The stable boundary layer - Turbulence in the Atmosphere - page 272

τ 3 / 2
k(g/θ 0 )θw .

(z) =− (12.5)

Nieuwstadt named this property local scaling. It implies that a turbulence quantity
in this set, made dimensionless with the local fluxes of temperature and momentum,
is a universal function of z/(z) . Nieuwstadt viewed local scaling as “an extension
of Monin-Obukhov similarity to the whole stable boundary layer.”
Nieuwstadt also pointed out that for z

the set should display the “ z -less scal-
ing” observed in the very stable surface layer (Chapter 10) . Physically, this means
that under very stable conditions the length scale of the turbulence is determined
by the local length scale rather than by z .
Nieuwstadt presented data taken under stable conditions along a meteorological
mast at Cabauw, the Netherlands. The results in “local-similarity” coordinates -
variables measured at a given height z , made dimensionless with the fluxes at that
height, and plotted against z/ - supported the local-scaling hypothesis and the
concept of a z -less limit.
The Monin-Obukhov length L is independent of height, so as we saw i n Chapter
10 we can use φ m (z/L), τ 0 , and Q 0 to calculate U(z) , for example. Bu t in local
scaling (z), so to obtain vertical structure one needs τ(z) and wθ(z) . With

the closure assumption that both the flux and gradient Richardson numbers are
constant at 0.2, Nieuwstadt ( 1984 ) found analytical solutions for these flux profiles
in stationary conditions:

=

u 2 z/h) 3 / 2 . wθ = Q 0 ( 1 − z/h), τ = ( 1 − (12.6)

∗

12.1.3 Large-eddy simulation of the SBL
Among the first applications of large-eddy simulation (LES) to the SBL was that
of Mason and Derbyshire ( 1990 ). They used a grid of 40

62 points in
a domain 1000 m deep over uniform, flat terrain; the horizontal resolution was
about 12 m. Finding great difficulty in starting runs from stable conditions (the
turbulence tended to decay), they began with a neutral turbulent boundary layer
and then applied cooling to the lower surface. Their three stably stratified cases ran
for about two hours after the onset of cooling. Case B had a rather small surface
temperature flux (

× 32 ×

0 . 01 m s − 1 K), as in some of the Minnesota runs (Figure 12.2) ;
Case C had a flux of three times that; and D had a constant cooling rate.
Their results did show the rapid decay of the friction velocity, as in Figure 12.3 ,
and the establishment of a thin, quasi-equilibrium, stably stratified boundary layer
within two hours. Case B had an equilibrium boundary layer depth of about 200m;
the flux and gradient Richardson numbers increased monotonically to about 0.2 at

−