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  • View in gallery

    Regional map of Monterey Bay, California, with inset of the study region showing the mooring cross-array oriented with the ~20 m isobath. Tidal ellipses are shown for C0, with surface and bottom ellipse directions shown separately.

  • View in gallery

    (a) Plan view of REMUS “mow the lawn” sampling pattern (gray) with transect numbers labeled T1–T14. Overlaid dark gray stars indicate observations of above background levels of dye. Each heavy black cross represents an instrument mooring, and the dye source location is labeled at the S2 mooring. (b) Background-removed concentration time series of Rhodamine WT dye, with transect labels. (c),(d) REMUS depth and concentration distribution as function of cross-shelf location over transect seven.

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    Physical structure on 7 Jul 2010 at C0 mooring: (a) temperature, (b) along-shelf currents, (c) cross-shelf currents, (d) cross-shelf vertical shear, and (e) representation of buoyancy to shear ratio for . The black contour line in (e) marks , below which shear is expected to dominate buoyancy. In each panel, the dye study period (1140–1400 PDT) is boxed in black with the dye release depth marked (dashed black). Dye was released continuously from the S2 mooring at 1140. REMUS began sampling at 1215, and the dye release window corresponding to REMUS observations of the plume was approximately 1154–1211 PDT.

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    (a) Variance-preserving spectra (°C2) of temperature records (23 Jun–12 Jul 2010) at bottom (gray), midcolumn (black), and surface thermistors (dashed). Temperature records are linearly detrended, segmented, zero padded, and windowed (Hamming) with 50% overlap following Emery and Thompson (2004). The spectra are smoothed via averaging in frequency bins of one-twentieth of a decade. Diurnal and semidiurnal tidal frequencies are identified for reference in addition to band of buoyancy frequencies (mean plus or minus two std dev) observed within the 12.35°–12.8°C range of isotherms over the dye study period on 7 Jul 2010. Cross-spectra are calculated in a similar manner and used for (b) the squared coherence between the 11 MAB thermistor record and thermistors located between 2 and 16 MAB. The 95% confidence level for the coherence analysis is 0.23.

  • View in gallery

    Scatterplot of SCAMP-derived (a) temperature diffusivity and (b) dissipation with median values represented by black crosses. In (a) and (b) the midwater column region is within the two lines at 10 and 13 mab depth, and histograms of data within this region are plotted in (c),(d). Median diffusivity and dissipation within the midwater column region are labeled with a dashed black line in (c) and (d), respectively.

  • View in gallery

    (a) Dye plume length scale (gray crosses) as a function of along-shelf distance from source, with light gray-shaded uncertainty bars from bootstrapping algorithm described in the appendix. The best-fit line with scale dependency (solid gray) was determined using the length-scale model of (13). The fixed result from three-dimensional turbulence theory is plotted for comparison (dashed black). (b) Scale-dependent lateral (cross shelf) dispersion coefficient using (12) for .

  • View in gallery

    Comparison of the lateral length scale derived from the shear-flow dispersion variance described by (22), represented by a dashed line, to the measured plume (solid gray best fit using ). The gray crosses are the moment-derived width of the measured plume.

  • View in gallery

    Particle-tracking model-derived widths compared to the measured plume widths. The field-measured results are presented in gray, with the solid gray line a best fit to (13), using . Black circles represent the time-averaged PTM width as function of along-shelf distance from source with plus or minus one std dev error bars indicating time variability. The best fit (dashed black line) uses (13), giving a scale dependency of .

  • View in gallery

    REMUS tow-yo depth as a function of cross-shelf distance for transect 2, with locations of measured dye indicated by black crosses. Labels are presented to supplement description of uncertainty estimates within text.

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Scale-Dependent Dispersion within the Stratified Interior on the Shelf of Northern Monterey Bay

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  • 1 Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental Engineering, Stanford University, Stanford, California
  • | 2 COBIA Lab, College of Engineering, The University of Georgia, Athens, Georgia
  • | 3 Department of Civil and Environmental Engineering, University of California, Berkeley, Berkeley, California
  • | 4 Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental Engineering, Stanford University, Stanford, California
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Abstract

Autonomous underwater vehicle measurements are used to quantify lateral dispersion of a continuously released Rhodamine WT dye plume within the stratified interior of shelf waters in northern Monterey Bay, California. The along-shelf evolution of the plume’s cross-shelf (lateral) width provides evidence for scale-dependent dispersion following the 4/3 law, as previously observed in both surface and bottom layers. The lateral dispersion coefficient is observed to grow to 0.5 m2 s−1 at a distance of 700 m downstream of the dye source. The role of shear and associated intermittent turbulent mixing within the stratified interior is investigated as a driving mechanism for lateral dispersion. Using measurements of time-varying temperature and horizontal velocities, both an analytical shear-flow dispersion model and a particle-tracking model generate estimates of the lateral dispersion that agree with the field-measured 4/3 law of dispersion, without explicit appeal to any assumed turbulence structure.

Corresponding author address: Ryan J. Moniz, Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental Engineering, Stanford University, Y2E2 473 Via Ortega, Stanford, CA 94305. E-mail: rjmoniz@stanford.edu

Abstract

Autonomous underwater vehicle measurements are used to quantify lateral dispersion of a continuously released Rhodamine WT dye plume within the stratified interior of shelf waters in northern Monterey Bay, California. The along-shelf evolution of the plume’s cross-shelf (lateral) width provides evidence for scale-dependent dispersion following the 4/3 law, as previously observed in both surface and bottom layers. The lateral dispersion coefficient is observed to grow to 0.5 m2 s−1 at a distance of 700 m downstream of the dye source. The role of shear and associated intermittent turbulent mixing within the stratified interior is investigated as a driving mechanism for lateral dispersion. Using measurements of time-varying temperature and horizontal velocities, both an analytical shear-flow dispersion model and a particle-tracking model generate estimates of the lateral dispersion that agree with the field-measured 4/3 law of dispersion, without explicit appeal to any assumed turbulence structure.

Corresponding author address: Ryan J. Moniz, Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental Engineering, Stanford University, Y2E2 473 Via Ortega, Stanford, CA 94305. E-mail: rjmoniz@stanford.edu

1. Introduction

Advection and diffusion processes govern the fate and transport of passive scalars in the nearshore coastal ocean and therefore influence the distribution and concentration of nutrients, contaminants, and nonmotile biological aggregates such as certain species of phytoplankton and zooplankton (e.g., Roberts 1999; Dekshenieks et al. 2001). Within the stratified coastal shelf waters of Monterey Bay, California, important transport and mixing mechanisms include wind-driven diurnal upwelling and internal waves and bores (Drake et al. 2005; Petruncio et al. 1998; Woodson et al. 2007; Walter et al. 2012); these reflect the complex interaction of regional-scale forcing on local coastal flows (C. B. Woodson and D. A. Fong 2013, unpublished manuscript). In effect, these processes act on a broad range of time and spatial scales and have the capacity to collectively enhance scalar transport and dispersion.

Richardson (1926) first introduced the concept of scale-dependent dispersion, in which the rate of effective diffusion (i.e., the dispersion coefficient) increases with the length scale of the scalar distribution. The nomenclature adopted in this paper defines dispersion as the combined processes by which turbulence ultimately produces irreversible mixing. The horizontal dispersion coefficient is defined as the growth rate of the scalar variance :
e1
In contrast to Fickian dispersion, which requires a linear growth rate in time of the scalar variance (Taylor 1921), scale-dependent dispersion implies time-dependent growth in the scalar variance that is faster than linear. Allowing for variable power-law dependence in time, the scalar variance takes the following form:
e2
where is a dimensional constant, and defines the power law in time. Note that Fickian dispersion requires . Combining (1) and (2) to remove the time dependency, the dispersion coefficient can be expressed generally as
e3
where is a scale for the length of the scalar distribution. After defining , (3) presents a scale dependence of the form . Several early studies of dispersion found that scale dependence holds, that is, that (Richardson and Stommel 1948; Stommel 1949). More recent dye release studies in the ocean have corroborated the 4/3 law (Okubo 1971; Vasholz and Crawford 1985; Stacey et al. 2000; Fong and Stacey 2003; Jones et al. 2008).
In fact, similarity theories of turbulence (Kolmogorov 1941; Batchelor 1950) predict this 4/3 law as being appropriate, in a strict sense, when the plume length scale falls within the inertial subrange of turbulence wavenumber spectra, which is otherwise only characterized by the turbulent dissipation rate . As shown by Okubo (1971), for this case the scalar variance and dispersion coefficient take the following forms:
e4
e5
where and are numerical constants and defines the characteristic length scale of the distribution for numerical constant . This model has been applied to observations of the 4/3 law on a wide range of scales [e.g., tens to hundreds of meters by Fong and Stacey (2003) and Jones et al. (2008) and upward of 1000 km by Okubo (1971)]. Previous work has suggested that the model extends beyond the inertial subrange and the local stratification to the scale of the overall water depth (Fong and Stacey 2003). However, Okubo (1971) cautions against attributing 4/3 law dispersion to turbulence theory because other theories similarly produce the equivalent 4/3 law.
Shear-flow dispersion is an alternative theory capable of producing a 4/3 law. In this case, horizontal dispersion arises from the interaction of vertical diffusion (driven in this case by turbulence) with a vertically sheared horizontal flow (Taylor 1953; Saffman 1962; Fischer et al. 1979). Models of turbulent shear-flow dispersion separate the spectrum of turbulence into two regimes: large-scale eddies that form due to shears in the mean velocity and small-scale eddies responsible for mixing and described by an eddy diffusivity (Ebbesmeyer and Okubo 1975). For an unbounded, vertically constant and steady shear , the shear-flow dispersion contribution to the horizontal variance and dispersion coefficient is formulated as (Okubo 1968; Kullenberg 1972; Smith 1982; Young and Rhines 1982)
e6
e7
where is the vertical eddy diffusivity, and numerical constants and are distinct from those in (4) and (5). However, if instead the shear is purely oscillatory, the horizontal variance is inversely proportional to the square of the oscillation frequency and grows linearly in time (Smith 1982; Young and Rhines 1982). Nevertheless, oscillatory shears are capable of producing scale-dependent growth if the time scale of interest is shorter than the period of oscillation.
For a vertically constant shear of arbitrary time dependence , the shear-flow dispersion contribution to the vertically integrated horizontal variance takes the following form (Smith 1982):
e8
where it is assumed that the distribution grows from a point source at and is a distortion factor that captures the time variability of the distribution’s tilt:
e9
Sundermeyer and Ledwell (2001), in a dye tracer study of lateral dispersion over the eastern United States continental shelf, used horizontal variances calculated from the first three horizontal moments of the tracer distribution, following Townsend (1951), to evaluate time-variable shear-flow dispersion as a candidate mechanism for the observed dispersion. More recently, Steinbuck et al. (2011) used this formulation to quantify horizontal dispersion driven by a time-variable internal wave shear in the Gulf of Aqaba, finding agreement between the Smith (1982) analytical model and that of a Lagrangian particle-tracking model driven with observed velocities mapped onto isopycnal surfaces.

It is important to make the distinction between the above theoretical models. Turbulence theory postulates that the fluctuating motions driving dispersion are described by the inertial subrange model, which only depends on the length scale and the rate of turbulent kinetic energy dissipation (Batchelor 1952). As the spread of the scalar distribution grows, the length scale of the turbulence it experiences also grows, increasing the dispersion coefficient as described by (5). Once the scalar distribution is broader than the largest turbulent scales, the inertial subrange model cannot explain any further scale-dependent behavior following the 4/3 law. However, at these larger scales, the constant and steady shear-flow dispersion model of (7) produces this type of scale-dependent growth in the horizontal dispersion coefficient. The steady shear persistently increases scalar gradients while the scalar distribution spans a broader range of the vertically sheared current as turbulent motions spread the distribution vertically. In the case of a time-variable shear, the shear-flow dispersion is highly dependent on the time scale of oscillations relative to the time scales of scalar growth and may or may not be scale dependent (Okubo 1968; Young and Rhines 1982; Smith 1982). In this study we consider these theoretical models to assess under which conditions each model is capable of representing measured dispersion.

Here we consider scale-dependent lateral dispersion using a dye tracer study on the northern Monterey Bay shelf and quantify lateral dispersion within the thermocline. Rhodamine WT dye was released continuously in the stratified interior of the water column and was mapped using an autonomous underwater vehicle (AUV) equipped with a fluorometer. In the analysis that follows, moored temperature and velocity data, coupled with the high spatial and temporal resolution concentration measurements from the AUV, are used to characterize the physical mechanisms responsible for the observed lateral dispersion.

In the following section we describe the field location, dye release configuration, plume sampling strategy, and data processing and analysis methods. We present the general hydrodynamic conditions and the dye plume measurements in section 3, followed by analyses of scale-dependent dispersion. We present the 4/3 law of dispersion from turbulence theory for context and compare field results to those from a time-variable, horizontally uniform, shear-flow dispersion model. In section 4 we draw comparisons between the direct observations and the results from a shear-flow dispersion-based particle-tracking model. Finally, we discuss in section 5 the implications of our results on scalar distributions on the shelf and the physical mechanisms responsible for their formation and maintenance.

2. Field program and methods

a. Experimental setup

Dye release studies were conducted as part of a larger field program [National Science Foundation (NSF)-funded 2010 Lidar for Lateral Mixing (LatMix)] targeted at assessing the physical dynamics responsible for horizontal dispersion using a suite of moored, shipboard, and AUV observations. Measurements were made on the shelf of northern Monterey Bay in the boxed region southeast of Santa Cruz, California (Fig. 1). An array of nine moorings was deployed from 23 June to 12 July 2010 centered near the 20-m isobath in a region characterized by a mildly sloping, sandy bottom and a local water depth that ranged from 20 to 22 m. Mooring spacing in the along-shelf direction was 200 m and aligned with isobaths, while the spacing in the cross-shelf direction was 100 m.

Fig. 1.
Fig. 1.

Regional map of Monterey Bay, California, with inset of the study region showing the mooring cross-array oriented with the ~20 m isobath. Tidal ellipses are shown for C0, with surface and bottom ellipse directions shown separately.

Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-12-0229.1

Moored instruments were deployed to measure the spatially and temporally variable current, temperature, salinity, and internal and surface wave fields. Each mooring was equipped with a bottom-mounted RDI-Teledyne 1200-kHz ADCP, sampling at 0.5 Hz with 0.25-m vertical bins, with the exception of the center mooring (C0), which sampled at a higher rate of 1 Hz. Thermistors [Sea-Bird Electronics (SBE), 39 s] in the midwater column sampled at 10 s, conductivity–temperature–depth meters (CTDs, SBE 37 s) at the top and bottom sampled at 20 s, and bottom-mounted Nortek acoustic Doppler velocimeters (ADV) sampled at 16 Hz. The C0, E2, and W2 moorings included an SBE 26 + wave gauge to measure waves with tidal measurements every 10 min at C0 and every 120 min at the E2 and W2 moorings. Finally, an SBE 16 + CTD with an oxygen sensor was deployed at C0, sampling at 20 s. Mooring instrumentation details are summarized in Table 1.

Table 1.

Mooring instrumentation.

Table 1.

The primary focus of the present work is the data acquired by mapping a continuously released Rhodamine WT dye plume within the thermocline on 7 July 2010. An autonomous dye source [Space and Naval Warfare Systems Center (SPAWAR Syscen), San Diego] released a diluted Rhodamine WT solution at approximately 40 ml min−1 through a diffuser hose that minimized the momentum of the dye at its release. Fastened to the inside of a weighted milk crate with frontal dimensions of 0.28 m × 0.48 m, the dye source was securely attached to a line equipped with a surface buoy and bottom weight to maintain tension. The source was collocated with the S2 mooring in an effort to ensure that the dye plume would be carried through the moored instrument array by the dominant along-shelf velocities oriented toward the northwest. The source depth was fixed at 10 m below the surface [11.5 m above the bottom (MAB) on average] following the identification of the thermocline with vertical temperature profiles (not shown) prior to the dye release.

In concert with the dye release, we deployed a Remote Environmental Monitoring Units (REMUS) 100 autonomous underwater vehicle (Kronsberg, Hydroid, Inc.) that was equipped with a Turner Designs Cyclops Rhodamine fluorometer (excitation wavelength 550 nm) to measure Rhodamine WT dye concentrations (emission wavelength 590–715 nm). REMUS was programmed to begin transecting following the development of the dye plume and directly measured the lateral extent of the plume at variable distances downstream of the source. This spatial “mow the lawn” through the plume comprised progressive transects spaced 50 m apart in the along-shelf direction, moving away from the source (Fig. 2a). Each transect spanned nearly 800 m cross-shelf and took the form of a vertical tow-yo with an ~60 m wavelength and 2-m vertical range centered at the source depth. REMUS was programmed to sample at 9 Hz, providing a nominal horizontal spatial resolution of approximately 0.17 m at typical transect speeds of approximately 1.5 m s−1. The complete survey through the array spanned nearly 105 min.

Fig. 2.
Fig. 2.

(a) Plan view of REMUS “mow the lawn” sampling pattern (gray) with transect numbers labeled T1–T14. Overlaid dark gray stars indicate observations of above background levels of dye. Each heavy black cross represents an instrument mooring, and the dye source location is labeled at the S2 mooring. (b) Background-removed concentration time series of Rhodamine WT dye, with transect labels. (c),(d) REMUS depth and concentration distribution as function of cross-shelf location over transect seven.

Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-12-0229.1

Prior to REMUS starting its survey through the array at 1215 Pacific daylight time (PDT), the source began to release dye at 1140 PDT. The dye propagated through the array at 0.10 m s−1—an estimate based on the along-shelf currents within the thermocline over the dye study period (1140–1400, defined as the start of the dye release to the end of the REMUS survey through the array). The dye advection speed and the REMUS effective along-shelf travel speed of ~0.12 m s−1 specified an ~17-min duration dye release window (estimated as 1154–1211) corresponding to dye that REMUS actually measured. Based on interpolation of the thermistor records at the S2 mooring to compute temperature at the dye source depth, the source released dye over this time window within a temperature range spanning 12.4°–12.75°C, which falls within the 12.35°–12.8°C range of temperatures measured by REMUS at dye observations. Note that the plume’s continuous release is an important factor because the along-shelf dispersion dynamics are drastically different in the case of a plug release (i.e., the continuous release permits the exclusion of along-shelf dispersion because along-shelf advection dominates). Instrument contamination of measurements is not expected because REMUS moves faster than the plume, never crossing previously sampled areas of the plume twice.

Deployed simultaneously with the REMUS transecting, a self-contained autonomous microstructure profiler (SCAMP) (Precision Measurement Engineering Inc., Carlsbad, California) recorded finescale measurements of temperature and salinity at the center of the array. These profiles were used to infer turbulent dissipation rates and vertical diffusivity throughout the water column with 6-min time resolution, utilizing Batchelor spectrum fitting (Batchelor 1959) following the methods of Steinbuck et al. (2009).

b. Data processing and analysis methods

For the following analysis, we defined a coordinate system with at the dye source location with as the cross-shelf direction (positive onshelf toward the northeast), as the along-shelf direction (positive out of the bay toward the northwest), and as the vertical component measured from the bottom upward. Velocity measurements from the ADCPs were block averaged on 1-min intervals to reduce noise and rotated to cross- and along-shelf components that are aligned with the REMUS pass orientation and the principal axes of the flow. The lateral axis of the plume is therefore aligned with the cross-shelf direction. TS plots (not shown) from the top- and bottom-mounted CTDs at C0 indicate that density variations were dominated by temperature variations, behavior that is expected during the summer upwelling months in Monterey Bay (Woodson et al. 2011). Therefore, in what follows we consider the thermocline and pycnocline to be synonymous and give preference to the former with the use of the thermistor records.

The continuous dye release provided a strong signal to measure the lateral length scale of the plume at progressive downstream distances from the source. The raw Rhodamine WT signal of each transect exhibited nonzero background levels and a linear trend of decreasing concentration with positive distance on shelf. We do not attribute these background levels to the generated dye plume, so we applied a calibration to the raw signal: first, we subtracted the linear trend from the raw distribution. Next, we applied a low-pass filter for smoothing and to remove noise far from the distribution’s center of mass that would influence moment calculations. The background-removed dye concentration time series is presented in Fig. 2b.

Next, following Stacey et al. (2000), Fong and Stacey (2003), and Jones et al. (2008), we use the zeroth, first, and second moments, , of each background-removed concentration distribution to define both the plume center of mass and the plume variance, respectively:
e10
e11
where the limits of the integrals span the cross-shelf extent of each REMUS transect. This defines the plume variance at discrete distances from the source. Note that, in a strict sense, Eq. (11) assumes the concentration varies only in the lateral direction. In reality, the stratification on the inner shelf will create vertical structure; this, coupled with the tow-yo sampling strategy of REMUS, complicates this calculation. Nevertheless, we use this framework as the best available method for determining the spread of a plume in a vertically integrated sense, and we therefore account for these factors in generating uncertainty estimates presented with the analysis shown. We employ a bootstrapping technique following Emery and Thomson (2004) and Efron and Tibshirani (1993); the details of this method are provided in the appendix.
Scale-dependent lateral dispersion can be quantified using the definition of the dispersion coefficient as given by (1) and (3), in addition to the moment-derived variances of (11). A modification to (3) accounts for the initial plume width b at the source, which in the present study is the width of the milk crate described in section 2a:
e12
where is the dimensional constant of proportionality. Following Brooks (1960), we define the lateral length scale of the plume as and arrive at a solution for the plume lateral length scale as function of longitudinal distance from the source:
e13
where represents a magnitude of dispersion accounting for a Galilean transformation using the mean along-shelf velocity within the stratified interior. We treat and as adjustable parameters and use a nonlinear least squares fit to (13) with the input parameter taken from observations.

SCAMP profiles are processed according to the methods of Steinbuck et al. (2009). Millimeter-scale measurements of temperature gradients are used to estimate the dissipation rate of temperature variance in 50% overlapping segments of 10 cm. Both and a vertical eddy dissipation rate are found by fitting a theoretical Batchelor spectrum to the measured temperature gradient spectrum for each segment (Batchelor 1959). A turbulent vertical diffusivity of temperature is calculated following Osborn and Cox (1972) using .

In the next section, the length scale model of (13) is used with (12) to quantify the dye plume’s scale-dependent lateral dispersion. These results, in conjunction with SCAMP-derived estimates of dissipation, are used to address the turbulence model [defined by (4) and (5)]. Following this analysis, we consider as a mechanism of the observed dispersion the time-varying shear model defined by (8) and (9) using ADCP measurements of shear within the 12.35°–12.8°C range of isotherms corresponding to the REMUS measured plume. Smith (1982) derives the variance growth definition of (8), which we now summarize.

Consider a point source released into a vertically linear, time-varying velocity profile, with the following governing advection–diffusion equation:
e14
where is the scalar concentration; and are the horizontal and vertical diffusivities, respectively; and the cross-shelf velocity is defined as
e15
where is a constant background cross-shelf velocity, and is a time-varying shear. Smith (1982) provides a coupled set of ordinary differential equations for the time evolution of the horizontal and vertical variances:
e16
e17
e18
where is a distortion factor that characterizes the time evolution of the scalar patch’s vertical tilt. We necessarily assume in this analysis that the initial distributions at the source, in addition to plume distributions downstream, take on an idealized Gaussian structure as a model for the actual plume, which may, in fact, be filamentous at any instant (cf. Crimaldi et al. 2002). We define as the time of a point release such that at the vertical variance has grown to by the action of vertical turbulent diffusion in the absence of shear. As a result, the distortion factor is zero for all time between and because there is no shear. Smith (1982) provided the solution for the distortion factor in (9) that holds under the above assumptions. Allowing for an initial scalar distribution at , solutions for the scalar variances follow from (16) and (17):
e19
e20
Equation (19) applies for a horizontal section through the plume. Since the moment-derived widths of the measured plume are meant to represent the vertically integrated distribution, we include an additional term, , to (19) that follows from the derivation of the horizontal variance as presented by Smith [1982, Eq. (5.3)]:
e21
The first two terms in (21) represent the initial lateral variance and the growth in lateral variance from horizontal turbulent diffusion, respectively. These terms are considered small relative to the sum of third and fourth terms in (21) that represents the growth in the horizontal variance by shear-flow dispersion and depends on the time evolution of the shear. The vertically integrated horizontal variance is therefore estimated with the two dominant terms:
e22
The variance growth is directly proportional to the vertical diffusivity, which is expected in an unbounded flow (Saffman 1962; Fischer et al. 1979). In the analysis of section 3e, we compare the observed horizontal variance, assumed here to be representative of the ensemble-averaged distribution, with that predicted by (22). Following this analysis, in section 4, we employ a Lagrangian particle-tracking model to assess the shear-flow dispersion mechanism while accounting for vertical variability in the shear field using the ADCP-measured velocity field.

3. Results

a. General hydrodynamic context

C. B. Woodson and D. A. Fong (2013, unpublished mansucript) examined regional and local hydrodynamics during the LatMix study period from 23 June to 12 July 2010; relevant results are summarized here. Analysis of the 10-min-averaged velocity field reveals that the mean along-shelf currents are largely in a thermal wind balance. Following removal of this geostrophic flow, an empirical orthogonal function analysis of the complex velocity field revealed that the first empirical mode is a barotropic response that is highly coherent with the tides. The second empirical mode is coherent with diurnal winds at a lag of approximately 1.26 h. The ADCP records indicate mean along-shelf currents O(0.1 m s−1) out of the bay to the northwest during the dye study on 7 July 2010 (Fig. 3b). Cross-shelf currents are highly coherent at low frequencies (~3 cpd) driven by regional-scale forcing and tidal dynamics, while coherence at higher frequencies (24–72 cpd) is representative of surface seiching, which produces a barotropic response to the diurnal wind forcing, and internal waves. Internal wave variability is observed in shear magnitude spectrograms as broadband increases in shear at frequencies between about 20 and 140 cpd, a band consistent with the dominant periods of internal waves. The vertically sheared cross-shelf velocity within the stratified interior during the dye study period on 7 July 2010 is characteristic of this forcing and is expected to be an important contributor to lateral dispersion processes (Figs. 3c,d).

Fig. 3.
Fig. 3.

Physical structure on 7 Jul 2010 at C0 mooring: (a) temperature, (b) along-shelf currents, (c) cross-shelf currents, (d) cross-shelf vertical shear, and (e) representation of buoyancy to shear ratio for . The black contour line in (e) marks , below which shear is expected to dominate buoyancy. In each panel, the dye study period (1140–1400 PDT) is boxed in black with the dye release depth marked (dashed black). Dye was released continuously from the S2 mooring at 1140. REMUS began sampling at 1215, and the dye release window corresponding to REMUS observations of the plume was approximately 1154–1211 PDT.

Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-12-0229.1

b. Dye study physical conditions

Although dye was released from the S2 mooring during the 7 July 2010 field study, we present a physical characterization from the center mooring because it had the most highly resolved measurements and thus best shows the structure of flow within the instrument array. The 1-m vertically resolved temperature structure (Fig. 3a) is dominated by stratification from persistent regional upwelling winds coupled with the warming of surface waters within the bay (C. B. Woodson and D. A. Fong 2013, unpublished mansucript). The stratification supports internal waves that are observed to propagate predominantly onshelf (median heading of 350°) at periods within a range of 12–15 min. These high frequency internal waves may be associated with the degeneration of the semidiurnal internal tide (Petruncio et al. 1998) or with wind forcing of the warm surface layer (Woodson et al. 2011) and are therefore not generated at the site. The variance-preserving spectrum of a midwater column temperature record supports this internal wave characterization given an increase of temperature variance between 100 and 101 cph that does not exist in the spectra for surface and bottom temperature time series (Fig. 4a). Furthermore, midcolumn thermistor records (~8–14 MAB) are significantly coherent with each other at the 95% confidence level to a high frequency of ~20 cph (Fig. 4b). Uncorrelated turbulent temperature fluctuations are expected to dominate temperature variability at higher frequencies, above the buoyancy frequency band in Fig. 4a.

Fig. 4.
Fig. 4.

(a) Variance-preserving spectra (°C2) of temperature records (23 Jun–12 Jul 2010) at bottom (gray), midcolumn (black), and surface thermistors (dashed). Temperature records are linearly detrended, segmented, zero padded, and windowed (Hamming) with 50% overlap following Emery and Thompson (2004). The spectra are smoothed via averaging in frequency bins of one-twentieth of a decade. Diurnal and semidiurnal tidal frequencies are identified for reference in addition to band of buoyancy frequencies (mean plus or minus two std dev) observed within the 12.35°–12.8°C range of isotherms over the dye study period on 7 Jul 2010. Cross-spectra are calculated in a similar manner and used for (b) the squared coherence between the 11 MAB thermistor record and thermistors located between 2 and 16 MAB. The 95% confidence level for the coherence analysis is 0.23.

Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-12-0229.1

The center mooring thermistor chain also captures the vertical expansion of the thermocline possibly produced by intermittent vertical mixing associated with shear instabilities. To characterize water column stability, we calculated the gradient Richardson number, , where is the buoyancy frequency and is the vertical shear in the horizontal velocity, computed from the 1-m scale temperature and 0.25-m scale velocity fields. Midcolumn values of Ri suggest regions of stable stratification and of potential shear-induced instabilities (Fig. 3e) since Ri varied between 0.01 and 2.7 × 103 in the interior of the water column on 7 July 2010. SCAMP measurements from the same day provide finer-scale characterization of the temperature structure and the vertical mixing within the stratified interior, as seen in Fig. 5. Vertical temperature diffusivity is lowest around the thermocline (with median value 4.7 × 10−6 m2 s−1), as expected, peaking in surface and bottom boundary layers (with maximum value 10−3 m2 s−1). Intermittent, high dissipation rates (near 10−5 m2 s−3), observed near the bottom of the thermocline, correspond to mixing events following the passage of internal waves.

Fig. 5.
Fig. 5.

Scatterplot of SCAMP-derived (a) temperature diffusivity and (b) dissipation with median values represented by black crosses. In (a) and (b) the midwater column region is within the two lines at 10 and 13 mab depth, and histograms of data within this region are plotted in (c),(d). Median diffusivity and dissipation within the midwater column region are labeled with a dashed black line in (c) and (d), respectively.

Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-12-0229.1

c. Length-scale model of scale-dependent dispersion

The measured horizontal dispersion scale dependency is determined using the length-scale model presented in (13), where conditions during the dye study period defined the input parameters m and m s−1, and was defined as in Brooks (1960) using the square root of the moment-derived scalar variance defined by (11). The plume width scale is plotted for each transect in Fig. 6a. The width scales are used in a nonlinear least squares fit to (13), with the magnitude of dispersion and the scale dependency treated as adjustable parameters. The resulting optimization with 95% confidence intervals produces the scale dependency of (), plotted in Fig. 6a. The gray uncertainty bounds in Fig. 6a reflect the combined effects of the tow-yo sampling strategy and measured concentration distribution along each transect on the width calculations. It should be noted that the parameters and are not affected by uncertainty in the plume widths [as suggested by a second bootstrapping procedure that resamples widths within the uncertainty bounds and finds and using nonlinear fits to (13)] or by straining from the dye being released on different isotherms (i.e., cross-shelf straining accounts for less than 6% of the total width). More details of the uncertainty estimates are presented in the appendix.

Fig. 6.
Fig. 6.

(a) Dye plume length scale (gray crosses) as a function of along-shelf distance from source, with light gray-shaded uncertainty bars from bootstrapping algorithm described in the appendix. The best-fit line with scale dependency (solid gray) was determined using the length-scale model of (13). The fixed result from three-dimensional turbulence theory is plotted for comparison (dashed black). (b) Scale-dependent lateral (cross shelf) dispersion coefficient using (12) for .

Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-12-0229.1

Using , (12) prescribes a scale-dependent lateral dispersion coefficient that grows to 0.5 m2 s−1 approximately 700 m downstream of the source (Fig. 6b). This scale dependency is similar to previously published values in the bottom mixed layer on the inner shelf. For example, Fong and Stacey (2003) mapped a dye plume from its source to a downstream distance of 1200 m and found prior to conducting a compound dispersion analysis in which the authors observed a 4/3 law in the near field (<800 m from the source) and a scale-squared law in the far field (>800 m). Within statistical certainty, the observed lateral dispersion in this study supports a 4/3 law, which motivates analysis of dispersion first using turbulence theory.

d. Scale-dependent dispersion from turbulence theory

Fixing (eliminating as a fitting parameter), a nonlinear least squares fit to (13) gives (also in Fig. 6a), a value of the same order as those previously found in the bottom mixed layers of other nearshore systems [i.e., in Stacey et al. (2000) and in Fong and Stacey (2003)]. Equation (12) can be collapsed to a form with a single-dimensional constant, , where is expected to fall within the range from 0.002 to 0.01 cm2/3 s−1 (Fischer et al. 1979). The lateral dispersion within the stratified interior measured here yields cm2/3 s−1.

If the scales of the plume fall within the inertial subrange prescribed by the similarity theory of turbulence, this value presents a scale for the dissipation acting on horizontal dispersion. Unfortunately, the ADCP data were not reliable for estimating both the dissipation and bounds on the inertial subrange within the stratified interior (Bluteau et al. 2011). However, SCAMP provided a reliable estimate of dissipation within the thermocline as a measure of the energy input feeding the smaller, dissipative scales. We can conclude that the foundational energetic argument for the 4/3 law of dispersion given by (5) requires a constant for the SCAMP measured median dissipation within the thermocline (6.6 × 10−10 m2 s−3, Fig. 5) to match that acting on horizontal dispersion within the thermocline. Brier (1950) and Batchelor (1952) found the following relationship describing the mean-square separation of all particle pairs within a passive scalar cloud:
e23
where is known as the Richardson constant. The above constant is equivalent to a Richardson constant equal to 0.88, which falls within the wide theoretical bounds (0.06–3.52) set for its expected value (Sawford 2001).

e. Scale-dependent shear-flow dispersion driven by time-variable shear

An alternative to the Okubo–Richardson turbulence model for scale-dependent dispersion is that based on shear-flow dispersion (SFD). In what follows, we examine the horizontal dispersion associated with a time-variable shear and compare the model prediction from (22) with the observed lateral dispersion within the stratified interior.

Recall from the model description in section 2b that the horizontal variance depends on the distortion factor time series defined by (9). The factor was found by numerically integrating (9) on intervals coincident with the 1-min time-averaged velocity field. Following a conversion of the S2 mooring shear field to temperature space, , this method uses the time series of rms shear within the 12.35°–12.8°C range of isotherms to define . The horizontal variance was determined from (22), evaluated numerically, and then was transformed to spatial coordinates using the mean along-shelf velocity within the above range of isotherms. The vertical diffusivity was left as a parameter to optimize the minimum sum-squared error between the plume-measured length scale and that from the SFD mechanism [, for from (22)]. The optimal vertical diffusivity, 7.5 × 10−6 m2 s−1, falls within range of values of vertical diffusivities measured by SCAMP within the stratified interior (Fig. 5).

The inferred plume widths calculated this way are comparable to the measured plume widths (Fig. 7) and suggest that this model of SFD is capable of replicating lateral dispersion within the stratified interior. The scale dependency associated with the SFD mechanism is found using the length-scale model of (13). The resulting nonlinear least squares fit produces a scale dependency of and , consistent with the 4/3 law of dispersion. It should be noted that the 4/3 law follows directly in the case of a constant shear as given by (7). However, given that the length scale depends on the time history of the shear shown in (9), the 4/3 law of dispersion is not required to hold for time-varying shear. In this case the rms time-variable shear comprises steady and fluctuating parts, both approximately 0.03 s−1. The steady shear component drives the scale-dependent dispersion reflected in the best fit to the length-scale model of (13), whereas the fluctuating shear evidently has little effect, notably producing no deviation from the 4/3 law of dispersion. Although the fluctuating shears have the same magnitude as the steady shear, a transition between linear in time to time-cubed growth in the scalar variance (i.e., a transition between scale-independent and scale-dependent dispersion) is not expected because the transition time is proportional to the high-frequency oscillation period (Young and Rhines 1982). This transition occurs prior to the first cross-shelf transect and was therefore not resolved.

Fig. 7.
Fig. 7.

Comparison of the lateral length scale derived from the shear-flow dispersion variance described by (22), represented by a dashed line, to the measured plume (solid gray best fit using ). The gray crosses are the moment-derived width of the measured plume.

Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-12-0229.1

4. Particle-tracking model for scale-dependent dispersion

To account for the vertical variability in the shear field, the observed lateral dispersion was also modeled using a simple Lagrangian particle-tracking model (PTM) like those of Visser (1997), Batchelder et al. (2002), or Ross and Sharples (2004). Neutrally buoyant particles were released at the dye source location, and the vertical and horizontal positions of the particles were advanced using observations from the S2 mooring. To account for the dye source discharging along multiple isotherms, the PTM interpolated the dye source temperature from the thermistor chain and released particles along a particular isotherm to the nearest 0.05°C. The vertical position of each particle was first updated to account for isotherm displacements, such that particles were displaced vertically on the isotherm on which they were released. Second, all particles underwent a vertical random walk step associated with a constant diffusivity. For each model time step, , the vertical positions were updated as follows:
e24
where is the initial position of the particle, is the isotherm displacement, is a random variable with unit variance, and 7.5 × 10−6 m2 s−1 is the constant vertical diffusivity found using the optimization procedure in the previous section. Note that spurious particle aggregation is not a concern under the assumption of a constant vertical diffusivity (Visser 1997; Ross and Sharples 2004). At each time step, the horizontal position of each particle, , was updated by numerically integrating the particle’s local velocity, , using the trapezoidal rule.

The model released a total of 33 600 particles over the dye release period defined in section 2a, with the total number of particles chosen to obtain statistically reliable results. The particle cloud variance was defined using the moment equations of (10) and (11), for which the histogram of the particle lateral position was used to define at along-shelf stations of finite along-shelf breadth equal to 2 m spaced every 50 m from the source. This sampling of the particle plume was chosen to best compare PTM results to direct measurements of the dye plume in the field. For each along-shelf station, the cross-shelf particle cloud width was recorded at each model time step to capture the time variability. The lateral dispersion of the particle cloud was then analyzed using the length-scale model of (13) with the time-averaged cross-shelf widths. The scale dependency associated with the PTM plume, and , is in agreement with the scale dependency bounds found for the observed dye plume (Fig. 8) and compares well to the SFD analysis of section 3e. The variability in the particle cloud width is observed to grow with distance downstream from the source (Fig. 8). The majority of the observed dye plume widths falls within the range of particle widths in the nearfield (<600 m downstream), but a minority exceeds the upper limits of the particle width range farther downstream. In consideration of the uncertainty in the observed widths (Fig. 6), the PTM results in Fig. 8 are consistent with plume observations. We attribute discrepancies to potential horizontal variations in the shear field that the present particle-tracking model is unable to capture given its sole use of the measured velocity at the S2 mooring location.

Fig. 8.
Fig. 8.

Particle-tracking model-derived widths compared to the measured plume widths. The field-measured results are presented in gray, with the solid gray line a best fit to (13), using . Black circles represent the time-averaged PTM width as function of along-shelf distance from source with plus or minus one std dev error bars indicating time variability. The best fit (dashed black line) uses (13), giving a scale dependency of .

Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-12-0229.1

5. Summary and discussion

A fluorometer-equipped AUV was used to make measurements of a continuously released dye plume with the goal of quantifying lateral dispersion within the thermocline on the shelf of Monterey Bay. Using a set of cross-shelf transects spaced 50 m apart downstream of the source, we observed growth in the plume’s lateral length scale from less than 1 m at the source to nearly 150 m at an along-shelf distance 700 m downstream. These measurements are indicative of a scale-dependent lateral dispersion coefficient that approached 0.5 m2 s−1 at 700 m from the source. The plume width model of (13) concretely supports scale-dependent dispersion in the stratified interior. Uncertainty exists in the power law given scatter in the field-measured plume widths; however, the scale dependency is statistically consistent with the 4/3 law of dispersion.

One explanation for the observed 4/3 behavior is that it is the direct result of turbulent dispersion. In this classical framework, presented here for context, the plume width falls within the inertial subrange so that the plume is dispersed by three-dimensional turbulence structures. As the plume grows in scale, it is dispersed by progressively larger eddies. Estimates of dissipation from the plume width data using (5) and (12) reveal , equivalent to 0.009 cm2/3 s−1, which falls within the expected range of values, 0.002–0.01 cm2/3 s−1, given by Fischer et al. (1979). The median SCAMP estimate of dissipation within the thermocline, 6.6 × 10−10 m2 s−3, implies that and the Richardson constant, , are both within the wide range of values found by previous studies. However, this model requires that the plume scale lies within the inertial subrange. In the present case, given that the plume is ~50–150 m wide (i.e., 2–7 times larger than the depth), the plume scales are probably outside of the inertial subrange. To check this we note that the “universal” Kaimal et al. (1972) turbulence spectrum (appropriate for boundary layers) shows significant deviation from the −5/3 law when , where is the frequency and is the mean velocity at height above the bottom. This is equivalent to a condition on the length scale that for the inertial subrange to apply. In our case so that would be required. Thus, it seems unlikely that similarity theory applies to our plumes, so we conclude that the observed scale-dependent dispersion is not the result of large-scale turbulence.

As an alternative to the three-dimensional turbulence model, we investigated shear-flow dispersion as the mechanism for the lateral dispersion we observed. As commonly seen in the coastal ocean, the stratified interior supports an active internal wave field characterized by a vertically sheared cross-shelf velocity field (Fig. 3); combining this shear with weak vertical mixing allows for the existence of shear-flow dispersion (Young and Rhines 1982). When applied to our data, this theory, as formulated by Smith (1982) and used by Steinbuck et al. (2011), accurately models our observations.

In one sense these two approaches have some strong similarities. Indeed, Okubo (1968) provides a thoughtful explanation linking the SFD model to the continuous spectrum of turbulence. A scalar patch of known width encounters momentary shears and turbulent dispersion associated with eddies on the scale of the patch as it grows. If these eddies lie within the inertial subrange, scaling of the shear and diffusivity with the patch’s characteristic velocity and length scale, assuming constant dissipation, results in the same form for the patch’s scalar variance as given by turbulence theory in (4). Therefore, the scales that characterize the shear are of critical importance. In the event the scales of the shear, or the length scale of the plume for that matter, exceed the low wavenumber limit of the inertial subrange, the 4/3 law is only expected to hold if a steady shear drives the SFD mechanism [as in (6) and (7)].

The SFD model of (22) represents small-scale turbulent fluctuations via the specified vertical diffusivity and effectively reduces the spectrum of shear scales to one large-scale shear whose strength varies in time. The observed 4/3 law of dispersion implies that the time-averaged, steady shear part dominates the contribution to the growth in lateral variance, that is, that high-frequency shears are less effective drivers of SFD than are low frequency or steady ones. This is not unexpected; indeed, as discussed in Fischer et al. (1979), the ability of oscillatory shear to produce dispersion depends on the ratio of time scale for mixing perpendicular to the direction of the sheared velocity. When the period of oscillation is short relative to the mixing time, the reversing shear produces little net dispersion.

In the present study, high frequency shear variability is caused by the intermittent internal wave field, which was observed from temperature records to have an approximately 15 min period. Although it is the steady shear over the ~2 h dye study that appears to dominate dispersion, the internal waves help to create and define the steady shear. Additionally, the internal waves are likely to create time variations in vertical diffusivity and dissipation (MacKinnon and Gregg 2003; S. K. Willis et al. 2014, unpublished manuscript). The effect of this variability is represented in our SFD model by an optimal vertical diffusivity that is larger than the observed average diffusivity within the stratified interior, suggesting that larger mixing events are disproportionately associated with large shear events. A detailed analysis of the covariability of mixing and shear is underway, but is beyond the scope of this manuscript. Nonetheless, while the high frequency shear associated with the internal waves is less effective than the time-averaged shear in driving SFD, the presence of internal waves remains critical to the dynamics of dispersion.

A more realistic version of the shear-flow model can be constructed using a particle-tracking model that includes observed vertical variability in the shear field and models vertical turbulence as a random walk. The assumption of a horizontally uniform velocity field still represents a simplification of the true dynamics driving dispersion, but the model’s results (Fig. 8) demonstrate the efficacy of the PTM as a tool for investigating lateral dispersion within the thermocline. In light of this, the dense instrument array employed in the field program presents a unique opportunity to expand the present PTM to incorporate horizontal variability in both temperature and velocity fields, as well as temporal and vertical variability in the vertical diffusivity. This is the subject on an ongoing investigation and will be reported elsewhere.

The inertial subrange model and SFD model both predict that lateral dispersion within the thermocline is scale dependent, but, as we argue above, the SFD model is more appropriate for assessing scale-dependent dispersion. The SFD model is also appealing in that it is based on clearly defined and measurable properties of the flow field, including a temporally variable shear field and vertical diffusivity, rather than on uncertain similarity assumptions. From a predictive standpoint, that is, for use in coastal circulation models, the application of both models may be problematic in that either requires parameterization of the local strength of a remotely generated internal wave field, something that is well beyond the current generation of coastal circulation models. In either case, as the plume approaches larger scales, neither model may be appropriate as observations suggest a transition to two-dimensional turbulence and a scale dependency of (e.g., Fong and Stacey 2003). The scale dependency of a measured dye plume may suggest that this transition is occurring; however, measurements farther downstream would have been required to conduct a reliable compound dispersion analysis.

Finally, the scale-dependent dispersion observed in this study has important implications for understanding what shapes passive scalar distributions within the thermocline of shelf waters. The results of the analysis suggest the importance of vertical shear within the stratified interior coupled with vertical mixing as a driver of dispersion. The recent study of a submerged buoyant discharge from the Point Loma Ocean Outfall offshore of San Diego, California (Rogowski et al. 2012), supports this conclusion. One implication of the scale-dependent dispersion in this context is that horizontal fluxes may be maintained even while concentration gradients decay because of the growing horizontal dispersion coefficient with plume size.

Acknowledgments

The authors thank the other members of the LatMix team, notably Margaret McManus, in addition to John Douglas (R/V Sheila B.) and Jim Christman (R/V Shana Rae). This work was supported by NSF Awards OCE-0926340, OCE-0926738, and OCE-0925916. Additional support was provided by the Stanford Graduate Fellowship (SGF) and by the Singapore–Stanford Program. The authors are grateful for the thorough and helpful comments provided by two anonymous reviewers that strengthened this contribution.

APPENDIX

Uncertainty Estimates

The moment-derived widths used in the length-scale model of (13) are sensitive to the concentration distribution measured along each transect in addition to potential bias associated with the REMUS tow-yo sampling strategy. We attempt to account for these complex and compounding factors by specifying liberal uncertainty bounds on the widths in Fig. 6. We describe here the process by which these bounds are determined and the effect that the uncertainty has on the scale dependency determined from field measurements.

The method described below is applied to each transect individually, and we present a visual representation of this process in Fig. A1. The first step requires locating the cross-shelf bounds (i.e., the minimum and maximum cross-shelf locations) and corresponding vertical locations where REMUS measures above zero levels of dye. In the worst-case scenario, it is possible that REMUS might have missed portions of the dye plume outside of these bounds because of its tow-yo sampling pattern. To account for such a possibility, we establish wider cross-shelf bounds by identifying the next REMUS crossing of the vertical locations identified in the first step. This is followed by a bootstrapping technique in which the concentration distribution within these wider bounds is reordered at random (this has the effect of resampling the same mass of dye but at different cross-shelf locations); we calculate the moment-derived width for this distribution and repeat the process of reordering 1000 times. The uncertainty bounds plotted in Fig. 6 represent plus or minus two standard deviations of the distribution of widths calculated under this worst-case scenario.

Fig. A1.
Fig. A1.

REMUS tow-yo depth as a function of cross-shelf distance for transect 2, with locations of measured dye indicated by black crosses. Labels are presented to supplement description of uncertainty estimates within text.

Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-12-0229.1

We employed a second bootstrapping to quantify the impacts of these uncertainty bounds on the field-measured scale dependency. We chose at random a plume width within the uncertainty bounds for each transect and subsequently conducted a nonlinear least squares fit of (13). This process was likewise repeated 1000 times. The outputted distributions of and were within the scale-dependency bounds (, ) derived from the field measurements. We conclude from this analysis that the parameters and , as determined using the field-measured moment-derived widths, are not significantly affected by uncertainty in the plume widths.

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