Tracking the Nearfield Evolution of an Initially Shallow, Neutrally Buoyant Plane Jet Over a Sloping Bottom Boundary

Understanding coastal plane jets which occur when a body of water discharges into an ocean or a lake through a channel or outlet is important, since they play a significant role in sediment, nutrient, and pollutant exchange. This study investigates the nearfield of initially shallow, neutrally buoyant plane jets, bounded by a free surface and a sloping bottom (Sloping Bottom Jet; SBJ) that issue into a laterally unconfined quiescent ambient both experimentally and numerically, and compares them with a plane jet flowing over a horizontal bottom (Horizontal Bottom Jet; HBJ). Results revealed that, different from the HBJ, the width and centerline velocity of SBJ decrease near the mouth. The SBJ width gradually increases after that as the transverse longitudinal velocity profile progressively transforms from a “top‐hat” into a Gaussian distribution. Once the Gaussian distribution is established, both jets diverge and centerline velocity decreases. Shear layers are generated on the sides of both jets with Kelvin Helmholtz‐type Coherent Structures (KHCS) developing inside. KHCS produce periodic velocity fluctuations with a Strouhal number of ∼0.079 and contribute significantly to momentum exchange and turbulent kinetic energy production. Since the thickness of the SBJ increases longitudinally, the vertical extent of KHCS also increases. When the two shear layers meet and merge at the centerline, they cause a flapping motion of the jet. This location is closer to the jet mouth for SBJs than for the HBJ. These findings demonstrate that a sloping bottom modifies the flow field from quasi‐2D for the HBJ to strongly 3D for SBJs.


Introduction
Discharge of a body of water into an ocean or a lake through a channel or an outlet that resembles a neutrally buoyant flow into a quiescent ambient is a common feature of coastal environments.Understanding the development of this exchange flow, particularly in the nearfield of an outlet, is important since the jet-like discharge plays a significant role in the exchange of sediments, nutrients, and pollutants from rivers, bays or harbors to oceans or lakes.This is crucial with respect to the biogeochemical evolution of marine and lacustrine ecosystems in the coastal environment (Bianchi & Allison, 2009;Brown et al., 2000;McKee et al., 2004;Yao et al., 2016).Being able to characterize this flow allows optimally managing the coastal environment on a large scale.This exchange flow is often complex.However, much can be learned about its hydrodynamics when it is approximated by a shallow plane-jet flowing into a quiescent ambient (Cohen, 2012;Ortega-Sánchez et al., 2008;Özsoy & Ünlüata, 1982;Rowland et al., 2009).Previous investigations of neutrally buoyant plane jets bounded by a solid flat-bottom boundary and an upper free surface issuing from a wide, rectangular cross-section opening into a quiescent ambient, simulated aspects of the exchange flow from an outlet into the coastal environment (Cohen, 2012;Jiménez-Robles et al., 2016;Ortega-Sánchez et al., 2008;Rowland et al., 2009).The hydrodynamics of this configuration (Figures 1a and 1b) will be investigated in the present study in the laboratory and by numerical methods in order to provide insight into the characteristics of the exchange flow; this may also serve as a guide for field studies in the coastal environment that are often difficult to carry out.
In previous laboratory studies, plane jets were bounded by parallel horizontal boundaries (free surface and/or solid horizontal wall(s); Cohen, 2012;Deo et al., 2007Deo et al., , 2008;;Dracos et al., 1992;Giger et al., 1991;Jirka, 2001;Rowland et al., 2009).Assuming homogeneity in the vertical direction, the jet could be treated as quasi two- dimensional (2D).The jet mean flow is longitudinal, and ambient fluid is entrained on both sides resulting in an increase of the jet width (Figure 1a).Shear layers develop on both sides of the jet, generating Kelvin Helmholtztype Coherent Structures (KHCS) with vertical axes.The transverse profile of the longitudinal jet velocity gradually transforms from a "top-hat" profile in the flow establishment zone into a Gaussian profile in the established flow zone (Pope, 2000).Thereafter, the jet becomes self-similar and the centerline velocity starts to decrease (Cohen, 2012;Deo et al., 2007;Heskestad, 1965;Pope, 2000;Rowland et al., 2009).At the location where the Gaussian profile is reached, the shear layers at the two sides of the jet join at the centerline and can cause the jet to meander, a phenomenon known as "jet flapping" (Cohen, 2012;Deo et al., 2007;Dracos et al., 1992;Landel et al., 2012;Thomas & Goldschmidt, 1986).Analytical solutions for the main characteristics of quasi-2D plane jets (i.e., width, centerline velocity, transverse velocity profile, turbulent viscosity) that predict their variations in the longitudinal direction are available (Cohen, 2012;Deo et al., 2007;Heskestad, 1965;Pope, 2000;Rowland et al., 2009).However, jets over a sloping bottom in coastal environments behave different from the horizontal bottom plane jets discussed above (Broekema et al., 2018;Jiménez-Robles et al., 2016;Ortega-Sánchez et al., 2008) since they gradually spread in the vertical direction (Figure 1b).Analytical models describing the development of jets flowing over sloping boundaries were established by Özsoy and Ünlüata (1982) and Ortega-Sánchez et al. (2008).Ortega-Sánchez et al. (2008) suggested that, unlike horizontal bottom plane jets, sloping bottom jets first undergo a horizontal contraction as a result of their vertical extension, before spreading laterally due to entrainment.A typical example of such a flow in the coastal environment is the ebb jet flowing through Naruto Strait in Japan (Figure 2).
In contrast to extensively investigated horizontal bottom plane jets, laboratory studies of initially shallow plane jets (initial aspect ratio: B 0 /H 0 > 5) flowing over a sloping bottom have yet to be reported, and validated numerical and analytical models for such a configuration are not available.Here, B 0 and H 0 denote the initial width and depth of the jet, respectively.In this study, we conducted the first combined investigation of laboratory experiments and numerical modeling of an initially shallow, free-surface neutrally buoyant plane jet flowing from a horizontal rectangular channel onto a sloping solid bottom (8°) boundary in an unconfined ambient.Certain findings are compared with those obtained from a numerical, horizontal bottom plane jet study with the same flow configuration.After validation, the numerical model was also applied to different slopes (0°< β < 8°), since globally, 85% of nearshore slopes in oceans are reported to be <8° (Athanasiou et al., 2019).The following questions are addressed: • What are the similarities and differences between a neutrally buoyant plane jet flowing over a sloping bottom and a quasi-2D plane jet flowing over a horizontal bottom in terms of mean flow development?• Can the features of a neutrally buoyant plane jet flowing over a sloping bottom, suggested by existing analytical models, be verified by our experiments and numerical modeling results?If not, what are the limitations of the analytical models?• Are the characteristics of the shear layers and Kelvin Helmholtz-type Coherent Structures (KHCS) affected by the bottom slope?• What contributions do KHCS make to the momentum exchange across the jet-ambient interface and the production of turbulent kinetic energy?
The Supporting Information (SI) provides texts and figures (with prefix S) with additional details and clarifications of certain topics discussed in the main text.

Laboratory Experiments
The laboratory experiments were conducted in the 1.2-m deep circular tank (diameter 13 m) of the LEGI Coriolis Platform (Université Grenoble Alpes, CNRS, Grenoble, France) (Figure 1c).The same experimental set-up and measurement techniques were used as in Shi et al. (2022), who investigated negatively buoyant plumes.Water was discharged from a horizontal 2-m wide (B 0 ), 4-m long and 0.08-m deep (H 0 ), straight inflow channel into an unconfined ambient (8-m wide × 4.75-m long) with an inclined bottom boundary (8°slope).The water depth was kept constant during the experiment (0.08 m in the inflow channel and 0.75 m in the circular tank) by extracting water at the same volume flux as the inflow (Q 0 ) through three outlet drains positioned at the sides and below the sloping boundary on the tank floor (Figure 1c; for more details, see Text S1 in Supporting Information S1).
The instantaneous velocity field in the horizontal plane at z/H 0 = 0.5 with a spatial resolution of ∼0.01 m × 0.01 m was obtained from Particle Image Velocimetry (PIV) measurements.The measurement error for the instantaneous velocities was estimated to be ∼3% (details in Text S1 of the Supporting Information S1).A Cartesian coordinate system (x: longitudinal, y: transversal, z: vertical) is applied (Figure 1c); its origin (0, 0, 0) is located at the water surface and at the centerline at the downstream end of the inlet channel.Velocities in the x, y and z directions are Two cases (hereinafter referred to as Case 1 and Case 2) with different inflow velocities (Table 1) were investigated resulting in two different inflow Froude numbers (Fr 0 = U 0 / ̅̅̅̅̅̅̅̅ gH 0 √ equal to 0.08 for Case 1 and 0.13 for Case 2), which are within the range of Froude numbers (0.08-0.18) reported in the literature for tidal ebb jets (Broekema et al., 2018;Cohen, 2012;Onishi, 1984).Here, U 0 is the bulk average velocity of the jet at the channel mouth and g is the magnitude of gravitational acceleration.

Numerical Model
To simulate the experiments, a three-dimensional (3D) numerical model using Large Eddy Simulation (LES) based on the open-source CFD package OpenFOAM (De Lorenzis & Düster, 2020;Jasak, 2009;Weller et al., 1998) was adapted using the validated model configuration of Shi et al. (2022).A simplified geometry consisting of a rectangular inlet channel connected to a receiving tank with an 8°sloping bottom (Figure 1d) was applied to represent the geometry of the laboratory experiment.Two different inflow configurations for the jet over the sloping bottom were investigated (Cases 1 and 2 in Table 1).For more details of the model configuration, see Text S2 in Supporting Information S1.Text S3 in Supporting Information S1 gives further verification and validation of the model including a grid convergence index analysis and a comparison with experimental data for a shallow plane jet flowing over a horizontal bottom reported by Cohen (2012).The numerical model provided 3D flow field data at different depths, and also extended the range of the investigated domain in the offshore direction, beyond that of the experimental investigations.The model was also applied for other bottom slopes, β = 0°, 1°, 2°, 4°(Table 1).Case rf (β = 0°) was used as a reference case to determine the similarities and differences between sloping bottom and horizontal bottom jets.

Mean Flow Characteristics and Their Comparison With Theories
Time-averaged quantities are similar for the two 8°sloping bottom cases.Therefore, the focus will mainly be on Case 1 and the comparison with horizontal bottom Case rf.The measured mean velocity field at z/H 0 = 0.5 shows that, in contrast to the classical quasi-2D horizontal bottom plane jet, the jet in Case 1 converges laterally near the channel mouth (Figure 3a).This agrees with the outlet discharge observed in the coastal environment (e.g., Figure 2).The longitudinal velocity profile along the centerline (y = 0) drops steeply until x/B 0 = 0.5 (Figure 3b).It then slightly increases and decreases again before reaching x/B 0 = 1.6 (Figure 3b).The transverse velocity distributions at x/B 0 = 0.4 and 0.8 present a "top-hat" profile with a central region of laterally uniform longitudinal velocity (Figure 3c).The velocity profile gradually transforms.At x/B 0 = 1.2, the "top-hat" region becomes much smaller and then disappears in the established flow zone at x/B 0 = 1.6, at the limit of the measured domain.The half width of the "top-hat" region (r) gradually decreases from 0.5B 0 at the channel mouth to zero at x/B 0 = 1.6.Different from Case 1, the jet in Case rf spreads laterally from the channel mouth (Figure 4a).With increasing offshore distance, a bottom boundary layer gradually develops (Figure 4b, velocity distribution in the central section, y/B 0 = 0).For sloping bottom Case 1, the numerical velocity distribution at z/H 0 = 0.5 (Figure 4c) is very similar to that observed in the experiment (Figure 3a).Beyond x/B 0 > 1.6, the jet starts to diverge.
The velocity field was obtained in an inclined plane that was parallel to the sloping bottom boundary and was 2 cm above it (Figure 4d).The velocity field features a region of low velocity inside the jet (dashed-lined triangle), resulting from the angle between the inflow velocity and the sloping bottom.A similar low velocity region at that location was also observed by Shi et al. (2022) for negatively buoyant plumes over a sloping bottom boundary.The depth-averaged velocity distribution for Case 1 (Figure 4e) is similar to the one close to the water surface (Figure 4c) but has a smaller magnitude.Unlike Case rf, the jet in Case 1 spreads vertically, and thus the velocity in the central section (y/B 0 = 0) decreases (Figure 4f).Profiles of transversal velocity show no clear transversal velocity in Case rf (Figure 5a), whereas there is strong evidence of transversal velocity toward the centerline in Case 1 (Figure 5c).In both cases, the transverse profile of longitudinal velocity gradually transforms from a "top-hat" profile into a Gaussian distribution (Figures 5b and  5d), matching well with the Gaussian profile for a quasi-2D plane jet suggested by e.g., Deo et al. (2008): The location where the Gaussian profile is formed defines the transition point (x T ) from the flow establishment zone to the established flow zone.This transition point is much closer to the channel mouth in Case 1 (x T /B 0 ≈ 1.6) than in Case rf (x T /B 0 ≈ 4).Cohen (2012) also obtained x T /B 0 ≈ 4 for the location of the transition point for the plane jet over a horizontal bottom from experiments.Based on the transverse profile of longitudinal velocity (Figures 5b and 5d), jet width b was calculated and is shown in Figure 5e; b = 2|y 0.5 | where y 0.5 denotes the transverse coordinate and where the longitudinal velocity is 50% of the centerline velocity U c .The thickness of jet h c in the central section (y/B 0 = 0) is calculated following Lee and Yu (1997): ) 2 dz, with u c being centerline velocity and z b , bottom depth.In Case rf, jet thickness is constant (≈H 0 ) while its width increases  (i.e., vertical extension and lateral entrainment) balance each other at a certain distance (here at x/B 0 ≈ 1.6).Thereafter, entrainment becomes the dominant mechanism and width b slightly increases (Figure 5e, red solid line) as in Case rf.A comparison between Case rf and Case 1 in terms of depth-averaged centerline velocity and discharge per unit width in the central section (q c = U c H) shows that in Case rf, U c , and q c remain almost constant (Figure 5f).In contrast, in Case 1, a sharp decrease in U c is observed in the range 0 < x/B 0 < 0.5 (Figure 5f, red solid line) due to the increase of jet thickness (Figure 5e, blue solid line).After that, it undergoes a slight increase in the range 0.5 < x/B 0 < 1.6, because q c increases.The increase of q c is triggered by transversal velocities directed toward the centerline (Figure 5c).For x/B 0 > 1.6, the transversal velocity becomes negligible, and thus, q c remains almost constant and U c decreases again.
Analytical models for the variation of jet width and centerline velocity for the sloping bottom case were proposed by Özsoy and Ünlüata (1982) and Ortega-Sánchez et al. (2008).Unlike the present study where y 0.5 is used to measure the jet width, they defined a jet half-width b as the distance from the centerline to the location where the longitudinal velocity is equal to zero.They proposed the following relationship for the transverse profile of the longitudinal velocity in the flow establishment zone:  2008), the analytical model is not sensitive to f for slopes >3°.In the flow establishment zone, where r > 0, the models of Özsoy and Ünlüata (1982) and Ortega-Sánchez et al. ( 2008) are identical, derived from Equation 2, and assume that the centerline velocity remains constant.For the established flow zone, Equation 2 is still used in the model of Özsoy and Ünlüata (1982) by setting r = 0, while Ortega-Sánchez et al. ( 2008) used a hyperbolic secant function.Our numerical results are compared with the analytical model results of Özsoy and Ünlüata (1982) in Figures 6c-6f.The analytical results underestimate jet width and overestimate centerline velocity.Differences are due to the model assumption that there is a constant centerline velocity in the flow establishment zone (x/B 0 < 1.6).However, our numerical results (Figures 6e and 6f) and measurements (Figure 3b) demonstrate that the centerline velocity is not constant.In the established flow zone (x/B 0 > 1.6), the model roughly captures the variation of the jet width and centerline velocity if the curves are shifted upwards to compensate for the initial errors.

Turbulence Characteristics
Figures 7a and 7b presents the distribution of horizontal Turbulent Kinetic Energy (TKE), k H = 0.5 u'u' + v'v') , at z/H 0 = 0.5 for Case rf and Case 1.The vertical component of TKE is not available from the experimental data, but numerical model results confirm that the TKE is dominated by the horizontal components (Text S4 in  Below, the focus is on the turbulence characteristics of the shear layers in the flow establishment zone before they merge with each other.The turbulent kinetic energy budget is (Panchapakesan & Lumley, 1993;Pope, 2000): Terms in Equation 3 are, respectively, production, advection, dissipation and diffusion.The TKE production rate (P = u i 'u j '∂u i /∂x j ) is generally positive and thus a source of TKE.It describes the energy transfer from the mean velocity gradients (working against the Reynolds stresses) to fluctuating velocity fields.P was determined based on the measured and modeled horizontal velocity fields, since vertical velocities were not measured and the horizontal components are the main source (Text S4 in Supporting Information S1).In Case rf and Case 1, P is strong inside the shear layer, in particular close to the channel mouth (Figures 8a-8c).Further away from the channel mouth as the shear layer develops, P decreases because of the decrease of the velocity gradient (Figures 4a and 4c).The TKE budget was analyzed along the jet-ambient interface (determined by the jet width; dashed lines in Figures 7a, 7b, and 8a-8c) and shown in Figures 8d and 8e.The advection term was calculated as defined in Equation 3, but neglecting the vertical components.The TKE dissipation rate ε was estimated using the inertial-dissipation technique based on the energy spectrum (Geyer et al., 2010;Shaw & Trowbridge, 2001;Siebert et al., 2006).The diffusion term is difficult to determine accurately (e.g., Pope, 2000).Therefore, it was calculated using Equation 3 after all other terms were calculated.Figures 8d and 8e shows that the TKE production is mainly dissipated locally while the rest is balanced by diffusion.The advection term is comparatively small and its value may be due to the uncertainty when calculating TKE gradients.Like TKE (Figures 7a and 7b), horizontal Reynolds stress ( u'v') at z/H 0 = 0.5 is high in the shear layer (Figures 9a-9c).In the region 0.75 < x/x T < 1.0 in Case 1 where the two shear layers are close to each other, the magnitude of the Reynolds stress is larger than near the channel mouth.This phenomenon is not observed in Case rf.Turbulent viscosity υ t = u'v'/ (∂u/∂y) is an important parameter that is often used for shear stress closure (e.g., Odier et al., 2012;Shih et al., 1995;Shiono & Knight, 1991).It can be nondimensionalized into a turbulent Reynolds number R T = (0.5b r) u c /υ t .Considering that measured/modeled values of turbulent viscosity are not reliable close to the lateral boundary of a shear layer (Odier et al., 2012), R T is averaged inside the shear layers between y 0.8 and y 0.2 (Figure 9d); y 0.8 and y 0.2 represent the locations of 80% and 20% of the centerline velocity u c , respectively.No clear difference between Case rf and Cases 1 and 2 is observed in terms of R T (Figure 9d).Combining all experimental and numerical data in Figure 9d, an averaged value of R T ≈ 47 ± 14 was obtained, which is larger than the theoretical value (R T ≈ 31) for a self-similar 2D plane jet (Pope, 2000).This may indicate that classical 2D theory overestimates the Reynolds stress in a jet flowing over a solid bottom as was already reported by Rowland et al. (2009) for a horizontal bottom jet.

Characterization of the Shear Layers
A new coordinate system (x n , y n ) is defined as one that follows the jet width curve (dashed lines in Figures 7a and  7b) with corresponding velocities u n = u n + u′ n tangent to the curve, and v n = v n + v′ n perpendicular to the curve.Directions of (x n , y n ) do not show substantial differences with the original coordinates (x, y) in Case rf.However, in Case 1, (x n , y n ) a curvilinear coordinate system with changing directions because the jet converges.Following the y n direction, velocity u n must increase from a small value u n a (at y n-a ) in the ambient water to a large value u n c inside the jet (at y n-c ).Typical profiles of measured u n in Case 1 are plotted in Figure 10a.
Momentum thickness θ is a characteristic length scale that describes the width of a shear layer (Ho & Huang, 1982;Rogers & Moser, 1992): where u n o is the arithmetic mean of u n a and u n c .The variation of θ along x was calculated using the experimental and numerical velocity fields (Figure 10b).The momentum thickness continuously increases with x in Case rf.In the sloping bottom Cases 1 and 2, it first increases and then becomes almost constant after x/x T > 1.The momentum thickness is comparatively smaller in Case 1 with larger Fr 0 and Re 0 than Case 2 (Table 1), which is consistent with previous findings in slowly spreading shear layers (Dimotakis, 1991).

Structure Identification and Strouhal Numbers
Kelvin Helmholtz-type Coherent Structures (KHCS) are often observed in shear layers (Brown & Roshko, 1974).KHCS were also seen in the discharge from an outlet in the coastal environment (Figure 2) and are therefore an important feature of the exchange flow between a water body passing through a channel and the receiving ocean or lake.The transient iso-concentration surface (c = 0.7) in numerical Case 1 indicates the occurrence of KHCS (Figure 11).Concentration c is determined by the degree of mixing of two fluids, that is, c = 1 inside the inflow channel, and c = 0 in the ambient water (see Text S2 in Supporting Information S1).Thus, the iso-concentration surface (c = 0.7) in Figure 11 is composed of 70% inflow water and 30% ambient water.The turbulent structures generate nearly periodic local velocity fluctuations in the velocity time series (Figures 12a and 12b).The Power Spectral Density (PSD) is obtained from the velocity signals at locations P1-P5 (blue circles in Figures 7a and 7b) on the jet-ambient interface; the Strouhal number St θ = f θ/ u n o is used as the nondimensional frequency (Ho & Huerre, 1984).All spectra have a strong peak close to St θ = 0.079 indicating the dominant KHCS frequency.This Strouhal number (∼0.079) is consistent with the typical value of KHCS in slowly spreading shear layers (Ho & Huang, 1982;Ho & Huerre, 1984).

Contribution of KHCS to Momentum Exchange and Energy Production
KHCS contribute to the momentum exchange and production of TKE in shear layers (Taborda et al., 2022;Truong & Uijttewaal, 2019;White & Nepf, 2008).The momentum exchange between the jet and ambient waters is mainly represented by the horizontal Reynolds stress u n 'v n ' at the interface (y n = 0) (Taborda et al., 2022;  Truong & Uijttewaal, 2019).The TKE production rate P n in the curvilinear coordinate system (neglecting vertical components) is given by, To compare the contribution of KHCS and small-scale turbulence to momentum exchange and TKE production, it is necessary to separate the velocity fluctuations associated with KHCS from those associated with small-scale turbulence.This analysis was only based on the experimental data of Case 1. Case rf could not be investigated because the LES model does not directly resolve small-scale turbulence.
Two different methods were proposed in the literature to quantify the contribution of KHCS: The first method uses a simple low-pass filter to separate low-frequency (associated with KHCS) from small-scale, turbulenceinduced high-frequency velocity signals (Li et al., 2022;Truong & Uijttewaal, 2019).Examples of filtered velocity signals are shown in Figures 12a and 12b (using a low-pass filter, f filter = 4f cs following Truong and Uijttewaal ( 2019)); f cs is the dominant frequency of KHCS, that is, peak frequency in Figure 12c before nondimensionalization.In the second method, the measured flow field is reconstructed based on conditional-averaging in order to smooth out small-scale turbulence (Shi et al., 2023;Taborda et al., 2022;Yuan & Horner-Devine, 2017).The conditionalaveraging approach identifies KHCS using three consecutive zero-crossings of low-pass filtered v n ′ (Taborda et al., 2022;White & Nepf, 2008), based, for example, on the velocity signal at P1 (Figure 12b).A typical KHCS (Figure 12b) lasts for a time period of T i and contains a sweep (Q4) and an ejection (Q2) event, as distinguished using quadrant analysis (Bagherimiyab & Lemmin, 2018;Huai et al., 2019;Nezu & Nakagawa, 1993;White & Nepf, 2008).Results of a quadrant analysis of velocity signals are given in Text of the Supporting Information S1.A series of KHCS is identified based on low-pass filtered data, which provide the starting and ending time of a given structure (Taborda et al., 2022).The measured flow field (before low-pass filtering) of the identified structures along line l-l during their periods T i is then rescaled into a new time coordinate t A = T d (t t 0 )/T i , where T d = 1/f cs .Here, t is the instantaneous raw time and t 0 is the time when the structure first appears (Taborda et al., 2022).Rescaled velocity signals of the identified structures are then averaged to obtain the conditionally averaged data.Note that the conditionally averaged velocity field obtained here is based on the "raw" measured data rather than on the low-pass filtered data; the low-pass filtered data were only used to identify coherent structures.According to Taylor's frozen hypothesis, the structures are advected mainly by the mean velocity.The conditionally averaged velocity field relative to a reference frame moving with a velocity u n o is obtained by subtracting this reference velocity.The same procedure is also applied to the velocity signals of P4 to determine the structures on line m-m.The conditionally averaged flow field provides streamline patterns of the KHCS (Figure 13).Compared with the structure at line l-l, the structure at m-m is larger in both the x n (L x ) and y n (L y ) directions.Since coherent structures are induced by shear and located inside the shear layer, the size of the structure defined by L x and L y scales with the width of the shear layer, that is, L x ∼ L y ∼ θ.Thus, the time period of the structure is , which explains why the Strouhal number remains almost constant (Figure 12c).
Low-pass filtered or conditionally averaged flow fields were used to calculate Reynolds stress and second-order terms in P n (i.e., u′ n u′ n , u′ n v′ n , v′ n v′ n ) associated with KHCS (Figure 14).Based on the low-pass filtering method, low-frequency signals contribute 78% and 79% of the total momentum exchange at P1 and P4, respectively.These values change, respectively, to 63% and 68% if the conditionally averaged method is applied.For the TKE production rate, the integral of P n from y n /B 0 = 0.15 to 0.15 was calculated.Based on the low-pass filtering method, the low-frequency signals contributed 72% and 79% to the total production rate at lines l-l and m-m, respectively.If the conditional-averaging method is applied, these values are 45% and 50%, respectively.The processed velocity data obtained by the two different methods mentioned above are compared in Figures 12a and  12b.Using the low-pass filter, the velocity signals are smoothed, but the variations of the KHCS are preserved, that is, the three KHCS identified in Figure 12b have slight differences in terms of time period and fluctuation amplitude.Such variability of KHCS is smoothed out when the conditional-averaging method is used.This explains why momentum exchange and TKE production rate are underestimated using conditionally averaged data.

Vertical Extent of KHCS
The vertical thickness of the plane jet for the horizontal-bottom case remains constant (equal to H 0 ; Figure 5e, blue dashed line).The jet over an 8°sloping bottom, on the other hand, spreads vertically with increasing water depth (Figure 5e, blue solid line), and as a result, the vertical thickness (h cs ) of KHCS increases with the increase of jet thickness (Figure 11).KHCS are characterized by regions with relatively small u n (sometimes even turning backwards; see yellow crosses in Figures 11b and 11c).Autocorrelations of the modeled concentration c at P1 and P4 (at z c /H 0 = 0.5), along with cross-correlations R zz with points at different depths were determined using (Bagherimiyab & Lemmin, 2018;Nezu & Nakagawa, 1993): where σ is the time lag, and c′ is concentration fluctuation.The cross-correlation decreases with increasing distance between the locations (Figures 15a and 15b), indicating that coherence is gradually lost.However, the loss of coherence at P4 (Figure 15b) is more gradual than at P1 (Figure 15a), confirming that the vertical thickness of the coherent structures increases with increasing distance from the channel Using R zz = 0.3 as a threshold, the KHCS thickness increases from 1.25H 0 (73% of local depth) at P1 to 2.75H 0 (72% of local depth) at P4.The correlation time scale at P4 is longer than at P1.Moreover, a progressively larger positive time shift is observed when z increases.A positive time shift indicates that the concentration signal at a higher elevation correlates with the signal at a lower elevation at a later time.This suggests that the coherent structures are inclined, since u n is comparatively larger close to the water surface (Figure 15c).

Merging of Lateral Shear Layers
When the plane jet reaches the established flow zone, the two shear layers meet and merge at the centerline (Figures 7a and 7b).After the shear layers merge, the jet may manifest a "flag-like" flapping motion (Antonia et al., 1983;Cohen, 2012;Oler & Goldschmidt, 1982;Rowland et al., 2009).Comparing the times series of depthaveraged concentration (c d ) at x/x T = 0.5 and x/x T = 1 for Case 1 and Case rf (Figure 16), it can be seen for both cases that at x/x T = 0.5, the central part of the jet is stable and the KHCS only generate fluctuations at the sides of the jet.At x/x T = 1, however, the jet flaps, showing a meandering pattern.The frequency of the jet flapping motion ( f j ) was obtained from spectral analysis of the velocity at the centerline (i.e., the peak of PSD of the depthaveraged transversal velocity signal at x/x T = 1, y = 0).It is usually made nondimensional as f j b/U c ; the points are marked by circles on the profiles (for location, see Figure 7b).(Jirka, 2001) or f j B 0 /U 0 (Cohen, 2012;Deo et al., 2007), and ranges between 0.08 and 0.5 (Cohen, 2012).The numerical results gave f j b/U c ∼ 0.3 for both Case 1 and Case rf, within the range reported in the literature.The jet flapping motion also brings ambient water to the centerline (Figure 16d), resulting in a decreased depth-averaged concentration at (x/x T = 1, y = 0) in Case 1 (equal to 0.86 after time-averaging).In contrast, the jet flapping motion in Case rf is not strong enough to influence the centerline concentration, and at (x/x T = 1, y = 0), the timeaveraged concentration is still equal to unity.The stronger jet-flapping in Case 1 also produces a region (0.75 < x/ x T < 1) with strong momentum exchange (Figures 9a and 9b); this is much less evident in Case rf (Figure 9c).Previous studies pointed out that the instability of a jet is related to aspect ratio b/H and bottom friction (Canestrelli et al., 2014;Jirka, 1994;Socolofsky & Jirka, 2004): that is, the strength of the jet flapping motion is expected to be stronger with a smaller aspect ratio and smaller bottom friction.The main difference between Case 1 and Case rf is the aspect ratio at x/x T = 1, which is only b/h c = 1.3 in Case 1, but can be as large as b/h c = 37.5 in Case rf.

Application of the Numerical Model to Other Slopes
To determine the effect of the bottom slope angle on jet characteristics, the numerical model was applied to jets flowing over three additional slopes: β = 1°, 2°, and 4°under the same inflow conditions as Case 1 and Case rf (Table 1).Increase of jet thickness, which is controlled by bottom slope (Figure 17a), leads to nearshore contraction of the jet (Figure 17b), even with a slope angle as small as 1°.A larger bottom slope leads to a more rapid thickness increase and width decrease in the flow establishment zone.As a result, the length of the flow establishment zone (x T ) decreases with increasing bottom slope (Figure 17c).For the 2°sloping bottom, the numerical model gives (x T /B 0 ≈ 2.4), which is close to the value obtained from Figure 2 in the coastal environment (x T /B 0 ≈ 2.7), suggesting that the outflow from a channel or an outlet in the coastal environment can be well characterized by a plane jet over a sloping bottom.The time and depth-averaged centerline concentration (c d ) at x/ x T = 1 also decreases with an increase of the bottom slope angle (Figure 17c).This indicates that the strength of the jet-flapping motion increases with an increasing bottom slope angle due to a smaller aspect ratio (b/h c ) at x/ x T = 1 (Figures 17a and 17b).With an increasing slope angle, the rate of change of the depth-averaged centerline velocity increases rapidly in the nearshore zone as shown in Figure 17d.These results emphasize the sensitivity of the mean flow parameters to the slope angle in the flow establishment zone, even for small slope angles.The detailed study of the flow over an 8°slope presented in the previous sections shows that the change in all the other hydrodynamic characteristics (compared to the horizontal bottom case) is related to the mean flow parameters discussed in this section.Therefore, it can be assumed that the remaining jet characteristics (i.e., development of shear layers, vertical extension of KHCS and jet flapping motion strength) also change with slope angle.

Summary of Similarities and Differences Between Sloping Bottom and Horizontal Bottom Jets
Comparing the results of the present study for a jet flowing over a sloping bottom with the quasi-2D plane jet over a horizontal bottom, the following similarities and differences in terms of mean flow field and turbulence characteristics (Section 3.7.1)and Kelvin Helmholtz-type coherent structures (Section 3.7.2) were found.

Mean Field Development and Turbulence Characteristics
Similarities: • In the nearfield, the transverse profile of the longitudinal jet velocity transforms from a "top-hat" shaped profile (flow establishment zone) into a Gaussian distribution (established flow zone).• Shear layers develop and grow on both sides of the jet due to the presence of the velocity gradient between the jet and ambient water.• Strong signals of turbulent kinetic energy are observed in the shear layers.

Differences:
• The jet over a sloping bottom spreads vertically.Under the combined influence of the vertical spreading and lateral entrainment, the jet contracts before spreading laterally as seen in observations in the coastal environment (Figure 2).• For the 8°sloping bottom jet, the transition to the Gaussian distribution occurs at x T ≈ 1.6B 0 , which is 0.4 times that of the horizontal bottom jet (x T ≈ 4B 0 ).Thus, the length of the flow establishment zone is significantly reduced.Results of the analysis for different slope angles show that the length of the flow establishment is sensitive to the slope angle.• The centerline velocity decreases near the channel mouth in sloping bottom jets but remains constant in the horizontal bottom jet in the flow establishment zone.• The momentum thickness of the shear layers increases with distance in the flow establishment zone and then remains almost constant thereafter in the 8°sloping bottom case.In the horizontal bottom case, however, the momentum thickness continuously increases in the flow establishment zone and in the established flow zone.

Kelvin Helmholtz-Type Coherent Structures
Similarities: • In the shear layers, Kelvin Helmholtz-type Coherent Structures (KHCS) are generated, whose horizontal size increases with increasing offshore distance.This agrees with observations in the coastal environment (Figure 2).• KHCS result in quasi-periodic velocity fluctuations at the interface between the jet and ambient water.A nearly constant Strouhal number of ∼0.079 is observed for these KHCS.• Jet flapping occurs after the shear layers merge at the centerline (at around x/x T = 1).

Differences:
• The vertical size of KHCS in the 8°sloping bottom jet increases with increasing distance from the channel mouth because of the vertical spreading of the jet.• The jet-flapping motion is strong enough to influence the centerline concentration in sloping bottom jets at x/ x T = 1, whereas it is not for the jet over the horizontal bottom.

Conclusion
This paper presents the first laboratory study that documents the evolution of an initially shallow, neutrally buoyant jet, bounded by an 8°sloping, solid bottom boundary and an upper free surface, that issues into a laterally unconfined, quiescent ambient.Such jet flow occurs in coastal environments when a body of water discharges into an ocean or a lake through a channel or an outlet.A 3D numerical model using Large Eddy Simulation (LES) was developed and validated with the experimental data.Our experimental and numerical results indicate that existing analytical models for plane jets over sloping bottoms underestimate the jet width and overestimate the centerline velocity.Furthermore, a comparison was made with a plane jet flowing over a horizontal bottom with the same flow conditions.Our study has demonstrated that unlike the horizontal bottom jet which behaves like a quasi-2D plane jet, the flow field of the sloping bottom jet is clearly 3D.
Vertical spreading of the sloping bottom jet causes a horizontal contraction in the nearshore region and a smaller x T , which means a narrower and shorter flow establishment zone than a quasi-2D horizontal bottom jet.This difference is important when evaluating the effective domain of a jet flow, especially when estimating the transport of materials (e.g., sediment, pollution, nutrients, fish larvae) by the jet.A sloping bottom also results in a decrease of centerline velocity near the channel mouth that has not been considered by existing analytical models.This behavior may cause deposition of large-sized suspended sediment and thus influence the evolution of nearshore geomorphology.
Kelvin Helmholtz-type Coherent Structures developing in shear layers on both sides of the jet are found to be the main contributor to momentum exchange and the production of turbulent kinetic energy.KHCS meet at the centerline after a certain distance (x/x T = 1) and cause the entire jet to flap.In sloping bottom jets, the jet-flapping motion is strong enough to influence the centerline concentration at x/x T = 1, which further decreases with increasing sloping bottom angle.These large-scale turbulent structures (KHCS and jet flapping) have a considerable impact on momentum exchange and energy production.Consequently, it is necessary to use a 3D model that can capture these structures when studying jet hydrodynamics.The LES-based model applied in the present study allowed reproduction of 3D details of jet flows that are not fully accessible in experimental studies and field measurements.
Although a simplified geometry with a uniform bottom slope was applied in this study, the present numerical model can be applied to more complex coastal geometries shaped by jet erosion and ebb delta deposit in future studies.It

Water Resources Research
10.1029/2023WR034826 also has the potential to be extended to simulate sediment behavior (deposition and erosion) in jet flows in the coastal environment, as well as to study the evolution of nearshore geomorphology.These features are important since they affect the environmental impact of discharge from a channel or an outlet into a coastal environment.
experiments allowed the first characterization of a neutrally buoyant plane jet over a sloping bottom • Sloping bottom jets converge laterally near the mouth and have narrower, shorter flow establishment zones than horizontal bottom jets • Coherent structures developing in the shear layers contribute significantly to turbulent kinetic energy production and momentum exchange Supporting Information: Supporting Information may be found in the online version of this article.

Figure 1 .
Figure 1.Three-dimensional (3D) depth-exaggerated sketch of a neutrally buoyant plane jet flowing over: (a) a horizontal bottom and (b) a sloping bottom.(c) Experimental setup inside the Coriolis Platform.The jet flow was generated in a 4-m long free-surface inlet channel, which entered the wide ambient water body with an 8°sloping bottom.The white arrows indicate the coordinate system.Its origin is in the center at the surface end of the inflow channel.(d) 3D sketch of the simplified geometry used in the numerical model.The jet is colored in black, and the ambient water in gray.

Figure 2 .
Figure 2. Ebb jet flow through Naruto Strait (Japan; aerial photograph modified after Onishi (1984) with permission from the publisher Elsevier).Note the contraction of the jet (highlighted by orange dash-dotted lines) and the formation of Kelvin-Helmholtz vortices at both sides of the jet.Arrows: outlet width and the length of the flow establishment zone.Flow is from left to right; bottom slope of the ambient: β ≈ 2°.

Figure 3 .
Figure 3. Velocity distributions in Case 1 at z/H 0 = 0.5.(a) Measured velocity distribution.The black solid line marks the boundary of the jet based on its width b calculated from the numerical model (see Figure 5e).Legend: range of velocity.(b) Measured (circles) and modeled (dashed-dotted line) longitudinal velocity profile along the centerline.(c) Measured (circles) and modeled (dashed-dotted lines) transverse profiles of the longitudinal velocity at x/B 0 = 0.4, 0.8, 1.2, and 1.6; r indicates the half-width of the "top-hat."

Figure 4 .
Figure 4. Time-averaged velocity distribution for Case rf: (a) in a horizontal plane at z/H 0 = 0.5 (solid lines show jet width) and (b) at the central-section (y/B 0 = 0).(ce) Time-averaged velocity distribution for numerical Case 1: (c) in a horizontal plane at z/H 0 = 0.5, (d) in an inclined plane 2 cm above the sloping bottom boundary (dashed-lined triangle denotes a region with relatively lower velocity inside the jet), and (e) depth-averaged velocity field.(f) Velocity distribution at the central section (y/B 0 = 0) for numerical Case 1.

Figure 5 .
Figure 5. Transverse distribution of depth-averaged transversal and longitudinal velocity for numerical Case rf (a and b) and Case 1 (c and d).Black dashed line in (a) and (c) is zero.Black solid line in (b) and (d): Gaussian profile (Equation 1).Legends for (a-d): symbols related to the longitudinal coordinates.(e) Variations of the jet width (red) and jet thickness (blue) in numerical Case 1 (solid lines) compared with reference Case rf (dashed lines).The red circles denote the jet width obtained from the measured velocity field at z/H 0 = 0.5.Gray dash-dotted line: increase of the local water depth for Case 1: H = x tanβ + H 0 .(f) Variations of the depth-averaged centerline velocity (U c /U 0 ; red) and discharge per unit width in the central section (blue; right vertical axis) for numerical Case rf (dashed lines) and Case 1 (solid lines).

Figure 6 .
Figure 6.Transverse profiles of the longitudinal velocity for: (a) Case 1 and (b) Case 2 at different locations (x/B 0 ) indicated by colored symbols above the panels.Symbols: numerical data.Solid lines: results of Equation 2. Three typical locations |y| = r, b, and b are identified by vertical dashed lines.Variation of jet width for: (c) Case 1 and (d) Case 2. Variation of centerline velocity for: (e) Case 1 and (f) Case 2. Solid lines: data from the numerical model.Dashed lines: predictions of the analytical model(Özsoy & Ünlüata, 1982).

Figure 7 .
Figure 7. Distribution of horizontal turbulent kinetic energy k H at z/H 0 = 0.5 for: (a) Case rf and (b) Case 1.For Case 1, in the region x/B 0 < 1.5, the upper half of the panel presents the numerical results (NUM), and the lower one, the experimental (EXP) data.In the region x/B 0 > 1.5, only numerical data are available.Dashed lines: jet width b.Dash-dotted lines: inner boundary of the shear layers, representing k H /U 0 2 = 0.01.Five points (P1-5) on the jet width curve are shown by the blue circles in (a) and (b).For Case rf, these points are located at x/B 0 = 0.7, 1.4, 2.1, 2.8, and 3.5; for Cases 1 and 2, they are located at x/B 0 = 0.2, 0.4, 0.6, 0.8, and 1.0.A coordinate system corresponding to the jet width curve is represented by the blue arrows (x n , y n ).In Case 1, lines l-l and m-m are two lines perpendicular to the jet width curve with intersections located at x/B 0 = 0.2 and 0.8, respectively.(c) Lateral distributions of k H at z/H 0 = 0.5 at x/B 0 = 0.4 (purple), 1.2 (orange) in Case 1. Symbols: experimental data.Dashed lines: numerical data.Numerical transverse profiles of u'u' (red), v'v' (blue) and w'w' (green) for: (d) Case rf at x/B 0 = 1 (solid lines) and x/ B 0 = 3 (dashed lines), and (e) Case 1 at x/B 0 = 0.4 (solid lines) and x/B 0 = 1.2 (dashed lines).In plots (c-e), only half of the jet is presented.

Figure 8 .
Figure 8. TKE production rate (P, Equation 3) at z/H 0 = 0.5 for: (a) Case 1 experimental (EXP) data, (b) Case 1 numerical (NUM) data, and (c) Case rf numerical data.Considering that the system is symmetric, only half of the domain (y < 0) is plotted.(d and e) TKE budget analysis along the jet-ambient interface (dashed lines in panels a-c) for Case 1 (d) and Case rf (e).Symbols represent experimental data while lines represent numerical results.Terms are nondimensionalized using U 0 and B 0 .

Figure 9 .
Figure 9. Distribution of u'v' at z/H 0 = 0.5 for: (a) Case 1 experimental data, (b) Case 1 numerical data, and (c) Case rf numerical data.Considering that the system is symmetric, only half of the domain (y < 0) is plotted.(d) Turbulent Reynolds number averaged over the shear layers at z/H 0 = 0.5.Black vertical dashed line in (d): average value of R T over all data sets in this panel (see legend).

Figure 10 .
Figure 10.(a) Typical velocity profiles along lines l-l (black) and m-m (brown) shown in Figure 7b for Case 1. Ambient waters (u n a ) and jet (u n c ) values are marked by circles.(b) Variations of the shear-layer momentum thickness.Circles: experimental data.Lines: numerical data.Red: Case 1. Blue: Case 2. Green: Case rf.

Figure 11 .
Figure 11.(a) Instantaneous field of concentration c for numerical Case 1.The translucent background color (from blue to in legend) illustrates the concentration distribution in the 3D field.The iso-concentration surface for c = 0.7 is highlighted in green.Arrows: inflow direction.Details of the iso-concentration surface c = 0.7: (b) in the region of line l-l, and (c) in the region of line m-m; see colorbar for u n .Regions with relatively small u n values (sometimes even turning backwards) of Kelvin Helmholtz-type Coherent Structures (KHCS) are highlighted by yellow crosses.The KHCS thickness is given by h cs .The sense of rotation and the size of the KHCS are indicated by the curved arrows.

Figure 13 .
Figure 13.Streamlines of the conditionally averaged velocity field relative to a reference frame moving with the Kelvin Helmholtz-type Coherent Structures (KHCS).(a) Structure passing line l-l.(b) Structure passing line m-m.Results are based on experimental data from Case 1; for locations of lines l-l and m-m, see Figure 7b.L x and L y give the dimensions of the KHCS along (red) and across (blue) the interface between the jet and the ambient waters.Note that L x and L y are scaled differently.

Figure 14 .
Figure 14.Horizontal Reynolds stress (top panels) and TKE production rate (bottom panels) induced by the Kelvin Helmholtz-type coherent structures compared to total horizontal Reynolds stress and TKE production rate for experimental Case 1. Left panels: along line l-l.Right panels: along line m-m (for location, see Figure 7b).Green solid lines: total values.Blue dashed-dotted lines: low-pass filtered values (filt).Red dashed lines: conditionally averaged values (CA).

Figure 15 .
Figure 15.Autocorrelation for concentration signals at z c /H 0 = 0.5 and cross-correlation values between z c /H 0 = 0.5 and elevations below z c /H 0 = 0.5, at: (a) P1 and (b) P4; colors defined in legend.The correlation values are calculated based on numerical data over 200 s (100 Hz) for Case 1. Horizontal dashed lines: threshold R zz = 0.3.(c) Vertical profiles of the velocity u n passing through points P1 (black line) and P4 (brown line); the points are marked by circles on the profiles (for location, see Figure 7b).

Figure 16 .
Figure 16.Time series of depth-averaged instantaneous concentration c d at x/x T = 0.5 (left panels) and x/x T = 1 (right panels) for Case rf (top panels) and Case 1 (bottom panels).Colorbars: concentration range.Yellow dash-dotted lines represent the centerline: y/B 0 = 0.

Figure 17 .
Figure 17.Variations of (a) jet thickness, (b) width in the flow establishment zone for numerical cases with different bottom slopes.Stars mark the end of this zone (x T ).(c) Comparison of x T (red; left vertical axis) and time and depth-averaged centerline concentration at x/x T = 1 (blue; right vertical axis) for cases with different bottom slopes.(d) Variations of the depth-averaged centerline velocity in the flow establishment zone in numerical cases with different bottom slopes.The x coordinate is nondimensionalized by x T which is different from the nondimensionalization in (a) and (b).Colors in (a), (b), (d) are defined by the top legends.

Table 1
Summary of Parameters Used in Experimental (EXP) and Numerical (NUM) Cases SHI ET AL.