Geoscience Reference
In-Depth Information
APPENDIX
B
COORDINATE SYSTEMS
Local Cartesian and Earth-centered spherical coordinate systems are defined
for expressing equations within the earth system. These are not the standard
Cartesian and spherical coordinate systems found in math text books. Rather,
they are designed to easily relate to latitude and longitude coordinates.
Local Cartesian Coordinates
When space scales are relatively small, so the curvature of the Earth can be ne-
glected, we can use the local Cartesian coordinate system shown in
Figure B.1.
Up
z
Figure B.1. The local Cartesian coordinate system.
Orthogonal coordinate axes are
x
,
y
, and
z
, with unit
vectors
t
,
t
, and
t
, respectively.
North
y
^
^
j
k
East
^
x
i
• The east-west velocity,
udxd
/
is called the
zonal wind
in the atmosphere
and the
zonal current
in the ocean. It is positive when it is directed in the
positive
x
-direction, which is eastward (in the direction of the rotation of the
earth). An eastward (westward) flow is also referred to as
westerly
(
easterly
)
flow.
• The north/south velocity,
/,
vdyd
/
is called the
meridional wind
or the
meridional current.
It is positive when it is directed in the positive
y
-direction, which is northward. A northward (southward) flow is also
referred to as
southerly (northerly)
flow.
• The
vertical velocity
,
/,
wdzd
/
is positive when it is directed in the positive
z
-direction, which is upward. Downward flow is sometimes referred to as
subsidence
, especially when sinking motion occurs on large space scales.
/,
As discussed in
chapter 2
,
z
can be replaced by
p
as a vertical coordinate,
in which case the vertical
p
- velocity,
/
is used to express vertical mo-
tion, and the
x-
and
y
-directions are parallel (orthogonal) to pressure surfaces
instead of elevation surfaces. In this case,
t
,
t
, and
t
are still used as the unit
vectors.
Vector Operations in Local Cartesian Coordinates:
dp dt
/,
|
2
/
++
|
2
|
2
Del operator:
d
i
j
k
(B.1)
2
x
2
y
2
z
