Geoscience Reference
In-Depth Information
heating and cooling. We'll assume that the spatial integral scale of the scalar field
is of the order of the turbulence scale
.
By this des
ign
C
, the mean part of the scalar, is independent of
x
and
t
,the
turbulent flux
cu
j
vanishes, and the conservation equation for the fluctuating scalar
field
c(
x
,t)
is
c
,t
+
(cu
j
)
,j
=
s(
x
,t)
+
γc
,jj
.
(16.2)
Multiplying
Eq. (16.2)
by 2
c
, ensemble averaging, rewriting the molecular term,
and dropping its diffusion part yields the scalar variance equation
∂
∂t
c
2
(c
2
u
j
)
,j
−
=
2
sc
−
2
γ c
,j
c
,j
=
0
.
(16.3)
By homogeneity, which is implied by isotropy, the second term on the right,
turbulent transport of variance, vanishes so the equation reduces to
2
sc
−
2
γ c
,j
c
,j
=
Pr
−
χ
c
,
(16.4)
with
Pr
the mean rate of production of variance by the fluctuating source term and
χ
c
its mean rate of destruction through molecular diffusion.
Now we introduce the Fourier-Stieltjes representations for the fluctuating
quantities
(Chapter 15)
:
e
i
κ
·
x
dZ(
κ
,t),
e
i
κ
·
x
dZ
i
(
κ
,t),
c(
x
,t)
=
u
i
(
x
,t)
=
(16.5)
e
i
κ
·
x
dS(
κ
,t),
e
i
κ
·
x
dF
j
(
κ
,t).
s(
x
,t)
=
cu
j
(
x
,t)
=
Hereafter we will not explicitly indicate the dependence on
t
.
Using these representations in
Eq. (16.2)
gives the evolution equation for the
Fourier-Stieltjes components:
∂
∂t
dZ(
κ
)
γκ
2
dZ(
κ
)
=−
iκ
j
dF
j
(
κ
)
−
+
dS(
κ
),
(16.6)
where
κ
2
κ
i
κ
i
.
The power spectral density
φ
of the fluctuating scalar field is defined through
=
φ(
κ
)d
κ
=
dZ(
κ
)dZ
∗
(
κ
),
(16.7)
and its time derivative is
∂
∂t
φ(
κ
)d
κ
=
dZ
∗
(
κ
)
∂
dZ(
κ
)
∂
∂t
dZ
∗
(
κ
).
∂t
dZ(
κ
)
+
(16.8)
