Turbulence and distributed chaos with spontaneously broken symmetry

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by  A. Bershadskii

Turbulence and distributed chaos with spontaneously broken symmetry
A. Bershadskii
ICAR, P.O. Box 31155, Jerusalem 91000, Israel
Itisshownthatinturbulentflowsthedistributedchaoswithspontaneouslybrokentranslational
space symmetry (homogeneity) has a stretched exponential spectrum exp−(k/k )β with β = 1/2.
β
Good agreement has been established between the theory and the data of direct numerical simula-
tionsofisotropichomogeneousturbulence(energydissipationratefield),ofachannelflow(velocity
field), of a fully developed boundary layer flow (velocity field), and the experimental data at the
plasma edges of different fusion devices (stellarators and tokamaks). An astrophysical application
tothelarge-scalegalaxiesdistributionhasbeenbrieflydiscussedandgoodagreementwiththedata
of recent Sloan Digital Sky Survey SDSS-III has been established.
6
1
0
”The mathematicians always want that their mathe-
2
maticsshouldbepure,thatis,strictandprovable,wher-
2
r everpossible. However,themostinterestingandrealistic
p
A problems could not usually be solved in that manner.” 1 exp-(k/k )3/4
˜ A.N. Kolmogorov β

7 ε
ε
E 0
] n
n l
y INTRODUCTION -1
d
u- It is known that anisotropic turbulence can emerge -2 Reλ=167
l from deterministic chaos [1]. In a recent paper Ref. [2] 0 0.2 0.4 0.6 0.8 1
f
. it was also shown for isotropic homogeneous turbulence. 3/4
s (k/kd)
c Theisotropichomogeneousturbulenceemergesfromdis-
i tributedchaos. Forthedistributedchaosthepowerspec- FIG. 1: Logarithm of power spectrum of the energy dis-
s sipation rate field ε at Re = 167 as function of (k/k )3/4.
y tra are linear weighted superposition of the exponentials λ d
ThestraightlineindicatestheexponentialdecayEq. (1)with
h converging into the stretched exponential:
β =3/4. The data are taken from the DNS [13].
p
(cid:90) ∞
[
E(k)(cid:39) P(κ) e−(k/κ)dκ∝exp−(k/kβ)β (1)
4 0
v where P(κ) is a probability distribution of κ, the κ is of the velocity correlation (where <...> denotes an en-
4 wavenumber of the waves (pulses) driving the chaos. An semble average) was allowed to be weakly depending on
6
asymptotic theory has been developed in the Ref. [2] in x. Spectrum of the state with the spontaneously broken
3
7 order to find β. In this theory the asymptotic (κ → ∞) translational symmetry was shown (numerically) to be
0 scaling of the group velocity of the waves driving the different from the Kolmogorov-Zakharov spectrum. The
1. chaos authors of the Ref. [3] argued that this phenomenon can
have a general nature.
0
υ(κ)∼κα (2)
6
1
is used in order to find the β:
: SPONTANEOUSLY BROKEN SPACE
v
2α SYMMETRY
i β = (3)
X 1+2α
r The main physical problem now is to determine a di- Therearetwoinvariantsinisotropichomogeneoustur-
a
mensional parameter controlling the scaling (2). bulence related to the fundamental space symmetries.
It was recently discovered [3] that spontaneous break- Let us consider integrals
ing of the space translational symmetry (homogeneity)
(cid:90)
in wave turbulence results in a crucial change of the sta- I = rn−2(cid:104)u(x,t)·u(x+r,t)(cid:105)dr (4)
n
tistical attractor. This phenomenon is related to the
Kolmogorov-Zakharov spectrum instability to the per-
at sufficiently rapid decay of the two-point correlation
turbations weakly breaking the space translational sym-
function of the fluid velocity field. The integrals I and
2
metry. Namely, the Fourier transform
I are the Birkhoff-Saffman and Loitsyanskii invariants.
4
(cid:90) These invariants are associated with the conservation
(cid:104)u(x)·u(cid:63)(x+r)(cid:105)eikrdr
laws of the linear and angular momentum, respectively
2
[4]-[8]. Theseconservationlawsthemselves(accordingto where integration is over the volume of motion, then we
Noether’s theorem) are consequences of the fundamen- obtain for the incompressible fluid
tal space symmetries: translational (homogeneity) and
d(cid:104)u(x,t)·u(x+r,t)(cid:105)
rotational (isotropy) [9]. The Birkhof-Saffman invariant V =−2(cid:104)ω(x,t)·ω(x+r,t)(cid:105) (8)
determines the scaling Eq. (2) for the isotropic homoge- d(νt) V
neous turbulence and provides α=3/2 [2]:
where ω(x,t) = ∇ × u(x,t) and νt is the difusive
(viscous) time appropriate for consideration of quasi-
υ(κ)(cid:39)a I1/2 κ3/2 (5)
2 invariants at the final stage of the turbulence decay (cf.
(cid:82)
Chapter 15 Ref. [4]). Finally let us take integral dr
where a is a dimensionless constant. This gives β =3/4 V
on both sides of the equation (8)
(Eq. (3)). If one uses the Loitsyanskii integral I
4
instead, then one obtains β =5/6. (cid:82)
d (cid:104)u(x,t)·u(x+r,t)(cid:105) dr
V V =−2γ (9)
d(νt)
TheBirkhoff-SaffmanandLoitsyanskiiinvariantswere
recentlygeneralisedforrotating,stratifiedandMHDtur- where
bulence [10],[11]. Therefore, the above mentioned values
(cid:90)
of the β can be also valid for these types of anisotropic γ = (cid:104)ω(x,t)·ω(x+r,t)(cid:105) dr (10)
V
turbulence. V
Inthisnotewewillconsideraturbulencethatemerges
Of course, the average over volume eliminates depen-
from distributed chaos with spontaneously broken space
dence on x even in the case of the broken translational
translational symmetry (homogeneity). First of all let
symmetry(cf. Introduction),butthisaverageisequalto
us note that, unlike the inertial range, the asymptotic
the ensemble average in the limit R → ∞ only. If the
Eq. (2) is under strong viscous influence. Therefore, in
Birkhof-Saffman integral is finite at this limit, then the
order to understand what dimensional parameter should
right hand side of the Eq. (9) (the parameter γ) equals
be used in the Eq. (2) at the spontaneous breaking of
to zero [6],[7] and we have
the translational symmetry let us consider a final stage
of decaying turbulence inside of a finite sphere of radius (cid:90) (cid:90)
lim (cid:104)u(x,t)·u(x+r,t)(cid:105) dr= (cid:104)u(x,t)·u(x+r,t)(cid:105)dr
R (with boundary conditions). One should not be con- V
R→∞ V
fused by difference in asymptotic k → 0 for the final
decay case and the asymptotic Eq. (2). The main ideas and
related to the viscosity effects are similar in these two (cid:90)
cases. For simplicity we will consider an incompressible (cid:104)u(x,t)·u(x+r,t)(cid:105)dr=const
fluid. In the final stage of decay only remnants of tur-
bulence are remained and one can neglect the nonlinear at the final stage of decay of homogeneous isotropic tur-
convection terms in the Navier-Stoks equation for veloc- bulence. The same procedure can be applied to the field
ity field u(x,t) [4],[7] (but we preserve the pressure term ω(x,t)itselfwithconclusionthatforR→∞theparam-
because the long-range interactions in the finite volume eterγ ischangingwiththediffusiontimeνtmuchslower
can still exist due to the pressure) than the integral (cid:82) (cid:104)u(x,t) · u(x + r,t)(cid:105) dr. All this
V V
indicates that the parameter γ Eq. (10) can be consid-
∂u(x,t) =−1∇p(x,t)+ν∇2u(x,t) (6). ered as the parameter characterising spontaneous break-
∂t ρ ingofthespacetranslationalsymmetry(letusrecallthat
theBirkhof-Saffmaninvariantisassociatedwiththecon-
Let us shift this equation by a constant vector r
servation law of momentum and, through this, with the
space translational symmetry).
∂u(x+r,t) 1
=− ∇p(x+r,t)+ν∇2u(x+r,t) (7) TheninsteadoftheEq. (5)weobtainfromthedimen-
∂t ρ
sional consideration
Let us multiply both sides of Eq. (6) by u(x+r,t) and υ(κ)(cid:39)a |γ|1/2 κ1/2 (11)
2
both sides of Eq. (7) by u(x,t) (the dot products), then
make a summation of the two equations and then inte- where a is a dimensionless constant. Equations (2) and
2
grate the both sides over volume of the motion on the (3) then give β = 1/2. Observation of the stretched
variablex. Ifwedenotethevolumeaverageofafunction exponential power spectrum with the β = 1/2 can
A(x,t) as be an indication of presence of the distributed chaos
with the spontaneously broken translational symmetry
(cid:82)
A(x,t)dx (homogeneity).
(cid:104)A(x,t)(cid:105)V = V (cid:82)
dx
V
3
2 0
1.5 exp-(k/kβ)1/2 -2 exp-(k/kβ)1/2
1 -4
)
Eεε 0.5 (kx -6
x
n 0 E -8
l n
-0.5 l-10
-1 Re=257 -12
λ +
y =99.75
-1.5 -14
0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6
1/2
(k/kd) (k δ )1/2
ν
x
FIG. 2: Logarithm of power spectrum of the energy dis- FIG. 3: Logarithm of the streamwise power spectrum of
sipation rate field ε at Reλ = 257 as function of (k/kd)1/2. streamwise component of the velocity at y+ =99.75 as func-
ThestraightlineindicatestheexponentialdecayEq. (1)with tion of (k δ )1/2. The straight line indicates the exponential
x ν
β =1/2. The data are taken from the same DNS [13]. decay Eq. (1) with β =1/2.
.
straight line in this figure indicates the Eq. (1) with
Whereshouldonesearchforthedistributedchaoswith β =3/4 (the Kolmogorov wavenumber k =((cid:104)ε(cid:105)/ν3)1/4,
d
spontaneously broken space symmetries? First of all it and k (cid:39) 0.234k ). Fig. 1 shows the data obtained at
β d
can be the energy dissipation rate. Being a measure of Reynolds number Re = 167. One can see that at this
λ
velocitygradientstheenergydissipationrateisnaturally value of Re the ε field exhibits good homogeneity. Fig-
λ
moresensitivetothedistortionsofthehomogeneitythan ure2showsthedataofthesameDNSatRe =257. The
λ
velocity field itself. One can also look to situation where straightlineinthisfigurecorrespondstotheEq. (1)with
the motion is bounded in one direction and free in the β = 1/2 (k (cid:39) 0.079k ) that indicates the spontaneous
β d
two other directions. This space configuration is rather breaking of the translational symmetry in the ε field.
appropriate for the spontaneous breaking of the space The energy dissipation rate plays crucial role in
symmetries (a boundary layer or a channel flow, for in- development of inertial range [4] near the range of
stance,cf. Ref. [12]). Theedgesofmagneticallyconfined the distributed chaos for high Reynolds numbers [2].
plasmas also provide an interesting and practically im- Therefore,thisspontaneousbreakingofthetranslational
portant example of the shear flows. Another interesting symmetry (homogeneity) in the ε field can have very
case is the finite expanding universe. significant consequences for the inertial range (espe-
cially taking into account that the wavenumbers of the
inertial range are smaller than those of the distributed
ISOTROPIC TURBULENCE chaos range and, therefore, are more vulnerable to the
homogeneity distortions). It seems that not only local
In the direct numerical simulations of presumably ho- isotropy [4], [14] but also local homogeneity should be
mogeneous isotropic turbulence the motion is always ap- the matter of concern for the inertial range (and may be
proximately homogeneous. The energy dissipation rate more significant one, cf. Introduction).
ε = 2νe2, where e2 = e e with e = (∂u /∂x +
ij ij ij i j
∂u /∂x )/2, is more sensitive to inhomogeneities than
j i
velocityfield. Therefore,theεfieldcanexhibitthespon-
taneous breaking of the translational symmetry in sit- CHANNEL AND BOUNDARY LAYER FLOWS
uations where the velocity field itself is still sufficiently
homogeneous. Figure1showspowerspectrumoftheen- In figures 3-5 we will use the data (taken from the
ergy dissipation rate field (E ) obtained in a DNS of site [15]) of a direct numerical simulation (DNS) of walls
εε
a statistically stationary incompressible turbulence with (planes)boundedturbulentflowwiththefrictionvelocity
periodic boundary conditions which can imitate homo- Reynolds number Re ∼ 1000 (channel flow, see also
τ
geneous isotropic turbulence (see description of the DNS Refs. [16],[17],[18]), and the data (taken from the site
in Ref. [13]). The scales in this figure are chosen in [19]) of a direct numerical simulation of fully developed
order to represent the Eq. (1) as a straight line. The boundary layer flow Re (cid:39) 1990 Re (cid:39) 6499 (see also
τ θ
4
0 -5
1/2
-2 exp-(k/k )
β
-10
-4
k)z -6 k)z -15 exp-(k/kβ)3/4
( (
Ex -8 E x
n -10 n -20
l -12 l
-25
-14 y+=99.75 y/δ99=1.301
-16 -30
0 0.2 0.4 0.6 0.8 0 50 100 150 200
(k δν)1/2 (kz Lz/2π)3/4
z
FIG. 4: Logarithm of the spanwise power spectrum of the FIG. 7: The same as in Fig. 6 but outside the boundary
streamwise component of the velocity at y+ =99.75 as func- layer (at y/δ = 1.301). The straight line indicates the ho-
99
tion of (kzδν)1/2. mogeneous value of β = 3/4. The data were taken from the
. same site [19].
0
-2
1/2 -3 k-5/3
-5 exp-(k/kβ) -4
) -5
x )
k kz -6
E(x -10 (x -7
E -8

n g -9
l -15 o -10
l
-11
δ
y+=999.7 -12 y/ 99=1.301
-20
-13
0 0.1 0.2 0.3 0.4 0.5 0.6
0 1 2 3
(kxδν)1/2 log (kz Lz/2π)
FIG. 5: The same as in Fig. 3 but for y+ =999.7. FIG. 8: The same as in Fig. 7 but in the log-log scales.
The dashed straight line indicates the homogeneous power-
law E(k)∝k−5/3.
0
1/2
exp-(k/k )
β Refs. [20],[21],[22]).
-5
)
z ThepowerspectrashownintheFigures3-5correspond
k
( tothedistancesy+ =99.75andy+ =999.7fromawallof
x -10
E the channel flow (belonging to the so-called ’logarithmic
n law’ range, i.e. far away from the near wall range with
l -15 its complex structures) [15]. The power spectrum shown
y+ = 100 in Fig. 6 corresponds to distance y+ (cid:39) 100 (the fully
-20 developed boundary layer flow [19]).
0 10 20 30 40
Figure3showsstreamwise(k )powerspectrumofthe
(k Lz/2π)1/2 streamwise component of the vexlocity. The scales in this
z
figure are chosen in order to represent the Eq. (1) with
FIG. 6: The same as in Fig. 4 but for direct numerical sim- β = 1/2 as a straight line. The δ = ν/u is the vis-
ν τ
ulations of fully developed boundary layer flow Reτ (cid:39)1990 ( cous length scale, uτ is the friction velocity. Figure 4
Re (cid:39)6499, L is the DNS box dimension along the z axis).
θ z showsspanwise(k )powerspectrumofthesamestream-
The data were taken from the site [19]. z
wise component of the velocity. The distance from the
wall is y+ = 99.75 (in viscous units) for the both cases.
5
k (cid:39)0.0021δ−1 for the streamwise (k ) power spectrum
β ν x
(Fig. 3), and k (cid:39) 0.0025δ−1 for spanwise (k ) power
β ν z
spectrum (Fig. 4). For the streamwise (kx) power spec- -14 exp-(f/fβ)1/2
trumthistypeofspectrumisobservedalsoattheendof
the ’logarithmic law’ range: y+ = 999.7, as one can see -16
atFig. 5. Theparameterk (cid:39)0.0012δ−1 forthestream- E
wise (k ) power spectrum aβt the distanνce y+ =999.7. n -18
x l
Forcomparison,thefigures7and8showdatafromthe
-20
same DNS as Fig. 6 but outside the boundary layer at
y/δ99 = 1.301 (where δ99 is the thickness of the velocity -22
boundary layer). For the ln-linear scales in the Figure 7
5 10 15 20 25 30
the straight line indicates the exponential decay Eq. (1)
1/2
withβ =3/4. Figure8showsthesamespectrumbutnow (λ f [kHz])
in the log-log scales. In the Fig. 8 the straight line in-
dicates the Kolmogorov’s power-law spectrum k−5/3. In
FIG. 9: Power spectrum of ion saturation current. The
thetermsoftheRef. [2]thepictureshownintheFigs. 7
fusion devices - JET: λ = 4.5, W7−AS(1) : λ = 3.5, W7−
and8istypicalforhomogeneousturbulence: coexistence AS(2) :λ=4.5, TJ-I: λ=1, TJ-IU: λ=3, The straight line
of the range of the distributed chaos dominated by the is drawn in order to indicate a stretched exponential decay
Birkhoff-Saffmanintegral(characterizedbyβ =3/4,Fig. Eq. (12).
7) and of the inertial (Kolmogorov) range (Fig. 8). This
is precisely what one can expect outside the boundary
layer.
ments were made at the plasma edges of several fusion
The exponent β = 1/2 provides rather good agree-
devices (see the legend). The authors of the Ref. [23]
ment with the data shown in the Figs. 3-6, that can be
(wherethesedatacamefrom)rescaledthefrequencywith
considered as an indication of strong presence of the dis-
the parameter λ. The cumulative data for different de-
tributed chaos with spontaneously broken translational
vices (used in the Fig. 9) were taken from the Ref. [24].
space symmetry (homogeneity). The value of k seems
β The scales in Fig. 9 are chosen in order to represent the
to be non universal in this case (cf. with the isotropic
Eq. (12) as a straight line.
homogeneous case [2]).
LARGE-SCALE DISTRIBUTION OF THE
THE EDGES OF MAGNETICALLY CONFINED
GALAXIES
PLASMAS
It is an old and rather natural idea that turbulence
Interesting and practically important example of the
played a crucial role in formation of galaxies and clus-
shear flows is now under vigorous experimental investi-
tersofgalaxies[25][26](seeformorerecentdevelopments
gation at the edges of magnetically confined plasmas in
Refs. [27],[28],[29] and references therein). In the fi-
different fusion devices. Chaotic motion at the edges de-
nite expanding universe one can expect the spontaneous
grades performance of these devices. Measurements of
breakingofthespacetranslationalsymmetry(homogene-
ionsaturationcurrentisusedinordertoobtaindynamic
ity) at certain stage. To check role of the distributed
informationthere. Itwasdiscoveredthatfrequencyspec-
chaos one can use the power spectrum of a distribution
traoftheionsaturationcurrentfluctuationsarecollapsed
of galaxies. This spectrum is the Fourier transform of
in a singular functional shape for the toroidal devices
the two-point correlation function ξ(r):
(stellarators and tokamaks) [23]. The frequency spectra
obtainedbyaprobewithafixedspacelocationinashear n (cid:90)
P(k)= ξ(r)eikrdr (13)
flow reflex the spatial spectra of the structures moving (2π)3/2
near the probe (the Taylor hypothesis [4]). Therefore,
the frequency stretched exponential spectrum with n representing the average density and the correla-
tion function ξ(r) is defined by equation
E(f)∝exp−(f/f )1/2 (12)
β
dp=n(1+ξ(r))dV (14)
reflex the wavenumber spectrum Eq. (1) with β = 1/2.
Inthiscasethedifferencesinthespectrabetweenthedif- wheredpisprobabilityoffindingagalaxyinavolumedV
ferent devices can be related to the difference in the pa- separated by a distance r (isotropic case) from a given
rameter f only (after normalization of the amplitudes). one (see Ref. [30], for instance).
β
Figure 9 shows the normalized and frequency rescaled In recent paper Ref. [31] the P(k) spectrum was
power spectra of ion saturation current. The measure- calculated for the data of the BOSS - Baryon Oscilla-
6
Sillero for sharing their data. I also thank P. A. David-
12 son, C. Meneveau, J. Schumacher and K. R. Sreenivasan
1/2
exp-(k/k ) for comments and encouragement.
3 β
h) 11
c/
p
M 10
(

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(
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withspontaneouslybrokenspacetranslationalsymmetry
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