Geoscience Reference
In-Depth Information
of the turbulence, taken as distance
z
from the surface; the velocity scale
u
of
the turbulence, taken as the friction velocity
u
∗
, the square root of the magnitude
of the mean kinematic surface stress; the mean temperature flux at the surface,
Q
0
; the mean surface flux of a conserved scalar constituent,
C
0
; and the buoyancy
parameter
g/θ
0
.By
surface-layer structure
we denote the mean vertical gradients
of wind, potential temperature, and the mixing ratio of conserved scalars, plus the
turbulence statistics.
An underlying assumption is that the outer part of the boundary layer does
not exert an important influence on the surface layer. Thus, taking
z
is
consistent with the near-surface kinematics of vertical velocity discussed in the
Appendix. The
friction velocity u
∗
is a natural choice for the turbulent velocity
scale because beginning just above the surface the mean shear stress is carried
essentially entirely by the turbulence. Likewise, the mean surface flux of a con-
served scalar divided by
u
∗
is an appropriate intensity scale for the fluctuations of
the scalar. The mean surface temperature flux
Q
0
is included because it is propor-
tional to the rate of buoyant production of TKE in the surface layer. Finally, the
buoyancy parameter
g/θ
0
is included because it appears in the equation of motion
(8.56)
.
In
Chapter 8
we defined virtual temperature as
∼
T
v
T(
1
=
+
˜
˜
q
is specific humidity. Decomposing into mean and fluctuating parts (and reverting
briefly to our previous notation),
0
.
61
q),
where
T
=
T
+
θ,
q
=
Q
+
q,
shows that the
fluctuations are related by
θ
v
0
.
61
qT
and the mean vertical turbulent flux of
virtual temperature is
(Problem 10.18)
θ
+
θ
v
w
θw
+
0
.
61
T qw.
(10.11)
The contribution from water-vapor flux can be very important, so in general
Q
0
is
taken to be the mean surface flux of virtual temperature.
Let's discuss why other plausible candidates are not included in the M-O
governing group:
• The boundary-layer depth
h
was not included on the erroneous assumption that it does
not directly influence surface-layer turbulence. We'll see shortly, however, that the largest
eddies in the CBL do contribute substantially to the horizontal velocity fluctuations near
the surface.
• The mean velocity is excluded because a description of turbulence must be
invariant to a
Galilean transformation
that moves the coordinate system at a constant
velocity, and the mean velocity is not
Galilean invariant
.
• The turbulent Coriolis force/mass (of order
fu
) is very small compared to the turbulent
inertia force/mass (of order
u
2
/
) when
∼
z
; i.e., the inverse turbulent Rossby number
fz/u
1. This can also hold when
∼
h
.
