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

Harris, Jamie; Connaughton, Colm; Bustamante, Miguel D.

doi:10.1088/1367-2630/15/8/083011

Cited by 10 publications

(22 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Second and more striking, small-scale intermittency is quickly reduced for D < 3 and it almost vanishes already at D = 2.98. As a consequence, the presence or absence of some of the Fourier modes strongly modify the fluctuations of all the others, suggesting the possibility that intermittency is the result of percolating dynamical properties across the whole Fourier lattice [33]. Because of the spectrum modification, the scaling exponent of the second order longitudinal structure function becomes ζ 2 + (D − 3), where ζ 2 is the measured in the D = 3 homogeneous and isotropic case.…”

Section: Parametersmentioning

confidence: 99%

Turbulence on a Fractal Fourier Set

et al. 2015

View full text Add to dashboard Cite

A novel investigation of the nature of intermittency in incompressible, homogeneous, and isotropic turbulence is performed by a numerical study of the Navier-Stokes equations constrained on a fractal Fourier set. The robustness of the energy transfer and of the vortex stretching mechanisms is tested by changing the fractal dimension D from the original three dimensional case to a strongly decimated system with D=2.5, where only about 3% of the Fourier modes interact. This is a unique methodology to probe the statistical properties of the turbulent energy cascade, without breaking any of the original symmetries of the equations. While the direct energy cascade persists, deviations from the Kolmogorov scaling are observed in the kinetic energy spectra. A model in terms of a correction with a linear dependency on the codimension of the fractal set E(k)∼k(-5/3+3-D) explains the results. At small scales, the intermittency of the vorticity field is observed to be quasisingular as a function of the fractal mode reduction, leading to an almost Gaussian statistics already at D∼2.98. These effects must be connected to a genuine modification in the triad-to-triad nonlinear energy transfer mechanism.

show abstract

Section: Parametersmentioning

confidence: 99%

Turbulence on a Fractal Fourier Set

et al. 2015

View full text Add to dashboard Cite

show abstract

“…In this new type of connectivity, the lower bound is n triads + 2 (dot-dashed curve, green online). [25]).…”

Section: Quasi-resonant Triadsmentioning

confidence: 99%

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

Bustamante¹,

Hayat²

2013

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

We consider the set of Diophantine equations that arise in the context of the partial differential equation called "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 wavenumbers come in "triads", i.e., groups of three wavevectors.We provide the full solution to the Diophantine equations in the physically sensible limit when the Rossby deformation radius is infinite. The method is completely new, and relies on mapping the unknown variables via rational transformations, first to rational points on elliptic curves and surfaces, and from there to rational points on quadratic forms of "Minkowski" type (such as the familiar space-time in special relativity). Classical methods invented centuries ago by Fermat, Euler, Lagrange, 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. Computationally speaking, our method has a clear advantage over brute-force numerical search: on a 10000 2 grid, the brute-force search would take 15 years using optimised C ++ codes on a cluster, whereas our method takes about 40 minutes using a laptop.Moreover, the method is extended to generate so-called quasi-resonant triads, which are defined by relaxing the resonant condition on the frequencies, allowing for a small mismatch. Quasi-resonant triads' distribution in wavevector space is robust with respect to physical perturbations, unlike resonant triads' distribution. Therefore, the extended method is really valuable in practical terms. We show that the set of quasi-resonant triads form an intricate network of connected triads, forming clusters whose structure depends on the value of the allowed mismatch. It is believed that understanding this network is absolutely relevant to understanding turbulence. We provide some quantitative comparison between the clusters' structure and the onset of fully nonlinear turbulent regime in the barotropic vorticity equation, and we provide perspectives for new research.

show abstract

“…Detailed study of this effect can be performed, using the approach developed in [7] for an elastic pendulum subject to the external force. Drawn from the resonance conditions provided in Equations (1) and (2), resonance detuning in the nonlinear case can be defined in a number of ways, e.g., as a phase detuning [8], or frequency detuning [9][10][11]. The frequency detuning, used in the present paper, is defined as…”

Section: Introductionmentioning

confidence: 99%

“…Existing studies are limited to kinematics, i.e., a study of the structure of many quasi-resonances depending on the magnitude of the detuning. Moreover, the structure under consideration is presented in a form that allows neither to restore corresponding dynamical system, nor to deduce any dynamical characteristics of a detuned resonance [10]:…”

Section: Introductionmentioning

confidence: 99%

Resonance Enhancement by Suitably Chosen Frequency Detuning

Dutykh

Tobisch

2020

Mathematics

View full text Add to dashboard Cite

The theory of exact resonances (kinematics and dynamics) is well developed while even the very concept of detuned resonance is ambiguous and only studies of their kinematic characteristics (that is, those not depending on time) are available in the literature. In this paper, we report novel effects enforced by the resonance detuning on solutions of the dynamical system describing interactions of three spherical planetary waves. We establish that the energy variation range can significantly exceed the range of the exact resonance for suitably chosen values of the detuning. The asymmetry of system’s solutions with respect to the sign of the detuning parameter is demonstrated. Finally, a non-monotonic dependence of the energy oscillation period with respect to detuning magnitude is discovered. These results have direct implications in physics of atmosphere, e.g., for prediction of weather extremes in the Northern Hemisphere midlatitudes (Proc. Nat. Acad. Sci. USA 2016, 133(25), 6862–6867). Moreover, similar study can be conducted for a generic three-wave system taken in the Hamiltonian form which makes our results applicable for an arbitrary Hamiltonian three-wave system met in climate prediction theory, geophysical fluid dynamics, plasma physics, etc.

show abstract

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

Cited by 10 publications

References 34 publications

Turbulence on a Fractal Fourier Set

Turbulence on a Fractal Fourier Set

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

Resonance Enhancement by Suitably Chosen Frequency Detuning

Contact Info

Product

Resources

About