INTRODUCTION - Hydrology: An Introduction - page 12

given parameterization is said to be more parsimonious than another one, when it needs
fewer variables and parameters to describe the phenomenon.
A parameterization can be called robust when the outcome is relatively insensitive
to its structure and to errors and uncertainties in the input variables and parameters.
In hydrology, a model usually refers to a combination of several parameterizations to
simulate more complicated phenomena and their interactions.

1.5 Conservation laws: the equations of motion

1.5.1 Rate of change of fluid properties

Consider a fluid in motion with a velocity field v = u i + v j + w k , in which ( u ,v,w ) are the
velocity components and ( i , j , k ) are the unit vectors in the ( x , y , z ) directions, respectively,
and let C ( x , y , z , t ) denote some property of this fluid. The rate at which this property
changes for a given particle of the fluid located at ( x , y , z ) at time t , can be determined by
tracking the particle to its new position ( x + u δ t , y + vδ t , z + wδ t ), a small distance away
at time t + δ t . The fluid property has then become

C ( x + u δ t , y + vδ t , z + wδ t ) = C + ∂

C
∂ x

u δ t + ∂

∂ y vδ t + ∂

C

∂ z wδ t + ∂

C C

∂ t δ t

Thus after the small displacement, the property of the fluid assumes the new value C +
( DC / Dt ) δ t . This shows that the rate of change of the property C of the moving fluid particle
is given by

DC
Dt =

∂ C

∂ t + u ∂ C ∂ x + v ∂ C ∂ y + w ∂ C

(1.3)

∂ z

DC / Dt is commonly referred to as the substantial time derivative, and is also variously
called the fluid mechanical time derivative, the time derivative following the motion ,or
the material or particle derivative. Physically, Equation (1.3) is the total rate of change
in the property, as seen by an observer moving with the fluid. The first term on the right
describes the changes taking place locally at ( x , y , z ). The last three terms describe the
changes observed while moving between locations with different values of C ; the rate of
change depends on the speed of the motion, ( u ,v,w ).

1.5.2 Conservation of mass: the continuity equation

Because hydrology is concerned with amounts of water observed at different times and
locations, conservation of mass is the main governing principle. There are several ways of
deriving a formulation that embodies this principle.

At a point
One way, after Euler's 1755 derivation (Lamb, 1932), is to consider an element of fluid
mass, which occupies a small volume δ ∀= ( δ x δ y δ z ) at time t , shown as ABCD in two
dimensions in Figure 1.3, and whose center moves at a velocity v = u i + v j + w k . If the
mass of fluid per unit volume, that is its density, is ρ , the mass of the element is given by
( ρδ ∀ ). In the absence of chemical reactions or sources and sinks, the mass of this element