Streaming Instability Research Articles

Generation of kinetic Alfvén wave and whistler waves by parametric decay of high power laser in laser–plasma interaction

In this paper, we report theoretical modeling for parametric decay instability of the high-intensity elliptically polarized laser beam [pump wave, (ω0)]. A wave–wave interaction model is investigated, based on the decay of the high-intensity elliptically polarized laser (ω0) into an oblique whistler wave (OWW, (ω1)) and a kinetic Alfvén wave (KAW, (ω2)). The importance of oblique whistler waves (OWWs, (ω1)) and kinetic Alfvén waves (KAWs, (ω2)) similar to solar wind spectra [Chatterjee et al., Nat. Commun. 8, 15970 (2017)] has been pointed out, as a means to understand the turbulent magnetic field amplification, implicating electron and ion dynamics [Chatterjee et al., Nat. Commun. 8, 15970 (2017); Tzeferacos et al., Nat. Commun. 9, 591 (2018); Meinecke et al., Proc. Natl. Acad. Sci. 112, 8211 (2015); Mondal et al., Proc. Natl. Acad. Sci. 109, 8011 (2012); Romagnani et al., Phys. Rev. Lett. 122, 025001 (2019); Perri et al., Phys. Rev. Lett. 109, 191101 (2012); and Adak et al., Phys. Rev. Lett. 114, 115001 (2015)]. In the nonlinear stage, the decay instability is expected to attain the turbulent state, via a cascade process or filamentation/modulation instability (oscillating two stream instability). Therefore, in the present paper, we have considered the first part of this research, namely, the beating mechanism (ω2=ω0−ω1), induced due to the nonlinear interaction of elliptically polarized laser velocity and oblique whistler wave density perturbation. The nonlinear saturation will be conferred in future investigations. Besides turbulence, the relevance of the present work to terahertz radiation generation [Singh et al., Europhys. Lett. 104, 35002 (2013); Dewan et al., Phys. Plasmas 25, 103105 (2018); Singh et al., Phys. Plasmas 18, 022304 (2011); M. Singh and R. P. Sharma, Contrib. Plasma Phys. 53(7), 540–548 (2013); Adak et al., Phys. Rev. Lett. 114, 115001 (2015); G. Brodin and L. Stenflo, Contrib. Plasma Phys. 54, 623 (2014); L. Stenflo, Phys. Scr. T50, 15–19 (1994); Li et al., Phys. Rev. E 84, 036405 (2011); L. Stenflo, Phys. Scr. T107, 262 (2004); and R. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008), Chap. 2] and fast ignition laser fusion [Kumar et al., arXiv:1804.02200 (2018)] by ion heating has been emphasized. The coefficients for the nonlinear coupling pertaining to this parametric decay process and the growth rate of the decay instability are investigated.

Pebble drift and planetesimal formation in protoplanetary discs with embedded planets

Nearly axisymmetric gaps and rings are commonly observed in protoplanetary discs. The leading theory regarding the origin of these patterns is that they are due to dust trapping at the edges of gas gaps induced by the gravitational torques from embedded planets. If the concentration of solids at the gap edges becomes high enough, it could potentially result in planetesimal formation by the streaming instability. We tested this hypothesis by performing global 1D simulations of dust evolution and planetesimal formation in a protoplanetary disc that is perturbed by multiple planets. We explore different combinations of particle sizes, disc parameters, and planetary masses, and we find that planetesimals form in all of these cases. We also compare the spatial distribution of pebbles from our simulations with protoplanetary disc observations. Planets larger than one pebble isolation mass catch drifting pebbles efficiently at the edge of their gas gaps, and depending on the efficiency of planetesimal formation at the gap edges, the protoplanetary disc transforms within a few 100 000 yr to either a transition disc with a large inner hole devoid of dust or to a disc with narrow bright rings. For simulations with planetary masses lower than the pebble isolation mass, the outcome is a disc with a series of weak ring patterns but there is no strong depletion between the rings. By lowering the pebble size artificially to a 100 micrometer-sized “silt”, we find that regions between planets get depleted of their pebble mass on a longer time-scale of up to 0.5 million years. These simulations also produce fewer planetesimals than in the nominal model with millimeter-sized particles and always have at least two rings of pebbles that are still visible after 1 Myr.

Cosmic ray transport in starburst galaxies

ABSTRACT Starburst galaxies are efficient γ-ray producers, because their high supernova rates generate copious cosmic ray (CR) protons, and their high gas densities act as thick targets off which these protons can produce neutral pions and thence γ-rays. In this paper, we present a first-principles calculation of the mechanisms by which CRs propagate through such environments, combining astrochemical models with analysis of turbulence in weakly ionized plasma. We show that CRs cannot scatter off the strong large-scale turbulence found in starbursts, because efficient ion-neutral damping prevents such turbulence from cascading down to the scales of CR gyroradii. Instead, CRs stream along field lines at a rate determined by the competition between streaming instability and ion-neutral damping, leading to transport via a process of field line random walk. This results in an effective diffusion coefficient that is nearly energy independent up to CR energies of ∼1 TeV. We apply our computed diffusion coefficient to a simple model of CR escape and loss, and show that the resulting γ-ray spectra are in good agreement with the observed spectra of the starbursts NGC 253, M82, and Arp 220. In particular, our model reproduces these galaxies’ relatively hard GeV γ-ray spectra and softer TeV spectra without the need for any fine-tuning of advective escape times or the shape of the CR injection spectrum.

Suppression of a spontaneous dust density wave by modulation of ion streaming

Synchronization of a self-excited dust density wave has been experimentally investigated in a strongly coupled dusty plasma. A dust density wave of frequency ∼78 Hz is spontaneously generated from the dust void boundary due to the ion streaming instability. The electric field in the dust void region is measured, and the electric field force and ion drag force on the dust particles at the void boundary are estimated to explain the mechanism of spontaneous dust density wave excitation. Synchronization occurring through the suppression mechanism is observed by modulating the ion streaming by applying an external sinusoidal signal to the dust void. At sufficiently high modulation amplitude, the onset of period-doubling bifurcation is observed. Fast Fourier transform spectral analysis is done using time-series data obtained from high-speed video imaging. The van der Pol equation with a force term is used to correlate the observed suppression phenomena.

The Tides of the Glacial Ocean and Their Possible Connection to Heinrich Event Instabilities of the Laurentide Ice Sheet

AbstractThe possibility that Heinrich event 1 (H1), which occurred during the most recent deglaciation event of the current ice age, may have been both triggered and subsequently continuously forced by the tides, is investigated. For the purpose of this investigation, a discontinuous Galerkin method is employed to solve the shallow water tidal equations on the sphere using a nonuniform icosahedral grid to enable high resolution of the Hudson Strait region that was the locus of the ice stream instability that caused H1. Since Heinrich events are known to have involved, for most if not all such events, an instability of the Hudson Strait Ice Stream, an initial representation of H1 has been investigated in which the ice loss is assumed to proceed up the strait to Southampton Island over a period constrained by observations. The tidal energy and dissipation associated with the dominant M2 semidiurnal tide were found to be extremely high relative to their modern strengths prior to H1 but dropped sharply after the instability occurred. A globally averaged (eustatic) sea level rise of approximately 1 m associated directly with this simple model is inferred. The high‐amplitude tides at the Atlantic outlet of the Hudson Strait are further demonstrated to track inland with the retreating ice up the strait as the instability of the ice stream migrates inland at a moderate speed consistent with estimates of the inferred instability event duration.

Nonlinear diffusion of cosmic rays escaping from supernova remnants: Cold partially neutral atomic and molecular phases

Aims. We aim to elucidate cosmic ray (CR) propagation in the weakly ionized environments of supernova remnants (SNRs) basing our analysis on the cosmic ray cloud (CRC) model. Methods. We solved two transport equations simultaneously: one for the CR pressure and one for the Alfvén wave energy density where CRs are initially confined in the SNR shock. Cosmic rays trigger a streaming instability and produce slab-type resonant Alfvén modes. The self-generated turbulence is damped by ion-neutral collisions and by noncorrelated interaction with Alfvén modes generated at large scales. Results. We show that CRs leaking in cold dense phases such as those found in cold neutral medium (CNM) and diffuse molecular medium (DiM) can still be confined over distances of a few tens of parsecs from the CRC center for a few thousand years. At 10 TeV, CR diffusion can be suppressed by two or three orders of magnitude. This effect results from a reduced ion-neutral collision damping in the decoupled regime. We calculate the grammage of CRs in these environments. We find that in both single and multi-phase setups at 10 GeV, CNM and DiM media can produce grammage in the range 10–20 g cm−2 in the CNM and DiM phases. At 10 TeV, because of nonlinear propagation the grammage increases to values in the range 0.5–20 g cm−2 in these two phases. We also present preliminary calculations in inhomogeneous interstellar medium combining two or three different phases where we obtain the same trends.

Simulation of Electron-Electron Two Stream Instability (ETSI)

Stream instabilities are widely studied due to their importance in understanding astrophysical phenomena such as acceleration of high velocity of solar wind. In this work, the simulation of electron two stream instability was performed using Vorpal Simulation (VSim) code to explore the kinetic energy of plasma that arises due to the interaction between two counter-streaming electron beams at different velocities as well as different electron densities. The electron beam velocity was varied in the range of 3.58 × 106 m/s - 7.98 × 106 m/s and the resulting kinetic energy of plasma increased from 19 × 10−6J - 210 × 10−6J respectively. Also, increasing the electron density at fixed beam velocity from 1.05 × 1014m−3 - 5.84 × 1014m−3, the kinetic energy was observed to increase from 100 × 10−6J - 200 × 10−6J .However, the kinetic energy of the electron increases more with increasing beam velocity than with increasing electron density. The electric field energy which arose due to the interaction of the streaming beams did not exceed the energy of the beams.

Electrical charging overcomes the bouncing barrier in planet formation

In protoplanetary disks, solid objects (so-called planetesimals) are formed from dust. Micrometre-sized dust grains grow into millimetre-sized aggregates. Once those aggregates have diameters exceeding a few centimetres, they become subject to concentration mechanisms such as the streaming instability, permitting the formation of self-gravitating clusters, which might eventually collapse into kilometre-sized planetesimals. However, for the streaming instability to set in, clumps spanning sizes from centimetres to decimetres are required in the centre of a protoplanetary disk. In the size range between millimetres and centimetres, aggregates bounce off each other rather than sticking together, and growth is stalled. Here we show in microgravity experiments that collisions between millimetre-sized grains lead to sufficient electrical charging for aggregation to bridge this gap between the bouncing barrier and the onset of the streaming instability. We computationally simulate aggregation and find that models agree with the experimental data only if electrical charging is present. We therefore propose that collisional charging may promote early growth in the size gap that current models of planetesimal formation cannot account for. In our understanding of planetary formation, it is still unclear how millimetre-sized dust grains grow into centimetre-sized aggregates. Microgravity experiments now show that electrical charging of the grains leads to the formation of larger clumps.

Characteristics of a Langmuir Soliton Observed in a Solar Type III Burst

We present the high time resolution in situ observations of a Langmuir wave packet obtained by the STEREO WAVES experiment in the source region of a solar type III radio burst. This wave packet is unique in the sense that it occurs as an intense localized one-dimensional magnetic field aligned wave packet with peak intensity well in excess of the threshold for the oscillating two stream instability (OTSI), soliton formation and related strong turbulence processes. Most importantly, the measured half-width of this wave packet is approximately equal to the expected half-width of a Langmuir soliton with the same peak intensity as that of the wave packet. Furthermore, a density cavity is created by the ponder-motive force of this wave packet as expected of a Langmuir soliton. These findings indicate that the observed wave packet is most likely a Langmuir soliton formed as a result of the balance between the non-linearity related self-compression and the dispersion related spreading. We discuss the implication of these observations for theories of solar type III radio bursts.

Demographics of Planetesimals Formed by the Streaming Instability

Abstract The streaming instability (SI) is a mechanism to aerodynamically concentrate solids in protoplanetary disks and facilitate the formation of planetesimals. Recent numerical modeling efforts have demonstrated the increasing complexity of the initial mass distribution of planetesimals. To better constrain this distribution, we conduct SI simulations including self-gravity with the highest resolution hitherto. To subsequently identify all of the self-bound clumps, we develop a new clump-finding tool, Planetesimal Analyzer. We then apply a maximum likelihood estimator to fit a suite of parameterized models with different levels of complexity to the simulated mass distribution. To determine which models are best-fitting and statistically robust, we apply three model selection criteria with different complexity penalties. We find that the initial mass distribution of clumps is not universal regarding both the functional forms and parameter values. Our model selection criteria prefer models different from those previously considered in the literature. Fits to multi-segment power-law models break to a steeper distribution above masses close to those of 100 km collapsed planetesimals, similar to observed size distributions in the Kuiper Belt. We find evidence for a turnover at the low-mass end of the planetesimal mass distribution in our high-resolution run. Such a turnover is expected for gravitational collapse, but had not previously been reported.

Neutrino-driven Instability of Ion Acoustic Waves in an Ultrarelativistic Degenerate Plasma

Abstract The coupling between ion acoustic waves (IAWs) and a neutrino beam undergoing flavor oscillations in a dense, relativistically degenerate plasma is analyzed. The neutrino-driven streaming instability of the IAWs is investigated with relevance to plasma conditions at the last stage of stellar evolution in a massive supernova progenitor. The influence of neutrino beam parameters such as the energy of the incident neutrino beam and eigenfrequency of the neutrino flavor oscillations on the instability growth rate is obtained numerically. It is observed that the neutrino flavor oscillations significantly affect the neutrino-driven instability of the IAWs. Our results also indicate that the time period for the onset of the streaming neutrino-driven instability is shorter than the typical time period of a core-collapse supernova explosion. The findings of this investigation may shed new light on the understanding of the underlying physical mechanism responsible for the core-collapse supernova.

The Mass and Size Distribution of Planetesimals Formed by the Streaming Instability. II. The Effect of the Radial Gas Pressure Gradient

The streaming instability concentrates solid particles in protoplanetary disks, leading to gravitational collapse into planetesimals. Despite its key role in producing particle clumping and determining critical length scales in the instability's linear regime, the influence of the disk's radial pressure gradient on planetesimal properties has not been examined in detail. Here, we use streaming instability simulations that include particle self-gravity to study how the planetesimal initial mass function depends on the radial pressure gradient. Fitting our results to a power-law, ${\rm d}N / {\rm d}M_p \propto M_p^{-p}$, we find a single value $p \approx 1.6$ describes simulations in which the pressure gradient varies by $\gtrsim 2$. An exponentially truncated power-law provides a significantly better fit, with a low mass slope of $p^\prime \approx 1.3$ that weakly depends on the pressure gradient. The characteristic truncation mass is found to be $\sim M_G = 4 \pi^5 G^2 \Sigma_p^3 / \Omega^4$. We exclude the cubic dependence of the characteristic mass with pressure gradient suggested by linear considerations, finding instead a linear scaling. These results strengthen the case for a streaming-derived initial mass function that depends at most weakly on the aerodynamic properties of the disk and participating solids. A simulation initialized with zero pressure gradient---which is {\em not} subject to the streaming instability---also yields a top-heavy mass function but with modest evidence for a different shape. We discuss the consistency of the theoretically predicted mass function with observations of Kuiper Belt planetesimals, and describe implications for models of early stage planet formation..

Instabilities and turbulence in low-β guide field reconnection exhausts with kinetic Riemann simulations

The role of turbulence in low-β, guide-field reconnection exhausts is explored in 2D reconnection and 2D and 3D Riemann simulations. The structure of the exhaust and associated turbulence is controlled by a pair of rotational discontinuities (RDs) at the exhaust boundary and a pair of slow shocks (SSs) that are generated by counterstreaming ions beams. In 2D, the exhaust develops large-amplitude striations at the ion Larmor radius scale that are produced by electron-beam-driven ion cyclotron waves. The electron beams driving the instability are injected into the exhaust from one of the RDs. However, in 3D Riemann simulations, the additional dimension (in the out-of-plane direction) results in strong Buneman and electron-electron streaming instabilities at the RD which suppress the electron beam formation and therefore the striations in the exhaust. The strength of the streaming instabilities at the RD is controlled by the ratio of the electron thermal speed to Alfvén speed, with the lower thermal speed being more unstable. In the 3D simulations, an ion-ion streaming instability acts to partially thermalize the counterstreaming ion beams at the SSs. This instability is controlled by the ratio of the sound speed to Alfvén speed and is expected to be stable in the low β solar corona. The results suggest that in a guide field reconnection exhaust with 1≫β>me/mi, the kinetic-scale turbulence that develops will be too weak to play a significant role in energy conversion and particle acceleration. Therefore, the energy conversion will be mostly controlled by laminar physics or multi-x-line reconnection.

Dust-vortex Instability in the Regime of Well-coupled Grains

Abstract We present a novel study of dust-vortex evolution in global two-fluid disk simulations to find out if evolution toward high dust-to-gas ratios can occur in a regime of well-coupled grains with low Stokes numbers (St = 10−3 − 4 × 10−2). We design a new implicit scheme in the code RoSSBi, to overcome the short time-steps occurring for small grain sizes. We discover that the linear capture phase occurs self-similarly for all grain sizes, with an intrinsic timescale (characterizing the vortex lifetime) scaling as 1/St. After vortex dissipation, the formation of a global active dust ring is a generic outcome confirming our previous results obtained for larger grains. We propose a scenario in which, regardless of grain size, multiple pathways can lead to local dust-to-gas ratios of about unity and above on relatively short timescales, <105 yr, in the presence of a vortex, even with St = 10−3. When St > 10−2, the vortex is quickly dissipated by two-fluid instabilities, and large dust density enhancements form in the global dust ring. When St < 10−2, the vortex is resistant to destabilization. As a result, dust concentrations occur locally due to turbulence developing inside the vortex. Regardless of the Stokes number, dust-to-gas ratios in the range 1–10, a necessary condition to trigger a subsequent streaming instability, or even a direct gravitational instability of the dust clumps, appears to be an inevitable outcome. Although quantitative connections with other instabilities still need to be made, we argue that our results support a new scenario of vortex-driven planetesimal formation.

A Particle Module for the PLUTO Code. III. Dust

Abstract Implementation of a new particle module describing the physics of dust grains coupled to a gas via drag forces is the subject of this work. The proposed particle–gas hybrid scheme has been designed to work in Cartesian as well as in cylindrical and spherical geometries. The numerical method relies on a Godunov-type second-order scheme for the fluid and an exponential midpoint rule for dust particles, which overcomes the stiffness introduced by the linear coupling term. Besides being time-reversible and globally second-order accurate in time, the exponential integrator provides energy errors that are always bounded, and it remains stable in the limit of arbitrarily small particle stopping times, yielding the correct asymptotic solution. Such properties make this method preferable to the more widely used semi-implicit or fully implicit schemes at a very modest increase in computational cost. Coupling between particles and grid quantities is achieved through particle deposition and field-weighting techniques borrowed from particle-in-cell simulation methods. In this respect, we derive new weight factors in curvilinear coordinates that are more accurate than traditional volume or area weighting. A comprehensive suite of numerical benchmarks is presented to assess the accuracy and robustness of the algorithm in Cartesian, cylindrical, and spherical coordinates. Particular attention is devoted to the streaming instability, which is analyzed in both local and global disk models. The module is part of the PLUTO code for astrophysical gas dynamics, and it is mainly intended for the numerical modeling of protoplanetary disks in which solid and gas interact via aerodynamic drag.

Three-dimensional simulations of non-resonant streaming instability and particle acceleration near non-relativistic astrophysical shocks

ABSTRACT We use particle-in-magnetohydrodynamics-cells to model particle acceleration and magnetic field amplification in a high-Mach, parallel shock in three dimensions and compare the result to 2D models. This allows us to determine whether 2D simulations can be relied upon to yield accurate results in terms of particle acceleration, magnetic field amplification, and the growth rate of instabilities. Our simulations show that the behaviour of the gas and the evolution of the instabilities are qualitatively similar for both the 2D and 3D models, with only minor quantitative differences that relate primarily to the growth speed of the instabilities. The main difference between 2D and 3D models can be found in the spectral energy distributions (SEDs) of the non-thermal particles. The 2D simulations prove to be more efficient, accelerating a larger fraction of the particles and achieving higher velocities. We conclude that, while 2D models are sufficient to investigate the instabilities in the gas, their results have to be treated with some caution when predicting the expected SED of a given shock.

Role of cosmic-ray streaming and turbulent damping in driving galactic winds

ABSTRACT Large-scale galactic winds driven by stellar feedback are one phenomenon that influences the dynamical and chemical evolution of a galaxy, redistributing material throughout the circumgalatic medium. Non-thermal feedback from galactic cosmic rays (CRs) – high-energy charged particles accelerated in supernovae and young stars – can impact the efficiency of wind driving. The streaming instability limits the speed at which they can escape. However, in the presence of turbulence, the streaming instability is subject to suppression that depends on the magnetization of turbulence given by its Alfvén Mach number. While previous simulations that relied on a simplified model of CR transport have shown that super-Alfvénic streaming of CRs enhances galactic winds, in this paper we take into account a realistic model of streaming suppression. We perform three-dimensional magnetohydrodynamic simulations of a section of a galactic disc and find that turbulent damping dependent on local magnetization of turbulent interstellar medium (ISM) leads to more spatially extended gas and CR distributions compared to the earlier streaming calculations, and that scale heights of these distributions increase for stronger turbulence. Our results indicate that the star formation rate increases with the level of turbulence in the ISM. We also find that the instantaneous wind mass loading is sensitive to local streaming physics with the mass loading dropping significantly as the strength of turbulence increases.

On the Growth and Saturation of the Gyroresonant Streaming Instabilities

Abstract The self-regulation of cosmic-ray (CR) transport in the interstellar and intracluster media has long been viewed through the lenses of linear and quasi-linear kinetic plasma physics. Such theories are believed to capture the essence of CR behavior in the presence of self-generated turbulence but cannot describe potentially critical details arising from the nonlinearities of the problem. We utilize the particle-in-cell numerical method to study the time-dependent nonlinear behavior of the gyroresonant streaming instabilities, self-consistently following the combined evolution of particle distributions and self-generated wave spectra in one-dimensional periodic simulations. We demonstrate that the early growth of instability conforms to the predictions from linear physics, but that the late-time behavior can vary depending on the properties of the initial CR distribution. We emphasize that the nonlinear stages of instability depend strongly on the initial anisotropy of CRs—highly anisotropic CR distributions do not efficiently reduce to Alfvénic drift velocities, owing to reduced production of left-handed resonant modes. We derive estimates for the wave amplitudes at saturation and the timescales for nonlinear relaxation of the CR distribution and then demonstrate the applicability of these estimates to our simulations. Bulk flows of the background plasma due to the presence of resonant waves are observed in our simulations, confirming the microphysical basis of CR-driven winds.

Characterization of Ring Substructures in the Protoplanetary Disk of HD 169142 from Multiwavelength Atacama Large Millimeter/submillimeter Array Observations

Abstract We present a detailed multiwavelength characterization of the multi-ring disk of HD 169142. We report new Atacama Large Millimeter/submillimeter Array (ALMA) observations at 3 mm and analyze them together with archival 0.89 and 1.3 mm data. Our observations resolve three out of the four rings in the disk previously seen in high-resolution ALMA data. A simple parametric model is used to estimate the radial profile of the dust optical depth, temperature, density, and particle size distribution. We find that the multiple ring features of the disk are produced by annular accumulations of large particles, probably associated with gas pressure bumps. Our model indicates that the maximum dust grain size in the rings is ∼1 cm, with slightly flatter power-law size distributions than the interstellar medium-like size distribution (p ∼ 3.5) found in the gaps. In particular, the inner ring (∼26 au) is associated with a strong and narrow buildup of dust particles that could harbor the necessary conditions to trigger the streaming instability. According to our analysis, the snowlines of the most important volatiles do not coincide with the observed substructures. We explore different ring formation mechanisms and find that planet–disk interactions are the most likely scenario to explain the main features of HD 169142. Overall, our multiwavelength analysis provides some of the first unambiguous evidence of the presence of radial dust traps in the rings of HD 169142. A similar analysis in a larger sample of disks could provide key insights on the impact that disk substructures have on the dust evolution and planet formation processes.

Possible origin of the slow-diffusion region around Geminga

ABSTRACT Geminga pulsar is surrounded by a multiTeV γ-ray halo radiated by the high-energy electrons and positrons accelerated by the central pulsar wind nebula (PWN). The angular profile of the γ-ray emission reported by High-Altitude Water Cherenkov Observatory indicates an anomalously slow diffusion for the cosmic-ray electrons and positrons in the halo region around Geminga. In the paper we study the possible mechanism for the origin of the slow diffusion. At first, we consider the self-generated Alfvén waves due to the streaming instability of the electrons and positrons released by Geminga. However, even considering a very optimistic scenario for the wave growth, we find this mechanism does not work to account for the extremely slow diffusion at the present day, if taking the proper motion of Geminga pulsar into account. The reason is straightforward as the PWN is too weak to generate enough high-energy electrons and positrons to stimulate strong turbulence at the late time. We then propose an assumption that the strong turbulence is generated by the shock wave of the parent supernova remnant (SNR) of Geminga. Geminga may still be inside the SNR, and we find that the SNR can provide enough energy to generate the slow-diffusion circumstance. The TeV haloes around PSR B0656+14, Vela X, and PSR J1826-1334 may also be explained under this assumption.