Comoving Frame Research Articles

Relativistic radiative transfer in a moving stratus irradiated by a luminous flat source

Abstract Relativistic radiative transfer in a geometrically thin stratus (sheet-like gaseous cloud with finite optical depth), which is moving at a relativistic speed around a luminous flat source, such as accretion disks, and is irradiated by the source, is examined under the special relativistic treatment. Incident radiation is aberrated and Doppler-shifted when it is received by the stratus, and emitted radiation is also aberrated and Doppler-shifted when it leaves the stratus. Considering these relativistic effects, we analytically obtain the emergent intensity as well as other radiative quantities in the purely scattering case for both infinite and finite strati. We mainly consider the frequency-integrated case, but also briefly show the frequency-dependent one. We also solve the relativistic radiative transfer equation numerically, and compare the results with the analytical solutions. In the infinite stratus, the mean intensity in the comoving and inertial frames decreases and becomes constant, as the stratus speed increases. The flux in the comoving frame decreases exponentially with the optical depth. The emergent intensity decreases as the speed increases, since the incident photons are redshifted at the bottom-side of the stratus. In the finite stratus, the mean intensity in the comoving and inertial frames quickly increases in the top-side region due to the aberrated photons. The flux in the comoving frame is positive in the range of 0 < β ≤ 0.4, while it becomes negative for β ≳ 0.5. The behavior of the emergent intensity is similar to that of the infinite case, although there is an irradiation effect caused by the aberrated photons.

Use of principal component analysis for identification of temporal and spatial patterns in the dynamics of ionospheric equatorial anomaly

In this paper we describe results of the principal components analysis of the dynamics of Total Electronic Content (TEC) data with the use of global maps presented by the Jet Propulsion Laboratory (NASA, USA) for the period of 2007-2011. We show that the result of the decomposition in principal components essentially depends on the method used for preprocessing the data, their representation (the used coordinate system), and the data centering technique (e.g., daily and seasonal components extracting). The use of momentarily co-moving frame of reference and other special techniques provide opportunity for the detailed analysis of the ionospheric equatorial anomaly. The covariance matrix of decomposition was calculated using Spearman's rank correlation coefficient that allows reducing statistical relationship between principal components.

Isotropic expansion of an inhomogeneous universe

We propose a cosmological model that describes isotropic expansion of inhomogeneous universe. The energy-momentum tensor that creates the spatial inhomogeneity may not affect the uniform expansion scaling factor $a(t)$ in the FLRW-like metrics. Such energy-momentum tensor may not be exotic; in fact any linear or non-linear $\sigma$-model has this feature. We show that the classical spatial inhomogeneity can be embedded in both inflation models and the traditional cosmological expansion by perfect fluid. The spatial inhomogeneity resembles the primordial quantum perturbation that was frozen in the co-moving frame. We obtain some exact inhomogeneous solutions with spherical or axial symmetries. We also show that some of our cosmological models can be viewed as the dynamical black hole formation.

Void asymmetries in the cosmic web: a mechanism for bulk flows

AbstractBulk flows of galaxies moving with respect to the cosmic microwave background are well established observationally and seen in the most recent ΛCDM simulations. With the aid of an idealised Gadget-2 simulation, we show that void asymmetries in the cosmic web can exacerbate local bulk flows of galaxies. The {\it Cosmicflows-2} survey, which has mapped in detail the 3D structure of the Local Universe, reveals that the Local Group resides in a “local sheet” of galaxies that borders a “local void” with a diameter of about 40 Mpc. The void is emptying out at a rate of 16 km s-1Mpc-1. In a co-moving frame, the Local Sheet is found to be moving away from the Local Void at ∼ 260 km s-1. Our model shows how asymmetric collapse due to unbalanced voids on either side of a developing sheet or wall can lead to a systematic movement of the sheet. We conjectured that asymmetries could lead to a large-scale separation of dark matter and baryons, thereby driving a dependence of galaxy properties with environment, but we do {\it not} find any evidence for this effect.

A three-dimensional finite element arbitrary Lagrangian–Eulerian method for shock hydrodynamics on unstructured grids

We present a three-dimensional (3D) finite element (FE) arbitrary Lagrangian–Eulerian (ALE) method for shock hydrodynamics on unstructured grids. The method is based on an FE Eulerian Godunov scheme for linear tetrahedra that has been extended to include mesh motion in an unsplit, flux-conservative formulation. The proposed method eliminates the splitting errors present in traditional Lagrange-plus-remap methods that occur during the remap phase. Unlike typical unsplit approaches, the mesh velocity is not determined by boundary motion but is instead based on the local fluid velocity. Smoothing operations are then applied to the mesh velocity to avoid mesh tangling. This approach allows the mesh to follow the fluid motion in a robust manner and leverage one of the primary advantages of Lagrangian schemes for shock hydrodynamics, namely that the resolution follows the flow. An approximate Riemann solver is used to calculate fluxes in the co-moving frame of the mesh. Results for a number of standard test problems are presented for 3D meshes of up to 107 tetrahedra. Global convergence rates of 0.8–1.0 are observed for shock dominated flows and 1.9 for smooth flows. We also demonstrate that the method satisfies the discrete geometric conservation law to truncation error, conserves total energy to machine precision, and preserves symmetry.

Radiation hydrodynamics integrated in the PLUTO code

The transport of energy through radiation is very important in many astrophysical phenomena. In dynamical problems the time-dependent equations of radiation hydrodynamics have to be solved. We present a newly developed radiation-hydrodynamics module specifically designed for the versatile MHD code PLUTO. The solver is based on the flux-limited diffusion approximation in the two-temperature approach. All equations are solved in the co-moving frame in the frequency independent (grey) approximation. The hydrodynamics is solved by the different Godunov schemes implemented in PLUTO, and for the radiation transport we use a fully implicit scheme. The resulting system of linear equations is solved either using the successive over-relaxation (SOR) method (for testing purposes), or matrix solvers that are available in the PETSc library. We state in detail the methodology and describe several test cases in order to verify the correctness of our implementation. The solver works in standard coordinate systems, such as Cartesian, cylindrical and spherical, and also for non-equidistant grids. We have presented a new radiation-hydrodynamics solver coupled to the MHD-code \PLUTO that is a modern, versatile and efficient new module for treating complex radiation hydrodynamical problems in astrophysics. As test cases, either purely radiative situations, or full radiation-hydrodynamical setups (including radiative shocks and convection in accretion discs) have been studied successfully. The new module scales very well on parallel computers using MPI. For problems in star or planet formation, we have added the possibility of irradiation by a central source.

A BAKE-OFF BETWEEN CMFGEN AND FASTWIND: MODELING THE PHYSICAL PROPERTIES OF SMC AND LMC O-TYPE STARS

The model atmosphere programs FASTWIND and CMFGEN are both elegantly designed to perform non-LTE analyses of the spectra of hot massive stars, and include sphericity and mass-loss. The two codes differ primarily in their approach towards line blanketing, with CMFGEN treating all of the lines in the co-moving frame and FASTWIND taking an approximate approach which speeds up execution times considerably. Although both have been extensively used to model the spectra of O-type stars, no studies have used the codes to independently model the same spectra of the same stars and compare the derived physical properties. We perform this task on ten O-type stars in the Magellanic Clouds. For the late-type O supergiants, both CMFGEN and FASTWIND have trouble fitting some of the He I lines, and we discuss causes and cures. We find that there is no difference in the average effective temperatures found by the two codes for the stars in our sample, although the dispersion is large, due primarily to the various difficulties each code has with He I. The surface gravities determined using FASTWIND are systematically lower by 0.12 dex compared to CMFGEN, a result we attribute to the better treatment of electron scattering by CMFGEN. This has implications for the interpretation of the origin of the so-called mass discrepancy, as the masses derived by FASTWIND are on average lower than inferred from stellar evolutionary models, while those found by CMFGEN are in better agreement.

Travelling waves in a neural field model with refractoriness

At one level of abstraction neural tissue can be regarded as a medium for turning local synaptic activity into output signals that propagate over large distances via axons to generate further synaptic activity that can cause reverberant activity in networks that possess a mixture of excitatory and inhibitory connections. This output is often taken to be a firing rate, and the mathematical form for the evolution equation of activity depends upon a spatial convolution of this rate with a fixed anatomical connectivity pattern. Such formulations often neglect the metabolic processes that would ultimately limit synaptic activity. Here we reinstate such a process, in the spirit of an original prescription by Wilson and Cowan (Biophys J 12:1–24, 1972), using a term that multiplies the usual spatial convolution with a moving time average of local activity over some refractory time-scale. This modulation can substantially affect network behaviour, and in particular give rise to periodic travelling waves in a purely excitatory network (with exponentially decaying anatomical connectivity), which in the absence of refractoriness would only support travelling fronts. We construct these solutions numerically as stationary periodic solutions in a co-moving frame (of both an equivalent delay differential model as well as the original delay integro-differential model). Continuation methods are used to obtain the dispersion curve for periodic travelling waves (speed as a function of period), and found to be reminiscent of those for spatially extended models of excitable tissue. A kinematic analysis (based on the dispersion curve) predicts the onset of wave instabilities, which are confirmed numerically.Electronic supplementary materialThe online version of this article (doi:10.1007/s00285-013-0670-x) contains supplementary material, which is available to authorized users.

In situ granular charge measurement by free-fall videography

We present the design and performance characterization of a new experimental technique for measuring individual particle charges in large ensembles of macroscopic grains. The measurement principle is qualitatively similar to that used in determining the elementary charge by Millikan in that it follows individual particle trajectories. However, by taking advantage of new technology we are able to work with macroscopic grains and achieve several orders of magnitude better resolution in charge to mass ratios. By observing freely falling grains accelerated in a horizontal electric field with a co-falling, high-speed video camera, we dramatically increase particle tracking time and measurement precision. Keeping the granular medium under vacuum, we eliminate air drag, leaving the electrostatic force as the primary source of particle accelerations in the co-moving frame. Because the technique is based on direct imaging, we can distinguish between different particle types during the experiment, opening up the possibility of studying charge transfer processes between different particle species. For the ∼300 μm diameter grains reported here, we achieve an average acceleration resolution of ∼0.008 m/s(2), a force resolution of ∼500 pN, and a median charge resolution ∼6× 10(4) elementary charges per grain (corresponding to surface charge densities ∼1 elementary charges per μm(2)). The primary source of error is indeterminacy in the grain mass, but with higher resolution cameras and better optics this can be further improved. The high degree of resolution and the ability to visually identify particles of different species or sizes with direct imaging make this a powerful new tool to characterize charging processes in granular media.

A Lagrangian analysis of a developing and non-developing disturbance observed during the PREDICT experiment

Abstract. The problem of tropical cyclone formation requires among other things an improved understanding of recirculating flow regions on sub-synoptic scales in a time evolving flow with typically sparse real-time data. This recirculation problem has previously been approached assuming as a first approximation both a layer-wise two-dimensional and nearly steady flow in a co-moving frame with the parent tropical wave or disturbance. This paper provides an introduction of Lagrangian techniques for locating flow boundaries that encompass regions of recirculation in time-dependent flows that relax the steady flow approximation. Lagrangian methods detect recirculating regions from time-dependent data and offer a more complete methodology than the approximate steady framework. The Lagrangian reference frame follows particle trajectories so that flow boundaries which constrain particle transport can be viewed in a frame-independent setting. Finite-time Lagrangian scalar field methods from dynamical systems theory offer a way to compute boundaries from grids of particles seeded in and near a disturbance. The methods are applied to both a developing and non-developing disturbance observed during the recent pre-depression investigation of cloud systems in the tropics (PREDICT) experiment. The data for this analysis is derived from global forecast model output that assimilated the dropsonde observations as they were being collected by research aircraft. Since Lagrangian methods require trajectory integrations, we address some practical issues of using Lagrangian methods in the tropical cyclogenesis problem. Lagrangian diagnostics are used to evaluate the previously hypothesized import of dry air into ex-Gaston, which did not re-develop into a tropical cyclone, and the exclusion of dry air from pre-Karl, which did become a tropical cyclone and later a major hurricane.

Numerical Approach for Analysis of Transient Scattering by an Accelerated Body

A numerical approach for analysis of transient scattering by an accelerated body based on the frame hopping method (FHM) is developed. First, transform formulae for coordinates and fields between an inertial laboratory frame and a non-inertial co-moving frame are derived. The solution of retarded potential equations in the accelerated frame is given. Then, the scattering analysis is performed in the co-moving accelerated frame using the time domain integral equation (TDIE) method. Last, the scattering fields found in the co-moving frame are ransformed back to the laboratory frame. A moving conducting sphere is used to validate the procedure. The proposed approach is unified in terms of suiting stationary, uniform and accelerated objects.

Instability of a uniformly collapsing cloud of classical and quantum self-gravitating Brownian particles

We study the growth of perturbations in a uniformly collapsing cloud of self-gravitating Brownian particles. This problem shares analogies with the formation of large-scale structures in a universe experiencing a "big-crunch" or with the formation of stars in a molecular cloud experiencing gravitational collapse. Starting from the barotropic Smoluchowski-Poisson system, we derive a new equation describing the evolution of the density contrast in the comoving (collapsing) frame. This equation can serve as a prototype to study the process of self-organization in complex media with structureless initial conditions. We solve this equation analytically in the linear regime and compare the results with those obtained by using the "Jeans swindle" in a static medium. The stability criteria, as well as the laws for the time evolution of the perturbations, differ. The Jeans criterion is expressed in terms of a critical wavelength λ(J) while our criterion is expressed in terms of a critical polytropic index γ(4/3). In a static background, the system is stable for λ<λ(J) and unstable for λ>λ(J). In a collapsing cloud, the system is stable for γ>γ(4/3) and unstable for γ<γ(4/3). If γ=γ(4/3), it is stable for λ<λ(J) and unstable for λ>λ(J). We also study the fragmentation process in the nonlinear regime. We determine the growth of the skewness, the long-wavelength tail of the power spectrum and find a self-similar solution to the nonlinear equations valid for large times. Finally, we consider dissipative self-gravitating Bose-Einstein condensates with short-range interactions and show that, in a strong friction limit, the dissipative Gross-Pitaevskii-Poisson system is equivalent to the quantum barotropic Smoluchowski-Poisson system. This yields new types of nonlinear mean-field Fokker-Planck equations, including quantum effects.

The non-thermal broadband spectral energy distribution of radio galaxies

AbstractI present a model for the non-thermal production of electromagnetic radiation in the jets of radio galaxies. The model goes far beyond the simple one-zone models usually applied to these sources. The transport equation is solved in the co-moving frame of the jet, taken into account the inhomogeneous structure of the outflow. Energy distributions for all types of particles are then obtained in a self-consistent way, including protons, electrons, and secondaries. The spectral energy distribution resulting from all relevant radiative processes is computed, including synchrotron radiation, relativistic Bremsstrahlung, proton-proton collisions and subsequent decays, photo-meson production, radiation from pairs formed by photon absorption and injection from decays, as well as direct pair production. Absorbing fields in the host galaxy are considered when computing the final SED. The model is applied to Centaurus A and compared with the available multi-wavelength data.

Linear Stability of Traveling Waves in First-Order Hyperbolic PDEs

In the analysis of traveling waves it is common that coupled parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example, this happens whenever a reaction diffusion equation with more than one non-diffusing component is considered in a co-moving frame. In this paper we analyze the stability of traveling waves in nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation (PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal rates. The linear stability result presented here is a major step in the proof of nonlinear stability.

A geometric classification of traveling front propagation in the Nagumo equation with cut-off

An important category of solutions to reaction-diffusion systems of partial differential equations is given by traveling fronts, which provide a monotonic connection between rest states and maintain a fixed profile when considered in a co-moving frame. Reaction-diffusion equations are frequently employed in the mean-field (continuum) approximation of discrete (many-particle) models; however, the quality of this approximation deteriorates when the number of particles is not sufficiently large. The (stochastic) effects of this discreteness have been modeled via the introduction of (deterministic) 'cut-offs' that effectively deactivate the reaction terms at points where the particle concentration is below a certain threshold. In this article, we present an overview of the effects of such a cut-off on the front propagation dynamics in a prototypical reaction-diffusion system, the classical Nagumo equation. Our analysis is based on the method of geometric desingularization ('blow-up'), in combination with dynamical systems techniques such as invariant manifolds and normal forms. Using these techniques, we categorize front propagation in the cut-off Nagumo equation in dependence of a control parameter, and we classify the corresponding propagation regimes ('pulled,' 'pushed,' and 'bistable') in terms of the bifurcation structure of a projectivized system of equations that is obtained from the original traveling front problem, after blow-up. In particular, our approach allows us to determine rigorously the asymptotics (in the cut-off parameter) of the correction to the front propagation speed in the Nagumo equation that is due to a cut-off. Moreover, it explains the structure of that asymptotics (logarithmic, superlinear, or sublinear) in dependence of the front propagation regime. Finally, it enables us to calculate the corresponding leading-order coefficients in the resulting expansions in closed form.

Nonlinear electrostatic shock waves in inhomogeneous dense dusty magnetoplasmas

The Linear and nonlinear propagation characteristics of drift dust acoustic waves were studied in an inhomogeneous electron–ion-dust (e–i–d) quantum magnetoplasma with neutrals in the background using the well-known quantum hydrodynamic (QHD) model. The Korteweg–de Vries–Burgers (KdVB) and Kadomtsev–Petviashvili–Burgers (KPB) equations are obtained. Furthermore, the solutions of the KdVB and KPB equations are presented using the tangent hyperbolic (tanh) method. The variation of the shock profile with the quantum Bohm potential, collision frequency, ratio of drift to shock velocity in the co-moving frame and effect of magnetic field is also investigated. The relevance of the present investigation to dense astrophysical environments is also highlighted.

Comoving frame models of hot star winds

We provide hot star wind models with radiative force calculated using the solution of comoving frame (CMF) radiative transfer equation. The wind models are calculated for the first stars, O stars, and the central stars of planetary nebulae. We show that without line overlaps and with solely thermal line broadening the pure Sobolev approximation provides a reliable estimate of the radiative force even close to the wind sonic point. Consequently, models with the Sobolev line force provide good approximations to solutions obtained with non-Sobolev transfer. Taking line overlaps into account, the radiative force becomes slightly lower, leading to a decrease in the wind mass-loss rate by roughly 40%. Below the sonic point, the CMF line force is significantly lower than the Sobolev one. In the case of pure thermal broadening, this does not influence the mass-loss rate, as the wind mass-loss rate is set in the supersonic part of the wind. However, when additional line broadening is present (e.g., the turbulent one) the region of low CMF line force may extend outwards to the regions where the mass-loss rate is set. This results in a decrease in the wind mass-loss rate. This effect can at least partly explain the low wind mass-loss rates derived from some observational analyses of luminous O stars.

Three-dimensional molecular line transfer: a simulated star-forming region

We present the first non-LTE, co-moving frame molecular line calculations of a star-forming cluster simulated using smoothed particle hydrodynamics (SPH), from which we derive high-resolution synthetic observations. We have resampled a particle representation onto an adaptive mesh and self-consistently solved the equations of statistical equilibrium in the co-moving frame, using TORUS, a three-dimensional adaptive mesh refined (AMR) radiative transfer (RT) code. We verified the applicability of the code to the conditions of the SPH simulation by testing its output against other codes. We find that the level populations obtained for optically thick and thin scenarios closely match the ensemble average of the other codes. We have used the code to obtain non-LTE level populations of multiple molecular species throughout the cluster and have created three-dimensional velocity-resolved spatial maps of the emergent intensity. Line profiles of cores traced by N2H+ (1-0) are compared to probes of low density gas, 13CO (1-0) and C18O (1-0), surrounding the cores along the line of sight. The relative differences of the line-centre velocities are shown to be small compared to the velocity dispersion, matching recent observations. We conclude that one cannot reject competitive accretion as a viable theory of star formation based on observed velocity profiles.

Computing the Wave Speed of Soliton-Like Solutions in SmC* Liquid Crystals

We present a novel method for numerically computing the wave speed of a soliton-like travelling wave in chiral smectic C liquid crystals (SmC*) that satisfies a parabolic partial differential equation (PDE) with a general nonlinear term [1]. By transforming the PDE to a co-moving frame and recasting the resulting problem in phase-space, the original PDE can be expressed as an integral equation known as an exceptional nonlinear Volterra-type equation of the second kind. This technique is motivated by, but distinct in nature from, iterative integral methods introduced by Chernyak [2]. By applying a simple trapezoidal method to the integral equation we generate a system of nonlinear simultaneous equations which we solve for our phase plane variable at equally spaced intervals using Newton iterates. The equally spaced phase variable solutions are then used to compute the wave speed of the associated travelling wave. We demonstrate an algorithm for performing the necessary calculations by considering an example from liquid crystal theory, where a parabolic PDE with a nonlinear reaction term has a solution and wave speed which are known exactly [3,4]. The analytically derived wave speed is then compared with the numerically computed wave speed using our new scheme.

Tropical cyclogenesis in a tropical wave critical layer: easterly waves

Abstract. The development of tropical depressions within tropical waves over the Atlantic and eastern Pacific is usually preceded by a "surface low along the wave" as if to suggest a hybrid wave-vortex structure in which flow streamlines not only undulate with the waves, but form a closed circulation in the lower troposphere surrounding the low. This structure, equatorward of the easterly jet axis, is identified herein as the familiar critical layer of waves in shear flow, a flow configuration which arguably provides the simplest conceptual framework for tropical cyclogenesis resulting from tropical waves, their interaction with the mean flow, and with diabatic processes associated with deep moist convection. The recirculating Kelvin cat's eye within the critical layer represents a sweet spot for tropical cyclogenesis in which a proto-vortex may form and grow within its parent wave. A common location for storm development is given by the intersection of the wave's critical latitude and trough axis at the center of the cat's eye, with analyzed vorticity centroid nearby. The wave and vortex live together for a time, and initially propagate at approximately the same speed. In most cases this coupled propagation continues for a few days after a tropical depression is identified. For easterly waves, as the name suggests, the propagation is westward. It is shown that in order to visualize optimally the associated Lagrangian motions, one should view the flow streamlines, or stream function, in a frame of reference translating horizontally with the phase propagation of the parent wave. In this co-moving frame, streamlines are approximately equivalent to particle trajectories. The closed circulation is quasi-stationary, and a dividing streamline separates air within the cat's eye from air outside. The critical layer equatorward of the easterly jet axis is important to tropical cyclogenesis because its cat's eye provides (i) a region of cyclonic vorticity and weak deformation by the resolved flow, (ii) containment of moisture entrained by the developing gyre and/or lofted by deep convection therein, (iii) confinement of mesoscale vortex aggregation, (iv) a predominantly convective type of heating profile, and (v) maintenance or enhancement of the parent wave until the vortex becomes a self-sustaining entity and emerges from the wave as a tropical depression. The entire sequence is likened to the development of a marsupial infant in its mother's pouch. These ideas are formulated in three new hypotheses describing the flow kinematics and dynamics, moist thermodynamics and wave/vortex interactions comprising the "marsupial paradigm". A survey of 55 named tropical storms in 1998–2001 reveals that actual critical layers sometimes resemble the ideal east-west train of cat's eyes, but are usually less regular, with one or more recirculation regions in the co-moving frame. It is shown that the kinematics of isolated proto-vortices carried by the wave also can be visualized in a frame of reference translating at or near the phase speed of the parent wave. The proper translation speeds for wave and vortex may vary with height owing to vertical shear and wave-vortex interaction. Some implications for entrainment/containment of vorticity and moisture in the cat's eye are discussed from this perspective, based on the observational survey.