arXiv subjects

Miguel D. Bustamante

Publications and source records attributed to Miguel D. Bustamante.

At least 37 records · Page 2Linked to original sources

arXiv2019

Nonlinear Rossby wave-wave and wave-mean flow theory for long term Solar cycle modulations

The Schwabe cycle of solar activity exhibits modulations and frequency fluctuations on slow time scales of centuries and millennia. Plausible physical explanations for the cause of these long-term variations of the solar cycle are still elusive, with possible theories including stochasticity of alpha effect and fluctuations of the differential rotation. It has been suggested recently in the literature that there exists a possible relation between the spatio-temporal structure of Solar cycle and the nonlinear dynamics of magnetohydrodynamic Rossby waves at the solar tachocline, including both wave-wave and wave-mean flow interactions. Here we extend the nonlinear theory of MHD Rossby waves presented in a previous article to take into account long term modulation effects due to a recently discovered mechanism that allows significant energy transfers throughout different wave triads: the precession resonance mechanism. We have found a large number of Rossby-Haurwitz wave triads whose frequency mismatches are compatible with the solar cycle frequency. Consequently, by analyzing the reduced dynamics of two triads coupled by a single mode (five-wave system), we have demonstrated that in the amplitude regime in which precession resonance occurs, the energy transfer throughout the system yields significant long-term modulations on the main $\sim 11$yr period associated with intra-triad energy exchanges. We further show that such modulations display an inverse relationship between the characteristic wave amplitude and the period of intra-triad energy exchanges, which is consistent with the Waldmeier's law for the solar cycle. In the presence of a constant forcing and dissipation, the five-wave system in the precession resonance regime exhibits irregular amplitude fluctuations with some periods resembling the Grand Minimum states.

astro-ph.SR↗

arXiv2019

On the convergence of the normal form transformation in discrete Rossby and drift wave turbulence

We study numerically the region of convergence of the normal form transformation for the case of the Charney-Hasagawa-Mima (CHM) equation to investigate whether certain finite amplitude effects can be described in normal coordinates. We do this by taking a Galerkin truncation of four Fourier modes making part of two triads: one resonant and one non-resonant, joined together by two common modes. We calculate the normal form transformation directly from the equations of motion of our reduced model, successively applying the algorithm to calculate the transformation up to $7^\textrm{th}$ order to eliminate all non-resonant terms, and keeping up to $8$-wave resonances. We find that the amplitudes at which the normal form transformation diverge very closely match with the amplitudes at which a finite-amplitude phenomenon called $precession$ $resonance$ (Bustamante $et$ $al.$ 2014) occurs, characterised by strong energy transfers. This implies that the precession resonance mechanism cannot be explained using the usual methods of normal forms in wave turbulence theory, so a more general theory for intermediate nonlinearity is required.

physics.flu-dyn↗

arXiv2018

Exact discrete resonances in the Fermi-Pasta-Ulam-Tsingou system

In systems of N coupled anharmonic oscillators, exact resonant interactions play an important role in the energy exchange between normal modes. In the weakly nonlinear regime, those interactions may facilitate energy equipartition in Fourier space. We consider analytically resonant wave-wave interactions for the celebrated Fermi-Pasta-Ulam-Tsingou (FPUT) system. Using a number-theoretical approach based on cyclotomic polynomials, we show that the problem of finding exact resonances for a system of N particles is equivalent to a Diophantine equation whose solutions depend sensitively on the set of divisors of N. We provide an algorithm to construct all possible resonances, based on two methods: pairing-off and cyclotomic, which we introduce to build up explicit solutions to the 4-, 5- and 6-wave resonant conditions. Our results shed some light in the understanding of the long-standing FPUT paradox, regarding the sensitivity of the resonant manifolds with respect to the number of particles N and the corresponding time scale of the interactions leading to thermalisation. In this light we demonstrate that 6-wave resonances always exist for any N, while 5-wave resonances exist if N is divisible by 3 and N > 6. It is known (for finite N) that 4-wave resonances do not mix energy across the spectrum, so we investigate whether 5-wave resonances can produce energy mixing across a significant region of the Fourier spectrum by analysing the interconnected network of Fourier modes that can interact nonlinearly via resonances. The answer depends on the set of odd divisors of N that are not divisible by 3: the size of this set determines the number of dynamically independent components, corresponding to independent constants of motion (energies). We show that 6-wave resonances connect all these independent components, providing in principle a restoring mechanism for full-scale thermalisation.

nlin.CD↗

arXiv2018

Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in 1D Burgers equation

We present a theoretical and numerical study of Fourier space triad phase dynamics in one-dimensional stochastically forced Burgers equation at Reynolds number $\mathrm{Re} \approx 2.7 \times 10^4$. We demonstrate that Fourier triad phases over the inertial range display a collective behaviour characterised by intermittent periods of synchronisation and alignment, reminiscent of Kuramoto model (Kuramoto 1984) and directly related to collisions of shocks in physical space. These periods of synchronisation favour efficient energy fluxes across the inertial range towards small scales, resulting in strong bursts of dissipation and enhanced coherence of Fourier energy spectrum. The fast time scale of the onset of synchronisation relegates energy dynamics to a passive role: this is further examined using a reduced system with the Fourier amplitudes fixed in time -- a phase-only model. We show that intermittent triad phase dynamics persists without amplitude evolution and we broadly recover many of the characteristics of the full Burgers system. In addition, for both full Burgers and phase-only systems the physical space velocity statistics reveals that triad phase alignment is directly related to the non-Gaussian statistics typically associated with structure-function intermittency in turbulent systems.

physics.flu-dyn↗

arXiv2018

Discrete resonant Rossby/drift wave triads: an explicit parameterisation and a fast direct numerical search algorithm

We report results on the explicit parameterisation of discrete Rossby-wave resonant triads of the Charney-Hasegawa-Mima equation in the small-scale limit (i.e. large Rossby deformation radius), following up from our previous solution in terms of elliptic curves (Bustamante and Hayat, 2013). We find an explicit parameterisation of the discrete resonant wavevectors in terms of two rational variables. We show that these new variables are restricted to a bounded region and find this region explicitly. We argue that this can be used to reduce the complexity of a direct numerical search for discrete triad resonances. Also, we introduce a new direct numerical method to search for discrete resonances. This numerical method has complexity ${\mathcal{O}}(N^3)$, where $N$ is the largest wavenumber in the search. We apply this new method to find all discrete irreducible resonant triads in the wavevector box of size $5000$, in a calculation that took about $10.5$ days on a $16$-core machine. Finally, based on our method of mapping to elliptic curves, we discuss some dynamical implications regarding the spread of quadratic invariants across scales via resonant triad interactions, in the form of sharp bounds on the size of the interacting wavevectors.

physics.flu-dyn↗

arXiv2018

Physically Feasible Decomposition of Engino$^{\circledR}$ Toy Models: A Graph Theoretic Approach

During the 125th European Study Group with Industry held in Limassol, Cyprus, 5-9 December 2016, one of the participating companies, Engino.net Ltd, posed a very interesting challenge to the members of the study group. Engino.net Ltd is a Cypriot company, founded in 2004, that produces a series of toy sets -- the Engino$^{\circledR}$ toy sets -- consisting of a number of building blocks which can be assembled by pupils to compose toy models. Depending on the contents of a particular toy set, the company has developed a number of models that can be built utilizing the blocks present in the set, however the production of a step-by-step assembly manual for each model could only be done manually. The goal of the challenge posed by the company was to implement a procedure to automatically generate the assembly instructions for a given toy. In the present paper we propose a graph-theoretic approach to model the problem and provide a series of results to solve it by employing modified versions of well established algorithms in graph theory. An algorithmic procedure to obtain a hierarchical, physically feasible decomposition of a given toy model, from which the assembly instructions can be recovered, is proposed.

cs.DM↗

arXiv2016

Derivation of the Biot-Savart equation from the Nonlinear Schrödinger equation

We present a systematic derivation of the Biot-Savart equation from the Nonlinear Schrödinger equation, in the limit when the curvature radius of vortex lines and the inter-vortex distance are much greater than the vortex healing length, or core radius. We derive the Biot-Savart equations in Hamiltonian form with Hamiltonian expressed in terms of vortex lines, $$ H= \frac{κ^2}{8 π}\int_{|{\bf s}-{\bf s}'|>ξ_*} \frac{d{\bf s} \cdot d {\bf s}'}{|{\bf s}-{\bf s}'|} \,,$$ with cut-off length $ξ_* \approx 0.3416293 /\sqrt{ρ_0},$ where $ρ_0$ is the background condensate density far from the vortex lines and $κ$ is the quantum of circulation.

physics.flu-dyn↗

arXiv2016

Phase and precession evolution in the Burgers equation

We present a phenomenological study of the phase dynamics of the one-dimensional stochastically forced Burgers equation, and of the same equation under a Fourier mode reduction on a fractal set. We study the connection between coherent structures in real space and the evolution of triads in Fourier space. Concerning the one-dimensional case, we find that triad phases show alignments and synchronisations that favour energy fluxes towards small scales --a direct cascade. In addition, strongly dissipative real-space structures are associated with entangled correlations amongst the phase precession frequencies and the amplitude evolution of Fourier triads. As a result, triad precession frequencies show a non-Gaussian distribution with multiple peaks and fat tails, and there is a significant correlation between triad precession frequencies and amplitude growth. Links with dynamical systems approach are briefly discussed, such as the role of unstable critical points in state space. On the other hand, by reducing the fractal dimension $D$ of the underlying Fourier set, we observe: i) a tendency toward a more Gaussian statistics, ii) a loss of alignment of triad phases leading to a depletion of the energy flux, and iii) the simultaneous reduction of the correlation between the growth of Fourier mode amplitudes and the precession frequencies of triad phases.

physics.flu-dyn↗

arXiv2015

Atypical late-time singular regimes accurately diagnosed in stagnation-point-type solutions of 3D Euler flows

We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point-type introduced by Gibbon et al. (1999). By employing the method of mapping to regular systems, presented in Bustamante (2011) and extended to the symmetry-plane case by Mulungye et al. (2015), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^2.$ This late-time regime has an atypical spectrum: it is Gaussian rather than exponential in the wavenumbers. The analyticity-strip width decays to zero in a finite time, albeit so slowly that it remains well above the collocation-point scale for all simulation times $t < T^* - 10^{-9000}$, where $T^*$ is the singularity time. Reaching such a proximity to singularity time is not possible in the original temporal variable, because floating point double precision ($\approx 10^{-16}$) creates a `machine-epsilon' barrier. Due to this limitation on the \emph{original} independent variable, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to the singularity time: $T^*- t \approx 10^{-140}.$

physics.flu-dyn↗

arXiv2015

A Robust Quantitative Comparison Criterion of Two Signals based on the Sobolev Norm of Their Difference

In this manuscript we present a method for the quantitative comparison of two surfaces, applicable to temporal and/or spatial extent in one or two dimensions. Often surface comparisons are simply overlaid graphs of results from different methodologies that are qualitative at best; it is the purpose of this work to facilitate quantitative evaluation. The surfaces can be analytical, numerical, and/or experimental, and the result returned by the method, termed surface similarity parameter or normalized error, has been normalized so that its value lies between zero and one. When the parameter has a value of zero, the surfaces are in perfect agreement, whereas a value of one is indicative of perfect disagreement. To provide insight regarding the magnitude of the parameter, several canonical cases are presented, followed by results from breaking water wave experimental measurements with numerical simulations, and by a comparison of a prescribed, periodic, square-wave surface profile and the subsequent manufactured surface.

physics.flu-dyn↗

arXiv2015

Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions

Motivated by the work on stagnation-point type exact solutions (with infinite energy) of 3D Euler fluid equations by Gibbon et al. (1999) and the subsequent demonstration of finite-time blowup by Constantin (2006) we introduce a one-parameter family of models of the 3D Euler fluid equations on a 2D symmetry plane. Our models are seen as a deformation of the 3D Euler equations which respects the variational structure of the original equations so that explicit solutions can be found for the supremum norms of the basic fields. The value of the model's parameter determines if there is finite-time blowup, and the singularity time can be computed explicitly in terms of the initial conditions and the model's parameter. We use a representative parameter value, for which the solution blows up in finite-time, as a benchmark for the systematic study of errors in numerical solutions. We compare numerical integrations of our "original" model with a "mapped" version of these equations. The mapped version is a globally regular (in time) system of equations, obtained via a bijective nonlinear mapping of time and fields from the original model equations (Bustamante 2011). We show that the mapped system's numerical solution increases accuracy in estimates of supremum norms and singularity time while entailing only a small additional computational cost. We study the Fourier spectrum of the model's numerical solution and find that the analyticity-strip width (a measure of the solution's analyticity) tends to zero as a power law in a finite time. This is in agreement with the finite-time blowup of the supremum norms, in the light of rigorous bounds stemming from the bridge (Bustamante & Brachet 2012) between the analyticity-strip method and the Beale-Kato-Majda type of theorems. We conclude by discussing the implications of this research on the analysis of numerical solutions of the 3D Euler fluid equations.

physics.flu-dyn↗

arXiv2014

Robust energy transfer mechanism via precession resonance in nonlinear turbulent wave systems

A robust energy transfer mechanism is found in nonlinear wave systems, which favours transfers towards modes interacting via triads with nonzero frequency mismatch, applicable in meteorology, nonlinear optics and plasma wave turbulence. We introduce the concepts of truly dynamical degrees of freedom and triad precession. Transfer efficiency is maximal when the triads' precession frequencies resonate with the system's nonlinear frequencies, leading to a collective state of synchronised triads with strong turbulent cascades at intermediate nonlinearity. Numerical simulations confirm analytical predictions.

physics.flu-dyn↗

arXiv2013

Counting of discrete Rossby/drift wave resonant triads (again)

The purpose of our earlier note (arXiv:1309.0405 [physics.flu-dyn]) was to remove the confusion over counting of resonant wave triads for Rossby and drift waves in the context of the Charney-Hasegawa-Mima equation. A comment by Kartashov and Kartashova (arXiv:1309.0992v1 [physics.flu-dyn]) on that note has further confused the situation. The present note aims to remove this obfuscation.

physics.flu-dyn↗

arXiv2013

Robust energy transfer mechanism and critically balanced turbulence via non-resonant triads in nonlinear wave systems

A robust energy transfer mechanism is found in nonlinear wave systems, which favours transfers towards modes interacting via non-resonant triads, applicable in meteorology, nonlinear optics and plasma wave turbulence. Transfer efficiency is maximal when the frequency mismatch of the non-resonant triad balances the system's nonlinear frequency: at intermediate levels of oscillation amplitudes an instability is triggered that explores unstable manifolds of periodic orbits, so turbulent cascades are most efficient at intermediate nonlinearity. Numerical simulations confirm analytical predictions.

physics.flu-dyn↗

arXiv2013

Counting of discrete Rossby/drift wave resonant triads

The purpose of this note is to remove the confusion about counting of resonant wave triads for Rossby and drift waves in the context of the Charney-Hasegawa-Mima equation. In particular, we aim to point out a major error of over-counting of triads in the paper "Discrete exact and quasi-resonances of Rossby/drift waves on beta-plane with periodic boundary conditions", by Kartashov and Kartashova, arXiv:1307.8272v1 [physics.flu-dyn] (2013).

physics.flu-dyn↗

arXiv2013

Complete classification of discrete resonant Rossby/drift wave triads on periodic domains

We consider the set of Diophantine equations that arise in the context of the barotropic vorticity equation on periodic domains, when nonlinear wave interactions are studied to leading order in the amplitudes. The solutions to this set of Diophantine equations are of interest in atmosphere (Rossby waves) and Tokamak plasmas (drift waves), because they provide the values of the spectral wavevectors that interact resonantly via three-wave interactions. These come in "triads", i.e., groups of three wavevectors. We provide the full solution to the Diophantine equations in the case of infinite Rossby deformation radius. The method is completely new, and relies on mapping the unknown variables to rational points on quadratic forms of "Minkowski" type. Classical methods invented centuries ago by Fermat, Euler, Lagrange and Minkowski, are used to classify all solutions to our original Diophantine equations, thus providing a computational method to generate numerically all the resonant triads in the system. Our method has a clear computational advantage over brute-force numerical search: on a 10000^2 grid, the brute-force search would take 15 years using optimised C++ codes, whereas our method takes about 40 minutes. The method is extended to generate quasi-resonant triads, which are defined by relaxing the resonant condition on the frequencies, allowing for a small mismatch. Quasi-resonances are robust with respect to physical perturbations, unlike exact resonances. Therefore, the new method is really valuable in practical terms. We show that the set of quasi-resonances form an intricate network of clusters of connected triads, whose structure depends on the value of the allowed mismatch. We provide some quantitative comparison between the clusters' structure and the onset of fully nonlinear turbulence in the barotropic vorticity equation, and provide perspectives for new research.

physics.flu-dyn↗

arXiv2013

Quadratic invariants for discrete clusters of weakly interacting waves

We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular (M times N) matrix A with entries 1, -1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J = N - M* (greater than or equal to N - M), where M* is the number of linearly independent rows in A. Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including the basic structures leading to M* < M. We illustrate our findings by presenting examples from the Charney-Hasegawa-Mima wave model, and by showing a classification of small (up to three-triad) clusters.

physics.flu-dyn↗

arXiv2012

Percolation transition in the kinematics of nonlinear resonance broadening in Charney-Hasegawa-Mima model of Rossby wave turbulence

We study the kinematics of nonlinear resonance broadening of interacting Rossby waves as modelled by the Charney-Hasegawa-Mima equation on a biperiodic domain. We focus on the set of wave modes which can interact quasi-resonantly at a particular level of resonance broadening and aim to characterise how the structure of this set changes as the level of resonance broadening is varied. The commonly held view that resonance broadening can be thought of as a thickening of the resonant manifold is misleading. We show that in fact the set of modes corresponding to a single quasi-resonant triad has a nontrivial structure and that its area in fact diverges for a finite degree of broadening. We also study the connectivity of the network of modes which is generated when quasi-resonant triads share common modes. This network has been argued to form the backbone for energy transfer in Rossby wave turbulence. We show that this network undergoes a percolation transition when the level of resonance broadening exceeds a critical value. Below this critical value, the largest connected component of the quasi-resonant network contains a negligible fraction of the total number of modes in the system whereas above this critical value a finite fraction of the total number of modes in the system are contained in the largest connected component. We argue that this percolation transition should correspond to the transition to turbulence in the system.

nlin.CD↗