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disk accretion is one of the most fundamental processes occurring in a variety of astrophysical objects like accretion powered x - ray pulsars , protostars , cataclysmic variables etc .
it is observed that the molecular viscosity is too small to account for the observed time scales and luminosities @xcite . to circumvent this problem , shakura and
sunyaev @xcite made a normalization of the coefficient of viscosity in the theory of turbulent viscosity proposed by heisenberg @xcite and introduced a dimensionless parameter @xmath0 , where @xmath1 and @xmath2 were the sound speed and the disk scale height respectively .
though , the @xmath3 model is quite successful in explaining many astrophysical observations , a degree of ad - hocness enters the model through the parameter @xmath3 .
the presence of rayleigh criterion of hydrodynamic stability , i.e. a flow with specific angular momentum increasing monotonically , being satisfied in accretion disk , rules out the hydrodynamic origin of turbulence , though there is literature available @xcite in its favor .
the origin of turbulence is still a question .
differential rotation can lead to an instability provided the specific angular momentum decreases outward , a condition generally not satisfied in the accretion disk , though there are reports suggesting such regions being realized locally @xcite .
a recent work @xcite numerically demonstrated the presence of such regions . though differential rotation alone
can not lead to instability , inclusion of magnetic field alters the scenario altogether .
it was well known from the classical works of velikhov @xcite and chandra @xcite that a differentially rotating flow with a negative angular velocity gradient in a weak magnetic field is unstable .
the presence of this instability in accretion disk was established by balbus & hawley @xcite after three decades that the subthermal fields in the presence of differential rotation can cause magnetorotational instability ( mri ) , which can provide the physical basis for the transport of angular momentum in accretion disk .
it has been computationally shown @xcite that mri is capable of introducing turbulence in the accretion disk .
it is to be noted that mri is essentially an instability of incompressible flow and its growth starts suppressing as the compressibility seeps in . the earliest attempt to understand the effect of compressibility on the growth of mri was made by blaes & balbus @xcite .
they concluded that the @xmath4 does not affect the instability .
later on , the behavior of compressible mri in the mhd flows was addressed by kim & ostriker @xcite .
it was demonstrated that when the magnetic field strength is superthermal , the inclusion of toroidal fields tends to suppress the growth of the mri , and that for quasi - toroidal field configurations no axisymmetric mri takes place in the limit @xmath5 . in an another work ,
pessah & psaltis @xcite studied the role of toroidal fields in compressible flows .
it was again demonstrated that the growth rate of mri is affected in the compressible flow and mri stabilizes in superthermal fields .
clearly , the compressibility plays an important role in the dynamics of instabilities operating in the accretion flow .
recently , bonanno and urpin @xcite ( henceforth paper i ) studied the effect of compressibility on the instabilities in the presence of magnetic field .
they considered the magnetic field with non vanishing radial and azimuthal components .
a new instability was observed which survived for all values of magnetic field , unlike mri which survives only in the weak field limit
. the maximum growth rate of the reported instability was @xmath6 where @xmath7 is the rotation frequency .
in a recent work @xcite , we investigated the behavior of above instability for a special case @xmath8 ( @xmath9 being epicyclic frequency , defined as @xmath10 ) , @xmath11 is the rotational frequency ) .
an instability with a high growth rate was observed . in an offshoot of this work
, we investigate the behavior of instability reported in @xcite in the keplerian flow .
it is to be noted that the only difference between the work carried by bonanno and urpin @xcite and us is the inclusion of parameter space which was not considered in their study .
the paper is organized as follows : the section ii investigates the growth rate of instability .
the section iii deals with the results and discussions . finally , we conclude by summarizing the new findings .
paper i considered an axisymmetric differentially rotating system in the presence of a magnetic field . a cylindrical coordinate system ( @xmath12 , @xmath13 , @xmath14 ) with s being the radial distance from the rotation axis was constructed .
unperturbed system was described by ( @xmath15 ) = ( @xmath16 ) .
furthermore , @xmath11 where @xmath11 being the angular velocity of the astrophysical flow , was taken to be approximately a function of s alone ; i.e. @xmath17 .
the isothermal flow for a compressible fluid was described by the mhd equations    @xmath18    @xmath19    @xmath20    @xmath21    @xmath22    where @xmath23 and @xmath24 are the density and fluid velocity , respectively ; @xmath25 is the gas pressure ; @xmath26 is gravity ; @xmath27 is the magnetic field , @xmath28 is the magnetic diffusivity , and @xmath29 is the adiabatic index .
magnetic field has non vanishing radial and azimuthal components .
quasistationary state for a differentially rotating flow with magnetic field was considered .
axisymmetric eulerian perturbations with space time dependence @xmath30 where @xmath31 were introduced in the unperturbed accretion disk and the dispersion relation was obtained for a special case of @xmath32 after neglecting the ohmic dissipation in the induction equation .
we take the dispersion relation of paper i for determining the stability of axisymmetric short wavelength perturbations .
the dispersion relation ( eq .
( @xmath33 ) of paper i ) is given by the @xmath34 where @xmath35    equation ( [ e : eq1 ] ) is a polynomial of degree five , so five non trivial roots exist .
we apply routh - hurwitz method ( @xcite ) ( see appendix ) to find the regions describing instabilities .
this method has been applied in the field of astrophysics very frequently ( @xcite ) instability exist if any of the conditions written below is satisfied .
@xmath36    from the above inequalities , it is clear that the instability will exist only when @xmath37    it is to be noted that only those perturbations are considered here in which the wavevector is perpendicular to the magnetic field , i.e. @xmath38 .    let us investigate the growth rate of the hydromagnetic instability . to calculate the growth rate of this instability , it is convenient to introduce the dimensionless quantities @xmath39 the polynomial given by eq .
( [ e : eq4 ] ) becomes @xmath40    this equation is solved numerically ( see @xcite for details ) by computing the eigen values of the matrix whose characteristic polynomial is given by equation ( [ e : eq5 ] ) for different values of @xmath41 , @xmath42 and @xmath43 parameters .
since we are interested in the instabilities , only real roots ( perturbations @xmath30 ) are considered .
we consider the parameter space constrained by @xmath44 and investigate the growth of instability for different values of @xmath42 and @xmath43 , keeping in mind @xmath45 .
it is to be noted that this parameter space was not considered by bonanno and urpin @xcite .
the fig.(1 ) , ( 2 ) , & ( 3 ) shows the dependence of re ( @xmath46 ) for @xmath47 .
the value of @xmath43 is fixed to @xmath48 , @xmath49 and @xmath50 and the values of the parameter @xmath42 is varied for @xmath51 .
similar trend is observed in all the cases , i.e. the growth rate of instability increases with the decrease in @xmath42 . for a value of @xmath52 , growth rate of the instability at @xmath53 increases from @xmath54 to @xmath55 and @xmath56 for @xmath57 respectively .
as @xmath58 increases further , at @xmath59 , the growth rate of instability increases from @xmath60 for @xmath61 to @xmath62 for @xmath63 to @xmath64 for @xmath65 .
it is evident that a vigorous instability is operating in the above described parameter space .
we revisit the problem of hydromagnetic instability in differentially rotating compressible flows and analyze the instability .
the parameter space described by @xmath45 was investigated and diskovered a vigorous instability .
the behavior of the instability for a typical value of @xmath43 and @xmath42 ( @xmath66 ) was studied .
it is observed that the instability growth rate increases with the increase of @xmath43 and it also depends on the wavenumber .
the growth rate of the instability becomes very high , as high as ( @xmath67 @xmath68 ) for @xmath69 at @xmath70 , @xmath52 ) .    in a recent work ,
bonanno and urpin @xcite carried out the studies for a compressible flow and showed the presence of a new instability .
the notable point of their instability was that it existed for any value of magnetic field , the only condition being @xmath37 .
the major difference between the studies carried out by kim & ostriker @xcite , pessah & psaltis @xcite and bonanno & urpin @xcite is that , in the later case , they considered a field with non vanishing radial and azimuthal component .
we investigated a parameter space and noticed a high growth rate instability in that region . recall that the mri does not exist for @xmath38 .
since present study was carried out for @xmath38 , mri does not operate in our case .
it should also be noted that for the incompressible case , i.e .
@xmath71 , polynomial ( [ e : eq4 ] ) becomes @xmath72 it is clear that the present instability appears only for compressible fluid .
it is to be noted that the radial component of magnetic field is non zero in the present case .
radial field in the presence of differential rotation shears into an azimuthal field resulting in an increase in the magnitude of @xmath4 . as an example
, desch @xcite describes that if @xmath4 = @xmath73 in a keplerian disk at the beginning of one orbit , by the end of the orbit @xmath4 increases to @xmath74 .
after two orbits , @xmath4 increases even more , to @xmath75 .
this problem , i.e. the increase in the azimuthal field in the present case was avoided by treating the state as a quasistationary state .
when @xmath28 is small , one can obtain from eq .
( 4 ) that the azimuthal field grows approximately linearly with time , @xmath76 where @xmath77 , and @xmath78 is the azimuthal field at @xmath79 . as long as the second term on the r.h.s . is small compared to the first one , and @xmath80 stretching of the azimuthal field does not affect significantly the basic state ; @xmath81 is the characteristic timescale of generation of @xmath82 . as a result
, the basic state can be treated as quasi - stationary during the time @xmath83 . in recent times
, bonanno and urpin has shown that this instability exists in many cases of interest @xcite .
the presence of this instability has also been demonstrated in the case of protostellar disks @xcite also .
it is clear that the instability discussed in the present report appears in many astrophysical situations including protoplanatery disk ( albeit with small growth rate ) .
this instability is different and independent of the magnetorotational instability .
the necessary and sufficient condition for this instability to occur is @xmath37 , i.e. @xmath84 ; therefore ,    \1 ) the instability exists for non vanishing radial and azimuthal component of magnetic field whereas the poloidal component 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 @xmath85 whereas mri exists only when the angular velocity in the disk decreases outwards , i.e. @xmath86 .
\3 ) the instability exists even in sufficiently strong magnetic field which suppresses the magnetorotational instability .
mri exists only in the weak field limit .
a special case was considered in the present report where wavevector is perpendicular to the magnetic field , i.e. @xmath32 . since the growth rate of mri is directly proportional to the @xmath87 , mri does not exists in this case .
this is one of the most important differences between the present instability and the mri .
it is observed that a high growth rate is reached for large values of @xmath88 ( @xmath89 ) . if @xmath90(typical in keplerian disk ) ,
the magnetic energy density is greater than the rotational energy density .
therefore , the present instability will be operating in a regime where `` magnetic energy density more that the rotational energy density '' can be realized . for an accretion disk , the rotational energy ( essentially by definition )
is larger than all other energies thermal , magnetic , radiation .
so , the rotationally supported accretion disk are out of question to envisage the regions where magnetic energy density is more than rotational energy density .
one possible realization of such regions might appear when one inches towards the inner most region of accretion disk , where the disk starts truncating / evaporating and the rotational support starts getting diminishing and magnetic field takes over .
a standard pulsar ( or even more a magnetar ) meets this condition .
consider a @xmath91 g surface field on a pulsar .
several scale heights away from the star the density is very small , so the rotational energy density is much less than the magnetic energy density .
if we consider the low - density regions far above the disk then any magnetic bubble that floats away after reconnection could temporarily have a larger magnetic energy density than rotational energy density .
the scenario closest to our regime was considered by begelman and pringle @xcite where magnetic pressure exceeds the combined gas + radiation pressure in the disk but not the rotational energy density . for magnetohydrodynamic instabilities in accretion disk ,
the balance is achieved via the joint action of rotation , pressure gradients and magnetic tension . for the regions mentioned above
, one can not count on rotation .
it is important to identify a robust equilibrium about which to perturb , perhaps gradients in pressure or radiation pressure could help in the inner disk regions .
one needs to solve such scenario theoretically .
we will consider this scenario in a future work .
in the present report , we have demonstrated the presence of very high growth rate instability in a differentially rotating compressible flows .
it has been shown that the growth rate of this instability is very high and for a special case considered above , the growth rate increases to as high as @xmath92 for @xmath59 .
it is observed that the present instability might be operating in regions where magnetic energy density is more than the rotational energy density .
the author thanks c. miller , v. urpin and b.p.pandey for the insightful discussions .
the author also thanks both the anonymous referees for their useful comments .
this research has made use of nasa s astrophysics data system and arxiv.org e - print archive .
let us consider an @xmath93 order polynomial p(x ) @xmath94 the hurwitz theorem states that the above polynomial will be unstable if any of the following inequalities is satified @xcite .
@xmath95 - } \nonumber \\ & & \quad \quad \quad \quad \quad \quad - a_{0 } ( a_{4 } a_{3 }   - a_{2})^{2 } < 0 \ ; , \label{13}\end{aligned}\ ] ]
the transport of angular momentum in the outward direction is the fundamental requirement for accretion to proceed in an accretion disk . 
 this objective can be achieved if the accretion flow is turbulent . 
 instabilities are one of the sources for the turbulence . 
 we study a differentially rotating compressive flow in the presence of non vanishing radial and azimuthal magnetic field and demonstrate the occurrence of a high growth rate instability . 
 this instability operates in a region where magnetic energy density exceeds the rotational energy density .