arXiv subjects

V. Zheligovsky

Publications and source records attributed to V. Zheligovsky.

15 recordsLinked to original sources

arXiv2018

Magnetic field generation by pointwise zero-helicity three-dimensional steady flow of incompressible electrically conducting fluid

We introduce six families of three-dimensional space-periodic steady solenoidal flows, whose kinetic helicity density is zero at any point. Four families are analytically defined. Flows in four families have zero helicity spectrum. Sample flows from five families are used to demonstrate numerically that neither zero kinetic helicity density, nor zero helicity spectrum prohibit generation of large-scale magnetic field by the two most prominent dynamo mechanisms: the magnetic $α$-effect and negative eddy diffusivity. Our computations also attest that such flows often generate small-scale field for sufficiently small magnetic molecular diffusivity. These findings indicate that kinetic helicity and helicity spectrum are not the quantities controlling the dynamo properties of a flow regardless of whether scale separation is present or not.

physics.flu-dyn↗

arXiv2015

The Cauchy-Lagrangian method for numerical analysis of Euler flow

A novel semi-Lagrangian method is introduced to solve numerically the Euler equation for ideal incompressible flow in arbitrary space dimension. It exploits the time-analyticity of fluid particle trajectories and requires, in principle, only limited spatial smoothness of the initial data. Efficient generation of high-order time-Taylor coefficients is made possible by a recurrence relation that follows from the Cauchy invariants formulation of the Euler equation (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430). Truncated time-Taylor series of very high order allow the use of time steps vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the accuracy of the solution. Tests performed on the two-dimensional Euler equation indicate that the Cauchy-Lagrangian method is more - and occasionally much more - efficient and less prone to instability than Eulerian Runge-Kutta methods, and less prone to rapid growth of rounding errors than the high-order Eulerian time-Taylor algorithm. We also develop tools of analysis adapted to the Cauchy-Lagrangian method, such as the monitoring of the radius of convergence of the time-Taylor series. Certain other fluid equations can be handled similarly.

math.NA↗

arXiv2013

Existence, uniqueness and analyticity of space-periodic solutions to the regularised long-wave equation

We consider space-periodic evolutionary and travelling-wave solutions to the regularised long-wave equation (RLWE) with damping and forcing. We establish existence, uniqueness and smoothness of the evolutionary solutions for smooth initial conditions, and global in time spatial analyticity of such solutions for analytical initial conditions. The width of the analyticity strip decays at most polynomially. We prove existence of travelling-wave solutions and uniqueness of travelling waves of a sufficiently small norm. The importance of damping is demonstrated by showing that the problem of finding travelling-wave solutions to the undamped RLWE is not well-posed. Finally, we demonstrate the asymptotic convergence of the power series expansion of travelling waves for a weak forcing.

nlin.CD↗

arXiv2013

A very smooth ride in a rough sea

It has been known for some time that a 3D incompressible Euler flow that has initially a barely smooth velocity field nonetheless has Lagrangian fluid particle trajectories that are analytic in time for at least a finite time (Ph. Serfati C.R. Acad. Sci. Série I 320, 175-180 (1995); A. Shnirelman arXiv:1205.5837 (2012)). Here an elementary derivation is given, based on Cauchy's form of the Euler equations in Lagrangian coordinates. This form implies simple recurrence relations among the time-Taylor coefficients of the Lagrangian map, used here to derive bounds for the C^{1,γ} Hölder norms of the coefficients and infer temporal analyticity of Lagrangian trajectories when the initial velocity is C^{1,γ}.

math.AP↗

arXiv2011

A priori bounds for Gevrey-Sobolev norms of space-periodic three-dimensional solutions to equations of hydrodynamic type

We present a technique for derivation of a priori bounds for Gevrey-Sobolev norms of space-periodic three-dimensional solutions to evolutionary partial differential equations of hydrodynamic type. It involves a transformation of the flow velocity in the Fourier space, which introduces a feedback between the index of the norm and the norm of the transformed solution, and results in emergence of a mildly dissipative term. To illustrate the technique, we derive finite-time bounds for Gevrey-Sobolev norms of solutions to the Euler and inviscid Burgers equations, and global in time bounds for the Voigt-type regularisations of the Euler and Navier-Stokes equation (assuming that the respective norm of the initial condition is bounded). The boundedness of the norms implies analyticity of the solutions in space.

math.AP↗

arXiv2010

Generation of a symmetric magnetic field by thermal convection in a plane rotating layer

We investigate numerically magnetic field generation by thermal convection with square periodicity cells in a rotating horizontal layer of electrically-conducting fluid with stress-free electrically perfectly conducting boundaries for Rayleigh numbers in the interval 5100\le R\le 5800. Dynamos of three kinds, apparently not encountered before, are presented: 1) Steady and time-periodic regimes, where the flow and magnetic field are symmetric about a vertical axis. In regimes with this symmetry, the global alpha-effect is insignificant, and the complex structure of the system of amplitude equations controlling weakly nonlinear stability of the system to perturbations with large spatial and temporal scales suggests that the perturbations are likely to exhibit uncommon complex patterns of behaviour, to be studied in the future work. 2) Periodic in time regimes, where magnetic field is always concentrated in the interior of the convective layer, in contrast to the behaviour first observed by St Pierre (1993) and often perceived as generic for electrically infinitely conducting boundaries. 3) A dynamo exhibiting chaotic behaviour of heteroclinic nature, where a sample trajectory enjoys excursions between a periodic magnetohydrodynamic regime and rolls. The rolls are amagnetic, but generate magnetic field kinematically. As a result, magnetic energy falls off almost to zero, while the rolls are approached.

nlin.CD↗

arXiv2010

Dependence of magnetic field generation by thermal convection on the rotation rate: a case study

Dependence of magnetic field generation on the rotation rate is explored by direct numerical simulation of magnetohydrodynamic convective attractors in a plane layer of conducting fluid with square periodicity cells for the Taylor number varied from zero to 2000, for which the convective fluid motion halts (other parameters of the system are fixed). We observe 5 types of hydrodynamic (amagnetic) attractors: two families of two-dimensional (i.e. depending on two spatial variables) rolls parallel to sides of periodicity boxes of different widths and parallel to the diagonal, travelling waves and three-dimensional "wavy" rolls. All types of attractors, except for one family of rolls, are capable of kinematic magnetic field generation. We have found 21 distinct nonlinear convective MHD attractors (13 steady states and 8 periodic regimes) and identified bifurcations in which they emerge. In addition, we have observed a family of periodic, two-frequency quasiperiodic and chaotic regimes, as well as an incomplete Feigenbaum period doubling sequence of bifurcations of a torus followed by a chaotic regime and subsequently by a torus with 1/3 of the cascade frequency. The system is highly symmetric. We have found two novel global bifurcations reminiscent of the SNIC bifurcation, which are only possible in the presence of symmetries. The universally accepted paradigm, whereby an increase of the rotation rate below a certain level is beneficial for magnetic field generation, while a further increase inhibits it (and halts the motion of fluid on continuing the increase) remains unaltered, but we demonstrate that this "large-scale" picture lacks many significant details.

nlin.CD↗

arXiv2009

The Monge-Ampere equation: various forms and numerical methods

We present three novel forms of the Monge-Ampere equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge-Ampere problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge-Ampere equation is a sum of such distributions.

nlin.CD↗

arXiv2009

Determination of a flow generating a neutral magnetic mode

The problem of reconstruction of a flow of conducting incompressible fluid generating a given magnetic mode is considered. We use the magnetic induction equation to derive ordinary differential equations along the magnetic field lines, which give an opportunity to determine the generating flow, if additional data is provided on a two-dimensional manifold transversal to magnetic field lines, and show that an arbitrary solenoidal vector field can not be a neutral magnetic mode sustained by any flow of conducting fluid.

nlin.CD↗

arXiv2009

Amplitude equations for weakly nonlinear two-scale perturbations of free hydromagnetic convective regimes in a rotating layer

Weakly non-linear stability of regimes of free hydromagnetic thermal convection in a rotating horizontal layer with free electrically conducting boundaries is considered in the Boussinesq approximation. Perturbations are supposed to involve large spatial and temporal scales. Applying methods for homogenisation of parabolic equations, we derive the system of amplitude equations governing the evolution of perturbations under the assumption that the alpha-effect is insignificant in the leading order. The amplitude equations involve the operators of anisotropic combined eddy diffusivity correction and advection. The system is qualitatively different from the system of mean-field equations for large-scale perturbations of forced convective hydromagnetic regimes. It is mixed: equations for the mean magnetic perturbation are evolutionary, all the rest involve neither time derivatives, nor the molecular diffusivity operator.

nlin.CD↗

arXiv2008

Mean-field equations for weakly nonlinear two-scale perturbations of forced hydromagnetic convection in a rotating layer

We consider stability of regimes of hydromagnetic thermal convection in a rotating horizontal layer with free electrically conducting boundaries, to perturbations involving large spatial and temporal scales. Equations governing the evolution of weakly nonlinear mean perturbations are derived under the assumption that the alpha-effect is insignificant in the leading order (e.g., due to a symmetry of the system). The mean-field equations generalise the standard equations of hydromagnetic convection: New terms emerge -- a second-order linear operator representing the combined eddy diffusivity, and quadratic terms associated with the eddy advection. If the perturbed CHM regime is non-steady and insignificance of the alpha-effect in the system does not rely on the presence of a spatial symmetry, the combined eddy diffusivity operator also involves a non-local pseudodifferential operator. If the perturbed CHM state is almost symmetric, alpha-effect terms appear in the mean-field equations as well. Near a point of a symmetry-breaking bifurcation, cubic nonlinearity emerges in the equations. All the new terms are in general anisotropic. A method for evaluation of their coefficients is presented; it requires solution of a significantly smaller number of auxiliary problems than in a straightforward approach.

nlin.CD↗

arXiv2007

Eddy diffusivity in convective hydromagnetic systems

An eigenvalue equation, for linear instability modes involving large scales in a convective hydromagnetic system, is derived in the framework of multiscale analysis. We consider a horizontal layer with electrically conducting boundaries, kept at fixed temperatures and with free surface boundary conditions for the velocity field; periodicity in horizontal directions is assumed. The steady states must be stable to short (fast) scale perturbations and possess symmetry about the vertical axis, allowing instabilities involving large (slow) scales to develop. We expand the modes and their growth rates in power series in the scale separation parameter and obtain a hierarchy of equations, which are solved numerically. Second order solvability condition yields a closed equation for the leading terms of the asymptotic expansions and respective growth rate, whose origin is in the (combined) eddy diffusivity phenomenon. For about 10% of randomly generated steady convective hydromagnetic regimes, negative eddy diffusivity is found.

nlin.CD↗

arXiv2006

Weakly nonlinear stability of convective magnetohydrodynamic systems without alpha-effect to perturbations involving large scales

I consider the problem of weakly nonlinear stability of three-dimensional convective magnetohydrodynamic systems, where there is no alpha-effect or it is insignificant, to perturbations involving large scales. I assume that the convective MHD state (steady or evolutionary), the stability of which I investigate, does not involve large spatio-temporal scales, and it is stable to perturbations involving the same small spatial scales, as the perturbed state. Mean-field equations, which I derive for the perturbation using asymptotic techniques for multiscale systems, are a generalization of the equations of magnetohydrodynamics (the Navier-Stokes and magnetic induction equations). The operator of combined eddy diffusivity emerges, which is in general anisotropic and not necessarily negatively defined, as well as new quadratic terms analogous to the ones describing advection.

nlin.CD↗

arXiv2006

Weakly nonlinear stability of magnetohydrodynamic systems with a center of symmetry to perturbations, involving large scales

I consider the problem of weakly nonlinear stability of three-dimensional parity-invariant magnetohydrodynamic systems to perturbations, involving large scales. I assume that the MHD state, the stability of which I investigate, does not involve large spatio-temporal scales, and it is stable to perturbations involving the same small spatial scales, as the perturbed MHD state. Mean-field equations, which I derive for the perturbation using asymptotic techniques for multiscale systems, are a generalization of the standard equations of magnetohydrodynamics (the Navier-Stokes equation with the Lorentz force and the magnetic induction equation). In them, the operator of combined eddy diffusivity emerges, which is in general anisotropic and not necessarily negatively defined, and new quadratic terms, analogous to the ones describing advection. A method for efficient computation of coefficients of the eddy diffusivity tensor and eddy advection terms in the mean-field equations is proposed.

nlin.CD↗

arXiv2005

On linear stability of steady space-periodic magnetohydrodynamic systems to perturbations involving large periods

I construct a complete asymptotic expansion of solutions to the problem of linear stability of three-dimensional steady space-periodic magnetohydrodynamic states to perturbations involving large periods. Eddy diffusivity tensor is derived for parity-invariant steady states. I present numerical evidences that if perturbations of the flow are permitted, then the effect of negative eddy diffusivity emerges at much larger magnetic molecular diffusivities than in the kinematic dynamo problem (where no perturbations of the flow are assumed).

nlin.CD↗