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

CHAPTER 2. THEORY 19
side which correspond to the transport of C by turbulent fluctuations. These
additional terms can be rewritten together with the diffusive transport D∆C
as
∂
∂x
�
D ∂C
∂x − u′ C ′
�
+ ∂
�
D
∂y
∂C
∂y − v′ C ′
�
+ ∂
�
D
∂z
∂C
∂z − w′ C ′
�
. (2.2.35)
All terms in the parentheses represent mass transport. The first of these
terms stands for the transport due to molecular diffusion (Fick’s law) and
the second is a turbulent flux that arises by virtue of the correlation between
�u ′ and C ′
where �u ′ = (u ′ , v ′ , w ′ ). Since the molecular diffusion coefficient is
usually a very small quantity (i.e. �u ′ C ′ ≫ D ∂C ), the molecular diffusion
∂x
terms can be neglected compared to turbulent flux. The terms denoting
the turbulent transport represent unknown quantities. Assuming that the
turbulent diffusion is a Fickian process, this closure problem is solved by
introducing a coefficient of eddy diffusivity for each direction. With this
assumption and the notation Ax, Ay for the horizontal eddy diffusivities and
ν ′ for the vertical eddy diffusivity, the closure scheme is
u ′ C ′ ∂C
= −Ax
∂x
v ′ C ′ ∂C
= −Ay
∂y
w ′ C ′ = −ν
′ ∂C
. (2.2.36)
∂z
The coefficients Ax, Ay, ν ′ are strongly flow dependent and vary within the
flow field. With the assumption that turbulent mixing is much stronger than
molecular diffusion the transport equation for turbulent flow is
∂C
∂t
+ ∇ · (�u C) = ∂
∂x
�
Ax
∂C
∂x
�
+ ∂
∂y
�
Ay
∂C
∂y
�
+ ∂
∂z
�
ν
2.2.4 Eulerian vs. Lagrangian perspective
′ ∂C
∂z
�
. (2.2.37)
Fluid motion and any constituent (temperature, salinity, concentration of an
arbitrary substance) transported by fluid motion can be described from two
frames of reference, from a stationary frame (Eulerian) and from one which
is moving along with the flow (Lagrangian). In the Eulerian perspective, the
flow and its constituents are described with respect to fixed spatial positions
�x = (x, y, z) and time t. Thus, they are written as e.g. �u(�x, t) and C(�x, t).
The Lagrangian perspective follows the flow and traces the history of individual
fluid particles. Unlike in the Eulerian description, spatial position is
not a fixed reference but another variable of the particle. The flow variables
are written with respect to time t and to a single, initial reference position,
e.g. �x0 the particle position at t = 0. In this case all variables are recorded
as e.g. �x(�x0, t) and C(�x0, t).