Geoscience Reference
In-Depth Information
temperature of this air (assume that the temperature of the water vapour is identical
to that of the air)?
b) Assume we have two air parcels (at the same temperature and pressure): one
dry
air
parcel, the other a parcel
with water vapour
. Which parcel will have the higher den-
sity?
3.3.5 Turbulent Kinetic Energy
Turbulence is moving luid (in our case air). Therefore, the intensity of turbulent
motion can best be characterized by considering the luctuations of the wind speed, in
particular by the kinetic energy of those luctuations. Because turbulence is a three-
dimensional phenomenon, we need to take into account motion in all three directions:
vertical (
w
) as well as two horizontal directions (
u
and
v
). This results in the follow-
ing deiniti
on
of the (speciic, i.e., per unit mass) turbulent kinetic energy (TKE, also
denoted by
e
):
1
e
≡
2
(
′′ ′′ ′′
)
(3.9)
uu
++
vv
ww
As the TKE is useful to describe the intensity of the turbulent motion, the next step is
to ind out which processes enhance turbulence (increase TKE) and which processes
suppress turbulence. To that end we make a small excursion to more advanced luid
mechanics. From the equations that describe the motion of a viscous luid in a situa-
tion with density stratiication
5
for a horizontally homogeneous situation, the follow-
ing budget equation for TKE can be derived (see, e.g., Wyngaard,
2010
):
u
z
g
=−
∂
∂
∂
∂
e
t
uw
′′
+
θ
w
′
θ
−−−
...
...
ε
(3.10)
IV. dissipation
v
I. time change
II. shear production
III. buoyancy
where we have omitted (
…)
two t
erms
that
do
matter, but that mainly redistribute
TKE in space. The terms
u
′′ and
w
′′
θ
v
are the turbulence luxes of momentum and
virtual heat, respectively (see
Section 3.4
). The terms shown in Eq. (
3.10
) have the
following interpretation:
I. Change in time of TKE. If it is positive this means that TKE increases, whereas a negative
value implies a decrease of TKE.
II. Production of TKE by wind shear. This term is generally positive, since
u
′′
(vertical
momentum transport) is generally negative, whereas the mean horizontal wind speed
increases with height.
5
Navier-Stokes equations in combination with the Boussinesq approximation
Search WWH ::
Custom Search