Finite Beta Research Articles

On the sausage magnetohydrodynamic waves in magnetic flux tubes: finite plasma beta and phase mixing

Abstract Over the past twenty years, there has been increasing evidence of the existence of sausage waves in the solar atmosphere. These observations make them useful tools in the context of atmospheric seismology. Here, we study sausage magnetohydrodynamic waves in a magnetic flux tube of non-zero plasma beta with a circular cross-section and a radially inhomogeneous plasma density. Solving numerically the equations of motion for an initial value problem, the spatio-temporal evolution of the velocity perturbations is obtained for different sets of parameters. We show that the ratio of the amplitudes of the longitudinal and radial perturbations is determined by the amount of plasma beta. Additionally, the longitudinal component of the velocity perturbation experiences phase-mixing within a layer surrounding the boundary of the flux tube with a rate depending on the amount of plasma beta. The results revealed that in the presence of a non-zero plasma beta, the flux tube exhibits oscillations in both the radial and longitudinal directions, characterized by a combination of two frequencies: one belonging to the slow continuum and the other to the Alfvén continuum. Also, the period of radial oscillation is obtained for different sets of parameters. The dependence of the period of the radial oscillation on the wavenumber confirms the results obtained in previous studies.

Generalized kinetic equation for tokamak plasma equilibrium distribution function

A generalized kinetic equation for the equilibrium distribution function in a finite beta, arbitrary tokamak plasma is derived. The equation is correct to second order in ρ/L (ρ is the particle Larmor radius and L is the system size). Resolving the finite Larmor radius length scales with no restriction on the ratio of poloidal to total equilibrium magnetic field, Bϑ/B, it generalizes the drift kinetic theory of Hazeltine [Phys. Plasmas 15, 77 (1973)] to the limit of Bϑ/B∼1 (e.g., to ensure validity for spherical tokamaks). Two cases are considered. The first provides the equilibrium distribution function, consistent with the generalized gyrokinetic formalism of Dudkovskaia et al. [Plasma Phys. Controlled Fusion 65, 045010 (2023)], derived specifically to capture neoclassical equilibrium currents in the gyrokinetic stability analyses in strong gradient regions. The second assumes short length scales in the direction perpendicular to the magnetic field, which can occur as a result of small coherent magnetic structures in the plasma, such as neoclassical tearing mode magnetic islands close to threshold. This then extends the drift island equations of Dudkovskaia et al. [Nucl. Fusion 63, 016020 (2023)] for the plasma response to magnetic islands to a spherical tokamak plasma configuration. Resolving ρ∼ρϑ (or Bϑ∼B), where ρϑ is the particle poloidal Larmor radius, is also expected to influence calculations of the magnetic island propagation frequency and the associated contributions to the island onset conditions.

Drift kinetic theory of neoclassical tearing modes in tokamak plasmas: polarisation current and its effect on magnetic island threshold physics

A nonlinear 4-dimensional drift island theory derived in (Imada et al 2019 Nucl. Fusion 59 046016 and references therein) provides qualitative predictions of the plasma response to a stationary neoclassical tearing mode (NTM) magnetic island in a low beta, large aspect ratio tokamak plasma. (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) refines a model for the magnetic drift frequency and exploits the limit of rare collisions, reducing this theory to 3-dimensional and thus providing a more accurate treatment of the trapped-passing boundary layer. The drift island theory is further improved in (Dudkovskaia et al 2023 Nucl. Fusion 63 016020) by introducing plasma shaping and finite beta effects. In the present paper, an improved model is adopted to resolve the drift island separatrix boundary layer, allowing one to investigate the polarisation current contribution that exists around the magnetic island separatrix, including in the presence of the background electric field. In particular, different magnetic topologies from both sides of the separatrix generate a radial discontinuity in the distribution function gradient there, when collisions are neglected. Allowing for collisional dissipation in the leading order distribution function around the separatrix resolves this discontinuity, smoothing the density distribution. The overall effect of the polarisation current on the NTM threshold is then combined from the outer contributions that exist outside the layer, as well as the separatrix layer piece, and self-consistently accounts for the electrostatic potential reconstructed from plasma quasi-neutrality. The corresponding NTM threshold is quantified and compared with previous predictions of (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001, Dudkovskaia et al 2023 Nucl. Fusion 63 016020).

Physics basis for the Wisconsin HTS Axisymmetric Mirror (WHAM)

The Wisconsin high-temperature superconductor axisymmetric mirror experiment (WHAM) will be a high-field platform for prototyping technologies, validating interchange stabilization techniques and benchmarking numerical code performance, enabling the next step up to reactor parameters. A detailed overview of the experimental apparatus and its various subsystems is presented. WHAM will use electron cyclotron heating to ionize and build a dense target plasma for neutral beam injection of fast ions, stabilized by edge-biased sheared flow. At 25 keV injection energies, charge exchange dominates over impact ionization and limits the effectiveness of neutral beam injection fuelling. This paper outlines an iterative technique for self-consistently predicting the neutral beam driven anisotropic ion distribution and its role in the finite beta equilibrium. Beginning with recent work by Egedalet al.(Nucl. Fusion, vol. 62, no. 12, 2022, p. 126053) on the WHAM geometry, we detail how the FIDASIM code is used to model the charge exchange sources and sinks in the distribution function, and both are combined with an anisotropic magnetohydrodynamic equilibrium solver method to self-consistently reach an equilibrium. We compare this with recent results using the CQL3D code adapted for the mirror geometry, which includes the high-harmonic fast wave heating of fast ions.

Comparison of combination methods to create calibrated ensemble forecasts for seasonal influenza in the U.S.

The characteristics of influenza seasons vary substantially from year to year, posing challenges for public health preparation and response. Influenza forecasting is used to inform seasonal outbreak response, which can in turn potentially reduce the impact of an epidemic. The United States Centers for Disease Control and Prevention, in collaboration with external researchers, has run an annual prospective influenza forecasting exercise, known as the FluSight challenge. Uniting theoretical results from the forecasting literature with domain-specific forecasts from influenza outbreaks, we applied parametric forecast combination methods that simultaneously optimize model weights and calibrate the ensemble via a beta transformation and made adjustments to the methods to reduce their complexity. We used the beta-transformed linear pool, the finite beta mixture model, and their equal weight adaptations to produce ensemble forecasts retrospectively for the 2016/2017, 2017/2018, and 2018/2019 influenza seasons in the U.S. We compared their performance to methods that were used in the FluSight challenge to produce the FluSight Network ensemble, namely the equally weighted linear pool and the linear pool. Ensemble forecasts produced from methods with a beta transformation were shown to outperform those from the equally weighted linear pool and the linear pool for all week-ahead targets across in the test seasons based on average log scores. We observed improvements in overall accuracy despite the beta-transformed linear pool or beta mixture methods' modest under-prediction across all targets and seasons. Combination techniques that explicitly adjust for known calibration issues in linear pooling should be considered to improve probabilistic scores in outbreak settings.

Estimating hepatitis B virus prevalence among key population groups for European Union and European Economic Area countries and the United Kingdom: a modelling study

BackgroundHepatitis B virus (HBV) epidemiology in Europe differs by region and population risk group, and data are often incomplete. We estimated chronic HBV prevalence as measured by surface antigen (HBsAg) among general and key population groups for each country in the European Union, European Economic Area and the United Kingdom (EU/EEA/UK), including where data are currently unavailable.MethodsWe combined data from a 2018 systematic review (updated in 2021), data gathered directly by the European Centre for Disease Control (ECDC) from EU/EEA countries and the UK and further country-level data. We included data on adults from the general population, pregnant women, first time blood donors (FTBD), men who have sex with men (MSM), prisoners, people who inject drugs (PWID), and migrants from 2001 to 2021, with three exceptions made for pre-2001 estimates. Finite Mixture Models (FMM) and Beta regression were used to predict country and population group HBsAg prevalence. A separate multiplier method was used to estimate HBsAg prevalence among the migrant populations within each country, due to biases in the data available.ResultsThere were 595 included studies from 31 countries (N = 41,955,969 people): 66 were among the general population (mean prevalence (stackrel{-}{p}) 1.3% [range: 0.0-7.6%]), 52 among pregnant women (stackrel{-}{p}=1.1% [0.1–5.3%]), 315 among FTBD (stackrel{-}{p}=0.3% [0.0-6.2%]), 20 among MSM (stackrel{-}{p}=1.7% [0.0-11.2%]), 34 among PWID (stackrel{-}{p}=3.9% [0.0-16.9%]), 24 among prisoners (stackrel{-}{p}=2.9% [0.0-10.7%]), and 84 among migrants (stackrel{-}{p}=7.0% [0.2–37.3%]). The FMM grouped countries into 3 classes. We estimated HBsAg prevalence among the general population to be < 1% in 24/31 countries, although it was higher in 7 Eastern/Southern European countries. HBsAg prevalence among each population group was higher in most Eastern/Southern European than Western/Northern European countries, whilst prevalence among PWID and prisoners was estimated at > 1% for most countries. Portugal had the highest estimated prevalence of HBsAg among migrants (5.0%), with the other highest prevalences mostly seen in Southern Europe.ConclusionsWe estimated HBV prevalence for each population group within each EU/EAA country and the UK, with general population HBV prevalence to be < 1% in most countries. Further evidence is required on the HBsAg prevalence of high-risk populations for future evidence synthesis.

Modeling of plasma beta effects on the island divertor transport in the standard configuration of W7-X

The influence of plasma beta effects on the edge plasma transport in the Wendelstein 7-X standard configuration is studied systematically by using EMC3-EIRENE combined with a 3D equilibrium code named HINT. The magnetic topology changes induced by plasma beta effects are significantly reflected in plasma transport behaviors and heat flux patterns on divertor targets. After validating the modeling strategy by comparisons with experimental data, the extended simulations for high performance plasmas show that the threshold separatrix density for accessing the power detachment is reduced in higher beta plasmas. Compared with the vacuum field case, the impurity radiation distributions with finite beta effects are modified in the magnetic island region. The divertor heat flux is distributed more evenly along the toroidal direction on the strike line at the vertical target. The strike line on the horizontal target moves towards the pumping gap with an increase in the plasma beta. In addition, the different pressure profiles with the same central beta also result in a modified heat flux pattern on the divertor targets.

A statistical model approach based on the Gaussian Mixture Model for the diagnosis and classification of bone fractures

ABSTRACT Medical imaging has significantly influenced the field of image processing in recent years. Several advancements have been made in medical imaging that is now being used, and all of them are geared at making disease diagnosis more accurate. Researchers recently explored how to diagnose, define, and classify bone fractures, and numerous approaches have been proposed. However, a uniform categorization has yet to be developed for all detected fractures. This article focuses on the methods for identifying realignments among the bone structures utilizing model-based approaches, and this methodology is highlighted here. The performance metrics are used to assess the findings, and the constructed model has shown substantial results in binary classification and multi-classification of bone fractures. The model that has been suggested is called the Finite Beta Gaussian Mixture Model (FBGMM), and its performance may be evaluated with the use of a confusion matrix. This project intends to construct an image processing system that can identify bone fractures promptly and reliably by including data from X-rays. FGBMM achieves good accuracy in both binary and multiclassification of fractures.

Drift kinetic theory of the NTM magnetic islands in a finite beta general geometry tokamak plasma

In (Imada et al 2019 Nucl. Fusion 59 046016 and references therein) a new 4D drift kinetic nonlinear theory, valid in the limit of a low beta, small inverse aspect ratio, circular cross section, toroidal geometry, to describe the plasma response to the neoclassical tearing mode (NTM) magnetic perturbation is derived. In (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) this theory is reduced in a low collisionality limit, which allows a dimensionality reduction to a 3D problem to efficiently resolve the collisional dissipation layer in the vicinity of the trapped-passing boundary. (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) adopts an improved model for the magnetic drift frequency, which reduces the threshold magnetic island half-width from , where is the trapped ion banana orbit width, to , making it in closer agreement with experimental observations for the large aspect ratio tokamak equilibrium. In the present paper, the theory is extended to a high beta, arbitrary tokamak geometry to capture the plasma shaping effects on the NTM threshold physics with the focus on the non-zero triangularity discharges that are known to have a strong impact on the plasma MHD stability. First, it is found that the higher triangularity plasma is more prone to NTMs which is in agreement with the tearing mode onset relative frequency measurements in DIII-D. Second, the NTM threshold dependence on the tokamak inverse aspect ratio obtained in (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) is refined and extended to a finite aspect ratio limit. Third, the NTM threshold dependence on poloidal beta is obtained and benchmarked against the EAST threshold island width measurements.

The synergetic effects of three-dimensional magnetic perturbations and finite beta on collisionless trapped electron mode in tokamak plasmas

The effects of three-dimensional (3D) magnetic perturbations (MPs) and finite beta (β, i.e., the ratio of plasma kinetic pressure to magnetic pressure) on the instability of collisionless trapped electron mode (CTEM) have been studied. Based on the local 3D equilibrium model, we have derived general expressions for longitudinal invariant and the corresponding precession drift frequency of trapped electrons, which include the synergetic effects of MPs and finite β. It is found that 3D effects can either stabilize or destabilize CTEM instability by analytically solving the linear dispersion relation of CTEM. These effects depend on the poloidal and toroidal mode numbers as well as the phase of 3D MPs. Specially, for the destabilizing phase of MPs, the stabilizing effect of finite β on CTEM can be even reversed when the displacement of magnetic flux surface exceeds a critical value. Moreover, the synergetic effects of 3D MPs with stabilizing phase and finite β can further reduce the required absolute value of negative magnetic shear to completely stabilize CTEM instability. This indicates that 3D MPs might be used as an actuator for lowing the level of anomalous electron heat transport, and thus facilitate the formation of electron internal transport barrier (eITB).

Theory and simulations of field strengths in magnetosonic shock waves in finite beta plasmas

Field strengths in an oblique magnetosonic shock wave in a collisionless, finite beta plasma are studied with theory and simulations. With the use of the warm, two-fluid model, the maximum values of the magnetic field, the transverse electric field, and the electric potential in a shock wave are analytically obtained as functions of the shock speed. One-dimensional, electromagnetic particle simulations are then carried out to measure the field strengths in shock waves. The theory and the simulation results are found to be consistent.

The synergetic effects of three-dimensional magnetic perturbations and finite beta on collisionless trapped electron mode in tokamak plasmas

Abstract The effects of three-dimensional (3D) magnetic perturbations (MPs) and finite beta (β, i.e., the ratio of plasma kinetic pressure to magnetic pressure) on the instability of collisionless trapped electron mode (CTEM) have been studied. Based on the local 3D equilibrium model, we have derived general expressions for longitudinal invariant and the corresponding precession drift frequency of trapped electrons, which include the synergetic effects of MPs and finite β. It is found that 3D effects can either stabilize or destabilize CTEM instability by analytically solving the linear dispersion relation of CTEM. These effects depend on the poloidal and toroidal mode numbers as well as the phase of 3D MPs. Specially, for the destabilizing phase of MPs, the stabilizing effect of finite β on CTEM can be even reversed when the displacement of magnetic flux surface exceeds a critical value. Moreover, the synergetic effects of 3D MPs with stabilizing phase and finite β can further reduce the required absolute value of negative magnetic shear to completely stabilize CTEM instability. This indicates that 3D MPs might be used as an actuator for lowing the level of anomalous electron heat transport, and thus facilitate the formation of electron internal transport barrier (eITB).

Expectation propagation learning of finite multivariate Beta mixture models and applications

Expectation propagation learning of finite multivariate Beta mixture models and applications

Simulation of ion temperature gradient mode in Chinese First Quasi-axisymmetric Stellarator

The Chinese First Quasi-axisymmetric Stellarator (CFQS) is now the only quasi-axisymmetric stellarator under construction in the world. In this work, ion temperature gradient (ITG) mode in CFQS is studied by using gyrokinetic Vlasov code GKV. The basic characteristics of the eletrtostatic ITG are separately given under the adiabatic condition and the non-adiabatic condition. There is a critical temperature gradient for ITG. The growth rate of ITG is proportional to the temperature gradient. Furthermore, the growth rate depends on not only the absolute value of density gradient, but also the plus or minus sign of the density gradient. The negative density gradient can strongly suppress the ITG. The kinetic electron can destabilize the ITG and the electron temperature gradient can also destabilize the ITG. For electromagnetic condition, the ITG modes can be suppressed by the finite plasma beta, and then a transition from ITG to Alfvenic ion temperature gradient mode/kinetic ballooning mode (AITG/KBM) comes into being. The maximum growth rate of KBM is linearly proportional to density gradient and temperature gradient when both gradients are large.

Topological sampling through windings

We propose a modification of the Hybrid Monte Carlo (HMC) algorithm that overcomes the topological freezing of a two-dimensional U(1) gauge theory with and without fermion content. This algorithm includes reversible jumps between topological sectors – winding steps – combined with standard HMC steps. The full algorithm is referred to as winding HMC (wHMC), and it shows an improved behaviour of the autocorrelation time towards the continuum limit. We find excellent agreement between the wHMC estimates of the plaquette and topological susceptibility and the analytical predictions in the U(1) pure gauge theory, which are known even at finite beta . We also study the expectation values in fixed topological sectors using both HMC and wHMC, with and without fermions. Even when topology is frozen in HMC – leading to significant deviations in topological as well as non-topological quantities – the two algorithms agree on the fixed-topology averages. Finally, we briefly compare the wHMC algorithm results to those obtained with master-field simulations of size Lsim 8 times 10^3.

Online variational inference on finite multivariate Beta mixture models for medical applications

Online variational inference on finite multivariate Beta mixture models for medical applications

Random Spanning Forests and Hyperbolic Symmetry

We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter beta >0 per edge. This is called the arboreal gas model, and the special case when beta =1 is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter p=beta /(1+beta ) conditioned to be acyclic, or as the limit qrightarrow 0 with p=beta q of the random cluster model. It is known that on the complete graph K_{N} with beta =alpha /N there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for alpha > 1 and all trees have bounded size for alpha <1. In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on {mathbb {Z}}^2 for any finite beta >0. This result is a consequence of a Mermin–Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.

Maximum Field Strengths in Shock Wave in Finite Beta Plasma

With use of the warm, two-fluid model, analytic expressions are obtained for the maximum values of the magnetic field, the transverse electric field, and the electric potential in a magnetosonic sh...

The combined kinetic effects of the ion temperature gradient and the velocity shear of a plasma flow parallel to the magnetic field on the drift-Alfven instabilities

The drift-Alfvén instabilities of the sheared flow along the magnetic field of a finite beta (1&amp;gt;β≫me/mi) plasma with comparable inhomogeneous ion temperature and homogeneous electron temperature are examined by the numerical analysis of the derived linear dispersion equation. Accounting for the electromagnetic ion kinetic response, which has been neglected in conventional discussions of the drift-Alfvén instabilities of a steady plasma, a new kinetic drift-Alfvén instability has been found. It develops in a plasma shear flow due to the coupled action of the ion temperature gradient, the flow velocity shear, and the inverse ion Landau damping.

Theoretical studies of equilibrium beta limit in LHD plasmas

This study deals with the equilibrium beta limit for high-beta plasmas of the Large Helical Device using a 3D equilibrium calculation code, HINT, which is an initial value solver based on the relaxation method without an assumption of nested flux surfaces. For the finite beta equilibrium, due to the nonlinear 3D equilibrium response, the flux surface breaking begins, and the magnetic field line becomes stochastic in the peripheral region. However, although the magnetic axis shifts until the conventional theoretical beta limit, the separatrix limiting the equilibrium does not appear in the plasma core. For the high beta equilibrium, breaking of the force balance begins in the strongly stochastized region. To keep the force balance in that region, the pressure gradient decreases and the fixed profile is relaxed. As a result, the volume averaged beta, which is equivalent to the plasma stored energy, saturates, although the peak beta increases. The beta value, where the force balance begins breaking in the inside of the stochastic region, is proposed as a new index of the equilibrium beta limit for the typical heliotron-type stellarator. According to this new index of the equilibrium beta limit, the achieved beta value in the experiment of the Large Helical Device is still lower than the equilibrium beta limit. This suggests that if the heating power can be increased, the achieved beta value may still increase.