Geoscience Reference
In-Depth Information
using horizontal homogeneity to eliminate the first term on the right for
α
1
and2.Thentakingthe
x
-axis along the mean-flow direction and assuming local
isotropy for the dissipative term (
Part III
), the component TKE equations are
=
∂u
2
∂t
=
∂wu
2
∂z
1
2
uw
∂U
1
2
1
ρ
p
∂u
3
,
0
=−
∂z
−
+
∂x
−
(5.55)
∂v
2
∂t
∂wv
2
∂z
1
2
1
2
ρ
p
∂v
1
3
,
=
=−
+
∂y
−
0
(5.56)
∂w
2
∂t
∂w
3
∂z
−
1
2
1
2
1
ρ
∂
∂z
pw
+
1
ρ
p
∂w
3
.
=
0
=−
∂z
−
(5.57)
The turbulent and pressure transport terms in
Eqs. (5.55)
-
(5.57
) are observed
not to be large near the surface in the
neutral
case - i.e., in the absence of buoy-
ancy effects. Then the steady balances of the TKE components near the surface
reduce to
1
2
∂u
2
∂t
=
uw
∂U
ρ
p
∂u
1
3
,
0
=−
∂z
+
∂x
−
(5.58)
∂v
2
∂t
=
1
2
ρ
p
∂v
1
3
,
0
=
∂y
−
(5.59)
∂w
2
∂t
1
2
ρ
p
∂w
1
3
.
=
=
−
0
(5.60)
∂z
Incompressibility implies that the pressure covariances in
Eqs. (5.58)
-
(5.60)
sum
to zero,
p
∂u
∂x
+
p
∂v
∂y
+
p
∂w
∂z
=
p
∂u
i
∂x
i
=
0
,
(5.61)
so we conclude that they represent
intercomponent TKE transfer
. Thus, we interpret
Eq
s. (
5.58)
-
(5.60)
as follows: the sole TKE source is the rate of shear production
of
u
2
/
2; some of that production rate is balanced by the rate of dissipation in the
u
2
/
2 equation and the remainder is transferred by the pressure covariances at equal
rates to the other two TKE components.
5.5.3.3 The role of turbulent pressure
Isotropic turbulence statistics are independent of translation, rotation, and reflec-
tion of the coordinate axes (
Part III
). Thus scalar fluxes vanish under isotropy, for a
nonzero flux would imply a preferred direction and, hence, anisotropy. In the flux
budget
(5.40)
the pressure term, as the principal loss mechanism for scalar flux,
