arXiv subjects

Michael Mond

Publications and source records attributed to Michael Mond.

At least 19 recordsLinked to original sources

arXiv2023

Lyapunov exponents and Lagrangian chaos suppression in compressible homogeneous isotropic turbulence

We study Lyapunov exponents of tracers in compressible homogeneous isotropic turbulence at different turbulent Mach number $M_t$ and Taylor-scale Reynolds number $Re_\lambda$. We demonstrate that statistics of finite-time Lyapunov exponents have the same form as in incompressible flow due to density-velocity coupling. Modulus of the smallest Lyapunov exponent $\lambda_3$ provides the principal Lyapunov exponent of the time-reversed flow, which usually is wrong in a compressible flow. This exponent, along with the principal Lyapunov exponent $\lambda_1$, determines all the exponents due to the vanishing of the sum of all Lyapunov exponents. Numerical results by high-order schemes for solving the Navier-Stokes equations and tracking particles verify these theoretical predictions. We found that: 1) The largest normalized Lyapunov exponent $\lambda_1 \tau_\eta$, where $\tau_\eta$ is the Kolmogorov time scale, is a decreasing function of $M_t$. Its dependence on $Re_\lambda$ is weak when the driving force is solenoidal, while it is an increasing function of $Re_\lambda$ when the solenoidal and compressible forces are comparable. Similar facts hold for $|\lambda_3|$, in contrast with well-studied short-correlated model; 2) The ratio of the first two Lyapunov exponents $\lambda_1/\lambda_2$ decreases with $Re_\lambda$, and is virtually independent of $M_t$ for $M_t \le 1$ in the case of solenoidal force but decreases as $M_t$ increases when solenoidal and compressible forces are comparable; 3) For purely solenoidal force, $\lambda_1 :\lambda_2 :\lambda_3 \approx 4:1:-5$ for $Re_\lambda > 80$, which is consistent with incompressible turbulence studies; 4) The ratio of dilation-to-vorticity is a more suitable parameter to characterize LEs than $M_t$.

physics.flu-dyn↗

arXiv2022

Compressible two-dimensional turbulence: cascade reversal and sensitivity to imposed magnetic field

We study the impact of compressibility on two-dimensional turbulent flows, such as those modeling astrophysical disks. We demonstrate that the direction of cascade undergoes continuous transition as the Mach number Ma increases, from inverse at zero Ma, to direct at infinite Ma. Thus, at Ma of order one comparable amounts of energy flow from the scale of the pumping to large and small scales, in accord with previous data. For supersonic turbulence with large Ma, the cascade is direct, as in three dimensions, which results in multifractal density field. For compressible flows of conducting fluids, imposing external magnetic field allows to manipulate the flow producing possibly large changes even at small Mach number. Thus Zeldovich antidynamo theorem, by which at zero Ma the magnetic field is zero in the steady state, must be used with caution. Real flows have finite Ma and, however small it is, for large magnetic flux through the disk, the magnetic field changes the flow appreciably, or rearranges it completely. For large Ma, relevant for astrophysical disks, the magnetic field energy is enhanced by a large factor as compared to estimates based on the mean field. We propose to use two-dimensional Burgers turbulence, whose three-dimensional counterpart is used for studies of the large-scale structure of the Universe, as a model for supersonic thin accretion disks.

physics.flu-dyn↗

arXiv2021

Linear and nonlinear hydromagnetic stability in laminar and turbulent flows

We consider the evolution of arbitrarily large perturbations of a prescribed pure hydrodynamical flow of an electrically conducting fluid. We study whether the flow perturbations as well as the generated magnetic fields decay or grow with time and constitute a dynamo process. For that purpose we derive a generalized Reynolds-Orr equation for the sum of the kinetic energy of the hydrodynamic perturbation and the magnetic energy. The flow is confined in a finite volume so the normal component of the velocity at the boundary is zero. The tangential component is left arbitrary in contrast with previous works. For the magnetic field we mostly employ the classical boundary conditions where the field extends in the whole space. We establish critical values of hydrodynamic and magnetic Reynolds numbers below which arbitrarily large initial perturbations of the hydrodynamic flow decay. This involves generalization of the Rayleigh-Faber-Krahn inequality for the smallest eigenvalue of an elliptic operator. For high Reynolds number turbulence we provide an estimate of critical magnetic Reynolds number below which arbitrarily large fluctuations of the magnetic field decay.

physics.flu-dyn↗

arXiv2021

Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles

The Navier-Stokes equations generate an infinite set of generalized Lyapunov exponents defined by different ways of measuring the distance between exponentially diverging perturbed and unperturbed solutions. This set is demonstrated to be similar, yet different, from the generalized Lyapunov exponent that provides moments of distance between two fluid particles below the Kolmogorov scale. We derive rigorous upper bounds on dimensionless Lyapunov exponent of the fluid particles that demonstrate the exponent's decay with Reynolds number $Re$ in accord with previous studies. In contrast, terms of cumulant series for exponents of the moments have power-law growth with $Re$. We demonstrate as an application that the growth of small fluctuations of magnetic field in ideal conducting turbulence is hyper-intermittent, being exponential in both time and Reynolds number. We resolve the existing contradiction between the theory, that predicts slow decrease of dimensionless Lyapunov exponent of turbulence with $Re$, and observations exhibiting quite fast growth. We demonstrate that it is highly plausible that a pointwise limit for the growth of small perturbations of the Navier-Stokes equations exists.

physics.flu-dyn↗

arXiv2019

Role of the sonic scale in the growth of magnetic field in compressible turbulence

We study the growth of small fluctuations of magnetic field in supersonic turbulence, the small-scale dynamo. The growth is due to the fastest turbulent eddies above the resistive scale. We observe that for supersonic turbulence these eddies are effectively incompressible which creates a robust structure of the growth. The eddies are localised below the sonic scale $l_s$ defined as the scale where the typical velocity of the turbulent eddies equals the speed of sound. Thus the flow below $l_s$ is effectively incompressible and the field growth proceeds as in incompressible flow. At large Mach numbers $l_s$ is much smaller than the integral scale of turbulence so the fastest growing mode of the magnetic field belongs to small-scale turbulence. We derive this mode and the associated growth rate numerically in a white noise in time model of turbulence. The relevance of this model relies on considering evolution time larger than the correlation time of turbulence.

physics.flu-dyn↗

arXiv2019

Density and tracer statistics in compressible turbulence: phase transition to multifractality

We study the statistics of fluid (gas) density and concentration of passive tracer particles (dust) in compressible turbulence. We raise the question of whether the fluid density which is an active field that reacts back on the transporting flow and the passive concentration of tracers must coincide in the steady state, which we demonstrate to be crucial both theoretically and experimentally. The fields' coincidence is provable at small Mach numbers, however at finite Mach numbers the assumption of mixing is needed, not evident due to the possibility of self-organization. Irrespective of whether the fields coincide we obtain a number of rigorous conclusions on both fields. As Ma increases the fields in the inertial range go through a phase transition from a finite continuous smooth to a singular multifractal distribution. We propose a way to calculate fractal dimensions from numerical or experimental data. We derive a simple expression for the spectrum of fractal dimensions of isothermal turbulence and describe limitations of lognormality. The expression depends on a single parameter: the scaling exponent of the density spectrum. We propose a mechanism for the phase transition of concentration to multifractality. We demonstrate that the pair-correlation function is invariant under the action of the probability density function of the inter-pair distance that has the Markov property implying applicability of the Kraichnan turbulence model. We use the model to derive an explicit expression for the tracers pair correlation that demonstrates their smooth transition to multifractality and confirms the transition's mechanism. Our results are of potentially important implications on astrophysical problems such as star formation as well as on technological applications such as supersonic combustion. As an example we demonstrate strong increase of planetesimals formation rate at the transition.

physics.flu-dyn↗

arXiv2016

Microscopic statistical description of incompressible Navier-Stokes granular fluids

Based on the recently-established Master kinetic equation and related Master constant H-theorem which describe the statistical behavior of the Boltzmann-Sinai classical dynamical system for smooth and hard spherical particles, the problem is posed of determining a microscopic statistical description holding for an incompressible Navier-Stokes fluid. The goal is reached by introducing a suitable mean-field interaction in the Master kinetic equation. The resulting Modified Master Kinetic Equation (MMKE) is proved to warrant at the same time the condition of mass-density incompressibility and the validity of the Navier-Stokes fluid equation. In addition, it is shown that the conservation of the Boltzmann-Shannon entropy can similarly be warranted. Applications to the plane Couette and Poiseuille flows are considered showing that they can be regarded as final decaying states for suitable non-stationary flows. As a result, it is shown that an arbitrary initial stochastic $1-$body PDF evolving in time by means of MMKE necessarily exhibits the phenomenon of Decay to Kinetic Equilibrium (DKE), whereby the $1-$body PDF asymptotically relaxes to a stationary and spatially-uniform Maxwellian PDF.

physics.flu-dyn↗

arXiv2015

Explosive Instability in Keplerian Disks

In this paper it is shown that deferentially 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 near-resonance three-wave interactions of a magnetorotational instability with stable Alfven-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 behavior: 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.

physics.plasm-ph↗

arXiv2014

Near-resonant interactions of the magnetorotational instability in thin Keplerian discs

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, explosively unstable near- resonance triads may grow much faster than their strict-resonance counterparts.

physics.plasm-ph↗

arXiv2014

Diffusive MHD Instabilities: Beyond the Chandrasekhar Theorem

The magnetohydrodynamic stability of axially unbounded cylindrical flows is considered which contain a toroidal magnetic background field with the same radial profile as the linear azimuthal velocity. Chandrasekhar (1956) has shown for ideal fluids the stability of this configuration if the Alfven velocity of the field equals the velocity of the background flow. It is demonstrated for magnetized Taylor-Couette flows at the Rayleigh line, however, that for finite diffusivity such flows become unstable against nonaxisymmetric perturbations where the critical magnetic Reynolds number of the rotation rate does not depend on the magnetic Prandtl number Pm if Pm much << 1. In order to study this new diffusive azimuthal magnetorotational instability, flows and fields with the same radial profile but with different amplitudes are considered. For Pm << 1 the instability domain with the weakest fields and the slowest rotation rates lies below the Chandrasekhar line of equal amplitudes for Alfven velocity and rotation velocity. We find that then the lines of marginal instability scale with the Reynolds number and the Hartmann number. The minimum values of the field strength and the rotation rate which are needed for the instability (slightly) grow for more and more flat rotation. Finally, the corresponding electric current of the background field becomes so strong that the Tayler instability (which even exists without rotation) also appears in the bifurcation map at small Hartmann numbers displacing after all the azimuthal magnetorotational instability.

astro-ph.GA↗

arXiv2014

Resonant instability of the nonlinearly-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- Alfv'een-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.

physics.plasm-ph↗

arXiv2013

The Magneto-Rotational Decay Instability in Keplerian Disks

The saturation of the magnetorotational (MRI) instability 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 Alfv'en-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).

physics.plasm-ph↗

arXiv2012

Non-Dissipative Saturation of the Magnetorotational Instability in Thin Disks

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.

astro-ph.SR↗

arXiv2011

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 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.

physics.plasm-ph↗

arXiv2011

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

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 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.

astro-ph.GA↗

arXiv2010

Thermomagnetic instability in hot discs

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.

astro-ph.GA↗

arXiv2009

Wind-wave stabilization by a foam layer between the atmosphere and the ocean

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.

physics.ao-ph↗

arXiv2009

Amplification of whistler waves in a thin Keplerian disc embedded in an axisymmetric magnetic field

The linear stability of thin Keplerian discs of weakly ionized polytropic plasma embedded in a mixed (toroidal-poloidal) axisymmetric magnetic field is studied. The effects of the central body on the equilibrium state of the discs are modeled by an axial total current and a dipole magnetic field. Studying the amplification of high-frequency waves shows that Keplerian discs can be destabilized due to combined effects of equilibrium density stratification and poloidal magnetic field. By considering a boundary value problem in the limit of strong Hall effect, it is shown that whistler waves which are free of direct influence of compressibility, rotation and gravity, are amplified along their trajectory while other modes of wave propagation do not change their amplitude.

physics.plasm-ph↗