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electron density profile as a function of normalized poloidal flux: psinorm ne(10^20/m^3) dne/dpsiN Pressure profile as a function of normalized poloidal flux: psinorm ptot(KPa) dptot/dpsiN scaled to oftain beta=5% at magnetic axis Magnetic safety factor profile as a function of normalized poloidal flux: psinorm q_profile All needed to perform the NOVA calculations.
Experiments have demonstrated that a Z-pinch can persist for thousands of times longer than the growth time of global magnetohydrodynamic (MHD) instabilities such as the m=0 sausage and m=1 kink modes. These modes have growth times on the order of ta=a/vi, where vi is the ion thermal speed and a is the pinch radius. Axial flows with duz/dr ≲ vi/a have been measured during the stable period, and the commonly accepted theory is that this amount of shear is sufficient to stabilize these modes as predicted by numerical studies using the ideal MHD equations. However, these studies only consider specific equilibrium profiles that typically have a modest magnitude for the logarithmic pressure gradient, qP≡d ln P/d ln r, and may not represent experimental conditions. Linear stability of the sheared-flow Z-pinch is studied here via a direct eigen-decomposition of the matrix operator obtained from the linear ideal MHD equations. Several equilibrium profiles with a large variation of qP are examined. Considering a practical range of k, 1/3 ≲ ka ≲ 10, it is shown that the shear required to stabilize m=0 modes can be expressed as duz/dr≥Cγ0/(ka)α. Here, γ0=γ0(ka) is the profile-specific growth rate in the absence of shear, which scales approximately with |qP|. Both C and α are profile-specific constants, but C is order unity and α≈1. It is further demonstrated that even a large value of shear, duz/dr=3vi/a, is not sufficient to provide linear stabilization of the m=1 kink mode for all profiles considered. This result is in contrast to the currently accepted theory predicting stabilization at much lower shear, duz/dr=0.1vi/a, and suggests that the experimentally observed stability cannot be explained within the linear ideal-MHD model.
Abstract not provided.
Abstract The successful application of three-dimensional (3D) magnetohydrodynamic (MHD) spectroscopy enables us to identify the multi-mode eigenvalues in DIII-D and KSTAR tokamak experiments with stable plasmas. The temporal evolution of the multi-modes’ stabilities have been detected. The new method is numerically efficient allowing the real time detection of MHD modes’ stabilities during the discharge. The method performs active detection of the plasma stability by utilizing the upper and lower rows of internal non-axisymmetric coils to apply a wide variety of 3D fields. Multi-mode eigenvalues are extracted using subspace system identification of the plasma response measured by 3D-field magnetic sensors distributed at different poloidal locations. The equivalence of this new method with the one introduced by Wang (2019 Nucl. Fusion 59 024001) has been numerically corroborated. The more robust and efficient calculation developed here will enable real time monitoring of the plasma stability based on the extracted eigenvalues of stable modes.
We discuss the time dependent behavior of some simple pulsed neutron simulations of subcritical problems in slab geometry. Our intent is to investigate the eigenvalue structure of the discretized neutron transport equation and to show under some reasonable assumptions that a dominant time eigenvalue exists that has a nonnegative eigenvector that determines the long time dependent behavior of the solution.
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Acoustic propagation in two-dimensional corrugated ducts is analyzed. A transform is made to a coordinate system that matches the duct walls. The resulting wave equation separates in space and time. The spatial differential equation is then solved by expanding in the eigenfunctions of a flat-walled duct. The prescription for this expansion is given by the appropriate application of time-independent perturbation theory. Calculations of first-order corrections for a periodic duct of both finite and infinite length are presented.
This report characterizes the sets of small disturbances possible in cylindrical and annular ducts with mean flow whose axial and tangential components vary arbitrarily with radius. The linearized equations of motion are presented and discussed, and then exponential forms for the axial, circumferential, and time dependencies of any unsteady disturbances are assumed. The resultant equations form a generalized eigenvalue problem, the solution of which yields the axial wavenumbers and radial mode shapes of the unsteady disturbances. Two numerical discretizations are applied to the system of equations: (1) a spectral collocation technique based on Chebyshev polynomial expansions on the Gauss-Lobatto points, and (2) second and fourth order finite differences on uniform grids. The discretized equations are solved using a standard eigensystem package employing the QR algorithm. The eigenvalues fall into two primary categories: a discrete set (analogous to the acoustic modes found in uniform mean flows) and a continuous band (analogous to convected disturbances in uniform mean flows) where the phase velocities of the disturbances correspond to the local mean flow velocities. Sample mode shapes and eigensystem distributions are presented for both sheared axial and swirling flows. The physics of swirling flows is examined with reference to hydrodynamic stability and completeness of the eigensystem expansions. The effect of assuming exponential dependence in the axial direction is discussed.
The statistical mechanics of Fourier models of ideal, homogeneous, incompressible magnetohydrodynamic (MHD) turbulence is discussed, along with their relevance for dissipative magnetofluids. Although statistical theory predicts that Fourier coefficients of fluid velocity and magnetic field are zero-mean random variables, numerical simulations clearly show that certain coefficients have a non-zero mean value that can be very large compared to the associated standard deviation, i.e., we have coherent structure. We use eigenanalysis of the modal covariance matrices in the probability density function to explain this phenomena in terms of `broken ergodicity', which is defined to occur when dynamical behavior does not match ensemble predictions on very long time-scales. We provide examples from 2-D and 3-D magnetohydrodynamic simulations of homogeneous turbulence, and show new results from long-time simulations of MHD turbulence with and without a mean magnetic field
A method for measuring the quantity of fluid in a tank may include the steps of selecting a match between a measured set of electromagnetic eigenfrequencies and a simulated plurality of sets of electromagnetic eigenfrequencies using a matching algorithm, wherein the match is one simulated set of electromagnetic eigenfrequencies from the simulated plurality of sets of electromagnetic eigenfrequencies, and determining the fill level of the tank based upon the match.
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We have developed a new global eigenvalue code, Multiscale Analysis for plasma Stabilities (MAS), for studying plasma problems with wave toroidal mode number (n) and frequency (ω) in a broad range of interest in general tokamak geometry, based on a five-field Landau-fluid description of thermal plasmas. Beyond keeping the necessary plasma fluid response, we further retain the important kinetic effects including diamagnetic drift, ion finite Larmor radius, finite parallel electric field (E||), ion and electron Landau resonances in a self-consistent and non-perturbative manner without sacrificing the attractive efficiency in computation. The physical capabilities of the code are evaluated and examined in the aspects of both theory and simulation. In theory, the comprehensive Landau-fluid model implemented in MAS can be reduced to the well-known ideal MHD model, electrostatic ion-fluid model, and drift-kinetic model in various limits, which clearly delineates the physics validity regime. In simulation, MAS has been well benchmarked with theory and other gyrokinetic and kinetic-MHD hybrid codes in a manner of adopting the unified physical and numerical framework, which covers the kinetic Alfv\'en wave (KAW), ion sound wave (ISW), low-n kink, high-n ion temperature gradient mode (ITG) and kinetic ballooning mode (KBM). Moreover, MAS is successfully applied to model the Alfv\'en eigenmode (AE) activities in DIII-D discharge #159243, which faithfully captures the frequency sweeping of reversed shear Alfv\'en eigenmode (RSAE), the tunneling damping of toroidal Alfv\'en eigenmode (TAE), as well as the polarization characteristics of kinetic beta-induced Alfv\'en eigenmode (KBAE) and beta-induced Alfv\'en-acoustic eigenmode (BAAE) being consistent with former gyrokinetic theory and simulation. With respect to the key progress contributed to the community, MAS has the advantage of combining rich physics ingredients, realistic global geometry and high computation efficiency together for plasma stability analysis in linear regime.
Recent JET deuterium experiments with an advanced tokamak scenario using an internal transport barrier (ITB) have been performed to clearly observe destabilized toroidicity-induced Alfvén eigenmodes (TAEs) by fast ions; interestingly, these also exhibit unstable electromagnetic (EM) perturbations in the sub-TAE frequency range. We identify such EM perturbations to be beta-induced ion temperature gradient (BTG) eigenmodes and not beta-induced Alfvén eigenmodes nor beta-induced Alfvén acoustic eigenmodes, which are often unstable in such high-beta plasmas with high-power neutral beam injection (NBI). The BTG modes are the most unstable modes due to the high thermal ion temperature gradient related to the ITB, high thermal ion temperature compared to thermal electron temperature (high T i /T e ), and a high ion beta regime. BTG mode experimental characteristics match the analytical theory, i.e., location in the vicinity of a rational magnetic surface with a low magnetic shear, mode frequency scaling with the ion diamagnetic frequency (ω$^*_i$), and a coupling among Alfvén and drift waves. We also perform linear gyrokinetic simulations with validated plasma profiles and equilibrium, and find a mode kinetically driven by thermal ions with similar characteristics as the experimental BTG modes.
The presence of discrete frequencies in the resonant response of the magnetosphere is considered. The eigenequation characterizing the eigenmodes of the hydromagnetic waves in a cylindrical model for the outer magnetosphere is shown to have two turning points along the radial axis. The turning point locations are related to the values of the eigenperiod and the associated east-west wavenumber of the eigenmode. It is also found that only the discrete set of the magnetospheric cavity eigenmodes can efficiently couple the perturbations excited on the boundary of the magnetosphere to the field-line resonant mode excited inside the turning point of the cavity eigenmode.