Geoscience Reference
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where as before d
dt is the total time derivative. Notice that the Laplacian oper-
ator in (2.15) acts on each velocity component. That is, the x component of the
viscous force is given by
/
2 u
F v ) x = η
(
x 2
0 is a good approximation, and (2.13)
is used almost exclusively for viscous forces. We must note, however, that in
the lower atmosphere, the coefficient
For large-scale flow patterns
∇·
U
=
is determined not by molecular collisions
but by the interaction between eddies in the flow. In other words, the effective
viscosity coefficient is much larger than the molecular coefficient calculated from
kinetic theory. Above the turbopause (near 100 km), where turbulent mixing
ceases, the classical molecular viscosity coefficient is appropriate.
We now consider the forces that can contribute to the external force F in
(2.15). The body force F includes the important gravitational term
η
ρ
g . Another
important body force for ionized species is due to electromagnetic effects. As is
well known, when a current I passes through a wire in a magnetic field, the elec-
tromagnetic I
B force is transferred from the charged particles to the material
in the wire. In continuous media the force density equivalent to this force is given
by J
×
×
B ,
F EM =
J
×
B
(2.16)
This force density transfers electromagnetic energy to kinetic energy, accelerating
or decelerating the fluid.
The upward flux of momentum
π w due to waves can be very important in the
upper atmosphere. Waves from the dense lower regions tend to grow in ampli-
tude as they propagate upward, since the mass density decreases exponentially
with altitude. If the waves are absorbed at any given height, then
0
and the local atmosphere will be accelerated. Ocean swimmers often experience
a phenomenon of this type when water waves break on the beach at an angle
that is not exactly perpendicular to the shoreline. Some of the wave momentum
is absorbed in the surf region, and a “long-shore” current results. This current
transports sand (and people) along the coast and is an important factor in the
formation, structure, and location of beaches.
This would complete the momentum equation for the neutrals except for the
fact that the earth is rotating. Newton's laws as expressed in (2.15) refer to
a frame of reference at rest or one moving with constant linear velocity. In a
reference frame ( R ) rotating with constant angular velocity
( ∇· π w ) =
, the time rate of
change of a vector is related to the time derivative in an inertial frame (I) by
d A
dt I = d A
dt R + ×
/
/
A
(2.17)
 
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