Surface Layer Scaling and Aerodynamic Resistance - Terrestrial Hydrometeorology - page 294

Table 20.1 Form of the flux-gradient relationships adopted in this text.

Flux gradient
relationship

Stable
conditions

Neutral
conditions Unstable conditions

1 4

−

zd
L

zd
L

⎛ − ⎞ ⎛ − ⎞ ⎡ zd

L

− ⎤

⎛ ⎞

φ φ =+ ⎜ 15 1 φ =− ⎜ 116

⎜ ⎟ ⎟

⎟

⎢ ⎥

M

⎝ ⎠

M

⎝ ⎠

M

⎝ ⎠

⎣ ⎦

1 2

−

zd
L

− zd

L

− ⎡ zd

L

− ⎤

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

φ φ =+ ⎜ 15 1 φ =− ⎜ 116

⎜ ⎟ ⎟ ⎟

⎢ ⎥

H

⎝ ⎠

H

⎝ ⎠

H

⎝ ⎠

⎣ ⎦

1 2

−

⎛ zd

L

− ⎞ zd

L

− ⎡ zd

L

− ⎤

⎛ ⎞ ⎛ ⎞

φ φ =+ ⎜ 15 1 φ =− ⎜ 116

⎜ ⎟ ⎟ ⎟

⎢ ⎥

V

⎝ ⎠

V

⎝ ⎠

V

⎝ ⎠

⎣ ⎦

for f M , f H and f V within the limits of experimental error and the underlying
assumptions of surface layer scaling theory. The most widely accepted form
for these functions is given in Table 20.1 and is adopted in this text. In the absence
of any better information, it is usually considered acceptable to assume the
flux-gradient relationships for the fluxes of other atmospheric variables such as
carbon dioxide flux are the same as for sensible heat and moisture.
The general form of the surface layer wind profile is given by integrating the
functional form of f M in Table 20.1 to give:

⎡ ⎤

⎛ ⎞

⎛⎞ u zd zd − ⎛ − ⎞

(20.13)

uz

*

() = ln + Ψ

⎢ ⎥

⎜ ⎟

⎜⎟ ⎜ ⎟

⎝⎠ ⎝ k z ⎝ L ⎠

⎠

⎢ ⎥

⎣ ⎦

0

where, for values of ( z - d )/ L ≥ 0, Ψ is given by:

zd zd
L

− 5( − )

⎛ ⎞

(20.14)

Ψ =

⎜ ⎟

L

⎝ ⎠

and for values of ( z - d )/ L < 0, Ψ is given by:

⎡

2

⎤

⎛ zd − ⎞ ⎡ 1 + x ⎤ 1 + x

Ψ = − 2ln − ln

⎜ ⎟ ⎢ ⎥

⎢ ⎥

⎝ ⎠

L 2 2

⎣ ⎦ (20.15)

⎣ ⎦

π ⎡ ⎛ zd − ⎞ ⎤

− 1

x

+ 2 tan [ ] − where x = φ

⎢ ⎜ ⎟ ⎥

M

2 ⎝ L ⎠

⎣ ⎦

Returning fluxes to natural units

In Chapter 15 the concept of kinematic units was introduced as a mechanism to
simplify the equations describing turbulent transfer, and to enhanced similarity
between equations giving opportunity to draw analogy between them. Having