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Papers by Michael Mond

In this paper it is shown that differentially rotating disks that are in the presence of weak axi... more In this paper it is shown that differentially rotating disks that are in the presence of weak axial magnetic field are prone to a new nonlinear explosive instability. The latter occurs due to the nearresonance three-wave interactions of a magnetorotational instability with stable Alfvén-Coriolis and magnetosonic modes. The dynamical equations that govern the temporal evolution of the amplitudes of the three interacting modes are derived. Numerical solutions of the dynamical equations indicate that small frequency mismatch gives rise to two types of behaviour: 1. explosive instability which leads to infinite values of the three amplitudes within a finite time, and 2. bounded irregular oscillations of all three amplitudes. Asymptotic solutions of the dynamical equations are obtained for the explosive instability regimes and are shown to match the numerical solutions near the explosion time.

The linear stability of thin vertically-isothermal density-stratified Keplerian discs in toroidal... more The linear stability of thin vertically-isothermal density-stratified Keplerian discs in toroidally-dominated magnetic fields is treated by asymptotic expansions in the small aspect ratio of the discs. The discs are found to be spectrally stable. The great variety of possible initial conditions leads to three regimes of non-exponential growth of perturbations, which are classified according to different relative levels of the in-plane and axial perturbed velocities. The first two regimes of instability are characterized by the decoupling of the magneto-sonic (MS) and inertia-Coriolis (IC) modes, as well as by algebraic temporal growth of the perturbations, which are driven by either MS or IC modes (hereafter MS-and IC-regimes of instability, respectively). The third, mixed IC-MS regime of non-exponential, non-algebraic growth is due only to nonaxisymmetric perturbations. The latter regime is characterized by high radial and azimuthal wavenumbers, and growth time of the order of tens of rotating periods. The mixed IC-MS regime most likely exhibits the maximal growth as compared with the IC-and MS-regimes. In the first two regimes of instability the compressible MS mode plays a principal role either as the driver of the growth or the driven growing mode, while the mixed IC-MS regime is described by the Bousinesq approximation for incompressible fluid. The latter is obtained as a natural limit of the expansion scheme. The presence of magnetic field in the mixed IC-MS regime may drastically increase the growth rates of the perturbations as compared with the pure hydrodynamic system.

The magneto-rotational decay instability (MRDI) of thin Keplerian discs threaded by poloidal magn... more The magneto-rotational decay instability (MRDI) of thin Keplerian discs threaded by poloidal magnetic fields is introduced and studied. The linear magnetohydrodynamic problem decouples into eigenvalue problems for in-plane slow-and fast-Alfvén-Coriolis (AC), and vertical magnetosonic (MS) eigenmodes. The magnetorotational instability (MRI) is composed of a discrete number of unstable slow AC eigenmodes that is determined for each radius by the local beta. In the vicinity of the first beta threshold a parent MRI eigenmode together with a stable AC eigenmode (either slow or fast) and a stable MS eigenmode form a resonant triad. The three-wave MRDI relies on the nonlinear saturation of the parent MRI mode and the exponential growth of two daughter linearly stable waves, slow-AC and MS modes with an effective growth rate that is comparable to that of the parent MRI. If, however, the role of the AC daughter wave is played by a stable fast mode, all three modes remain bounded.

Linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II) do... more Linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II) dominant magnetic field components are studied in thin vertically-isothermal Keplerian discs. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. In both equilibrium configurations the perturbations are decoupled into in-plane and vertical modes. For equilibria of type I those two modes are the Alfvén-Coriolis and sound waves, while for equilibria of type II they are the inertia-Coriolis and magnetosonic waves. Exact expressions for the growth rates as well as the number of unstable modes for type I equilibria are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. It is further shown that the axisymmetric MRI is completely suppressed by dominant toroidal magnetic fields (i.e. equilibria of type II). This renders the system prone to either non-axisymmetric MRI or nonmodal algebraic growth mechanisms. The algebraic growth mechanism investigated in the present study occurs exclusively due to the rotation shear, generates the inertia-Coriolis driven magnetosonic modes due to non-resonant or resonant coupling that induces, respectively, linear or quadratic temporal growth of the perturbations.

Physics of Plasmas, 2016

Differentially rotating disks under the effect of axial magnetic field are prone to a nonlinear e... more Differentially rotating disks under the effect of axial magnetic field are prone to a nonlinear explosive magnetorotational instability (EMRI). The dynamic equations that govern the temporal evolution of the amplitudes of three weakly detuned resonantly interacting modes are derived. As distinct from exponential growth in the strict resonance triads, EMRI occurs due to the resonant interactions of an MRI mode with stable Alfvén–Coriolis and magnetosonic modes. Numerical solutions of the dynamic equations for amplitudes of a triad indicate that two types of perturbations behavior can be excited for resonance conditions: (i) EMRI which leads to infinite values of the three amplitudes within a finite time, and (ii) bounded irregular oscillations of all three amplitudes. Asymptotic explicit solutions of the dynamic equations are obtained for EMRI regimes and are shown to match the numerical solutions near the explosion time.

The linear instability of thin, vertically-isothermal Keplerian discs, under the influence of axi... more The linear instability of thin, vertically-isothermal Keplerian discs, under the influence of axial magnetic field is investigated. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. It is shown that the perturbations are decoupled into in-plane and vertical modes. Exact expressions for the growth rates as well as the number of unstable modes are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. In addition, a weakly nonlinear analysis of the MRI is performed. It is shown that near the instability threshold the latter is saturated by the stable magnetoacoustic modes.

Wind shortwave instability of a foam layer between the atmosphere and the ocean is examined in or... more Wind shortwave instability of a foam layer between the atmosphere and the ocean is examined in order to reach greater understanding of the recent findings of the decrease in momentum transfer from hurricane winds to sea waves. The three-fluid configuration with the high contrasts in densities of the air, foam and water provides for an effective mechanism to stabilize the water surface.

The equations that govern the weakly nonlinear near-resonant interaction of a parent magnetorotat... more The equations that govern the weakly nonlinear near-resonant interaction of a parent magnetorotational instability with two linearly stable daughter modes in thin nonuniform magnetized Keplerian discs are derived. While the equations for the daughter modes resemble their first order classical uniform counterparts, the parent mode is governed by a second order forced Duffing equation. It is demonstrated that the solutions of those three coupled equations exhibit a wide and rich spectrum of dynamical behavior. In particular, it is shown that amplitudes of unstable triads may grow explosively with time, namely reaching infinite values in a finite time. Paradoxically, unstable near- resonance triads may explosively grow much faster than their strict-resonance counterparts.

The Astrophysical Journal, 2007

Cool weakly ionized gaseous rotating disks are considered by many models to be the origin of the ... more Cool weakly ionized gaseous rotating disks are considered by many models to be the origin of the evolution of protoplanetary clouds. Instabilities against perturbations in such disks play an important role in the theory of the formation of stars and planets. Thus, a hierarchy of successive fragmentations into smaller and smaller pieces as a part of the Kant-Laplace theory of formation of the planetary system remains valid also for contemporary cosmogony. Traditionally, axisymmetric magnetohydrodynamic (MHD) and, recently, Hall-MHD instabilities have been thoroughly studied as providers of an efficient mechanism for radial transfer of angular momentum and of radial density stratification. In the current work, the Hall instability against nonaxisymmetric perturbations in compressible rotating fluid in external magnetic field is proposed as a viable mechanism for the azimuthal fragmentation of the protoplanetary disk and, thus, perhaps initiates the road to planet formation. The Hall instability is excited due to the combined effect of the radial stratification of the disk and the Hall electric field, and its growth rate is of the order of the rotation period. This family of instabilities is introduced here for the first time in an astrophysical context.

The Astrophysical Journal, 2007

The stratification-driven Hall instability in a weakly ionized polytropic plasma is investigated ... more The stratification-driven Hall instability in a weakly ionized polytropic plasma is investigated in the local approximation within an equilibrium Keplerian disk of a small aspect ratio ǫ. The leading order of the asymptotic expansions in ǫ is applied to both equilibrium and perturbation problems. The equilibrium disk with an embedded purely toroidal magnetic field is found to be stable to radial, and unstable to vertical shortwave perturbations. The marginal stability surface was found in the space of the local Hall and inverse plasma beta parameters, as well as the free parameter of the model related to the total current. To estimate the minimal values of the equilibrium magnetic field that leads to the instability, it was constructed as a sum of a current free magnetic field and the simplest approximation for magnetic field created by the distributed electric current.

Physics of Fluids, 2008

The generalized Kelvin–Helmholtz (KH) and Miles mechanisms of the water wave generation by wind a... more The generalized Kelvin–Helmholtz (KH) and Miles mechanisms of the water wave generation by wind are investigated for two-layer piecewise linear model of the wind profile. It is shown by asymptotic expansions in small air-to-water density ratio that two mechanisms of the instability operate in quite different scales. Miles’ short waves are generated by weak winds, in particular, Miles’ regime is responsible for initiation of the instability at the minimum wind speed, while the generalized KH regime dominates at strong winds and raises moderately short waves.

Physical Review Letters, 2012

A new non-dissipative mechanism is proposed for the saturation of the axisymmetric magnetorotatio... more A new non-dissipative mechanism is proposed for the saturation of the axisymmetric magnetorotational (MRI) instability in thin Keplerian disks that are subject to an axial magnetic field. That mechanism relies on the energy transfer from the MRI to stable magnetosonic (MS) waves. Such mode interaction is enabled due to the vertical stratification of the disk that results in the discretization of its MRI spectrum, as well as by applying the appropriate boundary conditions. A second order Duffing-like amplitude equation for the initially unstable MRI modes is derived. The solutions of that equation exhibit bursty nonlinear oscillations with a constant amplitude that signifies the saturation level of the MRI. Those results are verified by a direct numerical solution of the full nonlinear reduced set of thin disk magnetohydrodynamics equations.

Physical Review Letters, 2013

The saturation of the magnetorotational (MRI) instability in thin Keplerian disks through threewa... more The saturation of the magnetorotational (MRI) instability in thin Keplerian disks through threewave resonant interactions is introduced and discussed. That mechanism is a natural generalization of the fundamental decay instability discovered five decades ago for infinite, homogeneous and immovable plasmas. The decay instability relies on the energy transfer from the MRI to stable slow Alfvén-Coriolis (AC) as well as magnetosonic (MS) waves. A second order forced Duffing amplitude equation for the initially unstable MRI as well as two first order equations for the other two waves are derived. The solutions of those equations exhibit bounded bursty nonlinear oscillations for the MRI as well as unbounded growth for the linearly stable slow AC and MS perturbations, thus giving rise to the magneto-rotational decay instability (MRDI).

Monthly Notices of the Royal Astronomical Society, 2014

The magneto-rotational decay instability (MRDI) of thin Keplerian discs threaded by poloidal magn... more The magneto-rotational decay instability (MRDI) of thin Keplerian discs threaded by poloidal magnetic fields is introduced and studied. The linear magnetohydrodynamic problem decouples into eigenvalue problems for in-plane slow-and fast-Alfvén-Coriolis (AC), and vertical magnetosonic (MS) eigenmodes. The magnetorotational instability (MRI) is composed of a discrete number of unstable slow AC eigenmodes that is determined for each radius by the local beta. In the vicinity of the first beta threshold a parent MRI eigenmode together with a stable AC eigenmode (either slow or fast) and a stable MS eigenmode form a resonant triad. The three-wave MRDI relies on the non-linear saturation of the parent MRI mode and the exponential growth of two daughter linearly stable waves, slow-AC and MS modes with an effective growth rate that is comparable to that of the parent MRI. If, however, the role of the AC daughter wave is played by a stable fast mode, all three modes remain bounded.

Monthly Notices of the Royal Astronomical Society, 2012

The linear stability of thin vertically isothermal density-stratified Keplerian discs in toroidal... more The linear stability of thin vertically isothermal density-stratified Keplerian discs in toroidally dominated magnetic fields is treated by asymptotic expansions in the small aspect ratio of the discs. The discs are found to be spectrally stable. The great variety of possible initial conditions leads to three regimes of non-exponential growth of perturbations, which are classified according to different relative levels of the in-plane and axial perturbed velocities. The first two regimes of instability are characterized by the decoupling of the magnetosonic (MS) and inertia-Coriolis (IC) modes, as well as by algebraic temporal growth of the perturbations, which are driven by either MS or IC modes (hereafter MS and IC regimes of instability, respectively). The third, mixed IC-MS regime of non-exponential, non-algebraic growth is due only to non-axisymmetric perturbations. The latter regime is characterized by high radial and azimuthal wavenumbers, and growth time of the order of tens of rotating periods. The mixed IC-MS regime most likely exhibits the maximal growth as compared with the IC and MS regimes. In the first two regimes of instability the compressible MS mode plays a principal role either as the driver of the growth or the driven growing mode, while the mixed IC-MS regime is described by the Boussinesq approximation for incompressible fluid. The latter is obtained as a natural limit of the expansion scheme. The presence of magnetic field in the mixed IC-MS regime may drastically increase the growth rates of the perturbations as compared with the pure hydrodynamic system.

Monthly Notices of the Royal Astronomical Society, 2009

A detailed solution of an initial value problem of a vertically localized initial perturbation in... more A detailed solution of an initial value problem of a vertically localized initial perturbation in rotating magnetized vertically stratified disc is presented. The appropriate linearized magnetohydrodynamics equations are solved by employing the Wentzel-Kramers-Brillouin (WKB) approximation and the results are verified numerically. The eigenfrequencies as well as eigenfunctions are explicitly obtained. It is demonstrated that the initial perturbation remains confined within the disc. It is further shown that thin enough discs are stable but as their thickness grows increasing number of unstable modes participate in the solution of the initial value problem. However, it is demonstrated that due to the localization of the initial perturbation, the growth time of the instability is significantly longer than the calculated inverse growth rate of the individual unstable eigenfunctions.

Monthly Notices of the Royal Astronomical Society, 2011

The linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II... more The linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II) dominant magnetic field components are studied in thin vertically isothermal Keplerian discs. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. In both the equilibrium configurations the perturbations are decoupled into in-plane and vertical modes. For equilibria of type I those two modes are the Alfvén-Coriolis and sound waves, while for equilibria of type II they are the inertia-Coriolis and magnetosonic waves. Exact expressions for the growth rates as well as the number of unstable modes for type I equilibria are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. It is further shown that the axisymmetric MRI is completely suppressed by dominant toroidal magnetic fields (i.e. equilibria of type II). This renders the system prone to either non-axisymmetric MRI or nonmodal algebraic growth mechanisms. The algebraic growth mechanism investigated in the present study occurs exclusively due to the rotation shear, generates the inertia-Coriolis driven magnetosonic modes due to non-resonant or resonant coupling that induces, respectively, linear or quadratic temporal growth of the perturbations.

Monthly Notices of the Royal Astronomical Society, 2010

A linear stability analysis of ionized discs with a temperature gradient and an external axial ma... more A linear stability analysis of ionized discs with a temperature gradient and an external axial magnetic field is presented. It is shown that both hydromagnetic and thermomagnetic effects can lead to the amplification of waves and make discs unstable. The conditions under which the instabilities grow are found and the characteristic growth rate is calculated. The regimes at which both the thermomagnetic and magnetorotational instabilities can operate are discussed.

Dynamics of Atmospheres and Oceans, 2010

The study is motivated by recent findings of the decrease in the momentum transfer from strong wi... more The study is motivated by recent findings of the decrease in the momentum transfer from strong winds to sea. The Kelvin-Helmholtz instability (KHI) of a three-fluid system of air, foam and water is examined within the range of intermediately short surface waves. The foam layer thickness necessary for effective separation of the atmosphere and the ocean is estimated. Due to high density contrasts in the three-fluid system, even a relatively thin foam layer between the atmosphere and the ocean can provide a significant stabilization of the water surface by the wavelength shift of the instability towards smaller scales. It is conjectured that such stabilization qualitatively explains the observed reduction of roughness and drag.

Astronomische Nachrichten, 2008

The magnetorotational instability (MRI) of thin, vertically-isothermal Keplerian discs, under the... more The magnetorotational instability (MRI) of thin, vertically-isothermal Keplerian discs, under the influence of an axial magnetic field is investigated near the instability threshold. The nonlinear interaction of Alfven-Coriolis (MRI) modes with stable magnetoacoustic waves is considered. The transition of the Alfven-Coriolis modes to instability occurs when the linearized system has zero eigenvalue of multiplicity two. As a result the nonlinear ordinary differential equation that describes the evolution of the amplitude of the MRI mode near the threshold is of second order. Solutions of that amplitude equation reveal that the MRI is saturated to bursty periodical oscillations due to the transfer of energy to the stable magnetosonic modes.

In this paper it is shown that differentially rotating disks that are in the presence of weak axi... more In this paper it is shown that differentially rotating disks that are in the presence of weak axial magnetic field are prone to a new nonlinear explosive instability. The latter occurs due to the nearresonance three-wave interactions of a magnetorotational instability with stable Alfvén-Coriolis and magnetosonic modes. The dynamical equations that govern the temporal evolution of the amplitudes of the three interacting modes are derived. Numerical solutions of the dynamical equations indicate that small frequency mismatch gives rise to two types of behaviour: 1. explosive instability which leads to infinite values of the three amplitudes within a finite time, and 2. bounded irregular oscillations of all three amplitudes. Asymptotic solutions of the dynamical equations are obtained for the explosive instability regimes and are shown to match the numerical solutions near the explosion time.

The linear stability of thin vertically-isothermal density-stratified Keplerian discs in toroidal... more The linear stability of thin vertically-isothermal density-stratified Keplerian discs in toroidally-dominated magnetic fields is treated by asymptotic expansions in the small aspect ratio of the discs. The discs are found to be spectrally stable. The great variety of possible initial conditions leads to three regimes of non-exponential growth of perturbations, which are classified according to different relative levels of the in-plane and axial perturbed velocities. The first two regimes of instability are characterized by the decoupling of the magneto-sonic (MS) and inertia-Coriolis (IC) modes, as well as by algebraic temporal growth of the perturbations, which are driven by either MS or IC modes (hereafter MS-and IC-regimes of instability, respectively). The third, mixed IC-MS regime of non-exponential, non-algebraic growth is due only to nonaxisymmetric perturbations. The latter regime is characterized by high radial and azimuthal wavenumbers, and growth time of the order of tens of rotating periods. The mixed IC-MS regime most likely exhibits the maximal growth as compared with the IC-and MS-regimes. In the first two regimes of instability the compressible MS mode plays a principal role either as the driver of the growth or the driven growing mode, while the mixed IC-MS regime is described by the Bousinesq approximation for incompressible fluid. The latter is obtained as a natural limit of the expansion scheme. The presence of magnetic field in the mixed IC-MS regime may drastically increase the growth rates of the perturbations as compared with the pure hydrodynamic system.

The magneto-rotational decay instability (MRDI) of thin Keplerian discs threaded by poloidal magn... more The magneto-rotational decay instability (MRDI) of thin Keplerian discs threaded by poloidal magnetic fields is introduced and studied. The linear magnetohydrodynamic problem decouples into eigenvalue problems for in-plane slow-and fast-Alfvén-Coriolis (AC), and vertical magnetosonic (MS) eigenmodes. The magnetorotational instability (MRI) is composed of a discrete number of unstable slow AC eigenmodes that is determined for each radius by the local beta. In the vicinity of the first beta threshold a parent MRI eigenmode together with a stable AC eigenmode (either slow or fast) and a stable MS eigenmode form a resonant triad. The three-wave MRDI relies on the nonlinear saturation of the parent MRI mode and the exponential growth of two daughter linearly stable waves, slow-AC and MS modes with an effective growth rate that is comparable to that of the parent MRI. If, however, the role of the AC daughter wave is played by a stable fast mode, all three modes remain bounded.

Linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II) do... more Linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II) dominant magnetic field components are studied in thin vertically-isothermal Keplerian discs. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. In both equilibrium configurations the perturbations are decoupled into in-plane and vertical modes. For equilibria of type I those two modes are the Alfvén-Coriolis and sound waves, while for equilibria of type II they are the inertia-Coriolis and magnetosonic waves. Exact expressions for the growth rates as well as the number of unstable modes for type I equilibria are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. It is further shown that the axisymmetric MRI is completely suppressed by dominant toroidal magnetic fields (i.e. equilibria of type II). This renders the system prone to either non-axisymmetric MRI or nonmodal algebraic growth mechanisms. The algebraic growth mechanism investigated in the present study occurs exclusively due to the rotation shear, generates the inertia-Coriolis driven magnetosonic modes due to non-resonant or resonant coupling that induces, respectively, linear or quadratic temporal growth of the perturbations.

Physics of Plasmas, 2016

Differentially rotating disks under the effect of axial magnetic field are prone to a nonlinear e... more Differentially rotating disks under the effect of axial magnetic field are prone to a nonlinear explosive magnetorotational instability (EMRI). The dynamic equations that govern the temporal evolution of the amplitudes of three weakly detuned resonantly interacting modes are derived. As distinct from exponential growth in the strict resonance triads, EMRI occurs due to the resonant interactions of an MRI mode with stable Alfvén–Coriolis and magnetosonic modes. Numerical solutions of the dynamic equations for amplitudes of a triad indicate that two types of perturbations behavior can be excited for resonance conditions: (i) EMRI which leads to infinite values of the three amplitudes within a finite time, and (ii) bounded irregular oscillations of all three amplitudes. Asymptotic explicit solutions of the dynamic equations are obtained for EMRI regimes and are shown to match the numerical solutions near the explosion time.

The linear instability of thin, vertically-isothermal Keplerian discs, under the influence of axi... more The linear instability of thin, vertically-isothermal Keplerian discs, under the influence of axial magnetic field is investigated. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. It is shown that the perturbations are decoupled into in-plane and vertical modes. Exact expressions for the growth rates as well as the number of unstable modes are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. In addition, a weakly nonlinear analysis of the MRI is performed. It is shown that near the instability threshold the latter is saturated by the stable magnetoacoustic modes.

Wind shortwave instability of a foam layer between the atmosphere and the ocean is examined in or... more Wind shortwave instability of a foam layer between the atmosphere and the ocean is examined in order to reach greater understanding of the recent findings of the decrease in momentum transfer from hurricane winds to sea waves. The three-fluid configuration with the high contrasts in densities of the air, foam and water provides for an effective mechanism to stabilize the water surface.

The equations that govern the weakly nonlinear near-resonant interaction of a parent magnetorotat... more The equations that govern the weakly nonlinear near-resonant interaction of a parent magnetorotational instability with two linearly stable daughter modes in thin nonuniform magnetized Keplerian discs are derived. While the equations for the daughter modes resemble their first order classical uniform counterparts, the parent mode is governed by a second order forced Duffing equation. It is demonstrated that the solutions of those three coupled equations exhibit a wide and rich spectrum of dynamical behavior. In particular, it is shown that amplitudes of unstable triads may grow explosively with time, namely reaching infinite values in a finite time. Paradoxically, unstable near- resonance triads may explosively grow much faster than their strict-resonance counterparts.

The Astrophysical Journal, 2007

Cool weakly ionized gaseous rotating disks are considered by many models to be the origin of the ... more Cool weakly ionized gaseous rotating disks are considered by many models to be the origin of the evolution of protoplanetary clouds. Instabilities against perturbations in such disks play an important role in the theory of the formation of stars and planets. Thus, a hierarchy of successive fragmentations into smaller and smaller pieces as a part of the Kant-Laplace theory of formation of the planetary system remains valid also for contemporary cosmogony. Traditionally, axisymmetric magnetohydrodynamic (MHD) and, recently, Hall-MHD instabilities have been thoroughly studied as providers of an efficient mechanism for radial transfer of angular momentum and of radial density stratification. In the current work, the Hall instability against nonaxisymmetric perturbations in compressible rotating fluid in external magnetic field is proposed as a viable mechanism for the azimuthal fragmentation of the protoplanetary disk and, thus, perhaps initiates the road to planet formation. The Hall instability is excited due to the combined effect of the radial stratification of the disk and the Hall electric field, and its growth rate is of the order of the rotation period. This family of instabilities is introduced here for the first time in an astrophysical context.

The Astrophysical Journal, 2007

The stratification-driven Hall instability in a weakly ionized polytropic plasma is investigated ... more The stratification-driven Hall instability in a weakly ionized polytropic plasma is investigated in the local approximation within an equilibrium Keplerian disk of a small aspect ratio ǫ. The leading order of the asymptotic expansions in ǫ is applied to both equilibrium and perturbation problems. The equilibrium disk with an embedded purely toroidal magnetic field is found to be stable to radial, and unstable to vertical shortwave perturbations. The marginal stability surface was found in the space of the local Hall and inverse plasma beta parameters, as well as the free parameter of the model related to the total current. To estimate the minimal values of the equilibrium magnetic field that leads to the instability, it was constructed as a sum of a current free magnetic field and the simplest approximation for magnetic field created by the distributed electric current.

Physics of Fluids, 2008

The generalized Kelvin–Helmholtz (KH) and Miles mechanisms of the water wave generation by wind a... more The generalized Kelvin–Helmholtz (KH) and Miles mechanisms of the water wave generation by wind are investigated for two-layer piecewise linear model of the wind profile. It is shown by asymptotic expansions in small air-to-water density ratio that two mechanisms of the instability operate in quite different scales. Miles’ short waves are generated by weak winds, in particular, Miles’ regime is responsible for initiation of the instability at the minimum wind speed, while the generalized KH regime dominates at strong winds and raises moderately short waves.

Physical Review Letters, 2012

A new non-dissipative mechanism is proposed for the saturation of the axisymmetric magnetorotatio... more A new non-dissipative mechanism is proposed for the saturation of the axisymmetric magnetorotational (MRI) instability in thin Keplerian disks that are subject to an axial magnetic field. That mechanism relies on the energy transfer from the MRI to stable magnetosonic (MS) waves. Such mode interaction is enabled due to the vertical stratification of the disk that results in the discretization of its MRI spectrum, as well as by applying the appropriate boundary conditions. A second order Duffing-like amplitude equation for the initially unstable MRI modes is derived. The solutions of that equation exhibit bursty nonlinear oscillations with a constant amplitude that signifies the saturation level of the MRI. Those results are verified by a direct numerical solution of the full nonlinear reduced set of thin disk magnetohydrodynamics equations.

Physical Review Letters, 2013

The saturation of the magnetorotational (MRI) instability in thin Keplerian disks through threewa... more The saturation of the magnetorotational (MRI) instability in thin Keplerian disks through threewave resonant interactions is introduced and discussed. That mechanism is a natural generalization of the fundamental decay instability discovered five decades ago for infinite, homogeneous and immovable plasmas. The decay instability relies on the energy transfer from the MRI to stable slow Alfvén-Coriolis (AC) as well as magnetosonic (MS) waves. A second order forced Duffing amplitude equation for the initially unstable MRI as well as two first order equations for the other two waves are derived. The solutions of those equations exhibit bounded bursty nonlinear oscillations for the MRI as well as unbounded growth for the linearly stable slow AC and MS perturbations, thus giving rise to the magneto-rotational decay instability (MRDI).

Monthly Notices of the Royal Astronomical Society, 2014

The magneto-rotational decay instability (MRDI) of thin Keplerian discs threaded by poloidal magn... more The magneto-rotational decay instability (MRDI) of thin Keplerian discs threaded by poloidal magnetic fields is introduced and studied. The linear magnetohydrodynamic problem decouples into eigenvalue problems for in-plane slow-and fast-Alfvén-Coriolis (AC), and vertical magnetosonic (MS) eigenmodes. The magnetorotational instability (MRI) is composed of a discrete number of unstable slow AC eigenmodes that is determined for each radius by the local beta. In the vicinity of the first beta threshold a parent MRI eigenmode together with a stable AC eigenmode (either slow or fast) and a stable MS eigenmode form a resonant triad. The three-wave MRDI relies on the non-linear saturation of the parent MRI mode and the exponential growth of two daughter linearly stable waves, slow-AC and MS modes with an effective growth rate that is comparable to that of the parent MRI. If, however, the role of the AC daughter wave is played by a stable fast mode, all three modes remain bounded.

Monthly Notices of the Royal Astronomical Society, 2012

The linear stability of thin vertically isothermal density-stratified Keplerian discs in toroidal... more The linear stability of thin vertically isothermal density-stratified Keplerian discs in toroidally dominated magnetic fields is treated by asymptotic expansions in the small aspect ratio of the discs. The discs are found to be spectrally stable. The great variety of possible initial conditions leads to three regimes of non-exponential growth of perturbations, which are classified according to different relative levels of the in-plane and axial perturbed velocities. The first two regimes of instability are characterized by the decoupling of the magnetosonic (MS) and inertia-Coriolis (IC) modes, as well as by algebraic temporal growth of the perturbations, which are driven by either MS or IC modes (hereafter MS and IC regimes of instability, respectively). The third, mixed IC-MS regime of non-exponential, non-algebraic growth is due only to non-axisymmetric perturbations. The latter regime is characterized by high radial and azimuthal wavenumbers, and growth time of the order of tens of rotating periods. The mixed IC-MS regime most likely exhibits the maximal growth as compared with the IC and MS regimes. In the first two regimes of instability the compressible MS mode plays a principal role either as the driver of the growth or the driven growing mode, while the mixed IC-MS regime is described by the Boussinesq approximation for incompressible fluid. The latter is obtained as a natural limit of the expansion scheme. The presence of magnetic field in the mixed IC-MS regime may drastically increase the growth rates of the perturbations as compared with the pure hydrodynamic system.

Monthly Notices of the Royal Astronomical Society, 2009

A detailed solution of an initial value problem of a vertically localized initial perturbation in... more A detailed solution of an initial value problem of a vertically localized initial perturbation in rotating magnetized vertically stratified disc is presented. The appropriate linearized magnetohydrodynamics equations are solved by employing the Wentzel-Kramers-Brillouin (WKB) approximation and the results are verified numerically. The eigenfrequencies as well as eigenfunctions are explicitly obtained. It is demonstrated that the initial perturbation remains confined within the disc. It is further shown that thin enough discs are stable but as their thickness grows increasing number of unstable modes participate in the solution of the initial value problem. However, it is demonstrated that due to the localization of the initial perturbation, the growth time of the instability is significantly longer than the calculated inverse growth rate of the individual unstable eigenfunctions.

Monthly Notices of the Royal Astronomical Society, 2011

The linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II... more The linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II) dominant magnetic field components are studied in thin vertically isothermal Keplerian discs. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. In both the equilibrium configurations the perturbations are decoupled into in-plane and vertical modes. For equilibria of type I those two modes are the Alfvén-Coriolis and sound waves, while for equilibria of type II they are the inertia-Coriolis and magnetosonic waves. Exact expressions for the growth rates as well as the number of unstable modes for type I equilibria are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. It is further shown that the axisymmetric MRI is completely suppressed by dominant toroidal magnetic fields (i.e. equilibria of type II). This renders the system prone to either non-axisymmetric MRI or nonmodal algebraic growth mechanisms. The algebraic growth mechanism investigated in the present study occurs exclusively due to the rotation shear, generates the inertia-Coriolis driven magnetosonic modes due to non-resonant or resonant coupling that induces, respectively, linear or quadratic temporal growth of the perturbations.

Monthly Notices of the Royal Astronomical Society, 2010

A linear stability analysis of ionized discs with a temperature gradient and an external axial ma... more A linear stability analysis of ionized discs with a temperature gradient and an external axial magnetic field is presented. It is shown that both hydromagnetic and thermomagnetic effects can lead to the amplification of waves and make discs unstable. The conditions under which the instabilities grow are found and the characteristic growth rate is calculated. The regimes at which both the thermomagnetic and magnetorotational instabilities can operate are discussed.

Dynamics of Atmospheres and Oceans, 2010

The study is motivated by recent findings of the decrease in the momentum transfer from strong wi... more The study is motivated by recent findings of the decrease in the momentum transfer from strong winds to sea. The Kelvin-Helmholtz instability (KHI) of a three-fluid system of air, foam and water is examined within the range of intermediately short surface waves. The foam layer thickness necessary for effective separation of the atmosphere and the ocean is estimated. Due to high density contrasts in the three-fluid system, even a relatively thin foam layer between the atmosphere and the ocean can provide a significant stabilization of the water surface by the wavelength shift of the instability towards smaller scales. It is conjectured that such stabilization qualitatively explains the observed reduction of roughness and drag.

Astronomische Nachrichten, 2008

The magnetorotational instability (MRI) of thin, vertically-isothermal Keplerian discs, under the... more The magnetorotational instability (MRI) of thin, vertically-isothermal Keplerian discs, under the influence of an axial magnetic field is investigated near the instability threshold. The nonlinear interaction of Alfven-Coriolis (MRI) modes with stable magnetoacoustic waves is considered. The transition of the Alfven-Coriolis modes to instability occurs when the linearized system has zero eigenvalue of multiplicity two. As a result the nonlinear ordinary differential equation that describes the evolution of the amplitude of the MRI mode near the threshold is of second order. Solutions of that amplitude equation reveal that the MRI is saturated to bursty periodical oscillations due to the transfer of energy to the stable magnetosonic modes.