Scaling the Mass Conservation Equation
In part, because the total density is dominated by the vertical stratifcation of the atmosphere, we
decompose (separate) the density into base state (mean) that is height dependent and perturbation
quantities which are a function of (x, y, z, t),
where
.
We know that
ρ
0
is a f(z) only
, thus the above equation can be
rewritten
multiplying by 1/
ρ
0
we have
When comparing the last two terms we know that
, thus the 4th term is much smaller
than the 5th term above. We now have what Holton has, namely
∂
∂
t
----
ρ′
ρ
0
+
(
)
u
∇ ρ′
ρ
0
+
(
)
ρ′
ρ
0
+
(
)∇
u
⋅
+
⋅
+
0
=
ρ
x y z t
,
, ,
(
)
ρ
0
z
( )
ρ′
x y z t
,
, ,
(
)
+
=
∂ρ
0
∂
t
⁄
∂ρ
0
,
∂
x
⁄
∂ρ
0
∂
y
⁄
,
0
=
(
)
∂
∂
t
----
ρ′
u
∇ρ′
w
d
ρ
0
dz
--------
ρ′
ρ
0
+
(
)∇
u
0
≈
⋅
+
+
⋅
+
1
ρ
0
-----
∂ρ′
∂
t
-------
u
∇ρ′
⋅
+
w
ρ
0
-----
d
ρ
0
dz
--------
ρ′
ρ
0
-----
∇
u
∇
u
0
≈
⋅
+
⋅
+
+
small
ρ′
ρ
0
⁄
0
«
1
ρ
0
-----
∂ρ′
∂
t
-------
u
∇ρ′
⋅
+
w
ρ
0
-----
d
ρ
0
dz
--------
∇
u
0
≈
⋅
+
+
∂ρ′
∂
t
-------
u
ρ
0
-----
∂ρ′
∂
x
-------
v
ρ
0
-----
∂ρ′
∂
y
-------
,
w
ρ
0
-----
∂ρ′
∂
z
-------
w
ρ
0
-----
d
ρ
0
dz
--------
∂
u
∂
x
-----
∂
v
∂
y
-----
,
∂
w
∂
z
------
ρ′
ρ
0
T
v
-----------
ρ′
ρ
0
T
h
-----------
ρ′
W
ρ
0
H
---------
ρ′
U
ρ
0
L
---------
ρ′
U
ρ
0
L
---------
ρ′
W
ρ
0
H
---------
d
ρ
0
dz
--------
W
ρ
0
-----
U
L
---
W
H
----
10
2
–
10
2
–
10
4
---------------------
10
2
–
10
10
6
----------------
10
-7
10
-8
10
10
6
-------
10
5
–
∼
10
2
–
10
4
----------
10
6
–
∼
ρ
0
ρ
0
0
( )
e
z
H
⁄
–
≈
1
ρ
0
-----
d
ρ
0
dz
--------
ρ
0
0
( )
e
z
H
⁄
–
H
ρ
0
0
( )
e
z
H
⁄
–
-------------------------------
∝
W
H
----
10
6
–
∼

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*