Tracing complex singularities with spectral methods

Sulem, Catherine; Sulem, P. L.; Frisch, H.

doi:10.1016/0021-9991(83)90045-1

Cited by 199 publications

(196 citation statements)

References 22 publications

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“…The logarithmic decrement technique [25] is now applied to quantify further the delay of the onset of the evolution as the amplitude of the imposed uniform field is increased. If the fields are regular, then the energy spectra must decay at least exponentially at large wavenumber k. Based on this assumption, the logarithmic decrement δ(t) is defined by the large k asymptotic of the energy spectra:…”

Section: Dynamical Slowing Down In the Presence Of A Uniform Magnmentioning

confidence: 99%

Alfvén waves and ideal two-dimensional Galerkin truncated magnetohydrodynamics

2011

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We investigate numerically the dynamics of two-dimensional Euler and ideal magnetohydrodynamics (MHD) flows in systems with a finite number of modes, up to 4096 2 , for which several quadratic invariants are preserved by the truncation and the statistical equilibria are known. Initial conditions are the Orszag-Tang vortex with a neutral X-point centered on a stagnation point of the velocity field in the large scales. In MHD, we observe that the total energy spectra at intermediate times and intermediate scales correspond to the interactions of eddies and waves, ET (k) ∼ k −3/2 . Moreover, no dissipative range is visible neither for Euler nor for MHD in two dimensions; in the former case, this may be linked to the existence of a vanishing turbulent viscosity whereas in MHD, the numerical resolution employed may be insufficient. When imposing a uniform magnetic field to the flow, we observe a lack of saturation of the formation of small scales together with a significant slowing-down of their equilibration, with however a cut-off independent partial thermalization being reached at intermediate scales.

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Section: Dynamical Slowing Down In the Presence Of A Uniform Magnmentioning

confidence: 99%

Alfvén waves and ideal two-dimensional Galerkin truncated magnetohydrodynamics

2011

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“…• Our goal in high resolution calculations will be to include spectral convergence tests, in particular the analyticity strip method (Sulem et al [12], see also [13,14] and references therein) which gives independent evidence of singular/nonsingular behavior of the flow and allows one to extrapolate the convergence of ω ∞ .…”

Section: Looking Forwardmentioning

confidence: 99%

3D Euler about a 2D symmetry plane

Bustamante

Kerr

2008

Physica D: Nonlinear Phenomena

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Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr (1993) and the lack thereof by Hou and Li (2006). Core profiles designed to reproduce the two results are presented, new more robust analysis is proposed, and new criteria for when calculations should be terminated are introduced and compared with classical resolution studies and spectral convergence tests. Most of the analysis is on a 512 × 128 × 2048 mesh, with new analysis on a just completed 1024 × 256 × 2048 used to confirm trends. One might hypothesize that there is a finite-time singularity with enstrophy growth like Ω ∼ (Tc − t) −γ Ω and vorticity growth like ω ∞ ∼ (Tc − t) −γ . The new analysis would then support γΩ ≈ 1/2 and γ > 1. These represent modifications of the conclusions of Kerr (1993). Issues that might arise at higher resolution are discussed.

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“…We note in passing that this property was the key technical point in the work by Frisch and Morf (1981) concerning a particular scenario for intermittency in dynamical systems. More relevant to the present work is the paper by Sulem et al (1983), who, to the best of our knowledge, were the first to make use of the Fourier transform to characterize the analyticity properties of computer-generated solutions of PDEs. They concentrated on the behaviour of the width δ(t) of the analyticity strip as a function of time; δ(t) was obtained as the logarithmic decrement of the exponential fall-off of the modulus of the Fourier transform at very large wavenumbers.…”

Section: Meromorphy and Fourier Transformmentioning

confidence: 99%

Meromorphy and topology of localized solutions in the Thomas–MHD model

Fournier

Galtier

2001

J. Plasma Phys.

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Abstract. The one-dimensional MHD system first introduced by J.H. Thomas [Phys. Fluids 11, 1245(1968] as a model of the dynamo effect is thoroughly studied in the limit of large magnetic Prandtl number. The focus is on two types of localized solutions involving shocks (antishocks) and hollow (bump) waves. Numerical simulations suggest phenomenological rules concerning their generation, stability and basin of attraction. Their topology, amplitude and thickness are compared favourably with those of the meromorphic travelling waves, which are obtained exactly, and respectively those of asymptotic descriptions involving rational or degenerate elliptic functions. The meromorphy bars the existence of certain configurations, while others are explained by assuming imaginary residues. These explanations are tested using the numerical amplitude and phase of the Fourier transforms as probes of the analyticity properties. Theoretically, the proof of the partial integrability backs up the role ascribed to meromorphy. Practically, predictions are derived for MHD plasmas.

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Tracing complex singularities with spectral methods

Cited by 199 publications

References 22 publications

Alfvén waves and ideal two-dimensional Galerkin truncated magnetohydrodynamics

Alfvén waves and ideal two-dimensional Galerkin truncated magnetohydrodynamics

3D Euler about a 2D symmetry plane

Meromorphy and topology of localized solutions in the Thomas–MHD model

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