Investigation on the scaling of magneto-Rayleigh–Taylor instability to the current rise time of Z-pinch plasmas

Understanding how the magneto-Rayleigh–Taylor instability (MRTI) scales to the current rise time is vital for Z-pinch dynamic hohlraum driven inertial confinement fusion. Wang et al discovered in prior theoretical work that the perturbation amplitude of MRTI before stagnation increases linearly with the current rise time when the implosion velocity of Z-pinch plasma is held constant. In the present work, three types of wire-array experiments with similar implosion dynamics and constant implosion velocity are performed on an 8 MA pulse power generator to investigate the scaling of MRTI to the rise time. It is successfully accomplished for the first time to obtain the similar wire-array Z-pinch implosions in which the current rise time is scaled up to three times on the generator by controlling the trigger time of its 24 modules. Both the experimental results, which include x-ray radiation pulses and x-ray images of imploding plasmas, and the related numerical analysis have shown that the MRTI before stagnation grows linearly with the rise time, as predicted by the theoretical model.


Introduction
Magneto-Rayleigh-Taylor instability (MRTI) always occurs at the outer surface of the Z-pinch plasma, which is accelerated by the azimuthal magnetic field induced by the axial current. Many applications of fast Z-pinches, such as inertial confinement fusion (ICF) [1][2][3][4][5], x-ray radiation sources [6,7] and neutron sources [8], are susceptible to the instabilities. In the particular case of z-pinch dynamic hohlraums (ZPDH) [2,4,9], the high temperature hohlraum radiation needed for driving ICF capsules, is strongly affected by the MRTIs. Hence it is very critical to control the instability for improving the radiation temperature. Besides stabilizing the instabilities by utilizing the axial magnetic field [10][11][12][13], the multi-shell load design [14][15][16], the dielectric coating [17] and other methods [18][19][20], it is also essential to prevent the instabilities from full development by optimizing load parameters and the electric current shape, such as the current rise time τ and the load radius r 0 , in advance.
It is known from linear theory that single mode MRTI is only related to the implosion distance for single-shell like implosions [21,22]. And many linear and weakly nonlinear theoretical models are developed to understand the MRTIs and possible stabilizing mechanisms [23][24][25]. The MRT instability, however, will develop into strongly nonlinear stage quickly in fast z-pinches [26]. Neither of the linear and the weakly nonlinear theories can predict the MRTI perturbation through the whole implosion process. Hussey et al developed a heuristic model for predicting the perturbation amplitude in z-pinch implosions [27]. And a theoretical model of nonlinear MRTI was also developed for gas-puff like implosions by coupling a 0D Z-pinch implosion model to a selfsimilar growth model of Rayleigh-Taylor instability (RTI) [28].
Numerical simulations [29][30][31][32][33][34] help understand the nonlinear MRTI better in z-pinch community for past decades. Researchers studied the MRT instability and their effects on the x-ray radiation by 2D and 3D radiation magnetohydrodynamic (MHD) simulations [33][34][35]. The simulations must consider the development of MRTI to reproduce the observed x-ray radiation in experiments. It is inferred from these simulations and the corresponding experiments that long rise times and large radius will lead to full development of MRTI. Recently, Wang et al studied the effects of the load radius and the current rise time on MRTIs by numerical simulation and theoretical analysis [36]. It is found that perturbation amplitude A of the imploding plasma before stagnation will increase linearly with the rise time τ (or the initial radius) if the implosion velocity v s is kept constant as well as the implosion constant Π, which is a dimensionless parameter determining the implosion dynamics and the coupling efficiency of Z-pinch loads with the pulse power generator [21,22]. Nevertheless, experimental investigation is also essential to prove the scaling relation for MRTIs in wire-array implosions because some assumptions were made in the theoretical models.
Recent experiments [17,[37][38][39] on Z investigated the MRTI and its control for the liner implosion which is used in magnetized liner inertial fusion [40][41][42]. The harmonic generation and inverse cascade of MRTIs was also observed directly in recent liner experiments on Z [43]. Usually, the MRTIs are less developed in this situation than that in ZPDH due to its small radius, the applied axial magnetic field, as well as the structure strength of materials [17,37,39]. Thin-foil cylindrical liner experiments on MAIZE facility clearly showed the process of bubble merging [44]. Both liner and gas-puff [45] experiments succeeded in reducing the growth of MRTI by the axial magnetic field. In some gas-puff experiments [46,47] the MRTIs was also mitigated by varying initial profile of mass density.
The growth of MRTIs could be successfully characterized by its detrimental effect on the x-ray radiation peak power and yield in Z-pinches for radiation source. Some wire-array experiments investigated the scaling of the current rise time (implosion time) and load radius on x-ray radiation yield [48][49][50][51]. It was found from these experiments that MRTIs developed more severe in the long-pulse implosion. However, the implosion velocity varied with the rise time and the load parameters in the experiments. In addition, the implosion constant Π, i.e. a characteristic parameter of implosion dynamics in some extent, also changed with either the load parameters or the current shape. The MRTIs in longer-pulse implosion from larger radius were also understood well in some gas-puff experiments [52,53]. Especially in [54,55] and references therein, short-pulse and long-pulse implosions of gaspuff pinches were compared on the same generator of ∼4 MA, Double Eagle, with the aim to achieve the same implosion velocity for the purpose of generating K-shell emission. Wirearray experiments with a shell-like implosion, other than a snowplow implosion in gas-puff Z-pinches, are still needed to be conducted under the same implosion constant and implosion velocity to investigate the scaling of MRTI to the current rise time.
In this paper, three types of wire-array experiments with the rise time of about 50 ns, 100 ns and 150 ns, are designed and carried out on an 8 MA pulse power generator, and the scaling of nonlinear MRTIs to the rise time is investigated. The present work is organized as follows. The experimental designs and arrangements are given in section 2. The experimental results are presented in section 3. Numerical analysis of experimental results and discussions about the development of MRTI are made in section 4. Final conclusions and remarks on future works are given in section 5.

Experimental design
To study the scaling of MRTI to the current rise time, three types of single wire-array experiments are designed carefully.
The key design ideas are as follows. First, the current rise time is scaled according to the theoretical scaling by adjusting the pulse shape of the generator. Secondly, the implosion velocity v s and the implosion constant Π must be kept constant. It is known that the implosion velocity is proportional to r 0 /τ at the same convergence ratio of Z-pinch plasmas for the same implosion constant. The load radius, therefore, must be scaled with the rise time in experiments. And the implosion constant is Π = µI 2 τ 2 /4πmr 0 2 , where I is the current peak and m is the load mass per unit length. Finally, the same current peak and load mass are utilized to keep Π constant for similar implosion dynamics.
The experiments are carried out on the 8 MA pulse power generator, which has 24 modules [56]. The current rise time (10%-90%) is usually about 60 ns for typical Z-pinch applications in synchronous discharge mode. And in this work, the current rise time is changed by changing the trigger time of laser triggered gas switches (LTGSs) and the gaps of water switches in our experiments. A full circuit model [57], coupling with a 0D thin-shell implosion model, is used to design the pulse shape of the generator and the load parameters based on above key points. The current rise time (10%-90%) is changed from 47 ± 5 ns, 98 ± 5 ns to 148 ± 5 ns for three type experiments respectively. The radius of wire array r 0 is scaled from 7 mm (R7), 14 mm (R14) to 21 mm (R21) corresponding to the rise time. The load mass m l is 1 mg cm −1 . The height of wire array h is 1.0 cm. The 0D implosions velocity is about 40 cm µs −1 when the same convergence ratio of the load is 10 for all types. Similar implosion dynamics are obtained as shown in figure 1. In order to achieve shell-like implosions, high wire number N wire is utilized for wire-arrays. We also choose the same wire space ∆r ∼ 0.33 mm for obtaining similar ablation dynamics of wire-arrays by adjusting wire diameter d and wire number. And tungsten wires are utilized in our experiments. Table 1 lists the designed parameters of the experiments. The trigger times of LTGS for each type are listed in table 2. In addition, the discharge voltage V m is also adjusted to keep the peak value of load current constant. Note that the module numbers listed in table 2 is not sequential around the azimuth of the machine. In fact, four modules with same trigger time are axis-symmetrical around the azimuth for considering the symmetry of discharge. For example, modules 1-2 are opposite to modules 3-4 across the insulation stack. B-dot probes placed symmetrically outside the wire-array measure the electric current. Diagnostics instruments of the radial x-ray radiation include x-ray diodes and bolometers. Xrays emitted from the imploding plasma are recorded using multi-frame, time-resolved (2 ns gate) pinhole imaging system and time-integrated pinhole imaging system. These x-ray selfemission images can present the characteristics of instability perturbation. Streak camera imaging technique is utilized to record the visual light from the imploding plasma, diagnosing the implosion trace. Figure 2 presents the schematic diagram of the load configuration and the experimental arrangements.

Wire-array implosion
Three shots are carried out for each type experiment. Figure 3 presents the load currents and x-ray radiation pulses. The 0 time is defined at the time of the radial radiation peak. It is found that the current shapes as well as the x-ray radiation pulses are consistent for each type. The measured current rise time are 52 ± 1.5 ns, 100 ± 2.5 ns and 159 ± 2.5 ns for the R7, R14 and R21 respectively. And the load peak current are 7.0 ± 0.1 MA, 7.1 ± 0.2 MA and 7.5 ± 0.4 MA. Table 3 gives the measured implosion parameters for all wire-array implosions. Figure 4 shows a radial streak image of visual light from R21 imploding plasma. The implosion trace of plasma is recorded by the outer edge of the streak image as the red lines shown in figure 4. The outer edge is defined where the intensity of the light is 50% of the maximum in each column. The implosion trajectory can be roughly divided into several stages, and the slope of the trajectory varies obviously between different stages. The maximum slope stage occurs before the final deceleration stage, and we take the average velocity of this maximum slope stage as the maximum implosion velocity. The averaged velocity is about 26.3 cm µs −1 for type R7, 24.6 cm µs −1 for R14 and 23.8 cm µs −1 for R21, showing nearly identical implosion as expected. The pinched size of plasma is given by the minimum radius of the implosion trace. The implosion velocity and pinched size of plasma are listed in table 3 for each shot.
It is well known that Z-pinch radiation source is susceptible to MRT perturbation, which can be indirectly used to demonstrate the variations of the perturbation. Figure 5 presents the radial x-ray radiation peak and full width at half-maximum (FWHM) as the function of the rise time. It is seen that the radiation power decreases monotonically with the rise time. On the contrary, the FWHM of the x-ray pulse almost increases linearly. It is known that the MRTIs will reduce the peak power of the x-ray and broaden the pulse width [34]. The latter is approximately proportional to w/v s [27], where w is the width of the disturbed imploding plasma and v s is the imploding velocity. In addition, the implosion velocities from streak images are approximately identical. It is thus possible that the perturbation amplitude increases with the rise time.
represents that the trigger time is the same as that of the synchronous discharge mode. '−10' represents that the trigger time is 10 ns earlier than that of the synchronous discharge mode. '+ 10' represents that the trigger time is 10 ns later than that of the synchronous discharge mode. Other data represent similarly.  Although the x-ray yield decreases with the rise time, its change is smaller comparing to the power peak and FWHM of the radiation as the figure 5(c) shows. The contribution to x-ray yield is possibly from the implosion kinetic and the PdV work by magnetic pressure. The former may be identical because the Π parameter is constant for three type experiments. The latter, however, is affected by the instability very much.   Figure 6 presents the time-resolved pinhole images of x-ray self-emissions for different type experiments. It is observed from the images that axis-symmetrical perturbations occur before the radiation peak, and the small-scale perturbations gradually develop into the large scale for each shot, i.e. the inverse cascade process. The perturbation structure cannot be recognized very clearly at early time of the implosion for R7 because the dominant wavelength of perturbation is smaller than the diagnostic resolution. It is observed that the diameter of x-rays from imploding plasmas before stagnation is ∼2.  figure 7, where the time-integrated images of x-ray self-emission and their axially averaged profiles are presented. Clearly, these profiles show that the axially averaged pinch diameter increases with the rise time from R7 to R21. It is also shown in figure 6 the finger-structure of the xray images also qualitatively increases with the rise time. But it can only present the perturbation amplitude of the plasma to some extent because the x-ray image presents the side view of the 3D imploding plasma. Numerical analysis is also needed to help understand the development of perturbations.

Development of implosion instability
A 2D MHD model in r-z section is used to simulate the wirearray implosion. The model can reproduce the perturbation well before the stagnation of aluminum liners in previous experiments [58]. In our simulation, the assumption of shelllike implosion was taken as the [33,50] done. It is found that larger than 90% of the plasma remains in the original region of ∼1.0 mm width prior to the acceleration by considering the ablation dynamics of wire-arrays for three types using a 2D MHD model in r−θ plane. So, we take a plasma shell of 1.0 mm width and 10 000 K as the initial conditions in simulations. The experimental currents of shots 679 (R7), 672 (R14), 675 (R21) are utilized to drive plasma implosion in the calculations. The initial radius of 7 mm, 14 mm and 21 mm are set for R7, R14 and R21 type respectively. And 0.1% random perturbation of mass density seeds the MRTI for type R14 and  R21. 1.0% random perturbation is utilized for type R7. The grid resolution is 25 µm in our calculations. Figure 8 presents the mass density contours of imploding plasmas before stagnation from 2D simulation for type R14 implosion. It is observed that the perturbation will grow gradually from the small scale to the large scale by bubble merging, qualitatively being consistent with the observation from x-ray images. Corresponding to figure 8, the simulated selfemission images of the imploding plasma from 2D simulation are also presented in figure 9 by solving steady-state radiative transport equations of 120 frequencies. We also compare the simulated images with the x-ray self-emission images from experiments in figure 9. The general features of perturbation from the simulated images are consistent with the observed from x-ray images very well. It is thought that our 2D simulation can reproduce the major features of nonlinear MRTIs for wire-array implosions.
Then we make a Fast Fourier Transform (FFT) analysis of the radially integrated mass as a function of axial position to understand the development of MRTIs as in our previous work [36]. The method is also same to that in [16,51]. Corresponding to figures 8 and 10 gives the perturbation amplitude and the dominant wavelength as a function of implosion distance L Imp . The dominant wavelength is defined as the wavelength of the mode with peak amplitude. Note that the wavelength as a function of time appears to be 'step functions' because of the definition of the dominant wavelength and the discontinuous spectrum from FFT analysis. In reality, the dominant wavelength will vary smoothly with time. The implosion distance is defined as L Imp = r 0 − r c, where is the mass-averaged plasma radius. It is seen that the perturbation grows exponentially with L imp at the beginning of the implosion when L imp < 0.26 cm, as marked with a in figure 10, because the initial amplitude A 0 is only 2.8 µm, which is far smaller than the initial dominant wavelength λ d of 160 µm.
In this stage, the instability is approximately in linear growth. Then the perturbation amplitude varies approximately linearly with L Imp after the amplitude is increased larger than the dominant wavelength when L imp > 0.26 cm. It is known from [36] that the self-similar growth of MRTI has where A s is the perturbation amplitude, A nl is the nonlinear saturated amplitude as marked with a in figure 10 and L nl is the corresponding implosion distance of A nl . α is a dimensionless constant related to the self-similar model of RT. It is therefore supposed that the self-similar growth is a good approximation in this stage for our three type experiments. It is should be noted that the dimensionless constant α is larger than that of classical nonlinear RT, where the RT occurs at the interface of two infinite thickness material, because the MRTI development is in thin-shell regime for most implosion time here. In the thin-shell regime, the dominant wavelength of MRTI λ d are larger than the width of the Zpinch plasma ∆r, as in the section L imp > 0.5 in figure 10. In the nonlinear growth regime, the RT in thin-shell implosion grows faster than the so-called classical RT (λ d < ∆r), as in the section 0.26 <L imp < 0.5 in figure 10. A possible reason is that there is no drag force to suppress the instability in nonlinear regime for thin-shell implosions. Therefore, α here is larger than the classical nonlinear RT in most studies.
In addition, the maximum implosion velocity v c , which is derived from r c , is about 23.1 cm µs −1 for type R7 in the calculations. And the maximum implosion velocities are 24.9 cm µs −1 and 25.4 for types R14 and R21. These implosion velocities are consistent with that from the streak images. In our calculations, it is interesting that the implosion trace of r c with considering MRT instabilities is consistent with that without instability.
In summary, it is found from the above simulation and our previous work that the width of the nonlinear Rayleigh-Taylor mixing zone scales as the free-fall distance, A ∝ 10 3 αL imp ∝ 5 3 αgt 2 , where g is the acceleration, t is the acceleration time. In our experiments, the implosion constant, load mass, and peak current are maintained constant, therefore the initial radius r 0 is proportional to τ , the acceleration scales as g ∝ r 0 /τ 2 ∝ 1/τ , and the acceleration time t scales as τ , implying that A is proportional to τ .

Scaling of MRTI to the rise time
Then the scaling of MRTI to the rise time is studied by comparing the x-ray images before stagnation of the imploding plasma for R7, R14 and R21 with the simulated self-emission images in figure 11. And the following features can be attained from these figures. First, the simulated images show similar perturbation features and the same size with the x-ray images for R14 and R21. Secondly, the dominant wavelength of the perturbation from the simulated images is similar to the observed from x-ray images for R7. The size of the simulated images, however, is far smaller than that from the x-ray images in experiments. One possible reason is that the experimental implosion velocity is larger than the calculated. The perturbation amplitudes are approximate when it is normalized to the same velocity as shown in the following. In addition, it is speculated that 3D effects may be important in the formation of wire-array plasmas, which should be studied carefully in future. It is also found that the size of the simulated self-emission images before stagnation (at the beginning of x-ray radiation rising) are approximately equal to the peak-valley (PV) amplitude of the perturbed plasma by comparison between figure 8(f ) and −10.5 ns in figure 10. Thus, the diameter of x-ray self-emission before stagnation can be utilized to evaluate the PV amplitude of the perturbation in experiments. The dominant wavelength is estimated by counting the finger structure of the x-ray self-emission images. Figure 12 gives the PV amplitude and wavelengths obtained from x-ray images as function of the rise time as well as those from the 2D simulations. It is found both the amplitude and dominant wavelength increase linearly with the rise time (initial radius). It is noted For comparison with the theoretical results, the FFT amplitudes from 2D simulation are also presented in figure 12. The prediction from the theoretical scaling, A ∼ 5 9 αv s τ with α = 0.11, agrees with the simulations, where α is a constant coefficient related to the self-similar model of RT. It should be noted that the rise time of 0%-100% current, which is about 1.7 times of that of 10%-90% current, is used in the theoretical model. And the implosion velocity of ∼24.0 cm µs −1 from the streak images is utilized here. It should also be noted that the PV amplitude is larger than the amplitude obtained by Fourier analysis in 2D simulation.
In our calculations, the effect of initial perturbation on the MRTIs is also studied by changing it from 0.001% to 1%. It is found that the initial perturbation has trivial effect on the amplitude of the instability before stagnation, of which the change is under ∼10%, if the initial random perturbation is larger than 0.1%. The reason is that the instability develops quickly into strong nonlinear regime (or self-similar growth) for larger initial perturbation. As a result, the implosion distance in linear regime of the instability is small. And the effects of initial perturbation can be approximately ignored when the MRTI is in self-similar growth [59]. It probably accounts for the similar property of MRTI before stagnation for three types, i.e. the amplitude increases linearly with both the rise time and initial radius, even though the initial perturbation is uncertain in experiments.

Conclusion
Three types of single wire-array experiments are designed and carried out on the 8 MA pulse power generator to study the scaling of nonlinear MRTI to the current rise time. The experiments present the following results. First, the current rise time for wire-array implosions is scaled from 52 ns, 100 ns to 159 ns (up to three times) successfully on the generator by carefully adjusting the trigger time of LTGS of 24 modules, with the current peak around 7.0 MA by controlling the charging voltage of the generator. Secondly, streak images of visual light from imploding plasma shows nearly identical implosion velocities for three type experiments. Thirdly, the power peak of x-ray pulses decreases monotonically with the rise time. On the contrary, the FWHM of x-ray pulse increases with the rise time. Finally, the x-ray self-emission images show the inverse cascade process of the perturbation in wire-array implosions. And it is also observed from the x-ray images that the MRTI perturbation before stagnation qualitatively increases with the current rise time (the initial radius).
Numerical simulations are performed in this paper for analyzing experimental results. The simulated self-emission images of imploding plasmas from 2D simulations show similar perturbation structures and the same size with the x-ray self-emission, indicating our simulations can reproduce the major features of MRTIs for wire-array implosions. It is found from x-ray images and the simulated images that the PV amplitude of MRTIs before stagnation indeed increases linearly with the rise time (and load radius) as predicted by the theoretical model. In addition, it should be noted that the load radius is scaled with the rise time simultaneously in experiments while the implosion velocity is invariant, so the MRTIs before stagnation also show similar property to some extent. In a word, the experimental results prove the scaling of MRTIs to the rise time A ∼ 5 9 αv s τ with α = 0.11 if the implosion constant are kept constant.
The scaling of MRTI perturbation will be true for large implosion distance although the initial seeding of the instability will change with load parameters and current shapes, because the effects of initial perturbation can be approximately ignored when the MRTI is in self-similar growth. And the instability will develop into the strong nonlinear regime and be in self-similar growth quickly for long enough implosion distance (or radius). Usually, the radius of wire-array are 10 ∼ 40 mm for most Z-pinch applications, especially the ZPDH. However, it should be noted the initial seeding could have more important effect on the MRTI for small radius loads due to less implosion distance in self-similar growth. And in this case the scaling of MRTI should be modified. The thin-shell assumption, used in this paper, may be questionable for wire-array implosion with low wire number, for which the precursor plasma is not trivial. The implosion is more snowplow-like, and the MRTI development may be very different. Therefore, our results are also not accurate enough in this case. In addition, the mass of the precursor plasma and the delaying of the implosion will decrease for higher wire number [60] so that the shell assumption is an appropriate approximation with respect to MRTI, but the effects of wire ablation dynamic also should be considered carefully with 3D model in future.
Long rise time larger than 100 ns may be applied on the 50 MA pulse power generator which can be used for fusion ignition research [61], where the MRTIs will be very severe. Therefore, a nested or multi-nested wire-array may be utilized to suppress the instability, and the scaling of MRTIs in this case should be studied in future.