R. Periäfiez et al. 
san be seen in Fig. 3. The area under study is divided into a number 
of small grid cells (in this example 1884 x 3712 m?), which allows a 
larger spatial resolution than in box models. 
Some examples of Eulerian radionuclide transport models are Bre- 
‚on and Salomon (1995), Harms (1997), Koziy et al. (1998), Preiler and 
<heng (1999), Cetina et al. (2000), Bailly du Bois and Dumas (2005), 
£stournel et al. (2012), Periäfiez et al. (2012), Maderich et al. (2016, 
2017), among many others, 
3.1.3. Lagrangian Models 
In Lagrangian models the released activity is represented by a num- 
zer of particles, each one equivalent to a given amount of activity (Bq). 
Many examples of Lagrangian models applied to radionuclide transport 
are found in literature (Schonfeld, 1995; Harms et al., 2000; Periäfiez 
and Elliot, 2002; Toscano-Jimenez and Garcfa-Tenorio, 2004; Periänez, 
2005b; Nakano et al., 2010; Kawamura et al., 2011; Kobayashi et al., 
2007; Min et al., 2013; Periäfiez et al., 2016a, among many others). 
[he path followed by each particle is calculated and radionuclide 
concentrations are obtained from the number of particles per volume 
or mass unit. The equations describing variations of particle (in state 
x) position over each time increment dr are given by the It6 (Protter, 
2004) stochastic differential equations: 
dx = u,dt + Sg + V2K,dW,, 
dy=v,dt+ hg + V2K,dW,, 
dz= w.dt + Sage+ V2K,dW,, 
where u, v, and w, are velocity components on coordinate axis (x, y, z) 
for state a; W,., WW, are independent components of the stochastic 
motion, which have zero mean and variance dt (dW2 = dw} =dW2 = 
dt), and for a finite time step 4r they can be simulated as AW, = 
VAR, AW, = VArR,, AW, = VArR,, where (R,, R,, R.) are normally 
distributed random variables having zero mean and standard deviation 
one. Derivatives of the diffusion coefficients above prevent the artificial 
accumulation of particles in regions of low diffusivity (Proehl et al., 
2005; Lynch et al., 2015). 
A method based on the solution of the Kolmogorov equation for the 
transfer probability is used in the stochastic approach for simulating 
'ransfers between different states of the radionuclide, If a particle at a 
»oint in time is in state «, then the probability p,yg of transfer to state £ 
during time dr is found from the solution of the Kolmogorov equation 
with initial conditions p„4(0) = 5.5, where ö,g is the Kronecker delta,* 
This equation, known as the Kolmogorov forward equation (Parzen, 
1962), is written as: 
dDx A 
a = zZ KygPay- 
Ror a two-state transfer between dissolved (state 1) and adsorbed ra- 
dionuclides (state 2), considering a one-step reaction and one sediment 
size (see Section 3.4), the equations for the transfer probabilities are: 
dpı2 
dr kıpı — KaPız, 
dp21 
Lk 
dt 1P21 + kaPazı 
Zpı 
ar 5 kp + kapız, 
dpx 
va kıPar — KaPazr 
(8) 
(9) 
(10) 
(11) 
4 This function is defined as: 
= { 1 a=ß 
87) 0 a*ß 
Environmental Modelling and Software 122 (2019) 104523 
where we have the following adsorption and desorption coefficients: 
kı = -kıı = kız and k, = -kzz = ka, From the solution of the system 
of Eqs. (8)-(9), the probabilities to change state during each time step 
will be equal to: 
Pia = FE CC + Ka), 
D = era — exp(-(k, + k2)4P)) 
whereas pıy = 1-pı2 and pa = 1-p2; are the probabilities to remain in 
the previous state. Similarly, “death” of particles due to the radioactive 
decay can be described. If more than two possible states (adsorption 
on multifractional sediments and/or two step kinetic reaction etc.) are 
considered, then the system of Eqs. (7) should be solved numerically. 
For small k„g4t, we can approximate equation (7) by the first order 
numerical scheme that yields 
n n 
Pag(At) — Pag OO X kygday(O)AF = X kypöayAt = kagdt. (14 
y=1 y=1 
As seen from (12)-(13) and (14), for small k„g47 solutions for p„,g(4f) 
coincide, but (14) is applicable for any number of states. 
A comparison of the main advantages and disadvantages of each 
model type is included in Section 5.5, where the particular situations 
in which each model type is advantageous are discussed, 
3.2. Model spatial and temporal resolution 
Models may adopt different structures depending on the physical 
characteristics of the area to be modelled. A one-dimensional model 
may be enough to simulate dispersion in an essentially one-dimensiona) 
structure, like a channel, where mixing in depth and in the transverse 
direction is fast. Such a model was applied to the Suez Canal (Abril 
et al., 2000). A simple model in which the conservation of mass, heat, 
salt and tracer are considered only in the vertical direction and inflows 
and outflows through straits act as forcing terms for the vertical cir- 
culation and feedback for the Mediterranean sub-basins was developed 
by Maderich (1998) to reconstruct 1%7Cs contamination in the period 
1960-2010 for the Mediterranean Seas chain. 
Two-dimensional depth-averaged models may be applied to es- 
tuaries, lakes, bays and coastal areas. They assume that the water 
column is homogeneous, ie., no vertical structure eXists either in wateı 
cireulation or in radionuclide concentrations (Periäfiez et al., 2013c). 
{In general, these conditions are more easily satisfied in shallow waters 
and in winter: wind-induced mixing is more intense during this season; 
although some buoyancy driven mixing due to surface cooling may also 
occur. Calm conditions in summer may lead to vertical stratification 
and, thus, the validity of a depth-averaged model would be seasonally 
dependent. An example where this happens is the North Sea (van 
Leeuwen et al., 2015). Winter cooling in the tropics is not enough to 
destroy stratification, which is permanent. In contrast, stratification is 
virtually non-existent in polar regions. 
A full three-dimensional model is required if there is vertical struc- 
ture in the water column. These models present the highest level 
of complexity (Harms et al., 2000; Hazell and England, 2003; Gao 
et al., 2004; Orre et al., 2007; Kobayashi et al., 2007; Maderich et al., 
2017; Min et al., 2013, etc.). In special situations other intermediate 
approaches are valid. For instance, two-layer models consist of two 
depth-averaged models, one over the other. These models can be 
applied if the marine area may be treated as two well-mixed water 
layers which move in different directions. This is the case in the 
Alborän Sea, western Mediterranean, (Periäfiez, 2008). A vertical two- 
dimensional model can also be applied in fjords, for instance, where 
transverse mixing is fast but a significant vertical structure exists due 
to stratification. 
Temporal resolution of the model (time step used for integration 
of the corresponding differential equations) is another key factor to