Reduced Magnetohydrodynamics Research Articles

Parametric description of intermittent probability distribution functions in solar wind and magnetohydrodynamic turbulence

ABSTRACT In this work, we find empirical evidence that the scale-dependent statistical properties of solar wind and magnetohydrodynamic (MHD) turbulence can be described in terms of a family of parametric probability distribution functions (PDFs) known as Normal Inverse Gaussian (NIG). Understanding these PDFs is one of the most important goals in turbulence theory, as they are inherently connected to the intermittent properties of solar wind turbulence. We investigate the properties of PDFs of Elsasser increments based on a large statistical sample from solar wind observations and high-resolution numerical simulations of MHD turbulence. In order to measure the PDFs and their corresponding properties, three experiments are presented: fast and slow solar wind for experimental data and a simulation of reduced MHD (RMHD) turbulence. Conditional statistics on a 23-yr-long sample of WIND data near 1 au and high-resolution pseudo-spectral simulation of steadily driven RMHD turbulence on a $2048^3$ mesh are used to construct scale-dependent PDFs. The empirical PDFs are fitted to NIG distributions, which depend on four free parameters. Our analysis shows that NIG distributions accurately capture the evolution of the PDFs, with scale-dependent parameters, from large scales characterized by a Gaussian distribution, turning to exponential tails within the inertial range and stretched exponentials at dissipative scales. We also show that empirically-measured NIG parameters exhibit well-defined scaling properties that are similar across the three empirical data sets, which may be indicative of universal behaviour.

Magnetic Topology and the Structure of the Slow Solar Wind

We present a 3D reduced magnetohydrodynamic (RMHD) model of reflection driven Alfvén wave turbulence in an open magnetic field positioned near the solar equator. The non-linear interactions between outward and inward propagating waves generate turbulence. The RMHD equations describing the turbulence include the effects of solar wind outflow velocity on the dissipation of waves. For Alfvén wave turbulence to be a viable mechanism for heating the corona and accelerating the slow solar wind, there must be sufficient counter-propagating waves to generate the turbulence. In our previous study of the fast wind, we showed that the Alfvén wave turbulence model creates the energy needed for accelerating the fast solar wind, when observed density fluctuations are included in the model. Here, we will explore whether the conditions of the equatorial corona, a source of slow solar wind, are capable of generating sufficient Alfvén wave turbulence to account for the acceleration of the slow wind. We show that for a specific sets of model parameters, the energy from the Alfvén wave turbulence model is smaller by an order of magnitude than the energy needed to heat and accelerate the slow solar wind.

Effects of Density Fluctuations on Alfvén Wave Turbulence in a Coronal Hole

Abstract We present a study of density fluctuations in coronal holes. We used a reduced magnetohydrodynamic (RMHD) model that incorporated observationally constrained density fluctuations to determine whether density fluctuations in coronal holes can enhance Alfvén wave reflection and dissipation, thereby heating coronal holes and driving the fast solar wind. We show results for two models of the background atmosphere. Each model is a solution of the momentum equation and includes the effects of wave pressure on the solar wind outflow. In the first model, the plasma density and Alfvén speed vary smoothly with height. Wave reflection is relatively weak in the smooth model, resulting in a low energy dissipation rate. In the second model, we include additional density fluctuations along the flux tube based on the observations. We find that density ρ fluctuations on the order of δρ/ρ ∼ 0.24 increase the Alfvén wave turbulence to levels sufficient to heat the open field regions in coronal holes.

Effect of local toroidal flow on double‐tearing modes in cylindrical geometry

AbstractThe effect of local toroidal flow (LTF) on double‐tearing modes (DTMs) is investigated in cylindrical geometry using the reduced magnetohydrodynamics (RMHD) model. The results indicate that the LTF between the two rational surfaces is the dominant suppression effect on DTMs. The suppression effect is enhanced with the flow width increasing, and the DTMs become more stable with the increase of the shear of the LTF between the two rational surfaces. So, in the reversed shear magnetic field configuration, the local flow is driven between the two rational surfaces, which can effectively suppress the development of DTMs and maintain the high‐performance state.

Simulation of the internal kink mode in visco-resistive regimes

We present numerical simulation results of the nonlinear evolution of the (1,1) internal kink mode in the presence of various kinds of equilibrium plasma flows. The present studies are carried out in the framework of a two fluid model to extend our past investigations done with a reduced magneto-hydrodynamic (RMHD) model. Two-fluid effects are found to significantly influence the mode dynamics in a number of different ways. In the linear regime diamagnetic effects in combination with flows provide a synergistic stabilizing influence that also carries over to the nonlinear regime. In addition one observes novel symmetry breaking phenomena in the linear growth rates as well as in the nonlinear saturated states of the mode. Our study also explores the influence of strong viscosity on the mode evolution and finds interesting modifications in the real frequency of the mode in the linear regime.

2D magnetofluid models constructed by a priori imposition of conservation laws

Motivated by a geometric method employed for the derivation of the Nambu bracket for ideal two-dimensional incompressible hydrodynamics, we reconstruct the reduced magnetohydrodynamic (RMHD) model by a priori imposition of its conservation laws. It turns out that there exists a more general model that conserves the same quantities with RMHD. The noncanonical Hamiltonian and Nambu description of this generic system are derived and in addition a canonical description is formed by Clebsch-parameterizing the vorticity and the magnetic flux function. The method for the construction of the dynamical equations is based on the imposition of the conservation laws as orthogonality conditions. Furthermore, this approach enabled us to construct three families of models that respect any combination of two out of the three conservation laws. Some of these models can serve as conservative regularizations of RMHD since under certain conditions they keep the enstrophy bounded without the need of introducing viscosity and may be candidates for incorporating small length scale physics into the RMHD framework.

A Model for Solar Wind Structure and Dynamics Based on Alfvén Wave Turbulence

We present a 3D reduced magnetohydrodynamic (RMHD) model of reflection-driven Alfvén wave turbulence in an open magnetic field at the center of a coronal hole. The non-linear interactions between outward (dominant) and inward (minority) propagating waves generate turbulence. The RMHD equations describing the turbulence include the effects of solar wind outflow velocity on the dissipation of waves. The Alfvén wave turbulence results in the transfer of energy to small perpendicular scales (“direct cascade”) or to larger scales (“inverse cascade”). We first show the result of our calculation for a model where there is smooth variation of plasma parameters such as Alfvén speed and density with height along the flux tube. However, this model does not produce the required energy to heat the plasma and accelerate the fast solar wind. Therefore, we introduce our second model, where we include additional density fluctuation along the open field. These density variations simulate the effects of compressive MHD waves in the solar wind. We find that these variations in density enhance the turbulence dissipation rates, and thereby increase the heating rate and the acceleration of the solar wind.

Analysis of a decoupled time‐stepping algorithm for reduced MHD system modeling magneto‐convection

In this article, we analyze the stability and error estimate of a decoupled algorithm for a magneto‐convection problem. Magneto‐convection is assumed to be modeled by a coupled system of reduced magneto‐hydrodynamic (RMHD) equations and convection‐diffusion equation. The proposed algorithm applies the second‐order backward difference formula in time and finite element in space. To obtain a noniterative decouple algorithm from the fully discrete nonlinear system, we use a second‐order extrapolation in time to the nonlinear terms such that their skew symmetry properties are preserved. We prove the stability of the algorithm and derive error estimates without assuming any stability conditions. The algorithm is unconditionally stable and requires the solution of one RMHD problem and one convection‐diffusion equation per time step. Numerical test is presented that illustrates the accuracy and efficiency of the algorithm.

Reduced MHD in Astrophysical Applications: Two-dimensional or Three-dimensional?

Abstract Originally proposed as an efficient approach to computation of nonlinear dynamics in tokamak fusion research devices, reduced magnetohydrodynamics (RMHD) has subsequently found application in studies of coronal heating, flux tube dynamics, charged particle transport, and, in general, as an approximation to describe plasma turbulence in space physics and astrophysics. Given the diverse set of derivations available in the literature, there has emerged some level of discussion and a lack of consensus regarding the completeness of RMHD as a turbulence model, and its applicability in contexts such as the solar wind. Some of the key issues in this discussion are examined here, emphasizing that RMHD is properly neither 2D nor fully 3D, being rather an incomplete representation that enforces at least one family of extraneous conservation laws.

DIRECT AND INVERSE CASCADES IN THE ACCELERATION REGION OF THE FAST SOLAR WIND

ABSTRACT Alfvén waves are believed to play an important role in the heating and acceleration of the fast solar wind emanating from coronal holes. Nonlinear interactions between the dominant waves and minority waves have the potential to transfer wave energy either to smaller perpendicular scales (“direct cascade”) or to larger scales (“inverse cascade”). In this paper we use reduced magnetohydrodynamic (RMHD) simulations to investigate how the cascade rates depend on perpendicular wavenumber and radial distance from the Sun center. For models with a smooth background atmosphere, we find that an inverse cascade ( ) occurs for the dominant waves at radii between 1.4 and and dimensionless wavenumbers in the inertial range ( ), and a direct cascade ( ) occurs elsewhere. For a model with density fluctuations, there are multiple regions with an inverse cascade. In both cases, the cascade rate varies significantly with perpendicular wavenumber, indicating that the cacsade is a highly nonlocal process. As a result of the inverse cascades, the energy dissipation rates are much lower than expected from a phenomenological model and are insufficient to maintain the temperature of the background atmosphere. We conclude that RMHD models are unable to reproduce the observed properties of the fast solar wind.

HEATING AND ACCELERATION OF THE FAST SOLAR WIND BY ALFVÉN WAVE TURBULENCE

ABSTRACT We present numerical simulations of reduced magnetohydrodynamic (RMHD) turbulence in a magnetic flux tube at the center of a polar coronal hole. The model for the background atmosphere is a solution of the momentum equation and includes the effects of wave pressure on the solar wind outflow. Alfvén waves are launched at the coronal base and reflect at various heights owing to variations in Alfvén speed and outflow velocity. The turbulence is driven by nonlinear interactions between the counterpropagating Alfvén waves. Results are presented for two models of the background atmosphere. In the first model the plasma density and Alfvén speed vary smoothly with height, resulting in minimal wave reflections and low-energy dissipation rates. We find that the dissipation rate is insufficient to maintain the temperature of the background atmosphere. The standard phenomenological formula for the dissipation rate significantly overestimates the rate derived from our RMHD simulations, and a revised formula is proposed. In the second model we introduce additional density variations along the flux tube with a correlation length of 0.04 R ⊙ and with relative amplitude of 10%. These density variations simulate the effects of compressive MHD waves on the Alfvén waves. We find that such variations significantly enhance the wave reflection and thereby the turbulent dissipation rates, producing enough heat to maintain the background atmosphere. We conclude that interactions between Alfvén and compressive waves may play an important role in the turbulent heating of the fast solar wind.

Inertial-range kinetic turbulence in pressure-anisotropic astrophysical plasmas

A theoretical framework for low-frequency electromagnetic (drift-)kinetic turbulence in a collisionless, multi-species plasma is presented. The result generalises reduced magnetohydrodynamics (RMHD) and kinetic RMHD (Schekochihinet al.,Astrophys. J. Suppl. Ser., vol. 182, 2009, pp. 310–377) to the case where the mean distribution function of the plasma is pressure-anisotropic and different ion species are allowed to drift with respect to each other – a situation routinely encountered in the solar wind and presumably ubiquitous in hot dilute astrophysical plasmas such as the intracluster medium. Two main objectives are achieved. First, in a non-Maxwellian plasma, the relationships between fluctuating fields (e.g. the Alfvén ratio) are order-unity modified compared to the more commonly considered Maxwellian case, and so a quantitative theory is developed to support quantitative measurements now possible in the solar wind. Beyond these order-unity corrections, the main physical feature of low-frequency plasma turbulence survives the generalisation to non-Maxwellian distributions: Alfvénic and compressive fluctuations are energetically decoupled, with the latter passively advected by the former; the Alfvénic cascade is fluid, satisfying RMHD equations (with the Alfvén speed modified by pressure anisotropy and species drifts), whereas the compressive cascade is kinetic and subject to collisionless damping (and for a bi-Maxwellian plasma splits into three independent collisionless cascades). Secondly, the organising principle of this turbulence is elucidated in the form of a conservation law for the appropriately generalised kinetic free energy. It is shown that non-Maxwellian features in the distribution function reduce the rate of phase mixing and the efficacy of magnetic stresses, and that these changes influence the partitioning of free energy amongst the various cascade channels. As the firehose or mirror instability thresholds are approached, the dynamics of the plasma are modified so as to reduce the energetic cost of bending magnetic-field lines or of compressing/rarefying them. Finally, it is shown that this theory can be derived as a long-wavelength limit of non-Maxwellian slab gyrokinetics.

INTERMITTENCY AND ALIGNMENT IN STRONG RMHD TURBULENCE

We develop an analytic model of intermittent, three-dimensional, strong, reduced magnetohydrodynamic (RMHD) turbulence with zero cross helicity. We take the fluctuation amplitudes to have a log-Poisson distribution and incorporate into the model a new phenomenology of scale-dependent dynamic alignment between the Elsässer variables . We find that the structure function scales as , where is the variation in across a distance perpendicular to the magnetic field. We calculate the value of β to be ≃ 0.69 based on our assumption that the most intense coherent structures are two-dimensional with a volume filling factor . Two consequences of this structure-function scaling are that the total-energy power spectrum is and the kurtosis of the fluctuations is . Our model resolves the problem that alignment angles defined in different ways exhibit different scalings. Specifically, we find that the energy-weighted average angle between the velocity and magnetic-field fluctuations is , the energy-weighted average angle between and is , and the average angle between and without energy weighting is when , where L is the outer scale. We also carry out a direct numerical simulation of RMHD turbulence. The scalings in our model are similar to the scalings in this simulation as well as the structure-function scalings observed in the slow solar wind.

Simulated annealing applied to two-dimensional low-beta reduced magnetohydrodynamics

The simulated annealing (SA) method is applied to two-dimensional (2D) low-beta reduced magnetohydrodynamics (R-MHD). We have successfully obtained stationary states of the system numerically by the SA method with Casimir invariants preserved. Since the 2D low-beta R-MHD has two fields, the relaxation process becomes complex compared to a single field system such as 2D Euler flow. The obtained stationary state can have fine structure. We have found that the fine structure appears because the relaxation processes are different between kinetic energy and magnetic energy.

ERRATUM: “DIRECT NUMERICAL SIMULATIONS OF REFLECTION-DRIVEN, REDUCED MAGNETOHYDRODYNAMIC TURBULENCE FROM THE SUN TO THE ALFVÉN CRITICAL POINT” (2013, ApJ, 776, 124)

We present direct numerical simulations of inhomogeneous reduced magnetohydrodynamic (RMHD) turbulence between the Sun and the Alfv\'en critical point. These are the first such simulations that take into account the solar-wind outflow velocity and the radial inhomogeneity of the background solar wind without approximating the nonlinear terms in the governing equations. RMHD turbulence is driven by outward-propagating Alfv\'en waves ($z^+$ fluctuations) launched from the Sun, which undergo partial non-WKB reflection to produce sunward-propagating Alfv\'en waves ($z^-$ fluctuations). We present ten simulations with different values of the correlation time $\tau_{\rm c\,\sun}^+$ and perpendicular correlation length $L_{\perp \sun}$ of outward-propagating Alfv\'en waves (AWs) at the coronal base. We find that between 15% and 33% of the $z^+$ energy launched into the corona dissipates between the coronal base and Alfv\'en critical point. Between 33% and 40% of this input energy goes into work on the solar-wind outflow, and between 22% and 36% escapes as $z^+$ fluctuations through the simulation boundary at $r=r_{\rm A}$. The $z^\pm$ power spectra scale like $k_\perp^{-\alpha^\pm}$, where $k_\perp$ is the wavenumber in the plane perpendicular to $\vec{B}_0$. In our simulation with the smallest value of $\tau_{\rm c\,\sun}^+$ ($\sim 2 \mbox{min}$) and largest value of $L_{\perp \sun}$ ($2\times 10^4 \mbox{km}$), we find that $\alpha^+$ decreases approximately linearly with increasing $\ln(r)$, reaching a value of 1.3 at $r=11.1 R_{\sun}$. Our simulations with larger values of $\tau_{\rm c\,\sun}^+$ exhibit alignment between the contours of constant $\phi^+$, $\phi^-$, $\Omega_0^+$, and $\Omega_0^-$, where $\phi^\pm$ are the Els\"asser potentials and $\Omega_0^\pm$ are the outer-scale parallel Els\"asser vorticities.

MAGNETIC FIELD LINE RANDOM WALK IN MODELS AND SIMULATIONS OF REDUCED MAGNETOHYDRODYNAMIC TURBULENCE

The random walk of magnetic field lines is examined numerically and analytically in the context of reduced magnetohydrodynamic (RMHD) turbulence, which provides a useful description of plasmas dominated by a strong mean field, such as in the solar corona. A recently developed non-perturbative theory of magnetic field line diffusion is compared with the diffusion coefficients obtained by accurate numerical tracing of magnetic field lines for both synthetic models and direct numerical simulations of RMHD. Statistical analysis of an ensemble of trajectories confirms the applicability of the theory, which very closely matches the numerical field line diffusion coefficient as a function of distance z along the mean magnetic field for a wide range of the Kubo number R. This theory employs Corrsin's independence hypothesis, sometimes thought to be valid only at low R. However, the results demonstrate that it works well up to R = 10, both for a synthetic RMHD model and an RMHD simulation. The numerical results from the RMHD simulation are compared with and without phase randomization, demonstrating a clear effect of coherent structures on the field line random walk for a very low Kubo number.

DIRECT NUMERICAL SIMULATIONS OF REFLECTION-DRIVEN, REDUCED MAGNETOHYDRODYNAMIC TURBULENCE FROM THE SUN TO THE ALFVÉN CRITICAL POINT

We present direct numerical simulations of inhomogeneous reduced magnetohydrodynamic (RMHD) turbulence between the Sun and the Alfven critical point. These are the first such simulations that take into account the solar-wind outflow velocity and the radial inhomogeneity of the background solar wind without approximating the nonlinear terms in the governing equations. RMHD turbulence is driven by outward-propagating Alfven waves (z + fluctuations) launched from the Sun, which undergo partial non-WKB reflection to produce sunward-propagating Alfven waves (z – fluctuations). We present 10 simulations with different values of the correlation time and perpendicular correlation length L ⊥☉ of outward-propagating Alfven waves at the coronal base. We find that between 15% and 33% of the z + energy launched into the corona dissipates between the coronal base and Alfven critical point. Between 33% and 40% of this input energy goes into work on the solar-wind outflow, and between 22% and 36% escapes as z + fluctuations through the simulation boundary at r = r A. The z ± power spectra scale like , where k ⊥ is the wavenumber in the plane perpendicular to B 0. In our simulation with the smallest value of (~2 minutes) and largest value of L ⊥☉ (2 × 104 km), we find that α+ decreases approximately linearly with increasing ln (r), reaching a value of 1.3 at r = 11.1 R ☉. Our simulations with larger values of exhibit alignment between the contours of constant +, –, , and , where ± are the Elsasser potentials and are the outer-scale parallel Elsasser vorticities.

PERPENDICULAR ION HEATING BY REDUCED MAGNETOHYDRODYNAMIC TURBULENCE

Recent theoretical studies argue that the rate of stochastic ion heating in low-frequency Alfv\'en-wave turbulence is given by $Q_\perp = c_1 [(\delta u)^3 /\rho] \exp(-c_2/\epsilon)$, where $\delta u$ is the rms turbulent velocity at the scale of the ion gyroradius $\rho$, $\epsilon = \delta u/v_{\perp \rm i}$, $v_{\perp \rm i}$ is the perpendicular ion thermal speed, and $c_1$ and $c_2$ are dimensionless constants. We test this theoretical result by numerically simulating test particles interacting with strong reduced magnetohydrodynamic (RMHD) turbulence. The heating rates in our simulations are well fit by this formula. The best-fit values of $c_1$ are $\sim 1$. The best-fit values of $c_2$ decrease (i.e., stochastic heating becomes more effective) as the grid size and Reynolds number of the RMHD simulations increase. As an example, in a $1024^2 \times 256$ RMHD simulation with a dissipation wavenumber of order the inverse ion gyroradius, we find $c_2 = 0.21$. We show that stochastic heating is significantly stronger in strong RMHD turbulence than in a field of randomly phased Alfv\'en waves with the same power spectrum, because coherent structures in strong RMHD turbulence increase orbit stochasticity in the regions where ions are heated most strongly. We find that $c_1$ increases by a factor of $\sim 3$ while $c_2$ changes very little as the ion thermal speed increases from values $\ll v_{\rm A}$ to values $\sim v_{\rm A}$, where $v_{\rm A}$ is the Alfv\'en speed. We discuss the importance of these results for perpendicular ion heating in the solar wind.

Reduced MHD and Astrophysical Fluid Dynamics

AbstractRecent work has shown a relationship between between the equations of Reduced Magnetohydrodynamics (RMHD), used to model magnetic fusion laboratory experiments, and incompressible magnetoconvection (IMC), employed in the simulation of astrophysical fluid dynamics (AFD), which means that the two systems are mathematically equivalent in certain geometries. Limitations on the modelling of RMHD, which were found over twenty years ago, are reviewed for an AFD audience, together with hitherto unpublished material on the role of finite-time singularities in the discrete equations used to model fluid dynamical systems. Possible implications for turbulence modelling are mentioned.

A MHD-turbulence model for solar corona

The disposition of energy in the solar corona has always been a problem of great interest. It remains an open question how the low temperature photosphere supports the occurence of solar extreme phenomena. In this work, a turbulent heating mechanism for the solar corona through the framework of reduced magnetohydrodynamics (RMHD) is proposed. Two-dimensional incompressible long time simulations of the average energy disposition have been carried out with the aim to reveal the characteristics of the long time statistical behavior of a two-dimensional cross-section of a coronal loop and the importance of the photospheric time scales in the understanding of the underlying mechanisms. It was found that for a slow, shear type photospheric driving the magnetic field in the loop self-organizes at large scales via an inverse MHD cascade. The system undergoes three distinct evolutionary phases. The initial forcing conditions are quickly “forgotten” giving way to an inverse cascade accompanied with and ending up to electric current dissipation. Scaling laws are being proposed in order to quantify the nonlinearity of the system response which seems to become more impulsive for decreasing resistivity. It is also shown that few, if any, qualitative changes in the above results occur by increasing spatial resolution.