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But buoyancy directly impacts only the energy-containing eddies significantly;
excessive stability extinguishes them, which shuts down the energy cascade and
extinguishes the rest of the turbulence.
We'll illustratewith the component kinetic energy budgets, Eq. (8.69) , for a quasi-
steady, horizontally homogeneous, stably stratified surface layer. Wyngaard and
Coté ( 1971 ) found that turbulent transport is small compared to shear production
and dissipation. The Kansas observations suggest that pressure transport is also
small, so we rewrite the pressure covariances and express the component energy
budgets to a first approximation as
∂u 2
∂t =
1
2
uw ∂U
ρ 0 p ∂u
1
3 ,
0
=−
∂z +
∂x
(12.1)
∂v 2
∂t =
1
2
ρ 0 p ∂v
1
3 ,
0
=
∂y
(12.2)
∂w 2
∂t
1
2
ρ 0 p ∂w
1
g
θ 0 θw
3 .
=
0
=
∂z +
(12.3)
The buoya ncy term in Eq. (12.3) , now being negative, represents the mean rate
of loss of w 2 / 2 by working against buoyancy forces (Chapter 9) . The spectral
dynamics (Chapter 16) indicates that this rate of loss occurs in the energy-containing
range of the w field. The viscous dissipation term in Eq. (12.3) is another mean
rate of loss, so the pressure-covariance term must provide the balancing mean
rate of input of kinetic energy that allows steady turbulence. Since the pressure-
cov ariance terms in the three budgets sum to zero by incompressibility, we see that
w 2 is maintained at a nonzero value solely by intercomponent energy transfer -
i.e., transfer of kinetic energy from horizontal velocity fluctuations to vertical ones
(Chapter 5) .
In Chapter 9 we introduced the gradient Richardson number; in surface-layer
coordinates it is
g
0
∂z
∂z 2 .
Ri
=
(12.4)
The surface-layer observations of Kondo et al . ( 1978 ) show that temperature and
velocity signals tend to be fully turbulent for Ri < 0 . 2, but formore stable conditions
intermittent turbulence appears, especially in the temperature field. As Ri increases
further the fluctuations become more intermittent, weaker, and finally disappear.
This intermittency makes the measurement, understanding, and parameterization
of the nocturnal SBL particularly challenging.
Nieuwstadt ( 2005 ) used direct numerical simulation of channel flow to study
the maintenance of turbulence in stable stratification. The flow was horizontally
 
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