Geoscience Reference
In-Depth Information
∂u
i
u
k
u
j
∂x
j
−
(
turbulent transport
)
u
k
∂p
(
pressure-gradient interaction
)
1
ρ
u
i
∂p
∂x
k
−
∂x
i
+
2
3
δ
ik
(
viscous dissipation
).
−
(5.41)
The interpretation of the terms here is analogous to that for the scalar flux budget.
The leading terms are of order
u
3
/
.
There was renewed interest in these equations in the late 1960s and early 1970s
because of their potential as turbulencemodels (
Daly andHarlow
,
1970
;
Donaldson
,
1971
). The form of the mean-gradient and tilting production terms in the scalar flux
equation (5.40)
, for example, suggests that the flux-gradient relation for scalars can
be more complicated than the usual eddy-diffusivity expression.
5.5 Applications
The conservation equations for turbulent fluxes were first studied observationally
in a comprehensive way in the 1968 Kansas experiment (
Haugen
et al
.
,
1971
). The
full suite of instrumentation and the quasi-steady, locally homogeneous conditions
enabled analyses of the budgets of turbulence kinetic energy (TKE), Reynolds shear
stress, and temperature flux in the surface layer. We'll focus here on their behavior
when buoyancy effects are negligible; we cover the general case in
Part II
.
5.5.1 The TKE budget
Contracting
i
on
k
in
Eq. (5.41)
and dividing by 2 yields the equation for the evo-
lution of the mean kinetic energy per unit mass of the turbulence, more commonly
called the TKE budget:
1
2
∂
∂t
u
i
u
i
=−
U
j
2
∂
∂x
j
u
i
u
i
−
u
i
u
j
∂U
i
1
2
∂
∂x
j
u
i
u
i
u
j
∂x
j
−
1
ρ
∂x
i
pu
i
−
ν
∂u
i
∂
∂u
i
∂x
j
.
−
(5.42)
∂x
j
The viscous-dissipation term is typically labeled
:
ν
∂u
i
∂x
j
∂u
i
∂x
j
.
≡
(5.43)
