Self-similar Spectrum Research Articles

Self-similar evolution of wave turbulence in Gross-Pitaevskii system.

We study the universal nonstationary evolution of wave turbulence (WT) in Bose-Einstein condensates (BECs). Their temporal evolution can exhibit different kinds of self-similar behavior corresponding to a large-time asymptotic of the system or to a finite-time blowup. We identify self-similar regimes in BECs by numerically simulating the forced and unforced Gross-Pitaevskii equation(GPE) and the associated wave kinetic equation(WKE) for the direct and inverse cascades, respectively. In both the GPE and the WKE simulations for the direct cascade, we observe the first-kind self-similarity that is fully determined by energy conservation. For the inverse cascade evolution, we verify the existence of a self-similar evolution of the second kind describing self-accelerating dynamics of the spectrum leading to blowup at the zero mode (condensate) at a finite time. We believe that the universal self-similar spectra found in the present paper are as important and relevant for understanding the BEC turbulence in past and future experiments as the commonly studied stationary Kolmogorov-Zakharov (KZ) spectra.

Band conductivity oscillations in a gate-tunable graphene superlattice

Electrons exposed to a two-dimensional (2D) periodic potential and a uniform, perpendicular magnetic field exhibit a fractal, self-similar energy spectrum known as the Hofstadter butterfly. Recently, related high-temperature quantum oscillations (Brown-Zak oscillations) were discovered in graphene moiré systems, whose origin lies in the repetitive occurrence of extended minibands/magnetic Bloch states at rational fractions of magnetic flux per unit cell giving rise to an increase in band conductivity. In this work, we report on the experimental observation of band conductivity oscillations in an electrostatically defined and gate-tunable graphene superlattice, which are governed both by the internal structure of the Hofstadter butterfly (Brown-Zak oscillations) and by a commensurability relation between the cyclotron radius of electrons and the superlattice period (Weiss oscillations). We obtain a complete, unified description of band conductivity oscillations in two-dimensional superlattices, yielding a detailed match between theory and experiment.

Self-Similarity Properties of Complex Quasi-Periodic Fibonacci and Cantor Photonic Crystals

In this paper, the influence of structural modifications on basic quasi-periodic (QP) photonic crystals (PhC’s) on self-similarity feature in their spectral responses is examined. Investigated crystals are chosen based on a present knowledge on the QP crystals, and are classified according to their structure. One of the QP crystals considered for the calculations is a concatenation, Fibonacci structure. It characterizes with a self-similar spectra for its different orders, which means, that the spectral shape repeats itself and can be partially identical for a different orders of the Fibonacci QP crystal. The calculations were also performed for the fractal structure, based on a Cantor QP crystal. Just as for the case of the Fibonacci structure, it characterizes with a self-similar spectra for different orders of the structure. Considered photonic devices are next put through simple modification operations by multiplication, conjugation or mirror reflection. Resulting, modified structures are used for the calculations of their spectral response. Results show, that the self-similarity of the spectra is not affected by performed modifications, and thus spectral response of QP PhC can be designed without losing this feature. Moreover the regular expansion of the repeated central part of the spectrum that appears in higher-order Fibonacci QP PhC spectra (due to the self-similarity) with the increase Fibonacci crystal order is presented here for the first time.

Topological states on fractal lattices

We investigate the fate of topological states on fractal lattices. Focusing on a spinless chiral p-wave paired superconductor, we find that this model supports two qualitatively distinct phases when defined on a Sierpinski gasket. While the trivial phase is characterized by a self-similar spectrum with infinitely many gaps and extended eigenstates, the novel "topological" phase has a gapless spectrum and hosts chiral states propagating along edges of the graph. Besides employing theoretical probes such as the real-space Chern number, inverse participation ratio, and energy-level statistics in the presence of disorder, we develop a simple physical picture capturing the essential features of the model on the gasket. Extending this picture to other fractal lattices and topological states, we show that the p+ip state admits a gapped topological phase on the Sierpinski carpet and that a higher-order topological insulator placed on this lattice hosts gapless modes localized on corners.

Driven Hofstadter butterflies and related topological invariants

The properties of the Hofstadter butterfly, a fractal, self-similar spectrum of a two-dimensional electron gas, are studied in the case where the system is additionally illuminated with monochromatic light. This is accomplished by applying Floquet theory to a tight-binding model on the honeycomb lattice subjected to a perpendicular magnetic field and either linearly or circularly polarized light. It is shown how the deformation of the fractal structure of the spectrum depends on intensity and polarization. Thereby, the topological properties of the Hofstadter butterfly in the presence of the oscillating electric field are investigated. A thorough numerical analysis of not only the Chern numbers but also the ${W}_{3}$ invariants gives the appropriate insight into the topology of this driven system. This includes a comparison of a direct ${W}_{3}$ calculation to the method based on summing up Chern numbers of the truncated Floquet Hamiltonian.

Self-similar spectra of point-source scalar plumes in a turbulent boundary layer

Measurements of concentration fluctuations in a passive scalar plume released within a turbulent boundary layer are utilised to ascertain the scaling of concentration spectra. It is observed that the concentration spectra in a narrow meandering plume has a self-similar behaviour in both transverse ($y$) and vertical ($z$, i.e. wall-normal) directions. Experimental data reveal self-similarity when the magnitude of concentration spectra is scaled by the local concentration variance whereas frequency is suitably scaled utilising the integral length scale of the streamwise velocity or the boundary layer thickness and the source velocity as length and velocity scales, respectively. Furthermore, our data show that at each frequency, the concentration energy is distributed across the$y$and$z$directions that is proportional to concentration variance at that location. These results are consistent with our non-dimensional analysis. Based on these observations, if the mean plume statistics are known, a model is proposed with which concentration spectrum at any position within the plume can be calculated using the spectrum at any another location as the input. The model is tested extensively for point-source plumes released at various heights and streamwise distances in a turbulent boundary layer, and is found to predict spectra at different$y$and$z$locations in close agreement with measurements.

Self-similar interstory drift spectrum and response distribution of flexural-shear beam with nonuniform lateral stiffness

This paper proposes the concept of self-similar interstory drift spectrum based on the dimensional analysis of flexural-shear beam with nonuniform lateral stiffness, and scrutinizes the effects of lateral stiffness reduction on the seismic responses and response distribution. Firstly, the finite element formulation for the combined beam is established, which is convenient for parametric analysis. Subsequently, the intrinsic length scale and time scale are proposed to conduct correctly the dimensional analysis for seismic responses of flexural-shear beam, indicating that the normalized maximum interstory drift ratio and normalized maximum floor acceleration present a complete similarity in the normalized building height and the beauty of order. Hence, the concept of self-similar interstory drift spectrum is suggested to avoid the use of empirical relationship between fundamental period and building height. Moreover, the effects of lateral stiffness reduction on the normalized responses and response distribution of flexural-shear beam are examined under the idealized pulses and near-fault ground motions. With the exception of significant stiffness reduction, the effect of lateral stiffness reduction is generally small. In particular, the vertical distribution of interstory drift mainly depends on the structural type, whereas the floor acceleration generally increases along the building height. Finally, the established regression model can sufficiently fit the mean self-similar interstory drift spectrum and mean self-similar floor acceleration spectrum, and the normalized seismic responses can be well predicted by the fitted curves.

The focusing problem for the Leith model of turbulence: a self-similar solution of the third kind

Time-dependend evolution of hydrodynamic turbulence corresponding to formation of a thermodynamic state at the large-scale part of the spectrum is studied using the inviscid Leith model. In the wave vector space, the evolution leads to shrinking of the zero-spectrum ‘hole’—the so-called focusing problem. However, in contrast with the typical focusing problem in the nonlinear filtration theory, the focusing time is infinite for the Leith model. Respectively, the evolution is described by a self-similar solution of the third kind (discovered in Nazarenko and Grebenev (2017 J. Phys. A: Math. Theor. 50 035501)), and not the second kind as in the case of the typical filtration problem. Using a phase-plane analysis applied to the dynamical system generated by this type of similarity, we prove the existence of a new self-similar spectrum to this problem. We show that the final stationary spectrum scales as the thermodynamic energy equipartition spectrum, .

The kinetic equation solutions and Kolmogorov spectra

The Hasselmann kinetic equation (KE) for stochastic nonlinear surface waves is studied numerically with the aim of searching for features of the Kolmogorov turbulence (KT). To this aim, solutions of the KE for the long-term wave-spectrum evolution are executed. As far as the total wave action N and wave energy E are not preserved simultaneously in the course of the KE solution, two versions of the numerical algorithm are used, preserving values of N or E in separate. In every case, the KE solutions result in formation of the self-similar spectrum shape, Ssf(ω), with the frequency tail Ssf(ω) ∼ ω-4, independently of the N- or E-fluxes generated by the nonlinear interactions. This urges us to state that the used KE does not obey to regulations of the KT. The reason of this fact resides in the mathematical feature of the kinetic integral, which, in any case of solving the KE, results in formation of the nonlinear energy-transfer tail of kind Nl(ω) ∼ - ω-4, what stabilizes the spectral tail in form SSf(ω) ∼ ω-4.

Dimensional response analysis of bilinear SDOF systems under near-fault ground motions with intrinsic length scale

This paper proposes an intrinsic length scale with unambiguous physical significance for dimensional response analysis of inelastic systems, and establishes a versatile regression model for the self-similar response spectra of bilinear single-degree-of-freedom (SDOF) under near-fault ground motions. Firstly, according to the dimensional analysis of elastic SDOF systems subjected to analytical pulses, the intrinsic length scale is proposed to compare with the widely used energetic length scale. It is shown that the normalized displacement with respect to the intrinsic length scale is compatible with the analytical solution, and the overall absolute dispersion in the normalized responses is effectively reduced and is more evenly distributed in the entire frequency-ratio range. Then, the intrinsic length scale is adopted in the case of bilinear SDOF systems under analytical pulses, and similar results are observed. Finally, 30 near-fault ground motion records are selected as seismic input of the bilinear SDOF systems. Results indicate that both overall absolute and relative dispersions associated with intrinsic length scale are smaller than those with energetic length scale, when the normalized yield displacement is large. Therefore, the intrinsic length scale is recommended as an alternative length scale for dimensional analysis of elastoplastic structures. Moreover, it is illustrated that there exist two states of complete similarity in the normalized yield displacement. Accordingly, a versatile regression model is suggested for self-similar response spectrum, which can predict structural response.

Floquet analysis of a fractal-spectrum-generating periodically driven quantum system

We employ Floquet analysis to study the spectral properties of a double-kicked top (DKT) system. This is a classically nonintegrable dynamical system, which also shows chaos. However, even for the underlying classically chaotic dynamics, the quantum quasienergy spectrum of this system does not follow the random matrix conjecture which was proposed for the quantum spectrum of any classically chaotic systems. Instead the quasienergy spectrum of the DKT system shows a butterfly-like self-similar fractal spectrum. Here we investigate the relation between the quasienergy spectrum and the energy spectrum of the corresponding time-independent Floquet Hamiltonian. This Hamiltonian is determined by factorizing the Floquet time-evolution operator into three terms: an initial kick and a final kick, and in between a time-independent evolution dictated by a time-independent Hermitian operator which is called the Floquet Hamiltonian. Like any other generic systems, the Floquet Hamiltonian of the DKT system is also not possible to determine exactly. We apply a recently proposed perturbation theory to obtain the approximate Floquet Hamiltonian at the high-frequency driving limit. We then study the parameter regime where the quasienergy spectrum of the Floquet time-evolution operator matches the energy spectrum of the approximate Floquet Hamiltonian. We have also done a comparative analysis of how the two butterfly spectra disappear with the variation of a system parameter. Finally, we also explore the self-similar property of the energy spectrum of the approximate Floquet Hamiltonian and find its connection with the Farey sequence. Unlike all previous studies, here we have extensively investigated the self-similar property of the whole DKT butterfly.

Generalized self-similar spectrum and the effect of large-scale in decaying homogeneous isotropic turbulence

In statistically stationary conditions, the turbulent energy spectrum in a high Reynolds number flow exhibits a k−5/3 (Kolmogorov) regime, with a faster decay at larger, dissipative wavenumbers. Here, we investigate how the energy spectrum of a turbulent flow evolves in time when turbulence decays freely, in the absence of forcing. Results from direct numerical simulation of decaying turbulence in a periodic box with several different initial conditions suggest a generalized self-similar spectrum, depending on and , where η(t) and L(t) are, respectively, the small (Kolmogorov) and large scales of the flow. A closure method allows us to obtain an explicit form of the spectrum, which reproduces the deviations from the Kolmogorov spectrum at small k observed numerically. The solution can also be used to determine the second and third order structure functions in decaying turbulent flows, and to study their scaling behavior. Our results compare favorably with high-Reynolds number wind tunnel data. Moreover, our theoretical results provide support to the interesting empirical observation by Pearson et al (2002 Phys. Fluids 14 1288–90) that, independent of the large scale structure of the flow, the dimensionless energy dissipation rate is a universal constant when scaled in terms of the turbulent kinetic energy of the flow, and of the length scale corresponding to the peak of the compensated energy spectrum.

Field Observations and Modeling of Surfzone Sensible Heat Flux

AbstractSurfzone sensible heat flux (HS,SZ) obtained through direct eddy-covariance estimates was measured at four different sandy beach sites along Monterey Bay, California. The HS,SZ source region is estimated from a footprint probability distribution function (pdf) model and is only considered when at least 70% of the footprint pdf occupies the surfzone. The measured HS,SZ is 2 times the modeled interfacial sensible heat (HS,int) using COARE3.5. A formulation for estimating sensible heat flux from spray droplets (HS,spray) generated during depth-limited wave breaking is developed. The sea-spray generation function for droplet radii ranging over 0.1 < ro < 1000 μm is based on self-similar spectra of spray droplets measured from the surfzone forced by the average depth-limited breaking wave dissipation across the surfzone. However, it is shown that the size of the spume droplets that contribute to HS,spray is limited owing to the relatively short residence time in air as the droplets fall to the sea surface during wave breaking. The addition of the surfzone-modeled HS,spray to the COARE3.5 HS,int gives values similar to the observed surfzone HS,SZ, highlighting the importance of depth-limited wave-breaking processes to sensible heat flux. Measured HS,SZ values are an order of magnitude larger than simultaneous open ocean observations.

Self-similar gravity wave spectra resulting from the modulation of bound waves

Experiments are performed in a large basin to investigate nonlinear gravity waves generated by a periodic or a slowly modulated forcing. High-frequency self-similar spectra ascribed to bound waves are reported, which share similarities with the ones observed in previous experiments of wave turbulence.

Interacting Multiscale Acoustic Vortices as Coherent Excitations in Dust Acoustic Wave Turbulence.

In this work, using three-dimensional intermittent dust acoustic wave turbulence in a dusty plasma as a platform and multidimensional empirical mode decomposition into different-scale modes in the 2+1D spatiotemporal space, we demonstrate the experimental observation of the interacting multiscale acoustic vortices, winding around wormlike amplitude hole filaments coinciding with defect filaments, as the basic coherent excitations for acoustic-type wave turbulence. For different decomposed modes, the self-similar rescaled stretched exponential lifetime histograms of amplitude hole filaments, and the self-similar power spectra of dust density fluctuations, indicate that similar dynamical rules are followed over a wide range of scales. In addition to the intermode acoustic vortex pair generation, propagation, or annihilation, the intra- and intermode interactions of acoustic vortices with the same or opposite helicity, their entanglement and synchronization, are found to be the key dynamical processes in acoustic wave turbulence, akin to the interacting multiscale vortices around wormlike cores observed in hydrodynamic turbulence.

Elements of fractal geometry in the 1H NMR spectrum of a copolymer intercalation-complex: identification of the underlying Cantor set.

Sequence-selective intercalation of pyrene into the chain-folds of a random, binary copolyimide under fast-exchange conditions results in the development of self-similar structure in the diimide region of the 1H NMR spectrum. The resulting spectrum can be described by the mathematics of fractals, an approach that is rationalised in terms of a dynamic summation of ring-current shielding effects produced by pyrene molecules intercalating into the chain at progressively greater distances from each "observed" diimide residue. The underlying set of all such summations is found to be a defined mathematical fractal namely the fourth-quarter Cantor set, within which the observed spectrum is embedded. The pattern of resonances predicted by a geometric construction of the fourth-quarter Cantor set agrees well with the observed spectrum.

Favourable and Unfavourable EMF Frequency Patterns in Cancer: Perspectives for Improved Therapy and Prevention

Carcinogenesis fits in a frequency pattern of electromagnetic field (EMF) waves, in which a gradual loss of cellular organization occurs. Such generation of cancer features can be inhibited by adequate exposure to coherent electromagnetic frequencies. However, cancer can also be initiated and promoted at other distinct frequencies of electromagnetic waves. Both observations were revealed by analyzing 100 different EMF frequency data reported in a meta-analyses of 123 different, earlier published, biomedical studies. The studied EM frequencies showed a fractal pattern of 12 beneficial (anti-cancer) frequencies, and 12 detrimental (cancer promoting) frequencies, that form the central pattern of a much wider self-similar EMF spectrum of cancer inhibiting or promoting activities. Inhibiting of the cancer process, and even curing of the disease, can thus be considered through exposure to the coherent type of EM fields. Stabilization of the disease can be understood by constructive resonance of macromolecules in the cancer cell with the externally appied coherent EMF field frequencies, called solitons/polarons. The latter, for instance, have been shown earlier to induce repair in DNA/RNA conformation and/or epigenetic changes. The field of EMF treatment of cancer disorders is rapidly expanding and our studies may invite further experimental and clinical studies in which systematically various potential EMF treatment protocols could be applied, with combined and modulated frequencies, to obtain even more efficient EMF anti-cancer therapies.

Self-Similar Interfacial Impedance of Electrodes in High Conductivity Media.

Electrode polarization (EP) happening due to accumulation of ions at the electrode/electrolyte interface is an inevitable phenomenon while measuring impedance spectrum in high conductivity buffers and at low RF spectrum. Well-characterized time scales elucidating the EP effect are important for the rational design of microfluidic devices and impedance sensors. In this Article, interfacial impedance at the electrode/electrolyte interface is investigated considering channel height and Debye length effects on characteristic time scale in a binary electrolyte solution using parallel plate electrode configuration. Experimental results reveal self-similarity of normalized electrical impedance as a function of the normalized frequency. The experimental results also match with numerical solutions obtained by finite element simulation of unsteady fully coupled Poisson-Nernst-Planck (PNP) equations. Furthermore, fractal shaped gold nanostructured electrodes are examined, and it has been proven that electric double layer (EDL) formed on porous electrode surfaces acts as a thick EDL and modifications to the characteristic time scale is necessary for porous electrodes. Finally, a constant phase element (CPE) model is proposed to account for the self-similar impedance spectrum, which can be used for different channel heights and solution conductivities.

Hofstadter butterfly evolution in the space of two-dimensional Bravais lattices

The self-similar energy spectrum of a particle in a periodic potential under a magnetic field, known as the Hofstadter butterfly, is determined by the lattice geometry as well as the external field. Recent realizations of artificial gauge fields and adjustable optical lattices in cold atom experiments necessitate the consideration of these self-similar spectra for the most general two-dimensional lattice. In a previous work, we investigated the evolution of the spectrum for an experimentally realized lattice which was tuned by changing the unit cell structure but keeping the square Bravais lattice fixed. We now consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the lattice is deformed between lattices with different point symmetry groups. We model the optical lattice by a sinusoidal real space potential and obtain the tight binding model for any lattice geometry by calculating the Wannier functions. We introduce the magnetic field via Peierls substitution and numerically calculate the energy spectrum. The transition between the two most symmetric lattices, i.e. the triangular and the square lattice displays the importance of bipartite symmetry featuring deformation as well as closing of some of the major energy gaps. The transition from the square to rectangular and from the triangular to centered rectangular lattices are analyzed in terms of coupling of one-dimensional chains. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the transitions. We use gap Chern numbers to identify distinct topological regions in the space of Bravais lattices.

Complementary remarks to properties of the energy spectrum in Leith's model of turbulence

In [1], we found that an anomalous exponent x∗ of the power-law asymptotic E(k,t)∼Ck−x∗ as t→t∗ for k≫1 in k-space representation occurs into the interval 5/3<x∗<x2, where t∗ is a finite singular time. Here 5/3 is the Kolmogorov index for the pure Kolmogorov spectrum and x2≈1.95 corresponds to a Hopf bifurcation which appears in the dynamical system associated with the Leith model in the form of self-similar representation. This is then births the transient self-similar spectrum of the kinetic energy of turbulence discovered in [2]. The upper estimate was justified only by numerical simulations [2]. We prove that the transient self-similar spectrum realised by a heteroclinic connection on the phase plane indeed corresponds to x∗<x2. Then, we study the behavior of the energy spectrum E(k,t) beyond t∗ as t→∞.