MHD Equations Research Articles

Sharp ill-posedness for the non-resistive MHD equations in Sobolev spaces

In this paper, we prove a sharp ill-posedness result for the incompressible non-resistive MHD equations. In any dimension d≥2, we show the ill-posedness of the non-resistive MHD equations in Hd2−1(Rd)×Hd2(Rd), which is sharp in view of the results of the local well-posedness in Hs−1(Rd)×Hs(Rd)(s>d2) established by Fefferman et al. (2017) [11]. Furthermore, we generalize the ill-posedness results from Hd2−1(Rd)×Hd2(Rd) to Besov spaces Bp,qdp−1(Rd)×Bp,qdp(Rd) and B˙p,qdp−1(Rd)×B˙p,qdp(Rd) for 1≤p≤∞,q>1. Different from the ill-posedness mechanism of the incompressible Navier-Stokes equations in B˙∞,q−1[3,21], we construct an initial data such that the paraproduct terms (low-high frequency interaction) of the nonlinear term make the main contribution to the norm inflation of the magnetic field.

High-Order ADER Discontinuous Galerkin Schemes for a Symmetric Hyperbolic Model of Compressible Barotropic Two-Fluid Flows

This paper presents a high-order discontinuous Galerkin (DG) finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) model of compressible two-phase flow, introduced by Romenski et al. in [59, 62], in multiple space dimensions. In the absence of algebraic source terms, the model is endowed with a curl constraint on the relative velocity field. In this paper, the hyperbolicity of the system is studied for the first time in the multidimensional case, showing that the original model is only weakly hyperbolic in multiple space dimensions. To restore the strong hyperbolicity, two different methodologies are used: (i) the explicit symmetrization of the system, which can be achieved by adding terms that contain linear combinations of the curl involution, similar to the Godunov-Powell terms in the MHD equations; (ii) the use of the hyperbolic generalized Lagrangian multiplier (GLM) curl-cleaning approach forwarded. The PDE system is solved using a high-order ADER-DG method with a posteriori subcell finite-volume limiter to deal with shock waves and the steep gradients in the volume fraction commonly appearing in the solutions of this type of model. To illustrate the performance of the method, several different test cases and benchmark problems have been run, showing the high order of the scheme and the good agreement when compared to reference solutions computed with other well-known methods.

Non-linear magnetic buoyancy instability and turbulent dynamo

ABSTRACT Stratified discs with strong horizontal magnetic fields, are susceptible to magnetic buoyancy instability (MBI). Modifying the magnetic field and gas distributions, this can play an important role in galactic evolution. The MBI and the Parker instability, in which MBI is exacerbated by cosmic rays, are often studied using an imposed magnetic field. However, in galaxies and accretion discs, the magnetic field is continuously replenished by a large-scale dynamo action. Using non-ideal MHD equations, we model a section of the galactic disc (we neglect rotation and cosmic rays considered elsewhere), in which the large-scale field is generated by an imposed α-effect of variable intensity to explore the interplay between dynamo instability and MBI. The system evolves through three distinct phases: the linear (kinematic) dynamo stage, the onset of linear MBI when the magnetic field becomes sufficiently strong and the non-linear, statistically steady state. Non-linear effects associated with the MBI introduce oscillations which do not occur when the field is produced by the dynamo alone. The MBI initially accelerates the magnetic field amplification but the growth is quenched by the vertical motions produced by MBI. We construct a 1D model, which replicates all significant features of 3D simulations to confirm that magnetic buoyancy alone can quench the dynamo and is responsible for the magnetic field oscillations. Unlike similar results obtained with an imposed magnetic field, the non-linear interactions do not reduce the gas scale height, so the consequences of the magnetic buoyancy depend on how the magnetic field is maintained.

Mean attractors and invariant measures of locally monotone and generally coercive SPDEs driven by superlinear noise

We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces. The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itô's energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic Bénard problem, Burgers type equations, 3D Leray α-model, convective Brinkman-Forchheimer equations, fractional (s,p)-Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others.

Global classical solutions for three-dimensional compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in bounded domains

In this paper, we study an initial-boundary value problem of three-dimensional (3D) compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in a bounded simply connected domain, whose boundary has a finite number of two-dimensional connected components. We derive the global existence and uniqueness of classical solutions provided that the initial total energy is suitably small. Our result generalizes the Cauchy problems of compressible Navier–Stokes equations with Coulomb force [S. Liu, X. Xu and J. Zhang, Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier–Stokes–Poisson equations subject to large oscillations and non-flat doping profile, J. Differ. Equ. 269 (2020) 8468–8508] and compressible MHD equations [H. Li, X. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal. 45 (2013) 1356–1387] to the case of bounded domains although tackling many surface integrals caused by the slip boundary condition are complex. The main ingredient of this paper is to overcome the strong nonlinearity caused by Coulomb force, magnetic field, and rotation effect of micro-particles by applying piecewise-estimate method and delicate analysis based on the effective viscous fluxes involving velocity and micro-rotational velocity.

The Impact of Inclined Magnetic Field on Streamlines in a Constricted Lid-driven Cavity

The influence of oriented magnetic field on the incompressible and electrically conducting flow is investigated in a square cavity with a moving top wall and a no-slip constricted bottom wall. Radial basis function (RBF) approximation is employed to velocity-stream function-vorticity formulation of MHD equations. Numerical results are shown in terms of streamlines for different values of Hartmann number M, orientation angle of magnetic field ϴ and the height of the constricted bottom wall hc with a fixed Reynolds number. It is obtained that the number of vortices increases as either hc or M increases. However, the increase in ϴ leads to decrease the number of vortices. Formation of vortices depends on not only the strength and the orientation of the magnetic field but also the constriction of the bottom wall.

Jeans Gravitational Instability of a Rotating Collisionless Magnetized Plasma with Anisotropic Pressure

The problem of self-gravitational instability of an astrophysical rotating plasma in a strong magnetic field with an anisotropic pressure tensor is studied on the basis of the Chew–Goldberger–Low (CGL) quasi-hydrodynamic equations modified by generalized polytropic laws. Using the general form of a dispersion relation obtained by the normal-mode perturbation method, a discussion is provided of the propagation of small-amplitude perturbation waves in an infinite homogeneous plasma medium for transverse, longitudinal, and oblique directions with respect to the magnetic field vector. It is shown that different polytropic indices and anisotropic pressures not only change the classical Jeans instability condition but also cause the appearance of new unstable regions. Modified Jeans instability criteria are obtained for isotropic MHD equations and anisotropic CGL equations owing to the influence of the polytropic indices on gravitational and firehose instabilities for astrophysical plasma. It is shown that in the case of a longitudinal mode of perturbation wave propagation, the Jeans instability criterion does not depend on uniform rotation. In the case of the transverse propagation regime, the presence of rotation reduces the critical wave number and exerts a stabilizing effect on the growth rate of the unstable regime.

New structure-preserving mixed finite element method for the stationary MHD equations with magnetic-current formulation

New structure-preserving mixed finite element method for the stationary MHD equations with magnetic-current formulation

Generalization to a wider class of entropy split methods for compressible ideal MHD

The high order entropy split methods of Sjögreen & Yee (2019, 2021), by entropy splitting of the compressible Euler (inviscid) flux derivatives for a thermally-perfect gas are based on Harten’s entropy function (Harten, 1983; Gerritsen and Olsson, 1996; Yee et al., 2000). Their formulation have been proven entropy conserving and stable by taking advantage of the homogeneity property of Euler flux, symmetrizable Euler flux derivatives and energy-norm stability in conjunction with high order classical spatial central, dispersion relation-preserving (DRP) (Bogey and Bailly, 2002; Brambley, 2016; Johansson, 2004) or Padé (compact) spatial discretizations (Hirsh, 1975) with summation-by-parts (SBP) operators (Strand, 1994). These high order entropy split methods not only preserve 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. The present work employs a new approach to obtain a wider class of high order entropy split methods that do not have the homogeneous property. The new method consists of a two-point numerical flux portion and a non-conservative portion in such a way that entropy conservation holds without requiring the homogeneity property of the compressible inviscid flux. For high order classical spatial central, DRP or Padé spatial discretizations, this new approach can be proven to be entropy conservative while at the same time allowing a wider class of symmetrizable inviscid flux derivatives. More importantly, we extend this new approach to derive a new entropy split method for the equations of ideal MHD. Representative test cases are illustrated with comparison among existing methods of the same high order of accuracy.

A decoupled, unconditionally energy-stable and conservative finite element algorithm for a constrained transport model of the incompressible MHD equations

This article mainly considers numerical approximation for a constrained transport magnetohydrodynamic (CT-MHD) model in three dimensions. The superiority of the algorithm is that the magnetic field and current density naturally satisfy divergence-free at the discrete level, the divergence of velocity is improved simultaneously, and the electromagnetic field and the fluid field are decoupled. Finally, we rigorously prove the unconditional energy stability of the proposed algorithm and display five simulations to confirm its accuracy and validity.

Energy conservation of the resistive and the non‐resistive magnetohydrodynamic equations

In this paper, we provide sufficient conditions on the incompressible viscous resistive MHD equations and the viscous non‐resistive MHD equations to ensure the energy equality. Part of our results also holds for the ideal MHD equations.

Unique local weak solutions of the non-resistive MHD equations in homogeneous Besov space

ABSTRACT In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution ( u , b ) of the non-resistive MHD equations for the initial data u 0 ∈ B ˙ p , 1 d p − 1 ( R d ) and b 0 ∈ B ˙ p , 1 d p ( R d ) with 1 ≤ p ≤ ∞ , and the uniqueness of the weak solution when 1 ≤ p ≤ 2 d . Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to 1 ≤ p ≤ ∞ from 1 ≤ p ≤ 2 d , but the uniqueness of the solution requires 1 ≤ p ≤ 2 d yet.

Solvability and trajectory controllability of impulsive stochastic MHD equations with Rosenblatt process

This manuscript is concerned with the existence and trajectory controllability of impulsive stochastic magnetohydrodynamics equation governed by the Rosenblatt process. In this work, the study is made without imposing the compactness conditions in the generator S(τ) of an analytic semigroup. The noise terms rely on both velocity and magnetic fields. Initially, we reformulated the considered system in the Hilbert space by using the Stokes operator and the existence of a mild solution is established by using Banach fixed point theorem. The trajectory controllability of the proposed model is studied in the presence of impulses using Gronwall’s inequality. An example is illustrated for the developed theoretical concepts with graphical representations.

The Role of Pickup Ions in the Interaction of the Solar Wind with the Local Interstellar Medium. I. Importance of Kinetic Processes at the Heliospheric Termination Shock

The role of pickup ions (PUIs) in the solar wind interaction with the local interstellar medium is investigated with 3D, multifluid simulations. The flow of the mixture of all charged particles is described by the ideal MHD equations, with the source terms responsible for charge exchange between ions and neutral atoms. The thermodynamically distinct populations of neutrals are governed by individual sets of gas dynamics Euler equations. PUIs are treated as a separate, comoving fluid. Because the anisotropic behavior of PUIs at the heliospheric termination shocks is not described by the standard conservation laws (a.k.a. the Rankine–Hugoniot relations), we derived boundary conditions for them, which are obtained from the dedicated kinetic simulations of collisionless shocks. It is demonstrated that this approach to treating PUIs makes the computation results more consistent with observational data. In particular, the PUI pressure in the inner heliosheath (IHS) becomes higher by ∼40%–50% in the new model, as compared with the solutions where no special boundary conditions are applied. Hotter PUIs eventually lead to charge-exchange-driven cooling of the IHS plasma, which reduces the IHS width by ∼15% (∼8–10 au) in the upwind direction, and even more in the other directions. The density of secondary neutral atoms born in the IHS decreases by ∼30%, while their temperature increases by ∼60%. Simulation results are validated with New Horizons data at distances between 11 and 47 au.

A decoupled, unconditionally energy-stable and structure-preserving finite element scheme for the incompressible MHD equations with magnetic-electric formulation

A decoupled, unconditionally energy-stable and structure-preserving finite element scheme for the incompressible MHD equations with magnetic-electric formulation

Steady states of the Parker instability

ABSTRACT We study the linear properties, non-linear saturation, and a steady, strongly non-linear state of the Parker instability in galaxies. We consider magnetic buoyancy and its consequences with and without cosmic rays. Cosmic rays are described using the fluid approximation with anisotropic, non-Fickian diffusion. To avoid unphysical constraints on the instability (such as boundary conditions often used to specify an unstable background state), non-ideal magnetohydrodynamic equations are solved for deviations from a background state representing an unstable magnetohydrostatic equilibrium. We consider isothermal gas and neglect rotation. The linear evolution of the instability is in broad agreement with earlier analytical and numerical models; but we show that most of the simplifying assumptions of the earlier work do not hold, such that they provide only a qualitative rather than quantitative picture. In its non-linear stage the instability has significantly altered the background state from its initial state. Vertical distributions of both magnetic field and cosmic rays are much wider, the gas layer is thinner, and the energy densities of both magnetic field and cosmic rays are much reduced. The spatial structure of the non-linear state differs from that of any linear modes. A transient gas outflow is driven by the weakly non-linear instability as it approaches saturation.

The Problem Parameters Effects on Transient Behavior of MHD Duct Flow

The present study focuses the effects of Reynolds number Re and magnetic Reynolds number Rm on the transient behavior of the MHD flow. The incompressible, electrically conducting and viscous fluid flows through a long pipe subjected to magnetic field B0(t)=B0f(t). B0 is the intensity and f(t) is the time varying function of the magnetic field which is chosen as polynomial, trigonometric, exponential and logarithmic function to illustrate the problem parameters effects. The Re and Rm effects on the behavior of the flow at transient levels are studied with these functions by taking Hartmann number Ha value as 20. The unsteady MHD equations in coupled form are treated by using the dual reciprocity boundary element method (DRBEM). The study reveals that, when Re or Rm increases the time level where the flow elongates is postponed to a further time level. Moreover, the increase in Re flattens the flow as in the increase of Hartmann number. However, the increase in Rm increases the flow magnitude. The transient flow and induced current contours are demonstrated for several Re and Rm values. After the flow elongates, the flow and induced current lines preserve the behavior for polynomial, exponential and logarithmic type f(t) while trigonometric type f(t) causes the flow to show periodic behavior.

A consistent projection finite element method for the non-stationary incompressible thermally coupled MHD equations

This paper introduces a consistent projection finite method for the non-stationary incompressible thermally coupled MHD equations, which buoyancy affects because temperature differences in the flow cannot be neglected. It is a fully discrete projection method. The unconditional stability and error estimates of the velocity, pressure, temperature, and magnetic field under some assumptions are given. Then some numerical experiments are prescribed to support the effectiveness of the consistent projection finite element method.

Local-in-time well-posedness theory for the inhomogeneous MHD boundary layer equations without resistivity in lower regular Sobolev space

In this paper, we consider the well-posedness theory in lower regular Sobolev space for the Prandtl-type equation arising from the inhomogeneous incompressible MHD equation in dimension two. Due to the degeneracy of horizontal dissipation, the well-posedness result of MHD boundary layer equation is established in higher regular Sobolev space rather than lower one generally. By using the anisotropic Sobolev inequality and characteristic line method, the well-posedness result of inhomogenous MHD Prandtl-type equation can be established under the H2−framework.

Global regularity of 2D MHD equations with almost Laplacian velocity dissipation

We obtain the global existence and global regularity for the 2D MHD equations with almost Laplacian velocity dissipation, which require the dissipative operators weaker than any power of the fractional Laplacian. The result can be regarded as a further improvement and generalization of previous works.