A sequence of vessel-based observations within Deep Cove were repeated over the course of the field campaign. Along-channel transects were aligned with the main river discharge, lateral transects cut through the plume perpendicular to its trajectory, and oblique (‘zig-zag’) transects captured both lateral and longitudinal plume evolution (Figure 1C). At least six consecutive lateral transects were repeated during each sampling period (Figure 2). Horizontal velocity estimates were obtained from a vessel-mounted ADCP (RDI Workhorse, 600 kHz) which sampled water velocities in 1 m vertical bins from 2.5 to 41.4 m. Near-surface velocities were obtained by applying a linear fit to the velocity data to extrapolate from 2.5 m to the surface. The extrapolated velocity profiles had compared well to in-situ near-surface velocity measurements from previous field campaigns (McPherson et al., 2019), and were in good agreement with surface plume velocities estimated by the Lagrangian GPS drifters. Currents were rotated according to the local bathymetry to determine along-channel (u) and across-channel (v) velocities. A weighted 10 m ‘bowchain’ was attached to the vessel which comprised temperature (RBRsolo) and CTD loggers (RBRconcerto) sampling at 2 and 5 Hz, respectively. High-resolution profiles of practical salinity and temperature were obtained from continuously profiling 'tow-yoed' CTD loggers (RBRconcerto). These data enabled estimation of the buoyancy frequency from the measured density profiles,

where ρ is the potential density.

The lateral transects of temperature from the bowchain were used to quantify vertical (K_z) and lateral (K_y) diffusion by estimating the change in thickness and width of the plume over distance, respectively. While salinity is generally used to identify and track freshwater plumes in coastal systems (Hetland, 2005), the persistent low-salinity surface layer observed throughout Deep Cove and the wider fjord region, resulting from the tailrace inflow and high rainfall (Gibbs, 2001), makes the distinction between the buoyant plume and freshwater surface layer less pronounced in the salinity field than in temperature (Figures 3C–E). Thus, temperature was used to define the plume boundaries in this study (Figures 3A,B). The diffusion components were therefore estimated by,

where t_t is the time taken for the plume to flow between each transect, y_p is the width of the plume at its base and z_p is the plume depth. The base of the plume is defined as the depth of maximum stratification. The depth of the plume is then defined as the distance from the base of the plume to the maximum height of the bounding isotherm.

Lagrangian measurements of near-surface currents were made during six GPS drifter experiments. The plume discharge rate and wind speeds for each experiment are detailed in Table 1, and wind direction was consistently up-fjord due to the surrounding steep topography. The headwaters of the fjord absorb the momentum of tidal oscillations (O'Callaghan and Stevens, 2015) and tides do not influence near-field mixing here (McPherson et al., 2019), thus the tidal impact on drifter trajectories is not considered.

The drifters each had a cylindrical drogue of height 0.5 m and diameter 0.2 m, and were ballasted to measure the upper 0.5 m of the water column by a small spherical float. Wind slippage was minimal as the float had little exposure to wind above the surface water level. Each drifter was equipped with a GPS receiver (Columbia V-900 GPS data logger) which recorded every 1 s. The GPS devices have a position accuracy up to 1.5 m, depending on satellite coverage. The drifters were released approximately 10 m apart across the width of the tailrace discharge point and were recovered after ~ 1 h. A total of 8 drifters were deployed in the first two experiments then, due to the loss of a GPS receiver, 7 drifters were deployed and retrieved in the subsequent four experiments.

The drifters were used to quantify the plume lateral spreading rate by estimating the change in plume width with distance from the discharge point, following a similar approach applied by Chen et al. (2009). The normalized plume width is given by W/W₀, where W is the plume width at a given location and W₀ is the width of the plume at a reference location. For drifter data, W is evaluated as the standard deviation of the distances between all drifters in the cross-plume direction at the given location. The plume width is then normalized by W₀. For each GPS drifter deployment, W₀ is taken as the first estimate of plume width (W) closest to the tailrace discharge point. This choice of W₀ enables W/W₀ for each drifter deployment to be compared, despite the variability in the deployment locations of the drifters in each experiment. For the lateral transects, W₀ = 100 m, which is the measured distance of the width of the tailrace discharge point (McPherson et al., 2020). Here, W is calculated along each drifter track at intervals of 10 m from the reference point. The uncertainty limits are the minimum and maximum standard deviations of all the subsets of the deployed drifters at each 10 m interval. The drifter method of estimating plume spreading is independent of drifter speed as plume width is evaluated as the distance between drifter tracks at a specified point from the reference point, and not at a given time.

Lateral dispersion is then quantified from the continuous convergence and divergence of drifters from the center of the plume. The standard deviation of the distance between the drifters and the plume centerline (σ_r), defined as the average drifter track, was used to calculate the diffusion coefficient,

where t_d is the diffusion time (i.e., the time elapsed since the individual drifter deployment) and l = 3σ_r is the scale of diffusion (Okubo, 1971). Horizontal plume spreading is generally anisotropic thus σr=(σx2+σy2), where σ_x, σ_y denote the standard deviations in the along and across-channel directions, respectively (Stocker and Imberger, 2003).

Contributions to results in section 5.3.1 were from three near-surface moorings deployed in September 2015 (Figure 1C). Further details about this field campaign can be found in McPherson et al. (2019). Velocity measurements were obtained from an upwards-facing Acoustic Doppler Current Profiler at 10 m (ADCP; RDI Workhorse, 600 kHz), set to record an ensemble every 3 min in 2 m vertical bins. Spectral analysis was conducted on the velocity observations (Emery and Thomson, 2001) in which the time series was split into half-overlapping intervals equivalent to the inertial frequency, and the spectrum was computed using Welch's periodogram method.

Plume lateral spreading rates can also be quantified using the control volume method, where measured quantities are connected to plume dynamics over a defined finite region of the flow field, termed a control volume. Freshwater conservation is applied to estimate plume width (b) in the control volume region (MacDonald and Geyer, 2004; Chen et al., 2009). Details of the control volume over the near-field plume region in Deep Cove can be found in McPherson et al. (2020). Horizontal scale-dependent dispersion is then determined from the growth of b with distance from the tailrace discharge point,

where β = 12α/(Ub₀) represents the magnitude of dispersion using the mean along-channel velocity U, and b₀ is the initial plume width (Fong and Stacey, 2003). A non-linear least squares fit of the control volume estimates of b to Equation (5) is taken and both β and n are treated as adjustable parameters. By optimising for n and α using this fit, these parameters can be compared to the empirical estimates from Equation (4). The method has been used to estimate horizontal dispersion in both near-coastal systems and the open ocean (Stacey et al., 2000; Fong and Stacey, 2003; Jones et al., 2008; Moniz et al., 2014).

Observations from the bowchain and tow-yoed CTD showed a highly-stratified upper water column with a 2 m thick freshwater (σ_t ≈ 1 kg m⁻³) surface layer overlaying a sharp density interface, and a dense, oceanic ambient (σ_t = 24 kg m⁻³) below 5 m (Figures 3B–E). The zig-zag temperature transects illustrate the evolving structure and path of the buoyant plume within the surface layer (Figure 3A). The 3 m thick plume was observed in the surface layer as a core of water approximately 1° C warmer than the 13.6°C ambient surface layer (Figures 3A–C). The plume boundary can thus be defined by an isotherm of 14.5°C. Maximum plume temperatures were found at the base of the surface layer, toward the core of the plume (~ 14.9°C) where warmer water was entrained from the ambient below (15.5°C) (Figure 3C). The plume was confined to the surface layer by strong salinity-induced stratification (N² = 10⁻¹ s⁻²) in the pycnocline (Figure 3F). Generally weaker values were observed within the plume layer (N² = 10⁻² s⁻²) and reduced toward zero with depth below the interface to 10 m.

The velocity structure of the near-field region was characterised by a fast-flowing surface plume with speeds over 1.5 ms⁻¹, which overlay a relatively stationary ambient below 5 m (Figure 3G). The ambient surface layer currents were weak (< 0.1 ms⁻¹), thus the outer plume boundary can also be defined by speeds of 0.2 ms⁻¹. The near-surface velocity field shows the plume decelerates as it propagates downstream. Maximum surface velocities at the plume centerline decreased from 1.05 ms⁻¹ to 0.2 km downstream of the tailrace discharge point to 0.81 and 0.65 ms⁻¹ at 0.5 and 1.5 km downstream, respectively, (Figure 4A). Flow speeds tended toward zero with lateral distance from the plume centerline, into the surrounding surface ambient. The near-surface plume measurements compared well to the velocities derived from the GPS drifters, estimated as the time derivative of the coordinate position. The drifters initially moved at speeds of ~ 1.7 ms⁻¹ near the tailrace discharge point and decreased to approximately 0.6 ms⁻¹ toward the seaward end of Deep Cove (Figure 5). Maximum speeds of over 2 ms⁻¹ were recorded near the discharge point.

The rate of plume lateral spreading was quantified using two different observational methods; Lagrangian GPS drifters which measured near-surface plume velocities directly (Figure 6) and lateral transects of temperature and velocity fields (Figures 4, 7). These directly observed results can then be compared to estimates of plume width determined by a control volume method. This control volume technique used hydrographic observations from along and across-fjord transects, and is detailed in McPherson et al. (2020).

Estimates of plume width derived from the across-channel hydrographic transects generally compared well to the estimates derived from the control volume method (Figure 9). The control volume results show that W/W₀ increased from 1 close to the tailrace discharge point to W/W₀ = 1.8 over 3 km downstream, indicating that the plume spread laterally in the near-field region as it propagated seaward. The estimates of W/W₀ derived from the repeated lateral transects of temperature and velocity, using 14.5 ° C and 0.2 ms⁻¹ as the definitions of the plume boundary, respectively, correspond well at the three downstream locations. They all show a consistent increase in W/W₀ over the along-channel distance and are in good agreement with the control volume plume width estimates. The estimates of W/W₀ derived from the average GPS drifter trajectory (Figure 6) are generally smaller than the other observed values, suggesting an underestimation of plume width using this method.

While a general increase in W/W₀ over the length of Deep Cove was observed in both the control volume and transect data, the variability in the measurements highlights an evolving along-channel lateral spreading rate (db/dx). Over the total 3 km near-field region, db/dx = 0.045 from the control volume estimates. However, this rate increases and decreases over different sections of Deep Cove. All methods show an agreement in the evolving pattern of db/dx in the initial 1.5 km, with weaker horizontal spreading rates in the first 0.5 km and an increase toward 1.5 km. Further downstream, there is a slowing of plume spreading toward the end of Deep Cove. The highest lateral spreading rates are generally derived from between the lateral transects of temperature and velocity, with db/dx ≈ 0.05 between 0.2 and 0.5 km increasing to between 0.04 and 0.09 between 0.5 and 1.5 km. Estimates of db/dx from the control volume method are at the lower range over the same intervals, peaking at 0.05 at 1.5 km. While the corresponding horizontal spreading rate from the GPS drifters also shows this evolution of db/dx, with an initial weaker db/dx and an increased rate between 0.5 and 1.5 km, the estimates of db/dx remain one order of magnitude smaller than the control volume estimates over the length of the track. The good agreement between the estimates of plume width and db/dx from the control volume and lateral vessel transects (Figure 9) indicates that either the drifters underestimated the plume width, or in fact quantified a different process that was locked to the near-surface.

This change in db/dx in space is not surprising as the spreading rate is determined by the initial momentum from the variable plume inflow (Figure 2), the density difference between the freshwater surface layer and ambient water below, and the degree of vertical mixing at the interface (Chen et al., 2009). Thus the effect of the enhanced shear-driven mixing observed at the base of the plume (McPherson et al., 2019) which weakens the density gradient at the interface as the plume propagates downstream (Figure 7) would in turn reduce the lateral spreading rate.

The discrepancy between the results can be examined by considering what the GPS drifters actually measured in this field setting. While estimates of lateral spreading from drifters in other river plume systems have generally compared well to results from numerical and other observational techniques (Hetland and MacDonald, 2008; Chen et al., 2009; McCabe et al., 2009), drifters are also susceptible to meteorological and other oceanographic factors. A drifter consisting solely of a surface float with no drogue provides velocities and a trajectory that are a combination of surface advection, Stokes drift (an extra wave-induced force at the water surface) and direct wind forcing (Lumpkin et al., 2017). However, the impact of these external forcings on the GPS drifters used in the experiments conducted in Deep Cove were minimized by their design. The effects of wind on the trajectories of the drifters in Deep Cove was reduced by having very little of the drifter visible above the water, by conducting the GPS drifter experiments when wind speeds were low (Table 1), and by including a ballast centered at 0.5 m beneath the surface. The ballast also ensured the drifter was largely unaffected by the motion due to drift beneath the surface-wave-driven Stokes layer.

In Deep Cove, the drifters tended to remain clustered together within the body of the plume with no evidence of horizontal spreading (Figure 5). Over the initial 1 km, all drifter trajectories tended to show a decrease in plume width, before increasing further again downstream (Figure 6). The high flow speeds measured by the drifters (Figure 5) indicate the drifters remained within the main flow. The clustering of the drifters near the plume centerline is likely caused by a convergence of surface flow that concentrated the drifters in regions of high velocity within the center of the plume (Hetland and MacDonald, 2008).

The density and pressure gradients in the surface layer suggest this convergence of surface water occurred in the initial 1 km. Density and pressure in the surface layer are greatest within the center of the plume (Figure 3E) due to the enhanced entrainment of high density ambient water into the low density surface plume, driven by strong vertical mixing (McPherson et al., 2019). This vertical mixing creates a horizontal density gradient at the base of the plume. The corresponding horizontal baroclinic pressure gradient component is therefore greatest at 3 m below the surface, and peaked on either side of the plume centreline at both 0.25 and 1.5 km downstream of the inflow (Figure 10). The baroclinic pressure gradients tends toward zero at the surface. The barotropic pressure gradient also displays corresponding peaks at eiher side of the plume centerline, though the peaks of either pressure component are of an opposite sign.

By combining the across-channel baroclinic and barotropic pressure gradients, the total across-channel pressure gradient determines the direction and magnitude of the near-surface flow (Figure 11). At the surface, the total pressure gradient is dominated by the barotropic component, as the baroclinic pressure gradient is weakest at 0.5 m (Figure 11A). The positive total pressure gradient to the left of the plume centerline and the negative to the right indicates a strong flow converging toward the center of the plume. With depth, the baroclinic component increases and begins to influence the total pressure gradient, thus the direction of near-surface flow. A lateral diverging flow from the centre of the plume is already apparent at 2 m, while there is still exists a stronger converging flow outside 50 m (Figure 11C). At 3 m, the baroclinic component dominates the total pressure gradient (Figure 11D). This results in a strong lateral flow of plume water away from the centerline, driving the lateral spreading of the plume. The lateral velocity gradient (dv/dy) also illustrates diverging velocities at 2.5 m below the surface, peaking at the edges of the plume (Figure 4C).

This balance of flow convergence toward the plume centerline at the near-surface and divergence from the centerline at the plume base can also be demonstrated in the Gaussian shape of the near-field plume in the lateral transects of temperature (Figure 7). The wider base illustrates the lateral spreading of the plume driven by the baroclinic pressure gradient, while the narrowing toward the water surface indicates the near-surface convergence.

Further downstream at 1.5 km, the role of the pressure gradient components are reversed (Figure 10B). The barotropic component drives a near-surface divergence of flow while the corresponding baroclinic pressure gradient shows a convergence of flow, the strength of which increases with depth. The barotropic pressure gradient tends to dominate throughout the surface layer which drives a total lateral spreading of the plume. This near-surface divergence of flow from the plume centerline is also reflected in the GPS drifter trajectories which show an increase of plume width from 1 to 1.5 km. The rate of plume spreading over this distance derived from the GPS drifters compared relatively well to db/dx from the control volume estimates (Figure 9). Toward the end of Deep Cove, the drifters measured a decrease in plume width (Figure 9) which suggests another reversal of the lateral pressure components and a convergence of near-surface flow.

These horizontal density, pressure and velocity gradients (Figures 3B, 4, 10, 11), and the structure of the plume (Figure 7), suggests that the GPS drifters, drogued to the upper 0.5 km, were concentrated within the plume core over the intial 1 km by near-surface convergence, driven by the barotropic pressure gradient. The clustering of drifters about the plume centerline and within the main flow (Figure 4) meant that both the plume width and lateral spreading rate were not accurately measured, but also that the drifters were then dominated by the strong along-channel advection, which was a dominant component of the plume dynamics along the whole 3 km near-field region (McPherson et al., 2020). This resulted in an underestimation of the overall lateral spreading rate by the drifters in comparison to the other observational methods (Figure 9). Further downstream, the barotropic pressure gradient acted to diverge the near-surface flow away from the centerline which produced a comparable lateral spreading rate to the other observational methods employed, before converging again toward the end of Deep Cove.

While the drifters converged toward the plume centerline at the near-surface and tended to underestimate the plume width, the results from the hydrographic transects and control volume also suggest a slower lateral spreading rate than typically found in near-field settings. The coastal river plumes conventionally studied generally show radial expansion and splaying streamlines (Hetland and MacDonald, 2008; McCabe et al., 2009; Kakoulaki et al., 2020), and adhere to a horizontal spreading rate for a two-layer gravity current of (g′H)/2 (Farmer et al., 2002). In the near-field region of Deep Cove, where g′ is calculated using an across-channel Δσ_t from Figure 3E, this equates to a bulk horizontal spreading rate of ~ 0.19 ms⁻¹. With a mean along-channel plume velocity of 0.8 ms⁻¹, the plume would theoretically spread by ~ 450 m over the 3 km near-field region (db/dx = 0.15). However, the observational results did not agree with these estimates. The plume width increased from 105 to 240 m over the length of Deep Cove, with a total spreading rate that was slower than theoretical rate by one order of magnitude (db/dx = 0.045 m⁻¹, Figure 9).

The difference between these lateral spreading rates could be attributed in part to the intense vertical mixing at the base of the plume in Deep Cove. Enhanced turbulent mixing has been shown to drive interfacial stress which is a dominant force acting to decelerate the near-field plume (Chen et al., 2009; McCabe et al., 2009; McPherson et al., 2020). The deceleration is controlled by entraining low-momentum ambient water into the surface layer, simultaneously mixing high-density ambient water into the freshwater surface layer, thus reducing the density gradient and slowing lateral spreading (Hetland, 2010). In comparison to typical estimates of ϵ in coastal environments, the measurements observed in the near-field region of Deep Cove are orders of magnitude greater. The highly stratified interface (N² = 10⁻¹ s⁻², Figure 3F) supports the intense shear generated by the high flow speeds (u > 1.5 ms⁻¹, Figure 3G), resulting in surface-intensified turbulence (ϵ > 10⁻³ W kg⁻¹) (McPherson et al., 2019). This corresponds to enhanced surface-intensified turbulence stress, over one order of magnitude greater than the peak stress in the Columbia River (Kilcher et al., 2012).

Comparing the balance of terms in a near-field momentum budget between Deep Cove and the Columbia River shows that the enhanced vertical mixing in Deep Cove is likely linked to the reduced corresponding lateral spreading rate. In the Columbia River, where internal stress was weaker, the role of lateral spreading in balancing the total momentum budget was much greater than in Deep Cove. The spreading term balanced the acceleration terms in the surface layer, and the internal stress divergence drove the deceleration. This translates to a greater lateral spreading rate in the Columbia River, db/dx = 0.16 (Kilcher et al., 2012), which was one order of magnitude larger than in Deep Cove. In Deep Cove however, where turbulence stress was high, the spreading term was negligible and the internal stress similarly controlled the plume deceleration. The balance of dynamics in each near-field setting suggests that the increase in internal stress reduces the spreading rate. When the internal stress increases relative to the other near-field processes, the enhanced turbulence at the interface drives a reduction in the density difference between the ambient and surface and therefore a reduced lateral spreading rate. Thus, the high ϵ in Deep Cove could be responsible for the slower horizontal spreading than typically observed for near-field plumes with weaker turbulent mixing.

While the enhanced turbulent mixing at the base of the plume in the near-field is likely to reduce the rate of lateral plume spreading, the surrounding topography and its impact on plume evolution is also relevant in this system. The trajectories of the GPS drifters highlight the proximity of the fjord sidewalls to the mean plume flow (Figure 5). While the role of the sidewalls on the lateral plume spreading rate cannot be quantified, they are able to explain the circulation pattern in the surface layer by driving the barotropic pressure gradient and convergence of near-surface flow. The plume spreads laterally, from high to low pressure, as it propagates downstream (Figure 10). However, the sidewalls act as a physical barrier which prevents the continued and uninhibited lateral spreading of the plume. As the strong vertical density gradients (Figures 3E,F) also prohibit the downwelling of the surface ambient water, the surface water is then pushed up against the fjord sidewalls and increases the water height there. This creates a barotropic pressure gradient across the width of Deep Cove which drives the water back toward the center of the fjord and the plume. However, as the baroclinic pressure gradient is zero at the surface but greater than the barotropic component at 3 m (Figure 11), the water converges at the surface but is forced to diverge laterally at the base of the plume.

However, a uniform convergence of near-surface flow along the fjord was not observed. The lateral pressure gradient components showed a transition from near-surface convergence in the initial 0.25 km to divergence further downstream at 1.5 km (Figure 10) which was reflected in the trajectories of the GPS drifters as they propagated downstream (Figure 6). This along-channel change in near-surface flow could also be attributed to the sidewalls driving a form of oscillatory motion, such as a seiche, which propagates throughout the fjord basin. The initial expansion of the plume as it enters the fjord, no longer confined by the tailrace channel, forces the surrounding ambient water toward the fjord sidewalls. The barotropic pressure gradient is formed as described above and drives the near-surface convergence. However, this returning flow toward the center of plume could also be an oscillation which continues to propagate downstream and move laterally, impacting the barotropic pressure gradient and thus the drivers of plume spreading, reflected in the trajectories of the GPS drifters. However, the current spatial and temporal resolution of the data is unable to fully resolve any basin-wide oscillatory motion and its impact on the plume. The change in the fjord topography also impacts the plume spreading. The fjord is wider between 0.25 and 1 km as a bay exists to the south of the inflow (Figure 1C), before narrowing further downstream. As described above, the sidewalls restrict lateral spreading of both the plume and the surface ambient, thus any narrowing or widening of the sidewalls relative to the location of the plume would also affect the lateral pressure gradients and spreading.

There have been no observational field studies of the spreading dynamics of topographically-constrained plumes; river plumes typically studied generally exhibit a discharge perpendicular to the coast with no lateral boundaries (Yankovsky, 2000; Hetland and MacDonald, 2008; Chen et al., 2009; Kakoulaki et al., 2020). However, a change in drifter patterns was observed when GPS and modelled drifters, deployed near the mouth of a weakly-stratified tidal inlet, left the constrained tidal channel and reached the mouth of the inlet (Spydell et al., 2015). Within the channel, the drifters were generally retained in the main flow while, upon exiting the inlet, the lateral spreading rates increased. While a decrease in flow velocities with downstream distance was observed, which could balance the increase in horizontal spreading, the impact of the channel walls on the drifter trajectories and spreading rate were not considered. It is not unlikely that the same barotropic pressure gradient, from the increased sea surface height at the sidewalls of the channel, produced a convergence of flow which contributed to the drifters clustering together in the main flow. Though the fjord sidewalls appear to influence the near-surface circulation in Deep Cove, the relative roles of the enhanced turbulence and topography on the reduced lateral spreading rate relative to other plume systems remains unclear.

While advection governs the evolution and transport of the along-channel flow (Kilcher et al., 2012; McPherson et al., 2020), diffusion processes determine the across-channel transport in the near-field. Thus the processes responsible for the observed lateral spreading of the plume (Figure 9) can be examined by quantifying horizontal dispersion. A number of theoretical models describe dispersion by defining different drivers. The plume growth is characterised by an exponent of n = 4/3 which is the value expected for three-dimensional turbulence (Fong and Stacey, 2003). Turbulence theory postulates that dispersion depends on the length scale of the motions and rate of turbulent dissipation (Batchelor, 1952), while shear-flow theory considers vertical diffusion in a horizontally sheared flow, driven by turbulence, as the governing process (Taylor, 1953; Fischer et al., 1979). Both theories are capable of producing a 4/3 power law, suggestive of scale-dependent growth (Stommel, 1949; Batchelor, 1950), and both are assessed here, to determine which is capable of representing the observed lateral spreading.

The scale-dependence of the observed lateral dispersion must be first examined to determine if the n = 4/3 power law is met, thus if these models are suitable. The slope of the best-fit of the lateral diffusion estimates, n, defines the scale-dependence of the dispersion. The diffusion diagram is first examined, which combines estimates of diffusion from the GPS drifters and hydrographic transects (Figure 8). Based on scale, the estimates of lateral diffusion observed here are comparable to other plume inflows (Yankovsky, 2000; Hunt et al., 2010). The line of best-fit applied to these estimates yielded n = 1.38 ± 0.23. This observed n is not inconsistent with, and indeed compares relatively well to, the theoretical n = 4/3 which suggests scale-dependent behaviour following the 4/3 law.

Furthermore, the plume width model (Equation 5), using estimates of b derived from the control volume, can also be used to quantify the plume's scale-dependent lateral dispersion. A non-linear least-squares fit of b is applied to Equation (5), setting b₀ = 128 m and U = 1.1 ms⁻¹, and optimising for n and β, gives a best-fit of n = 1.39 ± 0.35 and β = 0.024 ± 0.005 (Figure 12). Most of the plume width estimates fall within the 95% confidence intervals on either side of the predicted values given the optimised parameters. The close agreement of n between the diffusion diagram and plume width model results indicate that, within statistical certainty, n = 4/3, and independently supports the empirical dispersion coefficient from the diffusion diagram. Thus, both turbulence and shear-driven theoretical models can be examined in order to determine the drivers of lateral dispersion.