Geoscience Reference
In-Depth Information
An excellent discussion of this result is given by Goldstein (1950). In the rotating
frame where we live and take measurements, the time derivative of the velocity
in (2.15) must thus be replaced by
d
U
I
/
dt
I
=
d
U
I
/
dt
R
+
×
U
I
(2.18)
We have included a subscript on the velocity vector to show specifically that
even though we now have the proper time derivative in the rotating frame,
the vector
U
I
is still the velocity in inertial space. Since the goal is to describe
dynamics totally in the rotating coordinates,
U
I
must be expressed in that frame
also. Viewed from inertial space, an object moving on the earth's surface with
velocity
U
R
has an additional velocity
×
r
, where
r
is the position vector drawn
from the center of the earth—that is,
U
I
=
U
R
+
×
r
(2.19)
Since
is constant, the time derivative of (2.19) in the inertial frame is
d
U
I
/
dt
I
=
d
U
R
/
dt
I
+
×
d
r
dt
I
/
Now each of the derivatives on the right-hand side must be replaced by the
operation (2.18) in order to have the expression given entirely by quantities
measured in the rotating coordinates, and, thus,
d
U
I
/
dt
I
=
d
U
R
/
dt
R
+
×
U
R
+
×
d
r
dt
R
+
×
r
/
For a parcel of fluid moving across the surface,
d
r
dt
R
=
/
U
R
, so we have
d
U
I
/
dt
I
=
d
U
R
/
dt
R
+
2
×
U
R
+
×
(
×
r
)
2
cos
The second term is known as the Coriolis force. The last term is equal to
r
θ
,
where
is the latitude and has components both radially inward and equator-
ward. This term may be combined with
g
to describe an effective gravitational
field. We will use the symbol
g
for these combined terms and move the Coriolis
term to the right-hand side of (2.15), which yields the following equation of
motion of the neutral atmosphere in a rotating frame:
θ
2
U
ρ
d
U
/
dt
=−∇
p
+
ρ
g
+
η
∇
−∇·
π
w
−
2
ρ(
×
U
)
+
J
×
B
(2.20)
2.1.4 Momentum Equations for the Plasma
In Section 2.1.3 the only effect of a coexisting plasma embedded in the neutral
atmosphere was the
J
B
force transferred to the neutrals. We now explore the
fluid equations for the plasma species.
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