High-growth-rate magnetohydrodynamic instability in differentially rotating compressible flow

Mradul Sharma*Theoretical Astrophysics Section, Astrophysical Sciences Division, Bhabha Atomic Research Centre, Mumbai 400085, India

�Received 21 May 2010; revised manuscript received 11 August 2010; published 9 September 2010�

The transport of angular momentum in the outward direction is the fundamental requirement for accretion toproceed in an accretion disk. This objective can be achieved if the accretion flow is turbulent. Instabilities areone of the sources for the turbulence. We study a differentially rotating compressive flow in the presence ofnonvanishing radial and azimuthal magnetic field and demonstrate the occurrence of a high growth rateinstability. This instability operates in a region where magnetic energy density exceeds the rotational energydensity.

DOI: 10.1103/PhysRevE.82.037302 PACS number�s�: 47.20.�k, 47.65.�d, 95.30.Qd

I. INTRODUCTION

Disk accretion is one of the most fundamental processesoccurring in a variety of astrophysical objects such as accre-tion powered x-ray pulsars, protostars, cataclysmic variablesetc. It is observed that the molecular viscosity is too small toaccount for the observed time scales and luminosities �1�. Tocircumvent this problem, Shakura and Sunyaev �2� made anormalization of the coefficient of viscosity in the theory ofturbulent viscosity proposed by Heisenberg �3� and intro-duced a dimensionless parameter �csH, where cs and H werethe sound speed and the disk scale height respectively.Though, the � model is quite successful in explaining manyastrophysical observations, a degree of adhocness enters themodel through the parameter �. The presence of Rayleighcriterion of hydrodynamic stability, i.e. a flow with specificangular momentum increasing monotonically, being satisfiedin accretion disk, rules out the hydrodynamic origin of tur-bulence, though there is literature available �4,5� in its favor.The origin of turbulence is still a question. Differential rota-tion can lead to an instability provided the specific angularmomentum decreases outward, a condition generally not sat-isfied in the accretion disk, though there are reports suggest-ing such regions being realized locally �6–8�. A recent work�9� numerically demonstrated the presence of such regions.Though differential rotation alone cannot lead to instability,inclusion of magnetic field alters the scenario altogether. Itwas well known from the classical works of Velikhov �10�and Chandra �11� that a differentially rotating flow with anegative angular velocity gradient in a weak magnetic field isunstable. The presence of this instability in accretion diskwas established by Balbus and Hawley �12� after three de-cades that the subthermal fields in the presence of differentialrotation can cause magnetorotational instability �MRI�,which can provide the physical basis for the transport ofangular momentum in accretion disk. It has been computa-tionally shown �13,14� that MRI is capable of introducingturbulence in the accretion disk. It is to be noted that MRI isessentially an instability of incompressible flow and itsgrowth starts suppressing as the compressibility seeps in.The earliest attempt to understand the effect of compressibil-

ity on the growth of MRI was made by Blaes and Balbus�15�. They concluded that the B� does not affect the instabil-ity. Later on, the behavior of compressible MRI in the MHDflows was addressed by Kim and Ostriker �16�. It was dem-onstrated that when the magnetic field strength is superther-mal, the inclusion of toroidal fields tends to suppress thegrowth of the MRI, and that for quasitoroidal field configu-rations no axisymmetric MRI takes place in the limit cs→0.In an another work, Pessah and Psaltis �17� studied the roleof toroidal fields in compressible flows. It was again demon-strated that the growth rate of MRI is affected in the com-pressible flow and MRI stabilizes in superthermal fields.Clearly, the compressibility plays an important role in thedynamics of instabilities operating in the accretion flow.

Recently, Bonanno and Urpin �18� �henceforth paper I�studied the effect of compressibility on the instabilities in thepresence of magnetic field. They considered the magneticfield with nonvanishing radial and azimuthal components. Anew instability was observed which survived for all values ofmagnetic field, unlike MRI which survives only in the weakfield limit. The maximum growth rate of the reported insta-bility was �� where � is the rotation frequency. In a recentwork �19�, we investigated the behavior of above instabilityfor a special case �e

2�0 ��e2 being epicyclic frequency, de-

fined as �e2=4�2�1+ s

2�d�ds �, � is the rotational frequency�.

An instability with a high growth rate was observed. In anoffshoot of this work, we investigate the behavior of insta-bility reported in �19� in the Keplerian flow. It is to be notedthat the only difference between the work carried by Bon-anno and Urpin �18� and us is the inclusion of parameterspace which was not considered in their study.

The paper is organized as follows: the Sec. II investigatesthe growth rate of instability. The Sec. III deals with theResults and Discussions. Finally, we conclude by summariz-ing the new findings.

II. INSTABILITY CRITERIA

Paper I considered an axisymmetric differentially rotatingsystem in the presence of a magnetic field. A cylindricalcoordinate system �s ,� ,z� with s being the radial distancefrom the rotation axis was constructed. Unperturbed systemwas described by �vr ,v� ,vz�= �0,s� ,0�. Furthermore, �where � being the angular velocity of the astrophysical flow,*mradul@barc.gov.in

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was taken to be approximately a function of s alone; i.e. �=��s�. The isothermal flow for a compressible fluid wasdescribed by the following MHD equations

v� + �v� · ��v� = −�p

�+ g� +

1

4���� B� � B� , �1�

� + � · ��v�� = 0, �2�

p + v� · �p + p � · v� = 0, �3�

B� − � �v� B� � + � � �� B� � = 0, �4�

� · B� = 0, �5�

where � and v� are the density and fluid velocity, respectively;p is the gas pressure; g� is gravity; B� is the magnetic field, �is the magnetic diffusivity, and is the adiabatic index. Mag-netic field has non vanishing radial and azimuthal compo-nents.

Quasistationary state for a differentially rotating flow withmagnetic field was considered. Axisymmetric Eulerian per-turbations with space time dependence �exp� t− ik� ·r�� wherek� = �ks ,0 ,kz� were introduced in the unperturbed accretiondisk and the dispersion relation was obtained for a specialcase of k� .B� =0 after neglecting the Ohmic dissipation in theinduction equation.

We take the dispersion relation of paper I for determiningthe stability of axisymmetric short wavelength perturbations.The dispersion relation �Eq. �16� of paper I� is given by the

5 + 3��02 + �e

2� + 2�B�3 + ��e

2�02 + ��e

2�B�3 = 0,

�6�

where

�e2 = 2��2� + s

d�

ds�, �0

2 = k2�cs2 + cm

2 � , � = kz2/k2,

cm2 =

B2

4��, cs

2 =p

�, �B�

3 =k2B�Bss��

4��, �� =

d�

ds;

Equation �6� is a polynomial of degree five, so five nontrivial roots exist. We apply Routh-Hurwitz method �20� �seeAppendix� to find the regions describing instabilities. Thismethod has been applied in the field of astrophysics veryfrequently �18,21–25�. Instability exist if any of the condi-tions written below is satisfied.

��e2�B�

3 � 0, �B�3 � 0, ��B�

3 �2 � 0. �7�

From the above inequalities, it is clear that the instabilitywill exist only when �B�

3 �0It is to be noted that only those perturbations are consid-

ered here in which the wave vector is perpendicular to themagnetic field, i.e. k� ·B� =0.

Let us investigate the growth rate of the hydromagneticinstability. To calculate the growth rate of this instability, it isconvenient to introduce the dimensionless quantities

� =

�e, � =

1

x2

�02

�e2 , � =

1

x2

�B�3

�e3 , x = ks . �8�

The polynomial given by Eq. �8� becomes

�5 + �3��x2 + 1� + �2�x2 + ���x2 + ��x2 = 0 �9�

This equation is solved numerically �see �26� for details�by computing the eigen values of the matrix whose charac-teristic polynomial is given by Eq. �9� for different values of�, �, and � parameters.

Since we are interested in the instabilities, only real roots�perturbations �exp� t− ik� ·r��� are considered. We considerthe parameter space constrained by ��� and investigate thegrowth of instability for different values of � and �, keepingin mind ���. It is to be noted that this parameter space wasnot considered by Bonanno and Urpin �18�. The Figs. 1–3shows the dependence of Re ��� for �=0.3. The value of � isfixed to 1.0, 2.0, and 3.0 and the values of the parameter � isvaried for �=0.05,0.1,0.2. Similar trend is observed in allthe cases, i.e. the growth rate of instability increases with thedecrease in �. For a value of �=0.05, growth rate of theinstability at x2=500 increases from 3.5�e to 4.6�e and5.5�e for �=1.0,2.0,3.0, respectively. As x2 increases fur-ther, at x2=700, the growth rate of instability increases from3.8�e for �=1.0 to 5.1�e for �=2 to 6.0�e for �=3. It is

FIG. 1. The dependence of real parts of the root � on x2 for�=0.3, �=1.0 and �=0.2,0.5,0.05.

FIG. 2. The dependence of real parts of the root � on x2 for�=0.3, �=2.0 and �=0.2,0.5,0.05.

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evident that a vigorous instability is operating in the abovedescribed parameter space.

III. RESULTS AND DISCUSSION

We revisit the problem of hydromagnetic instability indifferentially rotating compressible flows and analyze the in-stability. The parameter space described by ��� was inves-tigated and discovered a vigorous instability. The behavior ofthe instability for a typical value of � and � ��=0.3, �=1.0,2.0,3.0, �=0.05,0.1,0.2� was studied. It is observedthat the instability growth rate increases with the increase of� and it also depends on the wave number. The growth rateof the instability becomes very high, as high as ��6.0�� forx2=700 at �=3, �=0.05�.

In a recent work, Bonanno and Urpin �18� carried out thestudies for a compressible flow and showed the presence of anew instability. The notable point of their instability was thatit existed for any value of magnetic field, the only conditionbeing �B�

3 �0. The major difference between the studies car-ried out by Kim and Ostriker �16�, Pessah and Psaltis �17�,and Bonanno and Urpin �18� is that, in the later case, theyconsidered a field with nonvanishing radial and azimuthalcomponent. We investigated a parameter space and noticed ahigh growth rate instability in that region. Recall that theMRI does not exist for k� ·B� =0. Since present study was car-ried out for k� ·B� =0, MRI does not operate in our case. Itshould also be noted that for the incompressible case, i.e.cs→�, polynomial �Eq. �8�� becomes

� 2 + ��e2� = 0. �10�

It is clear that the present instability appears only for com-pressible fluid. It is to be noted that the radial component ofmagnetic field is nonzero in the present case. Radial field inthe presence of differential rotation shears into an azimuthalfield resulting in an increase in the magnitude of B�. As anexample, Desch �27� describes that if B�=Br in a Kepleriandisk at the beginning of one orbit, by the end of the orbit B�

increases to �10Br. After two orbits, B� increases evenmore, to �20Br. This problem, i.e. the increase in the azi-muthal field in the present case was avoided by treating thestate as a quasistationary state. When � is small, one can

obtain from Eq. �4� that the azimuthal field grows approxi-mately linearly with time,

B��t� = B��0� + s��Bst , �11�

where ��=d� /ds, and B��0� is the azimuthal field at t=0.As long as the second term on the rhs is small compared tothe first one, and

t � �� =1

s��

B��0�Bs

, �12�

stretching of the azimuthal field does not affect significantlythe basic state; �� is the characteristic time scale of genera-tion of B�. As a result, the basic state can be treated asquasistationary during the time t���. In recent times, Bon-anno and Urpin has shown that this instability exists in manycases of interest �21,28�. The presence of this instability hasalso been demonstrated in the case of protostellar disks �29�also. It is clear that the instability discussed in the presentpaper appears in many astrophysical situations including pro-toplanatery disk �albeit with small growth rate�.

A. Difference between MRI and the present instability

This instability is different and independent of the mag-netorotational instability. The necessary and sufficient condi-tion for this instability to occur is �B�

3 �0, i.e. B�Bss���0,��= d�

ds ; therefore,�1� The instability exists for nonvanishing radial and azi-

muthal component of magnetic field whereas the poloidalcomponent of magnetic field is important for MRI to occur.MRI can exits for vanishing radial and azimuthal fields.

�2� The instability exists for any differential rotation, i.e.for any sign of d�

ds whereas MRI exists only when the angularvelocity in the disk decreases outwards, i.e. d�

ds �0.�3� The instability exists even in sufficiently strong mag-

netic field which suppresses the magnetorotational instabil-ity. MRI exists only in the weak field limit.

A special case was considered in the present report wherewave vector is perpendicular to the magnetic field, i.e. k� .B�=0. Since the growth rate of MRI is directly proportional tothe k� .B� , MRI does not exists in this case. This is one of themost important differences between the present instabilityand the MRI.

B. Where the instability operates

It is observed that a high growth rate is reached for largevalues of ��� ���1�. If s d�

ds ���typical in keplerian disk�,the magnetic energy density is greater than the rotationalenergy density. Therefore, the present instability will be op-erating in a regime where “magnetic energy density morethat the rotational energy density” can be realized. For anaccretion disk, the rotational energy �essentially by defini-tion� is larger than all other energies: thermal, magnetic, andradiation. So, the rotationally supported accretion disk is outof question to envisage the regions where magnetic energydensity is more than rotational energy density. One possiblerealization of such regions might appear when one inches

FIG. 3. The dependence of real parts of the root � on x2 for�=0.3, �=3.0 and �=0.2,0.5,0.05.

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toward the inner most region of accretion disk, where thedisk starts truncating/evaporating and the rotational supportstarts getting diminishing and magnetic field takes over. Astandard pulsar �or even more a magnetar� meets this condi-tion. Consider a 1012 G surface field on a pulsar. Severalscale heights away from the star the density is very small, sothe rotational energy density is much less than the magneticenergy density. If we consider the low-density regions farabove the disk then any magnetic bubble that floats awayafter reconnection could temporarily have a larger magneticenergy density than rotational energy density.

The scenario closest to our regime was considered by Be-gelman and Pringle �30� where magnetic pressure exceedsthe combined gas+radiation pressure in the disk but not therotational energy density. For magnetohydrodynamic insta-bilities in accretion disk, the balance is achieved via the jointaction of rotation, pressure gradients and magnetic tension.For the regions mentioned above, one cannot count on rota-tion. It is important to identify a robust equilibrium aboutwhich to perturb, perhaps gradients in pressure or radiationpressure could help in the inner disk regions. One needs tosolve such scenario theoretically. We will consider this sce-nario in a future work.

IV. CONCLUSION

In the present report, we have demonstrated the presenceof very high growth rate instability in a differentially rotatingcompressible flows. It has been shown that the growth rate ofthis instability is very high and for a special case consideredabove, the growth rate increases to as high as �6� for x2

=700. It is observed that the present instability might beoperating in regions where magnetic energy density is morethan the rotational energy density.

ACKNOWLEDGMENTS

The author thanks C. Miller, V. Urpin, and B. P. Pandeyfor the insightful discussions. The author also thanks both theanonymous referees for their useful comments. This researchhas made use of NASA’s Astrophysics Data System and arX-iv.org e-print archive.

APPENDIX: HURWITZ METHOD

Let us consider an fifth order polynomial P�x�

P�x� = a5x5 + a4x4 + a3x3 + a2x2 + . . . . . + a0. �A1�

The Hurwitz theorem states that the above polynomial willbe unstable if any of the following inequalities is satisfied�24�:

a0 � 0,

A1 � a4a3 − a2 � 0,

A2 � a2�a4a3 − a2� − a4�a4a1 − a0� � 0,

A3 � �a4a1 − a0��a2�a4a3 − a2� − a4�a4a1 − a0��

− a0�a4a3 − a2�2 � 0, �A2�

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