Nonlinear Magnetohydrodynamics Research Articles

Study of MHD Nanofluid Flow over a Horizontal Stretching Plate by Analytical Methods

The nonlinear two-dimensional forced-convection boundary-layer magneto hydrodynamic (MHD) incompressible flow of nanofluid over a horizontal stretching flat plate with variable magnetic field including the viscous dissipation effect is solved using the Variational iteration method (VIM), homotopy perturbation method (HPM) and Adomian decomposition method (ADM). In the paper, our results of the VIM, HPM and ADM are compared with the numerical method (Runge-Kutta fourth-rate). Also the influence of physical factors such as m, Eckert number (Ec) and the percentage of nanoparticles (ϕ) on the velocity and temperature profiles have been investigated. The comparisons of the results show that the HPM has the capability of solving the nonlinear boundary layer MHD flow of nanofluid with sufficient accuracy. The results show with increasing of the parameter m, velocity decreases but for temperature the reverse trend is observed and also with increasing of nanoparticles volume, the temperature value decreases and the velocity value increases. Also with increasing of Eckert number, the temperature value increases.

Non-linear damping of visco-resistive Alfvén waves in solar spicules

Interaction of Alfven waves with plasma inhomogeneities generates phase mixing which can lead to dissipate Alfven waves and to heat the solar plasma. Here we study the dissipation of Alfven waves by phase mixing due to viscosity and resistivity variations with height. We also consider nonlinear magnetohydrodynamic (MHD) equations in our theoretical model. Non-linear terms of MHD equations include perturbed velocity, magnetic field, and density. To investigate the damping of Alfven waves in a stratified atmosphere of solar spicules, we solve the non-linear MHD equations in the x-z plane. Our simulations show that the damping is enhanced due to viscosity and resistivity gradients. Moreover, energy variations is influenced due to nonlinear terms in MHD equations.

Numerical treatment of nonlinear MHD Jeffery–Hamel problems using stochastic algorithms

In this paper a new stochastic technique for solving the nonlinear Jeffery–Hamel flow equations is presented, taking into consideration the magnetohydrodynamics (MHD) effects. A feed-forward artificial neural network (ANN) trained with particle swarm optimization (PSO) and the active-set method (ASM) is used for solving the problem. We first transform the original two-dimensional MHD Jeffery–Hamel problem into an equivalent third order boundary value problem (BVP). A mathematical model of the BVP is developed using a neural network formulation in an unsupervised manner. Optimal weights of the networks are learned with the PSO algorithm first, which is used as a tool for global search; the active-set method is employed for rapid local convergence in the second step. The designed scheme is evaluated on different cases of the problem by varying the angles of the walls, and the Reynolds and Hartmann numbers. Accuracy, convergence and effectiveness of the approach is established through extensive analyses based on a large number of independent runs. Comparative studies are carried out with numerical results of fully explicit Runge–Kutta method, as well as recently reported analytic solvers including variants of Adomian decomposition, Homotopy Perturbation, differential transform, Homotopy analysis and variational iteration methods to validate the correctness of the proposed scheme.

Non-linear MHD simulation of ELM energy deposition

The mechanisms of edge-localized mode (ELM) energy deposition are studied by means of non-linear magnetohydrodynamic (MHD) simulation of ELMs. The footprint of the ELM heat flux at the divertor is found to increase approximately linearly with the total ELM energy loss for JET-scale plasmas, which is similar to the experimentally observed broadening of the ELM energy deposition with ELM energy loss. For these relatively large ELMs, in which conductive losses dominate, the divertor footprint broadening is due to an increase in the magnetic perturbation of the ballooning mode with increasing ELM energy loss, which results in a widening of the homoclinic tangles intersecting the target. The first results from ELM simulations in the ITER Q = 10 scenario indicate that on the ITER scale the broadening is similar for conductive and convective ELMs at least up to an ELM energy loss of 4 MJ. For the larger conductive-type ELMs the magnetic perturbation and its homoclinic tangles determine the pattern of the ELM heat flux at the divertor target similar to the JET-scale results. For the smaller convective ELMs, the ELM footprint is determined by the radial distance travelled by plasma filaments expelled by the ELM and the loss of the plasma energy in the filaments along the magnetic field lines.

Nonlinear MHD simulations of the gravitational ballooning mode close to marginal stability

The ballooning mode is thought to play a key role in the mechanism which drives the problematic edge localized modes in tokamak plasmas, and possibly in other plasma eruptions, such as in the magnetosphere. We investigate the essential nonlinear physics of this mode by simulating an instability driven by gravity in a slab of magnetized plasma; magnetic field curvature plays a similar role in toroidal confinement systems.Full ideal magnetohydrodynamics (MHD) simulations are performed, which exhibit three distinct phases of the mode's evolution: (I) a linear phase that agrees well with the predictions of linear ideal MHD; (II) a non-linear regime where the instantaneous growth rate evolves, and is somewhat lower than the linear value and (III) an explosive plasma eruption. These regimes are characterized and compared with the predictions of analytic nonlinear theory, considering cases that are close to and far from marginal stability. Evidence of a subcritical instability is demonstrated in phase II where, provided the mode's amplitude is large enough, it can develop into an eruption even for values of gravity that give linear stability. The drop in growth rate during phase II depends on the strength of the linear drive, demonstrating that the initial phases of the nonlinear evolution are also dependent on the strength of the linear drive. To extend the calculations deeper into the nonlinear regime a four-field reduced MHD model is developed, which reproduces the same features as the full ideal MHD system.

Numerical treatment for nonlinear MHD Jeffery–Hamel problem using neural networks optimized with interior point algorithm

In this paper new computational intelligence techniques have been developed for the nonlinear magnetohydrodynamics (MHD) Jeffery–Hamel flow problem using three different feed-forward artificial neural networks trained with an interior point method. The governing equation for the two-dimensional MHD Jeffery–Hamel flow problem is transformed into an equivalent third order nonlinear ordinary differential equation. Three neural network models using log-sigmoid, radial basis and tan-sigmoid activation functions are developed for the transformed equation in an unsupervised manner. The training of weights of each neural network is carried out with an interior point method. The proposed models are evaluated on different variants of the Jeffery–Hamel problem by varying the Reynolds number, angles of the walls and the Hartmann number. The accuracy, convergence and effectiveness of the designed models are validated through statistical analyses based on a sufficiently large number of independent runs. Comparative studies of the proposed solutions with standard numerical results, as well as recently reported solutions of analytic solvers illustrate the worth of the proposed solvers.

Self-consistent simulations of nonlinear magnetohydrodynamics and profile evolution in stellarator configurations

Self-consistent extended MHD framework is used to investigate nonlinear macroscopic dynamics of stellarator configurations. In these calculations, initial conditions are given by analytical 3-D vacuum solutions. Finite beta discharges in a straight stellarator are simulated. Vacuum magnetic fields are applied to produce stellarator-like rotational transform profiles with iota(0) ≤ 0.5 and iota(0) ≥ 0.5. The vacuum magnetic fields are either helically symmetric or spoiled by the presence of magnetic harmonics of incommensurate helicity. As heat is added to the system, pressure-driven instabilities are excited when a critical β is exceeded. These instabilities may grow to large amplitude and effectively terminate the discharge, or they may saturate nonlinearly as the configuration evolves. In all of these studies, anisotropic heat conduction is allowed with κ∥/κ⊥=104−107.

Retrograde versus Prograde Models of Accreting Black Holes

There is a general consensus that magnetic fields, accretion disks, and rotating black holes are instrumental in the generation of the most powerful sources of energy in the known universe. Nonetheless, because magnetized accretion onto rotating black holes involves both the complications of nonlinear magnetohydrodynamics that currently cannot fully be treated numerically, and uncertainties about the origin of magnetic fields that at present are part of the input, the space of possible solutions remains less constrained. Consequently, the literature still bears witness to the proliferation of rather different black hole engine models. But the accumulated wealth of observational data is now sufficient to meaningfully distinguish between them. It is in this light that this critical paper compares the recent retrograde framework with standard “spin paradigm” prograde models.

Internal disruptions and sawtooth like activity in Large Helical Device

Large Helical Device (LHD) inward-shifted configurations are unstable to resistive magnetohydrodynamic (MHD) pressure-gradient-driven modes. These modes drive sawtooth like events during LHD operation. In this work, we simulate sawtooth like activity and internal disruptions in order to improve the understanding of these relaxation events and their effect over the device efficiency to confine the plasma, with the aim to improve the LHD present and future operation scenarios minimizing or avoiding the disadvantageous MHD soft and hard limits. By solving a set of reduced non-linear resistive MHD equations, we have studied the evolution of perturbations to equilibria obtained before and after a sawtooth like event in LHD. The equilibrium β value is gradually increased during the simulation until it reaches the experimental value. Sawtooth like events and internal disruption events take place in the simulation for β0 values between 1% and 1.48%. The main driver of the sawtooth like events is the resonant and non-resonant effect of the (n = 1, m = 3) mode. The instability is stronger for resonant events, and they only appear when β0 = 1.48%. Internal disruptions are mainly driven by the (n = 1, m = 2) mode, and they extend throughout the whole plasma core. Internal disruption events do not show up when resonant sawtooth like events are triggered.

Towards Automated Memory Model Generation Via Event Tracing

The importance of memory performance and capacity is a growing concern for high performance computing laboratories around the world. It has long been recognized that improvements in processor speed exceed the rate of improvement in dynamic random access memory speed and, as a result, memory access times can be the limiting factor in high performance scientific codes. The use of multi-core processors exacerbates this problem with the rapid growth in the number of cores not being matched by similar improvements in memory capacity, increasing the likelihood of memory contention. In this paper, we present WMTools, a lightweight memory tracing tool and analysis framework for parallel codes, which is able to identify peak memory usage and also analyse per-function memory use over time. An evaluation of WMTools, in terms of its effectiveness and also its overheads, is performed using nine established scientific applications/benchmark codes representing a variety of programming languages and scientific domains. We also show how WMTools can be used to automatically generate a parameterized memory model for one of these applications, a two-dimensional non-linear magnetohydrodynamics application, Lare2D. Through the memory model we are able to identify an unexpected growth term which becomes dominant at scale. With a refined model we are able to predict memory consumption with under 7% error.

Multiple timescale calculations of sawteeth and other global macroscopic dynamics of tokamak plasmas

The M3D-C1 (Breslau et al 2009 Phys. Plasmas 16 092503) code is designed for performing three-dimensional nonlinear magnetohydrodynamics (MHD) calculations of a tokamak plasma that span the timescales associated with ideal and resistive stability as well as parallel and perpendicular transport. This requires a scalable fully implicit time advance where the time step is not limited by a Courant condition based on the MHD wave velocities or plasma flow but is chosen instead to accurately and efficiently resolve the physics. In order to accomplish this, we make use of several techniques to improve the effective condition number of the implicit matrix equation that is solved every time step. The split time advance known as the differential approximation (Caramana 1991 J. Comput. Phys. 96 484) reduces the size of the matrix and improves its diagonal structure. A particular choice of velocity variables and annihilation operators approximately splits the large matrix into three sub-matrices, each with a much improved condition number. A final block-Jacobi preconditioner further dramatically improves the condition number of the final matrix, allowing it to converge in a Krylov solver (GMRES) with a small number of iterations. As an example, we have performed transport timescale simulations of a tokamak plasma that periodically undergoes sawtooth oscillations (Von Goeler et al 1974 Phys. Rev. Lett. 33 1201). We specify the transport coefficients and apply a 'current controller' that adjusts the boundary loop-voltage to keep the total plasma current fixed. The short-time plasma response depends on the initial conditions, but the long-time behavior depends only on the transport coefficients and the boundary conditions applied.

Plasma response models for non-axisymmetric perturbations

The plasma response to non-axisymmetric perturbations arising from external coils or linear instabilities can be treated using various linear and nonlinear models, none of which are fully satisfactory. Linear models cannot provide the full response and the result can depend on the detailed physical model used. The nonlinear response can be treated as a dynamic stability problem or from a nearby perturbed equilibrium approach. The nearby equilibrium approach aims to bypass the detailed evolution and search for the appropriate final state. For these nonlinear models, there is no guarantee that the final state is the one chosen dynamically by the plasma among possible multiple states, or is even accessible. To assure accessibility of the final state, one needs to relate the two-dimensional and nearby three-dimensional system through some set of invariants. One implementation is to add a perturbation from an external field or obtained from a stability code to the equilibrium and solve for 3D force balance. In that case, the invariants are buried in the numerical details of the equilibrium code. An appropriate set of constraints is not presently known; they depend on whether the dynamic evolution should be considered adiabatic or not. It is proposed that a suitable set of invariants may be obtained from considering the magnetic helicity, which is conserved exactly in ideal magnetohydrodynamics (MHD) but is broken at rational surfaces by non-ideal effects. In general, constraints for the equilibrium approach, including magnetic helicity, can be validated using full nonlinear extended MHD calculations in the dynamic approach.

The elemental shear dynamo

AbstractA quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially growing, large-scale (mean) magnetic dynamo in the presence of a uniform parallel shear flow. It is a ‘kinematic’ theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magnetohydrodynamics, and it is rigorously derived in the limit of small magnetic Reynolds number, ${\mathit{Re}}_{\eta } \ll 1$. Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with a small seed wavenumber in the direction perpendicular to the mean shearing plane, a positive exponential growth rate $\gamma $ can occur for arbitrary values of ${\mathit{Re}}_{\eta } $, viscous Reynolds number ${\mathit{Re}}_{\nu } $, and random-force correlation time ${t}_{f} $ and orientation angle ${\theta }_{f} $ in the shearing plane. The value of $\gamma $ is independent of the domain size. The shear dynamo is ‘fast’, with finite $\gamma \gt 0$ in the limit of ${\mathit{Re}}_{\eta } \gg 1$. Averaged over random realizations of the forcing history, the ensemble-mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (magnetic energy). In the limit of small ${\mathit{Re}}_{\eta } $ and ${\mathit{Re}}_{\nu } $, the dynamo behaviour is related to the well-known alpha–omega ansatz when the force is slowly varying ($\gamma {t}_{f} \gg 1$) and to the ‘incoherent’ alpha–omega ansatz when the force is more rapidly fluctuating.

Simulation of Alfvén eigenmode bursts using a hybrid code for nonlinear magnetohydrodynamics and energetic particles

A hybrid simulation code for nonlinear magnetohydrodynamics (MHD) and energetic-particle dynamics has been extended to simulate recurrent bursts of Alfvén eigenmodes by implementing the energetic-particle source, collisions and losses. The Alfvén eigenmode bursts with synchronization of multiple modes and beam ion losses at each burst are successfully simulated with nonlinear MHD effects for the physics condition similar to a reduced simulation for a TFTR experiment (Wong et al 1991 Phys. Rev. Lett. 66 1874, Todo et al 2003 Phys. Plasmas 10 2888). It is demonstrated with a comparison between nonlinear MHD and linear MHD simulation results that the nonlinear MHD effects significantly reduce both the saturation amplitude of the Alfvén eigenmodes and the beam ion losses. Two types of time evolution are found depending on the MHD dissipation coefficients, namely viscosity, resistivity and diffusivity. The Alfvén eigenmode bursts take place for higher dissipation coefficients with roughly 10% drop in stored beam energy and the maximum amplitude of the dominant magnetic fluctuation harmonic δBm/n/B ∼ 5 × 10−3 at the mode peak location inside the plasma. Quadratic dependence of beam ion loss rate on magnetic fluctuation amplitude is found for the bursting evolution in the nonlinear MHD simulation. For lower dissipation coefficients, the amplitude of the Alfvén eigenmodes is at steady levels δBm/n/B ∼ 2 × 10−3 and the beam ion losses take place continuously. The beam ion pressure profiles are similar among the different dissipation coefficients, and the stored beam energy is higher for higher dissipation coefficients.

Are gravitational waves from giant magnetar flares observable?

Are giant flares in magnetars viable sources of gravitational radiation? Few theoretical studies have been concerned with this problem, with the small number using either highly idealized models or assuming a magnetic field orders of magnitude beyond what is supported by observations. We perform nonlinear general-relativistic magnetohydrodynamics simulations of large-scale hydromagnetic instabilities in magnetar models. We utilise these models to find gravitational wave emissions over a wide range of energies, from 10^40 to 10^47 erg. This allows us to derive a systematic relationship between the surface field strength and the gravitational wave strain, which we find to be highly nonlinear. In particular, for typical magnetar fields of a few times 10^15 G, we conclude that a direct observation of f-modes excited by global magnetic field reconfigurations is unlikely with present or near-future gravitational wave observatories, though we also discuss the possibility that modes in a low-frequency band up to 100 Hz could be sufficiently excited to be relevant for observation.

Nonlinear dynamo action in a cylindrical container driven by precession

Precession, which results simply from the composition of two rotations with distinct axes, is an efficient way to drive a 3D flow in a closed rigid container. Are such flows relevant to dynamo action in some astrophysical bodies? Positive answers are available for a spherical and a spheroidal containers, using parameters which are, however, not realistic. An experimental approach could be relevant to natural dynamos and seems within reach using a cylindrical container (cf. the experiment now planned at the DREsden Sodium facility for DYNamo and thermohydraulic studies in Germany (DRESDYN), F. Stefani, personal communication, 2011). Using a nonlinear magnetohydrodynamics (MHD) code (SFEMaNS), we numerically demonstrate that precession is able to drive a cylindrical dynamo.

HYDROMAGNETIC INSTABILITIES IN RELATIVISTIC NEUTRON STARS

We model the non-linear ideal magnetohydrodynamics of poloidal magnetic fields in neutron stars in general relativity assuming a polytropic equation of state. We identify familiar hydromagnetic modes, in particular the 'sausage/varicose' mode and 'kink' instability inherent to poloidal magnetic fields. The evolution is dominated by the kink instability, which causes a cataclysmic reconfiguration of the magnetic field. The system subsequently evolves to new, non-axisymmetric, quasi-equilibrium end-states. The existence of this branch of stable quasi-equilibria may have consequences for magnetar physics, including flare generation mechanisms and interpretations of quasi-periodic oscillations.

Nonlinear MHD simulations of edge-localized-modes in JET

Nonlinear magneto-hydrodynamic (MHD) simulations with the JOREK code may be used to improve our understanding of edge-localized-modes (ELMs) (Huysmans and Czarny 2007 Nucl. Fusion 47 659–66, Huysmans et al 2009 Plasma Phys. Control. Fusion 51 124012, Pamela et al 2010 Plasma Phys. Control. Fusion 52 075006). These H-mode related instabilities may cause some damage to the tungsten divertor of ITER (Bazylev et al 2007 Phys. Scr. T128 229–33), and it was demonstrated experimentally that the ELM energy losses increase with both machine size and decreasing collisionality (ITER Physics Basis Editors and ITER EDA 1999 Nucl. Fusion 39 2175, Loarte et al 2003 Plasma Phys. Control. Fusion 45 1549–69). In sight of producing simulations of ELMs in ITER, in order to give some predictions of ELM size and divertor heat fluxes in the future device, simulations first need to be quantitatively validated against the experimental data of present machines. This paper presents simulations of ELMs in the JET tokamak for the low-collosionality type-I ELMy H-mode pulse #73569. The simulation results are compared with experimental data to provide a qualitative validation of the simulations. This comparison comprises the dynamics of filaments and divertor heat fluxes, the effect of resistivity and collisionality on ELM energy losses and the observation of ELM precursors prior to the pedestal collapse.

The HPM Applied to MHD Nanofluid Flow over a Horizontal Stretching Plate

The nonlinear two‐dimensional forced‐convection boundary‐layer magneto hydrodynamic (MHD) incompressible flow of nanofluid over a horizontal stretching flat plate with variable magnetic field including the viscous dissipation effect is solved using the homotopy perturbation method (HPM). In the present work, our results of the HPM are compared with the results of simulation using the finite difference method, Keller′s box‐scheme. The comparisons of the results show that the HPM has the capability of solving the nonlinear boundary layer MHD flow of nanofluid with sufficient accuracy.

WINDS FROM LUMINOUS LATE-TYPE STARS. II. BROADBAND FREQUENCY DISTRIBUTION OF ALFVÉN WAVES

We present the numerical simulations of winds from evolved giant stars using a fully non-linear, time dependent 2.5-dimensional magnetohydrodynamic (MHD) code. This study extends our previous fully non-linear MHD wind simulations to include a broadband frequency spectrum of Alfv\'en waves that drive winds from red giant stars. We calculated four Alfv\'en wind models that cover the whole range of Alfv\'en wave frequency spectrum to characterize the role of freely propagated and reflected Alfv\'en waves in the gravitationally stratified atmosphere of a late-type giant star. Our simulations demonstrate that, unlike linear Alfv\'en wave-driven wind models, a stellar wind model based on plasma acceleration due to broadband non-linear Alfv\'en waves, can consistently reproduce the wide range of observed radial velocity profiles of the winds, their terminal velocities and the observed mass loss rates. Comparison of the calculated mass loss rates with the empirically determined mass loss rate for alpha Tau suggests an anisotropic and time-dependent nature of stellar winds from evolved giants.