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

CHAPTER 2. THEORY 31
Substituting Eq. (2.4.42) into Eq. (2.4.41) and only taking into account the
first two terms on the right-hand side yields
df(x(t)) = (A(x, t)dt + B(x, t)dW (t)) f(x(t))
∂x
+ 1
(A(x, t)dt + B(x, t)dW (t))2
2 � �� �
(dx(t)) 2
∂2f(x(t)) ∂x2 .
(2.4.43)
Again, dt is an infinitesimal time step so that (dt) 2 ≈ 0 and dW dt ≈ 0.
This allows us to write (dx(t)) 2 as
(dx(t)) 2 = (A(x, t) dt + B(x, t) dW (t)) 2
(2.4.44)
= A(x, t) 2 (dt) 2 + 2 A(x, t) B(x, t) dW (t) dt + B(x, t) 2 (dW (t)) 2
≈ B(x, t) 2 (dW (t)) 2
and Eq. (2.4.41) simplifies to
df(x(t)) = (A(x, t)dt + B(x, t)dW (t)) ∂f(x(t))
∂x
+ 1
2 B(x, t)2 (dW (t)) 2 ∂2f(x(t)) ∂x2 .
(2.4.45)
In order to include a probability distribution function, the mean of Eq.
(2.4.45) divided by dt is calculated
� � ��
� �
df(x(t))
dW (t) ∂f(x(t))
= A(x, t) + B(x, t) (2.4.46)
dt
+
�
1 (dW (t))2
B(x, t)2
2 dt
From the properties of W (t), it follows that
and Eq. (2.4.46) is reduced to
� � �
df(x(t))
= (A(x, t)
dt
∂f(x(t))
�
+
∂x
dt
∂ 2 f(x(t))
∂x 2
∂x
�
.
� dW (t)
dt
� �
d2W (t)
= 0 and dt
�
= 1
�
1
2 B(x, t)2 ∂2f(x(t)) ∂x2 �
. (2.4.47)
The mean of f(x(t)) can as well be expressed in integral form
�
〈f(x(t))〉 = f(x(t)) p(x, t|x0, t0)dx (2.4.48)