Geoscience Reference
In-Depth Information
Table 2.1 Latitude-
dependent Coriolis parameter
f in s
-1
for the northern
hemisphere. The values in
both columns are negative for
the southern hemisphere
Coriolis parameter in s
-1
Latitude (in degrees)
0.727 9 10
-4
30
0.935 9 10
-4
40
1.114 9 10
-4
50
1.260 9 10
-4
60
acceleration of the three wind components. In complete mass-specific form, they
read (mass-specific means that these equations are formulated per unit mass, the
mass-specific momentum has the physical dimension of a velocity. Therefore, we
say wind instead of momentum in the following):
!
o
u
ot
þ
!
r
u
þ
1
o
p
ox
fv
þ
f
w
v
þ
F
x
¼
0
ð
2
:
2
Þ
q
r
!
ov
ot
þ
!
r
v
þ
1
o
p
oy
þ
fu
u
þ
F
y
¼
0
ð
2
:
3
Þ
q
r
o
w
ot
þ
v
r
w
þ
1
o
p
oz
g
f
u
þ
F
z
¼
0
ð
2
:
4
Þ
q
I
II
III
IV
V
VI
VII
where u is the wind component blowing into positive x direction (positive in
eastward direction), v is the component into y direction (positive in northward
direction) and w is the vertical wind (positive upward). The wind vector is
!
¼ð
u
;
v
;
w
Þ
, the horizontal Coriolis parameter is f = 2X sinu where X is the
rotational speed of the Earth and u is the latitude (see Table
2.1
), the vertical
Coriolis parameter is f* = 2X cosu, r is the radius of curvature, and F
x
, F
y
, and F
z
are the three components of the frictional forces, which will be specified later. The
Eqs. (
2.2
)-(
2.4
), which are called Eulerian equations of motion in meteorology,
are a special form of the Navier-Stokes equations in hydrodynamics.
Term I in Eqs. (
2.2
)-(
2.4
) is called inertial or storage term, it describes the
temporal variation of the wind components. The non-linear term II expresses the
interaction between the three wind components. Term III specifies the above-
mentioned pressure force. Term IV, which is present in (
2.4
) only, gives the
influence of the Earth's gravitation. Term V denotes the Coriolis force due to the
rotating Earth. Term VI describes the centrifugal force in non-straight movements
around pressure maxima and minima (the upper sign is valid for flows around
lows, the lower sign for flows around high pressure systems). The last term VII
symbolizes the frictional forces due to the turbulent viscosity of air and surface
friction.
The terms in (
2.2
)-(
2.4
) may have different magnitudes in different weather
situations and a scale analysis for a given type of motion may lead to discarding
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