Compressible Hall magnetohydrodynamics in a strong magnetic fieldN. H. Bian and D. Tsiklauri Citation: Physics of Plasmas (1994-present) 16, 064503 (2009); doi: 10.1063/1.3159862 View online: http://dx.doi.org/10.1063/1.3159862 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/16/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the structure of guide magnetic field in the inertia-driven magnetic reconnection with the presence of shearflow Phys. Plasmas 20, 114501 (2013); 10.1063/1.4829039 Hall effect in a strong magnetic field: Direct comparisons of compressible magnetohydrodynamics and thereduced Hall magnetohydrodynamic equations Phys. Plasmas 17, 112304 (2010); 10.1063/1.3514203 Cathode spot motion in an oblique magnetic field Appl. Phys. Lett. 92, 011505 (2008); 10.1063/1.2832769 Ellipticity of axisymmetric equilibria with flow and pressure anisotropy in single-fluid and Hallmagnetohydrodynamics Phys. Plasmas 14, 062502 (2007); 10.1063/1.2741391 Slow mode waves and mirror instability in gyrotropic Hall magnetohydrodynamic model Phys. Plasmas 12, 122904 (2005); 10.1063/1.2141931

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Compressible Hall magnetohydrodynamics in a strong magnetic fieldN. H. Bian and D. TsiklauriInstitute for Material Research, University of Salford, Greater Manchester M5 4WT, United Kingdom

�Received 23 February 2009; accepted 8 June 2009; published online 29 June 2009�

Plasma dynamics becomes anisotropic in the presence of a strong background magnetic field, afeature that may be exploited to yield reduced fluid models. The reducedHall-magnetohydrodynamics model derived in a recent work by Gomez et al. �Phys. Plasmas 15,102303 �2008��, reflecting two-fluid effects such as the Hall current and the electron pressure, isextended to account for a crucial aspect of the role of the plasma compressibility, i.e., thecompression of the guide field. This reduced model constitutes therefore a description of thetwo-fluid plasma dynamics in a strong external magnetic field, which can be used also for values ofthe plasma pressure parameter � of the order of unity or smaller. © 2009 American Institute ofPhysics. �DOI: 10.1063/1.3159862�

Fluid models are valuable frameworks for the under-standing of the physics of magnetized plasmas. One-fluidmagnetohydrodynamics �MHD� is a standard description ofthe large-scale dynamics of plasmas, which fails however todescribe fluid phenomena with characteristic length scalessmaller than the ion skin depth, di=c /�pi, with �pi as the ionplasma frequency. A well known reason is that for scalesbelow di, it becomes inaccurate to make the approximationthat the magnetic field is carried by the bulk flow of ions, thelatter being instead frozen-in the electron component of theplasma flow. Therefore, two-fluid effects become importantat scales smaller than di and it is necessary to refine thedescription of the electron component of the fluid, as it isdone for instance in Hall MHD, which extends the validitydomain of MHD and accounts for the Hall current and elec-tron pressure effects.

In many situations of interest it is legitimate to considerthat the plasma is embedded in a strong and regular magneticfield B0. When the background field is only slightly per-turbed by the plasma motions, i.e., �B /B0�1 and �V /VA

�1, with VA being the Alfvén velocity, an important conse-quence is that the plasma tends to develop length scales thatare much smaller in the direction perpendicular to B0 thanthey are in the direction parallel to it, i.e., k� /k��1. Thisanisotropy of the plasma dynamics can be exploited to yieldreduced fluid models. The reduced-MHD �RMHD� model,first introduced in Refs. 1 and 2, can rigorously be obtainedfrom an asymptotic expansion of the compressible MHDequations under the ordering k� /k���B /B0��V /VA���1. In the reduction scheme, the fast time scale is the propa-gation time of the compressional Alfvén mode, which istherefore eliminated, while the low-frequency dynamics ofthe shear Alfvén and slow modes are retained, the slow modebecoming a sound wave in a low-� plasma. Within RMHD,the nonlinear shear Alfvénic disturbances evolve indepen-dently, being described by a stream and a flux function fromwhich the perpendicular perturbations �V� and �B� may bederived, the parallel perturbations �B� and �V� playing only apassive role. The self-consistency of the RMHD approxima-tion has been investigated and its validity, in the turbulent

regime, has been tested numerically by directly comparingits predictions with the ones of the compressible MHDequations.3

In a recent work,4 a system of reduced Hall-MHD�RHMHD� equations was derived from the incompressibleHall MHD following the same asymptotic procedure, whichis employed to obtain the conventional RMHD from MHD�see Ref. 4 for more references concerning RMHD, its vari-ous derivations, and applications�. A main limitation of thisRHMHD model is that because incompressibility of theplasma flow is assumed, the model is only valid for values ofthe plasma pressure parameter �, which are much larger thanunity, leaving the Hall effect as the sole mechanism for theproduction of parallel perturbations in the background mag-netic field. Although within the RMHD ordering the plasmacompressibility remains negligible in the dynamics of theperpendicular perturbations, leading to the decoupling of theshear Alfvén polarized perturbations, compressibility canmake important contribution to the dynamics of parallel per-turbations. The reason is that the effect of the small compres-sion of the large guide field produces a sizeable �B� andtherefore cannot be neglected in a plasma with pressure pa-rameter � of the order of unity or smaller. The aim of thiscommunication is to extend the RHMHD model derived inRef. 4 to also account for this effect of the plasma compress-ibility in the two-fluid description of plasmas embedded in astrong magnetic field.

Under the assumption of quasineutrality, ne=ni=n, withn as the plasma number density and considering that theelectrons have negligibly small inertia, the fluid equations ofmotion for the ions and the electrons are

nmi��tVi + Vi · �Vi� = − �Pi + ne�E + Vi � B� , �1�

0 = − �Pe − ne�E + Ve � B� , �2�

where Vi/e is the ion/electron velocity, mi is the ion mass, Pi,e

the ion/electron pressure, E is the electric field, and B is themagnetic field. Recalling that the current density j is definedas j=ne�Vi−Ve�, the system is supplemented by Maxwell’sequations: ��E=−�tB, ��B=�0j, and ��B=0. These

PHYSICS OF PLASMAS 16, 064503 �2009�

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equations are made dimensionless by introducing a typicallength scale L0, density n0, a typical value for the magneticfield B0, corresponding to the Alfvén velocity VA

=B0 /���0nmi�, a time scale L0 /VA, and the pressures arenormalized to the magnetic pressure B0

2 /�0.Equations �1� and �2� are then combined to give an ion

equation of motion,

�tV + V · �V = − �P + j � B , �3�

with P� Pi+ Pe, while the electron equation of motion isequivalent to the generalized Ohm’s law,

E + Ve � B = − di � Pe, �4�

with Ve=V−dij, �V�Vi�.Equation �4� is the statement of the frozen-in condition

of the magnetic field in the flow of electrons and not in thebulk flow of ions as it is assumed in the case of standardMHD. This difference of two-fluid MHD with standardMHD involves the additional nondimensional parameter di,which is the normalized ion skin depth di��c /�pi� /L0 with�pi=��ne2 /�0mi�. Assuming the plasma flow to be incom-pressible � ·V=0 and a uniform background density, Eq. �3�and the curl of Eq. �4�, i.e.,

�tB = � � ��V − dij� � B� = 0, �5�

constitute the set of incompressible Hall-MHD equations.Hall MHD extends the validity domain of MHD to �normal-ized� length scales of the order of di or smaller providedthese scales are also larger than de��c /�pe� /L0=�me /midi,otherwise, the effect of the electron inertia becomes impor-tant. Taking V=0 in the previous equations, i.e., assumingthe ions to be at rest, defines the electron MHD.

The existence of a strong background magnetic field B0zmakes the plasma dynamics anisotropic with ��k� /k� �1,i.e., structures have a size much smaller in the direction per-pendicular than parallel to the background field. We canwrite the normalized magnetic field as B=z+�B and makethe following ordering �B�� for its perturbation. The sole-noidal condition for the magnetic field perturbation providesthat its perpendicular component can be written in term of aflux function: �B���z+bzz. In the same way, the per-pendicular velocity is written in terms of a stream function,V��z+vzz, with the ordering V��B. Following thesame standard procedure employed to obtain RMHD fromthe MHD equations, the Hall-MHD equations �3�–�5� yield,at the order �2 in the asymptotic expansion, to

�t� = �z� − dibz� + � − dibz�� , �6�

�tbz = �z�vz − dijz� + �vz − dijz,�� + �,bz� − � · V , �7�

�t�z = �zjz + �jz,�� + �,�z� , �8�

�tvz = �zbz + �bz,�� + �,vz� , �9�

where the notation �A ,B�=z · ��A��B� is adopted and jz

=−�2�, �z=−�2 are, respectively, the z-component of thecurrent and of the vorticity. The last compressibility termmust be retained in Eq. �7� because it involves � ·V, which isof the order of �2, as can be understood by considering the

density equation under the ordering n�� for the normalizeddensity fluctuation.

The system can be closed by the pressure equation,

�tp = �,p� − � � · V , �10�

with the ordering, p��, for the normalized pressure pertur-bation. The plasma pressure parameter is defined as �=Cs

2 /VA2 =P0 / �B0

2 /�0�, with as the ratio of specific heatsand P0 as the background reference pressure. Let us note thatin Eqs. �6�–�10� it is possible to introduce the convenientnotations, ��A=�zA+ �A ,��, for the derivative along the fieldlines and dtA=�tA+ �A ,� for the convective derivative. Inderiving Eqs. �6�–�10�, we adopted the same conventions andapproximations as in Ref. 5 without the restriction to two-dimensional perturbations.

In the limit ��1 it then follows from Eq. �10� that theplasma flow becomes incompressible with � ·V=0 and theabove system is equivalent to the incompressible reduced-Hall MHD equations derived by Gomez et al.4 from the ex-pansion of Eqs. �3� and �5� at the order of �2. As the originalHall-MHD system, this reduced system conserves the total,kinetic plus magnetic energy

E = d3r����2 + vz2 + ����2 + bz

2� . �11�

For di=0, the incompressible RMHD equations are recov-ered with Eqs. �6� and �8� forming an independent system. Itdescribes the nonlinear dynamics of shear Alfvén wave po-larized perturbations, which is decoupled from the dynamicsof the slow wave polarized perturbations and, therefore, theenergies E1=�d3r����2+ ����2� and E2=�d3r�vz

2+bz2� can

be considered as separate conserved quantities.Linearizing Eqs. �6�–�9� with � ·V=0 and considering

that all the fields vary like exp�ik� ·r�+ ikzz− i�t�, the fol-lowing dispersion relation is obtained:4

�4 − 2kz2�1 + 1

2 �k�di�2��2 + kz4 = 0. �12�

In the limit k�di�1, we have

�2 =k�

2 di2kz

2

2 �1 � 1 −4

k�4 di

4�1/2� , �13�

which corresponds to the dispersion relation of the whistlerwaves with

� = � k�dikz �14�

and to the dispersion relation of the ion-cyclotron waves with

� = � kz/�k�di� . �15�

Taking V=0 in the previous RHMHD model leads to thereduced electron-MHD equations,

�t� = − di�zbz − di�bz,�� , �16�

�tbz = − di�zjz − di�jz,�� . �17�

This two-field model conserves the magnetic energy and itslinear modes are the whistler waves with ��= �k�dikz.Moreover, we notice that bz plays the role of a stream func-

064503-2 N. H. Bian and D. Tsiklauri Phys. Plasmas 16, 064503 �2009�

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tion in the dynamics of �, −dibz being the generalizedstream function in Eq. �6�.

Our main goal here is to relax the assumption of a large� allowing for the effect of a finite plasma compressibility.Since the perpendicular pressure balance equation, ���p+bz�=0, is satisfied at the order of � in the expansion of theion equation of motion, it follows that � ·V can be elimi-nated from Eqs. �10� and �7� using the fact that pbz,

�t� = �z� − dibz� + � − dibz,�� , �18�

�tbz = �

� + 1���z�vz − dijz� + �vz − dijz,��� + �,bz� ,

�19�

�t�z = �zjz + �jz,�� + �,�z� , �20�

�tvz = �zbz + �bz,�� + �,vz� . �21�

In a low � plasma, the role of the plasma compressibility isto make important contribution to the production of a bz, thelatter being produced solely by Hall effect in a large � in-compressible plasma. Following Fitzpatrick,5 we define Z=bz /c�, c�=�� /1+�, and d�=c�di, then with these nota-tions Eqs. �18�–�21� are rewritten as

�t� = �z� − d�Z� + � − d�Z,�� , �22�

�tZ = �z�c�vz − d�jz� + �c�vz − d�jz,�� + �,Z� , �23�

�t�z = �zjz + �jz,�� + �,�z� , �24�

�tvz = c��zZ + �,vz� + c��Z,�� . �25�

This reduced model conserves the total energy

E = d3r����2 + vz2 + ����2 + Z2� , �26�

linearizing it leads to the following dispersion relation:

�4 − kz2��1 + c�

2� + �k�d��2��2 + kz4 = 0. �27�

For k�d��1, the dispersion relations of the shear Alfvénwaves and the slow waves, respectively, ��= �kz and ��

= �c�kz, are recovered. These two waves have the samephase velocity only when ��1 �c��1�, the slow wave be-coming a sound wave when ��1. Taking the limit �→0�c�→0� while keeping d� finite in Eq. �12� gives the disper-sion relation of the “kinetic” Alfvén waves, ��

= �kz�1+ s

2k�2 , with s=��di being the ion sound Larmor

radius.When ��1 then c�=��, d�= s, Z=bz /��, and the

above set of equations is

�t� = �z� − sZ� + � − sZ,�� , �28�

�tZ = �z���vz − sjz� + ���vz − sjz,�� + �,Z� , �29�

�t�z = �zjz + �,�z� + �jz,�� , �30�

�tvz = ���zZ + �,vz� + ���Z,�� . �31�

This system is similar to the “four-field model” first derivedby Hazeltine et al.6,7 These four-field models can be ex-tended to account also for the effect of the electron inertia. Itis now easy to consider how the slow mode, which turns intoa sound wave with phase speed Cs�VA in a low � plasma, ismodified by two-fluid effects. In order for the kinetic Alfvénwaves to remain at zero amplitude, we assume � sZ, inwhich case �t�0 from Eq. �28� and we can eliminate jz

from the linearized Eqs. �29� and �30�, �t�1− s2��

2 �=�z��� svz�, which combined with Eq. �31� gives the fol-lowing wave equation for the sound wave: �tt�1− s

2��2 �

=�zz�.In the limit �→0, but s finite, the parallel flow dynam-

ics decouples, i.e., vz is passively advected and satisfies�tvz= � ,vz�. In this limit, the previous four-field model re-duces to

�t� = �z� − sZ� + � − sZ,�� , �32�

�tZ = − s�zjz − s�jz,�� + �,Z� , �33�

�t�z = �zjz + �jz,�� + �,�z� . �34�

Here, E3=�d3rvz2 and E4=�d3r����2+ ����2+Z2� are sepa-

rate conserved energies. In a uniform plasma this three-fieldmodel can be simplified further. Indeed, multiplying Eq. �33�by − s we see from Eqs. �33� and �34� that Z=− s�z, hencethe following two-field model is obtained:

�t� = �z� + s2�z� + � + s

2�z,�� , �35�

�t�z = �zjz + �,�z� + �jz,�� . �36�

This model also conserves E4=�d3r����2+ ����2+ s2�z

2�.Notice that it is very similar to the electron-MHD system fork� s, or if we suppose that the ions play the role of a neu-tralizing background, while it reduces to the conventionalRMHD description of shear Alfvén wave polarized perturba-tions for k� s. Linearizing this two-field model, Eq. �36�gives �t=�z�, which combined with Eq. �35� yields theequation, �tt=�zz�1− s

2��2 �, for the dispersive kinetic

Alfvén waves with dispersion relation

�� = � kz�1 + s

2k�2 . �37�

Some physical properties of the kinetic Alfvén waves, whichare believed to play a crucial role in the “dispersive range” ofMHD turbulence, were studied in Ref. 8. The kinetic Alfvénwave is a compressive mode which, unlike the Alfvén wave,involves a parallel component of the magnetic field pertur-bation such that the total pressure perturbation vanishes. Inthis sense, the kinetic Alfvén wave also takes on some prop-erties of the large k� limit of the slow mode. Also in Ref. 8,a two-fluid extension of the MHD dispersion relation wasderived: ��2 /kzVA

2 −1���2��2−k2VA2�−�k2VA

2��2−kz2VA

2��=�2��2−k2VA

2�k�2 s

2, explicitly showing that for ��1 thefast mode essentially cancels itself out of the dispersion re-lation.

We finally mention that the three-field model Eqs.�32�–�34� is the simplest fluid model that describes the non-

064503-3 Compressible Hall magnetohydrodynamics… Phys. Plasmas 16, 064503 �2009�

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linear interactions between kinetic Alfvén waves and driftwaves, if also the effect of a background plasma inhomoge-neity, say Z0=�Zx, is taken into account,

�t� = ��� − sZ� − s�Z�y� , �38�

dtZ = − �� sjz − �Z�y , �39�

dt�z = ��jz. �40�

The notations dtA=�t+ �A ,� and ��A=�zA+ �A ,�� havebeen employed. As done above for the description of thesound waves, assuming � sZ, we obtain from Eqs. �39�and �40� a familiar equation describing the nonlinear dynam-ics of electrostatic drift waves,

dt� − s2�2� − v��y = 0, �41�

with v�= s�Z,

� = −v�ky

1 + s2k�

2 , �42�

being the dispersion relation of the drift wave.In summary, we report the derivation of reduced models

that describe the two-fluid dynamics of a plasma embeddedin a strong uniform magnetic field. The starting point is theextension of the RHMHD system derived in Ref. 4, whichaccounts also for some aspects of the role of the plasmacompressibility. Therefore, the domain of validity of thisRHMHD system is not limited only to a large value of theplasma pressure parameter �. In a low-� plasma, the role ofthe plasma compressibility is to make important contributionto the production of a perturbation in the guide field strength,contrary to the case of a large-� incompressible plasmawhere the latter can solely be produced by Hall effect. Al-ready at the level of standard RMHD, it is essential to retainthis effect in order to account for how the slow wave be-comes acoustic in a low-� plasma. Moreover, compression is

an essential physical property of the kinetic Alfvén wave, atwo-fluid modification of the Alfvén mode, which involves amagnetic perturbation parallel to the background field and,hence, also takes on some properties of the large k� limit ofthe slow mode. The procedure, which has been followed hereto obtain the RHMHD model, is identical to the one in Ref.5 without the restriction to two-dimensional perturbationswith �z=0. The resulting model being very similar in its the-oretical structure to the incompressible RHMHD system ofRef. 4, this makes it a valuable computational tool for under-standing topics such as the small “dispersive scales” ofplasma turbulence and parallel electric field generation, forinstance, in an astrophysical context. We discuss the point ofcontact which exists between this RHMHD system and otherreduced Branginski models, which are used for investigatingthe dynamics of magnetically confined laboratoryplasmas9–11 �the latter being stratified, their dynamics alsoinvolve drift waves�. Among them, the “four-field model”6,7

plays a pivotal role and its Hamiltonian formulation, for thetwo-dimensional case, was discussed recently in Ref. 12.

1B. B. Kadomtsev and O. P. Pogutse, Sov. Phys. JETP 38, 283 �1974�.2H. R. Strauss, Phys. Fluids 19, 134 �1976�.3P. Dmitruk, W. H. Matthaeus, and S. Oughton, Phys. Plasmas 12, 112304�2005�.

4D. O. Gomez, S. M. Mahajan, and P. Dmitruk, Phys. Plasmas 15, 102303�2008�.

5R. Fitzpatrick, Phys. Plasmas 11, 3961 �2004�.6R. D. Hazeltine, M. Kotschenreuther, and P. J. Morrison, Phys. Fluids 28,2466 �1985�.

7R. D. Hazeltine and J. D. Meiss, Phys. Rep. 121, 1 �1985�.8J. V. Hollweg, J. Geophys. Res. 104, 14811 �1999�.9A. Zeiler, J. F. Drake, and B. Rogers, Phys. Plasmas 4, 2134 �1997�.

10A. Zeiler, D. Biskamp, J. F. Drake, and B. N. Rogers, Phys. Plasmas 5,2654 �1998�.

11B. D. Scott, Plasma Phys. Controlled Fusion 49, S25 �2007�.12E. Tassi, P. J. Morrison, F. L. Waelbroeck, and D. Grasso, Plasma Phys.

Controlled Fusion 50, 085014 �2008�.

064503-4 N. H. Bian and D. Tsiklauri Phys. Plasmas 16, 064503 �2009�

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