Session TM10: Mini-Conference: Mixing in Fusion Plasmas I

Basics of Turbulent Mixing

This talk will be a review of the work done by the speaker and his collaborators on the topic of mixing of substances by means of an underlying turbulent motion. Most attention will be placed on mixing in homogeneous flows and basic configurations. Phenomena covering large range of the diffusivity of the mixed substances will be covered. Where possible, comparisons between experiment, theory and computations will be made. Open problems will be pointed out.   

Turbulent transport in the flows affected by curvature and rotation

Navier Stokes equations for averaged flow parameters combined with a closure model for turbulent Reynolds stress tensor have been successfully applied for modeling of flows. Closure models for turbulent Reynolds stress tensor and their applicability conditions will be reviewed. Many flows can be correctly modeled using turbulent viscosity closure model. In this model, the turbulent stress tensor is proportional to the strain rate tensor. However, modifications of the turbulent viscosity closure models are required for flows affected by effects of curvature, rotation, or volumetric forces. The progress towards creating modification of the model applicable to several physical effects will be presented.   

The role of mix in inhibiting ignition in inertial confinement fusion schemes driven by either X-ray ablation or magnetized liners

We report on investigations into how mix can prevent the ignition of hotspot fuel in ICF schemes. In the mainstream approach using X-ray driven ablation, surface defects can be amplified by the Rayleigh-Taylor instability during ablation leading to feed-through of the ablator material which contaminates the hotspot material leading to radiative losses. Even without this feed through, asymmetries arising at the interface between the hotspot and the cold dense fuel layer are amplified by the Rayleigh-Taylor instability during the deceleration phase, leading to mixing of the dense cold fuel layer into the hotspot. This pulls material with low specific enthalpy into the hotpot, lowering the average hotspot temperature and quenching the burn. An alternative approach to achieving ignition is the Magnetized Liner Inertial Fusion scheme, where the fuel is rapidly compressed within a metallic liner using the magnetic pressure generated by a current of several mega-amps. Here the growth of the magneto-Rayleigh-Taylor instability can generate asymmetries in the target which promote the mix of the metallic liner with the fuel or mix of the cold dense regions of fuel with the central hotspot. In this talk we make use of large scale 3D radiation hydrodynamics and MHD models of X-ray driven and liner driven fusion schemes to highlight the roles played by mix in each case. We show that thermal conduction sets characteristic spatial scales, below which mix of the cold fuel with the hotspot becomes increasingly important.   

Anisotropic electron fluid closure relevant to collisionless dynamics in magnetized plasma

Spacecraft data show that electron pressure anisotropy develops in collisionless plasmas. This is in contrast to the results of ``Braginskii-type'' investigations, which suggest this anisotropy should be small. However, such theoretical studies exclude the effects of dynamic electron trapping, which is a non-linear effect and is, therefore, eliminated when linearizing the underlying kinetic equations. A general analytic model is derived for the electron guiding center distribution of an expanding flux tube including trapping as a principle driver of pressure anisotropy [1]. While the work is inspired by the problem of magnetic reconnection, the resulting closure obtained for the electron fluid equations is general and is likely applicable to fast dynamics in magnetically confined fusion plasmas. \\[1ex] [1] Egedal J, Le A, and Daughton W, ``A review of pressure anisotropy caused by electron trapping in collisionless plasma, and its implications for magnetic reconnection,'' (2013) Phys. Plasmas, 20, 061201.   

Nonlocal transport in the presence of transport barriers

There is experimental, numerical, and theoretical evidence that transport in plasmas can, under certain circumstances, depart from the standard local, diffusive description. Examples include fast pulse propagation phenomena in perturbative experiments, non-diffusive scaling in L-mode plasmas, and non-Gaussian statistics of fluctuations. From the theoretical perspective, non-diffusive transport descriptions follow from the relaxation of the restrictive assumptions (locality, scale separation, and Gaussian/Markovian statistics) at the foundation of diffusive models. We discuss an alternative class of models able to capture some of the observed non-diffusive transport phenomenology. The models are based on a class of nonlocal, integro-differential operators that provide a unifying framework to describe non- Fickian scale-free transport, and non-Markovian (memory) effects. We study the interplay between nonlocality and internal transport barriers (ITBs) in perturbative transport including cold edge pulses and power modulation. Of particular interest in the nonlocal ``tunnelling'' of perturbations through ITBs. Also, flux-gradient diagrams are discussed as diagnostics to detect nonlocal transport processes in numerical simulations and experiments.   

Intrinsic MHD mixing in magnetically confined fusion plasma

Magnetically confined toroidal plasmas for fusion research have proven far more complex than originally envisioned. Recent results from magnetohydrodynamics (MHD), based partly on large scale numerical simulation, show that much of the complexity is intrinsic, even when turbulent fluctuations are neglected. Perturbative theories based on simplified models provide much of the understanding of MHD, but ignore major nonlinear and mixing effects. At the plasma edge, the magnetic boundary surface introduces a geometrical source of magnetic stochasticity. X-points on the surface help create an optimal plasma shape. Nonlinearly, the Hamiltonian nature of the field leads to ``homoclinic'' magnetic tangles near the X-points that produce plasma mixing and stochasticity, which can penetrate deep into the plasma. New work shows that compressible MHD also contributes to complexity, by effectively breaking small parameter orderings in slowly growing instabilities. In the $m=1$, $n=1$ internal kink of the central plasma, higher order terms determine the growth rate. Nonlinearly, a fast, explosive growth phase with strong stochasticity leads to a sawtooth crash similar to experiment. Other plasma instabilities, including reconnection in space plasmas, can have analogous behavior.   

Experimental investigation of multi-scale non-equilibrium plasma dynamics

Lab experiments at Caltech resolve complex, detailed MHD dynamics spatially and temporally. Unbalanced forces drive fast plasma flows which tend to self-collimate via self-pinching. Collimation results from flow stagnation compressing embedded magnetic flux and so amplifying the magnetic field responsible for pinching. Measurements show that the collimated flow is essentially a dense plasma jet with embedded axial and azimuthal magnetic fields, i.e., a magnetic flux tube (flux rope). The measured jet velocity is in good agreement with an MHD acceleration model. Depending on how flux tube radius varies with axial position, jets flow into a flux tube from both ends or from just one end. Jets kink when the flux tube in which they are embedded breaches the Kruskal-Shafranov stability limit. The lateral acceleration of a sufficiently strong kink can produce an enormous effective gravity which provides the environment for an observed fine-scale, extremely fast Rayleigh-Taylor (RT) instability. The RT can erode the jet current channel to be smaller than the ion skin depth so there is a cascade from the ideal MHD scale of the kink to the non-MHD ion skin depth scale. This process can result in a magnetic reconnection whereby the jet and its embedded flux tube break.   

Nonuniformity Mitigation of Beam Illumination in Heavy Ion Inertial Fusion

In heavy ion inertial fusion wobbling heavy ion beam (HIB) illumination was proposed to realize a uniform implosion. The wobbling HIB axis oscillation is precisely controlled. The oscillating frequency may be several 100MHz $\sim$ 1GHz. In the wobbling HIBs illumination, the illumination nonuniformity oscillates in time and space on a HIF target. The oscillating-HIB energy deposition may contribute to the reduction of the HIBs' illumination nonuniformity. Three-dimensional HIBs illumination computations presented here show that the few percent wobbling HIBs illumination nonuniformity oscillates with the same wobbling HIBs frequency. In general a perturbation of physical quantity would feature the instability onset. Normally the perturbation phase is unknown so that the instability growth is discussed with the growth rate. However, if the perturbation phase is known, the instability growth can be controlled by a superposition of perturbations; the well-known mechanism is a feedback control to compensate the displacement of physical quantity. If the perturbation is induced by, for example, a HIB axis wobbling, the perturbation phase could be controlled and the instability growth is mitigated by the superposition of the growing perturbations.   

Mixing in phase--space due to the two-stream instability of ion and electron beams propagating in background plasma

Intense electron or ion beams propagating in plasmas are subject to the two-stream instability, which leads to a slowing down of the beam particles, acceleration of the plasma particles, and transfer of the beam energy to the plasma particles and wave excitations. Making use of the particle-in-cell codes EDIPIC and LSP, we have simulated two-stream instability interactions over a wide range of beam and plasma parameters. Typically, the instability saturates due to nonlinear wave-trapping effects of either the beam particles or plasma electrons. The saturation due to nonlinear wave-trapping effects limits the ``mixing'' in phase-space and may produce coherent structures in the electron velocity distribution function. For the case of an electron beam, simulations show that the two-stream instability is intermittent, with quiet and active periods. During the active periods of the two-stream instability, the beam interacts with the plasma most intensively at locations where the global frequency of the instability matches the local electron plasma frequency. These intense localized plasma oscillations produce peaks in the velocity distribution function similar to the ones measured in the experiment [1]. \\[4pt] [1] L. Xu et al., Appl. Phys. Lett. 93, 261502 (2008).