Geoscience Reference
In-Depth Information
15.4.1 The covariance, spectral density pair
With the Fourier-Stieltjes formalism we'll write our stationary, homogeneous
turbulent velocity field as
+∞
e i κ · x dZ i ( κ ),
u i ( x )
=
(15.44)
−∞
where the dZ i ( κ ) are randomvector functions. Aswith scalars, they have orthogonal
increments:
dZ i ( κ )dZ j ( κ )
κ = κ ;
dZ i ( κ )dZ j ( κ )
κ = κ . (15.45)
=
0 ,
=
φ ij ( κ )d κ ,
The transform relationships are
+∞
e i κ · r φ ij ( κ )d κ ,
u i ( x )u j ( x
+
r )
=
R ij ( r )
=
−∞
( 2 π) 3 +∞
(15.46)
1
e i κ · r R ij ( r )d r .
φ ij ( κ )
=
−∞
R ij is a covariance when i = j and a cross covariance when i = j . It is called the
correlation tensor . φ ij is the spectral density tensor .
By definition a homogeneous field is statistically unchanged under a translation
of its coordinate axes. From Eq. (15.46) this implies that R αα is an even function;
R αα ( r )
r ). Asaresult φ αα is purely real and an even function ( Problem
15.12) . This is not true for the off-diagonal terms, however; R 12 , for example, is
neither even nor odd, and φ 12 is neither real nor imaginary.
Cospectra and quadrature spectra are defined as for scalars. We first write
R ij
=
R αα (
O ij , the sum of even and odd parts. The Fourier transform of R ij
is, from (15.46) ,
=
E ij
+
( 2 π) 3 +∞
e i κ · r E ij ( r ) + O ij ( r ) d r
1
φ ij ( κ ) =
−∞
( 2 π) 3 +∞
( 2 π) 3 +∞
1
i
=
cos ( κ ·
sin ( κ ·
r )E ij ( r )d r
r )O ij ( r )d r
−∞
−∞
=
Co ij ( κ )
iQ ij ( κ ),
(15.47)
the sum of real and imaginary parts.
As for scalars, we can obtain the so-called one-dimensional spectra that can be
determined from measurements. If we measure u 1 along the x 1 axis, for example,
 
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