Content uploaded by Nils Basse

Author content

All content in this area was uploaded by Nils Basse on Aug 28, 2014

Content may be subject to copyright.

arXiv:astro-ph/0602452v4 11 Dec 2007

IEEE TRANSACTIONS ON PLASMA SCIENCE 1

A Study of Multiscale Density Fluctuation

Measurements

Nils P. Basse, Member, IEEE

Abstract—Intriguing parallels between density ﬂuctuation

power versus wavenumber on small (mm) and large (Mpc) scales

are presented. The comparative study is carried out between

fusion plasma measurements and cosmological data. Based on

predictions from classical ﬂuid turbulence theory, we argue that

our observations are consistent with 2D turbulence. The similar

dependencies of density ﬂuctuations on these disparate scales

might indicate that primordial turbulence has been expanded to

cosmological proportions.

Index Terms—Cosmology, density ﬂuctuations, fusion plasmas,

turbulence, wavenumber spectra.

I. INTRODUCTION

It is a very human trait to compare new observations to

previous experience. Our chance encounter with measurements

of the spectral power of density ﬂuctuations on Mpc scales

lead us to the conclusion that corresponding mm scale mea-

surements in fusion plasmas have surprisingly similar features

[1]. We are of the opinion that this correspondence could have

a signiﬁcant impact on current ideas regarding the formation

of the universe.

Let us brieﬂy present our reasoning: Fusion plasmas are

turbulent, whereas density ﬂuctuations on cosmological scales

are not. However, the cosmological ﬂuctuations might be what

has been dubbed ”fossilized turbulence” [2], [3], i.e. static

images of primordial turbulence. This original hot big bang

turbulence is in our picture represented by fusion plasma

turbulence. So the emerging understanding is as follows: (i)

turbulence was generated before the inﬂationary expansion of

the universe, (ii) as the universe cooled and expanded, the

primordial turbulence fossilized and is visible on cosmological

scales today. The theoretical basis of this hypothesis is outlined

in Refs. [4], [5].

We show in this paper that both sets of measurements ﬁt the

shape expected from 2D ﬂuid turbulence theory. According to

our interpretation, this implies that early turbulence was 2D.

The fusion plasma measurements presented in this paper are

of ﬂuctuations in the electron density. Phase-contrast imaging

(PCI) [6] is being used in the Alcator C-Mod tokamak [7] and

small-angle collective scattering (SACS) [8] was used in the

Wendelstein 7-AS (W7-AS) stellarator [9].

We speciﬁcally study density ﬂuctuation power P versus

wavenumber k (also known as the wavenumber spectrum) in

C-Mod and W7-AS. These wavenumber spectra characterize

Manuscript submitted August 23, 2007. This work was supported by the

U.S. Department of Energy, Ofﬁce of Fusion Energy Sciences.

N. P. Basse was with the Plasma Science and Fusion Center, Massachusetts

Institute of Technology, Cambridge, MA-02139, USA. He is now with ABB

Switzerland Ltd., Corporate Research, Segelhofstrasse 1, CH-5405 Baden-

D¨attwil, Switzerland (e-mail: nils.basse@ch.abb.com).

the nonlinear interaction between turbulent modes having dif-

ferent length scales. Our explicit assumption is that turbulence

in stellarators and tokamaks is comparable.

The second part of our measurements, a cosmological

wavenumber spectrum constructed from a variety of sources,

has been published in Ref. [10] and was subsequently made

available to us [11]. The measurements were used to constrain

cosmological variables, e.g. the matter density Ω

m

and neu-

trino masses - for further details see Refs. [10], [12].

The paper is organized as follows: In Sec. II we analyze

fusion plasma and cosmological wavenumber spectra. There-

after we treat the dimensionality of the measurements in Sec.

III. We discuss the hot big bang turbulence theory in Sec. IV

and conclude in Sec. V.

II. WAVENUMBER SPECTRA

We begin by studying the fusion plasma wavenumber spec-

trum shown in Fig. 1. The plot shows PCI measurements along

with a ﬁt to

P (kρ

s

) ∝ (kρ

s

)

−m

, (1)

where ρ

s

is the ion Larmor radius at the electron temperature

and m is a constant. The measurements were made in a

low conﬁnement mode C-Mod plasma, see Fig. 11 in Ref.

[13]. The wavenumbers measured have been multiplied by ρ

s

,

which for this case is 0.6 mm. This is the value at 80 % of

the plasma radius where the electron temperature is 400 eV,

the toroidal magnetic ﬁeld is 6.4 T and the working gas is

Deuterium.

Our ﬁt to the indicated PCI data yields m = 1.0 ± 0.03.

All ﬁts shown in this paper have a normalized χ

2

≤ 1,

ensuring a satisfactory quality. The error bars are standard

deviations and the semi-transparent rectangles indicate which

points are included to make the ﬁts.

In Fig. 2 we show SACS measurements at somewhat larger

wavenumbers compared to the PCI data. Again, the measured

wavenumbers have been multiplied by ρ

s

, which in this case

is 1 mm. This value is also at 80 % of the plasma radius where

the electron temperature is 300 eV, the toroidal magnetic ﬁeld

is 2.5 T and the working gas is Hydrogen.

The SACS measurements are ﬁtted to

P (kρ

s

) ∝

(kρ

s

)

−p

1 + (kρ

s

/(kρ

s

)

0

)

q

, (2)

where p = 2.8 ± 0.6 and q = 5.7 ± 1.3 are constants. The

functional form in (2) is taken from Ref. [15]. Basically this

equation describes two power-laws, where P ∝ (kρ

s

)

−p

=

(kρ

s

)

−2.8

for medium wavenumbers and P ∝ (kρ

s

)

−p−q

=

IEEE TRANSACTIONS ON PLASMA SCIENCE 2

Fig. 1. Wavenumber spectrum of broadband turbulence in C-Mod. Squares are measured points. The dashed line is a ﬁt to (1); the semi-transparent rectangle

indicates which points are included to make the ﬁt. The measurements are taken from Fig. 11 in Ref. [13].

Fig. 2. Wavenumber spectrum of broadband turbulence in W7-AS. Squares are measured points. The dashed line is a ﬁt to (2); all points are included to

make the ﬁt. The measurements are taken from Fig. 12 in Ref. [14].

(kρ

s

)

−8.5

for large wavenumbers. The transitional (kρ

s

)

0

is

in our case 3.7. The W7-AS data have been taken from Fig.

12 in Ref. [14].

It is at this point relevant to note that the medium wavenum-

ber fusion plasma exponent is not always three (or 2.8), it

typically varies between three and four depending on speciﬁc

plasma conditions [16], [17], [18]. Presumably this is due to

different instabilities driving turbulence for varying operating

conditions, leading to forcing centered at changing scales.

The cosmological wavenumber spectrum is shown in Fig.

3. The measurements are ﬁtted to (2), but using k instead

of kρ

s

; in this case, p = 1.2 ± 0.1 and q = 1.4 ± 0.05 are

constants. Here, P ∝ k

−p

= k

−1.2

for small wavenumbers

and P ∝ k

−p−q

= k

−2.6

for medium wavenumbers. The

transitional wavenumber k

0

is 0.3 h Mpc

−1

. Here, h =

H

0

/(100 km/s/Mpc) ≃ 0.7, where H

0

is the Hubble parameter

observed today.

IEEE TRANSACTIONS ON PLASMA SCIENCE 3

Fig. 3. Wavenumber spectrum of the combined cosmological measurements. Squares are measured points. The dashed line is a ﬁt to (2); the semi-transparent

rectangle indicates which points are included to make the ﬁt. The measurements are taken from Fig. 38 in Ref. [10].

III. DIMENSIONALITY OF THE MEASURED FLUCTUATIONS

We begin Sec. III by summarizing our ﬁndings on the

dependencies of power on wavenumber in Sec. II:

Small wavenumbers :

P (k) ∝ k

−1.0

(fusion) or P (k) ∝ k

−1.2

(cosmology).

Medium wavenumbers :

P (k) ∝ k

−2.8

(fusion) or P (k) ∝ k

−2.6

(cosmology).

Large wavenumbers :

P (k) ∝ k

−8.5

(fusion). (3)

Our measured density ﬂuctuation power is equivalent to the

d-dimensional energy spectrum F

d

(k) [19], [20], [21]

P (k) = F

d

(k) =

E(k)

A

d

A

1

= 2 A

2

= 2πk A

3

= 4πk

2

, (4)

where A

d

is the surface area of a sphere having radius k and

dimension d.

We can convert our results in (3) either under the 2D

turbulence assumption:

Small wavenumbers :

E(k) ∝ k

0.0

(fusion) or E(k) ∝ k

−0.2

(cosmology).

Medium wavenumbers :

E(k) ∝ k

−1.8

(fusion) or E(k) ∝ k

−1.6

(cosmology).

Large wavenumbers :

E(k) ∝ k

−7.5

(fusion). (5)

or under the 3D turbulence assumption:

Small wavenumbers :

E(k) ∝ k

1.0

(fusion) or E(k) ∝ k

0.8

(cosmology).

Medium wavenumbers :

E(k) ∝ k

−0.8

(fusion) or E(k) ∝ k

−0.6

(cosmology).

Large wavenumbers :

E(k) ∝ k

−6.5

(fusion). (6)

The established picture of 2D ﬂuid turbulence is: (i) tur-

bulence is forced on an intermediate scale k

f(2D)

, (ii) energy

is transferred to larger scales by the inverse energy cascade,

E(k) ∝ k

−5/3

[22], and enstrophy is transferred to smaller

scales by the forward enstrophy cascade, E(k) ∝ k

−3

[23],

and (iii) enstrophy is dissipated at the smallest scales [24].

For 3D turbulence the following process occurs: (i) turbu-

lence is forced on a large scale k

f(3D)

, (ii) energy is transferred

to smaller scales by the forward energy cascade, E(k) ∝

k

−5/3

, and (iii) energy is dissipated at the smallest scales.

It is interesting to note that in Ref. [25], two dependencies

of the 3D energy spectrum on wavenumber in the dissipation

range are considered: One is an exponential falloff, the other

claims that E(k) ∝ k

−7

and was proposed by W. Heisenberg.

This power-law is quite close to the one we found for fusion

plasmas at large wavenumbers.

To determine whether 2D or 3D turbulence is observed,

we consider the power-laws for medium wavenumbers: The

exponents should roughly be in the range [-3, -5/3] for 2D

turbulence and about -5/3 for 3D turbulence. Equations (5)

and (6) indicate that the observed 2D slopes are close to the

expected power-laws and that the 3D slopes are too shallow.

We note that the 2D slopes are closer to the value for the

IEEE TRANSACTIONS ON PLASMA SCIENCE 4

inverse energy cascade than for the forward enstrophy cascade.

The reason for this is not understood.

Turbulence in fusion plasmas is approximately 2D, since

transport along magnetic ﬁeld lines is nearly instantaneous.

For this reason, ﬂuctuations are measured parallel to the major

radius of the machine, i.e. perpendicular to the conﬁning

magnetic ﬁeld.

The reason we chose to analyze fusion plasma data was sim-

ply a matter of having available measurements and expertise

in that ﬁeld. Any turbulent 2D plasma should display similar

characteristics.

IV. HOT BIG BANG TURBULENCE

In Ref. [1] we suggested that the observed plasma turbu-

lence might originate during an early phase in the formation

of the universe. Recent theoretical work on the role of hot big

bang turbulence in the primordial universe [5] lends support

to this assumption:

In this theory, turbulence observed today was created before

cosmological inﬂation by inertial-vortex forces leading to an

inverse big-bang turbulence cascade with a -5/3 power-law

exponent. Turbulence in the plasma epoch has low Reynold’s

numbers ∼ 10

2

according to this picture and the preceding

quark-gluon plasma has a large gluon viscosity that acts to

damp the big bang turbulence. The claim is that hot big bang

temperature turbulence is fossilized before the universe cools

to the Grand Uniﬁed Theory strong force freeze-out tempera-

ture 10

28

K. Later, during nucleosynthesis, fossil temperature

turbulence was converted to fossil turbulence patterns in e.g.

density turbulence [4].

Analysis of power spectra of cosmic microwave background

radiation temperature anisotropies shows that they most likely

have a turbulent origin, supporting the idea of turbulence

generated in the big bang or the plasma epoch [26].

V. CONCLUSIONS

The fact that density ﬂuctuations on small (fusion plasma)

and large (cosmological) scales can be described by similar

functional dependencies, approximately consistent with 2D

ﬂuid turbulence, might indicate that (plasma) turbulence at

early times has been fossilized and expanded to cosmological

proportions.

Our conjecture concerning the primordial turbulence re-

ﬂected in our wavenumber spectra can be described as follows:

Forcing occurs at an intermediate scale k

f(2D)

. The inverse

energy cascade leads to spectral condensation at large scales

and the forward enstrophy cascade leads to enstrophy transfer

towards smaller scales. At very small scales, enstrophy is

dissipated.

Our interpretation of the measured wavenumber spectra is

consistent with the theoretical framework on hot big bang

turbulence presented in Ref. [5] and references therein.

ACKNOWLEDGMENT

This work was supported at MIT by the Department of En-

ergy, Cooperative Grant No. DE-FC02-99ER54512. We thank

M. Tegmark for providing the cosmological measurements

analyzed in this paper.

REFERENCES

[1] N.P. Basse, ”Density ﬂuctuations on mm and Mpc scales,” Phys. Lett. A,

vol. 340, p. 456, 2005.

[2] G. Gamow, ”On the formation of protogalaxies in the turbulent primordial

gas,” Proc. Natl. Acad. Sci., vol. 40, p. 480, 1954.

[3] C.H. Gibson, ”Fossil turbulence revisited,” J. Marine Systems, vol. 21, p.

147, 1999.

[4] C.H. Gibson, ”The ﬁrst turbulence and ﬁrst fossil turbulence,” Flow, Turb.

Combust., vol. 72, p. 161, 2004.

[5] C.H. Gibson, ”The ﬁrst turbulent combustion,” Combust. Sci. Tech., vol.

177, p. 1049, 2005.

[6] A. Mazurenko et al., ”Experimental and theoretical study of quasicoherent

ﬂuctuations in enhanced D

α

plasmas in the Alcator C-Mod tokamak,”

Phys. Rev. Lett., vol. 89, p. 225004, 2002.

[7] I.H. Hutchinson et al., ”First results from Alcator C-Mod,” Phys. Plasmas,

vol. 1, p. 1511, 1994.

[8] M. Saffman et al., ”CO

2

laser based two-volume collective scattering

instrument for spatially localized turbulence measurements,” Rev. Sci.

Instrum., vol. 72, p. 2579, 2001.

[9] H. Renner et al., ”Initial operation of the Wendelstein 7AS advanced

stellarator,” Plasma Phys. Control. Fusion, vol. 31, p. 1579, 1989.

[10] M. Tegmark et al., ”The three-dimensional power spectrum of galaxies

from the Sloan digital sky survey,” Astrophys. J., vol. 606, p. 702, 2004.

[11] M. Tegmark, private communication, 2005.

[12] M. Tegmark et al., ”Cosmological parameters from SDSS and WMAP,”

Phys. Rev. D, vol. 69, p. 103501, 2004.

[13] N.P. Basse et al., ”Characterization of core and edge turbulence in L-

and enhanced D

α

H-mode Alcator C-Mod plasmas,” Phys. Plasmas, vol.

12, p. 052512, 2005.

[14] N.P. Basse et al., ”Low- and high-mode separation of short wavelength

turbulence in dithering Wendelstein 7-AS plasmas,” Phys. Plasmas, vol.

9, p. 3035, 2002.

[15] T. Padmanabhan and S. Ray, ”Power transfer in nonlinear gravitational

clustering and asymptotic universality,” astro-ph/0511596.

[16] C. Honor´e et al., ”Small scale density ﬂuctuations in Tore Supra: Rupture

in the scaling law,” Proceedings of the 25th EPS Conference on Controlled

Fusion and Plasma Physics, Prague, European Physical Society, Petit-

Lancy, Switzerland, vol. 22C, p. 647, 1998.

[17] S. Zoletnik et al., ”Changes in density ﬂuctuations associated with

conﬁnement transitions close to a rational edge rotational transform in

the W7-AS stellarator,” Plasma Phys. Control. Fusion, vol. 44, p. 1581,

2002.

[18] P. Hennequin et al., ”Scaling laws of density ﬂuctuations at high-k on

Tore Supra,” Plasma Phys. Control. Fusion, vol. 46, p. B121, 2004.

[19] H. Tennekes and J.L. Lumley, ”A First Course in Turbulence,” MIT

Press, Cambridge, 1972.

[20] U. Frisch, ”Turbulence,” Cambridge Univ. Press, Cambridge, UK, 1995.

[21] G. Antar, Ph.D. Thesis,

´

Ecole Polytechnique, 1996.

[22] S. Chen et al., ”Physical mechanism of the two-dimensional inverse

energy cascade,” Phys. Rev. Lett., vol. 96, p. 084502, 2006.

[23] S. Chen et al., ”Physical mechanism of the two-dimensional enstrophy

cascade,” Phys. Rev. Lett., vol. 91, p. 214501, 2003.

[24] S. Chen et al., ”Far-dissipation range of turbulence,” Phys. Rev. Lett.,

vol. 70, p. 3051, 1993.

[25] J. von Neumann, ”Recent theories of turbulence,” in: A.H. Taub (Ed.),

Collected Works VI: Theory of Games, Astrophysics, Hydrodynamics and

Meteorology, Pergamon Press, Oxford, 1963.

[26] A. Bershadskii and K.R. Sreenivasan, ”Extended self-similarity of the

small-scale cosmic microwave background anisotropy,” Phys. Lett. A, vol.

319, p. 21, 2003.

Nils P. Basse received the B.Sc., M.Sc., and Ph.D.

degrees from the Niels Bohr Institute, University

of Copenhagen, Copenhagen, Denmark, in 1996,

1998, and 2002, respectively. He is a Scientist

with ABB Switzerland Ltd., Corporate Research.

He was a Postdoctoral Associate at the Plasma

Science and Fusion Center, Massachusetts Institute

of Technology (MIT), Cambridge, from 2002 to

2005. His present research interests include plasmas

in medium- and high-voltage circuit breakers.