Magnetohydrodynamic Equations Research Articles

A note on energy and cross‐helicity conservation in the ideal magnetohydrodynamic equations

In this paper, we are concerned with the conservation of total energy and cross‐helicity for the weak solutions in the ideal magnetohydrodynamic (MHD) equations. In the spirit of recent works of Berselli (J. Differ. Equ. 368 (2023), 350–375.) and Berselli‐Georgiadis (NoDEA Nonlinear Differ. Equ. Appl. 31 (2024), 33), by establishing a new generalized Constantin‐E‐Titi type commutator estimate to allow us to make full use of the total energy, we extend the previous classical results to a wider range of exponents. These results indicate the role of the time integrability, spatial integrability, and differential regularity of the velocity (magnetic field) in the conserved quantities of weak solutions in the ideal MHD equations.

On energy and magnetic helicity equality in the electron magnetohydrodynamic equations

On energy and magnetic helicity equality in the electron magnetohydrodynamic equations

Stability and exponential decay for the compressible viscous non-resistive MHD system

How to construct global solutions of the compressible viscous magnetohydrodynamic (MHD) equations without magnetic diffusion even with small initial data in R3 or T3 is still an extremely challenging open problem. The difficulty comes from the lack of magnetic diffusion and the fact that solutions to inviscid equations generally grow in time. Motivated by this open problem, the present paper focuses on a special case of this MHD system in T3 when the magnetic field is vertical. We establish the global existence and uniqueness of smooth solutions to this system near a steady-state solution. In addition, the solution is shown to be stable and decay exponentially in time. The proof discovers and makes use of the smoothing and stabilizing effect of the steady magnetic field on the perturbations.

A model for general relativistic magnetohydrodynamic spine jets

We study jets using a semi-analytical model of the general relativistic magnetohydrodynamic (GRMHD) equations in the Kerr metric that describes them near the rotation axis, assuming a steady state, and axisymmetry. The goal is to model the inner spine of a relativistic jet in order to solve for the bulk acceleration and the shape of the jet and understand how these quantities depend on the enthalpy and the magnetic field. The model is constructed by expanding the rotating black hole metric and forces with respect to the polar angle about the rotation axis. This results in a system of ordinary differential equations that determine the dependence on the radial distance. The difference with previous semi-analytical models that expand the metric around the rotation axis is that the flow is governed by a polytropic equation of state. The solutions in this work start from a stagnation surface very close to the event horizon and become highly relativistic, achieving large Lorentz factors at large distances.

Joint action of phase mixing and nonlinear effects in MHD waves propagating in coronal loops

The evolution of Alfv\'en waves in cylindrical magnetic flux tubes, which represent a basic model for loops observed in the solar corona, can be affected by phase mixing and turbulent cascade. Phase mixing results from transverse inhomogeneities in the Alfv\'en speed, causing different shells of the flux tube to oscillate at different frequencies, thus forming increasingly smaller spatial scales in the direction perpendicular to the guide field. Turbulent cascade also contributes to the dissipation of the bulk energy of the waves through the generation of smaller spatial scales. Both processes present characteristic timescales. Different regimes can be envisaged according to how those timescales are related and to the typical timescale at which dissipation is at work. We investigate the interplay of phase mixing and the nonlinear turbulent cascade in the evolution and dissipation of Alfv\'en waves using compressible magnetohydrodynamic numerical simulations. We consider perturbations in the form of torsional waves, both propagating and standing, or turbulent fluctuations, or a combination of the two. The main purpose is to study how phase mixing and nonlinear couplings jointly work to produce small scales in different regimes. We conducted a numerical campaign to explore the typical parameters, such as the loop length, the amplitude and spatial profile of the perturbations, and the dissipative coefficients. A pseudo-spectral code was employed to solve the three-dimensional compressible magnetohydrodynamic equations, modeling the evolution of perturbations propagating in a flux tube corresponding to an equilibrium configuration with cylindrical symmetry. We find that phase mixing takes place for moderate amplitudes of the turbulent component even in a distorted, nonaxisymmetric configuration, building small scales that are locally transverse to the density gradient. The dissipative time decreases with increasing the percentage of the turbulent component. This behavior is verified both for propagating and standing waves. Even in the fully turbulent case, a mechanism qualitatively similar to phase mixing occurs: it actively generates small scales together with the nonlinear cascade, thus providing the shortest dissipative time. General considerations are given to identify this regime in the parameter space. The turbulent perturbation also distorts the background density, locally increasing the Alfv\'en velocity gradient and further contributing to accelerating the formation of small scales. Our campaign of simulations is relevant for the coronal plasma where Reynolds and Lundquist numbers are extremely high. For sufficiently low perturbation amplitudes, phase mixing and turbulence work synergically, speeding up the dissipation of the perturbation energy: phase mixing dominates at early times and nonlinear effects at later times. We find that the dissipative time is shorter than those of phase mixing and the nonlinear cascade when individually considered.

Optimal error estimates of a decoupled finite element scheme for the unsteady inductionless MHD equations

Optimal error estimates of a decoupled finite element scheme for the unsteady inductionless MHD equations

Local existence of compressible magnetohydrodynamic equations without initial compatibility conditions

In this paper, we study the initial‐boundary value problem of three‐dimensional viscous, compressible, and heat conductive magnetohydrodynamic equations. Local existence and uniqueness of strong solutions are established with any such initial data that the initial compatibility conditions do not be required. The analysis is based on some suitable prior estimates for the strong coupling term and strong nonlinear term . Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations

Abstract In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $\beta =0$ and RPC-SAV-CN when $\beta \in (0,\frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-\alpha _{1}\varDelta t\nu \varDelta (\widetilde {\textbf {u}}^{n+1}-{\textbf {u}}^{n})$ and $\alpha _{2}\varDelta t\sigma ^{-1}\mbox {curl}\mbox {curl} (\textbf {H}^{n+1}-\textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.

Investigating the effects of numerical algorithms on global magnetohydrodynamics simulations of solar wind in the inner heliosphere

This paper explores the effects of numerical algorithms on global magnetohydrodynamics simulations of solar wind (SW) in the inner heliosphere. To do so, we use sunRunner3D, a 3-D magnetohydrodynamics model that employs the boundary conditions generated by CORHEL and the PLUTO code to compute the plasma properties of the SW with the ideal magnetohydrodynamics approximation up to 1.1 AU in the inner heliosphere. Mainly, we define three different combinations of numerical algorithms based on their diffusion level. This diffusion level is related to the way of solving the magnetohydrodynamics equations using the finite volume formulation, and, therefore, we set in terms of the divergence-free condition methods, Riemann solvers, variable reconstruction schemes, limiters, and time-steeping algorithms. According to the simulation results, we demonstrate that sunRunner3D reproduces global features of Corotating Interaction Regions observed by Earth-based spacecraft for a set of Carrington rotations that cover a period that lays in the late declining phase of solar cycle 24, independently of the numerical algorithms. Moreover, statistical analyses between models and in-situ measurements show reasonable agreement with the observations, and remarkably, the high diffusive method matches better with in-situ data than low diffusive methods.

Global well‐posedness of the 3D hydrostatic magnetohydrodynamics equations

In this paper, the three‐dimensional hydrostatic magnetohydrodynamic (HMHD) equations are considered on a thin domain. We showed the global existence and uniqueness (regularity) of strong solutions to the three‐dimensional incompressible HMHD equations without any small assumption on the initial data. More precisely, there exists a unique strong solution globally in time for any given ‐smooth initial data.

Nonlocal contributions to the turbulent cascade in magnetohydrodynamic plasmas.

We present evidence for nonlocal contributions to the turbulent energy cascade in magnetohydrodynamic (MHD) plasmas. Therefore, we revisit a well-known result derived directly from the MHD equations, i.e., the Politano and Pouquet law for the transfer of kinetic and magnetic energy in scale. We propose adding a term that accounts for nonlocal transfer and represents the influence of fluctuations from large scales due to the Alfvén effect. Supported by direct numerical simulations of homogeneous and isotropic MHD turbulence, we verify that in some plasma configurations, neglecting the additional nonlocal term might consistently overestimate energy dissipation rates and, thus, the contributions of turbulent energy dissipation potentially affecting solar wind heating, i.e., a central puzzle in space plasma physics that motivates the present work.

Global well‐posedness of strong solutions to the compressible magnetohydrodynamic equations with Coulomb force and non‐flat doping profile

Global well‐posedness of strong solutions to the compressible magnetohydrodynamic equations with Coulomb force and non‐flat doping profile

A structure preserving numerical method for the ideal compressible MHD system

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

Finite element iterative algorithm based on Anderson acceleration technique for incompressible MHD equations

In this paper, an Anderson’s acceleration algorithm for the finite element Oseen iterative scheme of the steady incompressible magnetohydrodynamics (MHD) problem is designed and analyzed. The Anderson acceleration algorithm based on solving an optimization problem in each iteration receives great attention to accelerate linear and nonlinear iterations. We prove that the proposed algorithm has higher convergence rate than that of the Oseen iterative method for the steady incompressible MHD, and give the accelerating convergence theory under different acceleration depths m. Numerical experiments including the singular solution and benchmark problems are carried out to verify the correctness and effectiveness of the algorithm. The tests show the marked improvement in the high physical parameters (including the hydrodynamic and magnetic Reynolds numbers, and the coupling number) when the Oseen iteration fails to converge.

Existence of energy-variational solutions to hyperbolic conservation laws

We introduce the concept of energy-variational solutions for hyperbolic conservation laws. Intrinsically, these energy-variational solutions fulfill the weak-strong uniqueness principle and the semi-flow property, and the set of solutions is convex and weakly-star closed. The existence of energy-variational solutions is proven via a suitable time-discretization scheme under certain assumptions. This general result yields existence of energy-variational solutions to the magnetohydrodynamical equations for ideal incompressible fluids and to the Euler equations in both the incompressible and the compressible case. Moreover, we show that energy-variational solutions to the Euler equations coincide with dissipative weak solutions.

On the Uniqueness of the Solution to the Inverse Problem of Determining the Diffusion Coefficient of the Magnetic Field Necessary for Constructing Analytical Models of the Magnetic Field of Mercury

This paper considers the problem of the uniqueness of the solution to the coefficient inverse problem for the system of equations of magneto-hydrodynamics, the solution to which allows more accurately describing the processes of generating the magnetic field of planets with a magneto-hydrodynamic dynamo. The conditions under which it is possible to determine three components of the magnetic induction vector and the magnetic field diffusion coefficient are determined.

A Crank–Nicolson leap-frog scheme for the unsteady incompressible magnetohydrodynamics equations

This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style’s scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm.

On the uniqueness of 3-D inhomogeneous viscous incompressible magnetohydrodynamic equations with bounded density

In this paper, we prove the uniqueness of the global solutions constructed by Xu et al. (2022).

Three-dimensional Modeling of the Solar Wind and Local Interstellar Medium Interaction with Pickup Ions in the Presence of Heliospheric Current Sheet

Our three-dimensional, time-dependent, multi-fluid model has been used to investigate the solar wind (SW)–local interstellar medium (LISM) interaction with pickup ions (PUIs) treated as a separate fluid. A non-zero, but fixed, angle between the Sun’s magnetic and rotation axis is adopted. The flow of the plasma mixture (thermal SW protons, PUIs, and electrons), is described by the system of ideal magnetohydrodynamic equations with the source terms responsible for charge exchange between ions and neutral atoms. Different populations of neutral atoms are governed by the individual sets of the Euler equations. As the standard Rankine–Hugoniot relations are not appropriate to describe the anisotropic behavior of PUIs at the termination shock, we use a kinetically-derived set of boundary conditions at it. We extend our previous work [1] and perform these new simulations on a Cartesian grid. This approach allows us to maintain a uniform grid resolution in all directions, without compromising resolution, at large distances from the Sun. The possibility of transition of the SW flow to a stochastic regime in the region between the termination shock and heliopause is further investigated.

Numerical simulations of turbulence in prominence threads induced by torsional oscillations

Context. Threads are the main constituents of prominences. They are dynamic structures that display oscillations, usually interpreted as magnetohydrodynamic (MHD) waves. Moreover, instabilities such as the Kelvin–Helmholtz instability (KHI) have also been reported in prominences. Both waves and instabilities may affect the thermodynamic state of the threads. Aims. We investigate the triggering of turbulence in prominence threads caused by the nonlinear evolution of standing torsional Alfvén waves. We study the heating in the partially ionized prominence plasma as well as possible observational signatures of this dynamics. Methods. We modeled a prominence thread as a radially and longitudinally nonuniform cylindrical flux tube with a constant axial magnetic field embedded in a much lighter and hotter coronal environment. We perturbed the flux tube with the longitudinally fundamental mode of standing torsional Alfvén waves. We numerically solved the three-dimensional (3D) MHD equations to study the temporal evolution in both ideal and dissipative scenarios. In addition, we performed forward modeling to calculate the synthetic Hα imaging. Results. The standing torsional Alfvén waves undergo phase-mixing owing to the radially nonuniform density. The phase-mixing generates azimuthal shear flows, which eventually trigger the KHI and, subsequently, turbulence. When nonideal effects are included, the obtained plasma heating is very localized in an annulus region at the thread boundary and does not increase the temperature in the cool core. Instead, the average temperature in the thread decreases owing to the mixing of internal and external plasmas. In the synthetic observations, first we observe periodic pulsations in the Hα intensity caused by the integration of the phase-mixing flows along the line of sight. Later, fine strands that may be associated with the KHI vortices are seen in the synthetic Hα images. Conclusions. Turbulence can be generated by standing torsional Alfvén waves in prominence threads after the triggering of the KHI, although this mechanism is not enough to heat such structures. Both the phase-mixing stage and the turbulent stage of the simulated dynamics could be discernible in high-resolution Hα observations.