The atmospheric surface layer - Turbulence in the Atmosphere - page 228

with θ the amplitude scale of the fluctuations in virtual potential temperature. The
r 1 / 3 dependence of Ri e means that the largest eddies feel buoyancy effects the
most strongly; the small eddies are dominated by inertia and (at the smallest scales)
viscous forces. Thus, as z/L becomes positive from zero the largest eddies are
damped first. In a stable surface layer the relative importance of buoyant destruction
increases with increasing height; at some height the eddies become limited in size
by the stability. Loosely speaking, at this height the turbulence does not “sense” z ,
the distance to the surface, as it does in an unstratified surface layer.
This suggests that under very stable conditions z drops out of the M-O governing
parameter group, leaving g/θ 0 ,Q 0 ,C 0 , and u ∗ . The scales in this limiting case are
then

velocity: u ∗ , temperature: T ∗ , scalar: c ∗ , length: L. (10.39)
Nondimensional quantities again become universal constants. This “local z -less
scaling,” as it is called ( Wyngaard , 1973 ; Chapter 12 ) indicates that, for example,

∂U
∂z ∼

u L , ∂

∂z ∼

T L ;

(10.40)

so that the M-O profile functions are

kz
u ∗

∂U
∂z

z
L ,

kz
T ∗

∂
∂z

z
L .

φ m =

∼

h =

∼

(10.41)

These predictions agree with measurements; as Figure 10.3 shows, φ m and φ h are
essentially linear over the entire stable range of the Kansas data.

10.4 Deviations from M-O similarity
The velocity scale of the energy-containing eddies in the convective boundary
layer with zero mean wind is expected to depend (at minimum) on g/θ 0 , Q 0 ,and
the boundary-layer depth z i . These parameters define the free-convection velocity
scale w ∗ :

g

θ 0 Q 0 z i 1 / 3

w ∗ = . (10.42)

0 . 2mKs − 1 and z i = 1000 m, typical values in fair weather over land,
Eq. (10.42) gives w ∗ ∼

For Q 0 =

2ms − 1 .
From the definitions of w ∗ and L we can write w ∗ /u ∗ = k − 1 / 3 ( z i /L) 1 / 3 . Con- −

−

vective boundary layers in the atmosphere have z i /L values up to several hundred,

so it is not unusual to find the free-convection-like state where w ∗ u ∗ .Since w ∗

contains z i , which is not an M-O parameter, the horizontal gusts in the surface layer
due to these large eddies are not M-O similar, as pointed out by Panofsky et al .