c. Viscous Force
As the air parcels are moving within the atmosphere, it movement leads to friction
close to the Earth’s surface. This friction forced the air parcel to slow down and also
changes its direction. There are at least two types of friction occur in the atmosphere; one
that occurs between two surfaces as in between the atmosphere and the
Earth’s surface; and molecular friction between air
molecules called viscosity. Friction caused by viscosity is much less significant than the one
caused between two surfaces. Molecular viscosity is negligible for the atmosphere
below 100 km due to very weak viscous force; it is only matters in a thin layer within a few
centimeters of the Earth’s surface where vertical shear is very large Holton 2004.
Figure 8 Vertical shearing stress on a volume of fluid element in the x direction.
The viscous force acting in the x direction of
δz δy
δx volume in Figure 8 is given by the
net difference of stresses over y and z directions to the x component of the force per
unit volume. To obtain the force per unit mass caused by vertical shear in the x direction, it is
divided by the mass
δz δy
δx ρ
of the element:
z u
z z
zx r
F
1
1
If the dynamic viscosity coefficient,
, is constant, the above equation may be
simplified to
2 2
z u
v
, where
v is
the kinematic viscosity coefficient. Since the wind movement may vary in all directions, it
is written in three Cartesian coordinate directions as:
u v
z u
y u
x u
v rx
F 2
2 2
2 2
2 2
2 2
2 2
2 2
2
v z
v y
v x
v v
ry F
w v
z w
y w
x w
v rz
F 2
2 2
2 2
2 2
2.2.2 Forces in a Rotating Reference
Frame
As it well known, Earth-based observer is considered to be in a non-inertial frame since
the Earth is spinning on its axis. Therefore, Newton’s second law of motion has to be
modified to describe the atmospheric motion in a rotating coordinate system. In spite of
small angular velocity of Earth’s rotation, the effects caused by rotation of the reference
frame are negligible; however, on some atmospheric motions at certain space and time
scales, the effect of apparent forces is important and must be accounted for Lynch
and Cassano 2006.
a. Centrifugal Force
As an object is moving in a circular motion with radius r, its direction is
continuously changing so that its velocity does not remain constant. To maintain its
circular path, a net force; called the centripetal force is directed toward the curvature of the
path. Thus, the centripetal acceleration is given by the rate of change of angular
velocity.
13 ..
..........
2 r
ce a
Where:
ce
a
= Centripetal acceleration kg m s
-2
= Angular velocity rad s
-1
r
= Radius of curvature m Newton’s third law of motion states that
for every action there is an equal and opposite reaction. The centrifugal force is the equal
counterpart force that is exerted in the opposite direction of the centripetal force; it is
an apparent force invoked to make Newto
n’s second law of motion work in a rotating
frame. The magnitude of centrifugal force acting on a body of mass m is given by:
14 ..
..........
2 r
m ce
ma sf
F
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