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

CHAPTER 2. THEORY 33
where BT is the transpose of matrix B. Setting n = 3, � A = (u, v, w),
⎡
⎤
Ax 0 0
⎢
⎥
⎣ 0 Ay 0 ⎦ and p(�x, t| �x0, t0) = C(�x, t) the Fokker-Planck
1
2BBT =
0 0 ν ′
equation becomes
∂C
∂t
+ ∇ · (�u C) = ∂2
∂x 2 (Ax C) + ∂2
∂y 2 (Ay C) + ∂2
∂z 2 (ν′ C) (2.4.54)
which is similar to the three-dimensional advection-diffusion equation (2.2.37)
∂C
∂t
�
∂
+ ∇ · (�u C) = Ax
∂x
�
∂C
+
∂x
∂
�
Ay
∂y
�
∂C
+
∂y
∂
�
ν
∂z
�
′ ∂C
. (2.4.55)
∂z
The apparent difference between both equations (at the right-hand side) and
its meaning for modelling diffusion with the Fokker-Planck equation (2.4.54)
will be discussed in the next section.
2.4.2.3 Modelling diffusion with random walk
A random walk model consisting of a large number of statistically independent
steps is suitable to represent the chaotic nature of turbulent diffusion.
The size of the diffusive step is determined by the stochastic differential
equation (2.4.35) whose unknown quantities have been determined in the
last section as
�A(�x, t) = (u(�x, t), v(�x, t), w(�x, t)) (2.4.56)
⎡ √
2 Ax 0 0
⎢
B = ⎣ 0 � ⎤
⎥
2 Ay 0 ⎦ . (2.4.57)
√
0 0 2 ν ′
Now, it is possible to write down Eq. (2.4.40) as follows
d�x(t) =
⎛
⎜
⎝
u(�x, t)
v(�x, t)
w(�x, t)
⎞
⎡
⎟ ⎢
⎠ dt + ⎣
√
2 Ax 0 0
0 � ⎤ ⎛
⎥ ⎜
2 Ay 0 ⎦ ⎝
√
0 0 2 ν ′
Z1
Z2
Z3
⎞
⎟
⎠ √ dt,
(2.4.58)
where Z1, Z2, Z3 are independent random numbers from the standard normal
distribution with zero mean and unit variance. Unfortunately, modelling
turbulent diffusion with the Fokker-Planck equation is not realistic. Visser