Magnetohydrodynamic waves in the presence of relative motion betweentwo populations of a plasma system

Subhash Kumara� and Harinder P. SinghDepartment of Physics and Astrophysics, University of Delhi, Delhi 110007, India

�Received 16 April 2010; accepted 5 August 2010; published online 28 September 2010�

The paper investigates the role of relative motion between the fluid components of a plasma model,which is simulated by concatenation of two anisotropic magnetohydrodynamic �MHD� fluids, on thepropagation of low-frequency waves and instabilities. The gyrotropic pressure of both the MHDcomponents is given by generalized polytropic laws that allow the system to reduce to a variety ofstates. The linearized analysis is carried out and dispersion relation is derived using the normalmode technique. The dispersion relation, which gives numerous earlier results as special cases, isdiscussed both analytically as well as numerically for parameters appropriate for the space plasma.It is found that the relative motion between the two components, besides modifying the condition forfire-hose instability, causes the concatenated system to exhibit different kinds of relative ordering ofthe magnitudes of phase speeds of the various MHD modes in the directions parallel and antiparallelto the magnetic field. The relative motion between the components also influences the phaserelationship between density and magnetic field fluctuations for the various compressive modes inthe background ambient plasma. © 2010 American Institute of Physics. �doi:10.1063/1.3483116�

I. INTRODUCTION

There are several astrophysical1–4 and magnetospheric5,6

situations where different plasma components, which arecoupled by the large scale ambient magnetic field coexist. Inthese situations, the plasma system may be appropriately de-scribed as concatenation of two magnetohydrodynamic�MHD� fluid components. Parker1 initiated the study of suchsystems in the context of interstellar cosmic rays and thismodeling has been used to investigate the propagation char-acteristics of hydromagnetic waves in a cold plasma mixedwith a hot plasma and the results were applied to the inter-pretation of Pc1 and Pc5 with right-hand polarizationsguided along the geomagnetic field line.5 Walker6 developeda hydromagnetic theory based on this concatenated modelfor describing a compressional pulsation with a large azi-muthal number. A significant outcome of the studies basedon a two-component MHD is that it allows an additionalmagnetosonic mode, which is the suprathermal mode, be-sides the usual slow, fast, and Alfvén MHD modes. Thismode originates due to the interaction between the two fluidsof the plasma system and has been found to be the fastestmode of propagation. These studies of two-populationplasma systems were undertaken in the plasma having iso-tropic pressure. However, it is well known that for a plasmasystem embedded in a strong magnetic field, the magneticfield suppresses the equilibration of pressures parallel andperpendicular to itself. For example, p� � p� in “cosmicpinches,”7 while in pulsar winds,8 the synchrotron-radiationlosses make p� � p�. There are large volumes of observa-tional data that suggest the existence of thermal anisotropy ofthe plasma in the solar wind,9 in the magnetosphere,10 and inthe ionosphere.11,12 Taking into account the ubiquitous pres-

sure anisotropy in the astrophysical and magnetospheric sys-tems, the two-component MHD has been extended to includepressure anisotropy.13,14

All the studies carried so far considered both the plasmacomponents of the system to be static in the steady state.However, a finite mean velocity is of great significance forcosmic rays since they must escape from the galactic disk.15

Cosmic rays are assumed to be produced anisotropicallythroughout the galactic disk by supernovae, etc. Thesestreaming anisotropic cosmic rays generate plasmawaves.16–20 The interaction between galactic cosmic rays andthe magnetosonic waves in the interstellar medium has beeninvestigated21,22 and suggested to be one of the likely mecha-nism to explain the observed isotropy and age of the cosmicrays. These studies concerning the galactic cosmic rays phe-nomenon were undertaken within the framework of kineticdescription. The fluid models of plasma system, besides pro-viding a much simpler description, are also appropriate spe-cially when considering the low-frequency phenomena andwhere bulk motions are involved. The pioneering study ofParker1 modeled the galactic cosmic rays in the interstellarmedium as a concatenation of two MHD fluids coupledthrough the ambient magnetic field. This modeling has beenutilized to study the effect of cosmic rays on hydromagneticwave propagation. This effect is significant not only in thegalactic region but also in the localized regions of solar at-mosphere, solar wind, solar flares, and interstellar spacewhere large a number of fast particles are present. This con-catenated modeling of a composite plasma system has beenextended to study the effect of pressure anisotropy in thenonrelativistic13,14 as well as relativistic4,23 domains. As re-marked earlier, this modeling has also been utilized to studymagnetospheric5,6 phenomena. However, the fluid descrip-tion for the concatenated plasma system has yet not includedthe effect of streaming of one of the components. This mo-a�Electronic mail: subhashkumar@rediffmail.com.

PHYSICS OF PLASMAS 17, 092104 �2010�

1070-664X/2010/17�9�/092104/8/$30.00 © 2010 American Institute of Physics17, 092104-1

tivated the present work to include a relative motion betweenthe components to study the propagation of low-frequency�characteristic frequency�gyrofrequency� MHD waves in atwo-population plasma system.

The main complication arising from the presence of an-isotropy is the fact that evolution of anisotropy, is by defini-tion a kinetic effect, and its proper description within fluidtheory remains a fundamental problem. This is because, toachieve closure, the hierarchy of moments of the Vlasovequation is artificially truncated based on assumptions. Oneapproach to this problem is the Chew, Goldberger, and Low�CGL� double-adiabatic theory.24 However there are evi-dences of unsuitability of the CGL laws in the Earth’s mag-netosphere, where the double polytropic laws25 with suitablepolytropic indices have been found to provide a viable clo-sure to the system. In the present investigation, the poly-tropic equations of state25 have been employed which, al-though heuristic, have been extensively used because theyare found suitable for a variety of physical situations. Thepaper is organized as follows. Section II gives the basicequations of the model and the dispersion relation. SectionIII discusses the dispersion relation analytically in the specialcases of wave propagation along and perpendicular to themagnetic field before carrying out numerical analysis ofpropagation characteristics of various modes for general di-rection of propagation. Friedrich 1 diagram that highlightsthe role of relative motion between the cosmic ray fluid mov-ing with respect to background plasma has been drawn. Sec-tion IV summarizes the conclusions of the study.

II. BASIC EQUATION AND DISPERSION RELATION

We consider a plasma system consisting of two idealanisotropic fluids �fluid 1 and fluid 2�. MHD equations thatdescribe an inviscid homogeneous plasma having gyrotropicpressure are used for both the components that are coupledtogether through an ambient magnetic field, which is uniformover a scale that is large compared to the wavelength ofdisturbance. One of the fluid �fluid 1� is moving with anuniform velocity V0 along the direction of ambient magneticfield with respect to the background anisotropic fluid �fluid2� in the steady state. The gyrotropic pressure of each of thefluids is described by polytropic laws.25 These fluids are as-sumed to be compressible, unbounded, and embedded in auniform magnetic field, which is aligned along the z-axis in aCartesian geometry. Under the assumption of perfect conduc-tion, the magnetic field constrains the two fluids to movetogether transverse to the field but it cannot prevent themfrom moving independently in the direction along the field.The idealized Ohm’s law together with Faraday’s law givesthe following linearized equation for the perturbed magneticfield �denoted by b�:

�b

�t= � � �v1 � B0 + V0 � b� = � � �v2 � B0� , �1�

where v1, v2 denotes the perturbed fluid velocities of the twoplasma components, respectively, and V0�=V0z� denotes thesteady state relative velocity of fluid 1 with respect to fluid 2�the background plasma�. It is evident from Eq. �1� that the

corresponding perpendicular components of the velocities ofthe two fluids are now related as

v1x = v2x + V0bx

B0, v1y = v2y + V0

by

B0. �2�

In the absence of any relative motion among the fluidcomponents, the perpendicular components of the two fluidsare equal.14 The linearized equations for the conservation ofmass are given as follows:

� �

�t+ V0

�

�z���1

�10� + � · v1 = 0, �3�

�

�t��2

�20� + � · v2 = 0, �4�

where �1 and �2 denotes the mass densities of the two plasmacomponents, respectively. The linearized equation of motionin direction perpendicular to the magnetic field may be writ-ten as

�10� �

�t+ V0

�

�z�v1 + �2

0�v2

�t+ � · �p1 + p2�

−�� � b� � B0

4�z = 0. �5�

Here the gyrotropic pressure of the plasma componentsis given by

p1,2 = �p�1,2− p�1,2

�nn + p�1,2I ,

where p�1,2, p�1,2

denote the pressure parallel and perpendicu-lar to the direction of magnetic field, respectively. I repre-sents the unit second order tensor and n denotes a unit vectoralong the direction of the magnetic field. There being noconstraints along the direction of magnetic field, the two flu-ids move independently. The linearized equation for the con-servation of momentum can, therefore, be written as

�10� �

�t+ V0

�

�z�v1z + �p�1

0 − p�1

0 ��

�z� bz

B0� +

�p�1

�z= 0, �6�

�20�v2z

�t+ �p�2

0 − p�2

0 ��

�z� bz

B0� +

�p�2

�z= 0. �7�

The equations of state of the respective plasma compo-nents are given by25

p�s

p�s

0 =�s

�s0 + ���s

− 1�� bz

B0� , �8�

p�s

p�s

0 = ��s

�s

�s0 − ���s

− 1�� bz

B0� , �9�

where ��sand ��s

;s=1,2 are the polytropic indices of thefluids. Any arbitrary perturbation in the system can beFourier analyzed to have the form of a plane wave�exp�i�t−k ·r��, where is the wave frequency and k isthe real wave vector, which is assumed to lie in the xz-plane:k= �k� ,0 ,k��. This space time dependence is then applied to

092104-2 S. Kumar and H. P. Singh Phys. Plasmas 17, 092104 �2010�

Eqs. �1�–�7�. Carrying out the usual normal mode analysis,we get the following dispersion relations:

��10 + �2

0�2

k2 − 2V0�10

kcos − B0

2

4�+ �

s=1

2

�p�s

0 − p�s

0 � − V02�1

0��cos2 = 0, �10�

�10�

k− V0 cos �2

− ��1p�1

0 cos2 ����2

02

k2 − ��2p�2

0 cos2 ���10 + �2

0�2

k2 −B0

2

4�

− 2V0�10

kcos − �

s=1

2

�p�s

0 − p�s

0 � − V02�1

0�cos2

− sin2 �s=1

2

��sp�s

o − �p�2

0 �2sin2 cos2 − ��2

02

k2 − ��2p�2

0 cos2 ��p�1

0 �2sin2 cos2 = 0. �11�

It may be noted that in the absence of relative motion�V0→0�, and for ��s

=3, ��s=2; s=1,2, corresponding to

the CGL double-adiabatic laws,15 one recovers the dispersionrelations for MHD waves in a plasma medium described bythe concatenation of two anisotropic MHD fluids.14 In the

limit of the single component plasma system ��10 , p�2

0 , p�2

→0�, the dispersion relation for MHD waves in an aniso-tropic plasma with polytropic indices is recovered.25

It is well known that in the MHD model consisting ofsingle component plasma, there exist six wave modes �threeforward propagating and three backward propagating distur-bances� propagating irrespective of whether the pressure isisotropic or anisotropic. The general dispersion relation ofthe present problem is eighth order in indicating the exis-tence of eight �four forward and four backward propagatingdisturbances� wave modes that is the outcome of the two-population MHD model. Thus the dispersion relations �Eqs.�10� and �11�� of the model under investigation indicates tothe existence of an additional mode, beside the usual slow,fast, and Alfvén hydromagnetic modes. This mode called thesuprathermal mode is excited due to interaction of the twofluids in a plasma system coupled by an ambient magneticfield. This is essentially an outcome of the concatenated fluidmodeling of the system for low-frequency phenomena.

III. DISCUSSION OF DISPERSION RELATION

Equation �10� is the dispersion relation of the modifiedAlfvén mode, which is factored out from the general disper-sion relation of the problem because it involves only trans-verse perturbations. The phase speed �V= /k� of the Alfvénmode is given by

V = �V0�10 � V0

2��10�2 + ��1

0 + �20� B0

2

4�+ �

s=1

2

�p�s

0 − p�s

0 � − V02�1

0�1/2

�10 + �2

0 �cos . �12�

Equation �12� shows that the phase speed of Alfvén mode isgoverned by the combined density of the fluids that has theeffect of increasing the inertia of the system, thereby reduc-ing the phase speed. Alfvén modes are modified by the pres-sure anisotropy as well as by the relative motion between thecomponents. The relative motion between the two compo-nents causes the forward and backward propagating Alfvénmodes to have different phase speeds. The greater the rela-tive speed between the fluid components, the greater is thedifference in phase speed in the direction parallel and anti-parallel to the magnetic field. For large enough relativespeeds, the analysis breaks down because their relativisticeffects become significant. It has been shown that, for thelow-frequency hydromagnetic modes in a fully relativisticbi-Maxwellian plasma, the relativistic effect tends to de-crease the magnitude of the Alfvén speed in the stableregime.26 Investigation of relativistic effects in the presentstudy of concatenated system is beyond the scope of thiswork as it involves consideration of relativistic basic equa-tions at the outset and this will be the subject matter of a

subsequent paper. The phase speed of the Alfvén mode �Eq.�12�� is independent of the polytropic indices ���s

,��s� be-

cause it is an incompressible mode. The Alfvén mode mayexhibit nonpropagating, pure exponential growth when

�s=1

2

�p�s

0 − p�s

0 � �B0

2

4�− V0

2� 1

�10 +

1

�20�−1

. �13�

This instability is called the fire-hose or ballooning instabil-ity. In the absence of a relative motion, V0=0 and suppress-ing the summation symbol for the case of an anisotropic,spatially uniform, nonrelativistic plasma embedded in an in-finite, homogeneous magnetic field of strength B0, the aboveexpression for the onset of fire-hose instability reduces to27

P�0 − P�

0 �B0

2

4�, �14�

where P�0 and P�

0 are the pressure components of the plasmain directions parallel and perpendicular to the magnetic field,respectively. In the low-frequency, long wavelength regime

092104-3 Magnetohydrodynamic waves in the presence… Phys. Plasmas 17, 092104 �2010�

and within the framework of relativistic Vlasov equation, thekinetic theory yields identical condition for the fire-hoseinstability28 for a relativistic plasma embedded in an infinitehomogeneous magnetic field. It is also shown that if morethan one species of particles are present, P�

0 and P�0 are the

total pressure components due to all species28 and the pres-ence or absence of fire-hose instability is strongly dependenton the choice of distribution function.29

Inspecting the condition for the fire-hose instability inthe concatenated system with relative motion among its fluidcomponents �Eq. �13��, we observe that to account for thisinstability, we have to not only consider the contribution ofall three pressures �p�s

0 , p�s

0 , and B02 /8�� but also include a

contribution arising due to relative motion between the fluidcomponents of the system. The relative motion sets a lowerthreshold for �s=1

2 p�s

0 in the concatenated system for the onsetof fire-hose instability as compared to a composite systemhaving no relative motion among its components.14 Thismeans that a longitudinal component of the relative velocitybetween the fluid components of the system supplements thepressure head in the direction of the ambient magnetic field.It is interesting to note that the Alfvén wave do not propagateeven if the pressure heads in the parallel and perpendiculardirections to the magnetic field equalize ��s=1

2 p�s

0 =�s=12 p�s

0 � if

V0 � �VA12 + VA2

2 , �15�

where VAs=B0 /�4��s0 ;s=1,2 is the classical �collisional�

Alfvén speed in the respective fluid components of the com-posite system. This means that even in the absence of any netpressure anisotropy in the system, if the relative velocitybetween the two populations of the concatenated system ex-ceeds a critical value �given by Eq. �15��, the system is proneto fire-hose instability. In other words, the presence of therelative motion among the two fluid components of the sys-tem can be a sufficient criterion for the onset of fire-hoseinstability. Analytical separation of the three modes given byEq. �11� in the most general form is not amenable to physicalinterpretation. So, we first discuss the modes in the specialcases when either k� =0 or k�=0 and then compute numeri-cally the phase speeds of various modes.

A. Propagation along the magnetic field„k�=0, k¸=k…

In the special case when the wave vector is parallel tothe direction of the magnetic field �=0�, Eqs. �10� and �11�show that all the four modes propagate; two of the modeshave the same phase speed given by

V = �V0�10 � V0

2��10�2 + ��1

0 + �20� B0

2

4�+ �

s=1

2

�p�s

0 − p�s

0 � − V02�1

0�1/2

�10 + �2

0 � . �16�

The other two modes propagate with phase speeds

V = � ��2p�2

0

�20 �1/2

, �17�

V = V0 � ��1p�1

0

�10 �1/2

. �18�

The double mode given by Eq. �16� is basically the Alfvénmode as affected by the pressure anisotropy and relative mo-tion between components. The double mode arises becausethe Alfvén mode and the fast mode have identical speedsalong the direction of magnetic field. These modes are trig-gered by velocity �magnetic field� perturbations transverse tothe direction of propagation. Each of these two modes isexcited corresponding to the perturbations along the two mu-tually perpendicular directions transverse to k. Since the twoperpendicular directions are indistinguishable, the symmetryconsideration requires both the modes to have identicalspeeds. The mode given by Eq. �17� is the slow mode andrepresents the sound mode propagating in the backgroundplasma �fluid 2�. The phase speed of this mode is unaffectedby the presence of the other fluid component, because there

are no constraints to bind the motion of the two fluids alongthis direction. Each fluid retains its identity and moves inde-pendently. The fourth mode given by Eq. �18� is the suprath-ermal mode that represents the sound speed propagating influid 1. The relative motion between the fluid componentscauses the asymmetry between the forward and backwardpropagating mode that now have different phase speeds.

B. Propagation transverse to the magnetic field„k�=k , k¸=0…

When the wave vector is perpendicular to the magneticfield direction, three of the four modes do not propagate. Theonly mode that propagates has a phase speed

V = � �B02/4� + �

s=1

2

��sp�s

0

�10 + �2

0 �1/2

. �19�

This is a stable mode that is a hybrid of the fast �correspond-ing to the single component MHD� and the suprathermalmode of the two component MHD. This mode is not affectedby relative motion between the fluid components, i.e., theforward and backward propagating modes have the same

092104-4 S. Kumar and H. P. Singh Phys. Plasmas 17, 092104 �2010�

phase speed. This is because the relative motion, being trans-verse to the direction of propagation, becomes ineffective. Inthis mode, the velocity perturbations are confined to the di-rection of wave normal so that wave propagation is analo-gous to the usual longitudinal sound wave. This is due to thefact that in this situation the tension in the magnetic lines offorce vanishes and the two components of the plasma behaveas single component. The presence of magnetic field onlyadds magnetic pressure to the plasma pressure.

C. Numerical computations

As mentioned earlier, the interstellar environment hasbeen simulated to be a concatenation of two anisotropicMHD fluids14 consisting of a cosmic ray fluid �fluid 1� andbackground plasma �fluid 2�. Cosmic rays are presumed tobe moving relative to the background plasma. To investigatethe general features of the various modes of propagation inthis system, it is necessary to discuss the modes when thewave vector is inclined at an arbitrary angle with respect tothe direction of ambient magnetic field. To this end, the dis-persion relations �Eqs. �10� and �11�� are solved numericallyfor the parameters appropriate for interstellar environ-ment:30 �1

�=p�1

0 /B02 /8��=0.025, �1

�=p�1

0 /B02 /8��=0.037,

�2�=p�2

0 /B02 /8��=0.498, �2

�=p�2

0 /B02 /8��=0.998, and

d�=�20 /�1

0�=0.5. The system does not support purelypropagating MHD modes for those parameters that satisfyEq. �13�, which in dimensionless form reads as

F = 1 + �s=1

2

�s�s

2 − s�s

2 � − �V0��2� d

1 + d� � 0. �20�

Figure 1 shows the plot of F versus the relative speed be-tween the components V0

��=V0 /VA� where VA�=B0 /�4��10� is

the classical Alfvén speed in fluid 1 for different densityratios of the two MHD components comprising the system.The plot shows that while the system with the above men-tioned parameters is stable for all values of density ratio�d=�1

0 /�20� in the absence of any relative motion between its

components, it is prone to fire-hose instability whenever therelative speed between the components exceeds a critical

value. This critical value of relative speed increases withincreasing density ratio.

Figure 2 depicts the polar plot of U, the phase speedsnormalized to the classical Alfvén speed VA in fluid 1 for theforward propagating slow �S�, fast�F�, and suprathermal �ST�mode and the forward and backward propagating Alfvénmodes �A+ and A−� in the composite system for polytropicindices ��s

=3, ��s=2. These polytropic indices simulate the

double-adiabatic �CGL� state in both the fluid components.The relative speed between the fluid components is assumedas V0

�=0.1. The figure shows that while the polar plot ofAlfvén mode remains symmetric, both along and transverseto the direction of the magnetic field, the other three magne-tosonic modes, i.e., slow, fast, and suprathermal modes nolonger remain symmetric with respect to direction perpen-dicular to the magnetic field. As compared to the case whenthere is no relative motion between the components,14 thesemodes show an increase in phase speeds parallel to the di-rection of magnetic field accompanied by a reduction inphase speeds along a direction that is antiparallel to the mag-netic field. For these numerical parameters, the reduction inphase speeds in the direction opposite to the magnetic fieldeven makes the fast mode slower than the forward propagat-ing Alfvén mode. The suprathermal mode is the only modethat propagates in a direction perpendicular to the magneticfield and has the fastest phase speed. Thus, the relative mo-tion of the components causes the concatenated system toexhibit two kinds of relative ordering of the magnitudes ofphase speeds of the various MHD modes in direction paralleland antiparallel to the magnetic field. The former has apseudo-MHD ordering �US�UA−

�UA+�UF�UST�, while

the later exhibit a new ordering �US�UA−�UF�UA+

�UST�. In the absence of the relative motion between thecomponents, these modes show a pseudo-MHD ordering forall direction of propagation for the same numericalparameters.14 The general propagation characteristics for thebackward propagating modes is similar to that discussedabove for the forward propagating modes and can be ob-tained by rotating Fig. 2 along the axis of symmetry perpen-

-4

-3

-2

-1

0

1

2

0 0.5 1 1.5 2 2.5 3

F

V ∗0

4 %

40 %

60 %

FIG. 1. Plot of F vs the relative speed between the components V0��

=V0 /VA� where VA�=B0 /�4��10� is the classical Alfvén speed in fluid 1 for

different density ratios d�=�10 /�2

0� of the two MHD components comprisingthe system.

� �

� �

� �

�

� �

� �

� �

� � � � � � � � � � � � � � � � � � �

�

�

�

�

�

�

� �

��

�

�

� �

�

�

�

FIG. 2. Polar plot of the phase speeds �normalized to the classical Alfvénspeed VA in fluid 1� of the forward propagating slow �S�, fast �F�, andsuprathermal �ST� mode and the forward and backward propagating Alfvénmodes �A+ and A−� for polytropic indices ��s

=3, ��s=2, which simulate the

double-adiabatic �CGL� state in both the fluid components.

092104-5 Magnetohydrodynamic waves in the presence… Phys. Plasmas 17, 092104 �2010�

dicular to the direction of magnetic field by an angle of �radians. The backward propagating modes show an increase�decrease� in phase speeds when the wave vector is antipar-allel �parallel� to the direction of magnetic field.

Taking note of the observational data31–33 that indicatethat the CGL laws are unsuitable in the space environment, anumerical study is carried out further with polytropic indices��s

�3, ��s�2. Active magnetospheric particle tracer

explorer/ion release module �AMPTE/IRM� plasma andmagnetic field measurements in the magnetosheath31 indicatethat polytropic indices are in the ranges 1���s

�1.5 and0.7���s

�1.1; s=1,2. The polar plot of the phase speeds�of the forward propagating modes� with the above men-tioned parameters and polytropic indices ��s

=1.1, ��s=1.5:s=1,2 are shown in Fig. 3�a� for V0

�=0.1.Figure 3�a� shows that while the role of relative motion

remains the same as discussed above, the relative ordering ofmagnitude of phase speeds of various modes changes fromthat mentioned above in directions parallel and antiparallel tothe wave vector. For this case, the forward as well as back-ward propagating Alfvén mode is faster than the fast mode inall directions of propagation. The suprathermal mode no

longer remains the fastest propagating mode in an angularregion opposite to the direction of magnetic field.

In order to understand the role of relative motion be-tween the fluid components on the MHD modes for poly-tropic indices appropriate for space plasma ��s

=1.1, ��s=1.5:s=1,2, the polar plot of the phase speed of the Alfvénand forward propagating slow, fast, and suprathermal modesare drawn for various values of relative speeds V0

� betweenthe fluid components. Figure 3�b� shows that the inclusion ofrelative motion between the fluid components causes the de-composition of the Alfvén mode into forward and backwardpropagating modes. The forward propagating Alfvén mode�A+� and the backward propagating Alfvén mode �A−� havephase speeds greater and lesser, respectively, than that of theAlfvén mode in the absence of relative motion. The differ-ence of speeds increases with increasing value of relativespeeds between the components. It may be expected thatwith large enough value of relative speed between the com-ponents, the backward propagating Alfvén mode may vanish.However a rigorous treatment of such situation requires arelativistic treatment of the problem.

Figure 4 shows the forward propagating slow �S� modesfor various values of relative speeds between the compo-nents. The inclusion of relative motion between the fluidcomponents causes an increase of phase speed of the slowmode in the direction along the magnetic field. The phasespeed of these slow modes may increase or decrease in thedirection opposite to the magnetic field depending on thechoice of V0

�. The fast �F� modes are plotted for variousvalues of relative speeds between the fluid components inFig. 5. The fast modes show an increase of phase speed inthe direction along the magnetic field for increasing value ofrelative speeds. This is accompanied with a progressive de-crease of phase speed in a direction opposite to the magneticfield with increase in the value of relative speed between thefluid components. Figure 6 shows the variation in phasespeed of the suprathermal �ST� mode with the relative speedbetween the fluid components. The suprathermal mode ex-hibits an increase in phase speed of the mode for −� /2��� /2 and a reduction in speed for � /2��3� /2 with

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(b)

FIG. 3. �a� Polar plot of the phase speeds �normalized to the classical Alfvénspeed VA in fluid 1� of the forward propagating slow �S�, fast �F�, andsuprathermal �ST� modes and the forward and backward propagating Alfvénmodes �A+ and A−� for observational data of the magnetosheath �polytropicindices ��s

=1.1, ��s=1.5�. �b� Polar plot of the phase speeds �normalized

to the classical Alfvén speed VA in fluid 1� of the forward and backwardpropagating Alfvén modes �A+ and A−� for observational data of the mag-netosheath �polytropic indices ��s

=1.1, ��s=1.5� for different values of

relative speed between the MHD components.

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FIG. 4. Polar plot of the phase speeds �normalized to the classical Alfvénspeed VA in fluid 1� of the forward propagating slow �S� modes for theobservational data of the magnetosheath �polytropic indices ��s

=1.1, ��s=1.5� for different values of relative speed between the MHD components.

092104-6 S. Kumar and H. P. Singh Phys. Plasmas 17, 092104 �2010�

increasing values of relative speed between the fluid compo-nents. However, the phase speed of this mode remains un-changed for = �� /2. This is because there is no compo-nent of relative velocity in these directions that can modifythe phase speed of this mode.

D. Density-magnetic field correlation

The phase relationship between density and magneticfield perturbations that excites various compressive modes isa useful tool for mode identification. Following the methodof Lou and Fan,3 one obtains the relationship between den-sity and magnetic field fluctuations for the respective fluidsof the concatenated system as follows:

bz

B0= � � − V0k��2 − c�1

2 k�2

k�� − V0k�1�2

V0k�2 + k�� − V0k��

− k�2�c�1

2 − c�1

2 /��1���1

�10 ,

�21�

and

bz

B0= � 2 − k�

2c�2

2

2 − k�2�c�2

2 − c�2

2 /��2���2

�20 , �22�

where c�s,�s

2 = ���s,�sp�s,�s

� /�s0 and /k is determined from the

dispersion relation for the compressive modes �Eq. �11��.The influence of the presence of the streaming suprath-

ermal particles �fluid 1� on the density-magnetic field corre-lation in the ambient background plasma �fluid 2� is investi-gated by computing

��=bz

B0� �2

�20� ,

numerically for the above mentioned parameters. Figure 7depicts the density-magnetic field correlation in the ambientbackground plasma for the suprathermal mode for variousvalues of the relative speed �V0

�� between the two fluid com-ponents of the concatenated system. This phase relationshipis found to be positive for all angles of propagation ��, withthe function � varying between 0 and 1 for various values ofV0

�, including the value V0�=0. Thus the inclusion of a plasma

component streaming along the ambient magnetic field in thebackground plasma does not effect the nature of phase rela-tionship among density and magnetic field perturbation en-hancements of the suprathermal mode.

Figure 8 is a plot of � versus for the fast mode of theconcatenated system. The plot shows that while � is negativefor V0

�=0, V0�=0.1 and V0

�=0.5 for all angles of propagation,it is positive for V0

�=1 in the forward direction �� /2��3� /2�. A similar nature is observed for the density-magnetic field correlation in the background plasma for theslow mode �Fig. 9�. This implies that in the case of the fastand slow modes, a large enough value of relative motionbetween the two fluid components of the composite systemcan change the nature of the phase relationship between thedensity and magnetic field perturbations in the backgroundplasma. The relationship changes from anticorrelation to cor-relation in the forward direction of propagation. As discussedin Sec. III C, the fast and slow modes also show an enhance-ment of phase speed in the forward direction, i.e., the relativemotion in the direction of ambient magnetic field between

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FIG. 5. Polar plot of the phase speeds �normalized to the classical Alfvénspeed VA in fluid 1� of the forward propagating fast �F� modes for observa-tional data of the magnetosheath �polytropic indices ��s

=1.1, ��s=1.5� for

different values of relative speed between the MHD components.

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FIG. 6. Polar plot of the phase speeds �normalized to the classical Alfvénspeed VA in fluid 1� of the forward propagating suprathermal �ST� modes forthe observational data of the magnetosheath �polytropic indices ��s=1.1, ��s

=1.5� for different values of relative speed between the MHDcomponents.

� � �

� � �

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FIG. 7. Density-magnetic field correlation in the ambient backgroundplasma for the suprathermal mode for various values of the relative speedbetween the two fluid components.

092104-7 Magnetohydrodynamic waves in the presence… Phys. Plasmas 17, 092104 �2010�

the components besides enhancing the phase speed, may alsobring about a change in the nature of density-magnetic fieldcorrelation in this direction.

IV. CONCLUSIONS

An ideal MHD model, which describes the mutual inter-action of two polytropic fluids in relative motion is used toinvestigate the propagation and stability of waves. The gen-eral dispersion relation is derived and discussed analyticallyas well as numerically for space plasma parameters. The re-sults are compared with the similar model that does not in-clude the relative motion between the fluid components. It isshown that the inclusion of relative motion among the fluidcomponents in the concatenated system modifies the condi-tion for fire-hose instability. It is found that the relative mo-tion among the components causes the concatenated systemto exhibit different kinds of relative ordering of the magni-tudes of phase speeds of the various MHD modes in direc-tions parallel and antiparallel to the magnetic field. The na-ture of ordering of magnitude of phase speeds of variousmodes is dependent on the choice of polytropic indices. Thepresence of relative motion between the components of the

concatenated system is found to modify the density-magneticfield correlation for the various magnetosonic modes in thebackground ambient plasma.

In the end, it may be remarked that the work presentedhere is an attempt to explore the effect of relative motionamong the fluid components of a concatenated anisotropicMHD system embedded in a uniform magnetic field, with alimited scope to include only nonrelativistic motion amongthe two fluid component comprising the system. A more re-alistic modeling of galactic cosmic rays requires inclusion ofa motion at relativistic speeds. This necessitates consider-ation of relativistic anisotropic magnetohydrodynamic equa-tions as the basic equations of the problem and shall be in-vestigated in a forthcoming paper. It is pertinent to point outthat although thermal anisotropy is a short lived effect but itmay become significant in local regions.

ACKNOWLEDGMENTS

The authors are grateful to the anonymous referee forhelpful suggestions that resulted in an improvement of thearticle.

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FIG. 8. Density-magnetic field correlation in the ambient backgroundplasma for the fast �F� mode for various values of the relative speed betweenthe two fluid components.

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FIG. 9. Density-magnetic field correlation in the ambient backgroundplasma for the slow �S� mode for various values of the relative speed be-tween the two fluid components.

092104-8 S. Kumar and H. P. Singh Phys. Plasmas 17, 092104 �2010�