Ideal Magnetohydrodynamic Equations Research Articles

Incompressible Limit of Ideal Magnetohydrodynamic Equations with a Perfectly Conducting Wall Condition

Incompressible Limit of Ideal Magnetohydrodynamic Equations with a Perfectly Conducting Wall Condition

Positivity-preserving entropy filtering for the ideal magnetohydrodynamics equations

In this work, we present a positivity-preserving adaptive filtering approach for discontinuous spectral element approximations of the ideal magnetohydrodynamics equations. This approach combines the entropy filtering method (Dzanic and Witherden, 2022) for shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-free magnetic field. Due to the inclusion of non-conservative source terms, an operator-splitting approach is introduced to ensure that the positivity and entropy constraints remain satisfied by the discrete solution. Furthermore, a computationally efficient algorithm for solving the optimization process for this nonlinear filtering approach is presented. The resulting scheme can robustly resolve strong discontinuities on general unstructured grids without tunable parameters while recovering high-order accuracy for smooth solutions. The efficacy of the scheme is shown in numerical experiments on various problems including extremely magnetized blast waves and three-dimensional magnetohydrodynamic instabilities.

Small Alfvén number limit for shallow water magnetohydrodynamics

Small Alfvén number limit of the shallow water magnetohydrodynamic equations with ill-prepared initial data is proved by applying fast averaging method. It is rigorously justified that the solutions of the original system tend to the solution of a one-dimensional system coupled with a linear system as the Alfvén number goes to zero. The approach is also used to study the small Alfvén number limit of the three-dimensional ideal incompressible magnetohydrodynamic equations.

Incompressible limit of nonisentropic ideal magnetohydrodynamic equations with periodic boundary conditions

This paper is concerned with the incompressible limit of the non-isentropic ideal magnetohydrodynamic equations with periodic boundary conditions and general initial data. Weak convergence results of the smooth solutions are established by filtering method in the torus.

Numerical path preserving Godunov schemes for hyperbolic systems

This paper primarily concerns the discontinuities capturing problems in nonconservative and nonconvex conservative hyperbolic systems. For the Godunov scheme of nonconservative hyperbolic systems, the numerical dissipation at discontinuous points in the simulation process is analyzed grid point by grid point through a new perspective. The numerical paths implied in the nonconservative variables represent different averaging and dissipating processes from the conservative cases. Unphysical dissipation of nonconservative variables ruins Rankine-Hugoniot relations, contributing to incorrect jumps, wrong propagation speed, and spurious fluctuations in other characteristic fields. For the discontinuities capturing problem of nonconservative hyperbolic systems, a novel numerical path preserving (NPP) method is proposed to modify the original Godunov schemes so that the dissipation of the numerical methods at discontinuities is carried out strictly following the consistent numerical path. Numerical simulations of the nonconservative systems are performed for isothermal and Euler equations. The results indicate that the NPP method can correctly capture the discontinuous structures and verify the correctness of our theoretical analysis. Additionally, the NPP method is extended to nonconvex hyperbolic conservation systems, and the Alfvénic wave (discontinuity) of the one-dimensional ideal magnetohydrodynamic (MHD) equations is simulated. It is found that the nonconvex nature of the flux causes unphysical compound wave structures while simulating the Alfvénic wave with the popular schemes nowadays, e.g., Roe of flux difference splitting method and WENO with flux vector splitting method. Modifying the original Godunov schemes with the NPP method proposed in this paper ensures correct simulation of Alfvénic discontinuity in MHD equations. It validates the effectiveness of the NPP method for nonconvex hyperbolic conservation systems.

Equilibriums of extremely magnetized compact stars with force-free magnetotunnels

We present numerical solutions for stationary and axisymmetric equilibriums of compact stars associated with extremely strong magnetic fields. The interior of the compact stars is assumed to satisfy ideal magnetohydrodynamic (MHD) conditions, while in the region of negligible mass density the force-free conditions or electromagnetic vacuum are assumed. Solving all components of Einstein's equations, Maxwell's equations, ideal MHD equations, and force-free conditions, equilibriums of rotating compact stars associated with mixed poloidal and toroidal magnetic fields are obtained. It is found that in the extreme cases the strong mixed magnetic fields concentrating in a toroidal region near the equatorial surface expel the matter and form a force-free toroidal magnetotunnel. We also introduce a new differential rotation law for computing solutions associated with force-free magnetosphere, and present other extreme models without the magnetotunnel.

A robust and contact resolving Riemann solver for the two-dimensional ideal magnetohydrodynamics equations

This paper presents a new cell-centered numerical method for ideal magnetohydrodynamics (MHD) that can be used in Lagrangian or Eulerian discretization. A two-dimensional Riemann solver based on the HLLC-type MHD method is established, which can be viewed as an extension from the HLLC-2D in hydrodynamics. The main feature of the algorithm is to introduce a nodal contact velocity and ensure the compatibility between edge fluxes and the nodal flow intrinsically. It transforms naturally from Lagrangian setting to the Eulerian setting in terms of grid nodal velocity, and gains benefits of the Lagrangian nature of the scheme. In the Lagrangian approach, the finite volume scheme itself can keep the magnetic field divergence-free strictly, while in the Eulerian case, a special constrained transport (CT) algorithm is constructed from the discontinuous fluxes on cell interfaces to ensure solenoidal nature again. Numerical tests are presented to demonstrate the performance of this new solver and compare the difference between the Lagrangian and Eulerian methods.

Provably Positive Central Discontinuous Galerkin Schemes via Geometric Quasilinearization for Ideal MHD Equations

In the numerical simulation of ideal magnetohydrodynamics (MHD), keeping the pressure and density always positive is essential for both physical considerations and numerical stability. This is however a challenging task, due to the underlying relation between such a positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. In this paper, we present the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and construct arbitrarily high-order provably PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approach, our analysis reveals that the PP property of standard CDG methods is closely related to a discrete magnetic DF condition, whose form was unknown prior to our analysis and differs from that for the noncentral discontinuous Galerkin (DG) and finite volume methods in [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124–2147]. The discovery of this relation lays the foundation for the design of our PP CDG schemes. In the one-dimensional (1D) case, the discrete DF condition is naturally satisfied, and we rigorously prove that the standard CDG method is PP under a condition that can be enforced easily with an existing PP limiter. However, in the multidimensional cases, the corresponding discrete DF condition is highly nontrivial, yet critical, and we analytically prove that the standard CDG method, even with the PP limiter, is not PP in general, as it generally fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence terms and then constructing new locally DF CDG schemes for Godunov’s modified MHD equations with an additional source term. The key point is to find out the suitable discretization of the source term such that it exactly cancels out all the terms in the discovered discrete DF condition. Based on the GQL approach, we prove in theory the PP property of the new multidimensional CDG schemes under a CFL condition. The robustness and accuracy of the proposed PP CDG schemes are further validated by several demanding 1D, two-dimensional, and three-dimensional numerical MHD examples, including the high-speed jets and blast problems with very low plasma beta.

Suppression mechanism of Richtmyer–Meshkov instability by transverse magnetic field with different strengths

The Richtmyer–Meshkov instability (RMI) is caused by an incident planar shock wave impinging on the heavy-gas-density interface. We have numerically investigated the RMI controlled by different transverse magnetic-field strengths based on the ideal compressible magnetohydrodynamics (MHD) equations. The MHD equations are solved by the corner transport upwind + constrained transport algorithm, which guarantees a divergence-free constraint on the magnetic field. We discuss the flow characteristics and shock patterns in both classical hydrodynamic and MHD situations and verify our conclusions by comparing the experimental results with the numerical results. The results show that the magnetic field modifies the pressure-gradient distribution, and the baroclinic vorticity splits and attaches to the MHD shock waves. In addition, the results indicate that the interaction of shock wave and density interface changes the distribution of magnetic-field energy and distorts the magnetic induction line in the region of magnetic-field energy accumulation. The distortion of the magnetic induction lines alters the magnetic field gradient and creates a magnetic tension that produces a torque opposing that generated by the shear force on the vorticity layer, so the Kelvin–Helmholtz instability is effectively suppressed and no Kelvin–Helmholtz vortex appears on the vorticity layer. The result is that the interface instability is suppressed.

The combined non-equilibrium diffusion and low Mach number limits of a model arising in radiation magnetohydrodynamics

We consider the combined non-equilibrium diffusion and low Mach limits of a model arising in radiation magnetohydrodynamics, which is described by the ideal compressible magnetohydrodynamic equations coupled to the radiation transfer equation. We study the case that the temperature has a large variation. In this situation, due to the complex asymmetric singular structure in the model, it is very hard to obtain uniform estimates of solutions in standard Sobolev spaces. To overcome the difficulties caused by the singular structure, we introduce two new weighted norms and construct new auxiliary equations. In the appropriate normed spaces, we show that the contribution of singular terms to the total energy is bounded by O(ϵ) with respect to the parameter ϵ, and then establish the uniform estimates of solutions. Moreover, we rigorously prove that, for the well-prepared initial data, the target system is a coupling of the nonhomogeneous incompressible magnetohydrodynamic equations and a diffusion equation.

Numerical Simulations of Oscillations in Solar Corona Excited by Vortex Shedding

We study transverse oscillations of plasma structures in the solar corona excited by vortex shedding solving the full set of the 3D ideal time-dependent magnetohydrodynamic equations. To present how the creation of vortices and excited oscillations depend on the initial flow speed and various values of the magnetic field, we performed a parametric study for two cases, an interaction of a steady flow with a rigid and spatially fixed cylindrical obstacle, and with a cylinder fixed at the boundaries. We show that generated vortices can excite oscillations, e.g., transverse (kink), in the solar corona, for example in coronal loops, filaments, flux ropes, or similar magnetic structures. It has been found that the oscillation period is close to the theoretically predicted value. Moreover, we study the dependence of the Strouhal number with respect to the magnetic field under the conditions of the solar atmosphere.

Zero surface tension limit of the free-boundary problem in incompressible magnetohydrodynamics* *XG was supported in part by NSFC Grant 12031006. CL was supported in part by the RGC Grant of Hong Kong CUHK-24304621 and CUHK Direct Grant for Research No. 4053457.

We show that the solution of the free-boundary incompressible ideal magnetohydrodynamic (MHD) equations with surface tension converges to that of the free-boundary incompressible ideal MHD equations without surface tension given the Rayleigh–Taylor sign condition holds initially. This result is a continuation of the authors’ previous works (Gu et al 2020 arXiv:2105.00596; Gu and Wang 2019 J. Math. Pures Appl. 128 1–41; Luo and Zhang 2021 SIAM J. Math. Anal. 53 2595–630). Our proof is based on the combination of the techniques developed in our previous works (Gu et al 2020 arXiv:2105.00596; Gu and Wang 2019 J. Math. Pures Appl. 128 1–41; Luo and Zhang 2021 SIAM J. Math. Anal. 53 2595–630), Alinhac good unknowns, and a crucial anti-symmetric structure on the boundary.

Recent advancement of entropy split methods for compressible gas dynamics and MHD

The entropy splitting of the compressible Euler flux derivatives based on Harten’s entropy function Harten (1983), Gerritsen and Olsson (1996), Yee et al. (2000) in conjunction with classical spatial central and DRP (dispersion relation-preserving) finite discretizations with summation-by-parts (SBP) operators Strand (1994) for both periodic and non-periodic boundary conditions is proven to be entropy conservative and stable for a thermally-perfect gas by Sjögreen and Yee (2019), Sjögreen et al. (2020), Sjögreen and Yee (2021). The various high order methods resulting from applying classical spatial central, DRP and Padé (compact) methods to the split form of the Euler flux derivative are referred to as entropy split methods as a function of the splitting parameter β. These entropy split methods are entropy conserving and stable but they are usually not conservative numerical methods without additional reformulation; e.g., those proposed in Sjögreen and Yee (2021).This is an expanded version of Sjögreen and Yee (2022) and is intended to present a more in depth study for both the compressible gas dynamics and ideal MHD equations. Here the same Harten entropy function is used for the ideal magnetohydrodynamic (MHD) governing equation set. This approach was not examined in great depth by Sjögreen & Yee and Yee & Sjögreen in two conference proceedings papers Sjögreen and Yee (2022), Yee and Sjögreen (2022), which also included the high order Padé (compact) spatial discretizations Hirsh(1975). For the extension of the entropy split method to the ideal MHD, the Godunov symmetrizable non-conservative form of the formulation is used. Due to the non-conservative portion (commonly referred to as a nonlinear source term vector) of the symmetrizable system Godunov (1972), there are variants in formulating the entropy split methods. Two different numerical treatments for these particular source terms of the symmetrizable MHD governing equation set are presented.Systematic 3D formulation in curvilinear grids with the corresponding numerical fluxes among method comparisons for the Euler and ideal MHD equations is included. The curvilinear formulation includes high order classical central, DRP and compact spatial discretizations. For readers interested in the implementation of these methods into their computer code, it is hope that these formulations and the corresponding numerical fluxes in 3D curvilinear grids will be helpful.The comparative studies concentrate on the Tadmor-type of discrete entropy conserving Tadmor (2003), momentum conserving Ducros et al. (2000), kinetic energy preserving Kennedy and Gruber (2008), Pirozzoli (2010), Coppola et al. (2019), Yee et al., Sjögreen & Yee entropy split methods Yee et al.(2000), Sjögreen and Yee (2018), Sjögreen and Yee (2019), Sjögreen et al. (2020), Sjögreen and Yee (2021), as well as the combination of these physical-preserving methods Ranocha (2020), Yee and Sjögreen (2020). All of these methods are not only preserved certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation for selected compressible flow types. Extensive error norm comparison with grid refinement were performed to show how well this methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows. In general the Tadmor-type entropy conserving methods are at least twice the CPU per time step than the rest of the skew-symmetric splitting methods.

Nonlinear Generation of Fluting Perturbations by Kink Mode in a Twisted Magnetic Tube

We study the excitation of fluting perturbations in a magnetic tube by an initially imposed kink mode. We use the ideal magnetohydrodynamic (MHD) equations in the cold-plasma approximation. We also use the thin-tube approximation and scale the dependent and independent variables accordingly. Then we assume that the dimensionless amplitude of the kink mode is small and use it as an expansion parameter in the regular perturbation method. We obtain the expression for the tube boundary perturbation in the second-order approximation. This perturbation is a superposition of sausage and fluting perturbations.

A 2D cell-centered Lagrangian scheme based on multi-state Riemann solver with exactly divergence-free magnetic fields

This paper presents a cell-centered Lagrangian method for the ideal magnetohydrodynamics (MHD) equations in two dimension. In order to compute the nodal velocity and the numerical fluxes through the cell interface, a 2D nodal approximate Riemann solver of HLLD-type is designed. The main new feature of the Riemann solver is two fast waves, two Alfvén waves and one entropy wave are considered for each Riemann problem, and thus the rotational discontinuities can be captured very well. In the Lagrangian scheme, the evolving of the magnetic field is proved to be consistent with the magnetic frozen principle and thus guarantee exactly the divergence-free constraint. In addition, a linear reconstruction method is applied to achieve second order spatial accuracy while the Runge-Kutta method is used to obtain second order temporal accuracy. Various numerical tests are presented to demonstrate the accuracy and robustness of the algorithm.

Monolithic parabolic regularization of the MHD equations and entropy principles

We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge–Kutta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.

Magneto-hydrodynamic eigenvalue solver for axisymmetric equilibria based on smooth polar splines

We present a new eigenvalue solver for the ideal magneto-hydrodynamics (MHD) equations in axisymmetric equilibria that enables the robust and accurate description of eigenfunctions near the magnetic axis. The algorithm is based on discrete differential forms in combination with C1-smooth polar splines in the framework of isogeometric analysis. The symmetric discretization leads to a Hermitian Alfvén part in the MHD force operator, mirroring the self-adjointness of the continuous operator. Moreover, eigenfunctions are continuous across the magnetic axis by use of the polar spline framework. We provide comparisons to the standard tensor product approach by selected numerical examples and show that a) the polar spline framework correctly reproduces the tensor product results sufficiently far away from the magnetic axis and that b) it gives better results close to the magnetic axis with regards to back transformations to the physical (Cartesian) domain as usually needed if a coupling to a particle-in-cell code is desired.

Quasimodes in the cusp continuum in nonuniform magnetic flux tubes

Context. The study of magnetohydrodynamic (MHD) waves is important both for understanding heating in the solar atmosphere (and in particular the corona) and for solar atmospheric seismology. The analytical investigation of wave mode properties in a cylinder is of particular interest in this domain because many atmospheric structures can be modeled as such in a first approximation. Aims. The aim of this study is to use linearized ideal MHD to investigate quasimodes (global modes that are damped through resonant absorption) with a frequency in the cusp continuum, in a straight cylinder with a circular base and an inhomogeneous layer at its boundary that separates two homogeneous plasma regions inside and outside. We are particularly interested in the damping of these modes, and therefore try to determine their frequency as a function of background parameters. Methods. After linearizing the ideal MHD equations, we found solutions to the second-order differential equation for the perturbed total pressure in the inhomogeneous layer in the form of (1) Frobenius series around the regular singular points that are the Alfvén and cusp resonant positions, and (2) power series around regular points. By connecting these solutions appropriately through the inhomogeneous layer and with the solutions of the homogeneous regions inside and outside the cylinder, we derive a dispersion relation for the frequency of the eigenmodes of the system. Results. From the dispersion relation, it is also possible to find the frequency of quasimodes, even though they are not eigenmodes. As an example, we find the frequency of the slow surface sausage quasimode as a function of the width of the inhomogeneous layer for values of the longitudinal wavenumber relevant for photospheric conditions. The results closely match findings by other authors who studied the resistive slow surface sausage eigenmode. We also discuss the perturbation profiles of the quasimode and the eigenfunctions of continuum modes.

Characteristic boundary conditions for magnetohydrodynamic equations

In the present study, a characteristic-based boundary condition scheme is developed for the compressible magnetohydrodynamic (MHD) equations in the general curvilinear coordinate system, which is an extension of the characteristic boundary scheme for the Navier–Stokes equations. The eigenstructure and the complete set of characteristic waves are derived for the ideal MHD equations in general curvilinear coordinates (ξ,η,ζ). The characteristic boundary conditions are derived and implemented in a high-order MHD solver where the sixth-order compact scheme is used for the spatial discretization. The fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme is also employed for the spatial discretization of problems with discontinuities. In our MHD solver, the fourth-order Runge–Kutta scheme is utilized for time integration. The characteristic boundary scheme is first verified for the non-magnetic (i.e., B=0) Sod shock tube problem. Then, various in-house test cases are designed to examine the derived MHD characteristic boundary scheme for three different types of boundaries: non-reflecting inlet and outlet, solid wall, and single characteristic wave injection. The numerical examples demonstrate the accuracy and robustness of the MHD characteristic boundary scheme.

Unusual divergence of magnetoacoustic beams

Two-dimensional magnetosonic beams directed along a line forming a constant angle θ with the equilibrium straight magnetic field are considered. Perturbations in a plasma are described by the system of ideal magnetohydrodynamic equations. The dynamics of perturbations in a beam are different in the cases of fast and slow modes, and it is determined by θ and equilibrium parameters of a plasma. In particular, a beam divergence may be unusual in the case of parallel propagation (θ = 0). Diffraction is more pronounced in the case of parallel propagation as compared to a flow without magnetic field, and less manifested in the case of perpendicular propagation. The beams propagating oblique to the magnetic field do not reveal diffraction. The dynamics of perturbations in a beam are analytically described in the cases of weak and strong nonlinearity compared to diffraction. Small magnitude perturbations at the axis of a beam in unusual cases propagate slower than that in the plane wave. Involving of thermal conduction leads to the coupling equations describing thermal self-action of a beam, which behaves differently in the ordinary and unusual cases. Self-focusing may occur in the presence of a magnetic field instead of conventional defocusing in gases.