Yuri M. Shtemler

Hall Instability of Thin Weakly Ionized Stratified Keplerian Disks

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 . The leading order of the asymptotic expansions in 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 perturbations and unstable to vertical short-wave perturbations. The marginal stability surface was found in the space of the local Hall and inverse plasma-β 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.

Hall equilibrium of thin Keplerian discs embedded in mixed poloidal and toroidal magnetic fields

Axisymmetric steady-state weakly ionized Hall-MHD Keplerian thin disks are investigated by using asymptotic expansions in the small disk aspect ratio †. The model incorporates the azimuthal and poloidal components of the magnetic flelds in the leading order in †. The disk structure is described by an appropriate Grad-Shafranov equation for the poloidal ∞ux function that involves two arbitrary functions of for the toroidal and poloidal currents. The ∞ux function is symmetric about the midplane and satisfles certain boundary conditions at the near-horizontal disk edges. The boundary conditions model the combined efiect of the primordial as well as the dipole-like magnetic flelds. An analytical solution for the Hall equilibrium is achieved by further expanding the relevant equations in an additional small parameter that is inversely proportional to the Hall parameter. It is thus found that the Hall equilibrium disks fall into two types: Keplerian disks with (i) small (Rd » ‐ 0 ) and (ii) large (Rd & ‐ ik , k > 0) radius of the disk. The numerical examples that are presented demonstrate the richness and great variety of magnetic and density conflgurations that may be achieved under the Hall-MHD equilibrium.

Nondissipative saturation of the magnetorotational instability in thin disks.

A new nondissipative 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 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.

Spectral and algebraic instabilities in thin Keplerian discs under poloidal and toroidal magnetic fields

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 Alfven― 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 non-modal 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.

An asymptotic model for the Kelvin–Helmholtz and Miles mechanisms of water wave generation by wind

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.

Resonant instability of the non-linearly saturated magnetorotational mode in thin Keplerian discs

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- Alfven-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.

Magnetorotational decay instability in Keplerian disks.

The saturation of the magnetorotational instability (MRI) in thin Keplerian disks through three-wave 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 as well as magnetosonic 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 Alfvén-Coriolis and magnetosonic perturbations, thus giving rise to the magnetorotational decay instability.

Regimes of the non-exponential temporal growth in thin Keplerian discs under toroidally dominated magnetic fields

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.

Generation of intermediately long sea waves by weakly sheared winds

The present study concerns the numerical modeling of sea-wave instability under the effect of logarithmic-wind profile in hurricane conditions. The central point of the study is the calculation of the wave growth rate, which is proportional to the fractional input energy from the weakly-sheared (logarithmic) wind to the wave exponentially varying with time. It is shown for hurricane conditions that the Miles-type stability model based on the Charnocks formula with the standard constant coefficient underestimates the growth rate ~5 to 50 times as compared with the model employing the roughness adopted from experimental data for hurricane winds. The drag reduction with wind speed at hurricane conditions coupled with the similar behavior of the dimensionless gravity acceleration, leads to the minimum in the maximal growth rate and the maximum in the most unstable wavelength.

The Hall instability of unsteady inhomogeneous axially symmetric magnetized plasmas

The Hall instability in cylindrically symmetric resistive magnetized plasmas in vacuum is investigated. The unperturbed self-similar equilibrium solutions for imploding Z-pinches with time-dependent total current It∼tS,S>1/3, are subjected by short-wave sausage perturbations. The instability criterion is derived in slow-time, frozen-radius approximation. In cylindrically symmetric configurations the instability is driven by the magnetic field curvature. The near-axis and near-edge branches of the neutral curve in the plane of the inverse Hall parameter and phase velocity with the frozen radial coordinate as a parameter are separated by the critical point, where the modified gradient from the unperturbed number density changes sign. The critical radius may be treated as a new characteristic size of the Z-pinch that emerges due to the instability: the pinch is envisaged restructured by the short-scale high-frequency Hall instability, in which a central stable core is surrounded by an outer shell. Such a mod...