Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 51

Following a similar procedure and assuming an incompressible fluid—an equa-
tion equivalent to Equation (3.8) on a Lagrangian reference can be obtained from
Equation (3.6):

+−

d
dt

β ∂

∂

∂
∂

β

= ϕ ( SS ) (3.9)

o i

x x

j j

Comparing Equation (3.8) and Equation (3.9) and using the fact that for an incom-
pressible fluid 0 (see Section 3.4.1.1), the relation between total and partial

time derivatives is obtained: †

∂ u i / ∂ x i =

d
dt

ββ β

= ∂
∂

∂
∂

+ u (3.10)

i

t x

i

This equation simply says that the property of a fixed elementary volume in a moving
fluid element can change through local changes in the fluid and advective changes
transported by the fluid into the elementary volume.
In the field of mathematical modeling, especially physical processes, there is a
strong tradition of obtaining the discretized equations starting from their differential
form using, for example, the Taylor series. In this text discretized equations will be
obtained using both a finite-volume approach and the Taylor series.

3.3.5 B OUNDARY AND I NITIAL C ONDITIONS

Partial differential equations relate values of properties to time and space derivatives.
Spatial derivatives relate values of the property in neighboring points, and conse-
quently the evaluation at boundaries requires the knowledge of information from
outside the study area called boundary conditions. Similarly, time derivative relates
values of the property in sequential instants of time and their evaluation requires
knowledge of the solution in the first instant of time called initial conditions.
Initial conditions have to be specified in terms of property values. In contrast,
spatial boundary conditions can be specified in terms of values or their derivatives.
Comparing differential and integral forms of evolution equations (Equation (3.7)
and Equation (3.8)) shows that imposing boundary conditions in terms of spatial
derivatives in differential equations is, in fact, equivalent to imposing fluxes across
the boundary in integral equations.
Physically boundaries can be divided into two main groups: solid boundaries
and open boundaries. Solid boundaries are impermeable and, consequently, there
is neither water flux nor advective transport across them. Diffusive flux across
solid boundaries can be neglected for most dissolved substances, but not for
momentum, where it is represented by bottom shear stress. Solid matter can be

† In fact, the condition of null velocity divergence is not required, only the continuity equation. What
happens is that the continuity equation becomes the condition of zero velocity divergence in the case of
incompressible flows.