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Front. Mar. Sci., 02 July 2019 |

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

  • 1CSIRO Land & Water, Waite Laboratories, Urrbrae, SA, Australia
  • 2Retired, Canberra, ACT, Australia

Trends in radium (Ra) activity in coastal seawater are frequently used to infer submarine groundwater discharge. In general, unlike in the deep oceans, Ra samples are only collected from the surface of the mixed layer in coastal areas. The assumption is that the water column is well mixed, as often evidenced by uniform temperature and salinity profiles. However, if the timescale for vertical mixing is similar to or less than the timescale for radioactive decay, the vertical profiles in Ra activity may not be uniform. In the present work, a two-dimensional dispersion model was developed to evaluate the potential effects of slow vertical mixing on Ra distribution in the mixed layer of an inner shelf. The variables considered were the vertical coefficient of solute dispersivity (Kz), the offshore coefficient of solute dispersivity (Kx), the coastal Ra flux (Fo), the benthic Ra flux (FB), and the slope of the seabed. The shorter-lived Ra isotopes (223Ra and 224Ra; t1/2 = 3.66 and 11.4 days, respectively) were sensitive to Kz when its value was low (<10–4 m2 s–1), resulting in complex activity patterns in the water column as a function of the other variables. Ra-228 (t1/2 = 5.75 years) was only moderately impacted by low Kz but the long-lived 226Ra (t1/2 = 1600 years) was insensitive to Kz. Surface water samples may not always be representative of water column Ra activity when Kz is low, which will need to be taken into account in future field programs for seawater Ra distribution in shelf environments.


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 (Kx), the tracer flux from the coastline (Fo), and the tracer flux from the seabed (FB) (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 Kx estimates, and Lamontagne and Webster (2019) showed how Kx 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 (Kz) on Ra distribution in the water column was evaluated. In a second step, the effect of other transport variables (Kx, Fo, and FB) 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. Conceptual representation of the cross-section for a sloping continental shelf with key boundary conditions controlling the Ra flux. Fo, coastal flux; FB, benthic flux; Fe, offshore boundary flux (set to 0). Solute exchange between cells occurs via dispersive processes.

A t = K x A 2 x + K z A 2 z - λ 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 2 A x 2 + K z 2 A z 2 = λ 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 Fo (Bq m–1 s–1) at the coastline and a constant benthic flux FB (Bq m–1 s–1) spread evenly between the cells on the seabed. A no-flux boundary condition (Fe = 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 Fe = 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 Kx, Kz, Fo, and FB for a shelf environment – these are summarized in Tables 1, 2. In a first step, Eq. (2) was solved for different Kz values for each Ra isotope using default values for Kx, Fo, and FB (Table 2). This process identified which isotope if any was sensitive to variations in Kz in terms of uneven activity distribution in the water column. For those isotopes sensitive to Kz, the additional effects of variations in Kx, Fo, FB, and seabed slope (m) on vertical Ra distribution were evaluated for a relatively low Kz value. The MATLAB script used to solve Eq. (2) is provided as Supplementary Material.


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


Table 2. Parameter range for Kz, Kx, Fo, and FB used in the simulations.



The range of Kz values found in the literature – including both coastal and deep ocean areas – spanned 7.5 × 10–6 to 10–1 m2 s–1. Thus, Kz values ranging from 10–6 to 10–2 m2 s–1 were used here. Variations in Kz did result in noticeable changes in the vertical and horizontal distribution of the shorter-lived tracers. For example, when using the default values for Fo and FB for 223Ra, Kz = 10–6 m2 s–1, and Kx = 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. Variations in 223Ra activity in the water column for a relatively low Kz (10–6 m2 s–1) and high Kx (10 m2 s–1).

The simulations showed that vertical patterns in Ra activity as a function of Kz were complex, with the highest activities either at the surface, the seabed, or at intermediate depths depending on the magnitude of Kz, 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 Kz values (Figure 3). In general, water column Ra activity was nearly constant when Kz > 10–4 m2 s–1, with the exception of the longest – lived 226Ra which was insensitive to Kz under the conditions tested. For the cases when Kz < 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 Kz 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 Kz 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 Kz values for 223Ra, 224Ra, and possibly 228Ra but not for 226Ra under the conditions evaluated.


Figure 3. Vertical variations in Ra activity for a range in Kz 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).


In addition to slow mixing of the water column, the complex distribution in Ra activity at low Kz 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 Kx was evaluated for 224Ra, 223Ra, and 228Ra for a relatively low Kz value (10–5 m2 s–1). Ra-226 was not evaluated because it was insensitive to Kz under the conditions tested, as demonstrated in the previous section.

The horizontal and vertical 223Ra and 224Ra distributions were substantially impacted by variations in Kx (Figure 4). In general, there was a tendency for the lowest 223Ra and 224Ra activities to be found near the seabed at low Kx values and near the surface at the higher Kx 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 Kx ≤ 1 m2 s–1 but ∼0.3 mBq L–1 near the surface and ∼1.5 mBq L–1 near the seabed for Kx = 100 m2 s–1. In other words, with a low Kz and a high Kx, 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 Kx did not noticeably change the vertical distribution in 228Ra.


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

Fo, FB, 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 Kz = 10–5 m2 s–1). Changing the magnitude of Fo 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 FB impacted the profiles at 2, 5, and 10 km with a tendency for 224Ra activity to be highest near the surface at higher FB (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 Fo and FB 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 Kx >> Kz in the simulations.


Figure 5. Effects of variations in Fo (A–C), FB (D–F), and seabed slope (G–I) on the vertical distribution of 224Ra for a relatively low Kz 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.


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, Kx, 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 Kx >> Kz. 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/Kz, where L (m) is a representative mixed layer thickness. If L2/Kz >> 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 Kz = 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 ∼Kz = 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.


A potentially mitigating or complicating factor in our analysis is that a constant Kz was assumed across the model domain, which may be unrealistic in some field settings (Koch-Larrouy et al., 2015). Kz 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, Kz should be anticipated to be scale-, and time-dependent. For example, Kz 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 Kz. This temporal variations in Kz could either mitigate or accentuate biases arising from surface sampling for tracers. As an example, if atmospheric fronts generating greater Kz 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 Kz 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 Kz varying from 10–6 to 10–4 m2 s–1 (and for Kx = 10 m2 s–1, Fo = 1 Bq m–1 s–1 and FB = 1 Bq m–1 s–1).

Consistent with previous analyses, the bias was greatest for Kz = 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 Kz = 10–5 m2 s–1 and negligible for Kz = 10–4 m2 s–1 (Figures 6B,C). For Kz = 10–6 m2 s–1, simulations were then run to better match the “observed” surface 224Ra activities using different Fo values (with Kx and FB remaining unchanged). The closest match to the surface 224Ra activities was for Fo ∼ 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 Kz is low could be important.


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


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.


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.


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:


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Keywords: submarine groundwater discharge, radium, dispersion, vertical mixing, seawater

Citation: Lamontagne S and Webster IT (2019) Theoretical Assessment of the Effect of Vertical Dispersivity on Coastal Seawater Radium Distribution. Front. Mar. Sci. 6:357. doi: 10.3389/fmars.2019.00357

Received: 01 March 2019; Accepted: 07 June 2019;
Published: 02 July 2019.

Edited by:

Henrietta Dulai, University of Hawaiʻi at Mānoa, United States

Reviewed by:

Joseph James Tamborski, Woods Hole Oceanographic Institution, United States
Jennifer Joan Verduin, Murdoch University, Australia

Copyright © 2019 Lamontagne and Webster. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Sébastien Lamontagne,