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

CHAPTER 2. THEORY 17
Substituting Eq. (2.2.18) into Eq. (2.2.11) and following the reasoning
which has lead to Eq. (2.2.14) yields
∂C
∂t + ∇ · � J = 0. (2.2.19)
Replacing � J with Eq. (2.2.17) gives the advection-diffusion equation as
∂C
∂t
+ ∇ · (�u C) = D ∆C (2.2.20)
The equation of continuity for incompressible flow is ∇ · �u = 0 and Eq.
(2.2.20) becomes
∂C
∂t
+ �u · ∇C = D ∆C. (2.2.21)
u
J x,in
y
∆y
z
∆z
∆x
Fig. 2.2.3: Schematic of a control volume CV with cross flow.
2.2.3 Turbulent diffusion
Molecular diffusion alone is entirely insufficient to produce the rate of mixing
observed in natural waters. The difference between the observed rate of
diffusion and the rate expected from molecular diffusion is the result of turbulent
diffusion. Thus, the advection-diffusion equation has to be extended
to include the influence of turbulence. A conservation equation for turbulent
flows can be derived from the advection-diffusion equation by decomposing
the velocity vector and the concentration into the sum of a time averaged
and a fluctuating part
C = C + C ′
, (2.2.22)
u = u + u ′ , (2.2.23)
v = v + v ′ , (2.2.24)
w = w + w ′
x
Jx,out
(2.2.25)