This document presents a theoretical model to describe the coupling between resistive wall modes (RWMs) and neoclassical tearing modes (NTMs) in a rotating tokamak plasma. The model derives a system of equations showing that an unstable NTM island can accelerate the growth of an RWM, removing the typical NTM island width threshold. Increasing plasma rotation at the island location can decouple the modes and restore a conventional RWM with a threshold island width.
This document discusses an ab initio density functional theory study of structural transitions and pseudoelastic behavior in copper nanowires under tensile strain. The study finds that for nanowires with diameters below 1.38 nm, surface stresses alone can cause the structure to transition from an initial face-centered cubic structure to a body-centered tetragonal structure. Under loading and unloading conditions, the structure reversibly transitions between body-centered tetragonal and face-centered tetragonal structures, explaining the observed pseudoelastic behavior. The mechanical properties of copper nanowires depend not only on diameter size but also on surface orientation.
Sergey Sibiryakov "Galactic rotation curves vs. ultra-light dark matter: Impl...SEENET-MTP
The document discusses ultra-light dark matter and its implications for galactic rotation curves. It begins by providing theoretical background on ultra-light dark matter and how it can form soliton cores within dark matter halos. It then discusses how the properties of these soliton cores, such as their mass and size, relate to the properties of the ultra-light dark matter particle. Finally, it discusses how measurements of galactic rotation curves could provide insights into ultra-light dark matter models by probing the presence and characteristics of these soliton cores.
Effect of cylindrical texture on dynamic characteristics of journal bearingijmech
Effect of cylindrical texture on dynamic characteristics of hydrodynamic journal bearing is presented in this paper. The Reynolds equation is discretized by finite difference method and solved numerically in an iterative scheme satisfying the appropriate boundary conditions. Stiffness and damping coefficients of fluid film and stability parameters are found using the first-order perturbation method for different eccentricity ratios and various texture parameters like texture depth and texture portion. From the present study, it has found that cylindrical texture exhibits better stability than plain journal bearing.
This document analyzes the effect of different directrix shapes (circular, parabolic, elliptical, inverted catenary) on membrane stresses in cylindrical shell roofs. Membrane theory is used to determine the normal forces (Nx, Nθ, Nxθ) and stresses under self-weight and live loads. Equations for each directrix are presented. An example problem is solved and results are shown in tables comparing stresses for each directrix. The analysis found that stresses are lowest with an inverted catenary directrix and highest with an elliptical directrix to cover the same area.
1) The document discusses variable elasticity effects that occur in rotating machinery systems where parameters affecting elastic behavior do not remain constant but vary over time.
2) Systems with variable elasticity are governed by differential equations with periodic coefficients known as Mathieu-Hill equations. These equations can exhibit important stability problems.
3) Analytical tools for solving Mathieu-Hill equations are presented, including Floquet theory, a matrix method, and converting to an integral equation. These tools can be used to analyze problems like a cracked rotor with reciprocating forces.
This document provides notes on light stops in supersymmetric models. It motivates light stops based on experimental constraints allowing non-degenerate squarks, naturalness of the Higgs mass, and mechanisms for baryogenesis. It reviews effective natural supersymmetry models focusing on the third generation squarks and gluino being accessible at the LHC to solve the hierarchy problem. The Higgs mass calculation including stop loops is also summarized.
This document discusses semi-transitive maps in dynamical systems. It begins by defining semi-transitive maps and showing that every transitive map is semi-transitive but not vice versa. It then proves several theorems about properties of semi-transitive maps, including that dense orbit implies semi-transitive for perfect spaces and semi-transitive implies dense orbit for separable second category spaces. It also shows that for infinite spaces, dense orbit and dense periodic points imply sensitive dependence on initial conditions for semi-transitive maps. The document aims to generalize results about topological transitivity to semi-transitive maps.
This document discusses an ab initio density functional theory study of structural transitions and pseudoelastic behavior in copper nanowires under tensile strain. The study finds that for nanowires with diameters below 1.38 nm, surface stresses alone can cause the structure to transition from an initial face-centered cubic structure to a body-centered tetragonal structure. Under loading and unloading conditions, the structure reversibly transitions between body-centered tetragonal and face-centered tetragonal structures, explaining the observed pseudoelastic behavior. The mechanical properties of copper nanowires depend not only on diameter size but also on surface orientation.
Sergey Sibiryakov "Galactic rotation curves vs. ultra-light dark matter: Impl...SEENET-MTP
The document discusses ultra-light dark matter and its implications for galactic rotation curves. It begins by providing theoretical background on ultra-light dark matter and how it can form soliton cores within dark matter halos. It then discusses how the properties of these soliton cores, such as their mass and size, relate to the properties of the ultra-light dark matter particle. Finally, it discusses how measurements of galactic rotation curves could provide insights into ultra-light dark matter models by probing the presence and characteristics of these soliton cores.
Effect of cylindrical texture on dynamic characteristics of journal bearingijmech
Effect of cylindrical texture on dynamic characteristics of hydrodynamic journal bearing is presented in this paper. The Reynolds equation is discretized by finite difference method and solved numerically in an iterative scheme satisfying the appropriate boundary conditions. Stiffness and damping coefficients of fluid film and stability parameters are found using the first-order perturbation method for different eccentricity ratios and various texture parameters like texture depth and texture portion. From the present study, it has found that cylindrical texture exhibits better stability than plain journal bearing.
This document analyzes the effect of different directrix shapes (circular, parabolic, elliptical, inverted catenary) on membrane stresses in cylindrical shell roofs. Membrane theory is used to determine the normal forces (Nx, Nθ, Nxθ) and stresses under self-weight and live loads. Equations for each directrix are presented. An example problem is solved and results are shown in tables comparing stresses for each directrix. The analysis found that stresses are lowest with an inverted catenary directrix and highest with an elliptical directrix to cover the same area.
1) The document discusses variable elasticity effects that occur in rotating machinery systems where parameters affecting elastic behavior do not remain constant but vary over time.
2) Systems with variable elasticity are governed by differential equations with periodic coefficients known as Mathieu-Hill equations. These equations can exhibit important stability problems.
3) Analytical tools for solving Mathieu-Hill equations are presented, including Floquet theory, a matrix method, and converting to an integral equation. These tools can be used to analyze problems like a cracked rotor with reciprocating forces.
This document provides notes on light stops in supersymmetric models. It motivates light stops based on experimental constraints allowing non-degenerate squarks, naturalness of the Higgs mass, and mechanisms for baryogenesis. It reviews effective natural supersymmetry models focusing on the third generation squarks and gluino being accessible at the LHC to solve the hierarchy problem. The Higgs mass calculation including stop loops is also summarized.
This document discusses semi-transitive maps in dynamical systems. It begins by defining semi-transitive maps and showing that every transitive map is semi-transitive but not vice versa. It then proves several theorems about properties of semi-transitive maps, including that dense orbit implies semi-transitive for perfect spaces and semi-transitive implies dense orbit for separable second category spaces. It also shows that for infinite spaces, dense orbit and dense periodic points imply sensitive dependence on initial conditions for semi-transitive maps. The document aims to generalize results about topological transitivity to semi-transitive maps.
Collapse propagation in bridge structures. A semi-analytical modelDCEE2017
Michele Brun.
We consider the advance of a transition flexural wave through a beam-like periodically supported slender structure. The
collapse of a bridge structure is modeled as a steady-state propagation of a transition wave within a slender structure. The problem is
governed by fourth-order partial differential equations and both propagating and evanescent waves are included in the general solution. It is
shown that the problem can be expressed within a class of functional equations of the Wiener-Hopf type . Three different propagation
regimes are found: subsonic, intersonic and supersonic and it is shown that propagation is restricted to the intersonic regime where part of the
energy is released to the damaged structure.
Applications to the study of the collapse of the San Saba bridge in Texas shows the validity of the model.
LES-DQMOM based Studies on Reacting and Non-reacting Jets in Supersonic Cross...Samsung Techwin
This document summarizes a presentation given at the 50th AIAA Aerospace Science Meeting on large eddy simulation (LES) studies of reacting and non-reacting transverse jets in supersonic crossflow. The presentation discusses the numerical methodology used, including the compressible flow solver and direct quadrature method of moments (DQMOM) combustion model. Results are presented for non-reacting and reacting jet in supersonic crossflow cases, including comparisons to experimental data. Key flow features like shock structures and vortical structures are analyzed.
The nuclear Overhauser effect (NOE) is an incoherent cross-relaxation process between two nuclear spins within approximately 5 angstroms of each other. The intensity of the NOE is proportional to r-6, where r is the distance between the spins, meaning it decays very quickly with increasing distance. NOE experiments can provide distance restraints for structure determination of biological macromolecules like proteins.
- Bound-entanglement, or non-distillable entanglement, is not a rare phenomenon for continuous variable Gaussian states.
- The document presents a class of Gaussian states for a 2+2 mode bipartite system that are provably positive partial transpose (PPT) entangled within a finite parameter range, demonstrating bound-entanglement is achievable.
- This PPT entangled Gaussian state can be experimentally realized using current linear optics and squeezing techniques, challenging the notion that bound-entanglement is inaccessible in practice for continuous variable systems.
The document provides an overview of the total potential energy (TPE) method and Rayleigh-Ritz method for structural analysis. It includes:
1) An introduction to the concepts of TPE, stationary value of TPE, and Rayleigh-Ritz method.
2) Examples of using an assumed displacement field and minimizing the TPE to determine deflections in simple structures like beams and cable networks.
3) The importance of the displacement field assumed being compatible with the boundary conditions for the solution to be accurate.
Nonclassical Properties of Even and Odd Semi-Coherent StatesIOSRJAP
Even and odd semi-coherent states have been introduced. Some of the nonclasscial properties of the states are studied in terms of the quadrature squeezing as well as sub-Poissonian photon statistics. The Husimi– Kano Q-function and the phase distribution in the framework of Pegg and Barnett formalism, are also discussed.
Nucleation and avalanches in film with labyrintine magnetic domainsAndrea Benassi
This document summarizes a phase field model used to study nucleation and avalanches in films with labyrinthine magnetic domains. The model uses a phase field approach with a power expansion energy functional to simulate the system. It produces two different limit behaviors depending on film thickness and disorder strength: multiple nucleation and coalescence, or expansion by a single branching domain. The model is used to analyze characteristic lengths, domain shapes, and avalanche statistics under variations of parameters like thickness, disorder strength, and temperature. Adding random field and anisotropy terms introduces additional parameters for random field strength and temperature noise.
The document summarizes a study on the effect of jet configuration on transverse jet mixing. Direct numerical simulations were performed to analyze the effect of jet velocity profile and exit shape. Results show that a parabolic velocity profile enhances mixing over a top-hat profile due to slower vortex breakdown. For exit shape, a circular jet exhibits the most efficient mixing while triangular jets display two counter-rotating vortex pairs that increase entrainment and mixing.
This document discusses solutions to the Emmons boundary layer problem with radiation heat loss. It presents the governing equations in Howarth transformed coordinates, which transforms the compressible flow equations into an incompressible form. It then discusses the Blasius solution approach using self-similarity to reduce the equations. With radiation included, the energy equation contains an additional term representing the radiation source. Extinction conditions are analyzed, showing that radiation lowers the flame temperature and weakens the flame as you move downstream. Self-similarity may break down for the energy equation when non-local radiation effects become significant.
Plastic deformation in model glass induced by oscillating inclusionNikolai Priezjev
Molecular dynamics simulations are performed to investigate the plastic response of a model glass to a local shear transformation in a quiescent system. The deformation of the material is induced by a spherical inclusion that is gradually strained into an ellipsoid of the same volume and then reverted back into the sphere. We show that the number of cage-breaking events increases with increasing strain amplitude of the shear transformation. The results of numerical simulations indicate that the density of cage jumps is larger in the cases of weak damping or slow shear transformation. Remarkably, we also found that, for a given strain amplitude, the peak value of the density profiles is a function of the ratio of the damping coefficient and the time scale of the shear transformation.
I am William T. I am a Solid Mechanics Assignment Expert at solidworksassignmenthelp.com. I hold a Bachlor's Degree in Engineering from McMaster University, Canada. I have been helping students with their Assignments for the past 10 years. I solve Assignments related to Solid Mechanics.
Visit solidworksassignmenthelp.com or email info@solidworksassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Solid Mechanics Assignments.
The document discusses the physical properties of polymers, including their elasticity. It describes two types of elastic deformation - instantaneous deformation (Doe) and time-dependent deformation (Dhe). Doe becomes very small at very short time scales, while Dhe is the primary type of deformation in polymers as it involves the uncoiling of polymer chains over time. The document then discusses the kinetic theory of rubber-like elasticity in polymers, modeling polymer chains as random walks using a Gaussian distribution to calculate the probability of a chain existing at a given distance.
1) The document analyzes the collision between a bouncing ball and a block. It shows that the quantity (v-V), where v is the ball's speed and V is the block's speed after each collision, remains constant.
2) It then derives an expression for the minimum distance Lmin the block reaches from the wall in terms of the mass ratio m/M of the ball and block.
3) An alternative solution is presented that uses conservation of momentum to derive expressions for the speeds of the ball and block after each collision in terms of the number of bounces N. This allows calculating the mass ratio m/M needed for the block to come to rest after a given number of bounces N
This study used finite element analysis to simulate cracking in a plain weave carbon fiber/silicon carbide composite under biaxial loading conditions. The objectives were to reveal damage patterns, study how they depend on loading path and initial stress, and develop simulation procedures to predict macro-level properties. A unit cell model was created with cohesive interfaces between tows and matrices. Various loading cases were applied and results showed that damage initiation was influenced by voids and tow intersections, while the damage pattern depended strongly on loading path and initial stress state. Future work could use extended finite element methods to reduce mesh sensitivity.
The document discusses instabilities that may arise in the Gribov-Levin-Ryskin (GLR) evolution equation for parton distribution functions (PDFs) at small values of x. It proposes that small initial azimuthal asymmetries could grow with decreasing x due to instabilities in the generalized linearized GLR equation. This could lead to azimuthally asymmetric PDFs, G(x,Q2,φ), and provide an initial state explanation for the azimuthal anisotropy coefficients vn observed in heavy ion collisions. Preliminary results using an ansatz solution show features like a weak dependence on x and near-decoupling of Fourier modes that could reproduce the observed scaling patterns in vn data.
Singular rise and singular drop of cutoff frequencies in slot line and strip ...ijeljournal
The complete frequency spectrum of the circularly- shielded slot line and strip line are determined using an efficient domain decomposition and mode matching method. Asymptotic analyses show that, when the gap width of the slot line or the width of the strip line are too small, the TE frequencies may drop singularly and the TM frequencies may rise singularly. These new properties greatly affect the cutoff frequencies.
SINGULAR RISE AND SINGULAR DROP OF CUTOFF FREQUENCIES IN SLOT LINE AND STRIP ...ijeljournal
The complete frequency spectrum of the circularly- shielded slot line and strip line are determined using an efficient domain decomposition and mode matching method. Asymptotic analyses show that, when the gap width of the slot line or the width of the strip line are too small, the TE frequencies may drop singularly and the TM frequencies may rise singularly. These new properties greatly affect the cutoff frequencies.
1) Streamwise vortices play an important role in sustaining wall turbulence by regenerating streaks through the lift-up effect.
2) In turbulent plane Couette flow at low Reynolds numbers, streamwise vortices that span the entire gap between plates have been observed.
3) The document proposes a two-step Galerkin projection method to derive a low-order model that can illustrate the dynamics and generation mechanism of these streamwise vortices, in a way that is analogous to what is observed in turbulent boundary layers.
New folderelec425_2016_hw5.pdfMar 25, 2016 ELEC 425 S.docxcurwenmichaela
The document discusses omnidirectional reflection from a one-dimensional photonic crystal structure. It presents the following key points:
1) A one-dimensional photonic crystal, such as a multilayer film, can exhibit complete reflection of light within a frequency range for all incident angles and polarizations, even without a full photonic bandgap.
2) The criterion for omnidirectional reflection is that there exists a frequency range where the projected band structures of the photonic crystal and surrounding medium do not overlap, rather than there being no propagating states within the crystal itself.
3) As an example, a multilayer film with refractive indices of n1 = 1.7 and n2 = 3.4
TIME-DOMAIN SIMULATION OF ELECTROMAGNETIC WAVE PROPAGATION IN A MAGNETIZED PL...John Paul
This document describes a time-domain transmission-line model for simulating electromagnetic wave propagation in a magnetized plasma. It discretizes the anisotropic Lorentzian conductivity function using bilinear Z-transforms for improved accuracy. It validates the model by comparing results to an analytic solution and applies it to simulate ionospheric propagation, including the dispersion of a pulse into a whistler wave.
Collapse propagation in bridge structures. A semi-analytical modelDCEE2017
Michele Brun.
We consider the advance of a transition flexural wave through a beam-like periodically supported slender structure. The
collapse of a bridge structure is modeled as a steady-state propagation of a transition wave within a slender structure. The problem is
governed by fourth-order partial differential equations and both propagating and evanescent waves are included in the general solution. It is
shown that the problem can be expressed within a class of functional equations of the Wiener-Hopf type . Three different propagation
regimes are found: subsonic, intersonic and supersonic and it is shown that propagation is restricted to the intersonic regime where part of the
energy is released to the damaged structure.
Applications to the study of the collapse of the San Saba bridge in Texas shows the validity of the model.
LES-DQMOM based Studies on Reacting and Non-reacting Jets in Supersonic Cross...Samsung Techwin
This document summarizes a presentation given at the 50th AIAA Aerospace Science Meeting on large eddy simulation (LES) studies of reacting and non-reacting transverse jets in supersonic crossflow. The presentation discusses the numerical methodology used, including the compressible flow solver and direct quadrature method of moments (DQMOM) combustion model. Results are presented for non-reacting and reacting jet in supersonic crossflow cases, including comparisons to experimental data. Key flow features like shock structures and vortical structures are analyzed.
The nuclear Overhauser effect (NOE) is an incoherent cross-relaxation process between two nuclear spins within approximately 5 angstroms of each other. The intensity of the NOE is proportional to r-6, where r is the distance between the spins, meaning it decays very quickly with increasing distance. NOE experiments can provide distance restraints for structure determination of biological macromolecules like proteins.
- Bound-entanglement, or non-distillable entanglement, is not a rare phenomenon for continuous variable Gaussian states.
- The document presents a class of Gaussian states for a 2+2 mode bipartite system that are provably positive partial transpose (PPT) entangled within a finite parameter range, demonstrating bound-entanglement is achievable.
- This PPT entangled Gaussian state can be experimentally realized using current linear optics and squeezing techniques, challenging the notion that bound-entanglement is inaccessible in practice for continuous variable systems.
The document provides an overview of the total potential energy (TPE) method and Rayleigh-Ritz method for structural analysis. It includes:
1) An introduction to the concepts of TPE, stationary value of TPE, and Rayleigh-Ritz method.
2) Examples of using an assumed displacement field and minimizing the TPE to determine deflections in simple structures like beams and cable networks.
3) The importance of the displacement field assumed being compatible with the boundary conditions for the solution to be accurate.
Nonclassical Properties of Even and Odd Semi-Coherent StatesIOSRJAP
Even and odd semi-coherent states have been introduced. Some of the nonclasscial properties of the states are studied in terms of the quadrature squeezing as well as sub-Poissonian photon statistics. The Husimi– Kano Q-function and the phase distribution in the framework of Pegg and Barnett formalism, are also discussed.
Nucleation and avalanches in film with labyrintine magnetic domainsAndrea Benassi
This document summarizes a phase field model used to study nucleation and avalanches in films with labyrinthine magnetic domains. The model uses a phase field approach with a power expansion energy functional to simulate the system. It produces two different limit behaviors depending on film thickness and disorder strength: multiple nucleation and coalescence, or expansion by a single branching domain. The model is used to analyze characteristic lengths, domain shapes, and avalanche statistics under variations of parameters like thickness, disorder strength, and temperature. Adding random field and anisotropy terms introduces additional parameters for random field strength and temperature noise.
The document summarizes a study on the effect of jet configuration on transverse jet mixing. Direct numerical simulations were performed to analyze the effect of jet velocity profile and exit shape. Results show that a parabolic velocity profile enhances mixing over a top-hat profile due to slower vortex breakdown. For exit shape, a circular jet exhibits the most efficient mixing while triangular jets display two counter-rotating vortex pairs that increase entrainment and mixing.
This document discusses solutions to the Emmons boundary layer problem with radiation heat loss. It presents the governing equations in Howarth transformed coordinates, which transforms the compressible flow equations into an incompressible form. It then discusses the Blasius solution approach using self-similarity to reduce the equations. With radiation included, the energy equation contains an additional term representing the radiation source. Extinction conditions are analyzed, showing that radiation lowers the flame temperature and weakens the flame as you move downstream. Self-similarity may break down for the energy equation when non-local radiation effects become significant.
Plastic deformation in model glass induced by oscillating inclusionNikolai Priezjev
Molecular dynamics simulations are performed to investigate the plastic response of a model glass to a local shear transformation in a quiescent system. The deformation of the material is induced by a spherical inclusion that is gradually strained into an ellipsoid of the same volume and then reverted back into the sphere. We show that the number of cage-breaking events increases with increasing strain amplitude of the shear transformation. The results of numerical simulations indicate that the density of cage jumps is larger in the cases of weak damping or slow shear transformation. Remarkably, we also found that, for a given strain amplitude, the peak value of the density profiles is a function of the ratio of the damping coefficient and the time scale of the shear transformation.
I am William T. I am a Solid Mechanics Assignment Expert at solidworksassignmenthelp.com. I hold a Bachlor's Degree in Engineering from McMaster University, Canada. I have been helping students with their Assignments for the past 10 years. I solve Assignments related to Solid Mechanics.
Visit solidworksassignmenthelp.com or email info@solidworksassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Solid Mechanics Assignments.
The document discusses the physical properties of polymers, including their elasticity. It describes two types of elastic deformation - instantaneous deformation (Doe) and time-dependent deformation (Dhe). Doe becomes very small at very short time scales, while Dhe is the primary type of deformation in polymers as it involves the uncoiling of polymer chains over time. The document then discusses the kinetic theory of rubber-like elasticity in polymers, modeling polymer chains as random walks using a Gaussian distribution to calculate the probability of a chain existing at a given distance.
1) The document analyzes the collision between a bouncing ball and a block. It shows that the quantity (v-V), where v is the ball's speed and V is the block's speed after each collision, remains constant.
2) It then derives an expression for the minimum distance Lmin the block reaches from the wall in terms of the mass ratio m/M of the ball and block.
3) An alternative solution is presented that uses conservation of momentum to derive expressions for the speeds of the ball and block after each collision in terms of the number of bounces N. This allows calculating the mass ratio m/M needed for the block to come to rest after a given number of bounces N
This study used finite element analysis to simulate cracking in a plain weave carbon fiber/silicon carbide composite under biaxial loading conditions. The objectives were to reveal damage patterns, study how they depend on loading path and initial stress, and develop simulation procedures to predict macro-level properties. A unit cell model was created with cohesive interfaces between tows and matrices. Various loading cases were applied and results showed that damage initiation was influenced by voids and tow intersections, while the damage pattern depended strongly on loading path and initial stress state. Future work could use extended finite element methods to reduce mesh sensitivity.
The document discusses instabilities that may arise in the Gribov-Levin-Ryskin (GLR) evolution equation for parton distribution functions (PDFs) at small values of x. It proposes that small initial azimuthal asymmetries could grow with decreasing x due to instabilities in the generalized linearized GLR equation. This could lead to azimuthally asymmetric PDFs, G(x,Q2,φ), and provide an initial state explanation for the azimuthal anisotropy coefficients vn observed in heavy ion collisions. Preliminary results using an ansatz solution show features like a weak dependence on x and near-decoupling of Fourier modes that could reproduce the observed scaling patterns in vn data.
Singular rise and singular drop of cutoff frequencies in slot line and strip ...ijeljournal
The complete frequency spectrum of the circularly- shielded slot line and strip line are determined using an efficient domain decomposition and mode matching method. Asymptotic analyses show that, when the gap width of the slot line or the width of the strip line are too small, the TE frequencies may drop singularly and the TM frequencies may rise singularly. These new properties greatly affect the cutoff frequencies.
SINGULAR RISE AND SINGULAR DROP OF CUTOFF FREQUENCIES IN SLOT LINE AND STRIP ...ijeljournal
The complete frequency spectrum of the circularly- shielded slot line and strip line are determined using an efficient domain decomposition and mode matching method. Asymptotic analyses show that, when the gap width of the slot line or the width of the strip line are too small, the TE frequencies may drop singularly and the TM frequencies may rise singularly. These new properties greatly affect the cutoff frequencies.
1) Streamwise vortices play an important role in sustaining wall turbulence by regenerating streaks through the lift-up effect.
2) In turbulent plane Couette flow at low Reynolds numbers, streamwise vortices that span the entire gap between plates have been observed.
3) The document proposes a two-step Galerkin projection method to derive a low-order model that can illustrate the dynamics and generation mechanism of these streamwise vortices, in a way that is analogous to what is observed in turbulent boundary layers.
New folderelec425_2016_hw5.pdfMar 25, 2016 ELEC 425 S.docxcurwenmichaela
The document discusses omnidirectional reflection from a one-dimensional photonic crystal structure. It presents the following key points:
1) A one-dimensional photonic crystal, such as a multilayer film, can exhibit complete reflection of light within a frequency range for all incident angles and polarizations, even without a full photonic bandgap.
2) The criterion for omnidirectional reflection is that there exists a frequency range where the projected band structures of the photonic crystal and surrounding medium do not overlap, rather than there being no propagating states within the crystal itself.
3) As an example, a multilayer film with refractive indices of n1 = 1.7 and n2 = 3.4
TIME-DOMAIN SIMULATION OF ELECTROMAGNETIC WAVE PROPAGATION IN A MAGNETIZED PL...John Paul
This document describes a time-domain transmission-line model for simulating electromagnetic wave propagation in a magnetized plasma. It discretizes the anisotropic Lorentzian conductivity function using bilinear Z-transforms for improved accuracy. It validates the model by comparing results to an analytic solution and applies it to simulate ionospheric propagation, including the dispersion of a pulse into a whistler wave.
1) The document describes a numerical simulation of the spherical collapse of dark matter perturbations in an expanding universe. It simulates both cold dark matter (CDM) and warm dark matter (WDM) cases.
2) For CDM, the simulation shows collapse times depend on the initial overdensity but are approximately symmetric around the turnaround time. Regions of all masses collapse as long as the initial overdensity exceeds a threshold.
3) For WDM, the simulation adds a pressure term to account for the thermal velocity of WDM in the early universe. This term slows collapse compared to CDM and can potentially prevent collapse of low-mass regions.
Mapping WGMs of erbium doped glass microsphere using near-field optical probe...NuioKila
This document discusses whispering gallery modes (WGMs) in erbium-doped glass microspheres. It begins by introducing a simple ray optics model to qualitatively describe WGMs. Light rays undergo total internal reflection along the microsphere circumference, forming standing wave resonances when the optical path length equals an integer number of wavelengths. A more accurate description is provided by Lorenz-Mie theory, which uses Maxwell's equations and vector spherical harmonics to model WGMs. Each WGM is characterized by radial, polar, and azimuthal mode numbers that describe the intensity distribution. The document focuses on using a near-field optical probe to map and couple light into the WGMs of erbium-
Breaking waves on the surface of the heartbeat star MACHO 80.7443.1718Sérgio Sacani
Massive astrophysical compact halo object (MACHO) 80.7443.1718 is a high mass, eccentric binary system exhibiting the largest-known-amplitude tidally excited oscillations. The system’s ±20% photometric amplitude, along with the high mass of the primary star, ~35 M⊙, make this the most extreme of the class of periodically perturbed ‘heartbeat stars.’ Here, we use a hydrodynamic simulation to demonstrate that with each periapse passage, an unseen companion star raises tidal waves so large that they break, shock-heating and dissipating energy and angular momentum on the surface of the star. The shock-heated material forms a rapidly rotating circumstellar atmosphere, which is stripped and reassembled with each periapse passage. The dissipation of nonlinear tides through surface wave breaking explains the super-synchronous rotation of the primary star, the evolution of spectral emission features and the observed decay of the binary orbital period. Connecting these features demonstrates that MACHO 80.7443.1718 is a natural product of massive binary star evolution, and that it provides an ideal laboratory for the direct study of nonlinear tidal dissipation.
1. The document defines key rock physics terms including density, porosity, saturation, velocity, impedance, Poisson's ratio, and reflection coefficients. Equations are provided for calculating these values from measured properties.
2. Methods of modeling reflection seismograms are described including normal reflection, reflection at an angle using Zoeppritz equations, AVO analysis, and impedance inversion.
3. Concepts of stress, strain, elasticity, elastic moduli, and their relationships to velocity are covered. The differences between static and dynamic moduli are also discussed.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
Investigation of the bandpass properties of the local impedance of slow wave ...Victor Solntsev
The properties of the local coupling impedance that determines the efficiency of the electron–wave interaction in periodic slow-wave structures are investigated. This impedance is determined (i) through the char- acteristics of the electromagnetic field in a slow-wave structure and (ii) through the parameters of a two-port chain simulating the structure. The continuous behavior of the local coupling impedance in the passbands of slow-wave structures, at the boundaries of the passbands, and beyond the passbands is confirmed with the help of a waveguide–resonator model.
A generalized linear theory of the discrete electron–wave interaction in slow...Victor Solntsev
A linear theory of the discrete interaction of electron flows and electromagnetic waves in slow-wave structures (SWSs) is developed. The theory is based on the finite-difference equations of SWS excitation. The local coupling impedance entering these equations characterizes the field intensity excited by the electron flow in interaction gaps and has a finite value at SWS cutoff frequencies. The theory uniformly describes the electron–wave interaction in SWS passbands and stopbands without using equivalent circuits, a circumstance that allows considering the processes in the vicinity of cutoff frequencies and switching from the Cerenkov mechanism of interaction in a traveling-wave tube to the klystron mechanism when passing to SWS stopbands. The features of the equations of the discrete electron–wave interaction in pseudoperiodic SWSs are analyzed.
End point of_black_ring_instabilities_and_the_weak_cosmic_censorship_conjectureSérgio Sacani
We produce the first concrete evidence that violation of the weak cosmic censorship conjecture can occur
in asymptotically flat spaces of five dimensions by numerically evolving perturbed black rings. For certain
thin rings, we identify a new, elastic-type instability dominating the evolution, causing the system to settle to
a spherical black hole. However, for sufficiently thin rings the Gregory-Laflamme mode is dominant, and the
instability unfolds similarly to that of black strings, where the horizon develops a structure of bulges connected
by necks which become ever thinner over time.
A Calabi-Yau manifold is a smooth space that represents a deformation which smooths out an orbifold singularity. This document discusses superstring theory and fermions in string theories. It introduces the spinning string action and shows that the Neveu-Schwarz model contains a tachyon ground state while the Ramond model contains massless fermions. Combining the two sectors using the Gliozzi-Scherk-Olive projection results in a model with N=1 supersymmetry in ten dimensions.
This document discusses the use of trapped atomic ions for quantum information processing and the creation of entangled states. It describes how ions can be trapped and laser cooled to suppress environmental perturbations and allow for coherent manipulation over long durations. Recent experiments have successfully generated entanglement between the internal states of pairs of trapped ions, implemented quantum logic gates like CNOT, and improved tools for high-precision measurement. Trapped ions provide a promising system for studying and applying concepts of quantum information processing.
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Resistive wall mode and neoclassical tearing mode coupling in rotating tokamak plasmas
1. Resistive wall mode and neoclassical tearing
mode coupling in rotating tokamak plasmas
Rachel McAdams1,2
, H R Wilson1
, and I T Chapman2
1
York Plasma Institute, Department of Physics, University of
York, Heslington, York, YO10 5DD, UK
2
EURATOM/CCFE Fusion Association, Culham Science Centre,
Abingdon, Oxon, OX14 3DB, UK
Abstract
A model system of equations has been derived to describe a toroidally
rotating tokamak plasma, unstable to Resistive Wall Modes (RWMs) and
metastable to Neoclassical Tearing Modes (NTMs), using a linear RWM
model and a nonlinear NTM model. If no wall is present, the NTM
growth shows the typical threshold/saturation island widths, whereas a
linearly unstable kink mode grows exponentially in this model plasma sys-
tem. When a resistive wall is present, the growth of the linearly unstable
RWM is accelerated by an unstable island: a form of coupled RWM-NTM
mode. Crucially, this coupled system has no threshold island width, giv-
ing the impression of a triggerless NTM, observed in high beta tokamak
discharges. Increasing plasma rotation at the island location can mitigate
its growth, decoupling the modes to yield a conventional RWM with no
threshold width.
1 Introduction
To operate ITER [1] in Advanced Tokamak (AT) scenarios, which are desir-
able for high fusion gain, control of performance limiting magnetohydrodynamic
(MHD) instabilities must be achieved. AT scenarios are characterised by high
bootstrap fraction and high βN = β(%)/(I/aB), β = 2µ0 p / B2
, (where p
is volume averaged pressure, B(T) is external magnetic field, I(MA) is toroidal
current, a(m) is minor radius)[2]. Neoclassical Tearing Modes (NTMs)[3] are
one such performance limiting instability [4], with a detrimental effect on the
achievable βN . NTMs are unstable if two criteria are satisfied: β must be suffi-
ciently high, and the plasma must typically be ”seeded” with a magnetic island
produced by another activity [5, 6], such as sawteeth[7, 8, 9, 10], fishbones [10]
or ELMs [11]. This initial seed island will only grow, and subsequently saturate,
1
arXiv:1302.4250v2[physics.plasm-ph]9May2013
2. if it exceeds a certain critical width [12, 13, 14]. The flattening of the pressure
profile caused by the presence of the island, and the consequential removal of
the pressure gradient-driven bootstrap current in the vicinity of the magnetic
island O-point enhances filamentation of the current density, providing a feed-
back mechanism for continued island growth. However, NTMs which do not
appear to have grown from a seed island, so-called triggerless NTMs, have also
been observed experimentally [10, 15, 16, 17], usually associated with high β or
near the RWM limit [18]. Another instability which is responsible for limiting
tokamak performance at βN > βno−wall
N is the Resistive Wall Mode (RWM);
a branch of the long-wavelength external kink mode, driven in a tokamak by
pressure but which has a growth rate slowed by the presence of finite resistivity
in surrounding conducting walls. RWMs are slowly growing on the timescale of
the wall diffusion time, and have a complex relationship with plasma rotation.
Analytical models predict that a modest amount of plasma rotation is sufficient
to stabilise RWM growth [17, 19, 20, 21]; and both experiment and theory show
that RWM drag can damp plasma rotation [17, 22].
Here, a model system of equations is derived containing linear RWM and
nonlinear NTM models and extended to include self-consistent nonlinear inter-
actions with toroidal plasma rotation. These equations are examined in a limit-
ing case relevant to the tokamak and applied as a possible model for exploring
the phenomenon of triggerless NTMs. Section 2 will outline the derivation of
the model system considered. This is followed by an analytic and numerical ex-
amination of the stability of the RWM-NTM coupled mode and its dependence
upon model parameters in Section 3. Conclusions are discussed in Section 4.
2 Theoretical Model
Suppose an ideal, toroidally rotating plasma is surrounded by a thin wall at
r = rw with finite conductivity σw, and contains a resistive layer in the vicinity
of an internal rational surface at radius r = rs where a magnetic island can form.
The regions of the plasma outside the resistive layer are assumed to be described
by linear ideal MHD. Within the layer we retain resistivity and the nonlinear
effects associated with magnetic islands. The plasma geometry is shown in
Figure 1. Due to skin currents in the resistive wall and at the rational surface,
the component of the vector potential parallel to the equilibrium magnetic field
has a discontinuous radial derivative at these locations. These discontinuities are
parameterised by ∆w and ∆L respectively. In a linear, ideal MHD, cylindrical
model employing a complex representation of the perturbed fields, they are
related through an expression of the form [23]
∆L =
1 − δ∆w
− + ∆w
. (1)
Here and δ depend on the equilibrium, and are related to the stability proper-
ties of the plasma in the limit of no wall and a superconducting wall respectively.
They can be derived for a given equilibrium by solving ideal MHD equations
2
3. Figure 1: The plasma is assumed to be ideal apart from the narrow layer near
a rational surface (the “rational layer”) where nonlinear effects and resistivity
are important. The plasma is surrounded by a resistive wall and vacuum.
outside the wall and rational layer with appropriate boundary conditions at
r = 0 and r = ∞, and noting the definition:
∆L,w =
rs,w
ψc
∂ψc
∂r
r+
s,w
r−
s,w
(2)
Here ψc is the complex representation of the magnetic flux function derived from
linear ideal MHD. When no wall is present, ∆w = 0 and for an ideally unstable
plasma (where an inertial response for the layer is appropriate and ∆L < 0
corresponds to instability to ideal MHD modes), is positive and small. For an
ideal superconducting wall at r = rw, ∆w → ∞ and δ > 0 provides stability
to the internal kink mode, i.e. ∆L < 0. Thus equilibria with small , δ > 0
are susceptible to a RWM but avoids the internal kink mode. Whilst we solve
linear equations in the outer region, we must anticipate a need to match to a
nonlinear layer solution at the rational surface. Nevertheless, we assume we can
separate the time dependence and write the outer region solution in the form:
ψc(r, t) = ˜ψ(r)eiξ
e pdt
(3)
where p(t) = γ(t) − iω(t), with γ the instantaneous growth rate, ω the mode
frequency relative to the wall and ξ = mθ−nφ the helical angle defined in terms
of the poloidal (θ) and toroidal (φ) angles. It will be assumed that the toroidal
mode number n = 1, with m the poloidal mode number. Using Amp`ere’s Law,
the wall response can be written as
∆w = pτ (4)
3
4. where τ = τw/τr and τw = µ0dσwrw is the wall diffusion time, with the wall
thickness d << rw, the minor radius at which the wall is located. The instan-
taneous eigenvalue p is normalised to τr = σµ0ars the resistive plasma diffusion
time where σ is the plasma conductivity at the rational surface. Combining
these results relates the layer response ∆L to the wall properties and the in-
stantaneous complex growth rate [23]:
∆L =
1 − δτp
− + τp
(5)
In the linearised ideal MHD region, a complex representation for the mag-
netic flux is required to ensure that relation (1) holds (a consequence of solving
a second order ordinary differential equation for ψc in this region). The physi-
cal flux is denoted ψL = Re[ψc]. In the rational layer where non-linear physics
is retained, the complex representation cannot be used, so we work with this
physical flux and assume a magnetic island exists at the rational surface. Let
us transform into the frame of reference where the rational surface is at rest, so
that the wall rotates. Analogous to the wall, Amp`ere’s law describes the current
perturbations
∂2
ψL
∂r2
= −µ0J|| (6)
where J|| is the component of current density perturbation parallel to the mag-
netic field. There are two key contributions to J|| to consider: an inductive
component proportional to ∂ψL/∂t and the perturbation in the bootstrap cur-
rent Jbs caused by the pressure flattening inside the island. We assume that
these current perturbations are localised within the layer and integrate across
that layer to derive the model equation:
∂ψL
∂r
rs+
rs−
= 2σµ0w
∂ψL
∂t
− 2µ0wJbs (7)
where w is the half-width of the magnetic island. The discontinuity on the
left of Equation (7), caused by the current sheet in the layer, can be expressed
in terms of ∆L as follows. Let ψL be of the form ψL = ˜ϕcos(ˆα + ζ), where
ˆα = ξ − ˆωdt, ζ(r) is a phase factor, and the mode frequency in this rotating
frame is ˆω = ω − ΩL, where ΩL is the toroidal plasma rotation frequency at
r = rs relative to the stationary wall. Matching the ideal solution provides
Re[ ˜ψ]e γdt
= ˜ϕcos(ΩLt + ζ) and Im[ ˜ψ]e γdt
= ˜ϕsin(ΩLt + ζ) (we assume that
the toroidal mode number n=1 and ˜ψ is the amplitude of the magnetic flux
function in Equation (3)). Thus, combining the results yields
( ˜ϕ/rs)(Re[∆L] cos(ˆα + ζ) − Im[∆L] sin(ˆα + ζ)) =
2σµ0w(∂ ˜ϕ
∂t cos(ˆα + ζ) + ˆω ˜ϕ sin(ˆα + ζ)) − 2µ0wJbs (8)
Multiplying by cos(ˆα + ζ) or sin(ˆα + ζ) and then integrating over ˆα yields
4
5. equations for ω and ∂ ˜ϕ
∂t . The island half width w is related to ˜ϕ by
w2
=
4rsLs ˜ϕ
BθqR
(9)
where Ls = Rq
s is the shear length scale, q is the safety factor, s is the magnetic
shear, R is the major radius and Bθ is the poloidal magnetic field. Normalising
the island half width w to the minor radius, and time to τr we obtain
4 ˙w = Re[∆L] +
ˆβ
w
1 −
w2
c
w2
(10)
2w(ω − ΩL) + Im[∆L] = 0 (11)
where ˙w refers to time derivatives with respect to ˆt = t/τr and Jbs = βθBθ
√
ˆε
Lpµ0
cos(α+
ζ), where βθ is the poloidal beta, ˆε the inverse aspect ratio, Lp = 1
p
dp
dr the pres-
sure length scale, and wc the seed island threshold width. We have defined
ˆβ = 8
√
ˆεβθrs
Lps . Note that γ = 2 ˙w/w, ω and ΩL are normalised to τr. We have
also introduced a heuristic NTM threshold factor (1 − w2
c /w2
), which could be
attributed to the polarisation current effects in the plasma [14, 24]. We have
neglected geometrical factors to yield a simpler model which nevertheless retains
the essential physics. Equation (10) is the same relation as that found in [25]
when rotation is not considered. The full expression for ∆L is given by
∆L =
(1 − δτγ)(− + τγ) − δτ2
ω2
(− + τγ)2 + τ2ω2
+ i
τω(1 − δ)
(− + τγ)2 + τ2ω2
(12)
To close the system, we require a torque balance relation. Ideal plasma is
torque-free [26]; the torque exerted on the plasma is a delta function at the
rational surface δ(r − rs).
Consider the perturbed MHD momentum balance equation in the non-ideal
layer, averaged over the flux surface.
δJ × δB + ρµ 2
v = 0 (13)
where angled brackets denote the average. Here ρ is the plasma density and
µ is the plasma viscosity. We assume that the processes under consideration
occur over many viscous diffusion times, and pressure is constant across the
thin layer. The pressure gradient is neglected as it will not contribute once we
integrate across the layer (as follows). Assume that v is continuous across the
layer but v possesses a discontinuity at r = rs, due to the localised torque at
that location. For a perturbation δB, × δB = − 2
ψLb. Only the toroidal
component is required. Integrating Equation (13) over the rational layer:
˜ϕ
Rµ0
sin(ˆα + ζ)
δψL
δr
r+
s
r−
s
+ ρµ
δvφ
δr
r+
s
r−
s
= 0 (14)
5
6. The discontinuity in the radial derivative of ψL = Re[ψc] is evaluated as above.
Integrating over ˆα, the terms in Re[∆L] disappear. Let vφ = ΩR and normalise
as before to derive
δΩ
δr
r+
s
r−
s
=
A
a
w4
Im[∆L] (15)
such that A = s2
ˆε2
a3
τV τr/512r3
sq2
τ2
A,with τV = a2
/µ the momentum confine-
ment time, a the minor radius, and τA =
a
√
µ0ρ
B the Alfv´en time and taking
the σ = T
3/2
e /1.65 × 10−9
ln λ, with Te expressed in keV. The safety factor q
at r = rs will be taken as q = 2. The electromagnetic torque is finite when
Im[∆L] = 0, and always acts to damp the plasma rotation [27]. Im[∆L] is re-
lated to the discontinuity in the derivative of the phase factor ζ [23]: the torque
only acts on the plasma in the rational layer when ∂ζ
∂r |
r+
s
r−
s
= 0.
In the outer ideal plasma regions, the momentum equation is simply
d2
Ω
dr2
= 0 (16)
The linear rotation profile in the ideal plasma regions, r < rs and rs < r < a, is
continuous at r = rs but has a discontinuity in the first radial derivative there,
caused by the torque. Imposing no slip boundary conditions at the outer edge
of the plasma r = a (viscous drag effects) and a driving force at the inner edge
at r = 0 that maintains dΩ
dr |r=0 = −Ω0/a, the toroidal rotation frequency profile
can be constructed in the region of ideal plasma.
Ω(r) = λ(a − rs) +
Ω0
a
(rs − r) 0 < r < rs (17)
Ω(r) = λ(a − r) rs < r < a (18)
for constant λ > 0. The discontinuity in the radial derivative is calculated to be
−λ + Ω0/a. Hence, at r = rs, using Equation (15), the damping of the toroidal
plasma rotation frequency is found to be
(Ω0 − fΩL) = Aw4
Im[∆L] (19)
where f = a
a−rs
.
3 Solutions For Coupled RWM-NTM Mode Sta-
bility
The full system of nonlinear equations (5), (10), (11), and (19) can be solved
numerically and evolved in time, with τ ≈ 10−2
corresponding to the choice of
a = 1.0m, rs = 0.5m, Te = 1keV, rw = 1.0m, and Coulomb logarithm lnΛ = 20.
We take = 0.1, δ = 5.
If there is no wall surrounding the plasma, then τ = 0. In this situation,
6
7. Figure 2: No RWM exists when the resistive wall is removed: only the NTM is a
solution. The dependence of the island width evolution on ˆβ and wc illustrates
the behaviour expected for a NTM.
the RWM is not a solution (an infinite growth rate is obtained, which can be
interpreted as the ideal kink mode). However, the NTM is a solution in this
limit (Figure 2), exhibiting its characteristic features: specifically a sufficiently
high ˆβ must be achieved for the NTM to become unstable, as well as a seed
island with a width w > wc. An NTM will saturate at a large island width,
proving detrimental to plasma confinement.
When τ is finite, both RWM and NTM branches are present. If the wall is
thin, or has a low conductivity then τ << 1 and we can solve the equations
analytically, as follows.
3.1 Analytical Solutions For Small Islands
We first consider the behaviour of the solutions when w < wc. For a standard
NTM, an island of this size would not grow as it is smaller than the threshold
width. The relevant ordering is τω << τγ ∼ . The bootstrap term (∼ ˆβ) in
Equation (10) is relatively large at small w < wc and negative, and this must
be balanced by either a large ˙w (i.e. γ) on the left hand side of Equation (10) or
by a large value of Re[∆L]. Neglecting τω in Equation (10), the branch when
γ is large (and negative) is the standard stable NTM root of the equations-as
illustrated in Figure 2. Specifically, in this limit Re[∆L] ∼ −δ and γ < 0 for
7
8. w < wc. The alternative is the RWM solution branch, in which the denominator
of Re[∆L] is small. Neglecting the left hand side of Equation (10), and balancing
Re[∆L] against the bootstrap term yields a RWM which grows at the rate:
τγ = −
w
ˆβ
1 −
w2
c
w2
−1
(20)
Thus, an island such that w < wc will have γ > 0 and will grow steadily
despite being below the NTM threshold width. When w approaches wc, the
growth rate is substantially enhanced by the coupling to the bootstrap term.
Indeed, as w → wc, τγ → ∞, but this is unphysical and the ordering then
breaks down. Similarly, for the assumptions that ω << ΩL and τω << 1, we
find the RWM frequency and toroidal rotation frequency:
τω =
2w3
Ω0
(1 − δ)ˆβ2
w2
c
w2
− 1
−2
(f + 2Aw5
)−1
(21)
ΩL =
Ω0
f + 2Aw5
(22)
We see from Equation (22) that the plasma rotation at the rational surface
ΩL decreases steadily in time from its initial value Ω0/f as the island grows.
The mode is initially locked to the wall (τω = 0 when w = 0), but as the island
grows, the mode frequency ω gradually increases. Nevertheless, it remains very
small, << ΩL. This is because Equation (20) for the growth rate means Im[∆L],
and therefore the electromagnetic torque, is very large. A consequence of this
is that there is no rotational stabilisation of the RWM exhibited in Equation
(20). This is different to the model presented in [28] where the bootstrap term
is not included. Then the RWM mode frequency rises faster with ΩL, leading
to a suppression of its growth rate as ΩL increases.
Note that ω is increasingly sensitive to w as w approaches wc, indicating
unlocking of the island and a dramatic spin-up. Indeed, as w → wc, Equation
(21) predicts τω → ∞ and the ordering is again broken. To recap, for w < wc
we find a slowly growing mode which is locked to the wall- the RWM. As the
island width w approaches wc, there is a substantial increase in the growth of
the island, and the mode begins to spin up. The small τγ, small τω ordering
then fails. Note from Equation (12) that Im[∆L] → 0 as τγ and τω grow. Thus,
the electromagnetic torque which locks the mode to the wall is reduced and the
mode will spin up to rotate with the plasma. We see from Equation (19) that
this forces Ω0 −fΩL ≈ 0, so the plasma also spins up towards the initial rotation
frequency profile. In Figure 3, we compare our analytic solutions in Equations
(20), (21) and (22) to numerical solutions of the full system, Equations (10),
(11) and (19). There is good agreement until w approaches wc, and then (as
expected) the analytic theory fails. Nevertheless, there is qualitative agreement,
both approaches showing a stronger growth and mode spin-up of the coupled
RWM-NTM system as w passes through wc. In the following subsection, we
employ our numerical solution to explore the coupled RWM-NTM mode in more
8
9. detail.
3.2 Dependence Of Numerical Solutions On ˆβ and Ω0
In the case with zero ˆβ, the NTM has no drive and we expect a conventional
RWM: a mode that rotates at a fraction of the plasma rotation, but acting to
slow the plasma, eventually locking to the wall. Rotation is expected to be
stabilising in this situation [28]. We have seen in the previous subsection both
analytically and numerically that at sufficiently large ˆβ the RWM couples to an
NTM, and takes on a different character. In this subsection we now explore the
essential physics of this coupling, considering the situation where a seed island
has a width smaller than the NTM threshold width wc, such that the NTM
solution is stable (i.e. wseed = wc/4).
Numerical solutions are shown for a range of ˆβ in Figure 4. We observe no
threshold seed island width for this coupled mode: a seed island of any size is
able to grow. In Figure 4a), the seed island grows steadily independent of ˆβ as
expected for a “classic” RWM until it reaches width wc. This is consistent with
the analytic solution, Equation (20), γ = /τ. As w increases above wc, the
island growth rate increases dramatically for high ˆβ as the bootstrap term then
becomes destabilising, providing an additional drive for the mode (this cannot
be captured in the analytic results). If ˆβ is reduced, then the NTM drive is
reduced and its effect on the island growth is either reduced or insignificant.
The plasma rotation frequency at r = rs, ΩL, is damped by torque exerted
by the unstable RWM, until the island width reaches wc (Figure 4c)). The
plasma then briefly spins up as the RWM couples to the NTM drive, before
again slowing to again lock to the wall at large w. Note that throughout the
period for which w ≥ wc, the plasma and mode are locked- the mode rotates
with the plasma. The magnitude of the spin up depends on ˆβ- increasing the
NTM drive clearly affects the RWM branch. The transient spin up takes place
over a time scale of 3-4 ms for the parameters we have chosen. (This may
invalidate the assumption that inertia can be neglected.) Decreasing ˆβ causes
the rotation spin up to reduce and eventually be removed, consistent with the
picture that the NTM drive is the underlying physics. It can be seen in Figure
4 that removing the NTM drive by reducing ˆβ causes the solution to behave
more as a classic RWM, with wall locking of both the mode and plasma.
The island evolution is also influenced by the amount of momentum injected
into the plasma core, as characterised by Ω0. This is shown in Figure 5. Note
that for our model the RWM is not sensitive to the plasma rotation, as evidenced
by the independence from Ω0 of the initial evolution of w until w = wc. This is
consistent with our analytic solution in Equation (20) but differs for models [28]
that do not include the NTM drive. The difference appears to be because the
coupled system has a stronger electromagnetic torque and the mode is locked to
the wall for even large rotation frequencies. It is a rotation of the mode relative
to the wall that provides the stabilisation in [28]. We find that the coupling
to the NTM is strongly influenced by flow, and the dramatic increase of island
9
10. growth rates for w > wc is only observed at the lowest Ω0. At higher Ω0, the
mode evolves as a classic RWM which is locked to the wall, with a substantially
reduced growth rate compared to the low Ω0 cases when w > wc.
4 Conclusion
In this paper, we have developed and analysed a simple model for the coupling
between a RWM and an NTM in tokamaks, retaining the effects of plasma
rotation. We find two branches- a mode which is essentially an NTM with a
threshold, and a coupled RWM-NTM which has no threshold.
While the RWM-NTM island width is small (below the NTM threshold),
the mode is a classic RWM, growing slowly (on a timescale characterised by
the wall resistive diffusion time) and locked to the wall. The plasma rotation
gradually slows during this phase. As this mode has small amplitude (w ∼ 1cm)
and is locked, it would be difficult to detect in an experimental situation. As
the island width exceeds the NTM threshold, there is a dramatic increase in
growth, particularly at high β as the mode couples to the NTM drive. At the
same time, the mode unlocks from the wall and instead rotates with the plasma.
The plasma rotation also increases at this time. The island continues to grow,
locked to the plasma flow, which gradually slows to lock to the wall at large
island width. This phase has the characteristics of an NTM.
Although the early evolution of the coupled RWM-NTM would be difficult
to detect experimentally, once the island width exceeds the NTM threshold wc,
and the mode spins up, it would then be detectable by Mirnov coils. Indeed,
at this time it would already have a width w ∼ wc (∼ 1cm) and so would have
the appearance of a triggerless NTM- a ∼ 1cm island which does not appear to
come from a seed island.
The phenomenon of triggerless NTMs near βno−wall
N , the ideal no-wall β
limit, can also be explained by invoking a pole in ∆ (which here is equal to
∆L/rs). As βN → βno−wall
N , ∆ → ∞, destabilising a classical tearing mode,
which in turn destabilises the neoclassical tearing mode when at a sufficiently
large amplitude [16]. In our model, this would be equivalent to letting ∆w → 0
and → 0, corresponding to the no-wall ideal MHD stability boundary. In
the model described here, the RWM is destabilised for βN > βno−wall
N but the
pole in ∆L is resolved by a dependence on the wall response. The physics
interpretation is therefore different to that provided in [16]. Our model has
some properties that can be tested experimentally, distinguishing it from the
model with a pole in ∆‘
. First, the plasma is expected to slow in the few 10’s
of milliseconds before the island width reaches wc. Then one would observe a
dramatic spin-up of the plasma, coincident with the appearance of the mode on
Mirnov coils, which would only last a few milliseconds before the plasma and
the mode slow and lock to the wall.
The emphasis of this paper is directed towards understanding the importance
of the RWM-NTM coupling in the presence of plasma rotation, not the physics
details of each mode. Thus, while we have retained the essential physics, we have
10
11. neglected more subtle effects such as the physics of rotation on the matching
of the layer solution to the external, ideal MHD solution [29]. Future work
should extend our model to include a more complete description of the RWM
and NTM physics in order to make quantitative predictions for the behaviour
of this couple RWM-NTM.
5 Acknowledgments
The support of the Engineering and Physical Sciences Research Council for
the Fusion Doctoral Training Network (Grant EP/K504178/1) is gratefully ac-
knowledged. This work was funded partly by the RCUK Energy Programme
under grant EP/I501045 and the European Communities under the contract
of Association between EURATOM and CCFE, and partly by EPSRC Grant
EP/H049460/1. The views and opinions expressed herein do not necessarily
reflect those of the European Commission.
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12
13. Figure 3: The comparison of the analytic solutions valid for w < wc with the
numerical solutions of the full system: a) and b) show both τγ and τω increasing
rapidly as w passes through wc. In c), the forced plasma spin up observed in
the numerical results is not captured in our analytic approximation. We have
fixed Ω0 = 100, wc = 0.02 and ˆβ = 1.0
13
14. Figure 4: Here w(t = 0) = 0.005 which is considerably smaller than the critical
seed island width wc = 0.02. The dependence of the island growth, growth rate
γ, mode frequency and plasma rotation frequency at the rational surface on β is
shown: increasing β (which increases the NTM drive) allows the island to grow
faster and to a larger size.
14
15. Figure 5: w(t = 0) = 0.005, ˆβ = 1.0 a) demonstrates that the initial plasma
rotation frequency has a noticeable effect on the island evolution, whereas b)
and c) contrast the mode growth rates and mode frequencies for differing values
of Ω0. d), e) and f) show the mode and rotation frequencies for the same values
of Ω0. Ω0 = 10000 corresponds to the plasma rotating at about 5% of the plasma
sound speed- this spin up only occurs for very small rotation frequencies.
15