COMPUTER SIMULATION OF THE SIDEBAND INSTABILITY

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Submitted on 1 Jan 1979

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COMPUTER SIMULATION OF THE SIDEBAND INSTABILITY

Magdi Shoucri

To cite this version:

Magdi Shoucri. COMPUTER SIMULATION OF THE SIDEBAND INSTABILITY. Journal de

Physique Colloques, 1979, 40 (C7), pp.C7-575-C7-576. �10.1051/jphyscol:19797278�. �jpa-00219265�

JOURNAL DE PHYSIQUE ColZoque C7, supple'ment au n07, Tome 40, JuiZZet 2979, page C7- 575

COMPUTER SIMULATION OF THE SIDEBAND INSTABILITY

Magdi M. _Shoucri.

Direction Sciences de base, I n s t i t u t de Recherche de Z 'Hydro-&&bee Varennes, Qkbec, Canada, JOL ZPO.

The p h y s i c a l o r i g i n s of t h e growth of s i d e - bands of a l a r g e a m p l i t u d e wave i s s t u d i e d numeri- c a l l y , by s o l v i n g t h e Vlasov e q u a t i o n i n phase- s p a c e by a s p l i t t i n g method r e c e n t l y r e p o r t e d i n t h e l i t e r a t u r e 1 .

The p e r t i n e n t e q u a t i o n s a r e t h e Vlasov- P o i s s o n system of e q u a t i o n s , which a r e w r i t t e n i n d i m e n s i o n l e s s form a s : .a

f i s t h e e l e c t r o n d i s t r i b u t i o n f u n c t i o n ; E t h e e l e c t r i c f i e l d and t h e o t h e r symbols have t h e i r c o n v e n t i o n a l meaning. The d i s t r i b u t i o n f u n c t i o n h a s been i n i t i a l i z e d a s :

f ( x , v , o ) = f o ( v ) ( l + ^a c o s km x m

3

r cx (cop kei x + c o s k p i x))

1 2

and f o ( v ) = (2r)-' exp(-v / 2 ) . The s u b s c r i p t s m, ti, p r e f e r r e s p e c t i v e l y t o t h e main wave, t h e lower s i d e b a n d s (k L i < k m ) and upper s i d e b a n d s ( k p i > km). The l e n g t h of t h e system is L = 80n, and km = 2n m/L = 0.3 w i t h m = 12. W e a r e u s i n g

F i g . 1 S p a t i a l l y a v e r a g e d d i s t r i b u t i o n f u n c t i o n .015

.g E .a-

0 3

C C 0

.-

f 005- 0 '

L C

i m i

o

128 p o i n t s i n s p a c e i n t h e p r e s e n t s i m u l a t i o n , and

- - - _{T = 165}

--- T = 380

I I I r l

a t i m e - s t e p A t = 1/8w -1 . The main wave i s e x c i t - P

2 3 4 5 6 7

Velocity

ed w i t h a l a r g e a m p l i t u d e a m = 0.18. T h i s c o r r e -

1

sponds t o a bounce f r e q u e n c y wb = (orm/2)' = 0.3.

A l l t h e s i d e b a n d s a r e e x c i t e d i n i t i a l l y w i t h t h e same l e v e l a = 2 x 1 0 -3 . For t h e lower s i d e b a n d s

kR1 = 2n x 11/L = 0.275, kL2 = 2x x 10/L = 0.25 and ke3 = 2n x 9/L = 0.225. For t h e upper s i d e - bands k = 2n x 13/L = 0.325, k = 2n x 14/L =

P 1 ?J2

0.35, k = 271 x 1 5 / L = 0.375. The measured asymp- v3

t o t i c f r e q u e n c i e s of t h e fundamental mode i s w = n 1.142. F i g u r e 1 shows t h e s p a t i a l l y averaged d i s - t r i b u t i o n f u n c t i o n , which a t T = 165w -1 h a s a l m o s t

P

s t a b i l i z e d t o a s h a p e having a s t a b l e p o s i t i v e

1

0 460 200 3 0 0 4 6 0

Time T

F i g . 2 Time v a r i a t i o n of t h e f u n d a m e n t a l mode w i t h k = 0 . 3

- 4 - ^I

I I

0 ?OO 2 0 0 300 400

I - I

Time T

F i g . 3 Time v a r i a t i o n o f t h e f i r s t h a r m o n i c w i t h

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797278

s l o p e , w h i l e t h e s i d e b a n d s a r e s t i l l growing (Figs.

(4-9)). The f r e q u e n c i e s of t h e lower s i d e b a n d s a r e r e s p e c t i v e l y wLl = 1.092, wR2 = 1.068 and wR3 = 1 , 0 3 3 and w = 1.196, w = 1.225, and w =

1.1 1 11 2 u3

1.256. W e n o t e t h a t w

$1

w l l l ⁼ 2.288, wi2 + ^{w =}

11 2 2.293 and wRg + ^{w =} 2.289, w h i l e t h e frequency

u3

of t h e f i r s t harmonic i s 2w m = 2.284. If i n addi- t i o n we n o t e t h a t t h e wavenumber of t h e f i r s t har- monic i s 2k m = 0.6, and t h a t k$, + ^{kUl = ki2} + ku2

= kR3 + ku3 = 0.6, our r e s u l t s c l e a r l y p o i n t s t o a n o n l i n e a r c o u p l i n g between t h e f i r s t harmonic and

I I I I 1

0 400 200 300 4 00

Time T

F i g . 4 Time v a r i a t i o n o f t h e l o w e r s i d e b a n d w i t h kQ1 = 0.275

t ^{I I I} i

0 400 200 300 400

Time T

F i g . 5 Time v a r i a t i o n o f t h e l o w e r s i d e b a n d w i t h kR2 = 0.25

I I I

I I r

0 100 200 300 100

Time r

F i g . 6 Time v a r i a t i o n of t h e l o w e r s i d e b a n d w i t h kL3 = 0.225

t h e sidebands2. We a l s o n o t e t h a t t h e lower s i d e - bands have t h e i r phase v e l o c i t i e s wii/kRi on t h e p o s i t i v e s i d e of t h e s l o p e of t h e d i s t r i b u t i o n f u n c t i o n , which enhances t h e i r growth. The s i d e - bands grow up t o a l e v e l where t h e s p a t i a l l y aver- aged d i s t r i b u t i o n f u n c t i o n i s d i s t o r t e d i n t o a s h a p e c l o s e t o a f l a t p l a t e a u , a s i s c l e a r from F i g . 1 a t T ' 380 -1 and t h e s i d e b a n d s a s w e l l a s

P

t h e fundamental mode s a t u r a t e .

I ^{I I I I}

0 100 200 300 400

Time T

F i g . 7 Time v a r i a t i o n of t h e u p p e r s i d e b a n d w i t h k = 0 . 3 2 5

1.1 1

I i

o roo 200 360 000

Time T

F i g . 8 Time v a r i a t i o n o f t h e u p p e r s i d e b a n d w i t h k = 0.35

u2

, ,

I I I I

0 400 200 300 400

Time T

F i g . 9 Time v a r i a t i o n o f t h e u p p e r s i d e b a n d w i t h k = 0.375

1.13 R e f e r e n c e s

1. R. GagnE and M.M. S h o u c r i , J . Comput. Phys. 24, 445 (1977).

2. M.M. S h o u c r i , Phys. F l u i d s 21, 1 3 5 9 (1978).