But many results obtained in these investigations can also be of i n t e r e s t f o r the p r o b l e m under consideration. Useful e x p r e s s i o n s a r e contained in [7, 8], in p a r t i c u l a r . LITERATURE 1.
2. 3. 4. 5. 6.
7. 8.
CITED
Yu. V. Mitikhin and R. A. P e r t s o v s k i i , t~lektrosvyaz t, No. 9, 27 (1972). V. A. Alimov and G. P. Komrakov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 15, No. 10, 1581 (1972). Ya. P. Al'pert, Propagation of Electromagnetic Waves and the Ionosphere, Plenum (1972). L S. Vsekhsvyatskaya, Statistical P r o p e r t i e s of Signals Reflected f r o m the Ionosphere [in Russian], Nauka, Moscow (1973). K. L. Afanas, ev, ]~. V. Bol'shakov, A. A. Garnaker, yan, et aL, Tr. Taganrog. Radiotekh. Inst., No. 22, 148 (1971). Yu. M. Polishchuk, Radiotekh. t~lektron., 15, No. 5, 891 (1970). V. F. Nesteryuk and N. N. P o r f i r ' e v a , Izv. Vyssh. Uchebn. Zaved., Radio~lektron., 14, No. 3 , 2 5 3 (1971). S. I. Pozdnyak and V. N. Melititskii, Introduction to the Statistical T h e o r y of the p o l a r i z a t i o n of Radio Waves [in Russian], Sovet-skoe Radio, Moscow (1974).
PROPAGATION
OF MAGNETOHYDRODYNAMIC
IN I N H O M O G E N E O U S
MEDIA
WITH
FINITE
WAVES CONDUCTIVITY
A. O. O v c h i n n i k o v
UDC 533.951 : 535.31
Using the methods of g e o m e t r i c a l optics, we investigate the propagation of magnetohydrodynamic waves in a cold inhomogeneous plasma. The magnetic viscosity is assumed to be a smoothly varying nonvanishing function of coordinates. We obtain equations that make is possible to calculate in the f i r s t approximation the amplitude and additional phase i n c r e m e n t of the wave p r o p a g a tion. The n e c e s s i t y is noted of taking into account the small magnetic viscosity in the g e o m e t r i c a l - o p t i c s description in the f r a m e w o r k of magnetohydrodynamics.
G e o m e t r i c a l optics is usually used in the p r o b l e m s of magnetohydrodynamic wave propagation in the assumption of p e r f e c t conductivity [1]. Ginsburg [2] e s t i m a t e d the role of conductivity and viscosity in the p a r ticular c a s e of uniform medium. Dubrovskii and Skuridin [3] took into account a constant finite conductivity o. by means of an asymptotic s e r i e s in i n v e r s e powers of o., and the solution has an essential singularity as o. --* r In the p r e s e n t work we investigate in the g e o m e t r i c a l - o p t i c s approximation the effect of a finite slowly varying conductivity on the propagation of perturbations in an inhomogeneous magnetohydrodynamic medium. The solution also makes it possible to take the limit o. -* 00. We shail c o n s i d e r monochromatic [exp(-iwt)] magnetohydrodynamic p r o c e s s e s in a cold p l a s m a which satisfy the following equations in the linear approximation (we adopt the SI s y s t e m of units): -- i ~oH = rot[;, H] -- rot(~ rot/f),
--i~v=~irot/~,
HI,
dtvH=0,
(1)
dtv H = 0. w h e r e H and v a r e small p e r t u r b a t i o n s of the magnetic field and of the velocity of motion of the medium, tI and p a r e spatialy slowly varying magnetic field and density in the absence of perturbations in the medium, and ~? = 1//zoo- is the magnetic viscosity. The unperturbed state is a s s u m e d to be steady, and the medium stationary. In the ray description of the p r o c e s s of small perturbation propagation, the quantities H and v a r e usually sought [4] in the f o r m of a s e r i e s in i n v e r s e powers of the frequency w: Leningrad State University. T r a n s l a t e d f r o m Izvestiya Vysshikh Uchebnykh Zavednii, Radiofizika, Vol. 21, No. 4, pp. 511-513, April, 1978. Original a r t i c l e submitted F e b r u a r y 15, 1977.
0033- 8443/78/2104-0349 $07.50 9 1978 Plenum Publishing Corporation
349
where t = r(r) is the equation of the wavefront at time t. We shall a s s u m e that the l o s s e s a r e sufficiently s m a l l and smoothly varying so that the following r e l a tion holds: ~ d w ) 2 << 1,
(3)
The validity of this inequality is due to the conditions of applicability of the magnetohydrodynamics (we neglect the displacement c u r r e n t s in c o m p a r i s o n with the conduction c u r r e n t s ) :
;~,lv ~1 ~ ~,
(4)
where ~ is the wavelength. We substitute (2) into (1), take into account that the t e r m s containing ~ and V~ should be included in the f i r s t - a p p r o x i m a t i o n equation due to inequalities (3) and (4), and solve the resulting equation with r e s p e c t to velocity. This gives in the z e r o and f i r s t approximation Vo - ~ {[(Vo• P v, --
N0 {[(vtXH)• P X H + rot H o •
•
• H = 0;
X H -- [rot(vo•
(5)
X
-- to2~q(V~)t(/-/0Xv~)X/-/+
+ i o~{[(WXHo)XV~IXw}Xlt = 0.
(6)
Equating to z e r o the determinant c o r r e s p o n d i n g to (5) we obtain two eikonal equations [1]: 1) (VrCA) 2 = 0. This c o r r e s p o n d s to the Alfv6n wave. 2) (VT)2C~ = 0. This c o r r e s p o n d s to the fast magnetic sound wave. Here CA =
H, a n d C A = I C A [ "
We shall now use the relationship between v 0 and H 0 [i], and the fact that because of (5), the sum of the f i r s t two v e c t o r s in (6) is a v e c t o r perpendicular to v 0. Thus, taking the Alfv6n eikonal and taking the s c a l a r p r o d u c t of (6) with v0, we obtain v0 (rot[v0, H l X w ) X H - - r o t
v0
XH~
(7) The solution (7) can be written in the f o r m h
\
2
h
JC~cos2~ 0
2CA 0
w h e r e h is the coordinate along the f o r c e line of the magnetic field, and c~ is the angle between H and VT. We have a s s u m e d that the g e o m e t r i c a l - o p t i c s solution v~ in the nonabsorbing c a s e 77 = 0 is known [I]. The second exponential f a c t o r in (8) c o r r e s p o n d s to the additional phase increment.
In fact, the equation
h
of the phase front at time t should be changed to i = ~ + ~ !,~v~) dh. However, the additional i n c r e m e n t o v e r a J 2C~ wavelength is v e r y s m a l l on account of (3) and (4). Here ~" plays as b e f o r e an important role, in p a r t i c u l a r v 0 and Ho• The inequality (3) can be written in the f o r m -~ << CAk COSz a for the Alfv6n wave| ~] (< CAk for the fast magnetic sound wave.
{9)
(i0)
F o r example, at the height 300 km in the ionosphere ~7 is s m a l l e r than CAk by four o r d e r s of magnitude
350
even in the c a s e of the h i g h - f r e q u e n c y l i m i t [5]. The inequality (10) is n e c e s s a r y so that the damping is small. In this c a s e (9) h a s to be viewed as d e t e r m i n i n g the r a n g e of angles a w h e r e the damping is small. In the i o n o s p h e r e at the height 200-300 km (9) is satisfied up to about 89 ~ In fact (9) does not allow us to investigate angles too close to 7r/2 w h e r e the m a g n e t o h y d r o d y n a m i c equations a r e not applicable [2]. F o r ~ >- k C A, the solution is damped even at the eikonal level, and the damping o v e r a wavelength is a p p r e c i a b l e . This v i o l a t e s the applicability conditions of the m a g n e t o h y d r o d y n a m i c approximation, and we t h e r e f o r e investigate only the c a s e when the inequalities (9) and (10) hold. In conclusion we note that s i m i l a r calculations f o r the f a s t magnetic sound wave lead to the e x p r e s s i o n
vo=v0ex p
c3]exp
i~j
0
2C,
)'
(11)
(}
w h e r e s is the coordinate along the ray. In a s i m i l a r fashion one can take into account o t h e r p r o c e s s e s which lead to energy dissipation. F o r example, to take into account the usual v i s c o s i t y , one has to r e p l a c e ~ by the s u m of V and the v i s c o s i t y coefficient, in the f a c t o r which d e t e r m i n e s the damping of the Alfv6n wave. F o r the f a s t magnetic sound wave, ~? should be r e p l a c e d by
(12)
P
w h e r e ~' and ~ a r e the f i r s t and second v i s c o s i t y coefficients. The author would like to e x p r e s s gratitude to V. N. K r a s f l ' n i k o v f o r his constant attention and help with c o m p l e t i o n of this work. LITERATURE 1o
2. 3. 4. 5.
CITED
J. B a z e r and J. Hurby, J. Geophys. Res., 68, No. 1, 147 (1963). V. L. Ginzburg, P r o p a g a t i o n of E l e c t r o m a g n e t i c Waves in P l a s m a [in Russian], Nauka, Moscow (1967). V. A. Dubrovskii and G. A. Skuridin, Geomagn~ Astron., 5, No. 2 , 2 3 4 (1965). V. M. Babich and A. S. Alekseev, in: P r o b l e m s of the D y n a m i c a l Theory of Seismic Waves P r o p a g a t i o n [in Russian], Vol. 5, LGU (1961). Ya. L. A l ' p e r t , Waves and Artificial Bodies in the P l a s m a around the Earth [in Russian], Moscow (1974).
351
