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

CHAPTER 3. IDEALISED TESTCASES 43
h n+1
1
with
C n+λ
k
and
h n+λ
k
C n+1
1
− hn 1C n 1
− ν
∆t
n 1
C n+λ
2
1
2 (hn+λ 2
− C n+λ
1
− h n+λ
1 ) − Fb = 0; k = 1 (3.2.21)
= λCn+1
k + (1 − λ)Cn k (3.2.22)
= λhn+1
k + (1 − λ)hnk. (3.2.23)
Here, upper indices denote time levels and lower indices stand for the vertical
discrete location. The Crank-Nicholson parameter is λ, such that for λ = 0
a fully explicit, for λ = 1 a fully implicit scheme and for λ = 0.5 the Crank-
Nicholson second-order in time scheme is obtained. This numerical scheme
leads to a system of linear equations in the form of a matrix equation.
The tridiagonal matrix of this equation is solved by means of the simplified
Gaussian elimination. The constant fall of sediment is modelled by using a
directional splitting method (i.e. applying a 1-D scheme in one direction at a
time) and a Total Variation Diminishing (TVD) advection scheme (Pietrzak
[1998], see Burchard and Bolding [2002] for details).
3.2.2.1 Modelling sediment suspension
For parameterisation, the sediment has a density of ρSed = 2650 kg/m 3 and
the sediment particles are assumed to be sphere-like with a diameter of
d = 62.5 · 10 −6 m, a volume of V = 4/3π(d/2) 3 = 1.28 · 10 −13 m 3 and a mass
of m = 3.3875 · 10 −10 kg. The fall velocity ws for sphere-shaped particles is
calculated using the equation proposed by Zanke (Zanke [1977])
ws = 10 ν
��
d
1 + 0.01 g′ d3 ν2 �
− 1
(3.2.24)
where ν is the molecular viscosity of water and g ′ is the reduced gravity
g ′ = g ρSed − ρ0
ρ0
(3.2.25)
with ρ0 being the standard reference density. First, the Rouse profile is
computed with the semi-implicit Eulerian scheme for diffusion and the TVD
scheme for advection in order to obtain the steady-state concentration of
suspended sediment. After a simulation time of 12 hours, the total sediment
concentration is Ctotal = 1.87 · 10 −2 kg/m 3 . This equals a number of