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

CHAPTER 2. THEORY 25
where ∆u = ur − ul and ∆x = xr − xl. In a next step the antiderivative
F (t) = −t ∆u
∆x to f(t) is found and the integration factor eF (t) is formed.
Then, Eq. (2.4.6) is multiplied with the integration factor which yields
d
� � � �
∆u
∆u
−t −t ∆u
xT e ∆x = e ∆x ul − xl . (2.4.7)
dt
∆x
Integration of both sides with respect to t gives
∆x
xT (t) = xl − ul
∆u
where C is the integration constant. C can be determined through the initial
condition xT (t0) = xT,0 (i.e. initial position of the particle)
∆x
xT (t0) = xT,0 = xl − ul
�
⇔ C =
xT,0 + ul
+ Cet ∆u
∆x (2.4.8)
∆x
∆u
∆u
− xl
+ Cet0 ∆u
∆x
�
∆u
−t0 e ∆x (2.4.9)
As a final step xT,0 is replaced in Eq. (2.4.9) by recognising
�
dx�
�
∆u
= uT,0 = xT,0
dt
∆x + ul
∆u
− xl
∆x
� t=t0
∆x
⇔ xT,0 = uT,0
∆u + xl
∆x
− ul
∆u
such that the particle position can be readily determined as
(2.4.10)
∆x ∆u
xT (t) = uT,0 e ∆x
∆u (t−t0)
. (2.4.11)
The distance ∆xT = xT (t) − xT,0 a particle travels during a time step
∆t = t − t0 is then
∆xT = uT,0 ∆x
� �
∆t ∆u
e ∆x − 1
∆u
(2.4.12)
In Appendix B a second way to obtain the solution to the movement equation
(2.4.12) is presented. In analogy to Eq. (2.4.12) the movement of a particle
in x and y-direction is
∆yT = vT,0 dy
� �
∆t ∆v
e dy − 1
∆v
∆zT =
(2.4.13)
wT,0 ∆z
� �
∆t ∆w
e ∆z − 1 .
∆w
(2.4.14)
For uniform flow (i.e., ∆u = ∆v = ∆w = 0) the particle motion is simply
calculated with
∆xT = uT,0 ∆t (2.4.15)
∆yT = vT,0 ∆t (2.4.16)
∆zT = wT,0 ∆t. (2.4.17)