I N S T I T U T D ' A E R O N O M I E S P A T I A L E D E B E L G I Q U E 3 - Avenue Circulaire
B - 1180 BRUXELLES
A E R O N O M I C A A C T A
A - N° 232 - 1981
Differences between solar wind plasmoids and ideal magnetohydrodynamic filaments
by
J. LEMAIRE and M. ROTH
B E L G I S C H I N S T I T U U T V O O R R U I M T E - A E R 0 N O M I E
3 - Ringlaan
B • 1180 BRUSSEL
FOREWORD
The paper : "Differences between solar wind plasmoids and ideal magnetohydrodynamic filaments" will be published in Planetary and Space Sciences.
AVANT-PROPOS
L'article : "Differences between solar wind plasmoids and ideal magnetohydrodynamic filaments" sera publié dans Planetary and Space Sciences.
VOORWOORD
Het artikel : "Differences between solar wind plasmoids and ideal magnetohydrodynamic filaments" zal verschijnen in Planetary and Space Sciences.
VORWORT
Das Artikel : "Differences between solar wind plasmoids and
ideal magnetohydrodynamic filaments" wird in Planetary and Space
Sciences herausgegeben werden.
D I F F E R E N C E S BETWEEN S O L A R WIND P L A S M O I D S A N D IDEAL M A G N E T O H Y D R O D Y N A M I C F I L A M E N T S
by
J. L E M A I R E and M. ROTH
Abstract
Plasma irregularities present in the solar wind are plasmoids (i.e. plasma-magnetic field entities). These actual plasmoids differ from ideal magnetohydrodynamic ( M H D ) filaments. Indeed, 1) their "skin" is not infinitely thin but has a physical thickness which is determined by the gyromotion of the thermal ions and electrons, 2) they are of finite extent and their magnetic flux is interconnected with the interplanetary magnetic flux, 3) when they penetrate into the magnetosphere their magnetic field lines become rooted in the ionosphere (i.e. in a medium with finite transverse conductivity), 4) the external Lorentz force acting on their boundary surface depends on the orientation of their magnetic moment with respect to the external magnetic field, 5) when their mechanical equilibrium is perturbed, hydromagnetic oscillations can be generated. It is also suggested that the front side of all solar wind plasmoids which have penetrated into the magnetosphere Is the inner edge of the magnetospheric boundary layer while the magnetopause is considered to be the surface where the magnetospheric plasma ceases to have a trapped pitch angle distribution.
Key words : Filaments, plasmoids, magnetopause.
R é s u m é
L e s i r r é g u l a r i t é s de p l a s m a p r é s e n t e s d a n s le v e n t s o l a i r e s o n t d e s p l a s m o ï d e s ( c ' e s t - à - d i r e d e s e n t i t é s d i s t i n c t e s de p l a s m a m a g n é t i s é ) . D a n s la r é a l i t é , ces p l a s m o ï d e s d i f f è r e n t d e s f i l a m e n t s i d é a u x m a g n é t o h y d r o d y n a m i q u e s ( M H D ) . En e f f e t , 1) l e u r " p e a u " n ' e s t pas i n f i n i m e n t m i n c e mais p o s s è d e u n e é p a i s s e u r p h y s i q u e q u i e s t d é t e r m i n é e p a r le m o u v e m e n t g i r a t o i r e d e s ions et d e s é l e c t r o n s t h e r m i q u e s , 2) i l s s o n t d ' e x t e n s i o n f i n i e et l e u r f l u x m a g n é t i q u e e s t i n t e r c o n n e c t é au f l u x m a g n é t i q u e i n t e r p l a n é t a i r e , 3) l o r s q u ' i l s p é n è t r e n t d a n s la m a g n é t o s p h è r e , l e u r s l i g n e s de f o r c e m a g n é t i q u e d e v i e n n e n t c o n n e c t é e s à l ' i o n o s p h è r e ( c ' e s t - à - d i r e à u n milifiii rip c o n d u c t i v i t é f i n i e ) , 4) la f o r c e de L o r e n t z e x t e r n e a g i s s a n t s u r l e u r s u r f a c e f r o n t i è r e d é p e n d de l ' o r i e n t a t i o n de l e u r moment m a g n é t i q u e p a r r a p p o r t au c h a m p m a g n é t i q u e e x t e r n e , 5) l o r s q u e l e u r é q u i l i b r e m é c a n i q u e e s t p e r t u r b é , d e s o s c i l l a t i o n s h y d r o m a g n é t i q u e s p e u v e n t ê t r e e n g e n d r é e s . O n s u g g è r e é g l a m e n t q u e la s u r f a c e a v a n t d e t o u s les p l a s m o ï d e s d u v e n t s o l a i r e q u i o n t p é n é t r é d a n s la m a g n é t o s p h è r e c o ï n c i d e a v e c le b o r d i n t e r n e de la c o u c h e f r o n t i è r e m a g n é t o s p h é r i q u e , t a n d i s q u e la m a g n é t o p a u s e e s t i d e n t i f i é e à la s u r f a c e à p a r t i r de l a q u e l l e le p l a s m a m a g n é t o s p h é r i q u e c e s s e d ' a v o i r u n e d i s t r i b u t i o n p i é g é e en a n g l e d ' a t t a q u e .
Mots c l é : f i l a m e n t s , p l a s m o ï d e s , m a g n é t o p a u s e .
Samenvatting
De plasma irregulariteiten in de zonnewind zijn "plasmoïden" : dit zijn welonderscheiden entiteiten van gemagnetiseerd plasma. Deze plasmoïden verschillen van de ideale magnetohydrodynamische filamenten. Inderdaad, (1) de dikte van hun "huid" is niet oneindig klein maar heeft een fysische afmeting bepaald door de gyrobeweging van de thermische ionen en elektronen; (2) zij zijn begrensd in de ruimte en hun magnetische flux is verbonden met de interplanetaire magnetische f l u x ; (3) wanneer zij in de magnetosfeer binnendringen dan worden hun magnetische veldlijnen verbonden aan de ionosfeer ( t . t . z . aan een milieu met eindige transversale geleidbaarheid); (4) de. uit- wendige Lorentzkracht die werkt op hun grensoppervlak is afhankelijk van de richting van hun magnetisch moment t . o . v . het uitwendige magneetveld; (5) wanneer hun mechanisch evenwicht verbroken wordt' kunnen hydromagnetische oscillaties optreden. Er wordt tevens ge- sugereerd dat de voorkant van al de zonnewind, plasmoïden die in de magnetosfeer binnendringen, samenvalt met de inwendige zijde van de magnetosferische randlaag, terwijl de magnetopauze beschouwd wordt als het oppervlak waar het magnetosferisch plasma geen gevangen "pitch angle" distributie meer bezit.
Trefwoorden : Filamenten, plasmoïden, magnetopauze.
- 3 -
Z u s a m m e n f a s s u n g
Die I n h o m o g e n e i t ä t e n des S o n n e n w i n d e s s i n d s o g e n a n t e Plasmoiden ( d . h . P l a s m a - M a g n e t i s c h e F e l d e r E i n h e i t e n ) . In W i r k l i c h k e i t e n t s c h e i d e n diese Plasmoiden s i c h m i t d i e idealen m a g n e t o h y d r o - d y n a m i s c h e n ( M H D ) Filamenten d u r c h : 1) i h r e " H a u t " d i e n i c h t u n e n d l i c h d ü n n i s t a b e r w e n i g s t e n s e i n i g e Ionen G y r o r a d i e n d i c k i s t ; 2 ) sie h a b e n n i c h t eine u n e n d l i c h e Länge o d e r B r e i t e , u n d i h r e n m a g n e t i s c h e F l u x u m i s t mit dem i n t e r p l a n e t a r e n m a g n e t i s c h e n F l u x u m i n t e r c o n n e c t i e r t ; 3 ) w e n n Sie in d e r M a g n e t o s p h ä r e e i n d r i g e n w e r d e n i h r e m a g n e t i s c h e F e l d l i n i e n m i t d e n F e l d l i n i e n d e r I o n o s p h ä r e i n t e r - c o n n e c t i e r t ; 4 ) d i e ä u s s e r e L o r e n t z K r a f t d i e a u f i h r e O b e r f l ä c h e w i r k t , h ä n g t v o n d i e R i c h t u n g i h r e s m a g n e t i s c h e Momentum a b ; 5 ) w e n n i h r e s m e c h a n i s c h e s G l e i c h g e w i c h t g e s t ö r t w i r d k ö n n e n h y d r o m a g n e t i s c h e Wellen e n t s t e h e n .
Es w i r d a u c h v o r g e s c h l a g e n dass d i e f o r d e r e O b e r f l ä c h e a l l e r S o n n e n w i n d e s Plasmoiden d i e in d e r M a g n e t o s p h ä r e e i n g e d r u n g e n s i n d , m i t d i e i n n e r G r e n z e d e r M a g n e t o s p h ä r i s c h e n " P l a s m a - B o u n d a r y L a y e r "
k o i n z i d i e r e n ; d i e M a g n e t o p a u z e k o i n z i d i e r t d a n n m i t d i e S t e l l e n wo d i e a n g u l a r e V e r t e i l l u n g e d e r T e i l c h e n g e s c h w i n d i g k e i t e n des m a g n e t o - s p h e r i s c h e Plasma seinen s y m e t r i s c h e n C h a r a k t e r ( g e f a n g e n h e i t s C h a r a k t e r ) v e r l i e r t .
1. I N T R O D U C T I O N
It was suggested in 1976, at the European Geophysical Society symposium on the Magnetopause regions, that the solar wind plasma carries small scale filamentary density irregularities which penetrate impulsively into the magnetosphere (Lemaire and Roth, 1978). These irregularities are plasmoids (i.e. plasma-magnetic field entities) according to the definition given by Bostik (1956).
Contrary to steady-state interaction models, this time- dependent penetration mechanism explains a wider range of magneto- spheric events which are typically impulsive phenomena (Lemaire, 1977).
Lemaire et al (1979) suggested also that the direction of the inter- planetary magnetic field controls the impulsive penetration of small scale magnetosheath plasma irregularities into the magnetosphere.
In this paper we elaborate further on this time-dependent impulsive penetration model, and emphasize in which respects it differs from ideal MHD models. It is indicated in section 2 that plasmoids of finite extent produce a magnetic field distribution which can inter- connect with any external magnetic field, even though such a magnetic coupling does not exist in the case of infinitely conducting M H D filaments. The total Lorentz force acting on the surface of a plasma element has been decomposed into an internal force and an external force (section 3). The external Lorentz force, as well as the " e n t r y condition", depend on the orientation of the magnetic moment of the plasmoid or of the current layer with respect to the external magnetic field. The mechanical equilibrium condition for a plasmoid is discussed in section 4. In the absence of mechanical equilibrium the front edge of a plasma element intrudes into the region of closed geomagnetic field lines. The inner edge of the magnetospheric "plasma boundary layer" is then defined in section 5 as the envelope of all these plasma fronts.
The outer edge of this transition region, which corresponds also to the
- 5 -
"magnetopause", is defined in terms of characteristic plasma properties instead of magnetic field signatures.
2. MHD AND NON-MHD FILAMENT MODELS
The plasma irregularities considered by Lemaire and Roth (1978) have dimensions which are smaller than the diameter of the magnetosphere. Furthermore, their finite volume does not necessarily coincide with a magnetic flux tube; their whole surface is not a tangential discontinuity as it is for the ideal MHD models recently studied by Schindler (1980).
Figure 1A represents a finite plasma element with a sketch of the magnetic field line topology associated with such a density
irregularity. By contrast, figure 1B shows an ideal MHD filament confined within a cylindrical magnetic flux tube of infinite length. In this MHD filament, plasma particles drift along (and "with") magnetic field lines which are equipotential lines as a consequence of the following assumed MHD conditions :
1) infinite parallel conductivity (CT
#= with the consequence that 2 . 3 = 0 ) ,
2) vanishingly small transverse Pedersen conductivity everywhere along the magnetic field lines (a
A= 0).
Strictly speaking these MHD conditions are almost never satisfied except in collisionless magnetized plasmas that have either infinite extent, or are surrounded by insulating walls.
In ideal MHD models the lateral plasma boundaries are
tangential discontinuities. Indeed, if this were not so, field lines would
dip across the surface of the plasma cloud (as, for instance, in the
case of classical double layers, oblique electrostatic shocks or rotational
discontinuities). Along the magnetic field lines the high-speed plasma
electrons would be able to run away out of the plasma element, unless
F i g . 1 a . - Representation of a collisionless plasmoid. T h e diamagnetic c u r r e n t s ( Jd) circulating at the edge of the plasma element p r o d u c e a magnetic field inside as well as outside its volume, which has dipole as well as multipole components. T h e c u r r e n t sheet is at least a few ion g y r o r a d i i t h i c k . T h e magnetic polarization M of the plasmoid is related to the total c u r r e n t d e n s i t y ( K = J ^ d N = l l x fl). T h e magnetic field lines t r a v e r s e the s u r f a c e of the cloud of particles. T h e magnetic field has a n o n - z e r o component in the direction of tf, the normal to this s u r f a c e ; this s u r f a c e is not e v e r y w h e r e a tangential d i s c o n t i n u i t y . T h e r m o - electric c h a r g e separation p r o d u c e s electrostatic potential differences a c r o s s the s u r f a c e of the plasmoid. T h e s e polarization electric fields h a v e ' components p e r p e n d i c u l a r and parallel to the magnetic field direction. When the external kinetic and magnetic p r e s s u r e s are too small, the volume of the plasma element e x p a n d s with a velocity v parallel to tf.
F i
9 - 1b.- Representation of ideal -MHD and infinitely long plasma filament. It is assumed that all magnetic field lines are parallel to the cylindrical filament surface. The magnetic field has no component perpendicular to this surface, and consequently there can exist no coupling or inter- connection between the magnetic flux inside and outside the filament.
This model filament can be compared to an infinitely long and super-
conducting solenoid (after Schindler, 1979).
t h e r e w e r e a p a r a l l e l e l e c t r i c p o t e n t i a l b a r r i e r p r e v e n t i n g t h e s e e l e c t r o n s f r o m d i f f u s i n g o u t w a r d f a s t e r t h a n t h e t h e r m a l i o n s . O b v i o u s l y , a p a r a l l e l p o t e n t i a l d i f f e r e n c e implies a p a r a l l e l c h a r g e s e p a r a t i o n e l e c t r i c f i e l d , and n o n - e q u i p o t e n t i a l m a g n e t i c f i e l d l i n e s ; b u t since t h i s w o u l d v i o l a t e one of t h e MHD c o n d i t i o n s ( I . B = 0 ) , i t is c o n c l u d e d t h a t i n t h e ideal MHD models t h e m a g n e t i c f i e l d lines m u s t i n d e e d be p a r a l l e l t o t h e plasma b o u n d a r i e s .
On t h e c o n t r a r y , f o r i r r e g u l a r i t i e s l i k e t h a t i l l u s t r a t e d in f i g u r e 1 A , t h e m a g n e t i c f i e l d lines a r e n o t e q u i p o t e n t i a l l i n e s , n o r are t h e y e v e r y w h e r e p a r a l l e l t o t h e plasma s u r f a c e . C o n s e q u e n t l y , t h e f i l a m e n t model of Lemaire et a l . (1979) is f u n d a m e n t a l l y d i f f e r e n t f r o m t h e ideal MHD m o d e l s .
F u r t h e r m o r e , in ideal MHD m o d e l s , i t is assumed t h a t t h e m a g n e t i c f i e l d v e c t o r c h a n g e s a b r u p t l y d i r e c t i o n a n d / o r i n t e n s i t y f r o m a c o n s t a n t v a l u e i n s i d e t h e f i l a m e n t , t o some d i f f e r e n t v a l u e B
r M o u t s i d e t h e f i l a m e n t . H o w e v e r , a c t u a l t a n g e n t i a l d i s c o n t i n u i t i e s a l w a y s
h a v e a f i n i t e t h i c k n e s s . A more r e a l i s t i c p i c t u r e is t h e n one w h e r e t h e m a g n e t i c f i e l d c h a n g e s g r a d u a l l y o v e r s e v e r a l ion g y r o - r a d i i as d e s c r i b e d in p a p e r s b y Lemaire and B u r l a g a ( 1 9 7 6 ) , R o t h (1976, 1978, 1979, 1980) and more r e c e n t l y b y Lee and Kan (1979, 1980). In o t h e r w o r d s , t h e " s k i n " of t h e plasma c l o u d is n o t i n f i n i t e l y t h i n b u t has a p h y s i c a l t h i c k n e s s , in t h e d i r e c t i o n p e r p e n d i c u l a r to ES, w h i c h is d e t e r m i n e d b y t h e g y r o m o t i o n o f t h e t h e r m a l ions a n d e l e c t r o n s .
T h e d i f f e r e n c e b e t w e e n ideal MHD f i l a m e n t s a n d n o n - M H D plasma c l o u d s is b e s t i l l u s t r a t e d b y t h e c o m p a r i s o n b e t w e e n , on t h e one h a n d , an i n f i n i t e l y long a n d i n f i n i t e l y c o n d u c t i n g solenoid ( o r m a g n e t i z e d r o d ) a n d , on t h e o t h e r h a n d , a solenoid ( o r a m a g n e t i c b a r ) of f i n i t e l e n g t h . I n d e e d , l i k e t h e solenoid of f i n i t e l e n g t h , a d i a m a g n e t i c plasma c l o u d p r o d u c e s a d i p o l e - l i k e m a g n e t i c f i e l d o u t s i d e i t s s u r f a c e (see f i g u r e 1 A ) . T h e m a g n e t i c f l u x t h r o u g h a n y c r o s s s e c t i o n o f t h e plasma
element is coupled to ( i . e . interconnected !) and equal in magnitude to the total flux outside the plasma cloud. T h e magnetic field vector h a s , in g e n e r a l , a component normal to the s u r f a c e of the plasma i r r e g u l - a r i t y ; magnetic field lines r u n a c r o s s the plasma b o u n d a r i e s . H o w e v e r , when this plasma element is stretched out infinitely in the magnetic field direction, the field intensity tends to zero outside the element, as for an infinitely long solenoid. F u r t h e r m o r e , the field inside infinitely long M H D filaments or solenoid is uniform; the magnetic field lines are e v e r y - where parallel to their s u r f a c e . A s for an infinitely long s u p e r - c o n d u c t i n g solenoid b r o u g h t into an external field, there is no magnetic connection (or c o u p l i n g ) between the inside and the outside of the ideal M H D filament. When the infinitely long and s u p e r c o n d u c t i n g solenoid is introduced into an external magnetic field, the total magnetic e n e r g y of the system does not depend on its orientation with respect to the external field. T h i s is not the case for a solenoid of finite length which can be either accelerated or decelerated in an external dipole magnetic field, d e p e n d i n g on the relative orientation of their respective magnetic moments. T h i s classical example indicates that modeling 3-dimensional s y s t e m s in terms of idealized two-dimensional, or one-dimensional solutions is not always appropriate.
Finally, it would not be realistic to c o n s i d e r that the Pedersen c o n d u c t i v i t y (a^) is nearly zero e v e r y w h e r e along geomagnetic field lines, indeed, magnetic field lines are u s u a l l y rooted, in the iono- s p h e r e (i.e in a medium with finite t r a n s v e r s e c o n d u c t i v i t y ) , and hence the second M H D condition is not satisfied :
I = I CT, dh = 0 P J J-
I
Because of these fundamental d i f f e r e n c e s , it seems difficult to draw valid c o n c l u s i o n s about the motion of n o n - M H D and finite plasma
filaments in an external magnetic field, from theoretical calculations based on ideal M H D models of infinitely long filaments. Similarly, it is not o b v i o u s that a stationary solution ( u s u a l l y p r o p o s e d for convenience because of its mathematical simplicity) is a p p r o p r i a t e to d e s c r i b e a physical mechanism which is i n h e r e n t l y based on time-dependent p r o c e s s e s .
Falthammar et al. (1978) d e s c r i b e many other situations in g e o p h y s i c a l and a s t r o p h y s i c a l plasmas where s u c h remarks can directly be applied.
3. L O R E N T Z F O R C E O N P L A S M O I D S
Let u s f i r s t c o n s i d e r an unmagnetized cloud of plasma in a v a c u u m devoid of a n y external magnetic field. T h e plasma p r e s s u r e inside the cloud is unbalanced since there is neither kinetic nor magnetic p r e s s u r e outside the filament, hence the p r e s s u r e g r a d i e n t force p u s h e s the plasma front o u t w a r d s , the volume occupied by the plasma e x p a n d s indefinitely and its d e n s i t y tends to zero. A n electric potential d r o p e x t e n d i n g o v e r a distance of a few D e b y e lengths a c r o s s the moving b o u n d a r y maintains local and global q u a s i - n e u t r a l i t y in the whole plasma blob. T h e height of this potential b a r r i e r is a few time k T / e , where T is the electron temperature. T h i s electric potential is p r o d u c e d b y thermoelectric c h a r g e separation at the f r o n t of the plasma d e n s i t y element.
Cosmic plasma clouds are u s u a l l y magnetized. T h e magnetic f l u x t h r o u g h a c r o s s - s e c t i o n of a plasma i r r e g u l a r i t y d e p e n d s on the intensity of the magnetic field in the source region in which the plasma has been formed b y ionization of the original neutral g a s e s .
T h e magnetic field d i s t r i b u t i o n inside and in the vicinity of the ionized cloud is generated b y magnetization c u r r e n t s , g r a d i e n t - B
and curvature currents (J^) circulating at the surface of the collision- less plasma element as illustrated in figure 1A. Such a plasma-magnetic field entity has been called "plasmoid" by Bostik (1956).
The net force (?) acting on the matter contained in a volume element (V) is the sum of the pressure gradient force, the Lorentz force, and the electric force.
F = - / div P dV + / (Jd x B) dV + / qE dV (1)
where ? is the kinetic pressure tensor for the electrons and ions; E is the electric field and q is the electric charge density satisfying Poisson's equation
div E = q (2)
From this equation and from Maxwell's equation
3 . = — curl B (3)
d »o
equation (1) becomes
F = - / div | + ( 5l_ + !o e2} f . e _ !_
2H0 2 . n0 dV (A)
C h a r g e separation electric fields can be large at t h e edges of the plasmoid. T h e y determine the d i s t r i b u t i o n s of the electrons and ions separately near the plasma s u r f a c e . B u t , since the net c h a r g e d e n s i t y q is extremely small, the electrostatic force on the whole plasma (electrons + ions) contributes little to In other terms
e E^ << — o (J
o r , what is e q u i v a l e n t , the electric d r i f t velocity v is much smaller than c t h e speed of light in vacuum :
3 = v « (po eo) = c
Using Stokes1 theorem, equation ( 4 ) can then be w r i t t e n as a surface integral
F = - f 8 . (? • | i f - u ) d s , 5 )
J ^HQ H0
S
where N is the u n i t vector normal to the surface ( S ) of t h e plasmoid.
T h i s equation has been used by Schindler ( 1 9 8 0 ) to show t h a t t h e r e is a force acting on a c u r v e d MHD filament which tends to reduce the c u r v a t u r e of the magnetic field lines and to s t r a i g h t e n them.
Equation ( 5 ) can, for instance, be used to evaluate the net force acting on t h e surface of a planar directional d i s c o n t i n u i t y . If ^
- 1 3 -
continuity, it can be shown that the force acting on a cylindrical mass element, with a unit cross section in the vicinity of this surface, is given by :
* = * • 4 - V + h r0 - Bo ) * - b 0 * • ®o - V ( 6 )
To obtain this equation, we have taken into account the fact that the normal component of continuous across the surface of dis- continuity; i . e .
a . 2 = ft. 2 . O 1
If P denotes the component of pressure component parallel to N, the N
normal component of ? is equal to
ft .
$
= P* - P* + j — (B* - Bj) (7)x o 2p0 1 0
The magnetic effect of the surface current with a density
K = J Jd dN (8)
is equivalent to that of a magnetized medium whose magnetic polarization is zero outside the plasmoid, and equal to inside the plasma element such that
K. = M x N (9)
I n t e g r a t i n g the three components of eq. ( 3 ) along 1 a normal direction N , one obtains :
- » K =
<»i X I (10)
T h e r e f o r e , we define th magnetic >olarization ins c 3 the plasmoid as
S i n c e the normal components of Bj and Bq are equal, it r e s u l t s that M.
is parallel to the s u r f a c e of d i s c o n t i n u i t y . Equation ( 7 ) can also be written :
N . F = (P - P ) + M + M . B (12) l o 2 o
T h e f i r s t term in equation (12) c o r r e s p o n d s to the e x c e s s of p e r p e n d i c u l a r kinetic p r e s s u r e inside the plasmoid, while the second term is the internal Lorentz force p r o d u c e d b y the plasma c u r r e n t s " ( J ^ ) on themselves :
1 x ( I - B ) dV = ^ Md o L 2 N (13)
Like surface tension acting on a droplet of w a t e r , this internal electromagnetic force maintains the cohesion of the plasma filament as an e n t i t y . This force normal to the surface of the plasma i r r e g u l a r i t y does not depend on the direction of the vector M.
T h e t h i r d term of equation ( 1 2 ) corresponds to the normal component of the e x t e r n a l Lorentz force :
f Jd x Bq dV = N (M . BQ) - M (N . Bq) (14)
V
Note t h a t equation ( 1 4 ) is identical to equation ( 5 ) of Lemaire et al. ( 1 9 7 9 ) . T h i s e x t e r n a l Lorentz force results from the interaction between t h e e x t e r n a l magnetic field Bq and the local plasma c u r r e n t s while t h e normal component of this force ( i . e . : (VI . BQ) corresponds to t h e t h i r d term in equation ( 1 2 ) . Note t h a t the last terms in equations ( 1 4 ) and ( 6 ) are identical and t h a t these t e r m s , ( N . Bq) , d i f f e r from zero only when Bq has a n o n - z e r o component p e r p e n d i c u l a r to the plasma boundary : for instance, in the case of a rotational d i s c o n t i n u i t y , when tf . t 0. For tangential discontinu- ities,
N . B = N . B. = 0 (15) o 1
and hence the last terms are zero.
In t h e case of f i n i t e plasmoids t h e internal and e x t e r n a l Lorentz forces contain additional multipolar terms depending upon t h e magnetic field g r a d i e n t s parallel to the surface of the plasma elements.
T h e contribution of these terms will be studied in a forthcoming p a p e r .
From equation (12), it can be concluded that the normal component of the total Lorentz force is the sum of an internal force (13) independent of the direction of M, and an external force (14) which depends on the angle between the vector M and the external magnetic field Bq. Consequently, for fixed values of | M | and | Bq |, the normal components of the external Lorentz force and of the total Lorentz force are maximum when tô and t?o are parallel. Note, however, that the normal component of the total Lorentz force does not change when the magnetic field inside the plasma filament is changed from + B.
to - Bj. This result can easily be seen from equation (7), and it has been used by Schindler (1980) to show that the motion of an ideal MHD filament is the same whether the fields inside and outside are parallel or antiparallei. This may appear to contradict the previous conclusion, but this is not so, because changing into - is not equivalent to changing M into - M.
4. CONDITION FOR MECHANICAL EQUILIBRIUM
When the condition
p N _ p N + ( b2 „ b2} = Q ( 1 6 )
i o 2|J n i oy
is satisfied or, what is equivalent, when
P N - P N + ^ M M2 + M . 2 = 0 (17) i o 2 ro o
it can be seen, from equation (7) or equation (12), that the normal component of ? is equal to zero as is also the acceleration of the plasma
-17-
b o u n d a r y . T h e plasma element is then in mechanical equilibrium. T h e s e equilibrium condition (16) or ( 1 7 ) can n e v e r be satisfied when the external magnetic field and the kinetic p r e s s u r e at large radial distance outside the plasmoid are both equal to zero, as in a vaccum.
In this case, the magnetized plasma cloud will e x p a n d adiabatically; the larger its magnetic p r e s s u r e , the larger its e x p a n s i o n rate. In g e n e r a l , h o w e v e r , the plasmoid is not in a vaccum and it will e x p a n d until the kinetic and magnetic p r e s s u r e s inside the plasma cloud reach the v a l u e s r e q u i r e d to s a t i s f y the equilibrium conditions ( 1 6 ) or ( 1 7 ) , i.e. until the total p r e s s u r e s on both s i d e s of the plasma s u r f a c e are balanced.
D u r i n g this p r o c e s s , the cloud b o u n d a r y will move o u t w a r d s and its volume will increase, while its d e n s i t y , magnetic p r e s s u r e and p e r p e n d i c u l a r kinetic p r e s s u r e will all decrease so as to reduce the total p r e s s u r e u n b a l a n c e . It can easily be e n v i s a g e d that o v e r s h o o t i n g due to the inertia of the mass element can lead to o v e r - e x p a n s i o n and to
s u b s e q u e n t contraction and e x p a n s i o n p h a s e s of the plasma element. A s a r e s u l t , h y d r o m a g n e t i c oscillations can be generated and p r o p a g a t e in the external plasma as A l f v e n w a v e s ; c o n t i n u o u s geomagnetic p u l s a t i o n s often o c c u r i n g as a train of separate Pc 3 - 4 wave packets like those recently a n a l y s e d b y Mier-Jedrzejowicz and H u g h e s ( 1 9 8 0 ) , can be initiated b y impulsive penetrations of separate solar wind plasmoids in the d a y s i d e magnetosphere.
5. A T I M E - D E P E N D E N T A N D I R R E G U L A R M A G N E T O P A U S E S U R F A C E
T h e plasma and field on both sides of a s t e a d y - s t a t e magneto- p a u s e in mechanical equilibrium s a t i s f y the equilibrium condition ( 1 6 ) o r ( 1 7 ) until the instant when the magnetosheath plasma p r e s s u r e and momentum d e n s i t y are s u d d e n l y enhanced b y the arrival of some new plasma cloud reaching the magnetopause region. O n l y solar wind plasma i r r e g u l a r i t i e s with an e x c e s s momentum are able to make their way t h r o u g h the magnetosheath and to reach the a v e r a g e magnetopause
position (Lemaire and R o t h , 1978). Because of t h e excess momentum and also because of t h e a d d i t i o n a l plasma p r e s s u r e , t h e s u r f a c e s e p a r a t i n g magnetosheath plasma and magnetospheric plasma moves t o w a r d s t h e E a r t h . Magnetospheric plasma can also be pushed ahead t o w a r d s t h e Earth b y t h i s magnetosheath plasma f r o n t a c t i n g as a p i s t o n . B u t small- scale solar w i n d i r r e g u l a r i t i e s i n t r u d i n g into the magnetosphere become permeated b y closed geomagnetic f i e l d lines, and t r a p p e d p a r t i c l e s of magnetospheric o r i g i n d r i f t into these plasma c l o u d s .
T h e a d m i x t u r e of m a g n e t o s h e a t h - l i k e and m a g n e t o s p h e r i c - l i k e p a r t i c l e populations is t y p i c a l of the " b o u n d a r y layer plasma" f o u n d at t h e o u t e r m o s t f r i n g e s of t h e magnetosphere (Eastman and Hones, 1979).
T h e f r o n t side of t h i s i n t r u d i n g plasma element is p a r t of the i n n e r edge of t h e "plasma b o u n d a r y l a y e r " , while the o u t e r edge of the so called "plasma b o u n d a r y l a y e r " is the magnetopause per se. E v e r y time an i n t r u d i n g plasma element p e r t u r b s t h e local geomagnetic f i e l d d i s t r i b u t i o n , t h e f i e l d lines w h i c h were o r i g i n a l l y closed e v e n t u a l l y become i n t e r c o n n e c t e d w i t h those of t h e solar w i n d . S i m i l a r l y , the i n n e r edge of t h e "plasma b o u n d a r y l a y e r " is formed b y the f r o n t sides of all t h e solar w i n d plasma i r r e g u l a r i t i e s t h a t have s u f f i c i e n t momentum to p e n e t r a t e into t h e closed magnetic f i e l d lines r e g i o n . T h i s excess of momentum is t r a n f e r r e d to t h e d a y s i d e cusp ionosphere as d e s c r i b e d b y Lemaire (1979). T h e excess k i n e t i c e n e r g y of t h e i n t r u d i n g plasma is d i s s i p a t e d b y Joule heating of t h e ionospheric plasma in t h e c l e f t regions ( L e m a i r e , 1977) where T i t h e r i d g e (1976) has detected a large t e m p e r a t u r e peak f r o m 400 km up to 1000 km a l t i t u d e . E v e r y time a new plasma d e n s i t y i r r e g u l a r i t y reaches the average magnetopause position and increases t h e plasma momentum and p r e s s u r e in t h e magnetosheath, new solar w i n d plasma b r e a k s into t h e geomagnetic f i e l d like an ocean wave b r e a k i n g on a sandy beach. T h e plasma b o u n d a r y layer can be compared to t h e t h i n t r a n s i t i o n layer where air b u b b l e s are e n g u l f e d below t h e s u r f a c e of an ocean p e r t u r b e d b y a g u s t y air flow ( L e m a i r e , 1978).
In t h e i m p u l s i v e p e n e t r a t i o n model, t h e m a g n e t o p a u s e is n o t a smooth s t a t i o n a r y s u r f a c e , b u t a r a t h e r i r r e g u l a r a n d t i m e - v a r y i n g b o u n d a r y b e t w e e n plasma of d i f f e r e n t o r i g i n s . F u r t h e r m o r e , in d i s - a g r e e m e n t w i t h c o n v e n t i o n a l i n t e r a c t i o n models, n o t all t h e solar w i n d p a r t i c l e s i m p i n g e on a s t a t i o n a r y a n d s h a r p l y d e f i n e d m a g n e t o p a u s e s u r f a c e ; i n d e e d , plasma elements t r a v e r s i n g t h e bow s h o c k w i t h a momentum d e n s i t y smaller t h a n t h e a v e r a g e w i l l n o t r e a c h t h e m a g n e t o - pause r e g i o n , b u t w i l l be d e f l e c t e d s i d e w a y s at a g r e a t e r d i s t a n c e f r o m t h e E a r t h a n d w i l l s l i p a r o u n d t h e f l a n k s of t h e m a g n e t o s p h e r e in t h e o u t e r m o s t l a y e r s o f t h e m a g n e t o s h e a t h .
F i n a l l y , in t h e i m p u l s i v e p e n e t r a t i o n m o d e l , t h e m a g n e t o p a u s e is d e f i n e d as t h e s u r f a c e w h e r e t h e m a g n e t o s p h e r i c plasma ceases t o h a v e a t r a p p e d p i t c h a n g l e d i s t r i b u t i o n . T h u s t h i s s u r f a c e s e p a r a t e s t h e closed g e o m a g n e t i c f i e l d lines f r o m t h o s e t h a t h a v e at least one
" f o o t " in t h e i n t e r p l a n e t a r y m e d i u m . T h i s d e f i n i t i o n of t h e m a g n e t o p a u s e is based on t h e d i f f e r e n c e b e t w e e n t h e plasma p r o p e r t i e s f o u n d on t h e t w o s i d e s , a n d n o t on t h e o b s e r v e d m a g n e t i c f i e l d s i g n a t u r e s . I n d e e d , t h e t y p e of m a g n e t i c f i e l d lines can c h a n g e f r o m " c l o s e d " t o " o p e n "
( i n t e r c o n n e c t e d ) w i t h o u t a s h a r p v a r i a t i o n in e i t h e r t h e i n t e n s i t y o r t h e d i r e c t i o n o f t h e m a g n e t i c f i e l d .
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