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DOE OSTI2023

Numerical Investigation of Wall-Cooling Effect on Aero-Optical Distortions for Hypersonic Boundary Layer

Compressible wall-modeled large-eddy simulations of Mach 8 turbulent boundary-layer flows over a flat plate were carried out for the conditions of the hypersonic wind tunnel at Sandia National Laboratories. The simulations provide new insight into the effect of wall cooling on the aero-optical path distortions for hypersonic turbulent boundary-layer flows. Four different wall-to-recovery temperature ratios, 0.3, 0.48, 0.71, and 0.89, are considered. Despite the much lower grid resolution, the mean velocity, temperature, and resolved Reynolds stress profiles from the simulation for a temperature ratio of 0.48 are in good agreement with those from a reference direct numerical simulation. The normalized root-mean-square optical path difference obtained from the present simulations is compared with that from reference direct numerical simulations, Sandia experiments, as well as predictions obtained with a semi-analytical model by Notre Dame University. Here the present analysis focuses on the effect of wall cooling on the wall-normal density correlations, on key underlying assumptions of the aforementioned model such as the strong Reynolds analogy, and on the elevation angle effect on the optical path difference. Wall cooling is found to increase the velocity fluctuations and decrease the density fluctuations, resulting in an overall reduction of the normalized optical path distortion. Compared to the simulations, the basic strong Reynolds analogy overpredicts the temperature fluctuations for cooled walls. Also different from the strong Reynolds analogy, the velocity and temperature fluctuations are not perfectly anticorrelated. Finally, as the wall temperature is raised, the density correlation length, away from the wall but inside the boundary layer, increases significantly for beam paths tilted in the downstream direction.

42 ENGINEERING↗

DOE OSTI2021

Velocity structure functions in multiphase turbulence: interpreting kinematics of Hα filaments in cool-core clusters

ABSTRACT The central regions of cool-core galaxy clusters harbour multiphase gas, with gas temperatures ranging from $10$ to $10^7\, \mathrm{K}$. Feedback from active galactic nuclei jets prevents the gas from undergoing a catastrophic cooling flow. However, the exact mechanism of this feedback energy input is unknown, mainly due to the lack of velocity measurements of the hot-phase gas. However, recent observations have measured the velocity structure functions (VSFs) of the cooler molecular (${\sim} 10\, \mathrm{K}$) and Hα filaments (${\sim} 10^4\, \mathrm{K}$) and used them to indirectly estimate the motions of the hot phase. In the first part of this study, we conduct high-resolution (3843–15363 resolution elements) simulations of homogeneous isotropic subsonic turbulence, without radiative cooling. We analyse the second-order velocity structure functions (VSF2) in these simulations and study the effects of varying spatial resolution, the introduction of magnetic fields, and the effect of projection along the line of sight (LOS) on it. In the second part of the study, we analyse high-resolution (7683 resolution elements) idealized simulations of multiphase turbulence in the intracluster medium from the companion study Mohapatra et al. We compare the VSF2 for both the hot ($T\sim 10^7\, \mathrm{K}$) and cold ($T\sim 10^4\, \mathrm{K}$) phases and find that their amplitude depends on the density contrast between the phases. They have similar scaling with separation, but introducing magnetic fields steepens the VSF2 of only the cold phase. We also find that projection along the LOS steepens the VSF2 for the hot phase and mostly flattens it for the cold phase.

Mohapatra, Rajsekhar (ORCID:0000000216007552)↗

DOE OSTI2020

Deep learning-based model for progress variable dissipation rate in turbulent premixed flames

A deep neural network (DNN) based large eddy simulation (LES) model for progress variable dissipation rate in turbulent premixed flames is presented. The DNN model is trained using filtered data from direct numerical simulations (DNS) of statistically planar turbulent premixed flames with n-heptane as fuel. Training data was comprised of flames with varying turbulence levels leading to a range of Karlovitz numbers. Through a-priori tests the DNN model is shown to predict the subfilter contribution to progress variable dissipation rate accurately over a range of filter widths and for all Karlovitz numbers examined in this study. Superior performance of the DNN model relative to an established physics-based model is also demonstrated. Additionally, transferability of the DNN model is highlighted by a-priori evaluation of the model using filtered DNS data from multiple cases with different Karlovitz numbers and fuel species than those that were used for training the model.

33 ADVANCED PROPULSION SYSTEMS↗

DOE OSTI2023

Role of heating and turbulent heat transfer in HE burning and detonation

Enhanced heat transfer model has been developed due to the turbulence in the cracks and pores in the burning fractured HE. This turbulence then serves in the initialization of the turbulence in the product gas. Enhanced heat transfer is also modeled in the turbulent product gas. Numerical simulations of the onset of HE burning and transition to detonation have been performed with and without the turbulent heat transfer model. Computational results indicate that onset of chemical reactions and transition to detonation is induced by the shockwave formed by the pressure rise caused by the heating. In other words, processes similar to shock-to-detonation transition (SDT) could become the driving mechanism for HE explosion and detonation in the case of heating rather than the temperature alone.

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗

DOE OSTI2025

Small-scale properties from exascale computations of turbulence on a $\mathbf{32\,768^3}$ periodic cube

To study the physics of small-scale properties of homogeneous isotropic turbulence at increasingly high Reynolds numbers, direct numerical simulation results have been obtained for forced isotropic turbulence at Taylor-scale Reynolds number R λ = 2500 on a 32 768 3 three-dimensional periodic domain using a GPU pseudo-spectral code on a 1.1 exaflop GPU supercomputer (Frontier). These simulations employ the multi-resolution independent simulation (MRIS) technique (Yeung & Ravikumar 2020, Phys. Rev. Fluids, vol. 5, 110517) where ensemble averaging is performed over multiple short segments initiated from velocity fields at modest resolution, and subsequently taken to higher resolution in both space and time. Reynolds numbers are increased by reducing the viscosity with the large-scale forcing parameters unchanged. Although MRIS segments at the highest resolution for each Reynolds number last for only a few Kolmogorov time scales, small-scale physics in the dissipation range is well captured – for instance, in the probability density functions and higher moments of the dissipation rate and enstrophy density, which appear to show monotonic trends persisting well beyond the Reynolds number range in prior works in the literature. Attainment of range of length and time scales consistent with classical scaling also reinforces the potential utility of the present high-resolution data for studies of short-time-scale turbulence physics at high Reynolds numbers where full-length simulations spanning many large-eddy time scales are still not accessible. A single snapshot of the 32 768 3 data is publicly available for further analyses via the Johns Hopkins Turbulence Database.

intermittency↗

DOE OSTI2025

On the role of ion temperature gradient turbulence in driving ion thermal transport in neutral beam injection-heated L-mode plasmas in a superconducting tokamak

In this paper, we report a detailed experimental study of the role of ion-temperature-gradient driven turbulence in driving ion thermal transport in neutral beam injection (NBI)-heated L-mode plasmas in Experimental Advanced Superconducting Tokamak (EAST) (Wan et al 2000 Nucl. Fusion 40 1057) coupled with linear and nonlinear gyrokinetic simulations. Significant ion-scale turbulence k ⟂ < cm −1 (k ⟂ < 1.5, where k ⟂ where is the perpendicular wavenumber and ρ s is the ion gyroradius calculated using local electron temperature T e ), measured with a microwave reflectometer, is observed in the plasma core at for r/a ≈ 0.265 and 0.5 (where r is half the diameter of the closed flux surface at a given radial position, and a is half the diameter of the last closed flux surface). Local linear stability analysis with experimental equilibrium quantities at these two radial locations using the GS2 gyrokinetic code shows that the most unstable ion-scale micro-instability is the ion temperature gradient (ITG) mode. Since the computed maximum ion-scale linear growth rates are larger than the local Waltz-Miller E × B shearing rate (Waltz and Miller 1999 Phys. Plasmas 6 4265), the E × B shear is unable to suppress ion-scale turbulence, consistent with the experimental observation of ion-scale turbulence. Experimental ion and electron thermal transports, calculated with power balance analysis, are both anomalous with ion thermal transport being the dominant channel. Nonlinear gyrokinetic simulations using the electrostatic global particle-in-cell gyrokinetic tokamak simulation (GTS) code (Weixing Wang et al 2010 Phys. Plasmas 17 072511) show good quantitative agreement between predicted ion thermal transport and that from the power balance analysis. These results demonstrate for the first time the important role of ITG turbulence in driving thermal transport in NBI-heated L-mode plasmas in EAST, showing the effectiveness of GTS code in simulating electrostatic turbulence in EAST and the need of suppressing ITG turbulence to improve energy confinement in EAST.

Sun, P. J. [Chinese Academy of Sciences (CAS), Hef↗

DOE OSTI2024

Turbulence link to L-mode, I-mode, and H-mode confinement in the DIII-D tokamak

Understanding the physics of low-confinement (L-), improved-confinement (I-), and high-confinement (H-) modes is critical for fusion reactors. The finding herein reports observations of two types of turbulence coexisting near the L-mode edge, one magnetohydrodynamic (MHD)-like and another micro-tearing mode (MTM)-like, linked to the H-mode and I-mode confinement in the DIII-D tokamak. Ion-scale magnetic and density turbulence is measured using a Faraday-effect radial-interferometer-polarimeter and beam-emission-spectroscopy (BES). Broadband turbulence spectra of up to ~600 kHz are observed in two discharges where transitions between L-mode, I-mode, and H-mode occurs. Turbulence is found to be inversely correlated with confinement, meaning lower turbulence power at higher confinement. Distinctively, the high-frequency (HF, >~100 kHz) magnetic turbulence power changes by the most (55%) during transitions primarily involving energy confinement change, whereas the low-frequency (LF, <~100 kHz) magnetic and density turbulence power changes by the most (80%) during transitions primarily involving particle confinement change. The LF turbulence amplitude oscillates with and leads to deuterium-alpha emission oscillations before an H-mode. These results imply that HF turbulence mainly affects energy confinement whereas LF turbulence can affect particle confinement. The magnetic and density turbulence exhibits coherence up to 0.6 and cross-phase magnitude close to π/2 in most cases, suggesting they have a common origin in both the LF and HF ranges. BES suggests that LF turbulence resides at the edge (ρ = 0.95) and HF turbulence can be at the outer core (ρ = 0.8) or edge (ρ = 0.95). Comparisons of measurements, theory, and gyrokinetic simulations suggest that HF turbulence is MTM-like in all cases, whereas LF turbulence is more consistent with MHD-like modes and the exact instability might change during transitions—except that a drift-wave origin is possible in a low collisionality H-mode. These results suggest that the H-mode involves suppressed MHD-like turbulence, whereas the I-mode mitigates MTM-like turbulence along with largely unchanged MHD-like turbulence.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

A Multi-physics Computational Investigation of Droplet Pathogen Transport Eminating from Synthetic Coughs and Breathing

In response to the global SARS-CoV-2 transmission pandemic, Sandia National Laboratories Rapid Lab-Directed Research and Development (LDRD) COVID-19 initiative has deployed a multi-physics, droplet-laden, turbulent low-Mach simulation tool to model pathogen-containing water droplets that emanate from synthetic human coughing and breathing. The low-Mach turbulent Eulerian/point-particle Lagrangian methodology directly couples mass, momentum, energy, and species to capture droplet evaporation physics that supports the ability to distinguish between droplets that deposit and those that persist in the environment. Additionally, the cough mechanism is modeled as a pulsed-spray with a prescribed log-normal droplet size distribution. Simulations demonstrate direct droplet deposition lengths in excess of three meters while the persistence of a droplet nuclei entrained within a buoyant plume is noted. Including the effect of protective barriers demonstrates effective mitigation of large-droplet transport. For coughs into a protective barrier, jet impingement and large-scale re-circulation can drive droplets vertically and back towards the subject while supporting persistence of droplet nuclei. Simulations in quiescent conditions demonstrate droplet preferential concentrations due to the coupling between vortex ring shedding and the subsequent advection of a series of three-dimensional rings that tilt and rise vertically due to a misalignment between the initial principle vortex trajectory and gravity. These resolved coughing simulations note vortex ring formation, roll-up and breakdown, while entraining droplet nuclei for large distances and time scales.

42 ENGINEERING↗

DOE OSTI2022

The role of breakup and coalescence in fine-scale bubble-induced turbulence. II. Kinematics

This second part of our research explores the kinematic aspect of fine-scale bubble-induced turbulence (BIT) to (i) present the effect of bubble breakup and coalescence and (ii) compare it against the universal kinematic fine-scale turbulence characteristics reported in the literature. To this end, we simulate a dilute bubbly system of 0.5% void fraction using two distinct numerical simulations. In the volume-of-fluid (VoF) simulation, bubbles undergo breakup and coalescence. In the immersed boundary method (IBM) simulation, however, they act as rigid spheres. We also perform a simulation of classical homogeneous isotropic turbulence (HIT). The first important outcome of this study is that BIT is radically different from HIT in terms of its kinematic fine-scale characteristics. In the vorticity-dominating regions, BIT exhibits a weak vortex stretching. This weak vortex stretching is due to (a) the intermediate strain-rate eigenvalues skewed weakly to positive and (b) the extensive strain-rate eigenvector aligning perpendicular to the vorticity vector. The BIT has, on average, not only a weak enstrophy production but also a weak strain production in strain-dominating regions. The weak strain production is due to (a) the presence of vortex stretching in highly strained fluid elements and (b) the absolute magnitude of compressive strain-rate eigenvalue being as close to the extensive strain-rate eigenvalue. Thus, none of the kinematic fine-scale HIT characteristics is noted for BIT. The second important conclusion is that bubble breakup and coalescence play little to no influence on the kinematics of fine-scale BIT as VoF and IBM simulations produce similar results.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Dynamic calibration of differential equations using machine learning, with application to turbulence models

We present a methodology for calibration of parametric ordinary and partial differential equation models, using off-the-shelf software for back-propagation in Neural Networks (NN). As a prototypical example, we consider calibration of a Reynolds-averaged Navier-Stokes (RANS) turbulence closure model, against ground truth data from direct numerical simulations (DNS) of two different turbulent flows. Numerical time integration is represented as a custom NN, where only the RANS model parameters are trainable. A loss function is defined to quantify the mismatch between the NN prediction and the ground truth over a predefined, finite time integration window. This loss function is then minimized using a gradient descent method utilizing the back-propagation algorithm. Furthermore, this dynamic approach to training is to be contrasted with a static approach, wherein a least square regression estimate for parameters is obtained in the limit of an infinitesimal time integration window. In a first test of static and dynamic approaches against ground truth data generated by the model, the former proves to be significantly faster and more accurate than the latter at recovering the parameters. When both calibration approaches are tested against DNS data, for which it is known that the model cannot achieve a perfect fit, the static approach yields a good prediction only for short times, while the dynamic approach results in physical and stable predictions over the entire integration window. After optimization of the dynamic approach for time step, spatial resolution, stability, and physics-based constraints, we obtain a 50% improvement of outcomes over those obtained from the existing, manually calibrated set of parameters, demonstrating the merits of this systematic and automated procedure.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Hierarchically Informed Engineering Models for Predictive Modeling of Turbulent Premixed Flame Propagation in Pre- chamber Turbulent Jet Ignition

The goal of the project is to improve the predictive accuracy and efficiency of turbulent combustion sub-models for pre-chamber turbulent jet ignition (TJI). This goal is achieved through the development of a hierarchically informed engineering model for turbulent combustion in TJI. The model development starts with the highest level of model description of turbulent combustion with direct numerical simulation (DNS) from which fundamental characteristics and scaling properties of turbulent premixed flame propagation under TJI relevant conditions are obtained.

42 ENGINEERING↗

DOE OSTI2025

Turbulent burning velocity of lean premixed hydrogen/air flames at engine conditions: Effects of turbulence intensity and length scale

For turbulent lean premixed hydrogen flames with strong thermodiffusively instabilities, most previous studies have focused on the influence of turbulence intensity, whereas the role of turbulence length scale is less well understood. Here, this study addresses this gap by conducting direct numerical simulations (DNS) of statistically planar turbulent premixed flames for a lean (ϕ=0.35) hydrogen/air mixture under independently varied turbulence intensity (u') and length scale (l T ) at engine-relevant thermodynamics conditions. Results show that as u' increases, the flame front becomes increasingly wrinkled, forming smaller cellular structures. In contrast, l T variations do not significantly alter the size of these structures. For the turbulent burning velocity (S T ), the normalized S T (i.e., S T /S L , where S L is the laminar flame speed) increases linearly with u', driven by both enhanced flame surface wrinkling (i.e., increased A T /A L ) and enhanced local burning rate (i.e., increased I 0 ). However, increasing l T reduces I 0 , despite a continued increase in A T /A L , resulting in only a marginal increase in S T /S L . To reveal the underlying mechanisms, especially the decreasing trend of I 0 with l T , local flame dynamics analyses are performed. It is found that as l T increases, the interaction between thermodiffusive effects and turbulence weakens due to the reduced tangential strain rate, while the flame curvature remains largely unchanged. This suppresses local reactivity enhancement and thus decreases I 0 , In contrast, an increase in u' enhances the interaction by amplifying both curvature fluctuation and tangential strain rate, leading to increased local reactivity (increased I 0 ). Finally, based on the DNS data, several new scaling models are proposed for the three global properties, S T /S L , A T /A L , and I 0 , and showed improvements compared to existing models. These findings provide new insights into the flame-turbulence interactions in thermodiffusively unstable hydrogen flames. The DNS dataset is also useful for the development of turbulent combustion models applicable to practical engine simulations.

Engine-relevant condition↗

DOE OSTI2022

Aerodynamic Analysis of Wind Loading on Parabolic Trough Collectors Using High-Fidelity CFD Modeling

Wind loading on parabolic solar collector support structures is the key driver for material costs. As identified by NREL's CSP Best Practices project [1], ability to predict wind-loading in deep-arrays and the impact of edge effects has potential to decrease levelized cost of electricity (LCoE) by 15%. In this study, we present a high-fidelity computational model that can model the flow around deep arrays of parabolic troughs under varying turbulent inflow conditions. Ability to accurately reproduce atmospheric boundary layer (ABL) profiles is first demonstrated through Large Eddy Simulation (LES) simulations and compared against wind tunnel measurements. Thereafter, unsteady turbulent flow past single row and six row configurations is computed using hybrid RANS/LES simulations. Investigations of wind loading, through parametric tilt angle variations of single row and on waked collectors in six-row configuration, are used to highlight design considerations for parabolic troughs only possible through high-fidelity unsteady calculations.

aerodynamics↗

DOE OSTI2024

Intrinsic Toroidal Rotation Driven by Turbulent and Neoclassical Processes in Tokamak Plasmas from Global Gyrokinetic Simulations

Gyrokinetic tokamak plasmas can exhibit intrinsic toroidal rotation driven by the residual stress. While most studies have attributed the residual stress to the parallel-momentum flux from the turbulent E × B motion, the parallel-momentum flux from the drift-orbit motion (denoted $Π^D_\parallel$) and the E × B-momentum flux from the E × B motion (denoted $Π_{E×B}$) are often neglected. Here, we use the global total-f gyrokinetic code XGC to study the residual stress in the core and the edge of a DIII-D H-mode plasma. Numerical results show that both $Π^D_\parallel$ and $Π_{E×B}$ make up a significant portion of the residual stress. In particular, $Π^D_\parallel$ in the core is higher than the collisional neoclassical level in the presence of turbulence, while in the edge it represents an outflux of countercurrent momentum even without turbulence. Using a recently developed “orbit-flux” formulation, we show that the higher-than-neoclassical-level $Π^D_\parallel$ in the core is driven by turbulence, while the outflux of countercurrent momentum from the edge is mainly due to collisional ion orbit loss. In conclusion, these results suggest that $Π^D_\parallel$ and $Π_{E×B}$ can be important for the study of intrinsic toroidal rotation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Advanced Turbulence Models for Large-Scale Atmospheric Boundary Layer Flows

We present high-fidelity large-eddy-simulation (LES) modeling approaches for the turbulent atmospheric boundary layer (ABL) flows. Wind energy is a prime example of an application driven by ABL. Generation of electrical energy from farms of wind turbines at night in the stable ABL is a particularly interesting situation. In this report, we consider the well-known GEWEX (Global Energy and Water Cycle Experiment) Atmospheric Boundary Layer Study (GABLS) stably stratified benchmark LES case. We use a high-order spectral element code Nek5000/RS, which is supported under the DOE's Exascale Computing Project (ECP) Center for Efficient Exascale Discretizations (CEED) project, targeting application simulations on various acceleration-device based exascale computing platforms. In our earlier ANL report, we demonstrated our newly developed subgrid-scale (SGS) models based on high-pass filter (HPF), mean-field eddy viscosity (MFEV), and Smagorinsky (SMG) with no-slip and traction boundary conditions, provided with low-order statistics, convergence and turbulent structure analysis. In this report, we extend the range of our SGS modeling approaches in the context of the mean-field eddy viscosity (MFEV), to include the solution of an SGS turbulent kinetic energy equation (TKE). We demonstrate the model fidelity of Nek5000/RS in comparison to that of AMR-Wind, a block-structured second-order finite-volume code with adaptive-mesh-refinement capabilities, with which we studied scaling performance for both codes in comparison on DOE's leadership computing platforms.

17 WIND ENERGY↗

DOE OSTI2026

Modification of ion-temperature-gradient turbulence by impurities in stellarator plasmas

Recent nonlinear gyrokinetic simulations have shown that impurities can strongly modify the turbulent heat flux in stellarator plasmas. Here, the ion-temperature-gradient (ITG) dispersion relation in a plasma containing impurities is analytically solved in certain limits and an expression for the modification of the ITG growth rate by impurities is derived. The analytical expression is the sum of three terms corresponding to three different physical causes (impurity density gradient, impurity temperature gradient and dilution) of the change in the growth rate. The scalings predicted analytically for the modification of the growth rate are shown to be reproduced by linear gyrokinetic simulations. The conditions for reduction or increase of the ITG growth by impurities are also correctly predicted by the analytical solution to the dispersion relation. Finally, a remarkable correlation is found between the analytical expression for the modification of the growth rate and the modification of the turbulent heat flux obtained from nonlinear gyrokinetic simulations.

Calvo, Iván [Research Centre for Energy, Environme↗

DOE OSTI2020

Simulating Real Atmospheric Boundary Layers at Gray-Zone Resolutions: How Do Currently Available Turbulence Parameterizations Perform?

Recent computational and modeling advances have led a diverse modeling community to experiment with atmospheric boundary layer (ABL) simulations at subkilometer horizontal scales. Accurately parameterizing turbulence at these scales is a complex problem. The modeling solutions proposed to date are still in the development phase and remain largely unvalidated. This work assesses the performance of methods currently available in the Weather Research and Forecasting (WRF) model to represent ABL turbulence at a gray-zone grid spacing of 333 m. We consider three one-dimensional boundary layer parameterizations (MYNN, YSU and Shin-Hong) and coarse large-eddy simulations (LES). The reference dataset consists of five real-case simulations performed with WRF-LES nested down to 25 m. Results reveal that users should refrain from coarse LES and favor the scale-aware, Shin-Hong parameterization over traditional one-dimensional schemes. Overall, the spread in model performance is large for the cellular convection regime corresponding to the majority of our cases, with coarse LES overestimating turbulent energy across scales and YSU underestimating it and failing to reproduce its horizontal structure. Despite yielding the best results, the Shin-Hong scheme overestimates the effect of grid dependence on turbulent transport, highlighting the outstanding need for improved solutions to seamlessly parameterize turbulence across scales.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Planck Dust Polarization Power Spectra Are Consistent with Strongly Supersonic Turbulence

Abstract The polarization of the cosmic microwave background is rich in information but obscured by foreground emission from the Milky Way’s interstellar medium (ISM). To uncover relationships between the underlying turbulent ISM and the foreground power spectra, we simulated a suite of driven, magnetized, turbulent models of the ISM, varying the fluid properties via the sonic Mach number, M S , and magnetic (Alfvén) Mach number, M A . We measure the power spectra of density ( ρ ), velocity ( v ), magnetic field ( H ), total projected intensity ( T ), parity-even polarization ( E ), and parity-odd polarization ( B ). We find that the slopes of all six quantities increase with M S . Most increase with M A , while the magnetic field spectrum steepens with M A . By comparing spectral slopes of E and B to those measured by Planck, we infer typical values of M S and M A for the ISM. As the fluid velocity increases, M S > 4 , the ratio of BB power to EE power increases to approach a constant value near the Planck-observed value of ∼0.5, regardless of the magnetic field strength. We also examine correlation coefficients between projected quantities, and find that r TE ≈ 0.3, in agreement with Planck, for appropriate combinations of M S and M A . Finally, we consider parity-violating correlations r TB and r EB .

Astronomy & Astrophysics↗