Theoretical Assessment of the Effect of Vertical Dispersivity on Coastal Seawater Radium Distribution

Introduction

The radium (Ra) quartet (²²³Ra, ²²⁴Ra, ²²⁶Ra, and ²²⁸Ra) are among the most commonly used environmental tracers for evaluating submarine groundwater discharge (SGD) (Charette and Scholten, 2008), including its terrestrial groundwater and recirculated seawater components (Burnett et al., 2003). One common experimental design to evaluate SGD with these tracers is to collect water samples along transects perpendicular to the shoreline, and use simple advective-dispersive transport models to quantify the offshore coefficient of solute dispersivity (K_x), the tracer flux from the coastline (F_o), and the tracer flux from the seabed (F_B) (Moore, 2000; Hancock et al., 2006). Several assessments have been made of the validity of this approach. For example, Knee et al. (2011) have shown how activity measurement error can bias the application of the Ra transport models, Moore (2015) highlighted some of the limitations in the use of ²²⁸Ra and ²²⁶Ra to evaluate mixing and advection rates, Li and Cai (2011) evaluated the effect of neglecting advection on K_x estimates, and Lamontagne and Webster (2019) showed how K_x could be tracer-dependent. However, unlike in deep oceans and shelf environments, where ²²⁸Ra, ²²⁶Ra, and ²²²Rn vertical profiles have been used to estimate vertical mixing (Broecker et al., 1967; Chung and Craig, 1973; Glover and Reeburgh, 1987; Koch-Larrouy et al., 2015), Ra measurement in shallower inner shelf environments are typically only taken from the top of the mixed layer (Moore, 2000; Dulaiova and Burnett, 2006; Lamontagne et al., 2008). The assumption is that vertical mixing is relatively rapid in the mixed layer so Ra activities should be approximately uniform vertically. However, few studies have been conducted to test this assumption (see below). A water column may appear to be well mixed as evidenced by near uniformity in temperature or salinity profiles but may appear to be less well mixed for radioactive tracers if the timescale of mixing is similar to, or longer, than the timescale of radioactive decay.

Where vertical Ra profiles have been measured on the inner shelf, some variations in activity have been reported. For example, Moore et al. (1995) observed noticeable vertical variations in ²²⁶Ra, ²²⁸Ra, and ²²⁴Ra on the Amazon shelf, but these were attributed in part to variations in salinity (i.e., mixing of different sources of water). Levy and Moore (1985) observed increases in ²²⁴Ra activity with depth in the mixed layer off the coast of South Carolina and Georgia, which they attributed to input from the seafloor. During an extensive survey of the South Atlantic Bight, occasional sampling near the surface and the bottom of the mixed layer showed that in most cases Ra activities were similar (Moore, 2007). Thus, vertical variations in Ra activity in the mixed layer may occur under some conditions in inner shelf environments.

Here, a theoretical assessment was made to show how variations in the vertical mixing rate could influence water column Ra activities in a shelf environment. As the focus here was on the short-lived Ra isotopes (²²³Ra and ²²⁴Ra), only dispersive transport was considered because advective transport in the offshore direction is usually not important for them (Lamontagne and Webster, 2019). The assessment was made by developing a two-dimensional model that included dispersion in the vertical direction, based on a previous one-dimensional offshore dispersive transport model by Hancock et al. (2006). In a first step, the effect of variations in the vertical coefficient of solute dispersivity (K_z) on Ra distribution in the water column was evaluated. In a second step, the effect of other transport variables (K_x, F_o, and F_B) on Ra distribution in the water column was evaluated.

In the following, the 2D model is described along with the range in parameter values considered. Whether the current experimental design for tracer surveys in coastal areas provides biased estimates of water column Ra activities is discussed.

Materials and Methods

In this study, the model domain for Ra transport was considered to be the cross-section of a sloping continental shelf (Figure 1). The sources of Ra to the water column were from the coast and the seabed. Tracer transport was assumed to occur via dispersive processes only, which can be described by the diffusion equation in the x (offshore) and z (depth) directions:

FIGURE 1

Figure 1. Conceptual representation of the cross-section for a sloping continental shelf with key boundary conditions controlling the Ra flux. F_o, coastal flux; F_B, benthic flux; F_e, offshore boundary flux (set to 0). Solute exchange between cells occurs via dispersive processes.

$$\frac{\partial A}{\partial t}=\frac{\partial K_x\partial A}{{\partial }^{2}x}+\frac{\partial K_z\partial A}{{\partial }^{2}z}-\mathrm{\lambda }A(1)$$

where A is the Ra activity, t is time and λ the decay constant for a given isotope. Considering the steady-state condition and anisotropic cross-section, Eq. (1) simplifies to:

$$K_x\frac{{\partial }^{2}A}{\partial x^2}+K_z\frac{{\partial }^{2}A}{\partial z^2}=\mathrm{\lambda }A(2)$$

This equation was solved numerically using the finite difference method. The numerical grid incorporated a single cell at the coastline and 200 vertical cells at the offshore boundary (Figure 1). Boundary conditions included a constant flux F_o (Bq m^–1 s^–1) at the coastline and a constant benthic flux F_B (Bq m^–1 s^–1) spread evenly between the cells on the seabed. A no-flux boundary condition (F_e = 0) was assigned to the offshore end. Cell dimensions were 200 m in the x and 2 m in the z directions, ensuring the offshore boundary was far enough away from the coast (40 km) in order to reasonably meet the F_e = 0 boundary condition at this boundary (vertical profiles were investigated for the first 10 km only).

The literature was scanned for a reasonable range in K_x, K_z, F_o, and F_B for a shelf environment – these are summarized in Tables 1, 2. In a first step, Eq. (2) was solved for different K_z values for each Ra isotope using default values for K_x, F_o, and F_B (Table 2). This process identified which isotope if any was sensitive to variations in K_z in terms of uneven activity distribution in the water column. For those isotopes sensitive to K_z, the additional effects of variations in K_x, F_o, F_B, and seabed slope (m) on vertical Ra distribution were evaluated for a relatively low K_z value. The MATLAB script used to solve Eq. (2) is provided as Supplementary Material.

TABLE 1

Table 1. Literature estimates for K_z in different marine environments.

TABLE 2

Table 2. Parameter range for K_z, K_x, F_o, and F_B used in the simulations.

Results

K_z

The range of K_z values found in the literature – including both coastal and deep ocean areas – spanned 7.5 × 10^–6 to 10^–1 m² s^–1. Thus, K_z values ranging from 10^–6 to 10^–2 m² s^–1 were used here. Variations in K_z did result in noticeable changes in the vertical and horizontal distribution of the shorter-lived tracers. For example, when using the default values for F_o and F_B for ²²³Ra, K_z = 10^–6 m² s^–1, and K_x = 10 m² s^–1, ²²³Ra distribution was clearly lower at the surface than at depth at a given distance from the shoreline, especially when farther offshore (Figure 2).

FIGURE 2

Figure 2. Variations in ²²³Ra activity in the water column for a relatively low K_z (10^–6 m² s^–1) and high K_x (10 m² s^–1).

The simulations showed that vertical patterns in Ra activity as a function of K_z were complex, with the highest activities either at the surface, the seabed, or at intermediate depths depending on the magnitude of K_z, the Ra isotope, and the offshore distance. To emphasize these, vertical variations in Ra activity were demonstrated at x = 2, 5, and 10 km for various K_z values (Figure 3). In general, water column Ra activity was nearly constant when K_z > 10^–4 m² s^–1, with the exception of the longest – lived ²²⁶Ra which was insensitive to K_z under the conditions tested. For the cases when K_z < 10^–4 m² s^–1, ²²³Ra and ²²⁴Ra activities were highest near the seabed at 2 km and in proximity of the surface at 5 and 10 km. The differences in activity through the water column were substantial at low K_z values for ²²³Ra and ²²⁴Ra. For example, ²²⁴Ra activities at 2 km were ∼2 mBq L^–1 near the surface and ∼ 5 mBq L^–1 near the seabed. The differences in activity through the water column were more subdued for ²²⁸Ra and only noticeable at the lowest K_z value. For example, ²²⁸Ra at 2 km varied from ∼0.30 to ∼0.35 mBq L^–1 through the water column (which would probably not be apparent in a field setting due to measurement uncertainty). Overall, only sampling for surface water would tend to bias the estimate of water column activity at low K_z values for ²²³Ra, ²²⁴Ra, and possibly ²²⁸Ra but not for ²²⁶Ra under the conditions evaluated.

FIGURE 3

Figure 3. Vertical variations in Ra activity for a range in K_z values for ²²³Ra (A–C), ²²⁴Ra (D–F), ²²⁸Ra (G–I), and ²²⁶Ra (J–L) at 2 km (left column), 5 km (middle column) and 10 km from the coastline (right column).

K_x

In addition to slow mixing of the water column, the complex distribution in Ra activity at low K_z also reflects the presence of two sources of Ra (coast and seabed) and how Ra from these sources is being dispersed through the water mass, especially as it becomes deeper offshore. Firstly, the effect of K_x was evaluated for ²²⁴Ra, ²²³Ra, and ²²⁸Ra for a relatively low K_z value (10^–5 m² s^–1). Ra-226 was not evaluated because it was insensitive to K_z under the conditions tested, as demonstrated in the previous section.

The horizontal and vertical ²²³Ra and ²²⁴Ra distributions were substantially impacted by variations in K_x (Figure 4). In general, there was a tendency for the lowest ²²³Ra and ²²⁴Ra activities to be found near the seabed at low K_x values and near the surface at the higher K_x values. For example, for ²²⁴Ra at 5 km (Figure 4B), activities were ∼0.5 mBq L^–1 near the surface and ∼0.1 mBq L^–1 near the seabed for K_x ≤ 1 m² s^–1 but ∼0.3 mBq L^–1 near the surface and ∼1.5 mBq L^–1 near the seabed for K_x = 100 m² s^–1. In other words, with a low K_z and a high K_x, there is a tendency for Ra produced via a benthic source to be exported in the offshore direction rather than to be mixed into the overlying water column (see also Figure 2). Whilst the overall magnitude of the activity changed, unlike for ²²³Ra and ²²⁴Ra, variations in K_x did not noticeably change the vertical distribution in ²²⁸Ra.

FIGURE 4

Figure 4. Variations in ²²⁴Ra (A–C), ²²³Ra (D–F), and ²²⁸Ra (G–I) activity as a function of K_x for a relatively low K_z value (10^–5 m² s^–1) at 2 km (left column), 5 km (middle column), and 10 km (right column).

F_o, F_B, and Seabed Slope

For simplicity, this part of the analysis only considered ²²⁴Ra (the worst-case scenario due to its shortest half-life). Variations in the magnitude of the coastal and seabed fluxes as well as seabed slope had a noticeable impact on the vertical ²²⁴Ra profiles (using K_z = 10^–5 m² s^–1). Changing the magnitude of F_o changed vertical ²²⁴Ra profiles when getting closer to the shoreline (Figures 5A,B) but not offshore (Figure 5C). This demonstrated that the coastal flux does not contribute any ²²⁴Ra to the water column at 10 km and beyond under the conditions tested (i.e., all the ²²⁴Ra originates from the seabed). In contrast, variations in F_B impacted the profiles at 2, 5, and 10 km with a tendency for ²²⁴Ra activity to be highest near the surface at higher F_B (Figures 5D–F) This may seem counterintuitive at first but can be explained by the sloping seabed. The highest ²²⁴Ra activities occur near the shoreline due the combined F_o and F_B fluxes and a smaller dilution potential in the shallower water column there (Figure 2). This ²²⁴Ra is then primarily dispersed laterally (that is, near the surface) due to K_x >> K_z in the simulations.

FIGURE 5

Figure 5. Effects of variations in F_o (A–C), F_B (D–F), and seabed slope (G–I) on the vertical distribution of ²²⁴Ra for a relatively low K_z value (10^–5 m² s^–1).

The assessment of the effect of seabed slope (m) required a small modification to the finite difference model, where the thickness of the cells on the z-axis was modified (to 1 m for m = 0.005 and 4 m for m = 0.02). These simulations suggested that a bias associated with slow vertical mixing would tend to become more significant along transects with deeper mixed layers (Figures 5G–I), especially farther offshore.

Discussion

Whilst the simulations were based on a simplistic representation of solute transport in coastal environments, especially for vertical mixing (Pacanowski and Philander, 1981; Durski et al., 2004), they clearly indicated that short-lived radiotracers like ²²³Ra and ²²⁴Ra may have a non-uniform vertical activity distribution in shelf environments when vertical dispersion is low. This is of concern for the design of sampling programs for SGD tracers in coastal environments, where only surface water samples are often collected. Moreover, it is not possible to generalize if there will be a tendency to under- or overestimate whole water column activities by surface water sampling because this will be in part determined by other factors, such as location along the transect, K_x, and the source and magnitude of the tracer flux.

Lietzke and Lerman (1975) made similar observations to this study when evaluating the role of anisotropy in diffusivity on the vertical distribution in ²²²Rn in Santa Barbara Basin. As for ²²³Ra and ²²⁴Ra distribution along a sloping seabed evaluated here, they found that much of the ²²²Rn in the water column of this basin was from benthic production at shallower depth moving laterally owing to K_x >> K_z. The basin also had a noticeable vertical gradient in ²²²Rn activity. Levy and Moore (1985) occasionally observed vertical ²²⁴Ra gradients in the mixed layer off the coast of North Carolina and Georgia, which they attributed to a ²²⁴Ra input from the seafloor. Such gradients may be more common than previously recognized for shelf environments.

Comparison With Characteristic Mixing Timescales

The simulation results are consistent with a qualitative assessment of the characteristic timescales of vertical mixing relative to radioactive decay. In general, transport processes tend to be less important in the mass-balance of a tracer when they occur more slowly than radioactive decay. For example, the characteristic timescale for vertical mixing by dispersion will be L²/K_z, where L (m) is a representative mixed layer thickness. If L²/K_z >> 1/λ, then vertical mixing is unlikely to contribute to differences in Ra activity in the water column. For example, assuming L = 20 m and K_z = 10^–5 m² s^–1, the characteristic timescale for vertical mixing will be 4 × 10⁷ s, whereas the timescales for radioactive decay (=1/λ) are 4.6 × 10⁵ s, 1.4 × 10⁶ s, 2.6 × 10⁸ s, and 7.2 × 10¹⁰ s for ²²⁴Ra, ²²³Ra, ²²⁸Ra, and ²²⁶Ra, respectively. This scaling is consistent with the numerical simulations performed here, which showed large vertical variations in activity due to slow mixing for ²²⁴Ra and ²²³Ra, small variations for ²²⁸Ra, and no effect for ²²⁶Ra.

Radon-222 is another common SGD tracer (Stieglitz et al., 2013; Tait et al., 2016) with a half-life (3.82 days) comparable to ²²⁴Ra. Thus, surface water sampling for ²²²Rn may similarly, not be representative of the whole water column activity when the vertical mixing rate is low. Being a dissolved gas, a further complication with ²²²Rn would be degassing, which has a characteristic timescale L/v (where v is the degassing velocity). The characteristic degassing timescale for ²²²Rn would typically be slightly slower than its timescale for radioactive decay. For example, for L = 20 m and v = 1 m day^–1 (∼1.2 × 10^–5 m s^–1), the characteristic timescale for degassing is ∼2 × 10⁶ s (relative to ∼5 × 10⁵ s for radioactive decay). This is also similar to the timescale of vertical mixing 4 × 10⁶ s for ∼K_z = 10^–4 m² s^–1 and L = 20 m. Thus, slow vertical mixing may impact on ²²²Rn activities in the water column because it can be slower than the timescales of both radioactive decay and degassing.

K_z

A potentially mitigating or complicating factor in our analysis is that a constant K_z was assumed across the model domain, which may be unrealistic in some field settings (Koch-Larrouy et al., 2015). K_z is an empirical parameter summing-up a range of potential dispersive processes induced by wind, tides, and other driving forces. As per other similar empirical dispersion parameters, K_z should be anticipated to be scale-, and time-dependent. For example, K_z can change significantly between calm and stormy conditions (Manucharyan et al., 2011) or following changes in wind direction (Kirincich and Barth, 2009). Some coastal embayments have periods with no tidal movements (de Silva Samarasinghe, 1998), which should also temporarily reduce K_z. This temporal variations in K_z could either mitigate or accentuate biases arising from surface sampling for tracers. As an example, if atmospheric fronts generating greater K_z have a characteristic timescale less then radioactive decay for a given tracer, the increased mixing of the water column they generate would tend to homogenize water column activities. Considering the paucity of data about vertical variations in Ra activity in inner shelf environments, the simple representation of K_z used here is probably sufficient on a preliminary basis.

Quantifying the Bias

Due to the paucity of information on vertical variability in Ra activity for inner shelves, it is not possible here to test the findings of this theoretical assessment with measurements under different field conditions. The potential magnitude of the bias can be evaluated here by comparing the longitudinal profiles for Ra from surface water relative to the depth-averaged values across the mixed layer generated from the simulations. For simplicity, this evaluation will be restricted to ²²⁴Ra (a worse-case scenario due to its short half-life). The evaluation will be made for K_z varying from 10^–6 to 10^–4 m² s^–1 (and for K_x = 10 m² s^–1, F_o = 1 Bq m^–1 s^–1 and F_B = 1 Bq m^–1 s^–1).

Consistent with previous analyses, the bias was greatest for K_z = 10^–6 m² s^–1 between surface and depth-average ²²⁴Ra activities, with a tendency for surface samples to underestimate activities inshore and slightly overestimate activities offshore (Figure 6A). The bias was less pronounced for K_z = 10^–5 m² s^–1 and negligible for K_z = 10^–4 m² s^–1 (Figures 6B,C). For K_z = 10^–6 m² s^–1, simulations were then run to better match the “observed” surface ²²⁴Ra activities using different F_o values (with K_x and F_B remaining unchanged). The closest match to the surface ²²⁴Ra activities was for F_o ∼ 0.25 to 0.50 Bq m^–2 s^–1, or 50–75% of the correct value (1 Bq m^–1 s^–1) (Figure 6D). Whilst only a crude evaluation in the absence of field measurements, this analysis suggests the bias caused by surface water sampling when K_z is low could be important.

FIGURE 6

Figure 6. Comparison of longitudinal Ra profile between surface samples and depth-averaged ²²⁴Ra activities for (A) K_z = 10^–6 m² s^–1, (B) K_z = 10^–5 m² s^–1, and (C) K_z = 10^–4 m² s^–1 (for K_x = 10 m² s^–1, F_o, and F_B = 1 Bq m^–1 s^–1). (D) Evaluation of the bias in F_o when only using surface relative to depth-average ²²⁴Ra activities.

Conclusion

There is a paucity of vertical Ra profiles over inner continental shelves. We demonstrated here that when vertical mixing is slow, vertical variations in short-lived radionuclides activity in a coastal mixed layer are possible. This is of concern for field programs aiming to measure SGD or other processes using these tracers. When the water column is not well mixed relative to a given tracer, only sampling near the surface (the current practice) will lead to biased estimates of the spatial distribution of the tracer and, consequently, of the inferred SGD estimates. Despite the significant effort required to collect Ra samples from seawater (Henderson et al., 2013), future field programs for SGD tracers over continental shelves should aim to collect at least one vertical profile in the mixed layer for short-lived tracers to evaluate the potential for slow vertical mixing. A potential benefit in measuring the vertical distribution in SGD tracers in the water column could be to help evaluate its recirculated component, which has a large component originating from the seabed.

Data Availability

No new dataset was generated in this study. The MATLAB script used to perform the simulations is available as Supplementary Material.

Author Contributions

SL scoped the theoretical analysis, did the numerical simulations, and drafted the manuscript. IW contributed to the theoretical development and drafting of the manuscript.

Funding

This manuscript was made possible by funding from CSIRO Land & Water to SL.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

We acknowledge the insightful comments from reviewers on a previous publication, who recommended that we also explore biases associated with vertical dispersivity.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fmars.2019.00357/full#supplementary-material

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