Nonlinear Magnetohydrodynamics Research Articles

Exact Nonlinear Decomposition of Ideal-MHD Waves Using Eigenenergies. II. Fully Analytical Equations and Pseudoadvective Eigenenergies

Physical insight into plasma evolution in the magnetohydrodynamic (MHD) limit can be revealed by decomposing the evolution according to the characteristic modes of the system. In this paper we explore aspects of the eigenenergy decomposition method (EEDM) introduced in an earlier study (ApJ, 967:80). The EEDM provides an exact decomposition of nonlinear MHD disturbances into their component eigenenergies associated with the slow, Alfvén, and fast eigenmodes, together with two zero-frequency eigenmodes. Here we refine the EEDM by presenting globally analytical expressions for the eigenenergies. We also explore the nature of the zero-frequency “pseudoadvective (PA) modes” in detail. We show that in evolutions with pure advection of magnetic and thermal energy (without propagating waves), a part of the energy is carried by the PA modes. Exact expressions for the error terms associated with these modes—commonly encountered in numerical simulations—are also introduced. The new EEDM equations provide a robust tool for the exact and unique decomposition of nonlinear disturbances governed by homogeneous quasi-linear partial differential equations, even in the presence of local or global degeneracies.

Entropy generation of Williamson hydromagnetic non-linear radiative nanofluid flow near a stagnation point over a porous stretching sheet

Purpose This study aims to analyze the multi-slip effects of entropy generation in steady non-linear magnetohydrodynamics thermal radiation with Williamson nanofluid flow across a porous stretched sheet near a stagnation point. Also, the qualities of viscous dissipation, Cattaneo–Christove heat flux and Arrhenius activation energy are taken into account. Thermophoresis, Brownian motion and Joule heating are also considered. Design/methodology/approach The Navier–Stokes equation, the thermal energy equation and the Solutal concentration equations are the governing mathematical equations that describe the flow and heat and mass transfer phenomena for fluid domains. By using the proper similarity transformations, a set of ordinary differential equationss are retrieved from boundary flow equations. The classical Runge–Kutta fifth-order algorithm along with the shooting technique is implemented to solve the obtained first order differential equations. Findings The study concludes that the temperature distribution boosting for thermal radiation, magnetic field and Eckert number where as the velocity and entropy generation escalate for the Williamson parameter, diffusion parameter and Brinkman number. The skin-friction and heat and mass transfer rate increases with the fluid injection. In addition, tabulated values of friction drag and rate of heat and mass transfer for various values of constraints are provided. Originality/value The comparison of the present results is carried out with the published results and noted a good agreement.

A divergence-free and [formula omitted]-conforming embedded-hybridized DG method for the incompressible resistive MHD equations

We present a divergence-free and Hdiv-conforming hybridized discontinuous Galerkin (HDG) method and a computationally efficient variant called embedded-HDG (E-HDG) for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. The proposed E-HDG approach uses continuous facet unknowns for the vector-valued solutions (velocity and magnetic fields) while it uses discontinuous facet unknowns for the scalar variable (pressure and magnetic pressure). This choice of function spaces makes E-HDG computationally far more advantageous, due to the much smaller number of degrees of freedom, compared to the HDG counterpart. The benefit is even more significant for three-dimensional/high-order/fine mesh scenarios. On simplicial meshes, the proposed methods with a specific choice of approximation spaces are well-posed for linear(ized) MHD equations. For nonlinear MHD problems, we present a simple approach exploiting the proposed linear discretizations by using a Picard iteration. The beauty of this approach is that the divergence-free and Hdiv-conforming properties of the velocity and magnetic fields are automatically carried over for nonlinear MHD equations. We study the accuracy and convergence of our E-HDG method for both linear and nonlinear MHD cases through various numerical experiments, including two- and three-dimensional problems with smooth and singular solutions. The numerical examples show that the proposed methods are pressure robust, and the divergence of the resulting velocity and magnetic fields is machine zero for both smooth and singular problems.

Exact Nonlinear Decomposition of Ideal-MHD Waves Using Eigenenergies

In this paper, we introduce a new method for exact decomposition of propagating, nonlinear magnetohydrodynamic (MHD) disturbances into their component eigenenergies associated with the familiar slow, Alfvén, and fast wave eigenmodes, and the entropy and field-divergence pseudoeigenmodes. First, the mathematical formalism is introduced, where it is illustrated how the ideal-MHD eigensystem can be used to construct a decomposition of the time variation of the total energy density into contributions from the eigenmodes. The decomposition method is then demonstrated by applying it to the output of three separate nonlinear MHD simulations. The analysis of the simulations confirms that the component wave modes of a composite wavefield are uniquely identified by the method. The slow, Alfvén, and fast energy densities are shown to evolve in exactly the way expected from comparison with known linear solutions and nonlinear properties, including processes such as mode conversion. Along the way, some potential pitfalls for the numerical implementation of the decomposition method are identified and discussed. We conclude that the exact, nonlinear decomposition method introduced is a powerful and promising tool for understanding the nature of the decomposition of MHD waves as well as analyzing and interpreting the output of dynamic MHD simulations.

MUSES: A nonlinear magnetohydrodynamics discontinuous Galerkin code for fusion plasmas

The article describes initial steps toward the development of our nonlinear magnetohydrodynamics code, MUSES. Unstructured grids and general coordinates are employed so that MUSES code is expected to have a capability to represent realistic geometries of fusion devices. Discontinuous Galerkin method is chosen as a code framework because of its high spatial accuracy and numerical stability on unstructured grids. The governing equation of the ideal magnetohydrodynamics in this work is Powell 8-wave model which can mitigate the divergence-free issue. Roe scheme is used as a numerical flux since the Powell 8-wave model is a nonconservative system. A cross code comparison is carried out with a linear magnetohydrodynamics code, MINERVA. Owing to discontinuous Galerkin method, a linear theory of an ideal internal kink mode is reproduced quantitatively although the upwind method is employed for numerical stability.

Non-linear dynamics of the double tearing mode

3D non-linear magnetohydrodynamics simulations of a double tearing mode with the JOREK code are presented in the context of trying to better understand the benign termination of runaway electron beams observed in some experiments. It is shown that the non-linear behavior qualitatively depends on the resistivity η via its effect on how fast secondary, non-linearly destabilized, tearing modes grow relative to the primary mode. Within a certain range of η, a violent and global relaxation is observed, consistent with the “Kadomtsev-predicted” reconnection region extending from almost the very center up to the edge of the plasma.

Formation Mechanism of Laser-driven Magnetized “Pillars of Creation”

The Pillars of Creation, one of the most recognized objects in the sky, are believed to be associated with the formation of young stars. However, so far, the formation and maintenance mechanism of the pillars are still not fully understood due to the complexity of the nonlinear radiation magnetohydrodynamics (RMHD). Here, assuming laboratory laser-driven conditions, we studied the self-consistent dynamics of pillar structures in magnetic fields by means of two-dimensional and three-dimensional (3D) RMHD simulations, and the results support our proposed experimental scheme. We find that only when the magnetic pressure and ablation pressure are comparable, the magnetic field can significantly alter the plasma hydrodynamics. For medium-magnetized cases (β initial ≈ 3.5), the initial magnetic fields undergo compression and amplification. This amplification results in the magnetic pressure inside the pillar becoming large enough to support the sides of the pillar against radial collapse due to pressure from the surrounding hot plasma. This effect is particularly pronounced for the parallel component (B y ), which is consistent with observational results. In contrast, a strong perpendicular (B x , B z ) magnetic field (β initial < 1) almost retains its initial distribution and significantly suppresses the expansion of blown-off gas plasma, leading to the inability to form pillar-like structures. The 3D simulations suggest that the bending at the head of “Column I” in the Pillars of Creation may be due to nonparallel magnetic fields. After similarity scaling transformation, our results can be applied to explain the formation and maintenance mechanism of the pillars, and can also provide useful information for future experimental designs.

Propagation of a dome-shaped, large-scale extreme-ultraviolet wave in the solar corona

Context. The first observation of a dome-shaped extreme-ultraviolet (EUV) wave was recorded by the EUVI instrument on board the STEREO-B spacecraft on January 17, 2010. This observation has allowed us to study the three-dimensional propagation of a large-scale EUV wave in the solar corona, which is considered to be a manifestation of a large-amplitude magnetohydrodynamic (MHD) wave. Aims. These unique observations by EUVI offer the opportunity to compare the theory of large-amplitude MHD waves with observations. Methods. Nonlinear MHD equations were employed for describing large-amplitude fast magnetosonic waves in terms of so-called simple waves. Results. Measuring the velocity of the EUV wave across the solar surface allows us to determine the quiet Sun’s magnetic field to be ≈3.2 G. This magnetic field can be extrapolated to the corona by means of magnetic flux conservation. Then, the height dependence of the Alfvén velocity can be calculated, adopting an isothermal, gravitationally stratified density model with a temperature of 1.4 MK for the quiet corona. The Alfvén velocity has a local maximum of ≈680 km s−1 at a height of ≈1030 Mm above the photosphere. The observations show that the EUV wave initially steepens and subsequently decays during its further evolution along the solar surface. This behavior can be aptly explained in terms of simple MHD waves. Initially, the wave front steepens due to nonlinear effects. Since the EUV waves are circularly or spherically propagating waves in the corona, their amplitudes are decreasing during the evolution. Hence, the wave steepening vanishes at the final state of the evolution of the EUV wave, which is consistent with the observations. In reality, the nature of the considered EUV wave is a combination of that of a circular and a spherical wave in the corona. Conclusions. The propagation of this dome-shaped EUV wave can be well described by the theory of large amplitude (simple) MHD waves.

Stabilized bi-cubic Hermite Bézier finite element method with application to gas-plasma interactions occurring during massive material injection in tokamaks

Development of a numerical tool based upon the high-order, high-resolution Galerkin finite element method (FEM) often encounters two challenges: First, the Galerkin FEMs give central approximations to the differential operators and their use in the simulation of the convection-dominated flows may lead to the dispersion errors yielding entirely wrong numerical solutions. Secondly, high-order, high-resolution numerical methods are known to produce high wave-number oscillations in the vicinity of shocks/discontinuities in the numerical solution adversely affecting the stability of the method. We present the stabilized finite element method for plasma fluid models to address the two challenges. The numerical stabilization is based on two strategies: Variational Multiscale (VMS) and the shock-capturing approach. The former strategy takes into account (the approximation of) the effect of the unresolved scales onto resolved scales to introduce upwinding in the Galerkin FEM. The latter adaptively adds the artificial viscosity only in the vicinity of shocks. These numerical stabilization strategies are applied to stabilize the bi-cubic Hermite Bézier FEM in the computational framework of the nonlinear magnetohydrodynamics (MHD) code JOREK. The application of the stabilized FEM to the challenging simulation of Shattered Pellet Injection (SPI) in JET-like plasma is presented. It is shown that the developed numerical stabilization model improves the stability of the underlying numerical algorithm and the computational cost required to reveal the complex physics is reduced. The physical and numerical models presented can be used to perform expensive simulations of the plasma applications in large computational domains such as JET, and ITER.

Gas kinetic schemes for solving the magnetohydrodynamic equations with pressure anisotropy

In many astrophysical plasmas, the Coulomb collision is insufficient to maintain an isotropic temperature, and the system is driven to the anisotropic regime. In this case, magnetohydrodynamic (MHD) models with anisotropic pressure are needed to describe such a plasma system. To solve the anisotropic MHD equation numerically, we develop a robust Gas-Kinetic flux scheme for non-linear MHD flows. Using anisotropic velocity distribution functions, the numerical flux functions are derived for updating the macroscopic plasma variables. The schemes are suitable for finite-volume solvers which utilize a conservative form of the mass, momentum and total energy equations, and can be easily applied to multi-fluid problems and extended to more generalized double polytropic plasma systems. Test results show that the numerical scheme is very robust and performs well for both linear wave and non-linear MHD problems.

On the frequency bifurcations of the MHD startup modes in NSTX

The observed bifurcations of the low frequency (&amp;lt;50 kHz) and low toroidal periodicity (n &amp;lt; 5) magnetohydrodynamic (MHD) activity often present in the initial part of the National Spherical Tokamak Experiment (NSTX) discharges can be explained by the evolution of the radial profile of the safety factor (q=rBϕ/RBθ) crossing multiple rational surfaces in the core. Important performance limiting instability mechanisms in the NSTX spherical tokamak are often linked to low frequency and low-n MHD activity. They are quite common in long-pulse NSTX plasmas. They can be present at the beginning of the plasma current flat-top, at the end of the discharge or during the whole duration, and they have been observed to deleteriously impact performance over a wide range of q95. An interesting feature observed in some NSTX discharges is the presence of a bifurcation in the frequency of the low n modes, as low as n = 1, that have frequencies comparable to the plasma core rotation divided by n. Equilibrium reconstructions constrained by magnetic diagnostics data and motional stark effect pitch angle radial profiles suggest that the observed bifurcations are linked to a fast evolving minimum value of q. 3D non-linear resistive MHD simulations show that these modes are ideal and exist as non-resonant before the correspondent rational surface enters the plasma.

Heat transport in magnetohydrodynamic physiological blood flow through a constricted artery

AbstractIn this paper, we study how the magnetohydrodynamic (MHD) pulsatile flow of blood and heat transfer works through a constricted artery with a flexible wall. The human circulatory network consists of veins and arteries that sometimes contain constrictions, allowing the impact of the applied magnetic field on flow fields to be observed. The walls of the flowing medium are considered to be a function of time. The flowing blood is hypothesized as shear‐thinning fluid, emulating Yeleswarapu's viscosity replica. Additionally, we consider the energy equation to understand the impact of a magnetic field on heat transfer rates for such flows. The vorticity transport equation along with the stream function equation is obtained using the vorticity–stream function technique. Numerical solutions of the governing nonlinear MHD equations and energy equation in addition to physically pertinent flow conditions were achieved by adapting a finite difference scheme. Considerable attention has been paid to ensure an accurate comparison between the current and previous results. The two sets of numbers appear to match closely. For an even deeper understanding of the flow and heat transport process, the effects of height of stenosis and diverse physiological parameters on time‐averaged wall shear stress (TAWSS), rate of heat transport, and so on are explored in depth through their graphical depiction. In the vicinity of the constriction, it is observed that the separation becomes longer with increasing constriction height. Higher magnetic force strength leads to a reduction in separation length. Newtonian fluids transfer heat more rapidly in their narrowing regions and downstream than fluids with non‐Newtonian behavior.

Probing non-linear MHD stability of the EDA H-mode in ASDEX Upgrade

Regimes of operation in tokamaks that are devoid of large edge localised modes have to be better understood to extrapolate their applicability to reactor-relevant devices. This paper describes non-linear extended magnetohydrodynamic (MHD) simulations that use an experimental equilibrium from an enhanced D-alpha (EDA) H-mode in ASDEX Upgrade. Linear ideal MHD analysis indicates that the operational point lies slightly inside of the stable region. The non-linear simulations with the visco-resistive extended MHD code, JOREK, sustain non-axisymmetric perturbations that are linearly most unstable with toroidal mode numbers of , but non-linearly higher and lower n become driven and the low-n become dominant. The poloidal mode velocity during the linear phase is found to correspond to the expected velocity for resistive ballooning modes. The perturbations that exist in the simulations have somewhat smaller poloidal wavenumbers (–) than the experimental expectations for the quasi-coherent mode in EDA, and cause non-negligible transport in both the heat and particle channels. In the transition from linear to non-linear phase, the mode frequency chirps down from approximately 35 kHz to 13 kHz, which corresponds approximately to the lower end of frequencies that are typically observed in EDA H-modes in ASDEX Upgrade.

Experiments and non-linear MHD simulations of hot vertical displacement events in ASDEX-Upgrade

Hot vertical displacement events (VDEs) are one of the worst case scenarios for high-current tokamaks as they are associated with large heat loads and electro-magnetic forces. Non-linear magneto-hydrodynamic (MHD) simulations of the thermal and current quench can help to understand their dynamics and consequences. In order to make predictions for future devices, the validation of codes against present machines is crucial. Dedicated experiments were performed in ASDEX Upgrade to provide a basis for simulations with the non-linear extended MHD code JOREK. 2D as well as non-axisymmetric simulations at realistic parameters can reproduce quantities like the edge safety factor q 95 at the thermal quench onset, the halo current magnitude and the level of vertical forces, while the exact width of the halo current area requires more sophisticated boundary conditions and is left for future work. Small horizontal forces are observed during the hot VDEs in the experiment as well as in the simulations.

Acoustic response in the transition region to transverse oscillations in a solar coronal loop

Context. Magnetohydrodynamic (MHD) waves play an important role in the dynamics and heating of the solar corona. Transverse (Alfvénic) oscillations of loops commonly occur in response to solar eruptions and are mostly studied in isolation. However, acoustic coupling has been shown to be readily observable in the form of propagating intensity variations at the loop footpoints. Aims. We extend the modelling of wave coupling between a transverse loop oscillation and slow magnetoacoustic waves in a structured loop to include a lower atmosphere. Methods. We achieve this with combined analytical modelling and fully non-linear MHD simulations. Results. Transverse loop oscillations result in the excitation of propagating slow waves from the top of the transition region and the lower boundary. The rate of excitation for the upward propagating waves at the lower boundary is smaller than for waves at the top of the transition region due to the reduced local sound speed. Additionally, slow waves are found to propagate downwards from the transition region, which reflect at the lower boundary and interfere with the upward propagating waves. Resonances are present in the normal mode analysis but these do not appear in the simulations. Due to the presence of the transition region, additional longitudinal harmonics lead to a narrower slow wave profile. The slow wave field is anti-symmetric in the direction of wave polarisation, which highlights the importance that the loop orientation has on the observability of these waves. The ponderomotive effect must be accounted for when interpreting intensity oscillations. Evidence is found for an additional short-period oscillation, which is likely a hybrid mode.

A multilevel block preconditioner for the HDG trace system applied to incompressible resistive MHD

We present a scalable block preconditioning strategy for the trace system arising from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block for which a system algebraic multigrid (AMG) is used to approximate the inverse. The complexity of the MHD equations together with the algebraic nature of the statically condensed HDG trace system makes the choice of smoother in the system AMG part critical for the convergence and performance of the block preconditioner. Our numerical experiments show GMRES preconditioned by ILU(0) of overlap zero as a smoother inside system AMG performs best in terms of robustness, time per nonlinear iteration and memory requirements. With several transient test cases in 2D and 3D including the island coalescence problem at high Lundquist number we demonstrate the robustness and parallel scalability of the block preconditioner. Additionally for the upper block a preliminary study of an alternate nodal block system solver based on a multilevel approximate nested dissection is presented. On a 2D island coalescence problem the multilevel approximate nested dissection preconditioner shows better scalability with respect to mesh refinement than the system AMG, but is relatively less robust with respect to Lundquist number scaling. In the Appendix B, we rigorously show: (1) the uniqueness of the solution of the nonlinear MHD system for small time, (2) the convergence of the Picard iterations for each backward Euler step, and (3) the convergence of the entire time stepping procedure.

Kinematic viscosity estimates in reversed-field pinch fusion plasmas

This paper concerns the kinematic viscosity in reversed-field pinch fusion plasmas, including both the study of numerical magneto-hydrodynamics (MHD) simulations and the analysis of RFX-mod experimental data.In the first part, we study the role of non-uniform time-constant radial viscosity profiles in 3D non-linear visco-resistive MHD simulations. The new profiles induce a moderate damp (for the velocity field) and a correspondent enhancement (for the magnetic field) of the spectral components resonating in the regions where the viscosity is higher.In the second part, we evaluate the kinematic viscosity coefficient on a wide database of RFX-mod shots according to the transport theories of Braginskii (considering parallel, perpendicular and gyro viscosity coefficients), considering the action on viscosity of ITG modes (ion temperature gradient) and according to the transport theory of Finn. We then exploit the comparison with the visco-resistive MHD simulations (where the visco-resistive dissipation rules the MHD activity) to show that the classical Braginskii perpendicular viscosity produces the best agreement between simulations and data, followed by the Braginskii gyro-viscosity.

Two-Level Finite Element Iterative Algorithm Based on Stabilized Method for the Stationary Incompressible Magnetohydrodynamics

In this paper, based on the stabilization technique, the Oseen iterative method and the two-level finite element algorithm are combined to numerically solve the stationary incompressible magnetohydrodynamic (MHD) equations. For the low regularity of the magnetic field, when dealing with the magnetic field sub-problem, the Lagrange multiplier technique is used. The stabilized method is applied to approximate the flow field sub-problem to circumvent the inf-sup condition restrictions. One- and two-level stabilized finite element algorithms are presented, and their stability and convergence analysis is given. The two-level method uses the Oseen iteration to solve the nonlinear MHD equations on a coarse grid of size H, and then employs the linearized correction on a fine grid with grid size h. The error analysis shows that when the grid sizes satisfy , the two-level stabilization method has the same convergence order as the one-level one. However, the former saves more computational cost than the latter one. Finally, through some numerical experiments, it has been verified that our proposed method is effective. The two-level stabilized method takes less than half the time of the one-level one when using the second class Nédélec element to approximate magnetic field, and even takes almost a third of the computing time of the one-level one when adopting the first class Nédélec element.

Utilizing time-reversibility for shock capturing in nonlinear hyperbolic conservation laws

In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or a priori knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of the conservation law which is advantageous for systems in which some components exhibit discontinuities while others do not. The efficacy of the method is shown in numerical experiments of multi-dimensional hyperbolic conservation laws such as nonlinear transport, Euler equations, and ideal magnetohydrodynamics using a high-order discontinuous spectral element method on unstructured grids.

Alfvén waves in reversed-field pinch and tokamak ohmic plasmas: nonlinear 3D MHD modeling and comparison with RFX-mod

The properties and possible triggering mechanisms of Alfvén waves in the reversed-field pinch (RFP) and circular tokamak configurations are discussed in the framework of nonlinear 3D magnetohydrodynamics (MHD) modeling. Numerical simulations are performed with the SpeCyl code (Cappello and Biskamp 1996 Nucl. Fusion) that solves the equations of the viscoresistive MHD model in cylindrical geometry. Configurations with increasing levels of complexity are analyzed. First, single-wave numerical solutions are compared with analytical ones in the simplest case of a uniform axial magnetic field: an excellent agreement is obtained for both the shear Alfvén wave (SAW) and the compressional Alfvén eigenmodes (CAEs). Then, tokamak and RFP configurations are studied. Phenomena such as phase mixing of SAW, resonant absorption of CAEs and the appearance of the global Alfvén eigenmode are described. Finally, the fully 3D RFP case with typical sawtoothing activity is investigated, showing for the first time in nonlinear RFP simulations the excitation of Alfvén waves by magnetic reconnection events. The modeling results appear to be consistent with the experimental characterization of Alfvénic activity observed in RFX-mod.