Geoscience Reference
In-Depth Information
In turbulence we usually write Fourier series in terms of the wavenumber
κ
n
=
2
πn/L
, so we write
Eq. (2.39)
as
N
N
a
0
2
+
f(x)
a
n
cos
(κ
n
x)
+
b
n
sin
(κ
n
x),
κ
n
=
2
πn/L.
(2.41)
n
=
1
n
=
1
It is a property of Fourier series that by increasing
N
we can approximate
f(x)
as
well as we like. Thus, we can formally write
∞
∞
a
0
2
f(x)
=
+
a
n
cos
(κ
n
x)
+
b
n
sin
(κ
n
x),
κ
n
=
2
πn/L.
(2.42)
n
=
1
n
=
1
Furthermore, in turbulence
f(x)
is a random function, different in every realiza-
tion
α
, so its Fourier coefficients
a
n
and
b
n
are also random. Thus we generalize
Eq. (2.42)
to include this randomness through the realization index
α
:
∞
∞
a
0
(α)
2
f(x,α)
=
+
a
n
(α)
cos
(κ
n
x)
+
b
n
(α)
sin
(κ
n
x),
κ
n
=
2
πn/L.
n
=
1
n
=
1
(2.43)
Today we use a computer to calculate Fourier coefficients, and so it is convenient
to express the series in exponential notation. By using the identities
e
iθ
e
−
iθ
e
iθ
e
−
iθ
2
i
+
−
cos
θ
=
,
sin
θ
=
(2.44)
2
we can write
(2.43)
as
a
n
−
e
iκ
n
x
a
n
+
e
−
iκ
n
x
.
∞
∞
a
0
2
ib
n
ib
n
f(x
;
=
+
+
α)
(2.45)
2
2
n
=
1
n
=
1
We can rewrite this as
a
n
−
e
iκ
n
x
a
−
n
+
e
−
iκ
−
n
x
.
(2.46)
∞
−∞
a
0
2
ib
n
ib
−
n
f(x
;
α)
=
+
+
2
2
n
=
1
n
=−
1
If we define
κ
−
n
=−
κ
n
we can express this more compactly as
∞
f(x
f(κ
n
;
α)e
iκ
n
x
;
α)
=
;
n
=−∞
a
n
−
ib
n
2
a
n
+
ib
n
2
f(κ
n
;
f(κ
n
;
α)
=
,n
+;
α)
=
,n
−
.
(2.47)
