Journal of Marine Systems 86 (2011) 28–33

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Journal of Marine Systems

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On the calculation of eddy diffusivity in the shelf water from radium isotopes:High sensitivity to advection

Chunyan Li a,b,⁎, Wei-Jun Cai c

a Department of Oceanography and Coastal Sciences, Coastal Studies Institute, Louisiana State University, United Statesb Marine Ecosystem and Environment Laboratory, College of Marine Sciences, Shanghai Ocean University, Chinac Department of Marine Sciences, University of Georgia, United States

⁎ Corresponding author at: Department of OceanoCoastal Studies Institute, Louisiana State University, Un3619; fax: +1 225 578 2520.

E-mail address: cli@lsu.edu (C. Li).

0924-7963/$ – see front matter © 2011 Elsevier B.V. Adoi:10.1016/j.jmarsys.2011.01.003

a b s t r a c t

a r t i c l e i n f o

Article history:Received 4 September 2010Received in revised form 27 January 2011Accepted 31 January 2011Available online 25 February 2011

Keywords:Eddy diffusivityAdvectionSensitivityRadioactiveContinental shelf

The concentrations of the radium isotopes have been used in previous studies to estimate eddy diffusivity onthe continental shelf. These studies assume that the advective transport of the radium isotopes is negligible. Atheoretical investigation using an analytic model with advection indicates, however, that the eddy diffusivitythus estimated is highly sensitive to the advection. It is shown that the error can be very large even for anadvective velocity of the order of 1 mm/s–1 cm/s. The sensitivity increases with the increase of the half life ofthe isotope. For a 1 mm/s advective velocity, the estimated eddy diffusivity for the radium isotope with theshortest half life (i.e. 224Ra) is almost doubled. In addition, we also conclude that (1) advection has moreimportant effects on smaller values of eddy diffusivity; (2) the effect of advection increases rapidly asadvection increases; and (3) offshore advection tends to increase the apparent eddy diffusivity and inshoreadvection tends to decrease the apparent eddy diffusivity, if the advection is ignored. Based on these facts, animproved model is presented to calculate both advection and eddy diffusivity using the activities of twoisotopes, which yields consistent diffusivity and advection from different isotope pairs.

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1. Introduction

Radium isotopes in seawater have been used as passive tracers ofcoastal waters to determine eddy diffusivity and submarine groundwater discharge in coastal ocean waters (Moore, 2000, 2007, 2010).The radium isotopes have a diverse range of decay constants, whichequal to 0.0608 day− 1, 0.1894 day− 1, 4.33×10− 4 year− 1, and0.12 year−1 for 223Ra, 224Ra, 226Ra, and 228Ra, respectively. Thesevalues of decay coefficient correspond to timescales (half life) of11.4 days (223Ra), 3.66 days (224Ra), 1601 years (226Ra), and5.75 years (228Ra), respectively. The first two are much shorter thanthe shelf water renewal timescale on the continental shelf of theSouth Atlantic Bight (SAB), which is on the order of 3 months(Atkinson et al., 1978), while the last two are much longer than that.In the study of Moore (2000), the cross shelf advection was assumednegligible and an analytic solution to the linear diffusion-radioactivedecay model was obtained, which was fitted to the radium isotopedata from the SAB through a log-linear regression to the activity of

individual isotopes. It was believed that advection would make thecross shelf distribution of the isotopes nonlinear on a log scale. Sincethe observational data from the SAB continental shelf did satisfy thelog-linear relation quite well, therefore the advection must have beennegligible (Moore, 2000). However, it appears that the eddydiffusivities calculated with different isotopes from the same surveyhave very different values. Besides, how advection would affect thelog-linear relation is unknown. These facts prompt us to examine thequestion under what conditions can the advection be neglected. Amore specific question is: assuming that the source is on the coast andthat its distribution along the shore is uniform, how sensitive is theeddy diffusivity (calculated from the diffusion-radioactive decaymodel) to cross shelf advection? To limit the complexity of theproblem, we pose this question in a mathematical way. In otherwords, under ideal conditions (e.g. uniform distribution along theshore; source distributed on the coast only, etc), if we can establish atheoretical relation between the actual eddy diffusivity Kh and theapparent eddy diffusivity K̃h obtained by ignoring the advection, thendifferent advection may correspond to different K̃h for a given Kh. Thesensitivity study examines the rate of change of K̃h to the change ofadvection u. Mathematically, we need to quantify ∂ K̃h(u,Kh)/∂u, withKh as a parameter in this relation. Only around those values of Kh and uthat result in “small” values of ∂ K̃h(u,Kh)/∂u, canwe use the diffusion-radioactive decay model, that ignores the advection, to reliably obtainthe approximate values of the eddy diffusivity.

Fig. 1. Effect of offshore (positive) cross shelf advection on the apparent eddy diffusivity(shown by the contours) for 223Ra, obtained from Eq. (8).

29C. Li, W.-J. Cai / Journal of Marine Systems 86 (2011) 28–33

Now the question becomes this: can we find the relation betweenK̃h and u, for a given Kh? The answer is yes because the diffusion-radioactive decay model can be extended to a broader applicationwhich includes the mean advection. In other words, if a constantadvective velocity can be used to approximate the cross shelf meanadvection, a similar analytic solution can be obtained. This will allowus to examine the effect of advection on the eddy diffusivity. In thefollowing sections, we will first develop such an extended model andprovide the solution. The sensitivity study will then be made,following which we will develop another model to calculate bothadvection and eddy diffusivity using two isotopes. Intuitively, using acombination of two isotopes to calculate the eddy diffusivity allowsthe use of more information and thus should present more reliableresults, given that the model assumptions are justified. For the modelto be valid, we assume that there is along-shelf uniformity and thatthe only source is on the coast.

2. Sensitivity analysis

In general, the diffusion–advection–decay process of a passive-tracer isotope in seawater can be described by the following equation

∂A = ∂t + u∂A= ∂x = Kh∂2A = ∂x2−λ A ð1Þ

where A, t, u, x, Kh, and λ are concentration (or activity), time, crossshelf advective velocity, cross shelf distance, horizontal eddydiffusivity, and radioactive decay coefficient, respectively. Amongthe parameters, λ is a known constant, Kh and u are unknown but areassumed constant here for simplicity. For a steady state, ∂A/∂ t = 0,and we have

∂2A = ∂x2− u= Khð Þ∂A= ∂x−λA = Kh = 0 ð2Þ

It is important to note that here the advective velocity u should beconsidered as a temporal mean value which does not include the tidalsignal. In other words, it should be obtained by averaging over a fewtidal cycles to filter out tides. Assume that the values of A on the coastare known and equal to A0 and it approaches to zero at the outer shelf.The boundary conditions for A are then:

A jx=0 = A0; A jx→∞ = 0 ð3Þ

The corresponding solution of Eq. (2) is then

A = A0eu−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λKh

p2Kh

x ð4Þ

Equivalently, the solution can be expressed as a log-linear format

log A = log A0 +u−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λKh

q2Kh

x ð5Þ

This shows that if the cross shelf advection (flow) is a constant, theconcentration or activity A will still be log-linearly distributed acrossthe shelf if the only source is at the coast. When u=0, the solutionbecomes that of Moore (2000), i.e.

log A = log A0−ffiffiffiffiffiffiλKh

sx ð6Þ

Therefore, Eq. (5) is a special case of Eq. (6). This indicates thateven if the distributions of the radium isotope across the shelf are log-linear, the advection is not necessarily negligible. Since both Eqs. (5)and (6) are log-linear, both can be used to fit radium isotope data asdescribed in Moore (2000). Assuming that there is a constantadvective velocity across the shelf, the method of Moore (2000) is

equivalent to the calculation of an “apparent” eddy diffusivity K̃h thatis related to the “true” eddy diffusivity Kh and the advective velocity uby

−ffiffiffiffiffiffiffiffiffiffiffiffiλ =K̃h

q= u−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λKh

q� �= 2Khð Þ ð7Þ

from which the “apparent" eddy diffusivity can be expressedexplicitly as

K̃h =λ

u−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λKh

q2Kh

0@

1A

2 ð8Þ

In the above equation, the apparent eddy diffusivity is dependenton the decay coefficient λ, advection u, and the actual eddy diffusivityKh. When u=0, the apparent eddy diffusivity is equal to the actualeddy diffusivity. If a small change in u does not result in a large changein K̃h, then wemay conclude that K̃h is not sensitive to the advection uand K̃h will be a good approximation of Kh. Our question is then: howsensitive is the apparent eddy diffusivity to the advective velocity? Inother words, how close are the apparent and true eddy diffusivitiesunder different cross shelf velocity values? To answer this question,we now discuss the relationship between Kh and K̃h under differentcross shelf velocity values by examining (1) the above equation and(2) the rate of change of K̃h with advection u, i.e. ∂ K̃h/∂u, for all theRadium isotopes. The rate of change of K̃h with respect to advection ucan be shown to be

∂K̃h

∂u =8λK2

h

u−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λKh

q� �2⋅

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λKh

q ð9Þ

While Eq. (9) provides a quantitative and formal sensitivity study,Eq. (8) provides some direct comparison between Kh and K̃h. We willtherefore first discuss Eq. (8) and then Eq. (9).

Figs. 1–4 show the results for 223Ra (Figs. 1 and 3) and 224Ra(Figs. 2 and 4) from Eq. (8). The results for 226Ra and 228Ra showmuchlarger range of variations of K̃h such that it is difficult to make contourplots and the results will be discussed later. They are, however, of lessinterest because of their rather long half life timescales (much longer

Fig. 2. Same as Fig. 1, except that it's for 224Ra. Fig. 4. Same as Fig. 3, except that it's for 224Ra.

30 C. Li, W.-J. Cai / Journal of Marine Systems 86 (2011) 28–33

than the shelf water renewal scales). Figs. 1 and 2 are for positiveadvection and Figs. 3 and 4 are for negative advection. In these figures,we only show the results for advective velocity between 0 and 6 cm/s,a range supported by some observations. As summarized in Moore(2000), the reasonable cross shelf advection on the South AtlanticBight have been found to between ~1 mm/s and 4 cm/s by numerousstudies (e.g. Windom and Gross, 1989; Pomeroy et al., 1993; Werneret al., 1993). For larger advective velocity values, the differencebetween Kh and K̃h is even larger. The contour lines show the values ofthe apparent eddy diffusivity K̃h. Obviously, at u=0, the contour linesintersect with the vertical axes at values equal to the values of thesecontours. For positive advection, as u increases, the actual eddydiffusivity decreases linearly if the apparent eddy diffusivity remainsunchanged. If the actual eddy diffusivity is given, the increase of theadvective velocity results in the increase of the apparent eddydiffusivity (Figs. 1 and 2). For negative advection, as the magnitudeof u increases, the actual eddy diffusivity increases linearly if the

Fig. 3. Effect of onshore (negative) cross shelf advection on the apparent eddydiffusivity (shown by the contours) for 223Ra, obtained from Eq. (8).

apparent eddy diffusivity remains unchanged. If the actual eddydiffusivity is given, the increase of the magnitude of the negativeadvective velocity results in the decrease of the apparent eddydiffusivity (Figs. 3 and 4).

As some examples, Tables 1–6 show the comparisons between thevalues of K̃h and Kh under different advection for all the four isotopes.The significance of the effect of advection on K̃h for all isotopes isquantified by the ratio between K̃h and Kh. In general, the larger theadvection, the larger the effect. The advection has the smallest effecton K̃h for 224Ra (Table 2 for positive u and Table 4 for negative u), theisotope with the shortest half life. The advection has a slightly largereffect on K̃h for 223Ra (Table 1 for positive u and Table 3 for negativeu), the isotope with the second shortest half life. The effects ofadvection on K̃h for the two isotopes with the longest half life, i.e.226Ra and 228Ra, are significantly higher (Table 5 for positive u andTable 6 for negative u).

Eq. (9) provides a clearer picture of the difference in sensitivity toadvection among different isotopes. Conceptually, we are moreinterested in examining the rate of change at near u=0with differentλ values. By letting u=0, Eq. (9) reduces to

∂K̃h

∂u ju=0

=

ffiffiffiffiffiffiKh

λ

rð10Þ

This relationship indicates that the rate of change with respect toadvection at u=0 is proportional to the square root of the half life ofthe isotope. This result is consistent with the above discussion andfurther quantifies the difference in sensitivity of the apparent eddydiffusivity to advection. From this we can also conclude that ifadvection is neglected, the apparent eddy diffusivity obtained from224Ra, which has the shortest half life of only 3.66 days, will be closest

Table 1Relation between Kh and K̃h for 223Ra (uN0).

u (cm/s) Kh (m2 s−1) K̃h (m2 s−1) K̃h/Kh

1 100 310 31 300 590 22 100 755 7.52 300 1086 3.63 100 1472 153 300 1830 6

Table 2Relation between Kh and K̃h for 224Ra (uN0).

u (cm/s) Kh (m2 s−1) K̃h (m2 s−1) K̃h/Kh

1 100 194 1.91 300 442 1.52 100 354 3.52 300 642 2.13 100 594 63 300 912 3

Table 4Relation between Kh and K̃h for 224Ra (ub0).

u (cm/s) Kh (m2 s−1) K̃h (m2 s−1) K̃h/Kh

−1 100 52 0.5−1 300 204 0.7−2 100 28 0.28−2 300 140 0.47−3 100 17 0.17−3 300 99 0.33

31C. Li, W.-J. Cai / Journal of Marine Systems 86 (2011) 28–33

to the actual value. The errors of estimate increase with 223Ra, 228Ra,and 226Ra in that order.

Figs. 5 and 6 are results from Eq. (9) for Kh=300 m2 s−1. Results forother values of Kh are similar and are omitted on the figures for clarity.The unit for ∂ K̃h/∂u ism. Fig. 5 is for positive uwhile Fig. 6 for negativeu.For positive u, ∂ K̃h/∂u is on the order of 104−105 m, 104 m, 109−1010 m, and 107 m for 223Ra, 224Ra, 226Ra, and 228Ra, respectively(Fig. 5). For negative u, ∂ K̃h/∂u is on the order of 103−104 m, 103−104 m, 10−2−106 m, and 1–106 m for 223Ra, 224Ra, 226Ra, and 228Ra,respectively (Fig. 6). Thesegradient values are extremely large, indicatinghigh sensitivity of the methodology of using the concentrations of theradioactive elements to calculate the eddy diffusivity without consider-ing the effect of advection. The difference between positive and negativeadvection is quantitative, not qualitative. The difference is by values, i.e.theconclusion about thehigh sensitivity is the samewhether advection ispositive or negative. Of course, there is no reason for us to believe that thepositive and negative advection must give exactly the same result.

In summary, the above results indicate that advection has largereffects on eddy diffusivity calculated from activities of 223Ra, 228Ra,and 226Ra than from 228Ra. Positive (offshore) advection causes largerapparent eddy diffusivity and negative (inshore) advection causessmaller apparent eddy diffusivity. The advection effects are moreimportant at small eddy diffusivity values. Note again, that the resultsfor 226Ra and 228Ra are included here for reference and comparison.They are of less practical interest because their decay timescales arelonger than the scales of the continental shelf dynamical processes.

3. Solving Kh and u simultaneously

Since advective velocity affects the apparent eddy viscosity, it isdesirable to calculate both the advective velocity and the true eddydiffusivity at the same time. With observational data for four differentisotopes that are from similar sources, this is possible. Thus we denotethe decay coefficients for two isotopes by λ1 and λ2. Obviously, Eq. (5)should hold for either isotope and the log-linear fit for these isotopeswill satisfy the following relations,

1L1

=u−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λ1Kh

q2Kh

ð11Þ

1L2

=u−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λ2Kh

q2Kh

ð12Þ

Table 3Relation between Kh and K̃h for 223Ra (ub0).

u (cm/s) Kh (m2 s−1) K̃h (m2 s−1) K̃h/Kh

−1 100 32 0.32−1 300 153 0.51−2 100 13 0.13−2 300 83 0.28−3 100 7 0.07−3 300 49 0.16

in which 1/L1 and 1/L2 are the slopes from the log-linear fit. Fortechnical convenience, here we use L1 and L2, which have a dimensionof length, instead of the slopes. Multiplying Eq. (11) by 2 Kh L1 andadding −L1u to both sides of the equation yield

2Kh−L1u = −L1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λ1Kh

qð13Þ

Similarly, multiplying Eq. (12) by 2 Kh L2 and adding −L2u to bothsides of the equation yield

2Kh−L2u = −L2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 + 4λ2Kh

qð14Þ

Multiplying each of the above equations with itself and with somestraightforward algebraic operations we get

Kh−uL1 = λ1L21 ð15Þ

Kh−uL2 = λ2L22 ð16Þ

We then subtract Eq. (15) from Eq. (16) to obtain u and thensubstituting the solution of u into the above equations to obtain Kh,

u =L21λ1−L22λ2

L2−L1ð17Þ

Kh = L22λ2 + L2u = L21λ1 + L1u ð18Þ

Before calculating the advection and eddy diffusivity using the aboveequations, we first summarize the log-linear fit to the cross shelfdistribution of the data between 0 and 50 km offshore in the SouthAtlantic Bight (Moore, 2000) for convenience of comparison. The log-linear slope for the activities of 223Ra, 224Ra, 226Ra, and 228Ra are−0.0439,−0.0724,−0.0061, and−0.0071 km-1, respectively. The log-linear fits yield R2 values of 0.973, 0.986, 0.922, and 0.942 for the fourisotopes, respectively. The standard errors are 0.116, 0.136, 0.072, and0.072 dpm 100 L-1 for the four isotopes. These R2 and standard errorvalues indicate that the data distributions across the shelf are very closeto log-linear. With these slope values, and without considering theadvection, the eddy diffusivity values would be 367, 418, 0.4, and75m2 s−1 for 223Ra, 224Ra, 226Ra, and 228Ra, respectively. These valuesdiffer by 3 orders of magnitude. One explanation is that the activities of

Table 5Relation between Kh and K̃h for 226Ra and 228Ra (uN0).

u (cm/s) Kh (m2 s−1) K̃h from 226Ra (m2 s−1) K̃h from 228Ra (m2 s−1)

1 100 7.3×1012 2.7×1010

1 300 7.3×1012 2.7×1010

2 100 2.9×1013 1.1×1011

2 300 2.9×1013 1.1×1011

3 100 6.6×1013 2.4×1011

3 300 6.6×1013 2.4×1011

Table 6Relation between Kh and K̃h for 226Ra and 228Ra (ub0).

u (cm/s) Kh (m2 s−1) K̃h from 226Ra (m2 s−1) K̃h from 228Ra (m2 s−1)

−1 100 1400 3.8×105

−1 300 12,400 3.3×106

−2 100 343 9.5×104

−2 300 3100 8.5×105

−3 100 153 4.2×104

−3 300 1400 3.8×105

32 C. Li, W.-J. Cai / Journal of Marine Systems 86 (2011) 28–33

226Ra and 228Ra do not vanish as x→∞ (or at the shelf break). This isbecause the half life of these two isotopes is much longer than thetimescales of the circulation especially that related to cross shelftransport. However, this cannot satisfactorily explain the 1000 timesdifference in the calculated eddy diffusivity (418 vs. 0.4) because thenon-zero activities of 226Ra, 228Ra at shelf break is only a fraction of thoseat the coast (Moore, 2000), therefore the general trend of theconcentrations of these isotopes is still →0 towards the shelf.

Ifwe choose any twoof the four isotopes,we canget the advection andeddy diffusivity as shown by Eqs. (17) and (18). The results of which aresummarized in Table 7. The advective velocity values are negative for allpairs ranging from −0.25 mm/s (from 223Ra and 228Ra) to −0.59 mm/s(from 223Ra and 224Ra), with an average value of−0.33±0.13 mm/s. Theeddy diffusivity values are between 422 and 507m2 s-1, with an averagevalue of 461±36m2 s-1. Apparently, different pairs of radium isotopesyield quite consistent advection and eddy diffusivity values. This is insharp contrast to the results calculated from individual isotopes withoutthe consideration of advection which yielded eddy diffusivities spanninga 3 orders of magnitude range. Since the calculated advective velocity isboth small and negative, the impact of advection in this case is not thatgreat and our results are larger than those without advection by as muchas 26%. This is not a bad approximation. With a positive advectionand even slightly larger than the present magnitude of advection inTable 7, the difference could have been much larger. The cross shelfadvective velocity can be a few cm and even much larger than that.

Fig. 5. Rate of change (derivative) of the apparent eddy diffusivity with respect to advectionobtained from Eq. (9). The unit is m. The four panels (a), (b), (c), and (d) are for 223Ra, 224

4. Concluding remarks

The sensitivity analysis shows that the apparent eddy diffusivityobtained from the log-linear fit based on the diffusion-decay model ishighly sensitive to changes in advection. Although the results showthat the isotope with the shortest half life, i.e. 224Ra (3.66 days),presents the smallest error of estimation for eddy diffusivity ifadvection is neglected, the sensitivity to the advection is so large thatthe previous method to calculate eddy diffusivity without includingadvection can be misinterpreted. The other three isotopes, 223Ra,228Ra, and 226Ra, present even bigger errors. This order of increasederror is the same as the order of increased half life. Apparently, this isdetermined by the mathematical relation, i.e. Eq. (8) with λ as aparameter. The rate of change of K̃h with respect to u increases withdecreasing λ. The sensitivity analysis also indicates that offshore(positive) advection causes a larger apparent eddy diffusivity than theactual values; and onshore (negative) advection causes a smallerapparent eddy diffusivity. The smaller the actual eddy diffusivity, thelarger the effect of advection on the apparent eddy diffusivity. Thelarger the advection, the larger difference between the apparent andactual eddy diffusivities.

In light of the results from the sensitivity analysis, we havedeveloped a model that uses two isotopes to calculate the eddydiffusivity. Using the data published in Moore (2000), we have testedthis method and the results are robust for different combinations ofisotopes. The calculated advection and eddy diffusivity are −0.33±0.13 cm s−1 and 461±36 m2 s−1, respectively.

It should be emphasized that the main point of this article is thatthe previous method of using the radium isotopes to calculate theeddy diffusivity can lead to significant errors due to the fact the resultsare very sensitive to advection, a parameter not considered in theconventional method. We caution the users of the eddy diffusivityfrom those results, or those who still apply the conventional method,the potential risk of misinterpretation of the results. In the newmethod the advection is taken into account, and it improves the

u for offshore (positive) advections and for the actual eddy diffusivity to be 300 m2 s−1,Ra, 226Ra, and 228Ra, respectively.

Fig. 6. Same as Fig. 5, except that here the derivative is for negative advections.

33C. Li, W.-J. Cai / Journal of Marine Systems 86 (2011) 28–33

consistency and eliminates the sensitivity of the eddy diffusivity to theadvective velocity. The new method still uses the one-dimensionalequation. As a general practice, such as in the original Moore's papersand in many other papers that followed Moore's, marine geochemistshave used one-dimensional eddy diffusion analytical model for suchtracer-based mixing modeling. It is thus appropriate that we use asimilar one-dimension analytical equation with advection to illustratethe issues with the diffusion only approach. The fact that the newmethod does give consistent results with no sensitivity to advectioncan be viewed as an improvement for finding an average eddydiffusivity. The study area is relatively small compared to the scale ofthe coastline. With the radium isotopes having decay timescales ofgreater than three days, the eddy diffusivity thus calculated should beinterpreted as an averaged value over that timescale, and it should bea reasonable first order estimate of the averaged value in that area.Wewould also like to emphasize that very few analytic solutions canresolve a realistic problem. Rather, the ease of use of a 1-D analyticmethod is convenient for the study and interpretation of themechanism, and an efficient way to use the cross shelf radium isotopedata. The capability of one-dimensional models, however, shouldnever be over-estimated, despite its usefulness in the understandingof some basic concepts. With a simple analytic model, it is the

Table 7Comparison of Kh from different pairs of isotopes.

Isotope pair u (cm s−1) Kh (m2 s−1)

223Ra and 224Ra −0.59 499223Ra and 226Ra −0.26 424223Ra and 228Ra −0.25 422224Ra and 226Ra −0.28 456224Ra and 228Ra −0.27 456226Ra and 228Ra −0.31 507Mean −0.33±0.13 461±36

simplification that is of value, while the user should be aware of thelack of details (such as a 2-D structure) but usually one cannot haveboth.

Acknowledgments

This work was sponsored by NASA award No. NAG5-10557, theNOAA Coastal Ocean Program through the South Carolina Sea GrantConsortium pursuant to National Oceanic and Atmospheric Admin-istration award No. NA960PO113, Georgia Sea Grant award No. NA06RG0029-RR746-007/7512067, R/HAB-13B, R/HAB-12-PD, R/HAB-18-PD, Georgia DNR Award No. RR100-279-9262764, and NSF OCE-0425153. We thank W.S. Moore for discussion of related subject andfor sharing the original data.

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