Environmental Engineering Reference
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and (
20
) we arrive at
2
c
s
∂
ʴ
2
H
∂ʴ
∂
2
+
t
−
4
ˀ
G
ˁʴ
−
a
2
∇
ʴ
=
0
.
(21)
t
2
∂
In this equation the second term is a
friction
component due to the background
expansion, the third term implies gravitational attraction, while the fourth is a pressure
term. Thus, it shows the important aspect of the competition between gravitational
attraction and pressure support.
Being the set of partial differential equations linear in the perturbations it is con-
venient to work instead in Fourier space,
2
arriving to ordinary differential equation
for which each Fourier mode evolve independently.
We define the variable
ʸ
as the divergence of the velocity in Fourier space, that is
i
a
k
ʸ
≡−
·
v
.
(22)
The factor
a
−
1
is a convention used since the size of a perturbation
k
−
1
grows
ʻ
∼
with
a
, and thus
k
a
becomes a comoving wave number.
In Fourier Space
/
∇ₒ−
i
k
, and Eqs. (
18
)-(
20
) can be written as
k
2
Ga
2
ˆ
=−
4
ˀ
ˁʴ,
(23)
dt
+
ʸ
=
d
0
,
(24)
k
2
a
2
ˆ
−
k
2
a
2
c
s
ʴ
=
d
dt
+
2
H
ʸ
−
0
.
(25)
To obtain the last equation we have taken the dot product of
i
k
a
with the Fourier
transform of Eq. (
20
) and used the definition (
22
). Note that in arriving at Eq. (
25
)
we have isolated the curl-free piece of the fluid peculiar velocity.
In Fourier space the Jeans equation for an expanding Universe (Eq.
21
) becomes
/
k
2
a
2
c
s
−
d
2
ʴ
dt
2
2
H
d
dt
+
+
4
ˀ
G
ˁ
ʴ
=
0
.
(26)
From this last equation it should be clear the interplay between gravitational
instability and pressure support. There exist a threshold scale, called the Jeans length
2
Our convention for a Fourier transform of a vector or a scalar function
f
is
f
(
k
)
=
d
3
xf
(
x
)
e
i
k
·
x
.
In this work, without worrying about confusions, we omit the tilde on Fourier transform quantities.
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