The Richtmyer-Meshkov instability of a double-layer interface in convergent geometry with magnetohydrodynamics

The interaction between a converging cylindrical shock and double density interfaces in the presence of a saddle magnetic ﬁeld is numerically investigated within the framework of ideal magnetohydrodynamics. Three ﬂuids of diﬀering densities are initially separated by the two perturbed cylindrical interfaces. The initial incident converging shock is generated from a Riemann problem upstream of the ﬁrst interface. The eﬀect of the magnetic ﬁeld on the instabilities is studied through varying the ﬁeld strength. It shows that the Richtmyer-Meshkov and Rayleigh-Taylor instabilities are mitigated by the ﬁeld, however, the extent of the suppression varies on the interface which leads to non-axisymmetric growth of the perturbations. The degree of asymmetry of the interfacial growth rate is increased when the seed ﬁeld strength is increased.


Introduction
The Richtmyer-Meshkov instability (RMI) refers to the instability of an interface between two fluids that is impulsively accelerated, usually by a shock

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wave [1,2].It occurs over wide ranges of length and time scales in technological applications and natural phenomena, such as supernova [3], combustion 5 [4], hypersonic air breathing engines [5] and inertial confinement fusion (ICF), a promising approach for fusion energy generation [6].In ICF, a small target filled with deuterium-tritium fuel mixture is heated by high-power lasers that drive an imploding shock into the target, compressing the fuel to a hotspot of sufficiently high temperature and pressure to initiate fusion reactions.In this high temperature and high energy-density scenario, the materials are expected to be in the plasma state, and thus may be influenced by a magnetic field.ICF experiments conducted on the Omega Laser showed that an external strong magnetic field might enhance the implosion performance by increasing the hotspot ion temperature and neutron yield [7,8].A two dimensional radiation-hydrodynamics numerical investigation found that the temperature and pressure of hotspots for ignition decreased under the influence of a strong magnetic field; it also found that the field might suppress the growth of hydrodynamic (HD) instabilities [9], such as the RMI and Rayleigh-Taylor instability (RTI) [10,11].The rapid growth of these instabilities on the target surface was responsible for the reduction of the energy production by breaking the spherical symmetry of the flow and severely degrading the final compression of the target [6].
In this investigation, we mainly focus on the evolution of RMI in converging flows under an external magnetic field within the framework of single-fluid magnetohydrodynamics (MHD), a fluid description of plasma dynamics.In planar geometry, the effects of an initial seed magnetic field on the RMI has been explored extensively.Samtaney [12] numerically studied the interaction between shock and inclined density interface in MHD, and found that the RMI was suppressed in the presence of a magnetic field.Wheatley et al. [13,14] investigated the case of the magnetic field perpendicular to the interface and showed that the mechanism of the suppression was attributed to the transport of the baroclinic voricity by MHD waves away from the density interface.For the case where the magnetic field was parallel [15] and oblique [16] to the interface, the RMI was also suppressed by the field.Samtaney [17] performed linear simula-

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tions of the RMI of double-interfaces in the presence of the magnetic field and showed that the growth rate of both interfaces decayed and oscillated around zero.In converging geometries, Bakhsh et al. [18,19] examined the evolution of instabilities via linear simulations in cylindrical geometry in the presence of normal and azimuthal fields and observed a transition from the early RMI phase to the RTI dominated phase (the distinction between RMI and RTI was most clearly seen in hydrodynamic cases).In addition, a significant suppression of the instability by a sufficiently strong magnetic field was observed.Mostert et al. [20] investigated the MHD RMI under the influence of two seed magnetic field configurations (uniform field and saddle field) in cylindrical and spherical converging flows.It showed that the extent of RMI suppression was not strongly dependent on field configuration, but that the saddle field resulted in a lower degree of implosion distortion or asymmetry.These prior nonlinear single fluid MHD investigations of the converging RMI culminated in a proposed octahedrally symmetric magnetic field configuration in 3D simulations [21].The octahedrally symmetric field suppresses the instability comparably to the other previously considered seed field configurations for light-heavy interface accelerations while results in a higher degree of symmetry of the underlying flow even at high field strengths.These results reveal that the applied field of higher symmetry degree helps maintain the symmetry of imploding flow of single interface, which is very possibly for double interfaces.
In this work, we continue the thread of these previous investigations in 2D and hence exclude the investigation of octahedral 3D fields.In our work, motivated by the presence of multiple density interfaces in ICF, we numerically investigate the RMI when a converging cylindrical shock interacts with two interfaces separating fluids of three different densities (referred to as a double density layer) in the presence of a saddle-topology seed magnetic field.The nonlinear interactions present in the double density layer are significantly more complex than that for a single density interface, and there is nonlinear coupling between the two interfaces.We believe it is important to determine whether the presence of a seed magnetic field still effectively suppresses the interfacial growth and how the essential physical suppression mechanism, vorticity transport via MHD waves, is influenced by the double density layer.It's worth mentioning that Mikaelian [22,23] has linearly investigated RTI and RMI in stratified cylindrical and spherical concentric shells with incompressible hydrodynamic models.
For double interface case, he noted that perturbations fed through from one interface to another when the shell was thin in the early stages of its evolution, while the coupling between interfaces decreased as the implosion thickened the shell, causing perturbations at each surface to grow independently.We observe the feedthrough phenomenon in the early stages when the shell is very thin, however we haven't observed the shell thickening by implosions.This difference may due to nonlinear effects which were not considered in Mikaelian's model.
Wheatley et al. [13,15] developed analytical MHD incompressible models in Cartesian geometry for impulsively accelerated interfaces.Incompressible models for RMI/RTI in converging geometry with double interfaces in MHD entail several technical difficulties and are outside the main scope of this paper.The remainder of this paper is organized as follows: In Section 2, MHD equations and initial setup of the problem are introduced, along with a brief description of the numerical method.In Section 3, simulation results and their interpretation are presented.Conclusions are presented in Section 4.

Physical setup
The initial physical setup of the problem is shown in Fig. 1: two density interfaces (DI1 & DI2) centered at the origin with mean radii r 1 = 1.0 and r 2 = 0.5, respectively, are perturbed with a single-mode of azimuthal wavenumber k = 32, and an amplitude equal to 4% of their wavelength λ.The perturbed interface radii ζ i for DI1 and DI2 are given as,

Numerical method
Following several of the previous investigations (e.g.Refs.[15,21]), we employ the ideal MHD model for this study.The dimensionless variables are defined as: where ρ, p, u and B are the density, pressure, velocity and magnetic field, respectively; and µ 0 is the permeability of free space.Neglecting the effect 115 of body forces and dissipation, the nondimensionalized ideal MHD equations with above notations can be written as follows [24], with the carets omitted for simplicity, where the specific heat ratio is fixed as γ = 5/3 throughout this study.In addition, we have divergence free constraint of the magnetic field, i.e., Because of the discretization errors, ∇ • B can be non-zero and may increase with time leading to unphysical results [25].Thus, this constraint should be 120 numerically satisfied all the time during the simulation.A second-order nonlinear compressible finite volume code developed by Samtaney [26] is applied to solve the ideal MHD equations expressed in strong conservation form, using an unsplit upwinding scheme with a Roe flux solver.A projection method is used to enforce the divergence free constraint of the magnetic field [25].In  clearly in Fig. 3, which shows the vorticity field near DI1 for two cases with perturbed interfaces (these slow waves are too weak to be seen in the x − t diagram in Fig. 2).Unlike the IFS, the ISS has little influence on DI2 since the waves generated during the ISS-DI1 interaction are too weak to produce any discernible interactions, but it shows an obvious impact on the banded structure which arises from the perturbations on the interfaces.Since the density in region II of case AL is higher than that of case BL, the TS travels faster in this region of case BL and converges to the origin at an earlier time.

Density and vorticity evolution
The density plots of case AL (t = 0.67) and case BL (t = 0.51) are shown in Fig. 4. At these times, the fast shocks have travelled across the two interfaces while the reflected waves generated from the TFS-DI2 interaction have not yet interacted with the converging DI1.A visual inspection of the density images make it apparent that the amplitude of the perturbations of the MHD case is smaller than that of the HD case, which suggests that the magnetic field does suppress the RMI, although not completely.Due to the lack of axisymmetry of the saddle field, the extent of the suppression varies on the interface, which leads to the non-axisymmetric growth of the perturbations.It appears that the perturbations are most diminished at ϕ ≈ π/4 where the field is nearly parallel to the interface and becomes the largest near region ϕ = 0 where the field is nearly perpendicular to the layer.We draw attention to the "bulge" formed on the CD at ϕ = π/4 under the seed field, which results from the non-axisymmetry of the ISS.Under the effect of the seed field, the radial velocity of the ISS is minimum at ϕ = π/4, which leads to the formation of a curvature singularity on the ISS at this point resulting in two reflected shocks.The high pressure behind these two shocks pushes the CD out and forms the "bulge" to balance the pressure [28].A stronger field strength results in a more pronounced "bulge".
The suppression of the instability is attributed to two related effects:one is the transport of baroclinically generated vorticity, created at the interface during the shock-interface interaction, away from the interface by MHD waves.This is illustrated by the vorticity distributions shown in Fig. 5

Effect of magnetic field
To quantitatively study the effect of magnetic field on the growth of amplitude of perturbations, it is convenient to consider the amplitude of perturbations in different azimuthal sectors demarcated by different intervals of ϕ.  is from a linear simulation of cylindrical RMI, the other is the initial growth rate calculated through Lombardini-Pullin (LP) model [29].To aid the quantification of the effect of the magnetic field and dependence on the angle, we denote the sector where ϕ ∈ [0, π/16] as "low-ϕ" region and the section where ϕ ∈ [3π/16, π/4] as "high-ϕ" region.Labels "a", "b", "c", "d" and "e" denote specially chosen time instances for both HD cases, also approximately coinciding with times for MHD cases, in the high-ϕ region.Label "a" represents the time when the IS interacts with DI1, "b" is the time when the reflected wave from TS-DI2 interaction hits DI1, "c" is the time when the reflected shock from origin interacts with DI1, "d" is the time when TS-DI2 interaction happens, and "e" represents the time when the reflected shock from origin interacts with DI2.In all cases, it shows that the amplitude of the perturbations is decreased by the magnetic field.
Because interface DI1 in case AL, and interfaces DI1 and DI2 of case BL are light-to-heavy interfaces, the evolutions of the perturbations appear similar.
After time instance "a" or "d", the amplitude of the perturbation grows rapidly to a peak followed by its decrease.This is attributed to the competition mechanism between RMI and RTI.The initial impulse due to the shock wave leads to the RMI but as the shock interface decelerates radially, the RTI manifests itself, driving growth in the opposite direction.In the same sector, a stronger field strength leads to a lower peak amplitude.For the same field strength, the peak amplitude value in the high ϕ region is less than that in the low ϕ region, since RMI is more highly suppressed where the field is close to parallel to the

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interface [20].In addition, the oscillation of the amplitude in high ϕ region indicates the existence of phase inversion, which is observed in Fig. 7(b).The small change of the amplitude at "b" comes from the interaction between DI1 and the reflected wave, which turns out to be the expansion wave for case AL and shock wave for case BL.Table 2 shows the maximum normalized amplitude of perturbations on DIs before time slot "c" or "e" in the low ϕ region.It shows that the extent of the suppression of case AL is smaller than that of case BL, especially for DI1.increases rapidly due to the interaction of the asymmetric IFS with the initially symmetric interface, while the mean radial position of DI almost remains unchanged.Consequently, the stronger magnetic field is, the larger asymmetry degree becomes, as indicated in Fig. 9.The degree of asymmetry is large under strong field strength.In such situations, ISS-DI interaction becomes strong, resulting in a severe distortion of the DI, as can be seen in Fig. 5(c).

Effect of layer thickness
After shock interactions have occurred in the MHD cases, the perturbation amplitudes are different in each sector due to the asymmetric RMI suppression induced by the magnetic field.We introduce another measure of the perturbation amplitude η defined as the root-mean-square of the perturbation amplitude in each sector.Fig.  Eventually the perturbation amplitude η reaches a peak and then decreases due to the dominance of the RTI, which drives the perturbations towards a phase inversion.We note that the duration of the RMI phase is positively correlated to the layer thickness, i.e., as d 12 increases the peak amplitude and subsequent decrease in amplitude occurs later in time.The competition between the RTI and the RMI phases results in a non-monotonic behavior, i.e., the peak amplitude is reached by the layer of medium thickness.When d 12 is small, the amplitude growth reinforcement due to the reflected rarefaction driven RMI from DI2 is also small.Although the extent of this RMI reinforcement for AL cases is stronger than that of AM cases, the duration over which RMI dominates RTI is reduced (since the RMI reinforcement happens), and this eventually makes the maximum amplitude of AL cases smaller than that of AM cases (see Table 3).
For finite β, the RMI growth is suppressed due to the magnetic field and the suppression mechanisms discussed above.The effect of the layer thickness follows the same trend as in the HD cases, i.e., the medium layer thickness exhibits the largest growth.For DI2 of case A in HD, the shock interacts with a heavy-to-light interface and leads to a phase inversion of the amplitude.The second interface being closer to the origin experiences a stronger shock (due to convergence effects) for the larger layer thickness cases.In Fig. 10(b), it is seen that the medium and large layer thickness cases show a similar growth in DI2 perturbation amplitude until τ ≈ 0.28, while the small layer case exhibits substantially small amplitude.The largest layer width case experiences the highest early DI2 perturbation growth rate, but quickly experiences a reshock at τ ≈ 0.28 (the reshock is seen as a sharp decrease in the amplitude in these plots).Consequently, the largest overall growth is experienced for the medium layer interface, as shown in Table 3.A similar behavior occurs for DI2 in the

Figure 1 :
Figure 1: Initial setup of the density interfaces (DI1 and DI2) and the Riemann interface (RI) that drives the converging shock.The contours of the initial seed saddle magnetic field are superimposed.

Figure 2 : 2 .
Fig. 5(d) Fig. 5(c) 125addition, our code has adaptive mesh refinement capability using the ChomboM A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPTframework[27]. From symmetry considerations, the computation is performed on a quarter-domain, for 0 < x, y < 2 .All the simulations use a coarsest mesh at resolution 256 2 with two levels of refinement, with the refinement ratio of 4 in each direction for an effective resolution of 4096 2 .The criterion for refinement, based on the local density gradient, is |∇ρ| > 0.02ρ.This mesh is sufficiently refined to resolve the perturbation amplitude according to the study by Mostert et al.[20].Hence we omit details of other convergence tests here.

Figure 4 :
Figure 4: Density plots of case AL (at t = 0.67) and case BL (at t = 0.51) overlaid with magnetic field lines.

Fig. 6 Figure 6 :Figure 7 :
Fig.6shows the vorticity plots of case BL-8 at different time.In this case, both DI1 and DI2 are light-to-heavy interfaces, thus, the behavior of DI2 is similar to that of DI1 if the influence of the ISS-DI1 interaction is not taken into account.At t = 0.51, the incoming fast waves have interacted with both DI1 and DI2.The slow shocks attached to each DI transport the vorticity away

Fig. 8
shows the evolution of the normalized amplitude η/η 0 of DI1 and DI2 in two sectors 255 for all cases.The results are compared with two reference amplitude curves: one M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Figure 8 :
Figure 8: Evolution of amplitude normalized by the initial amplitude of DI1 and DI2 for cases AL and BL at different time: (a) when IS interacts with DI1; (b) when the reflected wave from TS-DI2 interaction hits DI1; (c) when the reflected shock from the origin interacts with DI1; (d) of TS-DI2 interaction; (e) Time when the reflected shock from the origin interacts with DI2.For comparison, the figures show the amplitudes predicted by the linear impulse models for both interfaces.

Figure 9 :
Figure 9: Evolution of asymmetry of DI1 and DI2 of case AL and BL with different magnetic field strengths.Note that early (late) time corresponds to high values (low values) of the mean radial coordinate r.

For
the HD cases, the flow is symmetric, thus ζ(r) = 0 should be satisfied all the time.In our HD simulations, ζ(r) = 0 is only approximately true due to a mild asymmetry in the adaptive meshes.For MHD cases, the initial saddle magnetic field causes the IFS speed to vary depending on the local field conditions, breaking the axisymmetry.Thus during the IFS-DI interaction, ζ(r)

Figure 10 :
Figure 10: Evolution of root-mean-square perturbation amplitude of DI1 and DI2 of cases with different thicknesses.τ = t − t p , where t p is the time when a shock interacts with DI.
) was larger than that of case A (light-to-heavy for the first interface and heavy-to-light for the second).The saddle field increased the asymmetry degree of the interface by distorting it through the ISS-DI interaction.The effect varying the layer thickness was also examined and the case of medium layer thickness generally experienced a larger growth compared with smaller or larger layer thickness cases due to the competing effects of stronger initial shock interactions, as the inner density interface was moved closer to the origin, reflected wave driven RMI, and RTI onset.

Table 3 :
Peak root-mean-square perturbation amplitude of various cases.
amplitude.Fig.9comparesζ(r) for cases AL and BL with different magnetic field strengths before DI interaction with reflected shocks from origin (reshock).
10 compares the growth of relative root-mean-square am-