Geoscience Reference
In-Depth Information
term' on the right hand side of Equation (16.4) has contributions from diffusion
along all three axes, as follows:
⎛⎞
⎛
⎞
F
∂
u
∂
2
u
∂
2
u
∂
2
u
⎛⎞
x
=
=
u
+
+
(16.21)
⎜⎟
⎜⎟
⎜
⎟
⎝⎠
t
2
2
2
⎝⎠
r
∂
⎝
∂
x
∂
y
∂
z
⎠
a
viscosity
viscosity
Similar equations can also be readily derived for kinematic momentum (velocity)
in the Y and Z direction with analogous form, i.e.
F
⎛⎞
⎛
⎞
∂
v
∂
2
v
∂
2
v
∂
2
v
⎛⎞
y
=
=
u
+
+
(16.22)
⎜⎟
⎜⎟
⎜
⎟
⎝⎠
t
2
2
2
r
∂
⎝
∂
x
∂
y
∂
z
⎠
⎝⎠
a
viscosity
viscosity
⎛⎞
⎛
⎞
F
∂
w www
t
∂
2
∂
2
∂
2
⎛⎞
z
=
=
u
+
+
(16.23)
⎜⎟
⎜⎟
⎜
⎟
⎝⎠
2
2
2
⎝⎠
r
∂
⎝
∂
x
∂
y
∂
z
⎠
a
viscosity
viscosity
Were we to use vector algebra representation, the last three equations could be
written more concisely as a single vector equation thus:
⎛⎞
F
∂
⎛⎞
v
2
=
=
u
∇
v
(16.24)
⎜⎟
⎜⎟
⎝⎠
t
⎝⎠
r
∂
a
viscosity
viscosity
Note that the physical process that underlies the linear diffusion equation
describing transfer of momentum by molecular diffusion is very similar to those
that underlie the transfer by molecular diffusion of scalar quantities such as energy,
moisture, and other minor constituents of air. Later in this chapter conservation
equations similar to Equations (16.4), (16.5), and (16.6) are written for these scalar
quantities, and the contribution of molecular diffusion toward changes in the local
concentration of such scalar quantities must be included among the terms on the
right hand side of these conservation equations. It is possible to draw analogy with
the above analysis to define the required terms directly, but the diffusion coefficient
for each scalar quantity is different. Consequently, the molecular diffusion terms
to be included in the conservation equations for heat (which is framed in terms of
the virtual potential temperature,
q
v
), for moisture (which is framed in terms of
the specific humidity,
q
), and for an unspecified scalar quantity with concentration
c
are (
u
q
∇
2
q
v
), (
u
q
∇
2
q
), and (
u
c
∇
2
c
), respectively.
Axis-specific forces
In addition to the terms on the right hand side of Equations (16.4), (16.5), and
(16.6) associated with pressure gradients and viscosity, there are additional 'force'
terms that are specific to each axis. In the frame of reference we have adopted, the
axis-specific force in the Z direction can be immediately identified as the force of
