Turbulent Transport in the Atmospheric Surface Layer - Transport in the Atmosphere-Vegetation-Soil Continuum - page 78

This approximation is made under the ergodic hypothesis, that is that the ensemble
statistics at a given moment are identical to temporal (or spatial) statistics for a given
period (or space). Sometimes only weak ergodicity is assumed, restricting the ergodic
hypothesis to irst and second statistical moments only (Katul et al., 2005 ).
Thus if we have a time series with N observations X i , then the estimate of X would

1

N

∑

be X ≈ X i . This is a single number, valid for the entire time series of N obser-

N

i = 1

vations. From the time series X and its m ean X , a new time series can be determined,
containing the deviations X ′ ( XXX

′= − ). Thus the series X ′ has N values, like the
original time series X . The deviations we are interested in are the turbulent luctua-
tions, so the period over which averaging should take place should be long enough to
remove all turbulent luctuations from the mean (so that all turbulence signal is con-
tained in the luctuations), but short enough to prevent nonturbulent luctuations (such
as the diurnal cycle) to inluence the deviations X ′. The scale that separates the turbu-
lent from the nonturbulent luctuations is called the (co-) spectral gap. However, it is
often not as sharply deined as the word 'gap' suggests (Baker, 2010 )). Typical values
for the time scale of the (co-)spectral gap in the ASL are 10-30 minutes (Voronovich
and Kiely, 2007 ; see also Section 3.4.2 ).
A number of computational rules apply for Reynolds averaged quantities (strictly
valid only when ensemble means are used):

XY XY
aX

+=+
=

aX if is constant

if is constant

a

aa = a

(3.3)


 


 

∂
∂

X
x i

= ∂
∂

X
x

with is a space or time coordinate x

i

i

() =

X ′ 0

Statistics of a Single Variable
For a single variable (e.g., the time series of temperature shown in Figure 3.4 ) the
two statistical q uanti ties o f interest in the framework of this topic are the mean and
the variance, X and X ′ , respectively. One could think that, because the variance
involves averaging of luctuations, this should be zero (following the last rule in Eq.
( 3.3 )). But because a squared quantity ( X ′ X ′, always positive) is averaged, the result
is always positive. Besides the variance of X also the standard deviation ( σ X ) is often
used, which is the square root of the variance. The advantage of the standard deviation
is that it has the same unit as the quantity under consideration.