3D Ideal Magnetohydrodynamics Research Articles

Nambu Bracket for 3D Ideal Fluid Dynamics and Magnetohydrodynamics

Abstract The ideal magnetohydrodynamics (MHD) as well as the ideal fluid dynamics is governed by a Hamilton equation with respect to the Lie–Poisson bracket. The Nambu bracket manifestly represents the Lie–Poisson structure in terms of derivatives of the Casimir invariants. We construct a compact Nambu bracket representation for the 3D ideal MHD equations with the use of three Casimirs for the second Hamiltonians, the total entropy, and the magnetic and cross-helicities, whose coefficients are all constant. The Lie–Poisson bracket induced by this Nambu bracket does not coincide with the original one, but is supplemented by terms with an auxiliary variable. The supplemented Lie–Poisson bracket qualifies the cross-helicity as the Casimir. By appealing to Noether’s theorem, we show that the cross-helicity is an integral invariant associated with the particle-relabeling symmetry. Employing a Lagrange label function as the independent variable in the variational framework facilitates implementation of the relabeling transformation. By incorporating the divergence symmetry, other known topological invariants are put on the same ground as Noether’s theorem.

Improving CME evolution and arrival predictions with AMR and grid stretching in Icarus

Context.Coronal mass ejections (CMEs) are one of the main drivers of disturbances in interplanetary space. Strong CMEs, when directed towards the Earth, cause geomagnetic storms upon interacting with the Earth’s magnetic field, and can cause significant damage to our planet and affect everyday life. As such, efficient space weather prediction tools are necessary to forecast the arrival and impact of CME eruptions. Recently, a new heliospheric model called Icarus was developed based on MPI-AMRVAC, which is a 3D ideal magnetohydrodynamics model for the solar wind and CME propagation, and it introduces advanced numerical techniques to make the simulations more efficient. In this model the reference frame is chosen to be co-rotating with the Sun, and radial grid stretching together with adaptive mesh refinement (AMR) can be applied to the numerical domain.Aims.Grid stretching and AMR speed up simulation results and performance. Our aim is to combine the advanced techniques available in the Icarus model in order to obtain better results with fewer computational resources than with the equidistant grid. Different AMR strategies are suggested, depending on the purpose of the simulation.Methods.In this study, we model the CME event that occurred on July 12, 2012. A cone model was used to study the CME’s evolution through the background solar wind, and its arrival at and impact with the Earth. Grid stretching and AMR were combined in the simulations by using multiple refinement criteria, to assess its influence on the simulations’ accuracy and the required computational resources. We compare simulation results to the EUHFORIA model.Results.We applied different refinement criteria to investigate the potential of solution AMR for different applications. As a result, the simulations were sped up by a factor of ∼17 for the most optimal configuration in Icarus. For the cone CME model, we found that limiting the AMR to the region around the CME-driven shock yields the best results. The results modelled by the simulations with radial grid stretching and AMR level 4 are similar to the results provided by the original EUHFORIA and Icarus simulations with the ‘standard’ resolution and equidistant grids. The simulations with 5 AMR levels yielded better results than the simulations with an equidistant grid and standard resolution.Conclusions.Solution AMR is flexible and provides the user the freedom to modify and locally increase the grid resolution according to the purpose of the simulation. We find that simulations with a combination of grid stretching and AMR can reproduce the simulations performed on equidistant grids significantly faster. The advanced techniques implemented in Icarus can be further used to improve the forecasting procedures, since the reduced simulation time is essential to make physics-based forecasts less computationally expensive.

On the Gevrey regularity of solutions to the 3D ideal MHD equations

In this paper, similar to the incompressible Euler equation, we prove the propagation of the Gevrey regularity of solutions to the three-dimensional incompressible ideal magnetohydrodynamics (MHD) equations. We also obtain an uniform estimate of Gevery radius for the solution of MHD equation.

Local existence and blow-up criterion of 3D ideal magnetohydrodynamics equations

We investigate the local existence of smooth solutions of a 3D ideal magnetohydrodynamics (MHD) equations in a bounded domain and give a blow-up criteria to this equations with respect to vorticists.

A 3D cell-centered Lagrangian scheme for the ideal Magnetohydrodynamics equations on unstructured meshes

We propose a cell-centered Lagrangian numerical scheme for solving the 3D ideal Magnetohydrodynamics (MHD) equations on unstructured meshes. All the variables are cell-centered and the Lagrangian conservative system of ideal MHD equations is compatibly discretized in this scheme. The geometric conservation law (GCL) requirement is satisfied by using the discrete compatible Lagrangian framework of polyhedral meshes. A nodal solver for MHD equations is then constructed through the nodal pressure flux and magnetic field invoking total energy conservation and thermodynamic consistency. Besides, the magnetic divergence free constraint is fulfilled by a projection method after each time step. Various numerical tests are presented to assert the robustness and accuracy of our scheme.

Suppression of AGN-driven Turbulence by Magnetic Fields in a Magnetohydrodynamic Model of the Intracluster Medium

Abstract We investigate the role of active galactic nucleus (AGN) feedback in turbulent heating of galaxy clusters. Specifically, we analyze the production of turbulence by g-modes generated by the supersonic expansion and buoyant rise of AGN-driven bubbles. Previous work that neglects magnetic fields has shown that this process is inefficient, with less than 1% of the injected energy ending up in turbulence. This inefficiency primarily arises because the bubbles are shredded apart by hydrodynamic instabilities before they can excite sufficiently strong g-modes. Using a plane-parallel model of the intracluster medium (ICM) and 3D ideal magnetohydrodynamics (MHD) simulations, we examine the role of a large-scale magnetic field that is able to drape around these rising bubbles, preserving them from hydrodynamic instabilities. We find that while magnetic draping appears better able to preserve AGN-driven bubbles, the driving of g-modes and the resulting production of turbulence is still inefficient. The magnetic tension force prevents g-modes from transitioning into the nonlinear regime, suppressing turbulence in our model ICM. Our work highlights the ways in which ideal MHD is an insufficient description for the cluster feedback process, and we discuss future work such as the inclusion of anisotropic viscosity as a means of simulating high β plasma kinetic effects. These results suggest the hypothesis that other mechanisms of heating the ICM plasma such as sound waves or cosmic rays may be responsible for the observed feedback in galaxy clusters.

Three dimensional boundary displacement due to stable ideal kink modes excited by external n = 2 magnetic perturbations

In low-collisionality scenarios exhibiting mitigation of edge localized mode (ELMs), stable ideal kink modes at the edge are excited by externally applied magnetic perturbation (MP)-fields. In ASDEX Upgrade these modes can cause three-dimensional (3D) boundary displacements up to the centimeter range. These displacements have been measured using toroidally localized high resolution diagnostics and rigidly rotating MP-fields with various applied poloidal mode spectra. These measurements are compared to non-linear 3D ideal magnetohydrodynamics (MHD) equilibria calculated by VMEC. Comprehensive comparisons have been conducted, which consider for instance plasma movements due to the position control system, attenuation due to internal conductors and changes in the edge pressure profiles.VMEC accurately reproduces the amplitude of the displacement and its dependencies on the applied poloidal mode spectra. Quantitative agreement is found around the low field side (LFS) midplane. The response at the plasma top is qualitatively compared. The measured and predicted displacements at the plasma top maximize when the applied spectra is optimized for ELM-mitigation. The predictions from the vacuum modeling generally fails to describe the displacement at the LFS midplane as well as at the plasma top. When the applied mode spectra is set to maximize the displacement, VMEC and the measurements clearly surpass the predictions from the vacuum modeling by a factor of four. Minor disagreements between VMEC and the measurements are discussed. This study underlines the importance of the stable ideal kink modes at the edge for the 3D boundary displacement in scenarios relevant for ELM-mitigation.

Plasma response measurements of external magnetic perturbations using electron cyclotron emission and comparisons to 3D ideal MHD equilibrium

The plasma response from an external n = 2 magnetic perturbation field in ASDEX Upgrade has been measured using mainly electron cyclotron emission (ECE) diagnostics and a rigid rotating field. To interpret ECE and ECE-imaging (ECE-I) measurements accurately, forward modeling of the radiation transport has been combined with ray tracing. The measured data is compared to synthetic ECE data generated from a 3D ideal magnetohydrodynamics (MHD) equilibrium calculated by VMEC.The measured amplitudes of the helical displacement around the outboard midplane are in reasonable agreement with the one from the synthetic VMEC diagnostics. Both exceed the predictions from the vacuum field calculations and indicate the presence of a kink response at the edge, which amplifies the perturbation. VMEC and MARS-F have been used to calculate the properties of this kink mode. The poloidal mode structure of the magnetic perturbation of this kink mode at the edge peaks at poloidal mode numbers larger than the resonant components , whereas the poloidal mode structure of its displacement is almost resonant . This is expected from ideal MHD in the proximity of rational surfaces. The displacement measured by ECE-I confirms this resonant response.

Non-blow-up of the 3D ideal magnetohydrodynamics equations for a class of three-dimensional initial data in cylindrical domains

The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles.

Potato Chip Singularities of 3D Flows

A chip singularity forms when two distinct surfaces moving with a three-dimensional (3D) fluid coincide at a finite time. Potato chip singularities were suggested by a numerical study of 3D ideal magnetohydrodynamics. We prove that an incompressible flow satisfying a mild assumption on velocity growth cannot form a potato chip singularity.