R. Peridfiez et al. 
Environmental Modelling and Software 122 (2019) 104523 
Atmospheric deposition 
(dry and/or wet) 
Direct release 
Suspended * 
particles 
Dissolved. 
A; k we 
«—-  radionuclides 
ISettling 
Kinetic exchanges 
; * Deposition 
r 
X 
Kır 
1x 
Horizontal-vertical 
advection/diffusion 
Qasr 
Jpper sediment layer 
Deep sediment. 
Buria 
-radionuclides 4 suspended particles 
Fig. 2. Processes affecting the dispersion of non conservative radionuclides in a marine system. Kinetic rates k, and k, describe uptake and release of radionuclide by solid 
particles. Z is thickness of the upper sediment contaminated layer. 
L. Hydrodynamic model: it provides the water currents, which 
determine advective transport. Also, water currents and density 
stratification may be used to derive eddy diffusivities, which 
are used to evaluate turbulent mixing (see Section 3.3 for more 
details). 
Sediment transport model: it provides suspended matter con- 
centrations and erosion and deposition fluxes over the model 
domain. Details on the mathematical formulation of the physical 
processes are described for instance in Eisma (1993), van Rijn 
(1993), Winterwerp and van Kester (2004) and Lick (2008), but 
are not included in the present review. 
Radionuclide transport model: it includes the description of 
advection/diffusion processes and the description of adsorp- 
tion/desorption reactions between the dissolved and solid phases 
(Periäfiez, 2005a). Other relevant processes could be included 
as well, as for instance migration of radionuclides in the bot- 
tom sediments due to molecular diffusion and reworking of 
sediments by animals -bioturbation- (Maderich et al., 2017). 
Also redox reactions may be included in the case of plutonium 
(Periäfiez, 2003b), which presents a complex speciation and 
axidized and reduced species coexist in solution. 
The different ways in which equations describing these transport 
processes are written and numerically solved lead to different models. 
These are briefly described in the following section. 
3. Radionuclide transport models 
3.1. Box, Eulerian and Lagrangian models 
» 
Three types of transport models are used: Eulerian, Lagrangian and 
box-models. They are very briefly described below. 
3.1.1. Box models 
In these models the marine area under consideration is divided into 
a number of large boxes or compartments. These boxes are intercon- 
nected according to water circulation and it is assumed that the trans- 
port of dissolved radionuclides between boxes is proportional to the 
difference in radionuclide concentration between them. It is assumed 
that mixing of radionuclides within each box is uniform and instanta- 
neous, Other processes such as transfers of dissolved radionuclides to 
suspended matter, deposition of suspended matter and adsorption or 
radionuclides in bed sediments can be described. This is usually done 
using an equilibrium distribution coefficient k, (see Eq. (15) below for 
its definition). Box models are well suited for assessments of radionu- 
clide dispersion involving large spatial and temporal scales (thousand 
kilometres; years to decades). Some recent examples of box models are 
POSEIDON-R (Lepicard et al., 2004; Maderich et al., 2014a,b; Bezhenar 
et al., 2016), models developed in NRPA (losjpe et al., 2002, 2009) 
and others (Sänchez-Cabeza et al., 2002; Häkanson, 2005; Smith and 
Simmonds, 2009; for instance). 
The basic equation for box models provides the temporal evolution 
of radionuclide activity in the water column in box i, A, (Bq). The 
system of differential equations is (Nielsen, 1995): 
3 A, Rn N 
re Z ud = Z au —qA+Si mM 
where n is the number of boxes in the model, q;; (expressed in s71) is the 
transfer rate from box i to box /, q, (s71) is the rate of radionuclide loss 
from box / without transfer to another box (due to radioactive decay, 
sedimentation, etc.) and ‚SS; is the external source of radionuclides to 
box ; (Le., Bq per unit time which are released into the sea at that box 
due to atmospheric fallout and/or point sources). Transfer rates q,, are 
deduced from known oceanographic features of the region under study 
and they parameterize the advection and diffusion processes described 
in the previous section. They can also be obtained from the currents 
calculated by hydrodynamic models. 
It is assumed that, at any time, activity in the water column is par- 
titioned between the dissolved phase and suspended matter particles. 
This partition depends on the suspended matter concentration in box 
i, SM, and the radionuclide k, (equilibrium is assumed). If sedimen- 
tation rate of suspended particles is $R;, then the activity transfer rate 
q; from water to bed sediment due to particle sedimentation is 
_ KkaSR; 
TG TKSM)'