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Register a new userHydrodynamical instability with noise in the Keplerian accretion discs: Modified Landau equation
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Added by
Banibrata Mukhopadhyay
Publication date
2020
fields
Physics
and research's language is
English
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Origin of hydrodynamical instability and turbulence in the Keplerian accretion disc as well as similar laboratory shear flows, e.g. plane Couette flow, is a long standing puzzle. These flows are linearly stable. Here we explore the evolution of perturbation in such flows in the presence of an additional force. Such a force, which is expected to be stochastic in nature hence behaving as noise, could be result of thermal fluctuations (however small be), Brownian ratchet, grain-fluid interactions and feedback from outflows in astrophysical discs etc. We essentially establish the evolution of nonlinear perturbation in the presence of Coriolis and external forces, which is modified Landau equation. We show that even in the linear regime, under suitable forcing and Reynolds number, the otherwise least stable perturbation evolves to a very large saturated amplitude, leading to nonlinearity and plausible turbulence. Hence, forcing essentially leads a linear stable mode to unstable. We further show that nonlinear perturbation diverges at a shorter timescale in the presence of force, leading to a fast transition to turbulence. Interestingly, emergence of nonlinearity depends only on the force but not on the initial amplitude of perturbation, unlike original Landau equation based solution.
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We attempt to address the old problem of plane shear flows: the origin of turbulence and hence transport of angular momentum in accretion flows as well as laboratory flows, such as plane Couette flow. We undertake the problem by introducing an extra force in Orr-Sommerfeld and Squire equations along with the Coriolis force mimicking the local region of the accretion disk. For plane Couette flow, the Coriolis term drops. Subsequently we solve the equations by WKB approximation method. We investigate the dispersion relation for the Keplerian flow and plane Couette flow for all possible combinations of wave vectors. Due to the very presence of extra force, we show that both the flows are unstable for a certain range of wave vectors. However, the nature of instability between the flows is different. We also study the Argand diagrams of the perturbation eigenmodes. It helps us to compare the different time scales corresponding to the perturbations as well as accretion. We ultimately conclude with this formalism that fluid gets enough time to be unstable and hence plausibly turbulent particularly in the local regime of the Keplerian accretion disks. Repetition of the analysis throughout the disk explains the transport of angular momentum and matter along outward and inward direction respectively.
We examine the properties of strongly magnetized accretion discs in a global framework, with particular focus on the evolution of magnetohydrodynamic instabilities such as the magnetorotational instability (MRI). Work by Pessah and Psaltis showed that MRI is stabilized beyond a critical toroidal field in compressible, differentially rotating flows and, also, reported the appearance of two new instabilities beyond this field. Their results stemmed from considering geometric curvature effects due to the suprathermal background toroidal field, which had been previously ignored in weak-field studies. However, their calculations were performed under the local approximation, which poses the danger of introducing spurious behavior due to the introduction of global geometric terms in an otherwise local framework. In order to avoid this, we perform a global eigenvalue analysis of the linearized MHD equations in cylindrical geometry. We confirm that MRI indeed tends to be highly suppressed when the background toroidal field attains the Pessah-Psaltis limit. We also observe the appearance of two new instabilities that emerge in the presence of highly suprathermal toroidal fields. These results were additionally verified using numerical simulations in PLUTO. There are, however, certain differences between the the local and global results, especially in the vertical wavenumber occupancies of the various instabilities, which we discuss in detail. We also study the global eigenfunctions of the most unstable modes in the suprathermal regime, which are inaccessible in the local analysis. Overall, our findings emphasize the necessity of a global treatment for accurately modeling strongly magnetized accretion discs.
Hydromagnetic stresses in accretion discs have been the subject of intense theoretical research over the past one and a half decades. Most of the disc simulations have assumed a small initial magnetic field and studied the turbulence that arises from the magnetorotational instability. However, gaseous discs in galactic nuclei and in some binary systems are likely to have significant initial magnetisation. Motivated by this, we performed ideal magnetohydrodynamic simulations of strongly magnetised, vertically stratified discs in a Keplerian potential. Our initial equilibrium configuration, which has an azimuthal magnetic field in equipartion with thermal pressure, is unstable to the Parker instability. This leads to the expelling of magnetic field arcs, anchored in the midplane of the disc, to around five scale heights from the midplane. Transition to turbulence happens primarily through magnetorotational instability in the resulting vertical fields, although magnetorotational shear instability in the unperturbed azimuthal field plays a significant role as well, especially in the midplane where buoyancy is weak. High magnetic and hydrodynamical stresses arise, yielding an effective $alpha$-value of around 0.1 in our highest resolution run. Azimuthal magnetic field expelled by magnetic buoyancy from the disc is continuously replenished by the stretching of a radial field created as gas parcels slide in the linear gravity field along inclined magnetic field lines. This dynamo process, where the bending of field lines by the Parker instability leads to re-creation of the azimuthal field, implies that highly magnetised discs are astrophysically viable and that they have high accretion rates.
The extent of mixed regions around convective zones is one of the biggest uncertainties in stellar evolution. 1D overshooting descriptions introduce a free parameter ($f_{ov}$) that is in general not well constrained from observations. Especially in small central convective regions the value is highly uncertain due to its tight connection to the pressure scale height. Long-term multi-dimensional hydrodynamic simulations can be used to study the size of the overshooting region and the involved mixing processes. Here we show how one can calibrate an overshooting parameter by performing 2D Maestro simulations of Zero-Age-Main-Sequence stars ranging from $1.3$ to $3.5 M_odot$. The simulations cover the convective cores of the stars and a large fraction of the surrounding radiative envelope. We follow the convective flow for at least 20 convective turnover times, while the longest simulation covers 430 turnover time scales. This allows us to study how the mixing as well as the convective boundary evolve with time, and how the resulting entrainment can be interpreted in terms of overshooting parameters. We find that increasing the overshooting parameter $f_{ov}$ beyond a certain value in the initial model of our simulations, changes the mixing behaviour completely. This result can be used to put limits on the overshooting parameter. We find $0.010 < f_{ov} < 0.017$ to be in good agreement with our simulations of a $3.5 M_odot$ mass star. We also identify a diffusive mixing component due to internal gravity waves (IGW) that is active throughout the convectively stable layer, but likely overestimated in our simulations. Furthermore, applying our calibration method to simulations of less massive stars suggests a need for a mass-dependent overshooting description where the mixing in terms of the pressure scale height is reduced for small convective cores.
The intermittent small-scale structure of turbulence governs energy dissipation in many astrophysical plasmas and is often believed to have universal properties for sufficiently large systems. In this work, we argue that small-scale turbulence in accretion disks is universal in the sense that it is insensitive to the magnetorotational instability (MRI) and background shear, and therefore indistinguishable from standard homogeneous magnetohydrodynamic (MHD) turbulence at small scales. We investigate the intermittency of current density, vorticity, and energy dissipation in numerical simulations of incompressible MHD turbulence driven by the MRI in a shearing box. We find that the simulations exhibit a similar degree of intermittency as in standard MHD turbulence. We perform a statistical analysis of intermittent dissipative structures and find that energy dissipation is concentrated in thin sheet-like structures that span a wide range of scales up to the box size. We show that these structures exhibit strikingly similar statistical properties to those in standard MHD turbulence. Additionally, the structures are oriented in the toroidal direction with a characteristic tilt of approximately 17.5 degrees, implying an effective guide field in that direction.
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