Geoscience Reference
In-Depth Information
Turbulent motion in the atmosphere involves both viscous dissipation at rate
˜
Q
, so the entropy equation for turbulent dry air is
and heat transfer at rate
D θ
Dt
=
Q
T
.
D
s
Dt
=
˜
c
p
θ
T
−
˜
(8.23)
The local, instantaneous rate of dissipation per unit mass is
∂
∂
.
∂ u
j
∂x
i
∂ u
j
∂x
i
ν
2
u
i
˜
u
i
˜
˜
=
∂x
j
+
∂x
j
+
(7.32)
While viscous dissipation is always important in the TKE equation, it is important
in
Eq. (8.23)
only at flow speeds much higher than we typically find in the lower
atmosphere
(Problem 8.5)
; we neglect it. Thus with the heat transfer expressed as
the divergence of the heat fluxes due to conduction and radiation,
,
k
∂ T
1
ρ
∂
∂x
i
Q
R
i
=
−
∂x
i
+
(8.24)
with
k
the thermal conductivity,
Eq. (8.23)
becomes
.
D θ
Dt
=
θ
c
p
θ
T
∂x
i
∂x
i
+
∂ R
i
∂x
i
∂
2
D
s
Dt
=−
˜
−
k
(8.25)
ρc
p
T
˜
We can write this as
D θ
Dt
=
θ
T
θ
ρc
p
T
∂ R
i
2
T
α
∇
−
∂x
i
,
(8.26)
with
α
=
k/ρc
p
the thermal diffusivity.
According to the solution
(8.21)
the factor
θ/ T
multiplying the conduction term
in
(8.26)
is proportional to
p
−
R
c
p
. Beyond the energy-containing range the turbulent
pressure spectrum falls faster than the velocity and temperature spectra (
Chapter 7
),
so that at the small spatial scales where conduction is important the factor
θ/ T
varies
negligibly in space. Thus, we can take it inside the
˜
2
operator:
∇
2
θ
T
θ
T
∇
T
T
θ,
2
2
∇
=∇
(8.27)
