Marc-Etienne Brachet

Direct simulation of three-dimensional turbulence in the Taylor–Green vortex

The incompressible Navier-Stokes equations are numerically integrated on a Cray-2 machine with the periodic Taylor–Green initial data. Using a spectral method taking advantage of the symmetries of the flow, a resolution of 8643 and corresponding high Reynolds numbers (Rλ = 140) are obtained Visualisations of the resulting turbulent flow show that the turbulent activity is strongly correlated with low-pressure zones. We demonstrate that this behavior of the pressure field is linked to the fact that the square vorticity is spatially more concentrated than the energy dissipation.

Lack of universality in decaying magnetohydrodynamic turbulence.

Using computations of three-dimensional magnetohydrodynamic (MHD) turbulence with a Taylor-Green flow, whose inherent time-independent symmetries are implemented numerically, and in the absence of either a forcing function or an imposed uniform magnetic field, we show that three different inertial ranges for the energy spectrum may emerge for three different initial magnetic fields, the selecting parameter being the ratio of nonlinear eddy to Alfvén time. Equivalent computational grids range from 128(3) to 2048(3) points with a unit magnetic Prandtl number and a Taylor Reynolds number of up to 1500 at the peak of dissipation. We also show a convergence of our results with Reynolds number. Our study is consistent with previous findings of a variety of energy spectra in MHD turbulence by studies performed in the presence of both a forcing term with a given correlation time and a strong, uniform magnetic field. However, in contrast to the previous studies, here the ratio of characteristic time scales can only be ascribed to the intrinsic nonlinear dynamics of the paradigmatic flows under study.

Scaling laws for vortical nucleation solutions in a model of superflow

The bifurcation diagram corresponding to stationary solutions of the nonlinear Schrodinger equation describing a superflow around a disc is numerically computed using continuation techniques. When the Mach number is varied, it is found that the stable and unstable (nucleation) branches are connected through a primary saddle-node and a secondary pitchfork bifurcation. Computations are carried out for values of the ratio =d of the coherence length to the diameter of the disc in the range 1/5‐1/80. It is found that the critical velocity converges for =d ! 0 to an Eulerian value, with a scaling compatible with previous investigations. The energy barrier for nucleation solutions is found to scale as 2 . Dynamical solutions are studied and the frequency of supercritical vortex shedding is found to scale as the square root of the bifurcation parameter.

Stability and decay rates of nonisotropic attractive Bose-Einstein condensates

Nonisotropic attractive Bose-Einstein condensates are investigated numerically with Newton and inverse Arnoldi methods. The stationary solutions of the Gross-Pitaevskii equation and their linear stability are computed. Bifurcation diagrams are calculated and used to find the condensate decay rates corresponding to macroscopic quantum tunneling, two-three-body inelastic collisions, and thermally induced collapse. Isotropic and nonisotropic condensates are compared. The effect of anisotropy on the bifurcation diagram and the decay rates is discussed. Spontaneous isotropization of the condensates is found to occur. The influence of isotropization on the decay rates is characterized near the critical point.

Nonlinear oscillatory convection: A quantitative phase dynamics approach

Abstract The oscillatory instability of convective rolls, for both free-slip and rigid boundary conditions, is shown to be related to the translational and Galilean invariances of the Oberbeck-Boussinesq equations, which implies that the phase dynamics of the basic roll pattern is second order in time. In the free-slip case the invariance is exact and the instability comes in at zero wavenumber. It is argued that, for low Prandtl number fluids, the effect of the rigid boundaries is to weakly break the Galilean invariance, thereby shifting the instabilitys critical wavenumber to a finite value. We derive the equations governing the nonlinear phase dynamics and show that, when supercritical, the instability always saturates into travelling waves (as observed in experiments and numerical simulations). In the rigid case, the description is made quantitative by numerically computing both the linear and nonlinear coefficients of the phase equation. It is found that, depending of the values of Prandtl number P and basic roll pattern wavenumber α, the oscillatory instability can be supercritical or subcritical. In the case of mercury (P = 0.025), our model predicts transition from supercritical onset to subcritical onset, for α slightly below the critical wavenumber for convection onset.

Cascades, thermalization and eddy viscosity in helical Galerkin truncated Euler flows

The dynamics of the truncated Euler equations with helical initial conditions are studied. Transient energy and helicity cascades leading to Kraichnan helical absolute equilibrium at small scales, including a linear scaling of the relative helicity spectrum are obtained. Strong helicity effects are found using initial data concentrated at high wave numbers. Using low-wave-number initial conditions, the results of Cichowlas et.al. [Phys. Rev. Lett. 95, 264502 (2005)] are extended to helical flows. Similarities between the turbulent transient evolution of the ideal (time-reversible) system and viscous helical flows are found. Using an argument in the manner of Frisch et. al. [Phys. Rev. Lett. 101, 144501 (2008)], the excess of relative helicity found at small scales in the viscous run is related to the thermalization of the ideal flow. The observed differences in the behavior of truncated Euler and (constant viscosity) Navier-Stokes are qualitatively understood using the concept of eddy viscosity. The large scales of truncated Euler equations are then shown to follow quantitatively an effective Navier-Stokes dynamics based on a variable (scale dependent) eddy viscosity.

Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries

We investigate the ideal and incompressible magnetohydrodynamic (MHD) equations in three space dimensions for the development of potentially singular structures. The methodology consists in implementing the fourfold symmetries of the Taylor-Green vortex generalized to MHD, leading to substantial computer time and memory savings at a given resolution; we also use a regridding method that allows for lower-resolution runs at early times, with no loss of spectral accuracy. One magnetic configuration is examined at an equivalent resolution of 6144(3) points and three different configurations on grids of 4096(3) points. At the highest resolution, two different current and vorticity sheet systems are found to collide, producing two successive accelerations in the development of small scales. At the latest time, a convergence of magnetic field lines to the location of maximum current is probably leading locally to a strong bending and directional variability of such lines. A novel analytical method, based on sharp analysis inequalities, is used to assess the validity of the finite-time singularity scenario. This method allows one to rule out spurious singularities by evaluating the rate at which the logarithmic decrement of the analyticity-strip method goes to zero. The result is that the finite-time singularity scenario cannot be ruled out, and the singularity time could be somewhere between t=2.33 and t=2.70. More robust conclusions will require higher resolution runs and grid-point interpolation measurements of maximum current and vorticity.

The dynamics of unforced turbulence at high Reynolds number for Taylor-Green vortices generalized to MHD

We study decaying magnetohydrodynamics (MHD) turbulence stemming from the evolution of the Taylor–Green flow generalized recently to MHD, with equal viscosity and magnetic resistivity and up to equivalent grid resolutions of 20483 points. A pseudo-spectral code is used in which the symmetries of the velocity and magnetic fields have been implemented, allowing for sizable savings in both computer time and usage of memory at a given Reynolds number. The flow is non-helical, and at initial time the kinetic and magnetic energies are taken to be equal and concentrated in the large scales. After testing the validity of the method on grids of 5123 points, we analyze the data on the large grids up to Taylor Reynolds numbers of ≈2200. We find that the global temporal evolution is accelerated in MHD, compared to the corresponding neutral fluid case. We also observe an interval of time when such configurations have quasi-constant total dissipation, time during which statistical properties are determined after averaging over of the order of two turn-over times. A weak turbulence spectrum is obtained which is also given in terms of its anisotropic components. Finally, we contrast the development of small-scale eddies with two other initial conditions for the magnetic field and briefly discuss the structures that develop, and which display a complex array of current and vorticity sheets with clear rolling-up and folding.

Paradigmatic flow for small-scale magnetohydrodynamics: properties of the ideal case and the collision of current sheets.

We propose two sets of initial conditions for magnetohydrodynamics (MHD) in which both the velocity and the magnetic fields have spatial symmetries that are preserved by the dynamical equations as the system evolves. When implemented numerically they allow for substantial savings in CPU time and memory storage requirements for a given resolved scale separation. Basic properties of these Taylor-Green flows generalized to MHD are given, and the ideal nondissipative case is studied up to the equivalent of 2048;{3} grid points for one of these flows. The temporal evolution of the logarithmic decrements delta of the energy spectrum remains exponential at the highest spatial resolution considered, for which an acceleration is observed briefly before the grid resolution is reached. Up to the end of the exponential decay of delta , the behavior is consistent with a regular flow with no appearance of a singularity. The subsequent short acceleration in the formation of small magnetic scales can be associated with a near collision of two current sheets driven together by magnetic pressure. It leads to strong gradients with a fast rotation of the direction of the magnetic field, a feature also observed in the solar wind.

Two-fluid model of the truncated Euler equations

Abstract A phenomenological two-fluid model of the (time-reversible) spectrally-truncated 3 D Euler equation is proposed. The thermalized small scales are first shown to be quasi-normal. The effective viscosity and thermal diffusion are then determined, using EDQNM closure and Monte-Carlo numerical computations. Finally, the model is validated by comparing its dynamics with that of the original truncated Euler equation.