and:
e à ¼
cLE
qc p u Ã
ð2:41Þ
where / H , / V , q and c p are stability functions for the temperature profile, and water
vapor partial pressure, air density, and the specific heat of air at constant pressure,
respectively.
After conversion, the following coefficients of momentum diffusivity, heat, and
water vapor, respectively, are obtained
K M ¼ k u à z À d
ð
Þ/
À1
M
ð2:42Þ
K H ¼ k u à z À d
ð
Þ/
À1
H
ð2:43Þ
K V ¼ k u à z À d
ð
Þ/
À1
V
ð2:44Þ
The general equation for the diffusivity coefficient for any hypothetical gas is
derived in the same way. These last three equations make it possible to infer that the
ratios among stability functions are equal to ratios between the respective thermal
diffusivity coefficients. Therefore, turbulent changes for sensible heat and water
vapor, for example, are similar and this holds true for the exchange of any atmospheric scalar quantity.
The dimensionless Richardson number Ri is a measure of thermal stability of
atmospheric flow along the surface. A given volume of flowing air has a kinetic
energy component of mechanical origin, because of the interaction between eddies
with characteristic dimension l, with mean flow (ke) i , and an additional force due to
buoyancy, (ke) b . The gradient Richardson number expresses the ratio ((ke) b / (ke) i ).
Under thermal instability, Ri is negative, whereas under thermal stability Ri is
positive. A zero Ri value corresponds to thermal neutrality conditions.
If the temperature and airspeed are known at two heights, z 1 and z 2, the gradient
Richardson number Ri g , for the layer between these heights, given in terms of
finite-difference is (Thom 1975)
Ri g ¼
g
T
ðT 2 À T 1 Þðz 2 À z 1 Þ
ðu 2 À u 1 Þ
2
ð2:45Þ
The advantage of using the gradient Richardson number is that it depends on
mean gradient variables, which are easy to obtain.
Above a critical Richardson value Ri c (0.25 in inviscid flow), the flow changes
from laminar to turbulent. For values between 0 and Ri c , turbulence is mechanically
generated by tangential interactions between turbulent fluctuations and the mean
flow fields; for Ri g < 0, turbulence is of mixed mechanical and convective origin.
2.3 Aerodynamic Method Equations
25
e à ¼
cLE
qc p u Ã
ð2:41Þ
where / H , / V , q and c p are stability functions for the temperature profile, and water
vapor partial pressure, air density, and the specific heat of air at constant pressure,
respectively.
After conversion, the following coefficients of momentum diffusivity, heat, and
water vapor, respectively, are obtained
K M ¼ k u à z À d
ð
Þ/
À1
M
ð2:42Þ
K H ¼ k u à z À d
ð
Þ/
À1
H
ð2:43Þ
K V ¼ k u à z À d
ð
Þ/
À1
V
ð2:44Þ
The general equation for the diffusivity coefficient for any hypothetical gas is
derived in the same way. These last three equations make it possible to infer that the
ratios among stability functions are equal to ratios between the respective thermal
diffusivity coefficients. Therefore, turbulent changes for sensible heat and water
vapor, for example, are similar and this holds true for the exchange of any atmospheric scalar quantity.
The dimensionless Richardson number Ri is a measure of thermal stability of
atmospheric flow along the surface. A given volume of flowing air has a kinetic
energy component of mechanical origin, because of the interaction between eddies
with characteristic dimension l, with mean flow (ke) i , and an additional force due to
buoyancy, (ke) b . The gradient Richardson number expresses the ratio ((ke) b / (ke) i ).
Under thermal instability, Ri is negative, whereas under thermal stability Ri is
positive. A zero Ri value corresponds to thermal neutrality conditions.
If the temperature and airspeed are known at two heights, z 1 and z 2, the gradient
Richardson number Ri g , for the layer between these heights, given in terms of
finite-difference is (Thom 1975)
Ri g ¼
g
T
ðT 2 À T 1 Þðz 2 À z 1 Þ
ðu 2 À u 1 Þ
2
ð2:45Þ
The advantage of using the gradient Richardson number is that it depends on
mean gradient variables, which are easy to obtain.
Above a critical Richardson value Ri c (0.25 in inviscid flow), the flow changes
from laminar to turbulent. For values between 0 and Ri c , turbulence is mechanically
generated by tangential interactions between turbulent fluctuations and the mean
flow fields; for Ri g < 0, turbulence is of mixed mechanical and convective origin.
2.3 Aerodynamic Method Equations
25