Geography Reference
In-Depth Information
If we now let
Dx
Dt
≡
u,
Dy
v,
Dz
Dt
≡
Dt
≡
w
then u, v, w are the velocity components in the x, y, z directions, respectively, and
DT
Dt
=
u
∂T
∂T
∂t
+
v
∂T
w
∂T
∂z
∂x
+
∂y
+
(2.1)
Using vector notation this expression may be rewritten as
∂T
∂t
=
DT
Dt
−
U
·∇
T
where
U
T is called the
temperature
advection
. It gives the contribution to the local temperature change
due to air motion. For example, if the wind is blowing from a cold region toward
a warm region
=
i
u
+
j
v
+
k
w is the velocity vector. The term
−
U
·∇
T will be negative (cold advection) and the advection term
will contribute negatively to the local temperature change. Thus, the local rate
of change of temperature equals the rate of change of temperature following the
motion (i.e., the heating or cooling of individual air parcels) plus the advective rate
of change of temperature.
The relationship between the total derivative and local derivative given for tem-
perature in (2.1) holds for any of the field variables. Furthermore, the total deriva-
tive can be defined following a motion field other than the actual wind field. For
example, we may wish to relate the pressure change measured by a barometer on
a moving ship to the local pressure change.
Example
. The surface pressure decreases by 3 hPa per 180 km in the eastward
direction. A ship steaming eastward at 10 km/h measures a pressure fall of 1 hPa
per 3 h. What is the pressure change on an island that the ship is passing? If we
take the x axis oriented eastward, then the local rate of change of pressure on the
island is
−
U
·∇
u
∂p
∂x
where Dp/Dt is the pressure change observed by the ship and u is the velocity of
the ship. Thus,
∂p
∂t
=
Dp
Dt
−
10
km
h
−
∂p
∂t
=
−
1hPa
3h
3hPa
180 km
1hPa
6h
−
=−
so that the rate of pressure fall on the island is only half the rate measured on the
moving ship.
If the total derivative of a field variable is zero, then that variable is a conservative
quantity following the motion. The local change is then entirely due to advection.
As shown later, field variables that are approximately conserved following the
motion play an important role in dynamic meteorology.
