Geography Reference
In-Depth Information
In the Lagrangian system, however, it is necessary to follow the time evolution of
the fields for various individual fluid parcels. Thus the independent variables are
x
0,
y
0
, z
0
, and t , where x
0
, y
0
, and z
0
designate the position that a particular parcel
passed through at a reference time t
0
.
2.1
TOTAL DIFFERENTIATION
The conservation laws to be derived in this chapter contain expressions for the
rates of change of density, momentum, and thermodynamic energy following the
motion of particular fluid parcels. In order to apply these laws in the Eulerian frame
it is necessary to derive a relationship between the rate of change of a field variable
following the motion and its rate of change at a fixed point. The former is called
the
substantial
, the
total
,orthe
material
derivative (it will be denoted by D/Dt).
The latter is called the
local
derivative (it is merely the partial derivative with respect
to time). To derive a relationship between the total derivative and the local deriva-
tive, it is convenient to refer to a particular field variable (temperature, for example).
For a given air parcel the location (x, y, z) is a function of t so that x
=
x(t),
y
z(t). Following the parcel, T may then be considered as truly a
function only of time, and its rate of change is just the total derivative DT /Dt .
In order to relate the total derivative to the local rate of change at a fixed point,
we consider the temperature measured on a balloon that moves with the wind.
Suppose that this temperature is T
0
at the point x
0,
y
0
, z
0
and time t
0
. If the bal-
loon moves to the point x
0
+
=
y(t), z
=
δz in a time increment δt, then
the temperature change recorded on the balloon, δT , can be expressed in a Taylor
series expansion as
δx, y
0
+
δy, z
0
+
∂T
∂t
δt
∂T
∂x
δx
∂T
∂y
δy
∂T
∂z
δz
δT
=
+
+
+
+
( higher order terms)
Dividing through by δt and noting that δT is the change in temperature following
the motion so that
DT
Dt
≡
δT
δt
lim
δt
→
0
we find that in the limit δt
→
0
∂T
∂x
Dx
Dt
+
∂T
∂y
Dy
Dt
+
∂T
∂z
Dz
Dt
DT
Dt
=
∂T
∂t
+
is the rate of change of T following the motion.
