Magnetized liner inertial fusion experiments on the Z accelerator suffer from magneto-Rayleigh–Taylor instabilities (MRTI) that compromise integrity of the imploding cylindrical liner, limiting achievable fusion fuel conditions and ultimately reducing magneto-inertial fusion target performance. Dynamic screw pinches (DSP) provide a method to reduce MRTI in-flight via application of magnetic field line tension to the imploding liner outer surface. In contrast with z-pinches that drive implosions with an azimuthal magnetic field, dynamic screw pinches enforce an additional axial drive magnetic field component, making the overall drive magnetic field helical. As the liner implodes, cumulative MRTI development is reduced by dynamically shifting the orientation of the fastest growing instability modes. Three-dimensional magnetohydrodynamic simulations show that the DSP mechanism effectively stabilizes initially solid cylindrical liner implosions driven by Z-scale current pulses, indicating that MRTI mitigation increases with the ratio of axial to azimuthal drive magnetic field components (i.e., the drive field ratio). We also performed a spectral analysis of the simulated imploding density distributions, extracting wavelength and pitch angle of the simulated MRTI structures to study their dynamics during the implosion. Simulations of liners initially perturbed with drive-field-aligned sinusoidal structures indicate that MRTI mitigation in DSP implosions decreases with perturbation wavelength, once again suggestive of magnetic field line tension effects.

Improving implosion stability in magnetized liner inertial fusion (MagLIF)1,2 is predicted to improve fusion fuel compression and confinement. Magneto-Rayleigh–Taylor instabilities (MRTI) limit the compressive work that the imploding liner can do on the fuel, reducing achievable fuel temperatures and pressures and therefore thermonuclear yields. Additionally, instability of the imploding liner results in degradation of fuel confinement,3 reducing the time during which thermonuclear reactions can take place prior to disassembly.

Dynamic screw pinches (DSP)4–8 are designed to stabilize magnetically driven cylindrical implosions by employing a dynamic helical magnetic drive field. As the target implodes, the polarization of the magnetic field driving the implosion rotates, resulting in the dynamic reorientation of magnetic field line tension over the outer surface of the imploding liner, continuously shifting the fastest growing instability modes and reducing cumulative MRTI growth. Linear theory4 indicates how the DSP mechanism is predicted to mitigate MRTI and experiments studying DSP-driven thin foil implosions6 demonstrate stabilization compared to thin foil z-pinches. The numerical study of liner implosions more directly relevant to MagLIF experiments on the Z accelerator9,10 provides assessment of the DSP mechanism in the presence of physical effects not captured in the linear theory such as non-linear instability growth and mode merging,11,12 magnetic field diffusion into finite electrical conductivity materials, and the effects of shocks transmitting through liner material and material interfaces.

In this manuscript, we utilize three-dimensional magnetohydrodynamic (3D MHD) simulations to explore how effectively the DSP mechanism stabilizes magnetically driven cylindrical implosions. In Sec. II, we compare the stability of simulated implosions on the Z accelerator with respect to the initial ratio of the axial and azimuthal drive field components evaluated at the initial outer liner surface; a larger ratio of axial to azimuthal drive magnetic field components is predicted to result in more effective stabilization. We then compare the spectral characteristics of the imploding liner perturbations, focusing particularly on the evolution of the dominant wavelength and average pitch angle of instability structures and how these correspond to the effectiveness of the DSP mechanism. Section III briefly describes results from simulations exploring a special case: liner implosions with initial sinusoidal perturbations on the outer liner surface. By prescribing a dominant wavelength via a single mode sinusoidal perturbation, the dependence of the DSP mechanism on wavenumber can be explored without relying on instability structures growing from a white noise random MRTI seed with maximum wavenumber limited by the simulation mesh resolution.

To assess the stability of magnetically driven cylindrical implosions, we primarily used three-dimensional (3D) ALEGRA.13,14 ALEGRA is an arbitrary Lagrangian–Eulerian (ALE) hydrodynamics code that utilizes resistive MHD with thermal conduction. Simulations employed SESAME equation of state (EOS) and coupled thermal/electrical conductivity tables. The “baseline” simulation geometry used for this study (Fig. 1) was a 1-mm tall simulation region with periodic top and bottom boundary conditions and a base mesh resolution of 20  μm. Simulations employed a “radial trisection” mesh, with radial mesh cells (i.e., mesh cell edges oriented along the radial, azimuthal, and axial directions) converted to Cartesian geometry near the axis. Note that radial mesh cells correspond to increasing mesh cell resolution in the azimuthal direction as radius decreases (until the conversion to Cartesian geometry). The liner was modeled as beryllium with an initial outer radius R out 0 = 2.616 mm and an initial inner radius R in 0 = 2.325 mm, corresponding to an aspect ratio (AR) of 9, where AR = R in 0 / R out 0 R in 0. Simulations were driven with a prescribed 20-MA peak current, 100 ns rise time sine squared pulse: I t = 20 MA * sin 2 π t / τ, where τ = 2 t r = 200 ns and t r is the rise time. A “white noise” surface roughness was applied to the initial outer surface of the liner to seed MRTI growth by randomly perturbing mesh nodes conformal to R out 0 radially over a ±1.25  μm range (uniform distribution).

FIG. 1.

(Top) Cutaway view of initial ALEGRA simulation geometry with materials and dimensions labeled and the initial random mesh node surface perturbation to seed MRTI on the outer liner surface indicated by the dotted white. (Bottom) Simulation results from a later time when the liner has imploded toward the axis and MRTI has developed.

FIG. 1.

(Top) Cutaway view of initial ALEGRA simulation geometry with materials and dimensions labeled and the initial random mesh node surface perturbation to seed MRTI on the outer liner surface indicated by the dotted white. (Bottom) Simulation results from a later time when the liner has imploded toward the axis and MRTI has developed.

Close modal

While this MRTI seeding mechanism is consistent across the main ALEGRA simulations presented in this study (we explored several variations to a limited degree), we note that it is not rigorously benchmarked against prior implosion data from unmagnetized11 and magnetized15,16 initially solid liner implosion experiments on Z. This is partially because the mechanisms that give rise to the observed MRTI in prior experiments are not fully understood in terms of providing a self-consistent seeding protocol to reproduce the high levels of azimuthal correlation observed in unmagnetized experiments11 or the helical structures15–17 observed in magnetized implosions. We note that several hypotheses have been provided to explain the discrepancies such as electrothermal instabilities (ETI) exacerbating the effective surface roughness18,19 and local plasma induced flux compression20–22 in the target region giving rise to helical structures. Future work will focus on studying the DSP mechanism in the context of these MRTI seeding hypotheses, but we restrict our study here to a seeding protocol implemented in a resistive MHD numerical framework kept consistent across the main simulations presented in this study.

We focus our study on “empty” beryllium liners (in contrast with premagnetized, preheated fusion-fuel-filled liners associated with “integrated” MagLIF experiments1,2) and restrict our assessment of MRTI up to levels of convergence lower than those associated with peak compression in MagLIF. Specifically, we report on MRTI development up until the liner has imploded to CRin < 10 in this study vs CRin > 30 associated with peak compression in MagLIF experiments, where CRin is the convergence ratio of the inner liner surface, R in 0 / R in t. This is primarily due to the emphasis of this study on understanding acceleration driven MRTI development at the outer liner surface instead of instability feedthrough to the inner surface and its effects on fusion fuel conditions. Such development of inner surface perturbations is expected to be much less in our simulated system compared to MagLIF experiments due to the lower convergences studied and the lack of hydrodynamic pressure from the preheated/premagnetized fusion fuel that would induce deceleration of the inner liner surface at these higher convergences. The assessment of the development of inner surface perturbations resulting from feedthrough of MRTI from the outer liner surface to the inner liner surface during a fusion-fuel-filled liner implosion and the effects on fuel compression and confinement will be the subject of future study.

The initial drive magnetic field ratio evaluated at the liner outer surface, Ξ o = B z / B ϕ, was varied to explore how increasing Ξ o corresponds to improved stabilization. We study the Ξ o = 0 , 0.25 , 0.5 , 1.0 cases in this work. The drive magnetic field components vary with time according to
B z t = c 1 c 2 Ξ o μ o I t 2 π R out 0 ,
(1)
B ϕ t = c 1 c 3 μ o I t 2 π R out t ,
(2)
Ξ t = B z t B ϕ t = Ξ o R out t R out 0 .
(3)
In Eqs. (1)–(3), I t is the drive current, R out t is the outer radius of the liner, which decreases during implosion, the scale factors c 1 , c 2 , and c 3 are used to modify magnetic drive boundary conditions, specifically to enforce the desired initial drive magnetic field component proportionality and simultaneously maintain the implosion time for DSP simulations with different Ξ o (see Table I), and μ o is the permeability of free space. The implosion time was maintained for all simulations via application of the scale factors c 1 , c 2 , and c 3 to approximately preserve the acceleration time history for each simulated implosion and therefore to best compare MRTI growth for similarly accelerated liner implosion systems (see Fig. 2). For a z-pinch implosion, c 2 = 0, and therefore, B z t = 0: the liner is driven purely by the azimuthal magnetic field generated by axial current flowing in the liner.
TABLE I.

Scale factors applied to the drive magnetic field components. c 1 functions as a correction factor implemented to maintain implosion time between simulations with different initial drive magnetic field ratio Ξ o. c 2 and c 3 enforce the proportionality of the drive magnetic field components in accordance with Ξ o.

Ξ o = 0 Ξ o = 0.25 Ξ o = 0.5 Ξ o = 1.0
c 1  1.0  1.0  1.025  1.075 
c 2  0.243  0.447  0.707 
c 3  1.0  0.970  0.894  0.707 
Ξ o = 0 Ξ o = 0.25 Ξ o = 0.5 Ξ o = 1.0
c 1  1.0  1.0  1.025  1.075 
c 2  0.243  0.447  0.707 
c 3  1.0  0.970  0.894  0.707 
FIG. 2.

Plots of inner liner wall trajectory R in t for 1D radial mesh ALEGRA simulations of a z-pinch implosion ( Ξ o = 0) compared to DSP implosions with different initial drive field ratios ( Ξ o = 0.25 , 0.5 , 1.0). The inner wall trajectory is tracked by a Lagrangian tracer particle buried 2 mesh cells radially outward from the inner liner surface. These DSP simulations were driven with multiplication factors [ c 1 , c 2 , c 3, see Eqs. (1) and (2) and Table I] applied to the drive field components to approximately match implosion time across all simulations. The top plot shows the overlay of all four simulation trajectories from the time the inner liner surface initially begins to move to when the liner material impacts the inner radial mesh boundary (at r = 230  μm). The bottom plot zooms in to the time near impact (indicated by the green box in the left plot), showing a <1 ns spread in the trajectories.

FIG. 2.

Plots of inner liner wall trajectory R in t for 1D radial mesh ALEGRA simulations of a z-pinch implosion ( Ξ o = 0) compared to DSP implosions with different initial drive field ratios ( Ξ o = 0.25 , 0.5 , 1.0). The inner wall trajectory is tracked by a Lagrangian tracer particle buried 2 mesh cells radially outward from the inner liner surface. These DSP simulations were driven with multiplication factors [ c 1 , c 2 , c 3, see Eqs. (1) and (2) and Table I] applied to the drive field components to approximately match implosion time across all simulations. The top plot shows the overlay of all four simulation trajectories from the time the inner liner surface initially begins to move to when the liner material impacts the inner radial mesh boundary (at r = 230  μm). The bottom plot zooms in to the time near impact (indicated by the green box in the left plot), showing a <1 ns spread in the trajectories.

Close modal
For DSP implosions, we follow Ref. 4 and define the drive magnetic field polarization angle θ B t as the angle that the drive magnetic field lines make with the horizontal at the outer liner radius R out t,
θ B t = tan 1 Ξ t .
(4)
As the liner implodes, θ B t shifts from helical toward the horizontal (i.e., θ B t 0 as the liner implodes).
We quantitatively assess cumulative MRTI growth by computing the standard deviation of the areal density σ ρ r ,
σ ρ r = 1 N k = 1 N | ρ r k ρ r | 2 .
(5)
Here, ρ r is the average areal density value and N is the number of points in the 2D ρ r maps. For each simulation, we calculate σ ρ r as a function of the normalized distance moved by the inner liner surface Δ R(Figs. 3–5). Here, Δ R = 1 R in t R in ( 0 ) = 1 1 CR in. The inner liner radius was assigned at each simulation time step using an 1800 kg/m3 density contour. A 2D map of the areal density ρ r of the imploding liner was produced by radially integrating the mass density at each ( θ, z) coordinate in the simulation, where θ is the azimuthal coordinate and z is the axial coordinate. σ ( ρ r ) was calculated for each 2D ρ r map at select simulation times.
FIG. 3.

(Top) Example areal density ρ r maps extracted from a 3D ALEGRA simulation of a magnetically driven implosion with Ξ o = 0 (i.e., a z-pinch). (bottom) Example ρ r maps extracted from a 3D ALEGRA simulation of a magnetically driven implosion with Ξ o = 0.5. For all ρ r maps shown, the average areal density ρ r has been subtracted, so displayed is the areal density perturbed from the mean value. Time proceeds from top to bottom for both sets of plots, and normalized distance moved of the inner liner surface Δ R is noted next to each image. Vertical axes are in mm, and colorbar units are in units of kg/m2.

FIG. 3.

(Top) Example areal density ρ r maps extracted from a 3D ALEGRA simulation of a magnetically driven implosion with Ξ o = 0 (i.e., a z-pinch). (bottom) Example ρ r maps extracted from a 3D ALEGRA simulation of a magnetically driven implosion with Ξ o = 0.5. For all ρ r maps shown, the average areal density ρ r has been subtracted, so displayed is the areal density perturbed from the mean value. Time proceeds from top to bottom for both sets of plots, and normalized distance moved of the inner liner surface Δ R is noted next to each image. Vertical axes are in mm, and colorbar units are in units of kg/m2.

Close modal
FIG. 4.

(Top) ρ r maps from a 3D ALEGRA simulation of a Ξ o = 0 implosion and (bottom) from a 3D ALEGRA simulation of a Ξ o = 0.5 implosion identical to Fig. 3 except for the horizontal axis is in units of length along the circumference (instead of units of angle). The azimuthal angular extent for each ρ r map is converted to distance along the circumference C t by using the liner inner radius R in t at each time step via C t = 2 π R in t.

FIG. 4.

(Top) ρ r maps from a 3D ALEGRA simulation of a Ξ o = 0 implosion and (bottom) from a 3D ALEGRA simulation of a Ξ o = 0.5 implosion identical to Fig. 3 except for the horizontal axis is in units of length along the circumference (instead of units of angle). The azimuthal angular extent for each ρ r map is converted to distance along the circumference C t by using the liner inner radius R in t at each time step via C t = 2 π R in t.

Close modal
FIG. 5.

(Top) Plot of σ ρ r for 3D ALEGRA and 3D KRAKEN simulations of implosions with different drive field ratios, Ξ o = {0, 0.25, 0.5, 1.0}. Uncertainty bars were assigned based on the range of σ ρ r observed at the point of highest convergence for an ensemble of simulations with identical Ξ o and initial random surface roughness magnitude (for ALEGRA) or random temperature perturbation seed magnitude (for KRAKEN), but different random number seeds.

FIG. 5.

(Top) Plot of σ ρ r for 3D ALEGRA and 3D KRAKEN simulations of implosions with different drive field ratios, Ξ o = {0, 0.25, 0.5, 1.0}. Uncertainty bars were assigned based on the range of σ ρ r observed at the point of highest convergence for an ensemble of simulations with identical Ξ o and initial random surface roughness magnitude (for ALEGRA) or random temperature perturbation seed magnitude (for KRAKEN), but different random number seeds.

Close modal
In qualitative agreement with the linear theory, higher Ξ o results in reduced cumulative MRTI development (Fig. 5). To compare MRTI mitigation for DSP implosion simulations compared to the z-pinch case, we define G σ as the ratio of the cumulative MRTI growth inferred from σ ( ρ r ) for the z-pinch case ( Ξ o = 0) to that of a given DSP case denoted by Ξ o = x,
G σ , x = σ ρ r Ξ o = 0 σ ρ r Ξ o = x .
(6)
G σ can be compared at different Δ R to assess MRTI mitigation due to the DSP mechanism as a function of liner convergence. We focus on comparing G σ at the highest liner convergences captured in simulations to assess the difference in cumulative MRTI growth (Table II).
TABLE II.

Comparison of cumulative MRTI reduction G σ Eq. (6) computed based on the σ ( ρ r ) analysis Eq. (5) of 3D ALEGRA simulations and 3D KRAKEN simulations at Δ R = 0.8.

G σ , 0.25 G σ , 0.5 G σ , 1.0
ALEGRA  3.0  6.0  16.0 
KRAKEN  2.3  4.2  7.4 
G σ , 0.25 G σ , 0.5 G σ , 1.0
ALEGRA  3.0  6.0  16.0 
KRAKEN  2.3  4.2  7.4 

Simulated DSP implosions with Ξ o = 0.25, 0.5, and 1.0 demonstrate 3X, 6X, and 16X lower cumulative MRTI, respectively, once the liner has imploded to Δ R = 0.8 according to the σ ( ρ r ) analysis (Fig. 5, Table II). While the 16X reduction according to σ ( ρ r) analysis from Ξ o = 1 simulations does not match the 170X theoretical prediction4 for a λ = 200  μm instability mode, we emphasize that the theory is focused on linear mode amplitude for a single wavelength mode without accounting for effects like nonlinear instability saturation, mode coupling, material compression, and magnetic diffusion. The theoretical predictions4 for cumulative MRTI reduction of λ = 400 μm and λ = 1000 μm modes (based on the assessment of the reduction in MRTI amplitude growth), 18X and 3X, respectively, can be more closely compared to σ ( ρ r) analysis from simulation results in this study, potentially highlighting how the dynamic multi-mode instability spectrum in simulations influences the DSP mechanism. However, spectral analysis described in the next subsection indicates that the dominant instability wavelength reached in DSP implosion simulations is in the 80–200  μm range, so comparison with theoretical predictions of single mode MRTI reduction for λ > 200 μm remains dubious. The lack of quantitative agreement of cumulative MRTI reduction from theoretical predictions with simulation predictions does not diminish the fact that simulations agree with the qualitative prediction from the theory: the effectiveness of the DSP mechanism to reduce MRTI growth increases as the initial drive field ration Ξ o is greater.

To assess the robustness of the trends observed in the “baseline” simulations, we completed >80 additional 3D ALEGRA simulations focused on varying initial conditions and numerical modeling choices. Initial conditions and physics models we varied included the mesh resolution (12.5, 16, 20  μm), the surface roughness perturbation random seed, the uniform distribution surface roughness perturbation magnitude (±0.625, ±1.25, ±2.5  μm), the simulation region height (1 mm vs 1.5 mm), the vacuum treatment (i.e., the density, electrical conductivity, and Joule heating “floors” that serve to turn off physical mechanisms in mesh cells when density falls below a specified value), the type of mesh (i.e., Cartesian mesh vs radial trisection mesh), the type of MRTI seed (i.e., random temperature perturbation vs random density perturbation vs surface roughness perturbation), magnetic field effects on thermal conduction (i.e., anisotropic vs isotropic thermal conductivity), radiation transport (i.e., none vs gray diffusion), EOS tables, strength/yield models, and two-temperature physics. Throughout all the variations studied, the cumulative MRTI growth still decreased with Ξ o as we observed in the baseline simulations and qualitatively reproduce the trends observed in baseline simulations. Future exploration will focus on determining which simulation settings, MRTI seeding mechanism, mesh type, etc., best reproduce experimental data from Z implosions.

We also executed simulations including a background axial magnetic field similar in magnitude to that used in historical MagLIF experiments ( B z = 15 T) to compare to the unmagnetized baseline simulations ( B z = 0) of this study. Such background magnetic fields have been shown to influence implosion dynamics15,16,20–22 in z-pinch implosions. However, in the resistive MHD numerical system, we study here and without introduction of conductive plasmas or higher order transport terms (e.g., Hall), we observe no significant differences in implosion dynamics and MRTI development in magnetized simulations compared to unmagnetized simulations. Assessment of the DSP mechanism using extended MHD and/or particle-in-cell numerical tools should be the subject of future study, especially given the experimental evidence for the effects of background axial magnetic fields in solid liner z-pinch implosions.

To further explore how the DSP mechanism stabilizes implosions, we simulated the same target geometry driven with the same current pulse using KRAKEN, a 3D Eulerian resistive MHD code that implements the same system of electromagnetic equations from GORGON23 with finite volume hydrodynamics. KRAKEN's numerical algorithms are fundamentally different from ALEGRA's, providing an opportunity to test whether the trends extracted from ALEGRA DSP simulations are independent of the choice of simulation tool. KRAKEN simulations employed 20  μm Cartesian mesh cells and seeded MRTI via an initial ±1% random temperature perturbation applied to all beryllium cells in the simulation. In addition, KRAKEN's Cartesian mesh results in “stair stepping” of the mesh cell boundaries near the cylindrical outer liner initial surface, which imparts an additional, artificial, or numerical surface roughness component absent in ALEGRA simulations. KRAKEN simulations did not employ radiation transport or thermal conduction. The KRAKEN simulation region was 12 mm tall (without periodicity assigned to the top and bottom boundaries). As in ALEGRA simulations, Ξ o = 0 , 0.25 , 0.5 , 1.0 cases were studied, again with scale factors applied to the drive magnetic field boundary conditions [Eqs. (1) and (2)] to approximately match implosion time and acceleration time history. σ ( ρ r ) was extracted in the same manner as in ALEGRA simulations to assess MRTI cumulative growth throughout the simulated implosions.

KRAKEN simulations indicate levels of MRTI stabilization similar to ALEGRA for each DSP case compared to the z-pinch case (Figs. 5 and 6, Table II). The z-pinch KRAKEN simulation demonstrates lower cumulative MRTI development compared to ALEGRA, which is thought to be a result of the choice of MRTI seeding protocol. Although an exhaustive study of MRTI seeding protocols particular to ALEGRA and to KRAKEN has not been completed as a part of this study, future study could focus on improving agreement of the z-pinch cumulative MRTI results through fine-tuning of the seed (e.g., adjusting the amplitude and spectral content of the node perturbation in ALEGRA or adjusting the magnitude and spectral content of the temperature perturbation in KRAKEN). Despite the differences in numerical algorithms, MRTI seeding methods, and mesh geometry, the results from 3D ALEGRA and 3D KRAKEN display overall agreement. The largest discrepancy between ALEGRA and KRAKEN is a 27% difference for σ ( ρ r ) between the Ξ o = 0 z-pinch cases at Δ R = 0.8. This difference reduces to 18% for the Ξ = 0.25 case, 9% for the Ξ o = 0.5 case, and 20% for the Ξ o = 1.0 case. More importantly, the prediction of cumulative MRTI reduction G σ from 3D ALEGRA is comparable to 3D KRAKEN (Table II). ALEGRA predicts more DSP stabilization compared to KRAKEN for all of the DSP implosions compared to the z-pinch case (Table II). However, note that the largest source for the discrepancy in G σ between the codes comes from the observed difference in σ ( ρ r ) for the Ξ o = 0 z-pinch case, which functions as the baseline for calculating G σ.

FIG. 6.

Density slices extracted from 3D ALEGRA and 3D KRAKEN simulations of Ξ o = 0 and Ξ o = 0.5 simulations at four different times throughout the simulation corresponding to similar CRin. Density is in units of kg/m3. Note the presence of low density material inside of the liner cavity in KRAKEN simulations. This low density plasma is injected into the vacuum region inside of the liner when the shock breaks out at the inner surface. This injection of low density plasma does not occur in ALEGRA simulations, likely due to a difference in the numerical treatment of the vacuum–plasma interface between the two codes.

FIG. 6.

Density slices extracted from 3D ALEGRA and 3D KRAKEN simulations of Ξ o = 0 and Ξ o = 0.5 simulations at four different times throughout the simulation corresponding to similar CRin. Density is in units of kg/m3. Note the presence of low density material inside of the liner cavity in KRAKEN simulations. This low density plasma is injected into the vacuum region inside of the liner when the shock breaks out at the inner surface. This injection of low density plasma does not occur in ALEGRA simulations, likely due to a difference in the numerical treatment of the vacuum–plasma interface between the two codes.

Close modal
The feedthrough of MRTI to the inner surface of the liner is expected to be minimal in the simulation system we present since we are assessing modest convergences (CRin  10) compared to peak compression in MagLIF experiments (CRin > 30) and our simulated liners lack preheated/premagnetized fusion fuel to provide internal hydrodynamic pressure and deceleration-induced growth of inner surface perturbations. We assess the inner liner surface perturbation as a function of convergence by assigning R in t to an 1800 kg/m3 density contour nearest to the central axis. We then compute an average perturbation amplitude of the inner surface,
A = 1 N k = 1 N | R in t k R in t | .
(7)
Here, R in t is the average inner liner surface radius value at a particular time and N is the number of points in the 2D ( θ, z) map formed by the density contour. ALEGRA and KRAKEN simulations indicate that the perturbation amplitude remains below the 20 μm mesh resolution level throughout the implosion (up until the point of highest convergence, Δ R 0.8) for all DSP implosions (Fig. 7). In contrast, both z-pinch calculations indicate that the inner surface perturbation rises above the 20 μm mesh resolution level near Δ R 0.6 and continues to rise, suggesting that the DSP mechanism is not only reducing MRTI growth at the outer surface but also reducing the resultant feedthrough of MRTI to the inner liner surface.
FIG. 7.

Plot of inner liner surface perturbation amplitude A for 3D ALEGRA and 3D KRAKEN simulations of implosions with different initial drive field ratios, Ξ o = {0, 0.25, 0.5, 1.0}. The gray semi-transparent box indicates the mesh resolution, 20 μ m, below which values of perturbation amplitude were assumed to be heavily influenced by numerical noise and not necessarily reflective of a physical perturbation amplitude.

FIG. 7.

Plot of inner liner surface perturbation amplitude A for 3D ALEGRA and 3D KRAKEN simulations of implosions with different initial drive field ratios, Ξ o = {0, 0.25, 0.5, 1.0}. The gray semi-transparent box indicates the mesh resolution, 20 μ m, below which values of perturbation amplitude were assumed to be heavily influenced by numerical noise and not necessarily reflective of a physical perturbation amplitude.

Close modal

A 2D fast Fourier transform (FFT) was performed on the 2D maps of ρ r extracted at different points of convergence throughout each 3D ALEGRA simulation (Fig. 8). The 2D FFT provides the axial and azimuthal wavenumber information from which one can compute the dominant pitch angle of the instability structures and the wavelength of the dominant instability mode. The azimuthal angular coordinates for each ρ r map are converted to distance along the circumference C t by using the liner inner radius R in t at each time step via C t = 2 π R in t.

FIG. 8.

(Top) A 45  ° azimuthal extent is shown of a ρ r map from a 3D ALEGRA simulation of a Ξ o = 1 implosion. (bottom) The corresponding power spectrum map extracted from a fast Fourier transform analysis of the 2D areal density map shown in the top of the figure.

FIG. 8.

(Top) A 45  ° azimuthal extent is shown of a ρ r map from a 3D ALEGRA simulation of a Ξ o = 1 implosion. (bottom) The corresponding power spectrum map extracted from a fast Fourier transform analysis of the 2D areal density map shown in the top of the figure.

Close modal
Spectral analysis of ρ r maps indicates that the dominant instability wavelength in DSP implosions initially increases, reaches a peak, and then decreases as the liner implodes to higher Δ R (see Fig. 9, top). The dominant wavelength was computed by taking the inverse of the average wavenumber extracted from the 2D FFT analysis of the ρ r map,
λ t = 2 π k t = 2 π k z 2 t + k c 2 t .
(8)
FIG. 9.

(Top) Plot of dominant instability wavelength extracted from 3D ALEGRA simulations of magnetically drive cylindrical implosions with Ξ o = {0, 0.25, 0.5, 1.0} using 2D FFT. (Bottom) Plot of average instability pitch angle ϕ t calculated in Eq. (10).

FIG. 9.

(Top) Plot of dominant instability wavelength extracted from 3D ALEGRA simulations of magnetically drive cylindrical implosions with Ξ o = {0, 0.25, 0.5, 1.0} using 2D FFT. (Bottom) Plot of average instability pitch angle ϕ t calculated in Eq. (10).

Close modal
Here, k t is the average wavenumber extracted from a ρ r map at a given time, k z t is the average axial component of the wavenumber, and k c t is the average azimuthal component of the wavenumber. The average wavenumber components k i t were calculated via
k i 2 t = d 2 k F t , k k i 2 d 2 k F t , k ,
(9)
where F ( t , k ) is the 2D Fourier transform of a ρ r map. The peak in wavelength occurs earlier (i.e., at lower convergence or smaller Δ R) for DSP implosions with higher Ξ o. This is due to helical instability structures having both axial and azimuthal wavenumber components, k z and k c, respectively, and the azimuthal wavenumber components are subject to convergence effects.24 Specifically, the wavelength component in the azimuthal direction (i.e., along the circumference) λ c = 2 π / k c varies with the liner radius: λ c t R l t. Axially oriented “flute” modes have wavelengths that vary as R l t, while azimuthally oriented “sausage” modes have no direct dependence on R l t. Therefore, the peak in wavelength occurs earlier (at lower convergence) for simulations with instabilities that have larger λ c produced by larger Ξ o. As to why the instability structures initially increase in wavelength, this is due primarily to mode merging11,12 of the initial continuum spectrum of unstable Rayleigh–Taylor modes.

The z-pinch simulations indeed show increasing dominant instability wavelength as the liner converges (Fig. 9, top) as expected.11 In contrast, while mode merging occurs in the DSP simulations and results in the initial increase in wavelength during implosion, the R l t dependence of the azimuthal wavelength component later dominates and results in decreasing wavelength at a higher liner convergence. Furthermore, feedthrough of MRTI is approximately proportional to e k Δ r t, where Δ r t is the in-flight liner thickness and k is the magnitude of the perturbation wavevector, so larger primary k for a given in-flight liner thickness should generate much lower variations in ρ r. Importantly, the expected increase in dominant wavelength over time is not captured in the linear theory,4 so simulations of DSP implosions provide crucial information about the predicted effectiveness of the DSP mechanism in experiments.

Simulations show that the average pitch angle ϕ t of simulated instability structures (i.e., the angle between the helical structures and the equator) in DSP implosions increases as the liner implodes, suggesting that instability structures become imprinted into the density distribution at lower convergence and are subsequently unable to significantly reorient or shift on the timescale of the implosion (see Fig. 9, bottom). The average instability pitch angle ϕ t was computed via
ϕ t = tan 1 k c t k z t .
(10)
The helical structures rotating toward the vertical should result in the perturbation wavevector k and the drive magnetic vector field B dynamically becoming more orthogonal. Specifically, this corresponds to k · B = k B cos ψ k B increasing during implosion, enhancing stabilization according to the (planar geometry, ideal MHD) linear growth relation25 for MRTI: γ 2 = k g 2 k · B 2 μ o ρ, where γ is the exponential growth rate of the MRTI, g is the acceleration of the liner, ρ is the liner material density, and ψ k B is the angle between k and B . The second term in this relation is the principal motivation behind the DSP mechanism and corresponds to the stabilizing effects of magnetic field line tension; dynamically maximizing this term, particularly as k shifts throughout the implosion, corresponds to maximizing MRTI mitigation via the DSP mechanism.

To assess development of the most deleterious MRTI modes in z-pinches compared to dynamic screw pinches, namely, those aligned with the drive magnetic field lines, we used 3D ALEGRA to simulate liner implosions with Ξ o = 0 , 0.25 , 0.5 in which sinusoidal perturbations aligned with the initial magnetic drive field lines at R out t were initially applied to the outer liner surface (Fig. 10). A 6  μ m amplitude was used for the sinusoidal perturbation for all simulations, and four different wavelengths were simulated: λ = 100 , 150 , 200 , 250 μ m. While field-aligned MRTI modes tend to grow fastest in magnetically driven implosions, we expected lower cumulative MRTI growth in DSP simulations compared to z-pinch simulations. This is because the wavevector of the sinusoidal perturbations applied to the outer surface in the z-pinch case ( Ξ o = 0) remains always perpendicular to the azimuthal drive magnetic field throughout the implosion. In contrast, the wavevector of the helically oriented sinusoidal perturbation in the DSP calculations is only initially perpendicular with the direction of the drive magnetic field, and magnetic field line tension effects grow as the implosion progresses. As the DSP implosions proceed, the perturbation pitch angle ϕ t shifts toward the vertical and the drive magnetic field polarization θ B t shifts toward the horizontal, increasing stabilizing field-line tension effects and reducing MRTI cumulative growth. The dominant wavenumber (and thus wavelength) remains approximately constant throughout the simulation for all ϕ 0= 0 single-mode simulations, while the dominant wavelength in ϕ 0  0 single mode simulations decreases with R l t as expected due to convergence effects.

FIG. 10.

A subset of initial liner geometries for sinusoidally perturbed liner implosion simulations. The pitch angle of the sinusoidal perturbation ϕ is noted at the left for each row, and the perturbation wavelength λ is indicated at the top for each column.

FIG. 10.

A subset of initial liner geometries for sinusoidally perturbed liner implosion simulations. The pitch angle of the sinusoidal perturbation ϕ is noted at the left for each row, and the perturbation wavelength λ is indicated at the top for each column.

Close modal

The linear theory4 suggests that the DSP mechanism is more effective at mitigating growth of shorter wavelength instability structures. This essentially corresponds to stabilizing force imparted by magnetic field line tension scaling inversely with length scale: F T = μ o 1 ( B · ) B L 1, where F T is the force exerted perpendicular to B magnetic field lines (i.e., due to magnetic field line tension) and L is the characteristic length of the system (e.g., the dominant instability wavelength). Accordingly, the DSP mechanism more effectively stabilizes the simulated DSP implosions with shorter wavelengths (Fig. 11 and Table III), except for when comparing the 100  μ m wavelength perturbation case to the 150  μ m case. We postulate that this departure from the expected trend of more effective stabilization at shorter wavelengths arises from the increasing ratio of the perturbation amplitude to the wavelength ( 6 μ m / 150 μ m = 0.04 compared to 6 μ m / 100 μ m = 0.06), which causes the 100  μ m case to enter the nonlinear growth phase more quickly, diminishing the screw pinch mechanism's cumulative effectiveness. Future study will focus on normalizing for the perturbation amplitude to wavelength ratio to observe whether the trend of more effective DSP stabilization with shorter wavelength is preserved. More generally, this trend should also correspond to a changing effectiveness of the DSP mechanism as the dominant wavelength of instability structures changes during implosion. Simulated implosions initialized with a white noise surface roughness and implosions in reality that have a physical surface roughness are expected to demonstrate an increasing dominant wavelength over time, primarily due to processes like mode merging.11,12

FIG. 11.

(Top) Plot of σ ( ρ r ) vs Δ R for liner implosions with initial sinusoidal perturbations with wavelength λ = 100 μm. (bottom) Plot of σ ( ρ r ) vs Δ R for liner implosions with initial sinusoidal perturbations with wavelength λ = 250 μm.

FIG. 11.

(Top) Plot of σ ( ρ r ) vs Δ R for liner implosions with initial sinusoidal perturbations with wavelength λ = 100 μm. (bottom) Plot of σ ( ρ r ) vs Δ R for liner implosions with initial sinusoidal perturbations with wavelength λ = 250 μm.

Close modal
TABLE III.

Cumulative MRTI reduction G σ from 3D ALEGRA simulations of sinusoidally perturbed liner implosions.

G σ at Δ R = 0.4 λ = 100 μm G σ at Δ R = 0.4 λ = 150 μm G σ at Δ R = 0.4 λ = 200 μm G σ at Δ R = 0.4 λ = 250 μm
Ξ o = 0.25  1.29  1.30  1.23  1.09 
Ξ o = 0.5  1.50  1.65  1.47  1.38 
G σ at Δ R = 0.4 λ = 100 μm G σ at Δ R = 0.4 λ = 150 μm G σ at Δ R = 0.4 λ = 200 μm G σ at Δ R = 0.4 λ = 250 μm
Ξ o = 0.25  1.29  1.30  1.23  1.09 
Ξ o = 0.5  1.50  1.65  1.47  1.38 

3D simulations of dynamic screw pinch (DSP) implosions executed the ALEGRA and KRAKEN MHD codes demonstrate that DSP stabilization increases with the ratio of axial to azimuthal drive magnetic field components (i.e., the drive field ratio). Our results are in qualitative agreement with the linear theory.4 The observed stabilization by the DSP mechanism is encouraging, especially since simulations capture physical effects beyond those captured in the linear theory. Spectral analysis of the implosion instability structures indicates that the time-dependent dominant instability wavelength and average instability pitch angle contribute to the operation of the DSP mechanism, specifically by enabling magnetic field line tension to more effectively stabilize shorter wavelength structures compared to longer wavelength structures.

This study suggests that DSP stabilization is a powerful mechanism that can be leveraged to reduce cumulative MRTI development. Experiments to further determine whether and to what extent DSP implosions stabilize MRTI compared to z-pinch implosions will be necessary to improve models and determine best practices for establishing initial conditions such as MRTI seeding mechanisms. Potential improvements for modeling practices could focus on moving away from ad hoc MRTI seeding mechanisms such as artificial surface roughness initialization or volumetric temperature or density perturbations. Such improvements could involve finer mesh resolutions, perhaps enabled by local or adaptive mesh refinement to resolve known seeding processes such as the electrothermal instability that operate on the <1  μm scale.18 Additionally, the local plasma environment26 is not captured in the simulations shown in this study, but future examination of the effects of plasmas in the target region, such as those sourced by the pulsed power transmission lines, will provide information on whether and how the DSP mechanism is affected.

In experiments, DSP implosions will utilize helical return current geometries4,5 to produce the axial drive magnetic field component and will, therefore, introduce additional inductance into the pulsed power circuit. To enable a clean comparison, the implosion time was maintained for all simulations presented in this manuscript. However, a more complete evaluation of the effects of added inductance combined with the added magnetic drive pressure from the additional axial drive component should be pursued. A trade-off exists between added inductance from the axial drive magnetic field component, which will reduce circuit energy coupling from the generator to the imploding z-pinch liner, and the additional magnetic pressure from the axial magnetic drive component. Future circuit-coupled, 3D radiation-MHD simulations will focus on modeling the entirety of the MagLIF system with premagnetized, laser-preheated fusion fuel to evaluate thermonuclear performance in DSP implosions on Z.

The authors gratefully acknowledge helpful discussions with A. Porwitzky, W. Lewis, L. Stanek, J. Niederhaus, A.C. Robinson, E.P. Yu, M.R. Weis, D.C. Lamppa, T.J. Awe, A.J. Harvey-Thompson, and R.D. McBride. The authors also gratefully acknowledge programmatic support and encouragement from K. Beckwith, L. Shulenburger, B. Jones, D. Ampleford, K. Peterson, G. Rochau, and D. Sinars.

One of the authors (Shipley) was partially funded by the National Nuclear Security Administration through the Krell Institute via the Stewardship Science Graduate Fellowship (Grant No. DE-NA0003864). One of the authors (Shipley) was also funded in part by Sandia's Laboratory Directed Research and Development program via the Truman Fellowship, Project No. 226067. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC (NTESS), a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration (DOE/NNSA) under contract DE-NA0003525. This written work is authored by an employee of NTESS. The employee, not NTESS, owns the right, title, and interest in and to the written work and is responsible for its contents. Any subjective views or opinions that might be expressed in the written work do not necessarily represent the views of the U.S. Government. The publisher acknowledges that the U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this written work or allow others to do so, for U.S. Government purpose. The DOE will provide public access to results of federally sponsored research in accordance with the DOE Public Access Plan.

The authors have no conflicts to disclose.

Gabriel Alan Shipley: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (lead). Daniel Edward Ruiz: Conceptualization (supporting); Formal analysis (supporting); Methodology (supporting); Writing – review & editing (supporting). Christopher A. Jennings: Formal analysis (supporting); Methodology (supporting); Writing – review & editing (supporting). David Yager-Elorriaga: Conceptualization (supporting); Methodology (supporting); Writing – review & editing (supporting). Paul F. Schmit: Conceptualization (equal); Methodology (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available within the article.

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