Local Nonlinear Interaction Research Articles

Mind the gap: non-local cascades and preferential heating in high-β Alfvénic turbulence

ABSTRACT Characterizing the thermodynamics of turbulent plasmas is key to decoding observable signatures from astrophysical systems. In magnetohydrodynamic (MHD) turbulence, non-linear interactions between counter-propagating Alfvén waves cascade energy to smaller spatial scales where dissipation heats the protons and electrons. When the thermal pressure far exceeds the magnetic pressure, linear theory predicts a spectral gap at perpendicular scales near the proton gyroradius where Alfvén waves become non-propagating. For simple models of an MHD turbulent cascade that assume only local non-linear interactions, the cascade halts at this gap, preventing energy from reaching smaller scales where electron dissipation dominates, leading to an overestimate of the proton heating rate. In this work, we demonstrate that non-local contributions to the cascade, specifically large-scale shearing and small-scale diffusion, can bridge the non-propagating gap, allowing the cascade to continue to smaller scales. We provide an updated functional form for the proton-to-electron heating ratio accounting for this non-local energy transfer by evaluating a non-local weakened cascade model over a range of temperature and pressure ratios. In plasmas where the thermal pressure dominates the magnetic pressure, we observe that the proton heating is moderated compared to the significant enhancement predicted by local models.

A differential approximation model for passive scalar turbulence

Two-dimensional passive scalar turbulence is studied by means of a k-space diffusion model based on a third order differential approximation. This simple description of local nonlinear interactions in Fourier space is shown to provide a general expression, in line with previous seminal works, and appears to be suitable for various 2D turbulence problems. Steady state solutions for the spectral energy density of the flow are shown to recover the Kraichnan–Kolmogorov phenomenology of the dual cascade, while various passive scalar spectra, such as Batchelor or Obukhov–Corssin spectra are recovered as steady state solutions of the spectral energy density of the passive scalar. These analytical results are then corroborated by numerical solutions of the time evolving problem with energy and passive scalar injection and dissipation on a logarithmic wavenumber space grid over a large range of scales. The particular power law spectra that are obtained are found to depend mainly on the location of the kinetic and passive scalar energy injections. In particular, it is shown that by injecting the energy simultaneously at large and small scales, wave number spectra consistent with those of Nastrom–Gage can be obtained.

Testing macroscopic local realism using local nonlinear dynamics and time settings

We show how one may test macroscopic local realism where, different from conventional Bell tests, all relevant measurements need only distinguish between two macroscopically distinct states of the system being measured. Here, measurements give macroscopically distinguishable outcomes for a system observable and do not resolve microscopic properties (of order $\hbar$). Macroscopic local realism assumes: (1) macroscopic realism (the system prior to measurement is in a state which will lead to just one of the macroscopically distinguishable outcomes) and (2) macroscopic locality (a measurement on a system at one location cannot affect the macroscopic outcome of the measurement on a system at another location, if the measurement events are spacelike separated). To obtain a quantifiable test, we define $M$-scopic local realism where the outcomes are separated by an amount $\sim M$. We first show for $N$ up to $20$ that $N$-scopic Bell violations are predicted for entangled superpositions of $N$ bosons (at each of two sites). Secondly, we show violation of $M$-scopic local realism for entangled superpositions of coherent states of amplitude $\alpha$, for arbitrarily large $M=\alpha$. In both cases, the systems evolve dynamically according to a local nonlinear interaction. The first uses nonlinear beam splitters realised through nonlinear Josephson interactions; the second is based on nonlinear Kerr interactions. To achieve the Bell violations, the traditional choice between two spin measurement settings is replaced by a choice between different times of evolution at each site.

A competing-risks model explains hierarchical spatial coupling of measles epidemics en route to national elimination.

Apart from its global health importance, measles is a paradigm for the low-dimensional mechanistic understanding of local nonlinear population interactions. A central question for spatio-temporal dynamics is the relative roles of hierarchical spread from large cities to small towns and metapopulation transmission among local small population clusters in measles persistence. Quantifying this balance is critical to planning the regional elimination and global eradication of measles. Yet, current gravity models do not allow a formal comparison of hierarchical versus metapopulation spread. We address this gap with a competing-risks framework, capturing the relative importance of competing sources of reintroductions of infection. We apply the method to the uniquely spatio-temporally detailed urban incidence dataset for measles in England and Wales, from 1944 to the infection's vaccine-induced nadir in the 1990s. We find that despite the regional influence of a few large cities (for example, London and Liverpool), metapopulation aggregation in neighbouring towns and cities played an important role in driving national dynamics in the prevaccination era. As vaccination levels increased in the 1970s and 1980s, the signature of spatially predictable spread diminished: increasingly, infection was introduced from unidentifiable random sources possibly outside regional metapopulations. The resulting erratic dynamics highlight the challenges of identifying shifting sources of infection and characterizing patterns of incidence in times of high vaccination coverage. More broadly, the underlying incidence and demographic data, accompanying this paper, will also provide an important resource for exploring nonlinear spatiotemporal population dynamics.

Evidence of Loop Current Frontal Eddy Intensification Through Local Linear and Nonlinear Interactions with the Loop Current

AbstractDuring the 2010 Deepwater Horizon oil spill, an underestimated part of the oil was entrained into an intensified Loop Current Frontal Eddy (LCFE). These eddies, which are also known to play an essential role in the Loop Current eddy (LCE) shedding, are difficult to predict, and the dynamics involving their intensification are still not fully understood. The Loop Current (LC) and its strongest LCFEs were continuously tracked during 2009–2011 using sea surface height (SSH) from AVISO+. A mooring array provided complementary information about the internal structure of the LC‐LCFE interaction. The intensification of the tracked LCFEs presented similar characteristics, independent of their location: a steep increase in kinetic energy, a corresponding negative increase in SSH, and an increase in its area. As the LCFE grows, the flow at the interface with the LC becomes stronger and deeper and the horizontal density gradient between the features increases. The intensification of the front and the LCFEs is driven by the advection (nonlinear) term and the gradient pressure (linear) term in the momentum budget. Evidence of an inverse energy cascade suggests that LCFEs are extracting energy and mass from the submesoscale field to the zone of contact between the LC and the LCFE, strengthening the front and allowing the LCFEs to grow during periods of intensification. Understanding the physics driving the LCFE intensification is a key step to improve LC forecast models and to better predict LCE shedding events, as well as oil and particle transport around the LC.

Nonlinear interaction decomposition (NID): A method for separation of cross-frequency coupled sources in human brain

Cross-frequency coupling (CFC) between neuronal oscillations reflects an integration of spatially and spectrally distributed information in the brain. Here, we propose a novel framework for detecting such interactions in Magneto- and Electroencephalography (MEG/EEG), which we refer to as Nonlinear Interaction Decomposition (NID). In contrast to all previous methods for separation of cross-frequency (CF) sources in the brain, we propose that the extraction of nonlinearly interacting oscillations can be based on the statistical properties of their linear mixtures. The main idea of NID is that nonlinearly coupled brain oscillations can be mixed in such a way that the resulting linear mixture has a non-Gaussian distribution. We evaluate this argument analytically for amplitude-modulated narrow-band oscillations which are either phase-phase or amplitude-amplitude CF coupled. We validated NID extensively with simulated EEG obtained with realistic head modelling. The method extracted nonlinearly interacting components reliably even at SNRs as small as −15 dB. Additionally, we applied NID to the resting-state EEG of 81 subjects to characterize CF phase-phase coupling between alpha and beta oscillations. The extracted sources were located in temporal, parietal and frontal areas, demonstrating the existence of diverse local and distant nonlinear interactions in resting-state EEG data. All codes are available publicly via GitHub.

Making ICRF power compatible with a high-Z wall in ASDEX Upgrade

A comparison of the ASDEX Upgrade 3-strap ICRF antenna data with the linear electro-magnetic TOPICA calculations is presented. The comparison substantiates a reduction of the local electric field at the radially protruding plasma-facing elements of the antenna as a relevant approach for minimizing tungsten (W) sputtering in conditions when the slow wave is strongly evanescent. The measured reaction of the time-averaged RF current at the antenna limiters to the antenna feeding variations is less sensitive than predicted by the calculations. This is likely to have been caused by temporal and spatial fluctuations in the 3D plasma density distribution affected by local non-linear interactions. The 3-strap antenna with the W-coated limiters produces drastically less W sputtering compared to the W-coated 2-strap antennas. This is consistent with the non-linear asymptotic SSWICH-SW calculations for RF sheaths.

Chimera regimes in a ring of oscillators with local nonlinear interaction

One of important problems concerning chimera states is the conditions of their existence and stability. Until now, it was assumed that chimeras could arise only in ensembles with nonlocal character of interactions. However, this assumption is not exactly right. In some special cases chimeras can be realized for local type of coupling [1–3]. We propose a simple model of ensemble with local coupling when chimeras are realized. This model is a ring of linear oscillators with the local nonlinear unidirectional interaction. Chimera structures in the ring are found using computer simulations for wide area of values of parameters. Diagram of the regimes on plane of control parameters is plotted and scenario of chimera destruction are studied when the parameters are changed.

Fractional Schrödinger dynamics and decoherence

We study the dynamics of the Schrödinger equation with a fractional Laplacian (−Δ)α, and the decoherence of the solution is observed. Analytically, we obtain equations of motion for the expected position and momentum in the fractional Schödinger equation, equations that are the fractional counterpart of the well-known Newtonian equations of motion for the standard (α=1) Schrödinger equation. Numerically, we propose an explicit, effective numerical method for solving the time-dependent fractional nonlinear Schrödinger equation—a method that has high order spatial accuracy, requires little memory, and has low computational cost. We apply our method to study the dynamics of fractional Schrödinger equation and find that the nonlocal interactions from the fractional Laplacian introduce decoherence into the solution. The local nonlinear interactions can however reduce or delay the emergence of decoherence. Moreover, we find that the solution of the standard NLS behaves more like a particle, but the solution of the fractional NLS behaves more like a wave with interference effects.

Further tests of the local nonlinear interaction-based mechanism for simultaneous suppression of tone burst-evoked otoacoustic emissions

Tone burst-evoked otoacoustic emission (TBOAE) components measured in response to a 1 kHz tone burst (TB1) are suppressed by the simultaneous presence of an additional tone burst (TB2). This “simultaneous suppression of TBOAEs” has been explained in terms of a mechanism based on local nonlinear interactions between the basilar membrane (BM) travelling waves caused by TB1 and TB2. A test of this local nonlinear interaction (LNI)-based mechanism, as a function of the frequency separation (Δf, expressed in kHz) between TB1 and TB2, has previously been reported by Killan et al. (2012) using a simple mathematical model [Killan et al., Hear. Res. 285, 58–64 (2012)]. The two experiments described in this paper add additional data on the extent to which the LNI-based mechanism can account for simultaneous suppression, by testing two further hypotheses derived from the model predictions. Experiment I tested the hypothesis that TBOAE suppression is directly linked to TBOAE amplitude nonlinearity where ears that exhibit a higher degree of amplitude nonlinearity yield greater suppression than more linear ears, and this relationship varies systematically as a function of Δf. In order to test this hypothesis simultaneous suppression at a range of values of Δf at 60 dB peak-equivalent sound pressure level (p.e. SPL) and TBOAE amplitude nonlinearity from normal human ears was measured. In Experiment II the hypothesis that suppression will also increase progressively as a function of increasing tone burst level was tested by measuring suppression for a range of Δf and tone burst levels at 40, 50, 60 and 70 dB p.e. SPL. The majority of the findings from both experiments provide support for the LNI-based mechanism being primarily responsible for simultaneous suppression. However, some data were inconsistent with this view. Specifically, a breakdown in the relationship between suppression and TBOAE amplitude nonlinearity at Δf = 1 (i.e. when TB2 was reasonably well separated from, and had a higher frequency than TB1) and unexpected level-dependence, most notably at Δf = 1, but also where Δf = −0.5, was observed. Either the LNI model is too simple or an alternative explanation, involving response components generated at basal regions of the basilar membrane, is required to account for these findings.

Laboratory study of the nonlinear transformation of irregular waves over a mild slope

This paper considers the nonlinear transformation of irregular waves propagating over a mild slope (1:40). Two cases of irregular waves, which are mechanically generated based on JONSWAP spectra, are used for this purpose. The results indicate that the wave heights obey the Rayleigh distribution at the offshore location; however, in the shoaling region, the heights of the largest waves are underestimated by the theoretical distributions. In the surf zone, the wave heights can be approximated by the composite Weibull distribution. In addition, the nonlinear phase coupling within the irregular waves is investigated by the wavelet-based bicoherence. The bicoherence spectra reflect that the number of frequency modes participating in the phase coupling increases with the decreasing water depth, as does the degree of phase coupling. After the incipient breaking, even though the degree of phase coupling decreases, a great number of higher harmonic wave modes are also involved in nonlinear interactions. Moreover, the summed bicoherence indicates that the frequency mode related to the strongest local nonlinear interactions shifts to higher harmonics with the decreasing water depth.

A mechanism for simultaneous suppression of tone burst-evoked otoacoustic emissions

Tone burst-evoked otoacoustic emission (TBOAE) components in response to a 1 kHz tone burst are suppressed by the simultaneous presence of tone bursts at higher frequencies. To date, the underlying cause of this “simultaneous suppression” of TBOAEs is unclear. This paper describes a potential mechanism based on local nonlinear interactions between basilar membrane (BM) travelling waves, and tests the extent to which it is able to account for this specific suppression phenomenon. A simple mathematical model based on local nonlinear interactions was developed, and its predictions for a range of tone burst pairs were compared to corresponding TBOAE suppression data recorded from fourteen normally hearing human ears at a level of 60 dB p.e. SPL. Model predictions and mean TBOAE suppression data showed close agreement for all pairs of tone bursts. These results suggest that simultaneous suppression of TBOAEs can be explained solely in terms of the local nonlinear interaction-based mechanism. However, the involvement of other mechanisms, involving components generated at places basal to their characteristic place along the BM, cannot be excluded.

Collapse in the nonlocal nonlinear Schrödinger equation

We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schrödinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both Lyapunoff's method and virial identities. We find that in the one-dimensional case, i.e. for n = 1, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension n ⩾ 2 and singular kernel ∼1/rα, no collapse takes place if α < 2, whereas collapse is possible if α ⩾ 2. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of ∼1/r2 kernels for n = 3. Moreover, different evolution scenarios for the three-dimensional physically relevant case of Bose–Einstein condensates are studied numerically for both the ground state soliton and higher order toroidal states with, and without, an additional local repulsive nonlinear interaction. In particular, we show that the presence of local repulsive nonlinearity can prevent collapse in those cases.

A global-local approach to nonlinear system identification: A review

We present the basic components of a time-domain nonlinear system identification (NSI) method with a promise of applicability to a broad class of smooth and non-smooth dynamical systems. The proposed NSI method is based on the close correspondence between analytical and empirical slow-flow dynamics, and relies on the direct analysis of measured time series without any a priori assumptions on the system dynamics; i.e. it is purely an output-based approach. The central assumption of the methodology is that the measured time series can be decomposed in terms of a finite number of oscillating components that are in the form of ‘fast’ monochromatic oscillations modulated by ‘slow’ amplitudes. The empirical slow-flow model of the dynamics is obtained from empirical mode decomposition (EMD), and its correspondence to the analytical slow-flow model establishes a local nonlinear interaction model (NIM). A NIM consists of a set of intrinsic modal oscillators (IMOs) that can reproduce the measured time series over different time scales and can account for (even strongly) nonlinear modal interactions. Hence, it represents a local model of the dynamics, identifying specific nonlinear transitions. An IMO, whose response can reproduce the corresponding intrinsic mode function acquired from EMD analysis, represents the dynamical characteristics of the system under certain initial or parameter conditions (i.e. local aspects). By collecting energy-dependent frequency behaviors from all identified IMOs, a frequency-energy plot (in the modal space) can be constructed, which depicts global features of the dynamical system. This represents a global model of the dynamics. We demonstrate the efficacy of the outlined global–local NSI method with two-degree-of-freedom dynamical systems—one smooth, and the other non-smooth. Copyright © 2010 John Wiley & Sons, Ltd.

Local non-linear interactions in the visual cortex may reflect global decorrelation

The classical receptive field in the primary visual cortex have been successfully explained by sparse activation of relatively independent units, whose tuning properties reflect the statistical dependencies in the natural environment. Robust surround modulation, emerging from stimulation beyond the classical receptive field, has been associated with increase of lifetime sparseness in the V1, but the system-wide modulation of response strength have currently no theoretical explanation. We measured fMRI responses from human visual cortex and quantified the contextual modulation with a decorrelation coefficient (d), derived from a subtractive normalization model. All active cortical areas demonstrated local non-linear summation of responses, which were in line with hypothesis of global decorrelation of voxels responses. In addition, we found sensitivity to surrounding stimulus structure across the ventral stream, and large-scale sensitivity to the number of simultaneous objects. Response sparseness across voxel population increased consistently with larger stimuli. These data suggest that contextual modulation for a stimulus event reflect optimization of the code and perhaps increase in energy efficiency throughout the ventral stream hierarchy. Our model provides a novel prediction that average suppression of response amplitude for simultaneous stimuli across the cortical network is a monotonic function of similarity of response strengths in the network when the stimuli are presented alone.

On stochastic stability of regional ocean models with uncertainty in wind forcing

Abstract. A shallow-water model was used to understand model error induced by non-Gaussian wind uncertainty. Although the model was simple, it described a generic system with many degrees of freedom randomized by external noise. The study focused on the nontrivial collective behavior of finite-amplitude perturbations on different scales and their influence on model predictability. The error growth strongly depended on the intensity and degree of spatial inhomogeneity of wind perturbations. For moderate but highly inhomogeneous winds, the error grew as a power law. This behavior was a consequence of varying local characteristic exponents and nonlinear interactions between different scales. Coherent growth of perturbations was obtained for different scales at various stages of error evolution. For the nonlinear stage, statistics of prediction error could be approximated by a Weibull distribution. An approach based on the Kullback-Leibler distance (the relative entropy) and probability-weighted moments was developed for identification of Weibull statistics. Bifurcations of the variance, skewness and kurtosis of the irreversible predictability time (a measure of model prediction skill) were detected when the accepted prediction accuracy (tolerance) exceeded some threshold.

Well posedness and smoothing effect of Schrödinger-Poisson equation

In this work we take under consideration the Cauchy problem for the Schrödinger-Poisson type equation i∂tu=−∂x2u+V(u)u−f(∣u∣2)u, where f represents a local nonlinear interaction (we take into account both attractive and repulsive models) and V is taken as a suitable solution of the Poisson equation V=1∕2∣x∣∗(C−∣u∣2), C∊Cc∞ is the doping profile or impurities. We show that this problem is locally well posed in the weighted Sobolev spaces Hs≔{φ∊Hs(R):∫(1+x2)1∕2∣φ∣2&amp;lt;∞} with s⩾1, which means the local existence, uniqueness, and continuity of the solution with respect to the initial data. Moreover, under suitable assumptions on the local interaction, we show the existence of global solutions. Finally, we establish that for s⩾1 local in time and space, smoothing effects are present in the solution; more precisely, in this problem there is locally a gain of half a derivative.

Nonlinear coupling of Alfvén waves with widely different cross‐field wavelengths in space plasmas

Multiscale activity and dissipation of Alfvén waves play an important role in a number of space and astrophysical plasmas. A popular approach to study the evolution and damping of MHD Alfvén waves assumes a gradual evolution of the wave energy to small dissipative length scales. This can be done by local nonlinear interactions among MHD waves with comparable wavelengths resulting in turbulent cascades or by phase mixing and resonant absorption. We investigate an alternative nonlocal transport of wave energy from large MHD length scales directly into the dissipation range formed by the kinetic Alfvén waves (KAWs). KAWs have very short wavelengths across the magnetic field irrespectively of their frequency. We focus on the nonlinear mechanism for the excitation of KAWs by MHD Alfvén waves via resonant decay AW → KAW1 + KAW2. The resonant decay conditions can be satisfied in a rarified plasmas, where the gas/magnetic pressure ratio is less than the electron/ion mass ratio. The decay is efficient at low amplitudes of the magnetic field in the MHD waves, B/B0 ∼ 10−2. In turn, the nonlinearly driven KAWs have sufficiently short wavelengths for the dissipative effects to become significant. Therefore the cross‐scale nonlinear coupling of Alfvén waves can provide a mechanism for the replenishment of the dissipation range and the consequent energization in space plasmas. Two relevant examples of this scenario in the solar corona and auroral zones are discussed.

Interconnecting digital waveguides and finite difference structures: Recent advances

Digital waveguides and finite difference schemes have been generally considered as alternative techniques in physical modeling of spatially distributed systems. The main difference between these techniques is in the variables used in the computation; in digital waveguide simulations traveling-wave components are used, whereas physical variables are typical in finite difference schemes. Although both techniques are known to provide exact modeling of ideal one-dimensional band-limited wave propagation and both of them can be composed to approximate higher-dimensional mesh structures, they usually become computationally different in the presence of losses, dispersion, and nonlinear interactions. Despite these differences, successful examples of hybrid waveguide/finite-difference models have been proposed in the literature. In these references, the finite-difference sections typically account for local nonlinear interactions. More recently, a general, unifying, and DSP-oriented framework is introduced, where the functional equivalence of digital waveguides and finite difference schemes is systematically elaborated and the conditions of building mixed models are studied. The present lecture will summarize the new framework and provide demonstrations of hybrid models. Real-time sound examples will be presented. [Work supported by EU ALMA project (IST-2001-33059).]

Charging effect on transport through a nonlinear impurity

We use a generalized discrete nonlinear Schrödinger (DNLS) equation to describe tunnelling through a single magnetic impurity connected to two perfect leads, in the presence of a magnetic field. The charge build up at the impurity site is taken into account through the use of a self consistent Hartree approach. The transmission and conductance through the localized impurity is being considered in the presence of both local nonlinear interaction and a magnetic field which lifts the spin degeneracy at the impurity site.