Resistive Diffusion Time Research Articles

Dynamic Formation of a Transient Jet from Arched Magnetized Laboratory Plasma

A laboratory plasma experiment was built to explore the eruptive behavior of arched magnetized plasmas with dimensionless parameters relevant to the Sun’s photosphere (β ≈ 10−3, Lundquist number ≈104, plasma radius/ion gyroradius ≈20, ion–neutral collision frequency ≫ion cyclotron frequency). Dynamic formation of a transient plasma jet was observed in the presence of the strapping magnetic field. The eruption leading to the jet is unintuitive because the arched plasma is both kink- and torus-stable. The jet structure erupts within a few Alfvén transit times from the formation of the arched plasma. Extensive measurements of plasma temperature, density, magnetic field, and flows are presented. In its early stages, the jet plasma flows away from the arch with supersonic speeds (Mach 1.5). This high-speed flow persists up to the resistive diffusion time in the arched plasma and is driven by large gradients in the magnetic and thermal pressures near the birthplace of jets. There are two distinct electric current channels within the jet, one consisting of outgoing electrons and another composed of electrons returning to the anode footpoint. Significant current density around the jet is a consequence of the diamagnetic current produced by a large thermal pressure gradient in the jet. Ion–neutral charge-exchange collisions provide an efficient mechanism to produce the cross-field current and control the dynamics of the complex current channels of the jet.

Diffusive time evolution of the Grad–Shafranov equation for a toroidal plasma

We describe the evolution of a plasma equilibrium having a toroidal topology in the presence of constant electric resistivity. After outlining the main analytical properties of the solution, we illustrate its physical implications by reproducing the essential features of a scenario for the upcoming Italian experiment Divertor Tokamak Test Facility, with a good degree of accuracy. Although we find the resistive diffusion time scale to be of the order of $10^4$ s, we observe a macroscopic change in the plasma volume on a time scale of $10^2$ s, comparable to the foreseen duration of the plasma discharge by design. In the final part of the work, we compare our self-consistent solution to the more common Solov'ev one, and to a family of nonlinear configurations.

Two-fluid and finite Larmor radius effects on helicity evolution in a plasma pinch

The evolution of magnetic energy, helicity, and hybrid helicity during nonlinear relaxation of a driven-damped plasma pinch is compared in visco-resistive magnetohydrodynamics and two-fluid models with and without the ion gyroviscous stress tensor. Magnetic energy and helicity are supplied via a boundary electric field which initially balances the resistive dissipation, and the plasma undergoes multiple relaxation events during the nonlinear evolution. The magnetic helicity is well conserved relative to the magnetic energy over each event, which is short compared with the global resistive diffusion time. The magnetic energy decreases by roughly 1.5% of its initial value over a relaxation event, while the magnetic helicity changes by at most 0.2% of the initial value. The hybrid helicity is dominated by magnetic helicity in low-β pinch conditions and is also well conserved. Differences of less than 1% between magnetic helicity and hybrid helicity are observed with two-fluid modeling and result from cross helicity evolution. The cross helicity is found to change appreciably due to the first-order finite Larmor radius effects which have not been included in contemporary relaxation theories. The plasma current evolves towards the flat parallel current state predicted by Taylor relaxation theory but does not achieve it. Plasma flow develops significant structure for two-fluid models, and the flow perpendicular to the magnetic field is much more substantial than the flow along it.

Influence of toroidal rotation on resistive tearing modes in tokamaks

Influence of toroidal equilibrium plasma rotation on m/n = 2/1 resistive tearing modes is studied numerically using a 3D toroidal MHD code (CLT). It is found that the toroidal rotation with or without shear can suppress the tearing instability and the Coriolis effect in the toroidal geometry plays a dominant role on the rotation induced stabilization. For a high viscosity plasma (τR/τV ≫ 1, where τR and τV represent resistive and viscous diffusion time, respectively), the effect of the rotation shear combined with the viscosity appears to be stabilizing. For a low viscosity plasmas (τR/τV ≪ 1), the rotation shear shows a destabilizing effect when the rotation is large.

Numerical study of the plasma tearing instability on the resistive time scale

In this work, a new numerical scheme for the reduced resistive MHD system (RMHD) is presented. Numerical simulations of RMHD are notoriously challenging because of the disparate time-scales, encompassing the Alfvén wave period and the resistive diffusion time, and because of the formation of thin internal layers, especially in the nonlinear phase. The new scheme is specifically designed for the study of the long time scale dynamics with large time steps. The key difficulty, namely the singularity of the system matrix in the limit of an infinite time scale disparity, is overcome by techniques inspired by asymptotic preserving (AP) methods. The reformulated version of the fully-implicit RMHD scheme is based on a ‘micro–macro’ (MM) scheme with a stabilization term. The tearing mode evolution and the formation of a magnetic island are considered as a test case. The advantages of the MM scheme with respect to standard implicit and explicit schemes are demonstrated. Good agreement with known analytical results in the regime of nonlinear growth and saturation of the magnetic island are obtained.

Physics of forced magnetic reconnection in coaxial helicity injection experiments in National Spherical Torus Experiment

We numerically examine the physics of fast flux closure in transient coaxial helicity injection (CHI) experiments in National Spherical Torus Experiment (NSTX). By performing resistive Magnetohydrodynamics (MHD) simulations with poloidal injector coil currents held constant in time, we find that closed flux surfaces are formed through forced magnetic reconnection. Through a local Sweet-Parker type reconnection with an elongated current sheet in the injector region, closed flux surfaces expand in the NSTX global domain. Simulations demonstrate outflows approaching poloidally Alfvénic flows and reconnection times consistent with the Sweet-Parker model. Critical requirements for magnetic reconnection and flux closure are studied in detail. These primary effects, which are magnetic diffusivity, injector flux, injector flux footprint width, and rate of injector voltage reduction, are simulated for transient CHI experiments. The relevant time scales for effective reconnection are τV<τrec≈τAS(1+Pm)1/4<τR, where τV is the time for the injector voltage reduction, τA is the poloidal Alfvén transit time, τR is the global resistive diffusion time, and Pm and S are Prandtl and Lundquist numbers.

Neoclassical effects on the stabilization of tearing mode by current modulation

The neoclassical effects on the stabilization of tearing modes by current modulation have been investigated. Neoclassical effects enhance the resistivity and reduce the resistive diffusion time of the modulation current. Therefore, the oscillating current can penetrate deeper into the plasma. With an oscillating loop voltage, the plasma oscillates radially at the Ware-pinch velocity. These neoclassical effects improve the efficiency of tearing mode stabilization by the current modulation.

Improved criterion for sawtooth trigger and modelling

We discuss the role of neoclassical resistivity and local magnetic shear in the triggering of the sawtooth in tokamaks. When collisional detrapping of electrons is considered the value of the safety factor on axis, q(0, t), evolves on a new time scale, , where τη = 4πa2/[c2η(0)] is the resistive diffusion time, ν* = νe/(ϵ3/2ωte) is the electron collision frequency normalized to the transit frequency and ϵ = a/R0 is the tokamak inverse aspect ratio. Such an evolution is characterized by the formation of a structure of size around the magnetic axis, which can drive rapid evolution of the magnetic shear and decrease in q(0, t). We investigate two possible trigger mechanisms for a sawtooth collapse corresponding to crossing the linear threshold for the m = 1, n = 1 instability and non-linear triggering of this mode by a core resonant mode near the magnetic axis. The sawtooth period in each case is determined by the time for the resistive evolution of the q-profile to reach the relevant stability threshold; in the latter case it can be strongly affected by ν*.

Application of PDSLin to the magnetic reconnection problem

Magnetic reconnection is a fundamental process in a magnetized plasma at both low and high magnetic Lundquist numbers (the ratio of the resistive diffusion time to the Alfven wave transit time), which occurs in a wide variety of laboratory and space plasmas, e.g. magnetic fusion experiments, the solar corona and the Earth's magnetotail. An implicit time advance for the two-fluid magnetic reconnection problem is known to be difficult because of the large condition number of the associated matrix. This is especially troublesome when the collisionless ion skin depth is large so that the Whistler waves, which cause the fast reconnection, dominate the physics (Yuan et al 2012 J. Comput. Phys. 231 5822–53). For small system sizes, a direct solver such as SuperLU can be employed to obtain an accurate solution as long as the condition number is bounded by the reciprocal of the floating-point machine precision. However, SuperLU scales effectively only to hundreds of processors or less. For larger system sizes, it has been shown that physics-based (Chacon and Knoll 2003 J. Comput. Phys. 188 573–92) or other preconditioners can be applied to provide adequate solver performance. In recent years, we have been developing a new algebraic hybrid linear solver, PDSLin (Parallel Domain decomposition Schur complement-based Linear solver) (Yamazaki and Li 2010 Proc. VECPAR pp 421–34 and Yamazaki et al 2011 Technical Report). In this work, we compare numerical results from a direct solver and the proposed hybrid solver for the magnetic reconnection problem and demonstrate that the new hybrid solver is scalable to thousands of processors while maintaining the same robustness as a direct solver.

ELMy H-mode linear simulation with 3-field model on experimental advanced superconducting tokamak using BOUT++

H-mode plasmas with ELM (edge localized mode) have been realized on experimental advanced superconducting tokamak (EAST) with 2.45 GHz low hybrid wave at PLHW∼1 MW in 2010. Data from EAST experiments including magnetic geometry, measured pressure profiles, and calculated current profiles are used to investigate the physics of ELM utilizing the BOUT++ code. Results from linear simulations show that the ELMs in EAST are dominated by resistive ballooning modes. When the Lundquist number (dimensionless ratio of the resistive diffusion time to the Alfvén time) is equal to or less than 107, the resistive ballooning modes are found to become unstable in the ELMy H-mode plasma. For a fixed pedestal pressure profile, increasing plasma current generates more activities of low-n ELMs.

Analytic model for coaxial helicity injection in tokamak plasmas

Using a partial differential equation for the time evolution of the mean-field poloidal magnetic flux that incorporates resistivity η and hyper-resistivity Λ terms, an exact analytic solution is obtained for steady-state coaxial helicity injection (CHI) in force-free large aspect ratio tokamaks. The analytic mean-field Ohm’s law model allows for calculation of the tokamak CHI current drive efficiency and the plasma inductances at arbitrary levels of magnetic fluctuations, or dynamo activity. The results of the mean-field model suggest that CHI approaching Ohmic efficiency is only possible in tokamaks when the size of the effective current drive boundary layer, δ≡(Λ/η)1/2, becomes greater than half the size of the plasma, δ>a/2, with a the plasma minor radius. The electron thermal diffusivity due to magnetic fluctuation induced transport is obtained from the expression χe=Λ/μ0de2, with μ0 the permeability of free space and de the electron skin depth, which for typical tokamak fusion plasma parameters is on the order of a millimeter. Thus, the ratio of the energy confinement time to the resistive diffusion time in a tokamak plasma driven by steady-state CHI approaching Ohmic efficiency is shown to be constrained by the relation τE/τη<(de/a)2≈10-6. The mean-field model suggests that steady-state CHI can be viewed most simply as a boundary layer of stochastically wandering magnetic field lines.

On the application of electron cyclotron emission imaging to the validation of theoretical models of magnetohydrodynamic activity

Two-dimensional (2D) imaging of electron temperature perturbations provides a powerful constraint for validating theoretical models describing magnetohydrodynamic plasma behavior. In observation of Alfvén wave induced temperature fluctuations, electron cyclotron emission imaging provides unambiguous determination of the 2D eigenmode structure. This has provided support for nonperturbative eigenmode solvers which predict symmetry breaking due to poloidal flows in the fast ion population. It is shown that for Alfvén eigenmodes, and in cases where convective flows or saturated perturbations lead to nonaxisymmetric equilibria, electron plasma displacements oriented parallel to a gradient in mean temperature are well defined. Furthermore, during highly dynamic behavior, such as the sawtooth crash, highly resolved 2D temperature behaviors yield valuable insight. In particular, addressing the role of adiabatic heating on time scales much shorter than the resistive diffusion time through the additional diagnosis of local electron density allows progress to be made toward a comprehensive understanding of fast reconnection in tokamak plasmas.

A two-time-scale dynamic-model approach for magnetic and kinetic profile control in advanced tokamak scenarios on JET

Real-time simultaneous control of several radially distributed magnetic and kinetic plasma parameters is being investigated on JET, in view of developing integrated control of advanced tokamak scenarios. This paper describes the new model-based profile controller which has been implemented during the 2006–2007 experimental campaigns. The controller aims to use the combination of heating and current drive (H&CD) systems—and optionally the poloidal field (PF) system—in an optimal way to regulate the evolution of plasma parameter profiles such as the safety factor, q(x), and gyro-normalized temperature gradient, . In the first part of the paper, a technique for the experimental identification of a minimal dynamic plasma model is described, taking into account the physical structure and couplings of the transport equations, but making no quantitative assumptions on the transport coefficients or on their dependences. To cope with the high dimensionality of the state space and the large ratio between the time scales involved, the model identification procedure and the controller design both make use of the theory of singularly perturbed systems by means of a two-time-scale approximation. The second part of the paper provides the theoretical basis for the controller design. The profile controller is articulated around two composite feedback loops operating on the magnetic and kinetic time scales, respectively, and supplemented by a feedforward compensation of density variations. For any chosen set of target profiles, the closest self-consistent state achievable with the available actuators is uniquely defined. It is reached, with no steady state offset, through a near-optimal proportional-integral control algorithm. Conventional optimal control is recovered in the limiting case where the ratio of the plasma confinement time to the resistive diffusion time tends to zero. Closed-loop simulations of the controller response have been performed in preparation for experiments, and typical results are shown. Finally, in the last section of the paper, the first experimental results using this dynamic-model approach to control the plasma current and the safety factor profile on JET, either with the three H&CD systems or also with the PF system as an additional actuator, are presented and discussed.

A Constrained Tectonics Model for Coronal Heating

An analytical and numerical treatment is given of a constrained version of the tectonics model developed by Priest, Heyvaerts, & Title [2002]. We begin with an initial uniform magnetic field ${\bf B} = B_0 \hat{\bf z}$ that is line-tied at the surfaces $z = 0$ and $z = L$. This initial configuration is twisted by photospheric footpoint motion that is assumed to depend on only one coordinate ($x$) transverse to the initial magnetic field. The geometric constraints imposed by our assumption precludes the occurrence of reconnection and secondary instabilities, but enables us to follow for long times the dissipation of energy due to the effects of resistivity and viscosity. In this limit, we demonstrate that when the coherence time of random photospheric footpoint motion is much smaller by several orders of magnitude compared with the resistive diffusion time, the heating due to Ohmic and viscous dissipation becomes independent of the resistivity of the plasma. Furthermore, we obtain scaling relations that suggest that even if reconnection and/or secondary instabilities were to limit the build-up of magnetic energy in such a model, the overall heating rate will still be independent of the resistivity.

Development of steady-state scenarios compatible with ITER-like wall conditions

A key issue for steady-state tokamak operation is to determine the edge conditions that are compatible both with good core confinement and with the power handling and plasma exhaust capabilities of the plasma facing components (PFCs) and divertor systems. A quantitative response to this open question will provide a robust scientific basis for reliable extrapolation of present regimes to an ITER compatible steady-state scenario. In this context, the JET programme addressing steady-state operation is focused on the development of non-inductive, high confinement plasmas with the constraints imposed by the PFCs. A new beryllium main chamber wall and tungsten divertor together with an upgrade of the heating/fuelling capability are currently in preparation at JET. Operation at higher power with this ITER-like wall will impose new constraints on non-inductive scenarios. Recent experiments have focused on the preparation for this new phase of JET operation. In this paper, progress in the development of advanced tokamak (AT) scenarios at JET is reviewed keeping this long-term objective in mind. The approach has consisted of addressing various critical issues separately during the 2006–2007 campaigns with a view to full scenario integration when the JET upgrades are complete. Regimes with internal transport barriers (ITBs) have been developed at q95 ∼ 5 and high triangularity, δ (relevant to the ITER steady-state demonstration) by applying more than 30 MW of additional heating power reaching βN ∼ 2 at Bo ∼ 3.1 T. Operating at higher δ has allowed the edge pedestal and core densities to be increased pushing the ion temperature closer to that of the electrons. Although not yet fully integrated into a performance enhancing ITB scenario, Neon seeding has been successfully explored to increase the radiated power fraction (up to 60%), providing significant reduction of target tile power fluxes (and hence temperatures) and mitigation of edge localized mode (ELM) activity. At reduced toroidal magnetic field strength, high βN regimes have been achieved and q-profile optimization investigated for use in steady-state scenarios. Values of βN above the ‘no-wall magnetohydrodynamic limit’ (βN ∼ 3.0) have been sustained for a resistive current diffusion time in high-δ configurations (at 1.2 MA/1.8 T). In this scenario, ELM activity has been mitigated by applying magnetic perturbations using error field correction coils to provide ergodization of the magnetic field at the plasma edge. In a highly shaped, quasi-double null X-point configuration, ITBs have been generated on the ion heat transport channel and combined with ‘grassy’ ELMs with ∼30 MW of applied heating power (at 1.2 MA/2.7 T, q95 ∼ 7). Advanced algorithms and system identification procedures have been developed with a view to developing simultaneously temperature and q-profile control in real-time. These techniques have so far been applied to the control of the q-profile evolution in JET AT scenarios.

Observation of Instability-Induced Current Redistribution in a Spherical-Torus Plasma

A motional Stark effect diagnostic has been utilized to reconstruct the parallel current density profile in a spherical-torus plasma for the first time. The measured current profile compares favorably with neoclassical theory when no large-scale magnetohydrodynamic instabilities are present in the plasma. However, a current profile anomaly is observed during saturated interchange-type instability activity. This apparent anomaly can be explained by redistribution of neutral beam injection current drive and represents the first observation of interchange-type instabilities causing such redistribution. The associated current profile modifications contribute to sustaining the central safety factor above unity for over five resistive diffusion times, and similar processes may contribute to improved operational scenarios proposed for ITER.

ITER operation beyond its baseline scenario

The ‘improved H-mode’ regime, realized in ASDEX Upgrade in 1998 and confirmed by other devices, demonstrates the integration of advanced performance beyond the standard H-mode for confinement (confinement enhancement factor H98(y,2) ⩽ 1.6), stability (normalized beta βN ∼ 3–3.5) and, at densities close to the Greenwald density, exhaust in stationary discharges longer than 40 confinement times or up to six resistive diffusion times. The q-profile is characterized by low central magnetic shear and axis safety factor q0 > 1 that is obtained by particular discharge and heating ramp-up scenarios and maintained via fishbones or benign higher (m, n) instabilities without using elaborate current control. Core transport is still governed by drift-wave turbulence with stiff temperature profiles, but density profiles are more strongly peaked and contribute to the increase in global confinement. A further contribution manifests itself by enhanced pressures at the edge barrier pedestal top and at the ρ = 0.9 surface both increasing with the input power. (3, 2) NTMs remain small, enabling routine operation up to βN ∼ 3 (limited by (2, 1) NTMs) at ITER relevant collisionalities, for normalized Larmor radii down to four times the ITER value and for a broad range of q95 = 3–5. Tailored heat deposition including central wave heating allows for a compromise in density peaking for enhanced confinement and tolerable high-Z impurity concentrations even with tungsten coated structures.As far as the ITER relevance of this regime is concerned, its compatibility with significant central electron heating, low collisionality and even densities close to the Greenwald density combined with type-II ELMs and βN ∼ 3.5 is of importance. The GLF23 turbulence model still predicts peaked density profiles (R/Ln ∼ 3) and sufficient transport to avoid impurity accumulation. At low q95 ∼ 3 the fusion performance in terms of is more than doubled compared with the ITER baseline scenario (performance factor ∼ 0.2) extrapolating to long Q ≫ 10 pulses on ITER. At medium q's bootstrap current fractions up to 50% and performance factors close to 0.2 can be achieved resulting in long inductive pulse lengths of ∼1 h allowing ITER ‘hybrid’ operation at Q ⩽ 9.

Development of an ITER relevant advanced scenario at ASDEX Upgrade

The “improved H-mode,” realized in ASDEX Upgrade [A. Herrmann and O. Gruber, Fusion Sci. Technol. 44, 569 (2003)] in 1998, demonstrates that advanced requirements beyond the standard H-mode for confinement [confinement enhancement factor H98(y,2)>1], stability (normalized beta βN∼3–3.5) and, at densities close to Greenwald density, exhaust can be simultaneously met and maintained stationary for several resistive diffusion times. The q profile is characterized by low central magnetic shear and axis safety factor q0>1 that is obtained by particular heating and current ramp-up scenarios and maintained via benign instabilities. Core transport is still governed by drift-wave turbulence with stiff temperature profiles, but density profiles are more strongly peaked and contribute to the increase in confinement. Neoclassical tearing modes remain small, enabling routine operation up to βN∼3 at international thermonuclear experimental reactor (ITER) relevant collisionalities, for normalized Lamor radii down to four times the ITER value and for a broad range of q95=3.2–4.5. Using tailored heat deposition including central wave heating a compromise was found in density peaking for enhanced confinement and limiting the high-Z impurity concentrations even with a tungsten-coated first wall and divertor. As far as the ITER [ITER EDA Documentation Series No. 24, 2002] relevance of this regime is concerned, its compatibility with significant central electron heating, high edge densities, and type-II edge localized modes is of importance. The GLF23 turbulence model predicts still peaked density profiles and sufficient transport to avoid impurity accumulation. The fusion performance in terms of βNH98(y,2)∕q952 is nearly doubled compared with the ITER base-line scenario at low-q values, while at medium q’s bootstrap current fractions up to 50% and long inductive pulse lengths allow ITER “hybrid” operation.

Global Plasma Turbulence Simulations ofq=3Sawtoothlike Events in the RTP Tokamak

A two-fluid computer model of electromagnetic tokamak turbulence, CUTIE, is used to study the dynamic structure and turbulent transport in the Rijnhuizen Tokamak Project tokamak. A discharge with dominant, off-axis electron cyclotron heating is the main focus of the simulations which were extended over several resistive diffusion times. CUTIE reproduces the turbulent transport and MHD phenomena of the experiment. The noninductive components of the current density profile, viz., the dynamo current and the bootstrap current, are identified as key players in the turbulent transport and its suppression and in off-axis MHD events.

Particle pinch with fully noninductive lower hybrid current drive in Tore Supra.

Recently, plasmas exceeding 4 min have been obtained with lower hybrid current drive (LHCD) in Tore Supra. These LHCD plasmas extend for over 80 times the resistive current diffusion time with zero loop voltage. Under such unique conditions the neoclassical particle pinch driven by the toroidal electric field vanishes. Nevertheless, the density profile remains peaked for more than 4 min. For the first time, the existence of an inward particle pinch in steady-state plasma without toroidal electric field, much larger than the value predicted by the collisional neoclassical theory, is experimentally demonstrated.