Geoscience Reference
In-Depth Information
10.2.4.3 Diagnosing surface fluxes
Weather and climate models use the M-O mean profiles,
Section 10.2.4.1
,todiag-
nose surface fluxes. For surface momentum flux
τ
0
ρ
0
u
2
≡
, for example,
Eqs.
∗
(10.16)
and
(10.18)
indicate that
U
ref
≡
U(z
ref
)
depends on
τ
0
,ρ
0
,z
ref
,z
0
,L
.
Thus if we write
τ
0
=
f(U
ref
,z
ref
,z
0
,L,ρ
0
)
, then in the dimensional analysis
there are
m
−
1
=
5 governing parameters and
n
=
3 dimensions, M, L, and T.
The Pi Theorem tells us there are
m
3 independent dimensionless groups
that are functionally related. If we take these as
τ
0
/ρ
0
U
ref
,z
ref
/z
0
,z
ref
/L
,we
can write
−
n
=
τ
0
ρ
0
U
ref
≡
C
d
=
C
d
(z
ref
/z
0
,z
ref
/L).
(10.21)
C
d
ρ
0
U
ref
, with
C
d
the
drag coefficient
.
In this way one can also derive a
surface-exchange coefficient C
h
for the surface
temperature flux
(Problem 10.3)
. Thus, given
U
,
, and mean water vapor mixing
ratio
Q
at some height
z
ref
above the surface; the surface roughness length
z
0
and
the corresponding lengths for temperature and water vapor; and
and
Q
at these
inner heights, M-O similarity yields the surface fluxes of momentum, heat, and
water vapor. Virtually all large-scale meteorological models determine the surface
fluxesinthisway.
This gives a
drag law
for surface stress:
τ
0
=
10.2.4.4 Physical interpretation of L
In surface-layer coordinates the rates of shear and buoyant production of TKE are
uw
∂U
shear production rate
=−
∂z
,
g
θ
0
θw
g
θ
0
Q
0
.
buoyant production rate
=
(10.22)
Since the buoyant production rate is independent of
z
, very near the surface the
shear-production rate can greatly exceed it. If we define a height
z
e
at which the
buoyant production rate equals the value of the shear production rate under neutral
conditions,
u
∗
kz
e
g
θ
0
Q
0
,
u
2
=
(10.23)
∗
then
z
e
is
u
3
θ
0
kgQ
0
=|
∗
z
e
=
L
|
.
(10.24)
Thus
is a rough estimate of the height at which buoyancy effects become
dynamically important.
|
L
|
