Three-dimensional FLASH radiation-magnetohydrodynamics (radiation-MHD) modeling is carried out to study the hydrodynamics and magnetic fields in the shock-shear derived platform. Simulations indicate that fields of tens of Tesla can be generated via the Biermann battery effect due to vortices and mix in the counterpropagating shock-induced shear layer. Synthetic proton radiography simulations using MPRAD and synthetic X-ray image simulations using SPECT3D are carried out to predict the observable features in the diagnostics. Quantifying the effects of magnetic fields in inertial confinement fusion and high-energy-density plasmas represents frontier research that has far-reaching implications in basic and applied sciences.

When an inertial confinement fusion (ICF) capsule implodes, the material turns into dense plasmas and recent simulations have shown that such plasmas tend to be unstable and turbulence can develop.1 Even though it is debated whether turbulence is damped by the viscosity in the hot spot, the shocked interfaces as well as the interface between the shell and the hot spot can have very different dynamics and can indeed be unstable.2–7 It is believed that turbulence and the associated mixing process can be crucial for understanding ICF.

The Biermann battery effect8 is known to generate seed magnetic fields in laser driven plasma flows and has been studied extensively in high-energy-density (HED) laser-driven experiments,9–16 but the strength and importance of these fields in the close to or higher than solid density plasmas such as an ICF implosion are not well known. Three-dimensional extended-magnetohydrodynamic (extended-MHD) simulations of the stagnation phase of ICF including the Biermann battery term,8 Nernst term,17 and anisotropic heat conduction in the magnetic field indicate that self-generated magnetic fields can reach over 104 Tesla and can affect the electron heat flow.18 The simulations with premagnetization for ICF implosions show the significance of Lorentz force and α-particle trapping.19 In low density laser driven plasmas, the magnetic field can be amplified by turbulence and measured using temporal diagnostics by the B-dot probe20 and spatial diagnostics by proton radiography.21 The magnetic frequency spectrum in supersonic plasma turbulence has been measured in a recent experiment22 on the Vulcan laser. However, in those experiments20–22 the magnetic field is not high enough to change the dynamics of the hydrodynamical flow.

In this work, we use the shock-shear platform23,24 developed at the Los Alamos National Laboratory (LANL) to quantify the dynamics of magnetic fields in HED plasmas with instabilities and turbulence. The shock compression can achieve a regime where the density is around 1g/cc. The targets with large density can diffuse the proton beam and affect the interpretation of the proton image,25 but the simulations for the synthetic proton image including the stopping power and Coulomb scattering show that the deflection of the proton beam by magnetic fields is still detectable. Further improvements are still needed to make the fields high enough to change the dynamics of the small-scale evolution of vortices like those in a turbulent cascade and affect our understanding of turbulence.

The shock-shear platform,23,24 as a platform to isolatedly study the shear-induced instabilities and turbulence production under HED conditions, i.e., pressure larger than 1Mbar, has been used to investigate the turbulent mixing26,27 at material interfaces when subject to multiple shocks and reshocks or high-speed shear.23,28 The experiments29–33 using the shock-shear platform have been carried out on the OMEGA Laser Facility and National Ignition Facility (NIF). These experiments provide quantitative measurements to assist in validation efforts34–36 for mix models, such as the Besnard-Harlow-Rauenzahn (BHR) model.37,38 The experimental data and the validation efforts constrain models relevant to integrated HED experiments such as ICF or astrophysical problems. In the shock-shear targets, the Biermann battery (ne×Te) term8 can generate and sustain strong magnetic fields in the vortices due to the misalignment of the density gradient and temperature gradient caused by electron heat conduction. However, the magnetic fields in the shock-shear targets have not been quantified in simulations or experiments.

In this work, we use the radiation-MHD code FLASH39,40 to model the evolution of the shock-shear system on OMEGA.41 The experiment simulated in this paper uses 8 beams each with 500 J energy laser ablation in 1 ns on each side to drive strong adjacent contour-propagating shocks. Kelvin-Helmholtz instability laterally spreads across a thin layer of magnesium, copper, or plastic placed at the interface. The layer is cut with slots to seed the initial density perturbation, which can generate vortices during the evolution of the shock and shear. The temperature of the materials reaches tens of electron-volts, and simulations predict that the Mach number of the postshock flows in the experiment is around 2 on each side of the shear layer. The magnetic field is generated by the Biermann battery term8 and dissipated by the resistive term. The X-ray image42–44 and the proton radiograph9 are predicted and will be compared to the experimental data in a later paper.

This paper is organized as follows. Section II describes simulation methods and the configuration of the target system. In Sec. III, we show the results for hydrodynamics and MHD evolution from FLASH, the synthetic X-ray image using SPECT3D, and the synthetic proton radiography using MPRAD. The conclusions and discussions are given in Sec. IV.

The FLASH code39,40,45 is used to carry out the detailed physics simulations of our laser experiments to study the dynamics of the shock-shear system. FLASH is a publicly available, multiphysics, highly scalable parallel, finite-volume Eulerian code and framework whose capabilities include adaptive mesh refinement (AMR), multiple hydrodynamic and MHD solvers,46–49 implicit solvers for diffusion using the HYPRE library,50 and laser energy deposition. FLASH is capable of using the multi-temperature equation of states and multigroup opacities. To simulate laser-driven High-Energy-Density-Physics (HEDP) experiments, a 3 T treatment, i.e., TradTeleTion, is usually adopted. The equations which FLASH solves to describe the evolution of the 3 T magnetized plasma are

ρt+·(ρv)=0,
(1)
ρvt+(ρvv14πBB)+Ptot=0,
(2)
ρEtott+·(v(ρEtot+Ptot)14πB(v·B))14π·(B×(η×B))14π·(B×cePene)=·q+S,
(3)
Bt+·(vBBv)=×(ηB×B)+ce×Pene,
(4)

where the total pressure is given by Ptot=Pion+Pele+Prad+(1/8π)B2, and the total specific energy Etot=eion+eele+erad+(1/8π)B2+12v·v. The total heat flux q is the summation of electron heat flux qe=κTele and radiation flux qr, where κ is the Spitzer electron heat conductivity.51,52 The flux-limit used for electron thermal conduction is set to be 6% of the free streaming flux qFS=nekBTekBTe/me. The first term on the R.H.S of Eq. (4) contains the Spitzer magnetic resistivity ηB.51,52 The second term on the R.H.S of Eq. (4) is the Biermann battery term, which generates the magnetic field even if there is no seed magnetic field initially. The plasma has zero initial magnetic field in the simulations. Because plasma beta β is much larger than unity, the Hall term is neglectable and not included in the simulations. The Biermann battery term is turned off in the cells adjacent to the shock detected numerically.53 The magnetic field generation near the shock is not calculated because of the convergence problem54 for calculating the Biermann battery term on the Eulerian grid. The convergence problem might be resolved on a Lagrangian grid. On the other hand, the shock in this work is highly collisional and with a small thickness compared to the spatial resolution of proton radiography, thus the scale of the magnetic field near the shock is too small to be detectable. The energy equations for the three components are

t(ρeion)+·(ρeionv)+Pion·v=ρcv,eleτei(TeleTion),
(5)
t(ρeele)+·(ρeelev)+Pele·v=ρcv,eleτei(TionTele)·qele+QabsQemis+Qlas+Qohm,
(6)
t(ρerad)+·(ρeradv)+Prad·v=·qradQabs+Qemis,
(7)

where cv,ele is the electron specific heat and τei is the ion-electron Coulomb collision time. Qabs (absorption) and Qemis (emission) describe the energy transfer between the electron and the radiation, which is modeled using the multigroup flux-limited radiation diffusion. The laser absorption term Qlas is computed using ray-tracing in the geometric optics approximation via the inverse-Bremsstrahlung process. Qohm is the rate of electron energy increase due to Ohmic heating. The auxiliary equations Eqs. (5)–(7) are advanced in time such that the distribution of energy change due to the work and the total shock-heating is based on the pressure ratio of the components, which is a method implemented in FLASH inspired by the radiation-hydrodynamics code RAGE.55,56 We use the equation of state and opacity table from PROPACEOS57,58 for modeling all the material properties in our target system.

We initialize the FLASH simulation using the geometry and parameters of targets used for OMEGA experiments. The target system is composed of the shock tube, the gold cone for minimizing stray laser light, the foam filling the shock tube, and a plastic cap covering the end of the tube, as shown in Fig. 1. As shown in Fig. 1(c), a window is opened in the middle of the tube and along the path of the proton beam to make the proton beam less diffusive, i.e., less energy lost and scattering. However, the opened window can make the plasma squirt outwardly. We use the foam with density 62mg/cc, and the foam is divided by a layer with slanted or nonslanted slots, as shown in Figs. 1(e) and 1(f). The end cap is 1g/cc plastic. The shape of the slots, the material, and the thickness of the layer, and the material of the wall are listed in Table I. Some targets are built with a pepper-pot screen (PPS),59 as shown in Fig. 1(b). The PPS is used for a narrow view of the proton deflection signal in proton radiography, reducing the signal contamination from off-center line-of-sight. The 200μm diameter hole in the middle allows proton beams to go through the central part of the target. Other holes are used as references to register the position of protons. The PPS is a 40 μm thick tantalum foil.

FIG. 1.

The experimental setup. The shapes and dimensions of different parts of the target are used to initialize the FLASH simulations. (a) The far-view of the target system, including the shock tube, the gold cone for shielding, and the plastic end cap. The foam and the layer are not shown. The x axis extends through the window. (b) The target with a pepper-pot screen (PPS) for a narrow view proton radiography. The screen has five large holes with 200μm diameter and four small holes. The screen is at x=1.3mm plane, attached to the edge of the gold cone. The x axis extends through the window. (c) The dimension of the shock tube, the window, and the end cap. The beryllium shock tube has an oval-shaped window in the middle. The end cap is plastic. The foam and the layer are not in this figure. The inner radius of the tube is 250μm, and the outer radius of the tube is 350μm. (d) Same as (b) but the shock tube is plastic and thicker. The inner radius of the tube is 250μm, and the outer radius of the tube is 400μm. (e) The magnetism layer with 45° slanted slots. The wavelength of the slots is 150μm. (f) The plastic or copper layer with straight slots. The wavelength of the slots is 150μm. (g) A layer divides the low density foam into two half-cylinders to collimate the shock flow. The gold plugs hold back the shock at one end of each half-cylinder of foam.

FIG. 1.

The experimental setup. The shapes and dimensions of different parts of the target are used to initialize the FLASH simulations. (a) The far-view of the target system, including the shock tube, the gold cone for shielding, and the plastic end cap. The foam and the layer are not shown. The x axis extends through the window. (b) The target with a pepper-pot screen (PPS) for a narrow view proton radiography. The screen has five large holes with 200μm diameter and four small holes. The screen is at x=1.3mm plane, attached to the edge of the gold cone. The x axis extends through the window. (c) The dimension of the shock tube, the window, and the end cap. The beryllium shock tube has an oval-shaped window in the middle. The end cap is plastic. The foam and the layer are not in this figure. The inner radius of the tube is 250μm, and the outer radius of the tube is 350μm. (d) Same as (b) but the shock tube is plastic and thicker. The inner radius of the tube is 250μm, and the outer radius of the tube is 400μm. (e) The magnetism layer with 45° slanted slots. The wavelength of the slots is 150μm. (f) The plastic or copper layer with straight slots. The wavelength of the slots is 150μm. (g) A layer divides the low density foam into two half-cylinders to collimate the shock flow. The gold plugs hold back the shock at one end of each half-cylinder of foam.

Close modal
TABLE I.

The parameters and the maximum values of magnetic field and electron temperature for the three different targets/runs we use. Te and B are calculated by averaging over a (200μm)2 around the center of the target in the xz plane. PPS stands for the pepper-pot screen.

Target/run labelSlanted slotsLayer thicknessLayer materialWall thicknessWall materialTe (eV) at 10 nsB (kGauss) at 10 ns
Yes 15μm Mg 100μm Be 25 158 
No 6μm Cu 150μm CH 26 152 
No 6μm CH 150μm CH 28 86 
Target/run labelSlanted slotsLayer thicknessLayer materialWall thicknessWall materialTe (eV) at 10 nsB (kGauss) at 10 ns
Yes 15μm Mg 100μm Be 25 158 
No 6μm Cu 150μm CH 26 152 
No 6μm CH 150μm CH 28 86 

In the initialization, the pressure of all the solid regions is 5×103bar(=5×109erg/cm3), and the temperature is calculated self-consistently from the equation of state table. Using the same pressure instead of the same temperature among all the solid regions can prevent one solid region from expanding into another solid region and launching artificial shocks before the high-energy-density conditions are reached. Under HED conditions, the pressure is larger than 105bar(=1011erg/cm3); thus, the initial pressure is low enough to have a neglectable effect on the simulations. The vacuum region is initially filled with 106g/cc helium to avoid numerical problems in hydrodynamics or MHD solvers. The density is low enough that the effect of helium on the simulations is negligible.

A 3D cartesian grid with (240×240×464) zones is used to resolve a (1440μm×1440μm×2784μm) domain, corresponding to 6μm per cell width. Using AMR, each zone is adaptively refined to one leaf level, i.e., a resolution of 3μm or 23=8 zones, if the mass fraction of the layer material is larger than 10%. The refinement allows us to efficiently resolve the dynamics near the layer and reduce the computing time spent on the zones far away from the layer. Although we cannot resolve the turbulence dissipation scale with the current computing capability and neither do we use Reynolds-averaging Navier-Stokes (RANS) models such as the BHR model to resolve the small scale dissipation process of the fluid, FLASH is still a suitable tool for designing these experiments because the fabricated layers have low surface roughness.

To model the laser driven energy deposition, we use the spatial and temporal specifications of each of the 16 OMEGA driver beams. Ray tracing by solving the geometric optics and the inverse bremsstrahlung absorption is used. The 16 driver beams are turned on and turned off simultaneously with a 1 ns pulse duration and 8 beams on each side of the target. Each delivers 500 J of energy on a target. The radius of each beam is 283 μm and the intensity distribution we use is Gaussian.

For convention, t =0 is the time for laser turn on. The axis of the shock tube is the z axis. The layer dividing the foam is in the yz plane, i.e., the plane with x =0 everywhere. The center of the target is at x=y=z=0. The x axis extends through the window.

The primary diagnostic for a temporally and spatially resolved profile of the density and shock propagation in the experiments is the point projection X-ray radiography with a vanadium backlighter at 23× magnification. The backlighter source emits 5180 eV and 5205 eV helium like lines.60 The images are recorded on the X-ray framing camera (XRFC).42–44 We use SPECT3D61,62 to generate the synthetic ray-tracing X-ray image. The line of sight of XRFC is along the y axis, which captures the distortion of the layer.

Proton radiography,9 using D3He (14.7 MeV) protons from fusion, measures magnetic fields. The temporal resolution of the proton radiograph is typically 150ps and the spatial resolution is typically 45μm. The diffusion of the proton beam caused by Coulomb scattering63,64 and stopping power65–69 is significant for the targets we use. We use Monte Carlo code MPRAD25 to model the synthetic proton radiography, including the Lorentz force and the effects from Coulomb scattering and stopping power. The proton source stands at (0.75cm,0,0), while the image plate CR39 is located 27 cm from the center on the other side. The line of sight of the proton radiography is perpendicular to the line of sight of the X-ray image. The energy distribution of the proton source we use in the simulation is a Gaussian distribution with FWHM=0.25MeV centered at 14.7MeV.

We show the results from FLASH simulations and synthetic radiography to study the evolution and dynamics of the flows in the shock-shear targets in Figs. 2–4. In the synthetic radiographs, the spatial scales of the synthetic radiographs are divided by the magnification to align with the scales on the target system. The target we use in this work is different from previous shock-shear experiments28,30,34,35 mainly in two aspects: (1) cut slots in the layer for seeding density perturbation and (2) opened window on the wall for reducing the diffusion of proton beams.

FIG. 2.

Spatial distribution of different quantities at different times. The size of all plots is 1200μm×1200μm. From first to fourth rows are: density at the y =0 plane, electron temperature at the y =0 plane, X-ray flux normalized by the purely transparent flux, and magnetic field By in kGauss at y =0 plane (positive for into the plane). The plots in the second and the fourth rows are overlaid with magenta contours for the density of the wall material equal to 0.5g/cc. From fifth to the last rows are proton images for four different cases as labeled.

FIG. 2.

Spatial distribution of different quantities at different times. The size of all plots is 1200μm×1200μm. From first to fourth rows are: density at the y =0 plane, electron temperature at the y =0 plane, X-ray flux normalized by the purely transparent flux, and magnetic field By in kGauss at y =0 plane (positive for into the plane). The plots in the second and the fourth rows are overlaid with magenta contours for the density of the wall material equal to 0.5g/cc. From fifth to the last rows are proton images for four different cases as labeled.

Close modal
FIG. 3.

Same as Fig. 2 but for runB.

FIG. 3.

Same as Fig. 2 but for runB.

Close modal
FIG. 4.

Same as Fig. 2 but for runC.

FIG. 4.

Same as Fig. 2 but for runC.

Close modal

We show the evolution of density, electron temperature, and X-ray flux in the first three rows in Figs. 2–4. The gold plugs hold back the shock at one end of each half-cylinder of foam. Two shocks of roughly the same strength in the same material propagate from opposite directions toward the center of the tube. The layer placed in the middle between the two regions collimates the shocked flows and introduces a length scale through its thickness, which will influence the dominant modes of the resulting shear instability. The cut slots in the layer introduce alternating density gradients and cause magnetic field generation by the Biermann battery term, which is discussed in Sec. III C. Because the layer does not fully collimate the shocks, oblique shocks are launched into the opposite volumes of the tube. The shock front near the end of the tube travels further transversely. It takes roughly 8.5ns for the shocks to cross and create the pressure-balanced shear mixing region. The pressure in the two regions is roughly equal and the shocked material is the same on each side of the mixing layer, so that the mixing region does not experience a net translation away from the center of the shock tube. After 8.5ns, the oblique shock on either end of the tube gradually crosses the primary shock from the other direction. An oblique region of high density is developed by the reverse shock.

The ideally constructed target should be symmetric about a rotation of 180°. However, the different effective laser intensities on two ends of the target due to different laser incident angles cause the two shocks to move at slightly different speeds. The shock from the right side moves slightly faster as shown in the first rows in Figs. 2–4. This asymmetry does not affect the overall picture of the hydrodynamical and magnetic field evolution, but the asymmetry of the density distribution can affect the proton radiography, which is discussed in Sec. III D.

Because of the opened window on the wall, there are plasma plumes traveling outside the window. As shown in Fig. 5, the overall picture of hydrodynamical evolution is still similar to previous shock-shear experiments without a window,28,30,34,35 although the plasma plume carries mass and energy away from the tube. At later times, the shock can penetrate through the wall. This results in plumes outside the wall, which can then interact with the plume from the window.

FIG. 5.

Spatial distribution of different quantities for runA with or without window at 10 ns. The size of all plots is 1200μm×1200μm. From first to last rows are density at the y =0 plane, electron temperature at the y =0 plane, and magnetic field By in kGauss at y =0 plane (positive for into the plane). The plots in the third row are overlaid with magenta contours for the density of the wall material equal to 0.5g/cc.

FIG. 5.

Spatial distribution of different quantities for runA with or without window at 10 ns. The size of all plots is 1200μm×1200μm. From first to last rows are density at the y =0 plane, electron temperature at the y =0 plane, and magnetic field By in kGauss at y =0 plane (positive for into the plane). The plots in the third row are overlaid with magenta contours for the density of the wall material equal to 0.5g/cc.

Close modal

The transmitted X-ray flux is shown in the third rows in Figs. 2–4. In the X-ray flux, the location and the shape of the shock front is consistent with the density distribution and can be easily identified. The shocks in the wall can also be seen in the X-ray image. The plume launched from the wall or the window has low density and is not visible in the X-ray flux. The layer has high density and low X-ray transmission, leading to the low flux on the X-ray image. For runA and runB, where the layer material is magnesium and copper, respectively, the contrast of X-ray flux between the layer and the foam is high, while for runC where the layer material is CH, the X-ray contrast is low.

When the shock from one end of the tube passes, the temperature is high near the center of the half-cylinder as shown in the second rows in Figs. 2–4. A cold region is left behind the shock. The temperature gradient near the layer is perpendicular to the layer and pointing toward the shocked region, due to electron heat conduction. The density gradient is alternating, caused by the cut slots on the layer. Thus, the Biermann battery term generates the alternating magnetic field in the ±y direction, as shown in Fig. 6(a). However, the cold region left behind the shock has low electron temperature and thus high resistivity. The magnetic fields behind the shock diffuse very quickly. In the end, the only significant field left near the center of the tube is in the −y direction, because near the center of the tube, the layer is at high density instead of at a cut slot. On both sides of that high density layer, the field generation is in the −y direction. Two shocks from two ends of the tube cross amplify the magnetic field and create a doubly shocked, high temperature region, which has low resistivity and the field is less diffusive.

FIG. 6.

Schematics of the magnetic field generation by Biermann battery term(ne×Te). (a) Near the layer, the temperature gradient is perpendicular to the layer due to thermal conduction, the density gradient is alternating and along the layer due to the cut slots on the layer, so that the Biermann generated field is alternating into and out of the plane. (b) Outside the window, the density gradient points to the dense part of the plume, the temperature gradient along the outflow direction is small due to conduction, but the temperature gradient perpendicular to the outflow direction survives due to continuous launching of the plume from the shock tube; thus the field is into the plane on the right side and out of the plane on the left side.

FIG. 6.

Schematics of the magnetic field generation by Biermann battery term(ne×Te). (a) Near the layer, the temperature gradient is perpendicular to the layer due to thermal conduction, the density gradient is alternating and along the layer due to the cut slots on the layer, so that the Biermann generated field is alternating into and out of the plane. (b) Outside the window, the density gradient points to the dense part of the plume, the temperature gradient along the outflow direction is small due to conduction, but the temperature gradient perpendicular to the outflow direction survives due to continuous launching of the plume from the shock tube; thus the field is into the plane on the right side and out of the plane on the left side.

Close modal

The magnetic field in the plume traveling outside the window is generated in a similar way to the magnetic field generated in the ablation plume of a laser interaction with a solid target.9,10,12,13 The plume is continuously launched by the flow inside the shock tube and expands in all directions, with the density gradient to point toward the dense part of the plume, as shown in Fig. 6(b). The temperature gradient along the outflow direction is reduced due to electron thermal conduction, but the temperature gradient perpendicular to the outflow direction survives due to continuous launching of the plume from the shock tube. Thus, the magnetic field generated by the Biermann battery term is into the plane on the right side and out of the plane on the left side in Fig. 6(b).

The magnetic field evolution is shown in the fourth rows in Figs. 2–4. In the center of the tube, a field pointing in −y direction dominates. Outside the window, the field pointing in +y direction survives, while the field pointing in −y direction diffuses quickly due to low temperature and high resistivity. The total magnetic flux in the y =0 plane is conserved and vanishes. We are interested in the magnetic field near the center of the tube, which can potentially affect the mix. The magnetic field outside the window plays a role in the proton radiography as discussed in Sec. III D, but we are not interested in its dynamical importance because it is far away from the mix region. As shown in Fig. 5, the magnetic field near the center of the tube is similar between the runs with and without the window.

We use the MPRAD code25 to simulate the proton image by taking the output data from 3D FLASH simulations. In the simulations, we use a typical size 45μm for the proton source. We find that the features of the proton images are most prominent in 14.3MeV to 14.5MeV band, i.e., protons losing between 0.2MeV and 0.4MeV of kinetic energy. We compare the proton images with/without field, and with/without pepper pot screen (PPS) in the fifth to the last rows in Figs. 2–4. To quantify the asymmetry of the proton image, the averaged horizontal proton position in the blob at the center of the proton image is plotted in Fig. 7. The ideally constructed target should be symmetric about a rotation of 180° and the proton image should also be symmetric in the absence of magnetic field. The asymmetry of the proton image about the vertical axis can be interpreted as the existence of magnetic field.

FIG. 7.

The evolution of the averaged position of protons in the blob in final energy range 14.3MeV to 14.5MeV. The scale is divided by the magnification to align with the scales on the target system. The red curves are for runA, the black curves are for runB, and the blue curves are for runC. The dashed curves are for the MPRAD runs with magnetic field turned off, and the solid curves are for MPRAD runs with magnetic field turned on. (a) is for the no PPS case and (b) is for the with PPS case.

FIG. 7.

The evolution of the averaged position of protons in the blob in final energy range 14.3MeV to 14.5MeV. The scale is divided by the magnification to align with the scales on the target system. The red curves are for runA, the black curves are for runB, and the blue curves are for runC. The dashed curves are for the MPRAD runs with magnetic field turned off, and the solid curves are for MPRAD runs with magnetic field turned on. (a) is for the no PPS case and (b) is for the with PPS case.

Close modal

However, in the no PPS case, i.e., the fifth rows in Figs. 2–4, the blob in the middle of the image can be slightly asymmetric even without magnetic field. This asymmetry is not as large as the asymmetry in the images where there is field but no PPS, i.e., the six rows, which means that the proton deflection by magnetic field causes more asymmetry than by the density asymmetry due to the fact that the shock from the right side in Figs. 2–4, moves slightly faster. This slight difference is caused by the different effective laser intensities on two ends of the target due to different laser incident angles. In the simulations in this work, we do not take into account the unevenness of the foam and the power imbalance on two ends of the tube, which can potentially cause more asymmetry on the proton image than what we show in this work.

One advantage of using PPS is that the viewing of the surrounding holes is through the regions without the field and the viewing of the hole in the middle is only through the region with magnetic field, so that the net deflection caused by the magnetic field can be determined without another control shot using the same target. With PPS, the asymmetry in the no field case, i.e., the seventh rows in Figs. 2–4, is significantly less than the without field and without the PPS case, i.e., the fifth rows. The PPS is very efficient in reducing the asymmetry of the proton image caused by the intensity imbalance on two ends and the unevenness of the foam. As shown in Fig. 7(b), the asymmetry caused by the proton deflection is significantly larger than that caused by the nonuniform density. The blob has a positive net shift at early time, because of the field pointing in +y direction in the plume outside the window. At about 8.5ns, the proton deflection caused by the field pointing in +y direction in the plume outside the window and by the field in near the center of the tube pointing in −y direction cancels, resulting in zero net shift of the blob on the proton image. At a late time t>10ns, the field pointing in +y direction moves away from the z =0 plane, but the field near the center of the tube has no net advection, and the net shift of the blob is negative. The shift value on the image plate divided by the magnification can reach 50–70μm. The difference between the early time shift and late time shift can reach 70–90μm. The prediction for the net shift of the blob will be compared to the experimental data to validate the magnetic field model in FLASH.

We carried out the radiation-MHD simulations and predicted the X-ray and proton images by synthetic radiographs. The hydrodynamical evolution can be measured using XRFC and compared with the simulation results. The predicted proton radiography shows the direction and the amount of the shift of the proton beam going through the window and/or PPS. Although the target can diffuse the proton beam significantly, the evolution of the shift in the synthetic proton radiography is still consistent with the evolution of the magnetic fields in the target system and shows change between early time and late time. However, the prediction only shows the signal contribution from the mean magnetic fields from different columns along the line of sight. The signal from small scale fields always gets damped by the diffusion of the proton beam. The high energy proton beam accelerated by the Target Normal Sheath Acceleration (TNSA) mechanism using the OMEGA EP beam experiences less diffusion through the target.70 The Coulomb scattering angle is roughly proportional to Ep2, where Ep is the kinetic energy of the proton.25,63,64

The simulation shows that the design we use can achieve a regime with high plasma beta β. The Hall parameter χ, defined by the ratio of electron gyrofrequency to electron collision frequency, is small. The Reynolds number Re is high enough to ensure turbulence, and the magnetic Reynolds number Rm is around 50. Under the condition with these dimensionless parameters, the magnetic field remains dynamically unimportant. The magnetic energy density from Table II is 109erg/s, which is only 0.3% of the turbulent kinetic energy reported in the simulation in Ref. 35 for a previous mix modeling for shock-shear targets under similar conditions to this work. Thus, the magnetic field is also negligible for mix modeling in the shock-shear targets. It is desirable to optimize the measurable magnetic fields and improve the dynamical importance of the magnetic fields.

TABLE II.

Simulated plasma properties for runA. All quantities are in cgs units except temperature, which is expressed in eV. The length scale, L is approximately the diameter of the tube (500μm). The ne, ρ, Te, and Ti are calculated by averaging over a (2002μm)2 square around the center of the target in the xz plane, at t=10ns. The flow speed is u=7×106cm/s for each counter propagating flow.

Plasma propertyFormulaValue at r =0
Electron density ne(cm3 5.6×1022 
Mass density ρ(g/cm3 0.36 
Electron temperature Te(eV)  25 
Ion temperature Ti(eV)  25 
Magnetic field B (gauss)  1.6×105 
Average ionization Z  1.9 
Average atomic weight A  7.3 
Flow speed u(cm/s)  7×106 
Sound speed cs(cm/s) 9.8×105[ZTe+1.67Ti]1/2A1/2 3.4×106 
Mach number M u/cs 
Coulomb logarithm lnΛ min(23.5+ln(Te1.5/ne0.5/Z),25.3+ln(Te/ne0.5)) 1.4 
Hall parameter χ 6.1×1012Te3/2BZnelnΛ 8×104 
Plasma β 2.4×1012ne(Te+Ti/Z)B2/(8π) 5×103 
Péclet number Pe uL/(κe32nekB)(κe32nekB=5.5×1021Te5/2ne(3.3+Z)lnΛ) 8.3×103 
Magnetic Reynolds number Rm uL/η(η=8.2×105(0.33Z+0.18)lnΛTe3/2) 47 
Reynolds number Re uL/ν(ν=1.9×1019Ti5/2A1/2Z3nelnΛ) 8.6×106 
Plasma propertyFormulaValue at r =0
Electron density ne(cm3 5.6×1022 
Mass density ρ(g/cm3 0.36 
Electron temperature Te(eV)  25 
Ion temperature Ti(eV)  25 
Magnetic field B (gauss)  1.6×105 
Average ionization Z  1.9 
Average atomic weight A  7.3 
Flow speed u(cm/s)  7×106 
Sound speed cs(cm/s) 9.8×105[ZTe+1.67Ti]1/2A1/2 3.4×106 
Mach number M u/cs 
Coulomb logarithm lnΛ min(23.5+ln(Te1.5/ne0.5/Z),25.3+ln(Te/ne0.5)) 1.4 
Hall parameter χ 6.1×1012Te3/2BZnelnΛ 8×104 
Plasma β 2.4×1012ne(Te+Ti/Z)B2/(8π) 5×103 
Péclet number Pe uL/(κe32nekB)(κe32nekB=5.5×1021Te5/2ne(3.3+Z)lnΛ) 8.3×103 
Magnetic Reynolds number Rm uL/η(η=8.2×105(0.33Z+0.18)lnΛTe3/2) 47 
Reynolds number Re uL/ν(ν=1.9×1019Ti5/2A1/2Z3nelnΛ) 8.6×106 

The Biermann battery generated magnetic field is roughly ckBTe/eLu by balancing the Biermann battery term with the advection term. The plasma beta β is then proportional to neTe/(Te/Lu)2neu2/L2Te. If we keep the size of the target and the laser power, neu2 and L are roughly constants, and then β1/Te. Thus, increasing Te can reduce β and make the Lorentz force more important. The Hall parameter52χ is proportional to Te3/2/ne and the magnetic Reynolds number Rm is proportional to Te3/2. Both χ and Rm increase with temperature. For low Rm and low magnetic Prandtl number Prm, i.e., Prm=Rm/Re1, the power spectrum of the kinetic energy E(k) and the power spectrum of the magnetic energy M(k) are related by M(k)k2E(k), and M(k) is always softer than E(k), and the magnetic field remains dynamically unimportant even in small scales.20,71,72 High Rm is favorable for the amplification of magnetic fields and a hard power law for the magnetic energy spectrum.21,71,73 One way to achieve a higher temperature is to lower the density of the foam. However, making a low density foam in the target is challenging for target fabrication. It causes the unevenness in the foam, leads to the unevenness of the proton image, and makes it difficult to interpret the experimental data from proton radiography. In a low density foam, the flow may move too fast so that the time window for diagnostics is narrow.

Some experiments74 and theories75,76 show that around 10eV, the value of electrical resistivity [electrical resistivity η is related to magnetic resistivity ηB by ηB=(c2/4π)η] is different from the Spitzer resistivity. However, the electrical resistivity with temperature and density dependency under the condition of our experimental design is not well constrained. If the modeling in this work is correct in terms of electrical resistivity, then this would indicate that the magnetic field may not be dynamically important. However, if the electrical resistivity is significantly lower than the Spitzer resistivity that we use in this work, then the simulations in this work underestimates the magnetic fields, and the mix model could potentially cover up the magnetic field effects by the choice of the initial input conditions for the model. Future experiments executed at higher temperatures can potentially make magnetic fields start to play a more important role. In the future development of the simulations, the implementation of the implicit method for the magnetic diffusion equation is desirable for the case of large resistivity where a fully explicit method requires a small time step.

The research presented in this paper was supported by the Laboratory Directed Research and Development (LDRD) Program of the Los Alamos National Laboratory (LANL) under Contract No. 20180040DR. The simulations were performed with LANL Institutional Computing, which is supported by the U.S. Department of Energy National Nuclear Security Administration under Contract No. 89233218CNA000001, and with the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation (NSF) under Grant No. ACI-1548562.

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