Turbulent Transport in the Atmospheric Surface Layer - Transport in the Atmosphere-Vegetation-Soil Continuum - page 82

humidity to be well correlated. Indeed, in general they are, with correlation coefi-
cients between 0.8 and 1 (for the present data R θq = 0.82, see Figure 3.7b ).

Question 3.4: Given the following series of quantities X and Y :

X 4 1 5 2 3

Y 10 2 5 7 4

Com p ute th e following quantities:
a) X and Y
b) X ′ a nd Y ′
c) X ′ ′ and Y ′′
d) X ′′
e) R XY

Question 3.5: In the second panel of Figure 3.4 it can be observed that the temperature
seems to have a well-deined lower limit. Explain this lower limit, using the igure of
the daytime vertical temperature proile given in Figure 3.3a ( Figures 3.3 and 3.4 do not
refer to the same situation, but the proile in Figure 3.3 should be at least representative
of the proile that occurred during the observations depicted in Figure 3.4 ).

3.3.4 Buoyancy
Turbulent temperature luctuations (e.g., as shown in the previous section) give rise
to luctuations of air density. Those density luctuations, in combination with gravity,
in turn cause vertical acceleration of air (buoyancy). First we derive the link between
temperature luctuations and density luctuations, and subsequently buoyancy is
examined.
The equation of state of a perfect gas (such as air) provides the link between pres-
sure, temperature and density: p

= ρ . The presence of water vapour will change the
composition of air, and hence the gas constant R (through the molar mass) of the gas
mixture. To simplify the analysis, the effect of water vapour on density is moved from
the gas constant to the temperature: T is replaced by the virtual temperature T v and at
the same time R (which varies with moisture content) is replaced by the constant R d ,
giving the equation of state as: pRT

T

= ρ dv .
To analyse the equation of state for a turbulent environment, all variables need to be
decomposed into a mean and luctuating component. If it is assumed that ρ

′′ T << ρ T

v v

<< , then the following expressions can be derived for the mean
pressure and the pressure luctuations:

and ρ

′′ ′ T ρ ρ T or T ′

v v v

ρ

p R T
pppR T

=
=−=

d v

)

(3.5)

ρ ρ

′ ′

′ ( + T v

d v