# Browsing by Subject "Magnetohydrodynamics"

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Item Alfvén modes and wave-particle interaction in a tokamak(2017-09-28) Li, Meng, 1991-; Breizman, Boris N.; Berk, Herbert; Morrison, Philip; Fitzpatrick, Richard; Gamba, IreneShow more This work is motivated by the nonlinear wave-particle interaction problems. To build a self-consistent theory, we consider eigenmodes of the bulk plasma as well as the dynamics of the energetic particles. The modes of our particular interest are the Alfvén Cascades and the Toroidicity Alfven Eigenmodes (TAE), which we describe using Magnetohydrodynamic(MHD) analysis and the AEGIS codes. We investigate the stabilizing effect for the Alfvenic waves from continuum damping, especially near the TAE gap. For the kinetic description of the energetic particles, we propose new canonical straight field line coordinates to model the guiding center motion. We then formulate wave-particle interaction problem using the action-angle variables. In Chapter 2, we interpret Alfvén Cascades observed in Madison Symmetric Torus (MST). We do linear MHD calculations and find the mode frequency, structure, and stability boundary. We then perform MHD simulation using the AEGIS code, with the equilibrium reconstructed from experiment. The result is discussed and compared with the experimentally observed features. In Chapter 3, we analyze continuum damping for Alfvénic waves, especially in the extreme situation near the TAE gap. We find that the continuum tip absorption feature is actually related to the existing of TAEs in the gap. On the technical level, we improve the numerical scheme of AEGIS and resolve two closely-spaced singularities. As a result, the absorption features observed in the simulation show good agreement with our analytical calculation. In order to simulate the energetic particle guiding center motion in the Hamiltonian form, we propose a new set of straight magnetic field line coordinates. The new coordinates exist for general tokamak devices and facilitate both MHD calculations and energetic particles. The new coordinate system makes it very convenient to take the advantage of the Hamiltionian structure of the guiding center motion. We use a canonical transformation to action-angle variables to formulate the interaction model for particles. The action-angle variables allow us to resolve wave-particle resonances and describe the conserved quantities for resonance particles. The model can give us a complete picture for nonlinear stage of wave-particle interaction.Show more Item Aspects of relativistic Hamiltonian physics(2015-08) D'Avignon, Eric Cavell; Morrison, Philip J.; Shepley, Lawrence; Rindler, Wolfgang; Mahajan, Swadesh; Hazeltine, Richard; Shvets, GennadyShow more This dissertation presents various new results in relativistic Hamiltonian plasma physics. It begins with an overview of Hamiltonian physics, with an emphasis on noncanonical brackets, and presents various nonrelativistic systems to be generalized later on. There then follows an exposition on action principles for Hall and Extended MHD, which allow the derivation of the noncanonical Hamiltonian brackets for those systems. I next discuss the transition to relativistic Hamiltonian systems, and the special difficulties that arise in this step. A detailed exploration of relativistic Hamiltonian MHD follows, using a novel bracket formulation. This chapter also investigates alternative brackets, gauge degeneracies, and Casimir invariants. Next I lay out the connection between Lagrangian and Eulerian MHD (both in Hamiltonian forms), and present some early work on a bracket-based formulation of the relativistic Navier-Stokes equation. The next chapters develop various results using an antisymmetric relativistic spin tensor, and several unexpected and intriguing physical consequences of the Jacobi identity. I conclude with a program of future research and several useful appendices.Show more Item Global instabilities in rotating magnetized plasmas(2009-05) Pino, Jesse Ethan, 1981-; Mahajan, Swadesh M.; Hazeltine, R. D. (Richard D.)Show more The Magnetorotational Instability (MRI) is believed to be the primary mechanism for angular momentum transfer in astrophysical accretion disks. This instability, which exists in ionized disks in the presence of weak magnetic fields, can either transfer angular momentum directly, or give rise to anomalous viscosity via non-linear turbulence. While many previous analytical treatments are concerned with the local theory of the MRI, when the length scale of rotation shear is comparable to the length scale of the most unstable modes, a global analysis is necessary. In this dissertation we investigate the global theory of the linear MRI. In particular, we show how rotation shear can localize global modes and how the global growth rates can differ signicantly from the local approximation in certain cases. Changes in the equilibrium density are considered. In addition, the effects of Hall Magnetohydrodynamics on the MRI are studied in both the local and global cases.Show more Item Hamiltonian description of Hall and sub-electron scales in collisionless plasmas with reduced fluid models(2018-09-14) Miloshevich, George; Morrison, Philip J.; Hazeltine, Richard; Breizman, Boris; Fitzpatrick, Richard; Gamba, Irene MShow more In MHD magnetic helicity has been shown to represent Gauss linking numbers of magnetic field lines by Moffatt and others; thus it is endowed with topological meaning. The noncanonical Hamiltonian formulation of extended MHD models (that take two-fluid effects into account) has been used to arrive at their common mathematical structure, which manifests itself via the existence of two generalized helicities and two Lie-dragged 2-forms. The helicity invariants play an important role in the second part of thesis dedicated to understanding the directionality of turbulent cascades. Generally speaking, invariants (such as energy) can flow in two directions in a turbulent cascade: forward (towards small scales, leading to dissipation) and inverse (towards large scales), leading to the formation of a condensate. This directionality in extended MHD models is estimated using analytical considerations as well as tests involving 2D numerical simulations. The cascade reversal (transition) of the square magnetic vector potential is found, viz. when the forcing wavenumber exceeds the inverse electron skin depth the square magnetic vector potential starts to flow towards large wavenumbers, as opposed to the typical MHD behavior. In addition, the numerics suggest a simultaneous transition to the inverse cascade of energy in this inertial MHD regime. This is accompanied by the appearance of large scale structures in the velocity field, as opposed to the magnetic field as in the MHD case. Final chapters of the thesis are devoted to devising the action principle for the relativistic extended MHD. First the special relativistic version is discussed, where the covariant noncanonical Poisson bracket is found. This is followed by a short recourse towards describing relativistic collisionless reconnection mediated by the electron thermal inertia (purely relativistic effect). Next, 3+1 splitting inside the Poisson bracket is performed, while only non-relativistic terms are retained. Thus one arrives at nonrelativistic extended MHD bracket with arbitrary ion to electron mass ratio. In conclusion, it is outlined how the Hamiltonian 3+1 formalism can be developed for general relativistic Hall MHD using canonical Clebsch parametrization and some comments are added on possible issues regarding the quasi-neutrality assumption in the model that is used throughout the chapter.Show more Item Hybridized discontinuous Galerkin methods for magnetohydrodynamics(2018-11-02) Shannon, Stephen James; Bui-Thanh, Tan; Arbogast, Todd; Demkowicz, Leszek; Ghattas, Omar; Shadid, John; Waelbroeck, FrançoisShow more Discontinuous Galerkin (DG) methods combine the advantages of classical finite element and finite volume methods. Like finite volume methods, through the use of discontinuous spaces in the discrete functional setting, we automatically have local conservation, an essential property for a numerical method to behave well when applied to hyperbolic conservation laws. Like classical finite element methods, DG methods allow for higher order approximations with compact stencils. For time-dependent problems with implicit time stepping and for steady-state problems, DG methods give a larger globally coupled linear system than continuous Galerkin methods (especially for three dimensional problems and low polynomial orders). The primary motivation of the hybridized (or hybridizable) discontinuous Galerkin (HDG) methods is to reduce the number of globally coupled unknowns in DG methods when implicit time stepping or direct-to-steady-state solutions are desired. This is accomplished by the introduction of new “trace unknowns” defined on the mesh skeleton, the definition of one-sided numerical fluxes, and the enforcement of local conservation. This results in a globally coupled linear system where the local “volume unknowns” can be eliminated in a Schur complement procedure, resulting in a reduced globally coupled system in terms of only the trace unknowns. Magnetohydrodynamics (MHD) is the study of the flow of electrically conducting fluids under the influence of magnetic fields. The MHD equations are used to describe important physical phenomena including laboratory plasmas (plasma confinement in fusion energy devices), astrophysical plasmas (solar coronas, planetary magnetospheres) and liquid metal flows (metallurgy processes, the Earth’s molten core, cooling for nuclear reactors). Incompressible MHD, which is the main focus of this work, is relevant in low Lundquist number liquid metals, in high Lundquist number, large guide field fusion plasmas, and in low Mach number compressible flows. The equations of MHD are highly nonlinear, and are characterized by physical phenomena spanning wide ranges of length and time scales. For numerical methods, this presents challenges in both spatial and temporal discretization. In terms of temporal discretization, fully implicit numerical methods are attractive in their robustness; they allow for stable, high-order time integration over long time scales of interest.Show more Item Lie symmetry analysis of the 3-dimensional reduced Hall MHD equations(2022-05-09) Koutsomitopoulos, Panagiotis Tomonori; Hazeltine, R. D. (Richard D.); Fitzpatrick, RichardShow more To find symmetric Lie groups of a reduced form of the non-linear 3-dimensional Hall Magnetohydrodynamic equations, symmetry analysis was used to find determining equations and solve for the infinitesimal generating functions. Brian Cantwell’s Mathematica program was used to generate determining equations, and original Python code was written in order to simplify the process of reducing Mathematica output. Several symmetries were found, the most interesting of which are quadratic vortex tubes which extend through the plasma in the normal direction.Show more Item Magneto-hydrodynamics simulation study of high density thermal plasmas in plasma acceleration devices(2013-08) Sitaraman, Hariswaran; Raja, Laxminarayan L.Show more The development of a Magneto-hydrodynamics (MHD) numerical tool to study high density thermal plasmas in plasma acceleration devices is presented. The MHD governing equations represent eight conservation equations for the evolution of density, momentum, energy and induced magnetic fields in a plasma. A matrix-free implicit method is developed to solve these conservation equations within the framework of an unstructured grid finite volume formulation. The analytic form of the convective flux Jacobian is derived for general unstructured grids. A Lower Upper Symmetric Gauss Seidel (LU-SGS) technique is developed as part of the implicit scheme. A coloring based algorithm for parallelization of this technique is also presented and its computational efficiency is compared with a global matrix solve technique that uses the GMRES (Generalized Minimum Residual) algorithm available in the PETSc (Portable Extensible Toolkit for Scientific computation) libraries. The verification cases used for this study are the MHD shock tube problem in one, two and three dimensions, the oblique shock and the Hartmann flow problem. It is seen that the matrix free method is comparatively faster and shows excellent scaling on multiple cores compared to the global matrix solve technique. The numerical model was thus verified against the above mentioned standard test cases and two application problems were studied. These include the simulation of plasma deflagration phenomenon in a coaxial plasma accelerator and a novel high speed flow control device called the Rail Plasma Actuator (RailPAc). Experimental studies on coaxial plasma accelerators have revealed two different modes of operation based on the delay between gas loading and discharge ignition. Longer delays lead to the detonation or the snowplow mode while shorter delays lead to the relatively efficient stationary or deflagration mode. One of the theories that explain the two different modes is based on plasma resistivity. A numerical modeling study is presented here in the context of a coaxial plasma accelerator and the effect of plasma resistivity is dealt with in detail. The simulated results pertaining to axial distribution of radial currents are compared with experimental measurements which show good agreement with each other. The simulations show that magnetic field diffusion is dominant at lower conductivities which tend to form a stationary region of high current density close to the inlet end of the device. Higher conductivities led to the formation of propagating current sheet like features due to greater convection of magnetic field. This study also validates the theory behind the two modes of operation based on plasma resistivity. The RailPAc (Rail Plasma Actuator) is a novel flow control device that uses the magnetic Lorentz forces for fluid flow actuation at atmospheric pressures. Experimental studies reveal actuation ~ 10-100 m/s can be achieved with this device which is much larger than conventional electro-hydrodynamic (EHD) force based plasma actuators. A magneto-hydrodynamics simulation study of this device is presented. The model is further developed to incorporate applied electric and magnetic fields seen in this device. The snowplow model which is typically used for studying pulsed plasma thrusters is used to predict the arc velocities which agrees well with experimental measurements. Two dimensional simulations were performed to study the effect of Lorentz forcing and heating effects on fluid flow actuation. Actuation on the order of 100 m/s is attained at the head of the current sheet due to the effect of Lorentz forcing alone. The inclusion of heating effects led to isotropic blast wave like actuation which is detrimental to the performance of RailPAc. This study also revealed the deficiencies of a single fluid model and a more accurate multi-fluid approach is proposed for future work.Show more Item Measurement of the pressure drop of a liquid metal flowing through a packed bed of uniform conducting spheres with transverse magnetic field(1995-05) McWhirter, Jon David; Not availableShow more Item MHD spectroscopy of tokamak plasmas using Alfvén waves(2020-03-16) Oliver, Henry James Churston; Breizman, Boris N.; Sharapov, S. E. (Sergei E.); Hazeltine, Richard D; Mahajan, Swadesh M; Morrison, Philip JShow more Alfvén waves are electromagnetic waves that occur in magnetised plasmas. Alfvén waves are routinely observed in tokamaks — toroidal devices that confine plasma using magnets. Energetic ions that are used to heat the plasma within tokamaks can drive the Alfvén waves unstable. Additionally, alpha particles produced in fusion reactions may destabilise the wave. Alfvén waves with sufficiently high amplitudes can eject energetic particles from the plasma, damaging the reactor and decreasing fusion efficiency. When these waves are not strong enough to eject particles from the plasma, their benign behaviour can be used to diagnose the plasma. This technique is known as magnetohydrodynamic (MHD) spectroscopy. In this thesis, we outline three new techniques of MHD spectroscopy that we have developed. The first new method of MHD spectroscopy was developed in plasmas composed of hydrogen and deuterium in the Mega Ampere Spherical Tokamak (MAST). Compressional Alfvén eigenmodes (CAEs) and global Alfvén eigenmodes (GAEs) were suppressed in plasmas with high hydrogen concentrations. At the highest hydrogen concentrations investigated, high frequency ion-ion hybrid waves appeared. We used a 1D model of the refractive index and wave-particle resonances to explain these observations and estimate the relative ion concentration at which the spectrum of excited waves changed. These estimates agreed with experimental observations, suggesting the spectrum of excited waves can be used to diagnose the relative ion concentrations for plasmas with two ion species. The second new form of MHD spectroscopy was developed using observations of axisymmetric modes in experiments on the Joint European Torus (JET). Axisymmetric modes do not change in the toroidal direction and are driven unstable by energy gradients in the fast particle distribution function. Therefore, we can use observations of the axisymmetric mode to infer information about the gradient of the fast particle energy distribution function. We explained how these axisymmetric modes can exist without heavy damping. We also examined how the elongation of the plasma column modifies the mode using numerical and analytical tools. The final MHD spectroscopic technique was developed for JET plasmas injected with pellets of frozen deuterium, which are used to refuel the plasma core. We demonstrated how key pellet parameters can be inferred from dramatic changes to the Alfvén eigenfrequencies that we observed in JET. MHD spectroscopy of pellet injected plasmas was enabled by generalising two 3D MHD codes to incorporate 3D density profiles. 3D density profiles were generated using a model for the expansion of the pellet wake along a magnetic field line derived from the fluid equations. From the change in mode frequency, we estimate the density of the pellet wake and the time-scale for poloidal homogenisation of the wake. Before presenting these studies, we introduce the basics of fusion, tokamaks, and the models used to describe tokamak plasmas. We then discuss the MHD waves that we will use for MHD spectroscopy of tokamak plasmas, and how these waves are excited by fast particles. The three new methods of MHD spectroscopy are then discussed.Show more Item Rotating mirror plasmas in the quest of magnetofluid states(2006) Quevedo, Hernan Javier; Bengtson, Roger D.; Mahajan, Swadesh M.Show more The goal of this dissertation is to describe and discuss the first steps taken by the Magneto Bernoulli eXperiment (MBX) to create magnetofluid states in the laboratory using a rotating plasma in an external mirror magnetic field. The terminology magnetofluid has been introduced to characterize a plasma model, based on 2-fluid theory, that treats the flow and the magnetic field in a symmetrical way. Many interesting astrophysical and laboratory problems involve large flows and fall in this category. Based on the set of parameters where MBX should run, we set up the experiment, and added different probes to diagnose the rotating plasma. We have also installed a data acquisition system, and set up an archive system (to store the data) that can be accessed worldwide. Experimental results demonstrate that supersonic flows can be generated with biasing electrodes at the throat of the mirror magnetic field. Alfvenic flows needed for a transition to magnetofluid states could not be reached because the initial plasma density was too low. At low bias (slow rotational speed) the plasma has E × B/B2 drift rotation and the magnetic fields lines are equipotentials. With a higher bias, we observed large potential drops along the field lines. We also observed an asymmetry in the polarity of the bias which leads to constraints in the control of the sheared plasma flow. We present a model that captures many of these features. In conjunction with experimental efforts we develop a theory for a rotating plasma embedded in an external mirror magnetic field. An analytic solution that involves rigid rotation of the plasma shows important differences between a 2-fluid system and ideal MHD. We find high non equipotential magnetic lines and asymmetry to compare with the experimental results.Show more Item Tearing mode dynamics in tokamak plasmas(2016-05) Vergos, Nikolaos; Fitzpatrick, Richard, 1963-; Hazeltine, Richard; Breizman, Boris; Waelbroeck, Francois; Hallock, GaryShow more One of the most problematic instabilities in tokamak plasmas is tearing modes; they are driven by current and pressure gradients, and involve a reconfiguration of the magnetic and velocity fields localized into a narrow region located at a resonant magnetic surface. While the equilibrium magnetic field lines are located on concentric nested toroidal flux surfaces, the instability creates magnetic islands in which field lines connect flux tubes together, allowing for a high radial heat transport, and, thus, resulting in a loss of confinement, and, potentially, disruptions. In order for the magnetic field lines to break and reconnect, we need to take into account the resistivity of the plasma and solve the resistive magnetohydrodynamics (MHD) equations. The analytical solution consists of a boundary layer analysis (asymptotic matching) and takes advantage of the small radial width of the region where the perturbations vary significantly. Indeed, ideal magnetohydrodynamics can be used everywhere except in that narrow region where the full resistive problem must be solved. This dissertation addresses two related problems in the study of resistive tearing modes, and their interactions with externally induced resonant magnetic perturbations (error-fields). First, an in-depth investigation of the bifurcated states of a rotating, quasi-cylindrical, tokamak plasma in the presence of a resonant error-field is performed, within the context of constant-ψ resistive MHD theory. The response of the rotating plasma is studied in both the linear, and the nonlinear regime. In general, there is a "forbidden band" of tearing mode rotation frequencies that separates a branch of high-frequency solutions from a branch of low-frequency solutions. When a high-frequency solution crosses the upper boundary of the forbidden band there is a bifurcation to a low-frequency solution, and vice versa. Second, the analysis is extended to include the study of braking and locking of tearing mode rotation by the interaction of the mode with an error-field. It is found that this interaction can brake the plasma rotation, suppress magnetic island evolution and drive locked modes.Show more