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Two Dimensional Convection Patterns in Large Aspect Ratio Systems
Lecture Notes in Num. Appl. Anal., 5, 205-231 (1982) Nonlineur PDE in Applied Science U.S.-Jupun Seminur, Tokyo, 1982
Two Dimensional Convection Patterns i n Large Aspect Ratio Systems*
Alan C . N e w e l l Program i n Applied Mathematics University of Arizona, Tucson
85721
This work was supported i n part by DOA Contract DAAG29-82-C-0068.
205
206
c. NEWEI.1.
Alan
F i g u r e 1 below i s t a k e n f r o m a v i s u a l i z a t i o n of a n e x p e r i m e n t of G o l l u b a n d M c c a r r i a r 111 in which t h e y f o l l o w t h e t i m e development o f a c o n v e c t i n g h o r i z o n t a l l a y e r of f l u i d of d e p t h techniques.
h
t h r o u g h t h e u s e of l a s e r D o p p l e r
The d o t s a r e p o i n t s where, a t a depth of
h/4
from t h e t o p of t h e
l a y e r , t h e v e l o c i t y component p a r a l l e l t o t h e l a r g e r s i d e of t h e box is z e r o a n d mark t h e b o u n d a r i e s of t h e c o n v e c t i v e r o l l s .
The e x p e r i m e n t is c o n d u c t e d i n a
r a n g e of R a y l e i g h number where n o n l i n e a r s t a b i l i t y t h e o r y g u a r a n t e e s t h a t i n a h o r i z o n t a l l a y e r of i n f i n i t e e x t e n t , s t r a i g h t p a r a l l e l r o l l s b o t h e x i s t a n d a r e stable.
.. .. .. .. .. .. .. .. .. .. .. .. .. .<..'.' : ... ... ... ........... ... .! t.' ., *. .....'..' .. '...** . ...... .*:. . . . . . . .: .* ..; . ......... ... 1,;:::::
10
.
.
.
.
*
.
.
I
.
.
.
.
::::;!;;:*.s.:
... ...:...:...:...:...:...:...i .:. I. ... .,... ...: ..:. :;:,: . . . . .., .,,. .. .. .. .. .. ..... . . . . ........ .t*,..*'
::..;,t
2 .
... I ................. ............. ....... , ... .. . . ...... . . .r.t.! . i .
, . . !: , ::::.' :/a. ! :!:::::::;:I:.+, ,.* ..;:::;::;:::;::.+* . : : : : :: : : : : ;!::;!;/*. I
0
5
0
10
1s
x (cm) Figure 1 :
D o p p l e r Map Of The V e l o c i t y F i e l d F o r A S t a b l e C o n v e c t i v e Flow 256 Hrs. After R a y l e i g h Number Was I n c r e a s e d To 2.05 Rc. ( H o r i z o n t a l D i f f u s i o n Time, 40 HKS.)
However, t h e e x i s t e n c e of o r i e n t a t i o n a l d e g e n e r a c y a n d a band of s t a b l e r o l l wavenumbers t o g e t h e r w i t h t h e f a c t t h a t t h e rolls a l i g n t h e m s e l v e s p e r p e n d i c u l a r t o a l l l a t e r a l b o u n d a r i e s makes t h e s e s o l u t i o n s u n a t t a i n a b l e .
The p a t t e r n s t h a t
a r e s e e n a r e much more c o m p l i c a t e d n o t only c o n t a i n i n g Curved r o l l s b u t a l s o e x h i b i t i n g many d i s l o c a t i o n s .
F u r t h e r m o r e , i t is n o t c l e a r w h e t h e r a n d , i f s o ,
u n d e r what c i r c u m s t a n c e s t h e p a t t e r n a c h i e v e s a t i m e i n d e p e n d e n t e q u i l i b r i u m . I n d e e d , i n some cases, G o l l u b a n d M c c a r r i a r h a v e s e e n p a t t e r n s which are s l o w l y t i m e dependent o v e r many h o r i z o n t a l d i f f u s i o n t i m e s .
The f a i l u r e t o r e a c h a
207
Two Dimensional Convection Patterns s t e a d y s t a t e was a l s o noted e a r l i e r by Ahlers and Walden [Z] who observed t h a t the e f f e c t i v e thermal c o n d u c t i v i t y of a l a y e r of convecting f l u i d helium remained n o i s y f o r a l l Rayleigh numbers above
RC
f o r sufficiently large aspect
ratios. Our g o a l i n t h i s paper i s t o develop a t h e o r y t o d e s c r i b e t h e s e p a t t e r n s . We s t a r t with the o b s e r v a t i o n t h a t almost everywhere i n the convection f i e l d a l o c a l wavevector is d e f i n e d and v a r i e s slowly o v e r the box. the wavevector is tangent.
A t the boundaries,
Therefore we e x p e c t t h a t t h e r e e x i s t s l o c a l l y
p e r i o d i c s o l u t i o n s defined by f ( 0 ; A , R) when
f
is 2n - p e r i o d i c i n 0 and
VO = k(X, Y , T)
(1)
( V = (a/aX, a/aY)) is a slowly varying f u n c t i o n of p o s i t i o n c o o r d i n a t e s and t i m e .
X, Y, T, t h e h o r i z o n t a l
Indeed we know from the work of Busse 131 t h a t
such s o l u t i o n s a s f u n c t i o n s of 0 e x i s t and, a s we d e s c r i b e l a t e r , have c e r t a i n s t a b i l i t y properties.
I n o t h e r words, while t h e f i e l d v a r i a b l e
d
(wavevector), which t o g e t h e r with a knowledge of
f as f u n c t i o n of
0 d e s c r i b e t h e p a t t e r n , vary o v e r d i s t a n c e s of t h e s i z e of t h e box.
a s p e c t r a t i o c 2 , t h e r a t i o of t h e r o l l wavelength d dimension
L
of t h e box, is the
The Rayleigh number
RC,
R
only small
(2.
v a r i e s over
A (amplitude)
d i s t a n c e s of the o r d e r of the roll s i z e , t h e parameters and
f
The i n v e r s e
h) to the l i n e a r
parameter which e n t e r s t h e theory.
can be an o r d e r one amount above i t s c r i t i c a l value
the value a t which the p u r e l y conductive s t a t e becomes u n s t a b l e , although
i t must be less than t h e Rayleigh number a t which t h e s t r a i g h t p a r a l l e l r o l l
s o l u t i o n becomes u n s t a b l e .
The i d e a s we a r e about t o d e s c r i b e a r e c l o s e l y
r e l a t e d t o those of Whitham (41 who i n t h e l a t e s i x t i e s developed a theory t o d e s c r i b e f u l l y n o n l i n e a r , almost p e r i o d i c w a v e t r a i n s . A s a f i r s t a t t e m p t we s h a l l use model e q u a t i o n s , t h e use of which
f a c i l i t a t e s the a n a l y s i s by making c a l c u l a t i o n s e x p l i c i t b u t which c a p t u r e what
we b e l i e v e a r e some of the e s s e n t i a l f e a t u r e s of the Overbeck-Boussinesq equations.
We d e r i v e e q u a t i o n s f o r the slow v a r i a b l e s
d,
A which
( i ) show
t h a t a l l t h e s t a b i l i t y c r i t e r i a d e r i v e d by Busse and h i s c o l l e a g u e s [31 f o r
208
Alan C. N ~ W E L I . ( i i ) r e d u c e i n t h e l i m i t of small
s t r a i g h t p a r a l l e l r o l l s h o l d l o c a l l y and
R-RC
t o t h e Newell-Whitehead-Segel
[ 5 ] equations.
I n a d d i t i o n we show t h a t t h e
e f f e c t of c u r v a t u r e is t o d r i v e t h e r o l l p a t t e r n s toward a s t a t e in which t h e l o c a l wavenumber assumes a c o n s t a n t v a l u e a l m o s t e v e r y w h e r e , a r e s u l t c o n s i s t e n t w i t h a r e c e n t r e s u l t of Pomeau a n d M a n n e v i l l e [ 6 ] f o r a x i a l l y s y m m e t r i c r o l l s . T h i s is done by ( i ) p r o v i n g t h a t u n d e r c e r t a i n n a t u r a l boundary c o n d i t i o n s , t h e macroscopic e q u a t i o n s f o r t h e phase, v a l i d f o r times up to t h e h o r i z o n t a l d i f f u s i o n t i m e , are d e r i v e a b l e from a Lyapunov f u n c t i o n a l e v e n though t h e f u l l m i c r o s c o p i c e q u a t i o n s f o r t h e model n e e d n o t b e a n d ( i i )
examining t h e n a t u r e
of t h e s t a t i o n a r y s t a t e s which would be r e a c h e d on t h i s t i m e . But t h i s is n o t t h e whole s t o r y a s such p a t t e r n s i n a n d of t h e m s e l v e s c a n n o t s a t i s f y a l l t h e boundary c o n d i t i o n s .
I n between t h e p a t t e r n s ,
In
d i s l o c a t i o n s are formed and s o l u t i o n s d e s c r i b i n g t h e s e s t r u c t u r e s a r e g i v e n . l o o k i n g a t t h e s e s o l u t i o n s , i t is c l e a r t h a t t h e t i m e s c a l e which t h e t o t a l
s y s t e m would n e e d t o r e l a x to e q u i l i b r i u m is n o t s i m p l y t h e h o r i z o n t a l d i f f u s i o n time L
2
/V
b u t is t h i s time s c a l e m u l t i p l i e d by t h e a s p e c t r a t i o
L/d.
Finally,
we i n c l u d e i n t h e model a t e r m which t a k e s a c c o u n t of t h e "mean d r i f t " which o c c u r s i n s y s t e m s of low t o m o d e r a t e P r a n d t l number.
The p r e s e n c e of t h i s
e f f e c t was f i r s t p o i n t e d o u t by S i g g i a a n d Z i p p e l e u s 171, a n d c a n c a u s e marked change i n t h e b e h a v i o r of t h e s y s t e m .
I s h o u l d p o i n t o u t t h a t t h e i d e a s we d i s c u s s h e r e c a n b e e x t e n d e d t o i n c l u d e s i t u a t i o n s where t h e b a s i c f l o w
is c o n s i d e r e d t o b e a
f
is m u l t i p e r i o d i c .
2n p e r i o d i c f u n c t i o n i n e a c h of
p h a s e s 0 where 80 = $ a n d which a l s o depends on i i i p a r a m e t e r (or set of p a r a m e t e r s )
n
In t h a t case
f
n
amplitudes
Ai
and a
R.
A more comprehensive p a p e r is b e i n g w r i t t e n j o i n t l y w i t h my c o l l e a g u e Mike
C r o s s who s t a r t e d me t h i n k i n g a b o u t t h e s e problems a g a i n d u r i n g a most p l e a s a n t l e a v e s p e n t a t t h e I n s t i t u t e of T h e o r e t i c a l P h y s i c s a t t h e U n i v e r s i t y of C a l i f o r n i a i n Santa Barbara.
I am most g r a t e f u l f o t h e i r h o s p i t a l i t y a n d t o t h e
e n e r g e t i c l e a d e r s h i p of P i e r r e Hohenberg who was a c o n s t a n t s t i m u l u s .
209
Two Dimensional Convection Patterns 2.
The p r e d i c t i o n s of p r e s e n t t h e o r i e s :
Consider
w(X, Y , T )
g i v e n by
a ----w aT
on
-a)
<
X, Y
<
+
(V2
+
1)2w
-
Rw
+
w2w*
=
0
( 2)
-, The p r i n c i p a l r e a s o n f o r c h o o s i n g t h i s model is t h a t it
possesses a simple f u l l y nonlinear periodic s o l u t i o n e
io,
+ + X, X = ( X , Y ) .
+
0 = k
The " c o n d u c t i o n " s o l u t i o n
modes of t h e form
w(X,
whenever R
>
Y, T) = We
ic
(3)
2
p l a n e is g i v e n by
( R , k)
is u n s t a b l e t o
k
f i i n ( k - 1 ) 2 = 0 , which v a l u e i s r e a l i z e d when
n e u t r a l s t a b i l i t y c u r v e in t h e
w = 0
($1 R
=
k = 1.
The
We
(k2-1)2.
o b s e r v e t h e s y s t e m i s d e g e n e r a t e in t h e s e n s e t h a t any mode of t h e form ( 3 ) with
lcl
= 1 w i l l grow a t t h e same r a t e when R
>
0.
The n o n l i n e a r s a t u r a t i o n
of t h e l i n e a r l y u n s t a b l e modes i s d e s c r i b e d by t h e S t u a r t - W a t s o n [ 8 1 e q u a t i o n
- w%*
WT = Rhl
(4)
which in t h i s c a s e is a n e x a c t s o l u t i o n f o r ( 2 ) b u t which g e n e r a l l y i s o n l y t r u e f o r v a l u e s of
R
p r o p o r t i o n a l t o JR
near its c r i t i c a l value
- RC
. Because
RC
=
0
and f o r amplitudes
W
of t h e o r i e n t a t i o n a l d e g e n e r a c y , i t i s i n d e e d
n a t u r a l t o look f o r s o l u t i o n s of t h e form
a n d , in t h e small
R
limit, we c a n show
It is a n e a s y m a t t e r t o s e e t h a t t h e o n l y s t a b l e s o l u t i o n of t h i s set is t h e
single r o l l s o l u t i o n
- iZI
W1 = JR
e
k
,
w
j
= O , j # I .
(6)
Alan C. NEWELL
210
So, j u s t a s i n t h e case of the Rayleigh-Benard problem under a v e r t i c a l l y
symmetric, a p p l i e d temperature f i e l d with moderate P r a n d t l number u / K > 0 ( 1 ) , s i n g l e , p a r a l l e l r o l l s a r e p r e f e r r e d . pointed o u t , no d i r e c t i o n i s chosen a p r i o r i .
But, a s we have
Therefore when
R
is suddenly
r a i s e d from s u b c r i t i c a l t o s u p e r c r i t i c a l v a l u e s , the system a c t s a s a n o i s e A t any p a r t i c u l a r l o c a t i o n i n a l a r g e h o r i z o n t a l l a y e r , i t w i l l
amplifier.
choose among t h e wavelengths of the noise f o r one c l o s e t o n o t make any choice among d i r e c t i o n s .
kC
1
but i t
W i l l
Therefore u n l e s s the experiment i s
c a r e f u l l y c o n t r o l l e d ( a s i n t h e case of the Busse-Whitehead [ 9 1 and WhiteheadChen [ 101 e x p e r i m e n t s ) , r o l l s of approximately the c r i t i c a l wavelength b u t of d i f f e r e n t d i r e c t i o n s w i l l s p r i n g up in d i f f e r e n t p l a c e s .
This d i r e c t i o n a l
d i v e r s i t y is even more a c c e n t u a t e d due t o t h e i n f l u e n c e of a c l o s e d boundary.
(i is t h e
= 0
A t each boundary l o c t i o n ,
u n i t normal t o t h e boundary) and
t h e r e f o r e i f n i s continuous, r o l l s of a l l d i r e c t i o n s k a r e e x c i t e d .
Now, once
t h e r o l l d i r e c t i o n is chosen a t a p a r t i c u l a r l o c a t i o n , i t becomes more s t a b l e a g a i n s t l i n e a r d i s t u r b a n c e s of r o l l s of o t h e r d i r e c t i o n s t h e more i t grows So, f o r e a r l y times, one f i n d s a f l u i d l a y e r
towards i t s s a t u r a t i o n amplitude.
resembling a sea of q u a s i s t a b l e patches and somehow t h e f l u i d has t o f i n d a way t o resolve the i n c o m p a t i b i l i t i e s between them. There is a l s o a bandwidth degeneracy. f i n i t e bandwidth of wavenumbers
WT and W
+
JR
-
(k2
-
1)’
k
= (R
It i s e v i d e n t t h a t f o r R
can be e x c i t e d .
-
(k2
-
Indeed f o r
>
0, a
t 1,
- W%*
1)’)W
a s y m p t o t i c a l l y i n time.
We may t e s t t h e ( l i n e a r )
s t a b i l i t y of these s o l u t i o n s by s e t t i n g
w = eikX (W
+ b l e ILX+ IMY
+
b2e
-iLX
-
iMY
1
whereupon we f i n d a f t e r some c a l c u l a t i o n t h a t t h i s s o l u t i o n is u n s t a b l e i f
2A2(A2)’(L2~2)+”2(A2)’’(
2k2L2)+4k2L2(A2)’2 > 0
(5)
Two Dimensional Convection Patterns where A 2 = R
-
(k2
-
1)*
and
(A2)'
. The f i r s t
dA2/dk2
=
21 1 c l a s s of
i n s t a b i l i t i e s occurs f o r
L = 0,
M
f
0
2 2 ' B = (A ) ( A )
and
These correspond t o t h e zig-zag a wavevector with wavenumber k
>
0.
i n s t a b i l i t i e s discovered by Busse and a r i s e when
<
g i v e s i t s energy t o e i t h e r one of two modes ( k , f circle.
+
M = 0
>
+ k2(A2)"
2A2(A2)"k2
0) i n t e r a c t s with and
2 k ) l y i n g on the u n i t
41 -
The second c l a s s of i n s t a b i l i t i e s occur f o r A2(A2)'
>
1 (in which case ( A 2 ) '
and
0,
o r e q u i v a l e n t l y when $kB>O
-
where B = A 2 dA2/dk2 = -2(k2
1)(R
-
(k2
. This
-
(7)
is t h e Eckhaus
i n s t a b i l i t y and occurs when a r o l l has too s m a l l a wavelength.
It is useful t o
summarize these r e s u l t s by way of f i g u r e 2.
k8, A
I
I
I
I
I
R k Figure 2: Graphs of A , kB and R Busse Balloon
VS.
k. and t h e
212
Alan C . NEWELL
S o l u t i o n s can e x i s t f o r k
< k <
L
k
R
b u t a r e s t a b l e o n l y when kC
s h a d e d a r e a i s known a s t h e B u s s e b a l l o o n .
<
k
<$
When d e r i v e d i n t h e c o n t e x t of t h e
e q u a t i o n s i t i s somewhat more c o m p l i c a t e d .
f u l l Overbeck-Boussinesq
. The
The r i g h t
hand c u r v e which is t h e boundary w i t h t h e Eckhaus i n s t a b i l i t y i s r e p l a c e d f o r l a r g e P r a n d t l number by a boundary t o a n i n s t a b i l i t y t o r o l l s i n t h e
On t h e o t h e r h a n d , f o r s m a l l e r P r a n d t l numbers, t h e
perpendicular direction.
Eckhaus s t a b i l i t y boundary becomes l i n k e d w i t h t h e s k e w - v a r i c o s e i n s t a b i l i t y which i s a v a r i a n t of t h e f o r m e r when mean d r i f t e f f e c t s are i n c l u d e d . show how t o i n c l u d e t h e s e i n t h e model.
k
boundary need n o t remain a t
=
We w i l l
We a l s o remark t h a t t h e l e f t hand
kC = 1 b u t can bend l e f t w a r d d e p e n d i n g on
The B u s s e b a l l o o n ( o r Busse w i n d s o c k ) , t h e r e g i o n of s t a b l e
P r a n d t l number.
p a r a l l e l r o l l s i n the R, p Since f o r R
>
R
C'
-
v/K, k
p l a n e i s g i v e n in r e f e r e n c e [ 3 1 .
there i s an order JR
i n a direction parallel t o
-
$ a n d a n o r d e r 4JR
R
C
band o f a l l o w a b l e wavenumbers
- RC
band i n t h e p e r p e n d i c u l a r
d i r e c t i o n , i t i s n a t u r a l t o i n c l u d e a r i c h e r c l a s s of s o l u t i o n s which h a v e a n a l m o s t p a r a l l e l r o l l s t m c t u r e by l e t t i n g
-Y. -t ) e i x
w ( x , Y, T) = WC;,
2 4 where x = u X, y = uY, t = p T w i t h N
(8) 4
x
=
R
-
RC
<<
1. This g i v e s t h e N e w e l l -
Whitehead-Segel e q u a t i o n (51 which f o r t h e model ( 2 ) i s
aw a7
4.
(21
a+ a;
a
2
2 w 3)
=
xw - w%*
.
(9)
This e q u a t i o n i s c a n o n i c a l f o r d e s c r i b i n g s i t u a t i o n s in which t h e r o l l s a r e a l m o s t p a r a l l e l and
3.
R
i s c l o s e t o i t s c r i t i c a l value.
A new a p p r o a c h :
We w i l l now g i v e a d e s c r i p t i o n t h a t a l l o w s t h e l o c a l w a v e v e c t o r undergo o r d e r one changes c o n t i n u o u s l y b u t s l o w l y o v e r t h e box. w(X, Y , T) where
W
and
ie(X, = We
Y, T)
= VO a r e f u n c t i o n s of t h e s l o w v a r i a b l e s
to
Let ( 10)
213
Two Dimensional Convection Patterns 2
4
2
x = E X , ~ = E Y , ~ = E T
and I / E
2
is t h e a s p e c t r a t i o .
I t i s u s e f u l t o write 9 =
1 7 O(x,
y , t ) whence
E
V O = V+9 x G
T
=
x
= (m,
n)
(k cos JI, k sin J I )
=
(11)
2
2
= E B
k’
t
= E D
We w i l l a l s o f i n d i t u s e f u l t o i n t r o d u c e new c o o r d i n a t e s 9 = a(x, y ) ,
B = B(x, Y )
d e f i n e d by
ax = k Cos $
Bx = -P Sin J,
ay = k S i n JI
By = II Cos JI
The J a c o b i a n of t h e t r a n s f o r m a t i o n from (a, B )
to
( x , y)
is
kll
and
The curves a ( x , y ) = c o n s t a n t a r e . of course, t h e l o c i of c o n s t a n t phase 8 while the B c o o r d i n a t e s a r e the orthogonal t r a j e c t o r i e s and measure d i s t a n c e a l o n g t h e rolls.
by ll
The c u r v a t u r e K
$ . Tne
a
of the
a ( x , y)
=
c o n s t a n t curves is given
c u r v a t u r e of t h e c o n s t a n t curves B
is
K
B
=
-k
3.
I n t h e s e c o o r d i n a t e s , the c o m p a t i b i l i t y c o n d i t i o n s ( 1 1 ) g i v e us t h a t
and
Note t h a t
ma Aeie = e i e ( i e 2 u V 2 Aeie = eie(-k2
+
E4
&)A
+ ie2Dl +
,
r4D2)A
,
214
Alan C . NEWELL
where
a +
D ~ A=
=
2k2
=
2k
2n
+ Ak
+ a + (2+ % ) A ak
+ Akll
a$
,
,
2 aA ak 2 ae ~ + A k ~ - A k / L ~ ,
and
a2
a'
Q 2 A = 2 + 7 A
We now proceed t o determine the e q u a t i o n s s a t i s f i e d by the slow v a r i a b l e s and
the l a t t e r being t h e l e a d i n g approximation t o
A,
loss of g e n e r a l i t y , can be taken t o be r e a l .
u = a. R-A'
W
which, w i t h o u t
Let
... , = R + eS2 + ... . 0 +
E'O
+
We choose t h e sequences [ a n } , {R,) appearing i n the expansion f o r
W.
i n a manner so a s t o e l i m i n a t e s e c u l a r terms In t h i s c a s e , s e c u l a r means t h a t no s o l u t i o n
W2, W 4 e t c .
e x i s t s t o the a l g e b r a i c e q u a t i o n s f o r t h e 0 v a r i a b l e and expanded w = Aeie terms were removed, no 2ll
(20),
(21),
2 [ic a
+
c4
E2W2(o)
+
..., t h e n , u n l e s s
periodic solution f o r
( 1 6 ) , (17) i n t o ( 2 ) g i v e s
(22),
2 at +
+
(k
2
-
I f we had l e f t i n
2 2 2 i e (Dl(k - l ) * + ( k - l ) D l * )
w2
would e x i s t .
the secular Substituting
Two Dimensional Convection Patterns
+
- c4(D2(k2-l)*
-R
0
- r%, -
(k2-1)D2.
+ D12) + i E 6 ( D 1 - D 2 + D 2 * D I ) +
€k4 ....+ k 2 A W 2
What does t h i s e q u a t i o n t e l l u s ? R
0
+ W22)...](A
ic4(2AW4
At
+
215
68D 2
c?W2
+
2
...) = 0 .
O(1)
(24)
= ( k 2- 1 ) 2
and so, t o leading order, the amplitude
is d e t e r m i n e d from t h e " e i k o n a l "
A
equation A A t order
2
= R
E~
2
-
( k -1)
2
.
(25)
we have t h a t
+
(-Ro
(k2-1)2)W2
+ a%, =
R.H.S.
Note t h e f o l l o w i n g i n t e r e s t i n g f e a t u r e .
For
Ro = ( k 2 -
t h e r e f o r e , e v e n though
R
- RC
= 0(1),
A2
is f i n i t e and
t h e o n l y t e r m s on t h e RHS w h i c h
g i v e r i s e t o s e c u l a r b e h a v i o r a r e t h o s e which a r e p u r e l y i m a g i n a r y .
were s m a l l , we would h a v e t o remove a l l t h e terms on t h e
A2
o t h e r hand, i f
On t h e
RHS.
I n o t h e r words, t h e n u l l s p a c e of t h e l i n e a r i z e d e q u a t i o n i s c u t i n h a l f
when
A = O(1).
f o r the amplitude
What t h i s means i s t h a t i n s t e a d of h a v i n g t w o e q u a t i o n s , one
A
a n d t h e o t h e r f o r t h e p h a s e of t h e c o n v e c t i v e p a t t e r n , we
s i m p l y have a s i n g l e e q u a t i o n f o r t h e p h a s e . a l g e b r a l c a l l y from ( 2 5 ) . of small
R
-
RC
The a m p l i t u d e is d e t e r m i n e d
T h i s a l s o means, of c o u r s e , t h a t t h e l i m i t t o t h e c a s e
and small a m p l i t u d e
A
is f a i r l y s u b t l e .
[For workers i n
n o n l i n e a r wave t h e o r y , t h e r e i s a d i r e c t a n a l o g u e between t h i s l i m i t p r o c e s s and t h e l i m i t p r o c e s s one e n c o u n t e r s when one a t t e m p t s t o o b t a i n t h e n o n l i n e a r S c h r o d i n g e r e q u a t i o n from Whitham's t h e o r y . ]
We w i l l c a r r y o u t t h e p e r t u r b a t i o n
a n a l y s i s i n such a way s o a s t o f a c i l i t a t e t h i s l i m i t p r o c e s s .
What we do is t o
u s e t h e e x p a n s i o n ( 2 2 ) which i s a l s o a n a m p l i t u d e e x p a n s i o n t o r e e x p a n d n e c e s s a r y so t h a t we can s i m p l y set W t h e following equations,
2j
= 0, j
>
A
1. We f i n d d i r e c t l y f r o m
if
23)
Alan C. NEWELL
216 uA
- Dl(k2-l)A -
R-A*
(k2-1)
-
2
=
(k2-l)DIA
E4 ii ( A t -
-D$
+ E 4 (D1*D2 + D2*Dl)A = 2 (k -1)Df
+E'D~)
-
0,
(26)
2
D2(k - l ) A
.
Because of t h e s i m p l i c i t y of t h i s model, t h e s e e q u a t i o n s a r e e x a c t .
W e will
f i r s t examine t h e s e e q u a t i o n s w i t h a veiw t o making c o n t a c t w i t h known r e s u l t s a n d t h e n we w i l l discuss aome new consequences.
4.
Connections with previous t h e o r i e s . F o r v a l u e s of
and then
A
t h e phase u
of o r d e r u n i t y , we c a n n e g l e c t t h e RHS of e q u a t i o n ( 2 7 )
R
is g i v e n as f u n c t i o n of
-
k
by ( 2 5 ) .
Equation (26) tells u s about
4
8, a n d i g n o r i n g t h e O(E ) terms can be w r i t t e n as
Az Eae +
g m ~ +
an^=^,
7
or
where
B(k)
-
A2(k) dA2/dk2
.
(30)
I want t o rernark a t t h i s p o i n t t h a t t h e f a c t t h a t t h e s p a t i a l t e r m s h a v e c o n s e r v a t i o n form is n o t a c o n s e q u e n c e of t h i s p a r t i c u l a r model n o r t h e f a c t t h a t i t can be d e r i v e d from a Lyapunov f u n c t i o n a l . (a)
The Busse B a l l o o n h o l d s l o c a l l y .
E q u a t i o n ( 2 9 ) i s e l l i p t i c s t a b l e o r u n s t a b l e ( i n t i m e ) or h y p e r b o l i c u n s t a b l e d e p e n d i n g o n which one of t h e f o l l o w i n g f o u r c a s e s o b t a i n s : 1)
2)
B
<
d 0, =(kB)
B
>
d 0, =(kB)
< >
> <
d 0, x ( k B ) d 0, &kB)
3)
B
4)
B
0;
Elliptic stable
0;
E l l i p t i c unstable
<
0;
Hyperbolic unstable
>
0;
Hyperbolic unstable.
217
Two Dimensional Convection Patterns
These r e s u l t s are s i m p l y t h e same r e s u l t s which a r e d i s p l a y e d in F i g u r e 2,
x, y
e x c e p t now a l l t h e v a r i a b l e s a r e f u n c t i o n s of
and
t.
T h e r e f o r e w e can
s a y t h a t a l l t h e s t a b i l i t y f e a t u r e s we h a d f o u n d when l o o k i n g a t t h e s t a b i l i t y of s t r a i g h t p a r a l l e l r o l l s c o n t i n u e t o h o l d l o c a l l y .
Case (1) above i s t h e
Busse b a l l o o n ; c a s e ( 2 ) i n v o l v e s i n s t a b i l i t i e s which have wavenumber dependence
in b o t h t h e a l o n g a n d p e r p e n d i c u l a r t o t h e r o l l d i r e c t i o n s ; c a s e ( 3 ) i s t h e z i g zag i n s t a b i l i t y a n d c a s e ( 4 ) is t h e Eckhaus i n t a b i l i t y . by t a k i n g t h e l o c a l r o l l w a v e v e c t o r t o be
ae + xd (
A x
Hence f o r B (k, 0);
>
for B
a% +
k B ) T
ax
0, $(kB)>
<
0,
B
a% = -
0
aY
( k , 0)
T h i s can be e a s i l y s e e n
i n which c a s e ( 2 9 ) becomes
.
(32)
0, t h e i n s t a b i l i t y h a s a w a v e v e c t o r p e r p e n d i c u l a r t o
arrd kB >
0 t h e u n s t a b l e modes are p a r a l l e l t o
( k , 0).
The a d d i t i o n of t h e c 4 t e r m i n ( 2 6 ) which i n v o l v e s h i g h e r d e r i v a t i v e s o n l y s e r v e s t o c o n t r o l t h e growth of t h e i n s t a b i l i t i e s a f t e r t h e y b e g i n .
It d o e s n o t
i n h i b i t them a l t o g e t h e r nor d o e s it of i t s e l f t r i g g e r any new i n s t a b i l i t y . The r e a d e r might l i k e t o compare t h i s r e s u l t w i t h what h a p p e n s in n o n l i n e a r wavetrains.
x
and
t
T h e r e , t h e a n a l o g u e of e q u a t i o n ( 2 6 ) is a s e c o n d o r d e r s y s t e m i n and s o it i s t h e e l l i p t i c i t y o r h y p e r b o l i c i t y of t h e s e c o n d o r d e r
o p e r a t o r which d e t e r m i n e s i n s t a b i l i t y o r ( n e u t r a l ) s t a b i l i t y of t h e w a v e t r a i n .
For example, f o r a t r a i n of g r a v i t y waves on t h e sea s u r f a c e , t h e h y p e r b o l i c n a t u r e of ( 2 6 ) c h a n g e s t o e l l i p t i c when t h e r a t i o of d e p t h t o w a v e l e n g t h is l e s s t h a n 1.36. (b)
The Newell-klhitehead-Segel
limit.
To t h i s p o i n t , we have t a k e n v a r i a t i o n s i n t h e d i r e c t i v e s p a r a l l e l t o and
p e r p e n d i c u l a r t o t h e l o c a l r o l l t o be of t h e same o r d e r of m a g n i t u d e .
It i s
c l e a r t h a t i f f o r some r e a s o n t h e l o c a l wavenumber is f o r c e d t o s t a y a p p r o x i m a t e l y c o n s t a n t , t h e v a r i a t i o n s in wavenumber of o r d e r IJ p a r a l l e l
-
to (e.g.
a r e accompanied by v a r i a t i o n s of o r d e r J p i n t h e p e r p e n d i c u l a r d i r e c t i o n (kc
+ pL) 2 +
-
2
( J I J M ) = kc
2
)
.
Near k = 1,
we f i n d t h a t v a r i a t i o n s
p e r p e n d i c u l a r t o t h e r o l l a r e of a n o r d e r of magnitude g r e a t e r t h a n t h o s e p a r a l l e l t o t h e r o l l a n d t h i s l e a d s t o a b a l a n c e between t h e t e r m
1
V
+
k B and
218
Alan C . N F W LII
some of t h e
E~
terms i n th e phase e q u a tio n ( 2 6 ) .
T h i s s i t u a t i o n c e r t a i n l y o b t a i n s when
i s s u f f i c i e n t l y small, f o r t h e n
R
-
( s e e F i g u r e 2) t h e bandwidth of wavenumbers p a r a l l e l t o t h e r o l l i s O(JR) a n d
4 -
t h e bandwidth p e r p e n d i c u l a r t o t h e r o l l i s O( JR). A s we h a v e m e n t i o n e d , i n t h i s l i m i t t h e a m p l i t u d e n o l o n g e r f o l l o w s t h e p h a s e g r a d i e n t a s i n (25) b u t t h e
terms on t h e RHS of t h e a m p l i t u d e e q u a t i o n (27) became e q u a l l y i m p o r t a n t t o T h i s b a l a n c e i s a c h i e v e d when R =
t h e s e on t h e L.H.S.
E
4
x.
For r o l l s which a r e
and
e
a =
where x = direction.
E
2
=
x
+
E
2
7)
+(x,
X a s before and
(34) = y/s =
EY, t h e new s c a l i n g i n t h e p e r p e n d i c u l a r
It is now e a s y t o show from (11) t h a t
kaa
=
ax + $_a_ ,
La8
=
Y Y
K,
=
kq8 =
+yy
and
K
e
=
-kqa
1 / ;a~
=
,
-~+,;j-
(I
= at =
E+-$-
Y
E
2
$t
(35)
w'
where we have u s e d s u b s c r i p t s i n o r d e r t o d e n o t e p a r t i a l d e r i v a t i v e s . S u b s t i t u t i o n of (35) i n t o (26) a n d d i v i d i n g by
E
2
g i v e s (we d r o p t h e t i l d e on
Y)
A$t
-
1
2
2(+x + '2 6 y ) ( 2 a x
+
+
+yy)A
-
2(2ax
+
+(zax + q Y a y + $ y y ) ~ y , ,+ ay 2(2ax + z$ a + + y y ) ~= o Y Y
+
.
tJYY)(OX
+
1
2
7 $y)A
(36)
I t i s r e a d i l y shown t h a t , i f W = Aei$ i n ( 9 ) , e q u a t i o n (36) i s p r e c i s e l y t h e i m a g i n a r y p a r t of e q u a t i o n (9).
C a r r y i n g out t h e same c a l c u l a t i o n on ( 2 7 )
219
Two Dimensional Convection Patterns (recall A
+
2
A) g i v e s t h e r e a l p a r t of e q u a t i o n ( 9 ) .
E
T h e r e f o r e t h e e q u a t i o n s ( 2 6 ) , ( 2 7 ) c o n t a i n a l l t h a t was p r e v i o u s l y known about r o l l solutions.
5.
They a l s o c o n t a i n some new i n f o r m a t i o n .
New r e s u l t s ; some a n s w e r s , more q u e s t i o n s . I n what f o l l o w s we s h a l l t a k e
R
t o be of o r d e r one a n d t h e r e f o r e ( 2 7 ) can
be r e p l a c e d by ( 2 5 ) a l m o s t e v e r y w h e r e .
The e x c e p t i o n s a r e t h o s e r e g i o n s
where V = O ( E - ~ ) b u t t h e s e p o i n t s a r e i s o l a t e d .
We w i l l c o n c e n t r a t e on t h e
phase e q u a t i o n (26),
A
ae + si;1 V at
+
+
(kB)
E
4
( D 1 * D 2 + D2*D1)A
= 0
which may b e r e w r i t t e n i n a v a r i e t y of ways.
a
V(+kB) = k
+
kB
kBII
,
(37)
I n p a r t i c u l a r we may w r i t e
2
o r i n a more r e v e a l i n g way a s V(
applying k
= k2
a i
-aa-A ,
ak a t + k -(aa
a kB -aa L
.
(39)
a i 2 - - t o ( 3 7 ) g i v e s u s two e q u a t i o n s f o r ag A
ai A z a k ~ kkB B + ~+ e~~k~ G ): [ D l * D 2
k
and
+ D2*D1)A
J,
= 0
, (40)
and
For t h e f i r s t s t e p , l e t u s assume t h a t a l l d e r i v a t i v e s a r e of o r d e r one a n d consequently ignore the
E~
t h e Busse b a l l o o n 1 = kC
<
terms. k
<
W e w i l l assume t h a t e v e r y w h e r e
kE(R) a n d prove t h a t i n a r e g i o n
k R
belongs t o
with c e r t a i n
c o n d i t i o n s on t h e b o u n d a r y a R , t h e s y s t e m r e l a x e s t o a s t a t i o n a r y s t a t e w i t h wavenumber
k
t a k i n g on t h e v a l u e which makes
B = 0.
T h i s r e s u l t does n o t
depend c r i t i c a l l y on t h e f a c t t h a t t h e p r e s e n t model i s d e r i v e a b l e from a Lyapunov f u n c t i o n ; i n d e e d i t i s a l s o v a l i d f o r s y s t e m s which do n o t h a v e t h i s property.
The r e a d e r might l i k e t o v e r i f y t h a t t h e model
a (m
2 2 (V - l ) ) ( V -1)'w
+
( R - ww
*
* 2
2
- ww V ) V w
=
0 ,
which does n o t d e r i v e from Lyapunov f u n c t i o n a l , g i v e s t h e p h a s e e q u a t i o n
(k2
+
1)2A2etk2(1
-
vk2)
+
V*CB
+
O ( E ~ )= 0
(43)
220
Alan C . N E W E L L
-
.
4 2 2 k (1-uk ) a n d A 2 = R
- ( k 2 -t 1)3 I t is c r u c i a l , however, dk2 k2 t h a t t h e o r d e r one s p a t i a l d e r i v a t i v e t e r m s i n t h e phase e q u a t i o n h a v e
where B
A2
c o n s e r v a t i o n form a n d a l t h o u g h I h a v e n o t y e t c a t e g o r i z e d t h e c l a s s , t h i s We now prove o u r r e s u l t .
happens f o r a l a r g e c l a s s of problems.
For p o s i t i v e
J ~ ,l e t J
2
et + v
o
*(CB) =
(44)
and c o n s i d e r
F =
//
G(g) dxdy
(45)
R
where
2 G = -1/2,fk B ( k 2 ) d k 2
i s p o s i t i v e a s we i n s i s t
k
>
0
<
b e l o n g s t o t h e Busse b a l l o o n i n which B
0.
Then,
dF
- =
dt
n
-/
aR
Bt h . n
ds
- // R
2
J Bt
2
dxdy,
(47)
t h e outward u n i t normal t o t h e boundary a R , where we h a v e u s e d t h e f a c t
that V G =
i:
-b.T h e r e f o r e ,
if on a R e i t h e r (i) $*n = 0 ( t h e r o l l a x e s are
p e r p e n d i c u l a r t o t h e f i x e d b o u n d a r y ) o r (ii) B
fluid boundary where
of
G
k
decreases. f o r which
model ( 4 2 ) ,
0 ( a p o r t i o n of aR may b e a
B = O),
-dF
-
But
is minimum o n l y when
G
dA2/dk2 = 0.
+
B = 0
or
k = kC
For model ( 2 ) t h i s is t h e point
kC l i e s t o t h e l e f t of
The f a c t t h a t k
I
k
-
1
the value
k = 1; f o r
by a n amount d e p e n d i n g on v
.
kC on a f r e e boundary on p o r t i o n s of aR i s c o n s i s t e n t
w i t h t h e a n a l y s i s of t h e s t a t i o n a r y e q u a t i o n ,
This mans that the quantity 2 kB/e = -H ( 8 )
(49)
i s c o n s t a n t a l o n g t h e o r t h o g o n a l t r a j e c t o r i e s of t h e c o n s t a n t phase c o n t o u r s . R e c a l l in t h e i n t e r v a l 1
<
k
<
kE, kB
<
0. T h i s in t u r n means t h a t i f
t h e 6 c o n t o u r s c o n v e r g e , which t h e y w i l l do in p a t c h e s where t h e c u r v a t u r e of
22 1
Two Dimensional Convection Patterns t he phase contours i n c r e a s e s torwards a c e n t e r , L i n c r e a s e s .
h
t h e f l u x of
I t a l s o means t h a t
between two 6 c o n t o u r s is i n d e p e n d e n t o f a. I t may be u s e f u l f o r
t h e r e a d e r t o keep i n mind t h e a x i a l l y s y m m e t r i c c a s e where a = I k ( r ) d r ,
r
6
the r a d i a l coordinate,
c o o r d i n a t e , whence L =
. Now
=
JI
=
Q,
4 the azimuthal
i n o r d e r f o r t h e s o l u t i o n t o remain s t a b l e we
must have t h a t l < k < k E which imp Les t h a t 0
where IkB
i s t h e a b s o l u t e v a l u e of
0
<
LH2(6)
and t h e r e f o r e a s L
< +
-
before the aa
2
0
H (5) = kB
(E
E
c4
-2
K(B)
<
lkBIE T h e r e f o r e we must have t h a t
kE B(kE).
(50)
along a
6
8
contour,
H2(6) m u s t tend t o zero.
Since i t
c o n t o u r s , i t must become a s small a s i t c a n
terms i n ( 3 7 ) e n t e r t h e p i c t u r e , which t h e y w i l l do when and
)
2
lkBl
lkBIE
is c o n s t a n t a l o n g c o n s t a n t
a = -
<
L = O(E-’).
a n d hence
is small only n e a r
Thus, in o r d e r t h a t t h e i n e q u a l i t y (50) h o l d s , kB = O(E 2) e v e r y w h e r e t h a t
kc = 1,
we must have
k
=
1
L = O(1).
+
But, s i n c e
2 O ( E ). Near t h e s i n k , we
can f i n d s o l u t i o n s of ( 3 7 ) i t e r a t i v e l y in t h e form 2 2 2 k = 1 += + , ,-+ =- 3y + 1 where y 2 =
4R
L
..
2 r + ~ ~ 4v K ( B ) , which i n d i c a t e s t h a t a s r
v a l u e somewhere b e t w e e n
kC = 1
and
e2, k
+
g o e s from
kC = 1
to a
kE.
Thus o u r f i r s t p r e d i c t i o n i s t h a t on t h e time s c a l e
E - ~
,
the horizontal
d i f f u s i o n t i m e , p a t c h e s form which s a t i s f y t h e boundary c o n d i t i o n s a l o n g p o r t i o n s of t h e box b o u n d a r y , a n d in which
k
+
kc
k’
n = 0
almost everywhere.
Examine t h e n u m e r i c a l e x p e r i m e n t s of G r e e n s i d e , Coughran a n d S c h r y e r [ 111 ( f i g u r e 3 ) c a r r i e d o u t on e q u a t i o n ( 2 ) f o r r e a l b r g e [121 ( f i g u r e 4 ) .
J,
a n d t h e real e x p e r i m e n t s of
Alan C . NtwEl
222
F i g . 3:
Fig.
4:
I_
-
Numerical I n t e g r a t i o n of ( 2 ) , J, R e a l H o r i z o n t a l D i f f u s i o n Time F o r Time
.
From a n e x p e r i m e n t of P. Berge Contours of C o n s t a n t Downward V e l o c i t y . A s p e c t R a t i o of 1 6 . R 2Rc.
-
223
Two Dimensional Convection Patterns Note i n t h e r e c t a n g u l a r geometry of f i g u r e 3 , t h a t p a t c h e s w i t h c i r c u l a r symmetry form a b o u t t h e c o r n e r s A a n d C.
I n t h e c i r c u l a r geometry o f
Berge ' s r e a l e x p e r i m e n t , one a g a i n s e e s c i r c u l a r p a t c h e s f o r m i n g a b o u t s i n k s which a r e a t t a c h e d t o t h e boundary.
Moreover, i t i s a b u n d a n t l y c l e a r from t h e s e
f i g u r e s t h a t t h e box c a n n o t be t i l e d w i t h t h e s e p a t c h e s .
Certain areas, f o r
example t h e c o r n e r B a n d D i n f i g u r e 3 , a r e q u i t e i n c o m p a t i b l e . compensate f o r t h e s e mismatches,
the
In order to
terms of t h e phase e q u a t i o n ( 3 7 ) must
E~
be i n c o r p o r a t e d i n t he a n a l y s i s . One way i s t o t a k e a / a a , 3 / 3 8 t o the microscopic theory.
t o be
O(E-')
b u t t h i s simply b r i n g s u s back
A n o t h e r way, which r e t a i n s t h e f u n d a m e n t a l i d e a t h a t
t h e c o n v e c t i o n f i e l d c a n b e d e s c r i b e d by a s l o w l y v a r y i n g w a v e v e c t o r
c,
is t o
.
This
recognize t h a t t h e terms
can b a l a n c e when t h e
B
derivatives are
r e d u c e s b o t h terms i n ( 5 3 ) t o
O(E-')
and
k = 1
+
O ( E ~ ) a n d s i n c e w e saw t h a t t h e main e f f e c t of
t h e dynamics on t h e h o r i z o n t a l d i f f u s i o n time s c a l e i s t o d r i v e kc, if
t h i s approximation is not a t a l l unreasonable. R = 0(1),
k = 1
+
2
O(E )
O ( E ~ ) , then
A
is
k
towards
A l i t t l e a l g e b r a shows t h a t
fi t o w i t h i n
O ( E ~ )a n d t h e
s t a t i o n a r y p a t t e r n s which one might e x p e c t t o r e a c h on t h e c-6 t i m e s c a l e (which i s t h e h o r i z o n t a l d i f f u s i o n time s c a l e m u l t i p l i e d by t h e a s p e c t r a t i o
g i v e n by
and t h e c o m p a t i b i l i t y c o n d i t i o n s ( 1 1 ) a r e
c-')
are
224
Alan C. NEWELL
E q u a t i o n s (54), (55) show i t s e l f i s of o r d e r straight.
US
that
is a t most o r d e r o n e a n d t h e r e f o r e
JI
For t h e s e s o l u t i o n s , t h e n , t h e r o l l s a r e l o c a l l y a l m o s t
E.
S i n c e we a r e now working i n d i s t a n c e s of o r d e r
E
( a s measured in box
i n r o l l w a v e l e n g t h u n i t s ) we c a n make t h e f o l l o w i n g l o c a l u n i t s ; I/E a p p r o x i m a t i o n s , which a r e v e r y s i m i l a r t o t h o s e made when we were d e r i v i n g t h e Newell-Whitehead-Segel
equations (36).
Let
( 6 , ~ ) be l o c a l l y t h e ' a c r o s s a n d
along' r o l l cordinates and
a sa = - - i a E
as
1
a n d e q u a t i o n (53) i s ( u s i n g s u b s c r i p t s f o r p a r t i a l d e r i v a t i v e s )
which is p r e c i s e l y t h e Newell-Whitehead-Segel amplitude A h e l d constant.
equation ( 3 6 ) and (9) with
We a r e now g o i n g t o d i s c u s s s o l u t i o n s of t h i s
e q u a t i o n which l e n d some i n s i g h t i n t o F i g u r e 5 which is t h e a s e q u e l t o F i g u r e
3
225
Two Dimensional Convection Patterns
B
C
A
Figure 5:
Numerical I n t e g r a t i o n of (z), $J real For Time >> H o r i z o n t a l D i f f u s i o n Time.
Notice t h a t i n o r d e r t o compensate f o r t h e i n c o m p a t i b i l i t i e s i n t h e c o r n e r B of Figure 3, the c i r c u l a r r o l l s emanating from AD have undergone a change and have R o l l number 7, counting from A a l o n g
i n t r o d u c e d d i s l o c a t i o n s along t h e w a l l AB.
AD, doubles i n width a s i t approaches t h e s i d e AB and undergoes a d i s l o c a t i o n . Roll number 9 detaches from AB a l t o g e t h e r and forms a series of d i s l o c a t i o n s along AB (which we c a l l a g r a i n boundary) and then a t t a c h e s i t s e l f t o the upper
wall
BC.
R o l l s 10 through 25 t a k e on an
S
shape i n which t h e approximate
d i s t a n c e o v e r which s i g n i f i c a n t changes occur i s t h e square r o o t of the box dimension, o r the 'along the r o l l ' s c a l i n g in e q u a t i o n (58).
226
Alan
c. N E W E l 1
Figure 6:
Dislocation
W e f i r s t n o t e t h a t a property of t h i s e q u a t i o n is t h a t i f
$ ( 5 , 1 ; ) solves
Also observe t h a t
o
=
(581,
0 = ~
5 + e(c,s),
-
6
so does
-@(-L,T,).
(59)
~i s 9g i v e n by
= E
2?
5.
The shapes of the phase contours near d i s l o c a t i o n s s u g g e s t that w e search f o r s e l f s i m i l a r s o l u t i o n s of the form
227
Two Dimensional Convection Patterns
F(z)
s a t i s f i e s the equation,
F' =
which i s r e a l l y a n e q u a t i o n f o r
G
=
dz
'
F'.
It h a s t h e symmetry p r o p e r t y t h a t i f
s o l v e s ( 6 3 ) , s o does
( F ( z ) , G(z))
E q u a t i o n ( 6 3 ) h a s a one p a r a m e t e r f a m i l y of s o l u t i o n s F ( C ; z ) w i t h d e r i v a t i v e s 2 G(C,z) which decay as Ce-' a s z + -This can b e s e e n by l i n e a r i z i n g ( 6 3 )
.
which t h e n h a s e r r o r f u n c t i o n s o l u t i o n s .
For
C
v e r y small, t h e s e s o l u t i o n s
behave v e r y much l i k e t h e e r r o r f u n c t i o n s o l u t i o n s ; t h e y a r e s y m m e t r i c a b o u t z = 0
and l e a d t o a jump i n
nonlinear and the bigger (actually
F
-
F(z)
Iln
F
Co
of
C,
F
approaches its pole s o l u t i o n s
1 2-zo
(65) and 6 F f 2 F " .
Thus t h e r e i s a c r i t i c a l
which from n u m e r i c a l c a l c u l a t i o n s i s a p p r o x i m a t e l y . 5 6 4 ,
above which t h e s o l u t i o n s do n o t e x i s t o v e r t h e l i n e approaches
Co from below,
( z n)
v a l u e of 3.14 F(z),
Y(z)
for C =
0 =
5
However ( 6 3 ) i s
= 6 C .
has a logarithmic s i n g u l a r i t y )
r e p r e s e n t i n g a b a l a n c e between F"" value
AF = F(m) - F(-)
g e t s the c l o s e r
C
has the pole;
G
of
TI
at
AF
-m
<
z
<
i s very s e n s i t i v e to changes i n
C = .54 t o 7.93 a t C = .565.
m
.
As
C
C , g o i n g from a
I n f i g u r e 7, w e g r a p h
a n d i n F i g u r e 8 , we draw t h e c o n t o u r of c o n s t a n t p h a s e 0
- + P (%T)
.
228
Alan C. NEWELL
I
/---
-F(T,z),
Figure 7 :
Graphs of
Figure 8:
Constant Phase Contours
F(n,z).
0
.
Two Dimensional Convection Patterns
229
F o r v a l u e s of 0 s l i g h t l y g r e a t e r t h a n n, t h e c o n t o u r s are d e f i n e d f o r a l l values l e s s than
For
<
TI.
t h e phase c o n t o u r s i n t e r s e c t t h e
'5
7. For
= 0 a x i s a t the o r i g i n .
0, we u s e t h e symmetry p r o p e r t y (60) t o i n f e r t h a t t h e p h a s e c o n t o u r s i n
t h i s r e g i o n a r e s i m p l y a r e f l e c t i o n of t h o s e f o r
>
0 i n the
= 0 axis.
These
s o l u t i o n s seem t o g i v e a f a i r l y a c c u r a t e p i c t u r e of t h e r e a l d i s l o c a t i o n s seen i n experiments. F i n a l l y , w e i n d i c a t e how t o i n c l u d e mean d r i f t terms i n t h e model. Consider
$ + (V2+1) 2w where
-
Rw
+ w2w* + u
vw = 0
u = Vx TZ (z i s t h e u n i t v e c t o r p e r p e n d i c u l a r t o
X,Y)
and
F o l l o w i n g t h e p r e v i o u s a n a l y s i s , we f i n d t h a t t h e s l o w e q u a t i o n f o r t h e phase is
+
kt a kB -A % 3-
+ Akll 3aTT +
O(c4) = 0
where
2 a a uL p V ~ = k t = k l l ~
(70)
In (68), t h e p a r a m e t e r l i p mimics t h e e f f e c t of low P r a n d t l number s i t u a t i o n s where mean d r i f t i s c a u s e d by t h e n o n l i n e a r a d v e c t i o n t e r m s i n t h e momentum equations.
I n (70),
V
refers
t o t h e slow d e r i v a t i v e s wi t h r e s p e c t t o
6. SUMMARY. I n t h i s p a p e r we have p r e s e a t e d a m a t h e m a t i c a l framework f o r d e s c r i b i n g
230
Alan C. NEWELI
c o n v e c t i o n p a t t e r n s which i n c l u d e s a l l p r e v i o u s t h e o r i e s a n d from i t we have
In
made s e v e r a l p r e d i c t i o n s a b o u t t h e manner in which t h e p a t t e r n s e v o l v e . p a r t i c u l a r , we s u g g e s t t h a t on t h e h o r i z o n t a l d i f f u s i o n time s c a l e
TH, t h e
c o n v e c t i o n f i e l d d e v e l o p s p a t c h e s , o f t e n of a c i r c u l a r n a t u r e s u r r o u n d i n g a s i n k , i n which t h e wavenumber is c o n s t a n t .
The i n c o m p a t i b i l i t y of t h e s e p a t c h e s
is i r o n e d o u t o v e r t h e l o n g e r t i m e scale of t h e a s p e c t r a t i o t i m e s
TH
and the
p r o c e s s i n v o l v e s a g l i d i n g motion (compare F i g u r e s 3 a n d 5) i n which r o l l d i s l o c a t i o n s move i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e r o l l a x i s .
The c l i m b
m o t i o n , where t h e d i s l o c a t i o n s move a l o n g t h e r o l l a x i s , o c c u r on t h e s c a l e TH
c2
as t h e i r r o l e is t o a d j u s t w a v e l e n g t h , a l t h o u g h small a d j u s t m e n t s of o r d e r w i l l be made on t h e
E-%'~
scale.
While we b e l i e v e w e h a v e made a s t a r t , many q u e s t i o n s s t i l l remain open. Some of t h e s e are. 1.
F o r what c l a s s of models i s t h e f l o w on t h e h o r i z o n t a l d i f f u s i o n time s c a l e a g r a d i e n t o n e ; e q u i v a l e n t l y , f o r which models does ( 4 4 ) o b t a i n ?
2.
What is t h e e f f e c t of t h e mean d r i f t term?
What p a r a l l e l c o n c l u s i o n s can we
draw?
3.
Do t h e p a t t e r n s e v e r s e t t l e down o r do t h e y a l w a y s remain n o i s y ?
If the
f o r m e r is t h e c a s e , i s i t a consequence of geometry where t h e d i s l o c a t i o n s g e t stuck i n corners?
I n a c i r c u l a r g e o m e t r y , one m i g h t a r g u e t h a t t h e
g l i d e motion n e v e r s t o p s .
I f t h e l a t t e r is t h e c a s e , d o e s t h e r e s u l t i n g
c h a o t i c motion l i e on a low d i m e n s i o n a l s t r a n g e a t t r a c t o r , one w h i c h , f o r example, mimics t h e v e r y g e n t l e h e a v i n g of t h e g l i d e m o t i o n as i t r o t a t e s a r o u n d t h e box?
23 1
Two Dimensional Convection Patterns REFERENCES 1.
G o l l u b , I. P. a n d McCarriar A. R.
2.
A h l e r s G. a n d Walden R. W.
3.
Busse F. H. turbulence. Verlag
4.
Whitham G. B.
5.
N e w e l 1 A. C. a n d W hi t head J. A . 1969 J. F l u i d Mech. 203.
6.
Pomeau Y. a n d M a n n e v i l l e P.
7.
S i g g i a E. a n d Z i p p e l i u s A.
8.
S t u a r t J. T. 1960. 371-389. Mech.
9.
Busse F. H. a n d W hi t ehead J. A.
10.
Chen M. M. a n d Whi t ehead J. A .
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G r e e n s i d e H.
12.
Ekrge , P. 1980. S p r i n g e r - Ve r la g
.
1982.
Phys. Rev. A
1980 Phys Rev. T a t t .
44, -
z,347O.
445.
1980 Hydrodynamic i n s t a b i l i t i e s a n d t h e t r a n s i t i o n t o 97-136. Eds. H. L. Swinney a n d J. P. G ollub. Publ. S p r i n g e r -
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J. F l u i d Mech
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2,
9,
3, 373.
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S e g e l , L. A .
40,1067.
J. F l u i d Mech. t o a p p e a r .
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Watson J.
J. F l u i d Mech.
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J. F l u i d
47, 305-320.
J. F l u i d Mech.
S . , Coughran W . M. a n d S c h r y e r N . L.
.
38, 279.
J. F l u i d Mech.
2,1.
1982. P r e p r i n t .
Chaos a n d Order i n N a t u r e pp, 14-24.
Ed. H. Haken.
Publ.
Two Dimensional Convection Patterns i n Large Aspect Ratio Systems*
Alan C . N e w e l l Program i n Applied Mathematics University of Arizona, Tucson
85721
This work was supported i n part by DOA Contract DAAG29-82-C-0068.
205
206
c. NEWEI.1.
Alan
F i g u r e 1 below i s t a k e n f r o m a v i s u a l i z a t i o n of a n e x p e r i m e n t of G o l l u b a n d M c c a r r i a r 111 in which t h e y f o l l o w t h e t i m e development o f a c o n v e c t i n g h o r i z o n t a l l a y e r of f l u i d of d e p t h techniques.
h
t h r o u g h t h e u s e of l a s e r D o p p l e r
The d o t s a r e p o i n t s where, a t a depth of
h/4
from t h e t o p of t h e
l a y e r , t h e v e l o c i t y component p a r a l l e l t o t h e l a r g e r s i d e of t h e box is z e r o a n d mark t h e b o u n d a r i e s of t h e c o n v e c t i v e r o l l s .
The e x p e r i m e n t is c o n d u c t e d i n a
r a n g e of R a y l e i g h number where n o n l i n e a r s t a b i l i t y t h e o r y g u a r a n t e e s t h a t i n a h o r i z o n t a l l a y e r of i n f i n i t e e x t e n t , s t r a i g h t p a r a l l e l r o l l s b o t h e x i s t a n d a r e stable.
.. .. .. .. .. .. .. .. .. .. .. .. .. .<..'.' : ... ... ... ........... ... .! t.' ., *. .....'..' .. '...** . ...... .*:. . . . . . . .: .* ..; . ......... ... 1,;:::::
10
.
.
.
.
*
.
.
I
.
.
.
.
::::;!;;:*.s.:
... ...:...:...:...:...:...:...i .:. I. ... .,... ...: ..:. :;:,: . . . . .., .,,. .. .. .. .. .. ..... . . . . ........ .t*,..*'
::..;,t
2 .
... I ................. ............. ....... , ... .. . . ...... . . .r.t.! . i .
, . . !: , ::::.' :/a. ! :!:::::::;:I:.+, ,.* ..;:::;::;:::;::.+* . : : : : :: : : : : ;!::;!;/*. I
0
5
0
10
1s
x (cm) Figure 1 :
D o p p l e r Map Of The V e l o c i t y F i e l d F o r A S t a b l e C o n v e c t i v e Flow 256 Hrs. After R a y l e i g h Number Was I n c r e a s e d To 2.05 Rc. ( H o r i z o n t a l D i f f u s i o n Time, 40 HKS.)
However, t h e e x i s t e n c e of o r i e n t a t i o n a l d e g e n e r a c y a n d a band of s t a b l e r o l l wavenumbers t o g e t h e r w i t h t h e f a c t t h a t t h e rolls a l i g n t h e m s e l v e s p e r p e n d i c u l a r t o a l l l a t e r a l b o u n d a r i e s makes t h e s e s o l u t i o n s u n a t t a i n a b l e .
The p a t t e r n s t h a t
a r e s e e n a r e much more c o m p l i c a t e d n o t only c o n t a i n i n g Curved r o l l s b u t a l s o e x h i b i t i n g many d i s l o c a t i o n s .
F u r t h e r m o r e , i t is n o t c l e a r w h e t h e r a n d , i f s o ,
u n d e r what c i r c u m s t a n c e s t h e p a t t e r n a c h i e v e s a t i m e i n d e p e n d e n t e q u i l i b r i u m . I n d e e d , i n some cases, G o l l u b a n d M c c a r r i a r h a v e s e e n p a t t e r n s which are s l o w l y t i m e dependent o v e r many h o r i z o n t a l d i f f u s i o n t i m e s .
The f a i l u r e t o r e a c h a
207
Two Dimensional Convection Patterns s t e a d y s t a t e was a l s o noted e a r l i e r by Ahlers and Walden [Z] who observed t h a t the e f f e c t i v e thermal c o n d u c t i v i t y of a l a y e r of convecting f l u i d helium remained n o i s y f o r a l l Rayleigh numbers above
RC
f o r sufficiently large aspect
ratios. Our g o a l i n t h i s paper i s t o develop a t h e o r y t o d e s c r i b e t h e s e p a t t e r n s . We s t a r t with the o b s e r v a t i o n t h a t almost everywhere i n the convection f i e l d a l o c a l wavevector is d e f i n e d and v a r i e s slowly o v e r the box. the wavevector is tangent.
A t the boundaries,
Therefore we e x p e c t t h a t t h e r e e x i s t s l o c a l l y
p e r i o d i c s o l u t i o n s defined by f ( 0 ; A , R) when
f
is 2n - p e r i o d i c i n 0 and
VO = k(X, Y , T)
(1)
( V = (a/aX, a/aY)) is a slowly varying f u n c t i o n of p o s i t i o n c o o r d i n a t e s and t i m e .
X, Y, T, t h e h o r i z o n t a l
Indeed we know from the work of Busse 131 t h a t
such s o l u t i o n s a s f u n c t i o n s of 0 e x i s t and, a s we d e s c r i b e l a t e r , have c e r t a i n s t a b i l i t y properties.
I n o t h e r words, while t h e f i e l d v a r i a b l e
d
(wavevector), which t o g e t h e r with a knowledge of
f as f u n c t i o n of
0 d e s c r i b e t h e p a t t e r n , vary o v e r d i s t a n c e s of t h e s i z e of t h e box.
a s p e c t r a t i o c 2 , t h e r a t i o of t h e r o l l wavelength d dimension
L
of t h e box, is the
The Rayleigh number
RC,
R
only small
(2.
v a r i e s over
A (amplitude)
d i s t a n c e s of the o r d e r of the roll s i z e , t h e parameters and
f
The i n v e r s e
h) to the l i n e a r
parameter which e n t e r s t h e theory.
can be an o r d e r one amount above i t s c r i t i c a l value
the value a t which the p u r e l y conductive s t a t e becomes u n s t a b l e , although
i t must be less than t h e Rayleigh number a t which t h e s t r a i g h t p a r a l l e l r o l l
s o l u t i o n becomes u n s t a b l e .
The i d e a s we a r e about t o d e s c r i b e a r e c l o s e l y
r e l a t e d t o those of Whitham (41 who i n t h e l a t e s i x t i e s developed a theory t o d e s c r i b e f u l l y n o n l i n e a r , almost p e r i o d i c w a v e t r a i n s . A s a f i r s t a t t e m p t we s h a l l use model e q u a t i o n s , t h e use of which
f a c i l i t a t e s the a n a l y s i s by making c a l c u l a t i o n s e x p l i c i t b u t which c a p t u r e what
we b e l i e v e a r e some of the e s s e n t i a l f e a t u r e s of the Overbeck-Boussinesq equations.
We d e r i v e e q u a t i o n s f o r the slow v a r i a b l e s
d,
A which
( i ) show
t h a t a l l t h e s t a b i l i t y c r i t e r i a d e r i v e d by Busse and h i s c o l l e a g u e s [31 f o r
208
Alan C. N ~ W E L I . ( i i ) r e d u c e i n t h e l i m i t of small
s t r a i g h t p a r a l l e l r o l l s h o l d l o c a l l y and
R-RC
t o t h e Newell-Whitehead-Segel
[ 5 ] equations.
I n a d d i t i o n we show t h a t t h e
e f f e c t of c u r v a t u r e is t o d r i v e t h e r o l l p a t t e r n s toward a s t a t e in which t h e l o c a l wavenumber assumes a c o n s t a n t v a l u e a l m o s t e v e r y w h e r e , a r e s u l t c o n s i s t e n t w i t h a r e c e n t r e s u l t of Pomeau a n d M a n n e v i l l e [ 6 ] f o r a x i a l l y s y m m e t r i c r o l l s . T h i s is done by ( i ) p r o v i n g t h a t u n d e r c e r t a i n n a t u r a l boundary c o n d i t i o n s , t h e macroscopic e q u a t i o n s f o r t h e phase, v a l i d f o r times up to t h e h o r i z o n t a l d i f f u s i o n t i m e , are d e r i v e a b l e from a Lyapunov f u n c t i o n a l e v e n though t h e f u l l m i c r o s c o p i c e q u a t i o n s f o r t h e model n e e d n o t b e a n d ( i i )
examining t h e n a t u r e
of t h e s t a t i o n a r y s t a t e s which would be r e a c h e d on t h i s t i m e . But t h i s is n o t t h e whole s t o r y a s such p a t t e r n s i n a n d of t h e m s e l v e s c a n n o t s a t i s f y a l l t h e boundary c o n d i t i o n s .
I n between t h e p a t t e r n s ,
In
d i s l o c a t i o n s are formed and s o l u t i o n s d e s c r i b i n g t h e s e s t r u c t u r e s a r e g i v e n . l o o k i n g a t t h e s e s o l u t i o n s , i t is c l e a r t h a t t h e t i m e s c a l e which t h e t o t a l
s y s t e m would n e e d t o r e l a x to e q u i l i b r i u m is n o t s i m p l y t h e h o r i z o n t a l d i f f u s i o n time L
2
/V
b u t is t h i s time s c a l e m u l t i p l i e d by t h e a s p e c t r a t i o
L/d.
Finally,
we i n c l u d e i n t h e model a t e r m which t a k e s a c c o u n t of t h e "mean d r i f t " which o c c u r s i n s y s t e m s of low t o m o d e r a t e P r a n d t l number.
The p r e s e n c e of t h i s
e f f e c t was f i r s t p o i n t e d o u t by S i g g i a a n d Z i p p e l e u s 171, a n d c a n c a u s e marked change i n t h e b e h a v i o r of t h e s y s t e m .
I s h o u l d p o i n t o u t t h a t t h e i d e a s we d i s c u s s h e r e c a n b e e x t e n d e d t o i n c l u d e s i t u a t i o n s where t h e b a s i c f l o w
is c o n s i d e r e d t o b e a
f
is m u l t i p e r i o d i c .
2n p e r i o d i c f u n c t i o n i n e a c h of
p h a s e s 0 where 80 = $ a n d which a l s o depends on i i i p a r a m e t e r (or set of p a r a m e t e r s )
n
In t h a t case
f
n
amplitudes
Ai
and a
R.
A more comprehensive p a p e r is b e i n g w r i t t e n j o i n t l y w i t h my c o l l e a g u e Mike
C r o s s who s t a r t e d me t h i n k i n g a b o u t t h e s e problems a g a i n d u r i n g a most p l e a s a n t l e a v e s p e n t a t t h e I n s t i t u t e of T h e o r e t i c a l P h y s i c s a t t h e U n i v e r s i t y of C a l i f o r n i a i n Santa Barbara.
I am most g r a t e f u l f o t h e i r h o s p i t a l i t y a n d t o t h e
e n e r g e t i c l e a d e r s h i p of P i e r r e Hohenberg who was a c o n s t a n t s t i m u l u s .
209
Two Dimensional Convection Patterns 2.
The p r e d i c t i o n s of p r e s e n t t h e o r i e s :
Consider
w(X, Y , T )
g i v e n by
a ----w aT
on
-a)
<
X, Y
<
+
(V2
+
1)2w
-
Rw
+
w2w*
=
0
( 2)
-, The p r i n c i p a l r e a s o n f o r c h o o s i n g t h i s model is t h a t it
possesses a simple f u l l y nonlinear periodic s o l u t i o n e
io,
+ + X, X = ( X , Y ) .
+
0 = k
The " c o n d u c t i o n " s o l u t i o n
modes of t h e form
w(X,
whenever R
>
Y, T) = We
ic
(3)
2
p l a n e is g i v e n by
( R , k)
is u n s t a b l e t o
k
f i i n ( k - 1 ) 2 = 0 , which v a l u e i s r e a l i z e d when
n e u t r a l s t a b i l i t y c u r v e in t h e
w = 0
($1 R
=
k = 1.
The
We
(k2-1)2.
o b s e r v e t h e s y s t e m i s d e g e n e r a t e in t h e s e n s e t h a t any mode of t h e form ( 3 ) with
lcl
= 1 w i l l grow a t t h e same r a t e when R
>
0.
The n o n l i n e a r s a t u r a t i o n
of t h e l i n e a r l y u n s t a b l e modes i s d e s c r i b e d by t h e S t u a r t - W a t s o n [ 8 1 e q u a t i o n
- w%*
WT = Rhl
(4)
which in t h i s c a s e is a n e x a c t s o l u t i o n f o r ( 2 ) b u t which g e n e r a l l y i s o n l y t r u e f o r v a l u e s of
R
p r o p o r t i o n a l t o JR
near its c r i t i c a l value
- RC
. Because
RC
=
0
and f o r amplitudes
W
of t h e o r i e n t a t i o n a l d e g e n e r a c y , i t i s i n d e e d
n a t u r a l t o look f o r s o l u t i o n s of t h e form
a n d , in t h e small
R
limit, we c a n show
It is a n e a s y m a t t e r t o s e e t h a t t h e o n l y s t a b l e s o l u t i o n of t h i s set is t h e
single r o l l s o l u t i o n
- iZI
W1 = JR
e
k
,
w
j
= O , j # I .
(6)
Alan C. NEWELL
210
So, j u s t a s i n t h e case of the Rayleigh-Benard problem under a v e r t i c a l l y
symmetric, a p p l i e d temperature f i e l d with moderate P r a n d t l number u / K > 0 ( 1 ) , s i n g l e , p a r a l l e l r o l l s a r e p r e f e r r e d . pointed o u t , no d i r e c t i o n i s chosen a p r i o r i .
But, a s we have
Therefore when
R
is suddenly
r a i s e d from s u b c r i t i c a l t o s u p e r c r i t i c a l v a l u e s , the system a c t s a s a n o i s e A t any p a r t i c u l a r l o c a t i o n i n a l a r g e h o r i z o n t a l l a y e r , i t w i l l
amplifier.
choose among t h e wavelengths of the noise f o r one c l o s e t o n o t make any choice among d i r e c t i o n s .
kC
1
but i t
W i l l
Therefore u n l e s s the experiment i s
c a r e f u l l y c o n t r o l l e d ( a s i n t h e case of the Busse-Whitehead [ 9 1 and WhiteheadChen [ 101 e x p e r i m e n t s ) , r o l l s of approximately the c r i t i c a l wavelength b u t of d i f f e r e n t d i r e c t i o n s w i l l s p r i n g up in d i f f e r e n t p l a c e s .
This d i r e c t i o n a l
d i v e r s i t y is even more a c c e n t u a t e d due t o t h e i n f l u e n c e of a c l o s e d boundary.
(i is t h e
= 0
A t each boundary l o c t i o n ,
u n i t normal t o t h e boundary) and
t h e r e f o r e i f n i s continuous, r o l l s of a l l d i r e c t i o n s k a r e e x c i t e d .
Now, once
t h e r o l l d i r e c t i o n is chosen a t a p a r t i c u l a r l o c a t i o n , i t becomes more s t a b l e a g a i n s t l i n e a r d i s t u r b a n c e s of r o l l s of o t h e r d i r e c t i o n s t h e more i t grows So, f o r e a r l y times, one f i n d s a f l u i d l a y e r
towards i t s s a t u r a t i o n amplitude.
resembling a sea of q u a s i s t a b l e patches and somehow t h e f l u i d has t o f i n d a way t o resolve the i n c o m p a t i b i l i t i e s between them. There is a l s o a bandwidth degeneracy. f i n i t e bandwidth of wavenumbers
WT and W
+
JR
-
(k2
-
1)’
k
= (R
It i s e v i d e n t t h a t f o r R
can be e x c i t e d .
-
(k2
-
Indeed f o r
>
0, a
t 1,
- W%*
1)’)W
a s y m p t o t i c a l l y i n time.
We may t e s t t h e ( l i n e a r )
s t a b i l i t y of these s o l u t i o n s by s e t t i n g
w = eikX (W
+ b l e ILX+ IMY
+
b2e
-iLX
-
iMY
1
whereupon we f i n d a f t e r some c a l c u l a t i o n t h a t t h i s s o l u t i o n is u n s t a b l e i f
2A2(A2)’(L2~2)+”2(A2)’’(
2k2L2)+4k2L2(A2)’2 > 0
(5)
Two Dimensional Convection Patterns where A 2 = R
-
(k2
-
1)*
and
(A2)'
. The f i r s t
dA2/dk2
=
21 1 c l a s s of
i n s t a b i l i t i e s occurs f o r
L = 0,
M
f
0
2 2 ' B = (A ) ( A )
and
These correspond t o t h e zig-zag a wavevector with wavenumber k
>
0.
i n s t a b i l i t i e s discovered by Busse and a r i s e when
<
g i v e s i t s energy t o e i t h e r one of two modes ( k , f circle.
+
M = 0
>
+ k2(A2)"
2A2(A2)"k2
0) i n t e r a c t s with and
2 k ) l y i n g on the u n i t
41 -
The second c l a s s of i n s t a b i l i t i e s occur f o r A2(A2)'
>
1 (in which case ( A 2 ) '
and
0,
o r e q u i v a l e n t l y when $kB>O
-
where B = A 2 dA2/dk2 = -2(k2
1)(R
-
(k2
. This
-
(7)
is t h e Eckhaus
i n s t a b i l i t y and occurs when a r o l l has too s m a l l a wavelength.
It is useful t o
summarize these r e s u l t s by way of f i g u r e 2.
k8, A
I
I
I
I
I
R k Figure 2: Graphs of A , kB and R Busse Balloon
VS.
k. and t h e
212
Alan C . NEWELL
S o l u t i o n s can e x i s t f o r k
< k <
L
k
R
b u t a r e s t a b l e o n l y when kC
s h a d e d a r e a i s known a s t h e B u s s e b a l l o o n .
<
k
<$
When d e r i v e d i n t h e c o n t e x t of t h e
e q u a t i o n s i t i s somewhat more c o m p l i c a t e d .
f u l l Overbeck-Boussinesq
. The
The r i g h t
hand c u r v e which is t h e boundary w i t h t h e Eckhaus i n s t a b i l i t y i s r e p l a c e d f o r l a r g e P r a n d t l number by a boundary t o a n i n s t a b i l i t y t o r o l l s i n t h e
On t h e o t h e r h a n d , f o r s m a l l e r P r a n d t l numbers, t h e
perpendicular direction.
Eckhaus s t a b i l i t y boundary becomes l i n k e d w i t h t h e s k e w - v a r i c o s e i n s t a b i l i t y which i s a v a r i a n t of t h e f o r m e r when mean d r i f t e f f e c t s are i n c l u d e d . show how t o i n c l u d e t h e s e i n t h e model.
k
boundary need n o t remain a t
=
We w i l l
We a l s o remark t h a t t h e l e f t hand
kC = 1 b u t can bend l e f t w a r d d e p e n d i n g on
The B u s s e b a l l o o n ( o r Busse w i n d s o c k ) , t h e r e g i o n of s t a b l e
P r a n d t l number.
p a r a l l e l r o l l s i n the R, p Since f o r R
>
R
C'
-
v/K, k
p l a n e i s g i v e n in r e f e r e n c e [ 3 1 .
there i s an order JR
i n a direction parallel t o
-
$ a n d a n o r d e r 4JR
R
C
band o f a l l o w a b l e wavenumbers
- RC
band i n t h e p e r p e n d i c u l a r
d i r e c t i o n , i t i s n a t u r a l t o i n c l u d e a r i c h e r c l a s s of s o l u t i o n s which h a v e a n a l m o s t p a r a l l e l r o l l s t m c t u r e by l e t t i n g
-Y. -t ) e i x
w ( x , Y, T) = WC;,
2 4 where x = u X, y = uY, t = p T w i t h N
(8) 4
x
=
R
-
RC
<<
1. This g i v e s t h e N e w e l l -
Whitehead-Segel e q u a t i o n (51 which f o r t h e model ( 2 ) i s
aw a7
4.
(21
a+ a;
a
2
2 w 3)
=
xw - w%*
.
(9)
This e q u a t i o n i s c a n o n i c a l f o r d e s c r i b i n g s i t u a t i o n s in which t h e r o l l s a r e a l m o s t p a r a l l e l and
3.
R
i s c l o s e t o i t s c r i t i c a l value.
A new a p p r o a c h :
We w i l l now g i v e a d e s c r i p t i o n t h a t a l l o w s t h e l o c a l w a v e v e c t o r undergo o r d e r one changes c o n t i n u o u s l y b u t s l o w l y o v e r t h e box. w(X, Y , T) where
W
and
ie(X, = We
Y, T)
= VO a r e f u n c t i o n s of t h e s l o w v a r i a b l e s
to
Let ( 10)
213
Two Dimensional Convection Patterns 2
4
2
x = E X , ~ = E Y , ~ = E T
and I / E
2
is t h e a s p e c t r a t i o .
I t i s u s e f u l t o write 9 =
1 7 O(x,
y , t ) whence
E
V O = V+9 x G
T
=
x
= (m,
n)
(k cos JI, k sin J I )
=
(11)
2
2
= E B
k’
t
= E D
We w i l l a l s o f i n d i t u s e f u l t o i n t r o d u c e new c o o r d i n a t e s 9 = a(x, y ) ,
B = B(x, Y )
d e f i n e d by
ax = k Cos $
Bx = -P Sin J,
ay = k S i n JI
By = II Cos JI
The J a c o b i a n of t h e t r a n s f o r m a t i o n from (a, B )
to
( x , y)
is
kll
and
The curves a ( x , y ) = c o n s t a n t a r e . of course, t h e l o c i of c o n s t a n t phase 8 while the B c o o r d i n a t e s a r e the orthogonal t r a j e c t o r i e s and measure d i s t a n c e a l o n g t h e rolls.
by ll
The c u r v a t u r e K
$ . Tne
a
of the
a ( x , y)
=
c o n s t a n t curves is given
c u r v a t u r e of t h e c o n s t a n t curves B
is
K
B
=
-k
3.
I n t h e s e c o o r d i n a t e s , the c o m p a t i b i l i t y c o n d i t i o n s ( 1 1 ) g i v e us t h a t
and
Note t h a t
ma Aeie = e i e ( i e 2 u V 2 Aeie = eie(-k2
+
E4
&)A
+ ie2Dl +
,
r4D2)A
,
214
Alan C . NEWELL
where
a +
D ~ A=
=
2k2
=
2k
2n
+ Ak
+ a + (2+ % ) A ak
+ Akll
a$
,
,
2 aA ak 2 ae ~ + A k ~ - A k / L ~ ,
and
a2
a'
Q 2 A = 2 + 7 A
We now proceed t o determine the e q u a t i o n s s a t i s f i e d by the slow v a r i a b l e s and
the l a t t e r being t h e l e a d i n g approximation t o
A,
loss of g e n e r a l i t y , can be taken t o be r e a l .
u = a. R-A'
W
which, w i t h o u t
Let
... , = R + eS2 + ... . 0 +
E'O
+
We choose t h e sequences [ a n } , {R,) appearing i n the expansion f o r
W.
i n a manner so a s t o e l i m i n a t e s e c u l a r terms In t h i s c a s e , s e c u l a r means t h a t no s o l u t i o n
W2, W 4 e t c .
e x i s t s t o the a l g e b r a i c e q u a t i o n s f o r t h e 0 v a r i a b l e and expanded w = Aeie terms were removed, no 2ll
(20),
(21),
2 [ic a
+
c4
E2W2(o)
+
..., t h e n , u n l e s s
periodic solution f o r
( 1 6 ) , (17) i n t o ( 2 ) g i v e s
(22),
2 at +
+
(k
2
-
I f we had l e f t i n
2 2 2 i e (Dl(k - l ) * + ( k - l ) D l * )
w2
would e x i s t .
the secular Substituting
Two Dimensional Convection Patterns
+
- c4(D2(k2-l)*
-R
0
- r%, -
(k2-1)D2.
+ D12) + i E 6 ( D 1 - D 2 + D 2 * D I ) +
€k4 ....+ k 2 A W 2
What does t h i s e q u a t i o n t e l l u s ? R
0
+ W22)...](A
ic4(2AW4
At
+
215
68D 2
c?W2
+
2
...) = 0 .
O(1)
(24)
= ( k 2- 1 ) 2
and so, t o leading order, the amplitude
is d e t e r m i n e d from t h e " e i k o n a l "
A
equation A A t order
2
= R
E~
2
-
( k -1)
2
.
(25)
we have t h a t
+
(-Ro
(k2-1)2)W2
+ a%, =
R.H.S.
Note t h e f o l l o w i n g i n t e r e s t i n g f e a t u r e .
For
Ro = ( k 2 -
t h e r e f o r e , e v e n though
R
- RC
= 0(1),
A2
is f i n i t e and
t h e o n l y t e r m s on t h e RHS w h i c h
g i v e r i s e t o s e c u l a r b e h a v i o r a r e t h o s e which a r e p u r e l y i m a g i n a r y .
were s m a l l , we would h a v e t o remove a l l t h e terms on t h e
A2
o t h e r hand, i f
On t h e
RHS.
I n o t h e r words, t h e n u l l s p a c e of t h e l i n e a r i z e d e q u a t i o n i s c u t i n h a l f
when
A = O(1).
f o r the amplitude
What t h i s means i s t h a t i n s t e a d of h a v i n g t w o e q u a t i o n s , one
A
a n d t h e o t h e r f o r t h e p h a s e of t h e c o n v e c t i v e p a t t e r n , we
s i m p l y have a s i n g l e e q u a t i o n f o r t h e p h a s e . a l g e b r a l c a l l y from ( 2 5 ) . of small
R
-
RC
The a m p l i t u d e is d e t e r m i n e d
T h i s a l s o means, of c o u r s e , t h a t t h e l i m i t t o t h e c a s e
and small a m p l i t u d e
A
is f a i r l y s u b t l e .
[For workers i n
n o n l i n e a r wave t h e o r y , t h e r e i s a d i r e c t a n a l o g u e between t h i s l i m i t p r o c e s s and t h e l i m i t p r o c e s s one e n c o u n t e r s when one a t t e m p t s t o o b t a i n t h e n o n l i n e a r S c h r o d i n g e r e q u a t i o n from Whitham's t h e o r y . ]
We w i l l c a r r y o u t t h e p e r t u r b a t i o n
a n a l y s i s i n such a way s o a s t o f a c i l i t a t e t h i s l i m i t p r o c e s s .
What we do is t o
u s e t h e e x p a n s i o n ( 2 2 ) which i s a l s o a n a m p l i t u d e e x p a n s i o n t o r e e x p a n d n e c e s s a r y so t h a t we can s i m p l y set W t h e following equations,
2j
= 0, j
>
A
1. We f i n d d i r e c t l y f r o m
if
23)
Alan C. NEWELL
216 uA
- Dl(k2-l)A -
R-A*
(k2-1)
-
2
=
(k2-l)DIA
E4 ii ( A t -
-D$
+ E 4 (D1*D2 + D2*Dl)A = 2 (k -1)Df
+E'D~)
-
0,
(26)
2
D2(k - l ) A
.
Because of t h e s i m p l i c i t y of t h i s model, t h e s e e q u a t i o n s a r e e x a c t .
W e will
f i r s t examine t h e s e e q u a t i o n s w i t h a veiw t o making c o n t a c t w i t h known r e s u l t s a n d t h e n we w i l l discuss aome new consequences.
4.
Connections with previous t h e o r i e s . F o r v a l u e s of
and then
A
t h e phase u
of o r d e r u n i t y , we c a n n e g l e c t t h e RHS of e q u a t i o n ( 2 7 )
R
is g i v e n as f u n c t i o n of
-
k
by ( 2 5 ) .
Equation (26) tells u s about
4
8, a n d i g n o r i n g t h e O(E ) terms can be w r i t t e n as
Az Eae +
g m ~ +
an^=^,
7
or
where
B(k)
-
A2(k) dA2/dk2
.
(30)
I want t o rernark a t t h i s p o i n t t h a t t h e f a c t t h a t t h e s p a t i a l t e r m s h a v e c o n s e r v a t i o n form is n o t a c o n s e q u e n c e of t h i s p a r t i c u l a r model n o r t h e f a c t t h a t i t can be d e r i v e d from a Lyapunov f u n c t i o n a l . (a)
The Busse B a l l o o n h o l d s l o c a l l y .
E q u a t i o n ( 2 9 ) i s e l l i p t i c s t a b l e o r u n s t a b l e ( i n t i m e ) or h y p e r b o l i c u n s t a b l e d e p e n d i n g o n which one of t h e f o l l o w i n g f o u r c a s e s o b t a i n s : 1)
2)
B
<
d 0, =(kB)
B
>
d 0, =(kB)
< >
> <
d 0, x ( k B ) d 0, &kB)
3)
B
4)
B
0;
Elliptic stable
0;
E l l i p t i c unstable
<
0;
Hyperbolic unstable
>
0;
Hyperbolic unstable.
217
Two Dimensional Convection Patterns
These r e s u l t s are s i m p l y t h e same r e s u l t s which a r e d i s p l a y e d in F i g u r e 2,
x, y
e x c e p t now a l l t h e v a r i a b l e s a r e f u n c t i o n s of
and
t.
T h e r e f o r e w e can
s a y t h a t a l l t h e s t a b i l i t y f e a t u r e s we h a d f o u n d when l o o k i n g a t t h e s t a b i l i t y of s t r a i g h t p a r a l l e l r o l l s c o n t i n u e t o h o l d l o c a l l y .
Case (1) above i s t h e
Busse b a l l o o n ; c a s e ( 2 ) i n v o l v e s i n s t a b i l i t i e s which have wavenumber dependence
in b o t h t h e a l o n g a n d p e r p e n d i c u l a r t o t h e r o l l d i r e c t i o n s ; c a s e ( 3 ) i s t h e z i g zag i n s t a b i l i t y a n d c a s e ( 4 ) is t h e Eckhaus i n t a b i l i t y . by t a k i n g t h e l o c a l r o l l w a v e v e c t o r t o be
ae + xd (
A x
Hence f o r B (k, 0);
>
for B
a% +
k B ) T
ax
0, $(kB)>
<
0,
B
a% = -
0
aY
( k , 0)
T h i s can be e a s i l y s e e n
i n which c a s e ( 2 9 ) becomes
.
(32)
0, t h e i n s t a b i l i t y h a s a w a v e v e c t o r p e r p e n d i c u l a r t o
arrd kB >
0 t h e u n s t a b l e modes are p a r a l l e l t o
( k , 0).
The a d d i t i o n of t h e c 4 t e r m i n ( 2 6 ) which i n v o l v e s h i g h e r d e r i v a t i v e s o n l y s e r v e s t o c o n t r o l t h e growth of t h e i n s t a b i l i t i e s a f t e r t h e y b e g i n .
It d o e s n o t
i n h i b i t them a l t o g e t h e r nor d o e s it of i t s e l f t r i g g e r any new i n s t a b i l i t y . The r e a d e r might l i k e t o compare t h i s r e s u l t w i t h what h a p p e n s in n o n l i n e a r wavetrains.
x
and
t
T h e r e , t h e a n a l o g u e of e q u a t i o n ( 2 6 ) is a s e c o n d o r d e r s y s t e m i n and s o it i s t h e e l l i p t i c i t y o r h y p e r b o l i c i t y of t h e s e c o n d o r d e r
o p e r a t o r which d e t e r m i n e s i n s t a b i l i t y o r ( n e u t r a l ) s t a b i l i t y of t h e w a v e t r a i n .
For example, f o r a t r a i n of g r a v i t y waves on t h e sea s u r f a c e , t h e h y p e r b o l i c n a t u r e of ( 2 6 ) c h a n g e s t o e l l i p t i c when t h e r a t i o of d e p t h t o w a v e l e n g t h is l e s s t h a n 1.36. (b)
The Newell-klhitehead-Segel
limit.
To t h i s p o i n t , we have t a k e n v a r i a t i o n s i n t h e d i r e c t i v e s p a r a l l e l t o and
p e r p e n d i c u l a r t o t h e l o c a l r o l l t o be of t h e same o r d e r of m a g n i t u d e .
It i s
c l e a r t h a t i f f o r some r e a s o n t h e l o c a l wavenumber is f o r c e d t o s t a y a p p r o x i m a t e l y c o n s t a n t , t h e v a r i a t i o n s in wavenumber of o r d e r IJ p a r a l l e l
-
to (e.g.
a r e accompanied by v a r i a t i o n s of o r d e r J p i n t h e p e r p e n d i c u l a r d i r e c t i o n (kc
+ pL) 2 +
-
2
( J I J M ) = kc
2
)
.
Near k = 1,
we f i n d t h a t v a r i a t i o n s
p e r p e n d i c u l a r t o t h e r o l l a r e of a n o r d e r of magnitude g r e a t e r t h a n t h o s e p a r a l l e l t o t h e r o l l a n d t h i s l e a d s t o a b a l a n c e between t h e t e r m
1
V
+
k B and
218
Alan C . N F W LII
some of t h e
E~
terms i n th e phase e q u a tio n ( 2 6 ) .
T h i s s i t u a t i o n c e r t a i n l y o b t a i n s when
i s s u f f i c i e n t l y small, f o r t h e n
R
-
( s e e F i g u r e 2) t h e bandwidth of wavenumbers p a r a l l e l t o t h e r o l l i s O(JR) a n d
4 -
t h e bandwidth p e r p e n d i c u l a r t o t h e r o l l i s O( JR). A s we h a v e m e n t i o n e d , i n t h i s l i m i t t h e a m p l i t u d e n o l o n g e r f o l l o w s t h e p h a s e g r a d i e n t a s i n (25) b u t t h e
terms on t h e RHS of t h e a m p l i t u d e e q u a t i o n (27) became e q u a l l y i m p o r t a n t t o T h i s b a l a n c e i s a c h i e v e d when R =
t h e s e on t h e L.H.S.
E
4
x.
For r o l l s which a r e
and
e
a =
where x = direction.
E
2
=
x
+
E
2
7)
+(x,
X a s before and
(34) = y/s =
EY, t h e new s c a l i n g i n t h e p e r p e n d i c u l a r
It is now e a s y t o show from (11) t h a t
kaa
=
ax + $_a_ ,
La8
=
Y Y
K,
=
kq8 =
+yy
and
K
e
=
-kqa
1 / ;a~
=
,
-~+,;j-
(I
= at =
E+-$-
Y
E
2
$t
(35)
w'
where we have u s e d s u b s c r i p t s i n o r d e r t o d e n o t e p a r t i a l d e r i v a t i v e s . S u b s t i t u t i o n of (35) i n t o (26) a n d d i v i d i n g by
E
2
g i v e s (we d r o p t h e t i l d e on
Y)
A$t
-
1
2
2(+x + '2 6 y ) ( 2 a x
+
+
+yy)A
-
2(2ax
+
+(zax + q Y a y + $ y y ) ~ y , ,+ ay 2(2ax + z$ a + + y y ) ~= o Y Y
+
.
tJYY)(OX
+
1
2
7 $y)A
(36)
I t i s r e a d i l y shown t h a t , i f W = Aei$ i n ( 9 ) , e q u a t i o n (36) i s p r e c i s e l y t h e i m a g i n a r y p a r t of e q u a t i o n (9).
C a r r y i n g out t h e same c a l c u l a t i o n on ( 2 7 )
219
Two Dimensional Convection Patterns (recall A
+
2
A) g i v e s t h e r e a l p a r t of e q u a t i o n ( 9 ) .
E
T h e r e f o r e t h e e q u a t i o n s ( 2 6 ) , ( 2 7 ) c o n t a i n a l l t h a t was p r e v i o u s l y known about r o l l solutions.
5.
They a l s o c o n t a i n some new i n f o r m a t i o n .
New r e s u l t s ; some a n s w e r s , more q u e s t i o n s . I n what f o l l o w s we s h a l l t a k e
R
t o be of o r d e r one a n d t h e r e f o r e ( 2 7 ) can
be r e p l a c e d by ( 2 5 ) a l m o s t e v e r y w h e r e .
The e x c e p t i o n s a r e t h o s e r e g i o n s
where V = O ( E - ~ ) b u t t h e s e p o i n t s a r e i s o l a t e d .
We w i l l c o n c e n t r a t e on t h e
phase e q u a t i o n (26),
A
ae + si;1 V at
+
+
(kB)
E
4
( D 1 * D 2 + D2*D1)A
= 0
which may b e r e w r i t t e n i n a v a r i e t y of ways.
a
V(+kB) = k
+
kB
kBII
,
(37)
I n p a r t i c u l a r we may w r i t e
2
o r i n a more r e v e a l i n g way a s V(
applying k
= k2
a i
-aa-A ,
ak a t + k -(aa
a kB -aa L
.
(39)
a i 2 - - t o ( 3 7 ) g i v e s u s two e q u a t i o n s f o r ag A
ai A z a k ~ kkB B + ~+ e~~k~ G ): [ D l * D 2
k
and
+ D2*D1)A
J,
= 0
, (40)
and
For t h e f i r s t s t e p , l e t u s assume t h a t a l l d e r i v a t i v e s a r e of o r d e r one a n d consequently ignore the
E~
t h e Busse b a l l o o n 1 = kC
<
terms. k
<
W e w i l l assume t h a t e v e r y w h e r e
kE(R) a n d prove t h a t i n a r e g i o n
k R
belongs t o
with c e r t a i n
c o n d i t i o n s on t h e b o u n d a r y a R , t h e s y s t e m r e l a x e s t o a s t a t i o n a r y s t a t e w i t h wavenumber
k
t a k i n g on t h e v a l u e which makes
B = 0.
T h i s r e s u l t does n o t
depend c r i t i c a l l y on t h e f a c t t h a t t h e p r e s e n t model i s d e r i v e a b l e from a Lyapunov f u n c t i o n ; i n d e e d i t i s a l s o v a l i d f o r s y s t e m s which do n o t h a v e t h i s property.
The r e a d e r might l i k e t o v e r i f y t h a t t h e model
a (m
2 2 (V - l ) ) ( V -1)'w
+
( R - ww
*
* 2
2
- ww V ) V w
=
0 ,
which does n o t d e r i v e from Lyapunov f u n c t i o n a l , g i v e s t h e p h a s e e q u a t i o n
(k2
+
1)2A2etk2(1
-
vk2)
+
V*CB
+
O ( E ~ )= 0
(43)
220
Alan C . N E W E L L
-
.
4 2 2 k (1-uk ) a n d A 2 = R
- ( k 2 -t 1)3 I t is c r u c i a l , however, dk2 k2 t h a t t h e o r d e r one s p a t i a l d e r i v a t i v e t e r m s i n t h e phase e q u a t i o n h a v e
where B
A2
c o n s e r v a t i o n form a n d a l t h o u g h I h a v e n o t y e t c a t e g o r i z e d t h e c l a s s , t h i s We now prove o u r r e s u l t .
happens f o r a l a r g e c l a s s of problems.
For p o s i t i v e
J ~ ,l e t J
2
et + v
o
*(CB) =
(44)
and c o n s i d e r
F =
//
G(g) dxdy
(45)
R
where
2 G = -1/2,fk B ( k 2 ) d k 2
i s p o s i t i v e a s we i n s i s t
k
>
0
<
b e l o n g s t o t h e Busse b a l l o o n i n which B
0.
Then,
dF
- =
dt
n
-/
aR
Bt h . n
ds
- // R
2
J Bt
2
dxdy,
(47)
t h e outward u n i t normal t o t h e boundary a R , where we h a v e u s e d t h e f a c t
that V G =
i:
-b.T h e r e f o r e ,
if on a R e i t h e r (i) $*n = 0 ( t h e r o l l a x e s are
p e r p e n d i c u l a r t o t h e f i x e d b o u n d a r y ) o r (ii) B
fluid boundary where
of
G
k
decreases. f o r which
model ( 4 2 ) ,
0 ( a p o r t i o n of aR may b e a
B = O),
-dF
-
But
is minimum o n l y when
G
dA2/dk2 = 0.
+
B = 0
or
k = kC
For model ( 2 ) t h i s is t h e point
kC l i e s t o t h e l e f t of
The f a c t t h a t k
I
k
-
1
the value
k = 1; f o r
by a n amount d e p e n d i n g on v
.
kC on a f r e e boundary on p o r t i o n s of aR i s c o n s i s t e n t
w i t h t h e a n a l y s i s of t h e s t a t i o n a r y e q u a t i o n ,
This mans that the quantity 2 kB/e = -H ( 8 )
(49)
i s c o n s t a n t a l o n g t h e o r t h o g o n a l t r a j e c t o r i e s of t h e c o n s t a n t phase c o n t o u r s . R e c a l l in t h e i n t e r v a l 1
<
k
<
kE, kB
<
0. T h i s in t u r n means t h a t i f
t h e 6 c o n t o u r s c o n v e r g e , which t h e y w i l l do in p a t c h e s where t h e c u r v a t u r e of
22 1
Two Dimensional Convection Patterns t he phase contours i n c r e a s e s torwards a c e n t e r , L i n c r e a s e s .
h
t h e f l u x of
I t a l s o means t h a t
between two 6 c o n t o u r s is i n d e p e n d e n t o f a. I t may be u s e f u l f o r
t h e r e a d e r t o keep i n mind t h e a x i a l l y s y m m e t r i c c a s e where a = I k ( r ) d r ,
r
6
the r a d i a l coordinate,
c o o r d i n a t e , whence L =
. Now
=
JI
=
Q,
4 the azimuthal
i n o r d e r f o r t h e s o l u t i o n t o remain s t a b l e we
must have t h a t l < k < k E which imp Les t h a t 0
where IkB
i s t h e a b s o l u t e v a l u e of
0
<
LH2(6)
and t h e r e f o r e a s L
< +
-
before the aa
2
0
H (5) = kB
(E
E
c4
-2
K(B)
<
lkBIE T h e r e f o r e we must have t h a t
kE B(kE).
(50)
along a
6
8
contour,
H2(6) m u s t tend t o zero.
Since i t
c o n t o u r s , i t must become a s small a s i t c a n
terms i n ( 3 7 ) e n t e r t h e p i c t u r e , which t h e y w i l l do when and
)
2
lkBl
lkBIE
is c o n s t a n t a l o n g c o n s t a n t
a = -
<
L = O(E-’).
a n d hence
is small only n e a r
Thus, in o r d e r t h a t t h e i n e q u a l i t y (50) h o l d s , kB = O(E 2) e v e r y w h e r e t h a t
kc = 1,
we must have
k
=
1
L = O(1).
+
But, s i n c e
2 O ( E ). Near t h e s i n k , we
can f i n d s o l u t i o n s of ( 3 7 ) i t e r a t i v e l y in t h e form 2 2 2 k = 1 += + , ,-+ =- 3y + 1 where y 2 =
4R
L
..
2 r + ~ ~ 4v K ( B ) , which i n d i c a t e s t h a t a s r
v a l u e somewhere b e t w e e n
kC = 1
and
e2, k
+
g o e s from
kC = 1
to a
kE.
Thus o u r f i r s t p r e d i c t i o n i s t h a t on t h e time s c a l e
E - ~
,
the horizontal
d i f f u s i o n t i m e , p a t c h e s form which s a t i s f y t h e boundary c o n d i t i o n s a l o n g p o r t i o n s of t h e box b o u n d a r y , a n d in which
k
+
kc
k’
n = 0
almost everywhere.
Examine t h e n u m e r i c a l e x p e r i m e n t s of G r e e n s i d e , Coughran a n d S c h r y e r [ 111 ( f i g u r e 3 ) c a r r i e d o u t on e q u a t i o n ( 2 ) f o r r e a l b r g e [121 ( f i g u r e 4 ) .
J,
a n d t h e real e x p e r i m e n t s of
Alan C . NtwEl
222
F i g . 3:
Fig.
4:
I_
-
Numerical I n t e g r a t i o n of ( 2 ) , J, R e a l H o r i z o n t a l D i f f u s i o n Time F o r Time
.
From a n e x p e r i m e n t of P. Berge Contours of C o n s t a n t Downward V e l o c i t y . A s p e c t R a t i o of 1 6 . R 2Rc.
-
223
Two Dimensional Convection Patterns Note i n t h e r e c t a n g u l a r geometry of f i g u r e 3 , t h a t p a t c h e s w i t h c i r c u l a r symmetry form a b o u t t h e c o r n e r s A a n d C.
I n t h e c i r c u l a r geometry o f
Berge ' s r e a l e x p e r i m e n t , one a g a i n s e e s c i r c u l a r p a t c h e s f o r m i n g a b o u t s i n k s which a r e a t t a c h e d t o t h e boundary.
Moreover, i t i s a b u n d a n t l y c l e a r from t h e s e
f i g u r e s t h a t t h e box c a n n o t be t i l e d w i t h t h e s e p a t c h e s .
Certain areas, f o r
example t h e c o r n e r B a n d D i n f i g u r e 3 , a r e q u i t e i n c o m p a t i b l e . compensate f o r t h e s e mismatches,
the
In order to
terms of t h e phase e q u a t i o n ( 3 7 ) must
E~
be i n c o r p o r a t e d i n t he a n a l y s i s . One way i s t o t a k e a / a a , 3 / 3 8 t o the microscopic theory.
t o be
O(E-')
b u t t h i s simply b r i n g s u s back
A n o t h e r way, which r e t a i n s t h e f u n d a m e n t a l i d e a t h a t
t h e c o n v e c t i o n f i e l d c a n b e d e s c r i b e d by a s l o w l y v a r y i n g w a v e v e c t o r
c,
is t o
.
This
recognize t h a t t h e terms
can b a l a n c e when t h e
B
derivatives are
r e d u c e s b o t h terms i n ( 5 3 ) t o
O(E-')
and
k = 1
+
O ( E ~ ) a n d s i n c e w e saw t h a t t h e main e f f e c t of
t h e dynamics on t h e h o r i z o n t a l d i f f u s i o n time s c a l e i s t o d r i v e kc, if
t h i s approximation is not a t a l l unreasonable. R = 0(1),
k = 1
+
2
O(E )
O ( E ~ ) , then
A
is
k
towards
A l i t t l e a l g e b r a shows t h a t
fi t o w i t h i n
O ( E ~ )a n d t h e
s t a t i o n a r y p a t t e r n s which one might e x p e c t t o r e a c h on t h e c-6 t i m e s c a l e (which i s t h e h o r i z o n t a l d i f f u s i o n time s c a l e m u l t i p l i e d by t h e a s p e c t r a t i o
g i v e n by
and t h e c o m p a t i b i l i t y c o n d i t i o n s ( 1 1 ) a r e
c-')
are
224
Alan C. NEWELL
E q u a t i o n s (54), (55) show i t s e l f i s of o r d e r straight.
US
that
is a t most o r d e r o n e a n d t h e r e f o r e
JI
For t h e s e s o l u t i o n s , t h e n , t h e r o l l s a r e l o c a l l y a l m o s t
E.
S i n c e we a r e now working i n d i s t a n c e s of o r d e r
E
( a s measured in box
i n r o l l w a v e l e n g t h u n i t s ) we c a n make t h e f o l l o w i n g l o c a l u n i t s ; I/E a p p r o x i m a t i o n s , which a r e v e r y s i m i l a r t o t h o s e made when we were d e r i v i n g t h e Newell-Whitehead-Segel
equations (36).
Let
( 6 , ~ ) be l o c a l l y t h e ' a c r o s s a n d
along' r o l l cordinates and
a sa = - - i a E
as
1
a n d e q u a t i o n (53) i s ( u s i n g s u b s c r i p t s f o r p a r t i a l d e r i v a t i v e s )
which is p r e c i s e l y t h e Newell-Whitehead-Segel amplitude A h e l d constant.
equation ( 3 6 ) and (9) with
We a r e now g o i n g t o d i s c u s s s o l u t i o n s of t h i s
e q u a t i o n which l e n d some i n s i g h t i n t o F i g u r e 5 which is t h e a s e q u e l t o F i g u r e
3
225
Two Dimensional Convection Patterns
B
C
A
Figure 5:
Numerical I n t e g r a t i o n of (z), $J real For Time >> H o r i z o n t a l D i f f u s i o n Time.
Notice t h a t i n o r d e r t o compensate f o r t h e i n c o m p a t i b i l i t i e s i n t h e c o r n e r B of Figure 3, the c i r c u l a r r o l l s emanating from AD have undergone a change and have R o l l number 7, counting from A a l o n g
i n t r o d u c e d d i s l o c a t i o n s along t h e w a l l AB.
AD, doubles i n width a s i t approaches t h e s i d e AB and undergoes a d i s l o c a t i o n . Roll number 9 detaches from AB a l t o g e t h e r and forms a series of d i s l o c a t i o n s along AB (which we c a l l a g r a i n boundary) and then a t t a c h e s i t s e l f t o the upper
wall
BC.
R o l l s 10 through 25 t a k e on an
S
shape i n which t h e approximate
d i s t a n c e o v e r which s i g n i f i c a n t changes occur i s t h e square r o o t of the box dimension, o r the 'along the r o l l ' s c a l i n g in e q u a t i o n (58).
226
Alan
c. N E W E l 1
Figure 6:
Dislocation
W e f i r s t n o t e t h a t a property of t h i s e q u a t i o n is t h a t i f
$ ( 5 , 1 ; ) solves
Also observe t h a t
o
=
(581,
0 = ~
5 + e(c,s),
-
6
so does
-@(-L,T,).
(59)
~i s 9g i v e n by
= E
2?
5.
The shapes of the phase contours near d i s l o c a t i o n s s u g g e s t that w e search f o r s e l f s i m i l a r s o l u t i o n s of the form
227
Two Dimensional Convection Patterns
F(z)
s a t i s f i e s the equation,
F' =
which i s r e a l l y a n e q u a t i o n f o r
G
=
dz
'
F'.
It h a s t h e symmetry p r o p e r t y t h a t i f
s o l v e s ( 6 3 ) , s o does
( F ( z ) , G(z))
E q u a t i o n ( 6 3 ) h a s a one p a r a m e t e r f a m i l y of s o l u t i o n s F ( C ; z ) w i t h d e r i v a t i v e s 2 G(C,z) which decay as Ce-' a s z + -This can b e s e e n by l i n e a r i z i n g ( 6 3 )
.
which t h e n h a s e r r o r f u n c t i o n s o l u t i o n s .
For
C
v e r y small, t h e s e s o l u t i o n s
behave v e r y much l i k e t h e e r r o r f u n c t i o n s o l u t i o n s ; t h e y a r e s y m m e t r i c a b o u t z = 0
and l e a d t o a jump i n
nonlinear and the bigger (actually
F
-
F(z)
Iln
F
Co
of
C,
F
approaches its pole s o l u t i o n s
1 2-zo
(65) and 6 F f 2 F " .
Thus t h e r e i s a c r i t i c a l
which from n u m e r i c a l c a l c u l a t i o n s i s a p p r o x i m a t e l y . 5 6 4 ,
above which t h e s o l u t i o n s do n o t e x i s t o v e r t h e l i n e approaches
Co from below,
( z n)
v a l u e of 3.14 F(z),
Y(z)
for C =
0 =
5
However ( 6 3 ) i s
= 6 C .
has a logarithmic s i n g u l a r i t y )
r e p r e s e n t i n g a b a l a n c e between F"" value
AF = F(m) - F(-)
g e t s the c l o s e r
C
has the pole;
G
of
TI
at
AF
-m
<
z
<
i s very s e n s i t i v e to changes i n
C = .54 t o 7.93 a t C = .565.
m
.
As
C
C , g o i n g from a
I n f i g u r e 7, w e g r a p h
a n d i n F i g u r e 8 , we draw t h e c o n t o u r of c o n s t a n t p h a s e 0
- + P (%T)
.
228
Alan C. NEWELL
I
/---
-F(T,z),
Figure 7 :
Graphs of
Figure 8:
Constant Phase Contours
F(n,z).
0
.
Two Dimensional Convection Patterns
229
F o r v a l u e s of 0 s l i g h t l y g r e a t e r t h a n n, t h e c o n t o u r s are d e f i n e d f o r a l l values l e s s than
For
<
TI.
t h e phase c o n t o u r s i n t e r s e c t t h e
'5
7. For
= 0 a x i s a t the o r i g i n .
0, we u s e t h e symmetry p r o p e r t y (60) t o i n f e r t h a t t h e p h a s e c o n t o u r s i n
t h i s r e g i o n a r e s i m p l y a r e f l e c t i o n of t h o s e f o r
>
0 i n the
= 0 axis.
These
s o l u t i o n s seem t o g i v e a f a i r l y a c c u r a t e p i c t u r e of t h e r e a l d i s l o c a t i o n s seen i n experiments. F i n a l l y , w e i n d i c a t e how t o i n c l u d e mean d r i f t terms i n t h e model. Consider
$ + (V2+1) 2w where
-
Rw
+ w2w* + u
vw = 0
u = Vx TZ (z i s t h e u n i t v e c t o r p e r p e n d i c u l a r t o
X,Y)
and
F o l l o w i n g t h e p r e v i o u s a n a l y s i s , we f i n d t h a t t h e s l o w e q u a t i o n f o r t h e phase is
+
kt a kB -A % 3-
+ Akll 3aTT +
O(c4) = 0
where
2 a a uL p V ~ = k t = k l l ~
(70)
In (68), t h e p a r a m e t e r l i p mimics t h e e f f e c t of low P r a n d t l number s i t u a t i o n s where mean d r i f t i s c a u s e d by t h e n o n l i n e a r a d v e c t i o n t e r m s i n t h e momentum equations.
I n (70),
V
refers
t o t h e slow d e r i v a t i v e s wi t h r e s p e c t t o
6. SUMMARY. I n t h i s p a p e r we have p r e s e a t e d a m a t h e m a t i c a l framework f o r d e s c r i b i n g
230
Alan C. NEWELI
c o n v e c t i o n p a t t e r n s which i n c l u d e s a l l p r e v i o u s t h e o r i e s a n d from i t we have
In
made s e v e r a l p r e d i c t i o n s a b o u t t h e manner in which t h e p a t t e r n s e v o l v e . p a r t i c u l a r , we s u g g e s t t h a t on t h e h o r i z o n t a l d i f f u s i o n time s c a l e
TH, t h e
c o n v e c t i o n f i e l d d e v e l o p s p a t c h e s , o f t e n of a c i r c u l a r n a t u r e s u r r o u n d i n g a s i n k , i n which t h e wavenumber is c o n s t a n t .
The i n c o m p a t i b i l i t y of t h e s e p a t c h e s
is i r o n e d o u t o v e r t h e l o n g e r t i m e scale of t h e a s p e c t r a t i o t i m e s
TH
and the
p r o c e s s i n v o l v e s a g l i d i n g motion (compare F i g u r e s 3 a n d 5) i n which r o l l d i s l o c a t i o n s move i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e r o l l a x i s .
The c l i m b
m o t i o n , where t h e d i s l o c a t i o n s move a l o n g t h e r o l l a x i s , o c c u r on t h e s c a l e TH
c2
as t h e i r r o l e is t o a d j u s t w a v e l e n g t h , a l t h o u g h small a d j u s t m e n t s of o r d e r w i l l be made on t h e
E-%'~
scale.
While we b e l i e v e w e h a v e made a s t a r t , many q u e s t i o n s s t i l l remain open. Some of t h e s e are. 1.
F o r what c l a s s of models i s t h e f l o w on t h e h o r i z o n t a l d i f f u s i o n time s c a l e a g r a d i e n t o n e ; e q u i v a l e n t l y , f o r which models does ( 4 4 ) o b t a i n ?
2.
What is t h e e f f e c t of t h e mean d r i f t term?
What p a r a l l e l c o n c l u s i o n s can we
draw?
3.
Do t h e p a t t e r n s e v e r s e t t l e down o r do t h e y a l w a y s remain n o i s y ?
If the
f o r m e r is t h e c a s e , i s i t a consequence of geometry where t h e d i s l o c a t i o n s g e t stuck i n corners?
I n a c i r c u l a r g e o m e t r y , one m i g h t a r g u e t h a t t h e
g l i d e motion n e v e r s t o p s .
I f t h e l a t t e r is t h e c a s e , d o e s t h e r e s u l t i n g
c h a o t i c motion l i e on a low d i m e n s i o n a l s t r a n g e a t t r a c t o r , one w h i c h , f o r example, mimics t h e v e r y g e n t l e h e a v i n g of t h e g l i d e m o t i o n as i t r o t a t e s a r o u n d t h e box?
23 1
Two Dimensional Convection Patterns REFERENCES 1.
G o l l u b , I. P. a n d McCarriar A. R.
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A h l e r s G. a n d Walden R. W.
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Pomeau Y. a n d M a n n e v i l l e P.
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S i g g i a E. a n d Z i p p e l i u s A.
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S t u a r t J. T. 1960. 371-389. Mech.
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38, 279.
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