## 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reflect the complex interaction of regional-scale forcing on local coastal flows (C. B. Woodson and D. A. Fong 2013, unpublished manuscript).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.

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.

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.

Mooring instrumentation.

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.

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 *T*–*S* 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.

*b*at the source, which in the present study is the width of the milk crate described in section 2a:where

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 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.

## 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).

### 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 10^{0} and 10^{1} 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.

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, ^{3} 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} m^{2} s^{−1}), as expected, peaking in surface and bottom boundary layers (with maximum value 10^{−3} m^{2} s^{−1}). Intermittent, high dissipation rates (near 10^{−5} m^{2} s^{−3}), observed near the bottom of the thermocline, correspond to mixing events following the passage of internal waves.

### 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 ^{−1}, and

Using ^{2} 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

### d. Scale-dependent dispersion from turbulence theory

Fixing ^{2/3} s^{−1} (Fischer et al. 1979). The lateral dispersion within the stratified interior measured here yields ^{2/3} s^{−1}.

^{−10}m

^{2}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:where

### 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 ^{−6} m^{2} 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 ^{−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.

## 4. Particle-tracking model for scale-dependent dispersion

^{−6}m

^{2}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,

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

## 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 m^{2} 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 ^{2/3} s^{−1}, which falls within the expected range of values, 0.002–0.01 cm^{2/3} s^{−1}, given by Fischer et al. (1979). The median SCAMP estimate of dissipation within the thermocline, ^{−10} m^{2} s^{−3}, implies that

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

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.

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

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