Enhancement of Pressure Perturbations in Ablation due to Kinetic Magnetised Transport Effects under Direct-Drive ICF relevant conditions

Enhancement of Pressure Perturbations in Ablation due to Kinetic Magnetised Transport Effects under Direct-Drive ICF relevant conditions

Abstract

We present for the first time kinetic 2D Vlasov-Fokker-Planck simulations, including both self-consistent magnetic fields and ablating ion outflow, of a planar ablating CH-foil subject to nonuniform irradiation. Even for small hall parameters () self-generated magnetic fields are sufficient to invert and enhance pressure perturbations. The mode inversion is caused by a combination of the Nernst advection of the magnetic field and the Righi-Leduc heat-flux. Non-local effects modify these processes. The mechanism is robust under plasma conditions tested; it is amplitude independent and occurs for a broad spectrum of perturbation wavelengths, . The ablating plasma response to a dynamically evolving speckle pattern perturbation, analogous to an optically smoothed beam, is also simulated. Similar to the single mode case, self-generated magnetic fields increase the degree of nonuniformity at the ablation surface by up to an order of magnitude and are found to preferentially enhance lower modes due to the resistive damping of high mode number magnetic fields.

pacs:
\DeclareSIUnit\tei

\DeclareSIUnit\vthv_T \DeclareSIUnit\lmfp

Irradiation nonuniformity can be a major source of degrading target performance in direct-drive inertial confinement fusion (ICF) Li et al. (2004); Hu et al. (2016); Shah et al. (2017). In direct-drive, laser energy absorbed in the hot, low density, near-critical plasma must be effectively transported, predominantly via electron thermal conduction, towards the interface between hot expanding plasma and the cold capsule; the ablation surface. Irradiation nonuniformities imprint themselves onto the cold capsule surface during this ablation phase, where they can seed hydrodynamic instabilities, degrading target performance. The detrimental effect of nonuniform energy deposition is counteracted via thermal smoothing in the conduction zone, the region between the critical and ablation surfaces, and dynamic overpressure stabilisation Nuckolls et al. (1972); Goncharov et al. (1996); Sanz (1996) at the ablation surface. The conventional view of thermal smoothing is that the electrons transporting energy from the critical surface to the ablation surface conduct some heat sideways during transit. The lateral thermal conduction, according to the cloudy day model Brueckner and Jorna (1974), should result in an exponential attenuation of the pressure perturbation amplitudes, . This picture neglects both kinetic effects and magnetic fields, which can severely alter both the magnitudes and directions of heat fluxes Bell (1985); Braginskii (1965).

Self-generated magnetic fields have been measured in ablation phase ICF experiments Manuel et al. (2012); Igumenshchev et al. (2014) and are predicted to be important in a variety of ICF relevant conditions Walsh et al. (2017); Joglekar et al. (2014). Crossed number density, , and temperature, , gradients, that occur at perturbations in the laser energy deposition, generate magnetic fields through the Biermann battery mechanism, Stamper et al. (1971); Biermann and Schlüter (1950). The Nernst effect Nishiguchi et al. (1984) then advects these fields with the electron heat flux, , (at the velocity Haines (1986)) into the conduction zone and simultaneously, compressively amplifies them.

The temperature scale length within the conduction zone is typically on the order of the electron mean free paths. Under such conditions the classical (Braginskii) heat transport model Braginskii (1965) breaks down. At breakdown the transport becomes ‘non-local’ Bell (1985), it is no longer uniquely determined by the local temperature gradient. Experimental measurements Gregori et al. (2004); Gotchev et al. (2006); Hu et al. (2008) have demonstrated that non-local heat transport effects are important in nano-second time scale laser-solid interactions, and must be taken into account to align ICF simulations with experimental predictions of laser absorption Seka et al. (2008) and implosion dynamics Hu et al. (2008). It has been shown that there can be a significant interplay between the non-local heat flux effects and the magnetic field dynamics Joglekar et al. (2014); Ridgers et al. (2008). The magnetised heat transport effect dominant in this study is the Righi-Leduc heat flow, , where represents the magnetic field unit vector. This is the heat flow, with thermal conductivity , generated perpendicular to a temperature gradient, , due to the Lorentz force acting upon the heat carrying electrons. In this work, a combination of the Nernst advection and amplification of magnetic fields and the Righi-Leduc heat flow is found to invert and enhance perturbations within the conduction zone.

A key source of irradiation nonuniformity is irregularity within the laser beams. A variety of beam smoothing techniques (RPPKato et al. (1984),SSD Skupsky et al. (1989),ISI Lehmberg et al. (1987) etc.) are employed on laser systems to mitigate this. The smoothed beams are composed of a series of speckles, rapidly varying in time and space, such that they appear smooth over plasma response times and hydrodynamic length scales. Kinetic studies, neglecting magnetic fields, have been performed examining the degree of thermal smoothing for both single mode perturbations Epperlein et al. (1988) and optically-smoothed beam like perturbations Williams et al. (1991); Keskinen (2009, 2010). Full-physics hydrodynamic simulations of ICF targets subject to nonuniform irradiation have also been performed in two and three dimensions Demchenko et al. (2015); Igumenshchev et al. (2016, 2017), while the solid density target response to pressure perturbations is studied in Ishizaki and Nishihara (1997); Goncharov et al. (2000); Gotchev et al. (2006). The effects of magnetic fields on smoothing of single mode perturbations has also been studied by Bell et al.Bell and Epperlein (1986) and Sanz et al. Sanz et al. (1988) within a linearised hydrodynamic framework. Self-magnetisation of individual speckles has been predicted Dubroca et al. (2004); Thomas et al. (2009) and the collective magnetic field effects of a time evolving pattern of many speckles has been investigated with a reduced Braginskii transport model Rahman (1997) but has not been studied kinetically until now.

In this letter we aim to investigate the effect of magnetic fields and ablating ions on the degree of thermal smoothing. 2D kinetic simulations of a planar ablating target irradiated by a perturbed laser drive are carried out with the fully implicit Vlasov-Fokker-Planck code, IMPACT Ridgers et al. (2008); Kingham and Bell (2004). Two different types of heating perturbation are applied, a static single mode perturbation and a dynamically evolving pattern to mimic the speckles of an optically smoothed laser. Even for the small Hall parameters observed (, for an ablating plasma subject to a heating perturbation, ), self-generated magnetic fields have a significant effect on both the fluid and heat flow dynamics within the conduction zone. The magnetic fields cause an inversion and enhancement of the pressure perturbation amplitude for the single mode perturbation, displayed in Fig. 1. This inversion occurs, regardless of the hydrodynamic response of the plasma, and is distinct from perturbation oscillations that can occur as a result of dynamic overpressure stabilisation of the Rayleigh-Taylor instability at the ablation front Nuckolls et al. (1972). In the speckle pattern simulation, magnetic fields also increase the degree of nonuniformity at the ablation surface, resulting in up to an order of magnitude reduction in the degree of thermal smoothing.

Figure 1: (a) Perturbation amplitudes progressing down the temperature gradient, shown in (b), for a single mode perturbation simulation with and without magnetic fields at \SI372\pico\second. (c) is the profile after \SI372\pico\second, with dashed lines indicating slices from which the perturbation amplitude is sampled. (d), transverse gradient of the Righi-Leduc heat flow after \SI372\pico\second, dark region indicates Righi-Leduc induced heating causing the mode inversion.

The Vlasov-Fokker-Planck equation (VFP) for the electrons, Faraday’s and Ampère’s laws for the electromagnetic fields, and , and the magneto-hydrodynamic momentum equation for the cold ions are used to model the plasma. The electron distribution function is assumed to be weakly anisotropic and its Cartesian tensor expansion Shkarovsky et al. (1966) is truncated at .

Ion outflows, at upwards of \SI100\kilo\meter\per\second, are a key characteristic of ablating plasmas. These outflows are critical in correctly modelling the magnetic field dynamics within the conduction zone. B-field advection is a balance between frozen-in flow with the ions, the Nernst advection with the heat flux, and advection down resistivity gradients. The ablating plasma flows also alter the net energy flux, modifying the enthalpic heat flow, however this proves to be a less important factor. In order to include the ablation, inflow and outflow boundary conditions were implemented in the code. At the inflow, electrons are assumed to be in thermodynamic equilibrium; the isotropic part of the distribution function, , is forced to a Maxwellian with a constant number density, , and electron temperature, . Bulk plasma flow velocity, , normal to the boundary is set such that mass flux is conserved through the inflow. The internal boundary is assumed unmagnetised. At the outflow, linear extrapolations were used for and , while was extrapolated quadratically. An additional region of steady-state flow was added to the coronal plasma to ensure the outflow boundary did not impinge on the conduction zone physics. A 1D radiation-hydrodynamics simulation using the code HELIOS MacFarlane et al. (2006) of a planar ablating CH foil, with mean atomic number , was performed to simulate the earliest stages of ablation, in which ionisation and radiation transport physics are important. Profiles for , and , taken from this HELIOS simulation, displayed in Fig. 2, were used as initial conditions for the 2D IMPACT simulation. An inverse bremsstrahlung heating operator Langdon (1980) was used to model the perturbed laser drive, with mean intensity, \SI2.5e14\watt\per\centi\meter^2 and laser wavelength, = \SI351\nano\meter.

Figure 2: Initial temperature, , number density, and ion velocity profiles,, used to initiate the IMPACT simulation, between the vertical dashed black lines. is the laser absorption profile. Profiles are taken from \SI0.25\ps into a 1D radiation hydrodynamics simulation using the code HELIOS MacFarlane et al. (2006).
Figure 3: Ratio of the ablation surface nonuniformity to the critical surface nonuniformity as function of time for a static single mode \SI55\micro\meter perturbation and a dynamically evolving speckle pattern, \SI5\pico\second, with and without magnetic fields included.

The integrated smoothness of the ablation pressure, , introduced by Epperlein Epperlein et al. (1988) is defined as,

(1)

The ratio of this parameter’s values at critical, and ablation surfaces, , as a function of time for a \SI55μ\meter wavelength perturbation are displayed in Fig. 3. Magnetic fields assist smoothing before \SI111\pico\second in the static single mode case but have a detrimental effect on perturbations afterwards. For the dynamically evolving speckle pattern, the detrimental effect of the magnetic field sets in at an earlier time.

When B-field is included, the temperature perturbation inverts and grows through the weakly magnetised region. This can be seen in Fig. 1, in which the temperature perturbation amplitude is plotted for a selection of transverse slices along the temperature gradient for simulations with and without magnetic field after \SI372\pico\second. Once the perturbation has been seeded at the critical surface, magnetic field is advected into the conduction zone and amplified by the Nernst effect Nishiguchi et al. (1985). The conduction zone magnetic field generates an additional lateral heat flux towards the perturbation trough, the Righi-Leduc heat flow (). Once the B-field has developed significantly, sufficient energy is redirected by that the perturbation amplitude inverts and grows. This is clearly seen upon examination of the contribution towards . is large and negative in the centre indicating heating at the would-be perturbation trough, Fig. 1(d). Since the transport equations are not directly solved by a VFP code, the kinetic and (Fig. 5) have been reconstructed a priori from the distribution function. The derivation of the kinetic analogues of the classical Ohm’s law and heat flow equation Williams (2013); Luciani et al. (1985), that reproduce the correct classical expressions in the limit that tends to a Maxwellian, is presented in the Supplemental Material Sup ().

Mode inversion occurs regardless of amplitude modulation size. Inversion is exhibited for laser profile modulations down to the 1% level (the smallest tried). Inversion also occurs with no ion flow, when is forced to a Maxwellian (removing non-local effects), and for a broad selection of perturbation wavelengths, . Both B-field and temperature perturbation amplitudes are proportional to the degree of heating nonuniformity at the critical surface. Since a smaller temperature perturbation requires a proportionally smaller B-field modification of to achieve inversion, the mechanism is amplitude independent. For large modulations the lateral Nernst advection becomes important, compressing the B-field into the centre, increasing its peak value.

Figure 4: Numerical solution to reduced model, dashed lines, compared with simulation results, solid lines, for a selection of different perturbation wavelengths, , at the time of \SI223\pico\second. The kink in is the position of mode inversion.

A reduced mathematical model can be used to describe the mode inversion in the linear regime. Starting with the electron temperature equation and the induction equation, we assume perturbations of the form, . Perturbations in ion velocity and number density are neglected, and the time evolution of temperature is assumed negligible compared to the B-field evolution. To first order, the linearised equations are,

(2a)
(2b)
where

, , and are the dimensionless diffusive, Righi-Leduc heat flow and resistivity transport coefficients Epperlein and Haines (1986). On the right hand side of Eq. 2b, and represent resistive B-field diffusion and advection respectively. contains the Nernst amplification, resistive and hydrodynamic damping of the B-field, while is the Biermann battery source. is the electron-ion collision time, is the normalised collisionless skin depth, , and the asymptotic forms of the transport coefficients for small Hall parameters have been used Epperlein (1984). Zeroth order profiles are taken from the IMPACT simulations. Fig. 4, shows numerical solutions to the above equations (dashed lines) alongside the IMPACT perturbation amplitudes (solid lines) as a function of distance from the ablation surface, for a 1% perturbation after \SI223\pico\second. The boundary conditions used for Eq. 2 are, , and, , where and are the time averaged values of the perturbation amplitudes at the critical surface in IMPACT.

Figure 5: Lineouts of the kinetic and predicted classical heat flows and Nernst velocity magnitudes, , , and respectively, for a slice normal to the target surface in the singlemode, 100% perturbed, \SI55\micro\meter simulation. (a) displays the component, parallel to the bulk temperature gradient, (b) displays the transverse Nernst and heat flow components. The cross section was sampled from a transverse position \SI12\micro\meter into the simulation at a time of \SI223\ps.

The Nernst coefficient in the model has been suppressed by 35%, inline with the average reduction in the simulation results, which are compared with Braginskii predictions in Fig. 5. This results in a 4—5 fold reduction in peak B-field and brings the model into closer agreement with simulations in both amplitude and progression of B-field into the conduction zone. Kinetic modifications in the longitudinal and lateral components are both approximately proportional to the total deviation, that can be inferred from Fig. 5(a). and lateral nonlocal effects, therefore, approximately cancel on the right hand side of Eq. 2a to not significantly change the mode inversion threshold.

Qualitatively, the Nernst effect and heat flow deviate from the classical case in a similar fashion, compared in Fig. 5. The deviation of the Nernst velocity from its classical value tends to be more severe than for the heat flow. The peak longitudinal suppressions are  60% for the Nernst and  50% for the heat flow. The transverse heat flux is suppressed more severely than its longitudinal component at the top of the heat front Epperlein et al. (1988) and this is also the case for the transverse Nernst term. Hot electrons, with relatively long mean free paths, stream down the temperature gradient and preheat the cold dense plasma. This results in both heat flow and Nernst velocity values greater than classical predictions at the base of the heat front. The ratios between kinetic and classical calculations, subscripts and , take values between and , within the conduction zone.

Figure 6: (a), pressure perturbation amplitude, , as a function of wave number, , for a speckle run with and without magnetic fields, taken from a cross section from the ablation surface at \SI298\pico\second, the cloudy day model Brueckner and Jorna (1974) (C.d.) is also provided for comparison. (b), the difference in perturbation amplitudes between a speckle run simulation with and without magnetic fields, , as a function of distance from the ablation surface and wave number, at \SI298\pico\second. (c) Time evolution of the magnetic field for a dynamic speckle run. The speckle coherence time and typical radius are = \SI5\pico\second and \SI5\micro\meter, respectively.

Fig. 6(c) demonstrates the time evolution of the magnetic field for the simulation in which the heating is perturbed to mimic a dynamically evolving speckle pattern. A random set of electric field amplitudes, obeying Gaussian statistics, were generated at each time step then Fourier windowed in space and time domains Feugeas et al. (2008). The speckle coherence time was set at \SI5\ps and the typical speckle radius is \SI5\micro\meter.

Magnetic fields preferentially enhance lower wave number perturbations reaching the ablation surface, the amplitude wave number spectrum is compared in Fig. 6(a) alongside the cloudy day model (C.d.). The reason for this is two fold, the dominant contribution is the dependence of the resistive damping term in Eq. 2b, , a secondary cause is the scaling of the Biermann battery source term . This effect is also observed in the static single mode simulations, in which lower mode perturbations exhibit substantially higher B-fields, Fig. 4(b). It is therefore concluded that, although the earliest times are not simulated here, the mechanism presented may lengthen the decoupling time Goncharov et al. (2000) of medium to longer wavelength modes, .

In summary, 2D kinetic simulations of a planar-ablating CH foil have been performed, including, for the first time, both magnetic field effects and realistic ablating outflows. Once enough time has passed for the magnetic field to be advected into the conduction zone and amplified, the magnetic field enhances pressure perturbation amplitudes in both the case of a single mode perturbation and a time evolving speckle pattern. Even for the small Hall parameters seen here, the transverse Righi-Leduc heat flow is on the order of the transverse diffusive heat flow and is sufficient to cause an inversion of a static single mode perturbation and to distort the heat front. This mode inversion mechanism is robust, occuring over a wide range of laser non-uniformity amplitude. Magnetic fields are more detrimental to lower wave number perturbations as they are less susceptible to resistive damping. The effects of changing speckle pattern coherence times, different plasma regimes and how magnetic fields will interface with hydrodynamic instabilities at the ablation surface, will be the subject of further work. This work highlights the need for the inclusion of self-generated magnetic fields and kinetic effects in ICF design calculations. The mechanism presented may alter the required tolerances for beam nonuniformity in ICF implosions and could be measured by experiment.

Acknowledgements.
This work was supported by the Engineering and Physical Sciences Research Council through Grant No. EP/J500239/1.

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