Singular instabilities (collapses) and multifractality of structure functions in turbulence

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Singular instabilities (collapses) and multifractality of structure functions in turbulence

A. Bershadskii

To cite this version:

A. Bershadskii. Singular instabilities (collapses) and multifractality of structure functions in tur- bulence. Journal de Physique II, EDP Sciences, 1994, 4 (3), pp.413-418. �10.1051/jp2:1994136�.

�jpa-00247970�

Classification

Physics Abstracts 47.27

Short Communication

Singular instabilities (collapses) and multilbactality of structure functions in turbulence

A. Bershadskii

Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel.

(Received

4 November 1993, received in final 27 December 1993, accepted 17 January

1994)

Abstract. It is argued that the multifractality of higher order structure functions arises _{as a} result of local singular instabilities

(collapses)

in 3D-turbulence.The modified Landau approach

(with poles)

is used for studying supercritical instability in this _case. For the simplest situation

(pole

of order

two),

the asymptotic behavior of the structure functions is investigated. The arguments _are supported by comparison with experimental data from laboratory and geophysical experiments.

1. Introduction.

The internal intermittency of turbulent flow leads _to the existence of

subregions

with self-

generation

of turbulence. In the

subregions quasi-laminar

motion becomes unstable and _goes to chaos

by

_{one or} another scenario. The

properties

of this chaos

depend

_on the type ^of

scenario. This

phenomenon

is _one of the _sources of multifractal behavior of turbulence [1, 2].

The well-known Landau

approach

_[3] consists in the

description

of turbulence _near the critical

Reynolds

number with

taking

into account the most unstable mode with

complex frequency

uJ = WI + (al

/2,

here _WI » al Fluctuations of the

velocity

field _are

decomposed

as

~1 =

A(t)9(z, t)

(11

with

amplitude

A(t)

+~

exp(art/2 (wit) (2)

When _al > 0,

representation (2)

is valid

only

for small values of t. To

analyze

the nonlinear

stability problem

Landau

proposed

the

following amplitude equation

dlAl~

~

f

~

j~j2»

_~~j

d t

~_~

"

414 JOURNAL DE PHYSIQUE II N°3

under the

assumption

^{that the} _{average over} the time intervals smaller than

ap~

and

larger

than uJp~ was taken. The

simplest

nonlinear

situation,

for small (A(,

gives

_us

with

a

nontrivial

esult

(~(~ _~

(5)

Here al > 0 and

02 <

₀

chastic

media and ewrite

the

plitude

/

=

I...I + f(tl, (6j

where [...] denotes the Landau _expansion in

powers A and A*, _{(* is} the

complex

f(t) is the external

stochastic

force rom the _turbulent

environment. Let us

assume

correlation ^radius

of

external"

(turbulent)

forces is

_smaller

than

_typical times of dynamical

then (in

the

_first

approximation)

mean

value

equal

to

( f(t)

*(t'))

= 2a~&(t

- t')

(7)

We define

a

probability ^ensity

Pt

=

(&(lAl~ -1)). (8)

After

averaging

over the

quation

for

~~~ = -

t

I

^d I

Pm _{c~ exp}

~°~~

~

f~~~ 8(1), (10)

2a

where

8(1)

is the Heaviside function.

The maxima of P~x>(I)

correspond

to the local stable states while the minima to unstable

ones

(see

_{[4] and}

[5]). Figure1 (continuous line)

demonstrates the schematic

dependence

of Pm

on I. The

original

state, ¹ ₌ 0, is

statistically

unstable whereas the second state, Ii = -ai

/02,

stable

(cf.

with

dynamical

result

(5)). Similarly,

_{one can} obtain the Fokker-Plank equation and its

stationary

solution for the

general

_case, equation

(3).

2.

Singular

instabilities

(collapses).

It is known that the three-dimensional Euler equation ^{has solutions} which blow _up within

a finite time

(see,

for

example, [6-12]).

This

collapse

_can be described _{as a}

singular

_case of

supercritical instability.

Let _{us use} the

singular expansion

in

amplitude

_equation

(6) (and (3)),

instead of the

regular

one.The

simplest

nontrivial _case is the

expansion

with the pole of order two:

§

~ ~"~~ ~ ~

i~'

~~~~

p~(I)

: .,

:

",

. .

° °

./

o

lj

I

Fig.

1. Sketch of

Pm(I):

continuous line for regular supercritical Hopf bifurcation and dotes for

supercritical singular bifurcation

(collapse Pm(0)

=

0).

Indeed,

in the

vicinity

of1= 0, _we obtain

(cf.

with

regular

case

(10))

Pm _c~

exp[-

^~

(pI~~

₊

~tI~~/2)]8(1). (12)

Figure

1

(dotes)

shows the

dependence

of Pm _on I for

supercritical

situation

(p

_< 0 and

~t > 0

).

^{In this} _case, Pm

(0)

₌ 0, this

corresponds

to the

collapse

of the

original

solution.

In the

regular supercritical

_situation, Pm

(0) #

_0, and function Pm

(I)

has _a local minimum at point ¹ = 0 that corresponds to

instability

of the

original

solution

(1

₌

0).

This solution coexists with the stable

secondary

_one

(Ii

in the

regular supercritical

_case

(Fig.

1, continuous

line),

while in the

singular supercritical

situation _we have the

secondary

solution

only (since Pm(0)

=

0).

This solution

(Ii

"

-'f/P)

is stable because

Pm(I)

has its local maximum at

I = Ii

(see

condition

(13)

). ^{If the}

poles

of the orders _more than 2 _are taken into account more

than _one

equilibrium

states emerge

(ao

= 0 because of Pm

(I) divergency).

3.

Higher

order structure functions and local

collapses.

The

equilibrium

_states, defined

above,

are different from the

Kolmogorov

_{one [13].} The

equi-

librium states are determined

by

the condition that the

right

hand side of

(3)

_or

(11) equals

to _zero. Here the term

d(A(2/dt plays

the role of the _mean input of the energy of fluctuations

(f),

but in the

Kolmogorov approach

it is _a

governing

parameter. In the

vicinity

of

equilibrium Ii,

_f leads to _zero which _means that the

Kolmogorov approach

does not work here. There are two parameters,

p

and _~t, which _can be candidates for the role of the

governing

parameter

in this _case. The condition for statistical

stability

of the

equilibrium

state Ii is

q

^"

~P~l'f~

_< °.

(13)

As _{one can see} from

(13),

statistical

stability

of the

equilibrium

state

Ii

does _not

depend

on the

sign

of parameter fl and is

strongly dependent

_on parameter _~t. I,e., parameter _~t controls statistical

stability

of

equilibrium

Ii For this _reason, it is reasonable to _assume that

~t is the

governing

parameter ^{for this}

equilibrium

state.

416 JOURNAL DE PHYSIQUE II N°3

tanaQ0l5

~ tuna"013

4 8 12 16 20

P

Fig.

2. Scaling exponent (p of structure functions:

(x)

jet (Re> ₌ 852, [14]) and

(.)

large wind tunnel in Modane, (Re> ₌ 2720, [15]).

In the

Kolmogorov approach,

dimensional considerations lead to estimate [3, 13]

A~I _c~

f~/~r~/~, (14)

where A~I

= ~1(z +

r)

_~1(z), whereas in the

vicinity

of

equilibrium

Ii the _saute considerations lead to

AI~ _«

~ti/7ri/7 (isj

(the

dimension of parameter _~t is obtained from

Eq. (11)

_~t c~

[L]~[T]~~).

Numerical

experiments

show that the

collapses

_are localized in the _space

[7-11].

^Thus

scaling (15),

unlike the

Kolmogorov

_one

(14),

_can be observed

only

in the local

subregions.

One _can

separate ^the

subregions

with the

help

of

higher

order structure functions.

Indeed,

the structure functions of order _{p are}

((A~I)~)

₌

~°S~jl'~~~, ⁽¹⁶⁾

where A~

corresponds

to the

subregion

^with number I

(and

scale

r),

while N is the total number of these

r-subregions. If,

in the

subregions

with

relatively large

values of A~I,

scaling (15)

takes

place,

then for

large

_p,

((A~)~l

_CC

~~~~/~, (17)

where d is the dimension of the space

(or

of the experimental

signal)

and N _c~ r~~ For usual

experimental signals

d

= 1

(see

below and

Fig. 2). By introducing

the exponent

(p

via the relation [1]

1(a~)Pi

_cc

^r(p

₍₁₈₁

and

using

^relation

(17),

it is

straightforward

to obtain the

following

_asymptotic representation

(p

QS d ₊

p/7. (19)

The values of

(p,

^{obtained from} two

laboratory

experiments [14] ^and [15]

(at

rather

large

Reynolds numbers),

are shown in

figure

2, ^where the

straight

lines _are drawn for comparison with

(19).

As _{we can see,} the

asyptotic slopes

of the

experimental

lines _are

approximately

equal

to

1/7,

and d _ci 1

(the signals

_are one-dimensional in these

experiments).

4. Discussion.

There is _a number of factors which prevent the

collapses.

The first _one

is, naturally,

the vis- cous

dissipation _([16]

and

[17]).

^The dimension of motion _can also be _one of these factors.

The conservation laws

prohibit

the

collapses

in

strictly

2D-turbulence. It leads to an interest-

ing phenomenon

in real

quasi-2D-turbulence (as

is

known, strictly

2D-turbulence is unstable to

3D-disturbances).

The

3D-instability

_can be realized _as the

regular topological instability (see,

for

example,

_[18]) and _as the

singular instability (the

local

collapse)

because the pro- hibitive 2D-conservation laws _are invalid for 3D-disturbances. These local 3D-instabilities lead to the appearance of blobs characterized

by

active

mixing.

If there _are conditions for _numer-

ous appearances of such

blobs,

then the

phenomenon

controls

large-scale

diffusion of passive scalar.

Then,

in full

analogy

^{with the}

Kolmogorov approach [13],

^{it follows from} dimensional

consideration that

diffusivity, K~,

has the

following

form

K~ ₌

~

c~ ~t~/~l~/~,

(20)

where is the characteristic scale of _a cloud of the

passive

scalar

(in

the

Kolmogorov

_case K~ _c~

f~/~l~/~

_{[13] ).}

_Expression (20)

is different from both the _case of 3D-turbulence

(Richardson- Kolmogorov law)

and from the _case of

purely

2D-turbulence.

I/rom

(20),

_we obtain

c~

t"~ (21)

Relations

(20)

and

(21)

are in

good

agreement ^{with the results of the}

experiments

_on diffusion of

passive

scalar in the

troposphere

_[19] and _ocean [20], as can be _seen from

figures

3 and 4.

10~

jnv t7/6

~

~

X

I K~~~~~

7

z IQ

E /

~

/

fi / u

~ /

~

~

/ ^E

~ u

o ^° /

Q / ^~

~

/

^/

/

joo j~4 1~6 j0~ _{10~ 10~}

t

,

travel time (sec

j,

cm

Fig.

3

Fig.

⁴

Fig.

3. Observations of widths

(horizontal

standart deviation) of diffusing tracer _{as a} function of downwind travel time. Different symbols correspond to the results of different authors. The figure is

adapted from [19].

Fig. 4 Diffusivity _versus scale in the _ocean. Adapted ^from [20].

418 JOURNAL DE PHYSIQUE II N°3

Acknowledgments.

The author is

grateful

to J.F.

Musy

for

sending

data from reference

[15],

^A.

Mikishev,

K.R.

Sreenivasan,

A. Tsinober and N. Tsitverblit for the useful information and encouragement, ^and Referee for comments.

This work _was in part

supported by

Wolfson Foundation _as well _as

by

the Israeli

Ministry

of Absorbtion.

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