Growing pseudo-eigenmodes and positive logarithmic norms in rotating shear flows

Rotating shear flows, when angular momentum increases and angular velocity decreases as functions of radiation coordinate, are hydrodynamically stable under linear perturbation. The Keplerian flow is an example of such systems which appears in astrophysical context. Although decaying eigenmodes exhibit large transient energy growth of perturbation which could govern nonlinearity into the system, the feedback of inherent instability to generate turbulence seems questionable. We show that such systems exhibiting growing pseudo-eigenmodes easily reach an upper bound of growth rate in terms of the logarithmic norm of the involved nonnormal operators, thus exhibiting feedback of inherent instability. This supports the existence of turbulence of hydrodynamic origin in the Keplerian accretion disc in astrophysics. Hence, this enlightens the mismatch between the linear theory and experimental/observed data and helps in resolving the outstanding question of origin of turbulence therein.Comment: 12 pages including 4 figures; to appear in New Journal of Physic

Growing hydrodynamic modes in Keplerian accretion disks during secondary perturbations: Elliptical vortex effects

The origin of hydrodynamic turbulence, and in particular of an anomalously enhanced angular momentum transport, in accretion disks is still an unsolved problem. This is especially important for cold disk systems which are practically neutral in charge and therefore turbulence can not be of magnetohydrodynamic origin. While the flow must exhibit some instability and then turbulence in support of the transfer of mass inward and angular momentum outward, according to the linear perturbation theory, in absence of magnetohydrodynamic effects, it should always be stable. We demonstrate that the three-dimensional secondary disturbance to the primarily perturbed disk, consisting of elliptical vortices, gives significantly large hydrodynamic growth in such a system and hence may suggest a transition to an ultimately turbulent state. This result is essentially applicable to accretion disks around quiescent cataclysmic variables, in proto-planetary and star-forming disks, the outer region of disks in active galactic nuclei, where the gas is significantly cold and thus the magnetic Reynolds number is smaller than 10^4.Comment: 21 pages including 4 figures, aastex format; Accepted for publication in The Astrophysical Journa

Bypass to Turbulence in Hydrodynamic Accretion: Lagrangian Analysis of Energy Growth

Despite observational evidence for cold neutral astrophysical accretion disks, the viscous process which may drive the accretion in such systems is not yet understood. While molecular viscosity is too small to explain the observed accretion efficiencies by more than ten orders of magnitude, the absence of any linear instability in Keplerian accretion flows is often used to rule out the possibility of turbulent viscosity. Recently, the fact that some fine tuned disturbances of any inviscid shear flow can reach arbitrarily large transient growth has been proposed as an alternative route to turbulence in these systems. We present an analytic study of this process for 3D plane wave disturbances of a general rotating shear flow in Lagrangian coordinates, and demonstrate that large transient growth is the generic feature of non-axisymmetric disturbances with near radial leading wave vectors. The maximum energy growth is slower than quadratic, but faster than linear in time. The fastest growth occurs for two dimensional perturbations, and is only limited by viscosity, and ultimately by the disk vertical thickness. After including viscosity and vertical structure, we find that, as a function of the Reynolds number, R, the maximum energy growth is approximately 0.4 (R/log R)^{2/3}, and put forth a heuristic argument for why R > 10^4 is required to sustain turbulence in Keplerian disks. Therefore, assuming that there exists a non-linear feedback process to replenish the seeds for transient growth, astrophysical accretion disks must be well within the turbulent regime. However, large 3D numerical simulations running for many orbital times, and/or with fine tuned initial conditions, are required to confirm Keplerian hydrodynamic turbulence on the computer.Comment: 25 preprint pages, 2 figures, some modifications mainly to the Discussions section, Accepted for publication in Ap

Description of Pseudo-Newtonian Potential for the Relativistic Accretion Disk around Kerr Black Holes

We present a pseudo-Newtonian potential for accretion disk modeling around the rotating black holes. This potential can describe the general relativistic effects on accretion disk. As the inclusion of rotation in a proper way is very important at an inner edge of disk the potential is derived from the Kerr metric. This potential can reproduce all the essential properties of general relativity within 10% error even for rapidly rotating black holes.Comment: 5 Latex pages including 1 figure. Version to appear in Astrophysical Journal, V-581, N-1, December 10, 200

Dynamics of electromagnetic waves in Kerr geometry

Here we are interested to study the spin-1 particle i.e., electro-magnetic wave in curved space-time, say around black hole. After separating the equations into radial and angular parts, writing them according to the black hole geometry, say, Kerr black hole we solve them analytically. Finally we produce complete solution of the spin-1 particles around a rotating black hole namely in Kerr geometry. Obviously there is coupling between spin of the electro-magnetic wave and that of black hole when particles propagate in that space-time. So the solution will be depending on that coupling strength. This solution may be useful to study different other problems where the analytical results are needed. Also the results may be useful in some astrophysical contexts.Comment: 15 Latex pages, 4 Figures; Accepted for publication in Classical and Quantum Gravit

Behaviour of spin-1/2 particle around a charged black hole

Dirac equation is separable in curved space-time and its solution was found for both spherically and axially symmetric geometry. But most of the works were done without considering the charge of the black hole. Here we consider the spherically symmetric charged black hole background namely Reissner-Nordstrom black hole. Due to presence of the charge of black-hole charge-charge interaction will be important for the cases of incoming charged particle (e.g. electron, proton etc.). Therefore both gravitational and electromagnetic gauge fields should be introduced. Naturally behaviour of the particle will be changed from that in Schwarzschild geometry. We compare both the solutions. In the case of Reissner-Nordstrom black hole there is a possibility of super-radiance unlike Schwarzschild case. We also check this branch of the solution.Comment: 8 Latex pages and 4 Figures; RevTex.style; Accepted for Publication in Classical and Quantum Gravit

Bypass to Turbulence in Hydrodynamic Accretion Disks: An Eigenvalue Approach

Cold accretion disks such as those in star-forming systems, quiescent cataclysmic variables, and some active galactic nuclei, are expected to have neutral gas which does not couple well to magnetic fields. The turbulent viscosity in such disks must be hydrodynamic in origin, not magnetohydrodynamic. We investigate the growth of hydrodynamic perturbations in a linear shear flow sandwiched between two parallel walls. The unperturbed flow is similar to plane Couette flow but with a Coriolis force included. Although there are no exponentially growing eigenmodes in this system, nevertheless, because of the non-normal nature of the eigenmodes, it is possible to have a large transient growth in the energy of perturbations. For a constant angular momentum disk, we find that the perturbation with maximum growth has a wave-vector in the vertical direction. The energy grows by more than a factor of 100 for a Reynolds number R=300 and more than a factor of 1000 for R=1000. Turbulence can be easily excited in such a disk, as found in previous numerical simulations. For a Keplerian disk, on the other hand, similar vertical perturbations grow by no more than a factor of 4, explaining why the same simulations did not find turbulence in this system. However, certain other two-dimensional perturbations with no vertical structure do exhibit modest growth. For the optimum two-dimensional perturbation, the energy grows by a factor of ~100 for R~10^4.5 and by a factor of 1000 for R~10^6. It is conceivable that these two-dimensional disturbances might lead to self-sustained turbulence. The Reynolds numbers of cold astrophysical disks are much larger even than 10^6, therefore, hydrodynamic turbulence may be possible in disks.Comment: 39 pages including 9 figures; Final version to appear in The Astrophysical Journa

Stability of accretion disk around rotating black holes: a pseudo-general-relativistic fluid dynamical study

We discuss the solution of accretion disk when the black hole is chosen to be rotating. We study, how the fluid properties get affected for different rotation parameters of the black hole. We know that no cosmic object is static in Universe. Here the effect of the rotation of the black hole to the space-time is considered following an earlier work of the author, where the pseudo-Newtonian potential was prescribed for the Kerr geometry. We show that, with the inclusion of rotation of the black hole, the valid disk parameter region dramatically changes and disk becomes unstable. Also we discuss about the possibility of shock in accretion disk around rotating black holes. When the black hole is chosen to be rotating, the sonic locations of the accretion disk get shifted or disappear, making the disk unstable. To bring it in the stable situation, the angular momentum of the accreting matter has to be reduced/enhanced (for co/counter-rotating disk) by means of some physical process.Comment: 24 Latex pages including 7 figures; Accepted for publication in Astrophysical Journa

Scalar and Spinor Perturbation to the Kerr-NUT Spacetime

We study the scalar and spinor perturbation, namely the Klein-Gordan and Dirac equations, in the Kerr-NUT space-time. The metric is invariant under the duality transformation involving the exchange of mass and NUT parameters on one hand and radial and angle coordinates on the other. We show that this invariance is also shared by the scalar and spinor perturbation equations. Further, by the duality transformation, one can go from the Kerr to the dual Kerr solution, and vice versa, and the same applies to the perturbation equations. In particular, it turns out that the potential barriers felt by the incoming scalar and spinor fields are higher for the dual Kerr than that for the Kerr. We also comment on existence of horizon and singularity.Comment: 31 pages including 20 figures, RevTeX style: Final version to appear in Classical and Quantum Gravit

Stability of Accretion Disks in Presence of Nucleosynthesis

We study the effect of nuclear reaction on a thin, axisymmetric, differentially rotating, inviscid, steady accretion flow around a black hole from an analytical point of view. We find that for most of the reasonable disk parameters, when $p-p$-reaction, dissociation of deuterium and helium are taken into account, the transonic region of the disk continues to have the inner sonic point and if the temperature of the flow at the injection sonic point could be raised (by say, some heating processes) the flow would to pass through this inner sonic point. Otherwise, the flow may be unstable. We use the sonic point analysis to study the solution. In the rest of the disk parameters the inner sonic point is absent altogether and the flow will definitely be unstable.Comment: 18 Latex pages, 6 figures; Accepted for publication in ApJ; aaspp4.st