Geoscience Reference
In-Depth Information
On the surface
T
=
constant,
T
=
0. This means that the scalar product (
T
)
·
r
is zero on a
surface of constant
T
and hence that
∇
T
and
r
are perpendicular. Since
r
is parallel to the
surface and
T
=
constant,
∇
T
must be perpendicular, or normal, to that surface.
Divergence
The divergence of a vector field
V
,div
V
,isascalar field. It is written
∇·
V
. and is defined
as
∂
V
x
∂
x
+
∂
V
y
∂
y
+
∂
V
z
∂
div
V
≡
∇·
V
=
(A1.11)
z
where the components of
V
in Cartesian coordinates are (
V
x
,
V
y
,
V
z
). The divergence
represents a net flux, or rate of transfer, per unit of volume. If the wind velocity is
V
and the
air has a constant density
ρ
, then
(A1.12)
∇·
(
ρ
V
)
=
ρ
∇·
V
represents the net mass flux of air per unit volume. If no air is created or destroyed, then the
total mass flux entering each unit volume is balanced by that leaving it, so the net mass flux is
zero:
ρ
∇·
V
=
0
(A1.13)
Avector field for which
∇·
V
=
0iscalled
solenoidal
.
Curl
The curl of a vector field
V
, curl
V
,isavector function of position. It is written
∇
∧
V
,or
∇
×
V
, and is defined in Cartesian coordinates as
∂
V
z
∂
y
−
∂
V
y
,
∂
V
x
∂
z
−
∂
V
z
,
∂
V
y
∂
x
−
∂
V
x
curl
V
≡
∇
∧
V
=
(A1.14)
∂
z
∂
x
∂
y
It is related to rotation and is sometimes called
rotation
,orrot. For example, the differential
expression of
Ampere's law
for the magnetic field
H
due to a current
J
is
∇
∧
H
=
J
.
Alternatively, consider a body rotating with constant angular velocity
. Equation (A1.7)
expresses the velocity at
r
in terms of the angular velocity:
V
= ∧
r
Now, take the curl of
V
:
(A1.15)
∇
∧
V
=
∇
∧
(
∧
r
)
Since
is a constant, this equation can be simplified to
∇
∧
V
=
(
∇·
r
)
−
(
·∇
)
r
(A1.16)
=
3
−
=
2
