Stationary MHD Research Articles

Existence of Regular Solutions for a Class of Incompressible Non-Newtonian MHD Equations Coupled to the Heat Equation

We consider a system of PDE’s describing the steady flow of an electrically conducting fluid in the presence of a magnetic field. The system of governing equations composes of the stationary non-Newtonian incompressible MHD equations coupled to the heat equation wherein the influence of buoyancy is taken into account in the momentum equation and the Joule heating and viscous heating terms are included. We proved the existence of C1,γ(Ω¯)×W2,r(Ω)×W2,2(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }({\\bar{\\Omega }})\ imes W^{2,r}(\\Omega )\ imes W^{2,2}{(\\Omega )}$$\\end{document} solutions of the systems for 1<p<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1< p<2$$\\end{document} corresponding to a small data and we show that this solution is unique in case 6/5<p<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6/5< p < 2$$\\end{document}. Moreover, we also proved the higher regularity properties of this solution.

Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

Convergence Analysis of Some Finite Element Parallel Algorithms for the Stationary Incompressible MHD Equations

Convergence Analysis of Some Finite Element Parallel Algorithms for the Stationary Incompressible MHD Equations

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. III. Asymptotics of branching

The previous paper of this series presented the results of a numerical investigation of the dependence of the dominant growth rates of Bloch eigenmodes on the diffusivity parameters (the molecular viscosity ν and molecular magnetic diffusivity η) in three linear stability problems: the kinematic dynamo problem, and the hydrodynamic and MHD stability problems for steady spaceperiodic flows and MHD states. The dominant eigenmodes (i.e., the stability modes, whose growth rates are maximum over the wave vector q of the planar wave involved in the Bloch modes) comprise branches. In some branches, the dominant growth rates are attained for constant half-integer q. In all the three stability problems for parity-invariant steady states, offshoot branches, stemming from the branches of this type, were found, in which the dominant growth rates are attained for q depending on ν and/or η. We consider now such a branching of the dominant magnetic modes in the kinematic dynamo problem, where an offshoot stems from a branch of neutral eigenmodes for q = 0, and construct power series expansions for the offshoots and the associated eigenvalues of the magnetic induction operator near the point of bifurcation. We show that the branching occurs for the molecular magnetic diffusivities, for which the two eigenvalues of the eddy diffusivity operator become imaginary, and magnetic field generation by the mechanism of the negative eddy diffusivity ceases. The details of branching in the other linear stability problems under consideration are distinct.

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

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

We consider Bloch eigenmodes of three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states comprised of randomly generated Fourier coefficients and having energy spectra of three types: exponentially decaying, Kolmogorov with a cut off, or involving a small number of harmonics (“big eddies”). A Bloch mode is a product of a field of the same periodicity as the perturbed state and a planar harmonic wave, exp(iq · x). Such a mode is characterized by the ratio of spatial scales which, for simplicity, we identify with the length |q| &lt; 1 of the Bloch wave vector q. Computations have revealed that the Bloch modes, whose growth rates are maximum over q, feature the scale ratio that decreases on increasing the nondimensionalized molecular diffusivity and/or viscosity from 0.03 to 0.3, and the scale separation is high (i.e., |q| is small) only for large molecular diffusivities. Largely this conclusion holds for all the three stability problems and all the three energy spectra types under consideration. Thus, in a natural MHD system not affected by strong diffusion, a given scale range gives rise to perturbations involving only moderately larger spatial scales (i.e., |q| only moderately small), and the MHD evolution consists of a cascade of processes, each generating a slightly larger spatial scale; flows or magnetic fields characterized by a high scale separation are not produced. This cascade is unlikely to be amenable to a linear description. Consequently, our results question the allegedly high role of the α-effect and eddy diffusivity that are based on spatial scale separation, as the primary instability or magnetic field generating mechanisms in astrophysical applications. The Braginskii magnetic α-effect in a weakly non-axisymmetric flow, often used for explanation of the solar and geodynamo, is advantageous not being upset by a similar deficiency.

Линейные возмущения Блоховского типа пространственно-периодических магнитогидодинамических стационарных состояний. I. Математические вопросы

We consider Bloch eigenmodes in three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states. A Bloch mode is a product of a field of the same periodicity, as the state subjected to perturbation, and a planar harmonic wave, exp(iqx). The complex exponential cancels out from the equations of the respective eigenvalue problem, and the wave vector q remains in the equations as a numeric parameter. The resultant problem has a significant advantage from the numerical viewpoint: while the Bloch mode involves two independent spatial scales, its growth rate can be computed in the periodicity box of the perturbed state. The three-dimensional space, where q resides, splits into a number of regions, inside which the growth rate is a smooth function of q. In preparation for a numerical study of the dominant (i.e., the largest over q) growth rates, we have derived expressions for the gradient of the growth rate in q and proven that, for parity-invariant flows and MHD steady states or when the respective eigenvalue of the stability operator is real, half-integer q (whose all components are integer or half-integer) are stationary points of the growth rate. In prior works it was established by asymptotic methods that high spatial scale separation (small q) gives rise to the phenomena of the α-effect or, for parity-invariant steady states, of the eddy diffusivity. We review these findings tailoring them to the prospective numerical applications.

Liouville-type theorems for steady MHD and Hall-MHD equations in [formula omitted

In this paper, we study the Liouville-type theorems for three-dimensional stationary incompressible MHD and Hall-MHD systems in a slab with periodic boundary condition. We show that, under the assumptions that (uθ,bθ) or (ur,br) is axisymmetric, or (rur,rbr) is bounded, any smooth bounded solution to the MHD or Hall-MHD system with local Dirichlet integral growing as an arbitrary power function must be constant. This improves the result of Theorem 1.2 in Pan (2021) [33], where the Dirichlet integral of u is assumed to be finite. Inspired by Bang et al. (2022) [3], our proof relies on establishing Saint-Venant's estimates associated with our problem, and the result in the current paper extends that for stationary Navier-Stokes equations shown by [3] to MHD and Hall-MHD equations. To achieve this, more intricate estimates are needed to handle the terms involving b properly.

A Liouville-type theorem for the stationary MHD equations

We establish a new Liouville-type theorem for solutions of the stationary MHD equations imposing asymmetric oscillation growth conditions on the tensor-valued functions for the velocity and the magnetic field.

Remarks on the Liouville type theorems for the 3D stationary MHD equations

In this paper we study the stationary incompressible MHD equations on $ {\mathbb{R}}^3 $. We show that the velocity field $ u $ and the magnetic field $ b $, which vanish at infinity, must be identically zero, provided that $ \nabla u $ and $ \nabla b $ belong to $ L^q ({{\Bbb R}}^3) $ for some $ q \in (\frac{3}{2}, 2] $, and $ u $ belongs to $ \operatorname{BMO}^{-1}({{\Bbb R}}^3) $. Moreover, motivated by the recent work [19], we also obtain the Liouville type theorem by assuming that $ (u, b) $ is axisymmetric and $ u $ satisfies some decay property.

Liouville-type theorems for the stationary inhomogeneous incompressible MHD equations

In this paper, we investigate the Liouville-type problem for the three-dimensional stationary inhomogeneous incompressible MHD equations in the Lorentz spaces without any integrability condition for (∇u,∇B). More precisely, we prove that the velocity and magnetic field (u,B), being in some Lorentz spaces, must be zero provided that the density ρ is essentially bounded.

Free-plasma-boundary solver for axisymmetric ideal MHD equilibria with flow

An efficient iterative, free-plasma-boundary solver for the Grad–Shafranov–Bernoulli system of equations, that describes the ideal MHD equilibrium of a toroidally axisymmetric plasma with flow, is presented. The code implements a numerical scheme recently developed in the context of free-plasma-boundary solvers for ideal static MHD equilibria with magnetic islands and stochastic regions for stellarators. The shape of the plasma edge is permitted to change as needed until the total net force eventually vanishes en route to the equilibrium. Complex coil configurations can be treated in the toroidally axisymmetric approximation. The code opens the possibility of quantifying the changes that plasma flows may induce on important features of a tokamak equilibrium such as the shape of the plasma edge, the plasma confining volume, the position of the magnetic axis or the position of the X-point, among others. Some examples, selected for illustrative purposes, are shown for the ITER baseline magnetic configuration.

MHD flow with Navier type boundary conditions

We are concerned here with results of existence and regularity of solutions to a stationary MHD problem, with a possibly multi-connected 2D domain associated with Navier-type boundary conditions. The RHS f, corresponding to the external forces, is in Lp(Ω), with p≥2, without any restriction on its norm. Two main difficulties are encountered in solving these problems: the geometry of the domain and the non-linear term (u⋅∇)(u−αΔu). We prove the existence of a solution (u,B) in H2(Ω)×W2,r(Ω) for any finite r.

Erratum: “Unified nonlinear theory of spontaneous and forced helical resonant MHD states” [Phys. Plasmas 24, 040701 (2017)

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A Mixed Finite Element Method for Stationary Magneto-Heat Coupling System with Variable Coefficients.

In this article, a mixed finite element method for thermally coupled, stationary incompressible MHD problems with physical parameters dependent on temperature in the Lipschitz domain is considered. Due to the variable coefficients of the MHD model, the nonlinearity of the system is increased. A stationary discrete scheme based on the coefficients dependent temperature is proposed, in which the magnetic equation is approximated by Nédélec edge elements, and the thermal and Navier–Stokes equations are approximated by the mixed finite elements. We rigorously establish the optimal error estimates for velocity, pressure, temperature, magnetic induction and Lagrange multiplier with the hypothesis of a low regularity for the exact solution. Finally, a numerical experiment is provided to illustrate the performance and convergence rates of our numerical scheme.

Liouville type theorems for the stationary Hall‐magnetohydrodynamic equations in local Morrey spaces

This paper is concerned with the Liouville type theorems for the 3D stationary incompressible Hall‐magnetohydrodynamic (Hall‐MHD) equations. We establish that under some sufficient conditions in local Morrey spaces, solutions of the stationary Hall‐MHD equations are identically zero. In particular, we also prove Liouville type results for the stationary incompressible MHD equations on . Our theorems extend and generalize the classical results for the stationary incompressible Navier–Stokes equations.

Optimal Convergence Analysis of Two-Level Nonconforming Finite Element Iterative Methods for 2D/3D MHD Equations.

Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on a fine grid. A one-level Oseen iterative method on a fine mesh is further studied under a weak uniqueness condition. Moreover, the stability and error estimate are rigorously carried out, which prove that the proposed methods are stable and effective. Finally, some numerical examples corroborate the effectiveness of our theoretical analysis and the proposed methods.

Two-level Newton iterative method based on nonconforming finite element discretization for 2D/3D stationary MHD equations

In this paper, two-level Newton iterative method based on nonconforming finite element discretization is presented for solving 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix–Raviart type element for the velocity, and the conforming finite element for the magnetic field and pressure. The main idea of the proposed method is to solve MHD system by m Newton iterations on a coarse mesh, once correction by Stokes iteration on a fine mesh. The proposed method can save more computational time than one level method on the fine mesh with the same convergence rate. Moreover, the technical analysis of stability and optimal error estimates for two-level Newton iterative method are given. Finally, the applicability and efficiency of our proposed algorithm are illustrated by several numerical experiments.

On Liouville-type theorems for the stationary MHD and the Hall-MHD systems in $${ \mathbb {R} }^3$$

In this paper, we prove a Liouville-type theorem for the stationary MHD and the stationary Hall-MHD systems. Assuming suitable growth condition at infinity for the mean oscillations for the potential functions, we show that the solutions are trivial. These results generalize the previous results obtained by two of the current authors in Chae and Wolf (J Differ Equ 295:233–248, 2021). To prove our main theorems, we use a refined iteration argument.

On Liouville-type theorems for the 2D stationary MHD equations

We establish new Liouville-type theorems for the two-dimensional stationary magneto-hydrodynamic incompressible system assuming that the velocity and magnetic field have bounded Dirichlet integral. The key tool in our proof is observing that the stream function associated to the magnetic field satisfies a simple drift–diffusion equation for which a maximum principle is available.