Search NASA⌕ Search

SEARCH · Search NASA

Results for “Landau equation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 253 records · Page 14
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Moments of the Boltzmann collision operator for Coulomb interactions

Exact moments of the Boltzmann collision operator are calculated in the irreducible Hermitian moment expansion written in terms of the random-velocity variable of each species. The formulas are presented in closed, algebraic form and can be straightforwardly implemented in computer algebra systems. They are valid for two arbitrary masses, temperatures, and flow velocities, and hence include all other existing results derived for distribution functions expanded with respect to reference states of one temperature and flow velocity. In comparison, the Landau collisional moments are good approximations for large Coulomb logarithm and small relative flow velocity, but they fail to predict the correct behavior of most collisional moments for large relative flow even for weakly coupled plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1999

Hydrodynamic Instability and Thermal Coupling in a Dynamic Model of Liquid-Propellant Combustion

For liquid-propellant combustion, the Landau/Levich hydrodynamic models have been combined and extended to account for a dynamic dependence of the burning rate on the local pressure and temperature fields. Analysis of these extended models is greatly facilitated by exploiting the realistic smallness of the gas-to-liquid density ratio rho. Neglecting thermal coupling effects, an asymptotic expression was then derived for the cellular stability boundary A(sub p)(k) where A(sub p) is the pressure sensitivity of the burning rate and k is the disturbance wavenumber. The results explicitly indicate the stabilizing effects of gravity on long-wave disturbances, and those of viscosity and surface tension on short-wave perturbations, and the instability associated with intermediate wavenumbers for critical negative values of A(sub p). In the limit of weak gravity, hydrodynamic instability in liquid-propellant combustion becomes a long-wave, instability phenomenon, whereas at normal gravity, this instability is first manifested through O(1) wavenumbers. In addition, surface tension and viscosity (both liquid and gas) each produce comparable effects in the large-wavenumber regime, thereby providing important modifications to the previous analyses in which one or more of these effects was neglected. For A(sub p)= O, the Landau/Levich results are recovered in appropriate limiting cases, although this typically corresponds to a hydrodynamically unstable parameter regime for p << 1. In addition to the classical cellular form of hydrodynamic stability, there exists a pulsating form corresponding to the loss of stability of steady, planar burning to time-dependent perturbations. This occurs for negative values of the parameter A(sub p), and is thus absent from the original Landau/Levich models. In the extended model, however, there exists a stable band of negative pressure sensitivities bounded above by the Landau type of instability, and below by this pulsating form of hydrodynamic instability. Indeed, nonsteady modes of combustion have been observed at low pressures in hydroxylammonium nitrate (HAN)-based liquid propellants, which often exhibit negative pressure sensitivities. While nonsteady combustion may correspond to secondary and higher-order bifurcations above the cellular boundary, it may also be a manifestation of this pulsating type of hydrodynamic instability. In the present work, a nonzero temperature sensitivity is incorporated into our previous asymptotic analyses. This entails a coupling of the energy equation to the previous purely hydrodynamic problem, and leads to a significant modification of the pulsating boundary such that, for sufficiently large values of the temperature-sensitivity parameter, liquid-propellant combustion can become intrinsically unstable to this alternative form of hydrodynamic instability. For simplicity, further attention is confined here to the inviscid version of the problem since, despite the fact that viscous and surface-tension effects are comparable, the qualitative nature of the cellular boundary remains preserved in the zero-viscosity limit, as does the existence of the pulsating boundary. The mathematical model adopts the classical assumption that there is no distributed reaction in either the liquid or gas phases, but now the reaction sheet, representing either a pyrolysis reaction or an exothermic decomposition at the liquid/gas interface, is assumed to depend on local conditions there.

Margolis, S. B.↗
DOE OSTI2020

Perpendicular magnetofluid theory for magnetically confined plasmas in the collisionless limit

Note that the magnetically confined plasmas in thermonuclear fusion turn to be collisionless. Some particles are trapped by the inhomogeneous equilibrium field or by the waves, while some others keep circulating. One cannot expect the charged particles to be collectively localized in space by collisions along the magnetic field lines. Consequently, the fluid parameters relating to the parallel motion, such as parallel fluid velocity, become inadequate. In this paper, the perpendic- ular magnetofluid theory for magnetically confined plasmas in the collisionless limit is developed. It revises Braginskii’s transport theory by releasing the collisional dominance assumption. In the perpendicular direction, the particle spatial localization is solo resulted from the strong magnetic field, while the finite Larmor radius (FLR) effects are taken into account in the next order. In the parallel direction, the particle mobility feature is fully retained by constructing only the per- pendicular moments. Therefore, besides the FLR effects till order O($k$ $^{2}_{⊥}$ $ρ$ $^{2}_{i}$ ), the Landau damping, the trapped particle effects, and other parallel kinetic effects are all kept in the current formalism. Here, k ⊥ is the perpendicular wave number and ρ i is the ion Larmor radius. Here, the theory can be used both for the analytical and numerical studies of magnetically confined plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1983

Backscattering cascade of beam modes off ambient density fluctuations

The effects of a given nonthermal low-frequency density-fluctuation spectrum on the amplification of Langmuir waves by a 'bump-on-tail' beam of electrons are studied. The density-fluctuation spectrum is assumed to contain a uniform distribution of wavelengths ranging from much shorter than the beam mode wavelength to of the same order. This permits multiple large-angle (back) scattering to occur. One-dimensional numerical solutions of the kinetic equations are found which yield criteria for linear saturation of the beam instability of a cascade of backscattering to high wavenumber. The relevant time scales and spectral shapes are also determined in both the stable and unstable regions. Linear damping and Cerenkov emission by a possible nonthermal tail of electrons is taken into account. An application is made to the beam modes observed simultaneously with density fluctuations off the Jovian bow shock. It is shown that the observed level of density fluctuations is sufficient to saturate the unstable Langmuir waves, although nonthermal Landau damping may prevent a cascade to very high wavenumbers.

Russell, D. A.↗
DOE OSTI2021

Effect of surface disorder on the chiral surface states of a three-dimensional quantum Hall system

We investigate the effect of surface disorder on the chiral surface states of a three-dimensional quantum Hall system. Utilizing a transfer-matrix method, we find that the localization length of the surface state along the magnetic field decreases with the surface disorder strength in the weak disorder regime but increases anomalously in the strong disorder regime. In the strong disorder regime, the surface states mainly locate at the first inward layer to avoid the strong disorder in the outmost layer. The anomalous increase of the localization length can be explained by an effective model, which maps the strong disorder on the surface layer to the weak disorder on the first inward layer. Our work demonstrates that surface disorder can be an effective way to control the transport behavior of the surface states along the magnetic field. We also investigate the effect of surface disorder on the full distribution of conductances P(g) of the surface states in the quasi-one-dimensional (1D) regime for various surface disorder strengths. In particular, we find that P(g) is Gaussian in the quasi-1D metal regime and log-normal in the quasi-1D insulator regime. In the crossover regime, P(g) exhibits highly nontrivial forms, whose shapes coincide with the results obtained from the Dorokhov-Mello-Pereyra-Kumar equation of a weakly disordered quasi-1D wire in the absence of time-reversal symmetry. Our results suggest that P(g) is fully determined by the average conductance, independent of the details of the system, in agreement with the single-parameter scaling hypothesis.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

A collision operator for describing dissipation in noncanonical phase space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

Boltzmann equation↗
DOE OSTI2020

Equations of state, phase relations, and oxygen fugacity of the Ru-RuO 2 buffer at high pressures and temperatures

Experimental studies and measurements of inclusions in diamonds show that ferric iron components are increasingly stabilized with depth in the mantle. To determine the thermodynamic stability of such components, their concentration needs to be measured at known oxygen fugacities. The metal-oxide pair Ru and RuO 2 are ideal as an internal oxygen fugacity buffer in high-pressure experiments. Both phases remain solid to high temperatures and react minimally with silicates, only exchanging oxygen. To calculate oxygen fugacities at high pressure and temperature, however, requires information on the phase relations and equation of state properties of the solid phases. We have made in situ synchrotron X-ray diffraction measurements in a multi-anvil press on mixtures of Ru and RuO 2 to 19.4 GPa and 1473 K with which we have determined phase relations of the RuO 2 phases and derived thermal equations of state (EoS) parameters for both Ru and RuO 2 . Rutile-structured RuO 2 was found to undergo two phase transformations, first at ~7 GPa to an orthorhombic structure and then above 12 GPa to a cubic structure. The phase boundary of the cubic phase was constrained for the first time at high pressure and temperature. We have derived a continuous Gibbs free energy expression for the tetragonal and orthorhombic phases of RuO 2 by fitting the second-order phase transition boundary and P-V-T data for both phases, using a model based on Landau theory. The transition between the orthorhombic and cubic phases was then used along with EoS terms derived for both phases to determine a Gibbs free energy expression for the cubic phase. We have used these data to calculate the oxygen fugacity of the Ru + O 2 = RuO 2 equilibrium, which we have parameterized as a single polynomial across the stability fields of all three phases of RuO 2 . The expression is log 10 f O 2 (Ru – RuO 2 ) = (7.782 – 0.00996P + 0.001932P 2 – 3.76 × 10 –5 P 3 ) + (–13 763 + 592P – 3.955P 2 )/T + (–1.05 × 10 6 – 4622P)/T 2 , which should be valid from room pressure up to 25 GPa and 773–2500 K, with an estimated uncertainty of 0.2 log units. Our calculated fO 2 is shown to be up to 1 log unit lower than estimates that use previous expressions or ignore EoS terms.

Armstrong, Katherine↗
NASA NTRS1986

The wave magnetic field amplitude threshold for nonlinear trapping of energetic gyroresonant and Landau resonant electrons by nonducted VLF waves in the magnetosphere

A recently developed theory is used to calculate the wave magnetic field amplitude threshold, B-tau, necessary to allow the nonlinear trapping of energetic gyroresonant and Landau resonant electrons by VLF whistler mode waves in the magnetosphere propagating at an arbitrary angle, psi, with respect to the earth's magnetic field. A detailed raytracing study is carried out to establish the variation of psi with position along each magnetic shell in the range L = 2-5 and for frequencies 5-17.8 kHz. It is found that the minimum values of B-tau along each L shell generally occur at points of second-order resonance. In general, for nonducted fixed frequency signals there is only one point of second-order resonance on each L shell, and this is located within a few degrees of the magnetic equator. However, over a narrow range of L, there are as many as three points of second-order resonance and as many as three associated minima in B-tau. At least one of these points is located more than 10 deg from the magnetic equator.

Bell, T. F.↗
NASA NTRS1985

Accuracy of schemes for the Euler equations with non-uniform meshes

The effect of non-uniform grids on the solution of the Euler equations is analyzed. A Runge-Kutta type scheme based on a finite volume formulation is considered. It is shown that for arbitrary grids the scheme can be inconsistent even though it is second-order accurate for uniform grids. An improvement is suggested which leads to at least first-order accuracy for general grids. Test cases are presented in both two- and three-space dimensions. Applications to finite difference and implicit algorithms are also given.

Turkel, E.↗
NASA NTRS1986

Accuracy of schemes for the Euler equations with non-uniform meshers

The effect of nonuniform grids on the solution of the Euler equations is analyzed. A Runge-Kutta type scheme is considered based on a finite volume formuation. It is shown that for arbitrary grids the scheme can be inconsistent even though it is second-order accurate for uniform grids. An improvement is suggested which leads to at least first-order accuracy for general grids. Test cases are pesented in both two- and three-space dimensions. Applications to finite difference and impicit algorithms are also given.

Turkel, E.↗
NASA NTRS1999

Direct Numerical Simulation of Wetting and Spreading Behavior on Heterogeneous and Roughened Substrates

A method of calculation is presented that allows the simulation of the time-dependent three-dimensional motion of thin liquid layers on solid substrates for systems with finite equilibrium contact angles. The contact angle is a prescribed function of position on the substrate. Similar mathematical models are constructed for substrates with a pattern of roughness. Evolution equations are given, using the lubrication approximation, that include viscous, capillary and disjoining forces. Motion to and from dry substrate regions is made possible by use of a thin energetically-stable wetting layer. We simulate motion on heterogeneous substrates with periodic arrays of high contact-angle patches. Two different problems are treated for heterogenous substrates. The first is spontaneous motion driven only by wetting forces. If the contact-angle difference is sufficiently high, the droplet can find several different stable positions, depending on the previous history of the motion. A second simulation treats a forced cyclical motion. Energy dissipation per cycle for a heterogeneous substrate is found to be larger than for a uniform substrate with the same total energy. The Landau-Levich solution for plate removal from a liquid bath is extended to account for a pattern of roughness on the plate.

Schwartz, Leonard W.↗
DOE OSTI2022

Kinetic theory of particle-in-cell simulation plasma and the ensemble averaging technique

Abstract We derive the kinetic theory of fluctuations in physically and numerically stable particle-in-cell (PIC) simulations of electrostatic plasmas. The starting point is the single-time correlation at the start of the simulation between the statistical fluctuations of the weighted densities of macroparticle centers in the plasma particle phase-space. The fluctuations are associated with different initial conditions, typically due to the random initial conditions (in velocity space) of the macroparticles/simulation plasma, assigned according to their initial distribution of probability. The single-time correlations at all time steps and in each spatial grid cell are then determined from the Laplace–Fourier transforms of the discretized Klimontovich-like equation for the macroparticles and Maxwell’s equations for the fields, as computed by modern PIC codes. We recover the expressions for the electrostatic field and the plasma particle density fluctuation autocorrelation spectra as well as the kinetic equations describing the average evolution of PIC-simulated plasma particles, first derived by Langdon (1970b Proc. 4th Conf. Numerical Simulation of Plasmas ) using a test macroparticle approach perturbing a discretized Vlasovian plasma and then averaging the obtained physical quantity over the initial macroparticle velocity distribution. We generalize and extend these results to the modern algorithms in PIC codes using arbitrary macroparticle weights. Analytical estimates of statistical fluctuation amplitudes are derived as a function of the plasma simulation parameters, using the central limit theorem in the limit of a large number of macroparticles per cell. The theory is then used to analyze the ensemble averaging technique of PIC simulations where statistical averages are performed over ensembles of PIC simulations, modeling the same plasma physics problem but using different statistical realizations of the initial distribution functions of the macroparticles. This method is illustrated by linear Landau damping uncovering (from noise, which is usually considered numerical) the physical fluctuations driven by a single small amplitude electrostatic wave perturbing a PIC simulation plasma in equilibrium.

fluctuations correlations↗
DOE OSTI2024

The kinetic analog of the pressure–strain interaction

Energy transport in weakly collisional plasma systems is often studied with fluid models and diagnostics. However, the applicability of fluid models is limited when collisions are weak or absent, and using a fluid approach can obscure kinetic processes that provide key insights into the physics of energy transport. Kinetic diagnostics retain all of the information in 3D-3V phase space and thereby reach beyond the insights of fluid models to elucidate the mechanisms responsible for collisionless energy transport. In this work, we derive the Kinetic Pressure–Strain (KPS): a kinetic analog of the pressure–strain interaction, which is the channel between flow energy density and internal energy density in fluid models. Through two case studies of electron Landau damping, we demonstrate that the KPS diagnostic can elucidate kinetic mechanisms that are responsible for energy transport in this channel, just as the related field–particle correlation is known to identify kinetic mechanisms of transport between electromagnetic field energy density and kinetic energy density in particle flows. In addition, we show that resonant electrons play a major role in transferring energy between fluid flows and internal energy during the process of Landau damping.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

A projection method for particle resampling

Particle discretizations of partial differential equations are advantageous for high-dimensional kinetic models in phase-space due to their better scalability than continuum approaches with respect to dimension. Complex processes collectively referred to as particle noise hamper long time simulations with particle methods. One approach to address this problem is particle mesh adaptivity, or remapping, known as particle resampling and remeshing. Here, this work introduces a resampling method that projects particles to and from a (finite element) function space. The method is simple, using standard sparse linear algebra and finite element techniques, and it preserves all moments up to the order of a polynomial represented exactly by the continuum function space. It is distinguished from most other mesh-based methods in that new particle positions and number are decoupled from the mesh, allowing particle and continuum meshes to be adapted relatively independently. While this work is developed with structured particle and continuum phase-space grids on 1X + 1V Vlasov-Poisson models of Landau damping and two-stream instability, the method is well-suited to unstructured grids. Stable long time dynamics are demonstrated up to time T = 500. Reproducibility artifacts and data are publicly available.

Kinetic methods↗
DOE OSTI2025

Effects of artificial collisions, filtering, and nonlocal closure approaches on Hermite-based Vlasov–Poisson simulations

Kinetic simulations of collisionless plasmas are computationally challenging due to phase-space mixing and filamentation, resulting in fine-scale velocity structures. This study compares three methods developed to reduce artifacts related to limited velocity resolution in Hermite-based Vlasov–Poisson simulations: artificial collisions, filtering, and nonlocal closure approaches. We evaluate each method's performance in approximating the linear kinetic response function and suppressing recurrence in linear and nonlinear regimes. Numerical simulations of Landau damping demonstrate that artificial collisions, particularly higher orders of the Lenard-Bernstein collisional operator, most effectively recover the correct damping rate across a range of wavenumbers. Moreover, Hou-Li filtering and nonlocal closures underdamp high wavenumber modes in linear simulations, and the Lenard-Bernstein collisional operator overdamps low wavenumber modes in both linear and nonlinear simulations. This study demonstrates that hypercollisions offer a robust approach to kinetic simulations, accurately capturing collisionless dynamics with limited velocity resolution.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Squeezing quantum states in three-dimensional twisted crystals

Bloch's theorem provides a conventional starting point for describing wave propagation in periodic media, but in ordered materials where competing spatial periods coexist it is rendered ineffective, often with dramatic consequences. Here we develop an alternate approach that uses coherent free-particle vortex states to study quantum states in supertwisted crystals: three-dimensional stacks of atomically thin two-dimensional layers. Here, this formalism leads naturally to the representation of the spectrum using squeezed coherent states, and it reveals the crucial role of a Coriolis coupling in the equations of motion. This identifies an underlying noncommutative geometry and novel edge state structure in a family of complex ordered structures.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Crystalline solutions of the Kohn-Sham equations in the fractional quantum Hall regime

A Kohn-Sham density functional approach has recently been developed for the fractional quantum Hall effect, which maps the strongly interacting electrons into a system of weakly interacting composite fermions subject to an exchange correlation potential as well as a density dependent gauge field that mimics the "flux quanta" bound to composite fermions. To get a feel for the role of various terms, we study the behavior of the self-consistent solution as a function of the strength of the exchange correlation potential, which is varied through an ad hoc multiplicative factor. Here, we find that a crystal phase is stabilized when the exchange correlation interaction is sufficiently strong relative to the composite-fermion cyclotron energy. Various properties of this crystal are examined.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS2006

Self-Consistent Model of Magnetospheric Ring Current and Propagating Electromagnetic Ion Cyclotron Waves: Waves in Multi Ion Magnetosphere - 1

The further development of a self-consistent theoretical model of interacting ring current ions and electromagnetic ion cyclotron waves [Khazanov et al., 2003] is presented. In order to adequately take into account the wave propagation and refraction in a multi-ion plasmasphere, we explicitly include the ray tracing equations in our previous self-consistent model and use the general form of the wave kinetic equation. This is a major new feature of the present model and, to the best of our knowledge, the ray tracing equations for the first time are explicitly employed on a global magnetospheric scale in order to self-consistently simulate spatial, temporal, and spectral evolutions of the ring current and electromagnetic ion cyclotron waves. To demonstrate the effects of EMIC wave propagation and refraction on the EMIC wave energy distributions and evolution we simulate the May 1998 storm. The main findings of our simulation can be summarized as follows. First, due to the density gradient at the plasmapause, the net wave refraction is suppressed, and He(+)-mode grows preferably at plasmapause. This result is in a total agreement with the previous ray tracing studies, and very clear observed in presented B-field spectrograms. Second, comparison the global wave distributions with the results from other ring current model [Kozyra et al., 1997] reveals that our model provides more intense and higher plasmapause organized distributions during the May, 1998 storm period. Finally, the found He(+)-mode energy distributions are not Gaussian distributions, and most important that wave energy can occupy not only the region of generation, i. e. the region of small wave normal angles, but the entire wave normal angle region and even only the region near 90 degrees. The latter is extremely crucial for energy transfer to thermal plasmaspheric electrons by resonant Landau damping, and subsequent downward heat transport and excitation of stable auroral red arcs.

Khazanov, G. V.↗