2.1 Vorticity and magnetic field self-generation

The vorticity ( $\unicode[STIX]{x1D74E}$ ) equation can be derived by taking the curl of the velocity evolution equation, obtained via (2.10), thus giving,

From (2.12), where the divergence mitigation terms have been omitted ( $\unicode[STIX]{x1D6E4}_{B}=0$ ), we see that $(\unicode[STIX]{x2202}\boldsymbol{B}/\unicode[STIX]{x2202}t)=-c\unicode[STIX]{x1D735}\times \boldsymbol{E}$ which leads to

Here, the first term on the right-hand side is the so-called Biermann battery effect (Schluter & Biermann Reference Schluter and Biermann1950), considered to be the source of self-generated magnetic fields. This shows that self-generated magnetic fields are intrinsic to the two-fluid plasma equations, and that the Biermann battery source term is directly proportional to the baroclinic generation of fluid vorticity, as is the case in the single-fluid model with a generalised Ohm’s law (Evans Reference Evans1986). The link between fluid vorticity and self-generated magnetic fields in the two-fluid model is not explored in depth here as our focus is on the influence of two-fluid effects on the RMI.

2.2 Numerical implementation

Numerical solutions were obtained in one- and two-space dimensions using the adaptive mesh refinement (AMR) framework Chombo (Adams et al. Reference Adams, Colella, Graves, Johnson, Keen, Ligocki, Martin, McCorquodale, Modiano and Schwartz2015). The base numerical method consists of a spatially second-order accurate finite-volume scheme with a two-stage, second-order Runge–Kutta time integrator (Gottlieb, Shu & Tadmor Reference Gottlieb, Shu and Tadmor2001) with minmod limited interface values for flux calculation. Species and electromagnetic fluxes are calculated with the HLLC (Harten–Lax–van Leer–Contact) (Toro, Spruce & Speares Reference Toro, Spruce and Speares1994) and HLLE (Harten–Lax–van Leer–Einfeldt) (Einfeldt Reference Einfeldt1988) approximate Riemann solvers, respectively. The contribution from source terms is solved locally with the implicit method of Abgrall & Kumar (Reference Abgrall and Kumar2014). In order to track the density interface present in the RMI simulations a volume-of-fluid approach is used.

3.1 Spatial convergence

Attaining the prescribed spatial order of convergence of a numerical method is an important component in the verification of a numerical implementation. According to § 2.2, convergence of second order is expected in smooth regions of the flow for the solver used in this work. This convergence rate is verified by calculating solutions to the steepening wave problem of Landau & Lifshitz (Reference Landau and Lifshitz1987) and comparing these numerical solutions to the analytical solution. The steepening wave problem is an initially smooth nonlinear process whereby a sinusoidally perturbed flow field steepens to form a discontinuity at time $t_{b}=|2/(u_{0}\unicode[STIX]{x03C0}(\unicode[STIX]{x1D6FE}+1))|$ . The test case, as implemented, consists of a one-dimensional flow domain $-1\leqslant x\leqslant 1$ , periodic in $x$ and $y$ , with initial conditions

where $u_{0}=0.1$ , $\unicode[STIX]{x1D70C}_{0}=1$ , $p_{0}=1$ , $\unicode[STIX]{x1D6FE}=(5/3)$ and $a_{0}$ is the sound speed. Solutions for the velocity field at a time of $t=0.9t_{b}$ are shown in figure 2, where the convergence rate is shown to be slightly greater than two (2.06) when calculated over an order of magnitude change in grid resolution. Practically identical results are also obtained for the pressure and density fields and are thus not shown for the sake of brevity. From this result, the underlying fluid solver used to evolve the two-fluid system of equations is shown to perform as expected.

Figure 2.

Comparison of the velocity field to analytical solution of the nonlinear steepening wave problem at $t=0.9t_{b}$ .

3.2 Solver components

As the two-fluid system of equations is a combination of the electromagnetically forced Euler and Maxwell equations, it is desirable to verify each component of the solver in isolation as well as in combination. This process also allows a relatively rigorous approach due to the lack of accepted benchmarks for the two-fluid system of equations. By setting the charge of each fluid species to zero ( $q_{i}=q_{e}=0$ ), or by setting the plasma length scales to infinity ( $d_{L}=d_{D}=\infty$ ), the two-fluid equations reduce to Euler’s and Maxwell’s equations in isolation. We thus define a Riemann problem centred at $x=0$ for the fluid species with left state $S_{0}$ and right state $S_{1}$ (see (4.2)) while the initial conditions for the electromagnetic fields are

The non-dimensional speed of light for this test is $c=10$ while all other properties are as given in § 4.

Figure 3.

Comparison of simulation results to analytical solutions at $t=0.06$ .

The numerical solution to this initial value problem is compared to the analytical solution in figure 3 at a time of $t=0.06$ with 2048 cells per unit length. From these results the Euler and Maxwell solvers are shown to capture the analytical solution quite closely with mean $L_{2}$ error ${<}10^{-3}$ for all cases. The density profile showing the ion shock exhibits a slight deviation on the downstream side of the wave which is caused by the initial shock profile not matching the final numerical profile. The ion shock as well as all features in the electron fluid have, however, been captured by the numerical method with numerical dissipation accounting for the smoothing of any discontinuities. The numerical electromagnetic solution is also shown to match the analytical solution quite well although in this case the numerical dissipation appears to be somewhat stronger. The isolated solvers are thus shown to capture the physics of the system as intended.

3.3 Comparison to reference two-fluid solutions

To compare with an existing solution of the two-fluid system of equations, the solution to an electromagnetic shock by Loverich (Reference Loverich2003) is used. In this test the plasma lengths are given as $d_{L}=1/300$ and $d_{D}=d_{L}/100$ with $c=100$ and $(m_{e}/m_{i})=(1/1836)$ . The simulation was run on a uniform one-dimensional grid of 1024 cells over a domain of $0\leqslant x\leqslant 1$ . The initial conditions for this test are

where results are shown at $t=0.1$ in figure 4. In this figure the solution from the present solver is able to reproduce the solution by Loverich with the location and magnitude of all features in the combined density profile being captured. The present solver does appear to have lower numerical dissipation. This comparison gives confidence in the numerical method which is thus considered suitable for the following work.

Figure 4.

Comparison of simulation results from the present solver to the numerical electromagnetic shock solution of Loverich (Reference Loverich2003).

4.1 Grid convergence

In the full two-dimensional numerical solutions presented subsequently, pointwise convergence is not expected as, in the absence of a physical dissipation mechanism, the vortex sheets present in the flow will continue to develop finer-scale structures as the grid is refined. There are, however, physical plasma length scales associated with high frequency oscillations that must be adequately resolved. These plasma oscillations occur in the one-dimensional (all quantities being functions of only $x$ and $t$ ) base flow underlying the two-fluid RMI problem, thus their grid convergence can be studied without the confounding influence of vortex sheet roll up in this setting. To this end, the code base was used to simulate the one-dimensional base flow of the two-fluid RMI problem, equivalent to an initially flat interface, discretised with $N_{c}$ cells per unit length. Figure 5 shows the ion and electron density fields at $t=0.3$ for $N_{c}=1024,2048,4096,8192$ demonstrating grid convergence. The relatively low convergence rate of order 1.11 reported in this figure reflects the flow field being dominated by waves that activate the minmod limiter in the numerical implementation, causing the solver to locally revert to first-order accuracy. When the waves activating the limiter are widespread enough that the cumulative error in their vicinity dominates the global error norm, the global order of convergence can drop to near unity (Wheatley, Kumar & Huguenot Reference Wheatley, Kumar and Huguenot2010). From the expanded region displayed in figure 5 it may be seen that a resolution of $N_{C}=2048$ captures the essential details of the flow, with the mean $L_{2}$ -norm of the relative difference from the finest grid solution being equal to $5.6\times 10^{-5}$ . The minimum effective resolution adopted for the two-dimensional simulations is thus $N_{c}=2048$ .

Figure 5.

(a) Ion and electron number density at a fixed time of $t=0.3$ from 1-D simulation, illustrating grid convergence. (b) Zoom in of detail showing $n_{e}$ with increasing $N_{c}$ . (c) Convergence rate with increasing $N_{c}$ in $L_{2}$ norm. Note presence of separate electron and ion shocks and plasma oscillations.

6.1 Flow evolution

A square celled mesh with effective resolution of $2048$ cells per unit length was used for the 2-D simulation of the $d_{L}=d_{D}=0.1$ case. At this resolution, a grid convergence study (see appendix A) indicated that the interface width evolution is within 0.11 % of the Richardson extrapolated exact solution. The overall dynamics of the flow is illustrated in figure 7, which shows a time sequence of the ion and electron number density fields. A contour showing the evolution of the density interface in the single-fluid case, which was computed by the same solver on the same grid, is overlaid on the plots of $n_{i}$ . Comparing the interface behaviour between the single-fluid and two-fluid cases, it is apparent that two-fluid effects have given rise to a wealth of new flow physics. This has, amongst other things, resulted in a substantially greater interfacial growth rate compared to the single-fluid case and the occurrence of secondary instabilities.

Figure 7.

Electron and ion flow evolution with $d_{L}=d_{D}=0.1$ in a stationary reference frame. These frames illustrate how the electron motion significantly distorts the ion shock. Hydrodynamic interface location is overlaid on ion plots.

Figure 8.

Zoomed in contour plots of electron number density interface instabilities for $d_{L}=d_{D}=0.1$ . The plot region is fixed in the laboratory reference frame and shown in the context of the full electron interface by the inset plot in first frame of each panel. (a) Electron Rayleigh–Taylor instability at the head of the spike at early time. The bulk motion of the interface is shown by comparison between these frames. (b) Electron Kelvin–Helmholtz instability at the transition between bubble and spike at intermediate time.

Due to their low inertia, the electron dynamics is much faster than that of the ions. It can be seen in figure 7 that the electron density interface grows more rapidly and develops secondary instabilities sooner. These gradually influence the high inertia ions though electromagnetic coupling. Understanding the evolution of the electron fluid is therefore critical to understanding the two-fluid plasma RMI. The electron density oscillations between the interface and the electron shock are apparent from early $t$ . As in the one-dimensional base flow, these oscillations result in a time-varying electromagnetic acceleration of the density interfaces which drive a secondary RTI.

The presence of the RTI is confirmed in § 6.4, where it will be shown that vorticity is baroclinically generated at the density interfaces continuously after the initial shock interaction, substantially increasing the growth of the first-mode interface perturbation. The electron RTI is first visible at high wavenumbers near the head of the spike (leftmost part of the interface). The RTI occurs in this region initially because it is the first part of the interface processed by the electron shock, and therefore the first to develop charge separation, an induced $x$ -electric field and consequent $x$ -acceleration. A close up of the interface development in this region is provided in figure 8(a). At $t=0.05$ , the first high wavenumber feature to significantly grow is only evident immediately adjacent to the domain centreline and has a wavenumber of $k_{y}\approx 220$ . The wavenumber of this feature was the same in a simulation with half the effective resolution carried out as part of the grid convergence study documented in appendix A. Note that although the interface was perturbed with a single mode, by $t=0.04$ the vorticity deposited by the electron shock has distorted the interface to the extent that the spectrum of the interface perturbation has spread over a wide range of wavenumbers, as shown in figure 9. Thus all interface modes supported by the computational grid ( $k_{y}<1024$ ) are available for amplification.

By $t=0.06$ , figure 8(a) shows a more widespread lower wavenumber electron RTI mode growing on the head of the spike, with a wavenumber of $k_{y}\approx 50$ . As the bubbles and spikes associated with this mode grow, they interact with the primary-mode vorticity deposited on the interface by the electron shock. This scales with the slope of the interface in $y$ , and is thus zero at the centreline and edges of the domain. The primary-mode vorticity distribution both stretches the perturbations and, with increasing distance from the centreline (within the frame of figure 8 a), causes the evolution of the perturbations to increasingly become a combination of the RTI and the Kelvin–Helmholtz instability (KHI). By $t=0.08$ , the heads of the associated bubbles and spikes have rolled up into discrete vortices. The fine-scale flow features described above are regularised by numerical dissipation, and thus they are sensitive to the grid. As the grid is refined, in the absence of physical dissipation, the vortex sheets in this region will further roll up to generate even smaller scale flow features (Samtaney & Pullin Reference Samtaney and Pullin1996). The highly inclined portions of the electron interface, where the primary-mode vorticity from the RMI is strongest, begin to exhibit growth of a secondary KHI mode at a wavenumber of ${\approx}$ 33 by $t=0.1$ . A close up of the KHI development in one of these regions is shown in figure 8(b). In the grid convergence study documented in appendix A, we have verified that this most-amplified KHI mode has the same wavenumber on a grid with half the effective resolution. The most-amplified mode of the KHI is thus unlikely determined by the grid spacing in the present study. Increasing the interface thickness was also investigated (see appendix B) and was shown to delay the evolution of the KHI, but did not appear to significantly alter the wavenumber of the primary mode, at least within in the range of thicknesses investigated. By $t=0.12$ , this KHI mode has grown and rolled up into discrete vortices. Subsequently, the vortices generated by the RMI and secondary instabilities begin to interact, and by $t=0.2$ the electron flow field in the vicinity of the interface has become chaotic, thus flow symmetry is consequently broken.

Figure 9.

Distribution of wavenumbers, $k$ , present in the electron interface at a time of $t=0.04$ , immediately after the electron shock transition. All of these modes are then present for amplification in the forthcoming electron instabilities.

At $t=0.2$ , figure 7 shows that the ion shock has just traversed the ion density interface, which still remains relatively smooth, in stark contrast to the chaotic electron interface at this time. While the time constant for the response of the heavier ion fluid to the forces induced by charge-density oscillations is much longer than for the electron fluid, by $t=0.5$ two notable effects are evident: First, the RTI of the ion interface is well developed at $t=0.5$ , causing substantially greater perturbation growth than in the single-fluid RMI case. The combined action of the RMI and secondary RTI have caused the development of the interface to enter the nonlinear regime in the two-fluid case. The high-wavenumber secondary instabilities present in the electrons are also evident in the ions, but take approximately an order of magnitude greater time to develop. Second, the ion shock, symmetrically perturbed by passage through the ion interface, is heavily distorted by the influence of the chaotic electron fluid. The ion shock does not simply pass through the electron field, the convection velocity of the electron vortices is sufficiently large that they are able to catch the ion shock from behind, pass through it and re-enter from upstream. Consequently, the ion shock encounters a temporally and spatially varying charge field that heavily perturbs the shock through the action of the resulting self-generated electromagnetic fields. The curvature of the perturbations to the ion shock surface then steepen, leading to the rapid formation of triple points. These shock–shock interactions generate reflected waves resulting in the cellular pattern in the ion density field behind the shock that is visible in the bottom left-hand panel of figure 7. At the same time the electron shock has degenerated almost completely into an oscillatory wave packet. This weakening is due to the effects of the electromagnetic fields as discussed in § 5. By $t=1$ the general structure and extent of the electron and ion density fields are similar, although the electron field has more small-scale content, indicating that approximate large-scale charge neutrality has been restored.

Figure 10.

For the $d_{L}=d_{D}=0.1$ case, charge density, electric fields in $x$ and $y$ and magnetic field in $z$ . Acceleration in $x$ due to the Lorentz force acting on the ions ( ${\mathcal{L}}_{i}$ ) and electrons ( ${\mathcal{L}}_{e}$ ). Ion and electron interface locations shown for context. Note that the colour bar range has been truncated and centred on zero, the full range for each plot is $\sum _{\unicode[STIX]{x1D6FC}}q_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}=[-5.299,6.4642]$ , $E_{x}=[-0.0071,0.0089]$ , $E_{y}=[-0.0051,0.0053]$ , $B_{z}=[-0.0014,0.0014]$ , $({\mathcal{L}}_{i}/\unicode[STIX]{x1D70C}_{i})=[-6.6205,8.9296]$ , $({\mathcal{L}}_{e}/\unicode[STIX]{x1D70C}_{e})=[-8.798\times 10^{4},519.18]$ .

6.2 Field evolution

To illustrate the evolution of the electromagnetic fields, and their resulting action on the fluids, plots of the non-zero components of electric and magnetic fields are shown in figure 10, along with the $x$ -acceleration of the ions and electrons due to the Lorentz force ( ${\mathcal{L}}_{\unicode[STIX]{x1D6FC}}/\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}$ ). The evolution of charge density is also shown to aid in understanding the electric field. The acceleration in $y$ is omitted in figure 10 as it is the $x$ -acceleration that is of primary interest due to the resulting RTI. For completeness, the acceleration in each direction due to the individual fields are shown in figure 11 for one particular instant, $t=0.04$ . All of the fields are entirely self-generated, according to (2.12) and (2.13).

Figure 11.

Accelerations of ions and electrons at $t=0.04$ due to the action of the electromagnetic fields in $x$ and $y$ . Ion and electron interface locations shown for context. (a) Ion accelerations in $x$ and $y$ at $t=0.04$ . (b) Electron accelerations in $x$ and $y$ at $t=0.04$ .

The nature of the initial conditions, Riemann problems that generate differing speed shocks in each species, encourages initial charge separation which immediately drives the generation of a strong electric field aligned with the fluid motion due to the resulting current density, as shown in (2.13). In general, the evolution of the charge density and the $x$ -electric field follows that in the one-dimensional case described in § 5, with added complexity due to the interface perturbation and subsequent instability. Following the initial charge density and electric field distributions established by differential shock and interface motion up to $t\approx 0.02$ , the highly mobile electron fluid is accelerated backwards by the field. Note that this acceleration initially occurs at the interface spike, since it is the first portion of the interface processed by the electron shock, and is in the Rayleigh–Taylor unstable direction, from heavy to light fluid. Due to the lack of dissipation in the system, the electrons overshoot the distribution that would neutralise the plasma, creating a deficit of negative charge downstream of the electron shock at $t=0.04$ , and an inversion of the sign of the electric field at the interface. From this point on, two main factors contribute to the charge-density distribution; the motion of plasma waves, which generate widespread net change, and the relative position of the electron and ion density interfaces, which creates a narrow band of net charge between them. The evolving charge distribution indicates that the process of overshooting electron fluid motion continues, causing oscillations in charge density and the sign of $E_{x}$ , which broadly results in an oscillatory $x$ -acceleration of the fluid in the vicinity of the interfaces.

At $t=0.2$ , the ion shock has just finished processing the ion density interface, and the transmitted shock has compressed a layer of high density ion fluid. This corresponds to the layer of net positive charge density to the right of the ion interface in the final row of figure 10. This positive layer is long lived compared to the negative layer formed downstream of the electron shock owing to the fact that the strength of the ion shock is not significantly abated during the simulation time, which in turn is a consequence of the slow ion dynamics. The presence of a long lived layer of net positive charge to the right of the ion interface results in $E_{x}$ becoming strongly negative at the ion interface. The consequent Lorentz acceleration of the ion interface is thus consistently negative for the remainder of the simulation, which is in the Rayleigh–Taylor unstable direction. It is interesting to note that despite the direction of the electric field, the electron interface is also subject to a negative $x$ -acceleration, which is therefore primarily caused by the influence of the magnetic field.

While the ion Lorentz $x$ -acceleration appears approximately equal to a scaled version of $E_{x}$ , the acceleration of the electrons is significantly modified by the magnetic field. This is illustrated in figure 11, which compares the accelerations of each fluid due to the electric and magnetic fields at $t=0.04$ . As expected, the species accelerations due to the electric field components differ by a factor of $-100$ , the product of the mass and charge ratios, since the particle acceleration scales with $q_{\unicode[STIX]{x1D6FC}}\boldsymbol{E}/m_{\unicode[STIX]{x1D6FC}}$ . The accelerations due to the magnetic field, however, are six orders of magnitude greater for the electrons. This is because in addition to the electrons having lower mass, the magnetic component of the Lorentz force is dependent on the fluid velocity, which is much greater for the electrons. For the ion fluid, the magnetic component of the Lorentz acceleration is three orders of magnitude smaller than the electric component and thus has little influence. For the electron fluid, on the other hand, the magnetic component peaks at the same order of magnitude as the electrical component and contributes significantly to the overall acceleration, particularly on the sides of the spike.

The coupled nature of the electric and magnetic field evolution can be observed when comparing $E_{y}$ and $B_{z}$ in figure 10. At $t=0.04$ , for example, differential charge motion parallel to the electron density interface, caused by the shock-induced vorticity on the electron interface while the ions are comparatively undisturbed, leads to generation of $\boldsymbol{E}$ parallel to the interface (with the primary component being $E_{y}$ ) according to (2.13). Since the induced electron velocity due to the vorticity distribution decays away from the interface, so too does the current and the consequently generated $\boldsymbol{E}$ . Therefore this interface parallel $\boldsymbol{E}$ has an interface normal gradient, resulting in its curl being non-zero, which results in the generation of $B_{z}$ according to (2.13). This self-generated magnetic field is not aligned along a direction causing instability suppression in the MHD case (Wheatley et al. Reference Wheatley, Samtaney and Pullin2005, Reference Wheatley, Samtaney, Pullin and Gehre2014). It should be noted that while this self-generated magnetic field does not have a major impact on the initially unmagnetised two-dimensional RMI considered here, the presence of such fields will become increasingly important in the magnetised and three-dimensional settings.

Figure 12.

Ion and electron flow evolution with $d_{L}=d_{D}=0.01$ in a stationary reference frame. Hydrodynamic interface location overlaid on ion plots.

Figure 13.

For the $d_{L}=d_{D}=0.01$ case, charge density, electric fields in $x$ and $y$ and magnetic field in $z$ . Acceleration in $x$ due to the Lorentz force acting on the ions ( ${\mathcal{L}}_{i}$ ) and electrons ( ${\mathcal{L}}_{e}$ ). Note that the colour bar range has been truncated and centred on zero, the full range for each plot is $\sum _{\unicode[STIX]{x1D6FC}}q_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}=[-2.7383,3.7094]$ , $E_{x}=[-0.0064,0.0083]$ , $E_{y}=[-0.0056,0.0060]$ , $B_{z}=[-0.0072,0.0068]$ , $({\mathcal{L}}_{i}/\unicode[STIX]{x1D70C}_{i})=[-62.435,117.98]$ , $({\mathcal{L}}_{e}/\unicode[STIX]{x1D70C}_{e})=[-7.604\times 10^{5},3.732\times 10^{3}]$ .

6.3 Reduced plasma length scale case

As $d_{L}$ and $d_{D}$ decrease, the single-fluid MHD quasi-neutral limit is approached (although it is only formally approached if electron inertia is also neglected). To investigate how the two-fluid plasma RMI of a thermal interface behaves closer to this limit, a case with reference $d_{L}=d_{D}=0.01$ was simulated with an effective resolution of $4096$ per unit length. At this resolution, a grid convergence study (see appendix A) indicated that the interface width evolution is within 0.09 % of the Richardson extrapolated exact solution. The evolution of $n_{i}$ and $n_{e}$ for this case is illustrated in figure 12. It is apparent that the electron and ion dynamics are much more strongly coupled in this case, with the result that the structure of the ion and electron density fields are very similar at each time shown. The primary difference occurs at $t=0.1$ , where the electrons develop small-scale structures on the spike that only weakly affect the ion density. The similarity of the number density distributions implies that the extent of charge separation is much smaller. This is verified in figure 13, which shows the electromagnetic fields and resulting accelerations for the $d_{L}=d_{D}=0.01$ case. As the extent over which charge separation occurs is reduced, consistent with the reduced reference Debye length, the induced electric and magnetic fields become more localised to the shock and interface locations. We emphasise that the electromagnetic effects seen in the larger plasma length scale place are still present, they just occur over smaller distances. Plots of $x$ -acceleration due to the Lorentz force confirm that the interfaces are subject to the same order of magnitude acceleration as in the previous case, which again drives the RTI. The influence of the electromagnetically driven RTI is thus reduced, however, leading to the perturbation growth exceeding that seen in the single-fluid case to a lesser extent than for $d_{D}=0.1$ , especially for the electrons. Overall, reducing the plasma length scales in the two-fluid case has caused the ion interface to develop high wavenumber content sooner, as a result of tighter coupling to the more unstable electron fluid, while the instability of the electron interface is restrained by the tighter coupling to the high inertia ion fluid.

In contrast to figure 10, the significantly more localised plasma features visualised in figure 13 generate more complex wave structures. This is particularly evident for $E_{x}$ where the initially uniform and diffuse field for the $d_{L}=d_{D}=0.1$ case is replaced by a highly oscillatory field at very early time. This complex wave structure interacts with the density interfaces at approximately the same time but results in reflected waves of reduced, but comparable, strength. This interaction leads to high wavenumber oscillations in both the $E_{x}$ and $E_{y}$ field leading to fine-scale acceleration and perturbation of the density interfaces at early time. The magnetic field, $B_{z}$ , maintains a diffuse nature as the reference plasma length scales decrease, albeit with increased magnitude. This increase in strength leads to greater influence on the flow, as seen in figure 15, and will be discussed later. At later times the chaotic nature of the flow around the interfaces leads to significant mixing in sign of the fields and accelerations leading to a reduction in the effect of concerted acceleration and the associated RTI.

6.4 Interface statistics

The extent in the $x$ -direction (referred to as the interface width, $\unicode[STIX]{x1D702}$ ) of the density interfaces shown in figures 7–13, as well as for the hydrodynamic reference case, are shown over time in figure 14. The growth rates of the ion species interface widths are also shown. The extent of the interface in this case is defined as the region where $0.05\leqslant \unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\leqslant 0.95$ , where $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\in [0,1]$ is the volume-of-fluid tracer for each species. The interface width is calculated by reducing the data to a one-dimensional representation by integration across the domain, then determining the $x$ -locations where the values of $\unicode[STIX]{x1D71A}$ defining the interface occur.

Figure 14.

Interface width and growth rate over time for the plasma conditions simulated. Numbers shown in the legend indicate the pertinent plasma length scale, where $\infty$ represents neutral species.

Figure 15.

Ion and electron circulation ( $\unicode[STIX]{x1D6E4}_{z}^{2F}$ ) compared to the hydrodynamic equivalent ( $\unicode[STIX]{x1D6E4}_{z}^{H}$ ) along with cumulative time integral of Lorentz force magnetic ( $\unicode[STIX]{x1D70F}_{B}$ ) and electric ( $\unicode[STIX]{x1D70F}_{E}$ ) components over half of the domain ( $y<0.5$ ). Values computed on the interface of each species.

Since the ion species contains the vast majority of the fluid mass, it is the instability of the ion density interface that is of primary interest in most applications. In ICF, for example, the ion species is the fuel in which a fusion burn is to be ignited. Thus we primarily focus our attention on the ion interface statistics. The results presented in figure 14 show that, following the shock traversal and compression of the interfaces, the two-fluid interface width outpaces the reference hydrodynamic width in both magnitude and rate for all cases except electrons at small plasma reference length. For the $d_{L}=d_{D}=0.1$ and $d_{L}=d_{D}=0.01$ cases, the peak non-dimensional growth rates of the ion interface widths are 0.57 and 0.47, respectively, while the corresponding hydrodynamic growth rate is 0.24. Thus the instability of the interface is substantially amplified in the two-fluid case. Consequently, the ion interface widths at the final simulation time for the $d_{L}=d_{D}=0.1$ and $d_{L}=d_{D}=0.01$ cases are 2.2 $\unicode[STIX]{x1D702}_{0}$ and 2.0 $\unicode[STIX]{x1D702}_{0}$ , respectively, while the value for the hydrodynamic case is substantially lower at 1.5 $\unicode[STIX]{x1D702}_{0}$ . For the two plasma regimes investigated, the growth rate of the ion interface returned to close to the hydrodynamic level by the end of the simulation, while the electron interface behaviour remained significantly different. When $d_{L}=d_{D}=0.1$ significant charge separation is allowed to occur due to the relatively large reference Debye length, this allows for the electron interface to grow more freely in response to its greater acceleration with less moderating influence from the inertia bound ions. For $d_{L}=d_{D}=0.01$ the ions and electrons are coupled much closer and so the electrons are effectively restrained by the more massive species leading to the significantly reduced interface growth relative to the alternative case.

In § 6, the occurrence of electromagnetically driven RTI in the two-fluid plasma RMI flow was discussed. The enhanced growth in the ion interface width was attributed to this. Here, we attempt to quantify the relative importance of the RMI and RTI by examining the source of the circulation present on the interfaces throughout the simulations. The ion and electron circulation on the lower half of the interface is computed by numerically integrating,

where $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}(a(y),y)=0.01$ and $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}(b(y),y)=0.99$ . Once an interface becomes chaotic, such that vortices are convected between the upper and lower halves of the interface, this quantity ceases to be a useful measure of the sources of circulation. Fortunately, this does not occur significantly in the ion species in the cases under consideration until late in the simulation duration ( $t>0.5$ ). The histories of $\unicode[STIX]{x1D6E4}$ for the two-fluid and hydrodynamic simulations are shown in figure 15 along with the cumulative integral of the electric and magnetic components of the curl of Lorentz force ( $\unicode[STIX]{x1D70F}_{E}$ , $\unicode[STIX]{x1D70F}_{B}$ ) acting on each species interface, which generate vorticity according to (2.16). For both two-fluid cases, the ion circulation production is initially dominated by the deposition of vorticity on the interface by the traversing shock. This appears as an approximately linear increase in $\unicode[STIX]{x1D6E4}$ during the shock traverse, is responsible for the RMI and closely matches the hydrodynamic case. After the shock interaction is complete, the circulation on the two-fluid and hydrodynamic interfaces diverge, with continued generation of circulation in the two-fluid cases as opposed to gradual dissipation in the hydrodynamic case. In the $d_{D}=0.1$ case, the peak non-dimensional circulation (0.21 at $t=0.53$ ) is close to double that in the hydrodynamic case (0.12 at $t=0.175$ ), and is only slightly lower (0.19 at $t=0.76$ ) in the $d_{D}=0.01$ case. The direct circulation generation due to electromagnetic forces on the ions, through the curl of Lorentz force, is found to be negligible for both cases. The influence is larger in the $d_{D}=0.01$ case and is due to the curl of $\boldsymbol{B}$ , and mildly opposes the growth of the interface. Thus the primary post-shock-interaction circulation generation is baroclinic in nature, as this is the only other source term in the vorticity equation, and results from the linear acceleration of the interface, as in the RTI. The $x$ -acceleration of the interface was demonstrated to be due to electromagnetic effects in § 6. The circulation generated in this electromagnetically driven RTI adds to that generated in the shock-interaction process and consequently increases the growth rate of the interface.

The circulation on the electron interface follows a similar pattern, at least initially, with a notable delay of the onset of circulation growth relative to the hydrodynamic case, especially for $d_{L}=d_{D}=0.01$ . This delay is due to the interaction with the ions in the two-fluid case, through the fields, slowing the electron shock relative to the hydrodynamic case. Following an initial increase, associated with the shock transition, there is a rapid rise beyond the hydrodynamic circulation level due to the initial backwards acceleration of the electron interface shown in § 6. Subsequently, the acceleration oscillates in sign until at least $t=0.2$ , which is reflected in the minor variations in circulation. Unfortunately, the electron interface becomes chaotic by $t=0.2$ for the $d_{L}=d_{D}=0.1$ case, and the circulation metric on the lower half of the interface becomes disrupted by vortices being convected in and out of the domain of integration. Thus the metric is not useful for the electron interface beyond this time. It is interesting to note that for the $d_{L}=d_{D}=0.01$ case, the curl of the Lorentz force has a significant influence on the electron interface circulation. The magnetic term initially generates circulation with sign opposing the baroclinically generated circulation, before augmenting it later in the interface development. The curl of the electric field, on the other hand, produces a substantial fraction of the circulation on the interface after $t=0.2$ .