Full Magnetohydrodynamic Equations Research Articles

On steady state of viscous compressible heat conducting full magnetohydrodynamic equations

This paper is concerned with the study of equations of viscous compressible and heat-conducting full magnetohydrodynamic (MHD) steady flows in a horizontal layer under the gravitational force and a large temperature gradient across the layer. We assume as boundary conditions, periodic conditions in the horizontal directions, while in the vertical directions, slip-boundary is assumed for the velocity, vertical conditions for the magnetic field, and fixed temperature or fixed heat flux are prescribed for the temperature. The existence of stationary solution in a small neighborhood of a stationary profile close to hydrostatic state is obtained in Sobolev spaces as a fixed point of some nonlinear operator.

Global well-posedness and exponential decay rates of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with vacuum in some class of large initial data

In this paper, the initial-boundary value problem for the two-dimensional viscous, compressible, and heat conducting magnetohydrodynamics equations with vacuum is considered. When all the coefficients of viscosity, heat conductivity and magnetic diffusivity are constants, first, we show that for general initial data and any time T>0, the local strong solution in the sense of Sobolev norms will never blow-up provided that the density is bounded from above and the temperature belongs to space L∞(0,T;L2)∩L2(0,T;L∞). Second, based on the above blow-up criterion and the delicate analysis of the nonlinear strong coupling equations, we establish the global existence of strong solutions when the initial mass of the fluid and the initial energy of the magnetic field (i.e., ‖ρ0‖L1 and ‖H0‖L2) are suitably small, where the refined Zlotnik inequality and the decoupling of the full compressible MHD equations play an essential role in proving the uniform upper bound of the density and the higher-order a priori uniform in time estimates, respectively. It is worth mentioning that the initial velocity and temperature can be large. Moreover, we obtain exponential decay under the H1-norm of (u,θ,H) and L2-norm of (ρu˙,ρθ˙,Ht), where the long-time behaviors of the higher-order derivative of velocity are given and it is different from the known results.

Global well-posedness of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with large viscosity

Global well-posedness of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with large viscosity

SCELT (symbolic computation aided eigenvalue and linear code for tokamaks): A full MHD eigenvalue code in toroidal geometry developed with the use of a symbolic computation technique

In this work, we report the construction of an eigenvalue computer code largely in C++ language, SCELT, by using the symbolic computation technique for the first time to solve the linearized single fluid magnetohydrodynamic (MHD) eigenvalue problem in toroidal geometry. A symbolic vector analysis module is developed to function the automatic derivation of the tedious linearized full MHD equations in the magnetic flux coordinate system. Furthermore, another module is developed to implement the automatic numerical discretization. These two modules dramatically reduce the human workload and obviate the possibility of a mistake during code development. The tools provide a means of constructing matrices from differential operations and can be used for (generalized) linear problems, such as source driven and eigenvalue problems. Demo uses of both the symbolic vector analysis module and automatic numerical discretization module, such as the Poisson equation and tokamak equilibrium equation, are presented to demonstrate their advantages and potential broad applications. The full MHD eigenvalue code developed with these two modules is verified by the internal kink mode and tearing mode tests. Program summaryProgram Title: SCELT (Symbolic Computation aided Eigenvalue and Linear code for Tokamaks)CPC Library link to program files:https://doi.org/10.17632/35h3xmc28k.1Licensing provisions: GNU General Public License 3.0Programming language: C++, MATLABExternal routines/libraries:• EIGEN (https://eigen.tuxfamily.org/index.php?title=Main_Page)Nature of problem: The eigenvalue full MHD model determines a wide range of physical phenomena arising from the basic destabilizing forces, which are of considerable both fundamental and practical importance for magnetic fusion plasmas. From the viewpoint of physics, solving the model is equivalent to finding the spectrum of the system, and for realistic tokamak geometry, it becomes possible only with the help of computer simulation. From the viewpoint of mathematics, it is actually a problem of solving a set of partial differential equations in a general curvilinear coordinate system.Solution method: Different from the conventional method, a symbolic vector analysis module using the symbolic computation technique is developed to conduct the automatic expansion of the full MHD equations in the magnetic flux coordinate system. In addition, an automatic numerical discretization module is developed to automatically implement the numerical discretization. Facilitated by the two modules, the coefficients of the matrices of the eigenvalue equation will be automatically generated and passed to the linear eigenvalue solver (eig function of MATLAB). The eigenvalues and eigenvectors are resolved by the solver.Additional comments including restrictions and unusual features: As originally motivated by the development of an eigenvalue full MHD code, a symbolic vector analysis module by using symbolic computation technique and a closely related automatic numerical discretization module are developed and directly applied to physics code development for the first time at least in the fusion field. Many (generalized) linear physics problems can benefit from the standalone or integrated application of these two modules. The tokamak equilibrium and eigenvalue MHD problems presented in the paper can be considered two important and practical examples.

The dynamics and observability of circularly polarized kink waves

Context. Kink waves are routinely observed in coronal loops. Resonant absorption is a well-accepted mechanism that extracts energy from kink waves. Nonlinear kink waves are know to be affected by the Kelvin-Helmholtz instability. However, all previous numerical studies consider linearly polarized kink waves. Aims. We study the properties of circularly polarized kink waves on straight plasma cylinders, for both standing and propagating waves, and we compare them to the properties of linearly polarized kink waves. Methods. We used the code MPI-AMRVAC to solve the full 3D magnetohydrodynamic equations for a straight magnetic cylinder, excited by both standing and propagating circularly polarized kink (m = 1) modes. Results. The damping due to resonant absorption is independent of the polarization state. The morphology or appearance of the induced resonant flow is different for the two polarizations; however, there are essentially no differences in the forward-modeled Doppler signals. For nonlinear oscillations, the growth rate of small scales is determined by the total energy of the oscillation rather than the perturbation amplitude. We discuss possible implications and seismological relevance.

On the limitations of magneto-frictional relaxation

The magneto-frictional method is used in solar physics to compute both static and quasi-static models of the Sun's coronal magnetic field. Here, we examine how accurately magneto-friction (without fluid pressure) is able to predict the relaxed state in a one-dimensional test case containing two magnetic null points. Firstly, we show that relaxation under the full ideal magnetohydrodynamic equations in the presence of nulls leads necessarily to a non-force-free state, which could not be reached exactly by magneto-friction. Secondly, the magneto-frictional solutions are shown to lead to breakdown of magnetic flux conservation, whether or not the friction coefficient is scaled with magnetic field strength. When this coefficient is constant, flux is initially conserved, but only until discontinuous current sheets form at the null points. In the ensuing weak solution, we show that magnetic flux is dissipated at these current sheets. The breakdown of flux conservation does not occur for an alternative viscous relaxation scheme.

Enhanced phase mixing of torsional Alfvén waves in stratified and divergent solar coronal structures – Paper I. Linear solutions

ABSTRACT We derive a corrected analytical solution for the propagation and enhanced phase mixing of torsional Alfvén waves, in a potential magnetic field with exponentially divergent field lines, embedded in a stratified solar corona. Further we develop a code named TAWAS that calculates the analytic solution describing torsional Alfvén waves using idl software language. We then use TAWAS to demonstrate that both our correction to the analytic solution and the inclusion of wave reflection have a significant impact on Alfvén wave damping. We continue to utilize TAWAS by performing a parameter study in order to identify the conditions under which enhanced phase mixing is strongest. We find that phase mixing is the strongest for high frequency Alfvén waves in magnetic fields with highly divergent field lines and without density stratification. We then present a finite difference solver, Wigglewave, which solves the linearized evolution equations for the system directly. Comparing solutions from TAWAS and Wigglewave we see that our analytical solution is accurate within the limits of the WKB approximation but under-reports the wave damping, caused by enhanced phase mixing, beyond the WKB limit. Both TAWAS and Wigglewave solve the linearized governing equations and not the complete non-linear magnetohydrodynamics (MHD) equations. Paper II will consider simulations that solve the full MHD equations including important non-linear effects.

Numerical simulation of neoclassical tearing modes induced by resonant magnetic perturbations in tokamak plasmas

A model of neoclassical tearing mode (NTM) induced by resonant magnetic perturbation (RMP) in the presence of the bootstrap current in tokamak plasmas is established based on the full magnetohydrodynamic equations. The model is solved numerically by using a finite difference method and the numerical results are obtained. It is found that, the NTM can be induced by the given stable equilibrium, added the perturbed bootstrap current in ohm equation, and the applied RMP. Moreover, with the increasing amplitude of the static RMP, locked mode occurs, which is determined by the plasma flow and fraction of bootstrap current. The mode locking threshold increases with the increasing of plasma rotation frequency and decreasing of fraction of bootstrap current. Furthermore, the dynamic RMP is investigated, results show that the mode locking threshold is lower compared to the static RMP applied, which provides a better way of adjusting the NTM to the frequency of dynamic RMP and slightly suppressing the neoclassical magnetic island.

Effects of plasmoid formation on sawtooth process in a tokamak

For realistic values of the Lundquist number in tokamak plasmas, 1/1 magnetic islands lead to the formation of a secondary thin current sheet, which breaks up into a chain of small magnetic islands, called plasmoids. The role of plasmoid dynamics during the sawtooth reconnection process in fusion plasmas remains an unresolved issue. In this study, systematic simulations are performed to investigate the resistive internal kink mode using the full resistive magnetohydrodynamics equations implemented in the NIMROD code in a simplified tokamak geometry. For Lundquist number S≥1.6×107, the secondary current sheet is found to be unstable to plasmoids during the nonlinear resistive kink mode evolution with a critical aspect ratio of the current sheet of ∼70. The merging of small plasmoids leads to the formation of a monster plasmoid that can significantly affect the primary island evolution. This may provide an explanation for the partial reconnection observed in sawtooth experiments.

Chebyshev Spectral Collocation Method Approximation to Thermally Coupled MHD Equations

In this study, a Chebyshev spectral collocation method (CSCM) approximation is proposed for solving the full magnetohydrodynamics (MHD) equations coupled with energy equation. The MHD flow is two-dimensional, unsteady, laminar and incompressible, and the heat transfer is considered using the Boussinesq approximation for thermal coupling. The flow takes place in a square cavity which is subjected to a vertically applied external magnetic field, and the presence of the induced magnetic field is also taken into account due to the electrical conductivity of the fluid. The governing equations given in terms of stream function, vorticity, temperature, magnetic stream function, and current density, are solved iteratively using CSCM for the spatial discretisation, and an unconditionally stable backward difference scheme for the time integration. The induced magnetic field is obtained by means of its relation to the magnetic stream function. The behaviours of the flow and the heat transfer are investigated for varying values of Reynolds ($Re$), magnetic Reynolds ($Rem$), Rayleigh ($Ra$) and Hartmann ($Ha$) numbers.

On magnetostrophic mean-field solutions of the geodynamo equations. Part 2

A dynamo driven by motions unaffected by viscous forces is termed ‘magnetostrophic’, but cannot be found through today’s numerical simulations, which require substantial viscosity to stabilize solutions of the full magnetohydrodynamic (MHD) dynamo equations. By using an alternative numerical technique, proposed by Taylor (Proc. R. Soc. Lond.A, vol. 274, 1963, pp. 274–283), we recently obtained the first magnetostrophic dynamo solutions ever derived (Wu & Roberts,Geophys. Astrophys. Fluid Dyn., vol. 109, 2014, pp. 84–110). These were axisymmetric and of mean-field type. In an earlier paper (Roberts & Wu,Geophys. Astrophys. Fluid Dyn., vol. 108, 2014, pp. 696–715), we proposed an extension of Taylor’s method. Here we explore its numerical implications, comparing them to the consequences of Taylor’s original proposal. One of the differences between the two approaches is that our modification retains torsional waves but Taylor’s theory does not. A more important difference is that our extension of Taylor’s method is, for reasons presented here, the most general possible that does not suffer from the limitations imposed by viscosity on today’s numerical simulations.

Finite element analysis of effects of Joule heating, electro-osmosis and slip conditions on the performance of magnetohydrodynamic (MHD) micropumps

Magnetohydrodynamic (MHD) micropump which is energy saving and less pollution has been widely investigated by researchers in recent years. Based on the finite element analysis (FEA) of numerical model of 3D MHD micropump, we investigated the comprehensive effects of Joule heating, electro-osmosis and slip condition at walls on flow performance. This paper presents a new method for solving the full 3D MHD equations by coupling multi-physics fields, and gives a more comprehensive analysis of fundamental principles of MHD micropump. The temperature distribution under different flow channel geometries, the electro-osmotic velocity in MHD micropump, and the influence of slip condition at walls on flow velocity are analyzed in detail. The flow velocity slightly increases when taking the effect of Joule heating into account. In addition, the flow velocity is hardly affected by electron-osmosis in this work. It should be pointed out that the mean temperature of MHD micropump decreases with driven voltage increasing. With the advantage of fast and accurate analysis of MHD micropump performance, this study holds promising potential for the optimal design and fundamental research of MHD micropump.

Global strong solution for the full MHD equations with vacuum and large data

This paper deals with the compressible, viscous and heat-conducting magnetohydrodynamic (MHD) equations in dimension one. In the case when all transport coefficients are positive constant, we establish the global existence and uniqueness of strong solution for the initial–boundary-value problem. The initial data is large and vacuum state is allowed.

Stream function-velocity-magnetic induction compact difference method for the 2D steady incompressible full magnetohydrodynamic equations

In this paper, an effective and accurate numerical model that involves a suggested mathematical formulation, viz., the stream functions (ψ and A)-velocity-magnetic induction formulation and a fourth-order compact difference algorithm is proposed for solving the two-dimensional (2D) steady incompressible full magnetohydrodynamic (MHD) flow equations. The stream functions-velocity-magnetic induction formulation of the 2D incompressible full MHD equations is able to circumvent the difficulty of handling the pressure variable in the primitive variable formulation or determining the vorticity values on the boundary in the stream function-vorticity formulation, and also ensure the divergence-free constraint condition of the magnetic field inherently.A test problem with the analytical solution, the well-studied lid-driven cavity problem in viscous fluid flow and the lid-driven MHD flow in a square cavity are performed to assess and verify the accuracy and the behavior of the method proposed currently. Numerical results for the present method are compared with the analytical solution and the other high-order accurate results. It is shown that the proposed stream function-velocity-magnetic induction compact difference method not only has the excellent performances in computational accuracy and efficiency, but also matches well with the divergence-free constraint of the magnetic field. Moreover, the benchmark solutions for the lid-driven cavity MHD flow in the presence of the aligned and transverse magnetic field for Reynolds number (Re) up to 5000 are provided for the wide range of magnetic Reynolds number (Rem) from 0.01 to 100 and Hartmann number (Ha) up to 4000.

Magnetohydrodynamic equations (MHD) generation code

A program to generate codes in Fortran and C of the full magnetohydrodynamic equations is shown. The program uses the free computer algebra system software REDUCE. This software has a package called EXCALC, which is an exterior calculus program. The advantage of this program is that it can be modified to include another complex metric or spacetime. The output of this program is modified by means of a LINUX script which creates a new REDUCE program to manipulate the magnetohydrodynamic equations to obtain a code that can be used as a seed for a magnetohydrodynamic code for numerical applications. As an example, we present part of the output of our programs for Cartesian coordinates and how to do the discretization.

On the linear stability of magnetized jets without current sheets – relativistic case

In our prior papers, we considered the non-relativistic linear stability analysis of magnetized jets that do not have current sheet at the boundary. In this paper, we extend our analysis to relativistic jets. In order to find the unstable modes of current sheet-free, magnetized relativistic jets, we linearize full relativistic magnetohydrodynamics equations and solve them numerically. We find the dispersion relation of the pinch and kink mode instabilities. By comparing the dispersion relations of mildly relativistic jet (Lorentz factor 2) with moderately relativistic jet (Lorentz factor 10), we find that the jet with higher Lorentz factor is significantly more stable in both pinch and kink modes. We show that inclusion of the current sheet-free magnetic field in the jet further enhances the stability. Both pinch and kink mode instabilities become progressively more stable with increasing magnetization. We also show a scaling relation between the maximum temporal growth rate of the unstable mode and the Lorentz factor of the jet. The maximum temporal growth rates of the unstable modes are inversely proportion to the Lorentz factors for most of the modes that we study. However, for the fundamental pinch mode it is inversely proportional to the square of the Lorentz factor. This very beneficial scaling relation holds regardless of the presence of a magnetic field.

Effects of Magnetic and Kinetic Helicities on the Growth of Magnetic Fields in Laminar and Turbulent Flows by Helical Fourier Decomposition

Abstract We present a numerical and analytical study of incompressible homogeneous conducting fluids using a helical Fourier representation. We analytically study both small- and large-scale dynamo properties, as well as the inverse cascade of magnetic helicity, in the most general minimal subset of interacting velocity and magnetic fields on a closed Fourier triad. We mainly focus on the dependency of magnetic field growth as a function of the distribution of kinetic and magnetic helicities among the three interacting wavenumbers. By combining direct numerical simulations of the full magnetohydrodynamics equations with the helical Fourier decomposition, we numerically confirm that in the kinematic dynamo regime the system develops a large-scale magnetic helicity with opposite sign compared to the small-scale kinetic helicity, a sort of triad-by-triad α-effect in Fourier space. Concerning the small-scale perturbations, we predict theoretically and confirm numerically that the largest instability is achived for the magnetic component with the same helicity of the flow, in agreement with the Stretch–Twist–Fold mechanism. Vice versa, in the presence of Lorentz feedback on the velocity, we find that the inverse cascade of magnetic helicity is mostly local if magnetic and kinetic helicities have opposite signs, while it is more nonlocal and more intense if they have the same sign, as predicted by the analytical approach. Our analytical and numerical results further demonstrate the potential of the helical Fourier decomposition to elucidate the entangled dynamics of magnetic and kinetic helicities both in fully developed turbulence and in laminar flows.

Linear stability of certain translationally symmetric MHD equilibria with incompressible flow

In this paper, the linear stability analysis for ideal incompressible magnetohydrodynamic (MHD) flows is investigated. MHD flows in cylindrical domains with arbitrary cross-section as well as circular cross-section are considered. A solution method for obtaining solutions of the full MHD equations of incompressible flow is constructed. Two criteria around the linear stability of that flow are applied to several nonlinear equilibria with constant and non-constant Mach number. The constructed method is exploited in obtaining sufficient conditions for linear stability of the former equilibria.

Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data

The free boundary problem of planar full compressible magnetohydrodynamic equations with large initial data is studied in this paper, when the initial density connects to vacuum smoothly. The global existence and uniqueness of classical solutions are established, and the expanding rate of the free interface is shown. Using the method of Lagrangian particle path, we derive some L ∞ estimates and weighted energy estimates, which lead to the global existence of classical solutions. The main difficulty of this problem is the degeneracy of the system near the free boundary, while previous results (cf. [ 4 , 30 ]) require that the density is bounded from below by a positive constant.

Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code

Numerical simulations form an indispensable tool to understand the behavior of a hot plasma that is created inside a tokamak for providing nuclear fusion energy. Various aspects of tokamak plasmas have been successfully studied through the reduced magnetohydrodynamic (MHD) model. The need for more complete modeling through the full MHD equations is addressed here. Our computational method is presented along with measures against possible problems regarding pollution, stability, and regularity.The problem of ensuring continuity of solutions in the center of a polar grid is addressed in the context of a finite element discretization of the full MHD equations. A rigorous and generally applicable solution is proposed here.Useful analytical test cases are devised to verify the correct implementation of the momentum and induction equation, the hyperdiffusive terms, and the accuracy with which highly anisotropic diffusion can be simulated. A striking observation is that highly anisotropic diffusion can be treated with the same order of accuracy as isotropic diffusion, even on non-aligned grids, as long as these grids are generated with sufficient care. This property is shown to be associated with our use of a magnetic vector potential to describe the magnetic field.Several well-known instabilities are simulated to demonstrate the capabilities of the new method. The linear growth rate of an internal kink mode and a tearing mode are benchmarked against the results of a linear MHD code. The evolution of a tearing mode and the resulting magnetic islands are simulated well into the nonlinear regime. The results are compared with predictions from the reduced MHD model.Finally, a simulation of a ballooning mode illustrates the possibility to use our method as an ideal MHD method without the need to add any physical dissipation.