The relation between the true and observed fractal dimensions of turbulent clouds

Observations of interstellar gas clouds are typically limited to two-dimensional (2D) projections of the intrinsically three-dimensional (3D) structure of the clouds. In this study, we present a novel method for relating the 2D projected fractal dimension ($\mathcal{D}_{\text{p}}$) to the 3D fractal dimension ($\mathcal{D}_{\text{3D}}$) of turbulent clouds. We do this by computing the fractal dimension of clouds over two orders of magnitude in turbulent Mach number $(\mathcal{M} = 1-100)$, corresponding to seven orders of magnitude in spatial scales within the clouds. This provides us with the data to create a new empirical relation between $\mathcal{D}_{\text{p}}$ and $\mathcal{D}_{\text{3D}}$. The proposed relation is $\mathcal{D}_{\text{3D}}(\mathcal{D}_{\text{p}}) = \Omega_1 erfc ( \xi_1 erfc^{-1}[ (\mathcal{D}_{\text{p}} - \mathcal{D}_{\text{p,min}})/\Omega_2 ] + \xi_2 ) + \mathcal{D}_{\text{3D,min}}$, where the minimum 3D fractal dimension, $\mathcal{D}_{\text{3D,min}} = 2.06 \pm 0.35$, the minimum projected fractal dimension, $\mathcal{D}_{\text{p,min}} = 1.55 \pm 0.13$, $\Omega_1 = 0.47 \pm 0.18$, $\Omega_2 = 0.22 \pm 0.07$, $\xi_1 = 0.80 \pm 0.18$ and $\xi_2 = 0.26 \pm 0.19$. The minimum 3D fractal dimension, $\mathcal{D}_{\text{3D,min}} = 2.06 \pm 0.35$, indicates that in the high $\mathcal{M}$ limit the 3D clouds are dominated by planar shocks. The relation between $\mathcal{D}_{\text{p}}$ and $\mathcal{D}_{\text{3D}}$ of molecular clouds may be a useful tool for those who are seeking to understand the 3D structures of molecular clouds, purely based upon 2D projected data and shows promise for relating the physics of the turbulent clouds to the fractal dimension.Comment: 14 pages, 7 figures. Accepted 2019 May 1

Beyond Separation and Neutrality: “Non ‘Market Participant’” As the Central Metaphor of Religion Clause Jurisprudence

A metaphor has always dominated the Supreme Court\u27s religion clause jurisprudence. The metaphor prescribes the proper legal and political relationship between religion and government; it is a constitutional norm. This Article traces the historical development of this central metaphor—from separation to neutrality—and finds both metaphors lacking in normative adequacy. These inadequacies lead to the proposal of a new metaphor: government as a non \u27market participant.\u27 This metaphor essentially combines Justice Holmes\u27s marketplace of ideas with the market participant in the Court\u27s Dormant Commerce Clause jurisprudence, but with an important twist. The normative obligations of the new metaphor prevent government from regulating the behavior of citizen competitors in the religion market (free exercise) and from becoming a competitor or engaging in competitor behavior in that market (establishment). Both sets of governmental duties are aimed at achieving the single goal of maximal religious liberty for citizens

Filaments and striations: anisotropies in observed, supersonic, highly-magnetised turbulent clouds

Stars form in highly-magnetised, supersonic turbulent molecular clouds. Many of the tools and models that we use to carry out star formation studies rely upon the assumption of cloud isotropy. However, structures like high-density filaments in the presence of magnetic fields, and magnetosonic striations introduce anisotropies into the cloud. In this study we use the two-dimensional (2D) power spectrum to perform a systematic analysis of the anisotropies in the column density for a range of Alfv\'en Mach numbers ($\mathcal{M}_A=0.1$--$10$) and turbulent Mach numbers ($\mathcal{M}=2$--$20$), with 20 high-resolution, three-dimensional (3D) turbulent magnetohydrodynamic simulations. We find that for cases with a strong magnetic guide field, corresponding to $\mathcal{M}_A<1$, and $\mathcal{M}\lesssim 4$, the anisotropy in the column density is dominated by thin striations aligned with the magnetic field, while for $\mathcal{M}\gtrsim 4$ the anisotropy is significantly changed by high-density filaments that form perpendicular to the magnetic guide field. Indeed, the strength of the magnetic field controls the degree of anisotropy and whether or not any anisotropy is present, but it is the turbulent motions controlled by $\mathcal{M}$ that determine which kind of anisotropy dominates the morphology of a cloud.Comment: 20 pages, 11 figures; accepted for publication in MNRAS, 2019, Nov. 2

Growth or decay: universality of the turbulent dynamo saturation

The small-scale turbulent dynamo (SSD) is likely to be responsible for the magnetisation of the interstellar medium (ISM) that we observe in the Universe today. The SSD efficiently converts kinetic energy $E_{\rm kin}$ into magnetic energy $E_{\rm mag}$, and is often used to explain how an initially weak magnetic field with $E_{\rm mag} \ll E_{\rm kin}$ is amplified, and then maintained at a level $E_{\rm mag} \lesssim E_{\rm kin}$. Usually, this process is studied by initialising a weak seed magnetic field and letting the turbulence grow it to saturation. However, in this study, using three-dimensional, non-ideal magnetohydrodynamical turbulence simulations, we show that the same saturated state can also be achieved if initially $E_{\rm mag} \gg E_{\rm kin}$ or $E_{\rm mag} \sim E_{\rm kin}$. This is realised through a two-stage exponential decay (1. a slow backreaction that converts $E_{\rm mag}$ into $E_{\rm kin}$, and 2. Ohmic dissipation concentrated in anisotropic current sheets) into the saturated state, for which we provide an analytical model. This means that even if there are temporary local enhancements of $E_{\rm mag}$ in the ISM, such that $E_{\rm mag} > E_{\rm kin}$, e.g., through amplifications such as compressions, over a long enough time the field will decay into the saturated state set by the SSD, which is determined by the turbulence and magnetic dissipation. However, we also provide analytical models for the decay times and utilise wait-time statistics from compressive supernova events to show that if the magnetic field is enhanced above the saturated state, it will not have enough time to decay the field before the next supernova event. Hence, unless there exists a mechanism for destroying magnetic fields that is not in our non-ideal MHD models, the amplitudes of interstellar magnetic fields may also be a... (abridged).Comment: 18 pages. 14 figures. Submitted to MNRAS. Comments welcom

Fundamental MHD scales -- II: the kinematic phase of the supersonic small-scale dynamo

The small-scale dynamo (SSD) amplifies weak magnetic fields exponentially fast via kinetic motions. While there exist well-established theories for SSDs in incompressible flows, many astrophysical SSDs operate in supersonic turbulence. To understand the impact of compressibility on amplified magnetic fields, we perform an extensive set of visco-resistive SSD simulations, covering a wide range of sonic Mach number $\mathcal{M}$, hydrodynamic Reynolds number Re, and magnetic Prandtl number Pm. We develop robust methods for measuring kinetic and magnetic energy dissipation scales $\ell_\nu$ and $\ell_\eta$, as well as the scale at which magnetic fields are strongest $\ell_p$ during the kinematic phase of these simulations. We show that $\ell_\nu/\ell_\eta \sim$ Pm$^{1/2}$ is a universal feature in the kinematic phase of Pm $\geq 1$ SSDs, regardless of $\mathcal{M}$ or Re, and we confirm earlier predictions that SSDs operating in incompressible plasmas (either $\mathcal{M} \leq 1$ or Re $<$ Re$_{\rm crit} \approx 100$) concentrate magnetic energy at the smallest scales allowed by magnetic dissipation, $\ell_p \sim \ell_\eta$, and produce fields organised with field strength and field-line curvature inversely correlated. However, we show that these predictions fail for compressible SSDs ($\mathcal{M} > 1$ and Re $>$ Re$_{\rm crit}$), where shocks concentrate magnetic energy in large-scale, over-dense, coherent structures, with size $\ell_p \sim (\ell_{\rm turb} / \ell_{\rm shock})^{1/3} \ell_\eta \gg \ell_\eta$, where $\ell_{\rm shock} \sim \mathcal{M}^2 / [$Re $(\mathcal{M} - 1)^2]$ is shock width, and $\ell_{\rm turb}$ is the turbulent outer scale; magnetic field-line curvature becomes almost independent of the field strength. We discuss the implications for galaxy mergers and for cosmic-ray transport models in the interstellar medium that are sensitive to field-line curvature statistics.Comment: 25 pages, 15 figures, submitted to MNRAS, json-file w/ dat

AMPA Receptor Phosphorylation and Synaptic Colocalization on Motor Neurons Drive Maladaptive Plasticity below Complete Spinal Cord Injury.

Clinical spinal cord injury (SCI) is accompanied by comorbid peripheral injury in 47% of patients. Human and animal modeling data have shown that painful peripheral injuries undermine long-term recovery of locomotion through unknown mechanisms. Peripheral nociceptive stimuli induce maladaptive synaptic plasticity in dorsal horn sensory systems through AMPA receptor (AMPAR) phosphorylation and trafficking to synapses. Here we test whether ventral horn motor neurons in rats demonstrate similar experience-dependent maladaptive plasticity below a complete SCI in vivo. Quantitative biochemistry demonstrated that intermittent nociceptive stimulation (INS) rapidly and selectively increases AMPAR subunit GluA1 serine 831 phosphorylation and localization to synapses in the injured spinal cord, while reducing synaptic GluA2. These changes predict motor dysfunction in the absence of cell death signaling, suggesting an opportunity for therapeutic reversal. Automated confocal time-course analysis of lumbar ventral horn motor neurons confirmed a time-dependent increase in synaptic GluA1 with concurrent decrease in synaptic GluA2. Optical fractionation of neuronal plasma membranes revealed GluA2 removal from extrasynaptic sites on motor neurons early after INS followed by removal from synapses 2 h later. As GluA2-lacking AMPARs are canonical calcium-permeable AMPARs (CP-AMPARs), their stimulus- and time-dependent insertion provides a therapeutic target for limiting calcium-dependent dynamic maladaptive plasticity after SCI. Confirming this, a selective CP-AMPAR antagonist protected against INS-induced maladaptive spinal plasticity, restoring adaptive motor responses on a sensorimotor spinal training task. These findings highlight the critical involvement of AMPARs in experience-dependent spinal cord plasticity after injury and provide a pharmacologically targetable synaptic mechanism by which early postinjury experience shapes motor plasticity