Transport in chaotic systems

This dissertation addresses the general problem of transport in chaotic systems. Typical fluid problem of the kind is the advection and diffusion of a passive scalar. The magnetic field evolution in a chaotic conducting media is an example of the chaotic transport of a vector field. In kinetic theory, the collisional relaxation of a distribution function in phase space is also an advection-diffusion problem, but in a higher dimensional space.;In a chaotic flow neighboring points tend to separate exponentially in time, exp({dollar}\omega t{dollar}) with {dollar}\omega{dollar} the Liapunov exponent. The characteristic parameter for the transport of a scalar in a chaotic flow is {dollar}\Omega\ \equiv\ \omega L\sp2/D{dollar} where L is the spatial scale and D is the diffusivity. For {dollar}\Omega\ \gg\ 1{dollar}, the scalar is advected with the flow for a time {dollar}t\sb{lcub}a{rcub}\ \equiv{dollar} ln(2{dollar}\Omega{dollar})/2{dollar}\omega{dollar} and then diffuses during the relatively short period 1/{dollar}\omega{dollar} centered on the time {dollar}t\sb{lcub}a{rcub}{dollar}. This rapid diffusion occurs only along the field line of the {dollar}\rm \ s\sb\infty{dollar} vector, which defines the stable direction for neighboring streamlines to converge. Diffusion is impeded at the sharp bends of an {dollar}\rm \ s{dollar} line because of a peculiarly small finite time Lyapunov exponent, hence a class of diffusion barriers is created inside a chaotic sea. This result comes from a fundamental relationship between the finite time Lyapunov exponent and the geometry of the {dollar}\rm \ s{dollar} lines, which we rigorously show in 2D and numerically validated for 3D flows.;The evolution of a general 3D magnetic field in a highly conducting chaotic media is also related to the spatial-temporal dependence of the finite time Lyapunov exponent. The Ohmic dissipation in a chaotic plasma will become a dominate process despite a small plasma resistivity. We show that the Ohmic heating in a chaotic plasma occurs in current filaments or current sheets. The particular form is determined by the time dependence of spatial gradient of the finite time Lyapunov exponent along a direction in which neighboring point neither diverge nor converge

On the collisional damping of plasma velocity space instabilities

For plasma velocity space instabilities driven by particle distributions significantly deviated from a Maxwellian, weak collisions can damp the instabilities by an amount that is significantly beyond the collisional rate itself. This is attributed to the dual role of collisions that tend to relax the plasma distribution toward a Maxwellian and to suppress the linearly perturbed distribution function. The former effect can dominate in cases where the unstable non-Maxwellian distribution is driven by collisionless transport on a time scale much shorter than that of collisions, and the growth rate of the ideal instability has a sensitive dependence on the distribution function. The whistler instability driven by electrostatically trapped electrons is used as an example to elucidate such a strong collisional damping effect of plasma velocity space instabilities, which is confirmed by first-principles kinetic simulations

Runaway electron current reconstitution after a non-axisymmetric magnetohydrodynamic flush

Benign termination of mega-ampere (MA) level runaway current has been convincingly demonstrated in recent JET and DIII-D experiments, establishing it as a leading candidate for runaway mitigation on ITER. This comes in the form of a runaway flush by parallel streaming loss along stochastic magnetic field lines formed by global magnetohydrodynamic instabilities, which are found to correlate with a low-Z injection that purges the high-Z impurities from a post-thermal-quench plasma. Here we show the competing physics that govern the post-flush reconstitution of the runaway current in a ITER-like reactor where significantly higher current is expected. The trapped ``runaways'' are found to dominate the seeding for runaway reconstitution, and the incomplete purge of high-Z impurities helps drain the seed but produces a more efficient avalanche, two of which compete to produce a 2-3~MA step in current drop before runaway reconstitution of the plasma current

Electromagnetic turbulence simulation of tokamak edge plasma dynamics and divertor heat load during thermal quench

The edge plasma turbulence and transport dynamics, as well as the divertor power loads during the thermal quench phase of tokamak disruptions are numerically investigated with BOUT++'s flux-driven, six-field electromagnetic turbulence model. Here a transient yet intense particle and energy sources are applied at the pedestal top to mimic the plasma power drive at the edge induced by a core thermal collapse, which flattens core temperature profile. Interesting features such as surging of divertor heat load (up to 50 times), and broadening of heat flux width (up to 4 times) on the outer divertor target plate, are observed in the simulation, in qualitative agreement with experimental observations. The dramatic changes of divertor heat load and width are due to the enhanced plasma turbulence activities inside the separatrix. Two cross-field transport mechanisms, namely the $E\times B$ turbulent convection and the stochastic parallel advection/conduction, are identified to play important roles in this process. Firstly, elevated edge pressure gradient drives instabilities and subsequent turbulence in the entire pedestal region. The enhanced turbulence not only transports particles and energy radially across the separatrix via $E\times B$ convection which causes the initial divertor heat load burst, but also induces an amplified magnetic fluctuation $\tilde{B}$. Once the magnetic fluctuation is large enough to break the magnetic flux surface, magnetic flutter effect provides an additional radial transport channel. In the late stage of our simulation, $|\tilde{B}_r/B_0|$ reaches to $10^{-4}$ level that completely breaks magnetic flux surfaces such that stochastic field-lines are directly connecting pedestal top plasma to the divertor target plates or first wall, further contributing to the divertor heat flux width broadening.Comment: 20 pages, 12 figure

A mimetic finite difference based quasi-static magnetohydrodynamic solver for force-free plasmas in tokamak disruptions

Force-free plasmas are a good approximation where the plasma pressure is tiny compared with the magnetic pressure, which is the case during the cold vertical displacement event (VDE) of a major disruption in a tokamak. On time scales long compared with the transit time of Alfven waves, the evolution of a force-free plasma is most efficiently described by the quasi-static magnetohydrodynamic (MHD) model, which ignores the plasma inertia. Here we consider a regularized quasi-static MHD model for force-free plasmas in tokamak disruptions and propose a mimetic finite difference (MFD) algorithm. The full geometry of an ITER-like tokamak reactor is treated, with a blanket module region, a vacuum vessel region, and the plasma region. Specifically, we develop a parallel, fully implicit, and scalable MFD solver based on PETSc and its DMStag data structure for the discretization of the five-field quasi-static perpendicular plasma dynamics model on a 3D structured mesh. The MFD spatial discretization is coupled with a fully implicit DIRK scheme. The algorithm exactly preserves the divergence-free condition of the magnetic field under the resistive Ohm's law. The preconditioner employed is a four-level fieldsplit preconditioner, which is created by combining separate preconditioners for individual fields, that calls multigrid or direct solvers for sub-blocks or exact factorization on the separate fields. The numerical results confirm the divergence-free constraint is strongly satisfied and demonstrate the performance of the fieldsplit preconditioner and overall algorithm. The simulation of ITER VDE cases over the actual plasma current diffusion time is also presented.Comment: 43 page