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All sourcesNASA NTRSDOE OSTI
DOE OSTI2012

A GENERAL CIRCULATION MODEL FOR GASEOUS EXOPLANETS WITH DOUBLE-GRAY RADIATIVE TRANSFER

We present a new version of our code for modeling the atmospheric circulation on gaseous exoplanets, now employing a 'double-gray' radiative transfer scheme, which self-consistently solves for fluxes and heating throughout the atmosphere, including the emerging (observable) infrared flux. We separate the radiation into infrared and optical components, each with its own absorption coefficient, and solve standard two-stream radiative transfer equations. We use a constant optical absorption coefficient, while the infrared coefficient can scale as a power law with pressure; however, for simplicity, the results shown in this paper use a constant infrared coefficient. Here we describe our new code in detail and demonstrate its utility by presenting a generic hot Jupiter model. We discuss issues related to modeling the deepest pressures of the atmosphere and describe our use of the diffusion approximation for radiative fluxes at high optical depths. In addition, we present new models using a simple form for magnetic drag on the atmosphere. We calculate emitted thermal phase curves and find that our drag-free model has the brightest region of the atmosphere offset by {approx}12 Degree-Sign from the substellar point and a minimum flux that is 17% of the maximum, while the model with the strongest magnetic drag has an offset of only {approx}2 Degree-Sign and a ratio of 13%. Finally, we calculate rates of numerical loss of kinetic energy at {approx}15% for every model except for our strong-drag model, where there is no measurable loss; we speculate that this is due to the much decreased wind speeds in that model.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

Lyα heating of inhomogeneous high-redshift intergalactic medium

The intergalactic medium (IGM) prior to the epoch of reionization consists mostly of neutral hydrogen gas. Lyman-α (Lyα) photons produced by early stars resonantly scatter off hydrogen atoms, causing energy exchange between the radiation field and the gas. This interaction results in moderate heating of the gas due to the recoil of the atoms upon scattering, which is of great interest for future studies of the pre-reionization IGM in the H I 21 cm line. We investigate the effect of this Lyα heating in the IGM with linear density, temperature, and velocity perturbations. Perturbations smaller than the diffusion length of photons could be damped due to heat conduction by Lyα photons. The scale at which damping occurs and the strength of this effect depend on various properties of the gas, the flux of Lyα photons, and the way in which photon frequencies are redistributed upon scattering. To find the relevant length scale and the extent to which Lyα heating affects perturbations, we calculate the gas heating rates by numerically solving linearized Boltzmann equations in which scattering is treated by the Fokker-Planck approximation. We find that (1) perturbations add a small correction to the gas heating rate, and (2) the damping of temperature perturbations occurs at scales with comoving wavenumber k ≳ 10{sup 4} Mpc{sup –1}, which are much smaller than the Jeans scale and thus unlikely to substantially affect the observed 21 cm signal.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Fast penetration of megagauss fields into metallic conductors

Megagauss magnetic-field penetration into a conducting material is studied via a simplified but representative model, wherein the magnetic-diffusion equation is coupled with a thermal-energy balance. The specific scenario considered is that of a prescribed magnetic field rising (in proportion to an arbitrary power r of time) at the surface of a conducting half-space whose electric conductivity is assumed proportional to an arbitrary inverse power γ of temperature. We employ a systematic asymptotic scheme in which the case of a strong surface field corresponds to a singular asymptotic limit. In this limit, the highly magnetized and hot “skin” terminates at a distinct propagating wave-front. Employing the method of matched asymptotic expansions, we find self-similar solutions of the magnetized region which match a narrow boundary-layer region about the advancing wave front. The rapidly decaying magnetic-field profile in the latter region is also self similar; when scaled by the instantaneous propagation speed, its shape is time-invariant, depending only on the parameter γ. The analysis furnishes a simple asymptotic formula for the skin-depth (i.e., the wave-front position), which substantially generalizes existing approximations. It scales with the power γr + 1∕2 of time and the power γ of field strength, and is much larger than the field-independent skin depth predicted by an athermal model. The formula further involves a dimensionless O(1) pre-factor which depends on r and γ. It is determined by solving a nonlinear eigenvalue problem governing the magnetized region. Another main result of the analysis, apparently unprecedented, is an asymptotic formula for the magnitude of the current-density peak characterizing the wave-front region. Complementary to these systematic results, we provide a closed-form but ad hoc generalization of the theory approximately applicable to arbitrary monotonically rising surface fields. Our results are in excellent agreement with numerical simulations of the model, and compare favourably with detailed magnetohydrodynamic simulations reported in the literature.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

A fully coupled two-level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations

The neutron multigroup diffusion equations is an important approximation of the neutron transport equation, and has been widely used for studying the motion of neutrons and their interactions with the stationary background materials. Solving the multigroup diffusion equations faces challenging because many variables are coupled together and a lot of computing resources are required. In this paper, we focus on the development of a scalable parallel algorithm framework for the system of equations arising from the discretization of the multigroup diffusion equations using a finite element method in space and using an implicit scheme in time. The parallel agorithm framweork consists of an inexact Jacobian-free Newton for the nonlinear equations, a Krylov subspace method for the Jacobian system and a parallel Schwarz preconditioner for the acceleration of the algorithm convergence. A coarse space based on smoothed aggregation is introduced to construct a two-level method to improve the parallel performance of the Schwarz preconditioner. We numerically shown that compared with the one-level method, the proposed two-level method works significantly better in terms of the number of linear iterations and the compute time for a system of equations with millions of unknowns and a supercomputer with thousands of processors.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

COSMIC-RAY SMALL-SCALE ANISOTROPIES AND LOCAL TURBULENT MAGNETIC FIELDS

Cosmic-ray anisotropy has been observed in a wide energy range and at different angular scales by a variety of experiments over the past decade. However, no comprehensive or satisfactory explanation has been put forth to date. The arrival distribution of cosmic rays at Earth is the convolution of the distribution of their sources and of the effects of geometry and properties of the magnetic field through which particles propagate. It is generally believed that the anisotropy topology at the largest angular scale is adiabatically shaped by diffusion in the structured interstellar magnetic field. On the contrary, the medium- and small-scale angular structure could be an effect of nondiffusive propagation of cosmic rays in perturbed magnetic fields. In particular, a possible explanation for the observed small-scale anisotropy observed at the TeV energy scale may be the effect of particle propagation in turbulent magnetized plasmas. We perform numerical integration of test particle trajectories in low- β compressible magnetohydrodynamic turbulence to study how the cosmic rays’ arrival direction distribution is perturbed when they stream along the local turbulent magnetic field. We utilize Liouville’s theorem for obtaining the anisotropy at Earth and provide the theoretical framework for the application of the theorem in the specific case of cosmic-ray arrival distribution. In this work, we discuss the effects on the anisotropy arising from propagation in this inhomogeneous and turbulent interstellar magnetic field.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Jet wake from linearized hydrodynamics

We explore how to improve the hybrid model description of the particles originating from the wake that a jet produced in a heavy ion collision leaves in the droplet of quark-gluon plasma (QGP) through which it propagates, using linearized hydrodynamics on a background Bjorken flow. Jet energy and momentum loss described by the hybrid model become currents sourcing linearized hydrodynamics. By solving the linearized hydrodynamic equations numerically, we investigate the development of the wake in the dynamically evolving droplet of QGP, study the effect of viscosity, scrutinize energy-momentum conservation, and check the validity of the linear approximation. We find that linearized hydrodynamics works better in the viscous case because diffusive modes damp the energy-momentum perturbation produced by the jet. We calculate the distribution of particles produced from the jet wake by using the Cooper-Frye prescription and find that both the transverse momentum spectrum and the distribution of particles in azimuthal angle are similar in shape in linearized hydrodynamics and in the hybrid model. Their normalizations are different because the momentum-rapidity distribution in the linearized hydrodynamics analysis is more spread out, due to sound modes. Since the Bjorken flow has no transverse expansion, we explore the effect of transverse flow by using local boosts to add it into the Cooper-Frye formula. After including the effects of transverse flow in this way, the transverse momentum spectrum becomes harder: more particles with transverse momenta bigger than 2 GeV are produced than in the hybrid model. Although we defer implementing this analysis in a jet Monte Carlo, as would be needed to make quantitative comparisons to data, we gain a qualitative sense of how the jet wake may modify jet observables by computing proxies for two example observables: the lost energy recovered in a cone of varying open angle, and the fragmentation function. We find that linearized hydrodynamics with transverse flow effects added improves the description of the jet wake in the hybrid model in just the way that comparison to data indicates is needed. Our study illuminates a path to improving the description of the wake in the hybrid model, highlighting the need to take into account the effects of both transverse flow and the broadening of the energy-momentum perturbation in spacetime rapidity on particle production.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

An adaptive local reduced basis method for solving PDEs with uncertain inputs and evaluating risk

Many physical systems are modeled using partial differential equations (PDEs) with uncertain or random inputs. For such systems, naively propagating a fixed number of samples of the input probability law (or an approximation thereof) through the PDE is often inadequate to accurately quantify the "risk" associated with critical system responses. In this paper, we develop a goal-oriented, adaptive sampling and local reduced basis approximation for PDEs with random inputs. Our method determines a set of samples and an associated (implicit) Voronoi partition of the parameter domain on which we build local reduced basis approximations of the PDE solution. The samples are selected in an adaptive manner using an a posteriori error indicator. A notable advantage of the proposed approach is that the computational cost of the approximation during the adaptive process remains constant. We provide theoretical error bounds for our approximation and numerically demonstrate the performance of our method when compared to widely used adaptive sparse grid techniques. In addition, we tailor our approach to accurately quantify the risk of quantities of interest that depend on the PDE solution. We demonstrate our method on an advection-diffusion example and a Helmholtz example. Keywords: stochastic partial differential equation, adaptive sampling, reduced basis, a posteriori error estimation, coherent risk measure, conditional value-at-risk.

42 ENGINEERING↗
DOE OSTI2020

Dissolution of a cylindrical disk in Hele-Shaw flow: a conformal-mapping approach

Here we apply conformal mapping to find the evolving shapes of a dissolving cylinder in a potential flow. Similar equations can be used to describe melting in a flowing liquid phase. Results are compared with microfluidic experiments and numerical simulations. Shapes predicted by conformal mapping agree almost perfectly with experimental observations, after a modest (20 %) rescaling of the time. Finite-volume simulations show that the differences with experiment are connected to the underlying assumptions of the analytical model: potential flow and diffusion-limited dissolution. Approximate solutions of the equations describing the evolution of the shape of the undissolved solid can be derived from a Laurent expansion of the mapping function from the unit circle. Asymptotic expressions for the evolution of the area of the disk and the shift in its centre of mass have been derived at low and high Péclet number. Analytic approximations to the leading-order Laurent coefficients provide additional insight into the mechanisms underlying pore-scale dissolution.

58 GEOSCIENCES↗
DOE OSTI2012

THE LOCAL INTERSTELLAR SPECTRUM BEYOND THE HELIOPAUSE: WHAT CAN BE LEARNED FROM VOYAGER IN THE INNER HELIOSHEATH?

The local interstellar spectrum (LIS) is one of the most important but unknown parameters used in all modeling efforts to describe the modulation of Galactic cosmic rays on their way from the galaxy through a possible bow shock, heliosheath, and heliosphere toward the Earth. Because it has not been measured thus far, several LIS models derived from numerical simulations or data on Earth were developed. A new method to determine the LIS was introduced when the Voyager spacecraft crossed the termination shock and entered the heliosheath. Webber and Higbie derived a new LIS, which is lower than all previous LIS models over the entire energy range, on the basis of these measurements. Numerical simulations by Scherer et al. showed that particles already in the outer heliosheath (OHS) are modulated, suggesting that the LIS by Webber and Higbie is a heliopause spectrum (HPS) rather than the ''true'' LIS. By using the same simplified simulation model, we estimate the diffusion coefficient in the OHS to be consistent with several 10{sup 26} to 10{sup 27} cm{sup 2} s{sup -1} for all LIS models under consideration by mapping them to this HPS and conclude that the Voyager measurements will not be able to determine the LIS in the near future. We then discuss the circumstances under which the terrestrial archive can be used to at least exclude LIS models, unless one awaits a dedicated mission like e.g., the Interstellar Probe.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Mapping the particle acceleration in the cool core of the galaxy cluster RX J1720.1+2638

We present new deep, high-resolution radio images of the diffuse minihalo in the cool core of the galaxy cluster RX J1720.1+2638. The images have been obtained with the Giant Metrewave Radio Telescope at 317, 617, and 1280 MHz and with the Very Large Array at 1.5, 4.9, and 8.4 GHz, with angular resolutions ranging from 1'' to 10''. This represents the best radio spectral and imaging data set for any minihalo. Most of the radio flux of the minihalo arises from a bright central component with a maximum radius of ∼80 kpc. A fainter tail of emission extends out from the central component to form a spiral-shaped structure with a length of ∼230 kpc, seen at frequencies 1.5 GHz and below. We find indication of a possible steepening of the total radio spectrum of the minihalo at high frequencies. Furthermore, a spectral index image shows that the spectrum of the diffuse emission steepens with increasing distance along the tail. A striking spatial correlation is observed between the minihalo emission and two cold fronts visible in the Chandra X-ray image of this cool core. These cold fronts confine the minihalo, as also seen in numerical simulations of minihalo formation by sloshing-induced turbulence. All these observations favor the hypothesis that the radio-emitting electrons in cluster cool cores are produced by turbulent re-acceleration.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

Self-organization of local magnetoplasma structures in the upper layers of the solar convection zone

Self-organization and evolution of magnetoplasma structures in the upper layers of the solar convection zone are discussed as a process of diffuse aggregation of magnetic flux tubes. Equations describing the tube motion under the action of magnetic interaction forces, hydrodynamic forces, and random forces are written explicitly. The process of aggregation of magnetic flux tubes into magnetic flux clusters of different shapes and dimensions is simulated numerically. The obtained structures are compared with the observed morphological types of sunspot groups. The quantitative comparison with the observational data was performed by comparing the fractal dimensions of the photospheric magnetic structures observed in solar active regions with those of structures obtained in the numerical experiment. The model has the following free parameters: the numbers of magnetic flux tubes with opposite polarities on the considered area element (Nn and Ns), the average radius of the cross section of the magnetic flux tube (a), its effective length (l), the twist factor of the tube field (k), and the absolute value of the average velocity of chaotic tube displacements (d). Variations in these parameters in physically reasonable limits leads to the formation of structures (tube clusters of different morphological types) having different fractal dimensions. Using the NOAA 10488 active region, which appeared and developed into a complicated configuration near the central meridian, as an example, it is shown that good quantitative agreement between the fractal dimensions is achieved at the following parameters of the model: Nn = Ns = 250 ± 50; a = 150 ± 50 km; l ∼ 5000 km, and d = 80 ± 10 m/s. These results do not contradict the observational data and theoretical estimates obtained in the framework of the Parker “spaghetti” model and provide new information on the physical processes resulting in the origin and evolution of local magnetic plasma structures in the near-photospheric layers of the solar convection zone.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Fundamental study of key issues related to advanced sCO 2 Brayton cycle: Prototypic HX development and cavitation

Diffusion bonded heat exchangers are the leading candidates for the sCO 2 Brayton cycles in next generation nuclear power plants. Commercially available diffusion bonded heat exchangers utilize set of continuous semi-circular zigzag micro channels to increase the heat transfer area and enhance heat transfer through increased turbulence production. Such heat exchangers can lead to excessive pressure drop as well as flow maldistribution in the case of poorly designed flow distribution headers. The goal of the current project is to fabricate and test potential discontinuous fin patterns for diffusion bonded heat exchangers; which can achieve desired thermal performance at lower pressure drops. Prototypic discontinuous offset rectangular and Airfoil fin surface geometries were chemically etched on to 316 stainless steel plate and sealed against an un-etched flat pate using O-ring seal emulating diffusion bonded heat exchangers. Thermal-hydraulic performance of these prototypic discontinuous fin geometries was experimentally evaluated and compared to the existing data for the continuous zigzag channels. The data generated from this project will serve as the database for future testing and validation of numerical models.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2017

Three-dimensional spontaneous magnetic reconnection

Magnetic reconnection is best known from observations of the Sun where it causes solar flares. Observations estimate the reconnection rate as a small, but non-negligible fraction of the Alfvén speed, so-called fast reconnection. Until recently, the prevailing pictures of reconnection were either of resistivity or plasma microscopic effects, which was contradictory to the observed rates. Alternative pictures were either of reconnection due to the stochasticity of magnetic field lines in turbulence or the tearing instability of the thin current sheet. In this paper we simulate long-term three-dimensional nonlinear evolution of a thin, planar current sheet subject to a fast oblique tearing instability using direct numerical simulations of resistive-viscous magnetohydrodynamics. The late-time evolution resembles generic turbulence with a −5/3 power spectrum and scale-dependent anisotropy, so we conclude that the tearing-driven reconnection becomes turbulent reconnection. The turbulence is local in scale, so microscopic diffusivity should not affect large-scale quantities. This is confirmed by convergence of the reconnection rate toward ∼0.015v{sub A} with increasing Lundquist number. In this spontaneous reconnection, with mean field and without driving, the dissipation rate per unit area also converges to ∼0.006ρv{sub A}{sup 3}, and the dimensionless constants 0.015 and 0.006 are governed only by self-driven nonlinear dynamics of the sheared magnetic field. Remarkably, this also means that a thin current sheet has a universal fluid resistance depending only on its length to width ratio and to v{sub A}/c.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

Very Low-energy Supernovae: Light Curves and Spectra of Shock Breakout

The brief transient emitted as a shock wave erupts through the surface of a pre-supernova star carrying information about the stellar radius and explosion energy. Here, the CASTRO code, which treats radiation transport using multigroup flux-limited diffusion, is used to simulate the light curves and spectra of shock breakout in very low-energy supernovae (VLE SNe), explosions in giant stars with final kinetic energy much less than 10{sup 51} erg. VLE SN light curves, computed here with the KEPLER code, are distinctively faint, red, and long-lived, making them challenging to find with transient surveys. The accompanying shock breakouts are brighter, though briefer, and potentially easier to detect. Previous analytic work provides general guidance, but numerical simulations are challenging, due to the range of conditions and lack of equilibration between color and effective temperatures. We consider previous analytic work and extend discussions of color temperature and opacity to the lower energy range explored by these events. Since this is the first application of the CASTRO code to shock breakout, test simulations of normal energy shock breakout of SN 1987A are carried out and compared with the literature. A set of breakout light curves and spectra are then calculated for VLE SNe with final kinetic energies in the range 10{sup 47}--10{sup 50} erg for red supergiants with main-sequence masses of 15 and 25 M{sub ⊙}. The importance of uncertainties in stellar atmosphere model, opacity, and ambient medium is discussed, as are observational prospects with current and forthcoming missions.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

High-Fidelity Simulation of the Light-to-Dense Stratification Transient in the HiRJET Facility

Density stratification in a large enclosure is a crucial phenomenon to heat transfer and sustainable passive heat removal of a sodium fast reactor during reactivity transients. However, engineering turbulence models were identified to have unsatisfactory performance in predicting propagation of a stratified front. Yet, the scarcity of high-resolution data for stratification hampers the development of models. To explor e applications of leveraging direct numerical simulation (DNS) data to support turbulence model development, this work conducted DNS using NekRS to study a long stratification transient in the High-Resolution Jet (HiRJET) experimental facility. Here, this work considers an experiment run where light fluid is injected into a tank containing a denser fluid with a relative density difference of 1.5%. Formation of the stratified layer is identified as impingement of the buoyant jet promoting mixing of the two fluids. Based on the transient statistics, transport of the concentration can be characterized by regions with dominating effects of turbulent mixing, buoyant dissipation, and molecular diffusion, respectively, as moving away from the elevation of jet impingement. Concentration near the stratified front also exhibits oscillation at Brunt-Väisälä frequency. Preliminary validation of the simulation showed encouraging agreement of the concentration distribution with the reference experiment.

buoyant mixing↗
DOE OSTI2014

Numerical approach of the injection molding process of fiber-reinforced composite with considering fiber orientation

One of the most important challenges in the injection molding process of the short-glass fiber/thermoplastic composite parts is being able to predict the fiber orientation, since it controls the mechanical and the physical properties of the final parts. Folgar and Tucker included into the Jeffery equation a diffusive type of term, which introduces a phenomenological coefficient for modeling the randomizing effect of the mechanical interactions between the fibers, to predict the fiber orientation in concentrated suspensions. Their experiments indicated that this coefficient depends on the fiber volume fraction and aspect ratio. However, a definition of the fiber interaction coefficient, which is very necessary in the fiber orientation simulations, hasn't still been proven yet. Consequently, this study proposed a developed fiber interaction model that has been introduced a fiber dynamics simulation in order to obtain a global fiber interaction coefficient. This supposed that the coefficient is a sum function of the fiber concentration, aspect ratio, and angular velocity. The proposed model was incorporated into a computer aided engineering simulation package C-Mold. Short-glass fiber/polyamide-6 composites were produced in the injection molding with the fiber weight concentration of 30 wt.%, 50 wt.%, and 70 wt.%. The physical properties of these composites were examined, and their fiber orientation distributions were measured by micro-computed-tomography equipment μ-CT. The simulation results showed a good agreement with experiment results.

36 MATERIALS SCIENCE↗
DOE OSTI2022

A scalable DG solver for the electroneutral Nernst-Planck equations

The robust, scalable simulation of flowing electrochemical systems is increasingly important due to the synergy between intermittent renewable energy and electrochemical technologies such as energy storage and chemical manufacturing. The high Péclet regime of many such applications prevents the use of off-the-shelf discretization methods. In this work, we present a high-order Discontinuous Galerkin scheme for the electroneutral Nernst-Planck equations. The chosen charge conservation formulation allows for the specific treatment of the different physics: upwinding for advection and migration, and interior penalty for diffusion of ionic species as well the electric potential. Similarly, the formulation enables different treatments in the preconditioner: AMG for the potential blocks and ILU-based methods for the advection-dominated concentration blocks. Here we evaluate the convergence rate of the discretization scheme through numerical tests. Strong scaling results for two preconditioning approaches are shown for a large 3D flow-plate reactor example.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2013

Residual turbulence from velocity shear stabilized interchange instabilities

The stabilizing effect of velocity shear on the macroscopic, broad bandwidth, ideal interchange instability is studied in linear and nonlinear regimes. A 2D dissipative magnetohydrodynamic (MHD) code is employed to simulate the system. For a given flow shear, V Prime , linear growth rates are shown to be suppressed to below the shear-free level at both the small and large wavelengths. With increasing V Prime , the unstable band in wavenumber-space shrinks so that the peak growth results for modes that correspond to relatively high wavenumbers, on the scale of the density gradient. In the nonlinear turbulent steady state, a similar turbulent spectrum obtains, and the convection cells are roughly circular. In addition, the density fluctuation level and the degree of flattening of the initial inverted density profile are found to decrease as V Prime increases; in fact, unstable modes are almost completely stabilized and the density profile reverts to laminar when V Prime is a few times the classic interchange growth rate. Moreover, the turbulent particle flux diminishes with increasing velocity shear such that all the flux is carried by the classical diffusive flux in the asymptotic limit. The simulations are compared with measurements of magnetic fluctuations from the Maryland Centrifugal Experiment, MCX, which investigated interchange modes in the presence of velocity shear. The experimental spectral data, taken in the plasma edge, are in general agreement with the numerical data obtained in higher viscosity simulations for which the level of viscosity is chosen consistent with MCX Reynolds numbers at the edge. In particular, the residual turbulence in both cases is dominated by elongated convection cells. Finally, concomitant Kelvin-Helmholtz instabilities in the system are also examined. Complete stability to interchanges is obtained only in the parameter space wherein the generalized Rayleigh inflexion theorem is satisfied.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗