Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 58

The same result could have been derived by starting directly from Equation (3.8)
and again using ρ u i as the quantity to be conserved. When neglecting spatial varia-

in the diffusion terms (which is generally always a good approximation),
the Navier-Stokes equations directly result with the diffusivity in Equation (3.8)
being the viscosity just derived. As in the Euler equations the additional forces are
the pressure gradient and the gravitational acceleration.
It may be interesting to note that the physical process responsible for molecular
friction is the molecular diffusion of the fluid particles. If a faster fluid layer moves
above a slower fluid layer, some of the particles from the slower fluid layer will diffuse
into the faster layer, slowing down the upper layer. On the other hand, faster fluid particles
diffusing into the fluid layer below will accelerate the slower fluid layer. From both sides
of the layer this results in an effective friction, either slowing down or accelerating the
other layer. Therefore, the operator

tions of ρ

∂∂

2 x j

is also called the diffusion operator.

3.4.1.3 Conservation of Energy

Conservation of energy can be formulated as a conservation equation for temperature.
Simply put, the total change in temperature of a fluid volume is given by the rate of
heating of the volume. The change of temperature T of the fluid with time can be
expressed as

∂
∂

Q
x

1

s

ρ c

p 3

where Q s is the solar radiation [W/m 2 ] and c p is the specific heat of water [J kg −1 K −1 ].
The term on the right-hand side represents the source term of Equation (3.8).
The only other process that is changing the heat content of a fluid volume is
molecular diffusion. In this case fluid particles that are warmer and that diffuse into
colder fluid because of molecular motion will contribute to the warming up of the
colder fluid. As explained previously, the diffusion can be described through the
Laplace operator ∂∂

2 x j

, and the whole conservation energy of temperature can be

written as

2

∂
∂

Q
x

dT
dt

∂
∂

T
x

1

= ν +

s

(3.16)

T 2

ρ c

j p 3

where v T is the molecular diffusivity for temperature, a parameter that depends only
on the properties of the fluid. When compared to Equation (3.8) this is the conser-
vation of T with the external source given above.

3.4.1.4 Conservation of Salt

In the oceans and coastal lagoons the water has a certain salt content that varies
from nearly zero high up in the estuaries and rivers to values of about 10 psu in
brackish water and up to more than 30 psu in ocean waters. In lagoons where