Three-dimensional Lagrangian Tracer Modelling in Wadden Sea ...

CHAPTER 2. THEORY 27
i, j, k and not x,y,z. The position of a particle as well as its velocity is transformed
by dividing their components by the corresponding grid spacing (i.e.
�xT = (xT /∆x, yT /∆y, zT /∆z) = (iT , jT , kT ). The index (i, j, k) of the grid
box containing a particle is computed from the particle position as
i = int(iT + 0.5) (2.4.22)
j = int(jT + 0.5) (2.4.23)
k = int(kT + 0.5). (2.4.24)
Necessary for the interpolation of the velocity are the weighting factors which
are determined through
a = iT − real(i − 1) (2.4.25)
b = jT − real(j − 1) (2.4.26)
c = kT − real(k − 1) (2.4.27)
so that the tracer velocity can be obtained from Eq. (2.4.2) - (2.4.4). Finally,
the position is updated with respect to crossing of boundaries
i n+1
T
j n+1
T
k n+1
T
= inT + un � �
T ∆tT ∆u
e − 1
∆u
= jn T + vn T
∆v
= kn T + wn T
∆w
� e ∆tT ∆v − 1 �
(2.4.28)
(2.4.29)
� e ∆tT ∆w − 1 � . (2.4.30)
The gradients of u, v and w are discretised as
∆u =
∆v =
∆w =
1
1
n+
un+ 2 (i, j, k) − u 2 (i − 1, j, k)
∆x
1
1
n+
vn+ 2 (i, j, k) − v
∆y
2 (i, j − 1, k)
(2.4.31)
(2.4.32)
1
1
n+
wn+ 2 (i, j, k) − w 2 (i, j, k − 1)
, (2.4.33)
h(i, j, k)
since they are updated in GETM with an offset of 1/2∆t (see Fig. 2.1.1).
2.4.2 Modelling diffusion
2.4.2.1 The stochastic differential equation (SDE)
In 1908, Langevin considered the problem of the dynamical description of
molecular diffusion. He suggested that the equation of motion of a particle