Geoscience Reference
In-Depth Information
9.3.1 The sensitivity of turbulence to buoyancy
The ve
rti
cal comp
one
nt of our Navier-Stokes equation,
Eq. (8.57)
, has a buoyancy
term
g θ
v
/θ
0
, with
θ
v
the local deviation in virtual poten
tia
l temperature from the
background value. If we define a temperature scale
θ
∼
θ
v
, and take the turbulent
inertia terms in that equation as of order
u
2
/
, their ratio is a turbulence Richardson
number
Ri
t
:
buoyancy force
/
mass
inertia force
/
mass
gθ/θ
0
u
2
/
gθ
θ
0
u
2
.
Ri
t
=
∼
∼
(9.6)
1ms
−
1
has
Ri
t
An indoor air flow with
θ
1
/
30, and
turbulent buoyancy is unimportant. An ABL flow with the same
θ
,
u
but with
=
1K,
=
1m,
u
=
=
=
1000 m has
Ri
t
=
30; here turbulent buoyancy is quite important.
Atmospheric turbulence tends to have larger
and smaller
u
than engineering
turbulence, both of which tend to give larger
Ri
t
. As a result buoyancy effects are
typically much more important in atmospheric turbulence. Hereafter, with temper-
ature we will not generally use the modifier
virtual
, or its subscript v, but in general
temperature
is to be interpreted as
virtual temperature
.
9.3.2 Stable and unstable stratification
Equation (8.69)
says that when
∂/∂z
†
is negative, upward motion of a parcel
generates a positive potential temperature deviation
θ
and downward motion gen-
erates a negative one. In each case the resulting buoyancy force tends to amplify
the motion, and so we call this
unstable
stratification. When
∂/∂z
is positive,
parcels undergoing vertical displacements feel opposing buoyant accelerations;
this is
stable
stratification.
Through its strong buoyancy effects an elevated inversion can act as an effective
“lid” on the ABL. Consider, for example, an upward-moving air parcel of vertical
velocity
w
i
entering the inversion, a region of greater potential temperature. There
θ
, the difference between the temperatures of the parcel and the environment,
becomes negative; this causes a downward buoyancy force
gθ
/θ
0
on the parcel. If
that force brings the parcel to rest in a vertical distance
d
, then equating the work
done on the parcel,
gdθ
/θ
0
, and the change in parcel kinetic energy,
w
i
/
2, gives
gdθ
θ
0
w
i
w
i
θ
0
2
gθ
2
,
d
.
(9.7)
1ms
−
1
and
θ
=
For
w
i
=
15 m, a relatively small excursion
if the mean height of the capping inversion is 1000 m. Thus a surprisingly weak
1 K this gives
d
∼
†
Hereafter
θ
and
T
are, in general, virtual potential temperature and virtual temperature, respectively.
