PHYSICAL REVIEW E 90, 013106 (2014)
Observation of self-excited acoustic vortices in defect-mediated dust acoustic wave turbulence Ya-Yi Tsai and Lin I Department of Physics and Center for Complex Systems, National Central University, Jhongli, Taiwan 32001, Republic of China (Received 6 December 2013; published 28 July 2014) Using the self-excited dust acoustic wave as a platform, we demonstrate experimental observation of selfexcited fluctuating acoustic vortex pairs with ±1 topological charges through spontaneous waveform undulation in defect-mediated turbulence for three-dimensional traveling nonlinear longitudinal waves. The acoustic vortex pair has helical waveforms with opposite chirality around the low-density hole filament pair in xyt space (the xy plane is the plane normal to the wave propagation direction). It is generated through ruptures of sequential crest surfaces and reconnections with their trailing ruptured crest surfaces. The initial rupture is originated from the amplitude reduction induced by the formation of the kinked wave crest strip with strong stretching through the undulation instability. Increasing rupture causes the separation of the acoustic vortex pair after generation. A similar reverse process is followed for the acoustic vortex annihilating with the opposite-charged acoustic vortex from the same or another pair generation. DOI: 10.1103/PhysRevE.90.013106
PACS number(s): 52.35.Mw, 47.54.−r, 52.27.Lw
I. INTRODUCTION
Screw-type dislocation, with a helical waveform winding around a line with zero wave amplitude and undetermined phase, has been identified as an important phase singularity of 3D traveling-wave trains [1–13]. The number of 2π phase accumulations along a closed loop (or the number of helical twists in one wavelength) surrounding the singularity is the topological charge [1]. In acoustic waves [2–8] and optical waves [9–13], screw dislocations are also called acoustic vortices (AVs) and optical vortices, respectively. They carry angular momenta proportional to their topological charges [3,6,7,11–13] and have applications such as manipulating objects [3,6,7,12] and information [8]. Nevertheless, in AV studies, AVs were mainly passively generated in air or liquids, through the interference of waves excited from oscillating driver arrays with programmed phase lags, and monitored by spatially scanning the position of a single local transducer only for repeatable waveforms [4–7,13]. Whether and how AVs can be spontaneously excited in weakly disordered waves without periodic external driving remain open challenging issues. The self-organization of ordered nonlinear waves is a ubiquitous phenomenon in nonlinear open dissipative media such as biological, fluid, and plasma systems. Increasing excitation makes the ordered waveform unstable. It leads to the transition to the weakly disordered intermediate state (also called defect-mediated turbulence), with undulated waveforms before the transition to strong wave turbulence with a powerlaw spectrum [14–24]. In the weakly disordered wave, the amplitude and phase modulation through modulation instability induces waveform undulation and broadens the sharp peaks in the power spectrum. It consequently generates fluctuating defects traveling along the centers of low-amplitude holes or filaments, where phases are singular [17,18,20–24]. Namely, the dynamics can be built up from localized moving and interacting defects. In the past two decades, weakly disordered waves and the defect dynamics associated with low-amplitude holes or filaments have been studied in systems such as 1D traveling waves obeying Landau-Ginzburg equations [17,18], thermal and electrode convection systems [19], and chem1539-3755/2014/90(1)/013106(5)
ical reaction-diffusion systems [20–23]. Pitchfork, spiral, and scroll waveforms are the basic waveforms surrounding defects [17–23]. For 3D nonlinear traveling plane waves, only the recent experimental study on 3D nonlinear dust acoustic waves in a uniform dusty plasma background demonstrated the transition to the weakly disordered state with fluctuating defect pairs traveling along chaotic low-amplitude hole filaments in xyt space, through the spontaneous undulation of the 3D plane dust acoustic wave propagating in the -z direction [24]. Nevertheless, the following intriguing issues remain poorly understood. (a) What are those defects, are they AVs, and what are their stereo waveforms? (b) How can they be generated, evolve, and be annihilated through spontaneous undulation of the traveling plane wave? In this work, using the dusty plasma as a platform, these unexplored issues are experimentally addressed in xyt space by video imaging the temporal evolution of the particle density in an xy plane. The study demonstrates the pair generation, propagation, and annihilation of AVs with ±1 topological charges and helical waveforms with opposite helicity around the amplitude hole filament pair in xyt 2 + 1D space. The physical processes of AV excitation through local undulation instability-induced wave crest rupturing and reconnection, the precursor for AV generation, and the reversed process for AV pair annihilation are identified. The dust acoustic wave with longitudinal oscillation in the dusty plasma composed of negatively charged dust suspensions in the gaseous plasma is a fundamental longitudinal plasma wave governed by modulation-type dynamical equations [25–27]. It can be self-excited through the interplay of dust inertia, screened Coulomb interaction, and free energy from ion streaming [25–27]. The advantage of direct video imaging the dust density evolution also makes it a good platform to study the generic dynamical behaviors of nonlinear acoustic-type waves. Previous studies on shocks, solitary waves, wave-particle interaction, wave breaking, wave turbulence, and dislocation defects with pitchfork waveforms in nonuniform plasmas, mainly through imaging the side view waveforms in planes normal to wave fronts, are a few good examples [28–36]. Recent theoretical work has demonstrated
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that AVs can carry angular momenta and exist in linear dust acoustic waves, without investigating AV waveform dynamics [37]. II. EXPERIMENT
The experiment is conducted in a cylindrical rf dusty plasma system as described elsewhere [24,35]. Figure 1(a) shows a sketch of the experimental setup. A hollow square thin glass trap 25 mm in inner width and height is put at the center of the bottom electrode to confine the dusty plasma with suspended polystyrene particles 5 μm in diameter, 6.9 × 10−11 g in mass, 6 × 104 cm−3 in density. The weakly ionized glow discharge (ne ∼ 109 cm−3 ) supporting the self-excited defect-free waveform with straight and parallel wave crests can be generated in Ar gas at 220 mTorr using a 14-MHz rf power system at 2.8-W rf power [24]. The estimated Debye length λD and the estimated dust particle charges are on the order of a few tens of micrometers and a few to 10 000 electrons/particle, respectively. The wave propagates downward in the −z direction. The dust image in an xy plane illuminated by an expanded horizontal laser sheet (∼0.6 mm in width) is captured by a CCD at 400-Hz sampling rate. The normalized local dust density nd (x,y,t) = Id (x,y,t)/Id (x,y)t can be obtained by measuring the image brightness Id , coarse grained over a Gaussian weighted circle (0.3 mm in full width at half maximum), where Id (x,y)t is the time average of the local Id (x,y). Namely, nd = 1 when nd equals the temporally averaged dust density.
the amplitude A(x,t) and the phase φ(x,t) of the local nd oscillation are computed through Hilbert analysis [24,38]. We first obtain nd0 (x,t) by bandpass filtering the fundamental band (from 22 to 50 Hz) of nd (x,t) at the local point x and the imaginary part nˆ d0 (x,t) through the Hilbert ∞transform of nd0 (x,t), i.e., nˆ d0 (x,t) = H [nd0 (x,t)] = π1 −∞ [nd0 (x,t)/t − τ ]dτ [38]. Then, through the equation N (x,t) = nd0 (x,t) + i nˆ d0 (x,t) = A(x,t)eiφ(x,t) for N (x,t) in the complex plane, the amplitude A(x,t) and the phase φ(x,t) = arg[N (x,t)] of nd0 (x,t) oscillation can be obtained. Figure 2(a) shows the nd waveform and the corresponding plot of φ in xyt space. Basically, the side views of the xyt space plots of nd and φ show similar undulated waveforms, where φ = 0 (2π ) corresponds to the wave crest. Defects, labeled by crosses, are located at the vertices of pitchforkshaped waveforms (with connected or broken arms), where phases are undefined. They can also be easily identified since they concide with the low-amplitude holes, where the phases cannot be defined [18,21,24]. Figure 2(b) further shows two typical sequential plots of φ in the xy plane. The undulated waveform causes the presence of patches with varying φ. The intersection of a tilted wave crest surface (i.e., φ = 0 surface) with an xy plane at certain t causes the formation of curved lines with finite lengths along the interfaces of the white and the black patches. Those interface lines (ILs) evolve with time due
III. RESULTS AND DISCUSSION A. Identifying acoustic vortices from φ(x, y) and A(x, y) plots
Decreasing dissipation by decreasing pressure to 200 mTorr leads to the transition to the weakly disordered state with a 40-mm/s average wave speed [24]. Figures 2(b) and 2(c) show the typical local nd evolution and power spectrum, respectively. The temporal modulation of the local nd (the normalized dust density) fluctuation causes the broadening of the fundamental and the second harmonic peaks in its power spectrum [24]. In order to more clearly characterize the phase and the amplitude modulations, the temporal evolutions of
FIG. 1. (Color online) (a) Sketch of the experimental setup. (b) Temporal evolution and (c) power spectrum of the local nd [24].
FIG. 2. (Color online) (a) The nd waveform and the corresponding plot of φ in xyt space. (b) Two typical sequential plots of φ(x,y) at different times. The crosses label defect locations. In the xy plane, they are located at the ends of interface lines between the white and the black patches, where the phases have abrupt jumps. (c) Chaotic amplitude hole filaments in xyt space, showing their pair generation and annihilation, and propagation. (d) Stereo plot of φ(x,y) at a fixed t around a typical defect from the dashed box in (b), showing the 2π phase jump winding a circle around the defect. (e) Plots φ(x,y) and A(x,y) corresponding to the region of (d).
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to the evolving waveform [see Fig. 2(b) and Ref. [39]]. Defects are located at the ends of ILs, where phases are undefined. What are those defects? Figure 2(d) and the left panel of Fig. 2(e) show the stereo and contour plots of φ(x,y), respectively, for a small region centered at the lower defect in the dashed box of Fig. 2(b). Here φ changes by 2π after circulating one cycle around the defect. The defect is located at the center of the amplitude hole, defined as A(x,y) < 0.1 and labeled by the circled region of Fig. 2(e). The above findings clearly evidence that the defect is an AV with a +1 topological charge [1,5]. Obviously, the defect at the opposite end of the IL has a −1 topological charge. Figure 2(c) shows the trajectories of amplitude holes, the lines of silence, in xyt space, centered along defect trajectories. Under topological charge conservation, those chaotic hole filaments can be generated and annihilated pairwise and propagate with varying angles in xyt space.
further depict the decomposed waveforms. The two plots in the right column of Fig. 3(d) correspond to those on their left, but viewed at different angle. Obviously, the strong local undulation causes the sequential rupture of crest surfaces along finite lines, starting from a local region of crest 1. The sequential reconnections of the upward bending side of a ruptured crest i with the downward bending side of the trailing ruptured crest i + 1 causes the spontaneous generation of an AV pair, with helical waveforms oppositely spiraling along the hole filaments centered at the ±1 defects in xyt space [Figs. 3(a)–3(c)]. Increasing rupture increases the AV pair separation. Under the above helical waveform, it is easy to understand the evolution of φ(x,y) with t = nτ , where τ = 2.5 ms (the CCD sampling interval) in Figs. 4(a) and 4(b). The AV pair generation begins at n = 0, when the finite IL emerges. Note that the IL is the intersection line between an xy plane and a tilted crest surface. With increasing t, the screw-shape wave crest surfaces with opposite chirality make the two end sections of the ILs connected to the defect pair (located at the hole filament sites) rotate oppositely. This leads to the formation of a curved long IL, which is restraightened after one wave period (also see Ref. [39]). This process is repeated in the following few wave cycles until AV annihilation, associated with the appearance of the very short straight IL at n = 41 [Fig. 4(b)]. The stereo plot of Fig. 4(c) further depicts the opposite rotations of the end sections of ILs (as indicated by the shot black lines) around the filament pair. Is there a precursor leading to the onset of AV pair generation associated with the rupture of crest surface 1 and the reconnection with surface 2, around t = 0 s? Figure 3(e) evidences the emergence of the narrow kinked stripe, as indicated by the arrows, for crest surface 0. Figure 5(a) further
B. Stereo waveforms of AVs and their generic dynamical processes from pair generation to pair annihilation
What are the stereo waveforms of AVs in xyt space and how can they be formed through waveform undulation? Let us use the waveform evolution surrounding the filament loop, indicated by the green arrow in the lower right corner region of Fig. 2(c), from AV pair generation to pair annihilation, as an example. Figures 3(a) and 3(b) depict the wave crest surfaces with φ(x,y) = 0 [40], labeled by the sequence number i in xyt space, surrounding the two and the right filaments, respectively. For better viewing, Figs. 3(c) and 3(d)
FIG. 3. (Color online) Stereo waveforms of wave crest surfaces, labeled by the sequence number i, around the hole filament pair and the right filament in xyt space. (c)–(e) Decomposed waveforms viewed from different angles, which more clearly show how the rupture and reconnection of sequential wave crest surfaces lead to the formation of AV pairs with helical waveforms oppositely winding around the hole filaments at the two ends of the rupture. The two right panels of (d) from another angle clearly show the upward and the downward tilting of the ruptured crests. The arrows in (d) and (e) indicate the kinked strips leading to rupture.
FIG. 4. (Color online) (a) and (b) Evolution of φ(x,y) in the region around the AV pair of Fig. 3, with increasing t = nτ and τ = 2.5 ms. The AV pair generation and annihilation begin at n = 0 and 41, respectively. (c) Stereo plots in xyt space showing the oppositely spiraling end sections of the ILs around the two hole filaments. (d) Typical A(x,y) plots at different n showing low-amplitude scars that are gradually turned into the separated amplitude hole pair centered at the defect pair after defect formation.
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shows the plot of the sequential wave crests (blue lines) on top of the A(x ,t) plot, where x t is the surface normal to and intersecting the surface covering the filament loop of Fig. 3, along the center vertical line. The undulation instability causes the change from the smoothly tilted crest −1 to the strongly kinked crest 0, followed by the sequential rupture and reconnection of the trailing crests from crest 1. Figure 5(b) depicts the more detailed initial evolution from the smoothly tilted to the strongly kinked contour (equi-φ) lines, in the dashed box of Fig. 5(a). Figure 5(c) shows how the maximum slope Sm of each tilted contour line in x t space grows with increasing t. The solid curve is the best fit of the exponential growth with a 6-ms time constant. The increasing stretching with the growing kink also significantly reduces the local amplitude A. It is manifested by the emergences of low-amplitude scars that have decreasing A with increasing t in the boxed region of Fig. 5(a) and the n 0 panels of Fig. 4(d) and the low-amplitude hole filament tip below crest 1 in Fig. 3(a). It eventually leads to the crest surface rupture and reconnection [at the small circles of Fig. 5(a)] starting from crest 1, for defect pair generation. As shown in Figs. 3(a), 4(b), and 5(a), a similar reverse process occurs after the annihilation of the AV pair. The decreasing waveform undulation causes the emergence of a kinked strip without rupture for crest 5 and then the smoothly undulated crest 6, similarly to crests 0 and −1. A low A scar appears again around the rupture end of crest 4 and the kinked region of crest 5 [Fig. 5(a)]. Also note that the downward (upward) bending part of the ruptured crest surface 1 (4) does not touch crest surface 0 (5). There is no extra crest surface insertion between crests 0 and 5. After pair generation, an AV can also annihilate with an AV from the other pair generation. Figures 6(a) and 6(b) show another example of the stereo waveforms and the end sections of the ILs surrounding the W-shaped hole filaments, for the
FIG. 5. (Color online) (a) Plot of wave crests (blue lines) on top of the A(x ,t) plot, where the x t surface is the surface normal to and through the center of the surface covering the filament pair of Fig. 3. (b) Contour plot of φ(x ,t) corresponding to the dashed rectangular region of (a). (c) Plot showing the exponential growth of the maximum slope of each equi-φ line in the x t plane of (b) with increasing t. The solid curve is an exponential fit with a 6-ms time constant. The small circles in (a) indicate the local regions for the reconnection of adjacent ruptured crest surfaces.
annihilation of AVs α and β , generated from two different AV pairs α-α and β-β . Generic behaviors of the screwshape waveforms with opposite chirality through crest surface rupture and reconnection in the center regions of kink strips on crest surfaces for each AV pair, similar to the example of Fig. 3, are followed. Figure 6(c) shows the A(x ,t) and φ(x ,t) plots, where the x t surface is the surface normal to and intersecting the plane covering the filament pair α and α . The initial crest rupture is also led by the instability from the smoothly tilted to the strongly kinked crest surfaces [see the φ(x ,t) plot in Fig. 6(c)], associated with the strong local amplitude suppression [see the A(x ,t) plot in Fig. 6(c)]. Although AVs α and β are not generated in pairs, the opposite topological charges (chirality) allow their annihilation. Reference [39] provides clear examples of annihilations of oppositely charged defects from different pair generations. IV. CONCLUSION
We demonstrated the observation of fluctuating AV pairs with helical waveforms surrounding chaotic hole filament pairs as the fundamental spontaneous excitations in the self-excited weakly disordered 3D traveling acoustic wave. The physical processes for the spontaneous AV pair generation, propagation, and annihilation are identified. The rapid exponential growth of the kink strip formation through modulation instability leads to a strong amplitude reduction, followed by the sequential rupture and reconnection along finite lines on trailing crest surfaces. It forms two AVs with opposite chirality around the hole filament pair at the two ends of the rupture. The inverse process leads to the annihilation of an AV either with its twin brother or with another oppositely charged AV from another pair generation.
FIG. 6. (Color online) (a) and (b) Plots similar to those in Figs. 3(a) and 4(c), showing the waveform evolution from the generation of two AV pairs α-α and β-β to the annihilation of α and β with helical waveforms oppositely spiraling around their filaments, as indicated by arrows. (c) The A(x ,t) and φ(x ,t) plots demonstrating the strong kink formation with strong local amplitude reduction, from smoothly undulated equi-φ lines, leading to crest 1 rupture and reconnection with crest 2. The x t surface is the surface normal to and intersecting the plane covering the filament pair α and α around their initial growth region. The index i indicates the crest surface number.
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From the view of system symmetry, the undulation (modulation) instability induces spontaneous symmetry breaking and alters the topology of ordered plane waves, through crest rupture and reconnection and consequently AV pair excitation, in which topological charges are conserved. Since the dynamics of many nonlinear waves are also governed by the modulation-type nonlinear equations, the similar spontaneous symmetry breaking process for AV excitation through undulation instability induced crest surface rupture and reconnection is expected. Our study should be able to
shed light on understanding the generic waveform dynamics and lead to related theoretical and experimental studies in other weakly disordered waves self-excited from 3D longitudinal plane waves.
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[23] R. H. Clayton, E. A. Zhuchkova, and A. V. Panfilov, Prog. Biophys. Mol. Biol. 90, 378 (2006). [24] M. C. Chang, Y. Y. Tsai, and L. I, Phys. Plasma 20, 083703 (2013). [25] N. N. Rao, P. K. Shukla, and M. Y. Yu, Planet. Space Sci. 38, 543 (1990); P. K. Shukla, Phys. Scr. 45, 504 (1992). [26] V. E. Fortov, A. D. Usachev, A. V. Zobnin, V. I. Molotkov, and O. F. Petrov, Phys. Plasmas 10, 1199 (2003); P. Kaw and R. Singh, Phys. Rev. Lett. 79, 423 (1997); A. Piel, M. Klindworth, O. Arp, A. Melzer, and M. Wolter, ibid. 97, 205009 (2006); A. A. Mamun and P. K. Shukla, Phys. Plasmas 7, 4412 (2000). [27] W. M. Moslem, R. Sabry, S. K. El-Labany, and P. K. Shukla, Phys. Rev. E 84, 066402 (2011). [28] J. Heinrich, S.-H. Kim, and R. L. Merlino, Phys. Rev. Lett. 103, 115002 (2009); R. L. Merlino, J. R. Heinrich, S.-H. Hyun, and J. K. Meyer, Phys. Plasmas 19, 057301 (2012). [29] P. Bandyopadhyay, G. Prasad, A. Sen, and P. K. Kaw, Phys. Rev. Lett. 101, 065006 (2008). [30] P. K. Shukla and B. Eliasson, Phys. Rev. E 86, 046402 (2012). [31] M. Schwabe, M. Rubin-Zuzic, S. Zhdanov, H. M. Thomas, and G. E. Morfill, Phys. Rev. Lett. 99, 095002 (2007). [32] C.-T. Liao, L.-W. Teng, C.-Y. Tsai, C.-W. Io, and L. I, Phys. Rev. Lett. 100, 185004 (2008); M.-C. Chang, L.-W. Teng, and L. I, Phys. Rev. E 85, 046410 (2012). [33] L.-W. Teng, M.-C. Chang, Y.-P. Tseng, and L. I, Phys. Rev. Lett. 103, 245005 (2009). [34] J. Pramanik, B. M. Veeresha, G. Prasad, A. Sen, and P. K. Kaw, Phys. Lett. A 312, 84 (2003). [35] Y.-Y. Tsai, M.-C. Chang, and L. I, Phys. Rev. E 86, 045402(R) (2012). [36] K. O. Menzel, O. Arp, and A. Piel, Phys. Rev. E 83, 016402 (2011). [37] P. Shukla, Phys. Plasmas 19, 083704 (2012). [38] See D. Gabor, J. Inst. Elect. Eng. Part III, Radio Commun. 93, 429 (1946) for the Hilbert transform. [39] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevE.90.013106 for a movie (video 1) of the temporal evolution of φ(x,y) showing the generic behaviors of the pair generation, propagation, and pair annihilation of defect pairs with ±1 topological charges located at the ends of ILs. [40] Through interpolating the information of φ(x,y,t) at discrete times from a Hilbert analysis, the x,y,t coordinates with φ(x,y,t) = 0 are located to plot the wave crest surface, as depicted in Fig. 3.
ACKNOWLEDGMENT
This work was supported by the National Science Council of the Republic of China under Contract No. NSC102-2112M-008-017-MY3.
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Observation of self-excited acoustic vortices in defect-mediated dust acoustic wave turbulence Ya-Yi Tsai and Lin I Department of Physics and Center for Complex Systems, National Central University, Jhongli, Taiwan 32001, Republic of China (Received 6 December 2013; published 28 July 2014) Using the self-excited dust acoustic wave as a platform, we demonstrate experimental observation of selfexcited fluctuating acoustic vortex pairs with ±1 topological charges through spontaneous waveform undulation in defect-mediated turbulence for three-dimensional traveling nonlinear longitudinal waves. The acoustic vortex pair has helical waveforms with opposite chirality around the low-density hole filament pair in xyt space (the xy plane is the plane normal to the wave propagation direction). It is generated through ruptures of sequential crest surfaces and reconnections with their trailing ruptured crest surfaces. The initial rupture is originated from the amplitude reduction induced by the formation of the kinked wave crest strip with strong stretching through the undulation instability. Increasing rupture causes the separation of the acoustic vortex pair after generation. A similar reverse process is followed for the acoustic vortex annihilating with the opposite-charged acoustic vortex from the same or another pair generation. DOI: 10.1103/PhysRevE.90.013106
PACS number(s): 52.35.Mw, 47.54.−r, 52.27.Lw
I. INTRODUCTION
Screw-type dislocation, with a helical waveform winding around a line with zero wave amplitude and undetermined phase, has been identified as an important phase singularity of 3D traveling-wave trains [1–13]. The number of 2π phase accumulations along a closed loop (or the number of helical twists in one wavelength) surrounding the singularity is the topological charge [1]. In acoustic waves [2–8] and optical waves [9–13], screw dislocations are also called acoustic vortices (AVs) and optical vortices, respectively. They carry angular momenta proportional to their topological charges [3,6,7,11–13] and have applications such as manipulating objects [3,6,7,12] and information [8]. Nevertheless, in AV studies, AVs were mainly passively generated in air or liquids, through the interference of waves excited from oscillating driver arrays with programmed phase lags, and monitored by spatially scanning the position of a single local transducer only for repeatable waveforms [4–7,13]. Whether and how AVs can be spontaneously excited in weakly disordered waves without periodic external driving remain open challenging issues. The self-organization of ordered nonlinear waves is a ubiquitous phenomenon in nonlinear open dissipative media such as biological, fluid, and plasma systems. Increasing excitation makes the ordered waveform unstable. It leads to the transition to the weakly disordered intermediate state (also called defect-mediated turbulence), with undulated waveforms before the transition to strong wave turbulence with a powerlaw spectrum [14–24]. In the weakly disordered wave, the amplitude and phase modulation through modulation instability induces waveform undulation and broadens the sharp peaks in the power spectrum. It consequently generates fluctuating defects traveling along the centers of low-amplitude holes or filaments, where phases are singular [17,18,20–24]. Namely, the dynamics can be built up from localized moving and interacting defects. In the past two decades, weakly disordered waves and the defect dynamics associated with low-amplitude holes or filaments have been studied in systems such as 1D traveling waves obeying Landau-Ginzburg equations [17,18], thermal and electrode convection systems [19], and chem1539-3755/2014/90(1)/013106(5)
ical reaction-diffusion systems [20–23]. Pitchfork, spiral, and scroll waveforms are the basic waveforms surrounding defects [17–23]. For 3D nonlinear traveling plane waves, only the recent experimental study on 3D nonlinear dust acoustic waves in a uniform dusty plasma background demonstrated the transition to the weakly disordered state with fluctuating defect pairs traveling along chaotic low-amplitude hole filaments in xyt space, through the spontaneous undulation of the 3D plane dust acoustic wave propagating in the -z direction [24]. Nevertheless, the following intriguing issues remain poorly understood. (a) What are those defects, are they AVs, and what are their stereo waveforms? (b) How can they be generated, evolve, and be annihilated through spontaneous undulation of the traveling plane wave? In this work, using the dusty plasma as a platform, these unexplored issues are experimentally addressed in xyt space by video imaging the temporal evolution of the particle density in an xy plane. The study demonstrates the pair generation, propagation, and annihilation of AVs with ±1 topological charges and helical waveforms with opposite helicity around the amplitude hole filament pair in xyt 2 + 1D space. The physical processes of AV excitation through local undulation instability-induced wave crest rupturing and reconnection, the precursor for AV generation, and the reversed process for AV pair annihilation are identified. The dust acoustic wave with longitudinal oscillation in the dusty plasma composed of negatively charged dust suspensions in the gaseous plasma is a fundamental longitudinal plasma wave governed by modulation-type dynamical equations [25–27]. It can be self-excited through the interplay of dust inertia, screened Coulomb interaction, and free energy from ion streaming [25–27]. The advantage of direct video imaging the dust density evolution also makes it a good platform to study the generic dynamical behaviors of nonlinear acoustic-type waves. Previous studies on shocks, solitary waves, wave-particle interaction, wave breaking, wave turbulence, and dislocation defects with pitchfork waveforms in nonuniform plasmas, mainly through imaging the side view waveforms in planes normal to wave fronts, are a few good examples [28–36]. Recent theoretical work has demonstrated
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that AVs can carry angular momenta and exist in linear dust acoustic waves, without investigating AV waveform dynamics [37]. II. EXPERIMENT
The experiment is conducted in a cylindrical rf dusty plasma system as described elsewhere [24,35]. Figure 1(a) shows a sketch of the experimental setup. A hollow square thin glass trap 25 mm in inner width and height is put at the center of the bottom electrode to confine the dusty plasma with suspended polystyrene particles 5 μm in diameter, 6.9 × 10−11 g in mass, 6 × 104 cm−3 in density. The weakly ionized glow discharge (ne ∼ 109 cm−3 ) supporting the self-excited defect-free waveform with straight and parallel wave crests can be generated in Ar gas at 220 mTorr using a 14-MHz rf power system at 2.8-W rf power [24]. The estimated Debye length λD and the estimated dust particle charges are on the order of a few tens of micrometers and a few to 10 000 electrons/particle, respectively. The wave propagates downward in the −z direction. The dust image in an xy plane illuminated by an expanded horizontal laser sheet (∼0.6 mm in width) is captured by a CCD at 400-Hz sampling rate. The normalized local dust density nd (x,y,t) = Id (x,y,t)/Id (x,y)t can be obtained by measuring the image brightness Id , coarse grained over a Gaussian weighted circle (0.3 mm in full width at half maximum), where Id (x,y)t is the time average of the local Id (x,y). Namely, nd = 1 when nd equals the temporally averaged dust density.
the amplitude A(x,t) and the phase φ(x,t) of the local nd oscillation are computed through Hilbert analysis [24,38]. We first obtain nd0 (x,t) by bandpass filtering the fundamental band (from 22 to 50 Hz) of nd (x,t) at the local point x and the imaginary part nˆ d0 (x,t) through the Hilbert ∞transform of nd0 (x,t), i.e., nˆ d0 (x,t) = H [nd0 (x,t)] = π1 −∞ [nd0 (x,t)/t − τ ]dτ [38]. Then, through the equation N (x,t) = nd0 (x,t) + i nˆ d0 (x,t) = A(x,t)eiφ(x,t) for N (x,t) in the complex plane, the amplitude A(x,t) and the phase φ(x,t) = arg[N (x,t)] of nd0 (x,t) oscillation can be obtained. Figure 2(a) shows the nd waveform and the corresponding plot of φ in xyt space. Basically, the side views of the xyt space plots of nd and φ show similar undulated waveforms, where φ = 0 (2π ) corresponds to the wave crest. Defects, labeled by crosses, are located at the vertices of pitchforkshaped waveforms (with connected or broken arms), where phases are undefined. They can also be easily identified since they concide with the low-amplitude holes, where the phases cannot be defined [18,21,24]. Figure 2(b) further shows two typical sequential plots of φ in the xy plane. The undulated waveform causes the presence of patches with varying φ. The intersection of a tilted wave crest surface (i.e., φ = 0 surface) with an xy plane at certain t causes the formation of curved lines with finite lengths along the interfaces of the white and the black patches. Those interface lines (ILs) evolve with time due
III. RESULTS AND DISCUSSION A. Identifying acoustic vortices from φ(x, y) and A(x, y) plots
Decreasing dissipation by decreasing pressure to 200 mTorr leads to the transition to the weakly disordered state with a 40-mm/s average wave speed [24]. Figures 2(b) and 2(c) show the typical local nd evolution and power spectrum, respectively. The temporal modulation of the local nd (the normalized dust density) fluctuation causes the broadening of the fundamental and the second harmonic peaks in its power spectrum [24]. In order to more clearly characterize the phase and the amplitude modulations, the temporal evolutions of
FIG. 1. (Color online) (a) Sketch of the experimental setup. (b) Temporal evolution and (c) power spectrum of the local nd [24].
FIG. 2. (Color online) (a) The nd waveform and the corresponding plot of φ in xyt space. (b) Two typical sequential plots of φ(x,y) at different times. The crosses label defect locations. In the xy plane, they are located at the ends of interface lines between the white and the black patches, where the phases have abrupt jumps. (c) Chaotic amplitude hole filaments in xyt space, showing their pair generation and annihilation, and propagation. (d) Stereo plot of φ(x,y) at a fixed t around a typical defect from the dashed box in (b), showing the 2π phase jump winding a circle around the defect. (e) Plots φ(x,y) and A(x,y) corresponding to the region of (d).
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to the evolving waveform [see Fig. 2(b) and Ref. [39]]. Defects are located at the ends of ILs, where phases are undefined. What are those defects? Figure 2(d) and the left panel of Fig. 2(e) show the stereo and contour plots of φ(x,y), respectively, for a small region centered at the lower defect in the dashed box of Fig. 2(b). Here φ changes by 2π after circulating one cycle around the defect. The defect is located at the center of the amplitude hole, defined as A(x,y) < 0.1 and labeled by the circled region of Fig. 2(e). The above findings clearly evidence that the defect is an AV with a +1 topological charge [1,5]. Obviously, the defect at the opposite end of the IL has a −1 topological charge. Figure 2(c) shows the trajectories of amplitude holes, the lines of silence, in xyt space, centered along defect trajectories. Under topological charge conservation, those chaotic hole filaments can be generated and annihilated pairwise and propagate with varying angles in xyt space.
further depict the decomposed waveforms. The two plots in the right column of Fig. 3(d) correspond to those on their left, but viewed at different angle. Obviously, the strong local undulation causes the sequential rupture of crest surfaces along finite lines, starting from a local region of crest 1. The sequential reconnections of the upward bending side of a ruptured crest i with the downward bending side of the trailing ruptured crest i + 1 causes the spontaneous generation of an AV pair, with helical waveforms oppositely spiraling along the hole filaments centered at the ±1 defects in xyt space [Figs. 3(a)–3(c)]. Increasing rupture increases the AV pair separation. Under the above helical waveform, it is easy to understand the evolution of φ(x,y) with t = nτ , where τ = 2.5 ms (the CCD sampling interval) in Figs. 4(a) and 4(b). The AV pair generation begins at n = 0, when the finite IL emerges. Note that the IL is the intersection line between an xy plane and a tilted crest surface. With increasing t, the screw-shape wave crest surfaces with opposite chirality make the two end sections of the ILs connected to the defect pair (located at the hole filament sites) rotate oppositely. This leads to the formation of a curved long IL, which is restraightened after one wave period (also see Ref. [39]). This process is repeated in the following few wave cycles until AV annihilation, associated with the appearance of the very short straight IL at n = 41 [Fig. 4(b)]. The stereo plot of Fig. 4(c) further depicts the opposite rotations of the end sections of ILs (as indicated by the shot black lines) around the filament pair. Is there a precursor leading to the onset of AV pair generation associated with the rupture of crest surface 1 and the reconnection with surface 2, around t = 0 s? Figure 3(e) evidences the emergence of the narrow kinked stripe, as indicated by the arrows, for crest surface 0. Figure 5(a) further
B. Stereo waveforms of AVs and their generic dynamical processes from pair generation to pair annihilation
What are the stereo waveforms of AVs in xyt space and how can they be formed through waveform undulation? Let us use the waveform evolution surrounding the filament loop, indicated by the green arrow in the lower right corner region of Fig. 2(c), from AV pair generation to pair annihilation, as an example. Figures 3(a) and 3(b) depict the wave crest surfaces with φ(x,y) = 0 [40], labeled by the sequence number i in xyt space, surrounding the two and the right filaments, respectively. For better viewing, Figs. 3(c) and 3(d)
FIG. 3. (Color online) Stereo waveforms of wave crest surfaces, labeled by the sequence number i, around the hole filament pair and the right filament in xyt space. (c)–(e) Decomposed waveforms viewed from different angles, which more clearly show how the rupture and reconnection of sequential wave crest surfaces lead to the formation of AV pairs with helical waveforms oppositely winding around the hole filaments at the two ends of the rupture. The two right panels of (d) from another angle clearly show the upward and the downward tilting of the ruptured crests. The arrows in (d) and (e) indicate the kinked strips leading to rupture.
FIG. 4. (Color online) (a) and (b) Evolution of φ(x,y) in the region around the AV pair of Fig. 3, with increasing t = nτ and τ = 2.5 ms. The AV pair generation and annihilation begin at n = 0 and 41, respectively. (c) Stereo plots in xyt space showing the oppositely spiraling end sections of the ILs around the two hole filaments. (d) Typical A(x,y) plots at different n showing low-amplitude scars that are gradually turned into the separated amplitude hole pair centered at the defect pair after defect formation.
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shows the plot of the sequential wave crests (blue lines) on top of the A(x ,t) plot, where x t is the surface normal to and intersecting the surface covering the filament loop of Fig. 3, along the center vertical line. The undulation instability causes the change from the smoothly tilted crest −1 to the strongly kinked crest 0, followed by the sequential rupture and reconnection of the trailing crests from crest 1. Figure 5(b) depicts the more detailed initial evolution from the smoothly tilted to the strongly kinked contour (equi-φ) lines, in the dashed box of Fig. 5(a). Figure 5(c) shows how the maximum slope Sm of each tilted contour line in x t space grows with increasing t. The solid curve is the best fit of the exponential growth with a 6-ms time constant. The increasing stretching with the growing kink also significantly reduces the local amplitude A. It is manifested by the emergences of low-amplitude scars that have decreasing A with increasing t in the boxed region of Fig. 5(a) and the n 0 panels of Fig. 4(d) and the low-amplitude hole filament tip below crest 1 in Fig. 3(a). It eventually leads to the crest surface rupture and reconnection [at the small circles of Fig. 5(a)] starting from crest 1, for defect pair generation. As shown in Figs. 3(a), 4(b), and 5(a), a similar reverse process occurs after the annihilation of the AV pair. The decreasing waveform undulation causes the emergence of a kinked strip without rupture for crest 5 and then the smoothly undulated crest 6, similarly to crests 0 and −1. A low A scar appears again around the rupture end of crest 4 and the kinked region of crest 5 [Fig. 5(a)]. Also note that the downward (upward) bending part of the ruptured crest surface 1 (4) does not touch crest surface 0 (5). There is no extra crest surface insertion between crests 0 and 5. After pair generation, an AV can also annihilate with an AV from the other pair generation. Figures 6(a) and 6(b) show another example of the stereo waveforms and the end sections of the ILs surrounding the W-shaped hole filaments, for the
FIG. 5. (Color online) (a) Plot of wave crests (blue lines) on top of the A(x ,t) plot, where the x t surface is the surface normal to and through the center of the surface covering the filament pair of Fig. 3. (b) Contour plot of φ(x ,t) corresponding to the dashed rectangular region of (a). (c) Plot showing the exponential growth of the maximum slope of each equi-φ line in the x t plane of (b) with increasing t. The solid curve is an exponential fit with a 6-ms time constant. The small circles in (a) indicate the local regions for the reconnection of adjacent ruptured crest surfaces.
annihilation of AVs α and β , generated from two different AV pairs α-α and β-β . Generic behaviors of the screwshape waveforms with opposite chirality through crest surface rupture and reconnection in the center regions of kink strips on crest surfaces for each AV pair, similar to the example of Fig. 3, are followed. Figure 6(c) shows the A(x ,t) and φ(x ,t) plots, where the x t surface is the surface normal to and intersecting the plane covering the filament pair α and α . The initial crest rupture is also led by the instability from the smoothly tilted to the strongly kinked crest surfaces [see the φ(x ,t) plot in Fig. 6(c)], associated with the strong local amplitude suppression [see the A(x ,t) plot in Fig. 6(c)]. Although AVs α and β are not generated in pairs, the opposite topological charges (chirality) allow their annihilation. Reference [39] provides clear examples of annihilations of oppositely charged defects from different pair generations. IV. CONCLUSION
We demonstrated the observation of fluctuating AV pairs with helical waveforms surrounding chaotic hole filament pairs as the fundamental spontaneous excitations in the self-excited weakly disordered 3D traveling acoustic wave. The physical processes for the spontaneous AV pair generation, propagation, and annihilation are identified. The rapid exponential growth of the kink strip formation through modulation instability leads to a strong amplitude reduction, followed by the sequential rupture and reconnection along finite lines on trailing crest surfaces. It forms two AVs with opposite chirality around the hole filament pair at the two ends of the rupture. The inverse process leads to the annihilation of an AV either with its twin brother or with another oppositely charged AV from another pair generation.
FIG. 6. (Color online) (a) and (b) Plots similar to those in Figs. 3(a) and 4(c), showing the waveform evolution from the generation of two AV pairs α-α and β-β to the annihilation of α and β with helical waveforms oppositely spiraling around their filaments, as indicated by arrows. (c) The A(x ,t) and φ(x ,t) plots demonstrating the strong kink formation with strong local amplitude reduction, from smoothly undulated equi-φ lines, leading to crest 1 rupture and reconnection with crest 2. The x t surface is the surface normal to and intersecting the plane covering the filament pair α and α around their initial growth region. The index i indicates the crest surface number.
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From the view of system symmetry, the undulation (modulation) instability induces spontaneous symmetry breaking and alters the topology of ordered plane waves, through crest rupture and reconnection and consequently AV pair excitation, in which topological charges are conserved. Since the dynamics of many nonlinear waves are also governed by the modulation-type nonlinear equations, the similar spontaneous symmetry breaking process for AV excitation through undulation instability induced crest surface rupture and reconnection is expected. Our study should be able to
shed light on understanding the generic waveform dynamics and lead to related theoretical and experimental studies in other weakly disordered waves self-excited from 3D longitudinal plane waves.
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ACKNOWLEDGMENT
This work was supported by the National Science Council of the Republic of China under Contract No. NSC102-2112M-008-017-MY3.
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