⇦Chinese Physics Letters, 2019, Vol. 36, No. 2, Article code 024302 Pitch Shift in Exsomatized Cochlea Observed by Laser Interferometry * Zhang-Cai Long (龙长才)1, Tao Shen (沈涛)1, Yan-Ping Zhang (张艳平)2, Lin Luo (骆琳)3** Affiliations 1School of Physics, Huazhong University of Science and Technology, Wuhan 430074 2Affiliated Hospital, Huazhong University of Science and Technology, Wuhan 430074 3Department of Chinese language and literature, Huazhong University of Science and Technology, Wuhan 430074 Received 23 October 2018, online 22 January 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11374118 and 90820001.
**Corresponding author. Email: luolin@hust.edu.cn Citation Text: Long Z C, Chen T, Zhang Y P and Luo L 2019 Chin. Phys. Lett. 36 024302    Abstract Pitch is one of the most important auditory perception characteristics of sound; however, the mechanism underlying the pitch perception of sound is unclear. Although theoretical researches have suggested that perception of virtual pitch is connected with physics in cochlea of inner ear, there is no direct experimental observation of virtual pitch processing in the cochlea. By laser interferometry, we observe shift phenomena of virtual pitch in basilar membrane vibration of exsomatized cochlea, which is consistent with perceptual pitch shift observed in psychoacoustic experiments. This means that the complex mechanical vibration of basilar membrane in cochlea plays an important role in pitch information processing during hearing. DOI:10.1088/0256-307X/36/2/024302 PACS:43.66.Hg, 43.64.Kc, 87.85.fk © 2019 Chinese Physics Society Article Text Pitch is one of the most important perceptual attributes of sound. The pitch of a sound is the perceived position of the sound in musical scale and is described in frequency of pure tone with the same perceived pitch. In music, a sequence of pitches composes a melody, and simultaneous combinations of pitches compose harmony. In speech, the rise and fall of pitch contour compose prosody, which plays an important role in expressing meaning of words in tonal languages, such as Chinese, and improves speech intelligibility, especially in circumstance with competing sounds. For a long time, the mechanism of pitch perception has been a focus of research. However, how a hearing system processes and abstracts pitch information is still a mystery.[1] Due to the general deficit of pitch perception in cochlear implant and hearing aid users, the mechanism of pitch perception, especially the pitch processing mechanism in cochlea, is significantly important.[1] The scientific investigation of pitch perception mechanism can be dated back to Pythagoras. In modern times, Helmholtz connected this perception with physics in cochlea, who attributed different perceptual pitches of pure tones to sound sensors in hearing systems with different mechanical resonant frequencies.[2] Bekesy observed that sounds with different frequencies evoke mechanical vibrations at different positions of basilar membrane in cochlea, which is connected with different neural channels to higher hearing systems. This frequency-position topology in the cochlea can explain why sound with different frequencies has a different pitch corresponding to the frequency, and can also explain the phenomena that complex sound, composed of a fundamental component with frequency $f_{0}$ and its higher harmonic components with frequencies $2f_{0}$, $3f_{0}$, $nf_{0}$, has a pitch corresponding to its fundamental frequency $f_{0}$. However, when the fundamental component is removed, a complex sound comprised of only higher harmonic components still has perception pitch, and the pitch is that of the removed fundamental component. This pitch without corresponding physical frequency component is known as virtual pitch.[3] The virtual pitch phenomena had ever been explained by Helmholtz proposed combination-tone theory.[2] According to the theory, two tones with frequencies $f_{1}$ and $f_{2}$ respectively will produce components with combination frequencies $f_{\rm C}=nf_{1}+mf_{2}$ ($n$ and $m$ are integers) in a nonlinear auditory system, which are called combination tones. Among these combination tones, the component with frequency $f_{2}-f_{1}$ (and $f_{1} -f_{2}$) is called difference-frequency component. When a harmonic complex sound without fundamental frequency component is applied, the consecutive harmonic components will produce difference-frequency components with frequency of the missing fundamental component in a nonlinear auditory system, which would produce the pitch of the missing fundamental component. Combination tones in cochlear membrane vibration were finally observed experimentally in 1991.[4] However, combination-tone explanations of virtual pitch are still confronted with pitch shift observed by Boer and Schouten et al. in psychoacoustic experiments.[5,6] Boer[5] and Schouten et al.[6] used amplitude modulated signals in the form of $s(t)=\frac{A}{2}({1+\cos 2\pi f_{0} t})\cos 2\pi ({kf_{0} +\delta f})t$ ($k$ is an integer), which contains only three components with frequencies $\{({k-1})f_{0} +\delta f$, $kf_{0} +\delta f$, $({k+1})f_{0} +\delta f\}$. These components, centered at the carrier frequency $kf_{0} +\delta f$, are separated by modulating frequency $f_{0}$, and the frequency difference of consequent components is always $f_{0}$. When $\delta f=0$, these components are higher harmonics of missing fundamental with frequency $f_{0}$, and the signal has percept pitch of the missing fundamental, which is equal to the difference frequency of consequent components. When $\delta f\ne 0$, these components, which are transferred in frequency with a common value $\delta f$ and maintained difference frequency unchanged, are generally not harmonic. However, the signal still has perceptual pitch, and the percept pitch is not the difference frequency $f_{0}$, instead, with a shift from the difference frequency $f_{0}$ dependent on $k$ and $\delta f$. Boer described the perceived pitch of this signal in formula $f_{\rm pp}=f_{0} +\frac{\delta f}{k}$.[5] To explain the pitch shift phenomena, neuron stochastic resonance[7,8] and quasi-periodic signal driven three-frequency resonance in nonlinear vibration system[9,10] have been proposed. The former demonstrated theoretically that noise in neurons can produce spike train with frequency corresponding to the shift virtual pitch in stochastic resonance, a phenomenon in which optimal noise can enhance signal transmission.[11,12] The latter demonstrated theoretically that amplitude modulated signal adopted by Boer and Schouten et al. can produce so-called three-frequency nonlinear resonance with resonance frequency corresponding to the shift virtual pitch. Recently, Martignoli and Stoop demonstrated, in an electronic simulation device of cochlea consisting of a cascade of Hopf vibrators,[13,14] that pitch shift phenomena can originate from temporal structure of auditory peripheral nonlinear vibration.[15] However, there is no direct physiological experimental evidence for any of these theories. In this work, to demonstrate whether the pitch shift phenomenon is originated from cochlea and how the shift virtual pitch is expressed in cochlea, we measure and analyze basilar membrane vibration in cochlea by laser interferometer. For the first time, we observe pitch shift phenomena in exsomatized cochlear basilar membrane vibration, which is consistent with percept pitch observed in psychoacoustic experiments. The pitch that we observed in cochlea with shift characteristics of hearing perception comes from the temporal structure of basilar nonlinear vibration, instead of nonlinear three-frequency resonance. We used the cochlea of a guinea pig, which is similar to human cochlea, to conduct this research. The Guinea pigs that we adopted were about 400–500 g, which were provided by Experimental Animal Center under Hubei Provincial Center for Disease Control and Prevention. All Hubei provincial and national guidelines of China for the care and use of animals were followed. These animals were chosen without limit of gender and colour. Every Guinea pig was examined under microscope to make sure that it was without abnormality in ear channel and eardrum. After it was anesthetized by abdominal injection of pentobarbital sodium (concentration 3%, dose 30 mg/kg), cochlea with integer auricle was separated from guinea pig. Under microscope, the bulla was opened, and a hole about 200 µm was drilled at the top first turn of cochlea. Then, the prepared cochlea was placed on measuring platform of interferometry. Our previous researches had shown that after separated from body, exsomatized cochlea remains active for a short period of time.[16] To have an active cochlea to be measured in this research, the time taken for measurement was limited less than 30 min. In addition, the activity of a measured cochlea was checked according to whether its basilar membrane vibration contains combination-tone components.[16] The vibrations of basilar membrane were measured by our self-developed laser interfering measurement system.[17] In this system, we adopted new technology to achieve sub-nanometer resolution without nonlinear distortion.[17] In addition, this system was designed to measure vibration of object with weak refection, so that the vibration of basilar membrane was measured without reflecting micro beads projected into the cochlea. In this measurement system, displace wave forms of basilar membrane vibration, instead of vibration velocity, were recoded real time in computer. Detail of this system is available in our published articles[16,17] and dissertation.[18]
cpl-36-2-024302-fig1.png
Fig. 1. Stimulus signal: (a) stimulus wave form in the form of $s(t)=\frac{A}{2}({1+\cos 2\pi f_{0} t})\cos 2\pi ({kf_{0} +\delta f})t$, $f_{0}=200$ Hz, $k=4$, $\delta f=-50$ Hz. (b) Spectrum of stimulus with component amplitude ratio of 0.45:1:0.45, $f_{1}=550$ Hz, $f_{2}=750$ Hz and $f_{3}=950$ Hz.
Signals with the same form of Boer and Schouten et al. pitch shift psychoacoustic experiments[5,6] were used: $s(t)=\frac{A} {2} ({1+\cos 2\pi f_{0} t})\cos 2\pi ({kf_{0} +\delta f})t$, which were composed of pure tone components with frequencies: $\{({k-1})f_{0} +\delta f$, $kf_{0} +\delta f$, $( {k+1})f_{0} +\delta f\}$. Amplitude ratio of the three components was 0.45:1:0.45. Figure 1 is an example of input signal wave (Fig. 1(a)) and frequency spectrum (Fig. 1(b)). The signals were synthesized and produced digitally with 0.01 µs time resolution by digital signal generator RIGOL dG4062. The signals were introduced to CD/tape input of AC33 aduitometer (Denmark) and amplified, then drove an insert earphone (EARTONE-3A) inserted in guinea pig ear channel to produce sound stimulus. The sound level was 80 dB. We first checked whether three-frequency resonance occurs in cochlear membrane vibration. This was carried out by analyzing the frequency components of basilar membrane vibration. According to theory, for stimuli with frequencies $f_{1}$ and $f_{2}$, nonlinear three-frequency resonances would appear at frequencies $f_{\rm R}$ determined by both $f_{1}$ and $f_{2}$ in formula: $Rf_{\rm R}=pf_{1} +qf_{2}$ ($R$, $p$ and $q$ are integers). These resonances would exhibit resonance peaks in frequency spectrum. Compared with combination tones, which are only the parts of three-resonance for $R=1$, frequency components of three-frequency resonance are more populous, one of which would be with frequency identical to perception pitch. Our analysis showed that except peaks of stimuli and combination tones, there is no other peak in the frequency spectrum of basilar membrane vibration. Figure 2 is an example of observed basilar membrane vibration and frequency spectrum. This means that theory expected three-frequency resonance did not occur in the cochlea, and virtual pitch, which is shifted from difference frequency, cannot be attributed to three-frequency resonance of basilar membrane vibration.
cpl-36-2-024302-fig2.png
Fig. 2. Vibration response of basilar membrane. (a) Vibration wave form of basilar membrane. (b) Spectrum of basilar membrane vibration responding to signal stimulus of Fig. 1, which includes combination tones (200 Hz, 400 Hz, 600 Hz, etc.) in addition to stimulus frequency components ($f_{1}=550$ Hz, $f_{2}=750$ Hz, $f_{3}=950$ Hz). The characteristic frequency of the measuring position is 450 Hz.
cpl-36-2-024302-fig3.png
Fig. 3. Scheme of calculating temporal structure expressed pitch: (a) basilar membrane vibration wave form and peak-interval time, (b) distribution of peak-interval time versus inverse of peak-interval time. The most frequent peak-interval time is $T_{\rm P}$, and the inverse of which is temporal expressed pitch $f_{\rm P}$.
Our next work was to explore whether the temporal structure of the basilar vibration expresses pitch information. We accounted numbers of time interval $T$ between peaks (Fig. 3(a)) in basilar membrane vibration wave form to obtain inter-peak time interval distribution. Then we obtained the most frequent inter-peak time interval $T_{\rm P}$ of basilar membrane vibration, at which there is a maximum in the time-interval distribution (Fig. 3(b)). We use $1/{T_{\rm P}}$ as basilar membrane vibration temporal structure expressed pitch $f_{\rm P}$. This calculation of pitch $f_{\rm P}$ has been adopted in previous theoretical studies,[15] and some researchers suggested that this time information can be maintained in the following neuron transfer and can be extracted in a higher neuron system.[15]
cpl-36-2-024302-fig4.png
Fig. 4. Pitch shift of basilar membrane vibration. In this figure $f_{\rm car}$ is the carrier frequency, that is, the central frequency of stimulus signal. Stars are temporal structure expressed pitch measured in this research. Slope dashed line is pitch described by the Boer formula $f_{\rm PP}=f_{0} +{\delta f}/k$, $k=7$, and $f_{0}=200$ Hz. Frequencies of stimulus are $\{{1200+\delta f,1400+\delta f,1600+\delta f}\}$. When $\delta f\ne 0$, they shift apart from harmonic frequencies of 200 Hz, (1200 Hz, 1400 Hz, 1600 Hz), but difference frequency of neighbor components keeps unchanged in 200 Hz. The characteristic frequency of the measuring position is 500 Hz.
Keeping $k$ and $f_{0}$ constant, and changing $\delta f$, we obtained the situation where three components of stimulus changed around the harmonic. For example, when $k=7$, $f_{0}=200$ Hz, frequencies of three components, (1200 Hz$+\delta f$, 1400 Hz$+\delta f$, 1600 Hz$+\delta f$), changed around (1200 Hz, 1400 Hz, 1600 Hz), harmonic frequencies of 200 Hz. We measured temporal structure expressed pitch of these stimuli. Figure 4 is an example of this measurement. It can be seen that when $\delta f=0$, the temporal structure expressed pitch is 200 Hz, which is the missing fundamental frequency of stimulus frequencies (1200, 1400, 1600), and also the difference frequency of consequent components of the stimulus. With increasing (decreasing) $\delta f$, the temporal structure expressed pitch increases (decreases), and shifted from original missing fundamental frequency and difference frequency (200 Hz). The observed relation of pitch shift with $\delta f$ (expressed in stars in Fig. 4) is consistent with the Boer formula $f_{\rm pp}=f_{0} +\frac{\delta f}{k}$ (slope dashed line in Fig. 4), except a slight difference in the changing slope. This subtle difference in slope also existed in the psychoacoustic experimental result by Boer and Schouten, which is called the second pitch shift. Due to the limit of active time of an exsomatized cochlea for data collection, we could not obtain data for all different $k$ from one active cochlea. To check the general characteristics of basilar membrane vibration temporal structure expressed pitch and to compare it with psychoacoustic experimental result, we collected data from different cochleae. Although data from every cochlea were measured at the top first turn of a cochlea, measuring positions of different cochleae were not with the same characteristic frequency. We pool these data and show them in Fig. 5 (solid dots). In all this data in Fig. 5, except those for $k=2$, 11, 12, data for a certain integer $k$ from 3 to 10 were not from a same cochlea, but obtained from different cochleae.
cpl-36-2-024302-fig5.png
Fig. 5. Temporal structure expressed pitch observed in cochlear basilar membrane vibration. Ordinate ($f_{\rm p}$) is pitch, and abscissa ($f_{\rm car}$) is carrier frequency of stimulus signal. The stimulus signal form is $s(t)=\frac{A}{2}({1+\cos 2\pi f_{0} t})\cos 2\pi ({kf_{0} +\delta f})t$, $f_{\rm car}=kf_{0} +\delta f$, $f_{0}=200$ Hz. Dots are the measured data in this research. Slope dashed lines are pitch described by the Boer formula.
It can be seen that though these data were collected from different cochleae, and measuring positions did not have the same characteristic frequency, these data (solid dots in Fig. 5) still show ensemble characteristics. The data cluster according to $k$. For a certain $k$, it exhibits pitch shift depending on $\delta f$ (or center frequency, carrier frequency $f_{\rm car}$, $kf_{0} +\delta f$), which is consistent with psychoacoustic experimental based pitch shift formula (slope dashed line in Fig. 5). For different $k$, the slope of the pitch shift changes with $k$, which is also consistent with the psychoacoustic experimental based pitch shift formula (slope dashed lines with different $k$). The only difference from psychoacoustic result is that for $k$ from 3 to 10, the slope of pitch shift does not show a systematically slight difference from the Boer formula $f_{\rm pp}=f_{0} +\frac{\delta f}{k}$; that is, the second pitch shift. The fact that the pooled data from similar position of different cochleae exhibit ensemble pitch shift characteristics consistent with perception pitch characteristics indicates that temporal expressed pitch of basilar membrane vibration is general, unanimous, and robust, which is not influenced by individual difference of specimens. The absence of second pitch shift in the pooled data means that pitch shift is position-dependent. If pitch shift is slightly dependent on position in cochlea, then the subtle second pitch shift can be blurred by this position dependence in the pooled data. In conclusion, this research has revealed that basilar membrane vibration of active cochlea, in temporal structure, can express pitch consistent with perceptual virtual pitch observed in psychoacoustic experiments. We have not observed three-frequency resonance in cochlear basilar membrane vibration. This suggests that, in cochlea, basilar membrane vibration expresses virtual pitch of sound in temporal structure, instead of nonlinear three-frequency resonance. With this research, we can expect that there is temporal mechanism in neuron systems, in addition to frequency-channel topology mechanism and to abstract pitch information. Consequently, a hearing aid or cochlear implant that is endowed with nonlinearity of cochlea to transfer pitch information in temporal structure should produce a better effect in speech intelligibility and music appreciation than linear sound signal processing. References Revisiting place and temporal theories of pitchTwo-tone distortion in the basilar membrane of the cochleaPitch of Inharmonic SignalsPitch of the ResidueSubharmonic stochastic synchronization and resonance in neuronal systemsNews and views in briefNonlinear Dynamics of the Perceived Pitch of Complex SoundsPitch perception: A dynamical-systems perspectiveLower Hearing Threshold by NoiseFrequency Sensitivity in Nervous SystemsEssential Nonlinearities in HearingHopf Amplification Originated from the Force-Gating Channels of Auditory Hair CellsLocal Cochlear Correlations of Perceived PitchHigh resolution heterodyne interferometer based on time-to-digital converter
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⇦Chinese Physics Letters, 2019, Vol. 36, No. 2, Article code 025201 Successive Picket Drive for Mitigating the Ablative Richtmyer–Meshkov Instability * Meng Li (李蒙)1**, Wen-Hua Ye (叶文华)1,2** Affiliations 1Institute of Applied Physics and Computational Mathematics, Beijing 100094 2Center for Applied Physics and Technology, HEDPS, Peking University, Beijing 100871 Received 14 September 2018, online 22 January 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11675026, 11875091 and 11575034.
**Corresponding author. Email: li_meng@iapcm.ac.cn; ye_wenhua@iapcm.ac.cn Citation Text: Li M and Ye W H 2019 Chin. Phys. Lett. 36 025201    Abstract The ablative Richtmyer–Meshkov instability (ARMI) is crucial to the successful ignition implosion of the inertial confinement fusion (ICF) because of its action as the seed of the Rayleigh–Taylor instability. In usual ICF implosions, the first shock driven by various foots of the pulses plays a central role in the ARMI growth. We propose a new scheme for refraining from ARMI with a pulse of successive pickets. With the successive-picket pulse design, a rippled capsule surface is compressed by three successive shocks with sequentially strengthening intensities and ablated stabilization, and the ablative Richtmyer–Meshkov growth is mitigated quite effectively. Our numerical simulations and theoretical analyses identify the validity of this scheme. DOI:10.1088/0256-307X/36/2/025201 PACS:52.57.Fg, 52.57.-z, 47.20.-k, 52.50.Lp © 2019 Chinese Physics Society Article Text The Rayleigh–Taylor instability (RTI) and the Richtmyer–Meshkov instability (RMI) play important roles in ICF and many other physical phenomena.[1,2] With the ablation driven by laser beams or x-rays in ICF, the ablative Rayleigh–Taylor instability (ARTI) and ARMI differ from the classical ones. On account of ARMI, any perturbation from the outer surface of a capsule would grow right away and be amplified by the followed ARTI during the acceleration phase of the implosion, and it will then feed through the shell to the inner deuterium-tritium (DT) ice layer and even break up the imploding shell during the deceleration phase. Thus, the capsule must be designed to keep the growth of these instabilities at an acceptable level to prevent them from endangering ignition of the capsule. On the path to the capsule ignition using indirect drive (ID), designs and experiments have achieved significant progress. Initially, most attentions were paid to low foot (LF) drive. The experiments conducted on the National Ignition Facility (NIF) have achieved peak implosion velocity about 350 km/s, and the areal density of the DT fuel about 1.3 g/cm$^{2}$. Unfortunately, only $\sim10^{14}$ neutrons were achieved, which is far below the goal of ignition.[3,4] Further experiments indicated that there was a high degree of ablator mix into the DT fuel due to high instability growth at the ablation front.[5,6] Then, experiments with the high foot (HF) drive increased neutron yields up to 10$^{16}$. High neutron yields in HF implosion are attributed to growth mitigation of the ARMI and the followed ARTI at the ablation front. Compared to the LF pulse, the mitigation is due to a much higher picket and a higher foot plateau of the HF pulse. Although the HF implosion has high stabilization at the ablation front, the high adiabat of DT fuel sacrifices high DT compression.[7] Therefore, the adiabat shaping (AS)[8-14] implosion is performed as a trade-off between compression and stability, in which high DT compression has been obtained.[4,15-17] However, the implosion stability still needs to be improved further. According to hydrodynamic instability growth of the perturbation, the ICF implosion can be divided to the ARMI phase, the acceleration ARTI phase and the deceleration ARTI phase. Instability growth of the acceleration phase can be controlled by the HF and AS design with the high picket of the prepulse for the mitigation of ARMI growth.[18-20] Other attempts to suppress the ARTI are also on the way: the hybrid-drive ignition scheme is proposed to suppress implosion asymmetry and instabilities with indirect drive combined with direct drive,[21] and the decompression-recompression scheme is suggested to inhibit ARTI in the deceleration phase with the main pulse optimization.[22,23] No matter whether the LF drive or the HF drive is employed, the first shock has the largest contribution to the DT fuel adiabat, which dominantly determines the density profile of the shell. Thus, the first shock drives the ARMI that seeds for subsequent hydrodynamic instability of the implosion. Hence, the seed from the ARMI during the prepulse, especially during the period of the first shock, is one of most important issues for the ID implosion and ignition, and should be investigated in depth. For example, with the AS pulse or the HF pulse, when a rippled capsule surface is radiated by the high picket, the strong first shock is formed quickly and runs ahead of the ablation front and to perturb the capsule interior, which become seeds of the following ARTI.[17,19,20] In this Letter, we first propose a new pulse shape with successive pickets (SPs) to reduce the ARTI seeds during the early stage of the ID prepulse drive, which can compromise stability well with compression of the implosion.
cpl-36-2-025201-fig1.png
Fig. 1. (a) Sketch of the capsule, (b) radiation drives, and (c) ablation front GF spectra.
The cryogenic DT fusion capsule (Fig. 1(a)) is used in our numerical simulations. The capsule contains a 210-µm-thick CH ablator with density of 1.0 g/cm$^{3}$, an 80-µm-thick DT ice layer with density of 0.25 g/cm$^{3}$, and a DT-gas-filled cavity with a radius of 907 µm and density of 0.3 mg/cm$^{3}$. One-dimensional (1D) and two-dimensional (2D) single-mode perturbation simulations are performed with the radiation hydrodynamic code LARED-S.[24] The LARED-S code has been widely used in the study of ICF hydrodynamics instabilities.[21-27] It is a multidimensional massively parallel Eulerian code employing multi-group diffusion radiation transport, flux-limited Spitzer–Harm thermal conduction for electron and ion. With the fifth-order accurate WENO scheme, the 2D simulations mostly use $2000\times 40$ zones with a domain width of one full wavelength along $\theta$ direction. The drives of the SP pulse and the high picket (HP) pulse are plotted in Fig. 1(b). Both pulses have a peak temperature of 300 eV, and the troughs are kept at a relatively low level. By comparison, the sole HP pulse is presented with a red solid line and the SP pulse with a black dotted line. The only difference between them is that the sole high picket is replaced with three successive climbing up pickets. The shapes of the last SP picket and of the sole high picket are identical. The growth factor (GF) is defined by the ratio of the ablation front amplitude of given mode at a time to its initial amplitude. The RM growth factors are shown schematically in Fig. 1(c), the initial sinusoidally perturbed amplitudes are all 0.5 µm for comparison. The GFs of the SP case (the black symbols and line) are evidently less than those of the HP case (the red symbols and line), especially at the bottom of negative lobes near Legendre mode 100. These small lobes will be amplified during the following acceleration phase of the imploding capsule. It has been found that the negative lobe threatens the capsule ignition.[15,28,29] Certainly the implosion stability benefits from the squeezing negative lobe. The perturbation amplitudes of mode numbers 16 and 64 on the ablation front are both quite larger for the HP pulse than for the SP pulse (in Fig. 2(a)), these two selected modes are more sensitive for robustness of the hot spot surface and the ablation front, respectively. The overall oscillation frequencies of either of the two modes are close to each other. Via simulations with 40 and 220 grids (dash-dotted lines and dotted lines) in $\theta$ direction, the convergence of their numerical resolution is also identified in Fig. 2(a). Both of the perturbation amplitudes of the SP pulse (black lines and cyan lines) fade down during the first picket session, and then they dive downwards quickly and return to a platform during the second session. With the launch of the third picket, the perturbation amplitudes jump to a high level. In fact, they grow similar to that of the high picket in the HP case, but with a smaller increase. The shrinking oscillation amplitudes of the SP pulse are rooted in mitigations of the three pickets, especially the first picket.
cpl-36-2-025201-fig2.png
Fig. 2. Evolutions of (a) perturbation amplitudes of mode numbers 16 and 64 on the ablation front and (b) $V_{\rm a}$, $\rho_{\rm Max}$.
In the SP case, three sequentially strengthening shocks are generated, and the three shocks coalesce into one strong shock which is close to the shock driven by the sole high picket in the HP case. In Fig. 2(b), evolutions of the ablation velocity ($V_{\rm a}$), and the peak density of the shell ($\rho_{\rm Max}$) for the two pulses are compared. The two ablation velocities evolve quite different, the former grows very high almost immediately, the latter fluctuates at a lower level during the three pickets. The trajectories of $\rho_{\rm Max}$ indicate that the three compression ratios of the SP pulse at the early stage are lower than the compression ratio of the HP pulse, especially the first shock compression ratio. Neglecting the early stage, the evolutions of $V_{\rm a}$ and $\rho_{\rm Max}$ in the HP and SP cases are close to each other—the behaviors of the latter are just postponed by about 3.5 ns. Recall the formula of the perturbation amplitude on the ablation front[18] $$\begin{align} \frac{\eta_{\rm a} (k,t)}{\eta_{0} }=\,&\frac{\eta_{\rm a}^{\rm cl} (k,t)}{\eta_{0} }-1+C_{(0)} -C_{\rm t} kc_{\rm s} t\\ &+\eta_{\rm usu} (k,t)+\eta_{\nu } (k,t),~~ \tag {1} \end{align} $$ where $k=L/R$ is the wave number, $L$ is the Legendre mode number, $R$ is the capsule outer radius, $c_{\rm s}$ is the sound speed, and $\eta_{\rm a}^{\rm cl}$ is the perturbation evolution in the absence of mass ablation. The usual ablative RMI and vorticity convection terms are[19] $$\begin{align} \eta_{\rm usu} (k,t)=\,&e^{-2kV_{\rm a} t}({\alpha_{0} \cos (\omega t)+\beta_{0} \sin (\omega t)}), \\ \eta_{\nu } (k,t)\approx\,&-f(C)\frac{c_{\rm s} }{V_{\rm bl}}\Big\{e^{kV_{\rm a} t}\Big[1-\frac{1}{1+e^{-3({kV_{\rm a} t-0.55})}}\Big]\\ &+\frac{\sin ({\sqrt 3 kV_{\rm a} t})}{2\sqrt {kV_{\rm a} t} } \Big\}\\ =\,&\eta_{_{{\it \Omega} 1}} +\eta_{_{{\it \Omega} 2}}, \\ V_{\rm bl} \approx\,&V_{\rm a} /({2.4kL_{0} /\nu })^{1/\nu},~~ \tag {2} \end{align} $$ where $V_{\rm a}$ and $V_{\rm bl}$ are the ablation and blow-off velocities, respectively, oscillatory frequency $\omega =k\sqrt {V_{\rm a} V_{\rm bl}}$, and $C$ is the first shock compression ratio. The coefficients $C_{(0)}$, $C_{\rm t}$, $\alpha_{0}$ and $\beta_{0}$ are defined in Ref. [18]. The oscillation amplitude on the ablation front is closely related to the mode number and the ablation velocity. For a single mode, the oscillation amplitude is damped and polarizes. A low-mode perturbation experiences positive growth and a high-mode one undergoes phase reversal, just like the behaviors of mode 16 and 64. When $kV_{\rm a} t\ll 1$, $|{\eta_{_{{\it \Omega} 1}}}|\gg|{\eta_{_{{\it \Omega} 2}}}|$, the stabilization depends on the contribution of the vorticity convection term $\eta_{_{{\it \Omega} 1}}$. For a lower picket, the shock intensity and the ablation velocity are reduced, and the duration that satisfies $kV_{\rm a} t\ll 1$ becomes longer, so that the $\eta_{_{{\it \Omega} 1}}$ term plays a more active role. Meanwhile, for the RMI without mass ablation, the growth rate of RMI decreases relative to the Mach number of the incident shock.[30] Vorticity is considered as important quantity to determine the RMI growth rate in the linear stage. When the shock is weaker, the stored vorticity intensity is lower,[31] and then the RMI growth decreases. Hence, the $\eta_{\rm a} ^{\rm cl}$ term grows slowly due to the relatively weak shock. On the whole, the SP pulse leads to a considerable mitigation of the perturbation amplitude. The density contour diagrams of mode 64 are shown in Fig. 3, and they located at the beginning of shell acceleration and the moments of peak implosion velocity. All the black dash-dotted curves represent the interface between CH and DT ice. The green, magenta and white curves in Fig. 3(a) represent isolines of 0.5, 1.5 and 2.5 g/cm$^{3}$, respectively. The green and magenta curves in Fig. 3(b), represent isolines of 5.0 and 15.0 g/cm$^{3}$, respectively. At the two moments, the gap between the SP and HP cases widens out. The instability growth mitigations of the SP pulse emerge more significantly at the moment of peak implosion velocity. It is of note that the implosion and ignition process go on smoothly and achieve a yield-over-clean (YoC) of about 0.5 in the SP case, the HP's fusion reaction is truncated due to lack of confinement of the breakup shell. Evidence of the advantage of a lower foot radiation temperature was laid out by MacPhee et al.[32] A long toe ($\sim$100 eV) reduces perturbation of the fill tube embedded in the capsule. Further benefits of a lower toe can be expected.
cpl-36-2-025201-fig3.png
Fig. 3. Density contour diagrams of mode 64 in the HP and SP cases (a) at the beginning of acceleration phase and (b) at the time of peak implosive velocity.
In summary, the ARMI growth can be regarded as producing, relaxing and eradicating vortices. This can be significantly reduced with carefully tailored shocks when it is not sufficiently developed during the early linear stage. With our new SP pulse design, the ARMI growth is reduced considerably, meanwhile compression of the capsule shell is increased. Implosion performance is improved obviously in 2D simulations by the SP design due to the reduced seeds of the ARTI during the acceleration stage. Therefore, the SP pulse scheme enriches the hydrodynamic instability control strategy for the ignition capsule design. This deserves to be widely used in direct and indirect drive ICF implosion. References Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. IProgress towards ignition on the National Ignition FacilityMeasurements of the effects of the intensity pickets on laser imprinting for direct-drive, adiabat-shaping designs on OMEGAOnset of Hydrodynamic Mix in High-Velocity, Highly Compressed Inertial Confinement Fusion ImplosionsHot-spot mix in ignition-scale implosions on the NIFFuel gain exceeding unity in an inertially confined fusion implosionNumerical simulation of ablative Rayleigh–Taylor instabilitySuppression of Rayleigh-Taylor Instability in Z -Pinch Loads with Tailored Density ProfilesDirect-drive laser fusion: Status and prospectsImproved performance of direct-drive inertial confinement fusion target designs with adiabat shaping using an intensity picketTheory of laser-induced adiabat shaping in inertial fusion implosions: The decaying shockTheory of laser-induced adiabat shaping in inertial fusion implosions: The relaxation methodAnalisa Pengaruh Variasi Diameter Pipa Tekan PVC Pada Pompa Rotari Untuk Kecepatan Gaya Dorong AirA survey of pulse shape options for a revised plastic ablator ignition designDifferential ablator-fuel adiabat tuning in indirect-drive implosionsPerformance of indirectly driven capsule implosions on the National Ignition Facility using adiabat-shapingEarly stage of implosion in inertial confinement fusion: Shock timing and perturbation evolutionThe effects of early time laser drive on hydrodynamic instability growth in National Ignition Facility implosionsIndirect-drive ablative Richtmyer Meshkov node scalingA hybrid-drive nonisobaric-ignition scheme for inertial confinement fusionA scheme for reducing deceleration-phase Rayleigh–Taylor growth in inertial confinement fusion implosionsMain drive optimization of a high-foot pulse shape in inertial confinement fusion implosionsInertial fusion research in ChinaStabilization of ablative Rayleigh-Taylor instability due to change of the Atwood numberDestabilizing effect of density gradient on the Kelvin–Helmholtz instabilityPlastic ablator ignition capsule design for the National Ignition FacilityHigh-mode Rayleigh-Taylor growth in NIF ignition capsulesRichtmyer–Meshkov instability growth: experiment, simulation and theoryLinear theory of Richtmyer–Meshkov like flowsX-ray shadow imprint of hydrodynamic instabilities on the surface of inertial confinement fusion capsules by the fuel fill tube
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