3D Magnetohydrodynamic Equations Research Articles

Two-fluid sub-grid-scale viscosity in nonlinear simulation of ballooning modes in a heliotron device

A large eddy simulation (LES) approach is introduced to enable the study of the nonlinear growth of ballooning modes in a heliotron-type device, by solving fully 3D two-fluid magnetohydrodynamic (MHD) equations numerically over a wide range of parameter space, keeping computational costs as low as possible. A model to substitute the influence of scales smaller than the grid size, at sub-grid scale (SGS), and at the scales larger than it—grid scale (GS)—has been developed for LES. The LESs of two-fluid MHD equations with SGS models have successfully reproduced the growth of the ballooning modes in the GS and nonlinear saturation. The numerical results show the importance of SGS effects on the GS components, or the effects of turbulent fluctuation at small scales in low-wavenumber unstable modes, over the course of the nonlinear saturation process. The results also show the usefulness of the LES approach in studying instability in a heliotron device. It is shown through a parameter survey over many SGS model coefficients that turbulent small-scale components in experiments can contribute to keeping the plasma core pressure from totally collapsing.

On the Global Regularity for the 3D Magnetohydrodynamics Equations Involving Partial Components

In this paper, we study the regularity criteria of the three-dimensional magnetohydrodynamics system in terms of some components of the velocity field and the magnetic field. With a decomposition of the four nonlinear terms of the system, this result gives an improvement of some corresponding previous works (Yamazaki in J Math Fluid Mech 16: 551–570, 2014; Jia and Zhou in Nonlinear Anal Real World Appl 13: 410–418, 2012).

Global regularity for 2D magneto-micropolar equations with only micro-rotational velocity dissipation and magnetic diffusion

This paper studies the global regularity of classical solutions to 2D magneto-micropolar fluid equations with only micro-rotational velocity dissipation and magnetic diffusion. Here the micro-rotational velocity dissipation and magnetic diffusion are given by −ΔΩ and (−Δ)βb. Making use of several combined quantities, maximal regularity of heat operator and Littlewood–Paley decomposition theory, we establish a regularity criterion in terms of magnetic field for the case β=1 and the global regularity for β>1. The regularity criterion given here is also new even for the 2D magnetohydrodynamic equations. In addition, to prove these two main results, as preparation we establish a new global a priori estimate for magnetic field, namely Δb∈L∞(0,T;Lp(R2)) with p≥2 which also holds for the 2D magnetohydrodynamic equations as a particular case.

Refined regularity class of suitable weak solutions to the 3D magnetohydrodynamics equations with an application

By means of blow-up method and the special structure of the 3D viscous magnetohydrodynamics equations, we derive some interior regularity criteria in terms of horizontal part of the velocity with sufficiently small local scaled norm and both the vertical part of the velocity and the magnetic field with bounded local scaled norm for the suitable weak solutions to this system. As an application, this allows us to improve the previous limiting case for the regularity criterion about the MHD equations.

Global regularity for logarithmically critical 2D MHD equations with zero viscosity

In this article, the two-dimensional magneto-hydrodynamic (MHD) equations are considered with only magnetic diffusion. Here the magnetic diffusion is given by \({\mathfrak D}\) a Fourier multiplier whose symbol m is given by \(m(\xi )=|\xi |^2\log (e+|\xi |^2)^\beta \). We prove that there exists an unique global solution in \(H^s(\mathbb {R}^2)\) with \(s>2\) for these equations when \(\beta >1\). This result improves the previous works which require that \(m(\xi )=|\xi |^{2\beta }\) with \(\beta >1\) and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely \(m(\xi )=|\xi |^2\).

On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in $${\mathbb{R}^{3}}$$ R 3

In this paper, we establish a blow-up criterion of strong solutions to the 3D incompressible magnetohydrodynamics equations including two nonlinear extra terms: the Hall term (quadratic with respect to the magnetic field) and the ion-slip term (cubic with respect to the magnetic field). This is an improvement of the recent results given by Fan et al. (Z Angew Math Phys, 2015).

On regularity criterion to the 3D axisymmetric incompressible MHD equations

This paper is concerned with the regularity criterion for a class of axisymmetric solutions to 3D incompressible magnetohydrodynamic equations. More precisely, for the solutions that have the form of u = urer+uθeθ+uzez and b = bθeθ, we prove that if |ru(x,t)|≤C holds for −1≤t < 0, then (u,b) is regular at time zero. This result can be thought as a generalization of recent results in for the 3D incompressible Navier‐Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.

Remarks on the uniqueness of weak solution for the 3D viscous magneto-hydrodynamics equations in $${B^{1}_{\infty,\infty}}$$ B ∞ , ∞ 1

A uniqueness result of weak solution for the 3D viscous magneto-hydrodynamics equations in \({B^1_{\infty,\infty}}\) is proved by means of the Fourier localization technique and the losing derivative estimates.

Turbulent transport and coherent structures in 3D plasmas

Inertial transport of kinetic and magnetic enstrophy is shown to have a concentrative effect towards smaller scales from a range spanning the integral scale to a Kraichnan-type scale for the 3D magnetohydrodynamic equations. This is an improvement of the result from Bradshaw and Grujić (2013), using redesigned ensemble averages.

Strong solutions to 3D compressible magnetohydrodynamic equations with Navier‐slip condition

We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier‐slip condition. The Navier‐slip condition differs in many aspects from no‐slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax–Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier‐slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L∞ norm of the deformation tensor D(u). Copyright © 2015 John Wiley & Sons, Ltd.

Numerical simulations of transverse oscillations in radiatively cooling coronal loops

We aim to study the influence of radiative cooling on the standing kink oscillations of a coronal loop. Using the FLASH code, we solved the 3D ideal magnetohydrodynamic equations. Our model consists of a straight, density enhanced and gravitationally stratified magnetic flux tube. We perturbed the system initially, leading to a transverse oscillation of the structure, and followed its evolution for a number of periods. A realistic radiative cooling is implemented. Results are compared to available analytical theory. We find that in the linear regime (i.e. low amplitude perturbation and slow cooling) the obtained period and damping time are in good agreement with theory. The cooling leads to an amplification of the oscillation amplitude. However, the difference between the cooling and non-cooling cases is small (around 6% after 6 oscillations). In high amplitude runs with realistic cooling, instabilities deform the loop, leading to increased damping. In this case, the difference between cooling and non-cooling is still negligible at around 12%. A set of simulations with higher density loops are also performed, to explore what happens when the cooling takes place in a very short time (tcool = 100 s). We strengthen the results of previous analytical studies that state that the amplification due to cooling is ineffective, and its influence on the oscillation characteristics is small, at least for the cases shown here. Furthermore, the presence of a relatively strong damping in the high amplitude runs even in the fast cooling case indicates that it is unlikely that cooling could alone account for the observed, flare-related undamped oscillations of coronal loops. These results may be significant in the field of coronal seismology, allowing its application to coronal loop oscillations with observed fading-out or cooling behaviour.

Sunspot rotation

Context. Solar eruptions and high flare activity often accompany the rapid rotation of sunspots. The study of sunspot rotation and the mechanisms driving this motion are therefore key to our understanding of how the solar atmosphere attains the conditions necessary for large energy release. Aims. We aim to demonstrate and investigate the rotation of sunspots in a 3D numerical experiment of the emergence of a magnetic flux tube as it rises through the solar interior and emerges into the atmosphere. Furthermore, we seek to show that the sub-photospheric twist stored in the interior is injected into the solar atmosphere by means of a definitive rotation of the sunspots. Methods. A numerical experiment is performed to solve the 3D resistive magnetohydrodynamic (MHD) equations using a Lagrangian-Remap code. We track the emergence of a toroidal flux tube as it rises through the solar interior and emerges into the atmosphere investigating various quantities related to both the magnetic field and plasma. Results. Through detailed analysis of the numerical experiment, we find clear evidence that the photospheric footprints or sunspots of the flux tube undergo a rotation. Significant vertical vortical motions are found to develop within the two polarity sources after the field emerges. These rotational motions are found to leave the interior portion of the field untwisted and twist up the atmospheric portion of the field. This is shown by our analysis of the relative magnetic helicity as a significant portion of the interior helicity is transported to the atmosphere. In addition, there is a substantial transport of magnetic energy to the atmosphere. Rotation angles are also calculated by tracing selected fieldlines; the fieldlines threading through the sunspot are found to rotate through angles of up to 353 degrees over the course of the experiment.

Hopf-bifurcation theorem and stability for the magneto-hydrodynamics equations

This paper is devoted to the study of the dynamical behavior for the 3D viscous Magneto-hydrodynamics equations. We first prove that this system under smooth external forces possesses time dependent periodic solutions, bifurcating from a steady solution. If the time periodic solution is smooth, then the linear stability of the time periodic solution implies nonlinear stability is obtained in $\textbf{L}^p$ for all $p\in(3,\infty)$.

Formula omitted] asymptotic stability of the mild solution to the 3D MHD equation

In this paper, we investigate the L2− asymptotic stability of a global-in-time mild solution of the small initial value problem to the 3D magnetohydrodynamic (MHD) equations. We construct a weak solution to the MHD equation, and then we show this weak solution converges to the mild solution in the sense of L2 energy norm.

Global regularity to the 3D incompressible MHD equations

In this paper, we prove the global regularity to the 3D nonhomogeneous incompressible magnetohydrodynamic equations. Let ϱ0, m0 and H0 be the initial density, momentum and magnetic field, respectively. We establish a unique strong solution on R3×(0,T) for any 0<T<∞ under the assumption that the viscosity coefficient μ>0 is sufficiently large, or ‖|m0|2/ϱ0‖L12+‖H0‖L22 or ‖∇u0‖L22+‖H0‖H12 is small enough. Moreover, if the given data are more regular and satisfy an additional compatibility condition for the existence of strong solution, then we show that the strong solution is indeed a classical one. Moreover, the weak–strong uniqueness of solutions is also obtained, which generalizes the result in [13] to the case of vacuum.

An improved pressure regularity criterion of magnetohydrodynamic equations in critical Besov spaces

This paper is concerned with an improved pressure regularity criterion of the three-dimensional (3D) magnetohydrodynamic (MHD) equations in the largest critical Besov spaces. Based on the Littlewood-Paley decomposition technique, the weak solutions are proved to be smooth if the pressure lies in the largest critical Besov spaces, $\pi(x,t) \in L^{p}(0,T;\dot{B}^{0}_{q,r}(\mathbf{R}^{3}))$ for $\frac{2}{p}+\frac{3}{q}=2$ , $1\leq r\leq\frac{2q}{3}$ , $\frac{3}{2}< q\leq\infty$ .

Pullback attractors for non-autonomous 2D MHD equations on some unbounded domains

We study the 2D magnetohydrodynamic (MHD) equations for a viscous incompressible resistive fluid, a system with the Navier&#8211;Stokes equations for the velocity field coupled with a convection-diffusion equation for the magnetic fields, in an arbitrary

EVOLUTION OF TURBULENCE IN THE EXPANDING SOLAR WIND, A NUMERICAL STUDY

We study the evolution of turbulence in the solar wind by solving numerically the full 3D magneto-hydrodynamic (MHD) equations embedded in a radial mean wind. The corresponding equations (expanding box model or EBM) have been considered earlier but never integrated in 3D simulations. Here, we follow the development of turbulence from 0.2 AU up to about 1.5 AU. Starting with isotropic spectra scaling as $k^{-1}$, we observe a steepening toward a $k^{-5/3}$ scaling in the middle of the wavenumber range and formation of spectral anisotropies. The advection of a plasma volume by the expanding solar wind causes a non-trivial stretching of the volume in directions transverse to radial and the selective decay of the components of velocity and magnetic fluctuations. These two effects combine to yield the following results. (i) Spectral anisotropy: gyrotropy is broken, and the radial wavevectors have most of the power. (ii) Coherent structures: radial streams emerge that resemble the observed microjets. (iii) Energy spectra per component: they show an ordering in good agreement with the one observed in the solar wind at 1 AU. The latter point includes a global dominance of the magnetic energy over kinetic energy in the inertial and $f^{-1}$ range and a dominance of the perpendicular-to-the-radial components over the radial components in the inertial range. We conclude that many of the above properties are the result of evolution during transport in the heliosphere, and not just the remnant of the initial turbulence close to the Sun.

A Beale–Kato–Majda criterion for the 3D viscous magnetohydrodynamic equations

In this paper, we investigate the Cauchy problem for the 3D viscous incompressible magnetohydrodynamic equations and establish a Beale–Kato–Majda regularity criterion of smooth solutions in terms of the velocity vector in the homogeneous bounded mean oscillations space. Copyright © 2014 John Wiley &amp; Sons, Ltd.

On some new global existence results for 3D magnetohydrodynamic equations

This paper is devoted to the incompressible magnetohydrodynamic equations in . We prove that if the difference between the magnetic field and the velocity is small initially then it will remain forever, thus resulting in a global strong solution without the smallness restriction on the size of initial velocity or magnetic field. In other words, magnetic field can indeed regularize Navier–Stokes equations, due to cancellation.