Turbulent Closure, K Theory, and Mixing Length - Terrestrial Hydrometeorology - page 279

⎡ ⎤

2

⎛ ⎞ ⎛⎞

g ∂ g Δ q

q ∂ u

g

⎢ ⎥

g

R =

v

or R =

v

(19.3)

⎜ ⎟

⎜⎟

()

i

z z

i 2

q ∂ ∂ q

⎢ ⎝ ⎠ ⎝ ⎠ ⎥

Δ u

v

⎣ ⎦

v

Although often an acceptable estimate in the surface layer, where (as described later)
the turbulent fluxes are proportional to the gradient in mean values, the gradient
form of the Richardson number can be problematic when used elsewhere in the
ABL. This is shown in Fig. 19.1, which gives possible gradients of virtual potential
temperature and identifies the nature of the stability at different heights. The latter
are often inconsistent with the local gradient of virtual potential temperature.

Turbulent closure

Starting from the basic conservation equations for atmospheric flow, prognostic
equations were introduced in Chapter 17 that describe the evolution in space and
time of mean flow variables. These equations involved terms that include not only
mean flow variables but also turbulent fluxes, i.e., the time average of double cor-
relation coefficients. Then, in Chapter 18, prognostic equations were introduced
that describe the time average of turbulent variances in equations which involve
terms that include not only mean flow variables and turbulent fluxes, but also the
time averages of triple correlation coefficients. This process can be successively
repeated to derive prognostic equations for correlation coefficients of increasingly
higher complexity which contain terms that involved mean flow variables and cor-
relations of lower complexity. However, at each new level of complexity, the num-
ber of new unknown variables introduced into the prognostic equations derived
exceeds the number of equations available to describe them. Table 19.1 illustrates

Table 19.1 Progression of the sets of prognostic equations derived to describe
correlation coefficients of velocity with increasing complexity and the number
of new equations and new unknown combinations involved in these.

'Moment'
or 'order'

General form of
equations

Number of
new equations

New variables (and number
of new variables)

Zero u , v , and w , are specified

in space and time

0 u , v , w , (3)

( )

∂′ uu ′

∂ u

t

i j 2

u ,

2 2

First 3 ′ v ′ , w ′ , u ¢¢ , u ¢¢ , u ¢¢ , (6)

i

= .......

∂ ∂ x

j

( )

∂ uu

t

¢¢ ∂ uuu ¢¢¢

Second 6 u ¢

3

, v ¢

3

,

3

,

2
u ¢¢ ,

2
u w ¢ ,

2
v ¢¢ , w ¢ ¢

i j i j k

= .......

∂ ∂ x 2

vw

2
w ¢¢ ,

2
w ¢¢ , uvw ¢ ¢ , ¢¢ ¢ , (10)

j

etc. etc. etc. etc.