Geoscience Reference
In-Depth Information
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2
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2
Horizontal del operator
:
d
/
+
i
j
2
x
2
y
d
is also written
d
, where the subscript
z
indicates that
z
is held constant or,
when pressure is used as a vertical coordinate,
d
.
When the del operator acts on a scalar,
s
, the result is the gradient:
2
/
++
2
s
2
s
t
s
t
t
.
d
si
j
k
(B.2)
2
x
2
y
2
z
v
|
|
|
v
|
There are two ways to multiply two vectors,
AiAjA A
x
/
++
and
BiBjB B
x
/
++
y
z
|
|
BiBjB B
/
++
. The cross-product is defined as
y
z
|
|
|
i
A
B
j
A
B
k
A
B
v
v
|
|
|
AB
# =
= −+−+ −
iABABj AB AB kABAB
(
)
(
)
(
),
(B.3)
x
x
y
y
z
z
yz zy
zx xz
xy yx
and the result is a vector. Equivalently,
v
v
v v
ABAB
#
=
sin
θ
,
(B.4)
/
+
v
and q is the angle between
v
and
v
. The cross product
picks up the components of
v
and
v
that are perpendicular to each other.
The cross product is also used to calculate the “
curl
”:
2
2
2
where
AAAA
x
y
z
|
|
|
i
j
k
2
A
2
A
2
A
2
A
2
A
2
A
222
v
|
|
|
y
y
z
x
z
x
f
p
f
p
d
#
A
=
= −+−+ −
i
j
f
p
k
(B.5)
222
x
y
z
2
y
2
z
2
z
2
x
2
x
2
y
AAA
x
y
z
The second way to multiply two vectors is to use the dot product:
v
v
AB AB AB AB
xx yy zz
$ =++
.
(B.6)
Note that the result of the dot product is a scalar. Equivalently,
v v v v
(B.7)
The dot product picks up the components of
v
and
v
that are parallel to each
other. The dot product is used to calculate the divergence
AB AB
$
=
cos
θ
.
2
u
2
v
2
w
.
v
d
$
v
=++
(B.8)
2
x
2
y
2
z
Combinations are also possible, for example,
|
00
k
222
2222
2
v
2
u
|
v
k
$ #
(
d
v
)
=
= −
(B.9)
x
A
y
A
z
A
x
2
y
x
y
z
v
| | |
. In this case, the dot product is used to find the vertical
component of the curl of the velocity. This quantity is used frequently in earth
system dynamics and is known as the relative vorticity.
where
vw
vui
/
++
j
k
Earth-Centered Spherical Coordinates
For large-space scales, the curvature of the earth must be taken into account,
so the spherical coordinate system shown in
Figure B.2
is used. To locate point
