Geoscience Reference
In-Depth Information
15.3 Scalar functions of space and time
15.3.1 Extending the formalism
The concepts we have introduced apply also when our independent variable is
space rather than time. Homogeneity in a spatial coordinate then corresponds to
stationarity in time. We take as independent variables only those spatial coordinates
in which the stochastic function is homogeneous. Spatial coordinates in which it is
inhomogeneous, and time in nonstationary conditions, become parameters.
Consider a random scalar function
θ
(temperature, say) that is homogeneous in
three spatial coodinates,
x
i
≡
x
. Our Fourier representation is now
+∞
e
i
κ
·
x
dZ(
κ
;
θ(
x
;
t)
=
t),
(15.28)
−∞
where
κ
is a wavenumber vector. We write
0
,
κ
=
κ
dZ(
κ
;
t)dZ
∗
(
κ
;
t)
=
(15.29)
κ
=
κ
.
φ(
κ
,t)d
κ
,
φ(
κ
;
t)
is the power spectral density; it is the Fourier transformof the autocorrelation
function:
+∞
e
i
κ
·
r
φ(
κ
;
θ
2
(t)ρ(
r
;
t)
=
t)d
κ
,
−∞
(
2
π)
3
+∞
1
e
−
i
κ
·
r
θ
2
(t)ρ(
r
φ(
κ
;
t)
=
;
t)d
r
.
(15.30)
−∞
If
θ
were homogeneous in only two directions, as in the
x
1
-
x
2
plane in a
horizontally homogeneous turbulent boundary layer, for example, we would write
+∞
e
i(κ
1
x
1
+
κ
2
x
2
)
dZ(κ
1
,κ
2
;
θ(
x
,t)
=
x
3
,t),
−∞
with the Fourier-Stieltjes coefficients having the properties
dZ(κ
1
,κ
2
;
x
3
,t)dZ
∗
(κ
1
,κ
2
;
x
3
,t)
0
,
κ
1
,κ
2
=
κ
2
κ
1
=
=
(15.31)
κ
1
,κ
2
=
κ
2
φ(κ
1
,κ
2
;
x
3
,t)dκ
1
dκ
2
,
κ
1
=
The associated transform pair is
+∞
e
i(κ
1
r
1
+
κ
2
r
2
)
φ(κ
1
,κ
2
;
θ
2
(x
3
, t)ρ(r
1
,r
2
;
x
3
,t)
=
x
3
,t)dκ
1
dκ
2
,
−∞
