Quasi-linear Model Research Articles

A quasi-linear model of particle, momentum and heat transport in a stochastic magnetic field

We present a quasi-linear treatment of the drift-kinetic equation in the presence of a stochastic magnetic field, which provides a self-contained description of particle, parallel momentum and heat transport. Explicit analytical expressions, which satisfy the Onsager reciprocal relations, are obtained by approximating the distribution function by a local shifted Maxwellian. This theory completes previous formulations (Harvey et al., Phys. Rev. Lett., vol. 47, 1981, p. 102) by including the momentum transport and by generalizing the derivation from the cylindrical tokamak configuration to an arbitrary cylindrical pinch. Application to the reversed field pinch provides satisfactory results.

Application of linear electron Bernstein current drive models in reactor-relevant spherical tokamaks

Electron Bernstein current drive (EBCD) systems in spherical tokamaks are sensitive to plasma and launch conditions, and therefore require large parametric scans to optimise their design. One particular bottleneck in the simulation workflow is quasilinear modelling of current drive efficiency. Linear adjoint models are an attractive alternative, offering a ∼103 × speed-up compared to quasilinear codes. While linear models are well-tested and commonly used for electron cyclotron current drive (ECCD), they have seen little use in EBCD modelling. In this work, variants of the linear model are applied to EBCD and compared to quasilinear results in a reactor-relevant plasma, i.e. Spherical Tokamak for Energy Production (STEP). This comparison reveals it is important to accurately model the collision operator and finite Larmor radius effects in the linear model. When done properly, good agreement is found with quasilinear calculations, at least for normalised minor radii ρ < 0.7 and at low power densities. The power density threshold for quasilinear effects during EBCD is found to be significantly lower than that of ECCD. This is attributed to the much lower group velocity of the electron Bernstein wave (EBW). Thus, the linear model is only valid for EBCD modelling at low power densities (e.g. ≲ 1 MW launched EBW power in STEP). This may be satisfied in present-day experimental devices, but certainly not in reactors targeting non-inductive operation.

Core transport modelling of the DTT full power scenario using different fuelling strategies

A theory-based integrated modelling work of plasma response to deuterium fuelling in the new Divertor Tokamak Test facility (DTT) is performed, using the 1.5D transport code JETTO with the quasi-linear anomalous transport model QuaLiKiz for the core region. The full power DTT scenario E1 is investigated. It is characterised by 28.8 MW of Electron Cyclotron Resonance Heating, 10 MW of Neutral Beam Injection and 6 MW of Ion Cyclotron Resonance Heating to the plasma. Plasma density and temperature profile evolution is calculated up to the separatrix using two different fuelling methods, gas puffing and pellet injection, and two different seeding gases, argon and neon. To sustain the desired pedestal density level with gas puffing a big amount of neutral flux at the separatrix is needed. The feasibility limits of the pumping system are exceeded, regardless of the type of impurity introduced, thus making the use of pellets mandatory. The simulations performed with pellet injection as fuelling method predict that the pedestal density is well sustained with realistic parameters foreseen for the DTT pellet injector. Strong dependence of the core density on the electron cyclotron resonance (ECR) power deposition profile is found. Trapped Electron Modes dominance, low outward flux and strongly hollow density in the inner core region are foreseen with central peaked ECR power deposition profile. Ion Temperature Gradient modes dominance, inward flux and robust density sustainment on the whole radial interval are predicted for spread ECR power deposition, though with central density close to the ECR cut-off limit and with peaked impurity densities. An intermediate deposition extension is found to sustain the whole density profile and to obtain flatter core densities, as previously predicted for the reference full power DTT scenario by fixed pedestal simulations. The ECR deposition is negligibly modified by refraction changes both during a single pellet cycle and after several pellet cycles, indicating full compatibility between the ECR system and the pellet injection system.

Expanding-box Quasilinear Model of the Solar Wind

The expanding-box model of the solar wind has been adopted in the literature within the context of magnetohydrodynamics, hybrid, and full particle-in-cell simulations to investigate the dynamic evolution of the solar wind. The present paper extends such a method to the framework of self-consistent quasilinear kinetic theory. It is shown that the expanding-box quasilinear methodology is largely equivalent to the inhomogeneous steady-state quasilinear model discussed earlier in the literature, but a distinction regarding the description of wave dynamics between the two approaches is also found. The expanding-box quasilinear formalism is further extended to include the effects of a spiraling solar-wind magnetic field as well as collisional age effects. The present finding shows that the expanding-box quasilinear approach and the steady-state global-kinetic models may be employed interchangeably in order to address other more complex problems associated with the solar-wind dynamics.

A guidance and control design with reduced information for a dual-spin stabilized projectile

In this paper, an integrated guidance and control method based on an adaptive path-following controller is proposed to control a spin-stabilized projectile with only translational motion information under the constraint of an actuator, uncertainties in aerodynamic parameters and measurements, and control system complexity. Owing to the fairly high rotation speed, the dynamic model of this missile is strongly nonlinear, uncertain and coupled in pitch, yaw and roll channels.A theoretical equivalent resultant force and uncertainty compensation method are comprehensively used to realize decoupling of pitch and yaw. In response to the strong nonlinear and time-varying characteristics of the dynamic system, the quasi-linear model whose parameters are obtained by interpolation of points selected as the segmentation points in the trajectory envelope, is used for calculation in each step. To cope with the system uncertainty caused by model approximation, parameter uncertainty and ballistic interference, an extended state estimator is used to compensate the output feedback according to the test ballistic angle. In order to improve the tracking efficiency and ensure the tracking error convergence with only translational motion information, the virtual guide point, whose derivative is deduced according to the Lyapunov principle, is calculated in real time according to the projection relationship between the real-time position and the reference trajectory, and a virtual line-of-sight angle and the backstepping method are used for the design of the guidance and control system. In order to avoid the influence of control input saturation on the guidance and control performance due to the actuator limitation and improve the robustness of the system, an anti-saturation compensator is designed according to the two-step method.The feasibility and effectiveness of the path-following controller is verified through closed-loop flight simulations with measurement, control, and condition uncertainties. The results indicate that the designed controller can converge to the reference path and evidently decrease the distance between the impact point and target under different uncertainties.

Nonlinear mode coupling and energetics of driven magnetized shear-flow turbulence

To comprehensively understand the saturation of two-dimensional (2D) magnetized Kelvin–Helmholtz-instability-driven turbulence, energy transfer analysis is extended from the traditional interaction between scales to include eigenmode interactions, by using the nonlinear couplings of linear eigenmodes of the ideal instability. While both kinetic and magnetic energies cascade to small scales, a significant fraction of turbulent energy deposited by unstable modes in the fluctuation spectrum is shown to be re-routed to the conjugate-stable modes at the instability scale. They remove energy from the forward cascade at its inception. The remaining cascading energy flux is shown to attenuate exponentially at a small scale, dictated by the large-scale stable modes. Guided by a widely used instability-saturation assumption, a general quasi-linear model of instability is tested by retaining all nonlinear interactions except those that couple to the large-scale stable modes. These complex interactions are analytically removed from the magnetohydrodynamic equations using a novel technique. Observations are an explosive large-scale vortex separation instead of the well-known merger of 2D, a dramatic enhancement in turbulence level and spectral energy fluxes, and a reduced small-scale dissipation length scale. These show the critical role of the stable modes in instability saturation. Possible reduced-order turbulence models are proposed for fusion and astrophysical plasmas, based on eigenmode-expanded energy transfer analyses.

A conservative Galerkin solver for the quasilinear diffusion model in magnetized plasmas

The quasilinear theory describes the resonant interaction between particles and waves with two coupled equations: one for the evolution of the particle probability density function (pdf), the other for the wave spectral energy density (sed). In this paper, we propose a conservative Galerkin scheme for the quasilinear model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry.We construct an unconditionally conservative weak form, and propose a discretization that preserves the unconditional conservation property, by “unconditional” we mean that conservation is independent of the singular transition probability. The discrete operators, combined with a consistent quadrature rule, will preserve all the conservation laws rigorously. The technique we propose is quite general: it works for both relativistic and non-relativistic systems, for both magnetized and unmagnetized plasmas, and even for problems with time-dependent dispersion relations.We represent the particle pdf by continuous basis functions, and use discontinuous basis functions for the wave sed, thus enabling the application of a positivity-preserving technique. The marching simplex algorithm, which was initially designed for computer graphics, is adopted for numerical integration on the resonance manifold. We introduce a semi-implicit time discretization, and discuss the stability condition. In addition, we present numerical examples with a “bump on tail” initial configuration, showing that the particle-wave interaction results in a strong anisotropic diffusion effect on the particle pdf.

Tensor decomposition for modified quasi-linear viscoelastic models: Towards a fully non-linear theory

We discuss the decomposition of the tensorial relaxation function for isotropic and transversely isotropic (TI) modified quasi-linear viscoelastic (MQLV) models. We show how to formulate the constitutive equation using a convenient decomposition of the relaxation tensor into scalar components and tensorial bases. We show that the bases must be symmetrically additive, i.e., they must sum up to the symmetric fourth-order identity tensor. This is a fundamental property both for isotropic and anisotropic bases that ensures the constitutive equation is consistent with the elastic limit. We provide two robust methods to obtain such bases. Furthermore, we show that, in the TI case, the bases are naturally deformation-dependent for deformation modes that induce rotation or stretching of the fibres. Therefore, the MQLV framework allows to capture the non-linear phenomenon of strain-dependent relaxation, which has always been a criticised limitation of the original quasi-linear viscoelastic theory. We illustrate this intrinsic non-linear feature, unique to the MQLV model, with two examples (uni-axial extension and perpendicular shear).

A finite porous-viscoelastic model capturing mechanical behavior of human cervix under multi-step spherical indentation

The cervix is a soft tissue exhibiting time-dependent behavior under mechanical loads. The cervix is a vital mechanical barrier to protect the growing fetus. The remodeling of the cervical tissue, characterized by an increase in time-dependent material properties, is necessary for a safe parturition. The failure of its mechanical function and accelerated tissue remodeling is hypothesized to lead to preterm birth, which is birth before 37 weeks of gestation. To understand the mechanism of the time-dependent behavior of the cervix under compressive states, we employ a porous-viscoelastic material model to describe a set of spherical indentation tests performed on nonpregnant and term pregnant tissue. A genetic algorithm-based inverse finite element analysis is used to fit the force-relaxation data by optimizing the material parameters, and the statistical analysis of the optimized material parameters is conducted on different sample groups. The force response is captured well using the porous-viscoelastic model. The indentation force-relaxation of the cervix is explained by the porous effects and the intrinsic viscoelastic properties of the extracellular matrix (ECM) microstructure. The hydraulic permeability obtained from the inverse finite element analysis agrees with the trend of the value directly measured previously by our group. The nonpregnant samples are found significantly more permeable than the pregnant samples. Within nonpregnant samples, the posterior internal os is found significantly less permeable than the anterior and posterior external os. The proposed model exhibits the superior capability to capture the force-relaxation response of the cervix under indentation, as compared to the conventional quasi-linear viscoelastic framework (range of r2 of the porous-viscoelastic model 0.88–0.98 vs. quasi-linear model: 0.67–0.89). As a constitutive model with a relatively simple form, the porous-viscoelastic framework has the potential to be used to understand disease mechanisms of premature cervical remodeling, model contact of the cervix with biomedical devices, and interpret force readings from novel in-vivo measurement tools such as an aspiration device.

Constituent-based quasi-linear viscoelasticity: a revised quasi-linear modelling framework to capture nonlinear viscoelasticity in arteries

Arteries exhibit fully nonlinear viscoelastic behaviours (i.e. both elastically and viscously nonlinear). While elastically nonlinear arterial models are well established, effective mathematical descriptions of nonlinear viscoelasticity are lacking. Quasi-linear viscoelasticity (QLV) offers a convenient way to mathematically describe viscoelasticity, but its viscous linearity assumption is unsuitable for whole-wall vascular applications. Conversely, application of fully nonlinear viscoelastic models, involving deformation-dependent viscous parameters, to experimental data is impractical and often reduces to identifying specific solutions for each tested loading condition. The present study aims to address this limitation: By applying QLV theory at the wall constituent rather than at the whole-wall level, the deformation-dependent relative contribution of the constituents allows to capture nonlinear viscoelasticity with a unique set of deformation-independent model parameters. Five murine common carotid arteries were subjected to a protocol of quasi-static and harmonic, pseudo-physiological biaxial loading conditions to characterise their viscoelastic behaviour. The arterial wall was modelled as a constrained mixture of an isotropic elastin matrix and four families of collagen fibres. Constituent-based QLV was implemented by assigning different relaxation functions to collagen- and elastin-borne parts of the wall stress. Nonlinearity in viscoelasticity was assessed via the pressure dependency of the dynamic-to-quasi-static stiffness ratio. The experimentally measured ratio increased with pressure, from 1.03 pm 0.03 (mean pm standard deviation) at 80–40 mmHg to 1.58 pm 0.22 at 160–120 mmHg. Constituent-based QLV captured well this trend by attributing the wall viscosity predominantly to collagen fibres, whose recruitment starts at physiological pressures. In conclusion, constituent-based QLV offers a practical and effective solution to model arterial viscoelasticity.

Peer-to-Peer Transactive Energy Trading of a Reconfigurable Multi-Energy Network

This paper proposes a bi-level peer-to-peer (P2P) multi-energy trading framework for a coupled distribution network (DN) and district heating network (DHN). At the lower level, each nodal agent represents its intra-nodal prosumers to optimize the local energy scheduling and P2P trading strategies based on the modified Nash bargaining theory, and a distributed algorithm is then adopted to enable individual agents to make their strategies autonomously only with the sharing of trading information. Once the lower-level P2P bargaining is settled, each agent is required to submit its nodal net loads and trading adjustment tolerances to the network operators. At the upper level, the network operators minimize the line power losses while satisfying network operation constraints by reconfiguring the DN and DHN as well as enforcing necessary trading adjustments from the lower-level agents when the network violations incurred by the P2P trading cannot be fully solved by network reconfiguration. Mathematically, the DN operation is modelled based on the linearized DistFlow with a set of new radiality constraints to sufficiently ensure the radial structure of the DN. The DHN operation is formulated as a quasi-linear thermal flow model independent of mass flow rate and water temperature, by which the computation complexity and limitations associated with traditional DHN formulations are addressed. Finally, a multi-energy network consisting of an IEEE 33-bus DN and a 23-node DHN is used to demonstrate the effectiveness of the proposed P2P trading framework and the efficiency of the solution algorithms.

Mathematical modeling of complex heat transfer in the context of the endovenous laser ablation

An initial-boundary value problem for quasi-linear model of complex heat exchange (where the conductive and radiative contributions are simultaneously taken into account), describing the mechanism of endovenous laser ablation, is studied. The existence and uniqueness of a solution is proved. An iterative procedure to find a solution is constructed and its convergence is established. The efficiency of the iterative procedure is illustrated by conducted numerical simulations.

A novel regression method: Partial least distance square regression methodology

Partial Least Squares (PLS) is a multivariate linear statistical analysis method integrating principal component analysis (PCA), canonical correlation analysis (CCA), and multiple linear regression (MLR) analysis. It can effectively solve the problems of multiple correlation between variables and small sample size. However, PLS also has some imperfections, on the one hand, PLS extracts the latent variables based on the maximization of Pearson correlation coefficient of independent and dependent variables. Here, Pearson correlation coefficient usually cannot measure the nonlinear relationship between variables, so the latent variables cannot ensure the strongest interpretation. On the other hand, PLS applies MLR to the extracted latent variables, which cannot truly reflect the nonlinear relationship of data, so the regression function is usually under-fitting. The above two imperfect designs are the main reasons for the low regression accuracy and prediction performance of PLS for nonlinear data. According to the above problems, this paper proposes a Partial Least Distance Square (PLDS) regression method. PLDS reflects the original data information through distance variance in PCA, employs distance correlation coefficient to measure the correlation between independent and dependent variables in CCA, and performs quasilinear regression and model solution method to obtain the final regression equation in MLR. In order to verify the effectiveness of the proposed method, the UCI datasets and Chinese medicine datasets with dose-effect relationship was used to compare the new method with other classical regression methods. Finally, the experimental results show that PLSD has better prediction performance regardless of whether there is significant nonlinear relationship between variables. In addition, PLDS has high regression accuracy and low computation complexity for high-dimensional and small-sample data.

Multimodal convolutional neural networks for predicting evolution of gyrokinetic simulations

AbstractGyrokinetic simulations are required for the quantitative calculation of fluxes due to turbulence, which dominates over other transport mechanisms in tokamaks. However, nonlinear gyrokinetic simulations are computationally expensive. A multimodal convolutional neural network model that reads images and values generated by nonlinear gyrokinetic simulations and predicts electrostatic turbulent heat fluxes was developed to support efficient runs. The model was extended to account for squared electrostatic potential fluctuations, which are proportional to the fluxes in the quasilinear model, as well as images containing fluctuating electron and ion distribution functions and fluctuating electrostatic potentials in wavenumber space. This multimodal model can predict the time and electron and ion turbulent heat fluxes corresponding to the input data. The model trained on the Cyclone base case data successfully predicted times and fluxes not only for its test data, but also for the completely different and unknown JT‐60U case, with high accuracy. The predictive performance of the model depended on the similarity of the linear stability of the case used to train the model to the case being predicted.

Global ‘zero particle flux-driven’ gyrokinetic analysis of the density profile for a TCV plasma

The tokamak `a configuration variable (TCV) is a small-sized tokamak, where finite size effects (often called ‘rho-star’ or ‘global’ effects) could significantly impact the heat and particle fluxes, leading to discrepancies between gyrokinetic flux-tube results and global ones (McMillan et al 2010 Phys. Rev. Lett. 105 155001). The impact of global effects on the radial profile of the plasma density has been investigated in a previous study for a particular TCV discharge with negligible particle source, satisfying the ‘zero particle flux’ (ZPF) condition. A radially local flux-tube analysis, reconstructing the dependence of the peaking of the density profile on the main physical parameters, invoking the ZPF constraint, was pursued close to mid-radius in (Mariani et al 2018 Phys. Plasmas 25 012313). This analysis was followed by a global one (Mariani et al 2019 Plasma Phys. Control. Fusion 61 064005), where local quasi-linear (QL) and nonlinear (NL) results were compared with global simulations, showing small global effects on the density peaking. However, these gradient-driven (GD) global runs considered Krook-type heat and particle sources to keep temperature and density profiles fixed on average, which differ from the experimental radially localized sources. To remove this possible bias on the results, a different evaluation of the density peaking for the same case is performed here, based on global NL hybrid simulations where the temperature profiles are [still] kept fixed with the Krook-type sources, however the density profile relaxes in a flux-driven way (with zero particle source). The new hybrid simulations show a good agreement with the old GD runs. A global QL model is also developed and applied using the output from linear global runs, to estimate ratios of fluxes, showing a good agreement with the flux-tube results of global NL GD simulations. The effect of collisions on the results is also investigated, in order to evaluate their impact on the radial variation of the density peaking.

Experimentally Validated Nonlinear Delayed Differential Approach to Model Borehole Propagation for Directional Drilling

Abstract This paper presents the development and validation of a nonlinear delay differential equation (DDE) model for borehole propagation in the inclination plane. Most importantly, built upon a quasi-linear model, the nonlinear approach incorporates information pertaining to the floating stabilizers and bit tilt saturation by formulating a linear complementarity problem. As a result, the outputs of the nonlinear model were in better agreement with the field data when compared with the quasi-linear model. The maximum modeling error of the nonlinear DDE is less than 1 deg over a drilled depth of 600 feet.

The problem of capturing marginality in model reductions of turbulence

Reduced quasilinear and nonlinear (gradient-driven) models with scale separations, commonly used to interpret experiments and to forecast turbulent transport levels in magnetised plasmas, are tested against nonlinear models without scale separations (flux-driven). Two distinct regimes of turbulence—either above threshold or near marginal stability—are investigated with Boltzmann electrons. The success of reduced models hinges in particular on the reproduction of nonlinear fluxes. Good agreement between models is found above threshold, whilst reduced models significantly underpredict fluxes near marginality, overlooking mesoscale flow organisation and turbulence self-advection. Constructive prescriptions whereby to improve reduced models are discussed.

Random quasi-linear utility

We propose a random quasi-linear utility model (RQUM) where quasi-linear utility functions are drawn randomly via some probability distribution π, and utility ties are broken by a convenient lexicographic rule. We characterize RQUM and identify π uniquely in terms of stochastic choice data. McFadden's (1973) additive random utility model is obtained as a special case where utility ties have a zero probability in all menus. Another distinct case of RQUM captures finite populations and derives π with a finite support. Our main axioms are testable. They prohibit context and reference dependence, and also modify the non-negativity of Block-Marschack polynomials for monetary cost variations. We also characterize RQUM through a stronger version of McFadden and Richter's (1990) axiom of revealed stochastic preferences (ARSP). This approach extends to incomplete datasets.

Theory and phenomenology of stressed wave-dark-matter soliton

Soliton in the hostile turbulent wave dark matter ($\mathrm{\ensuremath{\Psi}}\mathrm{DM}$) halo of a galaxy agitates with various kinds of excitation, and the soliton even breathes heavily under great stress. A theory of collective excitation for a $\mathrm{\ensuremath{\Psi}}\mathrm{DM}$ soliton is presented. The collective excitation has different degrees of coupling to negative energy modes, where lower-order excitation generally necessitates more negative energy coupling. A constrained variational principle is developed to assess the frequencies and mode structures of small-amplitude perturbations. The predicted frequencies are in good agreement with those found in simulations. Soliton breathing at amplitudes on the verge of breakup is also a highlight of this work. Even in this extreme nonlinear regime, the wave function perturbation amplitudes are moderate. The simulation data show a stable oscillation with frequency weakly dependent on the oscillation amplitude and hint at a self-consistent quasilinear model for the wave function that accounts for modifications in the ground-state wave function and the equilibrium density. The mock solution, constructed from the simulation data, can shed light on the dynamics of the large-amplitude breathing soliton and support the quasilinear model, as evidenced by its ability to predict well the nonlinear eigenfrequency shifts and large-amplitude breathing frequency observed in simulations.

Multi-code estimation of DTT edge transport parameters

The main goal of the Divertor Tokamak Test facility (DTT) is to operate with a high value of power-exhaust-relevant parameter P S O L /R in plasma scenarios similar to those foreseen for the Demonstration Fusion Power Plant (DEMO) in terms of low collisionality and neutral opacity. For these unique characteristics, accurate modelling of the principal scenario is necessary for machine designing. In edge numerical codes, cross-field transport profiles have a high impact on modelling results. This work aims at providing a coherent set of transport parameters for DTT full-power (FP) single-null (SN) scenario edge modelling. To evaluate such parameters for DTT, a transport analysis on the current machine has been performed using SOLEDGE2D-EIRENE and SOLPS-ITER. The transport parameters to be used in the simulations of the DTT single-null scenario were selected using two complementary methods. The first is the modelling of JET and Alcator C-Mod (C-Mod) with SOLEDGE2D-EIRENE and SOLPS-ITER, validating transport parameters by comparing modelling results to experimental data from pulses which are considered DTT-relevant . JET pulses were selected with the highest auxiliary input power (from 29 to 33 MW), plasma current and toroidal field to better match DTT parameters; nitrogen and neon seeded pulses were selected to check possible seeding material dependencies. The considered C-Mod pulse better matches DTT plasma density and neutral opacity. Transport parameters are then scaled to DTT according to scaling laws. The second method derives the transport parameters by tuning their values inside the DTT separatrix to reproduce the pedestal profiles predicted by the EPED model via the Europed code and applied in DTT. The derived set of DTT transport parameters is consistent with the results obtained by modelling present machines, reproduces the expected heat flux decay length in detached conditions and, inside the separatrix, reproduces the predicted pedestal using transport parameters which are coherent with what is predicted by the quasi-linear turbulent model QuaLiKiz. • Edge and divertor transport parameters of the DTT single-null scenario were studied. • The edge transport parameters of JET and C-Mod pulses were derived by SOLEDGE2D-EIRENE and SOLPS modelling. • Transport parameters were scaled from present machines to DTT. • First principles core modelling with JETTO confirms the validity of the previous estimations.