Toroidal Ion Temperature Gradient Research Articles

Reduction of electrostatic turbulence in a quasi-helically symmetric stellarator via critical gradient optimization

We present a stellarator configuration optimized for a large threshold (‘critical gradient’) for the onset of the ion temperature gradient (ITG) driven mode, which achieves the largest critical gradient we have seen in any stellarator. Above this threshold, gyrokinetic simulations show that the configuration has low turbulence levels over an experimentally relevant range of the drive strength. The applied optimization seeks to maximize the drift curvature, leading to enhanced local-shear stabilization of toroidal ITG modes, and the associated turbulence. These benefits are combined with excellent quasi-symmetry, yielding low neoclassical transport and vanishingly small alpha particle losses. Analysis of the resulting configuration suggests a trade-off between magnetohydrodynamic (MHD) and ITG stability, as the new configuration possesses a vacuum magnetic hill.

Mitigation of resistive drift wave and ion temperature gradient instabilities by velocity shear

The effects of velocity shear on the resistive drift wave instability in the non-adiabatic limit and the toroidal ion temperature gradient instability are investigated for a plasma of inhomogeneous density/ion temperature, respectively. For the resistive drift wave, we find that the instability growth rate decreases monotonically with increasing magnitude of shear, but we find that complete stabilization is impossible. For the ion temperature gradient instability, we find that the standard WKB approximation is insufficient to describe the full behavior of the instability, and that an analysis of the localized eigenmode problem reveals two separate unstable solutions, which the WKB approximation does not predict. The impact of flow shear on these two new unstable solutions is discussed. In both resistive drift wave and ion temperature gradient instabilities, the sheared flow causes a shifting, tilting, and sharpening of the electrostatic potential eddies.

Global gyrokinetic simulation for toroidal impurity mode turbulence

In this study, global gyrokinetic simulations of the toroidal impurity mode (tIM) turbulence are performed. A linear analysis shows that the tIM is an instability that occurs in the bad curvature region when the density gradients of bulk ions and impurities are in opposite directions. The tIM can be unstable even when the temperature profiles are flat. In the presence of temperature gradients, the tIM and toroidal ion temperature gradient (tITG) mode could coexist. For the small temperature gradient, the tIM is found to be dominant. The tIM turbulence drives the large inward impurity and outward ion particle transports. Furthermore, the inward ion heat flux driven by the tIM turbulence causes the ion temperature profile to be more peaked than the initial one. For the large temperature gradient, while such inward ion heat flux does not occur because of the dominant tITG mode, the large inward impurity and outward ion particle fluxes are still observed due to the subdominant tIM. These results indicate that the tIM plays an important role in turbulent heat and particle transport when impurities are injected.

Vlasov equation, waves and dispersion relations in fractal dimensions: Landau damping and the toroidal ion temperature gradient instability problem

In this study, we have used the concept of ‘product-like fractal measure’ introduced by Li and Ostoja-Starzewski in their formulation of anisotropic and elastic media to introduce Vlasov equation in fractal dimension. We have been motivated by this new concept since it is safely applied to several systems with length scales bounded by lower and upper cutoffs via the dimensional regularization technique. We have considered through this study collisionless plasma in fractal dimension and we have analyzed the impacts of the Vlasov equation in some points in plasma physics starting from the Hamiltonian dynamics. Several results have been obtained and discussed accordingly. The main outcomes of the present study concern first the facts that waves in cold plasma are associated with low fractal dimensions and hence by low complexity and that the effective damping is affected by the sign of the fractal dimension and differs from the conventional Landau damping in magnitude and in sign for negative dimension. Besides, for a very low fractal dimension, the Landau damping is negligible or removed. This result was observed in kinetic turbulent plasmas and may solve the paradox problem emerging in the application of critical balance to a kinetic turbulence cascade.

Nonlinear functional relation covering near- and far-marginal stability in ion temperature gradient driven turbulence

A novel nonlinear functional relation of turbulence potential intensity, zonal flow potential intensity, and ion thermal diffusivity that accurately reproduces nonlinear gyrokinetic simulations of toroidal ion temperature gradient (ITG) driven turbulence is proposed. Applying mathematical optimization techniques to find extremal solutions in high-dimensional parameter space, the optimal regression parameters in the functional form are determined to be valid for both near- and far-marginal regime of the ITG stability including the Dimits-shift. Then, the regression error of ∼5% is accomplished. In addition, it is clarified that the intensity ratio of the zonal flow and turbulence potential intensity is a crucial factor to determine the reproduction accuracy.

Assessing physics of ion temperature gradient turbulence via hierarchical reduced-model representations

The saturation physics of ion temperature gradient (ITG) turbulence is probed by studying how amplitudes and scalings with key parameters vary across a hierarchy of reduced models. The models derive from nonlinear fluid equations for toroidal ITG turbulence under approximations to the mode coupling interactions in wavenumber space and the representation of turbulent decorrelation. Mode coupling approximations include local-in-wavenumber treatments like the spectral density of flux in quasilinear theory, a truncation to three nonlinearly interacting waves, and the interactions in a cascade to high radial wavenumber mediated by a single zonal flow. Turbulent decorrelation treatments are based on the triplet correlation time with and without eddy damping. Model fidelity is assessed by the scalings and magnitudes of the squared amplitudes of unstable mode, stable mode, and zonal flow with respect to the flow-damping rate and temperature gradient. It is shown that all models reproduce fundamental scalings, provided they incorporate the coupling of unstable mode, stable mode, and zonal flow. Accurate amplitude prediction requires eddy damping in the triplet correlation time and proper representation of the zonal-flow drive by interactions associated with the radial wavenumber cascade.

The effect of magnetic island on toroidal ion temperature gradient mode instability

In this work, we have investigated the influence of magnetic island (MI) on electrostatic toroidal ion temperature gradient (ITG) mode, where the ions are described by gyro-kinetic equations including MI, and adiabatic approximation is used for electrons. The eigen-equation for short-wavelength toroidal ITG mode in Fourier-ballooning representation is derived, and the corresponding eigenvalue as well as mode structure are solved. Both the flattening effects of MI on plasma pressure and MI-scale shear flow are considered. It is found that when only considering the flattening effects of MI, the ITG mode can be stabilized compared to the case without MI. Meanwhile, the effective toroidal ITG mode drive can be enhanced by including MI-scale flow, which indicates the dominant destabilizing by MI-scale flow over the stabilizing by flattening profile, which results in a higher growth rate than the case without MI. It is also found that the total flow shearing may prevent the ITG turbulence spreading from the X-point of MI but is not strong enough to prevent spreading from the separatrix across the O-point of larger MI via comparison between the flow shearing rate and the linear growth rate. Furthermore, the corresponding width of the lowest-order mode structure in the ballooning angle is slightly widened (narrowed) for the case without (with) MI-scale flow, compared to the case without MI. Moreover, the shifted even symmetry in the ballooning angle is not qualitatively influenced by the presence of MI. The mode structure is radially asymmetric but is symmetric with respect to the phase of MI at the O-point.

Formation of solitons and shocks in toroidal ion temperature gradient mode in the presence ofnon‐Maxwellianelectrons

AbstractLinear and nonlinear phenomena are investigated in toroidal ion temperature gradient (TITG)‐driven pure drift mode. The model includes inhomogeneity in background magnetic field, ion temperature, and density. Finite Larmor radius effect is incorporated to understand the effect of low‐frequency wave on ion dynamics. Electrons are assumed to follow nonthermal distribution, that is, kappa and Cairns distributions. Dispersion relation is obtained to analyse the linear behaviour of the TITG mode in the presence of non‐Maxwellian electron distribution. In the nonlinear regime, exact solutions (soliton and shocks) are obtained (in dispersive and dissipative medium respectively) by using functional variable method to solve the nonlinear partial differential equation obtained for the system under consideration. Graphical illustrations are used to exhibit the characteristics of linear and nonlinear structures and their dependence on different physical parameters. It is observed that for TITG‐driven pure drift mode, rarefactive solitons are formed for both thermal and nonthermal electron distributions. It is also observed that variation of electrons from standard thermal distribution affects the propagation characteristics of linear and nonlinear structures in TITG‐driven modes. Results of our investigations will be helpful to understand the low‐frequency waves in inhomogeneous plasmas in the presence of nonthermal electron distributions which are frequently observed by satellite missions and are also observed in laboratory plasmas.

The theoretical study on intermittency and propagation of geodesic acoustic mode in L-mode discharge near tokamak edge

Through a systematically developed theory, we demonstrate that the motion of Instanton identified in Zhang et al (2017 Phys. Plasmas 24 122304) is highly correlated to the intermittent excitation and propagation of geodesic acoustic mode (GAM) that is observed in tokamaks. While many numerical simulations have observed the phenomena, it is the first theory that reveals the physical mechanism behind GAM intermittent excitation and propagation. The preceding work is based on the micro-turbulence associated with toroidal ion temperature gradient mode, and slab-based phenomenological model of zonal flow. When full toroidal effect is introduced into the system, two branches of zonal flow emerge: the torus-modified low frequency zonal flow (TLFZF), and GAM, necessitating a unified exploration of GAM and TLFZF. Indeed, we observe that the transition from the Caviton to Instanton is triggered by a rapid zero-crossing of radial group velocity of drift wave and is found to be strongly correlated with the GAM onset. Many features peculiar to intermittent GAMs, observed in real machines, are thus identified in the numerical experiment. The results will be displayed in figures and in a movie; first for single central rational surface, and then with coupled multiple central rational surfaces. The periodic bursting first shown disappears as being replaced by irregular one, more similar to the intermittent characteristics observed in GAM experiments.

Generalized curvature modified plasma dispersion functions and Dupree renormalization of toroidal ITG

A new generalization of curvature modified plasma dispersion functions is introduced in order to express Dupree renormalized dispersion relations used in quasi-linear theory. For instance the Dupree renormalized dispersion relation for gyrokinetic, toroidal ion temperature gradient driven (ITG) modes, where the Dupree’s diffusion coefficient is assumed to be a low order polynomial of the velocity, can be written entirely using generalized curvature modified plasma dispersion functions: Knm's. Using those, Dupree’s formulation of renormalized quasi-linear theory is revisited for the toroidal ITG mode. The Dupree diffusion coefficient has been obtained as a function of velocity using an iteration scheme, first by assuming that the diffusion coefficient is constant at each v (i.e. applicable for slow dependence), and then substituting the resulting v dependence in the form of complex polynomial coefficients into the Knm's for verification. The algorithm generally converges rapidly after only a few iterations. Since the quasi-linear calculation relies on an assumed form for the wave-number spectrum, especially around its peak, practical usefulness of the method is to be determined in actual applications. A parameter scan of ηi shows that the form of the diffusion coefficient is better represented by the polynomial form as ηi is increased.

Correlation between zonal flow shearing and entropy transfer rates in toroidal ion temperature gradient turbulence

The transfer rate of the entropy variable in the wavenumber space perpendicular to the equilibrium magnetic field is evaluated by means of the gyrokinetic simulations of the toroidal ion temperature gradient turbulence. Fluctuations of the perturbed gyrocenter distribution function of the linearly unstable mode are transferred to those with higher radial wavenumbers through interactions with zonal flows driven by turbulence. The entropy transfer rate coincides with the zonal flow shearing rate in cases with weak ion temperature gradient where the zonal flows effectively regulate the turbulent transport. The obtained result verifies a correlation between the entropy transfer and the zonal flow shearing processes.

Enhanced stiffness weakening and confinement improvement by the linear electromagnetic effect in relatively high-q and low magnetic-shear plasmas

The linear electromagnetic stabilizing effect on the toroidal ion temperature gradient (ITG) mode is usually believed to be weak when plasma beta is low. It is shown that this effect can be significant even in such a plasma if the safety-factor (q) profile is in the regime with relatively high q and low magnetic-shear values. Furthermore, this effect is found to induce the profile stiffness weakening by reducing the ITG growth rate slope. These results imply that the linear electromagnetic effect can be an important player for the strong confinement improvement or stiffness weakening observed in negative or optimized shear mode, which typically starts to occur from low beta plasmas.

Effect of toroidal rotation on the linear stability of drift-resistive-inertial ballooning modes

The two-fluid drift-resistive-inertial-ballooning mode model [Rafiq et al., Phys. Plasmas 17, 082511 (2010)] developed by Rafiq et al. has been improved with the inclusion of the effects of local toroidal rotation and its shear. Linear fluid simulations have shown that the drift-resistive-inertial ballooning modes (DRIBMs) are destabilized by the toroidal rotation. It has been discovered that the effect of velocity shear on the DRIBMs strongly depends on the type of driving force. For the case of the density gradient driven DRIBMs, the modes can be stabilized by large velocity shear, while for the electron temperature gradient driven DRIBMs (ηe-driven DRIBMs), the velocity shear is shown to have a more effective stabilizing effect. However, in the situation of the ion temperature gradient driven DRIBMs (ηi-driven DRIBMs), the modes are further destabilized by the velocity shear. The behaviors of DRIBMs driven by the coexistence of ion and electron temperature gradients are shown to be similar to the ηe-driven DRIBMs, suggesting that the electron temperature gradient is one kind of strong driving force for DRIBMs. Besides, the toroidal ion temperature gradient modes are shown to be destabilized by the rotation and the velocity shear.

Derivation of the threshold condition for the ion temperature gradient mode with an inverted density profile from a simple physics picture

We show that the threshold condition for the toroidal ion temperature gradient (ITG) mode with an inverted density profile can be derived from a simple physics argument. The key in this picture is that the density inversion reduces the ion compression due to the ITG mode and the electron drift motion mitigates the poloidal potential build-up. This condition reproduces the same result that has been reported from a linear gyrokinetic calculation [T. S. Hahm and W. M. Tang, Phys. Fluids B 1, 1185 (1989)]. The destabilizing role of trapped electrons in toroidal geometry is easily captured in this picture.

Symmetry breaking induced by the parity change in global electromagnetic ion temperature gradient modes

We perform a computational study of the effects of finite plasma β(= plasma thermal energy/magnetic energy) on the residual stress in the context of the quasi-linear theory. The five-field electromagnetic (EM) toroidal ion temperature gradient (ITG) model is considered in the realistic tokamak geometry. Analyses show that the residual stress significantly increases and its radial profile changes from a dipolar to a unipolar shape with increasing β. The change in the mode parity is found to be responsible for the enhancement and the profile change of the residual stress driven by the global EM toroidal ITG mode. This finding highlights the importance of the global eigenmode structure in determining the radial profile of the residual stress.

Low frequency fully kinetic simulation of the toroidal ion temperature gradient instability

A fully kinetic ion model is useful for the verification of gyrokinetic turbulence simulations in certain regimes, where the gyrokinetic model may break down due to the lack of small ordering parameters. However, for a fully kinetic model to be of value, it must first be able to accurately simulate low frequency drift-type instabilities typically well within the domain of gyrokinetics. Here, a fully kinetic ion model is formulated with weak gradient drive terms and applied to the toroidal ion-temperature-gradient (ITG) instability for the first time. Implementation in toroidal geometry is discussed, where orthogonal coordinates are used for particle dynamics, but field-line-following coordinates are used for the field equation allowing for high resolution of the field-aligned mode structure. Variational methods are formulated for integrating the equation of motion allowing for accuracy at a modest time-step size. Linear results are reported for both the slab and toroidal ITG instabilities. Good agreement with full Vlasov and gyrokinetic theory is demonstrated in slab geometry. Good agreement with global gyrokinetic simulation is also shown in toroidal geometry.

Flux tube train model for local turbulence simulation of toroidal plasmas

A new simulation method for local turbulence in toroidal plasmas is developed by extending the conventional idea of the flux tube model. In the new approach, a train of flux tubes is employed, where flux tube simulation boxes are serially connected at each end along a field line so as to preserve a symmetry of the local gyrokinetic equations for image modes in an axisymmetric torus. Validity of the flux tube train model is confirmed against the toroidal ion temperature gradient turbulence for a case with a long parallel correlation of fluctuations, demonstrating numerical advantages over the conventional method in the time step size and the symmetry-preserving property.

Theoretical issues on the spontaneous rotation of axisymmetric plasmas

An extensive series of experiments have confirmed that the observed ‘spontaneous rotation’ phenomenon in axisymmetric plasmas is related to the confinement properties of these plasmas and connected to the excitation of collective modes associated with these properties (Coppi 2000 18th IAEA Fusion Energy Conf. (Sorrento, Italy, 2000) THP 1/17, www-pub.iaea.org/MTCD/publications/PDF/csp_008c/html/node343.htm and Coppi 2002 Nucl. Fusion 42 1). In particular, radially localized modes can extract angular momentum from the plasma column from which they grow while the background plasma has to recoil in the direction opposite to that of the mode phase velocity. In the case of the excitation of the plasma modes at the edge, the loss of their angular momentum can be connected to the directed particle ejection to the surrounding medium. The recoil angular momentum is then redistributed inside the plasma column mainly by the combination of an effective viscous diffusion and an inward angular momentum transport velocity that is connected, for instance, to ion temperature gradient (ITG) driven modes. The linear and quasi-linear theories of the collisionless trapped electron modes and of the toroidal ITG driven modes are re-examined in the context of their influence on angular momentum transport. Internal modes that produce magnetic reconnection and are electromagnetic in nature, acquire characteristic phase velocity directions in high temperature regimes and become relevant to the ‘generation’ of angular momentum. The drift-tearing mode, the ‘complex’ reconnecting mode and the m0 = 1 internal mode belong to this category, the last mode acquiring different features depending on the strength of its driving factor. Toroidal velocity profiles that reproduce the experimental observations are obtained considering a global angular momentum balance equation that includes the localized sources associated with the excited internal electrostatic and electromagnetic modes besides the appropriate diffusive and the inward angular momentum transparent terms.

Numerical computation of the modified plasma dispersion function with curvature

A particular generalization of the plasma dispersion function, which is linked to the regular plasma dispersion function via recurrence relations is discussed. The generalization allows a fast numerical implementation of a certain two-dimensional integral that appears in the description of the plasma dispersion in curved geometry, by reducing it to a single integral over a function involving the generalized plasma dispersion function. The local dielectric function of the toroidal ion temperature gradient driven mode can be written in terms of these integral functions. A matrix method is proposed to combine the consecutive integrals as a single 1D integral over a single integrand. The method allows two orders of magnitude speed up over the 2D integral implementation. Using various optimizations and an efficient implementation of the regular plasma dispersion function, further speed up is obtained.

Ion temperature gradient instability at sub-Larmor radius scales with non-zero ballooning angle

Linear gyro-kinetic stability calculations predict unstable toroidal ion temperature gradient modes (ITGs) with normalised poloidal wave vectors well above one (kθρi>1) for standard tokamak parameters with adiabatic electron response. These modes have a maximum amplitude at a poloidal angle θ that is shifted away from the low field side (θ≠0). The physical mechanism is clarified through the use of a fluid model. It is shown that the shift of the mode away from the low field side reduces the effective drift frequency which allows for the instability to develop. Numerical tests using the gyro-kinetic model confirm this physical mechanism. Furthermore, it is shown that modes localized away from the low field side can be important also for kθρi<1 close to the threshold of the ITG. In fact, modes with maximum amplitude at θ≠0 can exist for normalised temperature gradient lengths below the threshold of the ITG obtained for the case with the maximum at θ=0.