Axisymmetric plasma equilibrium
in gravitational and magnetic fields
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Krasheninnikov, S. I., and P. J. Catto. “Axisymmetric Plasma
Equilibrium in Gravitational and Magnetic Fields.” Plasma Physics
Reports 41.12 (2015): 1023–1027.
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http://dx.doi.org/10.1134/S1063780X15120065
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Pleiades Publishing
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Author's final manuscript
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http://hdl.handle.net/1721.1/105914
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1023
Axisymmetric Plasma Equilibrium
in Gravitational and Magnetic Fields
1S. I. Krasheninnikov
aand P. J. Catto
ba University of California San Diego, La Jolla, CA 92093, USA
b Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA e-mail: [email protected]
Received May 15, 2015
Abstract—Plasma equilibria in gravitational and open-ended magnetic fields are considered for the case of topologically disconnected regions of the magnetic f lux surfaces where plasma occupies just one of these regions. Special dependences of the plasma temperature and density on the magnetic f lux are used which allow the solution of the Grad–Shafranov equation in a separable form permitting analytic treatment. It is found that plasma pressure tends to play the dominant role in the setting the shape of magnetic field equilib-rium, while a strong gravitational force localizes the plasma density to a thin disc centered at the equatorial plane.
DOI: 10.1134/S1063780X15120065
1. INTRODUCTION
Magnetohydrodynamic (MHD) equilibria of plasma subject to a gravitational field were considered originally by Chandrasekhar [1]. In recent years, other MHD plasma equilibria studies retaining gravity effects appeared in the literature [2–8]. Without dis-cussing the validity of the approximations used in those papers, we notice that two of them [2, 8] are focused on disc-like plasma equilibria resembling the plasma shape of accretion discs. The goal of this work is to further study disc-like plasma equilibria in mag-netic and gravity fields. However, in the contrast to [2, 8], where the whole space is topologically connected by the magnetic field lines, we consider an axisymmet-ric disc-like plasma equilibrium with open magnetic field lines for the case where the space is divided into regions which are topologically disconnected by mag-netic f lux surfaces and plasma occupies just one of these regions or lobes (see Fig. 1). Somewhat similar topology of the magnetic field and plasma was consid-ered in [9] for the case of closed magnetic field lines corresponding to magnetic multipoles.
To demonstrate the possibility of the existence of such equilibria, we consider a model problem and find axisymmetric magnetic f lux surfaces , generated by the azimuthally symmetric equatorial plane current density
(1) 1The article is published in the original.
ψ ( 1) 0 ( ), ˆ ( , )R z = ϕj R− ν+ δ z j e
where and are the cylindrical coordinates, is a unit vector along the azimuthal direction, is a nor-malization constant, is a delta-function, and is an adjustable parameter. Then, using the expression for the magnetic field , from Ampere’s law we find
(2) Looking for the solution of Eq. (2) in a separable form by letting , where r and
R z eϕ 0 ˆj ( )z δ ν (∇ψ × ϕ)/R = e B ( 1) 0 2 ˆ 4 ( ). j R R z c R − ν+ π ∇ψ ⎛ ⎞ ∇ ⋅⎜ ⎟ = − δ ⎝ ⎠ 0 ˆ ( , )r (4 j / )c r−νh( ) ψ μ = − π μ
PLASMA
DYNAMICS
Fig. 1. Schematic view of plasma equilibrium in
gravita-tional and open-ended magnetic fields with topologically disconnected regions of magnetic f lux surfaces.
Plasma Gravitational center Separatrix Flux surfaces R Z
1024
PLASMA PHYSICS REPORTS Vol. 41 No. 12 2015 KRASHENINNIKOV, CATTO
( ) are spherical coordinates, we find the following equation for :
(3) We notice that Eq. (3) has the same solution if we let . To have the axial magnetic field dominate over poloidal field at the axis of symmetry, the solution of Eq. (3) should satisfy the boundary condition . Taking this into account and assuming that is a symmetric function of , the solution of Eq. (3) for a noninteger
can be written as
(4)
where is a Legendre function of the first kind and a homogeneous solution of Eq. (3) is . The jump condition, , associated with Eq. (3), is used to determine the coefficient along with . For positive even inte-gers , we can use Eq. (4), while for odd integer , we should replace in Eq. (4) with , which is a Legendre function of the second kind.
The functions for and are shown in Figs. 2 and 3, respectively. As we see, for , the function does not change its sign, which means that the whole space is topologically connected by magnetic field lines. For , how-ever, changes its sign at , which means that and the space is sepa-rated in three distinct magnetically disconnected regions or lobes with and . In the next
sec-cos( ) μ = ϑ − ≤ μ ≤1 1 ) ( hμ 2 2 ( 1) 2 ( ). 1 h h d dμ +ν ν + − μ = δ μ ( 1) ν → − ν + 2 1 ( )/ dh 1 0 h d μ→± ⎛ ⎞ μ ⎜ − μ ⎟ → μ ⎝ ⎠ ) ( hμ μ ν
( )
( )
1 1 ) ( ) , 2 (1 ) (0) ( (0) P P P h h P −ν ν ν −ν ν μ − μ μ μ = μ − − ν + 1 ( ) ( ) Pν μ = P−ν− μ 1 2 ( ) ( ) 1 ( )/(1 ) P−ν μ − μ μ =Pν − μ Pν μ + ν 0 / 1/2 dh dμμ= = 1 0 / (1 ) (0) dP−ν dμμ= = − − νP−ν ν ν ( ) Pν μ Qν( )μ ( ) hν μ ν =0.1 ν =2.1 0.1 ν = hν( )μ 2.1 ν = ( ) hν μ μ = μsep ≈0.064 sep ( ,r ) 0 ψ μ = μ = 0 ψ > ψ <0tion, we will exploit this three lobe solution for a self-consistent current satisfying the Shafranov–Grad equation.
The paper is organized as follows, in Section 2, we describe our model of plasma equilibrium in gravita-tional and magnetic fields and formulate the governing equation; in Section 3, we present analytic solutions of this equation for some extreme conditions; and in Sec-tion 4, we summarize our findings.
2. GOVERNING EQUATION
We consider plasma equilibrium in a gravitational potential and an axisymmetric mag-netic field , where and are the gravitational constant and the mass of the gravitational center located at , respectively. In the derivation of the equation governing our plasma equilibrium, we will follow [8] and assume that both electron, , and ion, , temperatures are functions of , as required for a drifting Maxwellian solution of the drift-kinetic or gyrophase-averaged Fokker–Planck equation in an axisymmetric system [10–12]. Then, expressing the plasma density in the form , from the force balance along the magnetic field lines,
we find , where
and M is the mass of plasma ions having the charge . From the force bal-ance equation across the magnetic field lines and Ampere’s law, we obtain the Grad–Shafranov equa-tion (5) 0 0 ( ) / G r = −G M r (∇ψ × ϕ)/R = e B G0 M0 0 r = e T i T ψ ) ( ) exp[ , )] ( , ( n r μ = η ψ κ ψ μ 2 ( , ) G r( )/Cs( ) κ ψ μ = − ψ 2 ( ) [ ( ) ( )]/ S i e i C ψ = Z T ψ +T ψ M i Z
{
}
2 2 2 2 2 2 4 ln( ) . S S S Mn R R R B C C C G ∇ψ π ⎛ ⎞ ∇ ⋅⎜ ⎟= − ∇ψ ⎝ ⎠ × ∇ η + ∇ + ∇κ + ∇Fig. 2. The function for .
0.6 0.5 0.4 0.3 0.2 0.1 0 0.7 0.2 0.4 0.6 0.8 1.0 0 μ ν = 0.1 hν(μ) ( )
hνμ ν =0.1 Fig. 3. The function for .
0 0.05 0.10 0.15 0.20 0.05 0.2 0.4 0.6 0.8 1.0 0 μ ν = 2.1 hν(μ) ( ) hνμ ν =2.1
Following [13], we will seek a solution of Eq. (5) in a separable form by taking
(6) where is an adjustable parameter which plays the role of an eigenvalue to find a solution of Eq. (5) (see, e.g., [13]), while and are normalization con-stants such that . One can show that ansatz (6) is compatible with Eq. (5) for the following depen-dences of and :
(7)
where and are normalization constants chosen in a such a way that . Substituting expressions (6) and (7) into Eq. (5), we find the gov-erning equation for to be
(8)
where
(9)
Equation (8) agrees with the different cases analyzed in [4, 8, 13–15]. The solution of Eq. (8) with the bound-ary conditions and can only be satisfied for some particular values of .
3. SOLUTIONS OF THE GOVERNING EQUATION
The plasma equilibria in gravitational and mag-netic fields considered by Krasheninnikov and Catto [4, 8], where within the framework of separable solu-tions (6) for the case where the whole space was topo-logically connected by the magnetic field lines. Here, however, we follow [9] and consider equilibria with open magnetic field lines and topologically discon-nected regions of the magnetic f lux surfaces where plasma occupies just one of these regions (recall Fig. 1). Based on the results in Section 2, it is easy to see that the topology of the magnetic f lux surfaces is determined by the magnitude of (see [9]). Indeed, symmetric ( ) vacuum ( ) solutions of Eq. (8), which in this case is identical to the left-hand
0 0 ( , )r h( )( / ) ,r r α ψ μ = ψ μ α 0 ψ r0 (0) 1 h = 2 ( ) s C ψ η ψ( ) 2 2 1/ 0 0 2 3/ 0 0 , ( ) ( / ) ( ) ( / ) , s C n C α + α ψ = ψ ψ η ψ = ψ ψ 2 0 C n0 ( , 0) 0 κ ψ μ = = ( ) hμ
{
}
2 2 2 1/ 1 4/ 1/ ( 1) 1 ( 2) exp ( 1) , 2 d h h d g h− α h+ α g h− α α α + + μ − μ ⎡ ⎤ = αβ − α + ⎣ − ⎦ 2 2 0 0 0 2 0 0 0 0 0 0 2 2 0 0 0 8 8 , . S S C r MnC Mn B G M G M g C r C r μ= μ= ⎛ ⎞ π β = π ⎜ ⎟ ≡ αψ ⎝ ⎠ = ≡ (0) 1 h = 2 1 ( )/ 1 0 h μ→± μ − μ → α α ( ) ( ) hμ = −μh β =0side of Eq. (3), for are given by the following expression:
(10)
where α a nonzero adjustable parameter ( for ), with the first form useful for α an even nega-tive integer, and the second form useful for α an odd positive integer. From Eq. (10) one can see that, for
or , the function
reaches zero only at the poles ( ). For other , additional roots of appear within the range , giving rise to additional lobes as was illustrated in Section 1. These roots become the
separatrices, , dividing the space into magnetically disconnected regions with different
signs of .
In [9] the Grad–Shafranov equation for plasma equilibrium in a multipolar magnetic field was consid-ered for closed magnetic field lines, corresponding to positive (with ). Here, we consider the case of plasma equilibria in gravitational and open magnetic fields corresponding to negative . In [4, 8], only were considered for and the whole space was topologically connected by the magnetic field lines. The simplest symmetric ( ) case with topologically disconnected magnetic f lux surfaces and open magnetic field lines can be found for
.
For and , the right-hand side of Eq. (8) is small and is close to the vacuum solu-tion corresponding to ,
(11) so that we have . The correction for the eigenvalue caused by the small, but finite right-hand side of Eq. (8), can be found by multiplying Eq. (8) by and integrating from to , to find
(12)
Substituting into Eq. (12), as a first approximation, (we discuss later the limits of the
appli-( )
0 1 h = α −α −α− α− α −α α− μ = μ μ − μ μ μ − μ μ ≡ = ( ) vac 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) , (0) (0) h h P P P P P P 1 h= ± μ 0 α = 1 α = −2 hvac(1)( ) h( 2)vac( ) 1 2 − μ = μ = − μ 1 μ = ± 0 α ≠ ( ) vac( ) 0 hα μ = 1 1 − ≤ μ ≤ sep) 0 ( ,r ψ μ = μ = ψ α 2< α <3 α 2 α > − G ≠ 0 ( ) ( ) hμ = −μh 4 3 − < α < − 1 β! βg !1 ( ) hμ 4 α = − ( 4) 2 2 vac( ) (1 )(1 5 ), h− μ = − μ − μ sep 1/ 5 μ ≈ 4 α = − ( 4) vac ( ) h− μ μ =0 μ =1{
}
sep 1 2 0 ( 4) 1/ 1 4 / vac 0 1/ ( 3)( 4) ( )(1 5 ) ( ) ( 2) 2 exp ( 1) . h d g h h h g h d μ − − α + α − α α − α + μ − μ μ = αβ μ − α + ⎡ ⎤ × ⎣ − ⎦ μ∫
∫
( 4) vac ( ) ( ) hμ = h− μ1026
PLASMA PHYSICS REPORTS Vol. 41 No. 12 2015 KRASHENINNIKOV, CATTO
cability of this approximation), we find the correction to the eigenvalue ,
(13)
From this form we can get some insight into what would happen if the right-hand side of Eq. (8) were to dominate. We would then expect the function to have a large negative second derivative about , which quickly reduces to zero, so that the region occupied by plasma shrinks and . In addition, we expect α to become greater than and, as the region becomes narrower, move toward . As a result, in the limit , in vacuum regions is given by Eq. (10) with . Expanding for small μ and , by considering vacuum solution (10), gives
(14) where is a new normalization constant and
(15) which becomes small for in the vicinity of .
When the magnitude of the right-hand side of Eq. (8) is large, we can neglect the second term on the left-hand side of Eq. (8) to obtain
(16)
where we let on the right-hand side of Eq. (8). Accounting for the boundary conditions and , multiplying Eq. (16) by , and integrating from gives
(17)
Integrating Eq. (17) from to , where , we obtain and find ,
(18)
Matching expressions (15) and (18), we find α to be
(19)
We can estimate the integral expression on the right-hand side of Eq. (19). For small and large g, we find
(20)
Thus, from Eqs. (19) and (20) we see that can only be close to for , giving
(21)
We also conclude that, for , stays close to and is close to for both small and large
g (although at , deviates significantly from in a small region in the vicinity of , so that Eq. (13) may not be applied). As a result, from Eq. (13) we find
(22)
with the limit a plasma disc equilibrium based on the localization of the density.
4. CONCLUSIONS
We have considered plasma equilibria in gravita-tional and open-ended magnetic fields for the case of topologically disconnected regions of the magnetic f lux surfaces where plasma occupies just the central lobe (see Fig. 1). Choosing the special dependences of both plasma temperature and density on the magnetic f lux as given by Eq. (7) allows us to search for the solu-tions of the Grad–Shafranov equation in separable form (6) that permits tractable analytic analysis. We find that the plasma pressure sets the shape of mag-netic equilibrium at high β, while a strong gravitational force ( ) localizes the plasma density to a disc about the equatorial plane, but does not alter the mag-netic equilibrium critically. At low β, the gravitational field affects the magnetic equilibrium when , but not for , and again a thin
α
(
)
{
}
(
)
{
}
sep 1/4 ( 4) ( 4) vac vac 0 1/4 ( 4) vac 3 4 ( ) ( ) 2 4 2 exp ( ) 1 . g h h g h d μ − − − α = − + β μ μ + ⎡ ⎤ × ⎢⎣ μ − ⎥⎦ μ∫
( ) hμ 0 μ ≈ ( ) hμ sep 1 μ ! 4 − 3 − 3 α → − h( )μ 3 α < − 3 α → − ( ) ( ) vac( ) ( sep ) , hα μ ≈ μα − μh h ( 3) sep 2(3 )/3, α→− μ ≈ − + α α −3(
)
2 1/3 1/3 1/3 2 3 1 exp ( 1) , 2 g d h h h g h d − ⎡ ⎤ = − β + ⎣ − ⎦ μ 3 α → − (0) 1 h = 0 / 0 dh dμμ= = dh dμ/ 0 μ =(
)
[
]
1/3 1/2 1 3 2 exp ( 1) . h dh g g d d ⎧ ⎫ ⎪ ⎪ = − β⎨ ξ + ξ ξ − ξ⎬ μ ⎪⎩∫
⎪⎭ 0 μ = μsep sep 0 ( ) hμ = h( )μ ( )sep α μ(
)
[
]
α − μ ⎧ ⎫ ⎪ ⎪ = ⎨ ξ + ξ ξ − ξ⎬ β∫ ∫
⎪⎩1/3 ⎪⎭ ( ) sep 1/2 1 1 0 1 2 exp ( 1) . 3 h g g d dh(
)
[
]
− α = − ⎧ ⎫ ⎪ ⎪ − ⎨ ξ + ξ ξ − ξ⎬ β∫ ∫
⎪⎩1/ 3 ⎪⎭ 1/2 1 1 0 3 1 2 exp ( 1) . 2 h g g d dh(
)
[
]
1/ 3 1/2 1 1 0 2 exp ( 1) 3 /4, 1, 1, 1. h g g d dh g g − ⎧ ⎫ ⎪ ξ + ξ ξ − ξ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ π ⎧ ≈⎨ ⎩∫ ∫
α 3 − β@1 1 , 1, 8 3 1 1 , 1. 6 g g π ⎧ + ⎪ β ⎪ α ≈ − ×⎨ ⎪ + ⎪ β ⎩ ! @ 1 β! α 4 − h( )μ hvac( 4)− ( )μ 1/ 1 g@ β@ h( )μ ( 4) vac( ) h− μ μ =0 ⎧ ⎪ α = − + β ×⎨ π ⎪ β ⎩ 1 24, 1, 25 5 4 3 1 , 1/ 1, 8 2 g g g ! @ @ 1/β@ @g 1 1 g @ 1/β@ @g 1 g !1plasma disc about the equatorial plane is obtained only when gravity is strong.
ACKNOWLEDGMENTS
This work was supported by the US Department of Energy, grant nos. DE-FG02-91ER-54109 at MIT and DE-FG02-04ER54739 at UCSD.
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