Geoscience Reference
In-Depth Information
q
S
Q
i
Q
o
x
Figure B2.3.1
A control volume with local storage S, inflows Q
i
, local source or sink q, output Q
o
and
length scale x in the direction of flow.
Thus the mass balance for the control volume may be written:
Sx
=
tQ
i
−
tQ
o
+
txq
=−
t
(
Q
o
−
Q
i
)
+
txq
=−
tQ
+
txq
where
S
is the incremental change of storage per unit length of channel over the time step
t
and
Q
is the incremental change in discharge across the control volume in the downstream
direction (i.e. if the inflow is greater than the outflow,
Q
will be negative and
−
Q
will be
positive). Dividing through by
tx
gives
Q
x
+
q
If we make the increments very small, this may be written in differential equation form as:
S
t
=−
∂Q
∂x
+
q
This type of equation is called a partial differential equation because it involves differentials
with respect to more than one variable (here
x
and
t
).
The control volume approach may be used to derive all the differential equations used in
process descriptions in hydrology (such as the channel flow momentum balance equation of
Box 5.6). If we extend this mass balance to a control volume in three spatial dimensions,
x, y,
and
z
, using local velocity
v
rather than discharge as the solution variable, we would have
∂S
∂t
=−
∂S
∂t
=−
∂v
∂x
−
∂v
∂y
−
∂v
∂z
+
s
where
s
is a local source or sink flux. This can also be written as
∂S
∂t
=−∇
v
+
s
where
is called the differential operator.
All these forms of the mass balance equation involve more than one unknown and cannot be
solved without more information. This is usually to assume that the local inflow or source/sink
flux is known and that understanding of the flow process can be used to relate
Q
to
S
, either
directly or through some intermediate variable. Many such process descriptions result in non-
linear partial differential equations that are not easily solved analytically but must be solved
by approximate numerical methods (see Box 5.3).
∇
