Free and layer turbulent percolation: topological instabilities and their suppression

HAL Id: jpa-00246971

https://hal.archives-ouvertes.fr/jpa-00246971

Submitted on 1 Jan 1994

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Free and layer turbulent percolation: topological instabilities and their suppression

A. Bershadskii, H. Branover

To cite this version:

A. Bershadskii, H. Branover. Free and layer turbulent percolation: topological instabilities and their suppression. Journal de Physique I, EDP Sciences, 1994, 4 (8), pp.1115-1120. �10.1051/jp1:1994241�.

�jpa-00246971�

Classification

Physics Abstracts

05.40 47.25C

Short Communication

Free and layer turbulent percolation: topological instabilities and their suppression

A. Bershadskii (~) ^{and H. Branover}

(2)

(~)P.O.Box

_{39953, Ramat-Aviv 61398, Tel-Aviv, Israel}

(~)Center

for MHD-Studies, Ben-GurionUniversity, Beer-Sheva, Israel

(Received

25 April 1994, revised 26 May 1994, accepted 7 June

1994)

Abstract. It is shown

(theoretically

and

experimentally)

that topological instabilities lead to free percolation of passive scalar in _quasi two-dimensional turbulence that is characterized

by trie three-dimensional value of trie cntical exponent _v = o.9 and by spectral exponent

"-4/3".

Suppression of these mstabilities transforms trie percolation to layer-type _process with

v =

4/3

and spectral exponent

"-7/3".

In trie last _case fractal dimension of trie passive scalar cluster equals

9/4

and fractal dimension of its perimeter equals

7/4 (1e,

is trie _{same as} fractal dimension of the huit of strictly two-dimensional percolation cluster). A good correspondence

is found between trie spectral and trie fractal scahng laws and the atmosphenc, numerical and

laboratory experimental data.

1. Introduction.

In

iii, (see

also

[2-5])

a

simple

model

analyzing

trie

properties

of fast components ^{of turbulent} diffusion _is

proposed.

These fast components _are related to critical

phenomena

_m turbulence and

relationship

v DP

"

~

Il)

3 _-1

connecting

critical exponent v, fractal dimension of trie passive scalar duster DP, and

passive

scalar

spectral

exponent _i

,

is obtained.

Using

trie

topological

condition

iii

mm[Da

_or

Dpi

= Du + DP d

(2)

(where

Du is the fractal dimension of

passive

scalar

surface,

[...] means the integer part ^of a

number,

d is the

topological

dimension of the

space)

and the Vassilicos

relationship

_[6,

ii

3-1=Dp (3)

ll16 JOURNAL DE PHYSIQUE I N°8

(where Dp

₌ Du i _is fractal diInension of

perimeter

of the

passive

scalar

duster)

the two

conjugated regimes

for quasi two-dimensional turbulence have been obtained in

iii il~~

=

5/3, D(~

₌

5/3, _D)~)

=

4/3, (4)

and

il~~

₌

4/3,

_D@~

=

4/3,

_D)~~

=

5/3. (5)

In this situation the critical exponent _{v appears} to be close to its universal value in the three- dimensional

percolation iv

_c~

0.9)

rather than to the _one for two-dimensional _case

iv

=

4/3)

181.

The main

goal

of this note is to show that two-dimensional-like turbulent

percolation (with

v =

4/3)

is also realized in the

quasi-two-dimensional

turbulence. For

understanding phys-

ical

(topological)

diiferences of these two-types of

quasi-two-dimensional

^turbulence _we wfll

use an idea about two diiferent types ^of instabilities of the two-dimensional turbulence in the three-dimensional _space: with and without spontaneous

helicity input [9-1Ii (it

is useful to _re- member that in the

strictly

two-dimensional turbulence

helîcîty equals

_zero and this is _a strong

topological

restriction of two-dimensional motion

[12]).

This attempt ^is also

supported by

_a

laboratory experiment

_as well _as

by

comparison with

geophysical

observations and numerical simulation.

2. Free

quasi

two-dimensional turbulence.

In [9] the

topological

mstabilities of two-dimensional turbulence in three-dimensional _space have been

suggested

_as a basis for

understanding properties

of

quasi

two-dimensional turbu- lence. The instabilities will

primarily

manifest themselves

by exciting

helical

travelling

waves

which bend the two-dimensional motion

planes. Input

of

helicity

into these _{waves is} the _main

physical (topological)

_process for this

phenomenon

and the parameter

((dh/dt() replaces

the

Kolmogorov

parameter

(si

=

((du2 /dt()

_as

governing

parameter ⁱⁿ a

corresponding

interval of scales

(cf.

also with [10,

12]).

In the classic

theory

of Corrsin-Obuchov [13]

scaling

spectrum of passive scalar concentration, _c, has the form

E~

_cc

(Ni (e)~~/~ k~~/~ (6)

N

= (dc~

/dt(),

while in the free quasi two-dimensional turbulence the

scaling spectral

law for E~ _is

f~ _~ j~rj

jj~~jj-i13 ^~-4/3 _j~)

~ àÎ

It is clear that the

percolation regime (5),(7)

is controlled

by

these three-dimensional

topological

mstabilities and three-dimensional value of _{u ci} 0.9

(see Introduction)

_can be understood _as

a consequence of this fact.

For

experimental

verification of the

itniuersality

of this

phenomenon

for _quasi two-dimensional turbulence

iii

_we used _a

laboratory quasi

two-dimensional turbulence, created in flow of _mer- cury m externat

(transversal) magnetic

field

Ill,

_14]

(see

also

[15]).

^Detailed

description

of the

experimental

installation used _can be found in [14]. ^{Since the} mercury is a Weil electri-

cally conducting fluid,

^externat

Inagnetic

field transforms the

original (without magnetic field)

three-dimensional turbulence mto quasi two-diInensional _une [9, II,

14].

^The temperature ^has been used _{as a} passive scalar in _Dur experiInent. The teInperature spectrum, E~, obtained in the flow without

Inagnetic

field is shown in

figure

la while the teInperature spectrum ^obtained

m the flow with externat

magnetic

field

(E

_ci 0.2

T)

is shown in

figure

16. The

straight

fines

are drawn for _coInpanson with

scahng

laws

(6)

^and

(7) (cf,

also

iii).

5

4/3

-6

5/3

_~

bo ù4

ÎÎ

°

-5/3

-7 ~

a b

~~

0 1 2

~ô _{1 2}

log

^k

_log

k

Fig. l.

a)

Spectrum of passive scalar

(temperature)

fluctuations

m laboratory flow of _mercury (Re _m 5). b) Temperature spectrum ^{for trie} same flow

as in figure 3a _m externat magnetic field Em 0.2 T.

3.

Layer

turbulence.

Another

approach

has been

suggested

in the paper

Ill].

The

travelling

_waves, which bend the

original

two-dimensional motion

planes,

will

bring

about fluctuations of turbulent energy

dissipation

_e. In the

corresponding

interval of scales average ^rate of _{space e}

fluctuations, ((de/dz() (where

_z is the coordinate

perpendicular

to two-diInensional turbulence

plane) replaces (si

_as the _governmg paraIneter

Ill].

Dilnensions of the both

governing

paraIneters

((dh/dt()

and

((de/dz()

_are the _saine, and kinetic energy spectrum ^{obtained froIn} diInensional considerations [9,

iii

Eu _{« Il}

()ll~/~ll)ll~/~ ^k-~/~

18)

has the _saine exponent

'-7/3'

in both these _cases.

However, these situations _are

essentially

diiferent for _passive scalar fluctuations. In the last

case

spectral

law

(7)

must be

replaced by Ec

_{« Il}

()

Il

11)11-~/~ ^k-~/~

¹⁹⁾

because of the _space fluctuations of N. This _quasi two-diInensional turbulence has _a

layer

nature and the Vassflicos considerations [6,

ii

_as well _as his

relationship (3)

_are not valid in

the _case. For the

layer

turbulence _we should _use the two-diInensional value of _v

=

4/3

_[8] to

ll18 JOURNAL DE PHYSIQUE I N°8

4800 4so wavelength km

ltÎ

Éi -7/3

6

s

A

B 4

-6 -5 4

log

^k

Fig. 2. Temperature ^{and kinetic} energy spectra ^from GASP flights _m stratosphere: A-kinetic energy, B-temperature. Adapted from [16].

obtain the fractal dimensions

(see

also

Discussion).

Then from

(1)

^and

(9)

_we ^{obtain for}

layer

turbulence:

DP #

9/4 (10)

and from the

topological

condition

(2)

Dp

₌

7/4. (Il)

Figure

2 shows one of the spectra of the temperature

(and energy)

fluctuations observed in

stratosphere

_[16] and follows

(9) (and (8)).

On the other hand, the nu1nericaI simulation of Moud forInation

by

_a cellular automaton mortel

(which

tutus out to have a strong resemblance tu

percolation-based

_grows

mortels)

_{[4] can} be used to obtain _a fractal information _on the

layer

turbulence. In

figure

3

(taken

froIn [4]) ^the crossover froIn

isotropic growth

of Mouds _{to re-} stricted

growth

in _a

layer

is shown. The dotted fine

gives Dp

=

4/3

at sInall scales while the solid fine

gives Dp

=

7/4

at

large

scales. The effective dimension of trie

large-scale

Moud penmeter ^obtained _{in [4]} ^for

layer Dp

₌ 1.74 + 0.05

(cf. iii

'

.~

DP " 7/4

«,

j %.~

~

. _É~ ~ 7,/3

1 ₃

0gP ~~

Fig. ³

-8

-9

o

Iogk

Fig. 4

Fig. 3. The _crossover from isotropic cloua growth ta restricted growth in _a layer (cf. with

iii).

Adapted tram [4].

Fig. 4. Temperature spectrum ^{for trie} _same flow _{as m} figures la, b in trie strong magnetic field B _m 1 T

(strong restriction).

4. Discussion.

The differences in the

properties

of free and of

layer

turbulent

percolation

canner be understood without

taking

into account the differences in their

topology.

For three-dimensional

percolation clusters,

due _to the mttlticonnected

topology

of their externat

boundary,

it _was

argued

in [8, 17]

thon the fractal dimensions of trie cluster and of ils externat httll _are the _same. On the other

nana,

for two-dimensional

percolation

cluster these fractal dimensions _are different. In the last

case the fractal dimension of the huit Dh _" I + I

lu [18],

^{1-e for} u =

4/3 Dh

"

7/4 Thus,

the

topological

restriction in

layer (with Dv

<

9/4)

_suppresses the

topological

instabilities

(Sect. 2)

^{and leads} to coincidence of

Dp

^for

layer percolation

and

Dh

for two-dimensional

percolation.

Following

this

logic,

_{one may} expect that

increasing

of the externat

magnetic

^field

(the topological

restriction in _our

experiment

with _{mercury, sec}

above),

should lead _to transition from spectrum

(7) (Fig. lb,

B _ci 0.2

T)

to spectrum

(9). Indeed, figure

4 shows trie spectrum of temperature fluctuations ai strong

magnetic

field

(B

_ce I

T)

in trie

experiment

and

straight

fine is drawn for

comparison

with

(9).

Thus, we believe that trie

paradox

of trie three-dimensional behavior of

quasi-two-dimensional

turbulent

percolation [ii

bas _a

topological

solution. This solution is

provided by

existence

i120 JOURNAL DE PHYSIQUE I N°8

of _two types ^of instabilities of two-dimensional turbulence in trie friree-d1nlensional _space:

with and without spontaneous

helicity input,

1-e- with and without trie

change

of the motion

topology.

The

layer-restriction

is _a necessary condition for the second

(two-dimensional-like)

type of

quasi-two-dimensional

^turbulent

percolation.

The spectrum and the fractal dimensions

obtained in this _case is diflerent from the spectrum and the fractal dimensions of the free

quasi-two-dimensional

turbulent

percolation

obtained in the

[Ii.

Finally,

ii should be noted that _we have net taken into account the multifractal

properties

of turbulent

percolation

_as ^weII _as ils relations with other types of critical

phenomena (see,

for instance, [8, 19,

20]).

It is aise

possible

that the

layered percolation

_{cari appear} in the three-dimensional turbulent diffusion toc

(see

result of _a recent experiment

[21]).

Acknowledgments.

Authors _are

grateful

to A.J.

Chorin,

K-R-

Sreenivasan,

D.

Staufler,

and J-C- Vassilicos for information, comments and encouragement, ^and to A. Eidelman and M. Kireev for

help

in the

experiment.

ILeferences

iii

Bershadskii A., Physica A 206

(1994)

120.

[2] Bershadskii A., Sov. Phys. Usp. 33

(1990)

1073.

[3] Chorin A.J., J. Star. Phys. 69

(1992)

67.

[4] Nagel K. and Raschke E., Physica A 182

(1992)

519.

[Si ^{Schneider M.} and Wohlke T., Physica A 189

(1992)

1.

[6] Vassilicos J.C., ^{Advances in} turbulence, 2, H-H- Femholds and H-E- Fiedler Eds.

(Spnnger,

Berlin,

1989)

_p. 404.

[7] ^Vassilicos J-K- and Hunt J-C-R-, Froc. Roy Soc. A 435

(1991)

505.

[8] ^{Staufser D.} and Aharony A., Introduction _ta Percolation Theory

(Taylor

and Francis, London

1992).

[9] Bershadskii A., Kit E. and Tsinober A., Froc. Roy. Soc. A 441

(1993)

147.

[10] Bershadskii A. and Tsinober A., Phys. Rev. E 48

(1993)

282.

[Il]

Branover H., Bershadskii A., Eidelman A. and Nagorny M., Boundary-Layer Meteorology 62

(1993) l17.

[12] ^{Mofsatt H-K and Tsinober} A., Annu. Rev. Fluid Mech. ₂₄

(1992)

281.

[13] ^{Monin A.S. and} Yaglom A.M., Statistical Fluid Mechanics 2

(MIT

Press, Cambridge,

1975).

[14] ^{Branover H. and} Sukoryansky S., Progr, _m Astro. and Aeronauticsl12

(1988)

87.

[15] Sreenivasan K-R-, Froc. Roy. Soc. A 434

(1991)

165.

[16] Gage K-S- and Nastrom C.D., J. Atm. Sci. 43 (1986) 729.

[17] ^Strenski P-N-, Bradley R-M- and Debierre J-M-, Phys. Rev. Lent. 66

(1991)

1330.

[18] Saleur H. and Duplantier B., Phys. Rev. Lent. 58

(1987)

2325.

[19] Nagatani T. and Stanley H-E-, J. Phys. Soc. Jpn 60

(1991)

1217.

[20] ^Sahimi M., Applications of percolation theory

(Taylor

and Francis, London, 1994).

[21] ^Wu X.-Z., Kadanofs L., Libracher A. and Sano M., Phys. Rev. Lent. 64

(1990)

2140.