Geoscience Reference
In-Depth Information
consideration of the required relationship between the resulting acceleration and
the change in velocity gives:
∂
⎛⎞
v
=−
2 i
wq
u
()
(16.31)
⎜
⎝⎠
t
∂
axis specific
For simplicity when writing Equations (16.30) and (16.31) we have considered
only simple specific cases, but these two equations apply in general and define the
axis-specific forces for the X and Y axes. For consistency with other terms on the
right hand side of Equations (16.4), (16.5) and (16.6), these three equations are
re-written as:
⎛⎞
F
x
=
fv
(16.32)
⎜
⎝
axis specific
r
a
⎛⎞
F
y
=−
fu
(16.33)
⎜
⎝
axis specific
r
a
⎛⎞
F
z
=−
g
(16.34)
⎜
⎝⎠
axis specific
r
a
where:
f
=
2 in()
ωθ
(16.35)
Combined momentum forces
As previously mentioned, to define the equations which together describe the
movement and evolution of momentum in the atmosphere the total rate of change
of momentum along each axis, i.e., Equations (16.4) to (16.6), are combined with
those that describe possible forces that may give rise to change, i.e., Equations
(16.10) to (16.12), Equations (16.21) to (16.23), and Equations (16.32) to (16.34),
as follows:
⎛
⎞
du
∂
u
∂
u
∂
u
∂
u
1
∂
P
∂
2
u
∂
2
u
∂
2
u
=+ + +
u
v w fv
=−
+υ
+ +
(16.36)
⎜
⎟
dt
t
x
y
z
x
2
2
2
∂
∂
∂
∂
r
∂
⎝
∂
x
∂
y
∂
z
⎠
a
⎛
⎞
dv
∂
v
∂
v
∂
v
∂
v
1
∂
P
∂
2
v
∂
2
v
∂
2
v
=+ + +
u
v w fu
=−−
+υ
+ +
(16.37)
⎜
⎟
dt
t
x
y
z
y
2
2
2
∂
∂
∂
∂
r
∂
⎝
∂
x
∂
y
∂
z
⎠
a
⎛
⎞
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u
∂
∂
∂
∂
1
∂
∂
2
∂
2
∂
2
=+ + +
v w g
−−
υ
+ +
(16.38)
⎜
⎟
dt
t
x
y
z
z
2
2
2
∂
∂
∂
∂
r
∂
⎝
∂
x
∂
y
∂
z
⎠
a
