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

CHAPTER 2. THEORY 35
2.4.2.4 Implementation
The random walk model implemented into GETM and GOTM is based on
the scheme proposed by Hunter et al. [1993]. A diffusive step is carried out
each macro time step ∆t for the internal mode of GETM and the particle
position is updated according to (2.4.63)-(2.4.65) with a slight modification:
the random numbers Z1, Z2, Z3 are not taken from the standard normal
distribution but from a uniform distribution with zero mean and a variance
of 1
3
(i.e. the uniform random numbers vary between +1 and −1). According
to the Lindeberg-Feller central limit theorem (Feller [1935]) the distribution
of a normal form variate
Znorm =
Z
� 〈(Z − 〈Z〉) 2 〉 , (2.4.66)
where Z is a random variable with zero mean and finite variance 〈(Z − 〈Z〉) 2 〉,
tends to the normal distribution with zero mean and unit variance N (0, 1).
In order to include Znorm = Z/ � 1/3 the random walk model (2.4.63)-
(2.4.65) becomes
xn+1 = xn + ∂Ax(�xn)
∂x
yn+1 = yn + ∂Ay(�xn)
∆t + Z2
∂y
zn+1 = zn + ∂ν′ (�xn)
∆t + Z3
∂z
�
∆t + Z1 6 Ax(�xn) ∆t (2.4.67)
�
6 Ay(�xn) ∆t (2.4.68)
� 6 ν ′ (�xn) ∆t. (2.4.69)
In the application of the random walk model to the Wadden Sea, horizontal
diffusion is not included. The horizontal diffusivities Ax, Ay are set to
zero since the shear dispersion (combination of vertical mixing and shear)
is the major horizontal dispersion mechanism. The vertical diffusivity ν ′ is
calculated by the turbulence model using the Kolmogorov-Prandtl relation
(Burchard and Bolding [2002])
ν ′ = c ′ k
µ
2
. (2.4.70)
ε
Here, k is the turbulent kinetic energy, ε is the dissipation rate, and c ′
µ is a
so-called stability function depending on shear, stratification and turbulent
time scale, τ = k/ε. The vertical eddy diffusivity is located on the Eulerian
grid at the position of the vertical velocity component w. Therefore it is
easy to linearly interpolate ν ′ to the instantaneous vertical position zn of a
tracer in the grid box with index (i, j, k) and update the position according