Order parameter and amplitude equations for the Rayleigh-Bénard convection

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Order parameter and amplitude equations for the Rayleigh-Bénard convection

W. Decker, W. Pesch

To cite this version:

W. Decker, W. Pesch. Order parameter and amplitude equations for the Rayleigh-Bénard convection.

Journal de Physique II, EDP Sciences, 1994, 4 (3), pp.419-438. �10.1051/jp2:1994137�. �jpa-00247971�

J- Pllys. II Hance 4

(1994)

419-438 MARCH 1994, PAGE 419

Classification

Physics Abstracts

47.10 47.20

Order parameter and amplitude equations for the

Rayleigh-B4nard convection

W. Decker and W. Pesch

Physikalisches Institut der Universitit Bayreuth, W-8580 Bayreuth, Germany

(Received

20 August 1993, revised 18 November 1993, accepted 6 December

1993)

Abstract. The reduced description of the roll patterns and their stability _near onset is

investigated in detail for Rayleigh-Bdnard convection. The starting point is _a novel order _pa- rameter equation

(OPE)

in Fourier space that is rigorous _up to cubic order in the amplitudes of the critical modes at threshold. Comparison with rigorous results from _a Galerkin analysis _ex-

hibits the range of validity of this order parameter description. In particular in the _case of _gases

(Prandtl

number

~ 1), the reduced description is fairly satisfactory. In

a next step the OPE _are

used to derive coupled amplitude equations in real space. In this _way

a complete description

of the _mean drift mode is achieved for the first time. It is shown that at least for intermediate Prandtl numbers _a large number of derivative terms is necessary ^to get good agreement with the rigorous results. I/rom the results _{one can} also judge, under what conditions _common model

equations _are likely to describe the real systems.

1. Introduction.

Rayleigh-Bdnard

convection

(RBC)

in fluid

layers

heated from below

provides

_a canonical _ex-

ample

for

pattern-forming

transitions in

nonequilibrium

systems [1, 2]. ^{It has been studied for}

many years without

loosing

its attractiveness

(besides general

reviews, _e.g. in

[3-7,

1, _{2] very}

recent

experimental

work may be

found,

_e.g. in

[8-10]).

^In a series of

important

_papers the

destabilization mechanisms of the

periodic

roll pattern ^{in ABC have been classified}

by

Busse and coworkers and the stable wavenumber

regimes (the

~'Busse balloon" have been

mapped

out _{as a} function of the two nondimensional parameters,

namely

the

Rayleigh

number R

(the

main external control

parameter)

and the Prandtl number Pr [3, 11].

Typical

destabilization mechanisms involve modulations of the roll

separation (Eckhaus instability (E) [12]),

undu- lations

along

the roll axis

(zigzag instability (ZZ))

and combinations of both types

(skewed

varicose

(SV))

_{[11, 13].}

The present _paper ^{is devoted} to

large-aspect-ratio

systems

slightly

above the onset of the convection

instability,

where _a small expansion parameter _e

(the

reduced distance of the _ex- ternal control parameter ^{R from threshold}

R~)

is available. One _can then

apply

the familiar

weakly

nonlinear

analysis,

which _appears in several variants in the literature [1, 2]

(or

with

more

emphasis

_on RBC

[14,

5, 6])-

The method _can be

phrased

_{as a} reduced

dynamical description

based _on the _space of the linear

modes~

which _grow

exponentially

_{or are}

only slightly damped

at threshold. With respect to the horizontal

directions~

in the idealized limit of

laterally

infinite

extension,

the modes _are characterized in Fourier space. The

simplest

_case is _a

periodic

roll pattern near onset with

wavevector q. Then the Fourier

amplitude A(q) plays

the role of _an order parameter, ^{which is}

determined

by expanding systematically

_up to cubic order in terms of

A(q).

The

application

to RBC has been initiated in [15, 16] and culminated in the

comprehensive

work in [17].

Roll patterns observed in nature have the

tendency

to

display

slow modulations in space

(and possibly

also slow in

time)

_apart from the _appearance of defects

(foci, dislocations)

_{[1~ 2].}

Such modulated patterns _can be described _{as wave}

packets

where several Fourier modes with

weights (A(q)(

interact.

Proceeding again

_up to cubic

order,

the

complex

Fourier

amplitudes A(q)

become

coupled by

a nonlinear

integral equation,

^the order parameter

equation (OPE) [18-20].

Our first main issue _was the

rigorous

derivation of the OPE for RBC.

By

construction it becomes exact in the limit _{e -} 0. But _{as an}

important

result _we found

by comparison

with the

rigorous

Busse

balloon,

that the order parameter concept remained

satisfactory

_up to

fairly large

values of _e in _{some cases}

(e.g.

for Pr _m 1

nearly

_up to _e

=

1).

That holds true for the

long-wavelength (E, ZZ, SV)

_as well _as for the

short-wavelength

cross-roll

(CR)

_[3]

instability

boundaries. The results _are

encouraging

to

apply

the order parameter

approach

_{as a} semi-

quantitative

concept to _more

complicated

pattern

forming

systems ^like

liquid crystals (21-23].

Here precise measurements _near threshold have revealed _a multitude of

interesting

scenarios, but _{a more}

rigorous

treatment

beyond

the OPE is

practically precluded by

the

complexity

of the system.

The OPE will _{serve as a} convenient

starting

point ^for an additional

approximation

scheme

in _{our case.} One scales the distance

(q~ q()

+~

e~/2 and introduces _a time scale

+~ e.

By

switching

to real _{space one} arrives _at

leading

order in

e at the famous

Newell-Whitehead-Segel (NWS) (or amplitude) equation

[24,

25], commonly applied

to the

description

of modulated patterns near threshold. The NWS

equation

_was obtained

originally by

_a multi-scale

analysis,

all terms up ^to order e~/2 _are balanced and _no

explicit e-dependence

is left.

It _was

recognized fairly early,

that in certain _cases

(at

least for small Prandtl

numbers)

_an

extension of the basic NWS equations is necessary

by taking

into account

higher

orders in _e

[26, 27].

Otherwise _a

quantitative description

of the

zigzag (ZZ) instability

_[28,3] is

impossible

and the skewed varicose

(SV) instability

does not appear ^at all. Of

particular importance

is the inclusion of the mean-flow

(mean-drift)

mode which is excited in second order, if the pattern deviates from strict

periodicity.

It is characterized

by

_a

typical length

scale much

larger

than the roll

spacing, by

_nonzero vertical

vorticity

and _a

nonvanishing spatial

_average of the

velocity

fields _across the fluid

layer.

Mean-flow effects in RBC in the context of

amplitude

equations

(AE)

have been considered at first for the

simplifying

situation of stress free

boundary

conditions

[26, 27].

After _some refinements [29]

existing discrepancies

between the results obtained in

[26,

27] ^{and others} based _on OPE

[30,

31] were reconciled.

For the realistic

rigid boundary

conditions the

appropriate procedure

has been outlined in

principle

in the literature [19, 32, 14,

6],

but has

surprisingly,

_as far _{as we} know, _never been worked _out in detail. Thus _we wished to fill this _gap for the standard ABC system, ^{in order}

to test the

approximation

scheme with _a view to _more

complicated

situations _a~~ well _as to

provide guidelines

_on the _way to

semi-quantitative

simulations of scenarios such _as the

spiral

turbulence [10]. ^{In the}

resulting coupled amplitude

equations the _mean flow

amplitude

is

N°3 AMPLITUDE EQUATIONS FOR THE MC 421

explicitly

isolated in contrast to the OPE equations. We find in

addition,

that also terms

arising

from the

elasticity

of the rolls

(higher

order derivative terms in the cubic part of the roll

amplitude equation ), play

an

important

role.

The _paper is

organized

in the

following

_manner. After this introduction _we sketch in section 2 the derivation of the OPE and _compare with

rigorous

results. Section 3 is devoted _to the

presentation

of the

coupled amplitude equations.

The

resulting stability

boundaries _are

briefly

discussed in section 4 and also

compared

with the OPE. After the conclusion in section 5 the

appendices

contain _some additional _more technical details.

2. Order parameter

equations.

2.I FORMULATION OF THE PROBLEM. Our

starting

point are the dimensionless

Boussinesq

equations

for the

velocity

field _u and the deviation of the temperature from the static distribution

(see

_{e-g. [28] ):}

V.u=0

,

(la)

V~6+Rk.u=u.V6+~6

,

(lc)

where k is the unit vector in z-direction

(opposite

to the direction of

gravity)

and _7r is the pressure.

We _use the conventional dimensionless quantities,

namely

the

Rayleigh

number

(R)

_as the main control parameter and the Prandtl number

(Pr)

and consider realistic

rigid boundary

conditions.

The

equations

_are reformulated

by

the introduction of two

velocity potentials f

and _{g ap-}

propriate

to the solenoidal vector field _u

ill,

_28]

u = 6

f

_{+ eg}

,

6 =

(b(~, b(~, -b(~ _b(~)

,

e =

(by, -b~, o) (2)

The

quiescent

state

corresponds

to

f

_{= g =} 6

= o. In the

following

_we will introduce the vector notation V

=

(6, f, g)

and _{use a}

symbolic

notation for the basic equations.

£V =

N2(V(V)

+

B~~

(3)

The

explicit

form of various matrix operators in

(3)

which contain

spatial

derivatives is clear from

equations ii)

and

(2).

2.2 WEAKLY _{NONLINEAR ANALYSIS.} The first step ^{of the}

analysis

is based _on the behavior at threshold which is described in detail in the literature

(see

_e-g-[33]

).

The onset of convection is determined with the

help

of the linear

eigenvalue problem

£V

=

B$f _(3).

_{The modal}

solutions have the

general

form:

viz,

_{g, Z,}

t)

#

vi;nl~,

_Z,

R)e~~~e~~,

_{q "}

jq, p),

_X

" IX,

g) j4)

The lowest value of R for which the linearized equation

(3)

has _a nontrivial solution for I = o

yields

the neutral surface

Ro(q),

and the minimum of

Ro(q)

with respect to q

gives

the

critical wavevector q~ which is

degenerate

_{on a} circle with radius _q~. Because of the rotational invariance we can fix the direction of _q~

arbitrarily.

The

corresponding

cut of the neutral surface

along

that direction is the neutral _curve

R(q).

The next step of the

weakly

nonlinear

analysis

_[18,

34, 6,

1, 2],

(for

_a recent _more detailed

presentation,

_see also [21, 22]

describing

the situation

slightly

above threshold _{is to} reduce the

dimension of the system

by choosing

an appropriate "basis" set of states, characterized _as the

"dynamical

active" _ones in [6]. One

approximates

the solutions for R >

R~

at lowest order

by

a wave

packet

of the linear

eigenmodes

Vo

(+ li;n(q,

_z, R

=

R(q))) (see (4)):

V Gt Vi _"

/dq ^{A(q, t) Vo(q,}

z)e~~l~ ⁺ c-c-

(5)

D

Here

A(q) plays

the role of the order parameter

(amplitude),

which vanishes at threshold. The

integration

domain D is _a small annulus centered around (q( = q~ which need not be

specified

in advance. The

amplitude A(q)

will be determined from

equation (3) by

_{a systematic}

expansion

of V which is small _near

threshold,

in the form V

=

Vi

+ V2 +

V3

+ _.. The second order solution V2

-~

A~ _is

explicitly

calculated _as the solution of the

corresponding inhomogeneous

linear system derived from

equation (3):

£V2

₌ N2

(Vi [Vi ). (6)

The solution V2 contains contributions with _wave vectors

ranging

^from (q( m o up ^to (q( m

2q~.

Proceeding

to third order the

equations

_are closed

by projecting

^the third-order solutions

V3

-J

A~ _onto the

subspace spanned by

the linear modes

(Vo(q, z),

_see

(5)).

One obtains the order parameter

equations (OPE)

in Fourier space:

al

(q)~A(q, ^t)

=

a2(q)A(q, t)

₊

/ _dqi / _dq2

£l3

(q,

_qi,

q2)A(qi, t)A(q2, t)A(q

_{qi q2,}

t), (7)

D D

Note that the coefficients

a~(q), (I

₌ 1, 2,

3) depend

_on the Prandtl

number;

their

explicit

calculation is done

numerically.

The

approach

could be

generalized

if _{one uses} the

eigenmodes

li;n

(Eq. (4),

1

#

o for _E >

o)

and avoids the adiabatic

approximation

involved in

equation (6) (see Appendix

C of

[21]),

but

we detected _no

significant changes

in _our results.

2.3 STABILITY DOMAINS FROM OPE. To start with _{we are} interested in

stationary

roll

solutions,

periodic

in space which _are characterized

by

_a wavevector qo. One _uses the ansatz:

Ar(q)

" CT

6(q qo)

_{+ Cl}

6(q

₊

qo) (8)

in

equation (7).

The double

integral

_on the

right

hand side of

(7)

then

collapses,

and the

amplitude

coefficient _{cr can}

easily

be calculated. It is also obvious that (cr(~ is

proportional

to

(R Ro(qo))/Ro(qo),

the reduced distance from the neutral

surface,

which _serves

essentially

as our small

expansion

_parameter.

The

stability analysis

of the stationary roll solutions is

performed by introducing

_a ^small

perturbation 6A(q, t)

of the

amplitude Ar (Eq. (8)):

6A(q,

_S,

t)

_{= (Ci}

iq,

_S)

6(q

_qo S) _{+ C2}

(q,

_S)

6(q

_{+ qo}

S))

^e~~

(9)

N°3 AMPLITUDE EQUATIONS FOR THE ABC ₄₂₃

and

linearizing equation (7).

The wavevector s denotes _a modulation of the

qo-periodic

pattern.

One arrives at _a linear

eigenvalue problem

for _cl and _c2 where the I with the maximal real part determines the

growth

rate a~~~j,~(qo, _s,

R).

When the maximum of _a~~~j,~

(with

respect

to

s)

_{crosses zero} the values

(qo, R)

_on the

stability

boundaries _are identified.

The

perturbations

6A

(9)

_are classified

by

the

magnitude

S and the orientation

angle

8 of the modulation _{wavevector s}

(see Eq. (9)), according

to s =

S(cos 8,

sin

8).

One

distinguishes

between

long-wavelength perturbations (S

< q~, an

analysis

_up to

O(s~)

is sufficient

),

_among which _are Eckhaus

(8

₌

o), zigzag (8

₌

7r/2)

and skewed-varicose

(o

< 8 <

7r/2) modes,

and

short-wavelength perturbations (S

_{m q~),} such _as the knot

(K)

and the cross-roll

(CR) instability

[3, II, 5]. For _a roll pattern with wavevector qo "

(q, o)

m q~, the cross-roll insta-

bility

is obtained with _s

=

(-q, p),

where _p is of order q~. There exists also another

important short-wavelength instability, namely

the

oscillatory

_one [3, 35,

36].

It appears

typically

at

higher

values of _E and cannot be described in the approximation scheme of the

OPE,

because decisive modes _are

neglected.

In the

following,

_we will consider

separately

three

representative

values of the Prandtl number and _compare with the

rigorous results,

which _are

briefly recapitulated.

I) ^{In the low Prandtl number}

regime (see

_{[36] we} took _an extreme _case

(Pr

=

o.ol),

which is realized e-g- in

liquid

sodium. The Busse balloon is limited _on the left hand side

by

the Eckhaus

(for

_very small

negative

values of _{q q~}

by

the

zigzag)

and _on the

right

side

by

the

skewed-varicose

instability.

Both

boundary

lines rise

steeply

with

E and the _upper limitation of the Busse balloon is of the

oscillatory

type

(not

accessible

by

the

OPE),

found at _E m o.08 for q = q~ [36]. ^{As shown in}

figure

la, ^{the OPE}

description gives acceptable quantitative

results

near onset, I-e- for _{E <} o.ol

(see

_e-g- the Eckhaus

line),

but the type ^of

destabilization,

and

more

generally

the orientation

angle

8 of the modulation wavevector, ^{is well}

represented by

the OPE in the whole _q

regime

covered in the

figure.

ii)

As _a

representative

_case for moderate Prandtl numbers _we have selected Pr

= o.71

(air,

see

[13]).

^{The results} are shown in

figure

16.

Turning

^from _q~ to the left _one has at first the

zigzag instability line,

which is _soon

preempted (q

5 2.95

by

the Eckhaus

instability.

For _{q > q~}

an SV-line

joins smoothly

into the E-line _near threshold. With

increasing

_E the

stability

line bends back to smaller _q and reaches the value _E

= o.46 for _q

= q~.

With

regard

to the

rigorous

results _{one sees} in

figure

16 that

stability boundaries,

in

partic-

ular the

SV-line,

_are well

reproduced

in _a

fairly large

_E-range- Moreover the knot instability below the Eckhaus _curve which it is _not

important

in the present case is well described.

iii)

For

large

Prandtl

numbers,

like Pr

= 7.o

(water)

shown in

figure

lc the

stability region

is bounded from the left

by

the

zigzag

line and from the

right by

the knot

instability (K),

which

now lies above the Eckhaus _curve

(see

_[13]

).

We have also included the

CR-line,

which resides

on the left side of the Busse

balloon,

and the SV-line

(with

small

angle 8,

and _very close to the Eckhaus

curve)

to the

right

of _q~. The latter is

represented nearly exactly by

the

OPE,

while the rather moderate

discrepancies

for the other _cases increase with _E.

In the context of the OPE

approach,

medium Prandtl numbers Pr

(m I)

are the most

interesting

_ones: otherwise

"upper"

limitations of the Busse balloon _are neither described

by

the OPE for small Pr, because

they

are of the

oscillatory

type, nor for

higher

Pr, since the destabilization _occurs at E-values _too

large

for the

application

_range of the OPE.

3. From order parameter to

amplitude equations.

In section 2 _we have

investigated periodic

roll solutions of the order parameter

equations (7)

in Fourier space. The

description

of modulated patterns is

preferably

done in real _space. It is then easier to

judge

the

importance

of mean-flow effects and the influence of _terms

originating

ooff

Pr = 0.01

I

0Pt 0.06

GAL

0.04 ,'

,/

0.02

Nc

o.oo

a)

~ ~ ~.° ~.~ q ^~

i o

Pr = 0.71 ', GAL Opt

I ',

~~ ,~ _{+ o}

o

~

".,

0 4

0 2 ^{"~ ~}

~

00 ~)

2.0 25 3.0 35 q ⁴⁰

o

~ ⁺ GAL ---0Pt

~ ",

u sv o sv

0fl '.

+ cR o cR

o ',

+ ',

0.6

~~

o

~g ⁺

° ~

~ k

+ v

0 2 ^~~

°

°

6 a S~

O ~

~

0.0

~ ~~

2.0 25 30 3.5 q ⁴⁰

Fig.

I.

a)

Stability domain for Pr

= o.01 calculated by OPE

(dashed line)

and by the Galerkin method

(GAL,

solid

line).

The dotted line represents the neutral _curve

(NC).

To the left of the band center (qc =

3.l16),

_we have E-destabilization, to the right SV, b) Stability domain for Pr

= 0.71

calculated by OPE

(dashed)

and by _a Galerkin method (GAL, solid line). From bandcenter to the left _one finds at first ZZ and for _q < 2.9 E-destabilization. The E-curve

on the right is met by the the SV-curve which bends back towards _qc. Included _are also the knot instability _curves (K) and the

neutral _curve

(NC,dotted).

c) Stability domain for Pr

= 7.oo calculated by OPE

(dashed)

and by _a Gaierkin method

(GAL,

solid

line).

Included _are the ZZ-line (q < qc) an the neutral _curve

(NC).

N°3 AMPLITUDE EQUATIONS FOR THE RBC 425

from the

elasticity

of the roll pattern. ^In this paper we confine ourselves to pattern

containing

wavevectors _{q near} the critical _{wavevector q~}

=

(q~,o).

For that purpose one introduces _a modulation

amplitude A(x)

defined _as:

A(x)

=

f _dq

A(q)e~~~~~CJ~

(10)

D+

The

integration

domain

D+

covers wave vectors with

6q

₌

(q

q~( ^small

compared

to (q~ and

correspondingly

the

amplitude A(x)

varies _{on a} scale of the order

6q~~

The

explicit

construction of _an

amplitude (or envelope)

equation is in

principle

done

by

_a

transcription

of the OPE

(7)

into real space. The various coefficients _a~ have to be

expanded

into

Taylor

series around _q~ and translated into

spatial

derivatives of

A(x) according

to:

(-ib~)~ (-iby)" A(x)

₌

fdq ^Q~

^P" A(q)e~~~~~CJ~

(II)

With _{q qc "~}

(Q,P).

This _can be done

immediately

for the coefficients _al and _a2 of the linear part ⁱⁿ equation

(7).

The

leading

contributions from the cubic part

a3(q

" qc, qi " +q~, _{q2 "}

~qc) reproduce

the

usual

(A(~A nonlinearity

ⁱⁿ real space. The direct attempt to generate derivatives in the cubic term

fails,

because the coefficient _a3 turns out to have

nonanalytic

contributions.

They

_are

immediately

identified

by divergencies,

if the

corresponding

derivative terms are constructed

numerically.

It is

possible

to trace back the

origin

of the

nonanalyticity [26,

27, 19, 37] to the second-order

terms

V2(q)

with _{q e s m} o

(see Eq.(6)).

It will become clear in the

following

that the _non-

analyticity

is

intimately

tied to the

velocity fields,

in

particular

to contributions characterized

by

_a

nonvanishing spatial

_{average across} the convection

cell,

the mean-flow terms.

They

need

a

special

treatment, ^which will be demonstrated in the

following.

CALCULATION _OF THE MEAN FLow PART. The

starting point

for the calculation of the

mean-flow contributions is the

slowly-varying

part of the

inhomogeneous

system

(6)

in the adiabatic

approximation (at

"

o).

This is needed to derive the OPE

(7).

Thus _we consider

the equation:

£(s)V2(s)

=

Inh(s)

_or

£(s)(62, f2,g2)

"

(lo, Ii, Ig) (12)

The

inhomogeneity Inh(s)

derives from N2

(Vi,

Vi

(see Eq. (6)) by pairwise superposition

of contributions with

nearly opposite

wavevectors

(m

q~),

resulting

in slow variations with small wavevector s.

The

explicit

form of

(12)

is

presented

in

Appendix

A

(27).

It is shown there that the horizontal velocities

(u~,

_My

(or correspondingly

the

potentials f, g)

behave

nonanalytically

in the limit _{s -} o. One finds

finally

that the

nonanalytic

terms can be derived from _a

"singular"

velocity potential

gj'~~(z, x)

=

B(x)(z~ 1/4)

with

B(x)

=

f _dsB(s)e~~~

,

(13)

where

B(s)

fulfills the equation:

s~B(s)

_=<

_Ig

_>

(14)

The

symbol

< > denotes _{an average}

weighted

with _a

Hagen-Poisseuille velocity profile

(see Eq. (40)).

After

isolating

the non-smooth part of the solution

V2(s)

and

subtracting

the

corresponding

contribution to the coefficient _{a3 a} smooth

gradient expansion

in cubic order exists.

One arrives at _a

complicated looking

system ^of two

coupled amplitude

equations:

(1 ⁺

lTl~~

+ T2~~~ + T3~(~

~tA

_#

E

(I ieib~ e2b] e3b(

+

ie4b(

₊

iesb~b()

^A

+

(~

b~

)b(

^~ _{+ in}

al

₊

r2bj

₊

r3b]b(

A

~~

(A(~A-iai(A(~b~A-ia2A~b~A*

a3(A(~b(A a4A~b]A*

_a5

(b~A)~

^A*

a6(b~A(~A a7(A(~b(A a8A~b(A*

ag

(byA)~

^A*

aio(byA(~A

+

iaii(A(~b~b(A

+

iai2A~b~b(A*

+ ia13

(b~byA) (byA)

A*

+ ia14

(b~byA) AbyA*

+

iai5A (byA) b~byA*

+ ia16

(b(A) (b~A)

A*

+ ia17

(b(A) ^Ab~A*

+

iai8A (b~A) b(A*

+

ia19(byA(~b~A

+ ia20

(byA)~ ^b~A*

+

ia21(A(~b(A

+

ia22A~b(A*

₊ ia23

(b(A) (b~A)

A*

+ ia24

(b(A)

Ab~A* + ia25A

(b~A) b(A*

+ ia26

(b~A(~b~A

isi

AbyB

_s2

(b~A) (byB)

_s3

(byA) (b~B) s4Ab~byB

,

(isa)

(b(

+

b()

B

=

qiby A

b~ ^~

b(

^A* + _c-c- +

q2b~ (iAb~byA*

+ _c-c-

2qc

+q3by (iA*b(A

+

c-c-)

+ q4

(A*b(byA

+

c-c-)

+ q5

(b~A*b(byA

+

c-c-)

+q6

(b(A*b~byA

+

c-c-)

+ q7

(b(A*byA

+

c-c-)

+ q8

(A*b~b(A

⁺

c-c-)

+ qg

(b~A*b(A

+

c-c-) (15b)

One should

keep

in mind that the derivative _terms

correspond partly

to the horizontal

gradients

in the

Boussinesq equation (1),

but _are also

generated by expanding

the linear

eigenvectors

in powers of

(q~ q]),

which _are involved in the derivation of the OPE

(7).

The numerical values of the coefficients have been calculated _as rational functions in the Prandtl number

using

Galerkin methods and _are available from the authors _upon request.

In the

following

_we shall refer from time to time to the

simplest

form of the

amplitude equation including

^mean-flow effects [27,

38],

which is obtained from

(15) by disregarding

all

higher

derivative terms I-e- with _{e~, rj, a~ e}

o), namely:

btA=EA+(~ _b~-

^~

_b(

2

A-(A(~A-isiAbyB

,

(16a)

2q~

(bj

+

b()

B

=

qiby

(A

b~ ~b(lA*

⁺

^{-c-j (16b)}

2qc The coefficients are

given

in that _case

by:

(

= o.38476, _{si "} o.01741 +

°'°°)~~,

~

-14.45439 Pr ~~~~

~ 0.27149

Pr~

_{0.00183 Pr +} 0.00323'

N°3 AMPLITUDE EQUATIONS FOR THE ABC 427

Note that after _a

rescaling (B

_-

B/si) only

the combination siqi determines the

coupling strength

to the mean-flow.

In order to

display

the

amplitude equations

in the usual form _we have rescaled _time in terms of _a characteristic time To (39, 34] ^{and the}

amplitude

A in terms of

(cr(qo

"

qc)( (see (8)).

^A

strictly periodic

roll pattern with _q

= q~

corresponds

to a solution

(A(x)(~

_{+ E} and

B(s)

₊ 0.

With the _use of that convention the convective heat transport H normalized with respect to the conductive _one

(H

+ Nu I, Nu: Nusselt

number,

_{[3]) near} threshold reads _as follows:

~

l ciP

~+

c2P-2

^{' ~~~~}

The numerical _constants obtained within _our calculational scheme _agree quite well with values

presented

in the literature (co " 2441.68, _{cl "} 6.74845 x

10~~,

_c2

" 1.18956 _x 10~~

[39,

3]),

The derivation of the

amplitude equations (15)

is based _{on a}

systematic expansion

_up to cubic order in the convection

amplitude A,

which behaves _near threshold _as E~/~. The fact that

we have

kept

_more derivative terms than is

usually

done needs

justification. Indeed,

if

lengths

in x-direction _are scaled like

E~~/~,

in

y-direction

like

E~~H

and time like E~~, _one obtains at

leading

order in _E the conventional

NWS-amplitude equation.

All terms

+~ T~,e~, r~, a~ would lead to

higher

order terms in _E and _are therefore

disregarded.

Also the

amplitude

B

plays

_no role at that order.

The

remaining higher-order

terms _are of various nature.

Firstly

_one has corrections to the time-derivative terms _on the left-hand side of

equation (Isa) (coefficients

_T~), which _are of rather minor importance.

Secondly

in the linear operator one has _on the

right-hand

side of

(Isa)

corrections

corresponding

to modifications of the

parabolically-shaped

neutral _curve

(coefficients

_e~ and _r~

). Especially

the terms

involving al

and

al

_are needed to describe

properly

the neutral _curve for wavevectors not

directly

_{near q~.} Furthermore derivative terms in the cubic terms of

(Isa)

_are

included,

which will be motivated below in _more detail. Of

particular importance

is the

coupling

to the mean-flow

amplitude

B

produced by

the four last terms in

(Isa),

where B is determined from

(lsb).

It is obvious that the inversion of the

Laplacian

_on

the left-hand side of

equation (lsb)

leads to

nonanalytic long-range velocity

fields.

In the

following

section 4 it will be shown that the

amplitude equations

in the form

presented

in equations

(15)

lead to _a

satisfactory

description of

stability regimes

of rolls with respect to

long-wavelength

disturbances when

compared

with the

rigorous

results

(see

Sect.

2).

More

specifically

it will turn out that

high-order

derivatives of the type

al, _b~b( (see

the coefficients all

a26)

_{are, e-g-, necessary} for the calculation of the

SV-instability boundaries,

_e-g-, ^for

Pr = 0.71.

Anticipating

^{the detailed discussion in} section

4,

_we want to

emphasize already

here that the

importance

^of

higher-order gradients

cannot be assessed

by simple

_power

counting

arguments with respect to _E.

There exist also

consistency

_reasons,

why

derivative terms in the cubic part ^of

equation (Isa)

should _appear. The

equation

for B contains

by

construction the

singular

contributions of the horizontal

velocity

fields. Their separation ^from the

analytic

_parts is _not unique. One could, e-g-, add

polynomials

in

b~, by multiplied

^with

(b]~

₊

b(~) ^applied

^to (A~( on the

right-hand

side of

(lsb).

The

corresponding nonsingular

contributions for B _can be

directly

calculated and inserted into

(Isa) leading

to modified derivative terms in the cubic

order, by

which the

changes

in the

B-equation

_are

compensated. By

this method _we could

partially

redistribute the coefficients. In any case terms _+~ a~ have to appear, as

they

describe also the internal

"elasticity"

of the rolls.

Let _us summarize what has been achieved _so far. An

amplitude equation systematically

_up

to cubic order in A without additional

assumptions

with respect to

length

and time scales has

been derived. The number of derivative _terms

kept

_can be

judged by comparison

with

rigorous stability

calculations and

by consistency

requirements. In that respect ^the

coupled amplitude equations

_{serve as a} kind of normal form

description [40-42]

^{of the}

stability regimes

of ABC

near threshold.

They

_can be used _{as a}

simplified

version of the

Boussinesq

equations in _some cases, but _{are not}

directly

accurate to some definite order of _a small parameter

[32].

4.

Stability analysis

of roll solutions.

In this section _we shall examine the

stability

of the roll solutions Ao

"

Fe~Q~

_{of the AE'S}

(15) against long-wavelength perturbations

of the form

6A

= e~Q~

(cie~(~~

^{~+~YYJ + c2e~~(~~}

+~YY))

_e"~

(19)

The

quantity Q

+ q qc denotes the distance from band center. The

stability analysis

_can

in

principle

at the best

reproduce

the

rigorous

Busse Balloon but

competing

destabilisation

mechanism, namely

mean-flow and

higher

derivative terms can more

easily

be

disentangled.

The calculation of the

amplitude

F is

straightforward

and when 6A is inserted into

(15),

the

growth

rate _{a can}

easily

be determined from _a 2 x 2

eigenvalue problem.

We _express the modulation wavevectors _s~ and sy ⁱⁿ

polar

coordinates

s~ = Scos

8,

_sy

= S sin

8, (20)

and

keep only

the

leading

terms in S The various

stability

boundaries _E~tab

(a

₌

0)

are then determined

by

the solutions of _a

quadratic equation

in _E

(S~

is factored _out ):

C(~J

(Q, 8)E~

+

C(~J(Q, 8)E

₊

C~°~(Q, 8)

= 0

(21)

The coefficients

C(°

can be calculated without

difficulty

from the

coupled amplitude

equations

(15):

C~°~

(Q, 8)

= -6 Q~

cos~8(~~ (22a)

C(~~(Q, 8)

₌ 2

cos~8(~

₊

Q(-2 j2 ^sin~8

~c

+

(6

ei

f~

6 _ri + 4 _al

f~) ^cos~8

+ 4

sin~8 cos~8si

_qi

f~) (22b)

C(~J(Q, 8)

=

(-2a7

2_{e3 +}

2a8) ^sin~8

+

(-2

_{a3 + 2 a4} 2 _{ei al} +

a2~ ei~ ai~ 2e2) ^cos~8

+ -2

sin~8

^{~~ ~~} ₊

(2

_{a2 si qi} 4 _{si q2} 2 _{al si qi} + 4_{q3 si} 2 _{ei si} qi

cos~8 sin~8

qc

+

Q(... (22c)

We have not detailed the

lengthy

and

complicated

contribution _to C(~~ linear in

Q,

^which

involves all coefficients of the AE'S

(15). They

_are needed _{to ensure} that the

stability

boundaries calculated from the OPE _are

reproduced rigorously

to order

O(S~)

and linear in

Q.

The standard destabilization mechanisms _are obtained

easily

from equation

(21)

and the

explicit expressions

^{for the} coefficients

C(°

N°3 AMPLITUDE EQUATIONS FOR THE ABC 429

The Eckhaus

stability boundary (8

=

0)

_E~tab +

EEck(Q),

which starts

quadratically

in

Q

from band _{center (E}

= 0,

Q

"

0)

is

given

_up to order

tJ(Q~) by:

c(I)(Q

e

~) ~j2Q2

~~~~~~~

C~~~(Q,

#

7r/2) (1 Q(~~3/~~

_2£l1

3el))

^~~~~

In comparison with the neutral _curve

(Eneutr(Q)

_"

f~Q~

+ the _curvature of _{EE~k at}

Q

= 0

is

larger by

the well-known factor three.

The

zigzag-instability

line

Ezz(Q) (8

₌

7r/2)

starts with _nonzero

slope

at bandcenter

(Q

=

o)

in contrast to the E~line and is

given by:

~~~~~~

~~~~~~

=

~~~ ~qisi

₊

qc(~+

a7

a8)

^~~~~

The last term in the denominator of

(24)

is

negligible

for not too

large

Prandtl numbers Pr 5 10 and _a calculation based _upon

equation (16)

is sufficient.

The

general

expression for _Ezz

(Q)

in

equation (24)

is

equivalent

to the criterion

given

in

[39],

when the numerical expressions for the coefficients _are introduced.

(q

_qc 10.76

0.073Pr~~

₊

0.128Pr~~

~~~

qc 0.166 +

23.04Pr~~

₊

6.196Pr~~

^~~~~

The

SV-instability

line

Esv(q),

which _emanates from

bandcenter,

is

given by

the maximum of the

following expression

with respect to

8,

which is

easily

obtained from the ratio of C~°~

and

C(~J

~~~~~'~~

cos2

8(1 3Q(3r3 /f~

a~~-~e~~~ ~iqi

sin~ 8)) 3Qqp~ ^{sin~ 8'}

^~~~~

It is evident that the SV-destabilization _can dominate the Eckhaus _process

only

for

Q

_> 0 _as

a result of contributions of order

tJ(Q~)

from the denominator.

An interesting question _concerns the

back-bending

of the SV-line for Prandtl numbers Pr _m I.

In

particular

the value and

slope

at q = q~

(Q

=

0)

is obtained

analogously

to

(24)

^from the relation

Esv(Q,8)

=

-C~~J(Q,8)/C(~J(Q,8). By

minimization with respect to 8 _we find

always

8 _to be _near ^~ for Pr _near I and

Q

_{near zero.} For the

simplified

version

(16)

_a

4

backbending

of the SV-curve cannot be achieved.

Because of the

interplay

of the

higher

derivative terms, ^which

decisively

_govern the

stability

boundaries for Pr _m I, it

might

be

questionable

if numerical model simulation of

amplitude equations

_or their

isotropic

counterparts

(Swift-Hohenberg equations) (see Appendix B),

where

typically

_many

higher

order-terms _are

neglected

_[43] can be

reliably

related to the

experimental

situation.

COMPARISON _{BETWEEN ORDER PARAMETER AND AMPLITUDE EQUATION.} In this Sectiou

we want to compare in _more detail the

stability

domains calculated from the

amplitude

_equa- tions

(15)

and the OPE

(7).

Our main _concern is the influence of the additional

approximations

involved in the derivation of

amplitude equations, namely

the truncated expansion ⁱⁿ _powers of

(q q~), leading

to the derivative terms in

(15). By

construction all three methods referred to in this paper

(rigorous

Galerkin [3], ^OPE and

AE)

are equivalent ^for small _E and _q m q~, if

one considers

long-wavelength

disturbancies (s( ^< qc of _a

periodic

pattern. It is therefore not

surprising

that

directly

above

threshold,

in _{a narrow}

region

around _{q~, a}

satisfying descrip-

tion of the

corresponding stability

boundaries

(E, ZZ,SV)

is

always

achieved

by

_means of the

amplitude equation.

For the detailed discussion it will be

advantageous

to consider

separately

the different values of the Prandtl number used in

figures

la-lc. The

figures

_may also be

consulted,

when

judging

in

principle

the _range of

validity

of the APE for

increasing

values of _E, because _one is limited in any case

by

the

reliability

of the OPE. One also should

keep

in mind that there _must

always

exist _a

description

in terms of the

AE,

which coincides _up to

arbitrary

_accuracy with the

OPE,

if

sufficiently

_many derivative terms _are

kept,

since all

nonanalytical

contributions _are absorbed in the

B-equation (lsb).

Let _{us start} with the _extreme low Prandtl number _case

(Pr

= 0.01, _see

Fig.

la for the

rigorous results).

The _upper part of the

rigorous stability regime

is not accessible

by

^{the OPE} and the AE. The lower part ^{of the}

stability

domain calculated from the AE is shown in

figure

2a

(solid line).

Included _are the results from the OPE

(triangles,

_see

Fig. la).

One _sees that _{near q~}

=

3.ll the

amplitude

equation works quite well. For (q q~( ^> 0.I the

stability boundary

starts to deviate downwards from the OPE

(triangles).

A _more detailed

analysis

shows that with

increasing

distance from _{q~ some} of the

higher

derivative terms have _a rather

disadvantageous

effect. The best agreement ^{between OPE} and the

amplitude

equation ^is

obtained,

if _some cubic derivative terms

(a~,

^I = 21

26)

ⁱⁿ

(Isa)

_are set _zero. One gets ^the short-dashed _curve, which coincides in the whole

q-regime

shown

perfectly

with the OPE results. In

comparison

to the OPE _one concludes that _some nonlinear derivative _terms

apparently

balance each

other,

_one would have to include _even fourth-order derivative terms in the cubic part ^{of the}

A-equation

to compensate ^the third-order

derivatives,

which lead to unwanted effects for _q < q~.

Finally

_we would like to

mention,

that

by keeping only

the

leading

terms _as done in

equation (3.),

one gets an E~line

(crosses)

in the whole

q-regime

in contrast to the

rigorous

^SV-line to

the

right.

One has to include at least the

rib(A

_term of the linear operator ^{and obtains} a

stability boundary

which deviates from the solid _curve

only quantitatively.

Let _{us now} turn our attention to intermediate Prandtl numbers

(Pr

= 0.71, _see

Fig.

lb for

rigorous results).

The boundaries

(solid line)

of the

stability

domain calculated from the AE

(15)

is shown in

figure

2b.

Clearly

_one has

satisfactory

agreement ^with

rigorous

results and the OPE

(triangles)

_up to

fairly high

values of _E

+~ 0.8 _near the bandcenter. In

particular

the

general

behavior of the

backbending SV-line,

which is

typical

for Prandtl numbers

-J I,

(see

e-g-

ii Ii)

fits well and the

slope

at q~ is

reproduced rigorously.

For the SV-curve the

higher

order derivative terms _are of

particular importance.

If for

example

all contributions from the

coefficients _a~, I

= 21,.

,

26 and q~,I

= 4,.

,

9 _are left out _one gets a

stability regime (long dash),

which is

closed,

in contrast to the

rigorous results,

but which agrees

satisfactorily

for

E > 0A _on the

right

and _E > 0.3 _on the left.

If the

higher

derivative _{terms are} left _out

totally,

_as in equation

(16),

the

stability

boundaries

are of the

E~type

for _{all q}

(short

dashed

line).

But _{for q < q~} the agreement with the OPE results is _even then

improved,

because

analogously

to the Pr

= 0.01 _case the _a~, I ₌ 21,.

,

26 contributions _are rather unfavorable for _{q~ q <} 0.5.

Finally

_{we come} to the

large

Prandtl number _case

(Pr

₌

7,

_see

Fig.

lc for the

rigorous results).

The solid line in

figure

2c describes

again

the lower

stability

limit obtained from the full

amplitude

equation

(15).

One should

keep

in mind that the

limiting

_curves

(E)

_{for q > q~}

are in

principle irrelevant,

because the

short-wavelength

CR

instability

dominates

(see Fig. lc).

The _new feature is introduced

by

the ZZ-line for _{q < q~.} The best

description

of the

rigorous

results is achieved

by

the

simplest

form of the

amplitude

equations in

(16), corresponding

to the

analytical expression (24).

The ZZ-line increases

nearly linearly

_as function of _q. Because with

increasing

E lower values of _{q come} into

play,

the inclusion of the

higher

derivative _terms