weakly dissipative plasma equilibria in kerr geometry
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Weakly dissipative plasma equilibria in Kerr geometryKlaus Elsässer and Yauhen Kot Citation: Physics of Plasmas (1994-present) 11, 278 (2004); doi: 10.1063/1.1630967 View online: http://dx.doi.org/10.1063/1.1630967 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/11/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Collisionless kinetic regimes for quasi-stationary axisymmetric accretion disc plasmas Phys. Plasmas 19, 082905 (2012); 10.1063/1.4748578 Recurrence plots and chaotic motion around Kerr black hole AIP Conf. Proc. 1283, 278 (2010); 10.1063/1.3506071 Response to “Comment on ‘Variational principles for stationary one- and two-fluid equilibria of axisymmetriclaboratory and astrophysical plasmas’” [Phys. Plasmas12, 064701 (2005)] Phys. Plasmas 12, 064702 (2005); 10.1063/1.1922107 Plasma physics in clusters of galaxies Phys. Plasmas 10, 1539 (2003); 10.1063/1.1558991 The spinning terrella plasma experiment: Initial results Phys. Plasmas 8, 1111 (2001); 10.1063/1.1355982
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Weakly dissipative plasma equilibria in Kerr geometryKlaus Elsassera) and Yauhen KotInstitut fur Theoretische Physik, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany
~Received 5 August 2003; accepted 13 October 2003!
Axisymmetric neutral plasma equilibria of electrons and ions near black holes can be constructedfrom solutions of ideal multifluid equations; the solutions for the ions are obtained from a modifiedGrad–Shafranov code for the stream function. The density and the flux functions are determinedself-consistently from the normalization of the Eulerian four-velocity and a simple dissipation modelfor the ions, respectively. Solutions with moderate or vanishing toroidal velocity exist, but the blackhole receives matter only in nonequilibrium situations. ©2004 American Institute of Physics.@DOI: 10.1063/1.1630967#
I. INTRODUCTION
Theory and observation of compact massive objectshave been of interest for several years~see, e.g., Ref. 1!.Besides the collapsed stars or galactic nuclei also the behav-ior of matter in their strong gravitational field may be stud-ied; for a rotating black hole one uses the metric found byKerr,2 an exact vacuum solution of Einstein’s field equations.Though violent processes can occur in active galactic nucleiwe may ask whether stationary states exist for the surround-ing matter, and how to calculate them possibly. A ‘‘strongly’’dissipative system would have vanishing entropy productionin equilibrium, and this would fix the possible motionsstrongly, e.g., to rigid rotation of a neutral fluid.3 But at hightemperatures or in the presence of an ionizing radiation weshould consider nearly fully ionized plasmas with many par-ticles in a Debye sphere; such systems are usually eitherturbulent or weakly dissipative. In the latter case we mayconsider ideal fluid equilibria which slide away slowly due tosmall forcing and damping terms. The solutions of ideal fluidequations like, e.g., the magnetohydrodynamic~MHD! equa-tions, depend not only on boundary conditions, but also onso-called flux functions, i.e., constants of motion along thepath of a fluid element. These ‘‘free’’ flux functions can, inthe case of laboratory plasmas, be determined principally bymeasuring the plasma profiles; theoretically they may be ob-tained from ‘‘solubility conditions’’ of higher-order solutionsin the presence of small forcing and damping terms.4,5 Weare, of course, in the astrophysical context restricted to thesecond possibility.
The curl of the electric fieldE should vanish if the mag-netic fieldB is stationary in time~Faraday’s law!. This con-dition depends obviously on the Ohmic law which deter-mines E. In ideal MHD this means that the curl ofv3Bshould vanish, wherev is the center-of-mass velocity. Butthis condition is not consistent with Kerr geometry, thereforewe replace the MHD description by multifluid equations asshown in Sec. II and Ref. 6, where the electron momentumbalance gives a generalized Ohm’s law. The key point is theKelvin–Helmholtz representation for an isothermal plasma
with a general form of weak forcing and damping terms.Section III describes a simple dissipation model for the ionfluid: Instead of using the usual dissipative part of theenergy-momentum tensor7 we assume a small toroidal fric-tion force due to collisions of ions with neutrals or photons;to maintain equilibrium, it is balanced by a small loop volt-age generated outside the computational area~which is aspherical shell outside the outer horizon of the black hole!.The ions are treated as an unmagnetized fluid, while the elec-tron fluid is coupled to it by the quasineutrality condition.This simplification may be relevant in the astrophysical case,where the ideal fluid equations allow a force balance of in-ertia, pressure and gravity without external electromagneticfields.
II. EQUILIBRIUM CONDITIONS
One of the traditional ways to describe a dissipative neu-tral fluid is the equation of motion which follows from Ein-stein’s field equation, namely,
~Tmn 1DTm
n ! ;n50,
whereTmn is the energy-momentum-stress tensor of an ideal
neutral fluid, andDTmn collects all terms due to dissipation7
and external forcing terms; the semicolon means covariantdifferentiation. For a multifluid plasma with electrons andions we use, for each particle species, a similar equation, butthe divergence ofDTm
n is put to the right-hand side, and it isreplaced by the sum of the Lorentz force~which, of course,is nondissipative! and a general expressionFm which com-prises the dissipative and external forces~as specified later!:
Tm;nn 5%
e
munFmn1Fm . ~1!
The ~scalar! mass density of the corresponding species in aco-moving inertial frame is denoted by%, e, andm are thecharge and mass of a single particle, respectively,un is theEulerian four-velocity with the normalization
umum51, ~2!
and Fmn is the electromagnetic field tensor. Axisymmetricideal fluid equilibria can be constructed from Eq.~1! witha!Electronic mail: [email protected]
PHYSICS OF PLASMAS VOLUME 11, NUMBER 1 JANUARY 2004
2781070-664X/2004/11(1)/278/8/$22.00 © 2004 American Institute of Physics
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Fm50; all physical quantities are then independent of time(x05t) and of the toroidal angle (x15w). Equation~1! canthen be integrated with respect to the directions of the corre-sponding two Killing vectors; this can be shown by theKelvin–Helmholtz representation of Eq.~1!, which is mosteasily evaluated in the following for an isothermal plasma~see also Ref. 6!.
The left-hand side of Eq.~1! can be written as follows:
~sumun2pdmn ! ;n5%unS s
%umD
;n
2p,m ,
wheres is the relativistic enthalpy density~divided byc2), pthe pressure, and the mass conservation law (%un) ;n50 hasbeen used@see also Eq.~7! below, a comma means partialderivative#. The exact meaning ofs is derived from the con-dition that both sides of Eq.~1! should be orthogonal toum;the left-hand side gives then
um%unS s
%umD
;n
5ump,m[dp
dt,
wheret is the proper time of a fluid element. Here we useEq. ~2! to obtain
d
dt S s
% D51
%
dp
dt.
The increments along the path of a fluid element may beobtained from the Gibbs–Duhem relation in the comovinginertial frame~m ands are the free enthalpy and the entropyper mass, respectively, andT the temperature!:
dS s
% D51
%dp5dm1sdT5d~m1sT!2Tds.
This leads to complete integrability ofs/% if the fluid isideal, i.e., either isentropic (ds50) or isothermal (dT50),otherwise an additional Clebsch potential would be needed.Here we assumedT50 and find thats is the relativistic freeenthalpy per volume~including the rest energy! in a co-moving inertial frame~with c51). The Kelvin–Helmholtzform of Eq. ~1! is then obtained by introducing the electro-magnetic four-potentialAm associated withFmn ~the force ofpossible external fields is included inFm), and the canonicalEulerian four-velocityVm :
Vm5s
%um1
e
mAm , ~3!
namely,
2%unVmn5Fm , ~4!
whereVmn is the curl ofVn :
Vmn5Vn,m2Vm,n . ~5!
Let us denote the ignorable coordinatesxm,m50,1, by par-ticular indicesr or s, and the other~poloidal! coordinates byindicesa or b. Then we have for single-valuedVn :
V rn52Vr ,n ;
unV rn52uaVr ,a[2dVr
dt. ~6!
Equation~6! shows that the componentsVr are arbitrary fora purely toroidal motion (ua50). In general, however, theyare constants of motion for an idealized fluid element withproper timet and with nonzero poloidal velocity; they arefunctions of the stream functionx which can be introducedto integrate the continuity equation:
05~A2g%un! ,n5~A2g%ua! ,a , ~7!
where g is the determinant of the metric tensor elementsgmn . To integrate Eq.~7! we use the permutation symbol«1ab in three-space~and%[mn) to obtain
n
n0ua5
1
A2g«1abx ,b ~8!
for any functionx (n0 is a normalizing constant!. From thiswe find
uax ,a50, ~9!
which shows that the linesx5const are the projections ofthe stream lines onto the poloidal plane (x2,x3). Note thatEqs. ~5!–~9! are valid for any axisymmetric equilibrium ofan isothermal fluid, with or without the forcing termFm inEq. ~4!. Note also thatFm is not entirely arbitrary; like anyforce it must conserve the normalization ofum, so it must beorthogonal to it@this follows also from Eq.~4!#:
umFm50. ~10!
A further restriction ofFm follows for each Killing vector ifwe want to find an ‘‘integrable’’ equilibrium, where the linesx5const in the poloidal plane are either closed or open toinfinity, but without ergodicity in any domainD2 . To see thiswe multiply Eq. ~4! for m5r by A2g and integrate eachside over some domainD2 . The left-hand side gives, withEqs.~6!, ~7! and the Gaussian law~with d2x[dx2dx3):
2ED2
d2xA2g%unV rn5 R]D2
d faA2g%uaVr , ~11!
where d fa is the one-dimensional surface element of theboundary]D2 of the domain, in the outward direction. Thecondition for anyFm and any domainD2 in the poloidalplane then reads as follows:
ED2
d2xA2gFr5 R]D2
d faA2g%uaVr . ~12!
In particular, if we chooseD2 to be the interior of a closedline x5const, we haveD25D2(x), and fromd fa;x ,a andEq. ~9! we see that the right-hand side of Eq.~12! vanishes.In this case we may choosex25x as a radial coordinate, andx35u as an angular coordinate running from 0 to 2p; differ-entiation of Eq.~12! with respect tox gives then
Ex5const
duA2gFr50. ~13!
If the asymptotic region contains matter we should define asuitable mass-weighted average^Fr& instead of the left-hand
279Phys. Plasmas, Vol. 11, No. 1, January 2004 Weakly dissipative plasma equilibria in Kerr geometry
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side of Eq.~13!; the details depend on the dissipation modeland are shown later@after Eqs.~19! and~20!#. It should alsobe noted thatA2g may vanish~e.g., at the horizon and at thesymmetry axis in Boyer–Lindquist coordinates9!; thesepoints, of course, should not lie within the domainD2 .
The general procedure to calculate a weakly dissipativeequilibrium is then as follows: First we solve Eq.~4! for thecaseFm50; for m5r this can be done by choosing someflux functions Vr5Vr(x) for any fluid component. Form5a we obtain an equation forx which can be foundelsewhere;6 it is solved numerically together with the nor-malization condition, Eq.~2! ~and Maxwell’s equations forAr , if necessary!. Then we insert these solutions into Eq.~13! for a particular dissipation model to obtain correspond-ing equations forVr(x); these new flux functions are thenused to solve thex-equation and the remaining set again, andwe try to iterate this process to obtain a self-consistent solu-tion. The result is then an ideal equilibrium configurationwhose flux functionsVr(x) are determined by the ‘‘solubil-ity condition,’’ Eq. ~13!, which is a necessary condition tosolve Eq.~4! for FrÞ0. In the following section, however,we use a dissipation model where the procedure is muchsimpler for a system with open field lines: The equation forVr(x) can be solved analytically, and the ‘‘inner iteration’’ ofsolving the x-equation and the normalization condition isreplaced by two independend steps, because thex-equationbecomes independent of the density.
III. A NEUTRAL PLASMA WITH FRICTION AND LOOPVOLTAGE
Here we explore the properties of the simplest possibleequilibrium in Kerr geometry. The gravitational force of theblack hole allows, in contrast to magnetized laboratory plas-mas, renunciation of equilibrium fieldsFmn for the ions. Thisis possible if no electric current flows, or if the particle fluxesof electrons (e) and simply charged ions (i ) are equal:
~num!(e)5~num!( i ). ~14!
Equation~14! with m5r 50,1 is characteristic for quasineu-tral plasmas where the ratio of the electron skin depthc/vpe
to the scale lengthL of fluid and metric quantities is a small-ness parameter,«.6 From Eq.~14! with m5a52,3 and Eq.~8! we obtainx (e)5x ( i )[x if the boundary values ofx arethe same for both species. This is consistent withx (e)5x ( i )
1O(«2) which follows from the equation forx ( j ), j 5e, ifno toroidal vacuum magnetic field is present@see Eq.~25! inthe second part of Ref. 6#. For an isothermal plasma withtemperatureT we have the well-known formula for theGibbs enthalpym ( j ) per mass for each speciesj , and thisgives the following equation of state:
S s
% D ( j )
511m ( j )
c2 511kBT
m( j )c2 lnS n
n0D ( j )
, ~15!
wherekB is Boltzmann’s constant. For relativistic tempera-tures we have different values ofs/% for different specieswith different massesm( j ). The particle densities, however,
are the same:n(e)5n( i )[n; this follows from Eq.~14! andthe normalization condition, Eq.~2!, for both species. FromEq. ~14! we obtain then
um(e)5um
( i )[um .
Using the same normalizing constantn0 for both electronsand ions, we can express the particle current in the planespanned by the Killing vectors from Eq.~3! as follows:
n( j )
n0ur
( j )5N( j )S Vr( j )~x!2
e( j )
m( j ) Ar D , ~16!
where
N( j )[n
n0S %
s D ( j )
5~n/n0!
11kBT
m( j )c2 lnS n
n0D .
Assuming thatN(e),N( i ) and Vr(e)(x),Vr
( i )(x) are of similarorder of magnitude, we may solve Eq.~16! for Ar to lowestorder inm(e)/m( i ), with the following result:
A r[ueu
m(e) Ar5N( i )
N(e) Vr( i )~x!2Vr
(e)~x!.
This solution is again of orderVr( j )(x), and the correspond-
ing last term in Eq.~16! for j 5 i is m(e)/m( i ) times smallerthan the preceding one, and it can be skipped: The ionsmove, to lowest order, independent ofAr and of the elec-trons; therefore we skip the index (i ) in the following, whereonly ion quantities are used. The densityn or N is deter-mined from the normalization condition, Eq.~2!. Here weuse the following abbreviations:
U2[S n
n0D 2
~2uaua!51
2g (a,b
@gabx ,ax ,b2gbb~x ,a!2#,
V2[S s
% D 2
urur5grsVr~x!Vs~x!,
y[s
%511v th
2 lnS n
n0D ,
v th2 [
kBT
m( i )c2 ,
where also Eqs.~8!, ~15!, and~16! for j 5 i have been used.Then we may write Eq.~2! for j 5 i as follows:
15urur1uaua5
V2
y2 2U2
~n/n0!2 ,
or
F~y,U2![y•A11@U2/~n/n0!2#5AV2, ~17!
n/n0[exp@~y21!/v th2 #.
If the stream functionx @andVr(x)] is given, we solve Eq.~17! for y, and then we known/n0 or N. The relevantx-equation can be obtained by varying the following func-tional with respect tox, keepingN and the metric fixed:6
280 Phys. Plasmas, Vol. 11, No. 1, January 2004 K. Elsasser and Y. Kot
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W~x!5E E dx2dx3A2gs
mn0•
1
2~uru
r2uaua!
5E E dx2dx3A2g•1
2 S NV211
NU2D . ~18!
The first variation ofW(x) with dx50 at the boundarygives, after a partial integration of the second term:
dW~x!5E d2xH A2gNVr~x!Vr8~x!
2F 1
~A2gN!(a,b
~gabx ,b2gbbx ,a!G,aJ dx,
where the prime ofVr8(x) means the derivative with respectto x. The resultingx-equation is then obtained by the re-quirement that the expression in the curly bracket vanishes.We are interested in solutions outside the horizon of theblack hole, so both summands in the bracket of Eq.~18! arepositive. The minimization ofW(x) can be done by usingfinite elements~a modified Grad–Shafranov code!, togetherwith an iteration to solve Eq.~17!; preliminary results havebeen obtained with arbitrarily given flux functions.8
The flux functionsVr(x) are now determined by thesolubility condition, Eq.~13!, so they depend on the forcingtermsFr in the symmetry plane. In fact, we have only to givethe toroidal componentF1 , the time-like componentF0 isthen fixed by Eq.~10! ~we neglect the poloidal componentsFa). The simplest way of defining an equilibrium flow is tochoose a friction force (2%nu1) and, like in tokamaks, theforce given by a loop voltageU. In the first case we mayconsider collisions of the ions with neutrals or with a gas ofphotons, with constant collision frequencyn. The secondcase, of course, requires a time-dependent magnetic fieldalong the symmetry axis, which may not exist in the astro-physical context, leading to a ‘‘slide-away’’ of the equilib-rium; here we assume it to compensate the friction force. Aconstant external electric field
F10ext[E1
ext52U5const
leads then to the following external forcing term for the ions:
Frext5%
ueum
UH u1 for r 50,
2u0 for r 51,
which is obviously perpendicular tour . The toroidal frictionforce of the ions leads, with Eqs.~2! and~10!, to the follow-ing force:
Rr52%nH ~u0u021!/u0 for r 50,
u1 for r 51,
with ur5(%/s)Vr(x) andur5grs(%/s)Vs(x). With the to-tal force for the ions
Fr5Rr1Frext
we obtain then from Eq.~13! for r 50 and r 51, respec-tively:
E duA2g%F2n~u0u021!/u01ueum
Uu1G50, ~19!
E duA2g%F2nu12ueum
Uu0G50. ~20!
The unknown flux functionsVr(x) are constant along theintegration path in Eqs.~19! and ~20!, therefore we obtainsimple algebraic relations for them. First we divide Eq.~20!by nV0(x) and solve it for the ratiov1(x)[V1(x)/V0(x).The result can be expressed by weighted averages~along thestream lines! which are defined as follows:
^~¯ !&[F E duA2gS %2
s D ~¯ !G Y E duA2gS %2
s D ,
and by a measure for the strength of the loop voltage~takento be positive!:
l 5ueumn
U, ~21!
namely,
v1~x
IV. NUMERICAL SOLUTIONS
The numerical solutions of this section have been ob-tained by a C11 program ~Ref. 10! which uses finiteelements;11 the procedure is similar to the calculation of dis-sipative Tokamak equilibria.5
We use the Kerr metric9 in the usual~Boyer–Lindquist!poloidal coordinates (r ,u), with total mass parameterM50.5 and total specific angular momentuma50.3. Then theouter horizon is a circle with radiusr 50.9, and the ‘‘proper’’inner boundary is the half-circler 50.9,0<u<p and theouter part of the symmetry axis,r>0.9,u50,p ~a discussionof this matter is given in the final section!, wherex should beconstant andVr(x) zero. For the actual computation, how-ever, we shifted this inner boundary slightly outwards tor51,u1<u<p2u1 and r>1,u5u1 ,p2u1 , respectively,
with u1.0.01, and used various nonzero boundary valuesfor x andVr(x). The outer boundary is chosen to ber 5r 2
5const,u1<u<p2u1, and the finite elements are generatedby a regular lattice in the (r ,u)-plane. To improve the nu-merical accuracy we used two additional circles and rays,respectively, near each of the boundaries. As a test problemwe solved the following equation:
D2x522 cos~2u!/r 4,
whereD2 is the ordinary Laplacian in polar coordinates, withthe solutionx5cos2 u/r2. Using the boundary values of thisanalytical solution, we obtained excellent agreement of thenumerical solution within the computational area 1<r<2,with a lattice of 39 circles and 104 values ofu per circle.Figure 1 shows the comparison ofx from the code~crosses!
FIG. 2. Stream function~vortex! in aclosed system above the (r ,u)-plane;flux functions according to Eq.~25!with k050.1 andk150.3.
FIG. 1. Test functionx above the po-loidal area r P@1;2#, uP@0;p#;crosses: numerical solution; lines: ana-lytical solution.
282 Phys. Plasmas, Vol. 11, No. 1, January 2004 K. Elsasser and Y. Kot
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and the analytical solution~lines! in the (r ,u)-plane; themeasured rms error was 0.31%, and no difference is visible.The same lattice has been used throughout this section.
To minimize our functionalW(x), Eq. ~18!, we have tospecify boundary conditions forx. If x is constant at theboundary, we have a closed system, and no matter enters orleaves the area. In this case we should calculate the fluxfunctions Vr(x) according to Eqs.~22! and ~23!. A moreinteresting case is an open system where all stream linescross the boundary; if they come from infinity and go toinfinity we may use Eq.~24! for Vr(x), in particularV0(x)51 andV1(x)50 for a purely poloidal flow.
A. Closed systems
A single vortex solution has been obtained for closedsystems~boundary value ofx:0.035! with various flux func-tions differing only slightly from the values of an open po-loidal flow (V0[1,V1[0). Instead of solving Eqs.~22! and~23! for each stream line, we used an expansion ofVr(x) forsmall x, with various values of the parametersk0 ,k1 :
V0~x!512k0x; V1~x!52k1x, ~25!
and withv th51. Figure 2 shows a typical example of thesevortex solutions: The fluid rotates in the (r ,u)-plane in theclockwise direction~becauseu3;2x ,2), and also in thew-direction @u1;V1(x),0#. A single vortex solution is re-stricted to sufficient small parametersk0 ,k1 ; it is destroyed,e.g., fork050.1,k1.0.43 andk050.2,k1.0.049, and mul-tivortex solutions may appear. The increasing values ofk0 ,k1 lead to higher maxima ofx and to higher velocities,and we may suppose that at some points a turbulent motionis generated.
The density according to Eq.~17! is obtained by the‘‘inner iteration’’ with the x-equation and is shown in Fig. 3;it exhibits an asymptotically flat region,n/n0.2, ranging
from the center of the vortex (r .1.3) up to the outer bound-ary. Near the axis (u.0,p) it drops down, and near thehorizon it increases sharply, indicating a singularity atr50.9. This leads to a poloidal fluid velocityuuau(.(n0 /n)ux ,bu) which is rather different from what wouldbe suggested by the linesx5const in Fig. 2; in Fig. 4 we seea large velocity in the asymptotic region and near the axis,while it remains small where the plasma is dense. Conse-quently we find in Fig. 5 that the total poloidal particle flux,(n/n0)@(u2)21(u3)2#1/2, is nearly symmetric with respect tothe center of the vortex, increasing at the horizon as well asin the asymptotic region: The vortex is tenuous, but fast ro-tating in the outer region, and dense and slowly rotating nearthe horizon.
B. Open systems
Systems whose stream lines come from infinity and goto infinity are of particular interest. They are characterized bynearly constant flux functionsV0(x).1,V1(x).2 l . Thex-equation becomes linear and homogeneous inx and inde-pendent of the density. The solutions are uniquely deter-mined by the boundary conditions which may be specified toproduce an inflow of particles in the equatorial plane (u2
,0,u350 for u5p/2) and an outflow along the axis (u2
.0,u350 for u50,p). Two examples have been used whichfulfill this requirement:
xuboundary5C11
2ksin~2u!S r
r 2D n
, ~26!
xuboundary5C11
2ksin~2u!S r 1
r D n u~p/2!2uu~p/2!
, ~27!
with constantsC, k, and n. For the resulting solutions weget nearly the same densities as for closed systems~Fig. 3!,and they are skipped here.
FIG. 3. Density above the (r ,u)-planefor the vortex in Fig. 2.
283Phys. Plasmas, Vol. 11, No. 1, January 2004 Weakly dissipative plasma equilibria in Kerr geometry
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FIG. 4. Poloidal velocityuuau abovethe (r ,u)-plane for the vortex inFig. 2.
FIG. 5. Poloidal particle flux abovethe (r ,u)-plane for the vortex inFig. 2.
FIG. 6. Stream lines in the(r ,u)-plane for an open system withV051, V150.02 and boundary condi-tion according to Eq.~26!, with C50.4, k52.5, andn53.
284 Phys. Plasmas, Vol. 11, No. 1, January 2004 K. Elsasser and Y. Kot
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The stream lines are shown in Figs. 6 and 7, respectively,in the poloidal (r ,u)-plane. The equatorial influx atu.1.5and the axial outflow atu.0,3.1 are clearly seen in bothcases. The second model exhibits, in addition, some streamlines which start and end near the horizon. This may be anartifact due to the lack of ‘‘real’’ inner boundary conditions;a more careful treatment of the inner boundary at the horizonrequires vanishing of the flux functions, and all stream lineswould then be closed before they reach the boundary.
V. SUMMARY AND DISCUSSION
Though violent processes can occur near black holes wehave explored the possibility of steady states for a weaklydissipative plasma. The momentum balance of an ideal neu-tral plasma in Kerr geometry has been solved exactly; thesolutions are defined by a set of flux functionsVr(x) andboundary conditions for the stream functionsx in theasymptotic region and near the outer horizon of the blackhole. These solutions minimize our functionalW(x), Eq.~18!, which is solved by finite elements, together with theequation for the densityN, Eq. ~17!. For a simple choice ofVr(x) we find equilibria with closed or open stream lines inthe poloidal plane, depending on the outer boundary condi-tion for x. The proper inner boundary condition is a delicatequestion becausegrs in Eq. ~17! diverges at the horizons ifthe usual formulas in Boyer–Lindquist coordinates are used~see, e.g., Ref. 9!. Moreover,g11 diverges at the rotation axis.To keep the right-hand side in Eq.~17! finite we shouldrequire thatVr(x) vanishes at the horizons, andV1(x) at therotation axis, too. Therefore our inner boundary in the poloi-dal plane should consist of the outer horizon and the rotationaxis, with the conditionx5const; this is then the innermoststream line, and the black hole will never receive any mate-
rial in the steady state. To avoid numerical complications wemoved this ‘‘true’’ inner boundary slightly to the outer re-gion.
The flux functionsVr(x) can be obtained by a model forexternal and dissipative forces which lead to ‘‘solubility con-ditions,’’ Eq. ~13!. We studied a simple model where the ionsare accelerated by a loop voltageU, and decelerated by afriction force arising from collisions with neutrals or pho-tons, both in the toroidal direction; the ratio of both the ac-celerating and decelerating force is characterized by the pa-rameterl , Eq.~21!, which has the dimension of a length@likeV1(x)] for c51. The resulting solubility conditions, Eqs.~22! and ~23!, are explicit equations forV1(x) and V0(x),respectively, which should be solved iteratively together withthe minimization ofW(x) and the normalization condition,Eq. ~17!. Particular solutions whose stream lines come fromand go to infinity are defined by Eq.~24!, with proper bound-ary conditions forx. Nearly poloidal flows (l !1) have con-stant flux functionsV0(x).1 and V1(x).2 l ; they havebeen shown in more detail, with strong condensation of theparticles near the horizon, and few particles escaping alongthe symmetry axis with high velocities.
1Black Holes: Theory and Observation, edited by F. W. Hehl, C. Kiefer,and R. J. K. Metzler~Springer-Verlag, Berlin, 1998!.
2R. P. Kerr, Phys. Rev. Lett.11, 237 ~1963!.3G. Neugebauer, in Ref. 1, p. 319.4H. Grad and J. Hogan, Phys. Rev. Lett.24, 1337~1970!.5A. Heimsoth, dissertation, Ruhr-Universita¨t Bochum, 1989.6K. Elsasser, Phys. Rev. D62, 044007~2000!; 65, 024015~2002!.7S. Weinberg,Gravitation and Cosmology~Wiley, New York, 1972!.8K. Elsasser, Phys. Scr., TT98, 24 ~2002!.9S. Chandrasekhar,The Mathematical Theory of Black Holes~Clarendon,Oxford, 1992!.
10Y. Kot, dissertation, Ruhr-Universita¨t Bochum, 2003.11H. R. Schwarz,Methode der Finiten Elemente~B. G. Teubner, Stuttgart,
1984!.
FIG. 7. Stream lines in the(r ,u)-plane for an open system withV051, V150.02 and boundary condi-tion according to Eq.~27!, with C50.4, k52.5, andn51.
285Phys. Plasmas, Vol. 11, No. 1, January 2004 Weakly dissipative plasma equilibria in Kerr geometry
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