Quasi-linear Model Research Articles

Time-dependent full-radius integrated modeling of the DTT tokamak main plasma scenarios

Abstract We use an integrated modeling workflow with the transport code ASTRA coupled with the quasi-linear transport model TGLF-SAT2, the neoclassical model NCLASS, the FACIT model for the neoclassical impurity transport and the IMEP routines for the pedestal calculations, in order to predict the evolution of the plasma profiles for the DTT tokamak main scenarios. The simulations cover the whole confined plasma radius, up to the separatrix, and the time evolution of the plasma including the early phase in limiter configuration, the whole current ramp-up phase in L-mode divertor configuration, the L-H transition and part of the stationary H-mode phase. Six fields are predicted, i.e. the ion and electron temperatures, the electron density, two impurity densities and the plasma current. The simulations indicate that the main DTT scenarios are within the technical capabilities of the machine. They also indicate that the DTT full power, full current, full field scenario will be able to operate in H-mode with a duration of the flat-top phase of the order of \sim30 seconds, and plasma parameters allowing a core-edge integrated study of the power exhaust, which is the main mission of the device. The simulations show also a strong flexibility of the DTT plasmas, that allows DTT to study reactor-relevant conditions unexplored by existing tokamaks.

ℒ2-gain performance of nonlinear sampled-data systems with gain-scheduled control

This paper presents new sufficient conditions for stabilisation of sampled-data nonlinear systems, recast as quasi-linear parameter-varying models. The adopted control strategy considers gain-scheduled state-feedback controllers subject to induced input delay. The L 2 -gain performance is investigated in the sense of Lyapunov. After the proposition of a Lyapunov function and with the application of Wirtinger's inequality, the derived conditions are rewritten in terms of linear matrix inequalities. Aiming at the enlargement of the aperiodic sampling time and at the minimisation of the L 2 -gain, the effectiveness of the developed approach is assessed through numerical simulations.

Nonlinear wave model of towed system and numerical method for its calculation

Towed systems have found wide practical applications. Notable underwater systems for the long-distance extraction of minerals (concretions) from the ocean floor extend over 5-10 km. The existing mathematical models for solving dynamic problems of such systems in various environments are not entirely accurate regarding the diversity of wave processes. This necessitates the development of refined wave models. This article presents a new quasi-linear mathematical model describing the nonlinear four-mode dynamics of a towed system in a spatially inhomogeneous field of mass and surface forces. It is described by a nonlinear system of twelve first-order partial differential equations. The principles of boundedness and hyperbolicity are satisfied. The validation of the two-mode reduction of the model is based on the numerical solution of the problem of the propagation of two waves: longitudinal and configurational. Using a numerical algorithm and a program based on the finite difference method, a comparison of two difference schemes – Crank-Nicolson and Euler – was conducted. The main limitations for applying the finite difference method used for numerical modeling of wave propagation and reflection in a towed system are the peculiarities of the defining quasi-linear equations, which are related to the necessity of simultaneous computation of variables responsible for fast and slow processes. For such systems of equations, the term "singularly perturbed system of equations" is used. These perturbations result from the significant differences in the propagation speeds of longitudinal and configurational waves at the physical level. Consequently, it is necessary to employ special time-stepping regularization and filtering methods for numerical results. This imposes certain restrictions on the ability to model real processes and on the accuracy of the obtained results, necessitating the use of implicit difference schemes and high-frequency filtering.

Large time behavior of solution to a quasilinear chemotaxis model describing tumor angiogenesis with/without logistic source

In this paper, we deal with the following Neumann-initial boundary value problem for a quasilinear chemotaxis model describing tumor angiogenesis: ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+μu−μu2,x∈Ω,t>0,vt=Δv+∇⋅(v∇w)−v+u,x∈Ω,t>0,0=Δw−w+u,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,in a bounded smooth domain Ω⊂Rn(n≤3), where the parameter χ,ξ>0,μ≥0, D(u) is supposed to satisfy the behind property D(u)≥(u+1)αwithα>0.Assume that either μ≥0,α>1 or μ=0,ξ≥λ1∗χ2, where the parameter λ1∗=λ1∗(u0,v0,Ω)>0, then the system admits a global classical solution (u,v,w) by subtle energy estimates. Moreover, when μ=0, it is asserted that the corresponding solution exponentially converges to the constant stationary solution (u0¯,u0¯,u0¯) provided the initial data u0 is sufficiently small, where u0¯=∫Ωu0|Ω|. Finally, when μ>0, it can be proved that the corresponding solution of the system decays to (1,1,1) exponentially for suitable large μ.

A quasilinear hyperbolic one-dimensional model of the lymph flow through a lymphangion with valve dynamics and a contractile wall

This paper presents a one-dimensional model that describes fluid flow in lymphangions, the segments of lymphatic vessels between valves, using quasilinear hyperbolic systems. The model incorporates a phenomenological pressure-cross-sectional area relationship based on existing literature. Numerical solutions of the differential equations align with known results, offering insights into lymphatic flow dynamics. This model enhances the understanding of lymph movement through the lymphatic system, driven by lymphangion contractions.

Overview of multiscale turbulence studies covering ion-to-electron scales in magnetically confined fusion plasma

Turbulent transport in magnetically confined fusion plasma has conventionally been analyzed at the ion gyroradius scale based on the microturbulence theory. However, ion-scale turbulence analysis sometimes fails to predict the turbulent transport flux observed experimentally. Microturbulence at the electron gyroradius scale and cross-scale interactions between disparate-scale turbulences are possible mechanisms to resolve this issue. This overview discusses the recent progress in multiscale turbulence studies and presents future perspectives from recent experimental, theoretical, and numerical investigations. The following aspects are highlighted: (1) the importance of electron-scale effects in experiments, (2) the physical mechanisms of cross-scale interactions, (3) modeling electron-scale effects in quasilinear transport models, and (4) the impacts of cross-scale interactions on burning plasmas. Understanding multiscale turbulence is necessary to improve performance prediction and explore optimal operations for future burning plasmas.

Complete quasilinear model for the acceleration-driven lower hybrid drift instability and a computational assessment of its validity.

A complete quasilinear model is derived for the electrostatic acceleration-driven lower hybrid drift instability in a uniform two-species low-beta plasma in which current is perpendicular to the background magnetic field. The model consists of coupled nonlinear velocity space diffusion equationsfor the volume-averaged ion and electron distribution functions. Each species' diffusion coefficient depends on a time-evolving spectral density of the electric-field energy per unit volume and a time-evolving dispersion relation. The dispersion relation is expressed analytically in integral form without the use of asymptotic limits and applies to arbitrary distribution functions, so long as they can be expressed as a function of one velocity coordinate, e.g., f(v_{y}) or f(v_{⊥}). The quasilinear model conserves energy and is complete in that it fully describes the evolution of the distribution functions, including resonant and nonresonant particle-wave interactions, while accounting for distribution-function-dependent mixed-complex frequencies. The quasilinear diffusion model is solved numerically and self-consistently using a Crank-Nicolson temporal discretization and a second-order finite-volume velocity-space discretization. Numerical solutions are compared to nonlinear fourth-order accurate continuum kinetic Vlasov-Poisson simulations. Evolution of electric-field energy, growth rates, distribution functions, and diffusion coefficients are shown to be in agreement with Vlasov simulations. The quasilinear model is shown to predict anomalous transport terms, like resistivity and heating, to within a factor of order unity. Discrepancies between the quasilinear model and Vlasov simulations are assessed and attributed primarily to lack of damping in the quasilinear description and to the use of unperturbed-orbit susceptibilities in the linear theory dispersion relation. The results illuminate the predictive accuracy of the quasilinear model, place approximate bounds on its validity, and provide much needed vetting of quasilinear theory's ability to predict the nonlinear state of a microturbulent plasma.

ATEP: an advanced transport model for energetic particles

In this paper we report on the implementation and verification of a phase-space resolved energetic particle (EP) transport model. It is based on a first-principle theoretical framework, i.e. the system of non-linear gyrokinetic equations and the related transport equations. Its focus is primarily directed toward understanding the meso-scale character of EPs and its consequences. Compared to the conventional description of thermal radial transport via a one-dimensional radial diffusion equation, the newly developed model is three-dimensional using canonical constants-of-motion (CoM) variables. The model does not assume diffusive processes to be dominant a priori, instead the EP fluxes are self-consistently calculated and directly evolved in CoM space. We use the EP-Stability workflow and the HAGIS code to determine the phase space fluxes explicitly either in the limit of constant mode amplitudes or an energy-conserving quasi-linear model. As an application of the model the transport of neutral-beam-generated EPs due to a toroidal Alfvén eigenmode in an ITER plasma is investigated. As there are no sources and collisions taken into account so far (for an extension of the model see the companion paper (Meng et al 2024 Nucl. Fusion accepted)), the results cannot be considered as an exhaustive study, but rather as a practical demonstration of the conceptual framework on the way to a comprehensive reduced description of burning plasmas.

Successful prediction of tokamak transport in the L-mode regime

A long standing shortfall in the predicted L-mode edge energy transport by reduced quasi-linear models of gyrokinetic turbulent transport has been resolved. The improved model TGLF-SAT2 has higher fidelity to gyrokinetic simulations of the electron-scale contribution to the electron energy transport and the ion-scale flux surface shape dependence of energy transport. The success of TGLF-SAT2 in predicting the L-mode and Ohmic edge profiles is critical to whole pulse simulation and opens the door to prediction of the H-mode power threshold.

Prediction of transport in the JET DTE2 discharges with TGLF and NEO models using the TGYRO transport code

The JET Deuterium-Tritium-Experiment Campaign 2 (DTE2) has demonstrated the highest-ever fusion energy production. To forecast the transport dynamics within these discharges, the TGLF and NEO models within the TGYRO transport code were employed. A critical development in this study is the new quasilinear transport model, TGLF-SAT2, specifically designed to resolve discrepancies identified in JET deuterium discharges. This model accurately describes the saturated three-dimensional (3D) fluctuation spectrum, aligning closely with a database of nonlinear CGYRO turbulence simulations, thereby enhancing the predictive accuracy of TGYRO simulations. In validating against the JET DTE2 discharges across two primary operating scenarios, TGYRO effectively predicted the temperature profiles within a broad radial window (ρ ∼ 0.2–0.85), though with minor ion temperature discrepancies near the core. However, a consistent underprediction of electron density profiles by 20% across the simulation domain was noted, indicating areas for future refinement. To achieve a self-consistent steady-state solution based on the JET DTE2 discharges, an integrated modeling workflow TGYRO-STEP within the OMFIT framework was introduced. This workflow iterates among the core transport, the pedestal pressure and the MHD equilibrium, ultimately yielding a converged solution that significantly reduces dependence on experimental boundary conditions for temperature and density profiles. The integrated simulation results show negligible differences in electron density and temperature profiles compared to standalone TGYRO modeling, while the ion temperature profile is lower due to the updated boundary condition in TGYRO-STEP. The application of the TGYRO-STEP workflow to JET DTE2 discharges serves as a crucial test to validate its robustness and highlights its limitations, providing valuable insights for its potential future application in ITER and Fusion Power Plant deuterium and tritium prediction modeling.

Fast ion relaxation in ITER mediated by Alfvén instabilities

We address the critical issue for future burning plasmas of whether high-energy fusion products or auxiliary heating-beam ions will be confined for a sufficiently long time to compensate for thermal plasma energy losses. This issue can be mitigated by one of the most deleterious collective phenomena—the instability of low, sub-cyclotron frequency Alfvén eigenmodes (AEs), such as toroidicity-induced AEs and reversed-shear AEs in the ITER steady-state scenario. Using a revised quasi-linear (QL) theory applied to energetic particle (EP) relaxation in the presence of AEs, we find that the AE instabilities can affect both neutral beam ions and alpha particles, although the resulting fast ion transport is expected to be modest if classical particle slowing down is assumed. On the other hand, the QL theory predicts that the AE amplitudes will be enhanced by the background microturbulence, although this topic remains outside our scope due to the significant numerical effort required to evaluate these effects. We report our results for EP relaxation dynamics obtained utilizing several tools: (i) a comprehensive linear stability study of the sub-cyclotron Alfvénic spectrum as computed by ideal magnetohydrodynamic NOVA simulations for the AE eigenproblem, (ii) drift kinetic NOVA-C calculations for wave–particle interaction and AE growth/damping rates, and (iii) predictive QL modeling coupled with the global transport code TRANSP to assess the EP relaxation on the equilibrium timescale.

Effect of rapid cooling, frozen storage, and thawing on the passive viscoelastic properties and structure of the rat aorta

Biological tissues decay over time after harvesting, which alters their biomechanical properties. This poses logistical challenges for studies investigating passive arterial biomechanics as tissues need to be characterized shortly after excision. Freezing and cryopreservation methods can help alleviate the need for biomechanical testing of fresh tissue in human ex vivo studies. However, these methods tend to eliminate or reduce arterial cell functionality and affect passive biomechanics. Furthermore, their impact on dynamic arterial biomechanics remains unknown despite arterial viscoelastic properties being an integral component contributing to arterial stiffness under in vivo loading conditions. The present study aims to investigate the impact of rapid cooling and subsequent storage at −80 °C on the passive viscoelastic properties of arterial tissue and aid in ascertaining whether this is a suitable method to delay tissue analysis for studies investigating passive arterial biomechanics. Control and frozen abdominal rat aorta segments were quasi-statically and dynamically tested using a biaxial testing set-up. The results were modeled using a constituent-based quasi-linear viscoelastic modeling framework, yielding directional stiffness parameters, individual constituent biomechanical contributions, and a quantification of viscoelastic stiffening under dynamic pressurization conditions. Frozen samples displayed significantly decreased wall thickness, viscoelastic dissipation, viscoelastic stiffening, and significantly decreased circumferential deformation with changes in luminal pressure. Furthermore, frozen samples displayed significantly increased circumferential stiffness, pulse wave velocity, and collagen load bearing. Consequently, these changes should be considered when utilizing this tissue preservation method to delay biomechanical characterization of rat aortic tissue.

Core performance predictions in projected SPARC first-campaign plasmas with nonlinear CGYRO

This work characterizes the core transport physics of SPARC early-campaign plasmas using the PORTALS-CGYRO framework. Empirical modeling of SPARC plasmas with L-mode confinement indicates an ample window of breakeven (Q > 1) without the need of H-mode operation. Extensive modeling of multi-channel (electron energy, ion energy, and electron particle) flux-matched conditions with the nonlinear CGYRO code for turbulent transport coupled to the macroscopic plasma evolution using PORTALS reveals that the maximum fusion performance to be attained will be highly dependent on the near-edge pressure. Stiff core transport conditions are found, particularly when fusion gain approaches unity, and predicted density peaking is found to be in line with empirical databases of particle source-free H-modes. Impurity optimization is identified as a potential avenue to increase fusion performance while enabling core-edge integration. Extensive validation of the quasilinear TGLF model builds confidence in reduced-model predictions. The implications of projecting L-mode performance to high-performance and burning-plasma devices is discussed, together with the importance of predicting edge conditions.

Energy exchange between electrons and ions in ion temperature gradient turbulence

Microturbulence in magnetic confined plasmas contributes to energy exchange between particles of different species as well as the particle and heat fluxes. Although the effect of turbulent energy exchange has not been considered significant in previous studies, it is anticipated to have a greater impact than collisional energy exchange in low collisional plasmas such as those in future fusion reactors. In this study, gyrokinetic simulations are performed to evaluate the energy exchange due to ion temperature gradient (ITG) turbulence in a tokamak configuration. The energy exchange due to the ITG turbulence mainly consists of the cooling of ions in the ∇B-curvature drift motion and the heating of electrons streaming along a field line. It is found that the ITG turbulence transfers energy from ions to electrons regardless of whether the ions or electrons are hotter, which is in marked contrast to the energy transfer by Coulomb collisions. This implies that the ITG turbulence should be suppressed from the viewpoint of sustaining the high ion temperature required for fusion reactions since it prevents energy transfer from alpha-heated electrons to ions as well as enhancing ion heat transport toward the outside of the reactor. Furthermore, linear and nonlinear simulation analyses confirm the feasibility of quasilinear modeling for predicting the turbulent energy exchange in addition to the particle and heat fluxes.

Experiments and modelling of negative triangularity ASDEX Upgrade plasmas in view of DTT scenarios

The paper presents experimental and modelling results of a comparison of negative (NT) and positive (PT) triangularity ASDEX Upgrade (AUG) discharges using the plasma shapes presently foreseen in the DTT tokamak, under construction in Italy. This work is part of a broader effort of investigation to understand whether the good properties observed in NT scenarios in DIII-D and TCV may be extrapolated to the DTT device and more generally to DEMO future operations. The experimental results have shown a practical gain of running these AUG plasmas with only ECRH and mixed NBI+ECRH phases in negative triangularity, even if they access the H-mode. Indeed, the NT electron kinetic profiles recover in all cases the PT electron pressures inside mid-radius due to reduced transport in the region ρtor=0.7−0.9 , while exhibiting lower individual ELM (Edge Localised Mode) energy losses. The ion pressure and expected fusion performance are comparable in the case of similar densities. Integrated modelling has been performed using the transport solver ASTRA and the quasi-linear turbulent model TGLF, investigating the transport properties of these discharges. The modelling reproduces the experiments qualitatively with reasonable accuracy. Nonetheless, the heat transport in NT cases is partially overestimated. This may be because TGLF uses the Miller equilibrium, which approximates the magnetic flux surfaces as up-down symmetric. In the caseof these asymmetric NT shapes, the simulated outer surfaces lose part of the tilt with respect to the z-axis, reducing the upper δ < 0 effect. A numerical test to discern the impact of the geometry by symmetrically flipping the shape has shown a beneficial effect of the negative triangularity on heat transport.

Recovering Quasi-Biennial Oscillations from Chaos

Abstract The Quasi-Biennial Oscillation (QBO) is understood to result from wave-mean-flow interactions, but the reasons for its relative stability remain a subject of ongoing debate. In addition, consensus has yet to be reached regarding the respective roles of different equatorial wave types in shaping the QBO’s characteristics. Here we employ Holton-Lindzen-Plumb’s quasilinear model to shed light on the robustness of periodic behavior in the presence of multiple wave forcings. A comprehensive examination of the various dynamical regimes in this model reveals that increased vertical wave propagation at higher altitudes favors periodicity. In the case of single standing wave forcing, enhanced vertical propagation is controlled by the wave attenuation length scale. The occurrence of non-periodic states at high forcing amplitudes is explained by the excitation of high vertical unstable modes. Increasing the attenuation length scale prevents the emergence of such modes. When multiple wave forcing is considered, the mean flow generated by a dominant primary wave facilitates greater vertical propagation of a perturbation wave. Raising the altitude where most of the wave damping occurs favors periodicity by preventing the development of secondary jets responsible for the aperiodic behavior. This mechanism underscores the potential role of internal gravity waves in supporting the periodicity of a QBO primarily driven by planetary waves.

К выбору периода дискретизации нелинейных гибридных систем управления

Hybrid systems are often used in various technical applications, such as robotics, aviation, space, energy, etc. The emergence of hybrid control systems is due to the use of computers to implement control laws, including nonlinear ones. Digital means cannot implement continuous control laws. However, the known methods of design, especially of nonlinear systems, lead precisely to continuous controls, which caused the need to discretize the continuous control with the largest possible period. Solving the problem of determining the maximum permissible sampling period of a nonlinear control system is a rather complex stage of its creation. In this article the problem of definition of the maximum allowed sample period of the nonlinear hybrid system control and its dependence on the module of the maximum eigenvalue of the functional matrix of quasilinear model of the continuous system is also considered. The nonlinear hybrid system is created by discretizing the control of the nonlinear continuous system. This continuous system is synthesized using the algebraic polynomial and matrix method of the nonlinear control systems design in which quasilinear models are used. It has been is established that the value of the maximum permissible sample period depends not only on the module of the maximum eigenvalue, but also on initial conditions and external influences. These dependences are complex and it is difficult to find them theoretically. Experimentally, based on the example of the specific nonlinear hybrid control system it is shown that eigenvalues smaller on the module lead to larger values of the maximum of the permissible sample period. The problem of choosing the sample period of control laws of the real hybrid control systems can be solved in the similar way based on a compromise between the system high-speed performance and the sample period.

Role of Microtubule Network in the Passive Anisotropic Viscoelasticity of Healthy Right Ventricle.

Cardiomyocytes are viscoelastic and key determinants of right ventricle (RV) mechanics. Intracellularly, microtubules are found to impact the viscoelasticity of isolated cardiomyocytes or trabeculae; whether they contribute to the tissue-level viscoelasticity is unknown. Our goal was to reveal the role of the microtubule network in the passive anisotropic viscoelasticity of the healthy RV. Equibiaxial stress relaxation tests were conducted in healthy RV free wall (RVFW) under early (6%) and end (15%) diastolic strain levels, and at sub- and physiological stretch rates. The viscoelasticity was assessed at baseline and after the removal of microtubule network. Furthermore, a quasi-linear viscoelastic (QLV) model was applied to delineate the contribution of microtubules to the relaxation behavior of RVFW. After removing the microtubule network, RVFW elasticity and viscosity were reduced at the early diastolic strain level and in both directions. The reduction in elasticity was stronger in the longitudinal direction, whereas the degree of changes in viscosity were equivalent between directions. There was insignificant change in RVFW viscoelasticity at late diastolic strain level. Finally, the modeling showed that the tissue's relaxation strength was reduced by the removal of the microtubule network, but the change was present only at a later time scale. These new findings suggest a critical role of cytoskeleton filaments in RVFW passive mechanics in physiological conditions.

Generalized Nash equilibrium problems with mixed-integer variables

AbstractWe consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido–Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

Comment on: “Terahertz rectangular waveguides with inserted graphene films biased by light and their quasi-linear electromagnetic modeling” [J. Comput. Electron. 20 (2021) 169

Comment on: “Terahertz rectangular waveguides with inserted graphene films biased by light and their quasi-linear electromagnetic modeling” [J. Comput. Electron. 20 (2021) 169