Castaing Scaling and Kernels for Theories of the Turbulent
Cascade
Miriam A. Forman
Department of Physics and Astronomy
State University of New York at Stony Brook NY11794-3800 USA
Abstract. The Castaing cascade kernel function G(u;L,L’) relates the pdf of velocity fluctuations on scale L’ to those on
scale L. The shape of G(u;L,L’) and how it depends on scale is a measure of intermittency alternative to measures such
as the exponent of the structure function or multifractal spectra. This paper provides analytical forms and graphic
illustrations of the Castaing kernels corresponding to six models of the turbulent cascade. The popular P-model and
Poisson models of turbulence have discrete Castaing kernels, which seems non-physical.
G(u). Such distributions can be conceptually useful
particularly for looking at the dependence of the
parameters on scale [5, 11, 13]. However, neither G(u)
nor PL(u) can be precisely Gaussian; if they were, the
scaling would be log-normal, and the third moment of
the parallel velocity increment would be zero, both of
which are refuted fundamentally by Frisch [6]. The
Castaing pdf is only approximate and should be used
with care. This paper deals only with Castaing scaling
which is much more general.
INTRODUCTION
Castaing scaling [1,2,3,4] describes the cascade of
turbulent energy in a fluid from scale L to smaller
scale L’ by relating the probability distribution
functions (pdfs) of the velocity increments δV at the
different scales with a cascade kernel G(u:L’,L) such
that
+∞
P L' (δV ) =
∫
(
)
G (u ; L' , L )e − u PL e − u δV du .
(1)
−∞
RELATION OF CASTAING KERNEL G,
TO THE STRUCTURE FUNCTION ζ
The kernel G(u;L’,L) represents the probability that
turbulence of amplitude e-uδV on scale L, cascades to
turbulence of amplitude δV on scale L’. Castaing
scaling is linear in δV, in the sense that the same G(u)
relates e-uδV at the larger scale to δV at the smaller
scale, for every -∞ < δV < ∞. Since Castaing kernels
directly describe the distribution of turbulent
amplitudes at larger scales that contribute to a given
amplitude at small scales, they provide a graphic
description of the cascade.
Multifractality and
intermittency appear in the cascade when the kernel
G(u) is anything other than a delta-function. The
details depend on the shape of G(u; L’,L) versus u, and
its dependence on L’.
Following equation 1, the scaling properties of all
the moments of the pdf arise from the scaling of
G(u;L,L). In particular, since the moment of order q at
scale L’ is
+∞
S(q , L' ) =
∫
∞
d(δV )(δV )q
−∞
∫
(
)
G (u ; L' , L )e − u PL e − u δV du
−∞
(2a)
∞


qu 
=  G (u ; L' , L )e du  S(q , L )


− ∞

∫
If both G(u) and PL(δV) are Gaussian functions of
their arguments, PL’(u) will be a winged “Castaing
distribution” characterized by the mean and width of
(2b)
all of the dependence on L’ (in other words, the
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edited by M. Velli, R. Bruno, and F. Malara
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550
unique parabola which has ζ(3) = 1, and ζ(q)⇒q/3
when µ⇒0. Some experts [e.g., 6] dismiss this model
in incompressible fluid turbulence because the ζ(q)
should not have a peak and its slope should be nonnegative at all q. A peculiarity of the KO62 kernel is
that at positive u it is non-zero (although small) and
increases with scale.
This allows a non-zero
probability for arbitrarily large fluctuations at smaller
scales to arise from smaller-amplitude fluctuations at
larger scales. These dominate the moments of large
positive order, and confuse their scaling. Whether this
is also strictly impossible in compressible and/or
magnetized fluids such as the solar wind is unclear.
Solar wind velocity increments do have ζ(q) with
negative slopes in some scale ranges at which the
intermittency is very large [7].
scaling) is contained in G(u:L’,L). Thus when the
scaling ζ(q) = d ln S(q , L ′) occurs on scales from L’ to
d ln L ′
L, (2) implies that the Fourier transform of G(u;L’,L)
is equal to [4]
ˆ (k ; L' , L ) = exp[− ln(L / L' )ζ (− ik )]
G
.
(3)
Scaling in actual data sets is usually described
with ζ(q), and turbulence theories with a mathematical
form for ζ(q). In practice ζ(q) can be calculated from
finite and imperfect real data sets only over a limited
range of q, because very large and very small δV will
be missing. If (1) applies, the scaling is completely
subsumed into the kernel G, and does not involve the
pdf at large scale. Also, the scaling of the moments of
the absolute value of δV is the same as the scaling of
the moments of δV for integer q ≥ 0. The odd
moments of the absolute values are not the same as the
odd moments of δV itself, but they scale the same way
if equation (1) is valid. The linear form of (1) assures
that moments of the absolute value of δV at odd, noninteger and negative q provide additional useful
information about G(u;L’,L).
The binomial and Poisson models [8,9,10] avoid
the negative slopes in ζ(q) and, since their kernels are
zero at u>0, never let small δV at large scale lead to
larger δV at smaller scales. Of these, DeBrulle’s [10]
2-parameter generalized Poisson model is the most
flexible, and can have D(hmin) = 2 (∆ = 0.4, β = 0.6
works well). However, the binomial and Poisson
kernels are uncomfortably discontinuous sums of delta
functions. In the binomial and Poisson models, δV at
small scale can arise only from certain discrete values
of δV at larger scale, which is implausible.
Although the idea of Castaing scaling underlies
turbulence theory already, it is instructive to examine
the actual kernels G(u;L’,L) for popular and
reasonable models of the turbulent cascade, to better
appreciate the differences between the models, and to
encourage further turbulence theories to describe
themselves with their kernels.
The Universal Multifractal model [13] is a
generalization of the KO62 model to a kernel that is a
continuous Levy function of u, near the most probable
values of u. Crucial truncations at extreme values of u
arise from plausible self-organized criticality and
limited data sets, preventing the slope of ζ(q) from
becoming negative. Such modifications could be
applied to the KO 1962 model as well.
COMPARISON OF CASTAING
KERNELS FOR TURBULENCE
MODELS
The intermittent models in Table 1 produce similarlooking ζ(q) forq-3<5 where ζ(q) might be reliably
estimated from data.
Table 1 lists in historical order seven models of
inertial turbulence, with their ζ(q) and G(u;L’,L),
minimum scaling exponent, and the dimension on the
minimum scaling exponent. All models have ζ(3) =1
in accord with the Kologorov 4/5 law [6] for inertial
turbulence. G(u;L’,L) was calculated by inverse
Fourier transform of eq. (3) [4]. The minimum scaling
exponent and dimension on the minimum scaling
exponent were calculated from ζ(q) by the method
given in [6].
7
Debrulle, delta=0.4, beta=0.6
6
5
binomial, P=0.7
logLevy, 1.5
She-Leveque
zeta(q)
4
3
lognormal, mu =0.2
2
1
0
-10
-5
0
5
10
15
20
25
30
-1
-2
q
-3
-4
Intermittent turbulence implies a Castaing kernel of
finite width; equivalently, a ζ(q) with negative
curvature. The Kolmogorov-Obkuhov 1962 (KO62)
form is the simplest extension of the K41 delta
function, to a normal distribution in u. Its ζ(q) is the
-5
FIGURE 1. Exponents of the structure function for six
models of the intermittent turbulent cascade. Parameters for
all models were chosen such that ζ(4)/4 -ζ(2)/2 = 0.022.
551
TABLE 1. ζ(q) and G(u) for models of the turbulent cascade
Turbulence model
ζ(q)
Exponent of structure
function <VLq> ∝ Lζ(q)
G(u,η
η)
Castaing kernel for L0 > L
u≡ln(δVL/ δVL 0) η≡ ln(L0/L)
Kolmogorov 1941
q
3
Delta function: δ(u + η/3)
Kolmogorov-Obukhov 1962
(log-normal)
2
q µ q  q  
+  −  
3 2 3  3  


Gaussian
q
 q
 3
3
1 − log2 P + (1 − P) 




binomial
n
1
2n
∑
0
m
n−m
n!


δ u − ln(P ) −
ln(1 − P )
3
3

 (n − m)! m!
.5<P<1
She and Leveque
1994
(Poisson)
De Brulle
1994
(Poisson)
Poisson
∞
∑
m =0





∑
δ(u + aη + mW )
m =0
ln
(bη)m e − bη
m!
1
1
log2  
3
P
2
1
b=
-∞
−α
Scale factor γ = 3 µ η
2(α − 1)
As UM, except straight line
for q > qD
∆
1− β
NA
1
µ 
 3 + 6(α −1)  η


Truncated Levy

1
q
µ
h min = + ( ) 1 − α D
3 6 α −1 
 3

552
3−
∆
1− β
Levy function of order α,
α
q
µ q q 

 −
+
3 2(α − 1)  3 3 


1<α<2
(most likely 1.5)
1− ∆
3
1
β
W=
3
Mean = −
S +L’s UM + self-organized
criticality and finite data set
NA
1
9
ln1.5  (2η)m − 2η
η

e
δ u + + m

9
3  m!

1− ∆
a=
3
(log-Levy)
-∞
Poisson
∞
0 < ∆, β < 1
Shertzer and Lovejoy’s
(1997)
Universal Multifractal
3
for (L0/L) = 2n
q

q
  2 3 
+ 21 −   
9
3
   


q

∆ 
1− ∆
3
−
β
q+
1

3
1− β 

1
3
2
 9 
3
η  µ  
exp −
u
1
+
+


 

3  2  
2πµη
 2µη 

µ ≈ 0.2
Meneveau and
Sreenivasan’s P-model
1991
(binomial)
hmin and D(hmin)



α −1 


D(hmin) =
3−
µ qD 
2  3 
α
She-Leveque
The non-linear processes causing self-organized
criticality of extreme fluctuations may make Eq. (1)
invalid at very extreme u anyway.
10
G(u), for L/L' = 4
Debrulle, delta=0.4, beta=0.6
binomial P=0.7
lognormal, mu =0.2
1
logLevy, alpha = 1.5, mu = 0.2
Kolmogorov 41
0.1
ACKNOWLEDGMENTS
0.01
It is a pleasure to acknowledge discussions with
Leeonard F. Burlaga
and colleagues at
NASA/Goddard Spaceflight Center, USA, and the
kindness of the Conference organizers. This work was
supported by NASA grant NAG 510995.
0.001
-2
-1.5
-1
-0.5
0
0.5
u
FIGURE 2. Castaing kernels (Eq. 1) for the models and
parameters in Figure 1, for a scale L/L’ = 4. The formulas in
Table 1 were used. The discrete models are shown as
isolated values scaled by the ∆u between points, to compare
with the continuous models. The integral under all curves =
1. The vertical line shows the location of the delta function
implied by the original Kolmogorov 1941 theory.
REFERENCES
Agreement is good in the central parts of the G(u)
for all models, corresponding to q-3<5, except for
the strange discreteness in the binomial and Poisson
models. Both (non-truncated) continuous models have
wings at u>0, but the logLevy has much bigger wings
than the lognormal. Shertzer et al.’s [13] assertion that
self-organized criticality will modify ζ(q) at its
extremes, will also modify the wings of the G(u) for
the lognormal and logLevy models.
1.
B. Castaing, Y. Gagne and E. Hopfinger, Physica D, 46,
177 (1990)
2.
Chaubaud, A. Naert, J. Peinke, F. Chilla, B. Castaing,
and B. Hebral, Phys. Rev. Lett., 73, 3277 (1994)
3.
Arneodo, S. Roux, and J.F. Muzy, J. Phys. II France, 7,
363 (1997)
4.
J.F. Muzy, J. Delour, and E. Bacry, arXiv:condmat/0005400 24 May 2000
5.
L. Sorriso-Valvo, V. Carbone, P. Veltri, G. Consolini,
and R. Bruno, Geophys. Rev. Lett., 26, 1801 (1999)
6.
U. Frisch, Turbulence, The Legacy of A. N.
Kolmogorov, Cambridge: Cambridge University press,
1995
7.
P. Veltri and A. Mangeney, in Solar Wind Nine, edited
by Habbal et al., AIP Conference Proceedings 471,
Woodbury, NY: American Institute of Physics, 1999,
pp. 543-546
8.
C. Meneveau and K.R. Sreenivasan, Phys. Rev. Lett.,
59, 1424 (1987)
9.
Z–S. She and E. Leveque, Phys. Rev. Lett, 72, 336
(1994)
CONCLUSIONS
The Castaing kernels describe the distribution of
amplitudes of turbulence on larger scales that give rise
to turbulence of a given amplitude on a smaller scale,
as a function of the ratio of amplitudes (ln(u)) and
scale. The Kolmogorov 1941 theory has a single delta
function as a kernel, but theories of intermittent
turbulence have more spread-out kernels. The popular
binomial (P) model and the Poisson models which
have well-behaved structure functions, have discrete
kernels that are sums of delta functions, implying that
small-scale turbulence of a given amplitude can arise
only from certain discrete amplitudes on larger scale.
This discreteness seems unphysical. On the other hand,
the full Kolmogorov 1962 lognormal theory and its
extension to logLevy, have continuous Castaing
kernels, but problematic wings and structure functions.
The lognormal or logLevy theories, modified by selforganized criticality and effect of finiteness of data
sets as proposed by Shertzer, et al. [13], avoids these
problems and seems the most acceptable among the
classes considered here.
10. B. Dubrulle, Phys. Rev. Lett, 73, 959 (1994)
11. C. Pagels and A. Balogh, Nonl. Proc. in Geophys., 8,
313 (2001)
12. Shertzer, D., S. Lovejoy, F. Schmitt, Y. Chigirinskya,
and D. Marsan, Fractals, 5, 427 (1997)
13. Forman, M.A., and L.F. Burlaga, in Solar Wind Ten,
edited by M. Velli, et al., AIP Conference Proceedings,
Woodbury, NY: American Institute of Physics (this
volume)
553