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the rate of loss through viscous dissipation (
). The principal source term, shear
production (
SP
), represents the rate of TKE input through the interaction of
Reynolds stresses and the mean velocity gradient. These can be the dominant
terms, in which case
SP
. Turbulence can survive a sufficiently small
rate of buoyant destruction, i.e.,
BD
BD
+
SP
is a possible steady
state only when there is another source term. We shall see that this can be
the case in the interfacial layer that caps the convective ABL; there the rate
of import of TKE from below can allow the turbulence to exist without shear
production.
This opens the possibility of a
critical state
of stably stratified turbulence inwhich
the rates of buoyant destruction and viscous dissipation approximately balance shear
production and the turbulence is in equilibrium, but an increase in the rate of buoyant
destruction extinguishes the turbulence. To gain some insight, let us examine the
implications of equating, in order of magnitude, the rates of shear production and
buoyant destruction of TKE. In terms of the scales
u, ,
and a characteristic potential
temperature fluctuation
θ
SP
.But
BD
∼
∂/∂z
we have
θ
0
u
∂
.
u
3
∼
uw
∂U
g
θ
0
wθ
g
∂z
∼
∼
(9.12)
∂z
This yields an order-of-magnitude estimate of the size
of the largest eddy that can
survive its rate of energy loss to stable stratification:
u
2
g
θ
0
1
/
2
∼
.
(9.13)
∂
∂z
Eddies much larger than
are extinguished by the stable stratification; much smaller
ones can survive. This length scale is sometimes defined with the vertical velocity
variance and given the subscript
B
,
w
2
g
θ
0
1
/
2
σ
w
N
,
B
=
=
(9.14)
∂
∂z
with
N
2
g/θ
0
∂/∂z
the square of the Brunt-Vaisala frequency.
B
is used
in models of stably stratified turbulence (
Brost and Wyngaard
,
1978
;
Nieuwstadt
,
1984
;
Hunt
,
1985
).
Mason and Derbyshire
(
1990
) found evidence for this scaling
in their LES studies of the stable boundary layer (SBL).
By selectively damping the largest eddies, stable stratification reduces the tur-
bulent diffusivity
K
∼
u
by decreasing both
and
u
. Measures of its strength
include the
flux Richardson number R
f
and the
gradient Richardson number Ri
:
=
