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On the accumulation-dispersion processes of the tracer 137Cs in the Italian seas
.I. Environ. Radimcriert,v.
Vol. 37. No. 2, pp. 155~ 173. 1997
0 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain ELSEVIER
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SO265-931X(96)00099-9
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On the Accumulation-Dispersion Processes of the Tracer 13’Cs in the Italian Seas
G. Buffoni & A. Cappelletti ENEA, (Received
C.P. 316, 19100 La Spezia, Italy
18 May 1996; accepted
6 November
1996)
ABSTRACT This paper addresses the study of accumulation-dispersion processes, in the Italian Seas, of radionuclides due to fall-out on to the surface of the sea. Two periods are considered: 1960-1984, during which the fall-out was caused by! large scale nuclear explosions in the atmosphere; 1986-1992, during which the ,fall-out was caused by the Chernobyl accident. We focus on the analysis of vertical dispersion processes by means of mathematical models: a lumped parameter model (a depth-averaged model) for the surface layer and a depth dependent modelfor the deep layer. A regression analysis performed by fitting the experimental data (fall-out and activities in sea waters) to the model equations allows us to estimate the parameters introduced in the models. Fallout data are used as an input flux for the lumped parameter model to simulate the dispersion processes in a surface layer. This model also produces the output ,jlux at the lower interface, which is used as input flux in a depth-dependent model for the deep layer. For sufficiently slow processes, as during the period 1960-1984, the experimental data are well described and interpreted by these models. On the contrary, for rapid processes, as during the period 19861992, and when the experimental data are not complete, the analysis carried out with these models cannot be satisfactory. c 1997 Elsevier Science Ltd
1. INTRODUCTION The fall-out of radionuclides on to the surface of the sea represents an input flux of radioactive tracers to the sea. This paper addresses the study of accumulation-dispersion processes of the tracers in the Italian Seas. 15.5
156
G. Buffooni, A. Cappelletti
The accumulation processes are due to the input flux and to different hydrodynamics characteristics between two adjacent sea regions. The dispersion processes are due to turbulent mixing and the transport by advection of the tracers. We will consider two periods with different mechanisms of release of radioactive elements to the atmosphere and consequent fall-out on to the earth’s surface. During 1960-1984, large scale atmospheric tests of nuclear weapons produced a fall-out of radionuclides on to the earth’s surface, which varied gradually in time and considerably by latitude and precipitation. However, in the area of interest, we can consider the fall-out approximately spatially uniform. During 19861992, the Chernobyl accident caused radioactive air plumes to spread over Europe and the Mediterranean Sea; the consequent fall-out was not uniform and varied rapidly in time. As a tracer of the accumulation-dispersion processes, we have chosen 137Cs. This radionuclide is presented in ionic form in aqueous solution, and it interacts weakly with organisms. Furthermore, fall-out and activity data for this radionuclide and for “Sr (ratio ‘37Cs/g0Sr= 1.31) are available in the literature. Fall-out and surface activity data for ‘37Cs have been obtained from 90Sr data in Giorcelli and Cigna (1975) for the period 1960-1974 and from the CNEN, ENEA and ANPA annual reports on the environmental radioactivity in Italy, for the period 1975-1992 (CNEN, 1978-1981; ENEA, 1982-1993; ANPA, 1995). A spatial averaged depth distribution of ‘37Cs in deep water, for the Mediterranean Sea, based on pre-1982 data, is shown in the UNEP report, 1992 (Fig. 1). The fall-out and surface activity data for ‘37Cs reported in Tables 1 and 2, for the periods 19601984 and 19861992, respectively, represent spatial averages from different sites of the Italian Seas. Because of the approximations (spatial and time averaging operations) and the time scale of the process, for the period 1960-1984 of 24 years, we have chosen yearly averages, whereas for the period 1986-1992, all the data available have been used. Many papers have addressed the role of radionuclides in the study of marine processes (Guary et al., 1988; Kershaw and Woodhead, 1991). In this paper, we focus on the analysis of vertical dispersion processes, because, at the present time, the data available in the literature do not allow us to simulate horizontal dispersion processes in the Mediterranean Sea. An analysis of the experimental data in Tables 1 and 2 are presented as well as a description and interpretation of these data and of the depth distribution in Fig. 1 using mathematical models. A depth-averaged model for the surface layer, and a depth-dependent model for the deep water are
Accumulation-dispersion processes of 13’Cs
157
I
Dm
,
(mt
,*
,,*’ I’ ,’ I
501
If ‘
f’
,
:
#’
f’
, , i t
tow
1 I , , I
i 0
: ts.1
I
zoo
Fig. 1. Spatially averaged depth distribution
of ‘37Cs in deep water, for the Mediterranean Sea, based on pre-1982 data. From UNEP (I 992) (modified).
used. In Section 2, the basic assumptions and the model equations are set out. In Sections 3 and 4, the results of the numerical simulations for the two periods 1960-1984 and 1986-1992 are presented and discussed. In Section 5, some concluding remarks are presented.
2. BASIC ASSUMPTIONS
AND MODEL EQUATIONS
The following assumptions are made about the data described in Section 1 and, consequently, about the delimitation of the marine system representing the area of interest:
G. Buffooni, A. Cappelletti
158
TABLE 1 Period 1960-1984. Experimental Averaged ‘s’Cs Activities in Coastal Sea Waters and Fall-out Data for the Italian Seas Time
Activity (Bq mm.‘)
1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984
6.12 5.23 36.26 4085 59.02 42.61 32.05 22.77 1187 11.71 7.36 3.38 4-61 2.07 2.19 2.52 3.55 2.77 2.48 2.66 2.70 2.43 5.60 5.10 4.74
Fall-out (Bq me2 year-‘) 177.6 145.8 652.7 1383.8 791.2 321.6 181.0 76.9 96.0 62.3 71.7 82.1 41.8 198 37.3 35.5 12.0 53.3 66.6 178 8.9 252 5.25 2.91 3.08
(i) The fall-out and surface activity data (Tables 1 and 2) represent spatial averages from different sites in the Italian Seas. Furthermore, the activity data also represent averaged values over a surface layer of unknown depth, H1 , which is, in general, time-dependent. (ii) The depth distribution of the activity (Fig. 1) represents a spatialaveraged depth distribution from different sites of the Mediterranean Sea, in a layer of depth HZ(H2 >> Hi), and at a given time (around the year 1980). (iii) The marine system is spatially homogeneous in the horizontal direction. The system may be considered as a water column, represented in Fig. 2, where the z-axis is assumed to be vertical upwards; it is made up of two layers: a surface and a deep layer, with different hydrodynamic characteristics, principally the turbulent mixing and vertical advection. Moreover, the accumulation-dispersion processes in the two layers have different space and time scales.
Accumulation-dispersion processes of 137C.~
159
TABLE 2a Period 1986-1992. Experimental Averaged ‘37Cs Activities in Coastal Sea Waters for the LigurianTyrrhenian Sea. The Time t = 0 Corresponds to April 30 1986 Time (days)
Activity (Bq m-“)
9 10 35 138 168 336 518 701 883 1066 1248 1431 1613 1978 2343
475 451 58 25.4 22.1 7.35 7.05 5.54 4.77 4.80 4.40 4.50 5.59 4.22 2.92
Therefore, the surface activity data will be described and interpreted by means of a lumped parameter model, representing an activity balance of the tracer in a layer of depth Hi; otherwise, the deep activity data are interpreted by means of a depth-dependent model. We will now make a final assumption about the mechanism generating the accumulation-dispersion processes of a tracer in sea water. (iv) Any perturbation of the system at a given time, t, is driven by a flux --h(t) at the air-sea interface, such that fo(t) > 0, for t - to > 0, andSO
-+ 0 for time t - to long enough.
Remark (1) Let Tf and T,,, be the characteristic time scales of the input flux -h(t) and turbulent mixing in the surface layer. Tf is of the order of
lf,l(df/d01 and G, of Hi*/(rc*D), where D is the vertical turbulent mixing coefficient. If Tf >> T,,,, then the turbulent mixing process is sufficiently rapid, so that we may consider a nearly homogeneous distribution of the tracer in the layer. 2.1. The lumped parameter model for the surface layer
The lumped parameter model for the surface layer is a depth-averaged model, and represents a global balance over the layer. To obtain this equation, let us first introduce the following depth-dependent model:
G. Buffooni,A. Cappelletti
160
TABLE 2b Period 1986-1992. Experimental Fall-out Data for the Ligurian-Tyrrhenian Sea. The Time t = 0 Corresponds to April 30 1986 Fall-out (Bg me2 day-‘)
Time (days) 0
39 29 28 1010 80 45 19 21 18 9 4.9 4.5 10.4 14.7 1.8 0.8 1.1 0.2 1.0 0.2 0.070 0.050 0.030 0.025 0.013 0.020 0.011 0.036 0.017
2 3 4 5 6 7 8 9 10 15 23 31 51 81 114 148 181 209 336 518 701 883 1066 1248 1431 1613 1978 2343
=e,
overthedepthrangeoto
-HI,
(1)
-Llg=-fo(t),
at z = 0,
(2)
-Ll$
at z = -HI,
(3)
over the depth range 0 to --HI,
(4)
= -yu,
u(fo,z) = uo(z),
Accumulation-dispersion processes of 13’Cs
161
where u(t, z) and ug(z) are the activity distributions of the tracer at times t and to, respectively; 1 is the specific depletion rate due to radioactivity decay, accumulation by organisms; D is the vertical turbulent mixing coefficient; y is a measure of the transfer process of the radionuclides, reaching the boundary z = -Hi, to the deep layer; e(t, z) is the tracer variation rate due to other processes not included in the previous terms, and due to stochastic variability of the environmental parameters (it may be considered as a noise term). Input to and output from the system are shown in Fig. 2. Assume now that, in general, Hi is time-dependent, and let ii(t) be the average concentration of the activity over the layer (-Hi, 0): 0 ii(t)
=
u(t,
2HI
z)dz.
J -H1
By averaging equations (1) over (-HI, 0) and taking into account the boundary conditions (2) and (3) we obtain the integrated balance dii fo z+jE=F+e,
I
where llw=2+g+$-$(l
.
z’o
(6)
-p), 1
I
z
-f,,(t)=input flux to the surface layer (fallout) -
air * sea surface layer
-)iu(t,z)= depletion rate -f,(t)= -y u(t,-HI) input flux to the deep layer -
z=-H, .
sea deep layer
-)cu(t,z)=depletion rate
1 flux to the sediments z=-Hz * bottom sediments
Fig. 2. Scheme of the marine system (not in scale).
G. Buffoni, A. Cappelletti
162
with p being the ratio (7) and i?(t) =
s
-!Hi
0
e(t,
z)dz.
-H,
The initial condition i&(t) =
uo
must be associated with eqn (5). Remark (2) When Tr > T,, it seems reasonable to assume that HI is time-independent; in this case, the parameter p is necessarily positive. The unknown parameters ,U and HI will be estimated by fitting the experimental data, activities and fluxes, to eqn (5). More precisely, we will assume p and H1 piecewise to be constant in time. In any time interval where p and HI are assumed to be constant, we will consider a discrete analog of eqn (5) in the form (a Crank-Nicholson scheme, Varga, 1965, p. 263): ~(t + At) = btu(t) + b~[f0(t) +fo(t + At)]/2 + e’(t), ii
= ZSO,
(8)
where b
=
’
(1 - PW2> (1 + &/2)’
At
b2 = H1(l + pAt/2)’
and Atz(t) e’(t) = (1 + pAt/2).
Equation (8) defines a discrete autoregressive process of the first order with a driving force. Let ii(t) andfo(t) be given by the experimental activities and surface fluxes, respectively. Thus, we can estimate the parameters b, and b2 by means of the least-squares method, i.e. by minimizing the function: 7-Z
@(h
> b2)
=
.I
e’2(t)dt. 7-I
In eqn (9) (Ti, T2) is the period of the perturbation of the system, for which fall-out and activity data are available. The operation of minimiza-
Accumulation-dispersion processes of’ ‘j7Cs
163
tion also gives the estimate of the standard deviation of the error e’(t). From the definitions of br and bz:
2 (1 -h)
At
‘=nt(l ____+b,)’ Remark
HI = b2(1 + /d/2)’
(3) Let us consider the case when the regression analysis leads
to a constant (time-independent) estimate of p and Hi. Then, from eqns for the deep layer as (3) (6) and (7) we obtain the input flux -f,(t) given by: -J(t)
= -yu(t,
-H1)
= -(p
- /l)H,u(t).
(11)
2.2. The depth-dependent model for the deep layer
The depth-dependent
model for the deep layer is written as follows:
du z+~u+$?u-D~) =e,
over the depth range -HI to -HP,
(12)
vu-DE=:f,(t),
at z= -HI,
(13)
(14)
U(to,z>= uo(z),
over the depth range --HI to -Hz,
(15)
Here, we have introduced a vertical velocity v (CO) to take into account both averaged (over space and over a long period of time) vertical movements of the water and, possibly, settling processes; the fluxft(t) in eqn (13) is given by eqn (11). All other symbols (u, uo, 2, D, e) have the same meanings as in Section 2.1, but now with reference to the deep layer. Remarks (4) Assume that ft(t) > 0 in eqn (13). Then, for any solution u(t, z) of eqns (12)-( 15), we can have au/az < 0 at z = -HI, i.e. ~(t, z) at z = -HI is increasing with the depth, if and only if v < 0. This situation is due to the accumulation of the tracer in a boundary layer near -HI, and it is verified when u(t, -HI) >fi(t)/lvl. (5) It is possible to show that, if any solution u( t, z) of eqns (12)-( 15) has a subsurface maximum for -Hz < z < -HI, then by necessity u(t,z) cannot be a steady-state solution. (6) The boundary condition (14) at the bottom z = -H2 is justitied only when the tracer is not transferred to the sediments; thus, the boundary
G. Buffoni, A. Cappelletti
164
condition (14) holds true if Hz >> H, and t < + m When the boundary condition (14) holds, the spatial distributions will be increasing with the depth near the bottom z = -Hz, over a sufficient length of time.
3. PERIOD
1960-1984
For this period, Tr = 0.83~ [l/a in eqn (17)]. A rough estimate of T,,, is obtained by assuming HI = 50m and D in the range 10e4 : 10e2 m2 s-l (Nihoul, 1975, p. 75); we have 0.0008 years < T, < 0.08 years. Thus, Tf > T,, and the surface layer may be considered as a well homogenized layer [see Remark (l)] and HI constant [see Remark (2)]. 3.1. Results for the surface layer We have tried a regression analysis based on eqn (8) by assuming p and HI to be constant over the whole period 196&1984, At = 1 year and with E(to) = 6.12Bqms3 We obtained b, = 0.45, to = 1960. b2 = 0.035 year m-’ ; thus, from eqn (lo), we have the estimates of p and HI : p =
0.76 year-‘,
HI = 20.89 m.
(16)
The error e’(t) has zero mean and a standard deviation 0~1= 5.3 Bqmw3; thus, CJ;= (1 + pAt/2)/At o,f = 7.3 Bq rnd3 year-‘. Remarks (7) The estimated value of p is much greater than the t3’Cs constant of radioactive decay Jo = 0.023 year-‘. Since p represents the specific rate of depletion in the layer (-HI, 0), consequently the half residence time in the layer of the tracer is given by t1/2(-H,,O)
= log2/p
= 0.91 years.
(8) The estimated depth, H,, seems to be low; however, it is consistent with the depths of coastal waters where the water samples have been drawn. By assuming eqn (16) the solution of eqn (5), with Z = 0 and ii( to) = 6.12 Bq me3 is in very good agreement with the experimental data (Fig. 3). The time integration of eqn (5) has been obtained for different time steps, At, producing the same results. A sensitivity analysis of the response of the model versus a variation of the values of the parameters p and HI has been performed. The analysis shows that, only with the values in eqn (16) does the model reproduce the position and the activity of the maximum and the trend of the decay process. Remarks (9) The input fluxes -fo(t) (from experimental data) for the
Accumulation-dispersion processes of ‘j7Cs
165
100 90 80 70 60-
.
0
““,““,““,““I”“1
. . . 1960
1965
1975
1970
1980
1985
Year
Fig. 3. Period 1960-1984. Results of the numerical simulations
averaged distribution
n(t), continuous
for the surface layer: depth line; experimental data, boxes.
surface layer and -f(t) (given by eqn (11)) for the deep layer are shown in Fig. 4. The fluxfr (t) reaches its maximum with a delay of about 1 year with respect to the maximum offo(t); the maximum value off,(t) is significantly less than the maximum value offo(t) (maxf,(t) = O-7maxfo(t)). (10) Since p and HI are constant, if we approximatefO(t), t 2 tl = 1964, by an analytical expression, then we can give an analytical expression for the decay of u(t). By assuming an approximately constant flux from 1970 to 1984, fo(t) can be well approximated by a sum of two negative exponential functions: fo(t) = Ae- a(r--I1)+ B,-w-Q)
+ c,
t 3 *I>
(17)
where A = 624.8, B = 137.04, C = 30, Bq m-* year-’ j? = 0.27 year-‘. Thus, from eqn (5), with B = 0, we obtain, for t 2 tl, A c ii(t) = PHI +H&-de +
qt,)
4-h)
+
B
and
cc= 1.20,
,-N-0)
HI(P--_) c
A
- PHI H,(P--a)-H,(P-D)
(18)
B
e-P’(’- II) I
G. Buffooni,A. Cappelletti
166
1960
1965
1970
1975
1980
1985
Year Fig. 4. Period 1960-1984. Input flux -fo(t) at the air-sea interface, boxes; output flux -f,(t) from the surface layer = input flux to the deep layer, continuous line.
(11) The function (18) has been obtained as a solution of model (5) with F = 0 and the driving force given by eqn (17). However, we can obtain different decay curves by directly fitting experimental activity data with either a sum of two or with one negative exponential function. The standard deviations of these approximating decay curves are nearly the same as in eqn (18). However, they are pure descriptions of the decay process, and we cannot give a physical interpretation of the time constants in the negative exponential functions, while a natural interpretation is given to the time constants of the curve (18). 3.2. Results for the deep layer The dynamics described by the model (12)-(15) depends greatly on the values of the parameters v, D and HI. The parameters v and D could be depth-dependent; however, since we have solved independently the problem for the surface layer, we will try numerical simulations by assuming v and D to be constant, with values in an appropriate range, characteristic of a deep layer. We want to compare the depth distribution ~(tZ,z), at time t2 = 1980,
Accumulation-dispersionprocesses L$ i’7C~
167
with the experimental depth distribution U*(Z)shown in Fig. 1. Therefore, we have to normalize u*(z) so that the total quantity of the tracer accumulated in the water column during the period (to, t2) is the same for the experimental and calculated distributions. The normalization constant for u*(z) is: -ffi 4=
[.I -H2
~(tz, z)dz + ti(QH, I o s -Hz
u*(z)dz
Due to the approximations (spatial and time averaging operations), leading to the depth distribution u*(z), the comparison of ~(t~, z) with q u*(z) will, by necessity, be qualitative. Since u*(z) shows a subsurface maximum. it follows, from remark (4), that we must have )v) > 0, because ~(t2, z) has a subsurface maximum. From remark (5), it follows that these depth distributions cannot be steady state distributions. The values of the parameters introduced in eqns (12)-( 15) for the numerical simulations are as follows: /z = Ilo = O-023year-’ = 7.3 lo-” SC’; to = 1960; u(t,,, z) = z+,(z)= 0; HZ = 2000 m; .fi (t) given by eqn (11); v = 1, 3, 5 IO-‘ms-‘; D = 17 2 10-4m2s-‘. D = 10d4 m* ss’; then, the time evolution of the Let 1?= 3 lo-‘ms-‘, depth distribution u(t, z) is shown in Fig. 5. The depth distributions u(tz, z) and qu*(z), for different values of v and D, are shown in Fig. 6a, b and c. For v = 3 x lo-‘ms-‘,
10e4m2 s-’ < D < 2 x lO-4 m2 s-‘,
the computed distribution u(tZ, z) reproduces the position (~10Om) and the concentration (M 5 Bq mb3) of the subsurface maximum of qu*(z). Remarks (12) Let H2 = 800m, v = 3 x lo-‘ms-‘, D = 1, 2 x 10-4m’ s -‘; then, over a long enough period of time, the distribution u(t,z) (Fig. 7) increases with the depth near the bottom z = -Hz [see remark (6)]. (13) The numerical integration of the problem (12)-(15) has been carried out as follows. A semi-discrete analogue of (12)-( 15) (continuous in time, discrete in space with constant grid step of 5m) was obtained by the finite volume method. The time integration of the semi-discrete equa-
168
G. Buffooni, A. Cappelletti
0
500
1000
1500
2000
Depth (m) Fig. 5. Period 1960-1984. Results of the numerical simulations for the deep layer, with Hz = 2000m, v = 3 x lo-‘ms-‘, D = 10~4m2s-‘: depth distributions u(r,z) in (a) 1965,
(b) 1970, (c) (1975), (d) 1980 and (e) 1985.
tion was performed using an implicit method (Varga, 1965, p. 263) with a time step of 432 000 s (= 5 days).
4. PERIOD
1986-1992
As previously stated, the regression analysis of eqn (8) implies the minimization of the function (9). We observe that fallout and activity measurements in the first year after the accident are scarce, so that the rapid evolution of the accumulation-dispersion process cannot be completely described. In particular, for the period of maximum fallout in Italian Seas (the first week of May, 1986) activity measurements in sea water are not available (Table 2). It follows that the integration period (Ti, T2) in (9) does not include the most significant part of the transient. In this situation, the function (9) does not show a well defined minimum, at least with respect to the parameter b2 [i.e. Hi from (lo)]. We have, nevertheless, tried a parametric study of eqn (5) by varying the parameters p and Hi and considering the first year after the accident. In the
Accumulation-dispersion processes of ‘37C.~
500
169
2000
1000
Depm WI
b
Bqm’
107 9B,6-
Fig. 6. Period 196Ck1984. Depth distributions u(t2, z), t2 = 1980, (D = lO-4 m’s_’ continuous line, D = 2 x 10-4m2s-’ dashed line), and qu*(z) (boxes); (a) v = IO-‘ms-‘. (b) v = 3 x IO-‘ms-‘, (c) v = 5 x IO-‘ms-‘.
170
G. Bufjbni, A. Cappelletti Bqlm” 10 94
0
100
200
300
400
500
600
700
800
Depth (m) 7. Period 1960-1984. Depth distribution v = 3 x lO_‘ms-‘, D = 10-4m2s-’ continuous
Fig.
~(t~,z), t2 = 1980, with Hz = 800m, line, D = 2 x 10e4 m2 SC’ dashed line.
following years, the fallout and the activity in sea water reach approximately the levels (slightly greater) before the accident. The fall-out is shown in Fig. 8a; we estimate the time Tr, as defined in remark (1) by assuming an exponential decay of the fall-out for a few days after the maximum. We obtain Tr = 0.5 days = 0.0014 years; thus, Tr is of the same order of magnitude of T, (see Section 3). Because of the rapid evolution of the transient, we cannot simulate the process in a satisfactory way by assuming ~1and HI time-independent (remark (1)). The solution of eqn (5) (Fig. 8b), with Z = 0 and U(0) = 2.52 Bq rnd3, describes well the experimental data by assuming two periods with different values for ,LL and HI as follows: Period (days) O-40 40-365
p (day-‘) o-13 0.013
HI (m) 1.5 20
A sensitivity analysis has shown that the model is highly sensitive to the values of the duration and of the parameters ,Uand HI of the first period. On the contrary, the values of the parameters p and HI relative to the second period may vary over a wide range, producing acceptable results.
Accumulation-dispersion
processes qf“37C.s
171
Bqlm’fd lO'-y
loz-
. . . c
lo'-
.
.
l
l
a
.
. 10°-
. .
lo4
/11,111,1,I,JII,II(,1’1(‘II,IJ’,I”,I’I,
0
20
40
60
80
100
120
140
160
180
200
180
200
Days
600-
b
0
20
40
60
80
100
120
140
160
Fig. 8. Period 1986-1992 (first 200 days). (a) Input flux -fo(t) at the air-sea interface, boxes; (b) results of the numerical simulations for the surface layer: depth averaged distribution u(r), continuous line and experimental data, boxes.
172
G. Buffooni. A. Cappelletti
5. CONCLUDING
REMARKS
Monitoring fall-out and activity evolution of tracers in sea waters, cannot be easily performed with high spatial and time resolution. It follows that the main characteristics of the accumulation-dispersion processes of the tracers can be described and interpreted by means of an analysis based on spatial and time averages of the experimental data. In this type of analysis, lumped parameter models can be successfully used. In particular, depthaveraged models, to study the surface layer, allow the estimation of the depth of the layer representative of the experimental data, the specific depletion rate and the flux to the deep layer. Depth-dependent models allow us to estimate the order of magnitude of the vertical transport and of the vertical mixing coefficient. Moreover, some characteristics of the vertical distributions of the tracers can be interpreted. We observe: (i) a subsurface maximum only in non-steady state distributions and when a vertical transport towards the bottom is present; and (ii) the increase of the activity level with the depth near the bottom when the flux of the tracer from water to sediment is low and when the time is long enough. From a comparison of the analysis performed for the two periods, we may observe that fall-out and activity measurements should be carried out at the same times and with sampling intervals small enough to describe completely the transients. This is particularly true in a rapid transient like that resulting from the Chernobyl accident.
ACKNOWLEDGEMENTS The authors would like to thank A. Cigna, R. Delfanti and C. Papucci for their suggestions about literature and the data available, and the referees for their criticisms and suggestions. This work was partially supported by the European Science Foundation ‘TAO’ (Transport in the Atmospheres and in the Ocean) project.
REFERENCES ANPA (1995) Rapporfo Annuale Sulla Radioattivita Ambientale in italia. For the year 1992. Agenzia Nazionale per la Protezione Ambientale. Report ARARAM/01/95 (in Italian). CNEN (from 1978 to 1981) Rapport5 Annuale Sulla Radioattivita Ambientale in Ztalia. For the years from 1974 to 1977. Comitato nazionale Energia nucleare. Reports
DISP-AMB/103/78,
/110/79, /117/79, /125/81 (in Italian).
Accumulation-dispersion processes of ‘37Cs
173
ENEA (from 1982 to 1993) Rapport0 Annuale Sulla Radioattivita Ambientale in Ztalia. For the years from 1978 to 1991. Ente Nazionale Energia ed Ambiente. Reports for the years from 1978 to 1991. Ente Nazionale Energia ed Ambiente. Reports DISP-ARA/O1/82, 107182, /09/83, 114183, 133184, ilO/ 85, 105186, 101187; DISP-ARA-RAM/O1/89, /01/90, /02/90, /01/92, /01/93 (in Italian). Giorcelli, F. G. and Cigna, A. A. (1975) Radiocontraminazione da Ricadute nei Mari Ztaliani daZl960 al 1974. Comitato Nazionale Energia Nucleare, Rome. Report RT/PROT(76)3 (in Italian). Guary, J. C., Guegueniat, P. and Pentreath, R. J. (editors) (1988) Radionuclides: a Tool for Oceanography. Elsevier Applied Science, London. Kershaw, P. J. and Woodhead, D. S. (editors) (1991) Radionuclides in the Study of Marine Processes. Elsevier Applied Science, London. Nihoul, J. C. J. (editor) (1975) ModelZing of Marine System. Elsevier, Amsterdam. UNEP (1992) Assessment of the State of Pollution of the Mediterranean Sea by Radioactive Substances. United Nations Environment Programme, MAP Technical Reports Series No. 62. Varga, R. S. (1965) Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, NJ.
Vol. 37. No. 2, pp. 155~ 173. 1997
0 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain ELSEVIER
PII:
SO265-931X(96)00099-9
0265-931X/97
$17.00 + 0.00
On the Accumulation-Dispersion Processes of the Tracer 13’Cs in the Italian Seas
G. Buffoni & A. Cappelletti ENEA, (Received
C.P. 316, 19100 La Spezia, Italy
18 May 1996; accepted
6 November
1996)
ABSTRACT This paper addresses the study of accumulation-dispersion processes, in the Italian Seas, of radionuclides due to fall-out on to the surface of the sea. Two periods are considered: 1960-1984, during which the fall-out was caused by! large scale nuclear explosions in the atmosphere; 1986-1992, during which the ,fall-out was caused by the Chernobyl accident. We focus on the analysis of vertical dispersion processes by means of mathematical models: a lumped parameter model (a depth-averaged model) for the surface layer and a depth dependent modelfor the deep layer. A regression analysis performed by fitting the experimental data (fall-out and activities in sea waters) to the model equations allows us to estimate the parameters introduced in the models. Fallout data are used as an input flux for the lumped parameter model to simulate the dispersion processes in a surface layer. This model also produces the output ,jlux at the lower interface, which is used as input flux in a depth-dependent model for the deep layer. For sufficiently slow processes, as during the period 1960-1984, the experimental data are well described and interpreted by these models. On the contrary, for rapid processes, as during the period 19861992, and when the experimental data are not complete, the analysis carried out with these models cannot be satisfactory. c 1997 Elsevier Science Ltd
1. INTRODUCTION The fall-out of radionuclides on to the surface of the sea represents an input flux of radioactive tracers to the sea. This paper addresses the study of accumulation-dispersion processes of the tracers in the Italian Seas. 15.5
156
G. Buffooni, A. Cappelletti
The accumulation processes are due to the input flux and to different hydrodynamics characteristics between two adjacent sea regions. The dispersion processes are due to turbulent mixing and the transport by advection of the tracers. We will consider two periods with different mechanisms of release of radioactive elements to the atmosphere and consequent fall-out on to the earth’s surface. During 1960-1984, large scale atmospheric tests of nuclear weapons produced a fall-out of radionuclides on to the earth’s surface, which varied gradually in time and considerably by latitude and precipitation. However, in the area of interest, we can consider the fall-out approximately spatially uniform. During 19861992, the Chernobyl accident caused radioactive air plumes to spread over Europe and the Mediterranean Sea; the consequent fall-out was not uniform and varied rapidly in time. As a tracer of the accumulation-dispersion processes, we have chosen 137Cs. This radionuclide is presented in ionic form in aqueous solution, and it interacts weakly with organisms. Furthermore, fall-out and activity data for this radionuclide and for “Sr (ratio ‘37Cs/g0Sr= 1.31) are available in the literature. Fall-out and surface activity data for ‘37Cs have been obtained from 90Sr data in Giorcelli and Cigna (1975) for the period 1960-1974 and from the CNEN, ENEA and ANPA annual reports on the environmental radioactivity in Italy, for the period 1975-1992 (CNEN, 1978-1981; ENEA, 1982-1993; ANPA, 1995). A spatial averaged depth distribution of ‘37Cs in deep water, for the Mediterranean Sea, based on pre-1982 data, is shown in the UNEP report, 1992 (Fig. 1). The fall-out and surface activity data for ‘37Cs reported in Tables 1 and 2, for the periods 19601984 and 19861992, respectively, represent spatial averages from different sites of the Italian Seas. Because of the approximations (spatial and time averaging operations) and the time scale of the process, for the period 1960-1984 of 24 years, we have chosen yearly averages, whereas for the period 1986-1992, all the data available have been used. Many papers have addressed the role of radionuclides in the study of marine processes (Guary et al., 1988; Kershaw and Woodhead, 1991). In this paper, we focus on the analysis of vertical dispersion processes, because, at the present time, the data available in the literature do not allow us to simulate horizontal dispersion processes in the Mediterranean Sea. An analysis of the experimental data in Tables 1 and 2 are presented as well as a description and interpretation of these data and of the depth distribution in Fig. 1 using mathematical models. A depth-averaged model for the surface layer, and a depth-dependent model for the deep water are
Accumulation-dispersion processes of 13’Cs
157
I
Dm
,
(mt
,*
,,*’ I’ ,’ I
501
If ‘
f’
,
:
#’
f’
, , i t
tow
1 I , , I
i 0
: ts.1
I
zoo
Fig. 1. Spatially averaged depth distribution
of ‘37Cs in deep water, for the Mediterranean Sea, based on pre-1982 data. From UNEP (I 992) (modified).
used. In Section 2, the basic assumptions and the model equations are set out. In Sections 3 and 4, the results of the numerical simulations for the two periods 1960-1984 and 1986-1992 are presented and discussed. In Section 5, some concluding remarks are presented.
2. BASIC ASSUMPTIONS
AND MODEL EQUATIONS
The following assumptions are made about the data described in Section 1 and, consequently, about the delimitation of the marine system representing the area of interest:
G. Buffooni, A. Cappelletti
158
TABLE 1 Period 1960-1984. Experimental Averaged ‘s’Cs Activities in Coastal Sea Waters and Fall-out Data for the Italian Seas Time
Activity (Bq mm.‘)
1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984
6.12 5.23 36.26 4085 59.02 42.61 32.05 22.77 1187 11.71 7.36 3.38 4-61 2.07 2.19 2.52 3.55 2.77 2.48 2.66 2.70 2.43 5.60 5.10 4.74
Fall-out (Bq me2 year-‘) 177.6 145.8 652.7 1383.8 791.2 321.6 181.0 76.9 96.0 62.3 71.7 82.1 41.8 198 37.3 35.5 12.0 53.3 66.6 178 8.9 252 5.25 2.91 3.08
(i) The fall-out and surface activity data (Tables 1 and 2) represent spatial averages from different sites in the Italian Seas. Furthermore, the activity data also represent averaged values over a surface layer of unknown depth, H1 , which is, in general, time-dependent. (ii) The depth distribution of the activity (Fig. 1) represents a spatialaveraged depth distribution from different sites of the Mediterranean Sea, in a layer of depth HZ(H2 >> Hi), and at a given time (around the year 1980). (iii) The marine system is spatially homogeneous in the horizontal direction. The system may be considered as a water column, represented in Fig. 2, where the z-axis is assumed to be vertical upwards; it is made up of two layers: a surface and a deep layer, with different hydrodynamic characteristics, principally the turbulent mixing and vertical advection. Moreover, the accumulation-dispersion processes in the two layers have different space and time scales.
Accumulation-dispersion processes of 137C.~
159
TABLE 2a Period 1986-1992. Experimental Averaged ‘37Cs Activities in Coastal Sea Waters for the LigurianTyrrhenian Sea. The Time t = 0 Corresponds to April 30 1986 Time (days)
Activity (Bq m-“)
9 10 35 138 168 336 518 701 883 1066 1248 1431 1613 1978 2343
475 451 58 25.4 22.1 7.35 7.05 5.54 4.77 4.80 4.40 4.50 5.59 4.22 2.92
Therefore, the surface activity data will be described and interpreted by means of a lumped parameter model, representing an activity balance of the tracer in a layer of depth Hi; otherwise, the deep activity data are interpreted by means of a depth-dependent model. We will now make a final assumption about the mechanism generating the accumulation-dispersion processes of a tracer in sea water. (iv) Any perturbation of the system at a given time, t, is driven by a flux --h(t) at the air-sea interface, such that fo(t) > 0, for t - to > 0, andSO
-+ 0 for time t - to long enough.
Remark (1) Let Tf and T,,, be the characteristic time scales of the input flux -h(t) and turbulent mixing in the surface layer. Tf is of the order of
lf,l(df/d01 and G, of Hi*/(rc*D), where D is the vertical turbulent mixing coefficient. If Tf >> T,,,, then the turbulent mixing process is sufficiently rapid, so that we may consider a nearly homogeneous distribution of the tracer in the layer. 2.1. The lumped parameter model for the surface layer
The lumped parameter model for the surface layer is a depth-averaged model, and represents a global balance over the layer. To obtain this equation, let us first introduce the following depth-dependent model:
G. Buffooni,A. Cappelletti
160
TABLE 2b Period 1986-1992. Experimental Fall-out Data for the Ligurian-Tyrrhenian Sea. The Time t = 0 Corresponds to April 30 1986 Fall-out (Bg me2 day-‘)
Time (days) 0
39 29 28 1010 80 45 19 21 18 9 4.9 4.5 10.4 14.7 1.8 0.8 1.1 0.2 1.0 0.2 0.070 0.050 0.030 0.025 0.013 0.020 0.011 0.036 0.017
2 3 4 5 6 7 8 9 10 15 23 31 51 81 114 148 181 209 336 518 701 883 1066 1248 1431 1613 1978 2343
=e,
overthedepthrangeoto
-HI,
(1)
-Llg=-fo(t),
at z = 0,
(2)
-Ll$
at z = -HI,
(3)
over the depth range 0 to --HI,
(4)
= -yu,
u(fo,z) = uo(z),
Accumulation-dispersion processes of 13’Cs
161
where u(t, z) and ug(z) are the activity distributions of the tracer at times t and to, respectively; 1 is the specific depletion rate due to radioactivity decay, accumulation by organisms; D is the vertical turbulent mixing coefficient; y is a measure of the transfer process of the radionuclides, reaching the boundary z = -Hi, to the deep layer; e(t, z) is the tracer variation rate due to other processes not included in the previous terms, and due to stochastic variability of the environmental parameters (it may be considered as a noise term). Input to and output from the system are shown in Fig. 2. Assume now that, in general, Hi is time-dependent, and let ii(t) be the average concentration of the activity over the layer (-Hi, 0): 0 ii(t)
=
u(t,
2HI
z)dz.
J -H1
By averaging equations (1) over (-HI, 0) and taking into account the boundary conditions (2) and (3) we obtain the integrated balance dii fo z+jE=F+e,
I
where llw=2+g+$-$(l
.
z’o
(6)
-p), 1
I
z
-f,,(t)=input flux to the surface layer (fallout) -
air * sea surface layer
-)iu(t,z)= depletion rate -f,(t)= -y u(t,-HI) input flux to the deep layer -
z=-H, .
sea deep layer
-)cu(t,z)=depletion rate
1 flux to the sediments z=-Hz * bottom sediments
Fig. 2. Scheme of the marine system (not in scale).
G. Buffoni, A. Cappelletti
162
with p being the ratio (7) and i?(t) =
s
-!Hi
0
e(t,
z)dz.
-H,
The initial condition i&(t) =
uo
must be associated with eqn (5). Remark (2) When Tr > T,, it seems reasonable to assume that HI is time-independent; in this case, the parameter p is necessarily positive. The unknown parameters ,U and HI will be estimated by fitting the experimental data, activities and fluxes, to eqn (5). More precisely, we will assume p and H1 piecewise to be constant in time. In any time interval where p and HI are assumed to be constant, we will consider a discrete analog of eqn (5) in the form (a Crank-Nicholson scheme, Varga, 1965, p. 263): ~(t + At) = btu(t) + b~[f0(t) +fo(t + At)]/2 + e’(t), ii
= ZSO,
(8)
where b
=
’
(1 - PW2> (1 + &/2)’
At
b2 = H1(l + pAt/2)’
and Atz(t) e’(t) = (1 + pAt/2).
Equation (8) defines a discrete autoregressive process of the first order with a driving force. Let ii(t) andfo(t) be given by the experimental activities and surface fluxes, respectively. Thus, we can estimate the parameters b, and b2 by means of the least-squares method, i.e. by minimizing the function: 7-Z
@(h
> b2)
=
.I
e’2(t)dt. 7-I
In eqn (9) (Ti, T2) is the period of the perturbation of the system, for which fall-out and activity data are available. The operation of minimiza-
Accumulation-dispersion processes of’ ‘j7Cs
163
tion also gives the estimate of the standard deviation of the error e’(t). From the definitions of br and bz:
2 (1 -h)
At
‘=nt(l ____+b,)’ Remark
HI = b2(1 + /d/2)’
(3) Let us consider the case when the regression analysis leads
to a constant (time-independent) estimate of p and Hi. Then, from eqns for the deep layer as (3) (6) and (7) we obtain the input flux -f,(t) given by: -J(t)
= -yu(t,
-H1)
= -(p
- /l)H,u(t).
(11)
2.2. The depth-dependent model for the deep layer
The depth-dependent
model for the deep layer is written as follows:
du z+~u+$?u-D~) =e,
over the depth range -HI to -HP,
(12)
vu-DE=:f,(t),
at z= -HI,
(13)
(14)
U(to,z>= uo(z),
over the depth range --HI to -Hz,
(15)
Here, we have introduced a vertical velocity v (CO) to take into account both averaged (over space and over a long period of time) vertical movements of the water and, possibly, settling processes; the fluxft(t) in eqn (13) is given by eqn (11). All other symbols (u, uo, 2, D, e) have the same meanings as in Section 2.1, but now with reference to the deep layer. Remarks (4) Assume that ft(t) > 0 in eqn (13). Then, for any solution u(t, z) of eqns (12)-( 15), we can have au/az < 0 at z = -HI, i.e. ~(t, z) at z = -HI is increasing with the depth, if and only if v < 0. This situation is due to the accumulation of the tracer in a boundary layer near -HI, and it is verified when u(t, -HI) >fi(t)/lvl. (5) It is possible to show that, if any solution u( t, z) of eqns (12)-( 15) has a subsurface maximum for -Hz < z < -HI, then by necessity u(t,z) cannot be a steady-state solution. (6) The boundary condition (14) at the bottom z = -H2 is justitied only when the tracer is not transferred to the sediments; thus, the boundary
G. Buffoni, A. Cappelletti
164
condition (14) holds true if Hz >> H, and t < + m When the boundary condition (14) holds, the spatial distributions will be increasing with the depth near the bottom z = -Hz, over a sufficient length of time.
3. PERIOD
1960-1984
For this period, Tr = 0.83~ [l/a in eqn (17)]. A rough estimate of T,,, is obtained by assuming HI = 50m and D in the range 10e4 : 10e2 m2 s-l (Nihoul, 1975, p. 75); we have 0.0008 years < T, < 0.08 years. Thus, Tf > T,, and the surface layer may be considered as a well homogenized layer [see Remark (l)] and HI constant [see Remark (2)]. 3.1. Results for the surface layer We have tried a regression analysis based on eqn (8) by assuming p and HI to be constant over the whole period 196&1984, At = 1 year and with E(to) = 6.12Bqms3 We obtained b, = 0.45, to = 1960. b2 = 0.035 year m-’ ; thus, from eqn (lo), we have the estimates of p and HI : p =
0.76 year-‘,
HI = 20.89 m.
(16)
The error e’(t) has zero mean and a standard deviation 0~1= 5.3 Bqmw3; thus, CJ;= (1 + pAt/2)/At o,f = 7.3 Bq rnd3 year-‘. Remarks (7) The estimated value of p is much greater than the t3’Cs constant of radioactive decay Jo = 0.023 year-‘. Since p represents the specific rate of depletion in the layer (-HI, 0), consequently the half residence time in the layer of the tracer is given by t1/2(-H,,O)
= log2/p
= 0.91 years.
(8) The estimated depth, H,, seems to be low; however, it is consistent with the depths of coastal waters where the water samples have been drawn. By assuming eqn (16) the solution of eqn (5), with Z = 0 and ii( to) = 6.12 Bq me3 is in very good agreement with the experimental data (Fig. 3). The time integration of eqn (5) has been obtained for different time steps, At, producing the same results. A sensitivity analysis of the response of the model versus a variation of the values of the parameters p and HI has been performed. The analysis shows that, only with the values in eqn (16) does the model reproduce the position and the activity of the maximum and the trend of the decay process. Remarks (9) The input fluxes -fo(t) (from experimental data) for the
Accumulation-dispersion processes of ‘j7Cs
165
100 90 80 70 60-
.
0
““,““,““,““I”“1
. . . 1960
1965
1975
1970
1980
1985
Year
Fig. 3. Period 1960-1984. Results of the numerical simulations
averaged distribution
n(t), continuous
for the surface layer: depth line; experimental data, boxes.
surface layer and -f(t) (given by eqn (11)) for the deep layer are shown in Fig. 4. The fluxfr (t) reaches its maximum with a delay of about 1 year with respect to the maximum offo(t); the maximum value off,(t) is significantly less than the maximum value offo(t) (maxf,(t) = O-7maxfo(t)). (10) Since p and HI are constant, if we approximatefO(t), t 2 tl = 1964, by an analytical expression, then we can give an analytical expression for the decay of u(t). By assuming an approximately constant flux from 1970 to 1984, fo(t) can be well approximated by a sum of two negative exponential functions: fo(t) = Ae- a(r--I1)+ B,-w-Q)
+ c,
t 3 *I>
(17)
where A = 624.8, B = 137.04, C = 30, Bq m-* year-’ j? = 0.27 year-‘. Thus, from eqn (5), with B = 0, we obtain, for t 2 tl, A c ii(t) = PHI +H&-de +
qt,)
4-h)
+
B
and
cc= 1.20,
,-N-0)
HI(P--_) c
A
- PHI H,(P--a)-H,(P-D)
(18)
B
e-P’(’- II) I
G. Buffooni,A. Cappelletti
166
1960
1965
1970
1975
1980
1985
Year Fig. 4. Period 1960-1984. Input flux -fo(t) at the air-sea interface, boxes; output flux -f,(t) from the surface layer = input flux to the deep layer, continuous line.
(11) The function (18) has been obtained as a solution of model (5) with F = 0 and the driving force given by eqn (17). However, we can obtain different decay curves by directly fitting experimental activity data with either a sum of two or with one negative exponential function. The standard deviations of these approximating decay curves are nearly the same as in eqn (18). However, they are pure descriptions of the decay process, and we cannot give a physical interpretation of the time constants in the negative exponential functions, while a natural interpretation is given to the time constants of the curve (18). 3.2. Results for the deep layer The dynamics described by the model (12)-(15) depends greatly on the values of the parameters v, D and HI. The parameters v and D could be depth-dependent; however, since we have solved independently the problem for the surface layer, we will try numerical simulations by assuming v and D to be constant, with values in an appropriate range, characteristic of a deep layer. We want to compare the depth distribution ~(tZ,z), at time t2 = 1980,
Accumulation-dispersionprocesses L$ i’7C~
167
with the experimental depth distribution U*(Z)shown in Fig. 1. Therefore, we have to normalize u*(z) so that the total quantity of the tracer accumulated in the water column during the period (to, t2) is the same for the experimental and calculated distributions. The normalization constant for u*(z) is: -ffi 4=
[.I -H2
~(tz, z)dz + ti(QH, I o s -Hz
u*(z)dz
Due to the approximations (spatial and time averaging operations), leading to the depth distribution u*(z), the comparison of ~(t~, z) with q u*(z) will, by necessity, be qualitative. Since u*(z) shows a subsurface maximum. it follows, from remark (4), that we must have )v) > 0, because ~(t2, z) has a subsurface maximum. From remark (5), it follows that these depth distributions cannot be steady state distributions. The values of the parameters introduced in eqns (12)-( 15) for the numerical simulations are as follows: /z = Ilo = O-023year-’ = 7.3 lo-” SC’; to = 1960; u(t,,, z) = z+,(z)= 0; HZ = 2000 m; .fi (t) given by eqn (11); v = 1, 3, 5 IO-‘ms-‘; D = 17 2 10-4m2s-‘. D = 10d4 m* ss’; then, the time evolution of the Let 1?= 3 lo-‘ms-‘, depth distribution u(t, z) is shown in Fig. 5. The depth distributions u(tz, z) and qu*(z), for different values of v and D, are shown in Fig. 6a, b and c. For v = 3 x lo-‘ms-‘,
10e4m2 s-’ < D < 2 x lO-4 m2 s-‘,
the computed distribution u(tZ, z) reproduces the position (~10Om) and the concentration (M 5 Bq mb3) of the subsurface maximum of qu*(z). Remarks (12) Let H2 = 800m, v = 3 x lo-‘ms-‘, D = 1, 2 x 10-4m’ s -‘; then, over a long enough period of time, the distribution u(t,z) (Fig. 7) increases with the depth near the bottom z = -Hz [see remark (6)]. (13) The numerical integration of the problem (12)-(15) has been carried out as follows. A semi-discrete analogue of (12)-( 15) (continuous in time, discrete in space with constant grid step of 5m) was obtained by the finite volume method. The time integration of the semi-discrete equa-
168
G. Buffooni, A. Cappelletti
0
500
1000
1500
2000
Depth (m) Fig. 5. Period 1960-1984. Results of the numerical simulations for the deep layer, with Hz = 2000m, v = 3 x lo-‘ms-‘, D = 10~4m2s-‘: depth distributions u(r,z) in (a) 1965,
(b) 1970, (c) (1975), (d) 1980 and (e) 1985.
tion was performed using an implicit method (Varga, 1965, p. 263) with a time step of 432 000 s (= 5 days).
4. PERIOD
1986-1992
As previously stated, the regression analysis of eqn (8) implies the minimization of the function (9). We observe that fallout and activity measurements in the first year after the accident are scarce, so that the rapid evolution of the accumulation-dispersion process cannot be completely described. In particular, for the period of maximum fallout in Italian Seas (the first week of May, 1986) activity measurements in sea water are not available (Table 2). It follows that the integration period (Ti, T2) in (9) does not include the most significant part of the transient. In this situation, the function (9) does not show a well defined minimum, at least with respect to the parameter b2 [i.e. Hi from (lo)]. We have, nevertheless, tried a parametric study of eqn (5) by varying the parameters p and Hi and considering the first year after the accident. In the
Accumulation-dispersion processes of ‘37C.~
500
169
2000
1000
Depm WI
b
Bqm’
107 9B,6-
Fig. 6. Period 196Ck1984. Depth distributions u(t2, z), t2 = 1980, (D = lO-4 m’s_’ continuous line, D = 2 x 10-4m2s-’ dashed line), and qu*(z) (boxes); (a) v = IO-‘ms-‘. (b) v = 3 x IO-‘ms-‘, (c) v = 5 x IO-‘ms-‘.
170
G. Bufjbni, A. Cappelletti Bqlm” 10 94
0
100
200
300
400
500
600
700
800
Depth (m) 7. Period 1960-1984. Depth distribution v = 3 x lO_‘ms-‘, D = 10-4m2s-’ continuous
Fig.
~(t~,z), t2 = 1980, with Hz = 800m, line, D = 2 x 10e4 m2 SC’ dashed line.
following years, the fallout and the activity in sea water reach approximately the levels (slightly greater) before the accident. The fall-out is shown in Fig. 8a; we estimate the time Tr, as defined in remark (1) by assuming an exponential decay of the fall-out for a few days after the maximum. We obtain Tr = 0.5 days = 0.0014 years; thus, Tr is of the same order of magnitude of T, (see Section 3). Because of the rapid evolution of the transient, we cannot simulate the process in a satisfactory way by assuming ~1and HI time-independent (remark (1)). The solution of eqn (5) (Fig. 8b), with Z = 0 and U(0) = 2.52 Bq rnd3, describes well the experimental data by assuming two periods with different values for ,LL and HI as follows: Period (days) O-40 40-365
p (day-‘) o-13 0.013
HI (m) 1.5 20
A sensitivity analysis has shown that the model is highly sensitive to the values of the duration and of the parameters ,Uand HI of the first period. On the contrary, the values of the parameters p and HI relative to the second period may vary over a wide range, producing acceptable results.
Accumulation-dispersion
processes qf“37C.s
171
Bqlm’fd lO'-y
loz-
. . . c
lo'-
.
.
l
l
a
.
. 10°-
. .
lo4
/11,111,1,I,JII,II(,1’1(‘II,IJ’,I”,I’I,
0
20
40
60
80
100
120
140
160
180
200
180
200
Days
600-
b
0
20
40
60
80
100
120
140
160
Fig. 8. Period 1986-1992 (first 200 days). (a) Input flux -fo(t) at the air-sea interface, boxes; (b) results of the numerical simulations for the surface layer: depth averaged distribution u(r), continuous line and experimental data, boxes.
172
G. Buffooni. A. Cappelletti
5. CONCLUDING
REMARKS
Monitoring fall-out and activity evolution of tracers in sea waters, cannot be easily performed with high spatial and time resolution. It follows that the main characteristics of the accumulation-dispersion processes of the tracers can be described and interpreted by means of an analysis based on spatial and time averages of the experimental data. In this type of analysis, lumped parameter models can be successfully used. In particular, depthaveraged models, to study the surface layer, allow the estimation of the depth of the layer representative of the experimental data, the specific depletion rate and the flux to the deep layer. Depth-dependent models allow us to estimate the order of magnitude of the vertical transport and of the vertical mixing coefficient. Moreover, some characteristics of the vertical distributions of the tracers can be interpreted. We observe: (i) a subsurface maximum only in non-steady state distributions and when a vertical transport towards the bottom is present; and (ii) the increase of the activity level with the depth near the bottom when the flux of the tracer from water to sediment is low and when the time is long enough. From a comparison of the analysis performed for the two periods, we may observe that fall-out and activity measurements should be carried out at the same times and with sampling intervals small enough to describe completely the transients. This is particularly true in a rapid transient like that resulting from the Chernobyl accident.
ACKNOWLEDGEMENTS The authors would like to thank A. Cigna, R. Delfanti and C. Papucci for their suggestions about literature and the data available, and the referees for their criticisms and suggestions. This work was partially supported by the European Science Foundation ‘TAO’ (Transport in the Atmospheres and in the Ocean) project.
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