In this work, a multifractal framework is proposed to investigate the effects of current sheets i... more In this work, a multifractal framework is proposed to investigate the effects of current sheets in solar wind turbulence. By using multifractal detrended fluctuation analysis coupled with surrogate methods and volatility, two solar wind magnetic field time series are investigated, one with current sheets and one without current sheets. Despite the lack of extreme-events intermittent bursts in the current sheet-free series, both series are shown to be strongly multifractal, although the current sheet-free series displays an almost linear behaviour for the scaling exponent of structure functions. Long-range correlations are shown to be the main source of multifractality for the series without current sheets, while a combination of heavy-tail distribution and non-linear correlations is responsible for multifractality in the series with current sheets. The multifractality in both time series is formally shown to be associated with an energy-cascade process using the p-model.
We present results of numerical investigation of regimes of steady thermal convective dynamo in a... more We present results of numerical investigation of regimes of steady thermal convective dynamo in a plane layer of electrically conducting fluid rotating about the vertical axis and subjected to large-scale perturbations.
The two-dimensional Navier-Stokes-Kuramoto-Sivashinsky equation on the Connection Machine
Computing Systems in Engineering, 1995
... Physics Fluids 10 (1967), p. 1417. Full Text via CrossRef. 2. B. Dubrulle and U. Frisch, Thee... more ... Physics Fluids 10 (1967), p. 1417. Full Text via CrossRef. 2. B. Dubrulle and U. Frisch, Theeddy-viscosity of parity-invariant flow. Physical Review A43 (1991), pp. 53555364. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (52). ...
Negative Isotropic Eddy Viscosity: A Common Phenomenon in two Dimensions
Navier—Stokes Equations and Related Nonlinear Problems, 1995
ABSTRACT We show the existence of two-dimensional flows having an isotropic and negative eddy vis... more ABSTRACT We show the existence of two-dimensional flows having an isotropic and negative eddy viscosity. Such flows, when subject to a very weak large-scale perturbation of wavenumber k, will amplify it isotropically with a rate proportional to k 2 .
Anisotropy and universality in Solar Wind turbulence. Ulysses spacecraft data
Springer Proceedings Physics
... All these functions have both isotropic and anisotropic contribution: 5gL(r,) = S^is°(rx) + S... more ... All these functions have both isotropic and anisotropic contribution: 5gL(r,) = S^is°(rx) + S^anis°(rx). ... This is consistent with the recovery-of-isotropy assumption in some MHD models However, it is important to also control higher order statistical objects, ie the whole shape of the ...
Magnetic fields of planets and other astrophysical objects are often sustained by con- ducting fl... more Magnetic fields of planets and other astrophysical objects are often sustained by con- ducting fluid motions, driven by convection in their interior. A common feature of such objects is rotation. This work is aimed at investigation of magnetic field generation in an idealized setup. We consider a rotating conducting fluid heated from below in a plane horizontal layer (often regarded as representing a segment of a spherical shell in the in- terior of a planet). Flows and magnetic fields in square periodicity cells are examined for the aspect ratio, for which the trivial hydrodynamic steady state becomes unstable to square patterns at the minimal Rayleigh number. Thermal convection of electrically conducting fluid in a plane horizontal layer is consid- ered here in the Boussinesq approximation. The fluid is heated from below in a plane hor- izontal layer rotating about the vertical axis, the stress-free isothermal horizontal bound- aries being perfect electrical conductors. In the dim...
We analyze low and high-latitude fast solar wind data from the Ulysses spacecraft from 1992 to 19... more We analyze low and high-latitude fast solar wind data from the Ulysses spacecraft from 1992 to 1994 using a a systematic method to analyse the anisotropic content of the magnetic field fluctuations. We investigate all available frequencies, 1−10 −6 Hz, for both high and low-latitudes datasets and are able to quantify the relative importance of the anisotropic versus the isotropic fluctuations. We analyse, up to sixth order, longitudinal, transverse and mixed magnetic field correlations. Our results show that strongly intermittent and anisotropic events are present in the solar wind plasma at high frequencies/small scales, indicating the absence of a complete recovery of isotropy. Anisotropic scaling properties are compatible for high and low-latitude data, suggesting a universal behaviour in spite of the different rate of evolution of the fast solar wind streams in the two environments.
Dependence of magnetic field generation on the rotation rate is explored by direct numerical simu... more 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, twofrequency 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.
Slow-down of nonlinearity in 2-D Navier-Stokes flow
Physica D: Nonlinear Phenomena, 1994
ABSTRACT The equations governing the dynamics of large-scale perturbations superimposed on incomp... more ABSTRACT The equations governing the dynamics of large-scale perturbations superimposed on incompressible small-scale flow driven by a force have, under suitable conditions, the same structure as Navier-Stokes equations. The breaking of Galilean invariance due to the presence of the small-scale flow will, in general, induce a `vertex renormalization': the constant a in front of the advective nonlinearity does not remain equal to unity. A class of basic flows where the calculation of a can be performed analytically is discussed. For finite Reynolds numbers, the constant a can indeed be very different from unity and can also vanish. The Reynolds number and the dynamics of a large-scale flow can then be quite different than predicted by setting a = 1.
The two-dimensional Navier-Stokes equations with a large-scale instability of the Kuramoto-Sivash... more The two-dimensional Navier-Stokes equations with a large-scale instability of the Kuramoto-Sivashinsky type, describing marginally negative eddy-viscosity situations, is simulated on a Connection Machine CM-2. Up to millions of time steps at the resolution 2562 and tens of thousands at the resolution 10242 are performed. Advantage is taken of a novel complex variable form of the twodimensional Navier-Stokes equations, which requires only two complex FFTs per time step. A linear growth phase, a disorganized inverse cascade phase, and a structured vortical phase are successively observed. In the vortical phase monopolar and multipolar structures are proliferating and display strongly depleted nonlinearities.
The existence of two-dimensional flows with an isotropic and negative eddy viscosity is demonstra... more The existence of two-dimensional flows with an isotropic and negative eddy viscosity is demonstrated. Such flows, when subject to a very weak large-scale perturbation of wavenumberkwill amplify it with a rate proportional tok2, independent of the direction.Specifically, it is assumed that the basic (unperturbed) flow is space-time periodic, possesses a centre of symmetry (parity-invariance) and has three- or six-fold rotational invariance to ensure isotropy of the eddy-viscosity tensor.The eddy viscosities emerging from the multiscale analysis are calculated by two different methods. First, there is an expansion in powers of the Reynolds number which can be carried out to large orders, and then extended analytically (thanks to a meromorphy property) beyond the disk of convergence. Secondly, there is a spectral method. The two methods typically agree within a fraction of 1%.A simple example, the ‘decorated hexagonal flow’, of a time-independent flow with isotropic negative eddy visco...
Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensio... more Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressible flow.As shown by Dubrulle & Frisch (1991), the eddy viscosity, which is in general a fourth-order anisotropic tensor, is expressible in terms of the solution of auxiliary problems. These are, essentially, three-dimensional linearized Navier–Stokes equations which must be solved numerically.The dynamics of weak large-scale perturbations of wavevector k is determined by the eigenvalues – called here ‘eddy viscosities’ – of a two by two matrix, obtained by contracting the eddy viscosity tensor with two k-vectors and projecting onto the plane transverse to k to ensure incompressibility. As a consequence, eddy viscosities in three dimensions, but not in two, can become complex. It is shown that this is ruled out for flow with cubic symmetry, the eddy viscosities of which may, however, become negative.An instance is the equilateral ABC-flow (A = B =...
Short-wave instabilities in the Benjamin-Bona-Mahoney-Peregrine equation: theory and numerics
Inverse Problems, 2001
ABSTRACT In this paper we discuss the nonlinear propagation of waves of short wavelength in dispe... more ABSTRACT In this paper we discuss the nonlinear propagation of waves of short wavelength in dispersive systems. We propose a family of equations that is likely to describe the asymptotic behaviour of a large class of systems. We then restrict our attention to the analysis of the simplest nonlinear short-wave dynamics given by U0ξτ = U0-3(U0)2. We integrate numerically this equation for periodic and non-periodic boundary conditions, and we find that short waves may exist only if the amplitude of the initial profile is not too large.
The two-dimensional Navier-Stokes equations with a large scale instability of the Kuramoto-Sivash... more The two-dimensional Navier-Stokes equations with a large scale instability of the Kuramoto-Sivashinsky type, describing marginally negative eddy-viscosity situations, is simulated on a Connection Machine CM-2. Up to millions of time steps at the resolution 2562 and tens of thousands at the resolution 1024 z are performed. A linear growth phase, a disorganized inverse cascade phase and a structured vortical phase are successively observed. In the vortical phase monopolar and multipolar structures are proliferating and display strongly depleted nonlinearities.
We present a novel method for estimating the circulations and positions of point vortices in a tw... more We present a novel method for estimating the circulations and positions of point vortices in a two-dimensional (2D) environment using trajectory data of passive particles in the presence of Gaussian noise. The method comprises two algorithms: the first one calculates the vortex circulations, while the second one reconstructs the vortex trajectories. This reconstruction is done thanks to a hierarchy of optimization problems, involving the integration of systems of differential equations, over time sub-intervals all with the same amplitude defined by the autocorrelation function for the advected passive particles' trajectories. Our findings indicate that accurately tracking the position of vortices and determining their circulations is achievable, even when passive particle trajectories are affected by noise.
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