Phase dynamics attractors in an extended cylindrical convective layer

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Phase dynamics attractors in an extended cylindrical

convective layer

A. Pocheau

To cite this version:

2059

Phase

_dynamics

attractors

in

an

extended

_cylindrical

convective

_layer

A. Pocheau

Laboratoire de Recherche en

_Combustion,

Université de Provence, Centre de Saint Jérôme,

S 252, 13397 Marseille, France

_(*)

(Reçu

le 20 mai

1988,

révisé le 13 novembre

_{1988, accepté}

le 19 avril

₁₉₈₉₎

Résumé. 2014 Nous étudions la

dynamique

de

_phase

des structures convectives

_après

extinction des

transitoires, en

_géométrie

_cylindrique

et à bas nombre de Prandtl. Nous observons de nombreux états

_{périodiques,}

stationnaires ou

_chaotiques.

Chacun d’eux est relatif à une fenêtre

_spécifique

en nombre de

_Rayleigh.

Les

_régimes

_périodiques

se différencient par des alternatives

_simples

_qui

portent sur le mouvement des défauts et _que nous attribuons aux instabilités des écoulements moyens. L’identification de ces instabilités

_d’après

les

_équations

de Cross-Newell nous _permet de reconstruire la route vers la turbulence.

Abstract. 2014 We

study

the

_{phase dynamics}

of convective structures after the

_decay

of transients, in

a

_cylindrical

container and at low Prandtl number. We observe numerous

_{stationary, periodic}

or

chaotic states. Each of them occurs at a

_specific

window in

_Rayleigh

number. The

_periodic

regimes

differ

_{by simple}

alternatives which concern defect motions and that we attribute to the

mean flow instabilities. The identification of these instabilities from the Cross-Newell

_equations

enables us to reconstruct the route to turbulence.

J.

_Phys.

France 50

₍₁₉₈₉₎

2059-2103 1er AOÛT _1989,

Classification

Physics

Abstracts 47.25 -

47.20

Introduction.

The

_{ergodic theory}

of

_dynamical

_systems

has determined various scenarios of transition to

deterministic chaos

_{[1, 2].}

These scenarios deal with the

_dynamics

in the

_vicinity

of the

_phase

space attractors i.e. a

_long

time after transients.

_They

relate to the different _ways

_by

which a

limit

_cycle

_{may destabilize}

_by

means other than an external noise and

they

involve few

effective modes.

_Ample

_experimental

evidence of these scenarios has been

obtained,

but

_only

in

_temporal

_systems

i.e. when the

_{spatial degrees}

of freedom are either frozen

_by

a

_spatial

quenching

or thermalized

_by

a

_{spatial mixing}

_[3].

_Accordingly,

a crucial

_step

in the

_study

of

dynamical

systems

would consist of

_improving

the

_{understanding}

of the mechanisms which

lead to chaos in

_{spatio-temporal}

_systems.

This _purpose has motivated a

_large

number of

studies,

both theoretical and

_{experimental,}

which have not

_yet

_given

a clear answer

[4].

The usual

_ergodic

_theory

is based on a

_{geometrical study}

of attractors in

_phase

_space. Within this

framework,

the modes of bifurcations _{may thus have} a clear mathematical

meaning

but have no direct

_physical

_{interpretation.}

This lack of contact with the

_physical

space is

likely

to increase the

_difficulty

of

_extending

the theories of

_dynamical

_systems

to

spatio-temporal

systems

and a

_fortiori

to

_{fully-developed}

turbulence. This statement has motivated us to

_study

in detail a

_{spatio-temporal}

convective

_system

and to

_understand,

both from an

_experimental

and a theoretical

_viewpoint,

how its

_spatial

features interfere with its

dynamics.

In _{this paper,} we

_report

observations of the various

dynamics

which arise in a

cylindrical

convective

_layer,

a

_long

time after transients have

_decayed.

The

_{periodic regimes}

are related to limit

_cycle

bifurcations which involve

_specific

defect motions.

_By

_analysing

the

_origin

of

gliding

motion,

we succeed in

identifying

the modes of bifurcations in terms of

_global

instabilities of the

_large

scale fields. The

_computation

of these instabilities within the framework of the convective

_pattern

_{theory finally}

leads us to reconstruct the route to

turbulence.

The paper is divided into six sections. In the first

section,

we _{survey the main visual}

observations of the transition to

_{time-dependence.}

In the next three

sections,

we

_report

our

observations of the route to turbulence with

_emphasis

on the

_length

of transients and on the

periodic regimes.

The last two sections are devoted to the

_analytical

identification of the modes of bifurcation and to the reconstruction of the route to turbulence.

1. Transition to

_{time-dependence}

at low Prandtl number.

Both

_experiments

and numerical studies of

_stability

show

that,

at Pr =

0.7, straight

rolls may

be stable and

_stationary

_up to e - =

1,5

[5-7],

where e =

(Ra -

Rac )/Rac

is the reduced

Rayleigh number,

Ra the

_Rayleigh

number and Rac its value at the onset of convection.

Surprisingly,

in a

_{cylindrical container,}

sustained

_{time-dependent}

states arise at values of e as

small as 0.13

[8-11].

Such a

_big

difference between onsets of

_{time-dependence,}

as

_large

as an

order of

_{magnitude depending}

on the roll

_pattern,

has been a

_puzzling

and

_stimulating

problem

for a

_long

time. Some advances have been achieved

_using

_pattern

visualization. In

particular,

visual observations have

_provided

a useful tool for

_pointing

out the

_strong

link

between

_spatial

distortion and

_{time-dependence}

_[8].

We summarize below the main results. In argon gas

(Pr

=

0.7),

below the onset of

time-dependence,

instead of

perfectly straight

rolls,

a

_slight

roll

_bending

is noticeable in a

_cylindrical

container

(Fig. la).

It results in two

opposite

centers of curvature of the

_phase

field

_(and

thus in two

_opposite

foci

₍₁₎₎

and in a

compression

of the central rolls. In the

_{stationary regime,}

the same

_stationary

but bent

pattern

is reached after

_transients,

_independent

of the initial condition. These

_equilibrium

patterns

therefore

_correspond

to attractors in

_phase

_space.

However,

as E

increases,

their

distortions

_(bending

and

_compression)

grow so

_that,

at the onset of

_{time-dependence,}

the

compression

becomes

_incompatible

with local

_equilibrium.

A roll

_pinching

then occurs at the

(1)

Since

bending

is often weak, the centers of curvature are

_usually

located outside the container.

The

_phase

field therefore

_displays

no

geometrical singularities.

However, its most curved

_points

will prove to be

_important

for the

_dynamical

behavior. These

points

are at the boundaries, on the line

joining

the curvature centers.

_{They correspond}

to an extremum of a

_geometrical

_property of the

_phase

field over the container, but not over an _open set around themselves. In

_spite

of this _restriction, we will

2061

a) t»

Fig.

1. -

a)

Stationary

slightly

bent roll _pattern at e = 0.12. Notice the

opposite

foci and the

slight

but

important

roll

_compression

at the center. We call these _patterns « basic distorted _{patterns »} and their distortions the basic distortions.

_b)

A roll

pinching

of the most

_compressed

rolls is

_occurring

at the pattern center for 8 = 0.13. This is the basic mechanism

giving

rise _to dislocation nucleation and then _to

time-dependence.

center of the

_container,

where the rolls are the shortest

_(Fig.1b).

This

_gives

rise to the nucleation of two dislocations. These defects then move towards the foci where

_they

disappear,

thus

_leading

to a defectless bent

_pattern

_again.

Since a roll

_pair

has been

_expelled

due to the elimination of the

_defects,

the central rolls are less constrained. One

_might

therefore

_expect

that a

_stationary

state could be reached. This is not the case however. The

foci feed the

_pattern

with new rolls so that the

_compression

increases

_again

until a new

pinching

occurs :

_{time-dependence}

is sustained.

Above the onset of

_{time-dependence,}

the Nusselt number measurements reveal that the

dynamics

is

_periodic

in a _{range of E}

[8-11].

Visual observations also show that the

_spatial

sequence described above is

repeated regularly

[9]

but we will see in section 4 that various

periodic regimes

may occur

_depending

on _{the way defects} move before elimination. The

pattern

state hence follows various limit

_cycles

in

_phase

_{space whose sequence}

_corresponds

to

the transition to turbulence. The

_experimental

_part

_{of this paper is devoted} to

_observing

these

cycles

[12].

Since,

due to the

_large

extent of the

_system,

_{spatial phenomena}

are

_expected

to

govern the

dynamics,

care will be taken of

accurately reporting

the

spatial

features.

In

_steady

_patterns

below the onset of

_{time-dependence}

as well as in the defectless

_phase

in

periodic regimes,

the main visible distortion

_(two

_{opposite bendings}

and a

_compression)

is such that all similar

_parts

of the

pattern

play

the same role. It thus

_displays

reflection

symmetries

with

respect

to the foci line and to the central roll line

_{(Fig. la).}

In the

_following,

we will refer to these distortions as the « basic distortions » and to the

_{corresponding}

_patterns

as the « basic distorted

patterns

». Basic distortions will prove to include the main

mechanisms of

_{time-dependence}

and will

_provide

us with a useful framework for

_analysing

the

2.

_{Expérimental procedure.}

We

_study

_{convection in argon gas and} we use the

_apparatus

described in

_[9].

Convective

patterns

are visualized

_by

_{shadowgraphy.}

In order both to enhance the contrast of the

convective

_picture

and to lower the

_temperature

difference AT across the fluid

_layer

at the onset of

convection,

the gas pressure P is increased to 40 bars. The fluid is heated from below

by

a resistive wire and cooled from above

_by

a water circulation

_surrounding

_{the upper}

sapphire

plate.

In order to measure the heat

_flux,

one must avoid radiative losses. This

condition is achieved

_{by surrounding}

both the

_heating

device and the cell

_by

a thermal shield.

The shield

_temperature

is

_regulated

within

10- 3

K and the

_temperature

of the inner device is

slaved to it within an _{accuracy of}

10- 5

K. At the threshold of

convection,

the

_temperature

difference AT is

_4Tc

= 5.1 K.

Finally,

the

temperature

fluctuations are estimated at 0.005 °C.

The Prandtl number Pr _{of argon gas is Pr} =

0.7,

nearly

independent

of the pressure

P.

_{When ATc}

is

_kept

constant,

the cell

_{depth d}

varies _{with the pressure}

_according

to

d3

oc

P - 2.

The vertical diffusion time r, =

d 2/ rc ,

where K is the thermal

diffusivity,

is then

proportional

to

P - 1/3

and _{decreases with the pressure. This}

_property

enables us to lower the

already

short characteristic times _{of argon gas and thus} to

_study

the

_pattern

behavior over

very

long

effective times. In this

_experiment,

_r, was reduced to 3.3 s.

The container is

_cylindrical,

its

_height

is d = 1.5 mm and its

_aspect

ratio r is 7.66

(F

= ratio of radius Rd to

depth

d =

R).

The horizontal diffusion time

Th

= R2

Tv is 191 s

which is

_{comparatively}

much shorter than in other

_{experiments [8,10,11,13,14].}

As

_reported

below,

the transient

_length

_{may be} as

_long

as 500 _Th. In the

_present

case, this

_corresponds

to a

day,

which is

_long

but tractable. In other visual

_experiments

however,

the

_waiting

time would have been so

_long

that the

_vicinity

of attractors would

_hardly

have been reached in real time.

During

the entire

_experiment,

the

_experimental

control

parameters

such as the

_heating

power, the pressure and various

temperature

measurements or controls were _{scanned every}

minute

_by

a

_computer

and recorded. This enabled us to

_compute

the heat flux at each time

and thus to

_provide

an accurate determination of the

_global

_dynamics.

The

_experiment

aims at

_scanning

the

_dynamical

behavior of the

_asymptotic

time

_regime

over a

_large

_{range of reduced}

_Rayleigh

numbers E

₍₀

e

1 ).

As the transients are

_long,

it

would have been tedious to reach the

_asymptotic

state at each

_Rayleigh

number

_starting

from

an

_arbitrary

initial condition. This is

_why

we started instead from the

_asymptotic

state obtained at a

_slightly

_higher

_Rayleigh

_number,

_{by only slightly decreasing}

Ra. More

_precisely,

the

_Rayleigh

number

_continuously

decreases

_during

the whole

_experiment.

The rate is

sufficiently

slow

_compared

to the

_{typical frequency}

of the

_pattern

_dynamics

to ensure an

adiabatic drift of e and thus to avoid

_{spurious triggering}

of

_pattern

transitions. This slow and constant decrease of the control

_parameter

was

_easily

obtained

_owing

to a _{very weak pressure}

leak of the

_pressurized

_apparatus,

at a rate of 2 x

10- 2

bars per

day.

Since the

Rayleigh

number is

_proportional

to

P 2,

the drift of e was about

10- 3

_per

_day.

Information on the slow

spatial

transformations of the

_patterns

was

_{provided by}

the

_{low-speed recording}

of the

_pattern

pictures.

3. Transient

_regimes.

The transient

_length

is

_highly

sensitive to the initial _{conditions and may be}

_quite

_large.

When

the evolution starts from a state close in

_phase

space to an _attractor, the transient is short

(

100 _7h i.e. few

_typical

_periods

of the

_dynamics).

This

_happens,

for

_instance,

when an

adiabatic decrease of E

_triggers

a bifurcation of the limit

_cycle,

the

_{asymptotic regime having}

already

been reached. In such a case, since the basin of attraction remains

unchanged

and

2063

contrary,

when the initial state is turbulent or

_displays

a

_{dynamics strongly}

different from that of the _{attractor, transients prove} to be much

_longer

and

_usually

last as

_long

as

500 _{T h.}

This

_great

_length

of transients raises the

_difficulty

of

_{distinguishing}

between chaos and transients. Each time a sustained

_aperiodic

state was

_encountered,

we

_lengthened

the

observations over much

_greater

times

(10 000 T h )

than the usual transient

_length

(500 Th ).

Although

we cannot exclude the

_possibility

of attractors

_being

encountered at a much later

time,

this

_procedure

enabled us to discriminate between

_strong

attractors

_(for

which transients last 500

_{T h)}

and chaotic states or

_perhaps

weak attractors. We

_emphasize

that no

transient

_longer

than 500 _{T h and smaller than} 10 000 _{T h} has ever been

observed,

even at the

boundaries of a chaotic window. This observation tends to

_reject

the existence of weak attractors.

We recall

_{that, during}

the observations at fixed

0394T,

the pressure leak causes a

_slight

scan of

e. In the

periodic

regimes,

this control

_parameter

drift _proves to be adiabatic

_compared

to the

dynamics.

Accordingly,

it is

_likely

to have no effect on the

_pattern

behavior,

unless it induces

the

_changing

of an attractor. If _so,

_spurious

chaotic evolutions

_might

be

_triggered,

for instance in the event of a short window of e

_containing

_many different attractors, as in a cascade.

Spurious

chaos may also arise if some transients much

_longer

than usual are encountered.

These

_{possibilities}

cannot be excluded as causes for chaotic

_{dynamics. However,}

since our

observations

show,

as described

further,

that the

_wandering

of the

_pattern

is

_likely

to be linked to the occurrence of

_strong

defect

interaction,

we

_expect

that its

_origin

is intrinsic. In addition to

_long

_transients,

_long

main

_periods

have also been observed. Both the transients and the main

_periods

lasted times of several

_{T Th}

to several

_{r2 Th.}

This observation

supports

the conclusions of Cross and Newell

_according

to which the earliest time to reach

equilibrium

should scale like

_rTh

_[15].

_{This time may} even be as

_large

as

T 2 Th.

For

_instance,

in one

_periodic

state

_(the

_regime

_4),

the slow relaxation of a defectless bent

pattern

resulted in defect nucleations after times as

_long

as 3

T 2

_Th. This statement shows that a

_slowing

down of the

_dynamics,

even on several Th’ cannot be a criterion of

_{stationarity.}

It also indicates that

experimental

studies of

_pattern

_dynamics

in

_large

_aspect

ratio cells must involve at least several hundred _Th of

_waiting

time in order to

_get

_beyond

transients. This leads us to

_question

the existence of

_{time-dependence}

in the

_vicinity

of the onset of convection at moderate Prandtl

number,

as stated in two

_experimental

studies

_{[13, 14] :}

in one of these

studies,

the

waiting

time was

_only

40 _{T h}

(Le. =

3

_rTh

since r =

14.0)

[13] ;

in the other _one, it _was

significantly larger

(200

T h Le.

0.9 r2 Th

since r =

15.0)

[14]

but the

_patterns

seemed _to

display

a forced rotation. It therefore would be worth

_ensuring

that these

_{time-dependences}

were indeed related to a non transient and intrinsic behavior.

4.

Long

time

dynamics

of the

_pattern.

In this

_section,

we

_report

observations of the

_pattern

_dynamics

above the onset of

time-dependence,

a

_long

time after the

_decay

of transients

_[12].

_Although

_{the pressure leak caused}

a decrease of e, we will

_describe,

for the sake of

_simplicity,

the observations for

increasing

e.

Many

windows of e were

found,

each of which

_corresponds

to a definite

_dynamics.

_They

are

sketched in

_figure

2. The onset of

_{time-dependence}

is at _Eo = 0.126. Below eo,

stationary

basic distorted

_patterns

are

_displayed.

In _{the range 0.126} : 8

_0.175,

five different

_periodic

states occur. For 0.175 E

0.346,

the behavior is chaotic

but,

for

slightly higher

_8, the

pattern

restabilizes towards

_{simpler dynamics :}

a

_stationary

state _{in the range}

0.346 e

0.37,

a

_{high frequency}

_periodic

state for 0.37 e 0.45 and a new

_stationary

Fig.

2. - Sketch of the route to turbulence : S _means

stationary regime

and P

_periodic

regime.

.

these

_{spatio-temporal regimes}

have

_proved

to be

_reproducible

and

_independent

of the initial condition :

_starting

from either a

_conductive,

a

_turbulent,

a

_stationary,

or a

_periodic

state, the

asymptotic regime

has

_always

been the same for a

_given

E.

Since the

_experiment

is

_performed

at fixed

OT,

the heat flux

_displays

fluctuations in order

to maintain AT constant. For

instance,

it increases when defects are

_present.

It is

_independent

of their

number,

however. The heat flux variations thus

_provide

a _{useful way} to detect the various

_phases

of the

_dynamics

but not to count the number of defects. In the

_recordings

reported

next, since the heat flux fluctuations are

_only

3 x

10- 3

of the

_{signal amplitude,}

we

present

them on a relative scale which is

_arbitrary

and linear.

4.1 FIVE CLOSE PERIODIC STATES. - In the

periodic regimes,

the main

_spatial

pattern

evolution is as follows : in the defectless

_phase,

patterns

look like basic distorted

_patterns

(Fig. la) ;

dislocations then nucleate in the center of the container and

_glide

towards the foci where

_{they disappear}

_(Fig.1b).

_{Nevertheless,}

_{differences appear in the way the dislocations}

move in the convective field. The defects

_glide

either to the same focus or not, either

simultaneously

or not, either

_quickly

or much more

_slowly.

In order to describe in a

_simple

and fruitful way all the various combinations that have

arisen,

it is useful to introduce the

following

codification for the dislocation motion :

The

_pattern

is seen from above. We call the line

_joining

the foci the y axis and its normal the x axis. Due to a

_{slight inhomogeneity}

in the

_{experiment, the y}

axis reaches

_nearly

the same

direction in the

_laboratory

_frame,

_regardless

of the initial condition. This

_property

enables us

to define the axis orientation in a common _{way for all the various}

periodic regimes.

Our

codification is then based on this axis orientation. It is sketched in

figures

3a and 3b.

We call

_Dl

_{(resp. D2)}

the dislocation which climbs the x axis towards the x > 0

_(resp.

x

₀₎

domain. When the dislocation

_Di

_glides

towards _{the y} > 0

(resp. y

0)

domain,

we call it

_{D{ (resp. Dl ).}

As an

_example,

the situation

_{corresponding}

to

_Dl

_D2

is sketched in the

figure

3b. The defect nucleation is

_designated

N and the defect elimination E. The same

codification also

_applies

to the

_grain

boundaries

_{generated by}

dislocations,

which are written

GB.

The five

_{periodic regimes}

are listed in table I and their most

_{representative phases}

are

2065

Fig.

3. - Codification for dislocation

gliding.

a)

The codification

specifies

the four

_gliding

domains.

b)

Sketch of the _pattern in the

_regime

_{corresponding}

to

_Di

_{D2 .}

Table 1. -

Listing

of

the

_five

_{periodic regimes.}

4.1.1

_Regime

1 : 0.125 s 0.136. -

This range of e contains the first

dynamics

encountered as the control

_parameter

is increased. It is

_{periodic but, surprisingly,}

its

_period

includes three basic

_cycles

of roll

_pinchings

and defect eliminations. In each

_cycle,

both dislocations start

_gliding

as soon as

_they

reach the sidewalls.

_However,

the dislocation

D2

moves

_quickly

towards a focus and

_{disappears, while,}

on the

_contrary,

the dislocation

Dl gives

rise to a

_{grain boundary}

at the sidewalls

_{(Fig. 5a).}

This

_{grain boundary}

then

_glides

towards the focus towards which

_Dl

started to

_glide

and

_{finally disappears.}

The

_cycles

differ

by

the

_gliding

direction of both dislocations. The same evolution is found

again only

after

three different consecutive

_cycles

have occurred.

Fig.

4. - Sketch of the

most

_{representative}

_{spatial phases}

of

_regimes

2 to 5.

_a)

In

_regime

2, both dislocations

glide

towards the same focus and

simultaneously.

b)

In

_regime

3,

_{they glide}

towards the

same focus but at different times.

_c)

In

_regime

_4,

_they

_glide

towards

_opposite

foci and at different times.

d)

In

_regime

5, the _pattern alternates rotations in either of the two _directions, the dislocation

D2 being trapped

towards either of the two foci

_{respectively.}

y > 0

(D- Dl

and the

_GBï).

_Finally,

in the last

_{cycle, Dl glides}

towards

_{y > 0}

and

D2 towards y

0

_{(D’ D-}

and then

_{GBi ).}

The

_cycle

which follows next is similar to the first

one. The

_{periodic regime}

thus

_corresponds

to :

Due to the

_strong

_{perturbation produced by}

the

_{grain boundaries,}

the

_pattern

orientation

changes slightly

at each

_cycle.

It decreases

_nearly

ten

_degrees

between the third and the first

cycles

and

_only

two or three

_degrees

between the first and the second

_cycle.

The

_{cycle lengths}

remain

_quite

constant but each kind of

_cycle

has its own

_{length :}

the first one lasts 6.7 _Th, the second 9.8 _’rh, and the third 8.9 _Th. (Fhe

_{slight spatial}

differences of the three

_cycles

thus also

_give

rise to

_temporal

shifts. These shifts are noticeable in the heat flux measurements

(Fig. 5b).

In

_phase

_{space, since} one

_period

of the

_pattern

evolution contains three basic

_cycles

of defect nucleations and

_{eliminations,}

a

_period

tripling

of the fundamental

_periodic

behavior

is involved.

At the lowest

_boundary

of this window of e, the

dislocations,

rather than

gliding

towards a

focus,

remain

_trapped

at the

sidewall,

near the x axis.

They

then become a nucleus for a third

2067

Fig.

5. - Periodic

_regime

1.

_a)

The dislocation

_Dl

has induced the

_generation

of a

_grain

_boundary.

b)

Heat flux

_{recording displaying high}

values when dislocations are _present. The

_periods

are shorter

Fig.

6. -

Transient

_regimes

towards a

_stationary

basic distorted _pattern.

_a)

A dislocation

_Dl

has been

trapped.

The interaction with the second dislocation will induce a third focus.

_b)

The _pattern

_dynamics

is

_{governed by}

the interaction between the three foci.

but

_they

are no

_{longer synchronized. Nevertheless,}

this

_apparently

chaotic

_regime

_proves to

be

_only

a transient which

_decays

towards the

_{previous stationary}

basic distorted

_patterns.

4.1.2

_Regime

2 : 0.136 e 0.143. - Both dislocations

glide simultaneously

towards the y > 0 domain

(Figs.

7a and

4a)

and then

disappear

at the foci within 1.5 _Th. This

corresponds

Fig.

7. - Periodic

regime

2.

_a)

Both dislocations

_glide

towards the same focus

_{(Di Di ). b)}

Heat flux

2069

to the sequence : N ==>

D 1 D2

E. Unlike the

_{previous regime,}

a

_periode

involves

_only

one

cycle

of defect nucleation and elimination. Its

_length

is stable to within an _{accuracy of} one

percent

for two consecutive

_cycles,

but it decreases from 33.0 _{Th at} E = 0.143 to

18.5 ïh

at e = 0.136. The

periodic

variations of the heat flux _are drawn in

figure

7b for

E = 0.138. Its Fourier power

_spectrum

in

figure

7c

displays

monoperiodic

features and

sharp

peaks.

In

_particular,

the Fourier

_amplitude

is small for

_vanishing

_frequencies.

4.1.3

_Regime

3 : 0.143 «-- e « 0.149. - The dislocation motions

are the same as in the

previous regime

except

that the

_gliding

motions are not simultaneous. The dislocation

Dl glides immediately

after

_climbing

while

_D2

_stays

on the x axis close to the sidewall

(Fig. 4b).

A short time after

_Dl

has

_disappeared

from the convective

domain,

D2

starts

_gliding

again

(Dr)

and is

_finally

eliminated at the focus.

_Compared

to the

_{previous regime,}

the

motions are

_only

shifted in time. The

_cycle

_{may be}

_{symbolized by}

The dislocation

_D2

remains _{4.7 Th in} the convective field and the

_period

of the

_cycle

decreases from 48.5 _Th at E = 0.149 to 41.5 _{Th at} E = 0.143. The heat flux

recordings

reproduce

well the

_periodic

features

_{(Fig. 8a)}

and

_give

rise in _{the power}

_spectrum

to

_sharp

peaks

and several harmonics of the fundamental

_frequency

_{(Fig. 8b).}

The level of coherence of the

_periodic

behavior is

_{high mainly}

because the

_period

remains

_nearly

constant over the

whole window.

_By

_comparison,

since the

_period

decreased more

_quickly

in the

_previous

regime,

the power

spectrum

was less

_sharp.

4.1.4

_Regime

4 : 0.149 E 0.154. - Unlike both the

previous

regimes,

the dislocations

glide

in

_opposite

directions,

Dl

towards the y > 0 domain

(Di )

and

D2

towards the

y 0 one

_{(D2 )}

_{(Fig. 4c).}

Both dislocations start

_gliding

_{simultaneously,}

but

_D2

_{pauses after}

having

glided only

the width of one roll

(Fig. 9a).

It

_begins

to

_{glide again nearly}

34 Th later and

finally disappears.

The

_cycle

_may be

_{symbolized by}

N =>

_D1

_D2

_{D2 =>}

E. Its

period

is

_quite

_{long : nearly}

150 _Th. Without such a

_{long wait,}

this

_{dynamical regime}

would

have been mistaken for a

_stationary

state.

The

_period

decreases

_slightly

with e: _{it goes from 157 Th at} e = 0.154 to 145 _{Th at}

E = 0.149. The different

_phases

of the

_cycle

_may be

_{distinguished}

in the heat flux

measurements

_{(Fig. 9b).}

Because of the

_length

of the

_period,

fewer

_cycles

than in the other

regimes

have been _{scanned. The power}

_spectrum

is thus less accurate but it does

_display

the

fundamental

_frequency

and two harmonics.

4.1.5

_Regime

5 : 0.154 E 0.175. - This

regime’s major original

feature is a

_global

rotation

_alternating

between the counterclockwise direction

_(symbolised

_by

_{R+ )}

and the

clockwise direction

_{(R- ).}

A similar

_global

_pattern

rotation has also been observed in

numerical simulation

_[21].

In both the R+ and R-

_cycle,

the dislocation motions are the same,

except

that the

_gliding

directions are

_opposite.

Both dislocations start

_{gliding simultaneously}

to

_opposite

directions. The dislocation

_Dl

travels

_quickly

to the focus

_(within

few

T-h)

and

_disappears,

while

_D2

_stops

after

_having

_glided

_only

the width of one roll. The whole

structure continues to rotate

_{(Fig. 4d)}

and the dislocation

_D2

starts

_gliding

_again

when the

angle

of rotation reaches

_nearly

90°

_{(Figs. 10a}

and

_c).

_Finally

_{D2 disappears}

at the focus. In the R-

_cycle

_{(Figs. 10a}

and

_{4d), Dl}

_glides

towards

_{the y >}

0 domain

_(Dl

_{D2 )}

so that the

cycle

is

_symbolized

_by

R- =:> N ==>

_{D 1}

_D2

+ R- =>

_D2

+ R- => E.

In the R+

_cycle

_{(Figs. 10c}

and

_4d),

_Dl

_glides

towards

_{the y}

0 domain

_{(D- D+ )}

so that the

2071

Fig.

8. - Periodic

regime

3.

a)

Heat flux

recording.

b)

Heat flux

_spectral

power

displaying

Fig.

9. - Periodic

_regime

4.

_a)

The dislocation

Dl

has

_{already glided}

towards the

_{y > 0}

domain

2073

In the next

_stage,

the R-

_regime

is

_{again displayed.}

The whole

pattern

thus alternates between clockwise and counterclockwise rotation. The

_period

of this evolution is twice that of the fundamental

_periodic

behavior. This

_corresponds

to a

_period

_doubling.

In each

_cycle,

the

_length

of the

_{phase displaying}

defects is constant with 8. In _contrast, the

relaxation

_phases

of the defectless

_pattern

last somewhat

_longer

as E increases.

_Although

the

spatial

features of both kinds of

_cycles

look

similar,

their

_dynamics

are

_slightly

_different,

however. We illustrate this

_{by analysing}

them at e = 0.16. In the first

cycle

(R- ),

the

relaxation of the defectless

_pattern

towards a new

_pinching

lasts 19.2 Th while the rotation

phase

with the

_trapped

defect lasts 42.3 Th. Its whole

length

is therefore 61.5 Th.

Comparatively,

the relaxation

_phase

of the second

_cycle

_{(R+ )}

towards

_pinching

last 7.7 Th and the rotation with defect lasts 30.7 Th. Its

total length,

38.4

’rh, is

thus smaller than that of the first

_cycle.

The

_dynamical

differences between

_cycles

is recovered in the heat flux measurements

(Fig. 10e).

Period two is noticeable both in the

_phase

_displaying

defects

_(high

heat flux

_values)

and in the defectless

_phase

_(low

heat flux

_values).

_{The power}

_spectrum

exhibits

_{sharp peaks}

which illustrate the

_{high dynamical}

coherence

_(Fig.10f).

The most

_{important peak}

occurs at

the mean

_{cycle frequency}

_(nearly

50

_Th).

It is followed

_by

three noticeable harmonics. The

dynamical

difference between the

_{cycles gives}

rise to the other

_peaks

which indicate the

occurrence of

_{period doubling.}

Since

_period

_doubling

has occurred in this

window,

one _{may wonder whether} an iteration of this

_phenomenon,

i.e. a subharmonic

cascad,

exists.

_{Looking carefully}

at _{the upper limit of}

the

present

window,

we did not _{find any evidence of it.}

_However,

one cannot rule out its

existence,

either because its range in e

_might

have been too narrow to be detected or because

the external noise in E

_produced

_by

the thermal

_{regulations might}

have

_suppressed

it.

4.2 A CHAOTIC REGIME : 0.175 8 0.346. - This window is

so

_large

that a continuous scan of 6 at the rate

_{provided by}

the

_present

_{pressure drift would have lasted} at least several months. We have then

_only

_performed

observations at various values of E. The observations

lasted 10 000 Th each and all showed chaotic

dynamics.

As in the

periodic regimes,

the

mechanisms of

_{time-dependence}

involve roll

_pinchings.

_However,

the dislocations do not

disappear

easily.

When

_they

are

_trapped

at the

_{sidewalls, they}

enable a new focus or a _range

of disinclinations to be

_generated

and to _grow, as additional defects appear

(Fig.11a).

Moreover,

grain

boundaries sometimes arise and move over the whole convective field

making

the various foci grow or

_decay

_{(Fig. llb).}

_Accordingly,

in the chaotic _states, no mean

spatial

structure can be defined over a

_long

time.

The erratic behavior of the convective flows is

_apparent

in the heat flux

_recordings

(Fig. llc)

and is also found in local

_optical

measurements of the roll motion at _any

_place

of the convective field. The chaotic

_dynamics

is not

_homogeneous

in the

_phase

_{space however.} When the

_spatial

structure is close to the

_stationary

_patterns

which will be encountered next,

at

_higher

e, the characteristic

_frequency

of the

phase dynamics

decreases

greatly.

The

pattern

state then looks

_{quasi-stationary.}

This means that the

pattern

already

« senses » that some

phase

space domains will

become,

at

_{slightly larger}

e, the basins of attraction of some

_point

attractors.

_Moreover,

some

_{nearly periodic}

_{sequences of roll}

_pinching

at the

_pattern

center,

similar to the

_{previous periodic regimes,}

sometimes occur, when two

_{diametrically}

_opposite

foci

_{predominate (Fig. lld).}

Since the mean structure is

_continually

_changing,

these

_regimes

remain

_only

for several

_periods,

however. These features are reminiscent of

_{intermittency.}

Is this behavior an indication that the

dynamics

is

actually

a

_tremendously

_long

transient towards

a

_stationary

or a

_periodic

state ?

_Although

we cannot rule this

_possibility

out, we

_emphasize

Fig.

10. - Periodic

regime

5. Similar states of

neighbouring cycles

in the rotation

regime.

This shows

_by

comparison

the

_global

_pattern rotation. We notice that the

_large

scale _pattern distortion

_changes

at each

cycle,

while the dislocation of the latter

_{figures keeps}

the same

_place.

This distortion can not then be

produced by

the dislocation and we attribute it to the intrinsic _asymmetry 2. This _asymmetry is sketched in

_figures

lOb and d and is in _agreement with

_figures

10a and c

_{respectively.}

_a)

The _pattern state

_just

before

the dislocation

D-

starts

_gliding

_again.

The _pattern is

_rotating

in the clockwise direction

(R- ). Notice

the tilt of the line

_joining

the

points

where the curvature is a maximum

_(previously

the y

axis).

Notice also the asymmetry of the

_phase

curvature, in agreement with

_figure

lOb

_(ô

>

0).

The rotation R- is thus linked with a

_perturbation

cp = 03B4x with 5 > 0, in _agreement with our derivation.

b)

Isophases

of the

_phase

field cp =

ko y (1 -

ax 2/R 2)

₊ 03B4x with 5 _> 0. The _curvature is

greater in the

xy 0 domains than in the xy > 0 domains. Notice the _agreement with

figure

10a.

c)

The _pattern state

just

before the dislocation

_D’

starts

_{gliding again.}

The pattern is

_rotating

in the counterclockwise direction

_{(R+ ).}

Notice the tilt of the line

_joining

the

_points

where the curvature is a maximum

(previously

the y

axis).

This tilt is

_opposite

to that of

figure

10a

_{(R- ).}

Notice also the asymmetry of the

phase

curvature, in agreement with

_figure

10d

_{(03B4 0 ).}

The rotation R+ is thus linked with a

perturbation

cp = 8x with 8

0,

in agreement with our derivation.

_d)

_Isophases

of the

_phase

field

cp

= ko y (1 -

ax2/R2)

+ 03B4x with 8 0. The curvature is _{greater in the xy > 0 domains than in the}

xy 0 domains. Notice the agreement with

_figure

10c. Heat flux

recording

e)

and its

_spectral

_power

_f).

e)

Notice the

_dynamical

differences between the two _types of

_cycles.

_f)

Notice the most

_{important peak}

2075

Fig.

11. - Chaotic

_regime.

_a)

A third focus has been

_{generated by}

dislocation interactions similar to

those of

figure

6a.

_b)

Three foci

_displaying

a

_{grain boundary.}

c)

Heat flux

_{recording displaying}

erratic

2077

Fig.

11

_(continued).

described

_{theoretically}

as almost attractors

_[1]

and confused in

_practice

with chaotic evolutions.

4.3 STATIONARY AND PERIODIC STATES BEYOND CHAOS. - For 0.346

E

_0.37,

a

stationary

state is observed

_{(Fig. 12).}

Instead of a continuous scan, the value of e was

_changed

abruptly by

suddenly varying

the

_temperature

difference AT. This did not break the

stationarity.

The whole observation lasted about 3 000 _Th i. e. six times

_longer

than the

_longest

transient. The

_pattern

_displays

two

_{diametrically opposite foci,}

as in the basic

distortion,

but

Fig.

12. -

in addition two dislocations are

_trapped

on the x axis near the sidewalls. In the

present

range of e, these defects are in

_equilibrium

with

_respect

to

_gliding.

This state therefore

_corresponds

to a

_locking

of the

_periodic

evolution

_by

the defects.

For 0.37 e

_0.45,

a

_periodic

state occurs. Its

_spatial

features are different from those of

the

_previous

limit

_cycles.

The

_pattern

is

_composed

of two

_{diametrically}

_opposite

_foci,

but one

of them has a short

_grain

_boundary.

Defect nucleation still occurs

_by

roll

_pinching

at the

pattern

center but both dislocations

_glide

towards the defectless focus and leave the remainder of the

_pattern

undisturbed and

_{quasi stationary}

_{(Fig. 13a).}

The

_period

is

_quite

constant

_{(Fig. 13b)}

but

_surprisingly

short : 4 mm i.e. 1.25 _Th

_{(Fig. 13c).}

This

_frequency

is so

high

that a new dislocation

_pair

is

_{generated only}

a short time after the

_{previous pair}

has

disappeared.

Nevertheless,

in this

_high

_frequency

state, since a

_{single pair}

of

_moving

dislocations is

_present

at a

time,

the defect interactions are weak and a mean structure exists.

For 0.45 a

0.657,

a new

stationary regime

is found. It was observed

_during

more than

10 000 _{Th. Two}

_slightly

different

_spatial

structures were

_{displayed depending}

on the initial

conditions

_{(Figs. 14a}

and

_14b).

This shows that several

_point

attractors _may coexist in the

phase

space. In both of these

patterns,

nearly straight

rolls are

_present

in the

bulk,

and short

foci,

dislocations and

_grain

boundaries exist at the sidewalls. These defects enable the

structure to

_satisfy

the

_{phase boundary}

condition at the sidewalls which

_requires

that the rolls end

_{perpendicularly}

to them.

At

_higher

but still moderate Prandtl number

₍₂

Pr

_{10 )}

and in a

_{cylindrical container,}

stationary

states _{may be} reached _up to

_high

values of e

(E S 3.5) [13, 14].

_They

often bear

strong

similarities with the former

_stationary

_patterns

observed in our

_experiment.

For

instance,

the

_stationary

_pattern

_displayed

_{in the range} 0.346 e 0.370 looks very much like

that which arises at e = 0.70 in the

experiment

of Heutmaker and Gollub

(Pr

=

2.5) [13],

Fig.

13. - Periodic state for 0.37

e 0.45.

a)

Notice the two dominant foci and the

grain boundary

on one of them. Roll

_pinchings

occur at thé _pattern center and both dislocations

_glide

towards the y 0 focus. A

single

dislocation

D2

is still visible. The

spatial

features are close to those of the

_previous

periodic

regimes. b)

Heat flux power spectrum. Notice the

_high

level of coherence of this state in

_spite

of the

_large

distance from the onset of

_{time-dependence. c)}

Heat flux

_recording.

Notice the

_high

2079

Fig.

14. -

Stationary

states for 0.45 E 0.657. Notice that in each _pattern

_(a

and

b)

the bulk is made of

_{nearly straight}

rolls, i.e. a

_stationary

structure which

_gives

rise to no mean flows. The

_equilibrium

is thus

mainly

related to that of the defects. Notice the _{greater pattern symmetry}

_displayed

in

_b)

_compared

to

_a).

while the

_{pseudo-stationary}

_patterns

observed

_by

_{Steinberg et}

al. in water

_(Pr

=

6.1 ) [14]

_are

quite

similar to those reached in _{the range 0.45} e 0.657. This

_agreement

indicates that the

stationary

structures observed

_beyond

the onset of

_{time-dependence}

are not related to a

particular

value of the

_aspect

ratio or to some other feature of our

_experiment

but are indeed

robust solutions of the

_{phase dynamics.}

4.4 THE THRESHOLD OF TURBULENCE : e = 0.657. -

Beyond

the

_{previous stationary}

regime,

no

_{periodic dynamics}

was ever found. Chaotic behavior was

_displayed,

whose

features are similar to those of the

_previous

chaotic

_regime.

No _{evidence of any other kind of}

dynamical

behavior was noticed up to e = 1.

4.5 CONCLUSION. - In this extended

_cylindrical

convective

_layer,

several different

_dynamics

are encountered a

_long

time after transients. Each of them is

_robust,

_reproducible

and is

observed in a definite window of the control

_parameter

e. These results are reminiscent of those obtained in

_liquid

helium at the same Prandtl number

[11],

but here the

_spatial

features

are identified. In

_{particular, although complex spatial}

features are

_involved,

the route to

turbulence exhibits few effective

_dynamical

modes and low dimensional non chaotic

attractors.

Though

a window of chaotic

_dynamics

occurs in the range 0.175 « s

0.346,

the threshold

of turbulence i.e. the value of e above which the

dynamics

is

_always

chaotic,

is at

e = 0.657. The transition _to turbulence

(0.126

e

0.657 )

thus takes

place

over a _{range of} 6

that is

_nearly

5 times

_greater

_{than the range} over which the transition to

_{time-dependence}

occurs

(0 e

0.126 ).

Since an adiabatic decrease of E leads to short transients between the

_periodic

states, and since these states

_display

similar

_{spatial features,}

one _may

_expect

that

_they

result from the

successive bifurcations of the same state. This statement leads us to look for an identification

2081

to this work. In the first one, we address the

_physical

mechanisms which

_give

rise to

time-dependence

with

_emphasis

on an

_analytical

solution valid for basic distortions. In the second

section,

we

_perform

a

_{stability analysis}

of this solution and we show that its modes of

instability

lead to bifurcations from basic distorted

_patterns

to various

_{time-periodic}

states.

This

_{stability analysis finally}

enables us to reconstruct the route to turbulence with

_good

qualitative

and

_quantitative

_agreement.

5. Phase

dynamics

within a

_coupling

with mean flows.

We summarize the main features of

_{phase dynamics}

when

_coupled

with mean flows and we

focus on an

_analytical

solution which describes

_stationary

basic

_distortions,

their

destabili-zation,

and some

_important

features of the

_previous

limit

_cycles.

This enables us to

_legitimize

quantitatively

a criterion for

_periodicity

or chaos and to address in the next section the limit

cycle

destabilization.

5.1 INTRINSIC AND NON-LOCAL NATURE OF THE DYNAMICS. - We claim that the

phase

dynamics

observed in this

_experiment

can

_only

be intrinsic. In

_particular,

since rolls are

_quite

large compared

to the

_typical

scale of the thermal

_{fluctuations,}

the external thermal noise is unable to

_modify

the convective motions. In

_addition,

when

_{spurious phenomena}

such as cell

rotation,

depth

ramp or external mean flows have been

_avoided,

the

_{phase dynamics}

is in

principle

derivable from the

_{Boussinesq equations.}

This is

_{supported by}

the fact that the transition to

_{time-dependence}

is

_strongly

related to an intrinsic

_parameter

(the

Prandtl

number)

and has been recovered both

_by

numerical simulations and

_by

an

_analytical

derivation of intrinsic models of convection

_[18,

20,

21].

As one focuses on finer and finer features of the

_{phase dynamics,}

however,

one _{may wonder} to what extent external disturbances may be

neglected.

In our

_experiment,

these disturbances would lead to

_forcings

independent

of the convective threshold and thus to

_regimes

robust in a

_large

_{range of E,}

contrary

to our observations.

_{Accordingly, though}

external effects may be

_present,

we

attribute the various

_changing

of

_regimes

in a narrow _{range of e} to intrinsic

_phenomena,

at

least for the

_{greater part.}

Experiments

have demonstrated that

_large

scale distortion of a roll

_pattern

is a

_quite

dangerous

mode of convection

_capable

of

_decreasing

the threshold of

_{time-dependence by}

at

least an order of

magnitude.

This « overdestabilization » can be related to the mean flows

which are

_{produced by}

_pattern

distortions

_{[16, 17].}

The

_amplitude

of these flows is

_negligible

compared

to the roll flows but their

_spatial

scale is

_{large compared}

to the roll size. The latter feature

_{produces non-locality}

which is

_responsible

for the overdestabilization

_[12].

Mean flows in convection involve non-local

_phenomena

for two rather different reasons. At

first,

since

_{they satisfy}

the mass

_{conservation,}

local distortion can

_produce

mean flows over

large

distances

_[16],

so that the link between distortions and the mean flows that

_{they produce}

is nonlocal.

_Secondly,

since mean flows

_spread

_{among many}

_rolls,

their effect on roll

_patterns

involves a collective feature : a

_large

scale distortion

_[17].

The link between mean flows and

the distortions that

_{they produce}

is thus also nonlocal. We

_emphasize

that this

_possibility

of

acting

over

_large

distances

_{explain why}

mean flows

_{produce huge}

effects in

spite

of their low

amplitude :

the weakness of the local actions is counterbalanced

_by

their accumulation from roll to roll

_[18].

5.2 A SOLUTION FOR THE BASIC DISTORTION. - Because distortion and mean flows

_generate

each

other,

the

_phase

field

_describing

the roll

_pattern

and the mean flow field are

_coupled

in a

non-local way. The

coupled equations describing

this interaction have been

_put

in a