Astrophys Space Sci (2012) 339:19–23DOI 10.1007/s10509-012-1010-0

L E T T E R

Relaxed magnetic field structures in multi-ion plasmas

M. Iqbal · P.K. Shukla

Received: 27 September 2011 / Accepted: 31 January 2012 / Published online: 16 February 2012© Springer Science+Business Media B.V. 2012

Abstract The steady state solution of a three species mag-netoplasma is presented. It is shown that relaxed magneticfield configuration results in a triple curl Beltrami equationwhich permits the existence of three structures. It is the con-sequence of inertial effects of the plasma constituents. Oneof the three vortices is of large scale while the remainingtwo relaxed structures are of small size of the order of elec-tron skin depth. The magnetic field profiles are given fordifferent Beltrami parameters. The study could be helpful tounderstand large magnetic field structures in three speciesplasmas found in space and laboratory.

Keywords Relaxation · Beltrami fields · Triple Beltramifields · Space plasma

1 Introduction

Multi-ion plasmas are found in abundance in space andastrophysical objects. Ionosphere and magnetosphere ofEarth, solar wind, bow shock in front of the magnetopauseboundary layers, heliosphere, Saturn’s magnetosphere andcometary tails contain multi-ion plasmas (Shemansky andHall 1992; Hultqvist et al. 1999; Stasiewics 2004; Schwennand Marsch 1990). The creation of fullerene and electron

M. Iqbal (�)Department of Physics, University of Engineeringand Technology, Lahore 54890, Pakistane-mail: iqbal_uot@yahoo.com

P.K. ShuklaRUB International Chair, International Centre for AdvancedStudies in Physical Sciences, Institut für Theoretische Physik,Fakultät für Physik und Astronomie, Ruhr-Universität Bochum,44780 Bochum, Germany

positron pairs in the laboratory have substantially increasedthe importance of multi-ion plasmas. The multi-ion plasmashave been studied extensively to investigate different modessuch as nonlinear Alfvén waves (Faria et al. 1998), acousticsolitary waves (Verheest et al. 2008), solitons (Sauer et al.2001) and electrostatic modes (Vranjes et al. 2008).

In the present work, we investigate the self-organizationof multi-ion magnetized plasmas. Self-organization is a uni-versal phenomenon and magnetofluids also self-organize tosome preferred states. Self-organization of magnetofluids isalso called as relaxation. It has been shown by the semi-nal work of Woltjer and Taylor (Woltjer 1958; Taylor 1974,1986) that plasmas relax to force-free states. Such type ofstates occur when the field and its vorticity are parallel. Themagnetic field structures observed in Reversed Field Pinchwere explained using the Taylor’s concept of relaxation. Butthe flow and pressure gradients being essential features of allpractical plasmas are missing in the relaxation theory pro-posed by Taylor. Employing different types of constraints,the investigators in the field of plasma relaxation have pre-sented different models mostly based on variational princi-ple, which lead to relaxed states characterized by strong flowand pressure gradients. One of such models presented byMahajan and Yoshida (Mahajan and Yoshida 1998; Yoshidaand Mahajan 2002) is mathematically very simple and cov-ers a large variety of solutions. This is a generalized modeland a variety of relaxed structures that is paramagnetic, aswell as diamagnetic could be obtained. Under certain condi-tions, one can see the glimpse of force free relaxed states andon the other hand, one can construct a non-force free, highβ relaxed states of a plasma system (Yoshida et al. 2001;Iqbal et al. 2001; Iqbal 2005). The relaxed state of this Hallmagnetohydrodynamics (HMHD) model is represented bydouble curl Beltrami equation. It has also been investigatedthat different multi-species plasma systems could relax to a

20 Astrophys Space Sci (2012) 339:19–23

state given by double curl Beltrami equation when the massof lighter constituents is ignored relative to heavier ones(Shukla and Mahajan 2004a, 2004b; Shukla 2004, 2005).

2 Mathematical model

In this letter, relaxation of three species plasma containingelectrons and two ions is considered. The electron equationof motion is given by

∂

∂t

(ve − eA

mec

)= ve ×

(∇ × ve − eB

mec

)

+ ∇(

eφ

me

− v2e

2− pe

ρe

), (1)

where E and B are the electric field and magnetic field re-spectively, ve, me, and ne are the velocity, mass and numberdensity of electrons, e is the charge of electrons, c is thespeed of light in vacuum, pe = neTe is the pressure, andTe is the temperature of electrons, φ and A are the scalarand vector potential, respectively and ρe = mene is the massdensity of electrons. The macroscopic evolution equationsof the two ion species are given by

∂V α

∂t+ zαe

mαc

∂A

∂t= V α ×

(∇ × V α + zαeB

mαc

)

− zαe

mα

∇(

φ + mαV 2α

2zαe

)− ∇pα

ρα

, (2)

where α is equal to l for the light ions and equal to h

for the heavy ions. Pressure of the ions with temperatureTα is given by pα = nαTα . The relations (vα · ∇)vα =12∇v2

α − vα × (∇ × vα) and E = −∇φ − c−1∂A/∂t areused to get (1) and (2). To introduce the dimensionless vari-ables, we normalize magnetic field to an arbitrary magneticfield B0, velocities to V 0 = B0/

√4πnee2, length to electron

skin depth λe = c√

me/4πnee2, time to inverse of electrongyrofrequency ω−1

ce = mec/eB0, pressure to B20/4π , vector

potential A to V 0mec/e and scalar potential φ to B20/4πnee.

The normalized equations of motion for electrons and α-species, respectively, read as

∂

∂t(ve − A) = ve × (∇ × ve − B)

+ ∇(

φ − v2e

2− pe

), (3)

∂

∂t(V α + μA) = V α × (∇ × V α + μB)

− ∇(

μφ + V2α

2+ ρe

ρα

pα

), (4)

where μ = zαme/mα . We are considering a quasineutral in-compressible three species plasma and all the species follow

the barotropic pressure [pα = pα(ρα)]. Taking curl of elec-trons and ions equations of motion, respectively, and omit-ting the normalized sign on variables, we obtain

∂

∂t(∇ × ve − B) = ∇ × [

ve × (∇ × ve − B)], (5)

∂

∂t(∇ × V α + μB) = ∇ × [

V α × (∇ × V α + μB)], (6)

where ∇ × A = B . The simplest possible steady state solu-tions of (5) and (6) are given as

∇ × ve − B = ave, (7)

∇ × V α + μB = bV α. (8)

These equations also represent the Beltrami condition thatis the flow and generalized vorticities are parallel. The elec-tron fluid velocity, using definition of current density J =∑

αzαnαeV α − eneV α , and ∇ × B = 4πc

J , is given by

ve = zlnl

ne

V l + zhnh

ne

V h − c

4πene

∇ × B, (9)

where zl (zh) represent the charge states of light (heavyions), nl (nh) represent the number densities of the light(heavy) ions, and V l (V h) represent the fluid velocities ofthe light (heavy) ions, respectively. Putting value of ve from(9) into (7), we obtain

∇ ×∑α

zαnα

ne

V α − ∇ × ∇ × B − B

= a∑α

zαnα

ne

V α − a∇ × B. (10)

Solving (10) and (8), we obtain

∇ × ∇ × B − a∇ × B + kB = (b − a)∑α

zαnα

ne

V α, (11)

where k = 1+∑α

z2αnαme

mαne. Putting value of

∑α

zαnα

neV α from

above equation into (10), V α is eliminated and the magneticfield B satisfies the equation written as follows

∇ × ∇ × ∇ × B − α1∇ × ∇ × B

+ α2∇ × B − α3B = 0, (12)

where α1 = a + b, α2 = ab + 1 + ∑α

z2αnαme

mαneand α3 =

b + a∑

αz2αnαme

mαne. Equation (12) is called triple curl Beltrami

equation. Superposition of three different Beltrami fields re-sults in triple curl Beltrami equation. Equation (12) can alsobe written as

(curl − λ1)(curl − λ2)(curl − λ3)B = 0, (13)

where the derivative “∇×” is written as curl. λ1, λ2 and λ3

are scale parameters and represent the eigenvalues of curloperator. The Beltrami parameters (a and b) and eigenvaluesare related by α1 = λ1 +λ2 +λ3, α2 = λ1λ2 +λ2λ3 +λ1λ3,and α3 = λ1λ2λ3. The operators are commutative, therefore,

Astrophys Space Sci (2012) 339:19–23 21

general solution to (13) can be written as a linear sum ofthree Beltrami fields. Let us consider a Beltrami field Gj

(where j = 1,2,3) and it satisfies the following conditions{∇ × Gj = λjGj (in ),

n · Gj = 0 (on ∂).(14)

Then

B = C1G1 + C2G2 + C3G3, (15)

solves (13). C′s are arbitrary constants and represent theamplitude of fields. The Beltrami field can be given byChandrasekhar-Kendall functions (Chandrasekhar andKendall 1957) in cylindrical geometry whereas in slab ge-ometry, the Beltrami field can be represented by ABC flow(Yoshida 2010). The eigenvalues of the curl operator (λj )are the solutions of the cubic equation

λ3 − α1λ2 + α2λ − α3 = 0. (16)

The eigenvalues may be real or complex (Yoshida and Giga1990) and are given by

λ1 = s − t + a + b

3, (17)

λ2 = −1

2(s − t) − i

√3

2(s + t) + a + b

3, (18)

λ3 = −1

2(s − t) + i

√3

2(s + t) + a + b

3, (19)

where t = (q2 + √

D)1/3, s = −(q2 − √

D)1/3, p = α2 − α21

3

and q = α1α23 − 2α3

127 − α3. All the λ′s are real and at least

two are equal, when D = q2

4 + p3

27 = 0. All the roots willbe real if D < 0. When D > 0, then one of the eigenval-ues (λj ) will be real while the other two will be complex

conjugate pair. Dimensionally, the eigenvalues are inverseof length. Therefore, a large eigenvalue will correspond to asmall scale structure and the small eigenvalue will give riseto a large scale structure. As we have three eigenvalues inthe system, it is therefore possible to obtain three vortices,one will correspond to system size and others will representmicroscopic structures of the order of electron skin depth. Ifwe ignore the mass of electrons relative to ions, the systemself-organizes to two vortices (Mahajan and Yoshida 1998;Shukla 2005). This shows that inertia plays a significant rolein creation of vortices (Iqbal et al. 2008) when the systemgets relaxed.

3 Analytical solution and field profiles

The Beltrami field in simple slab geometry is given as

B =⎛⎝ 0

C sinλx

C cosλx

⎞⎠ . (20)

The triple Beltrami field being the linear superposition ofthree different Beltrami fields as given in (15), therefore, canbe written as

B =⎛⎝ 0

C1 sinλ1x

C1 cosλ1x

⎞⎠ +

⎛⎝ 0

C2 sinλ2x

C2 cosλ2x

⎞⎠ +

⎛⎝ 0

C3 sinλ3x

C3 cosλ3x

⎞⎠ .

(21)

Applying boundary conditions, values of constants C1,C2 and C3 can be determined. For boundary conditions|Bz|x=0 = h, |(∇×B)z|x=0 = s and |(∇×B)y |x=d = t , weobtain

C1 = t (λ2 − λ3) − λ2(s − hλ3) sinλ2d + λ3(s − hλ2) sinλ3d

λ1(λ2 − λ3) sinλ1d − λ2(λ1 − λ3) sinλ2d + λ3(λ1 − λ2) sinλ3d,

C2 = t (λ1 − λ3) − λ1(s − hλ3) sinλ1d + λ3(s − hλ1) sinλ3d

−λ1(λ2 − λ3) sinλ1d + λ2(λ1 − λ3) sinλ2d + λ3(λ1 − λ2) sinλ3d,

C3 = t (λ1 − λ2) − λ1(s − hλ2) sinλ1d + λ2(s − hλ1) sinλ2d

λ1(λ2 − λ3) sinλ1d − λ2(λ1 − λ3) sinλ2d + λ3(λ1 − λ2) sinλ3d.

The Beltrami parameter a is equal to the ratio of vorticityof canonical momentum of electron fluid to electrons flowwhereas the Beltrami parameter b represents the ratio of vor-ticity of canonical momentum of α-species to their corre-sponding flows. In order to show the behavior of triple curlBeltrami magnetic field in terms of Beltrami parameters a

and b, we consider three species plasma composed of pro-tons [H+], oxygen ions [O+] and electrons [e] found at the

magnetic equator of Saturn at L = 5.5 (Shemansky and Hall1992). The typical densities of the constituents at the equatorare [H+] ≈ 3 cm−3, [O+] ≈ 30 cm−3 and [e] ≈ 33 cm−3.Figure 1 shows the profiles of magnetic field versus distancefor the Beltrami parameters a = b = 0. This figure shows thestrength and trend of magnetic field when the canonical vor-ticities of all the species involved are zero. Figure 2 showsthe magnetic field for a = 0.5 and b = 0.1. The Beltrami pa-

22 Astrophys Space Sci (2012) 339:19–23

Fig. 1 Magnetic field profiles for a = b = 0

Fig. 2 Magnetic field profiles for a = 0.5 and b = 0.1

Fig. 3 Magnetic field profiles for a = 2 and b = 10

rameters show that flow and canonical vorticities have samemagnitude while the flow of α-species is 10 times greaterthan the corresponding canonical vorticities. Figure 3 showsthe magnetic field for a = 2 and b = 10. In this case, thecanonical vorticities are greater than the flows. Canonicalvorticity of electron fluid is 2 times greater than its flow andthe canonical vorticities of α-species is 10 times to corre-sponding flows. Oscillatory magnetic field is obtained whichgoes on increasing slowly away from the center. We observethat the magnetic field increases away from the center inFigs. 1–3. If we take a = 3 and b = 2.5, an oscillatory mag-netic field results whose strength varies above and below itsinitial value of unity as shown in Fig. 4. The same boundaryconditions of h = 1, s = 0.5 and t = 1, and zh = zl = 1, aretaken for plotting all the graphs.

By considering different values of Beltrami parametersand boundary conditions, we can get a variety of different

Fig. 4 Magnetic field profiles for a = 3 and b = 2.5

profiles of magnetic field which are useful to study the fieldstructures in three species plasmas found in different spaceenvironments and laboratory. When any two scale parame-ters (λj ) are equal, the triple curl Beltrami system changesto double curl Beltrami system.

4 Conclusion

To conclude, self-organization of three species plasmas ispresented. The simplest steady state solutions are givenby two equations which represent the Beltrami conditionswhere flow becomes parallel to the corresponding gener-alized vorticity. Solving the equations, a triple curl Bel-trami equation for magnetic field is obtained. This showsthat magnetic field in three species plasmas relaxes to threestructures when the mass of all the plasma components aretaken into account. The analytical solution for slab geom-etry is presented and taking into account the parameters ofSaturn’s plasma, plots of magnetic field versus distance aregiven. The graphs show that a three species plasma, depend-ing on Beltrami parameters and eigenvalues of the curl op-erator, could relax to large magnetic field structures. Whenany two scale parameters are equal or neglecting the electroninertia as shown by Shukla (2005), we will have double curlBeltrami equation. The present work will help us understandthe relaxation of three species plasmas found in earth’s au-roral zone, magnetosphere, magnetotail, heliosphere, mag-netic equator of Saturn etc. and in laboratory.

References

Chandrasekhar, S., Kendall, P.C.: Astrophys. J. 126, 457 (1957)Faria, R.T. Jr., Mirza, A.M., Shukla, P.K., Pokhotelov, O.A.: Phys.

Plasmas 5, 2947 (1998)Hultqvist, B., Øieroset, M., Paschmann, G., Treumann, R.: Magneto-

spheric Plasma Sources and Losses. Kluwer Academic, Dordrecht(1999)

Iqbal, M.: J. Plasma. Phys. 71, 335 (2005)Iqbal, M., Mirza, A.M., Murtaza, G., Yoshida, Z.: Phys. Plasmas 8,

1559 (2001)Iqbal, M., Berezhiani, V.I., Yoshida, Z.: Phys. Plasmas 15, 032905

(2008)

Astrophys Space Sci (2012) 339:19–23 23

Mahajan, S.M., Yoshida, Z.: Phys. Rev. Lett. 81, 4863 (1998)Sauer, K., Dubinin, E., McKenzie, J.F.: Geophys. Res. Lett. 28, 3589

(2001)Schwenn, R., Marsch, E.: Physics of the Inner Helioshere: Large Scale

Phenomena. Springer, Berlin (1990)Shemansky, D.E., Hall, D.T.: J. Geophys. Res. 97, 4143 (1992)Shukla, P.K.: Phys. Plasmas 11, 5354 (2004)Shukla, P.K.: Phys. Lett. A 334, 205 (2005)Shukla, P.K., Mahajan, S.M.: Physica Scripta T113, 151 (2004a)Shukla, P.K., Mahajan, S.M.: Phys. Lett. A 328, 185 (2004b)Stasiewics, K.: Phys. Rev. Lett. 12, 125004 (2004)Taylor, J.B.: Phys. Rev. Lett. 33, 1139 (1974)

Taylor, J.B.: Rev. Mod. Phys. 58, 741 (1986)Verheest, F., Hellberg, M.A., Kourakis, I.: Phys. Plasmas 15, 112309

(2008)Vranjes, J., Petrovic, D., Pandey, B.P., Poedts, S.: Phys. Plasmas 15,

072104 (2008)Woltjer, L.: Proc. Natl. Acad. Sci. USA 44, 489 (1958)Yoshida, Z.: Nonlinear Science. Springer, Dordrecht (2010)Yoshida, Z., Giga, Y.: Math. Z. 204, 235 (1990)Yoshida, Z., Mahajan, S.M.: Phys. Rev. Lett. 88, 095001 (2002)Yoshida, Z., Mahajan, S.M., Ohsaki, S., Iqbal, M., Shatashvili, N.:

Phys. Plasmas 8, 2125 (2001)