Marine Environmental Research 22 (1987) 2L 5-232

A Simple Model for Pollutant Dispersal from a Sea-floor Source in the Presence of Bottom and Interior Scavenging

Christopher Garrett Department of Oceanography, Dalhousie University,

Halifax, NS. B3H 4Jl Canada

&

John Shepherd Ministry of Agricuhure, Fisheries and Food. Fisheries Laboratory.

Lowestoft, Suffolk NR33 0HT. Great Britain

(Received 9 July 1986; accepted 12 January 1987)

A B S T R A C T

We report the details and some extensions o/ a t/tree-dimensional model. tier'eloped hy GESA MP (1983), relevant to the oceanic dispersion of low- let, el radioactit~e waste escaping from a deep-sea dump site. The model includes simple parameterizations of physical dispersal, geochemical scat'engmg at the sea.[loor and in the water cohmm, and decav. A closed-form solution may he approximated asymptotically for t, arious physical regions to gire simple jbrmulae Jor the concentration. The results are o/ some operational value but are particularly use~fid for increasing intuition about the behatiour that may be expeeted j~'om more realistic models, bt particular, radionuelides may be located in a parameter space defned by two dimensionless numbers that show the importance of scat'enging and decay. Four d(fferent regions of this parameter space correspond to the radionuc/ides (or other pollutants) being primarily widespread at" cot~)qned near the source. and principally in the water column or on the sea-/toor sediments. It'e also show the relations~tip of" the three-dimensional model to a horizontally averaged one-dimensional model, and discuss the nature c~]the concentration fieM t 'eo' near the source where a constant. Fickian. difJilsit'ity is an incorrect parameterization of turbulent dispersion processes.

215

Marine Environ. Res. 0141-1136/87/$03"50 ~~ Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain

216 Christopher Garrett. John Shepherd

1 I N T R O D U C T I O N

Assessing the environmental impact of a pollutant escaping from a deep-sea dump site and the companion problem of regulating deep-sea waste disposal generally requires a prediction of the future pollutant concentration at any location in the ocean. The predicted concentration may then be incorporated into further calculations in which the effects on Man and marine biota are estimated for various biological pathways or modes of exposure.

This assessment process has been conducted with particular care and attention to detail for low-level radioactive wastes, with parallel research efforts being carried out by the International Atomic Energy Agency {e.g. IAEA, 1984) and by various regional or national agencies (e.g. OECD NEA, 1985).

A wide variety of oceanographic models have been used by different research groups to calculate the pollutant concentration field for a given release. To be useful a model must clearly allow for ocean circulation and mixing, and the decay rate {if any) of the pollutant. Biological and geochemical processes may also be important, as they are in determining the distribution of many naturally occurring substances in the ocean {e.g. Broecker & Peng, 1982). Some modelling groups have developed numerical models that are as elaborate and precise as modern computer power and our knowledge of the oceans permit. Other groups have opted for rather simple models, arguing that our limited knowledge of various oceanographic processes prevents the more elaborate models from producing significantly more accurate results. They also point out that simple models often lead to a greater understanding of the sensitivity of model predictions to various assumptions or the values of poorly known parameters.

This issue has been discussed further in a report (GESAMP, 1983), commissioned by the IAEA that recommended some rather simple models for use in the problem of low-level radioactive waste disposal. Even if it is argued that the use of more elaborate, and potentially more complete, models is required for this important problem, the simple GESAMP models provide very valuable benchmarks against which other models can be checked, thus reducing the chances of significant errors from programming mistakes in the more complicated numerical models.

The GESAMP models have also been very valuable "intuition builders', and it is mainly for this reason that we wish to discuss, in this paper, one of the GESAMP models, both in more detail than published in the original report and with further consideration of some shortcomings.

As already mentioned, biological and geochemical processes can be critically important in determining the fate of various substances in the

Pollutant dispersal l rom a sea4toor source' 217

ocean. In particular, various types of particulate material may have a high sorptive capacity for some chemical substances le.g. Broecker & Peng, 19821. These compounds may then be scavenged from the water column by sinking particles or taken up on existing sediments on the sea floor. A simple approximate parameterization of this process will be introduced in Section 2 of this paper.

In Section 3 we formulate the three-dimensional equations governing the concentration field for a sea-floor release of a pollutant, and show that for a fairly general class of three-dimensional models the horizontally averaged concentration is the same as that obtained from a one-dimensional {vertical) model. (This rules out the possibility that much more of an escaping pollutant is adsorbed on to scavenging bot tom sediments t`or a localized, rather than widespread, source.)

In Section 4 we briefly examine some simple results for one-dimensional models that are relevant to a discussion of three-dimensional models in Section 5. The latter are required, of course, l`or a calculation of the horizontal s t r u c t u r e {though not average) of the concentrat ion field.

While general solutions for the simple three-dimensional models we discuss can be obtained ['or a finite ocean, we shall concentrate, in Section 5, on solutions for an unbounded ocean. This is effectively one in which the pollutant is prevented, by decay or scavenging, from reaching the sides of an ocean basin.

This will lead us to the presentation, in Section 6, of an important two- dimensional parameter space from which the fate of a radionuc[ide with a particular decay rate and sorptive potential may be roughly assessed. The associated diagram (Fig. 3) on which we locate a few key radionuclides is one of the important results of the models.

The near-source concentrat ion of a particular pollutant can be a key factor, particularly in assessing the impact of a dumping practice on marine biota. While formulae for this concentrat ion will have been presented in Section 5, the models" parameterizat ion of mixing processes in the near field is incorrect. This issue, which was not addressed by G E S A M P (19831, is resolved in Section 7.

The model results presented in the paper assume a steady state and ignore advection by mean currents (with diffusion as the only physical transport process). These issues are discussed briefly in the concluding section.

2 THE P A R A M E T E R I Z A T I O N OF SCAVENGING

The uptake, or release, of a dissolved chemical by sinking particles in the ocean or by existing sediment on the sea floor is a complicated matter. For

218 Clwistopher Gurrett. John Shepherd

the purposes of the present paper we shall assume that the exchange rates between the particulate and dissolved phases or" a chemical are sufficiently high that an equilibrium state may be assumed, with the d is t r ibut ion coefficient', KD, defining the ratio of the chemical concentrat ion on the particles to that in the water column. (But see G E S A M P 11983) and literature cited therein t'or a discussion of the validitx or" the equilibrium assumption.)

2.1 Interior scavenging

For a particulate concen t ra t ion , f {volume of particles per unit volume of the water column, typically of order 10-s to 10--), and a particulate sinking speed, wp, a vertical flux. - ~{C, must be allowed for in the equat ion for the total amount , [ t - / + f K D ) C , of the substance at some position in the water column, with the 'interior scavenging speed'. I'~, given by I] =fKDw p. In practice t~] might involve summing over particles with different Kos and sinking speeds, and might vary with depth.

2.2 Boundary scavenging

The removal of a chemical on to sea-floor sediments occurs at a rate I~C, where C is the concentrat ion in the water above the sediments and lid is a "deposition velocity'. Clearly, V d includes a contr ibut ion. I,~ ~'"f~. associated with the net deposit ion ofpart icles sinking through the ocean: I~ bur~ cannot be more than the interior scavenging speed, I,;. but may be less if some of the chemical adsorbed on to sinking particles is remobilized at the sea floor, or if some of the particles are organic and dissolve at the sea floor. As discussed by G E S A M P (1983), there are two other contr ibut ions to the deposit ion velocity:

(i) A superficial layer of sediments of thickness, h, part iculate concentra t ion by volume, f ' , and distr ibution coefficient, K D, that is being rapidly turned over by bioturbation, or by resuspension and redeposit ion in benthic storms, has a chemical inventory h 'C , where h* = h ( f ' K D + 1 - J " ) and C is the concentra t ion in the water above the sediments. If the decay rate of the chemical is ). the corresponding deposit ion speed is ).h*.

(ii) A chemical may also diffuse th rough the pore water and be taken up on the underlying sediments. This is discussed quanti tat ively by G E S A M P (1983), who show that its contr ibut ion to the total deposit ion velocity is generally less than one of the other two contr ibutions.

P¢,llutant dispersal.lr~mt a sea-rioor source 219

We are thus left with a deposit ion velocity ~ = ~ ~bu~, + ).h*, with the first term dominant for long-lived substances: for a bioturbated laver or" thickness such t h a t f ' h = 50mm and a net sediment accumulation rate of t 0 m m per 103 years the two contr ibutions to ~,~ are equal for a decay time /. L >_ 5000 .,,ears.

As remarked by G E S A M P { 19831. for rather short decay times the uptake of a substance by the sediments may occur via pore-water diffusion and involve less than the ~ hole bioturbated layer. This leads to a corresponding reduction in the contr ibution to ~ , but we will not pursue the issue further here.

3 G E N E R A L F O R M U L A T I O N

Allowing for radioactive decay at rate 2, the interior scavenging process discussed in Section 2, a velocity field, u, and horizontal and vertical mixing coefficients, K u and Kv, the concentrat ion field, C, satisfies the equation

~ C * & + V(uC*) - ~( I~C)/~_- + / . C = Vu(KHVHC*) + c(Kv cC /c-), 'c- (1)

where V. = (?&v, 8/?v) and C* = (1 --f+./KD)C is the total concentration per unit volume. In practice, fKu<<l except for extremely reactive substances (but I/I is still important), so, for simplicity, we replace C* by C in eqn (1).

The boundary condition at the sea surface, also assuming fKD << 1 and ignoring exchange with the atmosphere, is

Kv (C/~z + I/IC = 0 (2)

At the sea floor lassumed flat) with a distributed source, Q/(x) (where Q is the total release rate), and deposit ion velocity. ~ , we have

-K, ~C,",P-- ~;c= Of!x) - ~,~c t3)

3.1 Horizontal averaging

If we assume a depth-independent area of the ocean (i.e. assume the sides of the ocean to be effectively vertical) and ignore removal of the substance at lateral ocean boundaries, we can integrate eqn (1) horizontally over the ocean basin at any depth and use the divergence theorem to eliminate the horizontal advective and diffusive fluxes. If we also ignore horizontal variations in w, K v and V~, we obtain

~C,& + ~(wC)./'& - ?( I,~ C)./?z + ;.C = ~iKv ~C,&)/& (4)

for the horizontally averaged concentrat ion C.

220 Christopher Garre:t, John Shepherd

and. assuming that boundary condition:

The surface boundary condition is

K,,~C ~z + ~.,{C = o (5)

I,~ is also horizontallv uniform, we have a sea-floor

- K , . ( C / d z - ~';C= Q/A - ~ C (61

where ,4 is the area of the sea floor. We have thus shown that the horizontally averaged concentration field

may be calculated from a one-dimensional model and is independent of the actual distribution of the source on the sea floor. This is an important conclusion: it rules out the hypothesis, which might have been put forward, that highly reactive sediments remove more of an escaping pollutant from a highly localized source, before it can diffuse far into the ocean, than would be removed for a more diffuse source. This conclusion is independent of the nature of the horizontal transport mechanisms, but does require neglect of adsorption at lateral boundaries and depends upon the lateral constancy of ", ~i- ~ and Kv.

There can, of course, still be very significant lateral variations in C. and we v~ill return to this in Section 5. However, in view of the appropriateness of a one-dimensional model tbr the horizontally averaged concentration, C, and to illustrate the relative importance of I,,, V~, )., Kv and the depth. H, of the ocean, we shall next briefly discuss some simple one-dimensional results.

4 O N E - D I M E N S I O N A L MODELS

It is clear from eqns (1) and (4) that the vertical distribution of C, or C, may be significantly affected by the relative magnitudes of the upwetling speed, ,:, and the interior scavenging speed, ~']. G E S A M P {1983) show that for typical oceanographic parameters V~ becomes larger than w for K D more than about 10 6"

In this section, for the sake of simplicity and to illustrate also the importance of). or Kv in one-dimensional models that aim mainly to set the scene for the three-dimensional models to be discussed in Section 5, we ignore w and take V~, Kv and V~ to be constant. We also restrict our attention to a steady state, for which the governing equations for C are then

dZC dC Kv d-~, + V~ -d-7-z - ),C = 0 (7)

with boundary conditions given by eqns (5) and (6).

Pollutant dispersal from a sea-floor source 221

4.1 Boundary scavenging

In the absence of interior scavenging Iso that I] = O), the horizontally averaged concentration is given by

C=Q{A2H)-IBZ(BsinhB+ S c o s h B ) - t c o s h [ B ( i - z H)] (8)

as a function of height, z, above the sea floor in an ocean of depth H, with B z= ;.HZ/Kv and S = VdH/Kv.

The scaling factor. Q(A2H)-t, in eqn (8) is just the concentrat ion that would result in a well-mixed ocean with no scavenging, the total oceanic inventory. Q;,.-~. being the release rate times the decay time.

The concentration profile is determined by the dimensionless constant, B'-, which may be interpreted as the ratio of the oceanic vertical mixing time. H'-;Kv, to the decay time, ).-t; for Bz<<I the concentrat ion is uniform, whereas for B 2 >> 1 the contaminant is confined near the sea floor by slow vertical mixing and or decay. The effect of boundary scavenging occurs through the dimensionless parameter, S. which may be written S = (VdH-, . . . . Kv)"Hand interpreted as the depth that may be scavew, ed~ at a speed, V, in the mixing time. HZ/Kv, compared with the depth, H.

While boundary scavenging does not change the shape of the concentration profile it does reduce its magnitude; this is because part of the total inventory. Q(A;.)- L now resides on the bottom sediments rather than in the water column. In general, the ratio of the contaminant inventory on the sediment to that in the water column is SB- t co th B. This may be interpreted as the ratio of an equivalent water depth, ~'~t).- t, of the scavenging sediment to the effective depth of the ocean, which is the ocean depth, H. for a long- lived pollutant (B<< 1) and the diffusion scale, (Kv/;.) ~ 2, for a short-lived pollutant (B>> 1).

4.2 Interior scavenging

If V is retained in eqns {7), (5) and (6), the solution is somewhat more complicated (see GESAMP, 1983); here we examine only the situations in which the contaminant is confined near the sea floor by either decay or the effect of interior scavenging. In this case we may write

where

C = (Q/A)( G - V~ + Kvr) -~ exp ( - r z ) (9)

r = ½( VJKv)[ 1 + (1 + 4K%)./V.,2) 1'2] (10)

and we see that the confinement is due to decay, with a scale height (Kv/,;.) 1'-',

222 Christopher Garrett. Jokr. Shepherd

for 4&\.,:. >> ~]:, and due to interior scavenging. ~i th a scale height Kv~ I,~. for 4Kv,;. << ~';:. In the latter case

C = t Q A ) ~ j - t e x p ! - ~] - Kvl t l l )

which shows the dependence of the magnitude of the sea-floor concentrat ion on ~ . As discussed in Section 2.2. the deposition xelocity may be written I~ = I~ b ' ~ -r- ;,.h*, so that, as remarked by G E S A M P 1~,,~.,, a long-lived pollutant (small 2) may reach a high concentrat ion if remobilization renders l~bu~ << ~,], even if I] is large enough to confine the pollutant near the sea floor.

5 A SIMPLE THREE-DI .MENSIONAL M O D E L

We now simpli~" the general model discussed in Section 3 by taking C* ~- C (as before) and assuming I,], Ku and Kv to be constant, We also ignore advection (i.e. take u = 0) for the sake of simplicity, but will discuss the effect of this further in Section 8. The steady-state equation for C is then

,- "~ ~ "~ ~ ; 2 KHVfiC+Kvc'-C., + I I ( C / ~ : - ) . C = O (12)

with eqns (2) and (3) as boundary conditions at the sea surtkme and (fiat) sea floor.

Specification of the problem would be completed by the choice of suitable lateral boundary conditions. These might allow for removal on to the bot tom sediments of the continental slope tas in Spencer et al., 1981) or require zero flux, as assumed in Section 3.1. However, the main purpose of this section is to illustrate the effects of scavenging and decay on the near- source concentrat ion field, and to determine whether the pollutant ever reaches the sea surface or lateral boundaries in significant quantities. We thus assume an unbounded ocean and solve eqn 1121 subject to the bottom boundary condition (eqn (3)),

We further simplify the problem by assuming cylindrical symmetry with (x 2 + yZ)t;2 = r. The governing equations are then

KH(f/2C/:(?r2+r-l~C/~r)+Kvc2C C 2 2 + Vi ( ~ C / { } : - , , : . C = 0 (13)

- K v 3 C / 3 z - I { C = O Q f f ' ( r ) - ~ C o n z = 0 114)

5.1 Point source, no scavenging

A useful benchmark is provided by the solution of eqns (l 3) and (14) with scavenging ignored and an isolated source at the origin, r = - = 0. This is

C = Q(2rc)-t[{gHlkv)r 2 + K~z'-] - t 2 exp [-{ ) .K~ tr2 "4- ).K~7 t..2)l 2] (15)

Pollutant dispersal/?ore a sea-floor so,lrce 223

obtained by treating tKwKv) t "-: as the vertical co-ordinate, so that. with I; = 0. eqn l t2) becomes the three-dimensional Helmholtz equation. The solution in eqn 115) shows an inverse dependence of the concentrat ion field on the scaled distance, combined with the decay appropriate for the time to diffuse from the source.

5.2 A general solution

We now assume that the source is confined to the region r < r,, and. for simplicity, assume a uniform distribution within this region, i.e.

./ '(r)={rc, ':)-t f o r r < r f l r ) = O f o r r > r ~ 16)

This may be written as the Bessel transform:

./'(r) = (rcr~)- 1 J l ( k ) Jo (k r ' ) dk 17)

where r ' = r:r~. and leads to the solution of eqns (13) and (14) as

C = Q(rcq) - t ( K H K v l - Z ' - C '

t"

to " ,~ , 2] C' = [(~a - ½~i) + (~,/4 + k~ + k-') I' - tJl(k)Jo(kr') 18)

exp [ - [ a , / 2 + {a~/:4 + ko + k 2 ) t 2 ] c '} d k

where -' = ( K w ' K v ) t 2(:/r ) o. i = V i r , ( K u K v ) - 1,2

(19) . . . . t Z = liar (K~ lKv) - 1 2 /<o : (/.r~-/'/'£11) °'d ,

This is a closed form for the solution which can be shown to have various asymptotic approximations in different regions. These are most easily understood if we examine the solution first in simplified situations without all three of the controlling factors of decay, interior scavenging and boundary scavenging.

5.3 No scavenging

We first take a~ = % = 0 in eqn (18). For r' and/or :'>> 1 the integral will converge within a small range of k away from zero, so that we may approximate J t l k ) by -[,k and use the identity (Gradshteyn & Ryzhik, 1980):

f O k(ko /,.2)-l,,2Jo(kl., ) [ _ ( k i f 3 dr - k z) l/2_,] dk = - t (-kop) (20) + exp p exp

where p = (r '2 + :,2)1.2. Hence, for r" a n d / o r : ' >> I, we ob ta in , as expected, the so lu t i on (eqn (15)) for a po in t source.

_;.4 Christopher Garretr. John Shepherd

At the origin (or very close to it such that r', -'<< 11 we have

C'(0, 0) = (k o + k"l- t "-j~k)dk (21) . 0

= 2k( ~ exp ( -½ko)s inh (½kot (22)

For k o >> 1 (i.e. for the radius of the source area large compared with the diffusion distance (Ka/x) - ) this gives

C(0, 0) = Q(=r~)- t(Kv,;.)- ~-" (23)

as obtained for the one-dimensional solution {eqn (9)) if V~ = 0. For a longer- lived radionuctide such that k o << 1 we have C'(0, 0) --- t and

C(0, 0) ~- Q(rwj-t(KHKv)-1.,_ (24)

which is the point-source solution evaluated at r = ½r~, - = 0. The solutions with no scavenging are thus much as we would have

expected from the solutions discussed earlier in the paper.

5.4 Boundary scavenging, no decay

To illustrate the effects of scavenging in the simplest possible way we take a i = 0, k o = 0 and only retain a d :~0. The discussion in Section 2 shows that this is, in fact, an unlikely situation as interior scavenging is likely to dominate boundary scavenging for long-lived radionuclides, but the solution is still useful. We have

;o C' --- (o" d + k)- tJllk)Jo(kr')exp ( - k - ' ) dk (25)

As before, we start by considering the solution for r' and/or -' >> 1, so that the integral is dominated by values of k sufficiently small that we may take Jr(k) ~- ~k. We then have two different regimes depending on the magnitude Of p = ( r '2 +z" ) t " - '- compared with ad -1 (assuming this to be >> 1 for the moment).

(i) l < < p < < a ~ t : W e c a n i g n o r e a d i n e q n ( 2 5 t a n d o b t a i n C ' ~ ½ p - ' , a s f o r a point source and no decay.

(ii) a~ - t << p: We can approximate eqn (25) by

C' = ½ka 2 IJo(kr'le-k: ' d/," (26) o

= - 4 a ~ -~ c ,,) . : - ; , [ ( r , Z + _ _ - 1:2] (27)

c..-

= ½a~ tz 'p- 3 (28)

Pollutant dispersal trorn a seafloor source 225

so that there is a t ransi t ion f rom a source-l ike behav iour for p << a a- t to a dipole-l ike behav iour for p >> a2 t, t hough a bet ter approxi- ma t ion than eqn (28) on the sea floor ( 5 = 0) is

C'(r', O) = ½rya 2r '--~ (29)

We comple te the analysis o f this sect ion by evaluat ing C' for r' and -' << i. We need j'o ~ {ad + k ) - t Jr(k) dk, which is approx imate ly 1 for a d << 1 and a a- t for cra >> 1, Hence, very close to the origin C is given by eqn (24) for weak scavenging (a,, << 1). For s t rong scavenging (% >> 1)

(30)

and in this case C'(r', : ') away f rom the source merges into the behav iour descr ibed by eqns (28) and (29) wi thou t the intermediate source-l ike behaviour.

The key result in this section is the identification of the t ransi t ion dis tance a a- ~, which, in d imensional terms, becomes (KuKv) ~/2 I/d- l hor izonta l ly and Kv ~ - t vertically, and beyond which the concen t ra t ion falls off rapidly.

5.5 Boundary scavenging with decay

If we allow for both b o u n d a r y scavenging and decay we expect some combina t ion o f the results with ei ther separately. Taking a~ = 0 in eqn (18) and proceeding as in Sections 5.2 and 5.3, it is s t ra igh t forward to derive the app rox ima t ions summar ized in Fig. 1.

We see that the main effect o f decay is to add the factor e x p ( - k o p ) to the solut ion with b o u n d a r y scavenging alone, but there are other factors involving k o for p >> a d- ~, and a possible change in the concen t ra t ion field at the source.

/" \ ~(ad + ko)-~z 'p-a(t + kop)e -k°p / \

l -- l~--kop

/ \ / \

/ \

tmin[l, (aa + ko)-'i-n ! r , - , e - ' o r ' / ~ I ½'~2'(~d ÷ k o ) - ' ~ ' - ~ ( 1 + kor '> -*o"

Fig. I. Approximate and asymptotic formulae for the normalized concentration. C', in the ocean and at the sea floor, for boundary scavenging and decay. The three regions to which the formulae apply, on the sca floor or in the water column, are p << I. l << 9 << %- ~ and

aa -t << p, where p is the scaled distance.

226 Christopher Garretr. John Shepherd

5.6 Interior scavenging

The structure of the general solution leqn (18)) shows that the solution incorporat ing the effects of interior scavenging max be obtained directly from the solution without it by:

(i) mult iplying by exp(-½aiz ' ) (ii) replacing k o by (a~,4 + ko) l -"

(iii) replacing a,i by ad-½a~

The exponential decay in the vertical is not surprising; interior scavenging restricts the vertical distance that a pol lutant can spread, independently of any removal at the sea floor.

/ ~" t % - 2 - ~ o z' - 4 o , o f \ ~ z ~ e ' e / \

I ½p-ie-~<"='e-~"'> \ I \

I \ / \

I rain{i, c'~- ~l- t ~r

~.":'.'-; ; - , ' U . , , ' , -I,'-"~7-~-',':i " O - ~ l ; . ' ; , " - ' i - - ' , ' , ' - ' . , ' , : - ' ' - - l , - - , " , , ; ' ~ - " " ; . . . . . . . . , " - " " : . . . . . . -

Fig. 2. A s in Fig. 1 b u t w i th i n t e r i o r s c a v e n g i n g , e; d = es~, a n d n o decay .

In particular, we show in Fig. 2 the asymptot ic approximat ions to the normalized concentra t ion field C' for the case of interior scavenging with k o = 0 (no decay) and ~s~=a~ (as for a long-lived pollutant with no remobilization). The dimensionless transition distance is now 2a~- i, and we note that the concentra t ion on the sea floor falls off more rapidly with radial distance than for the case of boundary scavenging and no decay.

As discussed by G E S A M P (1983) and in Section 4.2 of this paper, remobilization at the sea floor of a scavenged pol lutant may render V d less than V~. If I~ >½V~, the dimensionless transition distance increases to ( a , j - ½rs~)-~, but within this scale the concentrat ion field, away from the

- - | P origin, still behaves like ½p t exp [ - _va~(-- + p)] for k o = 0. Thus, even without the more rapid decay achieved at a greater distance, we still see that the pol lutant is confined within a dimensionless distance of order a~-

If a d -½~i < 0, as is possible, the si tuation becomes more complicated. In effect, significant amoun t s of pol lu tant that are scavenged from the water co lumn re-emerge f rom the sediments and act as a fresh source. There is thus a tendency for the pol lutant to spread cylindrically, with a vertical structure exp(-a~z') , and a dependence on the radial distance, r', propor t ional to - I n r'. The scavenging distance, beyond which the concentrat ion falls off

Pollutant dispersal.t?om a sea.floor so,~rce 227

much more rapidly, can be shown to be proportional to ( % ~ ) - 1 z, assuming this to be >> 1.

If % = 0, the concentrat ion field can be limited only by decay; with k 0 # 0 we again find cylindrical spreading out to a distance of order k o l assumed >>1.

6 P A R A M E T E R SPACE

Subject to the obvious limitations of the models, the results of Sections 4 and 5 provide useful approximate formulae for the concentrat ion field in the ocean. This will be discussed further in Section 8.

In this section we remark that the model results permit us to say whether the bulk of the pollutant inventory, which must be Q/',:. altogether, is to be found on the sediments or in the water column and whether it is located near the source in either medium. :The main results, presented in terms of the two dimensionless parameters B'-= ) .H2/Kv and S = VdH/Kv introduced in Section 4, are shown in Fig. 3.

To derive the conclusions shown in Fig. 3 we start with the results, from the one-dimensional boundary scavenging models of Section 4.1, showing that most of the pollutant is on the sea-floor sediments if S>>B 2 (i.e. Vj >> ),H) for long-lived substances ( B z << 1) or if S >> B (i.e. V d >> \/Kv).) for short-lived substances (B 2 << l).

The three-dimensional model with boundary scavenging, described in Section 5.4, shows that the sedimentary inventory of the contaminant is largely confined within a radius (KHKv)I<'-V~ t of the source. If this is significantly less than the horizontal scale, L, of the ocean the contaminant is thus confined near the source by scavenging. If we assume HZ/Kv = L,Z/KH, i.e. equal vertical and lateral mixing times, the condition for confinement becomes V~H/Kv >> I.

If the bulk of the contaminant is in the water column rather than on the sediments, the condition for confinement is clearly that the decay time should be less than the mixing time, i.e. that B-' = ).H2/Kv >> 1.

The models of Sections 4.2 and 5.6 show that the conclusions summarized in Fig. 3 remain valid with interior scavenging, provided that ~', is not significantly greater than V d. However, i fa scavenged contaminant is largely remobilized at the sea floor we can have V~ >> V d. In that case the results in Section 4.2 show that if V~ is large enough to confine the contaminant near the sea floor (V~H/Kv >> 1, i.e. S>> V~/V~) and to dominate decay (V~-" >> Kv)., i.e. S>> BV~/V~), the bulk of the contaminant inventory is confined to the sediments if S = VdH/K v >> B(Vd/V~) t~'z. Moreover, the results of Section 5.6 show that this sedimentary inventory is confined near the source if S>> (v~iv,) 11"-.

228 Chris topher Garrert. John Shepherd

S = C~H'ff,,- [ /

long-lived ~ , ~hort-lived

: lO -~ . . . . . . t O ~ : ~ " ~ t ' t ~ / 1 U) 2 10 4

B . . . . . . . . . . "ide sedl . . . . t.~ ~ ' * C ~ D: local . . . . . . / - -

t0-: i"

/ '°-t

Fig. 3. Regions of the pa ramete r space described by non-dimensional ized decay and scavenging rates. The bulk of the c o n t a m i n a n t inventory is associated with local or basin- wide sediments or water, as indicated for regions A, B, C and D: conf inement near the source is by scavenging for region A and by decay for region D. The boundar ies between the four regions are not sharp. The locations in pa ramete r space of six representat ive radionuclides are

shown: the pa ramete r values and their derivat ion are ,.z, iven in Table 1.

T A B L E 1 Parameter Values for Representat ive Radionucl idcs

Nuclide t t ,. 2 B z = 2 H "-,/Kv h Ko c 1~ S = VdH/Kv h ( y ) ( y - 1 ). ( m y - ' )"

6°Co 5"27 0"13 165 2 × 107 1"3 × 105 8"2 × 10 "~ 3I-[ 12"2 0"057 72 1 2"9 × 10 -3 1'8 × 10 - 3

t37Cs 30"0 0"023 29 5 x 105 58 3'7

'~C 5692 1"2 × 10 -~ 0'15 6 × l 0 s 0"096 0"061 Z39pt l 24 110 2"9 × 10 -5 0"037 105 1"l 0"73 - '3 'Th 1"4× 10 l° 5"0 × 10 - t ' 6"3 × 10 ' s 5 × 1 0 6 50 3 2

" 2 = it, ] In2, where / t 2 is the half-life. b Taking Kv = 10-4 m 2 S - I but H = 2 km (ra ther than 4 km) so tha t the mixing time is of order t03 years ra ther than 5 x 103 years. This part ial ly offsets the neglect of advect ion in the models. cValues from O E C D / N E A (1985) and Hill & Mobbs (1985).

Taking V d = V~ + ).h* = Kt)(f i% + ) f ' h ) with ./i% = 10 - s my - ' and f ' h = 0"05 m (see Section 2).

Pollutant dispersaltrorn a se,a-door source 229

In general, then, Fig. 3 may be used to assess the fate of a dispersing contaminant in a scavenging ocean. The deposition velocity, ~+~. is still the relevant scavenging parameter even if interior scavenging is allowed for, but if V~ is large enough, and if significant remobilization occurs at the sea floor, then more of the contaminant may be confined to the sediments than would be expected from an analysis based solely on I,~.

Ignoring this detail, we also show in Fig. 3 the location in the parameter space ofsix common radionuclides, chosen to illustrate the range of possible fates. (Placing t'tC on the diagram is perhaps misleading: oceanographically t~C would be widespread in the water, but in a time short compared with its half-life some of it would enter the atmosphere as the t~C t2C ratios achieved equilibrium. It is this atmospheric component of the t 'C, rather than the oceanic component, which leads, via terrestrial food consumption, to the greater radiological impact on Man/Bush et al., 1984).)

7 THE EFFECT OF NON-CONSTANT NEAR-SOURCE EDDY DIFFUSIVITY

We have assumed so far that oceanic mixing processes may be represented by constant Fickian coefficients KH and Kv. In tact, for a turbulent diffusion process this is only a valid approximation for times alter release greater than the Lagrangian autocorrelation time of the dispersive eddy field (e.g. Csanady, 1973). Close to a source this condition will not be satisfied, and the concentration field may be higher than that estimated in early sections.

We can examine the problem using the general formula (Csanady, 1973, p. 59)

9 9 - - C=2Q(2~c) -3"- exp[-½lx- ' ~.~+y-' a~+z-/~_-)](o~o~.~.) tdt (_,l)

for the concentration field in a half-space {hence the multiplier 2) due to a steady release at rate Q of a contaminant. The variances o.~, o~, o__-' give the mean square distances after t ime t of particles released at the origin.

For the anisotropic, but Fickian, diffusivities assumed so far, we would have

leading to

2 o~ 2KHt, O2 2Kvt (32)

C(r,z)= Q(27r)- [(KnKv)r" + K~z-] tl2 {_~)

as in eqn (15), with r 2 = X 2 + y 2 .

In the ocean vertical mixing is probably associated with small enough events that it is adequate to retain the Fickian representation, Kv, but in the

230 Christopher Garrett, John Shepherd

horizontal we must allow for the finite integral time scale, TL, of eddies with u~ u; as the mean square of each component of speed. For t >> TL, and hence at large distances from the source, it is adequate to use a Fickian diffusivity, KH = U 2 Te, but at short times, and hence dose to the source, o; ~ ~- U'-t'-. Hence, restricting our attention to the sea floor, where - = 0 .

C(r,O)~2Q(2x)-3"- f[expE-re/(2U'-t")]U-"[2K,.I-let-':dt [34)

(; 0 =Q(27z)-3'22t"~U-t,'2K\71'2 s-t '*expl-sld r -32 ~35)

Comparing this with eqn (33) we see a sharper singularity { r -3 , rather than r - t) than for the constant KH solution which applies well away from the source. The solution given by eqn (35) is asymptotically valid for small r, whereas C(r,0) from eqn (33) is asymptotically correct for large r. The two approximations are equal at r = 034KH/U = 0"34UTL so we expect higher concentrations that predicted by the Fickian model within a radius of this order around a source.

For a distributed source, as assumed in this paper, we must compare this critical distance with the source dimension (of the order of 100 km for a deep-sea dump site). Values of U and Tt. are spatially variable and not well known, but U ~ 10 -2 ms - t is reasonable (Dickson, 19831 and we assume T~. ~ 11 days as found by Freeland et aL (1975) in the Sargasso Sea. This implies 0.34UT L ~ 3 kin. We conclude that non-Fickian diffusion should not lead to significantly higher average concentrations at a dump site than predicted by a Fickian model, though, of course, special attention may need to be paid to the immediate vicinity of individual waste canisters.

8 DISCUSSION

It is hoped that the models described by G E S A M P (1983) and in this paper are of some direct practical value as well as being of considerable use for "intuition-building' and for comparison with more elaborate models. The three-dimensional model discussed here ignores lateral boundaries, but should give an adequate representation of the concentrat ion field in the near source region where the concentrat ion is greater than that predicted by horizontally averaged models of the type introduced in Section 4.

Indeed, the approach adopted by IAEA (1984) in its assessment and regulatory procedures was to use the higher of the concentrations predicted by either the three-dimensional or a one-dimensional model. The simplicity

Pollutant dispersal from a sea-floor source 231

of the models allows for a ready evaluation of the sensitivity of the results to the values of poorly known parameters (Hill & Mobbs, 1985).

Horizontal advection by mean currents was ignored in the three- dimensional model discussed in Section 5. This will certainly be valid within a distance of order KH/U (typically 100 kin) of the source where horizontal diffusion dominates advection by a mean current C: Outside this the principal effect ofadvection will be to change the shape of concentration field rather than to greatly modify the concentration levels to be found. Spatially varying advection will. of course, transport material from a sea-floor source to ocean boundaries more quickly than diffusion. As discussed in footnote b of Table 1, we have compensated for the neglect of advection in assigning parameter values to various radionuclides for Fig. 3 by taking an ocean depth of about 2000 m with a corresponding "mixing' time of order 103 years.

This paper has focused on steady-state solutions. These could be used to build time-dependent solutions by using Laplace transform techniques (the Laplace transform variable would merely be added to the decay rate ;3, or one could use a numerical approach. In practice, though, the form of the solution for a switched-on source, for example, is easy to assess: close to the source the steady-state solution is quickly reached~ whereas it is approached gradually, over a time comparable with the half-life of the contaminant, in the far field. The issue is discussed further by GESAMP (1983).

We conclude by iterating our view that a proper approach to complex pollution problems requires the development and use of a hierarchy of models. We hope that the models described in this paper are useful elements of that hierarchy.

A C K N O W L E D G E M E N T S

We are greatly indebted to our colleagues on the GESAMP Working Group responsible for GESAMP Report No. 19, and particularly to the working group chairman, Dr George Needler. We also thank Dr Amelia Hagen of IAEA, who facilitated and encouraged the work of the group and subsequent developments. C.G. is partially supported by the Natural Sciences and Engineering Research Council of Canada.

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232 Christopher Garrett, John Shepherd

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Spencer, D. W., Bacon, M. P. & Brewer, P. G. (1981). Models of the distribution of 2 topb in a section across the North Equatorial Atlantic Ocean. J. Marine Res.. 39, 119-38.