Incompressible Magnetohydrodynamics Research Articles

The Arrow–Hurwicz Iterative Finite Element Method for the Stationary Thermally Coupled Incompressible Magnetohydrodynamics Flow

The Arrow–Hurwicz Iterative Finite Element Method for the Stationary Thermally Coupled Incompressible Magnetohydrodynamics Flow

Weakly nonlinear surface waves on the plasma–vacuum interface

We consider the free boundary problem for a plasma–vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement, where the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Our aim is to construct highly oscillating surface wave solutions in weakly nonlinear regime to this plasma–vacuum interface problem.Under a necessary and sufficient stability condition for a piecewise constant background state, we construct approximate solutions at any arbitrarily large order of accuracy to the free boundary problem in three space dimensions when the initial discontinuity displays high frequency oscillations. Moreover, such approximate surface waves have nontrivial residual non-oscillatory components.

Regularity and Energy Conservation of the Nonhomogeneous Incompressible Ideal Magnetohydrodynamics Equations

In this paper, we study the regularity and local energy equation of the weak solutions for nonhomogeneous incompressible ideal magnetohydrodynamics system. The conditions given on the regularity of solutions guarantee the energy to be conserved. The main method we have employed relies on the commutator estimates.

CALLIOPE: Pseudospectral Shearing Magnetohydrodynamics Code with a Pencil Decomposition

Abstract The pseudospectral method is a highly accurate numerical scheme suitable for turbulence simulations. We have developed an open-source pseudospectral code, calliope, which adopts the P3DFFT library to perform a fast Fourier transform with the two-dimensional (pencil) decomposition of numerical grids. calliope can solve incompressible magnetohydrodynamics (MHD), isothermal compressible MHD, and rotational reduced MHD with parallel computation using very large numbers of cores (>105 cores for 20483 grids). The code can also solve for local magnetorotational turbulence in a shearing frame using the remapping method. calliope is currently the only pseudospectral code that can compute magnetorotational turbulence using pencil-domain decomposition. This paper presents the numerical scheme of calliope and the results of linear and nonlinear numerical tests, including compressible local magnetorotational turbulence with the largest grid number reported to date.

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.

Global well‐posedness for the three‐dimensional incompressible viscous nonresistive magnetohydrodynamics equations in an infinite slab

In this paper, we investigate the global well‐posedness of the system of incompressible viscous nonresistive magnetohydrodynamics (MHD) fluids in a three‐dimensional horizontally infinite slab with finite height. We reformulate our analysis to Lagrangian coordinates and then develop a new mathematical approach to establish global well‐posedness of the MHD system, which requires no nonlinear compatibility conditions on the initial data.

Rotational effect on the asymptotic stability of the MHD system

In this paper, we consider the large time behavior of solutions of the incompressible rotational magnetohydrodynamics equations. First, we prove the unique existence of a global smooth solution in the Sobolev spaces Hs(R3) for 1/2<s<3/2, when the rotation is sufficiently rapid. Second, we establish the temporal decay estimates for the solution under additional assumptions. It is observed that the rotation affects the decay rate of the velocity field.

Energy Transfer, Discontinuities, and Heating in the Inner Heliosphere Measured with a Weak and Local Formulation of the Politano–Pouquet Law

Abstract The solar wind is a highly turbulent plasma for which the mean rate of energy transfer ε has been measured for a long time using the Politano–Pouquet (PP98) exact law. However, this law assumes statistical homogeneity that can be violated by the presence of discontinuities. Here, we introduce a new method based on the inertial dissipation  I σ whose analytical form is derived from incompressible magnetohydrodynamics; it can be considered as a weak and local (in space) formulation of the PP98 law whose expression is recovered after integration in space. We used  I σ to estimate the local energy transfer rate at scale σ from the THEMIS-B and Parker Solar Probe data taken in the solar wind at different heliospheric distances. Our study reveals that discontinuities near the Sun lead to a strong energy transfer that affects a wide range of scales σ. We also observe that switchbacks seem to be characterized by a singular behavior with an energy transfer varying as σ −3/4, which slightly differs from classical discontinuities characterized by a σ −1 scaling. A comparison between the measurements of ε and  I σ shows that in general the latter is significantly larger than the former.

Transverse oscillations of the incompressible MHD mode in the visco-resistive plasmas: an explanation of Alfvénic to Landau-type characteristics

ABSTRACT In this paper, we characterize transverse oscillations as either Alfvénic or Landau-type in an incompressible non-ideal magnetohydrodynamic (MHD) fluid. We consider shear viscosity and magnetic diffusivity as dissipation mechanisms to derive a general dispersion relation for the incompressible MHD waves. The solutions of this dispersion relation for k as a function of ω – denoting by the source for any value of θ up to which magnetic tension acts as restoring force and dominates over internal friction forces – result in four roots, as follows. Two roots, which have a high phase velocity $c_{\rm A}\cos\theta $ are identified as almost undamped propagating Alfvén waves. The other two roots, which have a phase velocity $(2c_{\rm A}\cos\theta)/(\sqrt{\eta/\nu} + \sqrt{\nu/\eta})$, result in Alfvénic-type disturbances of a much shorter decay length than the wavelength. In contrast, when internal frictional forces start dominating over magnetic tension (i.e. for the propagation perpendicular to the background magnetic field, where the tension in the magnetic field becomes zero), the solutions of the dispersion are akin to Landau-type transverse oscillations. Transverse waves of this type were initially reported by Landau in an ordinary viscous fluid. However, our study corresponds to MHD visco-resistive fluid. The prediction for these lateral propagating transverse waves to be of Landau type may be very useful to explain the heating of observed filamentary structures across the magnetic field on a very small spatial scale in the solar coronal plasma, wherein the heating rate is directly proportional to the operating frequency of the driver, while its damping length is inversely proportional to the square root of the frequency.

Turbulence in the Sub-Alfvénic Solar Wind

Abstract The Parker Solar Probe (PSP) entered a region of sub-Alfvénic solar wind during encounter 8, and we present the first detailed analysis of low-frequency turbulence properties in this novel region. The magnetic field and flow velocity vectors were highly aligned during this interval. By constructing spectrograms of the normalized magnetic helicity, cross-helicity, and residual energy, we find that PSP observed primarily Alfvénic fluctuations, a consequence of the highly field-aligned flow that renders quasi-2D fluctuations unobservable to PSP. We extend Taylor’s hypothesis to sub- and super-Alfvénic flows. Spectra for the fluctuating forward and backward Elsässer variables ( z ±, respectively) are presented, showing that z + modes dominate z − by an order of magnitude or more, and the z + spectrum is a power law in frequency (parallel wavenumber) f −3/2 ( k ∥ − 3 / 2 ) compared to the convex z − spectrum with f −3/2 ( k ∥ − 3 / 2 ) at low frequencies, flattening around a transition frequency (at which the nonlinear and Alfvén timescales are balanced) to f −1.25 at higher frequencies. The observed spectra are well fitted using a spectral theory for nearly incompressible magnetohydrodynamics assuming a wavenumber anisotropy k ⊥ ∼ k ∥ 3 / 4 , that the z + fluctuations experience primarily nonlinear interactions, and that the minority z − fluctuations experience both nonlinear and Alfvénic interactions with z + fluctuations. The density spectrum is a power law that resembles neither the z ± spectra nor the compressible magnetic field spectrum, suggesting that these are advected entropic rather than magnetosonic modes and not due to the parametric decay instability. Spectra in the neighboring modestly super-Alfvénic intervals are similar.

Dyadic models for ideal MHD

We study two dyadic models for incompressible ideal magnetohydrodynamics, one with a uni-directional energy cascade and the other one with both forward and backward energy cascades. Global existence of weak solutions and local well-posedness are established for both models. In addition, solutions to the model with uni-directional energy cascade associated with positive initial data are shown to develop blow-up at a finite time. Moreover, a set of fixed points is found for each model. Linear instability about some particular fixed points is proved.

Viscous energy dissipation of kink waves due to phase mixing in twisted coronal flux tubes

ABSTRACT We have studied viscous energy damping of kink Magnetohydrodynamic (MHD) waves in weakly twisted magnetic flux tubes. The flux tube has been modelled as a homogeneous internal region with a straight magnetic field, surrounded by a radially non-uniform and magnetically twisted transitional layer embedded in a homogeneous ambient with a straight field. Using a modal expansion technique, we have solved an initial value problem for the incompressible viscous MHD equations and obtained spatio-temporal behaviour of the perturbations of the kink mode in linear regime. In the transitional layer, the perturbations are subject to phase mixing which enhances the viscous effects in the region. We show that magnetic twist can increase or decrease the efficiency of viscous damping of the phase-mixed perturbations in the non-uniform transitional layer. Using the temporal evolution of the total energy, we have obtained the viscous dissipation time as a power function of the Reynolds number. Our results show that magnetic twist could decrease or increase the viscous dissipation time.

A new global existence result for the 3D magneto-hydrodynamics equations

This manuscript is devoted to the 3D incompressible magneto-hydrodynamics (MHD) equations. The existence of the global strong solution is obtained, provided that the initial data satisfy an additional condition which allows to extend the unique local solution to a global one.

Parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

Parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

A finite element method for MHD that preserves energy, cross-helicity, magnetic helicity, incompressibility, and div B = 0

We construct a structure-preserving finite element method and time-stepping scheme for inhomogeneous, incompressible magnetohydrodynamics (MHD). The method preserves energy, cross-helicity (when the fluid density is constant), magnetic helicity, mass, total squared density, pointwise incompressibility, and the constraint divB=0 to machine precision, both at the spatially and temporally discrete levels.

On the vanishing dissipation limit for the incompressible MHD equations on bounded domains

In this paper, we investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magnetohydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial boundary value problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher-order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD equation as the dissipations tend to zero.

The Global Solvability of 3-D Inhomogeneous Viscous Incompressible Magnetohydrodynamic Equations with Bounded Density

The Global Solvability of 3-D Inhomogeneous Viscous Incompressible Magnetohydrodynamic Equations with Bounded Density

Asymptotic behavior of the stationary magnetohydrodynamic equations in an exterior domain

In this paper, we study the asymptotic behavior of solutions to the incompressible magnetohydrodynamic (MHD) equations in an exterior domain. We will show that, under some assumption, any nontrivial velocity field u and magnetic field h obey a minimal decaying rate exp(−C|x|2 log |x|) at infinity. Our proof is based on appropriate Carleman estimates. As a consequence, we establish a Liouville-type result for the three dimensional incompressible MHDs in an exterior domain.

A fully decoupled linearized finite element method with second-order temporal accuracy and unconditional energy stability for incompressible MHD equations

For highly coupled nonlinear incompressible magnetohydrodynamic (MHD) system, a well-known numerical challenge is how to establish an unconditionally energy stable linearized numerical scheme which also has a fully decoupled structure and second-order time accuracy. This paper simultaneously reaches all of these requirements for the first time by developing an effective numerical scheme, which combines a novel decoupling technique based on the “zero-energy-contribution” feature satisfied by the coupled nonlinear terms, the second-order projection method for dealing with the fluid momentum equations, and a finite element method for spatial discretization. The implementation of the scheme is very efficient, because only a few independent linear elliptic equations with constant coefficients need to be solved by the finite element method at each time step. The unconditional energy stability and well-posedness of the scheme are proved. Various 2D and 3D numerical simulations are carried out to illustrate the developed scheme, including convergence/stability tests and some benchmark MHD problems, such as the hydromagnetic Kelvin-Helmholtz instability, and driven cavity problems.

Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations

We study the well-posedness theory for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substitute the magnetic field into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, we linearize this equation with respect to the position vector field whose time derivative is the velocity, and obtain the local-in-time well-posedness of the solution by using energy estimates of the tangential derivatives and the curl with the help of Lie derivatives and the smooth-out approximation.