Compressible Magnetohydrodynamic Simulations Research Articles

Joint action of phase mixing and nonlinear effects in MHD waves propagating in coronal loops

Context. The evolution of Alfvén waves in cylindrical magnetic flux tubes, which represent a basic model for loops observed in the solar corona, can be affected by phase mixing and turbulent cascade. Phase mixing results from transverse inhomogeneities in the Alfvén speed, causing different shells of the flux tube to oscillate at different frequencies, thus forming increasingly smaller spatial scales in the direction perpendicular to the guide field. Turbulent cascade also contributes to the dissipation of the bulk energy of the waves through the generation of smaller spatial scales. Both processes present characteristic timescales. Different regimes can be envisaged according to how those timescales are related and to the typical timescale at which dissipation is at work. Aims. We investigate the interplay of phase mixing and the nonlinear turbulent cascade in the evolution and dissipation of Alfvén waves using compressible magnetohydrodynamic numerical simulations. We consider perturbations in the form of torsional waves, both propagating and standing, or turbulent fluctuations, or a combination of the two. The main purpose is to study how phase mixing and nonlinear couplings jointly work to produce small scales in different regimes. Methods. We conducted a numerical campaign to explore the typical parameters, such as the loop length, the amplitude and spatial profile of the perturbations, and the dissipative coefficients. A pseudo-spectral code was employed to solve the three-dimensional compressible magnetohydrodynamic equations, modeling the evolution of perturbations propagating in a flux tube corresponding to an equilibrium configuration with cylindrical symmetry. Results. We find that phase mixing takes place for moderate amplitudes of the turbulent component even in a distorted, nonaxisymmetric configuration, building small scales that are locally transverse to the density gradient. The dissipative time decreases with increasing the percentage of the turbulent component. This behavior is verified both for propagating and standing waves. Even in the fully turbulent case, a mechanism qualitatively similar to phase mixing occurs: it actively generates small scales together with the nonlinear cascade, thus providing the shortest dissipative time. General considerations are given to identify this regime in the parameter space. The turbulent perturbation also distorts the background density, locally increasing the Alfvén velocity gradient and further contributing to accelerating the formation of small scales. Conclusions. Our campaign of simulations is relevant for the coronal plasma where Reynolds and Lundquist numbers are extremely high. For sufficiently low perturbation amplitudes, phase mixing and turbulence work synergically, speeding up the dissipation of the perturbation energy: phase mixing dominates at early times and nonlinear effects at later times. We find that the dissipative time is shorter than those of phase mixing and the nonlinear cascade when individually considered.

On the Interpretation of the Scalings of Density Fluctuations from In Situ Solar Wind Observations: Insights from 3D Turbulence Simulations

Solar wind turbulence is often perceived as weakly compressible and the density fluctuations remain poorly understood both theoretically and observationally. Compressible magnetohydrodynamic simulations provide useful insights into the nature of density fluctuations. We discuss a few important effects related to 3D simulations of turbulence and in situ observations. The observed quantities such as the power spectrum and variance depend on the angle between the sampling trajectory and the mean magnetic field due to anisotropy of the turbulence. The anisotropy effect is stronger at smaller scales and lower plasma beta. Additionally, in situ measurements tend to exhibit a broad range of variations, even though they could be drawn from the same population with the defined averages, so a careful averaging may be needed to reveal the scaling relations between density variations and other turbulence quantities such as turbulent Mach number from observations.

A finite-volume scheme for modeling compressible magnetohydrodynamic flows at low Mach numbers in stellar interiors

Fully compressible magnetohydrodynamic (MHD) simulations are a fundamental tool for investigating the role of dynamo amplification in the generation of magnetic fields in deep convective layers of stars. The flows that arise in such environments are characterized by low (sonic) Mach numbers (ℳson ≲ 10−2). In these regimes, conventional MHD codes typically show excessive dissipation and tend to be inefficient as the Courant–Friedrichs–Lewy (CFL) constraint on the time step becomes too strict. In this work we present a new method for efficiently simulating MHD flows at low Mach numbers in a space-dependent gravitational potential while still retaining all effects of compressibility. The proposed scheme is implemented in the finite-volume SEVEN-LEAGUE HYDRO (SLH) code, and it makes use of a low-Mach version of the five-wave Harten–Lax–van Leer discontinuities (HLLD) solver to reduce numerical dissipation, an implicit–explicit time discretization technique based on Strang splitting to overcome the overly strict CFL constraint, and a well-balancing method that dramatically reduces the magnitude of spatial discretization errors in strongly stratified setups. The solenoidal constraint on the magnetic field is enforced by using a constrained transport method on a staggered grid. We carry out five verification tests, including the simulation of a small-scale dynamo in a star-like environment at ℳson ~ 10−3. We demonstrate that the proposed scheme can be used to accurately simulate compressible MHD flows in regimes of low Mach numbers and strongly stratified setups even with moderately coarse grids.

An Arbitrarily High-order Spectral Difference Method with Divergence Cleaning (SDDC) for Compressible Magnetohydrodynamic Simulations on Unstructured Grids

This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains.

Superdiffusion of cosmic rays in compressible magnetized turbulence

ABSTRACT Owing to the complexity of turbulent magnetic fields, modelling the diffusion of cosmic rays is challenging. Based on the current understanding of anisotropic magnetohydrodynamic (MHD) turbulence, we use test particles to examine the cosmic rays’ superdiffusion in the direction perpendicular to the mean magnetic field. By changing Alfvén Mach number MA and sonic Mach number MS of compressible MHD simulations, our study covers a wide range of astrophysical conditions including subsonic warm gas phase and supersonic cold molecular gas. We show that freely streaming cosmic rays’ perpendicular displacement increases as 3/2 to the power of the time travelled along local magnetic field lines. This power-law index changes to 3/4 if the parallel propagation is diffusive. We find that the cosmic rays’ parallel mean free path decreases in a power-law relation of $M_\mathrm{ A}^{-2}$ in supersonic turbulence. We investigate the energy fraction of slow, fast, and Alfvénic modes and confirm the dominance of Alfvénic modes in the perpendicular superdiffusion. In particular, the energy fraction of fast mode, which is the main agent for pitch-angle scattering, increases with MA, but is insensitive to MS ≥ 2. Accordingly, our results suggest that the suppressed diffusion in supersonic molecular clouds arises primarily due to the variations of MA instead of MS.

Anisotropic Turbulence in Position–Position–Velocity Space: Probing Three-dimensional Magnetic Fields

Abstract Direct measurements of three-dimensional magnetic fields in the interstellar medium are not achievable. However, the anisotropic nature of magnetohydrodynamic (MHD) turbulence provides a novel way of tracing the magnetic fields. Guided by the advanced understanding of turbulence’s anisotropy in the position–position–velocity (PPV) space, we extend the structure-function analysis to measure both the three-dimensional magnetic field orientation and Alfvén Mach number M A, which provides the information on magnetic field strength. Following the theoretical framework developed in Kandel et al., we find that the anisotropy in a given velocity channel is affected by the inclination angle between the three-dimensional magnetic field direction and the line of sight as well as media magnetization. We analyze the synthetic PPV cubes generated by incompressible and compressible MHD simulations. We confirm that the PPV channel’s intensity fluctuations measured in various position angles reveal plane-of-the-sky magnetic field orientation. We show that by varying the channel width, the anisotropies of the intensity fluctuations in PPV space can be used to simultaneously estimate both magnetic field inclination angle and strength of total magnetic fields.

Anisotropies in Compressible MHD Turbulence: Probing Magnetic Fields and Measuring Magnetization

Abstract Probing magnetic fields in the interstellar medium (ISM) is notoriously challenging. Motivated by the modern theories of magnetohydrodynamic (MHD) turbulence and turbulence anisotropy, we introduce the Structure Function Analysis (SFA) as a new approach to measure the magnetic-field orientation and estimate the magnetization. We analyze the statistics of turbulent velocities in three-dimensional compressible MHD simulations through the second-order structure functions in both local and global reference frames. In the sub-Alfvénic turbulence with the magnetic energy larger than the turbulent energy, the SFA of turbulent velocities measured in the directions perpendicular and parallel to the magnetic field can be significantly different. Their ratio has a power-law dependence on the Alfvén Mach number M A, which is inversely proportional to the magnetic-field strength. We demonstrate that the anisotropic structure functions of turbulent velocities can be used to estimate both the orientation and strength of magnetic fields. With turbulent velocities measured using different tracers, our approach can be generally applied to probing the magnetic fields in the multiphase ISM.

Modelling the imposed magnetospheres of Mars-like exoplanets: star–planet interactions and atmospheric losses

ABSTRACT Based on 3D compressible magnetohydrodynamic simulations, we explore the interactions between the magnetized wind from a solar-like star and a Mars-like planet – with a gravitionally stratified atmosphere – that is either non-magnetized or hosts a weak intrinsic dipolar field. The primary mechanism for the induction of a magnetosphere around a non-magnetized conducting planet is the pile-up of stellar magnetic fields in the day-side region. The magnetopause stand-off distance decreases as the strength of the planetary dipole field is lowered and saturates to a minimum value for the case of a planet with no magnetic field. Global features such as bow shock, magnetosheath, magnetotail, and strong current sheets are observed in the imposed magnetosphere. We explore variations in atmospheric mass loss rates for different stellar wind strengths to understand the impact of stellar magnetic activity and plasma winds – and their evolution – on (exo)planetary habitability. In order to simulate a case analogous to the present-day Mars, a planet without atmosphere is considered. Our simulations are found to be in good agreement with observational data from Mars Global Surveyor and Mars Atmosphere and Volatile EvolutioN missions and is expected to complement observations from the Emirates (Hope) Mars Mission, China's Tianwen-1 and NASA's Mars 2020 Perseverance mission.

Full compressible 3D MHD simulation of solar wind

ABSTRACT Identifying the heating mechanisms of the solar corona and the driving mechanisms of solar wind are key challenges in understanding solar physics. A full three-dimensional compressible magnetohydrodynamic (MHD) simulation was conducted to distinguish between the heating mechanisms in the fast solar wind above the open field region. Our simulation describes the evolution of the Alfvénic waves, which includes the compressible effects from the photosphere to the heliospheric distance s of 27 solar radii (R⊙). The hot corona and fast solar wind were reproduced simultaneously due to the dissipation of the Alfvén waves. The inclusion of the transition region and lower atmosphere enabled us to derive the solar mass-loss rate for the first time by performing a full three-dimensional compressible MHD simulation. The Alfvén turbulence was determined to be the dominant heating mechanism in the solar wind acceleration region (s > 1.3 R⊙), as suggested by previous solar wind models. In addition, shock formation and phase mixing are important below the lower transition region (s < 1.03 R⊙) as well.

Shear-driven Transition to Isotropically Turbulent Solar Wind Outside the Alfvén Critical Zone

Abstract Motivated by prior remote observations of a transition from striated solar coronal structures to more isotropic “flocculated” fluctuations, we propose that the dynamics of the inner solar wind just outside the Alfvén critical zone, and in the vicinity of the first surface, is powered by the relative velocities of adjacent coronal magnetic flux tubes. We suggest that large-amplitude flow contrasts are magnetically constrained at lower altitude but shear-driven dynamics are triggered as such constraints are released above the Alfvén critical zone, as suggested by global magnetohydrodynamic (MHD) simulations that include self-consistent turbulence transport. We argue that this dynamical evolution accounts for features observed by Parker Solar Probe (PSP) near initial perihelia, including magnetic “switchbacks,” and large transverse velocities that are partially corotational and saturate near the local Alfvén speed. Large-scale magnetic increments are more longitudinal than latitudinal, a state unlikely to originate in or below the lower corona. We attribute this to preferentially longitudinal velocity shear from varying degrees of corotation. Supporting evidence includes comparison with a high Mach number three-dimensional compressible MHD simulation of nonlinear shear-driven turbulence, reproducing several observed diagnostics, including characteristic distributions of fluctuations that are qualitatively similar to PSP observations near the first perihelion. The concurrence of evidence from remote sensing observations, in situ measurements, and both global and local simulations supports the idea that the dynamics just above the Alfvén critical zone boost low-frequency plasma turbulence to the level routinely observed throughout the explored solar system.

Three-dimensional Simulation of the Fast Solar Wind Driven by Compressible Magnetohydrodynamic Turbulence

Abstract Using a three-dimensional compressible magnetohydrodynamic (MHD) simulation, we have reproduced the fast solar wind in a direct and self-consistent manner, based on the wave/turbulence-driven scenario. As a natural consequence of Alfvénic perturbations at the coronal base, highly compressional and turbulent fluctuations are generated, leading to heating and acceleration of the solar wind. The analysis of power spectra and structure functions reveals that the turbulence is characterized by its imbalanced (in the sense of outward Alfvénic fluctuations) and anisotropic nature. The density fluctuation originates from the parametric decay instability (PDI) of outwardly propagating Alfvén waves and plays a significant role in the Alfvén-wave reflection that triggers turbulence. Our conclusion is that the fast solar wind is heated and accelerated by compressible MHD turbulence driven by PDI and resultant Alfvén-wave reflection.

Modeling Star–Planet Interactions in Far-out Planetary and Exoplanetary Systems

Abstract The magnetized wind from a host star plays a vital role in shaping the magnetospheric configuration of the planets it harbors. We carry out three-dimensional (3D) compressible magnetohydrodynamic simulations of the interactions between magnetized stellar winds and planetary magnetospheres corresponding to a far-out star–planet system, with and without planetary dipole obliquity. We identify the pathways that lead to the formation of a dynamical steady-state magnetosphere and find that magnetic reconnection plays a fundamental role in the process. The magnetic energy density is found to be greater on the nightside than on the dayside, and the magnetotail is comparatively more dynamic. It is found that stellar wind plasma injection into the inner magnetosphere is possible through the magnetotail. We further study magnetospheres with extreme tilt angles, keeping in perspective the examples of Uranus and Neptune. High dipole obliquities may also manifest due to polarity excursions during planetary field reversals. We find that global magnetospheric reconnection sites change for large planetary dipole obliquity, and more complex current sheet structures are generated. We discuss the implications of these findings for atmospheric erosion, the introduction of stellar and interplanetary species that modify the composition of the atmosphere, auroral activity, and magnetospheric radio emission. This study is relevant for exploring star–planet interactions and its consequence on atmospheric dynamics and habitability in solar system planets and exoplanets.

Thermal responses in a coronal loop maintained by wave heating mechanisms

A full 3-dimensional compressible magnetohydrodynamic (MHD) simulation is conducted to investigate the thermal responses of a coronal loop to the dynamic dissipation processes of MHD waves. When the foot points of the loop are randomly and continuously forced, the MHD waves become excited and propagate upward. Then, a 1-MK temperature corona is produced naturally as the wave energy dissipates. The excited wave packets become non-linear just above the magnetic canopy, and the wave energy cascades into smaller spatial scales. Moreover, collisions between counter-propagating Alfv\'{e}n wave packets increase the heating rate, resulting in impulsive temperature increases. Our model demonstrates that the heating events in the wave-heated loops can be nanoflare-like in the sense that they are spatially localized and temporally intermittent.

DENSITY-MAGNETIC FIELD CORRELATION IN MAGNETOHYDRODYNAMIC TURBULENCE DRIVEN BY DIFFERENT DRIVING SCHEMES WITH DIFFERENT CORRELATION TIMES

ABSTRACT Turbulent motions naturally produce density and magnetic-field fluctuations. Correlation between the two fluctuations is important for interpretation of observations, such as observations of the rotation measure (RM). In this paper, we study the effect of driving schemes on the density-magnetic-field correlation. In particular, we numerically investigate how the correlation time of driving affects the correlation between density and magnetic field. We perform compressible magnetohydrodynamic turbulence simulations at different sonic Mach numbers (M s ), using two different driving schemes—a finite-correlated driving and a delta-correlated driving. In the former, the forcing vectors change continuously with a correlation time comparable to the large-eddy turnover time. In the latter, the direction (and amplitude) of driving changes in a very short timescale. The finite-correlated driving results in strong anti-correlation between two fields when the sonic and the Alfvénic Mach numbers are similar to unity (i.e., when M s ∼ 1 and M A ∼ 1, respectively). However, the anti-correlation becomes weaker and approaches zero for higher values of M s or M A. The delta-correlated driving produces virtually no correlation between two fields when M s ∼ 1 and M A ∼ 1, and produces more and more positive correlations as M s or M A increases. We conjecture that two competing effects, tendency for achieving balance between the gas and the magnetic pressure and simultaneous compression of fluid and magnetic field, determine the correlation behavior. We also investigate how different driving schemes affect the Probability Density Function of three-dimensional density, dispersion measure, and RM.

The distribution of mean and fluctuating magnetic fields in the multiphase interstellar medium

Abstract We explore the effects of the multiphase structure of the interstellar medium (ISM) on galactic magnetic fields. Basing our analysis on compressible magnetohydrodynamic simulations of supernova-driven turbulence in the ISM, we investigate the properties of both the mean and fluctuating components of the magnetic field. We find that the mean magnetic field preferentially resides in the warm phase and is generally absent from the hot phase. The fluctuating magnetic field does not show such pronounced sensitivity to the multiphase structure.

MAGNETIC GAMES BETWEEN A PLANET AND ITS HOST STAR: THE KEY ROLE OF TOPOLOGY

Magnetic interactions between a star and a close-in planet are postulated to be a source of enhanced emissions and to play a role in the secular evolution of the orbital system. Close-in planets generally orbit in the sub-alfv\'enic region of the stellar wind, which leads to efficient transfers of energy and angular momentum between the star and the planet. We model the magnetic interactions occurring in close-in star-planet systems with three-dimensional, global, compressible magneto-hydrodynamic numerical simulations of a planet orbiting in a self-consistent stellar wind. We focus on the cases of magnetized planets and explore three representative magnetic configurations. The Poynting flux originating from the magnetic interactions is an energy source for enhanced emissions in star-planet systems. Our results suggest a simple geometrical explanation for ubiquitous on/off enhanced emissions associated with close-in planets, and confirm that the Poynting fluxes can reach powers of the order of $10^{19}$ W. Close-in planets are also showed to migrate due to magnetic torques for sufficiently strong stellar wind magnetic fields. The topology of the interaction significantly modifies the shape of the magnetic obstacle that leads to magnetic torques. As a consequence, the torques can vary by at least an order of magnitude as the magnetic topology of the interaction varies.

ON THE DIVERSITY OF MAGNETIC INTERACTIONS IN CLOSE-IN STAR-PLANET SYSTEMS

Magnetic interactions between close-in planets and their host star can play an important role in the secular orbital evolution of the planets, as well as the rotational evolution of their host. As long as the planet orbits inside the Alfv\'en surface of the stellar wind, the magnetic interaction between the star and the planet can modify the wind properties and also lead to direct angular momentum transfers between the two. We model these star-planet interactions using compressible magneto-hydrodynamic (MHD) simulations, and quantify the angular momentum transfers between the star, the planet, and the stellar wind. We study the cases of magnetized and non-magnetized planets and vary the orbital radius inside the Alfv\'en surface of the stellar wind. Based on a grid of numerical simulations, we propose general scaling laws for the modification of the stellar wind torque, for the torque between the star and the planet, and for the planet migration associated with the star-planet magnetic interactions. We show that when the coronal magnetic field is large enough and the star is rotating sufficiently slowly, the effect of the magnetic star-planet interaction is comparable to tidal effects and can lead to a rapid orbital decay.

The supernova-regulated ISM – II. The mean magnetic field

Abstract The origin and structure of the magnetic fields in the interstellar medium of spiral galaxies is investigated with 3D, non-ideal, compressible magnetohydrodynamic simulations, including stratification in the galactic gravity field, differential rotation and radiative cooling. A rectangular domain, 1 × 1 × 2 kpc3 in size, spans both sides of the galactic mid-plane. Supernova explosions drive transonic turbulence. A seed magnetic field grows exponentially to reach a statistically steady state within 1.6 Gyr. Following Germano (1992), we use volume averaging with a Gaussian kernel to separate magnetic field into a mean field and fluctuations. Such averaging does not satisfy all Reynolds rules, yet allows a formulation of mean-field theory. The mean field thus obtained varies in both space and time. Growth rates differ for the mean-field and fluctuating field and there is clear scale separation between the two elements, whose integral scales are about 0.7 and 0.3 kpc, respectively.

Turbulent coronal heating mechanisms: coupling of dynamics and thermodynamics

Context. Photospheric motions shuffle the footpoints of the strong axial magnetic field that threads coronal loops giving rise to turbulent nonlinear dynamics characterized by the continuous formation and dissipation of field-aligned current sheets where energy is deposited at small-scales and the heating occurs. Previous studies show that current sheets thickness is orders of magnitude smaller than current state of the art observational resolution (~700 km). Aim. In order to understand coronal heating and interpret correctly observations it is crucial to study the thermodynamics of such a system where energy is deposited at unresolved small-scales. Methods. Fully compressible three-dimensional magnetohydrodynamic simulations are carried out to understand the thermodynamics of coronal heating in the magnetically confined solar corona. Results. We show that temperature is highly structured at scales below observational resolution and nonhomogeneously distributed so that only a fraction of the coronal mass and volume gets heated at each time. Conclusions. This is a multi-thermal system where hotter and cooler plasma strands are found one next to the other also at sub-resolution scales and exhibit a temporal dynamics.

Two-dimensional magnetohydrodynamic simulations of poloidal flows in tokamaks and MHD pedestal

Poloidal rotation is routinely observed in present-day tokamak experiments, in particular near the plasma edge and in the high-confinement mode of operation. According to the magnetohydrodynamic (MHD) equilibrium theory [R. Betti and J. P. Freidberg, Phys. Plasmas 7, 2439 (2000)], radial discontinuities form when the poloidal velocity exceeds the poloidal sound speed (or rather, more correctly, the poloidal magneto-slow speed). Two-dimensional compressible magnetohydrodynamic simulations show that the transonic discontinuities develop on a time scale of a plasma poloidal revolution to form an edge density pedestal and a localized velocity shear layer at the pedestal location. While such an MHD pedestal surrounds the entire core, the outboard side of the pedestal is driven by the transonic discontinuity while the inboard side is caused by a poloidal redistribution of the mass. The MHD simulations use a smooth momentum source to drive the poloidal flow. Soon after the flow exceeds the poloidal sound speed, the density pedestal and the velocity shear layer form and persist into a quasi steady state. These results may be relevant to the L-H transition, the early stages of the pedestal and edge transport barrier formation.