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

CHAPTER 3. IDEALISED TESTCASES 41
∂CSed/∂t + w ∂CSed/∂z = 0 so that Eq. (3.2.8) reduces to
�
∂
ν
∂z
′ ∂CSed
t
∂z + wSed
�
CSed = 0
⇔ ν ′ t
∂CSed
∂z + wSed CSed = const. = 0. (3.2.9)
Eq. (3.2.9) represents a balance between the rate of settling wSed CSed
and the rate of turbulent diffusion ν ′ t ∂CSed/∂z. Integration of the balance
equation results in the expression for the concentration profile C(z), first
derived by Hunter Rouse (Rouse [1937])
� � z �
1
C(z) = Ca exp
dz
(3.2.10)
−wSed
a ν ′ t
where Ca is the reference concentration at the arbitrary level za which is
usually chosen to be close to the sediment bed. The reference concentration
Ca has to be known for the calculation of the concentration profile. It is
assumed that the eddy diffusivity ν ′ t is proportional to the eddy viscosity νt
and can be calculated from
ν ′ t = νt
σt
(3.2.11)
where σt ≈ 0.7 is the constant Prandtl number. The sediment profile (3.2.10)
with respect to (3.2.7) and (3.2.11) is
�
wSed
C(z) = C0 exp −σt
κu∗ � z
1
b z0 (z + z0) � 1 − z
�
� dz
(3.2.12)
D
�
wSed
= C0 exp −σt
κu∗ � ��
z + z0 D − z0
ln
.
b
z0 D − (z + zc)
The reference concentration C0 is computed at the roughness length at the
bottom z0
z0 = 0.1 ν
ub + 0.03h
∗
b 0
(3.2.13)
where ν is the eddy viscosity and h b 0 represents the height of the bottom
roughness elements. Following Smith and McLean [1977], the bottom sediment
concentration C0 is a function of the bottom friction velocity and the
critical friction velocity u c ∗ (at which sediment particles begin to go into
suspension)
� � �
b 2 � � �
c 2
u∗ u∗ 1 −
C0 = γ1
u c ∗
u b ∗
(3.2.14)
where γ1 = 1.56 · 10−3 is a constant. The steady state solution (3.2.12) is
the so-called Rouse profile and wSed/(κu∗ b ) is the Rouse number.