Shock wave refraction at a sharp density interface is a classical problem in hydrodynamics. Presently, we investigate the strongly planar refraction of a magnetohydrodynamic (MHD) shock wave at an inclined density interface. A magnetic field is applied that is initially oriented either perpendicular or parallel to the motion of incident shock. We explore flow structure by varying the magnitude of the magnetic field governed by the non-dimensional parameter $\beta \u2208(0.5,106)$ and the inclination angle of density interface $\alpha \u2208(0.30,1.52)$. The regular MHD shock refraction process results in a pair of outer fast shocks (reflected and transmitted) and a set of inner nonlinear magneto-sonic waves. By varying magnetic field (strength and direction) and inclination interface angle, the latter waves can be slow shocks, slow expansion fans, intermediate shocks, or slow-mode compound waves. For a chosen incident shock strength and density ratio, the MHD shock refraction transitions from regular (all nonlinear waves meeting at a single point) into irregular when the inclined density interface angle is less than a critical value. Irregular refraction patterns are not amenable to an analytical solution, and hence, we have obtained irregular refraction solutions by numerical simulations. Since the MHD shock refraction is self-similar, we further explore by converting the initial value problem into a boundary value problem (BVP) by a self-similar coordinate transformation. The self-similar solution to the BVP is numerically solved using an iterative method and implemented using the p4est adaptive mesh framework. The simulation shows that a Mach stem occurs in an irregular MHD shock refraction, and the flow structure can be an MHD equivalent to a single Mach reflection irregular refraction and convex-forwards irregular refraction that occur in hydrodynamic case. For Mach number *M* = 2, both analytical and numerical results show that perpendicular magnetics fields suppress the regular to irregular transition compared to the corresponding hydrodynamic case. As Mach number decreased, it is possible that strong perpendicular magnetics promote the regular to irregular transition, while moderate perpendicular magnetics suppress this transition compared to the corresponding hydrodynamic case.

## I. INTRODUCTION

Understanding magnetohydrodynamic (MHD) shock refraction is important for any application involving shock waves and variable density flows, such as inertial confinement fusion (ICF),^{1} as well as astrophysical phenomena.^{2} In ICF, the surface of a spherical capsule is rapidly ablated, driving a converging shock into deuterium–tritium fuel contained within the capsule. When this shock interacts with the density interfaces within the capsule, the shock refraction process occurs and vorticity is deposited on these interfaces. This leads to the generation of Richtmyer–Meshkov instability (RMI) that promotes mixing between the capsule material and the fuel, which limits the possibility of achieving energy break-even or production.^{1} Moreover, the fluids involved become rapidly ionized and hence will interact with magnetic fields. In the astrophysical context, the knowledge of shock refraction has implications for phenomena shock interactions with the interstellar medium. The present work contributes to understand the MHD shock refraction process. A canonical physical setup to investigate shock refraction is shown in Fig. 1. The flow is characterized by the following parameters: the incident shock sonic Mach number *M* (fast magnetosonic Mach number for fast mode MHD shocks), the density ratio of the interface $\eta =\rho b/\rho 0$, the ratio of specific heats *γ*, the angle between the incident shock normal and interface *α*, and the nondimensional strength of the initially applied magnetic field $\beta \u22121=B2/2p0$, where *B* and *p*_{0} denotes the dimensionless magnitude of the applied magnetic field and the pressure of the gas.

In hydrodynamics, the regular shock refraction results in a transmitted *T* and a reflected *R* shock.^{3,4} On the other hand, the shock refraction becomes irregular as the inclination angle of interface is decreased below a certain critical angle and frequently accompanied by the presence of a Mach stem.^{5,6} Where all waves resulting from the refraction process meet at a point (triple point) and are planar, this is known as regular shock refraction; otherwise, the shock refraction is irregular. Shock refraction, both regular and irregular, have been explored in depth in hydrodynamics. At a fast–slow interface, experiments and simulations have shown that the irregular refraction structure includes a centered expansion type of refraction (CER), single Mach reflection type of refraction (MRR), and concave-forwards type of refraction (CFR).^{3,7} For strongly planar ideal MHD, Samtaney^{8} numerically investigated the shock refraction with the presence of a magnetic field that was initially oriented parallel to the motion of incident shock. The parameters in this investigation were: $M=2,\eta =3,\gamma =1.4,\alpha =\pi /4$, and *β* = 2 (see Fig. 1). Here, we define a flow to be planar if there are no derivatives in the out-of-plane (*z*) direction, and strongly planar if there is also a reference frame in which there is no vector component in the *z*-direction. In the investigation by Samtaney, there are a pair of reflected shocks and a pair of transmitted shocks as shown in Fig. 2(a). RS and RF are the slow-reflected and fast-reflected magneto-sonic shocks, respectively, whereas TS (slow or intermediate shock) and TF (fast shock) are the transmitted magneto-sonic shocks. Samtaney noted that the growth of the Richtmyer–Meshkov instability is suppressed in the presence of a magnetic field, since the vorticity is transported away from the density interface onto a pair of slow or intermediate magneto-sonic shocks $(RS$ and $TS)$. Consequently, the density interface is devoid of vorticity and its growth and associated mixing are suppressed. Subsequently, Wheatley *et al.*^{9} developed analytical solutions to the planar and strongly planar regular MHD shock refraction by fixing the inclination angle $\alpha =\pi /4$ and varying $\beta \u2208(2,107)$. They focused on regular refraction where all the waves meet at a single point. Furthermore, they showed that the resulting refraction structure might consist of five, six, or seven waves. The wave pattern may include fast, intermediate, and slow MHD shocks, slow compound waves, 180° rotational discontinuities (RDs), and slow-mode expansion fans. As in hydrodynamics, even for ideal MHD when all the waves do not meet at a single point, the refraction pattern is deemed irregular. In Fig. 2(b), we show the case of an irregular MHD shock refraction obtained numerically with the same parameters as the case in Ref. 8 except $\alpha =\pi /6$. A Mach stem separates RF and IS from the point that the other waves meet and the resulting refraction wave pattern is irregular. Note that, in the irregular refraction case, the Mach stem is not straight; that is, it is curved (similar to irregular shock refraction in hydrodynamics, which involves a Mach stem).^{3,7} Hence, the wave structure is too complicated to be amenable to an analytical solution, and we need to obtain the solution numerically. Later on, in the present work, we will discuss the numerical solution of a set of partial differential equations arising from a self-similar transformation.

One of the main challenges in the MHD shock refraction is the appearance of inadmissible waves, especially those of the intermediate type. In the three-dimensional MHD system of equations, the evolutionary condition^{10–12} restricts physically admissible discontinuities to fast shocks, slow shocks, contact discontinuities, and 180° RDs. The evolutionary of intermediate shocks has been extensively studied and is somewhat controversial. Falle and Komissarov^{13} demonstrated that a shock is physical only if it satisfies both the viscosity admissibility condition and the evolutionary condition. In their framework, for a planar system, fast and slow shocks are evolutionary and have unique structurally stable dissipative structures, and RDs are considered admissible, while all intermediate shocks are non-evolutionary and can be destroyed by interactions with Alfvén waves. This is in contradiction to the conclusion obtained via numerical tests by Wu.^{14–16} For a strongly planar system, Falle and Komissarov^{13} concluded that $1\u21923$ and $2\u21924$ intermediate shocks along with slow (noted as *C*_{1}) and fast (noted as *C*_{2}) compound waves are shown to be evolutionary and have unique dissipative structures. Both $2\u21923$ intermediate shocks and 180° RDs are found to be non-evolutionary. These results are in agreement with those of Myong and Roe.^{17} We choose the conditions from the work of Falle and Komissarov due to its completeness in order to develop our MHD shock refraction analytical solutions.

The previous work by Samtaney^{8} and Wheatley *et al.*^{9} focused on regular refraction wherein the magnetic field is initially parallel to the direction of propagation of the incident shock. In the present work, based on the method presented by Wheatley *et al.*,^{9} we present analytical solutions to the problem of strongly planar regular shock refraction in the presence of a magnetic field, which is initially perpendicular $B=(0,By,0)$ to the motion of incident shock $V=(vx,0,0)$ (hereinafter referred to as perpendicular magnetic field, whereas the case in Ref. 8 is referred to the parallel magnetic field case). While the regular refraction solutions can be obtained by analytical means, the only option to obtain the irregular refraction solution is via numerical simulations. Noting that, in the absence of viscosity, conductivity, and resistivity, the MHD shock refraction is self-similar (irrespective of whether the refraction pattern is irregular or regular) we further explore the phenomenon of MHD shock refraction by numerical solutions of a self-similar formulation of the governing partial differential equations. For this, we first convert the equations of ideal MHD governing the initial value problem (IVP) into a boundary value problem (BVP) by a self-similar coordinate transformation. The main advantage is that the IVP for inviscid flow problems with vortex sheets is ill-posed, whereas the self-similar BVP system is well-posed.^{18} The IVP does not appear to converge to a weak solution as the mesh is refined, while the self-similar solution seems to converge with decreasing mesh size to a weak solution. The numerical results exhibit a very high resolution around discontinuities (here, high resolution refers to a sharper or higher gradient approximation to a discontinuity). An interesting outcome of the self-similar transformation concerns solutions involving contact slip lines (aka vortex sheets). The self-similar transformation and the numerical method to solve the resulting BVP have been described in detail in the recent paper by Chen and Samtaney,^{19} wherein the authors employed the generalized Lagrangian multiplier GLM–MHD equations. Presently, the self-similar GLM–MHD equations are employed to investigate both regular and irregular refractions. To briefly summarize, a knowledge gap exists in the literature in that the irregular refraction of shocks at a density interface in MHD have not been explored. Moreover, we also expand the parameter space by investigating two magnetic field orientations, parallel and perpendicular to the incident shock propagation direction. The outline of this paper is as follows. In Sec. II, we briefly introduce the numerical and analytical approaches. We present the results and discussion for regular refraction in Sec. III, including different wave configurations resulting from varying strength of magnetic field *β* and inclination angle of interface *α* focusing more on the solutions obtained by analytical means. In Sec. IV, we present numerical simulations for irregular MHD shock refraction patterns and identify a couple of MHD shock refraction patterns that are similar to the MRR and CFR configurations noted in irregular shock refractions in hydrodynamics. Finally, a summary is presented in the concluding Sec. V.

## II. EQUATIONS, ANALYTICAL METHOD, AND SELF-SIMILAR FORMULATION

### A. Ideal MHD equations

The ideal MHD equations, appropriately non-dimensionalized, in multiple spatial dimensions are

*E*denote the fluid density, hydrodynamic pressure, and total energy per unit volume, respectively; and $v$ and $B$ denote the velocity vector and the magnetic induction. The ideal gas is considered in the present work, and the gas hydrodynamic pressure

*p*and the total pressure

*p*are given by

_{t}respectively. We seek solutions to the strongly planar ideal MHD equations for two-dimensional (2D) conditions.

### B. Analytical method

The MHD Rankine–Hugoniot (RH) relations govern weak solutions to the steady-state form of equations of ideal MHD [Eq. (1)] corresponding to discontinuous changes from one state to another. We assume that all dependent variables vary only in the direction normal to the shock front, which is denoted with the subscript *n*. We also assume that all velocities and magnetic fields are coplanar, as we are seeking strongly planar ideal solutions. Under these assumptions, the set of jump relations for a stationary discontinuity separating two uniform states is^{20}^{,}

*t*denotes the component of a vector tangential to the shock, and $[X]=X2\u2212X1$ denotes the difference in the quantity $X$ between the states upstream (subscript 1) and downstream (subscript 2) of the shock. We introduce a set of normalized variables

^{21}as follows:

where *θ*_{1} is the angle between the upstream magnetic field and the shock normal. We rewrite the Eq. (3) to obtain the following algebraic equations in *r* and *b*:^{22}

where detailed definitions of $A,B,C,X$, and *Y* are given in Wheatley *et al.*^{9} The intersections of the curves defined by $F=0$ and *Z* = 0 are the locations in (*r*, *b*) space where all jump conditions are satisfied. Subsequently, we can exactly calculate the downstream state of a discontinuity with given upstream state. The wave configuration of the MHD shock refraction can be shown as in Fig. 3. There are four unknown angles $\varphi 1,\varphi 2,\varphi 3$, and $\varphi 4$, which identifies the location of RF, RS, TF, and TS, respectively. If discontinuity is an expansion fan or compound wave, $\varphi $ defines the location of the trailing edge (or tail) of an expansion fan. The states 3 and 5 are the conditions to the left and right of the SC, and then, the following matching conditions must hold across the SC:

Here, adopting the nomenclature from Wheatley *et al.*,^{9} $|K|=K=\beta \u22121/2,\u2009Kn=K\u2009cos\u2009\theta $, and $Kt=K\u2009sin\u2009\theta $. The main procedure to find the solution is as follows. First, we postulate a wave configuration including four plane waves. Mathematically, this corresponds to selecting which root of the Rankin–Hugoniot relations will be used to compute the jumps across each shock. Therefore, the guessed wave angles and the given state 0 and *b* allow us to compute the conditions on either side of the SC. An approximate solution to the strongly planar MHD shock refraction problem is then obtained by iterating on the wave angles until the matching conditions Eq. (5) are satisfied to six significant figures. The full solution technique for the MHD regular shock refraction problem is documented in Ref. 9. In the strongly planar system, note that the fast and slow waves (shocks or expansion fans), contact discontinuities, $1\u21923$ and $2\u21924$ intermediate shocks, slow and fast compound waves are considered admissible according to Falle and Komissarov.^{13}

### C. Self-similar GLM–MHD method

In the approach of Dedner *et al.*,^{23} the divergence constraint of the magnetic field (Gauss law) is coupled to the Faraday induction equation by introducing a new scalar field function or generalized Lagrangian multiplier (GLM) *ψ*. The resulting GLM–MHD system in full dimensional system $x=(x,y,z)$ can be rewritten in conservative form as

where the conservative variables vector ** U**, associated fluxes vector $Fj$, and source terms

**are defined as below:**

*S*where $i=x,y,z$ stands for the different vector components, *δ _{ij}* is the Kronecker symbol, and $ch\u2208(0,\u221e)$ and $cp\u2208(0,\u221e)$. The additional equation of unphysical variable

*ψ*implies that divergence errors are propagated to the domain boundaries at finite speed

*c*and damped at a rate given by $ch/cp$. Note that the only source term occurs in the equation for the unphysical variable

_{h}*ψ*through the mixed hyperbolic/parabolic correction.

For the hyperbolic system of conservation laws Eq. (6) (without source term), $U\u2261U(x,t):\u211cm\xd7\u211c\u2192\u211cn$ and $F(U):\u211cn\u2192\u211cn$. Under the self-similar transformation for 2D $\xi (\xi ,\zeta )\u2261x(x,y)/t$,^{24} we eliminate the independent variable time *t* and transform the system that depends only upon the self-similar coordinates $U\u0303(\xi )\u2261U(x,t):\u211cm\u2192\u211cm,m=2$. The Eq. (6) system becomes

where the self-similar flux vector is defined as $F\u0303j=Fj\u2212\xi jU$. We add a source term of the same form $\u2212ch2/cp2\psi $ to the self-similar system in an *ad hoc* fashion because adding this source term does not change the nature of physics, and inclusion of this source term allows for additional control of the divergence errors. The unsteady IVP Eq. (6) is thus transformed into the steady BVP Eq. (8). The self-similar solution to the BVP Eq. (8) is solved using an iterative method and implemented using the p4est adaptive mesh refinement (AMR) framework.^{25,26} Existing Riemann solvers (e.g., Roe and HLLD) can be modified in a relatively straightforward manner and used in the present method. The details of the numerical scheme are presented in Chen and Samtaney.^{19}

## III. RESULTS: REGULAR REFRACTION PARALLEL FIELD CASES

In this section, we present numerical and analytical solutions for regular refraction of a fast MHD shock at an inclined density interface for the case of a magnetic field that is initially oriented perpendicular to the shock propagation direction. Two reference cases, denoted as $R1$ and $R2$, are discussed in detail below. The reference case $R1$ is characterized by $M=2,\gamma =1.4,\eta =3,\alpha =\pi /4,\beta =2$,^{8} wherein the magnetic field (*B _{y}*) is initially perpendicular to the motion of incident shock (aligned in the

*x*direction). The reference case $R2$ is the same as $R1$ except that $\alpha =\pi /6$. After the discussion of reference case $R1$, we examine the evolution of the wave structures for decreasing

*β*, that is, by increasing the strength of the initially applied magnetic field. Thereafter, we examine the effect of changing the inclination angle

*α*by first examining the reference case $R2$ in detail. The discussion on irregular refraction in the presence of perpendicular and parallel magnetic fields is deferred until Sec. IV.

### A. Reference case *R*_{1}

We now examine the solution to the reference case $R1$ ($M=2,\gamma =1.4,\eta =3,\alpha =\pi /4,\beta =2$) where the magnetic field (*B _{y}* is present) is initially perpendicular to the motion of incident shock. Figure 4(a) shows the graphical solution of the RH relations for the conditions upstream of incident shock IS, with the exception of

*r*= 1 (corresponding to the upstream state), and there is only one real root $r0=0.559$, which corresponds the downstream of IS (state 1 in Fig. 3). It shows that the incident shock is a fast magnetosonic MHD shock instead of a hydrodynamic shock. For RF, the unique real root

*r*

_{1}is 0.872 (discarding the trivial root

*r*= 1), while it is 0.512 for TF, as shown in Figs. 4(b) and 4(c), respectively. It indicates that RF and TF are both fast mode MHD shocks. However, Fig. 4(d) shows the unique real root $r2>1$ (again discarding

*r*= 1), which implies that the reflected wave RS is either an expansion fan or a compound wave. Additionally, the sign of magnetic field does not change across this wave, indicating that the RS is an expansion fan through which only the magnitude of magnetic field changes. The same observation for TS is found in Fig. 4(e); hence, we conclude that TS is also an expansion fan. It demonstrates that the solution consists of three fast shocks and two slow-mode expansion fans. The four waves RF, RS, TF, and TS are found to lie at $\varphi 1=0.370\u2009255,\u2009\varphi 2=0.765\u2009571,\u2009\varphi 3=1.232\u2009414,$ and $\varphi 4=1.044\u2009445$, respectively. We also analytically compute the angular width of expansion fan $\Delta \varphi ERS=0.330\u2009408$ for RS, whereas $\Delta \varphi ETS=0.180\u2009417$ for TS, respectively.

In Fig. 5, we overlay the analytical wave structure (the locations of waves) on the numerical results. Figure 5(a) shows that the density field clearly displays the location of shocked contact (denoted as SC), over which the magnetic field remains continuous. The x-component of the magnetic field, *B _{x}*, clearly displays the locations of the weaker waves RF and RS that have small density jumps associated with them and these waves are not so clearly discerned in the density plot in Fig. 5(c). Both RS and TS increase the magnitude of magnetic field without changing its sign, but it is not obvious from the 2D field plots of density and magnetic fields that RS and TS are expansion fans. To show that both are indeed expansion fans, we plot $\rho ,Bx$, and

*B*profiles along a horizontal line at $\zeta /L=0.6667$ in Fig. 6. From left to right in Fig. 6(a), the waves are as follows: fast-mode shock RF, slow-mode expansion fan RS, shocked contact SC, slow-mode expansion fan TS, and fast-mode shock TF, respectively. The first two waves propagate to left, the remaining three waves move to the right. We also note that the magnetic field does not change its sign passing through all waves, and hence, there are neither intermediate shocks nor compound waves for this strongly planar solution of the reference case $R1$. Hence, the planar solution is seen to be identical to the strongly planar solution for the case $R1$. It is interesting to contrast this with the case of a parallel magnetic field,

_{y}^{8,9}where it was shown that TS is $2\u21924$ intermediate shock and RS is slow shock for strongly planar solution, whereas TS is replaced by a rotational discontinuity followed downstream by a slow shock for the planar solution. With the specific parameters of case $R1$, the wave structure resulting from perpendicular magnetic field is unique, while the wave structure is not unique for the parallel magnetic field case, which exhibits differences between planar and strongly planar situations. We further note, from Figs. 5 and 6, there is close agreement between the analytical solution and the numerical results insofar as the wave structures is concerned.

### B. Evolution of wave structure with *β*

We now examine how solutions to the regular MHD shock refraction problem vary as magnetic field strength (characterized by *β*) is varied. The problem is characterized with the same parameters in the case $R1$ except $\beta \u2208(0.5,106)$. The solutions for strong magnetic fields $\beta \u2208(0.5,10)$ are presented in Sec. III B 1, while the solutions for large *β* are shown in Sec. III B 2.

#### 1. Solution behavior for $\beta \u2208(0.5,10)$

In hydrodynamics, the regular refraction (for the chosen parameters) consists of a triple point; that is, there are three shocks (incident, reflected and transmitted) that meet at a single point. The angles of shocks *R* and *T* in hydrodynamic triple-point solution to the shock refraction problem are computed with $\beta \u22121=0,\u2009M=2,\alpha =\pi /4,\gamma =1.4$, and *η* = 3. We plot the deviation of the angles of the fast-mode shocks from their corresponding triple point values in Fig. 7(a). Here, $\Delta \varphi 1$ corresponds to the angle of *RF* minus the angle of *R*, while $\Delta \varphi 3$ corresponds to the angle of *T* minus the angle of TF, respectively. As the field strength is decreased, or as *β* is increased, the angles of the fast-mode shock in MHD tend toward the triple-point values from hydrodynamics. For $\beta <3$, the angle deviation from the triple-point values increases as *β* is decreased with the change of the RF angle being stronger than that of TF. Beyond *β* = 3, the evolution of angle deviation changes more gradually and tends to be linear at larger *β* values. Meanwhile, the location angles $\varphi 2$ (for RS) and $\varphi 4$ (for TS) tend toward the SC with increasing of *β*, as shown in Fig. 7(b). $\Delta \varphi 2$ ($\Delta \varphi 4$) denotes the difference between $\varphi 2$ ($\varphi 4$) and angular location of SC. The angular difference $|\Delta \varphi 2|$ decreases from 0.3 to around 0.1 with increase in *β*, while $\Delta \varphi 4$ decreases more slightly from 0.11 to 0.04. These variations imply that the reflected waves are more strongly affected than the transmitted waves by the variation in the strength of initial magnetic field. In addition, we note that there is a close agreement between the analytical solutions and the numerical results for these strongly planar shock refractions. Note that there is no transition of wave configuration with increasing of *β*, and the solution consists of three fast shocks and two slow-mode expansion fans. In contrast, for the case with the presence of parallel magnetic fields, as *β* increased the RS and TS transition from slow shocks to $2\u21924$ intermediate shocks, and then become slow compound waves. Such transitions were noted by Wheatley *et al.*^{9} for the parallel field case.

In Fig. 8(a), we show the variation of the angular width of expansion fans, RS and TS, as *β* is increased. Here, we plot only the analytical solutions since it is quite challenging to effectively resolve angular widths of the order of $10\u22123$ from numerical results. At $\beta =0.5$, the angular width for *RS* is $\Delta \varphi ERS=1.344\xd710\u22123$, whereas it is $\Delta \varphi ETS=3.191\xd710\u22124$ for TS. When a strong magnetic field is present, the angular width of both expansion fans significantly increases as *β* is increased. The angular width of the reflected expansion reaches a maximum value $\Delta \varphi ERS=6.50\xd710\u22122$ at $\beta =2.3627$, while the angular width of the transmitted expansion fan reaches a maximum value of $\Delta \varphi ETS=7.596\xd710\u22123$ at $\beta =2.2673$. Until $\beta \u22482.4$ (the location of the maxima), the angular width of the expansion fans is also accompanied by a significant increase in the angular locations of RS and TS [Fig. 7(b)]. The increase in the RS location angle is larger than the location angle of TS, suggesting that the angular width (defined as the angle from the leading wave in the *RS* wave group to the leading wave in the *TS* wave group) of the inner layer decreases as *β* increased [see Fig. 8(b)]. After reaching the maximum values of angular width $\Delta \varphi ERS$ and $\Delta \varphi ETS$, the angular location of $RS(\varphi 2)$ and $TS(\varphi 4)$ continues to increase with *β*. It leads to a continuous smooth and monotonic decrease in the angular width of the inner layer with increasing *β*, the angular width of the inner layer scales as $\beta \u22121/2$ for $\beta >5$. To summarize, in the range of strong magnetic fields $\beta \u2208(0.5,2.4)$, the wave configurations change significantly and the angular width of expansion fans $\Delta \varphi ERS$ and $\Delta \varphi ETS$ increase as *β* increased. In the range of moderate magnetic fields, the wave configurations change slightly and the angular width of expansion fans $\Delta \varphi ERS$ and $\Delta \varphi ETS$ decrease as *β* increased.

#### 2. Solution behavior at large β

For cases with parameters $M=2,\alpha =\pi /4,\gamma =1.4,\eta =3$, and $\beta >0.5$, the strongly planar solution consists of three fast shocks and two slow-mode expansion fans (Fig. 6), and there is no transition of wave configuration with an increase in *β*. Presently, we are concerned with the evolution of wave structure at large *β* up to 10^{6}. We plot only analytical angular widths since it is very challenging to obtain the numerical results that resolve well angular widths of the order of $10\u22124$.

Figure 9(a) shows that as the magnetic field weakens, the angular width of the inner layer diminishes. The slope of the angular width of the inner layer vs $\beta \u22121$, when plotted on a logarithmic scale, reveals that the angular width of the inner layer scales as $\beta \u22121/2$; that is, the inner layer angular width is directly proportional to the applied magnetic field magnitude *B*. Figure 9(b) shows the deviation of *RF* and *TF* from triple-point shock angles vs $\beta \u22121$. It reveals that as magnetic field weakens $\beta \u22121<10\u22124$, the deviation of two fast shocks from triple-point shock angles diminish, and both two scales as $\beta \u22121/2$. This scaling is also found for the evolution of angular width of two expansion fans if $\beta \u22121<10\u22124$ [see Fig. 9(c)]. These observations suggest that, in the limit as $\beta \u2192\u221e$, the location of RF (TF) tends to be identical to *R* (*T*) of the corresponding hydrodynamic case. The jumps across the inner layer, which are equal to those across the hydrodynamic SC in the limit, are supported by two expansion fans within the layer. In other words, as $\beta \u2192\u221e$ the MHD solution is identical to the corresponding hydrodynamic triple-point solution, with the exception that the hydrodynamic SC is replaced by the inner layer surrounded by the two slow expansion fans. This observation is consistent with the conclusion of the case in the presence of a parallel magnetic field.^{9}

### C. Evolution of wave structure with change of *α*

For a given set of parameters, the MHD shock refraction will transform regular into irregular refraction as the inclination interface angle *α* is decreased. First, we present the detailed wave structure of the case characterized with $M=2,\beta =2,\gamma =1.4,\eta =3$, and $\alpha =\pi /6$, denoted as reference case $R2$. For this case, we follow the same analysis process for the case $R1$ to show that RS becomes a slow compound wave; TS is a slow-mode expansion fan; and IS, RF, and TF are fast shocks. The compound wave *RS* relevant here consists of a $2\u21923=4$ intermediate shock, for which $vn2=cSL2$, followed immediately downstream by a slow-mode expansion fan, where $cSL2$ denotes the slow magnetosonic speed. Note that the case in the presence of parallel magnetic field and the hydrodynamic case, the refraction is irregular with $\alpha =\pi /6$, whereas for the present case of the perpendicular magnetic field, the refraction pattern is regular. Therefore, we are able to also obtain the analytical result in addition to numerical simulation results: both sets of results are plotted in Fig. 10 with the white lines depicting the location of the waves from the analytical solution. The density field clearly displays the location of TF, TS, and SC because of the strong density jump over these waves. The first two waves TF and TS can also be seen in images of the x- and y-component of the magnetic field. On the other hand, the reflected waves RF and RS are weak, and we only discern these well in the image of the x-component of the magnetic field shown in Fig. 10(b). The flow is compressed and the sign of *B _{x}* is changes in passing through the leading wavefront, followed immediately a slow-mode expansion fan which changes only the magnitude (but not the sign) of the magnetic field and leads to state 3 at the trailing edge of the expansion fan, which is part of the slow compound wave. The magnetic field lines are also overlaid in Fig. 10 to show how the various shocks in the system deflect the field.

In Fig. 11, we plot $\rho ,Bx$, and *B _{y}* profiles along a vertical line at $\xi /L=0.6667$. From left to right in Fig. 11(a), the discontinuities are as follows: fast shock TF, slow-mode expansion fan TS, SC, slow compound wave RS, and fast shock RF, respectively. In the analytical solution, we note the fine scale features of the slow compound wave RS, the leading wavel front of which is at $\xi /L=0.459$ for $\zeta /L=0.667$. There appears to be good agreement between the analytical solution and the numerical results.

We now investigate the wave pattern by varying $\alpha \u2208(0.378,1.521)$ and fixing $M=2,\beta =2,\gamma =1.4$, and *η* = 3. In Fig. 12(a), we show that the location angles $\varphi 1$ of RF and $\varphi 3$ of TF increase as *α* is increased. The transition of RS occurs at $\alpha =0.559$, the black (respectively, red) curve in the figure corresponds to RS being a slow-mode expansion fan (respectively, slow compound wave). The angle $\varphi 3$ diminishes smoothly as *α* decreased over the range of *α* that was examined. The RS transition from a slow mode expansion to a compound wave is associated with a change in the slope of the curve [at the black-red transition in Fig. 12(a)] so that the change of the angular width between RF and RS increases more rapidly when the RS is a compound wave. For RF, $\varphi 1$ decreases linearly as *α* decreased when the RS is slow-mode expansion fan, whereas it decreases more strongly and the RF location changes the sign ($\varphi 1<0$) when RS is a slow-mode compound wave ($\alpha <0.559$). The reason for this subtle change in slope of $\varphi 1$ vs *α* is probably due to the nonlinear nature of the algebraic equations governing the analytical solution of regular refraction, but we have not uncovered the exact reason for this. We further note that RF location angle increases strongly toward the positive y-axis direction up to $\varphi 1=\u22120.736\u2009439$ (negative meaning the angle is now measured clockwise from the negative x axis) at $\alpha =0.378$. In Fig. 12(b), the location angle $\varphi 2$ of RS and $\varphi 4$ of TS diminishes smoothly as *α* is decreased, and there is no obvious change of angle evolution in the transition region. For the maximum value $\alpha =1.521$ in this study, $\varphi 2$ tends to be close to $\varphi 4$ as shown in the zoomed-in region in Fig. 12(b).

Figure 13(a) shows that as *α* decreased, the angular width of slow-mode expansion fan $\Delta \varphi ETS$ increases smoothly. For the reflected RS wave, the angular width $\Delta \varphi ERS$ increases as *α* decreases if RS is a slow-mode expansion fan. On the other hand, $\Delta \varphi ERS$ decreases as *α* decreases when RS becomes a slow compound wave. The angular width of inner layer increases smoothly with decreasing of *α* as shown in Fig. 13(b). For $\alpha <0.378$, the analytical solution does not exist for the chosen parameters ($M=2,\beta =2,\gamma =1.4$, and *η* = 3). This is because shock refraction transitions to an irregular pattern. We now explore the critical angle *α _{crit}* of regular–irregular transition by varying the magnetic strength.

For the chosen parameter set ($M=2,\eta =3,\gamma =1.4$), the critical angle at which transition from regular to irregular refraction takes place is *α _{crit}* is around 35.5° for hydrodynamic shock refraction $(\beta \u22121=0)$. In the studied range, we find that the presence of an initially perpendicular magnetic field delays the transition to irregular refraction compared to the hydrodynamic case, as seen in Fig. 14(c). Note that

*α*tends to the corresponding $\alpha crit,hydro$ as

_{crit}*β*is increased, and the limiting behavior for large

*β*tends to the hydrodynamic case, for

*M*= 2, as discussed in Appendix A. In the presence of a strong magnetic field,

*α*increases strongly as magnetic field weakens until $\beta =2.7$. It is found that the transition angle of RS from slow-mode expansion fan to compound wave decreases as

_{crit}*β*is increased, and the reflected wave RS is a slow compound wave just before the wave pattern transitions to an irregular pattern in this range. In range of $\beta \u2208(2.7,3.406)$,

*α*suddenly increases and is identical to the RS transition angle which in turn also deceases as

_{crit}*β*is decreased. This range is somewhat unique where the

*α*vs

_{crit}*β*plot is discontinuous, the local minimum of

*α*is found at $\beta 0=3.406$. Beyond the above ranges, the

_{crit}*RS*transition angle cannot be analytically calculated, since it is less than

*α*so that RS is a slow-mode expansion fan when the wave pattern becomes irregular. To explore this somewhat anomalous behavior of critical angle

_{crit}*α*vs $\beta (\u22652)$, we also compute the critical angle for three other Mach numbers, $M=1.35,\u20091.5$, and 5, for which the critical angle is plotted in Figs. 14(a), 14(b), and 14(d), respectively. In the critical angle

_{crit}*α*vs

_{crit}*β*evolution, the discontinuous region is expanded and the local minimum occurs at larger

*β*as Mach number decreases (see $M=1.35,\u20091.5$ cases). The local minimum of

*α*occurs at $\beta =4.207$ and 6.106 for

_{crit}*M*= 1.5 and 1.35, respectively. It is interesting to examine the local minimum in the

*α*value as a function of Mach number. This is plotted in Fig. 15 along with the corresponding value of

_{crit}*β*at this critical transition angle. Aside from the branch seen for low values of

*β*[lower left branch in Fig. 14(c)], this critical angle value represents the smallest value at which the transition from regular to irregular refraction takes place. Beyond

*M*= 3.4, the local minimum no longer exists [as for example in Fig. 14(d) for

*M*= 5], and the lower left branch [as in Fig. 14(c)] takes over for all values of

*β*investigated. It is evident that the transition from regular to irregular refraction is a complex phenomenon, a more thorough exploration of the parameter space is left for the future.

## IV. RESULTS: IRREGULAR REFRACTION

Presently, we solve the self-similar formulation of ideal GLM–MHD equations [Eq. (8)] to investigate the irregular MHD shock refraction, including both the perpendicular and parallel orientations of the initially applied magnetic fields, as well as the hydrodynamic case. We note here that, in the absence of viscosity, conductivity, and resistivity, there is no inherent length scale in the shock refraction at an inclined straight density interface (before the shock reaches the point where the interface intersects the top boundary). During this shock traversal phase, the solution is self-similar (see, for example, previous work by Samtaney and Pullin^{18} in the context of Euler equations). To the best of our knowledge, there are no non-self-similar cases of shock refraction in the context of ideal MHD or Euler equations. Hence, even if the shock refraction is irregular, the solution is self-similar in $x/t,y/t$ coordinates. We compute the specific cases with parameters $M=2,\eta =3,\gamma =1.4$, and vary *β* and *α*. For the self-similar solution, the boundary conditions and the initial guess are depicted in Fig. 16(a). The pressure on either side of the contact discontinuity is unity. The state behind the incident shock is obtained by the Rankine–Hugoniot jump condition, and the shock is moving with the speed of $u0=M\u2009cf$, where *c _{f}* is the fast magnetosonic sound speed. The computational domain (now in velocity coordinates after the self-similar transformation) $\Omega =[0,3]\xd7[0,2]$ is discretized with mesh refinement utilizing nine levels (base mesh cell size is unity), yielding an effective uniform mesh resolution of 1536 × 1024. An example of AMR mesh structure is shown in Fig. 16(b).

The self-similar solutions with $\alpha =\pi /6$ are shown in Fig. 17 in which the left/right sequence is the case of the perpendicular, parallel magnetic field $(\beta =2)$, and hydrodynamic case $(\beta \u22121=0)$, respectively. For the hydrodynamic case, the Mach stem *m* connects the reflected shock *R* to the point where *m* intersects the density interface. A second shear layer *s* emerges from the triple-point where *IS*, *R,* and the Mach stem *m* intersect. This system is called single Mach-reflection irregular refraction MRR, which has been observed in the hydrodynamic experiments.^{3} The irregular shock structure produced in the parallel magnetic field case is the MHD equivalent of the single Mach reflection MRR. The reflected fast shock RF no longer intersects the density interface and is connected to the density interface, transmitted shocks, and RS by the MHD equivalent of a Mach stem *m*. The changes in flow properties across the MHD Mach stem *m* are consistent with it being a fast mode MHD shock. Figure 18(b) shows that it compresses the flow without changing sign of the magnetic field. Furthermore, the vorticity generated on the vicinity of shocked contact *SC* is transported onto the MHD waves, as in the case of regular refraction. This transport of baroclinic vorticity from the interface prevents the shear instability that causes the roll-up of the interface in the hydrodynamic case. These results indicate that the mechanism by which a magnetic field suppresses the MHD RMI is valid independent of whether regular or irregular refraction occurs at the density interface (for a discussion of this, we refer to the work of Samtaney^{8} and Wheatley *et al.*^{9}). Moreover, the second shear layer *s* is not present, and instead, the fast mode Mach stem *m* is a vortex sheet. We note that the ideal MHD does not support shear layers without mass flux (i.e., unless the shear layers are shocks or expansions). On the other hand, the resulting shock structure of the case in the presence of perpendicular magnetic field is regular, and the analytical and numerical solution is shown in detail in Figs. 10 and 11. The vorticity on the vicinity of shocked contact SC is transported onto the MHD waves (slow compound wave RS and slow expansion fan TS), as in the parallel magnetic field case. For this specific set of parameters, it is clearly shown that the presence of perpendicular magnetic field delays the regular–irregular transition, which is consistent with the analytical conclusion presented in Sec. III C, whereas the presence of parallel magnetic field suppresses the second shear layer and leads to the presence of a Mach stem with shear.

We now compute the above three cases with the inclination angle $\alpha =0.4$ and *β* = 8. The parameter combination of *β* = 8 and $\alpha =0.4$ leads the refraction pattern to be irregular for all three cases. Figure 18 shows the irregular refraction wave pattern with this specific choice of parameters. For the hydrodynamic case $(\beta \u22121=0)$, the incident and Mach-stem shocks IS and *m* appear as a continuous wave. which is convex-forwards along the segment formerly occupied by the incident shock IS. The reflected shock appears to disperse into a band of wavelets. This phrase “dispersive” is not technically appropriate here to this solution of Euler equations because the equations are mathematically characterized as hyperbolic and are not dispersive. However, we use “disperse” somewhat loosely because it has been used previously by Abd-El-Fattah and Henderson.^{3} In actuality, the reflected shock rather weakens into a band of compressive waves. In addition to the usual second shear layer *s* stemming from the triple point where IS, *R,* and the Mach stem *m* intersect, there is another band of shear layers visible in the hydrodynamic case. This system is called as convex-forwards irregular refraction CFR by Henderson *et al.*^{3} In the MHD cases, the band of shear layers that emerges from Mach stem completely vanishes for both orientations (parallel and perpendicular) of the magnetic field. Here, *s* is identified as a fast mode MHD shock, since the flow across this wave is compressed without changing sign of magnetic field. Hence with the specific parameters $\alpha =0.4$ and *β* = 8, the irregular shock structures produced in the presence of perpendicular or parallel magnetic fields cases appear also to be the MHD equivalent of the single Mach reflection refraction (MRR), while it is CFR type for the hydrodynamic case. The presence of a magnetic field weakens the Mach stem compared to the hydrodynamic case: the perpendicular magnetic field attenuates these features more effectively than the parallel field case. The Mach stem in the perpendicular magnetic field case lags behind the other two cases. Furthermore, the vorticity deposited on the vicinity of SC is also transported on the MHD waves for irregular refraction in the perpendicular magnetic field case. This has implications on the suppression of the RMI, in the sense that even for irregular refraction the vorticity is not on the density interface and one expects the instability suppression to still take place.

We further decrease the inclination interface angle to $\alpha =0.3$. Now three irregular flow structures appear as shown in Fig. 19. For the hydrodynamic case, the band of shear layers emerges from Mach stem *m,* which is convex-forwards, and the irregular wave structure is identified to the CFR type (similar to the one for $\alpha =0.4$). In the presence of a magnetic field (both the parallel and perpendicular orientations), the wave structure is considered as the MHD equivalent of CFR type that occurs in hydrodynamic case. These hydrodynamic CFR wave structures have been observed in experiments (see Figs. 11 and 12 in Abd-El-Fattah and Henderson^{3}). In the MHD equivalent of CFR irregular structure, the Mach stem is less convex-forwards than the corresponding case with $\alpha =0.4$ (MRR type). The changes in flow properties across the MHD Mach stem and waves, which appear similar to the band of shear layers in the hydrodynamic case, are consistent with these being fast mode MHD shocks. As the conclusion for the cases with $\alpha =0.4$, it is found that the presence of perpendicular magnetic field weakens more efficiently the Mach stem than the parallel magnetic field case.

## V. CONCLUSION

In summary, the present work investigates the ideal MHD flow structure produced by the refraction of a shock at an oblique planar density interface assuming the flow to be strongly planar (i.e., no velocity or component of the magnetic field in the z-direction). We consider the cases in the presence of magnetic fields, which are initially perpendicular and parallel to the motion of incident shock. We employ an iterative procedure to obtain the analytical solution to MHD regular shock refraction. The analysis is restricted to cases where the initially applied magnetic field is oriented perpendicular to the direction of shock wave propagation. For computing the flow fields numerically, we solve a boundary value problem stemming from a self-similar transformation of ideal GLM–MHD equations. The numerical simulations are used to obtain both regular and irregular shock refraction solutions (both regular and irregular refractions are self-similar). Specifically, we analytically investigate the regular solution to the cases with the presence of perpendicular magnetic fields and discover that the wave structure consists of three fast shocks (IS, RF, and TF), a slow-mode expansion fan (TS), and a slow-mode expansion fan or slow compound wave for RS by varying the magnetic field magnitude and the inclination interface angle. The inclination interface angle of RS transition from a slow-mode expansion fan to slow compound wave decreases as magnetic field magnitude decreases. In general, the analytical solutions agree well with the numerically computed ones. We plot the deviation of the wave angles from the corresponding hydrodynamic regular refraction case. The angular extents of the slow mode expansion fans reach a maximum for a particular value of the magnetic field ($\beta \u22482.3$) and then gradually decrease with increase in *β*. The angular width of the inner layer, the angles of the fast mode shocks (RF and TF), and the angular extents of the slow mode expansion fans (RS and TS) scale with $\beta \u22121/2$, that is, linearly with the strength of the applied magnetic field for large *β*.

At *M* = 2, the perpendicularly oriented initial magnetic field suppresses the transition compared to the corresponding hydrodynamic case, although the critical angle vs *β* plot is not monotonic. We quantified the critical angle where the transition from regular to irregular refraction takes place and specifically quantified the local minimum of the critical angle as a function of Mach number *M*. It is found that the transition from regular to irregular refraction is sufficiently complex that a more thorough investigation of the parameter space is warranted. We leave such an exploration of the parameter space for the future.

Due to the large parameter space in MHD shock refraction, it is somewhat challenging to provide a complete taxonomy of all the irregular refraction patterns. We present a sampling of the parameter space in which interesting irregular patterns occur. The self-similar transformation yields “steady” solutions and hence enables us to focus our attention in which we can concentrate the adaptive meshes where the refraction is taking place. The numerical results show that there are two types for irregular MHD shock refraction, the first one is an MHD variant of single Mach reflection MRR with the appearance of a Mach stem. The second type is an MHD variant of convex-forwards irregular refraction CFR, with the reflected shock has apparently dispersed into a band of wavelets. The Mach stem and the band of wavelet emerging from Mach stem are classified as fast mode MHD shocks in MHD irregular refraction. The vorticity deposited on the vicinity SC is transported on the MHD waves for regular and irregular shock refractions, indicating that the mechanism by which a magnetic field suppresses the MHD RMI is valid independent of whether regular or irregular refraction occurs at the density interface, and whether a parallel or perpendicular magnetic field is present. Moreover, the presence of magnetic field allows to weaken or completely vanish the second shear layer emerging from the triple-point between the incident shock, reflected fast shock and the Mach stem, indicating that the vorticity is transported from the vicinity of the density interface, preventing its instability. Perpendicular magnetic fields suppress more effectively this vorticity than the corresponding parallel magnetic fields.

## ACKNOWLEDGMENTS

This research was supported by funding from King Abdullah University of Science and Technology (KAUST) under Grant No. BAS/1/1349-01-01.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors declare that they have no conflict of interest.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: VERIFICATION OF LIMITING LARGE *β* MHD CASES VS HYDRODYNAMICS

Presently, we verify that the MHD solution converges to the hydrodynamics solution in the limit of large *β*. We compute the critical angle *α _{crit}* of regular to irregular transition for the MHD shock refraction problem for

*M*= 2 case, for the perpendicular magnetic field orientation. As seen in Fig. 20, that for large values of

*β*, the critical angle for the MHD shock refraction problem tends to the value of the hydrodynamic case.

### APPENDIX B: NUMERICAL CONVERGENCE TEST

Figure 21 shows the density profiles along a vertical line at $\xi /L=0.6667$ for the case *R*_{2}, where the numerical results obtained with the maximum mesh refinement 8, 9, and 10 levels (base mesh cell size is unity). In the expansion region of compound wave, the *L*_{2} norm is around $1.9\xd710\u22122,6.3\xd710\u22123$, and $6.0\xd710\u22123$ for these three resolutions, respectively. There is no evident difference (see in TF, TS, and CD regions, etc.) between *l*_{9} and *l*_{10} cases, except when we see a magnification of the compound wave region. We consider the maximum mesh refinement of nine levels is sufficient for our work. The primary reason is that we have demonstrated that at this resolution, the agreement between the analytical solution to the regular MHD shock refraction, and numerical results shows a strong agreement. Second, in the section of irregular MHD shock refraction, we quantitatively show the different wave patterns with different initial orientations of magnetic fields and inclination angles of density interface. Thus, the mesh resolution at maximum mesh refinement of nine levels is sufficient to draw physical conclusions, and further resolving the wave structures will not change the nature of the waves or the refraction patterns.