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Submitted on 1 Jan 1979
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INFLUENCE OF TOROIDAL MAGNETIC FIELD ON CONVECTION IN ROTATING STAR
V. Čadež
To cite this version:
V. Čadež. INFLUENCE OF TOROIDAL MAGNETIC FIELD ON CONVECTION IN ROTATING STAR. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-707-C7-708. �10.1051/jphyscol:19797342�.
�jpa-00219336�
JOURNAL DE PHYSIQUE CoZloque C7, suppZ6ment au no?, Tome 40, J u i l Z e t 1979, page C7- 707
WLUENCE OF TOROIDAL MAGNETIC FIELD ON CONVECTION IN R O T A N G STAR
V.M. Z a d e ~ .
I n s t i t u t e o f Physics, P.O. Box 57, Yu-11001 Beograd.
W e p r e s e n t a m o d i f i e d l i n e a r t h e o r y of c o n v e c t i o n i n a d i f f e r e n t i a l l y r o t a t i n g m a g n e t i c s t a r and some, r e l a t i v e l y s i m p l e ,
a n a l y t i c a l e x p r e s s i o n s r e g a r d i n g t h e c r i - t e r i a f o r t h e i n s t a b i l i t y o n s e t were d e r i - ved.
The problem of c o n v e c t i o n i n t e r m s of s t a n d a r d l i n e a r t h e o r y and t h e normal mode a n a l y s i s was w i d e l y t r e a t e d under v a r i o u s a p p r o x i m a t i o n s and r e s t r i c t i o n s i n t h e li- t e r a t u r e s o f a r / I / , / 3 / .
.
I n t h i s p a p e r , i n o r d e r t o i n v e s t i a a t e t h e c o n v e c t i v e i n s t a b i l i t y i n p r e s e n c e of a t o r o i d a l m a u n e t i c f i e l d and when a s t a r d f f f e r e n t l a l l p r o t a t e s , we s h a l l a p p l y t h e p e r t u r b a t i o n method and c o n s i d e r o n l v tho- s e p e r t u r b a t i o n s which a r e of t h e form of narrow l o o p s / 2 / . The r e a s o n why such p e r - t u r b a t i o n s a r e t r e a t e d i s t h e f a c t t h a t t h e y , b e i n q i n l a t e r a l t o t a l p r e s s u r e e q u i - l i b r i u m , become u n s t a b l e most e a s i l y .
The f l u i d , we s h a l l c o n s i d e r , i s a s s u - med c o m p l e t e l y i o n i z e d , i d e a l l y e l e c t r i c a l -
l y c o n d u c t i v e , i n v i s c i d and c o m p r e s s i b l e . I t s a t i s f i e s t h e p e r f e c t g a s law, i t s tem- p e r a t u r e f i e l d i s l e f t u n s p e c i f i e d and i t i s a s u b j e c t of s e l f - q r a v i t a t i o n a l a t r a c - t i o n . I n a d d i t i o n , t h e f l u i d i s permeated by a t o r o i d a l m a g n e t i c f i e l d + Bo and d i f f e - r e n t i a l l y r o t a t e s i n such a way t h a t t h e r e s u l t i n s t a n q e n t i a l f l u i d v e l o c i t y compo- n e n t Vo remains c o n s t a n t a l o n q t h e f i e l d l i n e . The whole system i s t h e r e f o r e a x i a l - l y symmetric w i t h r e s w e c t t o t h e a x i s o f r o t a t i o n . The f l u i d i s i n i t i a l l y s t a t i o n a - r y , i n t h e s t a t e of e q u i l i b r i u m d e s c r i b e d by:
I
spa+
Gowen ae
--
r a e - + L O - P I - r r a e q,S;rci,,.
0 ( 2 )Here a s p h e r i c a l c o o r d i n a t e s y s t e m i s i n t r o d u c e d s o t h a t i t s o r i a i n c o i n c i d e s w i t h t h e c e n t e r o f t h e s t a r w h i l e t h e po- l a r a x i s i s o r i e n t e d toward a p o l e .
Tn f u r t h e r a n a l y s i s we s h a l l i n v e s t i q a - t e i n s t a b i l i t y c h a r a c t e r i s t i c s i n t h e equa- t o r i a l r e q i o n
(f3
4 7 / 2 1 o n l y . T h i s r e s t r i c - t i o n , however, d o e s n o t e x c l u d e any o f t h e b a s i c p h y s i c a l p r o c e s s e s i n v o l v e d i n t h e i n s t a b i l i t y .From t h e r e a s o n of symmetry a l l t h e ba- sic s t a t e q u a n t i t i e s e k e P r e s s u r e Po, t h e f l u i d d e n s i t y
9
t h e t e m p e r a t u r e To, t h e a z i m u t h a l m a g n e t i c f i e l d i n t e n s i t y Bo and t h e a n q u l a r speed of r o t a t i o n A 1 have t o be f u n c t i o n s o n l y of t h e c o o r d i n a t e r i n t h e r e g i o n n e a r t h e e a u a t o r . Consequently;t h e eq. ( 2 ) i s i d e n t i c a l l y s a t i s f i e d t h e r e . To i n v e s t i q a t e t h e b e h a v i o r of t h e f l - u i d when s l i a h t l y p e r t u r b e d a d i a b a t i c a l l y , we s t a r t from s t a n d a r d MHD e q u a t i o n s :
d2
(c= a r l i a b a t i c sound s p e e d )&=,L-
dk dt
A l l p h y s i c a l q u a n t i t i e s e n t e r i n g a s t h e unknowns i n t h e above e q u a t i o n s , have t o b e c o n s i d e r e d a s a sum of t h e i r unper- t u r b e d , b a s i c s t a t e , v a l u e s and a s m a l l a d d i t i o n a l v a l u e due t o p e r t u r b a t i o n s .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797342
S i n c e t h e b a s i c s t a t e i s s t a t i o n a r y and a x i a l l y symmetric, t h e p e r t u r b e d q b a n t i t i e s can be e x p r e s s e d l o c a l l y i n t h e f o l l o w i n g form:
A
(~,.tt;cp,t)
= 't;Cr,e)
a p Qt)I f now w e c o n s i d e r o n l y t h o s e p e r t u r h a - t i o n s which axe g e o m e t r i c a l l y of narrow s c a l e i n t h e m e r i d i o n a l d i r e c t i o n , i . e . i f :
t h e n t h e i n i t i a l s e t of MBD e q u a t i o n s re- duces t o t h e f o l l o w i n g system of l i n e a r i z - ed a l g e b r a i c e q u a t i o n s :
where: ~=Alf:n s p e e d ,
o*? a'-mSL.
F i n a l l y , t h e d i s p e r s i o n e q u a t i o n t h a t f o l l o w s from e q s . ( 4 )
-
(9) is:where :
a,
=,4.+cZ
% =
i A 2 [ ? 'La +(3*- g ) $ r ~ ~ n o r -
Simple a n a l y s i s o f eq. ( 1 0 ) shows t h a t t h e c o n d i t i o n
can be t a k e n a s a c r i t e r i o n f o r c o n v e c t i v e i n s t a b i l i t y when a t o r o i d a l m a g n e t i c f i e l d Bo i s p r e s e n t and t h e system d i f f a r e n t i a l -
l y r o t a t e s . Indeed, when BO,Q+O t h e c r i t e - r i o n (11) r e d u c e s t o t h e known c o n d i t i o n f o r c o n v e c t i o n i n a non magnezited and non r o t a t i n g system.
I t can be concluded from t h e c o n d i t i o n (11) t h a t t h e t o r o i d a l m a g n e t i c f i e l d w i l l t e n d t o r e d u c e t h e c o n v e c t i o n p r o v i d e d t h a t t h e u n e s u a l i t y
h o l d s l o c a l l y . F o r example, t h i s can b e achieved i n o u t e r r e g i o n s of t h e s t e l l a r i n t e r i o r ( r > 2 c /g*) i f Bo i n c r e a s e s w i t h 2 r o r i f i t d e c r e a s e s . w i t h r but.,slower t h a n
qor.
I t h a s a l s o t o b e mentioned t h a t t h e p r e s e n c e of m a g n e t i c f i e l d a f f e c t s t h e s t a - b i l i t y of t h e system by promoting o k h e r , p u r e l y m a g n e t i c , i n s t a b i l i t i e s which w i l l n o t be c o n s i d e r e d i n t h i s work.A s t o t h e i n f l u e n c e o f r o t a t i o n , w e s e e t h a t i f
2
A?> o t h e c o n v e c t i o n w i l l be r e - duced. For example, a r i g i d body r o t a t i o n ar t e d n s t o s t a b i l i z e t h e c o n v e c t i v e i n s t a b i - l i t y ./1/ "Problems o f S t e l l a r c o n v e c t i o n " , Proc.
Nice, 1976, E d i t e d by E.A.Spiege1 and J.P.Zahn. Springel;rVerlag B e r l i n .
/2/ P.A.Gilman, Ap.J.
162
(1019) 1970./ 3 / C.H.Sung.Astron.Astroph.