Geoscience Reference
In-Depth Information
Substituting Eqs. 5.13 into Eqs. 5.11 and 5.12 and then taking the time mean
we get
7
A
Hc wT wT wT wT cwTwT
p
=
ρ
+ ++ =
l
l
ll
ρ
7
+
ll
A
(5.16)
S
p
and
HLwq wq wq wq cwqwq
p
=
ρ
9
+ ++=
l
l
ll
C
ρ
7
+
ll
A
(5.17)
L
where r is assumed to be constant in time. Close to the surface, where
w
,
,
0
Hc wT
p
=
ρ
ll
,
(5.18)
S
and
H wq
L
=
ρ
ll
.
(5.19)
According to Eq. 5.18, sensible heat is transferred vertically in the atmo-
sphere when
w
and
T
are correlated. When
w
> 0 and
T
> 0, upward pertur-
bation velocities are
corre
lated with warm temperature anomalies, so warm air
is rising. In this case,
w
ll
> 0, and there is a positive (upward) sensible heat flux
from the surface to the atmo
sph
ere. When
w
< 0 and
T
< 0, cool air is sinking.
This is also a case in which
H
S
> 0. If cold a
ir
is rising (
w
> 0 and
T
< 0) or
warm air is sinking (
w
< 0 and
T
> 0), then
H
S
< 0. Similarly, according to Eq.
5.19, rising moist air carries latent heat from the surface into the atmosphere.
Observations show that the time scales over which
w
and
T
, and
w
and
q
,
are correlated are on the order of seconds and that they vary strongly on small
space scales as well. So while Eqs. 5.18 and 5.19 are accurate expressions, they
are not practical because values of
w
,
T
, and
q
cannot be measured on small
enough space and time scales over the globe to provide even estimates of the
turbulent fluxes.
As a replacement for the exact expressions for the sensible and latent heat
fluxes,
physical parameterizations
have been developed. These parameteriza-
tions are based on the underlying physics of the turbulent fluxes and they relate
H
L
and
H
S
to commonly measured large-scale quantities. For example, the
bulk
aerodynamic formulas
are used to represent sensible and latent heat fluxes
from the surface in computer models and observational analyses:
(5.20)
Hc CVTT
p
=
ρ
(
−
)
S
DHS
A
−
(5.21)
H CVqq
L
=
ρ
(
)
DL
S
A
where
C
DH
and
C
DL
are
drag coefficients,
, often taken to be 0.001 over water
and 0.003 over land.
V
is the surface horizontal wind speed,
T
S
is the surface
temperature,
q
A
is the specific humidity of the surface air, and
q
S
is the satura-
tion specific humidity at temperature
T
S
. Note that the density of air, r, is used
in these parameterizations, not the density of the surface material.
NONEQUILIBRIUM SURFACE HEAT BALANCE
Equation 5.9 assumes the net heating of the surface is zero and that there are
no horizontal and vertical fluxes of heat into and out of the surface material. A
more general surface heat balance equation allows for these processes.