a ,theb), middleeastern (ca,,d b),), andmiddle western (c, d), ( eand,f) sidewestern of the (e, channel,f) side of respectively. the channel, respectively. Water level elevationWater level in elevation the deep channelin the deep is shown channel in the is bottom-leftshown in the small bottom-left panel, with small vertical panel, lines with showing vertical casting lines times.showing casting times. 3.4. Temporal Variation of Stratification in the Barataria Pass 3.4. TemporalIn order Variation to study theof Stra variationtification of in stratification the Barataria over Pass a diurnal cycle, a 25.6 h time series from the FVCOMIn order simulation, to study starting the variation from 06:00 of stratification 31 July to 07:40 over 1 a August diurnal 2008, cycle, was a 25.6 chosen h time to perform series from further the analysis.FVCOM simulation, Time series ofstarting water from level, 06:00 depth-averaged 31 July to 07:40 along-channel 1 August velocity 2008, was and chosen salinity to difference perform betweenfurther analysis. bottom and Time surface series at threeof water cross-channel level, depth-averaged locations are shown along-channel in Figure6 velocity. The water and elevation salinity anddifference depth-averaged between bottom velocity and are surface not in-phase. at three Based cross-channel on the deep locations channel are station, shown flood in Figure tide lasts 6. The for ~12water h withelevation maximum and depth-averaged vertically averaged velocity flood are velocity not in-phase. ~0.8 m/s, Based while on the ebb deep tide lastschannel for ~13.6station, h andflood maximum tide lasts verticallyfor ~12 h averagedwith maximum ebb velocity vertically ~1.2 averaged m/s. This flood tidal velocity asymmetry ~0.8 ism/s, mainly while caused ebb tide by 3 upstreamlasts for ~13.6 Davis h Pond and diversionmaximum discharge, vertically which averaged is ~200 ebb m velocity/s during ~1.2 this m/s. period This of tidal time. asymmetry Stratification is evolutionmainly caused at these by threeupstream locations Davis shows Pond distinct diversion cross-channel discharge, variation. which is Within~200 m 23/s h ofduring early this flood period tides, stratificationof time. Stratification at all three evolution locations decreasesat these three and reacheslocations a well-mixedshows distinct condition. cross-channel Then stratification variation. startsWithin increasing. 2 h of early First flood the tides, station stratification at the western at all three shoal locations quickly decreases reaches the and maximum reaches a (Figurewell-mixed6a), followedcondition. by Then the deepstratification channel stationstarts increasing. (Figure6b). First The stationthe station at the at easternthe western shoal shoal has a morequickly moderate reaches increasethe maximum rate (Figure (Figure6c). 6a), During followed the remaining by the deep period channel of flood station tide, (Figure the western 6b). shoalThe station experiences at the variationeastern shoal between has a well-mixed more moderate and stratified increase conditions.rate (Figure The 6c). deep During channel the remaining and the eastern period shoal of flood are alwaystide, the stratified, western shoal and the experiences latter has variation the largest between stratification well-mixed except and near stratified the end conditions. of flood. During The deep ebb tidechannel stratification and the eastern at the westernshoal are shoal always is thestratified, weakest and and the remains latter has almost the largest well-mixed stratification for the wholeexcept ebbnear tide. the end Stratification of flood. During at the deepebb tide channel stratificati and easternon at the shoal western have shoal similar is the evolution, weakest decreasing and remains in thealmost beginning, well-mixed reaching for the well-mixed whole ebb condition tide. Stratification 5–6 h after at ebbing, the deep and channel increasing and again eastern 3 h shoal before have the slacksimilar water. evolution, decreasing in the beginning, reaching well-mixed condition 5–6 h after ebbing, and increasing again 3 h before the slack water.

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Figure 6.6.Time Time series series (06:00 (06:00 31 July 31 to July 07:40 to 1 August07:40 1 2008) August of water 2008) level of elevation water level (red), elevation depth-averaged (red), along-channeldepth-averaged velocity along-channel (green), and velocity bottom-top (green salinity), and differencebottom-top at (salinitya) western difference shoal, (b )at deep (a) channel,western andshoal, (c )(b eastern) deep shoal.channel, Shaded and ( areac) eastern represents shoal. flood Shaded tide. area Arrows represents show flood different tide. stages Arrows during show a tidaldifferent cycle. stages during a tidal cycle.

3.5. Residual Currents in the BaratariaBarataria PassPass overover OneOne TidalTidal CycleCycle Tidal, ebb-,ebb-, andand flood-flood- averagedaveraged along-channelalong-channel velocities for the same 25.6 h timetime periodperiod are shownshown inin FigureFigure7 7.. TheThe locationlocation ofof thethe transecttransect isis shownshown inin FigureFigure1 b1b and and the the western western shoal shoal is is on on the the leftleft sideside ofof Figure7 7.. TheThe magnitudemagnitude ofof cross-sectionallycross-sectionally averaged ebb tides tides is is −−0.460.46 m/s, m/s, and and that that of ofcross-sectionally cross-sectionally averaged averaged flood flood tides tides is is 0.33 0.33 m/s. m/s. The The transverse transverse structure structure of of the along-channelalong-channel residualresidual currentcurrent differs differs significantly significantly between between ebb ebb and and flood flood tides. tides.For ebb For tide ebb, the tide, maximum the maximum outflow isoutflow at the is surface at the nearsurface the near western the shoal,western inclining shoal, inclining against theagainst western the bankwestern of thebank deep of the channel. deep Forchannel. flood For tide ,flood the maximumtide, the maximum inflow is nearinflow the is mid-depthnear the mid-depth in the central in the deep central channel. deep channel. Vertical velocityVertical shearvelocity is much shear larger is much along larger the easternalong the bank eastern than along bank the than western along bank.the western The magnitude bank. The of spatiallymagnitude averaged of spatially residual averaged current residual during thecurrent 25.6 hduring period the is − 25.60.09 h m/s, period which is − is0.09 in them/s, ebb which direction. is in Thethe ebb maximum direction. residual The maximum current is residual near the current western is shoal near withthe western the magnitude shoal with close the to magnitude−0.3 m/s. close to −0.3Using m/s. idealized numerical experiments, Cheng and Valle-Levinson [35] studied the sensitivity of estuarineUsing exchange idealized flow numerical pattern experiments, on two nondimensional Cheng and parameters,Valle-Levinson the [35] Rossby studied number the Rsensitivity0 (U/ f B, where U is the estuarine circulation velocity, f is the Coriolis parameter, and B is the width of the of estuarine exchange flow pattern on two nondimensional parameters, the Rossby number 𝑅 2 channel)(𝑈/𝑓𝐵, where and the U Ekmanis the estuarine number Ecirculationk (Az/ f H velocity,, where A 𝑓z is is the the vertical Coriolis eddy parameter, viscosity, andH isB wateris the depth,width f andof the ischannel) the Coriolis and the parameter). Ekman number They demonstrated𝐸 (𝐴/𝑓𝐻 , where that the 𝐴 exchange is the vertical flow iseddy vertically viscosity, sheared H is water depth, and 𝑓 is the Coriolis parameter). They demonstrated that the exchange flow is

vertically sheared at large 𝑅, and horizontally sheared at large 𝐸. In our case (𝑅 ≈1.72,𝐸 ≈

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J. Mar.J. Mar. Sci. Sci. Eng. Eng. 2018 2018, 6,, 6 x, xFOR FOR PEER PEER REVIEW REVIEW 10 10of 23 of 23 at large R0, and horizontally sheared at large Ek. In our case (R0 ≈ 1.72, Ek ≈ 2.06), the residual current2.062.06), ),the isthe bothresidual residual vertically current current and is is both horizontallyboth verticallyvertically sheared, and horizontallyhorizontally which issimilar sheared, sheared, to which theirwhich caseis issimilar similar shown to intotheir Figuretheir case case5 c

(shownR0shown= 2.63, in in FigureE Figurek = 1 )5c of5c ( [𝑅(35𝑅].=2.63,𝐸=2.63,𝐸 For a triangular =1=1)) ofof shaped [35]. ForFor cross aa triangulartriangular section, Wong shaped shaped [36 ]cross showedcross section, section, that theWong Wong estuarine [36] [36] circulationshowedshowed that that is outwardthe the estuarine estuarine at the circulation surfacecirculation and isis inward outwardoutward at theatat thethe bottom surfacesurface of and the and deepinward inward channel at at the the duebottom bottom to the of ofthe interaction the deep deep betweenchannelchannel baroclinicdue due toto forcethethe interaction andinteraction triangular betweenbetween bathymetry. barocl Fieldinicinic force observationsforce andand triangulartriangular at Barataria bathymetry. bathymetry. Pass captured Field Field this characteristicsobservationsobservations at [31at Barataria ],Barataria in which Pass Pass the capturedcaptured inflow is this very characteristics weak. Our model [31], [31], alsoin in which which reproduces the the inflow inflow a weak is isvery inflowvery weak. weak. near theOurOur bottom model model of also thealso reproduces eastern reproduces slope aa (Figureweakweak inflowinflow7c). The near asymmetry thethe bottombottom of estuarine–oceanof of the the eastern eastern slope slope exchange, (Figure (Figure i.e.,7c). 7c). inflowThe The tendingasymmetryasymmetry to the of of right estuarine–ocean estuarine–ocean side of the channel exchange,exchange, and outflowi.e., inflow to the tendingtending left side to to the (whenthe right right looking side side of up-estuary)of the the channel channel may and and be attributedoutflowoutflow to to the the left left Coriolis side side (when (when force looking [looking37]. up-estuaryup-estuary)) maymay be be attributed attributed to to the the Coriolis Coriolis force force [37]. [37].

Figure 7. Transverse distribution of (a) flood-, (b) ebb-, and (c) tidally-averaged along-channel FigureFigurevelocities, 7.7.Transverse Transverse looking up-estuary distribution distribution (unit: of( aof )cm/s). flood-, (a) Redflood-, (b )isolines ebb-, (b) and ebb-,represent (c) tidally-averagedand flood (c) tidally-averaged velocities, along-channel while black along-channel isolines velocities, lookingvelocities,ebb velocities. up-estuary looking (unit:up-estuary cm/s). (unit: Red isolinescm/s). Red represent isolines flood represent velocities, flood while velocities, black isolines while ebbblack velocities. isolines ebb velocities. TheThe transverse transverse structure structure of of cross-channel cross-channel residualresidual current also also differs differs significantly significantly between between ebb ebb andand floodThe flood transverse tides. tides.The The structure maximummaximum of flood-averagedcross-channel flood-averaged residual and and ebb-averaged ebb-averaged current also velocity differs velocity can significantly reach can reach 0.2 m/s 0.2between and m/s 0.16and ebb 0.16andm/s, m/sflood respectively., respectively.tides. The The maximum The ebb-averaged ebb-averaged flood-averaged velocity velocity exhibits and exhibits ebb-averaged no horizontal no horizontal velocitydivergence divergence can or reach convergence or 0.2 convergence m/s andin the 0.16 in them/s,deep deep respectively. channel channel (Figure (Figure The ebb-averaged 8b,8b,e), e), while while the thevelocity flood-averaged flood-averaged exhibits no velocity velocity horizontal shows shows divergence a astrong strong convergence or convergence convergence close close in to the to thedeepthe surface channelsurface in in the(Figure the mid-channel mid-channel 8b, e), while (Figure (Figure the8 flood-averaged8a,a,d). d). However, However, velocity nono closedclosed shows cell circulation a strong convergenceexists exists for for the the floodclose flood to averagetheaverage surface (Figure (Figure in the8d), mid-channel8d), which which is is different (Figuredifferent from8a, from d). that thatHowever, described described no closed inin NunesNunes cell andcirculation Simpson exists [1]. [ 1 ].The for The tidallythe tidally flood averagedaverageaveraged (Figure cross-channel cross-channel 8d), which velocity velocity is diff is relatively erentis relatively from weaker that weaker described (Figure (Figure8c), in and Nunes8c), it displaysand and it Simpsondisplays a clockwise [1].a clockwise circulationThe tidally cellaveragedcirculation in the westerncross-channel cell in channel the western velocity and an channel even is relatively weaker and an counterclockwise weakereven weaker (Figure counterclockwise cell 8c), in theand eastern it displays cell channel in the a (Figure clockwiseeastern8 f). channel (Figure 8f). circulation cell in the western channel and an even weaker counterclockwise cell in the eastern channel (Figure 8f).

FigureFigure 8. 8.Transverse Transverse distribution distribution ofof (a,,d d)) flood-,flood-, ( b,, ee)) ebb-,ebb-, and and ( (cc, ,f)) tidally tidally averaged averaged cross-channel cross-channel velocitiesvelocities (unit: (unit: cm/s, cm/s, positive positive isis eastward,eastward, negative is is westward) westward) and and lateral lateral circulation. circulation.

Figure 8. Transverse distribution of (a, d) flood-, (b, e) ebb-, and (c, f) tidally averaged cross-channel velocities (unit: cm/s, positive is eastward, negative is westward) and lateral circulation.

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3.6. Time Series of Cross-Sectional Salinity and Currents Structures Figure9 shows the temporal evolution of along-channel velocity, lateral circulation, salinity, and turbulent vertical eddy viscosity for the Barataria Pass transect during flood tide. Each row represents a time instance indicated by an arrow in Figure6. One hour after the flood starts ( T1 in Figure6), stratification is weak (bottom-top salinity difference ~1) across the deep channel and eastern shoal, and the western shoal is almost well-mixed (Figure9c). Strong lateral circulation mainly occurs in the mid-depth with a convergence zone below 10 m in the deep channel (Figure9b). The along-channel velocity is ~0.4 m/s, extending almost the whole deep channel. Thus, vertical and lateral velocity shear is weak. The vertical eddy viscosity is mostly smaller than 0.005 m2/s, probably due to the small flood current magnitude and its shears. Two hours later (T2 in Figure6), the whole water column becomes more or less well-mixed across the deep channel and eastern shoal, while in the western shoal a sharp salinity stratification develops with bottom-top salinity difference ~3 in 2 m water column (Figure9g). The distribution of salinity, and thus density, in the cross-channel direction is such that lateral baroclinic pressure gradients are directed from the central part of the deep channel towards the shoals. This is similar to the situation pointed out in Nunes and Simpson [1]. Based on their theory, such pressure gradient force will induce a lateral circulation with convergence at the surface and divergence near the bottom. This indeed occurs in our numerical results. Surface lateral flow at the west half of the channel changes from ~0.1 m/s westward to ~0.2 m/s eastward between T1 and T2. A pair of counter-rotating circulations can be clearly seen at T2 with strong convergence 2 m below the surface. Contrast to the idealized case in Lerczak and Geyer [2], the two circulation cells are not closed at this time. The maximum along-channel tidal velocity reaches ~0.8 m/s, confined at the mid-depth of the central channel (Figure9e). The vertical shear in along-channel velocity is relatively weak, while the horizontal shear is great, which is, over the western slope, ~0.6 m/s within 200 m distance. It follows that Nunes and Simpson’s argument is also applicable here in that differential advection of along-channel current is at least one of the mechanisms to generate the lateral salinity gradient. Strong vertical mixing (maximum eddy viscosity ~0.05 m2/s, Figure9h) occurs at the mid-depth and bottom boundary layer, where either tidal currents or bottom friction are strong (Figure9e). Strong turbulence mixing tends to destratify the water column, which explains the relatively uniform salinity distribution in the deep channel and east shoal (Figure9g). At the maximum flood (T3), the along-channel velocities intensify. Maximum along-channel velocity, located at channel thalweg, reaches ~1.2 m/s and extends to the surface (Figure9i), which is quite different from T2 (Figure9e). The lateral shear of along-channel velocity, i.e., differential advection, is 1.0 m/s across 300 m distance on both sides. However, salinity distribution changes drastically. The western shoal is completely well-mixed at this time, while the eastern shoal and part of the east channel have surface stratification with a salinity difference of ~3 within 5–6 m depth (Figure9k). The stratification in the deep channel, especially below 10 m, is weak, because the tidal mixing is relatively high (vertical eddy viscosity ~0.03 m2/s) at the bottom boundary (Figure9l). The lateral circulation pattern is similar to that at T2, but with intensified strength. The convergent zone rises to the ocean surface and the right circulation cell is now a complete circle (Figure9j). Four hours before the end of flood (T4), differential advection still persists, although the maximum along-channel tidal current has reduced to ~1 m/s and lowered below the surface (Figure9m). Turbulence mixing weakens (Figure9p). Hence, a weak stratification (bottom–surface salinity difference ~1.5) develops at the western shoal (Figure9o). The salinity distribution shows a more symmetric pattern relative to the axis of the channel compared with T1, T2, and T3. As a result, the counter-rotating lateral circulation cells are more symmetric and fully developed (Figure9n). At the flood slack (T5), the surface water column becomes stratified in the upper 6 m (salinity difference ~1) while the deep channel is almost well-mixed (Figure9s). The lateral circulation almost completely disappears in the deep channel (Figure9r). J. Mar. Sci. Eng. 2018, 6, 159 12 of 23

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Figure 9.FigureCross-sectional 9. Cross-sectional profiles profiles of currents of currents(u, (𝑢,𝑣,𝑤)v, w),, salinity salinity andand verticalvertical viscosity viscosity during during flood flood tide. tide. The first column (a, e, i, m, and q) is along-channel velocity, the second column (b, f, j, n, and r) The first column (a,e,i,m,q) is along-channel velocity, the second column (b,f,j,n,r) secondary circulation, secondary circulation, the third column (c, g, k, o, and s) salinity, and the last column (d, h, l, p, and the thirdt) vertical column viscosity. (c,g,k,o ,Thes) salinity,velocity contour and the interval last column is 0.2 m/s, (d, hpositive,l,p,t) verticalis up-estuary. viscosity. The salinityThe velocity contourcontour interval interval is 0.2 is m/s,0.5. Each positive row corresponds is up-estuary. to a time The instance salinity indicated contour by an intervalarrow in Figure is 0.5. 6. Each row corresponds to a time instance indicated by an arrow in Figure6. At the beginning of ebb tide (T6), although the along-channel ebb current increases to ~0.4 m/s At(Figure the beginning 10a), the ofsalinity ebb tidedistribution (T6), although and vertical the stratification along-channel (Figure ebb 10c) current are almost increases the same to as ~0.4 m/s (Figure 10thata), of the T5. salinityThere is a distribution weak (less than and 0.1 vertical m/s) eastward stratification lateral flow (Figure below 10 6 c)m are(Figure almost 10b). the same as that of T5. ThereTwo is a hours weak later (less (T7), than the 0.1 along-ch m/s)annel eastward ebb current lateral reaches flow below above 1.0 6 m m/s, (Figure locating 10 b).mostly on the western slope and reaching the sea surface. The magnitude of along-channel velocity across the Twomajority hours of later the cross (T7), section the along-channel is ~0.8 m/s (Figure ebb 10e), current which reaches induces above large tidal 1.0 m/s, mixing locating (maximum mostly on the westernvertical slope eddy and viscosity reaching ~0.16 the m sea2/s, surface.Figure 10h). The Thus, magnitude the whole of along-channel water column velocityis vertically across the majoritywell-mixed of the cross (Figure section 10g). isHowever, ~0.8 m/s a horizontal (Figure salinity10e), which gradient induces exists, with large higher tidal salinity mixing located (maximum verticalnear eddy the viscosity channel ~0.16axis, fresher m2/s, water Figure on 10 bothh). Thus,shoals. theThewhole western water shoal columnis fresher is than vertically the eastern well-mixed (Figure shoal10g). (Figure However, 10g). This a horizontal is because salinityfreshwater gradient is flushed exists, out of with the estuary higher through salinity the located western near the shoal, as shown in Figure 7b,c. The lateral circulation shows mostly eastward currents in the deep channel axis, fresher water on both shoals. The western shoal is fresher than the eastern shoal channel across the whole water column, while the eastern shoal has a convergence area (Figure 10f). (Figure 10g). This is because freshwater is flushed out of the estuary through the western shoal, as shown in Figure7b,c. The lateral circulation shows mostly eastward currents in the deep channel across the whole water column, while the eastern shoal has a convergence area (Figure 10f). During the next five hours, this cross section is always vertically well-mixed and salinity decreases constantly due to freshwater outflow. The turbulence mixing remains intense in the deep channel during this period. The maximum vertical eddy viscosity can reach up to 0.2 m2/s, which results in the water J. Mar. Sci. Eng. 2018, 6, 159 13 of 23 column in the east half of the channel vertically and horizontally well-mixed (Figure 10k, T8). The water column in the west half of the channel is also vertically well-mixed, but has a weak (~1) horizontal salinity decrease westward. The structure of lateral circulation and along-channel velocity remains the same, althoughJ. Mar. the Sci. maximum Eng. 2018, 6, ebbx FOR velocity PEER REVIEW has decreased from 1.0 m/s at T7 to 0.8 m/s at T8. 13 of 23

Figure 10.FigureCross-sectional 10. Cross-sectional profiles profiles of of currents currents( u(𝑢,𝑣,𝑤), v, w,) salinity, salinity and and vertical vertical viscosity viscosity during during ebb tide. ebb tide. The firstThe column first column (a,e,i,m (a,q, e),is i, along-channelm, and q) is along-channel velocity, the velocity, second the column second ( bcolumn,f,j,n,r )(b secondary, f, j, n, and circulation, r) secondary circulation, the third column (c, g, k, o, and s) salinity, and the last column (d, h, l, p, and the third column (c,g,k,o,s) salinity, and the last column (d,h,l,p,t) vertical viscosity. The velocity t) vertical viscosity. The velocity contour interval is 0.2 m/s, positive is landward. The salinity contourcontour interval interval is 0.2 is 0.5. m/s, Each positive row corresponds is landward. to a time The instance salinity indicated contour by an intervalarrow in Figure is 0.5. 6. Each row corresponds to a time instance indicated by an arrow in Figure6. During the next five hours, this cross section is always vertically well-mixed and salinity Laterdecreases during constantly the ebb due period to freshwater (T9), the outflow. vertical Th stratificatione turbulence mixing returns remains (Figure intense 10o), in probably the deep due to greatly reducedchannel during turbulence this period. vertical The eddymaximum viscosity vertical (Figure eddy viscosity10p). The can vertical reach up salinity to 0.2 m difference2/s, which is ~2 in the deepresults channel in the and water on thecolumn eastern in the shoal. east half Lateral of the circulation channel vertically shows and a weak horizontally flow divergence well-mixed close to (Figure 10k, T8). The water column in the west half of the channel is also vertically well-mixed, but the surface near the western slope (Figure 10n). The maximum along-channel ebb current moves to has a weak (~1) horizontal salinity decrease westward. The structure of lateral circulation and the surfacealong-channel central channel velocity andremains its magnitudethe same, although decreases the maximum to 0.6 m/s ebb (Figure velocity 10 hasm). decreased from At1.0 the m/s ebb at slackT7 to 0.8 (T10), m/s salinityat T8. distribution go back to similar to T1 situation. The western shoal is almost verticallyLater during uniform. the ebb Strong period stratification (T9), the vertical (salinity stratification difference returns ~2(Figure over 10o), 6 m probably depth) due occurs to in the upper watergreatly columnreduced turbulence of the deep vertical channel eddy andviscosity on the (Figure eastern 10p). shoal.The vertical The salinity lower waterdifference column is ~2 in the deep channelin the deep has channel a weak and stratification on the eastern (Figure shoal. 10Laterals). Lateral circulation circulation shows a weak is greatly flow divergence reducedcompared close to other time instances of the ebb tide (Figure 10r).

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to the surface near the western slope (Figure 10n). The maximum along-channel ebb current moves to the surface central channel and its magnitude decreases to 0.6 m/s (Figure 10m). At the ebb slack (T10), salinity distribution go back to similar to T1 situation. The western shoal is almost vertically uniform. Strong stratification (salinity difference ~2 over 6 m depth) occurs in the upper water column of the deep channel and on the eastern shoal. The lower water column in the J. Mar.deep Sci. channel Eng. 2018 has, 6, 159a weak stratification (Figure 10s). Lateral circulation is greatly reduced compared14 of 23 to other time instances of the ebb tide (Figure 10r). 4. Discussion 4. Discussion 4.1. Depth-Averaged Momentum Balance 4.1. Depth-Averaged Momentum Balance The Coriolis force, wind stress, and horizontal diffusion in Equations (1) and (2) are at least one The Coriolis force, wind stress, and horizontal diffusion in Equations (1) and (2) are at least one order of magnitude smaller than the other terms during the 25.6 h period. Thus, these three terms are order of magnitude smaller than the other terms during the 25.6 h period. Thus, these three terms notare shown. not shown. Figure Figure 11 shows 11 shows time time series series of sixof six terms terms (DDT, (DDT, ADV, ADV, DPBP, DPBP, DPBC, DPBC, FRIC, FRIC, andand AV2D)AV2D) in Equationsin Equations (1) and (1) (2)and at (2) seven at seven locations locations across acro thess channel the channel (Figure (Figure1c). The 1c). momentumThe momentum balance balance across thisacross narrow this channel narrow ischannel very complex, is very complex, as various as various locations locations have different have different characteristics. characteristics.

FigureFigure 11. 11.Time Time series seriesof of verticallyvertically averagedaveraged mo momentummentum equation equation terms terms in in the the cross- cross- (a– (ag–) gand) and along-channelalong-channel ( h(–hn–)n) directions directions duringduring thethe samesame 25.625.6 h tidal cycle. cycle. The The left left column column is isfor for the the cross-channelcross-channel direction. direction. TheThe rightright columncolumn is for the along-channel direction. direction. DDT DDT (dash (dash black) black) representsrepresents the thelocal local acceleration, AVD AVD (red) (red) the the non-linear non-linear advection, advection, COR (pink) COR (pink)the Coriolis the Coriolisforce, force,DPBP DPBP (green) (green) the barotropic the barotropic pressu pressurere gradient, gradient, DPBC (blue) DPBC the (blue) baroclinic the baroclinic pressure pressuregradient, gradient,AV2D (purple) the difference between 2-D and 3-D nonlinear terms, FRIC (orange) the bottom friction, and AV2D (purple) the difference between 2-D and 3-D nonlinear terms, FRIC (orange) the bottom friction, HDIF (yellow) the horizontal diffusion. Shaded areas indicate flood tide. Stations (C1–C7) from top and HDIF (yellow) the horizontal diffusion. Shaded areas indicate flood tide. Stations (C1–C7) from to bottom are located from the west to the east shown in Figure 1c. Note that the y-axis scales for top to bottom are located from the west to the east shown in Figure1c. Note that the y-axis scales for Figure (i,j) are different from others. Figure (i,j) are different from others.

In the along-channel momentum balance, the dominant balance is between the barotropic pressure gradient and the nonlinear advection, especially during ebb tide. The magnitudes of these two terms for stations on the western side (C2 and C3, Figure 11i,j) of the channel are at least twice as greater as that of other stations. This is because maximum ebb currents flush out of the Barataria Bay near these two stations. The sign of barotropic pressure and nonlinear advection at stations on the west channel (C1–C3, Figure 11h–j) is opposite to those on the east channel (C4–C7, Figure 11k–n). There is a J. Mar. Sci. Eng. 2018, 6, x FOR PEER REVIEW 15 of 23

In the along-channel momentum balance, the dominant balance is between the barotropic pressure gradient and the nonlinear advection, especially during ebb tide. The magnitudes of these two terms for stations on the western side (C2 and C3, Figure 11i,j) of the channel are at least twice as greater as that of other stations. This is because maximum ebb currents flush out of the Barataria Bay near these two stations. The sign of barotropic pressure and nonlinear advection at stations on the west channel (C1–C3, Figure 11h–j) is opposite to those on the east channel (C4–C7, Figure 11k–n). There is a spike during ebb tide, similar to that in Huang et al. [33]. The reason for this spike is not J. Mar. Sci. Eng. 2018, 6, 159 15 of 23 clear. During flood tide, the magnitudes of all terms at stations in the western channel (C1–C3, Figure 11h–j) are relatively small. This is because the sign of terms in the upper layer is opposite to spikethat of during the lower ebbtide, layer similar during to flood that in tide. Huang They et al.offset [33 ].each The other reason after for thisintegrating spike is notover clear. depth. During The floodbalance tide, is among the magnitudes the DDT, of nonlinear all terms advection, at stations barotropic in the western pressure channel gradient, (C1–C3, baroclinic Figure 11 pressureh–j) are relativelygradient, small.and the This AV2D. is because the sign of terms in the upper layer is opposite to that of the lower layer duringIn floodthe cross-channel tide. They offset momentum each other balance, afterintegrating the characteristics over depth. are similar The balance to that is of among along-channel,the DDT, nonlineari.e., the dominant advection, balance barotropic is pressurebetween gradient,advectio baroclinicn term and pressure the barotropic gradient, and pressure the AV2D. gradient. ExcludingIn the station cross-channel C1, the momentum signs of advection balance, and the ba characteristicsrotropic pressure are similar terms toat thatthe ofwestmost along-channel, station i.e.,(C2, the Figure dominant 11b) and balance the eastmost is between station advection (C7, Figure term and 11g) the are barotropic the same, pressure but opposite gradient. to the Excluding stations stationof the C1,deep the channel signs of (C3, advection C4, Figure and barotropic 11c,d). Du pressurering flood terms tides, at the the westmost balance station is among (C2, Figure the DDT, 11b) andnonlinear the eastmost advection, station barotropic (C7, Figure pressure 11g) are gradient, the same, baroclinic but opposite pressure to the gradient, stations and of the the deep AV2D. channel Note (C3,that, C4,the baroclinic Figure 11 c,d).pressure During is great flood at stations tides, the in the balance deep ischannel, among and the the DDT, magnitude nonlinear is larger advection, than barotropicthat of along-channel. pressure gradient, This baroclinicindicates pressurethe lateral gradient, salinity and gradient the AV2D. play Note an that,important the baroclinic role in pressuremomentum is great balance. at stations in the deep channel, and the magnitude is larger than that of along-channel. This indicates the lateral salinity gradient play an important role in momentum balance. 4.2. Driving Mechanism of Lateral Circulation 4.2. Driving Mechanism of Lateral Circulation Lerczak and Geyer [2] pointed out that lateral advection plays an important role in the estuarine dynamicsLerczak when and lateral Geyer [flows2] pointed are strong out that enough lateral to advection advect playswater anparcels important relative role to in 0.5 the times estuarine the dynamicsbreadth of when the channel lateral flows(4〈|𝑣|〉 are/𝜎𝐵 strong ≥1, enoughwhere | to𝑣| advect is the absolute water parcels value relative of lateral to 0.5 velocity times theamplitude, breadth of𝜎 is the the channel semidiurnal (4|v|/σ tidalB ≥ 1frequenc, wherey,|v |andis the B is absolute the channel value width) of lateral in velocitya tidal cycle. amplitude, As shownσ is the in semidiurnalprevious results, tidal frequency,lateral circulation and B is in the the channel Barataria width) Pass in is a strong tidal cycle. both Asduring shown maximum in previous flood results, and lateralmaximum circulation ebb. Thus, in the lateral Barataria advect Passion is strong is expected both during to be maximum an important flood andterm maximum in the momentum ebb. Thus, lateralbalance. advection The 3-D ismomentum expected to Equations be an important (3) and term(4) ar ine used the momentum to explore the balance. generation The 3-D mechanisms momentum of Equationsthe lateral (3)circulation. and (4) are used to explore the generation mechanisms of the lateral circulation.

Figure 12.12. TransverseTransverse distributions distributions of of terms terms (m/s (m/s2) in2) in along-channel along-channel momentum momentum equation equation at flood at flood (T3). Terms(T3). Terms include include (a) local (a acceleration,) local acceleration, (b) lateral (b) advection, lateral advection, (c) along-channel (c) along-channel advection, advection, (d) barotropic (d) pressurebarotropic gradient, pressure (e) baroclinic gradient, pressure (e) baroclinic gradient, (pressuref) total pressure gradient, gradient, (f) (totalg) horizontal pressure stress gradient, divergence, (g) (horizontalh) Coriolis stress force, anddivergence, (i) vertical (h stress) Coriolis divergence. force, and Red isolines(i) vertical represent stress positivedivergence. values, Red while isolines blue isolinesrepresent negative positive values. values, Dash while line showsblue contourisolines 0.ne Thegative contour values. intervals Dash are line shown shows in parenthesis.contour 0. The contour intervals are shown in parenthesis. Transverse distributions of various momentum terms of along- and cross-channel components at T3 (flood tide) are shown in Figures 12 and 13, respectively. In the along-channel momentum equation, baroclinic pressure gradient (Figure 12e), Coriolis force (Figure 12h) and vertical stress divergence (Figure 12i) are at least two orders of magnitude less than other terms at this time, thus are less important. The total pressure gradient comes from the barotropic pressure gradient (Figure 12d), and the largest terms are the nonlinear advection terms (Figure 12b,c). Lateral advection of along-channel momentum (Figure 12b, −uvx − wvz) is one of the largest terms in the momentum J. Mar. Sci. Eng. 2018, 6, x FOR PEER REVIEW 16 of 23

Transverse distributions of various momentum terms of along- and cross-channel components at T3 (flood tide) are shown in Figures 12 and 13, respectively. In the along-channel momentum equation, baroclinic pressure gradient (Figure 12e), Coriolis force (Figure 12h) and vertical stress divergence (Figure 12i) are at least two orders of magnitude less than other terms at this time, thus

J.are Mar. less Sci. important. Eng. 2018, 6, 159The total pressure gradient comes from the barotropic pressure gradient (Figure16 of 23 12d), and the largest terms are the nonlinear advection terms (Figure 12b,c). Lateral advection of

along-channel momentum (Figure 12b, −𝑢𝑣 −𝑤𝑣) is one of the largest terms in the momentum balance.balance. ThisThis term term has has a similar a similar pattern pattern but opposite but opposite sign with sign the with along-channel the along-channel momentum momentum advection

(advection−vvy). The (−𝑣𝑣 main). force The balance main force at this balance time is betweenat this time non-linear is between advection non-linear and the barotropicadvection pressureand the gradient,barotropic which pressure is consistent gradient, with which the is result consistent in 2-D with momentum the result analysis. in 2-D momentum analysis.

FigureFigure 13.13. TransverseTransverse distributions distributions of of terms terms (m/s (m/s2) in2) cross-channelin cross-channel momentum momentum equation equation at flood at flood (T3). Terms(T3). Terms include include (a) local (a acceleration,) local acceleration, (b) lateral (b advection,) lateral advection, (c) along-channel (c) along-channel advection, advection, (d) barotropic (d) pressurebarotropic gradient, pressure (e) baroclinic gradient, pressure (e) baroclinic gradient, (fpressure) total pressure gradient, gradient, (f) (totalg) horizontal pressure stress gradient, divergence, (g) (horizontalh) Coriolis force,stress anddivergence, (i) vertical (h stress) Coriolis divergence. force, and Red isolines(i) vertical represent stress positive divergence. values, Red while isolines blue isolinesrepresent negative positive values. values, Dash while line shows blue contourisolines 0.ne Thegative contour values. intervals Dash are line shown shows in parenthesis.contour 0. The contour intervals are shown in parenthesis. In the cross-channel momentum balance, Coriolis force (Figure 13h), although small, has exactly the sameIn the pattern cross-channel as the transverse momentum distribution balance, ofCoriolis along-channel force (Figure velocity 13h), (Figure although9i). Differential small, has advectionexactly the of same the flood pattern current as atthe this transverse time is clearly distribution shown of in thealong-channel picture. The velocity baroclinic (Figure pressure 9i). gradientDifferential (Figure advection 13e) has of almostthe flood the current same magnitude at this time as is the clearly barotropic shown pressure in the picture. gradient The (Figure baroclinic 13d), especiallypressure gradient near the (Figure bottom, 13e) due has to almost its magnitude the same increasing magnitude with as the depth. barotropic The baroclinic pressure pressuregradient gradient(Figure 13d), is mostly especially positive near in the the bottom, whole channel, due to its except magnitude near the increasing easternshoal. with depth. Combining The baroclinic with the salinitypressure distribution gradient is atmostly the transect positive (Figure in the9 k),whole it indicates channel, that except the conceptualnear the eastern model shoal. of cross-channel Combining baroclinicwith the salinity pressure distribution gradient induced at the bytransect differential (Figure advection 9k), it indicates is applicable that here. the conceptual However, the model simple of diagnosticcross-channel model baroclinic proposed pressure by Nunes gradient and Simpson induced [1] toby explain differential the mechanism advection ofis lateralapplicable circulation here. generation,However, the in which simple the diagnostic momentum model balance propos is betweened by cross-channel Nunes and pressureSimpson gradient [1] to andexplain vertical the stressmechanism divergence, of lateral seems circulation to be oversimplified. generation, Our in numerical which the experiment momentum shows balance that vertical is between stress divergencecross-channel is mostlypressure confined gradient to and the nearvertical bottom stress layer divergence, of the eastern seems slopeto be (Figure oversimplified. 13i). Several Our othernumerical terms—nonlinear experiment shows advection that (Figure vertical 13 stressb,c) and divergence horizontal is stress mostly divergence confined (Figureto the near13g)—are bottom as importantlayer of the as eastern the terms slope suggested (Figure in13i). [1]. Several other terms—nonlinear advection (Figure 13b,c) and horizontalDuring stress ebb tide divergence (i.e., at T7), (Figure the main 13g)—are momentum as important balance as in the the terms along-channel suggested direction in [1]. is among the nonlinearDuring ebb advection tide (i.e., terms at andT7), thethe barotropicmain momentum pressure balance gradient in (Figure the along-channel 14), which is thedirection same asis duringamong thethe floodnonlinear tide (Figureadvection 12). terms and the barotropic pressure gradient (Figure 14), which is the sameIn as the during cross-channel the flood momentum tide (Figure balance, 12). the barotropic pressure gradient (Figure 15d) dominates the baroclinicIn the cross-channel pressure gradient momentum (Figure 15balance,e). Thus, the contour barotropic lines of pressure the total cross-channelgradient (Figure pressure 15d) gradientdominates (Figure the 15baroclinicf) are very similarpressure to thatgradient of the former.(Figure It seems15e). thatThus, at thiscontour time thelines main of momentum the total balancecross-channel is between pressure along-channel gradient (Fig advectionure 15f) ofare cross-channel very similar momentumto that of the (Figure former. 15 c)It andseems pressure that at gradient (Figure 15f) over most of the cross-section area, both of which have alternative positive and negative vertical stripes and, when added together, almost cancel out each other. At the eastern shoal, all terms, excluding Coriolis force (Figure 15h) and baroclinic pressure gradient (Figure 15e), are needed in order to balance the cross-channel momentum equation. Because of the insignificant contribution from the baroclinic pressure gradient, no obvious convergence or divergence occurs in J. Mar. Sci. Eng. 2018, 6, x FOR PEER REVIEW 17 of 23 J. Mar. Sci. Eng. 2018, 6, x FOR PEER REVIEW 17 of 23 this time the main momentum balance is between along-channel advection of cross-channel momentumthis time the (Figure main 15c) momentum and pressure balance gradient is betw (Figureeen 15f)along-channel over most of advection the cross-section of cross-channel area, both ofmomentum which have (Figure alternative 15c) and positive pressure and gradient negative (Figure vertical 15f) stripes over mostand, whenof the addedcross-section together, area, almost both cancelof which out have each alternative other. At positivethe eastern and shoal,negative all veterms,rtical excludingstripes and, Coriolis when addedforce (Figure together, 15h) almost and barocliniccancel out pressureeach other. gradient At the (Figureeastern shoal,15e), areall terms,needed excluding in order Coriolis to balance force the(Figure cross-channel 15h) and J.momentumbaroclinic Mar. Sci. Eng. pressure 2018equation., 6, 159 gradient Because (Figure of the 15e), insignificant are needed contribution in order fromto balance the baroclinic the cross-channel pressure17 of 23 gradient,momentum no obviousequation. convergence Because of or divergencethe insignificant occurs incontribution this transect from during the the baroclinic ebb period pressure (Figure 10,gradient, second no column). obvious Thisconvergence is in clear or divergencecontrast to occursthe near in surfacethis transect convergence during the that ebb happens period (Figureduring thisflood10, second transect tide (Figure column). during 9, second theThis ebb is column). periodin clear (Figure contrast 10 to, second the near column). surface Thisconvergence is in clear that contrast happens to the during near surfaceflood tide convergence (Figure 9, second that happens column). during flood tide (Figure9, second column).

Figure 14. Transverse distributions of terms (m/s2) in along-channel momentum equation at ebb (T7). FigureTerms include 14. TransverseTransverse (a) local distributions acceleration, ofof (b termsterms) lateral (m/s(m/s advection,22)) in in along-channel along-channel (c) along-channel momentum momentum advection, equation equation (d) at atbarotropic ebb ebb (T7). (T7). Termspressure includeinclude gradient, ((aa)) locallocal (e) baroclinic acceleration, pressure ((b)) lateral gradient, advection, (f) total (c) pressurealong-channel gradient, advection, (g) horizontal ( d) barotropic stress pressuredivergence, gradient, gradient, (h) Coriolis (e )( baroclinice) baroclinic force, pressure and pressure ( gradient,i) vertical gradient, (f )stress total (f pressure) ditotalvergence. pressure gradient, Red gradient, (g )isolines horizontal ( grepresent) horizontal stress divergence, positive stress (values,divergence,h) Coriolis while force,( hblue) Coriolis and isolines (i) verticalforce, negative and stress (values.i) divergence. vertical Dash stress line Red dishows isolinesvergence. contour represent Red 0. isolines positiveThe contour represent values, intervals while positive blue are isolinesshownvalues, in negativewhile parenthesis. blue values. isolines Dash negative line shows values. contour Dash 0. The line contour shows intervalscontour are0. The shown contour in parenthesis. intervals are shown in parenthesis.

Figure 15.15. Transverse distributions distributions of termsof terms (m/s (m/s2) in2) cross-channelin cross-channel momentum momentum equation equation at ebb at (T7). ebb Terms (T7). includeTermsFigure include15. (a) Transverse local (a acceleration,) local distributions acceleration, (b) lateral of (b terms) advection, lateral (m/s advection,2) ( cin) along-channel cross-channel (c) along-channel advection,momentum advection, (d equation) barotropic (d) at barotropic ebb pressure (T7). gradient,pressureTerms include gradient, (e) baroclinic (a) local (e) baroclinic pressureacceleration, gradient, pressure (b) lateral (fgradient,) total advection, pressure (f) total (c gradient,) pressurealong-channel ( ggradient,) horizontal advection, (g) stresshorizontal (d) divergence, barotropic stress (divergence,pressureh) Coriolis gradient, force,(h) Coriolis and (e) (baroclinici) verticalforce, and stresspressure (i) divergence. vertical gradient, stress Red (f) ditotal isolinesvergence. pressure represent Red gradient, isolines positive ( grepresent) values, horizontal while positive stress blue isolinesvalues,divergence, negativewhile ( hblue) values.Coriolis isolines Dash force, negative line and shows (values.i) contourvertical Dash 0.stress The line contour dishowsvergence. intervalscontour Red are0. isolines The shown contour inrepresent parenthesis. intervals positive are shownvalues, inwhile parenthesis. blue isolines negative values. Dash line shows contour 0. The contour intervals are 4.3. Flood–Ebb Asymmetry shown in parenthesis. Here, we follow Lerczak and Geyer [2] using the cross-channel average of depth-averaged velocity amplitude < |u| > = 1 |u|dA (|u| is the absolute value of depth-averaged cross-channel velocity, A s A is the cross-channel area) to represent the strength of the lateral flow. The result is shown in Figure 16. The maximum < |u| > during flood is 0.32 m/s, while the maximum during ebb is 0.34 m/s. Generally, cross-channel current amplitude during ebb is comparable, even slightly greater than that during J. Mar. Sci. Eng. 2018, 6, x FOR PEER REVIEW 18 of 23

4.3. Flood–Ebb Asymmetry Here, we follow Lerczak and Geyer [2] using the cross-channel average of depth-averaged J. Mar. Sci. Eng. 2018, 6, 159 18 of 23 velocity amplitude |𝑢| = ∬ |𝑢|𝑑𝐴 (|𝑢| is the absolute value of depth-averaged cross-channel velocity, A is the cross-channel area) to represent the strength of the lateral flow. The result is shown flood, whichin Figure is different16. The maximum from the |𝑢| idealized during case flood [2 is], 0.32 in which m/s, whil laterale the maximum circulation during is about ebb is four0.34 times strongerm/s. during Generally, flood cross-channel tides than duringcurrent ebbamplitude tides. during However, ebb is this comparable, inconsistence even slightly is also greater observed in than that during flood, which is different from the idealized case [2], in which lateral circulation is James River estuary [14], where the lateral circulation shows no flood–ebb asymmetry during spring about four times stronger during flood tides than during ebb tides. However, this inconsistence is tides, andalso a observed reversed in asymmetry James River during estuary neap [14], tides,where thatthe islateral stronger circulation during shows ebb no than flood–ebb during flood. Li et al. [asymmetry14] attribute during negligible spring tides, flood–ebb and a reversed variations asymmetr duringy during spring neap tides tides, to that turbulent is stronger mixing, during which in their numericalebb than during experiment flood. Li et was al. [14] simulated attribute via negligible a turbulence flood–ebb closure variations model. during It reducesspring tides the to vertical shear andturbulent the flood–ebb mixing, which asymmetry in their in numerical the vorticity experiment generation. was simulated In comparison, via a turbulence the idealized closure case of model. It reduces the vertical shear and the flood–ebb asymmetry in the vorticity generation. In Lerczak and Geyer [2] used a constant eddy viscosity. In our numerical experiment, turbulent mixing comparison, the idealized case of Lerczak and Geyer [2] used a constant eddy viscosity. In our is simulatednumerical using experiment, the Mellor-Yamada turbulent mixing 2.5 turbulence is simulated closure using model the Mellor-Yamada and the 25.6 h2.5 diurnal turbulence tidal cycle is close toclosure tropic model tide. and Maximum the 25.6 h turbulentdiurnal tidal vertical cycle is viscosityclose to tropic is twice tide. Maximum as great duringturbulent ebb vertical (Figure 10, last column)viscosity as duringis twice floodas great (Figure during9 ebb, last (Figure column). 10, last Therefore, column) as our during simulation flood (Figure result 9, last is more column). similar to Li et al. [Therefore,14] than toour Lerczak simulation and result Geyer is more [2]. similar to Li et al. [14] than to Lerczak and Geyer [2].

Figure 16.FigureCross-sectional 16. Cross-sectional average average of lateralof lateral velocity velocity magnitude for for the the 25.6 25.6 h diurnal h diurnal tidal period. tidal period.

DuringDuring flood flood tide, tide, lateral lateral circulation circulation shows shows asymmetry asymmetry across across the section the (Figure section 9f,j,n). (Figure The9 f,j,n). counterclockwise circulation on the east side is stronger, which is due to the presence of lower The counterclockwise circulation on the east side is stronger, which is due to the presence of lower salinity water close to and over the eastern shoal (Figure 9k,o). This fresher water increases the salinity waterlateral closebaroclinic to and pressure over gradient the eastern on the shoal east(Figure side, and9k,o). thus Thisenhances fresher the waterstrength increases of the lateral the lateral barocliniccirculation pressure cell gradient at this side. on theThe east low side,salinity and water thus is enhancesfrom the Mississippi the strength River of theplume lateral based circulation on cell at thisanalyses side. Theof in-situ low observational salinity water data is fromand numerical the Mississippi simulation River result plume [22]. based on analyses of in-situ observationalAnother data andasymmetry numerical between simulation flood and result ebb [tides22]. lies in that maximum flood currents are Anotheralways asymmetry located, more between or less, at flood the central and ebb part tides of the lies deep in channel that maximum (Figure 9, flood first column), currents while are always maximum ebb currents, when they are greater than 0.8 m/s, are located closer to the western shoal or located, more or less, at the central part of the deep channel (Figure9, first column), while maximum over the western slope (Figure 10e,i). Only when maximum ebb currents are below 0.6 m/s, they ebb currents,move back when to theythe middle are greater of the channel than 0.8 (Figure m/s, 10a,m,q). are located closer to the western shoal or over the western slope (Figure 10e,i). Only when maximum ebb currents are below 0.6 m/s, they move back to the middle5. Conclusions of the channel (Figure 10a,m,q). Barataria Pass is a tidal inlet that connects the Barataria Bay to the continental shelf. Previous 5. Conclusionsinvestigation has introduced the tidal straining effect on density stratification during the same 25.6 h period along the same transect [22]. In this study, we conduct a numerical model simulation and Barataria Pass is a tidal inlet that connects the Barataria Bay to the continental shelf. Previous investigation has introduced the tidal straining effect on density stratification during the same 25.6 h period along the same transect [22]. In this study, we conduct a numerical model simulation and illustrate that the lateral variations in the salinity and velocity fields are comparable to or even larger than the vertical variations within a diurnal tidal cycle. The density distribution within any estuary is a result of both advective and mixing processes. In Barataria Pass, the turbulent mixing is closely related to the magnitude of ebb/flood current and the strength of the tidal bottom boundary layer. Characteristics of horizontal advection processes in the inlet are that maximum flood currents are located at the central part of the deep channel for a large part of the flood period. This differential advection [1], when acting upon the along-channel density gradient, produces a distinct density difference between the shoal and channel waters. In addition, J. Mar. Sci. Eng. 2018, 6, 159 19 of 23 the advection of Mississippi River water to the eastern channel during part of the flood period further enhances the density difference. On the contrary, maximum ebb currents swing between the western slope and central surface of the channel during the ebb. When maximum ebb flows are at the western slope, the differential advection mechanism does not work. When they go back to the channel center, the salinity contour lines are mostly horizontal due to weak vertical turbulence mixing. Thus, both situations are not favorable to produce an extreme density near the middle of the channel. During flood period, when density distribution is high near the channel center and low at both shoals, the horizontal pressure gradient drives a lateral circulation with two counter-rotating cells and surface or near surface convergence. This result from the Barataria Pass is similar to that reported by Nunes and Simpson [1]. However, detailed analysis of momentum equations indicates that, in addition to the pressure gradient and vertical stress divergence, nonlinear advection and horizontal stress divergence are also important terms. During ebb period, the lateral circulation is mostly eastward for the whole water column and persisting for almost the whole period. The surface divergence suggested by the differential advection mechanism is either non-existent or lasting for a very short period. The main momentum balance across most of the transect is between the along-channel advection of cross-channel momentum and pressure gradient. In addition, the sectional averaged lateral velocity magnitude during ebb is comparable to that during flood, which is different from the idealized numerical experiment [2]. Interactions among lateral circulation, along-channel tidal currents, and density stratification are complex processes. Lateral circulation can play a critical role in the dynamics of estuaries. It can act as a driving term for the estuarine exchange flow [2], and thus alter the along-channel momentum budget [38]. During flood tide, lateral circulation transported near-bed sediment out of the channel toward the shoal [37]. During ebb tide, it created a convergence in suspended sediment at the transition between channel and shoal [39]. Trapping by lateral circulation alters the lateral distribution of bed sediment. The idealized numerical experiment with constant eddy coefficients [2] demonstrated that the lateral circulation can be significantly different over a spring-neap cycle and density stratification can inhibit lateral circulation. Our study can be a starting point for further investigations of interactions among lateral circulation, estuarine circulation, and estuarine stratification in the partially stratified tidal inlet, the Barataria Pass.

Supplementary Materials: FVCOM simulation data are publicly available through the Gulf of Mexico Research Initiative Information and Data Cooperative (GRIIDC) at https://data.gulfresearchinitiative.org (DOI: 10.7266/n7-1qe9-mg27). Author Contributions: Conceptualization, L.C. and H.H.; Investigation, L.C., H.H., C.L.; Writing-Original Draft Preparation, L.C.; Writing-Review & Editing, H.H., C.L., D.J.; Visualization, L.C.; Funding Acquisition, D.J., H.H. Funding: This research was made possible in part by a grant from The Gulf of Mexico Research Initiative. L.C. was also funded by the Economic Development Assistantship from the Graduate School, Louisiana State University. Acknowledgments: Portions of this research were conducted with high performance computational resources provided by Louisiana State University (http://www.hpc.lsu.edu) and the Louisiana Optical Network Initiative (http://www.loni.org). Conflicts of Interest: The authors declare no conflict of interest.

Appendix A. Decomposition of Vectors into Along- and Cross-Channel Directions In x-y-σ coordinates, where x-direction is defined to the east and y-direction to the north, the FVCOM x- and y-axis 3-D momentum equations are written as:

  0  2 Z ∂uD ∂u D ∂uvD ∂uω ∂ζ gD ∂ 0 ∂D = − − − + f vD −gD −  D dσ + σρ  ∂t ∂x ∂y ∂σ |{z} ∂x ρ0 ∂x ∂x | {z } | {z }| {z } | {z } CORX| {z } σ DUDT ADVUX ADVVX ADVWX DPBPX | {z } DPBCX (A1) 1 ∂  ∂u  + Km + Fx D ∂σ ∂σ |{z} | {z } HVISCX VVISCX J. Mar. Sci. Eng. 2018, 6, x FOR PEER REVIEW 20 of 23

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A. Decomposition of Vectors into Along- and Cross-Channel Directions In 푥-푦-𝜎 coordinates, where 푥-direction is defined to the east and 푦-direction to the north, the FVCOM 푥- and 푦-axis 3-D momentum equations are written as: 휕푢퐷 휕푢2퐷 휕푢푣퐷 휕푢휔 휕휁 푔퐷 휕 0 휕퐷 = − − − + 푓푣퐷⏟ −푔퐷 − [ (퐷 ∫ 푑𝜎′ + 𝜎𝜌 )] ⏟휕푡 ⏟ 휕푥 ⏟ 휕푦 ⏟ 휕𝜎 ⏟ 휕푥 𝜌 휕푥 휕푥 퐶푂푅푋 ⏟ 0 휎 퐷푈퐷푇 퐴퐷푉푈푋 퐴퐷푉푉푋 퐴퐷푉푊푋 퐷푃퐵푃푋 퐷푃퐵퐶푋 J. Mar. Sci. Eng. 2018, 6, 159 1 휕 휕푢 20(A1) of 23 + (퐾 ) + 퐹⏟ 퐷⏟ 휕𝜎 푚 휕𝜎 푥 퐻푉퐼푆퐶푋 푉푉퐼푆퐶푋   0  2 Z ∂vD ∂uvD ∂v D ∂vω ∂ζ gD ∂ 0 ∂D = − − − − f uD−gD −  D dσ + σρ  휕푣퐷∂t 휕푢푣퐷∂x 휕푣2∂퐷y 휕푣휔∂σ | {z } ∂y휕휁 ρ0푔퐷∂y휕 0 ∂y 휕퐷 |{z } | {z }| {z } | {z } CORY | {z } σ ′ DVDT= − ADVUY− −ADVWY⏟ − 푓푢퐷 −푔퐷 − [ (퐷 ∫ 푑𝜎 + 𝜎𝜌 )] ⏟휕푡 ⏟ 휕푥 ⏟ ADVVY 휕푦 ⏟ 휕𝜎 ⏟DPBPY 휕푦 | 𝜌 휕푦 {z 휕푦} (A2) 퐶푂푅푌  ⏟ 0 DPBCY 휎 퐷푉퐷푇 퐴퐷푉푈푌 퐴퐷푉푉푌 퐴퐷푉푊푌1 ∂ ∂v퐷푃퐵푃푌 퐷푃퐵퐶푌 + Km + Fy (A2) 1 휕 D휕푣∂σ ∂σ |{z} + (퐾푚| ) +{z 퐹⏟푦 } HVISCY 퐷⏟ 휕𝜎 휕𝜎 VVISCY 퐻푉퐼푆퐶푌 푉푉퐼푆퐶푌 where (u, v) are the velocity components in the (x, y) directions, σ is the vertical coordinate. whereIn ( order푢, 푣) toare quantify the velocity along- components and cross-channel in the (푥 momentum, 푦) directions balance,, 𝜎 is the we choosevertical thecoordinate along-channel. 0 directionIn order (y )to to quantify be aligned along with- and cross the channel.-channel momentum It is positive balance, when we pointing choose the into along the-channel estuary. 0 directionThe cross-channel (푦′) to be direction aligned with(x ) is the defined channel. to be It positiveis positive when when pointing pointing to intothe eastern the estuary. boundary The 0 0 0 cross(Figure-channel A1). The direction relationship (푥′) is betweendefined to(u be, v positive) and (u ,whenv), (x pointing, y0) and to(x ,they) areeastern as following: boundary (Figure A1). The relationship between (푢′, 푣′) and (푢, 푣), (푥′, 푦′) and (푥, 푦) are as following: u0 = ucosθ + vsinθ (A3) 푢′ = 푢푐표푠휃 + 푣푠푖푛휃 (A3) v0 = −usinθ + vcosθ (A4) 푣′ = −푢푠푖푛휃 + 푣푐표푠휃 (A4) 0 x′ = (x − x0)cosθ + (y − y0)sinθ (A5) 푥 = (푥 − 푥0)푐표푠휃 + (푦 − 푦0)푠푖푛휃 (A5) 0 y =′ −(x − x )sinθ + (y − y )cosθ (A6) 푦 = (−(푥 −0푥0) + (푦 − 푦0)0푐표푠휃 (A6) 0 wwherehere θ휃is is the the angle angle between between the thex 푥-direction′-direction (cross-channel) (cross-channel) and and the thex-direction. 푥-direction.

FigureFigure A1. A1. IllustrationIllustration of of xx--yy coordinatecoordinate transformed transformed to to xx’’--y’’ coordinate. coordinate.

InIn x푥0′-y푦0′ coordinates,coordinates, the the momentum momentum equations equations are are written written as: as: 휕푢′퐷 휕푢′2퐷 휕푢′푣′퐷 휕푢′휔 휕휁 푔퐷 휕 0 휕퐷 = − − − + 푓푣′퐷 −푔퐷 − [  (퐷∫ 푑0𝜎′ + 𝜎𝜌 )]  ⏟ 0 ⏟ 0′ 2 0 ′ 0 ⏟ 0 ⏟ ⏟ ′ ′ Z ′ ∂휕푡u D ∂휕u푥 D⏟ ∂ 휕u푦 v D ∂휕𝜎u ω 0′ 휕푥∂ζ⏟ 𝜌 0gD 휕 푥 ∂ 휎 0 휕 푥∂ D ′ = − ′ − − ′ +퐶푂푅푋 − ′ − + 퐷푈퐷푇 퐴퐷푉푈푋 0 ′0 퐴퐷푉푊푋 f v D퐷푃퐵푃푋gD 0  0 D ′ dσ σρ 0  ∂t ∂x 퐴퐷푉푉푋∂y ∂σ | {z } ∂x ρ0 ∂x 퐷푃퐵퐶푋 ∂x | {z } | {z } | {z } 0| {z } σ 0 0 | {z } 0 CORX 0 (A7) DUDT ADVUX ADVVX0 ADVWX 1 휕 휕푢DPBPX′ | {z } + (퐾 ) + 퐹⏟ DPBCX0 (A7) 퐷휕𝜎 푚 휕𝜎 푥′ 1 ∂⏟ ∂u 0 퐻푉퐼푆퐶푋′ + Km ′ + Fx0 D ∂σ 푉푉퐼푆퐶푋∂σ |{z} ′ ′ ′ ′2 |′ {z } HVISCX0 0 휕푣 퐷 휕푢 푣 퐷 휕푣 퐷 휕푣 휔 0 휕휁 푔퐷 휕 휕퐷 = − − − −VVISCX푓푢′퐷 −푔퐷 − [ (퐷 ∫ 푑𝜎′ + 𝜎𝜌 )] ⏟ ⏟ ′ ′ ⏟ ⏟ ′ ′ ′ (A8) 휕푡 휕푥 ⏟ 휕푦 휕𝜎 ′ ⏟ 휕 푦 ⏟ 𝜌 0 휕 푦 휎 휕 푦 ′ ′ ′ 퐶푂푅푌   0  퐷푉퐷푇0 퐴퐷푉푈푌0 0 퐴퐷푉푉푌0′2 퐴퐷푉푊푌0 퐷푃퐵푃푌′ 퐷푃퐵퐶푌Z ′ ∂v D ∂u v D ∂v D ∂v ω 0 ∂ζ gD ∂ 0 ∂D = − 0 − 0 − − f u D−gD 0 −  0 D dσ + σρ 0  ∂t ∂x ∂y ∂σ | {z } ∂y ρ0 ∂y ∂y | {z } | {z } | {z } 0 σ 0 0 | {z } 0 CORY | {z } DVDT ADVUY ADVVY0 ADVWY DPBPY0 | {z } DPBCY0 (A8) 1 ∂  ∂v0  + Km + Fy0 D ∂σ ∂σ |{z} | {z } HVISCY0 VVISCY0 To project the momentum equations into the cross- and along-channel directions, we treat each term in the momentum equations as a vector in the (x, y) direction and then apply the same J. Mar. Sci. Eng. 2018, 6, x FOR PEER REVIEW 21 of 23

J. Mar. Sci. Eng. 2018, 6, 159 1 휕 휕푣′ 21 of 23 + (퐾푚 ) + 퐹⏟푦′ ⏟퐷 휕𝜎 휕𝜎 퐻푉퐼푆퐶푌′ 푉푉퐼푆퐶푌′ 0 0 decompositionTo project the as Equationsmomentum (A3) equations and (A4). into Thusthe cross terms- and in alongx -y coordinates-channel directions, can be calculatedwe treat each by correspondingterm in the momentum terms in x -y equationscoordinates as as a vectorfollows: in the (푥, 푦) direction and then apply the same ′ decomposition as Equations (A3) and0 (A4)(. Thus+ terms) in 푥′-푦 coordinates can be calculated by DUDT0 = ∂u D = ∂ ucosθ vsinθ D = ∂uD cosθ + ∂vD sinθ corresponding terms in 푥-푦 coordinates∂t as follows:∂t ∂t ∂t 휕푢′퐷 =휕(DUDTcos 푢푐표푠휃 + θ푣푠푖푛휃+ DVDTsin)퐷 휕푢퐷θ 휕푣퐷 퐷푈퐷푇′ = = = 푐표푠휃 + 푠푖푛휃 휕푡 휕푡 휕푡 휕푡 0 ∂v0 D ∂(−usinθ+ucosθ)D ∂uD ∂vD DVDT = ∂t = = 퐷푈퐷푇푐표푠휃∂t + 퐷푉퐷푇푠푖푛휃= − ∂t sin θ + ∂t cosθ 휕푣′퐷 =휕−(−DUDTsin푢푠푖푛휃 + θ푢푐표푠휃+ DVDTcos)퐷 θ휕푢퐷 휕푣퐷 퐷푉퐷푇′ = = = − 푠푖푛휃 + 푐표푠휃 휕푡 휕푡 휕푡 휕푡 Terms ADVUX’+ADVVX’, ADVUY’+ADVVY’, ADVWX’, ADVWY’, CORX’, CORY’, DPBPX’, = −퐷푈퐷푇푠푖푛휃 + 퐷푉퐷푇푐표푠휃 DPBPY’, DPBCX’, DPBCY’, VVISCX’, VVISCY’, HVISCX’, and HVISCY’ can be calculated with the Terms ADVUX’+ADVVX’, ADVUY’+ADVVY’, ADVWX’, ADVWY’, CORX’, CORY’, DPBPX’, sameDPBPY’, method. DPBCX’, While DPBCY’, ADVUX’, VVISCX’, ADVVX’, VVISCY’, ADVUY’, HVISCX’, and ADVVY’ and HVISCY’ should can be calculatedbe calculat byed thewith finite the volume difference. For example, ADVUX’ can be calculated as: same method. While ADVUX’, ADVVX’, ADVUY’, and ADVVY’ should be calculated by the finite volume difference. For0 2example, ADVUX’I can be calculatedI as: 0 ! ∂u D 0 0 0 0 0 0 UIJ1 0 2dx dy = u u Ddy = UIJ × × DIJ × dy (A9) x ∂x휕0푢′ 퐷 UIJ푈퐼퐽12′0 ∬ 푑푥′푑푦′ = ∮ 푢′푢′퐷푑푦′ = ∮ 푈퐼퐽′ × ( ) × 퐷퐼퐽 × 푑푦′ (A9) 휕푥′ 푈퐼퐽2′ where UIJ, UIJ10 and UIJ20 are shown in Figure A2. where 푈퐼퐽, 푈퐼퐽1’ and 푈퐼퐽2’ are shown in Figure A2.

Figure A2. IllustrationIllustration of local coordinate used to calculate the horizontal advection terms.

With Equations (A3)–(A6),(A3)–(A6), wewe have:have:

′ 푑dy푦 0 ==−−푠푖푛휃sinθ++푐표푠휃푑푦cosθdy (A(A10)10) 푈퐼퐽′ = 푈퐼퐽푐표푠휃 + 푉퐼퐽푠푖푛휃 (A11) UIJ0 = UIJcosθ + VIJsinθ (A11) 푈퐼퐽′1 = 푈퐼퐽1 × 푐표푠휃 + 푉퐼퐽1 × 푠푖푛휃 (A12) 021 2 1 2 1 UIJ 2 = UIJ2 × cosθ + VIJ2 × sinθ (A12) Substituting Equations (A10)–(A12) into Equation (A9), Substituting′2 Equations2 (A10)–(A12) into Equation (A9), ′ 휕푢 퐷 휕푢 퐷 2 휕푢푣퐷 2 휕푣푢퐷 2 퐴퐷푉푈푋 = ′ = 푠푖푛휃푐표푠 휃 + 푠푖푛 휃푐표푠휃 + 푠푖푛 휃푐표푠휃 휕푦 휕푦02 2 휕푦 휕푦 0 = ∂u D = ∂u D 2 + ∂uvD 2 + ∂vuD 2 ADVUX2 ∂y0 2 ∂y sinθcos θ ∂y sin θcosθ ∂y sin θcosθ 2 휕푣 퐷 2 휕푢 퐷 2 휕푢푣퐷 휕푣푢퐷 휕푣 퐷2 + 푠푖푛+3휃∂+v D sin3 푐표푠+ 3∂휃u D+cos3 +푠푖푛휃∂uvD푐표푠sin2휃cos+ 2 + ∂푠푖푛휃vuD sin푐표푠2cos휃 +2 + ∂v 푠푖푛D sin2휃푐표푠휃2 cos 휕푦 ∂y 휕푥 θ ∂x 휕푥θ ∂x θ θ휕푥 ∂x θ θ 휕푥 ∂x θ θ 퐴퐷푉푉푋’, 퐴퐷푉푈푌’ 퐴퐷푉푉푌’ With the same method, 0 , and0 are0 given as: With the same′ ′ method,2ADVVX , ADVUY , and ADVVY are given as: ′ 휕푢 푣 퐷 휕푢 퐷 2 휕푢푣퐷 2 휕푣푢퐷 3 퐴퐷푉푉푋 = ′ =0 −0 푠푖푛휃푐표푠 휃 − 푠푖푛 휃푐표푠휃 + 푐표푠 휃 0휕푦= ∂u v D =휕푦− ∂u2D 2 −휕푦∂uvD 2 + 휕푦∂vuD 3 ADVVX ∂y0 ∂y sinθcos θ ∂y sin θcosθ ∂y cos θ 2 2 2 2 2 2 휕푣 퐷+ ∂v D sinθcos2θ휕+푢 퐷∂u D sin2θcosθ +휕푢푣퐷∂uvD sin3θ −휕푣푢퐷∂vuD sinθcos2θ − ∂v휕D푣sin퐷 2θcosθ + 푠푖푛휃∂y 푐표푠2휃 + ∂푠푖푛x 2휃푐표푠휃 + ∂x 푠푖푛3휃 − ∂x 푠푖푛휃푐표푠2휃 −∂x 푠푖푛2휃푐표푠휃 휕푦 휕푥 휕푥 휕푥 휕푥 0 0 2 0 ′ ′ ∂u v D 2 ∂u D 2 ∂uvD 2 ∂vuD 3 ADVUY = 0 = − sin θcosθ + sinθcos θ − sin θ ′ 휕푢 푣 퐷 ∂x 휕푢 퐷 ∂y 2 휕푢푣퐷∂y 2 휕푣푢퐷∂y 3 퐴퐷푉푈푌 = =2 − 푠푖푛 휃푐표푠휃2 + 푠푖푛휃푐표푠 휃 − 푠푖푛 휃 2 휕푥′ ∂v D 2휕푦 ∂u D 휕푦2 ∂uvD 3 휕푦∂vuD 2 ∂v D 2 + ∂y sin θcosθ − ∂x sinθcos θ + ∂x cos θ − ∂x sin θcosθ + ∂x sinθcos θ

J. Mar. Sci. Eng. 2018, 6, 159 22 of 23

02 0 = ∂v D = ∂u2D 2 − ∂uvD 2 − ∂vuD 2 ADVVY ∂y0 ∂y sin θcosθ ∂y sinθcos θ ∂y sinθcos θ ∂v2D 3 ∂u2D 3 ∂uvD 2 ∂vuD 2 ∂v2D 2 + ∂y cos θ − ∂x sin θ + ∂x sin θcosθ + ∂x sin θcosθ − ∂x sinθcos θ

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