High-Resolution Nonhydrostatic Outfall Plume Modeling: Cross-Flow Validation

Roberts, Snyder, and Baumgartner developed the RSB model, an empirical length scale–derived model based on towed tank experiments and applied to ocean outfalls (Roberts et al. 1989a, b, c; Frick et al. 2003). The experimental studies were done on multiport T-shaped diffusers in linearly density-stratified conditions using a line source. Complete experimental details and diagrams of their experiments and diffusers were given by Roberts et al. (1989a). The parameters examined in these studies were current speed, $u$, current direction, $\theta$, port spacing, s, effluent density, ${\rho }_0$, and horizontal jet velocity from either side of a T-shaped diffuser, $u_j$. The relevant parameters for this study are presented. Discharge is characterized by the source flux per diffuser length, $q$ (${\mathrm{m}}^2/\mathrm{s}$), momentum, m (${\mathrm{m}}^3/{\mathrm{s}}^2$), and buoyancy flux, b (${\mathrm{m}}^3/{\mathrm{s}}^3$)where $Q$ = total discharge (${\mathrm{m}}^3/\mathrm{s}$); $L$ = diffuser length(m); and ${\rho }_a$ = ambient density at port level (${\mathrm{kg/}\mathrm{m}}^3$). The most important parameter controlling multiport near-field diffuser dynamics is given by the plume Froude numberand relates the current speed to the buoyancy flux of the source. A length scale, $l_b$ (m)is used to relate buoyancy flux to buoyancy frequency, or Brunt–Väisälä frequency, $N$ (1/s) (Wright et al. 1982)

Parameter $l_b$ can be used to nondimensionalize the horizontal and vertical scales of different domains. Additionallyrelates the momentum to buoyancy. Empirical relationships then can be characterized as a dimensional analysis by four independent parameterswhere $S_m$ = minimum dilution; $z_e$ = height of top of plume; $h_e$ = thickness of plume; and $z_m$ = height to minimum dilution. These parameters are normalized to the buoyancy flux and $l_b$.

The laboratory experiments of Roberts, Snyder, and Baumgartner were chosen for two primary reasons: (1) multiple far-field models have used the RSB model to initialize the established wastefield, including ROMS (Zhang and Adams 1999; Kim et al. 2001; Uchiyama et al. 2014), and (2) the EPA has adopted the RSB model (based on the RSB experiments) to meet wastewater treatment National Pollutant Discharge Elimination System (NPDES) dilution permits (Roberts et al. 2011). Additionally, Orange County Sanitation District (OCSD) relies upon the model to examine the impact of water conservation (i.e., reducing total effluent volume by extraction freshwater) on outfall plume characteristics.

RSB series 3 and 4 experiments were chosen as validation data for ROMS because of low jet momentum flux, strong dependence on Froude number, and genesis of RSB data fit curves. These series correspond to perpendicular (cross) flow experiments at Froude numbers 0, 0.1, 1, 10, and 100. The source is moved along the length of the tank at speeds corresponding to the appropriate Froude number to mimic cross-flow. Detailed experimental descriptions were given by Roberts et al. (1989a). The specific parameters in these experiments have $l_m/l_b<0.2$, the lowest momentum fluxes, and port spacing $s=5\mathrm{cm}$. These experiments are categorized as a line plume in which port spacing and jet momentum flux are negligible, and plume development is dominated by buoyancy; $S_m$, $z_e$, $h_e$, and $z_m$ have the greatest dependence on Froude number compared with the other setups (i.e., parallel currents or oblique currents over linear diffuser), and therefore the perpendicular current experiments provide a rigorous test for the ROMS model. Furthermore, Series 3 and 4 line plume experimental results exclusively are shown in the RSB height characterization figures [Roberts et al. (1989a), Figs. 10–12] and are the basis of the RSB curves and equations.

ROMS is a terrain-following-coordinate, split-explicit time-stepping oceanic model that solves the hydrostatic, free-surface primitive equations in a rotating environment and uses a K-profile parameterization (KPP) for turbulence closure (Large et al. 1994) with Boussinesq approximations (Shchepetkin and McWilliams 2003, 2005). The model solves the three-dimensional momentum (Reynolds-averaged Navier–Stokes paradigm), continuity, and tracer equations. The modeling system uses the Arakawa C-grid (Arakawa and Lamb 1977) for model state variables (Haidvogel et al. 2008). Separate barotropic and baroclinic modes are calculated using the split-explicit time-stepping method (Shchepetkin and McWilliams 2003, 2005). To discretize advection, ROMS utilizes the third-order upstream, or upwind, biased advection scheme by a Uniformly Third-Order Polynomial Interpolation Algorithm (UTOPIA)-like algorithm (Shchepetkin and McWilliams 1998, 2005). Horizontal tracer advection is constructed through this finite-volume method (Haidvogel et al. 2008). This scheme accounts for eddy diffusivity by a numerical hyperdiffusion related to the horizontal advection with an effective diffusivity coefficient that decreases with the grid scale (Uchiyama et al. 2014). Vertical tracer advection is handled by a conservative parabolic spline (Haidvogel et al. 2008).

A nonhydrostatic discretization is applied to the model (Guillaume et al. 2017). No explicit eddy diffusivity parameterizations are used; therefore, KPP is turned off. The necessary small-scale viscosity and diffusivity is purely the result of the diffusive discretization error of the upwind third-order advection schemes that are not strictly monotonic (Shchepetkin and McWilliams 1998, 2005). Limitations to this advection scheme were discussed and revised by Marchesiello et al. (2009) and Lemarié et al. (2012). However, several of the issues these schemes specifically improved upon were not applicable to this study because of the idealized conditions (i.e., flat bottom, linearly stratified, and no coastal boundaries). ROMS has many applications, and a majority of simulations are large-scale $O(10–100\mathrm{km})$ (Marchesiello et al. 2003; Penven et al. 2005). Recent ROMS simulations have moved from mesoscale to submesoscale (Gula et al. 2015; Molemaker et al. 2015; Dauhajre et al. 2017) and focused on local and coastal regions (Howard et al. 2014; Uchiyama et al. 2014).

Validation was conducted by nondimensionalizing the ROMS parameters of cross-flow velocity, $u$, buoyancy frequency, $N$, and buoyancy flux, $b$, into the Froude number relationship F [Eq. (4)] and length scale $l_b$ [Eq. (5)]. Typical effluent temperature, salinity, and ammonium were chosen as 26.9°C, 1.2 practical salinity units (PSU), and 1,500 mmol N ${\mathrm{m}}^{-3}$. A nonreactive passive tracer (${\mathrm{C}}_p$) was used to represent ammonium and measure dilution. Table 1 lists all ROMS simulations conducted.

A linearly stratified, or stably stratified, vertical profile was implemented uniformly in ROMS. The RSB experiments discharged negatively buoyant effluent into linearly density-stratified water created by filling their tank with saltwater. All RSB experiments used a buoyancy flux between $9.8\times {10}^{-5}$ and $1.1\times {10}^{-4}{\mathrm{m}}^3{\mathrm{s}}^{-3}$, and buoyancy frequency of $0.3{\mathrm{s}}^{-1}$. The linearly stratified density profile in ROMS was forced through both temperature and salinity profiles, and the buoyancy frequency was $0.011{\mathrm{s}}^{-1}$ for all experiments. ROMS buoyancy flux was $0.003{\mathrm{m}}^3{\mathrm{s}}^{-3}$, and $l_b$ was calculated from Eq. (5) as 13.09 m.

The flow speed for each experiment was calculated by selecting Froude numbers 0.1, 1, 10, and 100 and solving $u=(Fb{)}^{1/3}$. The corresponding velocities were 0.067, 0.144, 0.31, and $0.67\mathrm{m}{\mathrm{s}}^{-1}$, respectively, assuming that the buoyancy is calculated from Eq. (3) with ${\rho }_a=\mathrm{1,025.54}\mathrm{kg}{\mathrm{m}}^{-3}$. Typical ocean shelf flow velocities are between 0.05 and $0.30\mathrm{m}·{\mathrm{s}}^{-1}$ ($\sim F=0.1-10$).

ROMS experiments were run on a cluster in the UCLA Center for Earth Systems Research laboratory using 128–256 cores with message passing interface (MPI) parallelization. Each experiment was run for 11.75 h to give sufficient time for the region of interest to reach steady state. Time steps ranged between 0.3 and 7.5 s depending on cross-flow velocity. Total wall clock time for experiments was between 2 and 96 h. Output was saved every 15 min.

Results were normalized using the dimensional analyses presented by Roberts et al. (1989a, b); $F=0$ and 0.1 model runs were averaged over the last 1.25 h of the experiment, whereas $F=1$, 10, and 100 were averaged over the last 3.5 h for the diagnostics presented in the “Results” section. Lower Froude numbers require more time to reach a steady state in the region of interest. Model validation results also are averaged over the length of the pipe.

The ROMS domain was designed with realistic ocean length and height scales representative of wastewater pipe placement. Most experiments were computed using a $\mathrm{1,024}\times 512\times 64$ numerical grid. Unless specifically mentioned, the horizontal resolution used was 3 m in the horizontal directions and slightly less than 1 m in the vertical direction to resolve intermediate- and far-field dynamics. ROMS requires centimetric grid resolution to completely resolve immediate near-field mixing. The physical size of the computational domain was $3\mathrm{km}\times 1.5\mathrm{km}\times 60\mathrm{m}$. Additional experiments at 1- and 10-m horizontal resolutions were conducted for certain Froude numbers. Open boundary conditions were chosen with a zero-gradient eastern boundary ($\mathrm{d}u/\mathrm{d}x=0$) and 100-m-wide sponge layers on all boundaries to reduce reflections of internal waves at the open boundaries. The magnitude of the mixing parameter in the sponge layer was $0.1{\mathrm{m}}^2{\mathrm{s}}^{-1}$. Drag at the oceanic bottom was turned off to allow a uniform current from the ocean surface to bottom. A volume flux of nearly fresh water was implemented at the bottom of the vertical domain to mimic a bottom-mounted pipe. Because the objective was to compare directly with RSB laboratory experiments in which Earth’s rotation played no role, we set the Coriolis force equal to zero.

A constant effluent volume flux of $Q_p=10{\mathrm{m}}^3{\mathrm{s}}^{-1}$ was forced uniformly over an area 30 m wide and 900 m long in the 3- and 10-m-resolution experiments. Additional simulations with different forcing widths also were performed (Table 1). An ideal source width would parameterize initial momentum–dominated mixing by injecting effluent over multiple grid cells. The additional grid cells used to force the effluent discharge take advantage of the uniform mixing at the input to reach a dilution that is similar to a dilution after jet momentum–dominated near field mixing. A random input of volume flux on the order of ${10}^{-6}{\mathrm{m}}^3{\mathrm{s}}^{-1}$ was applied to each effluent input grid cell in all experiments to induce low-level noise that facilitates the occurrence of natural instabilities in the plume which otherwise could be repressed for an unrealistic period. The resolution and present configuration of the ROMS experiments were unable to represent the horizontal momentum of the diffuser ports. The 900-m-long pipe was chosen to represent a typical length of the diffuser portion of a marine outfall. Source flux calculated from Eq. (1) was $q=Q_p/L=1/90$. The 1-m-resolution experiments had shorter pipe length because the maximum length of the pipe with the $\mathrm{1,024}\times 512$ grid was 512 m. A 500-m-long pipe was chosen to prevent unwanted boundary interactions, and the volume flux was reduced proportionally to $5.56{\mathrm{m}}^3{\mathrm{s}}^{-1}$. The sponge layer width was reduced in the 1-m-resolution simulations to 33 m at $F=100$ and to 5 m at $F=0.1$. The 10-m simulation was run on a smaller grid of $256\times 128$ to capture the same area as the 1-m resolution for comparison and to decrease computing time. Vertical resolution was kept at 64 grid points in a 60-m vertical domain regardless of horizontal resolution.

For all cross-flow experiments, the tracer input was placed one-quarter of the distance downstream into the domain (i.e., 256 grid cells). In the case of the zero cross-flow experiment ($F=0$) the tracer is placed in the center of domain to to allow western and eastern plume spread. Fig. 1 shows a schematic of tracer input placement in the domain for zero cross-flow and cross-flow experiments.

Effluent source forcing was parameterized and mixed uniformly within the input grid cells. Incoming temperature, salinity, and passive tracer concentration are subject to grid size parameterization and become uniformly mixed within the grid cells that have effluent input. The immediate turbulent mixing that occurs when the effluent first exits the pipe, therefore, was not resolved here. Roberts et al. (1989a) noted that source momentum flux and port spacing do not significantly affect normalized dilution across all experiments and parameters tested. The individual plumes from each port merged to approximate a line source. Increased momentum flux caused decreased plume rise height, although dilution was nearly constant.

Effluent tracer source forcing was shaped by diffuser geometry and was dependent on resolution. Details of the implementation were presented by Uchiyama et al. (2014). Relevant equations are the nondimensional tracer concentration equation with equivalent source $P$ (1/s)withwhere $c$ = pollutant concentration normalized by input pollutant concentration $C_p$; and $\mathbf{F}$ advection-mixing flux associated with resolved flow and upwind advection scheme mentioned previously in the ROMS description. Pollutant species can be represented by multiplying $c$ fields by inflow concentration $C_p$. In Eq. (10), $A$ and $H$ are the spatial functions mimicking unresolved near field mixing above the diffusers, and have integrals equal to the source area and depthandwhere $A_s$ = horizontal area of diffuser; $H_s$ = vertical size; and $V_s=A_s{H}_s$ = volume. In Eq. (10), $P_s$ is determined by effluent volume flux and model source and is equal to $Q_p/V_s$, where $Q_p$ is volume flux (${\mathrm{m}}^3/\mathrm{s}$). All tracers were introduced to the domain at the bottom grid cell, i.e., $H_s=H(z)=1$ at $z=-60\mathrm{m}$. No shape function was fitted to H in order to mimic bottom-mounted pipes. Parameter $A=1$ in the horizontal grid cells of the diffuser pipe that tile the bottom of the domain to uniformly force effluent volume flux over pipe area. Parameter $A_s=N_{s}d{x}^2$, where $N_s$ is the number of cells that make up the diffuser, where $N_s=\mathrm{3,000}$ for $dx=3\mathrm{m}$, $N_s=\mathrm{15,000}$ for $dx=1\mathrm{m}$, and $N_s=270$ for $dx=10\mathrm{m}$. The $F=0.1$ experiments with different pipe widths at 3-m resolution had $N_s=900$ and 6,000, and the $F=100$ simulation with half the source width had $N_s=\mathrm{1,500}$.

Roberts et al. (1989a) observed two different flow regimes for buoyant plumes: forced entrainment and free plume. Clear forced entrainment was observed in simulations $F=10$ and 100, and free plumes were observed in $F=0$ and 0.1. Additionally, in the $F=1$ experiment, Roberts et al. (1989a) observed forced entrainment, free plume, and internal wave features. Model results were consistent with these observations.

Cross-pipe wastefields beginning at the first horizontal cell with tracer input with various Froude numbers and concentration gradients are shown in Fig. 2. The forced entrainment flow regime [Fig. 2(d)] was observed when current speed was high ($F\ge 10$) and the bottom of the wastefield remained at diffuser level. A buoyant plume flow pattern entraining oncoming flow could not be maintained and there was efficient mixing near the source (Cederwall 1971). These plumes exhibited evolving concentration fields and transient filaments with high concentration.

The $F=0$ simulation had the pipe in the middle of the domain and thus had less downstream distance from the pipe than the other experiments. The $F=0$ plume spread westward and eastward, but only one side of the plume is depicted. Stable stratification suppressed the vertical motion after sufficient dilution and prevents tracer from reaching the surface. At $F=0.1$, no lateral spreading upstream occurred and all effluent was advected downstream. As expected, thickness and plume rise height decreased with increasing F. The free plume pattern [Fig. 2(b)] occurred during low current speed and had normal plumelike characteristics, with the plume curved downstream and entrainment of the ambient flow. The $F=1$ simulation accurately exhibited both forced ($<750\mathrm{m}$) and free ($>750\mathrm{m}$) characteristics [Fig. 2(c)]. Froude number 1 had a strong signal of an internal wave between fluid layers called a lee wave [Fig. 2(c)]. Lee waves are generated from stably stratified flow over an obstacle, with the obstacle here consisting of the constant intrusion of a buoyant plume, and oscillate at the Brunt–Väisälä frequency (Marshall and Plumb 2008). Roberts et al. (1989a) also noted a pronounced internal wave in their $F=1$ experiments.

Tracer concentration decreased as current speed increased. At $F=0$, continuous entrainment of effluent discharge above the pipe caused lateral spreading of the plume, and some mixing occurred, but the core of the plume was relatively uniform in concentration; $F=0.1$ and 1 had lower concentrations from ambient flow forcing. Filaments of high concentration tracer moved through the plumes. The forced entrainment flow regime at $F>1$ had incidental holes with little to no effluent concentration. Portions of the plume with no tracer adjacent to high concentrations, most notably in Fig. 2(e) at $x=500\mathrm{m}$, were the result of an overshoot artifact in the presence of sharp gradients in concentration. Although these overshoots led to minor visual artifacts in which passive tracer concentrations temporarily were slightly negative, the overall implications were negligible. The tracer advection schemes were strictly conservative as a result of the control volume approach used (Shchepetkin and McWilliams 1998).

As Froude number increased, some distance was required before the plume was able to lift off the bottom of the domain, which was not observed in the RSB experiments. This distance is most noticeable in Fig. 2(e), where effluent is carried approximately 300 m downstream in the bottom cell before observable entrainment occurs. Consequently, nearly all of the tracer, and therefore the minimum dilution $S_m$, was trapped at the bottom of the domain, and distance for the plume to rise and develop was overestimated considerably. The plume was expected to rise off the bottom cell closer to the diffuser, which was realized in the $F=100$ 1-m-resolution simulation.

Table 2 lists quantitative comparisons of ROMS experiments to RSB validation data. Generally, minimum dilution was well predicted, whereas plume rise height, height to plume top, and consequently plume thickness were more variable. Top of plume height percentage divergence converged to approximately 20% for $F\ge 1$. The largest deviation between ROMS and RSB experiments was 42%, and the smallest was less than 1%.

Fig. 3 shows ROMS nondimensional dilution versus length results (solid line), plotted with RSB experimental data (symbols) for various Froude numbers. Roberts et al. (1989b) defined the established wastefield as the end of the initial mixing region $x_i$, which is the distance to the point at which the limiting value of dilution is obtained. The dashed line in Fig. 3 signifies the location of established wastefield of the RSB laboratory experiments given in Roberts et al. (1989b) as a function of F

Fig. 3 is used to determine the end of the initial mixing region. The established wastefield locations for $F=0,0.1,1,10$, and 100 were $x/l_b=12$, 12, 15, 32, and 82, respectively. These locations were used to represent the established wastefield for this metric and other metrics, $z_m,z_e$, and $h_e$, and are represented by the dash-dot-dot-dash line. Dilution increased as the effluent was advected in the down-pipe direction and was mixed by ambient forcing until a nearly constant dilution was reached. Generally, the minimum dilution was well modeled. Good agreement was observed at low Froude number flows ($\le 10$). However, at high velocities ($F=100$), the high-resolution ROMS underpredicted dilution at 3-m resolution. The 1-m resolution improved on this underprediction. Lower cross-flow experiments (Froude numbers 0.1 and 1) still were developing at the intersection with the RSB established wastefield estimate (dashed line). Dilution initially was overestimated for $F=0.1$, 1, and 10, and then converged to a steady state in the down-pipe direction. Notably, some experimental data (i.e., $F=1$) suggested that this higher dilution near the pipe may be occurring. It is unclear if higher dilution is physical or an artifact of the parameterized effluent forcing or limited resolution. For Froude numbers 10 and 100, the distance required before the plume became established was longer and did not intersect the RSB established wastefield estimate. Contrary to Roberts et al. (1989a), dilution was affected by current, and dilution at $F=0.1$ was not equal to the dilution at $F=0$.

Fig. 4 shows normalized minimum dilution in the established wastefield as a function of Froude number. Minimum dilution values were taken at the $x/l_b$ values listed previously. Lower dilution was observed for $F=100$ and higher dilution was observed for low F than that of the experimental data. The scale of the $x$-axis of F on all graphs is linear from 0 to ${10}^{-2}$ and logarithmic above ${10}^{-2}$. Symbols represent RSB experimental data for minimum dilution at the established wastefield (Roberts et al. 1989c). The dashed curve in Fig. 4 plots dilution data fit as a function of Froude number presented by Roberts et al. (1989a)and

Generally, model results and experimental data were in good agreement. Minimum dilutions for Froude numbers 1 and 10 coincided with experimental results. The RSB data fit curve [Eq. (15)] overestimated the $F=10$ data points. Model results indicated lower dilution with high cross-flow ($F=100$) and slightly higher minimum plume dilution at lower Froude number (0.1).

Vertical plume characteristics of rise height to minimum dilution, $z_m$, top of the plume, $z_e$, and thickness, $h_e$, at the end of the initial mixing region were found for all experiments. Roberts et al. (1989a) used 5% of the maximum concentration to delineate the plume top ($z_e$) and thickness ($h_e$). The same method was employed here.

Fig. 5 shows the normalized height of minimum dilution (or maximum concentration) in the established wastefield. The RSB empirical equations plotted areand[Roberts et al. (1989a), Eqs. (16c) and (18), respectively]. Similar to minimum dilution, ROMS overpredicted the rise height of minimum dilution at low Froude numbers and underpredicted it at high Froude numbers. Minimum dilution height at the end of the intial mixing region changed by 6% between $F=0$ and 0.1. Between $F=0$ and 0.1 in the RSB experiments, the change was 11% and 15% for the two data points. The height of the Froude number 0 case between ROMS and RSB was overestimated by 20% and 25%. Although minimum dilutions were well predicted for $F=1$ (Figs. 3 and 4), the height to the minimum dilution was underestimated by 15% for the lowest data point. In addition, $F=10$ closely matched experimental values, whereas $F=100$ deviated significantly from RSB experimental results (Table 2).

Normalized height to top of the plume is shown in Fig. 6, and normalized plume thickness is shown in Fig. 7. RSB estimates [Roberts et al. (1989a), Eq. (16) and (17)] for these characteristics are given byandand are shown in Figs. 6 and 7, respectively. ROMS simulation height to top of the plume ($z_e$) was consistent with experimental data for $F=0.1$. For higher cross-flows ($F>0.1$), ROMS deviated by 20% or more, whereas overprediction loccurred for no cross-flow (Table 2).

RSB experimental results for plume thickness, $h_e$, were dependent on the Froude number as $h_e$ decreased at $F\ge 10$. Except for $F=0.1$, the ROMS model results for the plume thickness were consistent with the RSB experimental results in their dependence on $h_e$. ROMS underpredicted $h_e$ for $F\ge 10$ compared with the RSB experimental results, which likely was a result of resolution; there was limited convergence with the $F=100$ 1-m experiment studied in the next section.

Despite good agreement of $z_e$ for $F=0.1$, the difference between experimental and model $h_e$ was significant (Table 2), indicating that the bottom of the plume was represented poorly. Fig. 2(b) shows a thick plume of about 35 m between 100 and 700 m from the pipe, and decreasing plume thickness farther from the pipe. The velocity fields indicate a current reversal under the $F=0.1$ plume with converging flow from the northern and southern open boundaries of the domain. The reasons for this current reversal and the likely effect on the plume dilution and thickness are explored in the “Discussion” section.

In ROMS, $F=1$ underpredicted the top and minimum dilution of the plume but was close to the laboratory experiments for plume thickness. This indicates that $z_m$ and $z_e$ were underestimated by similar heights and gave nearly the expected plume thickness. The $z_e$ and $h_e$ were virtually the same as each other for $F=10$ and 100 due to the forced entrainment plume regime.

Additional simulations and analyses considered the differences between physical experiments and model results along with model resolution and parameter sensitivities. These investigations included quantifying top of the plume height variability, the impacts of effluent source pipe resolution, and grid resolution. Figs. 8–10 summarize the supplementary simulations and analyses for the parameters top of the plume height, minimum dilution, and dilution as a function of distance from the pipe, respectively.

The top of the plume, $z_e$, is of particular interest. This parameter determines whether the plume reaches a critical height in the water column, such as the mixed layer or euphotic zone. Model results were averaged over both over time and along pipe. Additionally, $z_e$ and $h_e$ were based on 5% of the maximum concentration at the established wastefield as calculated by Roberts et al. (1989a). The variability of $z_e$ for all Froude numbers was calculated across time and along the pipe to determine whether the experimental values were within this range. The standard deviations were calculated from all height values of $z_e$ along the length of the pipe at the established wastefield for five saved outputs, equaling 1.25 h; e.g., the heights to the top of the plume were found at the distance from the pipe that was the established wastefield for every grid cell along the pipe i.e., 300 cells, for 5 saved outputs, for a total of 1,500 $z_e$ values used to calculate standard deviations. The $z_e$ at the 1% maximum concentration level also was calculated.

Fig. 8 shows two standard deviations of $z_e$ over time and distance along the pipe at 1% and 5% of the maximum concentration at the established wastefield. The $z_e$ RSB experimental values of $F\ge 0.1$ were within the variability across time and space of the ROMS experiments at the 5% line. The 1% line has a higher $z_e$ across all F numbers and further increases the range of variability. The no-cross-flow ($F=0$) $z_e$ value was greater than the RSB experimental $z_e$ values. The ROMS experiment was run for multiple hours, and the continuous buoyant influx likely caused higher plume rise by erosion of the background stratification. This zero-current-speed scenario is unlikely to occur in realistic environments. Calculating $z_e$ as 1% of the minimum dilution may be more appropriate because any amount of effluent reaching higher in the water column could affect ocean chemistry and biology in that zone.

Various effluent source widths were tested. Effluent source widths at $F=0.1$ (a typical flow velocity) were tested at 9 and 60 m (3 and 20 grid points, respectively) while input pipe length was held constant at 300 grid points (denoted $3\times 300$ and $20\times 300$). Additionally, a 15-m-wide pipe ($5\times 300$) was tested at a high cross-flow velocity, $F=100$. The established wastefield locations of these $F=0.1$ and 100 experiments were chosen to be the same as those in the previously presented experiments, $x/l_b=12$ and 82, respectively.

Roberts et al. (1989c) stated that changing jet momentum flux ($u_j$) primarily affects rise height and thickness, whereas dilution remains similar. Changing source pipe width in ROMS changed initial input mixing and is analogous to changing $u_j$. ROMS similarly showed that changing source pipe width primarily affected rise height, whereas dilution generally was insensitive to pipe width. Fig. 8 shows established wastefield $z_e$ with changing source widths. As expected, the 3 grid cell (9-m-wide) pipe had higher $z_e$ because of less initial mixing, and therefore greater buoyancy force.

Fig. 9 shows the minimum dilutions at the established wastefield for the source width experiments and the resolution experiments. At typical current speeds ($F=0.1$), the minimum dilution reached was insensitive to the increased resolution or source pipe width. Fig. 10 shows the minimum dilutions at down-pipe distances for the additional experiments. A prominent spike in dilution above the pipe occurred in both width extremes for $F=0.1$. The effluent was more diluted above both of these pipe widths, potentially for two different reasons. The 60-m-wide pipe had an initial input effluent spread in 2 times volume of that in the 30-m-wide pipe, and therefore had a spike in dilution closest to the pipe compared with the other experiments. The effluent from the 9-m-wide pipe was subject to greater mixing from the increased buoyancy force. Across all parameters, the 15-m-wide pipe at $F=100$ had no differences from the validation results. At high Froude numbers, the cross-flow dominated plume dilution and height characteristics regardless of pipe width.

The 3-m-resolution ROMS model agreed reasonably well with the RSB experiments. However, the potential impacts of grid resolution are explored with additional simulations. The effect of increasing the resolution to 1 m for $F=0.1$ and 100 was quantified with a 30-m-wide pipe. A coarser grid resolution of 10 m for $F=100$ also was considered. The 1- and 10-m locations of the established wastefield for $F=100$ are chosen to be at the end of their respective domains. The location of the established wastefield for $F=0.1$ at 1-m resolution was the same as that in the other $F=0.1$ experiments. Fig. 9 shows the minimum dilution for the $F=1$ and 100 experiments at different resolutions. The 1-m-resolution results had an overall higher dilution and top of the plume for both Froude numbers. Higher resolution substantially improved the model data comparison at the highest cross-flow velocity ($F=100$) because more mixing was resolved. The 1-m $F=100$ experiment (Fig. 10) increased dilution in less distance than the other $F=100$ experiments due to better resolved mixing near the effluent input. The 10-m resolution $F=100$ simulation began mixing the input effluent at a much greater distance from the pipe compared with the other $\mathit{F}=100$ experiments. The coarse resolution did not allow the plume to reach an established wastefield in the current domain. Despite meaningful differences in dilution for $F=100$ resolution experiments, the differences in plume top rise heights in Fig. 8 for these experiments were negligible. The 1-m $F=0.1$ simulation had a similar pattern as the other $F=0.1$ experiments in dilution as the plume moved away from the pipe (Fig. 10), but the initial overdilution amplitude near the pipe was less pronounced. The dilution remained relatively close to the value of that above the pipe.