Geoscience Reference
In-Depth Information
Entirely analogous arguments can be used to define the total derivative of the
v
and
w
components of velocity with time. Moreover, when applied to a unit volume
of air, Newton's second law of motion requires that the acceleration along each of
the three axes must be equal to the total force along each axis divided by the
density of air, thus:
du
∂ ∂ ∂ ∂
=+ + +
u
u
u
u
F
(16.4)
x
u
v w
=
dt
t
x
y
z
∂
∂
∂
∂
r
a
F
dv
∂
v
∂
v
∂
v
∂
v
(16.5)
y
=+
u
+ +
v w
=
dt
∂
t
∂
x
∂
y
∂
z
r
a
F
dw
∂ ∂ ∂ ∂
=+ + +
w
w
w
w
(16.6)
u
v w
=
Z
dt
∂
t
∂
x
∂
y
∂
z
r
a
where
F
x
,
F
y
, and
F
Z
are (at this point in time unspecified) axis-specific forces
acting on the parcel of air of unit volume. These three equations describe the
conservation of momentum along the three axes. To include them among the
suite of equations describing the movement and evolution of the atmosphere,
the next step is to identify all the possible 'force' terms whose sum causes
change in each velocity component. This procedure is described in the next
section.
In the discussion above, the change in velocity with time along three axes was
used to illustrate how the conservation equations that describe the movement
and evolution of the atmosphere are put together. In this case, the starting point
was the conservation of momentum, but the conservation equations for scalar
quantities (i.e., mass, energy, moisture and other atmospheric constituents) can
also be used as starting points. However, specifying these conservations is easier
than formulating the momentum conservation equations, and can be done by
analogy. For this reason, it is the derivation of the equations for momentum
conservation in the atmosphere which is described in more detail in the next
section.
Conservation of momentum in the atmosphere
To write the equations describing momentum conservation it is necessary to
identify all the possible 'forces' that can give rise to changes in the momentum in
the X, Y, and Z directions, to formulate these in mathematical form, and then to
include them as terms on the right hand side of Equations (16.4), (16.5) and (16.6).
In the next three sections the three terms that can change momentum are
considered separately.
