The convective boundary layer - Turbulence in the Atmosphere - page 250

Figure 11.9 Observations of the budgets of kinematic stress, Eqs. (11.14) and
(11.15) , in the CBL. S is shear production; T , turbulent transport; B , buoyant
production; P , pressure destruction. T he u nits are 10 − 3 m 2 s − 3 .Left: vw budgets
from theMinnesota experiment. Right: uw budget fromAMTEX. From Wyngaard
( 1984 ).

budget. From these data the S , T ,and B terms were calculated directly, allowing P
to be obtained from Eqs. (11.14) and (11.15) by difference. The results, shown in
Figure 11.9 , indicate that in each budget the principal gain term is shear production
and the principal loss term is the pressure covariance. †
We can define Rotta ( 1951 ) time scales for the pressure-covariance terms,
allowing for different values for the two components:

w ∂p w ∂p

1
ρ 0

u ∂p
∂z

uw
τ s ,

1
ρ 0

v ∂p
∂z

vw
τ .

−

∂x +

=− −

∂y +

=− (11.16)

Figure 11.10 (left) reveals no strong differences between the τ s and τ values
deduced from the measured stress budgets in Figure 11.9 . Thus we'll write them
simply as τ , and the stress budgets (11.14) and (11.15) can then be written as
expressions for eddy diffusivities,

w 2 τ 1

T + B ∂U

∂z

∂U
∂z ,

K s m

− + =−

uw (11.17)

S

uw

w 2 τ 1

T + B ∂V

∂z =−

∂V
∂z .

K m vw − + (11.18)

S

vw

Figures 11.2 and 11.10 suggest that w 2 τ 0 . 05 w ∗ z i in mid-CBL.

† A gain term on the right side of Eq. (11.14) or ( 11.15) is one that produces stress of the observed sign. I n ru ns
2A1 and 2A2 in Figure 11.9 , for example, vw is positive, so gain terms are positive. But in run 5A1 vw is
negative and therefore gain terms are negative.