Reduced MHD Model Research Articles

Analysis of the ECH effect on EPM/AE stability in Heliotron J plasma using a Landau closure model

The aim of the present study is to analyze the effect of the electron cyclotron heating (ECH) on the linear stability of Alfvén eigenmodes (AEs) and energetic particle modes (EPMs) triggered by energetic ions in Heliotron J plasma. The analysis is performed using the FAR3d code that solves a reduced MHD model to describe the thermal plasma coupled with a gyrofluid model for the energetic particle (EP) species. The simulations reproduce the AE/EPM stability trends observed in the experiments as the electron temperature (T e ) increases, modifying the thermal plasma β, EP β and EP slowing-down time. Particularly, the EPM and Global AE are stabilized in the low-bumpiness (LB) configuration due to an enhancement of the continuum, finite Larmor radius and e-i Landau damping effects as the thermal β increases. On the other hand, a larger ECH injection power cannot stabilize the AE/EPM in medium-bumpiness and high-bumpiness (HB) configurations because the damping effects are weaker compared to the LB case, unable to balance the further destabilization induced by an enhanced EP resonance as the EP slowing-down time and EP β increases with T e .

Simulation of the TAEs’ saturation phase in the Large Helical Device: MHD burst

The aim of the present study is to analyze the saturation regime of the toroidal Alfven eigenmodes (TAE) in the Large Helical Device plasma, particularly the MHD burst. The linear and nonlinear evolution of the TAEs are simulated by the FAR3d code that uses a reduced MHD model for the thermal plasma coupled with a gyrofluid model for the energetic particle (EP) species. The linear simulations indicate the overlapping of 1/2 − 1/1, 2/3–2/4 and 3/5–3/6 TAEs in the inner-middle plasma region and frequency range of 45–75 kHz, triggered by EPs with an energy of T f = 45 keV and EP β = 0.022. The nonlinear simulations show that 2/3–2/4 and 3/4–3/5 TAEs are further destabilized due to the energy transfer from the 1/1–1/2 TAE, leading to broad TAE radial overlapping and triggering of the MHD burst. The energy of the 1/1–1/2 TAE is also nonlinearly transferred to the thermal plasma destabilizing the 0/0 and 0/1 modes, inducing the generation of shear flows and zonal currents, as well as large deformations in the thermal pressure and EP density radial profiles. The nonlinear simulation reproduces the same succession of instabilities and the same frequency range with respect to the experiment. The instability propagates outward during the bursting phase, showing a large decrease of the EP density profile between the middle-outer plasma, indicating the loss of part of the EP population that explains the decrease in the plasma heating efficiency observed during the MHD burst.

JOREK3D: An extension of the JOREK nonlinear MHD code to stellarators

Although the basic concept of a stellarator was known since the early days of fusion research, advances in computational technology have enabled the modeling of increasingly complicated devices, leading up to the construction of Wendelstein 7-X, which has recently shown promising results. This recent success has revived interest in the nonlinear 3D MHD modeling of stellarators in order to better understand their performance and operational limits. This study reports on the extension of the JOREK code to 3D geometries and on the first stellarator simulations carried out with it. The first simple simulations shown here address the classic Wendelstein 7-A stellarator using a reduced MHD model previously derived by us. The results demonstrate that stable full MHD equilibria are preserved in the reduced model: the flux surfaces do not move throughout the simulation and closely match the flux surfaces of the full MHD equilibrium. Furthermore, both tearing and ballooning modes were simulated, and the linear growth rates measured in JOREK are in reasonable agreement with the growth rates from the CASTOR3D linear MHD code.

Theoretical analysis of the saturation phase of the 1/1 energetic-ion-driven resistive interchange mode

The aim of the present study is to analyze the saturation regime of the energetic-ion-driven resistive interchange mode (EIC) in the LHD plasma. A set of nonlinear simulations are performed by the FAR3d code that uses a reduced MHD model for the thermal plasma coupled with a gyrofluid model for the energetic particle (EP) species. The hellically trapped EP component is introduced through a modification of the averaged drift velocity operator to include their precessional drift. The nonlinear simulation results show similar 1/1 EIC saturation phases with respect to the experimental observations, reproducing the enhancement of the n/m = 1/1 resistive interchange modes (RIC) amplitude and width as the EP β increases, the EP β threshold for the 1/1 EIC excitation, the further destabilization of the 1/1 EIC as the population of the helically trapped EP increases and the triggering of burst events. The frequency of the 1/1 EIC calculated during the burst event is 9.4 kHz and the 2/2 and 3/3 overtones are destabilized, consistent with the frequency range and the complex mode structure measured in the experiment. In addition, the simulation shows the inward propagation of the 1/1 EIC due to the nonlinear destabilization of the 3/4 and 2/3 energetic particle modes, leading to the partial overlapping between resonances during the burst event. Finally, the analysis of the 1/1 EIC stabilization phase shows the excitation of the 1/1 RIC as soon as the flattening induced by the 1/1 EIC in the pressure profile vanishes, leading to the retrieval of the pressure gradient at the plasma periphery and the overcoming of the RIC stability limit.

3D simulations of vertical displacement events in tokamaks: A benchmark of M3D-C1, NIMROD, and JOREK

In recent years, the nonlinear 3D magnetohydrodynamic codes JOREK, M3D-C1, and NIMROD developed the capability of modeling realistic 3D vertical displacement events (VDEs) including resistive walls. In this paper, a comprehensive 3D VDE benchmark is presented between these state-of-the-art codes. The simulated case is based on an experimental NSTX plasma but with a simplified rectangular wall. There are differences between the physics models and numerical methods, and the VDE evolution leads to sensitivities on the initial conditions that cannot be avoided as can be done in edge localized modes (ELM) and sawtooth simulations (due to the non-cyclical nature of VDEs). Nonetheless, the comparison serves to quantify the level of agreement in the relevant quantities used to characterize disruptions, such as the 3D wall forces and energy decay. The results bring confidence regarding the use of the mentioned codes for disruption studies, and they distinguish aspects that are specific to the models used (e.g., reduced vs full MHD models). The simulations show important 3D features for a NSTX plasma, such as the self-consistent evolution of the halo current and the origin of the wall forces. In contrast to other reduced MHD models based on an ordering in the aspect ratio, the ansatz-based JOREK reduced MHD model allows capturing many aspects of the 3D dynamics even in the spherical tokamak limit considered here.

Analytical model for quasi-linear flow response to resonant magnetic perturbation in resistive-inertial and viscous-resistive regimes

In this work, a quasi-linear model for the plasma flow response to the resonant magnetic perturbation (RMP) in a tokamak has been self-consistently developed in the resistive-inertial and viscous-resistive regimes purely from the two-field reduced MHD model. Models for the plasma response to RMP are commonly composed of equations for the resonant magnetic field response (i.e., the magnetic island) and the torque balance of plasma flow. However, in previous plasma response models, the magnetic island and the torque balance equations are often derived separately from reduced MHD and full MHD equations, respectively. By contrast, in this work, we derive both the magnetic island response and the torque balance equations in a quasi-linear model for the plasma flow response entirely from a set of two-field reduced MHD equations. Such a quasi-linear model can recover previous plasma flow response models within certain limits and approximations. Furthermore, the theoretical origins and physical meanings of quasi-linear forces and moments in the flow response equation are also clarified.

A reduced MHD model for ITG-NTM interplay

A six-field reduced-MHD model is derived for plasma dynamics. The new model describes coherently both ion temperature gradient (ITG) mode and tearing mode and includes neoclassical effects. The model allows the construction of an energylike quantity with a linear pressure contribution that is conserved except for dissipative, finite Larmor radius, and neoclassical terms. This model may be used to study the nonlinear interaction between ITG microturbulence and neoclassical tearing mode, which is responsible for large-scale magnetic islands in tokamaks, and opens the way to a coherent description of turbulent impurity transport in magnetic islands.

Suppressive effects of diamagnetic drift on neoclassical double tearing modes based on four-field reduced MHD model

The explosive behavior of neo-classical double tearing mode (NDTM) is numerically investigated by a reduced four-field-drift-magnetohydrodynamic code [Ye et al., Nuclear Fusion 59, 096044 (2019)] with the bootstrap current effect. It is numerically observed that the explosive burst of NDTM can be absolutely suppressed by diamagnetic drift flow. The dependence of the critical value of diamagnetic drift flow V0*c for avoiding the explosion on typical plasma parameters is numerically studied in detail. It is found that V0*c decreases with increasing Δrs, while it increases with an increasing bootstrap current fraction. The plasma viscosity and resistivity can raise the threshold through their effects on the rotation of magnetic islands and the instability of system, respectively. Furthermore, the perpendicular transport and parallel transport, which are significant to the driven effects of the bootstrap current, have the opposite effects on V0*c. The underlying mechanisms are discussed.

Turbulent Heating in the Accelerating Region Using a Multishell Model

Recent studies of turbulence-driven solar winds indicate that fast winds are obtained only at the price of unrealistic bottom boundary conditions: too large wave amplitudes and small frequencies. In this work, the incompressible turbulent dissipation is modeled with a large-scale von Karman–Howarth–Kolmogorov-like phenomenological expression ( $Q_{\text{K41}}^{0}$ ). An evaluation of the phenomenology is thus necessary to understand if unrealistic boundary conditions result from physical or model limitations. To assess the validity of the Kolmogorov-like expression, $Q_{\text{K41}}^{0}$ , one needs to compare it to exact heating, which requires describing the cascade in detail. This has been done in the case of homogeneous MHD turbulence, including expansion, but not in the critical accelerating region. To assess the standard incompressible turbulent heating in the accelerating region, we use a reduced MHD model (multishell model) in which the perpendicular turbulent cascade is described by a shell model, allowing to reach a Reynolds number of $10^{6}$ . We first consider the homogeneous and expanding cases, and find that primitive MHD and multishell equations give remarkably similar results. We thus feel free to use the multishell model in the accelerating region. The results indicate that the large-scale phenomenology is inaccurate and it overestimates the heating by a factor at least 20, thus invalidating earlier studies of winds driven by incompressible turbulence. We conclude that realistic 1D wind models cannot be based solely on incompressible turbulence, but probably need an addition of compressible turbulence and shocks to increase the wave reflection and thus the heating.

Interplay between fluctuation driven toroidal axisymmetric flows and resistive ballooning mode turbulence

An interplay between fluctuation driven toroidal axisymmetric flows (convective cell modes) and resistive ballooning mode turbulence after the pedestal collapse is numerically studied by a four-field reduced MHD model in the BOUT++ framework. The strong flow shear suppresses the radial transport of pressure filaments, and the pressure profile in the pedestal region is partially recovered. As a result, a secondary instability is quasilinearly excited, which yields a secondary collapse. The subsequent damped oscillation is also analyzed by phase diagram analysis.

Enhancement of Elliptic Flow of π− under Intense Magnetic Field in √ sNN = 200 GeV Au+Au Collisions: A (2 + 1)-Dimensional Reduced-MHD Model Study

We investigate the effect of intense magnetic fields on the ( 2 + 1 ) -dimensional reduced- magnetohydrodynamical (MHD) expansion of hot and dense quark–gluon plasma (QGP) produced in s NN = 200 GeV Au+Au collisions. For the sake of simplicity, we consider the case in which the magnetic field points in the direction perpendicular to the reaction plane. We also consider this field to be external, with energy density parametrized as a two-dimensional Gaussian. The width of the Gaussian along the directions orthogonal to the beam axis varies with the centrality of the collision. The dependence of the magnetic field on proper time ( τ ) is parametrized for the case of zero and finite electrical conductivity of the QGP. We solve the equations of motion of ideal hydrodynamics for such an external magnetic field. For collisions with a non-zero impact parameter we observe a considerable increase in the elliptic-flow coefficient v 2 of π − in the presence of an external magnetic field, and the increment in v 2 is found to depend on the evolution and the initial magnitude of the magnetic field.

Ionospheric Alfvén resonator and aurora: Modeling of MICA observations

AbstractWe present results from a numerical study of small‐scale, intense magnetic field‐aligned currents observed in the vicinity of the discrete auroral arc by the Magnetosphere‐Ionosphere Coupling in the Alfvén Resonator (MICA) sounding rocket launched from Poker Flat, Alaska, on 19 February 2012. The goal of the MICA project was to investigate the hypothesis that such currents can be produced inside the ionospheric Alfvén resonator by the ionospheric feedback instability (IFI) driven by the system of large‐scale magnetic field‐aligned currents interacting with the ionosphere. The trajectory of the MICA rocket crossed two discrete auroral arcs and detected packages of intense, small‐scale currents at the edges of these arcs, in the most favorable location for the development of the ionospheric feedback instability, predicted by the IFI theory. Simulations of the reduced MHD model derived in the dipole magnetic field geometry with realistic background parameters confirm that IFI indeed generates small‐scale ULF waves inside the ionospheric Alfvén resonator with frequency, scale size, and amplitude showing a good, quantitative agreement with the observations. The comparison between numerical results and observations was performed by “flying” a virtual MICA rocket through the computational domain, and this comparison shows that, for example, the waves generated in the numerical model have frequencies in the range from 0.30 to 0.45 Hz, and the waves detected by the MICA rocket have frequencies in the range from 0.18 to 0.50 Hz.

Non-Ideal Ballooning Mode Instability with Real Electron Inertia

Impacts of electron inertia with an electron skin depth (ESD) longer than the realistic value used in early numerical studies on non-ideal ballooning modes (NIBMs) are numerically investigated by a linearized 3-field reduced MHD model. In this paper, 4 different ESDs,are used for an resistivity dependence study of the growth rate of NIBMs, where de s the real ESD and d*e = 10 corresponds to an order of ESD used in a numerical study on collisionless ballooning mode (CBM) reported in [Kleva and Guzdar Phys. Plasmas 6, 116 (1999)]. In the case with the real ESD d*e = de, a transition from resistive ballooning mode (RBM) to CBM occurs in the edge relevant resistivity regime, while the electron inertia effect is overestimated and the growth rate is almost independent of resistivity in the cases with d∗e =√10de and 10de. These results indicate that the real ESD is one of key factors for the edge stability and turbulence analysis.

DYNAMICAL FIELD LINE CONNECTIVITY IN MAGNETIC TURBULENCE

Point-to-point magnetic connectivity has a stochastic character whenever magnetic fluctuations cause a field line random walk, but this can also change due to dynamical activity. Comparing the instantaneous magnetic connectivity from the same point at two different times, we provide a nonperturbative analytic theory for the ensemble average perpendicular displacement of the magnetic field line, given the power spectrum of magnetic fluctuations. For simplicity, the theory is developed in the context of transverse turbulence, and is numerically evaluated for the noisy reduced MHD model. Our formalism accounts for the dynamical decorrelation of magnetic fluctuations due to wave propagation, local nonlinear distortion, random sweeping, and convection by a bulk wind flow relative to the observer. The diffusion coefficient DX of the time-differenced displacement becomes twice the usual field line diffusion coefficient Dx at large time displacement t or large distance z along the mean field (corresponding to a pair of uncorrelated random walks), though for a low Kubo number (in the quasilinear regime) it can oscillate at intermediate values of t and z. At high Kubo number the dynamical decorrelation decays mainly from the nonlinear term and DX tends monotonically toward 2Dx with increasing t and z. The formalism and results presented here are relevant to a variety of astrophysical processes, such as electron transport and heating patterns in coronal loops and the solar transition region, changing magnetic connection to particle sources near the Sun or at a planetary bow shock, and thickening of coronal hole boundaries.

Convective radial energy flux due to resonant magnetic perturbations and magnetic curvature at the tokamak plasma edge

With the resonant magnetic perturbations (RMPs) consolidating as an important tool to control the transport barrier relaxation, the mechanism on how they work is still a subject to be clearly understood. In this work, we investigate the equilibrium states in the presence of RMPs for a reduced MHD model using 3D electromagnetic fluid numerical code with a single harmonic RMP (single magnetic island chain) and multiple harmonics RMPs in cylindrical and toroidal geometry. Two different equilibrium states were found in the presence of the RMPs with different characteristics for each of the geometries used. For the cylindrical geometry in the presence of a single RMP, the equilibrium state is characterized by a strong convective radial thermal flux and the generation of a mean poloidal velocity shear. In contrast, for toroidal geometry, the thermal flux is dominated by the magnetic flutter. For multiple RMPs, the high amplitude of the convective flux and poloidal rotation are basically the same in cylindrical geometry, but in toroidal geometry the convective thermal flux and the poloidal rotation appear only with the islands overlapping of the linear coupling between neighbouring poloidal wavenumbers m, m – 1, and m + 1.

Simulation Study of Nonlocal Transport from Edge to Core in Tokamak Plasmas

AbstractThe simulation study of nonlocal transport from edge to core in tokamak plasmas is performed using the 4‐field reduced MHD model. A toroidally‐elongated cylindrical particle source is applied in the plasma edge, after saturation of the resistive ballooning turbulence is attained. After a short time, the source is switched off and plasma response is investigated in detail. The nonlocal transport appears at the location far from the edge source. It is found that the particle source induces (0,0) and (±1, 0) modes of density fluctuations, where (m, n) indicates the set of poloidal mode number m and the toroidal mode number n. These modes interact with each other by the nonlinear and/or toroidal couplings. The symmetry of (±1, 0) modes breaks after switching‐off the source and the formation of the spiral structure with poloidal rotation is observed, which yields a connection between core and edge regions. In this simulation, the convective cell mode such as (1,0) mode contributes the nonlocal transport. The simulation result indicates that two dimensional transport plays an essential role to produce the nonlocal transport. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A 3D finite volume scheme for the simulation of edge plasma in Tokamaks

A finite volume method in cylindrical coordinates for the simulation of the edge region in tokamaks is proposed. In contrast to standard methods, which require the projected form of the equations in curvilinear coordinates, a new method is proposed. This technique is based on the discretization on the vector form of the equations and then on the projection on the basis of the curvilinear system associated to each control volume. The proposed methodology has been validated on several simple test cases and applied on a 3D simulation of a reduced MHD model.

Stability of Double Tearing Mode in the Presence of Shear Flows

The stability of two eigen states of double tearing mode (DTM) with symmetric or antisymmetric islands in the presence of shear flows is numerically simulated based on a reduced MHD model in slab geometry. For given antisymmetric flow profile, a degenerated state is observed at a critical flow amplitude vc. Below vc, the shear flow stabilizes the DTM with antisymmetric islands and destabilizes the other one through distorting the magnetic flux mainly governed by the global effect of flow profile. Above vc, the degenerated state bifurcates into two eigen states with the same growth rate but opposed propagating direction. These two eigen modes show

Nonlinear dispersive scale Alfvén waves inmagnetosphere-ionosphere coupling: Physical processes and simulation results

Dispersive Alfven waves (DAWs) have been demonstrated to play a significant role in auroral generation of the magnetosphere-ionosphere coupling system. Starting from a two fluid reduced MHD model, we summarize the frequency, temporal and spatial characteristics of magnetospheric DAWs. Then, the nonlinear kinetic and inertial scale Alfven waves are studied, and we review some theoretical aspects and simulation results of dispersive Alfven waves in Earth’s magnetosphere. It is shown that dispersive standing Alfven waves can generate the field-aligned currents which transport energy into the auroral ionosphere, where it is dissipated by Joule heating and energy lost due to electron precipitation. The Joule dissipation can heat the ionospheric electron and produce changes in the ionospheric Pedersen conductivity. As a feedback, the conducting ionosphere can also strongly affect the magnetospheric currents. The ponderomotive force can cause the plasma to move along the field line, and generate ionospheric density cavity. The nonlinear structuring can lead to a dispersive scale to accelerate auroral particle, and the Alfven waves can be trapped within the density cavity. Finally, we show the nonlinear decay of dispersive Alfven waves related to two anti-propagating electron fluxes observed in the auroral zone.

Excitation of zero-frequency magnetic field-aligned currents by ionospheric heating

Abstract. Time-dependent, three-dimensional numerical simulations of the reduced MHD model describing shear Alfvén waves in the magnetosphere provide an interesting prediction superficially similar to results of several ionospheric heating experiments conducted at high altitudes. In these experiments, heating of the ionospheric F-region with a constant/zero-frequency beam of HF waves causes luminous structures in the ionosphere in the form of a ring or a solid spot with a characteristic size comparable to the size of the heated spot. Simulations suggest that spots/rings or similar optical appearance might be associated with a magnetic field-aligned current system produced by the ionospheric heating. Two of the most interesting features of this current system are (1) strong localization across the ambient magnetic field and (2) distinctive non-symmetrical luminous signatures (ring/spot) in magnetically conjugate locations in the ionosphere.