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III. Effect of buoyancy (density luctuations in a gravity ield).
This
term can either be
positive or negative, depending on the sign of virtual heat lux
w
′
θ
, that is, the covari-
ance of vertical wind and virtual potential temperature. The virtual heat lux can be
approximated as
6
(Stull,
1988
):
ww
q
. In some applica-
′′ ′′
θ θ
v
=
[
1061
+
.
]
+
061
.
θ
w
′′
q
tions the entire term III is denoted as the buoyancy lux.
IV. Dissipation of TKE due to molecular friction at the smallest scales. This is always a loss
term for TKE.
Terms II and III require some extra attention. The mechanism of shear production
(term II) can be understood as follows. The upward displacement (
w
′
>
0) of an air par-
∂
∂
>
u
z
0
cel in a situation with a mean vertical velocity gradient
produces a deceler-
ation of the air at the level to which the parcel is displaced:
−
∂
∂
u
z
′
. The displacement
w
per unit time along the direction of the acceleration is
u
′. Hence, the mean work per
unit time is (acceleration times displacement speed) is
−
∂
∂
u
z
′′
. This work results in
uw
the production of turbulent kinetic energy at the expense of the mean kinetic energy
of the low (the terms in Eq. (
3.10
) can be interpreted as work per mass per time).
Likewise, term III can be analysed. If a parcel experiences a positive accel-
eration due to a higher temperature (i.e., lower density), and the accompanying
displacement per unit time is a positive vertical velocity luctuation
w
′, positive
work is done on the parcel by the Archimedes force, and TKE is produced. If the
correlation between
w
′ and
θ
v
′ is negative (downward buoyancy transport), TKE
is destroyed.
If we neglect the contribution of moisture to
θ
v,
,
w
′′
θ
v
is proportional to the sensi-
ble heat lux (see
Section 3.4
). Thus when the sensible heat lux is positive (upward
heat transport), the buoyancy term produces TKE, whereas when the sensible heat
lux is negative, TKE is destroyed.
7
To characterize the role of buoyancy in the production of turbulence, often the ratio
of the buoyancy production term and the shear production term is used. This ratio
6
In terms of heat lux and the Bowen (
β
) ratio (ratio of sensible heat lux to latent heat lux, see
Section 7.1
) this
c
L
q
θc
p v
is of the order of 0.07, the
inluence of moisture on buoyancy becomes relevant already at Bowen ratios as high as 0.5.
7
A deviation of the local temperature at a given height from the mean temperature at that height can be interpreted
as potential energy. If the stratiication is such that the potential temperature decreases with height, this potential
energy will be immediately released and converted to TKE. If the potential temperature increases with height
potential energy may be converted into TKE and vice versa. Hence the terminology that buoyancy 'destroys' TKE
is only partly correct: under stable conditions TKE is converted into potential energy which is partly released back
as TKE and partly dissipated due to dissipation of temperature luctuations. This is the concept of total turbulent
energy where turbulent kinetic energy and turbulent potential energy are considered together (TTE, see, e.g.,
Zilitinkevich et al.
2007
).
becomes:
ww
p
−
1
. Note that, because
061
′′ ′′
θ θ
=
1061
+
.
+
061
.
θ β
.
/
v
v
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