Table 1. Predicted behavior of the Deacon and Richardson numbers in very stable conditions

Figure 5. Various suggestions for the functional form of the nondimensional wind speed (φm) and scalar (φh) gradients for stable conditions

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13

Pole, Lettau (1979) deduced the nondimensional gradi-

Traditionally, Ricr is assumed to be 0.200.25 (Busch

ent functions

1973, Businger 1973). But Mahrt (1981) and Heine-

mann and Rose (1990) have reported that a larger value

φm (ζ) = (1 + 4.5ζ)3 / 4

(81)

is sometimes indicated. Woods (1969) may have ex-

plained this range of values by demonstrating that there

and

is a hysteresis in Ricr. He concluded that a turbulent

flow becomes laminar when Ri exceeds 1, but a lami-

φh (ζ) = φm (ζ)2 = (1 + 4.5ζ)3 / 2

(82)

nar flow does not become turbulent until Ri falls below

0.25 (see also Plate 1971, p. 76). In his observations at

which are also depicted in Figure 5. Lettau's φh func-

South Pole, however, Lettau (1979) frequently found

tion predicts a very steep scalar gradient with increas-

turbulence to exist even when Ri exceeded unity; he

ing ζ that has not been confirmed by independent ob-

thus concluded that, in stable conditions, there was no

servations.

critical Richardson number. Monin and Yaglom (1971,

Although Lettau (1979) developed his functions es-

p. 440 f.) also argued that there seems to be no critical

pecially to treat very stable conditions, the log-linear

Richardson number in stable conditions. In light of this

form (eq 80) is also often applied in very stable condi-

controversy, I show in Table 1 what the four formula-

tions, though it has been tested only for 0 ≤ ζ <1 (e.g.,

tions for φm and φh depicted in Figure 5 predict as the

Dyer 1974, Hicks 1976, Yaglom 1977). As an alterna-

behavior of Ri in very stable conditions.

tive to the log-linear form for 0 ≤ ζ ≤10, Holtslag and

de Bruin (1988) and Beljaars and Holtslag (1991) pro-

Table 1. Predicted behavior of the Deacon and

posed the function

Richardson numbers in very stable conditions accord-

ing to the four gradient formulation shown in Figures

4 and 5.

φm (ζ) = φh (ζ)

limit ζ → ∞

φm, φh

(83)

= 1 + 0.7ζ + 0.75ζ(6 - 0.35ζ) exp(-0.35ζ).

1 + 5ζ

0.20

1 + 7ζ

I will refer to this as the Dutch formulation.

0.14

In Figure 5, we see that there is little difference

φm = (1 + 4.5ζ)3/4

among the four functions suggested for φm and φh when

1/2

0 ≤ ζ ≤ 0.5. Thus, because the log-linear form is much

φh = (1 +

4.5ζ)3/2

simpler mathematically than Lettau's (1979) or the

1 + 0.7 ζ + 0.75ζ(6 0.35ζ) exp(0.35ζ)

1.4

Dutch formulation, it might be preferable for weakly

stable conditions. Some other considerations, however,

might help us decide which of the four formulations is

Lettau (1979; see also Viswanadham [1979, 1982])

best in more stable conditions.

described two other parameters that characterize

The gradient Richardson number is

surface-layer profiles, the Deacon numbers for wind

speed and potential temperature, Dm and Dh, respec-

Θ / dz

tively. These quantify profile curvature. For wind speed

Ri =

(84)

(

U / dz)

-z(d 2U / dz 2 )

Dm ≡

(87)

From eq 78 and the definition of ζ (eq 58) we can write

dU / dz

this as

With eq 78a, it is easy to show that

(

)

g t* / kz φh (ζ) ζ φh (ζ)

ζ dφm (ζ)

Dm (ζ) = 1 -

(88)

Ri =

φm (ζ) dζ

(85)

(

)

Tv u / kz 2 φ2 (ζ) φ2 (ζ)

Similarly, for potential temperature

In stable conditions, turbulence ceases when the

-z(d 2Θ / dz 2 )

Dh ≡

Richardson number exceeds a critical value, Ricr. Thus,

(89)

dΘ / dz

we should expect accurate profile functions to yield this

critical value through eq 85; that is

which, with eq 78b, becomes

lim Ri = Ricr .

ζ dφh (ζ)

(86)

Dh (ζ) = 1 -

ζ→∞

(90)

φh (ζ) dζ

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