Geoscience Reference
In-Depth Information
For the background data assume θ = 295 K, q = 0.010 kg kg -1 , ρ = 1.15 kg m -3 ,
c p = 1015 J kg -1 K -1 . L v = 2.45·10 6 J kg -1 .
Experimental Determination of Similarity Relationships:
Spurious Correlations
The dimensionless groups used to derive the similarity relationships are not fully
independent. For instance, φ m and
z
L
both contain the surface friction, and φ h and
z
L
both contain surface friction and surface heat lux (see Eqs. ( 3.19 ) and ( 3.20 )).
In the derivation of similarity relationships from observations this fact is important.
The presence of shared variables implies that errors in the observed surface luxes
(both systematic errors and random errors) will result in a simultaneous variation of
both dimensionless groups. This may give rise to spurious self-correlation or spurious
scatter: correlation or decorrelation between dimensionless groups that is not physi-
cal but only due to the sharing of variables.
The direction of this variation - roughly along the similarity relationship or perpen-
dicular to it - depends on whether the shared variable occurs in either the numerator
or denominator of both dimensionless groups, or in the numerator of the one and the
denominator of the other. If we take the occurrence of u * in φ m and z
L
as an example,
we see that it occurs in the denominator of both dimensionless groups. Thus a posi-
tive deviation in u * will result in a negative deviation both in φ m and in the absolute
value of z
L
(see Figure 3.17 ). On the unstable side this implies a variation more or less
normal to the expected similarity relationship, whereas on the stable side the error in
u * causes a variation along the expected relationship. Hence, measurement errors in
-plots on the unstable side, whereas on the
stable side the data will spuriously conirm the expected relationship. The reverse
z
L
u * will only show up as scatter in φ m
: those result in
scatter on the stable side. See Baas et al. ( 2006 ) and Andreas and Hicks ( 2002 ) for
more details.
The preceding discussion was stated in terms of relative errors. However, under
stable conditions the luxes are small and disturbing factors such as instationarity and
intermittency (Klipp and Mahrt, 2004 ) may cause signiicant errors in the observed
luxes, hence increasing the relative error. This makes the usefulness of a lux-based
similarity theory (like MOST) questionable. Because gradients are usually large dur-
ing stable conditions (due to the lack of mixing) the use of similarity relationships
based on gradients (in terms of Richardson numbers) is sometimes advantageous
(Baas et al., 2006 ).
z
L
holds for the effect of errors in heat lux measurements on φ h
 
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