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Correlation Velocities.

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78
Annalen der Physik
tt
7. Folge
tt
Band 18, Heft 1-2
Y
1966
Correlation Velocifiesl)
By KURTSUCHY~)
Whith 4 Figures
Dedicated to Professor DT.Erich Huckel on his 70th birthday
Abstract
With the coefficients of a TAYLOR
expansion of the correlation function C(gt.)up to the
+
second order in [, t. three (or four) velocities can be defined to describe the space-time behaviour in a satisfying manner. They have a simple geometrical relationship among each other.
The necessary measurements for the computation of these coefficients are discussed.
Introduction
The relatedness of two physical quantities, y ( r ,t ) and y ( r , t ) , depending
both on the same variables r , t (e. g. space and time), is usually expressed by
their “correlation”
+&
-f
B(E,z)= < y * ( r , t ) y ( r
+ m , t
3 limT
1 Jdr, 1 J d l y * ( r , t ) y ( r + g t + z ) ,
V+-
(14
T
Tern
where y* = Rev - i Imy. It represents the mean deviations between v and y
-+
and depends on the deviations 6, z of their variables r , t .
Sometimes a statistical (ensemble) average over some (or all) variables r,t is
to be added. After this averaging the integrand y* ( r , t )y( r
t
z) may be
independent of some (or all) of these variables and the corresponding integrations
may be omitted. Thus the space-time averaging ( ) in calculating the correlation
B ( g z ) ,Eq. (l.l),may be replaced partially or wholly by statistical (ensemble}
averaging [l].This possibly mixed average is denoted by a bar in the following.
As an example for our general investigations we consider the behaviour of the
velocity vector u ( r ,t ) in a turbulent medium. If u ( r, t ) is its statistical average,
i.e. its mean velocity which may depend on r and t , then the turbulence can be
described by the correlation of the “fluctuating velocity” Au = u - U,viz.
+
-
B(2z) = A u ( r , t )A u ( r
+ z t + z).
+
(1.2).
1) This research was supported in part by the National Aeronautics and Space Administration under Grant NsG-220-62.
2) On leave of absence from the University of Marburg (Germany), hstitutefor theoretical Physics.
79
K. SUCHY:Correlation Velocities
This is a tensor of second rank. Its diagonal terms are the “autocorrelations”
of the velocity componentb:Au,; the off-diagonal terms are the L‘crosscorrelations ”
*
= d u , ( r , t )dzhv(r 5,t
r)
(1.4)
+ +
B~~(L~)
*
of these components. The correlation tensor B ( 5 , ~gives
)
a quantitative description of the behaviour of the “eddies” in a turbulent medium and is therefore of
great importance for all investigations about turbulence. We shall come back t o
this example repeatedly in this paper.
Another important correlation tensor is that of the electric micro field E ( r ,t )
in an electrolyte or in a plasma [2,3] :
B(L)
=~ * ( r , t ) +
~ (Krt + t).
(1.5).
For the description of the incoherent scatter of electromagnetic radiation b y
density fluctuations AN in a plasma the density correlation
is needed [4].It can be measured for the ionospheric plasma by echoes of radio
waves. Thus information about the electron density above its height maximum
and about electron and ion temperatures can be obtained by RADAR measurements from the ground [5]. Echoes of low frequency radio waves from the lower
boundary of the ionosphere yield information about density fluctuations in this
region. “Drifts” of these fluctuations have been observed which may be considered in analogy to the movement of eddies in the lower atmosphere. Because this
movement is related to the wind velocity u, see Sect. 10, one speaks of “winds”
in connection with those ionospheric drifts. The prerequisites for the equalization
of winds and drifts will be discussed in Sect. 10.
2. TAYLOR
expansion of the correlation
For the mathematical investigation of correlations the FOURIER
transformation is mostly used. This is due to the simple relationship between the FOURIER
transform of the correlation
B ( L )= < q * ( r , t y) ( r
E
lim
V-+T+m
1
1
+ Zt +
t)>
+
J d r T J d t q * ( r , t )y ( r + t , t
T’
T
+ t)
and the FOURIER
transforms of qj and y ) expressed by WIEKERStheorem
7 V T T q * ( r , t ) y ( r +&
+
t)>= 7 * q ( r , t ) 7 y ( r , t )
which can be deduced from the convolution theorem [6]
3 V T (~6r , t - t ) y ( r , ~=
> ~ q ( r ,y (tr ,~t ) .
v.11
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Band 18, Heft 1-2
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1966
+
But we will restrict ourselves to a. limited range of small t,zand in this case in-+
stead of a FOURIER
transformation a TAYLORexpansion in t,zis obvious :
+4.
The double scalar product of the second rank tensor E E with the tensor
written in Cartesian coordinates
a2
at at
B is
The double derivatives might be replaced by correlations of single derivatives
+
and hence their values for 5 = 0 = z by averages of products of single derivatives. The latter are sometimes directly measurable.
Differentiating
B ( K z ) = < y * ( r , t )Y ( r
+Kt +
7))
with respect t o r and t and substituting
+
r=r,, r + t = r 2 , t-tl,
t+.c=t,
we obtain
Because of a possible tensor character of y and ly we avoided to shift the vector
operator a p r . If we now resubstitute
-+
r, - r, = 5, t, - t, = z
we get
a2
at
-,
7 B ( t y z=
)
I n a completely analogous manner one finds
81
K. SUCHY:
Correlation Velocities
(2.5)
[$B(&)]
PTJ
0,o
= - -at a t ’
Eqs. (2.5), (2.3) and (2.7) may serve to replace the last three TAYLORcoefficients in Eq. (2.1) by averages of products. - As an example for the correct
placement of the operator alar we take y and y as vectors. Then we have, e.g., in
Cartesian coordinates
3. Kormalization of the correlation
For the investigation of the behaviour of the TAYLOR
expanded correlation,
-+
Eq. (2.1), with respect t o 6,t we try to make it independent of the special type of
the measured quantities y and 7y by a proper normalization with B (0,O). If y and
-+
y are tensors of ranks m and ?z their correlation B (E, t)is a tensor of rank m n.
+
Writing the TAYLOR
expansion, Eq. (2.1), for each component B,,,...v,,,+n(E,~)
we
normalize it by division by Bv,...vwn(O,
0). Thus we obtain its corresponding “correlation function”
+
-+
C(6,z) = 1
+
+ a + b z - 1 +t . A . f +
+
- + f
. 6t -
1
+ O([E+ +
y ta
213),
(3.1)
with (we omit in the following the tensor indices v 1 - a ’v,,,+,J
Because of the normalizationit is possible - and may be sometimes useful - to write the
correlation function (3.1) in the form [7]
+ br - -I2 -E +. A .+E +*fi -+5 r- -12 y z 2 ] + O ( 2 + ~1’).
-+
(3.3)
We have in general for each tensor component of B(6,t)another correlation
+
function C ([,t).
But if Bis symmetric with respect to some indices the corresponding correlation functions are identical. This is e.g. the case for the correlation
of second rank
6
Ann. Physik. 7. Folge, Bd. 18
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1966
which is, of course, symmetric. - The correlation tensor
+
+ +
B(5,z) = E * ( r , t ) E ( r g t
z)
u.53
is Hermitean and hence B,, = B$. The same relation holds for the corresponding correlation functions : C,, = C$. - I n both cases the correlation tensors of
second rank are determined by six different correlation functions.
Instead of the components of the correlation tensor B we might use other
significant quantities of B and normalize these dividing them by the initial term
in their TAYLORexpansion. For a’correlation tensor B of second rank we migth
take, e.g., its three eigenvalues and the directions of the three eigenvectors a s
quantities t o be normalized.
4. Assumptions about turbulence
For the symmetric correlation tensor
~ ( g z=) ~ u ( r , At )u ( r + gt + t)
11-23
which is determined by six different correlation functions, one assumption has
been made to reduce this variety to only one significant correlation function;
another assumption deals with a connection between the parameters A and y of
each correlation function.
The first assumption classifies a medium as a “chaotic turbulent medium” if
B degenerates into a scalar B multiplied with the unit tensor IJ[8] :
+& +
~ u ( r , du(r
t)
z) = ~ ( z U.
t )
(4.1)
The other assumption deals with the diagonal terms By, of B, i.e. the autocorrelations (1.3) of the three components A u , of Au. TAYLOR’S
hypothesis postulates for sufficiently high speed 1 U I
IAu 1 [9, 101
aAu 2 = ( 5 . aAu 2 .
(4.2)
>
(r*)
+)
Using Eqs. (2.5) (2.7) for the autocorrelation B, (1.3) we can rewrite this hypothesis as
and after division by B,(O,O) with Eq. (3.2)
y = is. A * U.
(4.3)
It must be kept in mind t h a t for each component Au, we may have another
autocorrelation and hence other parameters A, y etc. TAYLORS
hypothesis gives
the connection (4.3) separately for each of these autocorrelations.
From a merely formal point of view there are no objections to extend TAYLORS
hypothesis to the cross correlations B,, (1.4) between different componentsA u , and Au,, i.e. the offdiagonal elements of B (1.2).Thenit would hold for all tensor components of B (1.2).As mentioned above the correlation parametersmay still be different for each tensor component.
I n the special case of a “chaotic turbulent medium” the correlation parameters of the diagonal terms B,, (1.3) coincide while the off-diagonal terms B,, (1.4)
vanish.
83
I(.SUCHY:
Correlation Velocities
5. Correlation time
We will now investigate the physical meaning of the various correlation para-+
meters a, b, A, Byand y. At first we study b and y which are responsible for the
pure temporal behaviour. Then the parameters a and A describing the spatial
variation will be studied. As a link between both kinds of parameters a “fading
*
velocity” can be defined. The remaining parameter p gives rise to the introduction of two additional velocities.
The mean temporal deviation between ~1 and ly for
*
6 = 0 is given by
The constant z, gives the situation of the maximum
the factor y is proportional t o the
radius of curvature a t the maximum,
see Fig. 1.The reciprocal y-l is a measure for the decay of C(0, z) from its
maximum ralue CM(0, z). This can
best be demonstrated by the point of
intersection of the approximating
parabolawith the abscyssa, see Fig. 1.
The “correlation time” y - l / 2 is therefore a measure for the life time of the
+
temporal deviations a t = 0.
Fig. 1. Approximating parabola of C(0, t)
and the osculatine: circle at its maximum.
tldenote;; the same distance from 0
F~~an observer moving with the The time
at the abscyssa a.8 1at the ordinate
velocity v the mean temporal devia-+
*
tion C(0,z) a t 6 = 0, Eq. (5.1), corresponds to that a t 6 = v z :
*
1
C(vz,z)M 1 a .vz
b z - - v . A . v t 2 B vz2 - 1 y z 2
2
-+
1
=1
7(V . A . v - 2 8 . v f Y ) ( Z : - (z - zu)’]
(5.4)
2 .
a.v+b
-a . v + b
with z, 3 --+
+
+
+ -
+
v.A.v-2@.v+y
yu
The constant zv gives now the situation of the temporal maximum
*
1
C ~ ( v t , z=
) 1 g (V A * V - 2#? . V
y ) t:
C(Vtv,tu),
+
-
p
+ 7)
and because of
(V
6*
.A
*
V-
2~
*
+
(Z
1
- $) = 2 [ C M ( U ~ ,Z )C ( V ~ , Z ) ]
(5.5)
(5.6)
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*
Band 18, Heft 1-2
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1966
*
+
+
vis YS’’~=(V. A - v - 2 v .fi + ~ ) - l ’ ~The
. originalequations ( 5 . 1 ) ( 5 . 2 ) ( 5 . 3 ) are
the factor yv z ( v A . v - 2 v . fi y ) is now proportional to the radius of the curvature a t the maximum. Hence the correlation time for an observer moving with
immediately obtained from (5.4) (5.5) (5.6) by putting v = 0.
The correlation time
+
yt1I2 = ( a . A . a - 2a.p + y)-W
(5.7)
for an observer moving with the mean velocity ti and the “integral scale”
parallel to U
* +
1
L,, = y J d E S ( E x
a)c(Zo)
(5.8)
furnish FRENKIEL’S
condition [lOa] for the validity of TAYLOR’S
hypothesis
(4.3) :
6. Correlation tensor A and correlation ellipsoids
We repeat now the procedure of the preceding section with the mean spatial
deviation of 9 and y for z = 0 which is represented by
+
1*
+
C ( ~ O ) w I + o ~ E - ~ t ~ A ~ 5
+ ,1t o** A * t o - y*( 51
+
=1
with
+
+
-+
+
+
- Eo)*A.(E - 5 0 )
to A-l
*
(6.1)
U.
The constant vector todenotes the position of the maximum
+
C ~ ( t , o=
) 1
+ y1’Eo. A
+
+
+
Eo = C(Eo7 0).
(6.2)
The symmetric “correlation tensor”A, Eq. (3.2),generates a family of concentric
L‘correlationellipsoids” [111
(i?-g)-A*(Z-$J=
2[Cia(Z30)-C(~O)l
(6.3)
around
with C(;, 0) as parameter of the
family, Fig. 2. The roots ail2 of the eigenvalues a1 are proportional to the radii of
curvature a t the principal axes. Their reciprocals, the “principal correlation lengths”
an1I2, are proportional to the principal diameters and are therefore a measure for the
&
-t
Fig. 2. Family of similar correlation
*
ellipsoids with C ( & 0) as parameter.
The unit vector g1describes the direction of the eigenvector corresponding
to the eigenvalue a1
decay of C(t,O) from its maximum value
CM(gO)= C(Ti,O) at the center the ellipids, see Fig. 2.
We have now physically interpreted the
correlation parameters y (5.3), A (6.2), and
the constants
4.
to
y-16, to A-l
u [ 5 . 2 ] [6.2]
85
K. Sucw: Correlation Velocities
deduced with them. We can rewrite the expression
+.
introducing toand toand obtain
c (6,TO)
-
-
1-t
(E -to).
+
+ - +
1
+ +
A * (5 - 60) + p
*
Et.
- -2y ( t - q J * .
(6.4)
+
If C (6,
t)describes an autocorrelation function (9,= y) it must have a “total”
- + +
maximum with respect to all variables 5,ta t 5 = 0 = z. I n this case the parameters a and b vanish, according to Eqs. (3.1) (5.1) (6.1), and so do and to.Hence
+
for an autocorrelation function C(to,zo)equals unity and Eq. (6.4) becomes simplified.
6
7. Fading velocity
VF
We look now for a link between the temporal maximum C ~ ( 0 , of
t ) C ( 0 , t )a t
toand the spatial maximum
(Z
*
+
-+
CM(5,O)of C (5,O) a t 6.
For this we use the relations
- To) y ( Z
- t o ) = 2 [ c M ( o , T ) - c(027)l
P.31
-+
describing the decay of C(0,z) and C(&,O)from their maximum values C ~ ( 0 , t )
+ - +
and CM(z,0), respectively. Asking for the distance 16 - toI in a given direction
- 1, for which the spatial decay equals the temporal decay during
t - to,
we obtain for this “fading velocity” VF [12] by equalizing the right-hand
sides of Eqs. (5.3) (6.3)
V~*A*VF=Y.
(7.1)
The endpoints of V F form a particular member of the correlation ellipsoids
(6.3) which we call “characteristic correlation ellipsoid” [13]. The square roots
(y/an)-’/’
of the reciprocal eigenvalues y / a l are the “principal fading velocit,ies”
along the principal axes & of the characteristic correlation ellipsoid, cf. Fig. 2.
For an observer moving with the velocity v the fading velocity V F ( V ) is
obtained by equalizing the right-hand sides of
(z- g)/12
(t- z,)
Z0
+
+
( v - A - v- 2 ~ - p y )
(F- 6
A
(Z
- r0)= 2 [CM(VZ,Z)
- C ( ~ t , t ) ][5,6]
(g- 6)= 2 [ C M ( ~ O-) C(z,O)l
f6.31
leading to the relation
-+
V F ( V ) * A - V F =( Vv)- A * v - ~ v * / ~ + Y - Y , , .
(7.2)
For a given v the endpoints of * ( v ) form another member of the correlation
ellipsoids, Fig. 2, which has, however, the same shape as the characteristic correlation ellipsoid (7.1). Because just this shape is the representation of the aniso-
86
Annalen der Physik
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* Band 18, Heft 1-2 *
1966
tropy of the correlation we cannot gain additional information about the anisotropy by using observations with different velocities v.
zf we observe the autocorrelation function
B,(Et) =&(r,t) duu(r
*
+ t’t
+
moving with v = is we have with (7.2)
-+
u.31
7)
+
*(ti) - A * ~ F ( i i=
) U * A * U - 2U - B
7 3 yz.
(7.3)
Thus FEENKKEL’S
condition (5.9) for the validity of TAYLOR’S
hypothesis (4.3)
can be written
%$
U
=
.,,(?
(7.4)
U
8. Drift velocity VD
We have discussed the physical meaning of y and A and their ratio which
+
-+
represent the decay of C(0,z) and C ( t , O ) . We shall now find the vector B as
describing the velocities of the partial maxima of C(5,z).
At first we find the spatial maximum of C(5,z) for a given t from.
-b
-b
ac
_ -- - A (t+- t +o ) + B+ t = 0
+
(8.1)
a
aE
at
c~(z)
+
+ zA-l .?.
= to
Therefore the constant vector
-+
-b
VD
EMIT) -&I
=~
- 1 p.
~3.31
T
-b
represents the spatial motion of the spatial maximum of C( 6,z) during a given
time t.Hence it is the “drift velocity” of the spatial deviations [14].
If we replace in the expression (6.4) for the correlation function C(gz) by
+
p
=A-vD
~3.31
we can express C ( 5 , ~completely
)
by means of physically interpreted constants :
-b
=1
.+
+
+ to
A-5+
*
to~t -
+
+
(5 - V D Z ) .A . (t- VDT) - (7 - V D . A
*
VD) t2.
(8.5)
+
The first form (8.4) is convenient for vanishing drift velocity vD -B, the
+
second (8.5) for vanishing toN a and toN b, e.g. for an autocorrelation function
~151.
87
K. SUCHY:Correlation Velocities
9. Spreading velocity v s
+
-c
We nowlook for the temporal maximum of C(t,z) for a given 5. It.arises from
ac
-
at
-
=
--L
p
a
at
-+
[
-+
zd5) = 7,
+ T.y - q
’
The endpoints of the vector
vs =
tM(&
(9.1
-y(t-zo)=O
-+
-t o
(9.3)
in v s . B = y
s
form a plane perpendicular to the direction B. They describe the temporal motion
s
of the temporal maximum of C z) along a given direction [. Therefore they may
be called the “spreading velocities” of the temporal deviations. These and also
their minimum value
(g
have been called “apparent velocity” v, [16, 171.
It must be emphasized that the spreading velocity vs (9.3) is not a velocity
in the usual physical seme which is defined RS the (directed) path traveled during
+
a given time interval. In the definition of vs, however, the path t is given and the
-+
travel time z~ ( 5 ) - zois measured. Hence this travel time is the genuine physical
+
quantity and is called the “lag time” for a given El81 :
Similar conditions are valid for the group velocity
(9.5)
i n a rapidly varying (or strongly absorbing) inhomogeneous medium. It cannot be measured
inside this medium. But its travel time
t2 - t, =
Jr* d r - -a&
1,
between the two boundaries of the medium at rl andr, is a measurable quantity. Comparing
Eqs. (9.2) and (9.6) we find a correspondence of the two vectors f i y and a k p o with the
dimension of a reciprocal velocity. These are actually the quantities rendering measurable
physical information.
With the expressions
-b
B = A vD,
-+
y = v s - B = vs * A . vD
[8.3] [9.3]
FRENKIEL’S
condition for the validity of TAYLOR’S
hypothesis (4.3) can be written
(9.7)
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Annalen der Physik
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Band 18, Heft 1-2
1966
10. Connection between yp, VD, and v g
The connection of the fading velocity V F defined by
c7.11
with the drift velocity
18.31
and the spreading velocity
19-33
can be given by a simple geometrical picture. Combining (8.3) and (9.3) we find
Vs
* A * VD = Y -
(10.1)
A comparisonwithEq. (7.1) for the characteristic correlation ellipsoid which is
formed by the endpoints of VF shows the plane of the endpoints of vs (9.3) to be
Fig. 3. Endpoints of the spreading velocity w, cover
the polar plane of the drift velocity v, with respect to the charecteristic correlation ellipsoid
which is covered by the endpoints of the fading
velocity w,
the polar plane of the-endpoint
of V D (8.3) with respect to the
characteristic correlation ellipsoid (7.1).
With the correlation para+
meters A, p, y we can therefore
construct the characteristic correlation ellipsoid (7.1) for the
drift velocity up, the vector V D
(8.3) of the drift velocity, and
the polar plane of its endpoint
with respect to the characteristic
correlation ellipsoid as plane for
the endpoints of the spreading
velocity YS (9.3), cf. (Fig. 3).
Although the geometrical
relation between up, VD, and q3
is rather simple their algebraic
connectioninvolves the tensor A
and further simplifications am
desirable. We will discuss three possible cases.
If the directions6 and 8s coincide with the fixed direction &,the comparison
of (7.1) and (10.1) leads to the result 1191
U S U D = ~ j ) for &6D = 6s.
(10.2)
The same relation holda for an isotropic correlation with a correlation tensor
A proportional to the unit tensor U:
UD = t&
for A = a U .
(10.3)
A complete coalescence of VD and vs (with respect to direction and absolute
value) can only occur if the endpoint of the vector V D lies at the characteristic
correlation ellipsoid
v~*A*v~=Y.
17-13
89
K. SUCHY:
Correlation Velocities
Then its polar plane
VS.A
. VD = y
[10.1]
is the tangential plane of the characteristic correlation ellipsoid in the point
see Fig. 3.
A still more special case is the coincidence of V D with
B.
(vs)min=
VD,
[9*43
This can happen only if the vector V D is in the direction & of a principal correlation axis, see Fig. 3, i.e. V D coincides with a principal fading velocity. If the
corresponding eigenvalue is we obtain with Eq. (10.1)
= V D * A - V D = V$&,
=
Eliminating VD =
-+
= V D - @ = VD@.
(10.4)
we find as condition for the coincidence
-
= 1/o(a y for
VD
(vS),,,in.
(10.5)
Such a coincidence seems in fact to exist for
+ +
+
-b
B , (5,z) = Au, ( r, t ) Au,, ( r E , t t).
11.33
I n the lower atmosphere the equality of the minimum spreading velocity
and the mean velocity ti has been observed [20] as well as a correlation ellipsoid
stretched in the direction of U.Comparing
V ~ . A * V D
=y
[10.1]
with TAYLOR’S
hypothesis
u.A.u=y
14-31
we find with vs = (v&in = is also the equality of the drift velocity V D with U.
Thus we have vD = (VR),~,, (= U )in this case and Eq. (10.5) holds as an additional connection between the autocorrelation parameters of Au,.
A further consequence of the equality VD = U can be seen if we put
+
@=A*vD,
with
VD
~=vs.A*vD
= U = vs into
[8.3] [9.33
-+
- 2 U .@ + y Y E
[7.31
According to FRENKIEL’S
condition (9.7) TAYLOR’S
hypothesis (4.3) holds and
~F(ii)
A
*
VF ( U )
=U * A .U
we find
= 0.
U&i)
(10.6)
Thus an observer moving with the mean velocity U does not see a fading of the
autocorrelation of Au,. I n the language of the meteorologists : “The eddies travel
with the wind”.
For the same autocorrelation the picture of the spreading velocity VQ forming a plane with its endpoints (Fig. 3) according to
+
vs
*@
=y
19.31
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has been investigated [21]. It turned out that the “plane” behaviour holds only
in a finite range about
Y *
(Vs)rnin = 7 B.
P.41
A
This range is thegreater the less stretched the correlation ellipsoid is, i.e. the less
distinct the anisotropy is. This result may be due to the neglected terms
0
([z+ 213) in the TAYLORexpansion (3.1) for the correlation.
11. Velocity 01 random changes vc
In Eq. (10.6) we have found a particular value of the fading velocityfor an
observer moving with the drift velocity VD if VD coincides with vs. We drop this
very special last condition and ask quite generally for that velocity v of a moving
observer which minimizes the correlation ellipsoid for the fading velocity
VF ( v ) :
*
VF ( v ) * A * v F ( v ) =v * A VD - 2~ * B
y 3 yy
~7.21
-
+
=V*A.~-~V.A*VD$-~
From
~8.31
a
(v - V D ) = 0
av [VF ( V ) * A * VF (V)] = 2 A
we find the drift velocity V D answering our question. The corresponding
fading velocity VF (vD) describes the particular correlation ellipsoid
VF (VD)
A
*
VF (VD)
=y
- VD * A
*
VD
*
+
- VD] ‘ A * v D= [(vS)min - VD] * B = - B’A-’*B
+
= [(%)min
(11.1)
which reduces t o a null ellipsoid if VD coincides with (v&in. Under this very
special condition the correlation function shows no fading for an observer
moving with V D . For an observer a t rest the fading velocity V F ( O ) = VF is given by the characteristic correlation ellipsoid
V F * A * V p = 7.
l7.11
Both VF (vD) and VF (0) = VF can be used as a measure for the fadings. The
former has been called “velocity of random changes’’ [23]
VC
VF (VD).
(11.2)
Its dependence on the drift velocity VD should be kept in mind.
Eqs. (11.1) (7.4) (11.2) yield for a given direction CF = ?c the relation
(11.3)
between the speeds vc and up.
i.e. the
For two particular cases this relation becomes simplified: If 6D =
direction of the drift velocity coincides with that of the fading velocity VF (and
therefore also with &), then from Eq. (11.3) follows [24]
V; = V;
- V&
for
GD
= CF.
(11.4)
91
K. SUCHY:Correlation Velocities
If furthermore 6s = 6F then we have
V S WD = 0%
for 6F
1
.
.
[10.2]
= VD =VS.
This yields combined with (11.4)
V:
=(
-
~ g wD)vD
for i& = 6D = 9s.
(11.5)
If the correlation is isotropic the correlation tensor A degenerates into a
scalar and drops out of the relation (11.3):
vt = wt
- vb
for A
= 01
U.
(11.6)
U
[10.3]
Together with
vs W D = V$
for A = 01
we find also
~8 = (VQ - VD)VD
for A
= or
U.
(11.7)
With the relation
Vc
*
A * Vc= y - V D * A
*
[11.1]
VD
for the velocity of random changes vc the form (8.5) of the correlation function
can be written
12. Computation of the correlation parameters using measurements
At last we w i l l discuss the necessary correlation measurements for the deter+
mination of the correlation parameters a, b, A, /3, and y. We proceed in the order
of the preceding sections, starting with b and y for which only time measurements
are necessary, continuing with a and A determined by pure space measurements,
and closing with /?which requires combined space-time measurements.
-b
From the graph of C(O,z), Fig. 1, the parameter b could be obtained as the
slope of C (0,t)at t = 0, see Eq. (3.2). But the slope of an experimentally given
curve taken a t one point only is a rather uncertain value. Therefore we determine
from this curve the situation zoof the maximum and its value C ~ ( 0 , zwhich
)
can
be done with less uncertainty. With these two values the parameter
can be found as twice the mean slope of C ~ ( 0 , t-) C(0,z) against a (z- zJascale. If necessary the parameter b can be computed with toand y using the relation
b
f5.11
= Z,Y.
For an autocorrelation function we need only y because then
C ~ ( 0 , t=
) 1, see end of Sect. 6.
to=
0 and
92
Annalen der Physik
7. Folge
t
*
Band 18, Heft 1-2
1966
+
For the determination of the parameter a as the gradient ot C (t,0) at the
*
point 5 = 0 the same objections hold as for the analogous determination of b
above. Therefore one should try to find the position vector
and its value CM$0).
CE,
+
toof the
maximum
But this is a much more laborous task than the determination of zo and C ~ ( 0 , z above
)
because we
+
have to look for the maximum of C(6, 0) in
a three-dimensional -space. This can be
done by a systematic subdivision of the
+
5 -space. Fig. 4 shows the beginning of such
g
*
a procedure for a two dimensional 6-space
if cartegian coordinates with unit vectors
= 1, 2, 3) are used. At first we invest i g a t e ~ $ , , ~ in
) the two coordinate directions. Having found the quadrant with the
Fig. 4. Correlation function C ( 2 0 ) maximum, say the first quadrant, we bisect
+
it and determine the octant with the maxifor a two-dimensional &space
mum, etc., We must continue this bisecting
+
*
until we have determined 6 and C
, (f ,0) with the required accuracy.
We do not need this cumbersome procedure for an autocorrelation function,
*
because then 6 = 0 and CM( 6 , O ) = 1.
After the determination of andC,
we can approach the tensor A. We
have to select a certain coordinate system for the representation of the tensor
components acv.This must not be a Cartesian one but can .beadapted to the measuring procedures. (For turbulence measurements in a wind tunnel cylindrical
coordinates suit best.) Let g, (v = 1, 2, 3) be its covariant basis vectors.and g v
*
the contravariant ones, then the vector 6 and the tensor A can be written
g,(v
-b
c0
+
l=?Pg,,
(2,O)
A=.Z2apgpg".
c
(12.1)
v
Intentionally we do not use EINSTEIN'S
summation convention. For cartesian coordinates the distinction between co- and contravariance, i.e. between
subscripts and superscripts, may be dropped and the basis vectors are unit
vectors.
The quadratic form
(z- T o ) (z
go) = 2[cM(z,o) - c(;,o)l
2
2 a p v (5" - 6;)
c 1 v
('? - l;)= 2 [CM
A
reads with (12.1)and g c . g v = S.,
~6.31
(2,0) - c(2,o)]
For the determination of the. diagonal components
coordinate axis g , (v = 1, 2, 3) a fixed point
k vwe
(12.2)
select on each
"
*
5'
=i'g,,
v = 1, 2, 3
(12.3)
93
K. SUCHY:Correlation Velocities
"
*
see"Fig. 4. For these" three values
we have to measure the correlation function
-+
C(zv,O).Since t h e & lie on the coordinate axes we obtain
from Eq. (12.2) b y
"
subsequently inserting the three pairs of values
1251
ay, = 2
C,(X 0)
-+
-+
tVand C(&,0) (with v = 1 , 2 , 3 )
+
- c (&O)
(12.4)
( P - m2
To obtain the off-diagonal elements a,, we fix three additional points
"
*
"
-+
tx, =q x t
-
with arbitrary constants q x
-+
"
.
x
+ q& *5, = q x E " g x + q,
"
"
5,gp
(12.5)
+ 0 + 7,. For the special case q1 = 1 = q2the point-
*
Elzis shown for Cartesian coordintes in Fig. 4.
If we insert these
three values 5'
"
*
and their corresponding measured correlation functions C ( t , , 0) subsequently
into Eq. (12.2) we find
*
C , KO)
"
*
- c(E,,o)-
1
"
= a,, =
ax,
rl,rl,
"
7; (P - E ; ) ~ax, -
1
7;
(2'
-
(i"
x:,
(& -
(12.6)
For the computation of the covariant components of the vector
2 = +BY&?
(12.7)
"
-+
one has" to measure the time dependence of the three functions C (5,,z)a t the
points
*
E,(Y
maxima
=
"
1, 2, 3), Eq. (12. 3), on the coordinate axes until their temporal
-+
z M ( E V ) , the
"lag times", Eq. (9.3), have been found:
*
-GM
(it,)
= 70
+ Py-lBv.
(12.8)
This yields [26]
"
(12.9)
*
Having completed the computation of A, B, and y we summarize the determination of the different velocities which we have defined. With A and y the
fading velocity V F describes the characteristic correlation ellipsoid
* A *VF = y .
determine the vector of the drift, velocity
VF
The parameters A and
VD
= A-'
~7.11
+
*
r8.31
94
Annalen der Physik
t
7. Folge
+t
Band 18, Heft 1-2
+t
1966
whose polar plane with respect to the characteristic correlation ellipsoid is covered by the endpoints of the spreading velocity vs :
-+
vs*p =y.
[9-31
Finally the velocity of random changes decribes the particular correlation
ellipsoid
vc * A *vc = y - V D A . V D .
[11.1]
13. Influence of missing measnrments
I n the preceding section we have seen in detail which measurements are needed for the complete determination of the correlation parameters in the TAYLOR
expansion (3.2) up to the second order. We will now discuss briefly the lack of
certain measurements and their influence on the determination of the correlation
parameters.
-+
If time measurements are not possible, then to,
y , and cannot be computed
and therefore none t.he of velocities v p , VD, vg, and VC.
If spatial measurements in one direction, say g,, are not available, then ti,
4 , s = cisv, and p3 cannot be determined and hence none of the velocities V F , vD,
v g , VC. But if the assumption can be justified that the correlation is independent
+
of t3,then &, 01,s = cisv, and p 3 can be put equal to zero. Hence the vectors p and
V D are perpendicular to g,. For Cartesian coordinates the correlation ellipsoids
degenerate into enveloping elliptical cylinders with g, as common axis and the
vg-plane, i.e. the polar plane of V D , becomes parallel to the g,-axis. [27]
Turbulence measurements in a wind tunnel usually show no dependence on
the circumferential angle about the axis. Then the corresponding correlation
ellipsoids have rotational symmetry about this axis.
For the autocorrelation
-
+ t, t + d
-f
BW(Z z)= dUY(I,t) &(r
TAYLOR’Shypothe sis
11.31
y=ij.A.u
[4-33
can replace some measurements. If A has been determined by spatial measurements one does not need additional time measurements t o obtain y. If, on the
other hand, one has computed y from time measurements TAYLOR’S
hypothesis
(4.3) yields the diagonal element au of the tensor A with iI in the direction gi.
Because the tensor ellipsoids are stretched in this direction 0 1 io~ an eigenvalue
of A. The other two eigenvalues have still to be computed from spatial measurements which cannot; be replaced by TAYLOR’S
hypothesis.
14. Conclusion
We have seen that each of the three velocities V F , V D , vs describes a different
physical property of the correlation function and that we really need all three for
+.
the description of a correlation function C (6, t).This is sufficient for tl scalar cor+
relation B(E, 7) or for a separate description of each component of a tensorial
K. SUCHY:Correlation Velocities
95
correlation. In the latter case each component has its own correlation parame+
ters a, b, A, p, y , and hence its own triple V F , V D , V S . Nothing has been said about
possible connections among the parameters (or velocities) belonging t o different
components of the same tensorial correlation. It would be an interesting task t o
look for such connections, be it only in a hypothetical form. For instance, an
extension of TAYLOR'S
hypothesis (4.3) for each separate diagonal element Bw
(1.3) of B (1.2) towards a more comprehensive statement connecting all elements
of B would be very valuable. Perhaps it can be gained by a numerical comparison
of the corresponding correlation parameters after these have been computed
from measuring data.
N o t e a d d e d i n p r o of : Averages of drift measurements of the density
correlation (1.6) in the lower ionosphere (E-region) over Puerto Rico have
shown the proportionality
VC = 1 . 1 1 v D
and the coincidence of GD with the direction of the major axis of the correlation ellipsoid [T. J. KENESHEA,
M. E. GARDNER,W. PFISTER,Journ. Atmosph.
Terr. Phys. 27 (1965) 7-30, Figs.5 t o 9, 17, 201. Therefore we can use Eq.
(11.5) and find the relation
( V S )=
~ [(Lll)'
~ ~
between (us)*
+ 1 3 V D = 2.23 VD
and VD. With Eq. (10.2) we obtain furthermore
v2.23~~
=1
. 5 ~for~ CF = G D .
If we assume the mean velocity U also i n the direction i ) =
~ (i)&iu
TAYLOR'S
hypothesis (4.3) compared with (10.1) would require
VF =
then
13 = 1.5 VD = 0.33 (vs),,,in.
Inserting this into FRENKIEL'S
condition (9.7) we obtain
which cannot be true since the integral scale L (5.8) is a t least as large as the
principal correlation length aT1" in the direction of the major axis. Hence
TAYLOR'S
hypothesis (4.3) does not hold for t,he density correlation (1.6) in
the E-region of the ionosphere over Puerto Rico.
References
[l] BOOXER,
H. G., J. A. RATCLIFFE
and D. H. SHINN,Phil. Trans Roy. SOC.A 242 (1950)
579-609, $ 5. (a).
K., and R. W. LARENZ,
Z. Physik 163 (1961) 245-261, $5.
[2] HUNGER,
[3] MONTGOMERY,
D. C. and D. A. TIDMAN,Plasma Kinetic Theory McGrsw-Hill (New
York) 1964, Sect. 8.
[4] Ref. [3] $ 14.2.
[5] Ref. [3] $15.2.
[6] Ref. [l] $ 4.
[7] Ref. [l] $ 5. (d).
[8] STEENBECK,
M., Monatsber. Deut. b a d . Wiss. 6 (1963) 625-629, Eqs. (2) and (5).
[9] TAYLOR,
G . I., Proc. Roy. SOC.London A 16* (1938) 476-490, Eq. (8).
[lo] LUMLEY,
J. L., and H. A. PANOFSKY,
The Structure of Atmospheric Turbulence, Interscience Publishers (New York) 1964, $1.17.
96
Annalen der Physik
it
7. Folge
*
Band 18, Heft 1-2
it
1966
[lOa] FRBNKIEL,
F. N., gtude statistique de la turbulence. Fonctions spectrales et coefficients de correlation. Office National d'lhudes et des RBcherches ABronautiques
(0.
N. E. R. A.), Rapport Techique No. 34,1948. English translation: Statistical study
of turbulence - Spectral functions and correlation coefficients. National Advisory
Committee for Aeronautics (NACA). Technical Memorandum 1436, Washington,
D. C.,July1958.
[ll]P w s , G. J., and M. SPENCER,
Proc. Phys. SOC.LondonB 68 (1955) 481-492, J 2.1.
[l2] BRIWS,B. H., G. J. -LIPS
and D. H. SHI", Proc. Phys. SOC.London B 63 (1950)
106-121, Eq. (6).
[13] Ref. [ll]J 2.2.
[14] Ref. [12] Eq. (8).
[l5] DOUOHERTY,
J. P., Phil. Mag. (1960) 553-570, Eq. (6).
1161 Ref. [12] Eq. (11).
[17] Ref. [ll]Fig. 7 (b).
[18] Ref. [lo] pp. 192/193.
1191 Ref. El21 Eq. (15).
[20] Ref. [lo] p. 193.
1211 Ref. [l2] pp. 193/194.
1221 Ref. [12] J 4. (iii).
[23] GUSEV,V. D., and P. F. MIRKOTAN,URSI-CIG Inospheric Symposium, Nice (France),
December 1961, Eq. (25).
[24] Ref. [12] Eq. (14).
[25] Ref. [ll] J 2.2.
1261 Ref. [ll] J 3.1.
[27] Ref. [23] end of 8 3.
I have to thank Dr. K. RAWERwho gave me the impetus for this work and
Dr. J. TATJBENHEIN
who encouraged its publication. For stimulating help I am
indebted to Dr. D. A. "IDMAN,
Dr. J. R. WESKE
andMr. J. W. WRIGHT.
C o 11e g e P a r k , Institute for Fluid Dynamics and Applied Mathematics
of the University of Mary Land.
Bei der Redaktion eingegangen am 20. Oktober 1965.
Verantwortlich
ftir die Schriitleitung: Prof. Dr. G. Richter, 1199 Berlin-Adlerahof, Rudower Chaussee 6; fIir den Anzeigenteil: DEWAG-Werbuug Leipzig, 701 Leipzig, Friedrich-Ebert-Str. 110, Euf 7851. Z. 2. gilt Anzeigenpreisliste4.
Verlag: Johaun Ambrosius Barth, 701 Leipzig. Salomonstr. 18 B, F e r n ! : 27681,27682. VerMfentlicht unter
der Lizenz-Nr. 1396 des Presseamtes beim Vomitsenden des Ministerrates der DDR
&Druck: Paul Dtinnhaupt, ECithen (IV/5/1) L 151166
Printed In Qermany
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