The Effect of Stratification on the Aerodynamic Roughness Length - Meteorological and Air Quality Models for Urban Areas - page 54

(7.2) realistically describes the velocity profile in the surface layer and allows cal-
culating u T ∼ u ∗ through U ( z ) measured or modelled at any height z between 5 h 0

and 10 − 1 h .
Over very rough surfaces, the downward transfer of momentum is performed by
pressure forces caused by flow-obstacle interactions, and characterized by h 0 and the
maximal velocity in the roughness layer: U R ∼

u ∗ . Assuming that z 0 u depends only

on these two parameters yields z 0 u ∼ h 0 ( u ∗ drops out for dimensionality reasons).

Classical experiments with the sand roughness confirmed this conclusion and gave
z 0 u ≈

1

30 h 0 (Monin and Yaglom, 1971). For actual land surfaces, z 0 u /

h 0 varies from
1/10 to 1/30 and exhibits dependences on the shape of and the distance between the
roughness elements, but no systematic dependence on u ∗

. Accordingly, land sur-
faces are traditionally characterized by their roughness lengths considered as con-
stant parameters independent on the wind speed and stratification.
A few authors did observe that z 0 u could depend on stratification (Arya, 1975;
Joffre, 1982; Wood and Mason, 1991; Hasager et al., 2003; Grachev et al., 1997)
or discuss a possibility of increases in z 0 u due to convective plumes developing
between warm roughness elements (Coelho and Hunt, 1989; Zilitinkevich et al.,
2006a,b). Nevertheless the traditional consensus was not shaken and, to the best of
the authors' knowledge, neither theoretical models nor systematic empirical analy-
ses of the stratification effects on z 0 u have been developed.
To analyze stratified atmospheric flows over rough land surfaces we, in the
regular way, employ the Monin-Obukhov length scale (Monin and Obukhov,
1954):

u 3
∗

F − 1
b

L =− , (7.3)

Where F b is the vertical turbulent flux of buoyancy (defined as b = g ρ/ρ 0 , ρ is fluid

density, ρ 0 is its reference value, and g is the acceleration of gravity). In the atmo-

sphere, F b ≈ ( g / T ) F θ , where T is a reference value of the absolute temperature and

F θ . As a general rule, the role of stratification

is negligible at z<<L ; but becomes crucial as z

is the flux of potential temperature,

L or larger (Monin and Yaglom,
1971). Especially for urban and woodland canopies with h 0 ∼

∼

20-50m, comparable
values of L are quite often observed, so that one can expect a pronounced effect of
stratification upon z 0 u .
We emphasize that physically z 0 u is not a geometric feature of the surface but a
measure of the depth of a sub-layer within the roughness layer with, say, 90% of
the velocity drop from its maximal value, U R ∼ h 0 .Giventhe

roughness-layer eddy viscosity scale ( K M 0 ) the above definition allows estimation
of z 0 u from the conventional formula

u , approached at z ∼

∗

u 2
∗

τ = K M 0 ∂ U /∂ z taking τ ∼ and ∂ U /∂ z ∼

z 0 u , which gives 2

U R / z 0 u ∼ u

∗ /

2 Equation (7.4) is principally similar to the familiar “smooth-surface roughness length” formula-
tion: z 0 u ∼ ν/ u ∗ , with the only difference that K M 0 is substituted to ν .