Geoscience Reference
In-Depth Information
The simplest mean momentum balance in the steady, barotropic mixed layer,
Eqs. (11.3)
, and its mean shear balance,
Eqs. (11.8)
, are easily upset by horizontal
advection or nonstationarity. If the stress divergence in the CBL is of the order
of
u
2
10
−
4
ms
−
2
,thena1ms
−
1
change in a 10 m s
−
1
mean wind over
100 km horizontal distance and a 0.4 m s
−
1
change in mean wind speed per hour
give horizontal advection and time-change terms of that magnitude. Likewise, the
mean shear
equations (11.8)
are easily upset by advection and time change. Some
of the baroclinic CBLs observed by
Lemone
et al
.
(
1999
) displayed substantial
mean wind shear that they argued was caused by the large time-change term in the
mean horizontal momentum balance
(Problem 11.19)
.
/z
i
∗
11.2.3.3 Direct testing of eddy-diffusivity models
The simplest turbulence closure assumes the
deviatori
c
turb
ulent kinematic stress,
the d
ifference between the turbulent kinematic stress
u
i
u
j
and its isotropic form
u
k
u
k
δ
ij
/
3
=
2
eδ
ij
/
3
(Chapter 14)
, is proportional to the mean strain-rate tensor:
u
i
u
j
−
3
δ
ij
e
K
m
∂U
i
,
2
∂U
j
∂x
i
−
=
∂x
j
+
(11.9)
u
i
u
i
/
2. The proportionality factor, the eddy diffusivity,
†
has a subscript
m
, signifying momentum, to distinguish it from the eddy diffusivity for a conserved
scalar. In a horizontally homogeneous ABL the diagonal components of the
right
side of
Eq. (11.9)
vanish, so this model makes the diagonal components of
u
i
u
j
equal to 2
e/
3; this is typically not observed.
For the principal off-diagonal components in the horizontally homogeneous,
quasi-steady ABL,
Eq. (11.9)
implies
with
e
=
K
m
∂U
K
m
∂V
uw
=−
∂z
,
vw
=−
∂z
.
(11.10)
In the surface layer the set
(11.10)
serves as a way of readily inferring
K
m
(z)
from
measurements of
∂U/∂z
and
u
2
:
∗
u
2
uw
∂U/∂z
τ
0
/ρ
0
∗
K
m
(z)
=−
∂U/∂z
=
∂U/∂z
.
(11.11)
∂U/∂z
in the surface layer has been well documented and expressed through
the Monin-Obukhov similarity function
φ
m
(Chapter 10)
.The
φ
m
form
(10.17)
Turbulence theoreticians have traditionally criticized the representation of a turbulent flux as
−
K
times a mean
gradient, but the late Les Kovasznay (Johns Hopkins University) reportedly enjoyed pointing out that it was his
civil right to
define
such an eddy diffusivity. We discuss it here in that spirit.
†
