Diamagnetic Flow Research Articles

Limit cycle oscillations in the zonal-flow-catalyzed interactions of ion-temperature-gradient turbulence

Limit-cycle oscillations are studied for ion temperature gradient turbulence, which, in the absence of large diamagnetic (mean) shear flows, saturates through energy transfer from unstable modes to large-scale stable modes via zonal-flow intermediary modes. Oscillations of zonal flow and turbulence levels are strongly constrained by the reactive, largely non-dissipative character of the zonal flows. Since existing predator–prey models for observed oscillations in experiments do not include energy transfer through zonal flows to stable modes, low-order fluid models with this physics are constructed and investigated. A simple three-wave truncation produces low-amplitude zonal flows that slowly oscillate around a zero mean, with turbulence oscillations between coupled wavenumbers that exceed linear frequencies by orders of magnitude. This inconsistency with experimental observations is caused by the weak non-linear drive of zonal flows in three-wave systems and the lack of multiple-wavenumber turbulent interactions. A more comprehensive model that preserves multiple wavenumber interactions within the context of conservative zonal-flow-mediated energy transfer to stable modes accurately reflects observed dynamics when the phase between stable and unstable modes is occasionally randomized.

Fluid simulations of resistive drift-wave turbulence with diamagnetic flow in ZPED experiments

We derive a diamagnetic resistive fluid model (DRF) and develop an associated two-dimensional fluid simulation code (DRF-2D) to explore the dynamics of resistive drift modes within the plasmas of the Zheda Plasma Experiment Device (ZPED). The validation of the linear dispersion relation for the DRF-2D code revealed a harmonious agreement between analytical theory and linear numerical simulations. Leveraging plasma parameters obtained from the ZPED experiments, we conducted a comprehensive series of nonlinear simulations using the DRF-2D code. Our simulations successfully replicate the nonlinear trends in turbulent fluctuations and transport observed in the ZPED experiments, particularly demonstrating a remarkably accurate alignment of the turning point in the magnetic field. Notably, the DRF model sheds light on the observed frequency sign reversal from the electron diamagnetic direction to the ion diamagnetic direction in the ZPED experiments. This is demonstrated through well-matched turning points in the confining magnetic field between the nonlinear simulations and ZPED experiments. The fidelity of our model in capturing these phenomena underscores its efficacy in providing valuable insights and predictive capabilities for the intricate dynamics observed in the ZPED plasmas.

Effects of diamagnetic drift on nonlinear interaction between multi-helicity neoclassical tearing modes

A numerical study of the diamagnetic drift effect on the nonlinear interaction between multi-helicity neoclassical tearing modes (NTMs) is carried out using a set of four-field equations including two-fluid effects. The results show that, in contrast to the single-fluid case, 5/3 NTM cannot be completely suppressed by 3/2 NTM with diamagnetic drift flow. Both modes exhibit oscillation and coexist in the saturated phase. To better understand the effect of the diamagnetic drift flow on multiple-helicity NTMs, the influence of typical relevant parameters is investigated. It is found that the average saturated magnetic island width increases with increasing bootstrap current fraction f b but decreases with the ion skin depth δ. In addition, as the ratio of parallel to perpendicular transport coefficients χ ∥/χ ⊥ increases, the average saturated magnetic island widths of the 3/2 and 5/3 NTMs increase. The underlying mechanisms behind these observations are discussed in detail.

The performance of equilibrium radial electric field shear on microturbulence with different magnetic shears in tokamak plasmas

Global gyrokinetic particle simulations show that equilibrium radial electric field (E r ) shear reduces the linear growth rate, ion heat conductivity, and nonlinear turbulence amplitude for both the ion temperature gradient (ITG) and kinetic ballooning mode (KBM) microturbulence by tilting the poloidal mode structure. The increase in the magnetic shear enhances the stabilizing performance of the E r shear on linear growth rate for the ITG case but has no effect on that for the KBM case. The radial correlation length of the ITG turbulence is decreased by increasing the magnetic shear in a weak ion diamagnetic flow shear condition with low β, leading to a reduction in the effective E × B shearing rate, which weakens the suppression performance of the E r shear on the ITG turbulence amplitude. In contrast, under a larger ion diamagnetic shear flow for higher β, an increase in magnetic shear strengthens the suppression performance of the E r shear on the KBM turbulence amplitude due to an increase in the effective shearing rate by increasing the radial correlation length of the turbulence.

The effect of plasma beta on error-field penetration in radio-frequency wave heated plasmas in EAST

The plasma-beta effect on the n = 1 resonant magnetic perturbation (RMP) field penetration in purely radio-frequency (RF) wave heated discharges has been investigated in EAST. The experimental results show that the dependence of the threshold RMP coil current for field penetration, , on the total absorbed power P tot scales as approximately , indicating that the error-field tolerance is improved with increasing RF power. This is benefited by the increased electron perpendicular flow dominated by a counter-current electron diamagnetic flow with increasing RF power. However, theoretical scaling in cylindrical geometry overestimates the power index. Assuming an additional term for the normalized beta in the scaling, it is shown that the fitted from the experimental observation is around −1, indicating a degradation effect of plasma beta. To clarify the underlying physics of the plasma-beta effect that was not included in the theoretical scaling in cylindrical geometry, the MARS-Q code with full toroidal geometry is employed for simulation of nonlinear field penetration (Liu et al 2013 Phys. Plasmas 20 042503). The MARS-Q simulation results reproduce the dependence well, and hence the P tot scaling of the threshold current in experimental observations. The main reason for this is that the net total torque, which is mainly contributed by the neoclassical toroidal viscosity (NTV), increases with increasing plasma . The results demonstrate that the nonlinear toroidal coupling effect via NTV torque plays an important role in determining field penetration, even in cases with relatively low , which is far less than the no-wall beta limit.

Drift-ordered fluid vorticity equation with energy consistency

Although drift-ordered fluid models are widely applied in tokamak edge turbulence simulations, the models used are acknowledged not to conserve energy or even electrical charge. The present paper aims to remove many of the existing pitfalls in drift-fluid models, however, with the objective of finding a solution simple enough to be implemented in numerical applications. Our main result is an improved version of the drift-Braginskii equations involving a generalized vorticity function. In the new drift-Braginskii system, the quasi-neutrality condition translates into a transport equation for a generalized vorticity, expressed in conservation form, and related to the total mass-weighted circulation. It is found that kinetic energy conservation can be achieved if the polarization flow is defined recursively. The resulting model conserves the kinetic energy associated with E×B and diamagnetic flows and retains the associated perpendicular kinetic energy flux.

Screening effect of plasma flow on the resonant magnetic perturbation penetration in tokamaks based on two-fluid model

Numerical simulation on the resonant magnetic perturbation penetration is carried out by the newly-updated initial value code MDC (MHD@Dalian Code). Based on a set of two-fluid four-field equations, the bootstrap current, parallel, and perpendicular transport effects are included appropriately. Taking into account the bootstrap current, a mode penetration-like phenomenon is found, which is essentially different from the classical tearing mode model. To reveal the influence of the plasma flow on the mode penetration process, E × B drift flow and diamagnetic drift flow are separately applied to compare their effects. Numerical results show that a sufficiently large diamagnetic drift flow can drive a strong stabilizing effect on the neoclassical tearing mode. Furthermore, an oscillation phenomenon of island width is discovered. By analyzing it in depth, it is found that this oscillation phenomenon is due to the negative feedback regulation of pressure on the magnetic island. This physical mechanism is verified again by key parameter scanning.

Finite magnetic well effects on resistive and drift-resistive ballooning modes in a shaped tokamak

The impact of plasma shaping through magnetic well modifications on the stability of resistive ballooning modes (RBMs) in tokamaks is analysed, also including finite diamagnetic flows. Various limiting cases of the dispersion relation, obtained by matching the ballooning equation across the ideal and resistive layers, are analysed. It is found that stability is generally improved by the combination of vertical elongation and positive triangularity, although, in some cases, the growth rate of the unstable mode can be enhanced by these effects. Usually, vertically elongated plasmas with no triangularity are prone to exhibit worse stability properties. A value for the critical β above which RBMs are driven unstable is identified, and a connection with type-III ELM activity is established.

Role of the pedestal current on the stability of non-ideal ballooning modes

On the basis of a three-field flute-reduced magnetohydrodynamic model, which mainly describes the edge instabilities by shielding a major part of the J × B force in the flute reduction, we study the stability of ballooning modes in the edge pedestal, highlighting the role of an equilibrium parallel current gradient. This effect, which is designated as the current gradient driven (CGD) term in this paper, is shown to have an influence on the stability of finite-n pedestal ballooning modes due to the existence of a highly localized bootstrap current. An analysis in the ideal limit shows that the CGD term destabilizes the ballooning modes regardless of the sign of its gradient, especially near the stability boundaries. An inclusion of the finite Larmor radius (FLR) effect via ion diamagnetic flow and finite resistivity results in a coupling of the FLR effect and the current gradient. In this particular regime where the deviation from the ideal stability is considerable, this coupling effect is shown to dominate stability in intermediate n (20<n≤40) modes. Here, n is the toroidal mode number. Stability analyses using a series of model pedestal equilibria indicate that an increase in a bootstrap current can move the most unstable position from the pedestal middle to the bottom and the negative gradient of the bootstrap current at the pedestal bottom leads to further destabilization of intermediate n modes.

Comparing spontaneous and pellet-triggered ELMs via non-linear extended MHD simulations

Injecting frozen deuterium pellets into an ELMy H-mode plasma is a well established scheme for triggering edge localized modes (ELMs) before they naturally occur. This paper presents non-linear simulations of spontaneous type-I ELMs and pellet-triggered ELMs in ASDEX Upgrade performed with the extended MHD code JOREK. A thorough comparison of the non-linear dynamics of these events is provided. In particular, pellet-triggered ELMs are simulated by injecting deuterium pellets into different time points during the pedestal build-up described in A Cathey et al (2020 Nuclear Fusion 60 124007). Realistic ExB and diamagnetic background plasma flows as well as the time dependent bootstrap current evolution are included during the build-up to accurately capture the balance between stabilising and destabilising terms for the edge instabilities. Dependencies on the pellet size and injection times are studied. The spatio-temporal structures of the modes and the resulting divertor heat fluxes are compared in detail between spontaneous and triggered ELMs. We observe that the premature excitation of ELMs by means of pellet injection is caused by a helical perturbation described by a toroidal mode number of n = 1. In accordance with experimental observations, the pellet-triggered ELMs show reduced thermal energy losses and a narrower divertor wetted area with respect to spontaneous ELMs. The peak divertor energy fluence is seen to decrease when ELMs are triggered by pellets injected earlier during the pedestal build-up.

Transition from no-ELM response to pellet ELM triggering during pedestal build-up—insights from extended MHD simulations

Pellet edge localized mode (ELM) triggering is a well-established scheme for decreasing the time between two successive ELM crashes below its natural value. Reliable ELM pacing has been demonstrated experimentally in several devices, increasing the ELM frequency considerably. However, it was also shown that the frequency cannot be increased arbitrarily due to a so-called lag-time. During this time, after a preceding natural or triggered ELM crash, neither a natural ELM crash occurs nor is it possible to trigger an ELM crash by pellet injection. For this article, pellet ELM triggering simulations are advanced beyond previous studies in two ways. Firstly, realistic E × B and diamagnetic background flows are included. And secondly, the pellet is injected at different stages of the pedestal build-up. This allows us to recover the lag time for the first time in simulations and investigate it in detail. A series of nonlinear extended MHD simulations is performed to investigate the plasma dynamics resulting from an injection at different time points during the pedestal build-up. The experimentally observed lag-time is qualitatively reproduced. In particular, a sharp transition is observed between the regime where no ELMs can be triggered and the regime where pellet injection causes an ELM crash. Via variations of pellet parameters and injection time, the two regimes are studied and compared in detail, revealing pronounced differences in the nonlinear dynamics. The toroidal mode spectrum is significantly broader when an ELM crash is triggered, enhancing the stochasticity and therefore also the losses of thermal energy along magnetic field lines. In the heat fluxes to the divertor targets, pronounced toroidal asymmetries are observed. In the case of high injection velocities leading to deep penetration, the excitation of core modes like the 2/1 neoclassical tearing mode is also observed.

Ion velocity shear‐induced drift wave and momentum transport in non‐Maxwellian magnetoplasma flows

AbstractWe have investigated the diamagnetic flow in a non‐uniform partially ionized plasma with non‐Maxwellian electron population to explain the dynamics of ion velocity shear‐induced low‐frequency drift mode and associated instabilities. The dispersion relations are found, and instability threshold conditions are pointed out along with ion‐parallel momentum transport due to drift motion with relative phase shift of fluctuating quantities in a non‐conservative system. The real frequencies and instability growth rates are studied numerically and illustrated for typical space and laboratory plasmas. This study should be useful in understanding some aspects of low‐frequency time‐delayed perturbations with sheared flow leading to drift instabilities and cross‐field parallel ion momentum transport in nonuniform magnetoplasmas containing a non‐Maxwellian electron population.

Pedestal collapse by resonant magnetic perturbations

Pedestal collapse (i.e., the complete loss of the edge transport barrier (ETB)) in DIII-D H-mode plasmas occurs when resonant magnetic perturbations (RMPs) penetrate the steep gradient region at the plasma edge. Normally, RMP driven magnetic islands can occur at the top and bottom of the H-mode pedestal and these islands generate conditions consistent with edge-localized-mode (ELM) suppression and density pump-out, respectively, based on nonlinear two-fluid MHD simulations. In contrast, MHD simulations show that the steep pressure gradient region between the top and bottom of the DIII-D pedestal is generally immune to resonant field penetration due to large local E × B and diamagnetic flows. By this fortuitous circumstance, the edge-transport-barrier and H-mode confinement can be maintained while achieving ELM suppression. However, pedestal collapse can occur in DIII-D when the screening flows are inadequate to prevent field penetration in the steep gradient region of the pedestal. Non-linear two-fluid MHD simulations support the role of resonant field penetration in pedestal collapse for DIII-D H-mode plasmas with weak edge E × B and diamagnetic screening flows. ITER will likely have weaker edge screening flows than present experiments due to its much larger size, making it more susceptible to resonant field penetration in the steep gradient region of the pedestal. Analysis of model ITER equilibria demonstrates that resonant field penetration in the steep pressure gradient region is possible for RMP levels of the order required for ELM suppression. The effect of such penetration on the ITER pedestal will depend sensitively on the resulting degree of island overlap.

Zonally dominated dynamics and Dimits threshold in curvature-driven ITG turbulence

The saturated state of turbulence driven by the ion-temperature-gradient instability is investigated using a two-dimensional long-wavelength fluid model that describes the perturbed electrostatic potential and perturbed ion temperature in a magnetic field with constant curvature (a $Z$ -pinch) and an equilibrium temperature gradient. Numerical simulations reveal a well-defined transition between a finite-amplitude saturated state dominated by strong zonal-flow and zonal temperature perturbations, and a blow-up state that fails to saturate on a box-independent scale. We argue that this transition is equivalent to the Dimits transition from a low-transport to a high-transport state seen in gyrokinetic numerical simulations (Dimits et al., Phys. Plasmas, vol. 7, 2000, 969). A quasi-static staircase-like structure of the temperature gradient intertwined with zonal flows, which have patch-wise constant shear, emerges near the Dimits threshold. The turbulent heat flux in the low-collisionality near-marginal state is dominated by turbulent bursts, triggered by coherent long-lived structures closely resembling those found in gyrokinetic simulations with imposed equilibrium flow shear (van Wyk et al., J. Plasma Phys., vol. 82, 2016, 905820609). The breakup of the low-transport Dimits regime is linked to a competition between the two different sources of poloidal momentum in the system – the Reynolds stress and the advection of the diamagnetic flow by the $\boldsymbol {E}\times \boldsymbol {B}$ flow. By analysing the linear ion-temperature-gradient modes, we obtain a semi-analytic model for the Dimits threshold at large collisionality.

Critical velocity shear flow for triggering L-H transition and its parametric dependence in the HL-2A tokamak

The dynamics of shear flows and turbulence in the pedestal region during the L-H transition have been studied in the HL-2A tokamak. It has been shown that ion diamagnetic shear flow plays a key role in triggering the L-H transition. A critical velocity shear has been found for the L-H transition. Statistical results indicate that the velocity shear threshold is independent of the plasma density and the total heating power.

The role of edge resonant magnetic perturbations in edge-localized-mode suppression and density pump-out in low-collisionality DIII-D plasmas

Two-fluid nonlinear MHD simulations using the TM1 code demonstrate that the formation of magnetic islands at the top and bottom of the H-mode pedestal, together with the strong screening of resonant fields in the gradient region of the pedestal, can account for ELM suppression and density pump-out by n = 2 resonant magnetic perturbations (RMPs) in low-collisionality DIII-D ITER similar shape (ISS) plasmas. Using experimentally relevant transport coefficients, neoclassical resistivity, electron collisionality, and RMP amplitudes, nonlinear MHD simulations reproduce the observed level of density reduction (density pump-out) in DIII-D due to the formation of narrow magnetic islands in the resistive foot of pedestal. For large amplitude RMPs (Br/Bt > 1×10−4) simulations predict resonant field penetration and resulting in pressure reduction at the top of the pedestal, consistent with experimental observations at the onset of ELM suppression. The predicted reduction in the height and width of the pedestal by magnetic island enhanced collisional transport provides a quantitative mechanism for the stabilization of the peeling-ballooning mode (PBM). Importantly, these simulations predict strong screening of resonant fields in the steep gradient region of the pedestal due to strong E×B rotation and diamagnetic flows. However, if the plasma resistivity is made artificially larger (∼ 10X) than neoclassical, the simulations predict magnetic stochasticity throughout the plasma edge and the collapse of the pedestal due to the reduction in the penetration threshold with increasing resistivity. A scaling relation for resonant field penetration at the pedestal top, using several hundred nonlinear simulations, reproduces the density and E×B rotation dependence of the ELM suppression threshold observed in DIII-D. Simulations using ITER model equilibria suggest that the penetration threshold at the top of the ITER pedestal will be lower than in DIII-D due to the anticipated lower perpendicular flow velocities in ITER, resulting in the weaker screening of resonant fields.

Structure bifurcation induced by wide magnetic islands

Magnetic islands resulting from magnetic reconnection play significantly important roles in critical issues of magnetic fusion plasmas, because they essentially change magnetic topology and significantly degrade plasma confinement. In this paper, we report that the wide islands can lead to a structure bifurcation of flow parity inside islands, then induce a state with a large vortex flow. The mechanism is identified as an instability in the configuration with magnetic islands, in which the eigenmode parity is contrary to the conventional tearing mode and the radial profile of zonal flow also changes. The critical values of the width of magnetic island for the bifurcation are obtained and discussed. The dependence of bifurcation on various physical parameters is studied in detail. It is found that tearing instability parameter and diamagnetic drift flow have significant impacts on the bifurcation. Moreover, the characteristics of the bifurcation show the qualitative agreement with the experimental observation on the Large Helical Device.

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.

Effects of diamagnetic drift on magnetohydrodynamic explosive bursts in reversed magnetic shear tokamak plasmas

The explosive bursts, induced by self-organized double tearing modes (DTMs) in reversed magnetic shear tokamak plasmas, are numerically investigated based on a set of reduced two-fluid equations. It is found that the diamagnetic drift has a destabilizing effect on the mode in the Rutherford phase but can effectively stabilize the explosive burst of the mode in the following nonlinear phase. The plasma viscosity µ is found to have an important effect on the critical ion skin depth for completely suppressing the explosive bursts. In the low viscosity regime (i.e. , where is the critical viscosity), increases with increasing µ, because the increase of µ can effectively reduce the velocity difference of islands between the two rational surfaces, which mainly result from zonal flow and electron diamagnetic flow. In the high viscosity regime (i.e. ), however, decreases with increasing µ, because the increase of µ can greatly stabilize the DTM instability. In addition, the dependences of critical viscosity on viscosity and resistivity are obtained and the relevant mechanisms are analyzed.

Guiding-centre theory for kinetic-magnetohydrodynamic modes in strongly flowing plasmas

A kinetic-magnetohydrodynamic model with kinetic pressure closure is derived from a consistent guiding-centre framework. Higher-order (gyroviscous) corrections to the pressure tensor are derived in complex geometry from a reduced kinetic equation. The proposed model allows for flows of the order of the thermal ion velocity, taking into account important centrifugal effects due to the E × B flow, as well as the effects of diamagnetic flows associated with finite Larmor radius corrections to both ion fluid inertia and long mean-free path contributions. Wave–particle interactions, such as toroidal drift-resonance, are retained. Furthermore, the linearised model includes a quasi-neutrality equation, allowing the effects of a parallel electric field to be studied in fast rotating tokamak plasmas.