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Stationary incompressible MHD perturbationsgenerated by a current source in a moving plasma
\ PERGAMON
Planetary and Space Science 36 "0888# 434Ð444
Stationary incompressible MHD perturbations generated by a current source in a moving plasma T[I[ Woodwarda\b\ \ J[F[ McKenzieb\c a
Space and Atmospheric Physics\ Imperial Colle`e\ London SW6 1BZ b Physics Department\ University of Natal\ Durban\ South Africa c Max!Planck!Institut fur Aeronomie\ D!26078 Katlenbur`!Lindau\ Germany Received 6 April 0887^ received in revised form 18 October 0887^ accepted 3 November 0887
Abstract We consider the inductive interaction between a conducting body and a magnetized incompressible plasma in relative uniform motion\ which has application to the IoÐJupiter system\ for example[ An incompressible plasma only supports one mode of propagation\ namely the Alfven mode[ In the case of free oscillations\ this mode propagates the perturbations in the magnetic _eld and in the plasma velocity unattenuated along the direction of the background _eld\ while the plasma pressure balances the magnetic pressure[ The situation changes in the presence of source currents and in a ~owing plasma[ In particular\ the parallel plasma vorticity and parallel plasma current are propagated unattenuated along the familiar Alfven characteristics\ while the _eld and velocity perturbations su}er Laplacian decay in the near _eld[ We study these perturbations in the frame of the body and compare them to the case of no source terms[ Þ 0888 Elsevier Science Ltd[ All rights reserved[
0[ Introduction The interaction between a conducting body and a mag! netised plasma in relative motion arises in a variety of space plasma situations] for example\ the IoÐJupiter sys! tem "e[g[ Goldreich and Lynden!Bell\ 0858^ Goertz and Deift\ 0862^ Neubauer\ 0879^ Southwood et al[\ 0879^ Wright and Southwood\ 0876^ Wright\ 0876^ Linker et al[\ 0877\ 0880^ Wright and Schwartz\ 0889#\ or spacecraft and tethered satellite systems "e[g[ Drell et al[\ 0854^ Belcastro et al[\ 0871^ Barnett and Olbert\ 0875^ Dob! rowolny and Veltri\ 0875^ Estes\ 0877^ Hastings et al[\ 0877#[ The basic idea is that the motional electric _eld seen by a conducting body moving relative to a mag! netised plasma drives a current in or on the body which generates a stationary pattern of perturbations in the surrounding plasma[ Barnett and Olbert "0875# present the most com! prehensive and general treatment of this problem\ tack! ling it in a fully self!consistent manner and including wave frequencies up to the upper hybrid frequency[ How! ever\ they arrive at an intractable integral equation for the current through the body and resort to physical reasoning combined with approximation techniques in order to obtain a solution to the wave _elds[ In this paper we Corresponding author[ E!mail] t[woodwardÝic[ac[uk
consider only wave frequencies well below the proton gyrofrequency ði[e[ in the magnetohydrodynamic "MHD# regimeŁ\ as well as restricting ourselves to an incom! pressible plasma[ In this limit\ the only wave mode to propagate is the shear Alfven mode\ which carries a par! allel "to the background magnetic _eld# plasma current and parallel plasma ~uid vorticity away from the source region "e[g[ Lighthill\ 0859^ McKenzie\ 0880#[ These per! turbations are carried downstream in the ~owing plasma along the Alfven characteristics "e[g[ McKenzie et al[\ 0882^ Woodward and McKenzie\ 0882# and form the well!known Alfven wings "e[g[ Drell et al[\ 0854#[ Fur! thermore\ in the current work we avoid the di.cult prob! lem of the integral equation for the body "source# current by simply specifying a given source current and cal! culating the stationary wave pattern thereby induced in the ~ow[ The problem is formulated in an MHD frame! work and we are able to fully specify the ~ow and mag! netic perturbations as well as the electric potential "and therefore\ also the electric _eld# analytically for a wide variety of ~ow situations[ The layout of the paper is as follows[ Following the approach taken by McKenzie "0880# and Woodward and McKenzie "0883a\ b# we formulate the problem in the frame\ Sp\ of the plasma in Section 1[ Here we show how the imposition of a source current leads to a source of momentum\ which causes the total pressure P "plasma
9921Ð9522:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved[ PII] S 9 9 2 1 Ð 9 5 2 2 " 8 7 # 9 9 0 2 3 Ð 1
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plus magnetic# to su}er Laplacian decay away from the source region\ in contrast to the case of no source currents where P is constant everywhere[ Furthermore\ we derive wave equations for the ~ow u\ magnetic B and plasma current J perturbations and derivatives thereof and thereby characterise the nature of their propagation[ Thus\ we show that u and B are propagated as Alfven waves in addition to su}ering Laplacian decay as a result of a total pressure driving term\ while J and the Laplacian of the parallel magnetic _eld perturbation are propagated unattenuated as Alfven waves[ In Section 2 we transform into the frame\ Ss\ of the source and calculate the station! ary wave _elds observed in that frame[ Since there is no time dependence in this frame\ the electric _eld is deriv! able from a scalar potential\ f[ This is found to be of hybrid nature\ possessing Alfvenic characteristics as well as a decay associated with P[ Following this in Section 3 we assume a speci_c form for the source current and calculate exact analytic expressions for P\ u\ B\ J as well as f\ which we discuss and illustrate[ Finally\ we summarise and discuss our _ndings in Section 4[
1[ The governing wave equations In this section we formulate the interaction between a plasma and an electromagnetic _eld in the presence of a source current in the frame\ Sp\ of the plasma[ Neglecting the displacement current as well as all dissipative e}ects such as viscosity\ heat conduction and Ohmic heating\ a highly conducting magnetized incompressible plasma is governed by the dissipationless hydrodynamical equa! tions coupled with Maxwell|s equations in the form "e[g[ Cowling\ 0846^ Lighthill\ 0859^ Dungey\ 0857^ McKenzie\ 0880#] r const r9\ say c div u 9 r
Du −9p¦J×B Dt
"0#
r
Du B1 0 −9 p¦ ¦ B = 9B−Js×B Dt 1m9 m9
0
r r9^ div u 9 r9
B9 1u −9P¦ 9>B¦Ms 1t m9
1B B99>u 1t
curl B m9"J¦Js#
"3#
91P div Ms
"4#
"6# "7# "8#
"09#
This is just Poisson|s equation for P\ indicating that for non!zero source terms "Ms#\ P decays away from the source region[ This is in contrast to the case of no source currents "e[g[ Cowling\ 0846# where P is constant every! where] i[e[ the particle pressure p is exactly balanced by the hydrostatic pressure arising from the magnetic _eld B9B>:m9\ leaving the {magnetic tension| B99>B:m9 to bal! ance the rate of change of momentum[ Equations for u and B may be obtained by taking 1:1t of the linearized momentum equation "7# and using the linearized form of Faraday|s law "8#[ Thus\ we derive 0 1 "M −9P# r9 1t s
"00#
B9 9 "M −9P# r9 > s
"01#
LAB
Here r 0 r9\ u\ J and p are\ respectively\ the plasma density "constant#\ velocity\ current density and pressure\ Js represents source currents and B and E are the mag! netic and electric _elds[ The convective derivative D:Dt is given by 1:1t¦u = 9[ The frozen!in _eld line condition\ E¦u×B 9\ may be combined with Faraday|s law ðeqn "2#Ł to give
"5#
in which we have denoted the momentum source term Ms 0 −Js×B9 and total pressure P 0 p¦B9B>:m9[ Here the subscripts > refers to directions parallel to the back! ground magnetic _eld\ B9[ Taking the divergence of eqn "7# yields a time inde! pendent equation for the total pressure P in terms of the momentum source term\ namely
LAu "2#
1
Thus\ eqns "0#\ "4# and "5# form a closed set of equations for p\ u and B\ given Js[ Linearizing these equations about a uniform back! ground state speci_ed by u9 9\ p9\ B9 and Js9 9\ with _rst!order quantities being denoted\ respectively\ u\ p\ B and Js\ we derive the following set of equations]
"1#
1B −curl E 1t
DB B = 9u−B div u Dt
Further\ using Ampere|s law ðeqn "3#Ł to replace J in the momentum equation "1# we obtain
where LA is the Alfven wave operator de_ned as LA 0
11 1t1
−V1A91>
"02#
in which VA 0 B9:zm9r9 is the Alfven speed[ Thus\ u and B will possess Alfvenic characteristics "i[e[ will be propa! gated along the magnetic _eld#\ in addition to those characteristics arising from the source terms on the right! hand!sides of eqns "00# and "01#[ The source terms pro! portional to 9P give rise to decay[ u and B will\ therefore\
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
exhibit a hybrid of one!dimensional propagation as well as decay away from the source in the near _eld region\ while at large distances from the source only Alfvenic _eld!aligned perturbations will remain[ In the case of no source currents\ the driving terms in eqns "00# and "01# vanish[ In that case u and B are propagated unattenuated along the magnetic _eld\ with the magnetic _eld lines acting as {plucked strings| and the transverse and longi! tudinal motions are coupled by the divergence equations "div B 9 and div u 9#[ Furthermore\ straightforward operations on eqn "01# yields the following equations for the magnetic _eld] LA91B> −
B9 1 9> div Ms r9
LA curl> B −
B9 9> curl> Ms r9
B9 9> curl Ms−LAJs m9r9
frame Ss of the source[ For the sake of de_niteness and without too much loss of generality\ we consider a steady current source relative to which the plasma moves with velocity u9 "taken to be along the z!axis# perpendicular to the ambient magnetic _eld B9 "taken parallel to the x! axis#*see Fig[ 0[ In order to e}ect the transformation from the plasma frame Sp to the frame Ss of the source\ in which the plasma moves relative to the source which is at rest\ we use the following Galilean transformation] 1 1 : u9 = 9 u9 1t 1z
"06#
"03#
Then the Alfven wave operator LA ðeqn "02#Ł becomes
"04#
LA : V1AL?A^ L?A 0 M1
Thus\ 91B> and curl> B "i[e[ the parallel plasma current# are propagated along the magnetic _eld[ In addition\ taking the curl of eqn "01# and using Ampere|s law\ we may obtain equations describing the propagation of the plasma current perturbation J\ namely LAJ
436
"05#
Equations similar to eqns "03# and "04# for B may also be derived for the plasma velocity u by making use of the linearized form of Faraday|s law ðeqn "8#Ł[ To summarize this section then\ the presence of a _rst! order source current "Js# perturbation in addition to any induced plasma current perturbations "J# contributes a _rst!order source term "Ms# to the momentum equation[ The parallel momentum is una}ected by this form of source term\ while the perpendicular momentum com! ponents su}er an in~uence depending on the form of the source current Js[ Ms acts to drive perturbations in the magnetic _eld B and plasma velocity u\ which propagate away from the source region[ In particular\ the parallel curl of the magnetic _eld "curl> B#\ the Laplacian of the parallel magnetic _eld perturbation "91B>#\ the parallel vorticity "curl> u#\ as well as the plasma current per! turbation "J# are propagated one!dimensionally by the Alfven mode[ In contrast\ the propagation of the total magnetic _eld "B# and velocity "u# perturbations is hybrid in nature\ consisting of a one!dimensional Alfvenic propagation as well as decay away from the source region[
11
11 − 1z1 1x1
"07#
in which M 0 u9:VA is the Alfven Mach number[ Making these transformations in the wave equations developed in the previous section in the frame Sp of the plasma yields the corresponding equations describing the stationary perturbations in the frame Ss of the source[ Moreover\ in Ss the perturbation electric _eld E is derivable from a scalar potential f and is given by E −u9×B−u×B9 "u9By\ −u9Bx−uzB9\ uyB9# −9f
"08#
where u 0 "ux\ uy\ uz# and B 0 "Bx\ By\ Bz# are the plasma velocity and magnetic _eld perturbations\ respectively[ The equation for f follows from the y!component of the linearized momentum equation "7# combined with eqn "08#\ namely
2[ Stationary waves in a moving plasma generated by a moving current source In this section we use the above development to for! mulate the wave equations characterizing the stationary wave system generated by a moving current source in the
Fig[ 0[ The relative orientations of the plasma ~ow u9 and the back! ground magnetic _eld B9[
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T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
B9 1P L?Af −Msy m9m9 1y
"19#
where L?A is the Alfven wave operator in Ss ðeqn "07#Ł[ This equation displays the Alfven wave operator on the left!hand!side and shows that f is driven by the "y!deriva! tive of the# total pressure P and by the y!component of the momentum source term[ Thus\ f will be hybrid in nature\ exhibiting both Alfvenic characteristics\ as well as that arising from the pressure term[ The latter is a Laplacian decay away from the source region\ as we discussed in the previous section[ Regarding the nature of the current source\ we shall simply assume a given current distribution Js"r#[ For the case of a conducting body and a magnetized plasma in relative motion with the con_guration de_ned in Fig[ 0\ we expect that the motional electric _eld −u9×B9 −u9B9y¼ will drive a current in the y!direction on the body[ Thus\ we take Js Jsy¼[
As in McKenzie "0880# we assume that the stationary wave pattern which emerges at large distances from the body should be well!approximated by integrating the Green|s function solution for a point source over a source distribution of _nite size[ Therefore\ we consider the spec! i_c case of a d!function source current^ i[e[ we take "10#
corresponding to a point source of strength J9 pointing in the y!direction anti!parallel to the motional electric _eld "−u9×B9#[ Note that if the source is of purely induc! tive origin\ the sign of J9 should be reversed[ With this explicit form of Js we can derive exact ana! lytic solutions for P\ curl> B\ B\ J\ curl> u\ u\ as well as f[ The details of the calculations are provided in the Appendix for reference[ Here we present the _nal solu! tions and describe their behaviours in the context of the model[ The variation of the total pressure P is described by eqn "09# in both Sp and Ss[ The solution to this equation may be developed using Fourier analysis to yield
01
B9J9 1 0 ^ −z m 9 3p 1z r
P 3
Bx\y −
"11#
with r 0 zx1¦y1¦z1[ Thus\ P decays away from the source region "origin# in a dipole!like fashion\ both upstream and downstream in the ~ow[ This is shown in Fig[ 1 in the plane containing the source\ B9 and u9 "i[e[ the y 9!plane#[ Thus\ P varies quickly close to the source\ while in the far!_eld region P is approximately constant "zero#[ Hence\ it is only far from the source that the particle pressure p approximately balances the magnetic pressure B9Bx:m9[
$
%
m9J9 1 1 1 \ log ð"r−Mz:k#1−x1:k1Ł 7pMk 1x 1x 1y "12#
m9J9
"IBz0¦IBz1#
"13#
M1 d"y#ðd"z¦#¦d"z−#Ł 3
"14#
Bz −
M1
where IBz0
IBz1 −
3[ Properties of the stationary wave pattern
Js J9d"x#d"y#d"z#y¼
We can now feed this compact solution for P into the source frame version of eqn "01# for B and use a method identi_ed by McKenzie "0880# to solve for the x! and y!components of B\ while the solution for Bz may be developed using Fourier analysis[ The details are explai! ned in the Appendix[ The solutions may be written as follows]
0 7pMg1
6
1pd"y#ðd"z¦#−d"z−#Ł
−
0 11 ðlog "y1¦z1¦:g1#−log "y1¦z1−:g1#Ł 1g 1y1
¦
0 1 0 ^ MŁ0 3p 1x r
01
7 "15#
in which r is as de_ned above and z2 0 x2z:M
"16#
k 0 z0¦M1
"17#
g 0 z0¦M−1
"18#
The solutions for Bx and By may also be developed using Fourier analysis of the form of eqn "01# corresponding to Ss[ It is only possible\ however\ to obtain exact analytic solutions in this way for very sub! and very super!Alfv! enic ~ows "i[e[ for M ð 0 and M Ł 0\ respectively#[ Whereas\ the solutions in eqns "12# are valid in all ~ow situations[ Nevertheless\ such a Fourier analysis high! lights the characteristics of the variation of Bx and By\ which are embodied in a less apparent fashion in the solutions in eqns "12#[ The Bx perturbation consists of two contributions\ one of which is singular on the Alfven lines "z2 9\ y 9# and the other which corresponds to the Laplacian decay away from the source "origin# in a two!dimensional dipole!like fashion[ A knowledge of Bx and P allows the determination of the particle pressure p P−B9Bx:m9[ This is shown in Fig[ 2\ where it is seen that p is singular "and negative# on the Alfven lines as well as at the origin[ In the near! _eld region of the origin\ p decays away from the origin in a dipole!like fashion[ The By perturbation is zero along the x 9 and y 9
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
438
Fig[ 1[ The variation of the total pressure P on the plane y 9 in the near!_eld region of the source\ which is located at the origin[ Note that the units of pressure are B9J9:3p[
Fig[ 2[ The variation of the particle pressure p on the y 9!plane for M 9[0[ For visual reasons\ we plot −p "in units of B9J9:3p#[
planes and is anti!symmetric along the x! and y!direc! tions[ It consists of two _nite disturbances of opposite polarity centred on lines parallel to the Alfven lines and displaced away from y 9 on the y!axis[ This is indicated in Fig[ 3[ From eqn "13# it is seen that Bz is singular on the Alfven lines and decays away from them in a two!dimensional dipole!like fashion[ Further\ Bz decays away from the source in a dipole!like fashion\ by virtue of the term proportional to 1r−0:1x[
Solutions for the plasma current J in Ss may be straightforwardly developed from the Sp form of eqn "05# using Fourier analysis[ The results show that Jx
J9 d?"y#ðd"z¦#−d"z−#Ł 3M
Jy − Jz 9
J9 d"y#ðd?"z¦#−d?"z−#Ł 3M
"29# "20# "21#
449
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
Fig[ 3[ The variation of By on the y 0!plane for M 9[4[ The upper panel is a surface plot of By on the y 0!plane[ The lower two panels are sections through this surface] the bottom left!hand panel indicates the z 4 section through this surface^ while the bottom right!hand panel represents the x 09 section[ In all panels the units of By are m9J9:7pk1[
where the prime on the d!functions indicate the _rst derivative of the function[ Thus\ the parallel current Jx curl> B:m9 and Jy are propagated down the Alfven lines as two pulses of opposite polarity[ The derivatives of the d!functions in these solutions means that current loops in the xy!planes centred on each characteristic are propagated down the characteristics with opposite cir! culations[ This is depicted in Fig[ 4\ which is an xy!plane in Ss for some z × 9[ The large solid dots labelled z¦ 9 and z− 9 are the Alfven characteristics[ The direction of the current circulations around these are indicated by the arrowheads on the circle around each characteristic[ Solutions for the plasma velocity perturbations u may be derived in a similar way to those obtained for the magnetic _eld B[ We simply quote the solutions here ux\y −
$
%
m9u9J9 1 1 1 \ log ð"r−Mz:k#1−x1:k1Ł 7pB9Mk 1z 1x 1y "22#
Fig[ 4[ The current loops around the Alfven characteristics "labelled z¦\− 9#[ The arrows on the circles around the characteristics indicate the direction of ~ow of the current[
uz − where
m9u9J9 B9M1
"Iuz0¦Iuz1#
"23#
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440
Iuz0 −03 d"y#ðd"z¦#¦d"z−#Ł
"24#
Iuz1 I1a¦I1b
"25#
tials in accordance with its de_nition E −9f[ Thus\ we see that the electric _eld is dipole!like\ centred on the Alfven lines[
"26#
4[ Conclusion
with I1a
I1b
01 6
0 1 0 ^ MŁ0 3p 1z r 0 7pk1
1pd"y#ðd"z¦#¦d"z−#Ł
−
0 11 s log "y1¦z1i :g1# 1g 1y1 i ¦\−
7
"27#
k\ g and r are as de_ned earlier[ These are clearly struc! turally similar to the solutions for B ðeqns "12# and "13#Ł[ The ux perturbation is singular on the Alfven lines and of opposite sign on each characteristics\ ~owing towards the source along the z¦ 9 characteristic and away from the source along the z− 9 characteristic[ The uy perturbation is similar to the By perturbation ðeqn "12# and Fig[ 3Ł[ It is anti!symmetric in y and zero on the y 9!plane[ Its variation is indicated in Fig[ 5[ The uz disturbance is singular on the Alfven lines and decays away from them in a two!dimensional dipole!like fashion[ It is also singular at the origin[ Further\ Fourier analysis may be used to derive an expression for the parallel vorticity "curl> u# whose wave equation is similar to that for curl> B ðeqn "04#Ł[ This may be written as curl> u
m9u9J9 3M1B9
d?"y#ðd"z¦#¦d"z−#Ł
"28#
Thus\ the parallel vorticity is propagated down the Alfven lines as four pulses] two above the y 9!plane of positive vorticity and two of negative vorticity below this plane[ Finally\ feeding the solution for P ðeqn "11#Ł into the wave equation describing the behaviour of the electric potential f ðeqn "19#Ł we may derive an exact analytic solution for f using the method of McKenzie "0880#\ in a manner similar to that used for Bx "see the Appendix#[ Thus\ we derive that f
m9u9J9 1 log ð"r−Mz:k#1−x1:k1Ł 7pMk 1y
"39#
where k and r are de_ned as above[ This is singular on the Alfven lines and decays away slowly from them[ Figure 6 shows the equipotential surfaces corresponding to this form for f[ They are cylinders of opposite potentials above "y × 9\ positive# and below "y ³ 9\ negative# the y 9!plane aligned with the Alfven characteristics and bent at the origin[ The electric _eld lines are perpendicular to these surfaces and point from higher to lower poten!
We have analysed the stationary wave system gen! erated by the inductive interaction between a conducting body and a magnetised\ incompressible plasma in relative motion in the framework of a non!dispersive MHD for! mulation[ As is well!known\ an incompressible MHD plasma supports only the shear Alfven mode of propa! gation[ The magnetoacoustic modes of compressible MHD have either become evanescent "fast mode# or coincident with the shear mode "slow mode#[ This behav! iour is evident in the solutions we derive\ which on the whole exhibit singular behaviour on the Alfvenic charac! teristics "lines# and Laplacian decay away from the source region[ Consequently\ the analysis presented here adequately describes the incompressible response of the inductive interaction\ but is not appropriate for com! parison with the compressive behaviour in such a system[ Furthermore\ we consider a purely inductive interaction\ achieved by way of the use of a source current[ We leave for the subject of future work\ processes such as mass loading\ which provide\ in particular\ momentum source terms of di}erent character to that associated with an inductively generated source current as we describe in this article[ Thus\ our results are applicable to interactions in high b plasmas in which the e}ects of mass loading are not signi_cant[ In order to make the system analytically tractable\ we have prescribed a source current\ rather than calculating it self!consistently and furthermore\ we have assumed a point source current[ Thus\ we present the fully analytic Green|s function solutions for the ~ow\ magnetic and electric _elds as well as the accompanying current per! turbations of the interaction[ It is noted that these Green|s functions may be integrated over extended source current distributions such as would be expected in the various applications to the IoÐJupiter system or the experimental tethered satellite system[ Previous papers have discussed the non!dispersive ðe[g[ McKenzie et al[\ 0882# and dispersive "e[g[ Barnett and Olbert\ 0875^ Estes\ 0877^ Woodward and McKenzie\ 0883a\ b^ Sanmartin and Estes\ 0886# Alfvenic perturbations\ but do not pro! vide exact analytic solutions for all _eld variables\ as we do in this article[ The presence of a source current "Js# in this interaction provides a source of momentum for the plasma by virtue of the Lorentzian interaction "−Js×B9# between this source current and the background magnetic _eld "B9# which is frozen into the ~ow[ Consequently\ only the perpendicular momentum is a}ected[ Hence\ only source
441
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
Fig[ 5[ The variation of uy on the y 0!plane for M 9[4[ The upper panel is a surface plot of uy on the y 0!plane[ The lower two panels are sections through this surface] the bottom left!hand panel indicates the z 4 section through this surface^ while the bottom right!hand panel represents the x 09 section[ In all panels the units of uy are m9u9J9:3pk1B9[
currents perpendicular to the background magnetic _eld are of signi_cance in this type of interaction "Estes\ 0877#[ A further consequence of the source current and the incompressibility of the plasma is that the total pressure P "plasma plus magnetic# is not constant\ but decays away from the source current region in a dipole!like fashion[ This implies that in the near!_eld\ the particle pressure is not balanced by the magnetic pressure\ while in the far!_eld region this balance becomes better approximated[ This is in contrast to the case of no source currents\ where P is constant "Cowling\ 0846#[ The plasma current\ parallel circulation of the mag! netic _eld\ parallel vorticity and the Laplacians of the parallel magnetic _eld and plasma velocity perturbations are propagated as Alfven waves\ which are swept down! stream in the ~ow of the plasma to form the well!known Alfven lines "e[g[ Drell et al[\ 0854^ McKenzie\ 0880^ McKenzie et al[\ 0882#[ The pure magnetic _eld and
plasma velocity perturbations as well as the electric potential perturbation "in the frame of the source# are driven by the total pressure as well as directly by the momentum source term and therefore\ possess both Alfv! enic propagation characteristics and decay away from the source region[ We consider the speci_c case of a plasma ~ow per! pendicular to the magnetic _eld\ as would be found in the IoÐJupiter system\ for example[ The motional electric _eld induced in the conducting body would then drive a current perpendicular to both the ~ow and magnetic _eld[ Consequently\ we choose to impose a d!function source current parallel to this motional electric _eld and this has the following consequences for the perturbation _elds[ The components of the magnetic _eld and plasma vel! ocity perturbations in the plane of the background ~ow and magnetic _eld "i[e[ Bx\z and ux\z# are hybrid in nature\ being singular on the Alfven lines although not narrowly
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
442
Fig[ 6[ The equipotential surfaces for M 9[4 "a# Four surfaces in the shape of cylinders bent at the origin have been drawn] the two drawn in solid lines are for positive potentials^ while the adjacent two surfaces drawn in small!dashed line!type are for negative potentials[ The positive and negative surfaces are on either side\ in the y!direction of the Alfven characteristics "z2 9\ y 9#] i[e[ positive "negative# potential surfaces are on the y × 9 "y ³ 9# side of the y 9!plane[ Furthermore\ surfaces of higher absolute potential are smaller cylinders lying against the Alfven characteristics] these are represented by the smaller inner cylinders in the diagram[ "b# This is an z 0!section through the equipotential surfaces sketched in part "a#[ The solid lines indicate positive potentials\ while the dashed lines are for negative potentials[ The values of the potentials are indicated by the values for f "in units of m9u9J9:3pMk# on the diagram[
con_ned to them\ as well as su}ering a dipole!like decay away from the source in the near!_eld region[ The com! ponents "By and uy# perpendicular to the plane of the background ~ow and magnetic _eld are anti!symmetric about the y 9!plane and hence\ are zero on the y 9!
plane[ In this way the plasma ~ow and magnetic _eld\ which is frozen into the ~ow\ are de~ected around the source:body[ Finally\ the electric _eld is singular on the Alfven lines and of dipole nature in planes perpendicular to the Alfven lines[
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Acknowledgements
where k is a constant[ This requires that
J[F[McK[ wishes to thank FRD!CSIR for their partial support and T[I[W[ acknowledges _nancial support from the FRD\ South Africa\ as well as PPARC\ U[K[
f¦
Appendix] Solution for the magnetic _eld B perturbation The source frame wave equation for B is given by eqn "01# under the transformation "07#\ where P is given by eqn "11# and Ms B9Jsz¼ "Js J9d"x#d"y#d"z##[ The x! and y!components of this equation then yield
0 1 0 1
11 1 1 −0 \ r 1x 1z 1x 1y
0 0
"A0#
1
m9 1 1P B J− B9 1x 9 s 1z
1
11 1 1 1 Bx C 1 r−0 ¦M 1x 1z 1x 1z
1
"A2#
1
1 1 log "r¦f¦"x\ z## ¦M 1x 1z
6
0
1 7
x¦Mz 1 0 1 f¦ ¦ ¦M r¦f¦ r 1x 1z
in which f¦"x\ z# is an arbitrary function of x and z[ Now chose f¦"x\ z# such that the right!hand!side of eqn "A2# equals k r
"A6#
0
C 11 1 1 1 Bx ¦M log "r¦f−# 1x 1z k 1x1 1z
1
"A7#
Boundary conditions at =z= and causality con! siderations require the choice of the lower sign in "A5# "i[e[ k −z0¦M1 −=k=# and the upper sign in "A7# "i[e[ k =k=#[ Subtracting "A5# from "A7# and factoring out 1:1z yields the fundamental solution for Bx] C 11 log ð"r−Mz:k#1−x1:k1Ł 1Mk 1x1
"A8#
Solution for Bz usin` Fourier analysis The Fourier transform of eqn "A1# for Bz may be written Bzk −
m9J9
ikx
M1 "k1z −k1x :M1#
6
0¦
k1z
7
"k1x ¦k1y ¦k1z #
"A09#
where we denote the Fourier transform of Bz by Bzk[ Bz may then be reconstructed by calculating the inverse transformation\ namely
First\ note that
0
x−Mz ^ k 2z0¦M1 k
in which k 0 z0¦M1[ Similarly\ we can construct the solution ðeqn "12#Ł for B y[
Equations "A0# may be solved using the method adopted by McKenzie "0880# in his solution for the elec! tric potential[ Consider the equation for Bx and write it in the following form]
10
"A5#
since operating on "A5# with 1:1x¦M 1:1z retrieves the original eqn "A2#[ Similarly\ we can choose
"A1#
Solutions for Bx\y
0
1
Bx
J9 11r−0 1 m9 Js¦ 1x 3p 1z1
1 1 −M 1x 1z
0
C 11 1 1 1 Bx −M log "r¦f¦# 1x 1z k 1x1 1z
so that
in which C 0 −"m9J9:3p#\ r 0 zx1¦y1¦z1 and L?A is the Alfven wave operator in Ss ðeqn "07#Ł[ The z!com! ponent may be written as L?ABz
"A4#
Hence\ the original eqn "A2# is satis_ed if we have
f−"x\ z#
m9 1 1 1 L?A"Bx\ By# − P \ B9 1x 1x 1y −C
x¦Mz ^ k 2z0¦M1 k
"A3#
Bz
0 "1p#
2
g g g
−
−
−
×Bzk exp ði"kxx¦kyy¦kzz#Ł dkx dky dkz
"A00#
To this end\ let IBz0 0
i "1p#
g g g
2
−
kx
−
1 1 1 − "kz −kx :M #
×exp ði"kxx¦kyy¦kzz#Ł dkx dky dkz
"A01#
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
IBz1 0
i
g g g
2
"1p#
−
−
kxk1z
1 x
1 y
1 z
1 z
1 x
I1b
1
"k ¦k ¦k #"k −k :M #
−
×exp ði"kxx¦kyy¦kzz#Ł dkx dky dkz
"A02#
"IBz0¦IBz1#
"A03#
Then Bz −
m9J9 M1
M1 d"y#ðd"z¦#¦d"z−#Ł 3
"A04#
where z2 0 x2z:M are the Alfven characteristics\ as before[ The determination of IBz1 is a little more complicated[ First\ using Cauchy|s residue theorem we evaluate the kz! integral noting three poles at kz 2kx:M and kz izk1x ¦k1y \ to obtain "A05#
IBz1 I1a¦I1b where I1a −
I1b
0 1
05p M
i 7p
1
g
g
k1x
−
1 1 x
"g k ¦k #
kxzk1x ¦k1y
−
1 y
1 1 x
1 y
"g k ¦k #
"eikxz¦−eikxz−# eikyy dkx dky 1
1
e−zzkx ¦ky ei"kxx¦kyy# dkx dky
with g1 0¦M−1[ Noting that k1x "g1k1x ¦k1y #
0 g1
0
0−
k1y "g1k1x ¦k1y #
1
I1a may be split into two integrals over the two terms in the above expression[ The _rst one is straightforward\ while the second can be cast in the form of a standard sine transformation "Erdelyi et al[\ 0843#[ Thus\ we obtain I1a −
−
0 7pMg1
6
1pd"y#ðd"z¦#−d"z−#Ł
0 11 ðlog "y1¦z1¦:g1#−log "y1¦z1−:g1#Ł 1g 1y1
7
"A06#
To evaluate I1b\ we _rst transform to polar coordinates "kx\y k"cos f\ sin f## to obtain I1b
0 1 7p1 1x
"A08#
This then con_rms solution "13# of the main text for Bz[ References
Cauchy|s residue theorem may be used straight! forwardly in IBz0 to obtain IBz0
01
0 1 0 ^ r 0 zx1¦y1¦z1M Ł 0 3p 1x r
444
1p
9
9
gg
e−kz eik"xcosf¦ysinf# "g1 cos1 f¦sin1 f#
df dk
"A07#
In the limit of M Ł 0\ g1 ¹ 0[ In this case the integral over f in I1b can be written in terms of the zero!order Bessel function "Erdelyi et al[\ 0843#\ while the following integral over k is of standard form[ Hence\ we obtain
Barnett\ A[\ Olbert\ S[\ 0875[ Radiation of plasma waves by a con! ducting body moving through a magnetized plasma[ J[ Geophys[ Res[ 80\ 09\006[ Belcastro\ V[\ Veltri\ P[\ Dobrowolny\ M[\ 0871[ Radiation from long conducting tethers moving in the near earth environment[ Nuovo Cimento\ Soc[ Ital[ Fis[ C 4 "4#\ 420[ Cowling\ T[G[\ 0846[ Magnetohydrodynamics[ Interscience\ New York[ Dobrowolny\ M[\ Veltri\ P[\ 0875[ Structure of Alfven wings associated with a conductor moving through a magnetoplasma[ Astron[ Astro! phys[ 056\ 068[ Drell\ S[D[\ Foley\ H[M[\ Ruderman\ M[A[\ 0854[ Drag and propulsion in the ionosphere] an Alfven engine in space[ J[ Geophys[ Res[ 69\ 2020[ Dungey\ J[W[\ 0857[ In] Matsushita\ S[\ Campbell\ W[H[ "Eds[#\ Physics of Geomagnetic Phenomena[ Academic Press\ New York\ p[ 802[ Erdelyi\ A[\ Magnus\ W[\ Oberhettinger\ F[\ Tricomi\ F[G[\ 0843[ Tables of Integral Transforms\ vol[ 0[ McGraw!Hill\ New York[ Estes\ R[D[\ 0877[ Alfven waves from an electrodynamic tethered sat! ellite system[ J[ Geophys[ Res[ 82\ 834[ Goldreich\ P[\ Lynden!Bell\ D[\ 0858[ Io\ a Jovian unipolar inductor[ Astrophys[ J[ 045\ 48[ Goertz\ C[K[\ Deift\ P[A[\ 0862[ Io|s interaction with the magnetos! phere[ Planet[ Space Sci[ 10\ 0288[ Hastings\ D[E[\ Barnett\ A[\ Olbert\ S[\ 0877[ Radiation from large space structures in low Earth orbit with induced alternative currents[ J[ Geophys[ Res[ 82\ 0834[ Lighthill\ M[J[\ 0859[ Studies on magneto!hydrodynamic waves and other anisotropic wave motions[ Phil[ Trans[ Roy[ Soc[ London\ Ser[ A 141\ 280[ Linker\ J[A[\ Kivelson\ M[G[\ Walker\ R[J[\ 0877[ An MHD simulation of the ~ow past Io] Alfven and slow mode perturbations[ Geophys[ Res[ Lett[ 04\ 0200[ Linker\ J[A[\ Kivelson\ M[G[\ Walker\ R[J[\ 0880[A three!dimensional MHD simulation of the ~ow past Io[ J[ Geophys[ Res[ 85\ 10\926[ McKenzie\ J[F[\ 0880[ Stationary MHD waves generated by a source in a moving plasma[ J[ Geophys[ Res[ 85\ 8380[ McKenzie\ J[F[\ Woodward\ T[I[\ Inhester\ B[\ 0882[ Magnetoacoustic and Alfven potentials for stationary waves in a moving plasma[ J[ Geophys[ Res[ 87\ 8190[ Neubauer\ F[M[\ 0879[ Nonlinear standing Alfven wave current system at Io] theory[ J[ Geophys[ Res[ 74\ 0060[ Sanmartin\ J[R[\ Estes\ R[D[\ 0886[ Alfven wave far!_eld from steady current tethers[ J[ Geophys[ Res[ 091\ 03\514[ Southwood\ D[J[\ Kivelson\ M[G[\ Walker\ R[J[\ Slavin\ J[A[\ 0879[ Io and its plasma environment[ J[ Geophys[ Res[ 74\ 4848[ Woodward\ T[I[\ McKenzie\ J[F[\ 0882[ Stationary MHD structures[ Planet[ Space Sci[ 30\ 106[ Woodward\ T[I[\ McKenzie\ J[F[\ 0883a[ Stationary MHD waves modi! _ed by Hall current coupling*I[ Cold compressible ~ow[ Planet[ Space Sci[ 31\ 352[ Woodward\ T[I[\ McKenzie\ J[F[\ 0883b[ Stationary MHD waves modi! _ed by Hall current coupling*II[ Incompressible ~ow[ Planet[ Space Sci[ 31\ 370[ Wright\ A[N[\ 0876[ The structure of ULF waves produced by a tethered satellite system[ Astron[ Astrophys[ 075\ 243[ Wright\ A[N[\ Schwartz\ S[J[\ 0889[ The equilibrium of a conducting body embedded in a ~owing plasma[ J[ Geophys[ Res[ 84\ 3916[ Wright\ A[N[\ Southwood\ D[J[\ 0876[ Stationary Alfvenic structures[ J[ Geophys[ Res[ 81\ 0056[
Planetary and Space Science 36 "0888# 434Ð444
Stationary incompressible MHD perturbations generated by a current source in a moving plasma T[I[ Woodwarda\b\ \ J[F[ McKenzieb\c a
Space and Atmospheric Physics\ Imperial Colle`e\ London SW6 1BZ b Physics Department\ University of Natal\ Durban\ South Africa c Max!Planck!Institut fur Aeronomie\ D!26078 Katlenbur`!Lindau\ Germany Received 6 April 0887^ received in revised form 18 October 0887^ accepted 3 November 0887
Abstract We consider the inductive interaction between a conducting body and a magnetized incompressible plasma in relative uniform motion\ which has application to the IoÐJupiter system\ for example[ An incompressible plasma only supports one mode of propagation\ namely the Alfven mode[ In the case of free oscillations\ this mode propagates the perturbations in the magnetic _eld and in the plasma velocity unattenuated along the direction of the background _eld\ while the plasma pressure balances the magnetic pressure[ The situation changes in the presence of source currents and in a ~owing plasma[ In particular\ the parallel plasma vorticity and parallel plasma current are propagated unattenuated along the familiar Alfven characteristics\ while the _eld and velocity perturbations su}er Laplacian decay in the near _eld[ We study these perturbations in the frame of the body and compare them to the case of no source terms[ Þ 0888 Elsevier Science Ltd[ All rights reserved[
0[ Introduction The interaction between a conducting body and a mag! netised plasma in relative motion arises in a variety of space plasma situations] for example\ the IoÐJupiter sys! tem "e[g[ Goldreich and Lynden!Bell\ 0858^ Goertz and Deift\ 0862^ Neubauer\ 0879^ Southwood et al[\ 0879^ Wright and Southwood\ 0876^ Wright\ 0876^ Linker et al[\ 0877\ 0880^ Wright and Schwartz\ 0889#\ or spacecraft and tethered satellite systems "e[g[ Drell et al[\ 0854^ Belcastro et al[\ 0871^ Barnett and Olbert\ 0875^ Dob! rowolny and Veltri\ 0875^ Estes\ 0877^ Hastings et al[\ 0877#[ The basic idea is that the motional electric _eld seen by a conducting body moving relative to a mag! netised plasma drives a current in or on the body which generates a stationary pattern of perturbations in the surrounding plasma[ Barnett and Olbert "0875# present the most com! prehensive and general treatment of this problem\ tack! ling it in a fully self!consistent manner and including wave frequencies up to the upper hybrid frequency[ How! ever\ they arrive at an intractable integral equation for the current through the body and resort to physical reasoning combined with approximation techniques in order to obtain a solution to the wave _elds[ In this paper we Corresponding author[ E!mail] t[woodwardÝic[ac[uk
consider only wave frequencies well below the proton gyrofrequency ði[e[ in the magnetohydrodynamic "MHD# regimeŁ\ as well as restricting ourselves to an incom! pressible plasma[ In this limit\ the only wave mode to propagate is the shear Alfven mode\ which carries a par! allel "to the background magnetic _eld# plasma current and parallel plasma ~uid vorticity away from the source region "e[g[ Lighthill\ 0859^ McKenzie\ 0880#[ These per! turbations are carried downstream in the ~owing plasma along the Alfven characteristics "e[g[ McKenzie et al[\ 0882^ Woodward and McKenzie\ 0882# and form the well!known Alfven wings "e[g[ Drell et al[\ 0854#[ Fur! thermore\ in the current work we avoid the di.cult prob! lem of the integral equation for the body "source# current by simply specifying a given source current and cal! culating the stationary wave pattern thereby induced in the ~ow[ The problem is formulated in an MHD frame! work and we are able to fully specify the ~ow and mag! netic perturbations as well as the electric potential "and therefore\ also the electric _eld# analytically for a wide variety of ~ow situations[ The layout of the paper is as follows[ Following the approach taken by McKenzie "0880# and Woodward and McKenzie "0883a\ b# we formulate the problem in the frame\ Sp\ of the plasma in Section 1[ Here we show how the imposition of a source current leads to a source of momentum\ which causes the total pressure P "plasma
9921Ð9522:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved[ PII] S 9 9 2 1 Ð 9 5 2 2 " 8 7 # 9 9 0 2 3 Ð 1
435
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
plus magnetic# to su}er Laplacian decay away from the source region\ in contrast to the case of no source currents where P is constant everywhere[ Furthermore\ we derive wave equations for the ~ow u\ magnetic B and plasma current J perturbations and derivatives thereof and thereby characterise the nature of their propagation[ Thus\ we show that u and B are propagated as Alfven waves in addition to su}ering Laplacian decay as a result of a total pressure driving term\ while J and the Laplacian of the parallel magnetic _eld perturbation are propagated unattenuated as Alfven waves[ In Section 2 we transform into the frame\ Ss\ of the source and calculate the station! ary wave _elds observed in that frame[ Since there is no time dependence in this frame\ the electric _eld is deriv! able from a scalar potential\ f[ This is found to be of hybrid nature\ possessing Alfvenic characteristics as well as a decay associated with P[ Following this in Section 3 we assume a speci_c form for the source current and calculate exact analytic expressions for P\ u\ B\ J as well as f\ which we discuss and illustrate[ Finally\ we summarise and discuss our _ndings in Section 4[
1[ The governing wave equations In this section we formulate the interaction between a plasma and an electromagnetic _eld in the presence of a source current in the frame\ Sp\ of the plasma[ Neglecting the displacement current as well as all dissipative e}ects such as viscosity\ heat conduction and Ohmic heating\ a highly conducting magnetized incompressible plasma is governed by the dissipationless hydrodynamical equa! tions coupled with Maxwell|s equations in the form "e[g[ Cowling\ 0846^ Lighthill\ 0859^ Dungey\ 0857^ McKenzie\ 0880#] r const r9\ say c div u 9 r
Du −9p¦J×B Dt
"0#
r
Du B1 0 −9 p¦ ¦ B = 9B−Js×B Dt 1m9 m9
0
r r9^ div u 9 r9
B9 1u −9P¦ 9>B¦Ms 1t m9
1B B99>u 1t
curl B m9"J¦Js#
"3#
91P div Ms
"4#
"6# "7# "8#
"09#
This is just Poisson|s equation for P\ indicating that for non!zero source terms "Ms#\ P decays away from the source region[ This is in contrast to the case of no source currents "e[g[ Cowling\ 0846# where P is constant every! where] i[e[ the particle pressure p is exactly balanced by the hydrostatic pressure arising from the magnetic _eld B9B>:m9\ leaving the {magnetic tension| B99>B:m9 to bal! ance the rate of change of momentum[ Equations for u and B may be obtained by taking 1:1t of the linearized momentum equation "7# and using the linearized form of Faraday|s law "8#[ Thus\ we derive 0 1 "M −9P# r9 1t s
"00#
B9 9 "M −9P# r9 > s
"01#
LAB
Here r 0 r9\ u\ J and p are\ respectively\ the plasma density "constant#\ velocity\ current density and pressure\ Js represents source currents and B and E are the mag! netic and electric _elds[ The convective derivative D:Dt is given by 1:1t¦u = 9[ The frozen!in _eld line condition\ E¦u×B 9\ may be combined with Faraday|s law ðeqn "2#Ł to give
"5#
in which we have denoted the momentum source term Ms 0 −Js×B9 and total pressure P 0 p¦B9B>:m9[ Here the subscripts > refers to directions parallel to the back! ground magnetic _eld\ B9[ Taking the divergence of eqn "7# yields a time inde! pendent equation for the total pressure P in terms of the momentum source term\ namely
LAu "2#
1
Thus\ eqns "0#\ "4# and "5# form a closed set of equations for p\ u and B\ given Js[ Linearizing these equations about a uniform back! ground state speci_ed by u9 9\ p9\ B9 and Js9 9\ with _rst!order quantities being denoted\ respectively\ u\ p\ B and Js\ we derive the following set of equations]
"1#
1B −curl E 1t
DB B = 9u−B div u Dt
Further\ using Ampere|s law ðeqn "3#Ł to replace J in the momentum equation "1# we obtain
where LA is the Alfven wave operator de_ned as LA 0
11 1t1
−V1A91>
"02#
in which VA 0 B9:zm9r9 is the Alfven speed[ Thus\ u and B will possess Alfvenic characteristics "i[e[ will be propa! gated along the magnetic _eld#\ in addition to those characteristics arising from the source terms on the right! hand!sides of eqns "00# and "01#[ The source terms pro! portional to 9P give rise to decay[ u and B will\ therefore\
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
exhibit a hybrid of one!dimensional propagation as well as decay away from the source in the near _eld region\ while at large distances from the source only Alfvenic _eld!aligned perturbations will remain[ In the case of no source currents\ the driving terms in eqns "00# and "01# vanish[ In that case u and B are propagated unattenuated along the magnetic _eld\ with the magnetic _eld lines acting as {plucked strings| and the transverse and longi! tudinal motions are coupled by the divergence equations "div B 9 and div u 9#[ Furthermore\ straightforward operations on eqn "01# yields the following equations for the magnetic _eld] LA91B> −
B9 1 9> div Ms r9
LA curl> B −
B9 9> curl> Ms r9
B9 9> curl Ms−LAJs m9r9
frame Ss of the source[ For the sake of de_niteness and without too much loss of generality\ we consider a steady current source relative to which the plasma moves with velocity u9 "taken to be along the z!axis# perpendicular to the ambient magnetic _eld B9 "taken parallel to the x! axis#*see Fig[ 0[ In order to e}ect the transformation from the plasma frame Sp to the frame Ss of the source\ in which the plasma moves relative to the source which is at rest\ we use the following Galilean transformation] 1 1 : u9 = 9 u9 1t 1z
"06#
"03#
Then the Alfven wave operator LA ðeqn "02#Ł becomes
"04#
LA : V1AL?A^ L?A 0 M1
Thus\ 91B> and curl> B "i[e[ the parallel plasma current# are propagated along the magnetic _eld[ In addition\ taking the curl of eqn "01# and using Ampere|s law\ we may obtain equations describing the propagation of the plasma current perturbation J\ namely LAJ
436
"05#
Equations similar to eqns "03# and "04# for B may also be derived for the plasma velocity u by making use of the linearized form of Faraday|s law ðeqn "8#Ł[ To summarize this section then\ the presence of a _rst! order source current "Js# perturbation in addition to any induced plasma current perturbations "J# contributes a _rst!order source term "Ms# to the momentum equation[ The parallel momentum is una}ected by this form of source term\ while the perpendicular momentum com! ponents su}er an in~uence depending on the form of the source current Js[ Ms acts to drive perturbations in the magnetic _eld B and plasma velocity u\ which propagate away from the source region[ In particular\ the parallel curl of the magnetic _eld "curl> B#\ the Laplacian of the parallel magnetic _eld perturbation "91B>#\ the parallel vorticity "curl> u#\ as well as the plasma current per! turbation "J# are propagated one!dimensionally by the Alfven mode[ In contrast\ the propagation of the total magnetic _eld "B# and velocity "u# perturbations is hybrid in nature\ consisting of a one!dimensional Alfvenic propagation as well as decay away from the source region[
11
11 − 1z1 1x1
"07#
in which M 0 u9:VA is the Alfven Mach number[ Making these transformations in the wave equations developed in the previous section in the frame Sp of the plasma yields the corresponding equations describing the stationary perturbations in the frame Ss of the source[ Moreover\ in Ss the perturbation electric _eld E is derivable from a scalar potential f and is given by E −u9×B−u×B9 "u9By\ −u9Bx−uzB9\ uyB9# −9f
"08#
where u 0 "ux\ uy\ uz# and B 0 "Bx\ By\ Bz# are the plasma velocity and magnetic _eld perturbations\ respectively[ The equation for f follows from the y!component of the linearized momentum equation "7# combined with eqn "08#\ namely
2[ Stationary waves in a moving plasma generated by a moving current source In this section we use the above development to for! mulate the wave equations characterizing the stationary wave system generated by a moving current source in the
Fig[ 0[ The relative orientations of the plasma ~ow u9 and the back! ground magnetic _eld B9[
437
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
B9 1P L?Af −Msy m9m9 1y
"19#
where L?A is the Alfven wave operator in Ss ðeqn "07#Ł[ This equation displays the Alfven wave operator on the left!hand!side and shows that f is driven by the "y!deriva! tive of the# total pressure P and by the y!component of the momentum source term[ Thus\ f will be hybrid in nature\ exhibiting both Alfvenic characteristics\ as well as that arising from the pressure term[ The latter is a Laplacian decay away from the source region\ as we discussed in the previous section[ Regarding the nature of the current source\ we shall simply assume a given current distribution Js"r#[ For the case of a conducting body and a magnetized plasma in relative motion with the con_guration de_ned in Fig[ 0\ we expect that the motional electric _eld −u9×B9 −u9B9y¼ will drive a current in the y!direction on the body[ Thus\ we take Js Jsy¼[
As in McKenzie "0880# we assume that the stationary wave pattern which emerges at large distances from the body should be well!approximated by integrating the Green|s function solution for a point source over a source distribution of _nite size[ Therefore\ we consider the spec! i_c case of a d!function source current^ i[e[ we take "10#
corresponding to a point source of strength J9 pointing in the y!direction anti!parallel to the motional electric _eld "−u9×B9#[ Note that if the source is of purely induc! tive origin\ the sign of J9 should be reversed[ With this explicit form of Js we can derive exact ana! lytic solutions for P\ curl> B\ B\ J\ curl> u\ u\ as well as f[ The details of the calculations are provided in the Appendix for reference[ Here we present the _nal solu! tions and describe their behaviours in the context of the model[ The variation of the total pressure P is described by eqn "09# in both Sp and Ss[ The solution to this equation may be developed using Fourier analysis to yield
01
B9J9 1 0 ^ −z m 9 3p 1z r
P 3
Bx\y −
"11#
with r 0 zx1¦y1¦z1[ Thus\ P decays away from the source region "origin# in a dipole!like fashion\ both upstream and downstream in the ~ow[ This is shown in Fig[ 1 in the plane containing the source\ B9 and u9 "i[e[ the y 9!plane#[ Thus\ P varies quickly close to the source\ while in the far!_eld region P is approximately constant "zero#[ Hence\ it is only far from the source that the particle pressure p approximately balances the magnetic pressure B9Bx:m9[
$
%
m9J9 1 1 1 \ log ð"r−Mz:k#1−x1:k1Ł 7pMk 1x 1x 1y "12#
m9J9
"IBz0¦IBz1#
"13#
M1 d"y#ðd"z¦#¦d"z−#Ł 3
"14#
Bz −
M1
where IBz0
IBz1 −
3[ Properties of the stationary wave pattern
Js J9d"x#d"y#d"z#y¼
We can now feed this compact solution for P into the source frame version of eqn "01# for B and use a method identi_ed by McKenzie "0880# to solve for the x! and y!components of B\ while the solution for Bz may be developed using Fourier analysis[ The details are explai! ned in the Appendix[ The solutions may be written as follows]
0 7pMg1
6
1pd"y#ðd"z¦#−d"z−#Ł
−
0 11 ðlog "y1¦z1¦:g1#−log "y1¦z1−:g1#Ł 1g 1y1
¦
0 1 0 ^ MŁ0 3p 1x r
01
7 "15#
in which r is as de_ned above and z2 0 x2z:M
"16#
k 0 z0¦M1
"17#
g 0 z0¦M−1
"18#
The solutions for Bx and By may also be developed using Fourier analysis of the form of eqn "01# corresponding to Ss[ It is only possible\ however\ to obtain exact analytic solutions in this way for very sub! and very super!Alfv! enic ~ows "i[e[ for M ð 0 and M Ł 0\ respectively#[ Whereas\ the solutions in eqns "12# are valid in all ~ow situations[ Nevertheless\ such a Fourier analysis high! lights the characteristics of the variation of Bx and By\ which are embodied in a less apparent fashion in the solutions in eqns "12#[ The Bx perturbation consists of two contributions\ one of which is singular on the Alfven lines "z2 9\ y 9# and the other which corresponds to the Laplacian decay away from the source "origin# in a two!dimensional dipole!like fashion[ A knowledge of Bx and P allows the determination of the particle pressure p P−B9Bx:m9[ This is shown in Fig[ 2\ where it is seen that p is singular "and negative# on the Alfven lines as well as at the origin[ In the near! _eld region of the origin\ p decays away from the origin in a dipole!like fashion[ The By perturbation is zero along the x 9 and y 9
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
438
Fig[ 1[ The variation of the total pressure P on the plane y 9 in the near!_eld region of the source\ which is located at the origin[ Note that the units of pressure are B9J9:3p[
Fig[ 2[ The variation of the particle pressure p on the y 9!plane for M 9[0[ For visual reasons\ we plot −p "in units of B9J9:3p#[
planes and is anti!symmetric along the x! and y!direc! tions[ It consists of two _nite disturbances of opposite polarity centred on lines parallel to the Alfven lines and displaced away from y 9 on the y!axis[ This is indicated in Fig[ 3[ From eqn "13# it is seen that Bz is singular on the Alfven lines and decays away from them in a two!dimensional dipole!like fashion[ Further\ Bz decays away from the source in a dipole!like fashion\ by virtue of the term proportional to 1r−0:1x[
Solutions for the plasma current J in Ss may be straightforwardly developed from the Sp form of eqn "05# using Fourier analysis[ The results show that Jx
J9 d?"y#ðd"z¦#−d"z−#Ł 3M
Jy − Jz 9
J9 d"y#ðd?"z¦#−d?"z−#Ł 3M
"29# "20# "21#
449
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
Fig[ 3[ The variation of By on the y 0!plane for M 9[4[ The upper panel is a surface plot of By on the y 0!plane[ The lower two panels are sections through this surface] the bottom left!hand panel indicates the z 4 section through this surface^ while the bottom right!hand panel represents the x 09 section[ In all panels the units of By are m9J9:7pk1[
where the prime on the d!functions indicate the _rst derivative of the function[ Thus\ the parallel current Jx curl> B:m9 and Jy are propagated down the Alfven lines as two pulses of opposite polarity[ The derivatives of the d!functions in these solutions means that current loops in the xy!planes centred on each characteristic are propagated down the characteristics with opposite cir! culations[ This is depicted in Fig[ 4\ which is an xy!plane in Ss for some z × 9[ The large solid dots labelled z¦ 9 and z− 9 are the Alfven characteristics[ The direction of the current circulations around these are indicated by the arrowheads on the circle around each characteristic[ Solutions for the plasma velocity perturbations u may be derived in a similar way to those obtained for the magnetic _eld B[ We simply quote the solutions here ux\y −
$
%
m9u9J9 1 1 1 \ log ð"r−Mz:k#1−x1:k1Ł 7pB9Mk 1z 1x 1y "22#
Fig[ 4[ The current loops around the Alfven characteristics "labelled z¦\− 9#[ The arrows on the circles around the characteristics indicate the direction of ~ow of the current[
uz − where
m9u9J9 B9M1
"Iuz0¦Iuz1#
"23#
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
440
Iuz0 −03 d"y#ðd"z¦#¦d"z−#Ł
"24#
Iuz1 I1a¦I1b
"25#
tials in accordance with its de_nition E −9f[ Thus\ we see that the electric _eld is dipole!like\ centred on the Alfven lines[
"26#
4[ Conclusion
with I1a
I1b
01 6
0 1 0 ^ MŁ0 3p 1z r 0 7pk1
1pd"y#ðd"z¦#¦d"z−#Ł
−
0 11 s log "y1¦z1i :g1# 1g 1y1 i ¦\−
7
"27#
k\ g and r are as de_ned earlier[ These are clearly struc! turally similar to the solutions for B ðeqns "12# and "13#Ł[ The ux perturbation is singular on the Alfven lines and of opposite sign on each characteristics\ ~owing towards the source along the z¦ 9 characteristic and away from the source along the z− 9 characteristic[ The uy perturbation is similar to the By perturbation ðeqn "12# and Fig[ 3Ł[ It is anti!symmetric in y and zero on the y 9!plane[ Its variation is indicated in Fig[ 5[ The uz disturbance is singular on the Alfven lines and decays away from them in a two!dimensional dipole!like fashion[ It is also singular at the origin[ Further\ Fourier analysis may be used to derive an expression for the parallel vorticity "curl> u# whose wave equation is similar to that for curl> B ðeqn "04#Ł[ This may be written as curl> u
m9u9J9 3M1B9
d?"y#ðd"z¦#¦d"z−#Ł
"28#
Thus\ the parallel vorticity is propagated down the Alfven lines as four pulses] two above the y 9!plane of positive vorticity and two of negative vorticity below this plane[ Finally\ feeding the solution for P ðeqn "11#Ł into the wave equation describing the behaviour of the electric potential f ðeqn "19#Ł we may derive an exact analytic solution for f using the method of McKenzie "0880#\ in a manner similar to that used for Bx "see the Appendix#[ Thus\ we derive that f
m9u9J9 1 log ð"r−Mz:k#1−x1:k1Ł 7pMk 1y
"39#
where k and r are de_ned as above[ This is singular on the Alfven lines and decays away slowly from them[ Figure 6 shows the equipotential surfaces corresponding to this form for f[ They are cylinders of opposite potentials above "y × 9\ positive# and below "y ³ 9\ negative# the y 9!plane aligned with the Alfven characteristics and bent at the origin[ The electric _eld lines are perpendicular to these surfaces and point from higher to lower poten!
We have analysed the stationary wave system gen! erated by the inductive interaction between a conducting body and a magnetised\ incompressible plasma in relative motion in the framework of a non!dispersive MHD for! mulation[ As is well!known\ an incompressible MHD plasma supports only the shear Alfven mode of propa! gation[ The magnetoacoustic modes of compressible MHD have either become evanescent "fast mode# or coincident with the shear mode "slow mode#[ This behav! iour is evident in the solutions we derive\ which on the whole exhibit singular behaviour on the Alfvenic charac! teristics "lines# and Laplacian decay away from the source region[ Consequently\ the analysis presented here adequately describes the incompressible response of the inductive interaction\ but is not appropriate for com! parison with the compressive behaviour in such a system[ Furthermore\ we consider a purely inductive interaction\ achieved by way of the use of a source current[ We leave for the subject of future work\ processes such as mass loading\ which provide\ in particular\ momentum source terms of di}erent character to that associated with an inductively generated source current as we describe in this article[ Thus\ our results are applicable to interactions in high b plasmas in which the e}ects of mass loading are not signi_cant[ In order to make the system analytically tractable\ we have prescribed a source current\ rather than calculating it self!consistently and furthermore\ we have assumed a point source current[ Thus\ we present the fully analytic Green|s function solutions for the ~ow\ magnetic and electric _elds as well as the accompanying current per! turbations of the interaction[ It is noted that these Green|s functions may be integrated over extended source current distributions such as would be expected in the various applications to the IoÐJupiter system or the experimental tethered satellite system[ Previous papers have discussed the non!dispersive ðe[g[ McKenzie et al[\ 0882# and dispersive "e[g[ Barnett and Olbert\ 0875^ Estes\ 0877^ Woodward and McKenzie\ 0883a\ b^ Sanmartin and Estes\ 0886# Alfvenic perturbations\ but do not pro! vide exact analytic solutions for all _eld variables\ as we do in this article[ The presence of a source current "Js# in this interaction provides a source of momentum for the plasma by virtue of the Lorentzian interaction "−Js×B9# between this source current and the background magnetic _eld "B9# which is frozen into the ~ow[ Consequently\ only the perpendicular momentum is a}ected[ Hence\ only source
441
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
Fig[ 5[ The variation of uy on the y 0!plane for M 9[4[ The upper panel is a surface plot of uy on the y 0!plane[ The lower two panels are sections through this surface] the bottom left!hand panel indicates the z 4 section through this surface^ while the bottom right!hand panel represents the x 09 section[ In all panels the units of uy are m9u9J9:3pk1B9[
currents perpendicular to the background magnetic _eld are of signi_cance in this type of interaction "Estes\ 0877#[ A further consequence of the source current and the incompressibility of the plasma is that the total pressure P "plasma plus magnetic# is not constant\ but decays away from the source current region in a dipole!like fashion[ This implies that in the near!_eld\ the particle pressure is not balanced by the magnetic pressure\ while in the far!_eld region this balance becomes better approximated[ This is in contrast to the case of no source currents\ where P is constant "Cowling\ 0846#[ The plasma current\ parallel circulation of the mag! netic _eld\ parallel vorticity and the Laplacians of the parallel magnetic _eld and plasma velocity perturbations are propagated as Alfven waves\ which are swept down! stream in the ~ow of the plasma to form the well!known Alfven lines "e[g[ Drell et al[\ 0854^ McKenzie\ 0880^ McKenzie et al[\ 0882#[ The pure magnetic _eld and
plasma velocity perturbations as well as the electric potential perturbation "in the frame of the source# are driven by the total pressure as well as directly by the momentum source term and therefore\ possess both Alfv! enic propagation characteristics and decay away from the source region[ We consider the speci_c case of a plasma ~ow per! pendicular to the magnetic _eld\ as would be found in the IoÐJupiter system\ for example[ The motional electric _eld induced in the conducting body would then drive a current perpendicular to both the ~ow and magnetic _eld[ Consequently\ we choose to impose a d!function source current parallel to this motional electric _eld and this has the following consequences for the perturbation _elds[ The components of the magnetic _eld and plasma vel! ocity perturbations in the plane of the background ~ow and magnetic _eld "i[e[ Bx\z and ux\z# are hybrid in nature\ being singular on the Alfven lines although not narrowly
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
442
Fig[ 6[ The equipotential surfaces for M 9[4 "a# Four surfaces in the shape of cylinders bent at the origin have been drawn] the two drawn in solid lines are for positive potentials^ while the adjacent two surfaces drawn in small!dashed line!type are for negative potentials[ The positive and negative surfaces are on either side\ in the y!direction of the Alfven characteristics "z2 9\ y 9#] i[e[ positive "negative# potential surfaces are on the y × 9 "y ³ 9# side of the y 9!plane[ Furthermore\ surfaces of higher absolute potential are smaller cylinders lying against the Alfven characteristics] these are represented by the smaller inner cylinders in the diagram[ "b# This is an z 0!section through the equipotential surfaces sketched in part "a#[ The solid lines indicate positive potentials\ while the dashed lines are for negative potentials[ The values of the potentials are indicated by the values for f "in units of m9u9J9:3pMk# on the diagram[
con_ned to them\ as well as su}ering a dipole!like decay away from the source in the near!_eld region[ The com! ponents "By and uy# perpendicular to the plane of the background ~ow and magnetic _eld are anti!symmetric about the y 9!plane and hence\ are zero on the y 9!
plane[ In this way the plasma ~ow and magnetic _eld\ which is frozen into the ~ow\ are de~ected around the source:body[ Finally\ the electric _eld is singular on the Alfven lines and of dipole nature in planes perpendicular to the Alfven lines[
443
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
Acknowledgements
where k is a constant[ This requires that
J[F[McK[ wishes to thank FRD!CSIR for their partial support and T[I[W[ acknowledges _nancial support from the FRD\ South Africa\ as well as PPARC\ U[K[
f¦
Appendix] Solution for the magnetic _eld B perturbation The source frame wave equation for B is given by eqn "01# under the transformation "07#\ where P is given by eqn "11# and Ms B9Jsz¼ "Js J9d"x#d"y#d"z##[ The x! and y!components of this equation then yield
0 1 0 1
11 1 1 −0 \ r 1x 1z 1x 1y
0 0
"A0#
1
m9 1 1P B J− B9 1x 9 s 1z
1
11 1 1 1 Bx C 1 r−0 ¦M 1x 1z 1x 1z
1
"A2#
1
1 1 log "r¦f¦"x\ z## ¦M 1x 1z
6
0
1 7
x¦Mz 1 0 1 f¦ ¦ ¦M r¦f¦ r 1x 1z
in which f¦"x\ z# is an arbitrary function of x and z[ Now chose f¦"x\ z# such that the right!hand!side of eqn "A2# equals k r
"A6#
0
C 11 1 1 1 Bx ¦M log "r¦f−# 1x 1z k 1x1 1z
1
"A7#
Boundary conditions at =z= and causality con! siderations require the choice of the lower sign in "A5# "i[e[ k −z0¦M1 −=k=# and the upper sign in "A7# "i[e[ k =k=#[ Subtracting "A5# from "A7# and factoring out 1:1z yields the fundamental solution for Bx] C 11 log ð"r−Mz:k#1−x1:k1Ł 1Mk 1x1
"A8#
Solution for Bz usin` Fourier analysis The Fourier transform of eqn "A1# for Bz may be written Bzk −
m9J9
ikx
M1 "k1z −k1x :M1#
6
0¦
k1z
7
"k1x ¦k1y ¦k1z #
"A09#
where we denote the Fourier transform of Bz by Bzk[ Bz may then be reconstructed by calculating the inverse transformation\ namely
First\ note that
0
x−Mz ^ k 2z0¦M1 k
in which k 0 z0¦M1[ Similarly\ we can construct the solution ðeqn "12#Ł for B y[
Equations "A0# may be solved using the method adopted by McKenzie "0880# in his solution for the elec! tric potential[ Consider the equation for Bx and write it in the following form]
10
"A5#
since operating on "A5# with 1:1x¦M 1:1z retrieves the original eqn "A2#[ Similarly\ we can choose
"A1#
Solutions for Bx\y
0
1
Bx
J9 11r−0 1 m9 Js¦ 1x 3p 1z1
1 1 −M 1x 1z
0
C 11 1 1 1 Bx −M log "r¦f¦# 1x 1z k 1x1 1z
so that
in which C 0 −"m9J9:3p#\ r 0 zx1¦y1¦z1 and L?A is the Alfven wave operator in Ss ðeqn "07#Ł[ The z!com! ponent may be written as L?ABz
"A4#
Hence\ the original eqn "A2# is satis_ed if we have
f−"x\ z#
m9 1 1 1 L?A"Bx\ By# − P \ B9 1x 1x 1y −C
x¦Mz ^ k 2z0¦M1 k
"A3#
Bz
0 "1p#
2
g g g
−
−
−
×Bzk exp ði"kxx¦kyy¦kzz#Ł dkx dky dkz
"A00#
To this end\ let IBz0 0
i "1p#
g g g
2
−
kx
−
1 1 1 − "kz −kx :M #
×exp ði"kxx¦kyy¦kzz#Ł dkx dky dkz
"A01#
T[I[ Woodward\ J[F[ McKenzie : Planetary and Space Science 36 "0888# 434Ð444
IBz1 0
i
g g g
2
"1p#
−
−
kxk1z
1 x
1 y
1 z
1 z
1 x
I1b
1
"k ¦k ¦k #"k −k :M #
−
×exp ði"kxx¦kyy¦kzz#Ł dkx dky dkz
"A02#
"IBz0¦IBz1#
"A03#
Then Bz −
m9J9 M1
M1 d"y#ðd"z¦#¦d"z−#Ł 3
"A04#
where z2 0 x2z:M are the Alfven characteristics\ as before[ The determination of IBz1 is a little more complicated[ First\ using Cauchy|s residue theorem we evaluate the kz! integral noting three poles at kz 2kx:M and kz izk1x ¦k1y \ to obtain "A05#
IBz1 I1a¦I1b where I1a −
I1b
0 1
05p M
i 7p
1
g
g
k1x
−
1 1 x
"g k ¦k #
kxzk1x ¦k1y
−
1 y
1 1 x
1 y
"g k ¦k #
"eikxz¦−eikxz−# eikyy dkx dky 1
1
e−zzkx ¦ky ei"kxx¦kyy# dkx dky
with g1 0¦M−1[ Noting that k1x "g1k1x ¦k1y #
0 g1
0
0−
k1y "g1k1x ¦k1y #
1
I1a may be split into two integrals over the two terms in the above expression[ The _rst one is straightforward\ while the second can be cast in the form of a standard sine transformation "Erdelyi et al[\ 0843#[ Thus\ we obtain I1a −
−
0 7pMg1
6
1pd"y#ðd"z¦#−d"z−#Ł
0 11 ðlog "y1¦z1¦:g1#−log "y1¦z1−:g1#Ł 1g 1y1
7
"A06#
To evaluate I1b\ we _rst transform to polar coordinates "kx\y k"cos f\ sin f## to obtain I1b
0 1 7p1 1x
"A08#
This then con_rms solution "13# of the main text for Bz[ References
Cauchy|s residue theorem may be used straight! forwardly in IBz0 to obtain IBz0
01
0 1 0 ^ r 0 zx1¦y1¦z1M Ł 0 3p 1x r
444
1p
9
9
gg
e−kz eik"xcosf¦ysinf# "g1 cos1 f¦sin1 f#
df dk
"A07#
In the limit of M Ł 0\ g1 ¹ 0[ In this case the integral over f in I1b can be written in terms of the zero!order Bessel function "Erdelyi et al[\ 0843#\ while the following integral over k is of standard form[ Hence\ we obtain
Barnett\ A[\ Olbert\ S[\ 0875[ Radiation of plasma waves by a con! ducting body moving through a magnetized plasma[ J[ Geophys[ Res[ 80\ 09\006[ Belcastro\ V[\ Veltri\ P[\ Dobrowolny\ M[\ 0871[ Radiation from long conducting tethers moving in the near earth environment[ Nuovo Cimento\ Soc[ Ital[ Fis[ C 4 "4#\ 420[ Cowling\ T[G[\ 0846[ Magnetohydrodynamics[ Interscience\ New York[ Dobrowolny\ M[\ Veltri\ P[\ 0875[ Structure of Alfven wings associated with a conductor moving through a magnetoplasma[ Astron[ Astro! phys[ 056\ 068[ Drell\ S[D[\ Foley\ H[M[\ Ruderman\ M[A[\ 0854[ Drag and propulsion in the ionosphere] an Alfven engine in space[ J[ Geophys[ Res[ 69\ 2020[ Dungey\ J[W[\ 0857[ In] Matsushita\ S[\ Campbell\ W[H[ "Eds[#\ Physics of Geomagnetic Phenomena[ Academic Press\ New York\ p[ 802[ Erdelyi\ A[\ Magnus\ W[\ Oberhettinger\ F[\ Tricomi\ F[G[\ 0843[ Tables of Integral Transforms\ vol[ 0[ McGraw!Hill\ New York[ Estes\ R[D[\ 0877[ Alfven waves from an electrodynamic tethered sat! ellite system[ J[ Geophys[ Res[ 82\ 834[ Goldreich\ P[\ Lynden!Bell\ D[\ 0858[ Io\ a Jovian unipolar inductor[ Astrophys[ J[ 045\ 48[ Goertz\ C[K[\ Deift\ P[A[\ 0862[ Io|s interaction with the magnetos! phere[ Planet[ Space Sci[ 10\ 0288[ Hastings\ D[E[\ Barnett\ A[\ Olbert\ S[\ 0877[ Radiation from large space structures in low Earth orbit with induced alternative currents[ J[ Geophys[ Res[ 82\ 0834[ Lighthill\ M[J[\ 0859[ Studies on magneto!hydrodynamic waves and other anisotropic wave motions[ Phil[ Trans[ Roy[ Soc[ London\ Ser[ A 141\ 280[ Linker\ J[A[\ Kivelson\ M[G[\ Walker\ R[J[\ 0877[ An MHD simulation of the ~ow past Io] Alfven and slow mode perturbations[ Geophys[ Res[ Lett[ 04\ 0200[ Linker\ J[A[\ Kivelson\ M[G[\ Walker\ R[J[\ 0880[A three!dimensional MHD simulation of the ~ow past Io[ J[ Geophys[ Res[ 85\ 10\926[ McKenzie\ J[F[\ 0880[ Stationary MHD waves generated by a source in a moving plasma[ J[ Geophys[ Res[ 85\ 8380[ McKenzie\ J[F[\ Woodward\ T[I[\ Inhester\ B[\ 0882[ Magnetoacoustic and Alfven potentials for stationary waves in a moving plasma[ J[ Geophys[ Res[ 87\ 8190[ Neubauer\ F[M[\ 0879[ Nonlinear standing Alfven wave current system at Io] theory[ J[ Geophys[ Res[ 74\ 0060[ Sanmartin\ J[R[\ Estes\ R[D[\ 0886[ Alfven wave far!_eld from steady current tethers[ J[ Geophys[ Res[ 091\ 03\514[ Southwood\ D[J[\ Kivelson\ M[G[\ Walker\ R[J[\ Slavin\ J[A[\ 0879[ Io and its plasma environment[ J[ Geophys[ Res[ 74\ 4848[ Woodward\ T[I[\ McKenzie\ J[F[\ 0882[ Stationary MHD structures[ Planet[ Space Sci[ 30\ 106[ Woodward\ T[I[\ McKenzie\ J[F[\ 0883a[ Stationary MHD waves modi! _ed by Hall current coupling*I[ Cold compressible ~ow[ Planet[ Space Sci[ 31\ 352[ Woodward\ T[I[\ McKenzie\ J[F[\ 0883b[ Stationary MHD waves modi! _ed by Hall current coupling*II[ Incompressible ~ow[ Planet[ Space Sci[ 31\ 370[ Wright\ A[N[\ 0876[ The structure of ULF waves produced by a tethered satellite system[ Astron[ Astrophys[ 075\ 243[ Wright\ A[N[\ Schwartz\ S[J[\ 0889[ The equilibrium of a conducting body embedded in a ~owing plasma[ J[ Geophys[ Res[ 84\ 3916[ Wright\ A[N[\ Southwood\ D[J[\ 0876[ Stationary Alfvenic structures[ J[ Geophys[ Res[ 81\ 0056[