Magnetohydrodynamic Turbulence

7.4 Phenomenological models of intermittency 7.4.1 The log-normal model 7.4.2
The log-Poisson model 7.5 Intermittency in MHD turbulence 7.5.1 Log-Poisson
models for MHD turbulence 7.5.2 The effect of the mean magnetic field 8 ...

Magnetohydrodynamic Turbulence

Author: Dieter Biskamp

Publisher: Cambridge University Press

ISBN: 9781139441674

Page:

View: 391

This book presents an introduction to, and modern account of, magnetohydrodynamic (MHD) turbulence, an active field both in general turbulence theory and in various areas of astrophysics. The book starts by introducing the MHD equations, certain useful approximations and the transition to turbulence. The second part of the book covers incompressible MHD turbulence, the macroscopic aspects connected with the different self-organization processes, the phenomenology of the turbulence spectra, two-point closure theory, and intermittency. The third considers two-dimensional turbulence and compressible (in particular, supersonic) turbulence. Because of the similarities in the theoretical approach, these chapters start with a brief account of the corresponding methods developed in hydrodynamic turbulence. The final part of the book is devoted to astrophysical applications: turbulence in the solar wind, in accretion disks, and in the interstellar medium. This book is suitable for graduate students and researchers working in turbulence theory, plasma physics and astrophysics.

Study on Magnetohydrodynamic Turbulence and Its Astrophysical Applications

A significant progress has been achieved in the understanding of the
magnetohydrodynamic (MHD) turbulence since the turn of the century, and this
enables us to better describe turbulence in magnetized and partially ionized
plasmas.

Study on Magnetohydrodynamic Turbulence and Its Astrophysical Applications

Author: Siyao Xu

Publisher: Springer

ISBN: 9811375151

Page: 157

View: 188

Turbulence and magnetic fields are ubiquitous in the Universe. Their importance to astronomy cannot be overestimated. The theoretical advancements in magnetohydrodynamic (MHD) turbulence achieved during the past two decades have significantly influenced many fields of astronomy. This book provides predictive theories of the magnetic field generation by turbulence and the dissipation of MHD turbulence. These fundamental non-linear problems were believed to be tractable only numerically. This book provides complete analytical descriptions in quantitative agreement with existing numerics, as well as theoretical predictions in physical regimes still unreachable by simulations, and explanations of various related observations. It also discusses and promotes the astrophysical applications of MHD turbulence theories, including (i) the particle acceleration and radiation in high-energy phenomena, e.g., Gamma-Ray Bursts, supernova remnants, cosmic rays; (ii) interstellar density fluctuations and the effect on observations, e.g., Faraday rotation, scattering measurements of Galactic and extragalactic radio sources; (iii) density and magnetic field structure in molecular clouds toward star formation. In closing, this book demonstrates the key role of MHD turbulence in connecting diverse astrophysical processes and unraveling long-standing astrophysical problems, as foreseen by Chandrasekhar, a founder of modern astrophysics.

Study on Magnetohydrodynamic Turbulence and Its Astrophysical Applications

In closing, this book demonstrates the key role of MHD turbulence in connecting diverse astrophysical processes and unraveling long-standing astrophysical problems, as foreseen by Chandrasekhar, a founder of modern astrophysics.

Study on Magnetohydrodynamic Turbulence and Its Astrophysical Applications

Author: Siyao Xu

Publisher: Springer

ISBN: 9789811375149

Page: 157

View: 219

Turbulence and magnetic fields are ubiquitous in the Universe. Their importance to astronomy cannot be overestimated. The theoretical advancements in magnetohydrodynamic (MHD) turbulence achieved during the past two decades have significantly influenced many fields of astronomy. This book provides predictive theories of the magnetic field generation by turbulence and the dissipation of MHD turbulence. These fundamental non-linear problems were believed to be tractable only numerically. This book provides complete analytical descriptions in quantitative agreement with existing numerics, as well as theoretical predictions in physical regimes still unreachable by simulations, and explanations of various related observations. It also discusses and promotes the astrophysical applications of MHD turbulence theories, including (i) the particle acceleration and radiation in high-energy phenomena, e.g., Gamma-Ray Bursts, supernova remnants, cosmic rays; (ii) interstellar density fluctuations and the effect on observations, e.g., Faraday rotation, scattering measurements of Galactic and extragalactic radio sources; (iii) density and magnetic field structure in molecular clouds toward star formation. In closing, this book demonstrates the key role of MHD turbulence in connecting diverse astrophysical processes and unraveling long-standing astrophysical problems, as foreseen by Chandrasekhar, a founder of modern astrophysics.

Broken Symmetry in Ideal Magnetohydrodynamic Turbulence

Broken Symmetry in Ideal Magnetohydrodynamic Turbulence

Author: John V. Shebalin

Publisher:

ISBN:

Page: 17

View: 983

A numerical study of the long-time evolution of a number of cases of inviscid, isotropic, incompressible, three-dimensional fluid and magneto-fluid turbulence has been completed. The results confirm that ideal magnetohydrodynamic turbulence is non-ergodic if there is no external magnetic field present. This is due essentially to a canonical symmetry being broken in an arbitrary dynamical representation. The broken symmetry manifests itself as a coherent structure, i.e., a non-zero time-averaged part of the turbulent magnetic field. The coherent structure is observed, in one case, to contain about eighteen percent of the total energy. Isotropic turbulence, Ideal magnetohydrodynamics, Broken symmetry.

Instability driven Magnetohydrodynamic Turbulence

Instability-driven turbulence is ubiquitous in astrophysical and laboratory plasmas, where it is an important component to how these systems transport energy, momentum, particles, etc.

Instability driven Magnetohydrodynamic Turbulence

Author: Justin Wiiliam Walker

Publisher:

ISBN:

Page: 89

View: 126

Instability-driven turbulence is ubiquitous in astrophysical and laboratory plasmas, where it is an important component to how these systems transport energy, momentum, particles, etc. This thesis is concerned with how two instabilities drive and interact with magnetohydrodynamic turbulence. The magnetorotational instability (MRI) is the best candidate for driving turbulence in well-ionized accretion disks. In this thesis, the turbulence driven by the MRI is studied with a particular focus on how it compares to strong, driven, incompressible MHD turbulence. High-resolution, high- Reynolds number setups are analyzed to determine the existence and character of an inertial range of scales where a nonlinear cascade dominates the dynamics. In contrast to previous studies, systems with an imposed magnetic field- that activate the linear MRI-provide evidence for the existence of an inertial range when one considers the dynamics perpendicular to a strong, large-scale axial magnetic field that develops in the system. The outer scale of the turbulence is determined by balance between the linear shear, present at all scales, and the turbulent shear. In the case of a system without an imposed magnetic field-where the MRI dynamo is subcritical-evidence is found for self-sustained turbulence at magnetic Prandtl number Pm = 1. Previous work was not able produce such turbulence for systems at lower Reynolds numbers. A turbulent state is found to be easier to self-sustain in these systems in high-aspect-ratio domains, with angular momentum transport also highly dependent on aspect ratio. Vertically-extended domains exhibit higher transport. Azimuthally- extended domains show increased transport until a certain aspect ratio, beyond which the transport decreases. A phenomenological explanation is proposed by which the separation of toroidal magnetic flux vertically allows for increased transport and drive of the subcritical MRI dynamo; as the toroidal dimension is extended, the tearing mode acts to break up such large scale flux, reducing transport. Additionally, two-dimensional MHD turbulence is studied in order to test the predictions of new theories that hypothesize competition between the tearing mode and turbulent fluctuations in flows with high Lundquist number. A method is introduced to simulate evolution of single, critically- balanced eddy, thereby allowing the direct investigation of this competition. Results demonstrate the disruption of the correlated eddy structure on the shorter of the [Alfven] or tearing time. Steepening of the energy spectrum is observed, while decreased alignment is not. Generally, these results support the proposed theory, with further investigation warranted.

Magnetohydrodynamic Turbulence

After a brief outline of magnetohydrodynamic theory, this introductory book discusses the macroscopic aspects of MHD turbulence, and covers the small-scale scaling properties.

Magnetohydrodynamic Turbulence

Author: Dieter Biskamp

Publisher: Cambridge University Press

ISBN: 9780521810111

Page: 312

View: 135

After a brief outline of magnetohydrodynamic theory, this introductory book discusses the macroscopic aspects of MHD turbulence, and covers the small-scale scaling properties. Applications are provided for astrophysical and laboratory systems. Magnetic turbulence is the natural state of most astrophysical systems, such as stellar convection zones, stellar winds or accretion discs. It is also found in laboratory devices, most notably in the reversed field pinch.

Theory and Simulation of Real and Ideal Magnetohydrodynamic Turbulence

Incompressible, homogeneous magnetohydrodynamic (MHD) turbulence consists of fluctuating vorticity and magnetic fields, which are represented in terms of their Fourier coefficients.

Theory and Simulation of Real and Ideal Magnetohydrodynamic Turbulence

Author: John V. Shebalin

Publisher: BiblioGov

ISBN: 9781289284251

Page: 34

View: 992

Incompressible, homogeneous magnetohydrodynamic (MHD) turbulence consists of fluctuating vorticity and magnetic fields, which are represented in terms of their Fourier coefficients. Here, a set of five Fourier spectral transform method numerical simulations of two-dimensional (2-D) MHD turbulence on a 512(sup 2) grid is described. Each simulation is a numerically realized dynamical system consisting of Fourier modes associated with wave vectors k, with integer components, such that k = k less than or equal to k(sub max). The simulation set consists of one ideal (non-dissipative) case and four real (dissipative) cases. All five runs had equivalent initial conditions. The dimensions of the dynamical systems associated with these cases are the numbers of independent real and imaginary parts of the Fourier modes. The ideal simulation has a dimension of 366104, while each real simulation has a dimension of 411712. The real runs vary in magnetic Prandtl number P(sub M), with P(sub M) is a member of {0.1, 0.25, 1, 4}. In the results presented here, all runs have been taken to a simulation time of t = 25. Although ideal and real Fourier spectra are quite different at high k, they are similar at low values of k. Their low k behavior indicates the existence of broken symmetry and coherent structure in real MHD turbulence, similar to what exists in ideal MHD turbulence. The value of PM strongly affects the ratio of kinetic to magnetic energy and energy dissipation (which is mostly ohmic). The relevance of these results to 3-D Navier-Stokes and MHD turbulence is discussed.

Turbulence and Magnetic Fields in Astrophysics

This book contains review articles of most of the topics addressed at the conf- ence on Simulations of Magnetohydrodynamic turbulence in astrophysics: recent achievements and perspectives which took place from July 2 to 6, 2001 at the ...

Turbulence and Magnetic Fields in Astrophysics

Author: Edith Falgarone

Publisher: Springer

ISBN: 354036238X

Page: 468

View: 619

This book contains review articles of most of the topics addressed at the conf- ence on Simulations of Magnetohydrodynamic turbulence in astrophysics: recent achievements and perspectives which took place from July 2 to 6, 2001 at the Institut Henri Poincar ́e in Paris. We made the choice to publish these lectures in a tutorial form so that they can be read by a broad audience. As a result, this book does not give an exhaustive view of all the subjects addressed during the conference. The main objective of this workshop which gathered about 90 scientists from di?erent ?elds, was to present and confront recent results on the topic of t- bulence in magnetized astrophysical environments. A second objective was to discuss the latest generation of numerical codes, such as those using adaptive mesh re?nement (AMR) techniques. During a plenary discussion at the end of the workshop discussions were held on several topics, often at the heart of vivid controversies. Topics included the timescale for the dissipation of magneto-hydrodynamical (MHD) turbulence, the role of boundary conditions, the characteristics of imbalanced turbulence, the validity of the polytropic approach to Alfv ́en waves support within interst- lar clouds, the source of turbulence inside clouds devoid of stellar activity, the timescale for star formation, the Alfv ́en Mach number of interstellar gas motions, the formation process for helical ?elds in the interstellar medium. The impact of small upon large scales was also discussed.

Wavy Magnetohydrodynamic Turbulence

The goal is to explore the theory of resistivity quenching in a regime for which the mean-field theory can be rigorously constructed at large magnetic Reynolds number Rm. This is achieved by extending the simple 2D problem to include body ...

Wavy Magnetohydrodynamic Turbulence

Author: Shane Richard Keating

Publisher:

ISBN:

Page: 118

View: 447

Nonlinear closure models of the 2D magnetohydrodynamic equations predict that the turbulent diffusivity of magnetic fields in high magnetic Reynolds number flows will be strongly suppressed below the value predicted by simple kinematic models. The consequences of such `resistivity quenching' for models of dissipation and transport in magnetized flows are profound. However, to date, there has been little examination of the underlying assumptions implicitly made by such models --- (i) that the quenching is associated with a reduction in the cross-phase between the velocity and the magnetic potential, rather than a suppression of the turbulence itself, and (ii) that transport results from molecular collisions alone. In this dissertation, we revisit the 2D problem in an attempt to address these issues. To address (i), we examine the normalized cross-phase and its dependence on the initial magnetic field strength. We present the results of numerical simulations that are consistent with the current picture of resistivity quenching as primarily a suppression of transport of magnetic potential rather than turbulence intensity. To address (ii), the theory of turbulent resistivity in `wavy' magnetohydrodynamic turbulence in 2D is presented. The goal is to explore the theory of resistivity quenching in a regime for which the mean-field theory can be rigorously constructed at large magnetic Reynolds number Rm. This is achieved by extending the simple 2D problem to include body forces, such as buoyancy or the Coriolis force, that convert large scale eddies into weakly interacting dispersive waves. Remarkably, adding an additional restoring force to the already tightly constrained system of high Rm magnetohydrodynamic turbulence in 2D can actually increase the turbulent resistivity, by admitting a spatial flux of magnetic potential, driven by wave interactions, that is not quenched at large $\Rm$. In the final chapter we address a closely related topic: the effect of magnetic linkage on the homogenization of vorticity in closed streamline flow. It is found that magnetic stresses acting on the bounding streamline can maintain a cross-stream gradient in the vorticity, thus disrupting the homogenization process and profoundly altering the nature of the turbulent enstrophy cascade in such flows.

Turbulence in Magnetohydrodynamics

A sound knowledge is needed to tackle these problems. This work gives the basic information on turbulence in nature, comtaining the needed equations, notions and numerical simulations.

Turbulence in Magnetohydrodynamics

Author: Andrey Beresnyak

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110392240

Page: 286

View: 305

Magnetohydrodynamics describes dynamics in electrically conductive fluids. These occur in our environment as well as in our atmosphere and magnetosphere, and play a role in the sun's interaction with our planet. In most cases these phenomena involve turbulences, and thus are very challenging to understand and calculate. A sound knowledge is needed to tackle these problems. This work gives the basic information on turbulence in nature, comtaining the needed equations, notions and numerical simulations. The current state of our knowledge and future implications of MHD turbulence are outlined systematically. It is indispensable for all scientists engaged in research of our atmosphere and in space science.

Small Scale Structures in Three Dimensional Hydrodynamic and Magnetohydrodynamic Turbulence

The aim here is a better understanding of the similarities and differences between vortex and current dynamics, and of the influence of these structures on the statistical and transport properties of hydrodynamic and magnetohydrodynamic ...

Small Scale Structures in Three Dimensional Hydrodynamic and Magnetohydrodynamic Turbulence

Author: Maurice Meneguzzi

Publisher: Springer

ISBN: 9783662140246

Page: 421

View: 977

Small-scale structures in turbulent flows appear as a subtle mixture of order and chaos that could play an important role in the energetics. The aim here is a better understanding of the similarities and differences between vortex and current dynamics, and of the influence of these structures on the statistical and transport properties of hydrodynamic and magnetohydrodynamic turbulence, with special concern for fusion plasmas, and solar or magnetospheric environments. Special emphasis is given to the intermittency at inertial scales and to the coherent structures at small scales. Magnetic reconnection and the dynamo effect are also discussed, together with the effect of stratification and inhomogeneity. The impact of hydrodynamic concepts on astro and geophysical observations are reviewed.

Magnetohydrodynamic Turbulence and Angular Momentum Transport in Accretion Disks

It is currently believed that angular momentum transport in accretion disks is mediated by magnetohydrodynamic (MHD) turbulence driven by the magnetorotational instability (MRI).

Magnetohydrodynamic Turbulence and Angular Momentum Transport in Accretion Disks

Author: Martin Elias Pessah

Publisher:

ISBN:

Page: 336

View: 815

It is currently believed that angular momentum transport in accretion disks is mediated by magnetohydrodynamic (MHD) turbulence driven by the magnetorotational instability (MRI). More than 15 years after its discovery, an accretion disk model that incorporates the MRI as the mechanism driving the MHD turbulence is still lacking. This dissertation constitutes the first in a series of steps towards establishing the formalism and methodology needed to move beyond the standard accretion disk model and incorporating the MRI as the mechanism enabling the accretion process. I begin by presenting a local linear stability analysis of a compressible, differentially rotating flow and addressing the evolution of the MRI beyond the weak-field limit when magnetic tension forces due to strong toroidal fields are considered. Then, I derive the first formal analytical proof showing that, during the exponential growth of the instability, the mean total stress produced by correlated MHD fluctuations is positive and leads to a net outward flux of angular momentum. I also show that some characteristics of the MHD stresses that are determined during this initial phase are roughly preserved in the turbulent saturated state observed in local numerical simulations. Motivated by these results, I present the first mean-field MHD model for angular momentum transport driven by the MRI that is able to account for a number of correlations among stresses found in local numerical simulations. I point out the relevance of a new type of correlation that couples the dynamical evolution of the Reynolds and Maxwell stresses and plays a key role in developing and sustaining the MHD turbulence. Finally, I address how the turbulent transport of angular momentum depends on the magnitude of the local shear. I show that turbulent MHD stresses in accretion disks cannot be described in terms of shear-viscosity.

Nonlinear Processes in Geophysics

Merrifield , J. A. , Arber , T. D. , Chapman , S. C. , and Dendy , R. O .: The scaling
properties of two dimensional compressible magnetohydrodynamic turbulence ,
Phys . Plasmas , 13 , 012305 , 2006 . Merrifield , J. A. , Chapman , S. C. , and ...

Nonlinear Processes in Geophysics

Author:

Publisher:

ISBN:

Page:

View: 576

Phenomenology Treatment of Magnetohydrodynamic Turbulence with Non equipartition and Anisotropy

Magnetohydrodynamics (MHD) turbulence theory, often employed satisfactorily in astrophysical applications, has often focused on parameter ranges that imply nearly equal values of kinetic and magnetic energies and length scales.

Phenomenology Treatment of Magnetohydrodynamic Turbulence with Non equipartition and Anisotropy

Author:

Publisher:

ISBN:

Page: 8

View: 677

Magnetohydrodynamics (MHD) turbulence theory, often employed satisfactorily in astrophysical applications, has often focused on parameter ranges that imply nearly equal values of kinetic and magnetic energies and length scales. However, MHD flow may have disparity magnetic Prandtl number, dissimilar kinetic and magnetic Reynolds number, different kinetic and magnetic outer length scales, and strong anisotropy. Here a phenomenology for such ''non-equipartitioned'' MHD flow is discussed. Two conditions are proposed for a MHD flow to transition to strong turbulent flow, extensions of (1) Taylor's constant flux in an inertial range, and (2) Kolmogorov's scale separation between the large and small scale boundaries of an inertial range. For this analysis, the detailed information on turbulence structure is not needed. These two conditions for MHD transition are expected to provide consistent predictions and should be applicable to anisotropic MHD flows, after the length scales are replaced by their corresponding perpendicular components. Second, it is stressed that the dynamics and anisotropy of MHD fluctuations is controlled by the relative strength between the straining effects between eddies of similar size and the sweeping action by the large-eddies, or propagation effect of the large-scale magnetic fields, on the small scales, and analysis of this balance in principle also requires consideration of non-equipartition effects.

Scale locality of Magnetohydrodynamic Turbulence

We explain the disagreement with numerical studies that have claimed conversion non locally between disparate scales. We present supporting data from a 10243 simulation of forced MHD turbulence.

Scale locality of Magnetohydrodynamic Turbulence

Author:

Publisher:

ISBN:

Page:

View: 383

We investigate the scale-locality of cascades of conserved invariants at high kinetic and magnetic Reynolds numbers in the 'inertial-inductive range' of magnetohydrodynamic (MHD) turbulence, where velocity and magnetic field increments exhibit suitable power-law scaling. We prove that fluxes of total energy and cross-helicity - or, equivalently, fluxes of Elsaesser energies - are dominated by the contributions of local triads. Corresponding spectral transfers are also scale-local when defined using octave wavenumber bands. Flux and transfer of magnetic helicity may be dominated by nonlocal triads. The magnetic stretching term also may be dominated by non-local triads but we prove that it can convert energy only between velocity and magnetic modes at comparable scales. We explain the disagreement with numerical studies that have claimed conversion non locally between disparate scales. We present supporting data from a 10243 simulation of forced MHD turbulence.

Introduction to Modern Magnetohydrodynamics

An introductory text on magnetohydrodynamics for advanced students, covering its broad range of applications in nature and in the laboratory.

Introduction to Modern Magnetohydrodynamics

Author: Sébastien Galtier

Publisher: Cambridge University Press

ISBN: 1107158656

Page: 288

View: 735

An introductory text on magnetohydrodynamics for advanced students, covering its broad range of applications in nature and in the laboratory.

Advances in Turbulence 7

Shebalin , J . V . , Matthaeus , W . H . & Montgomery , D . ( 1983 ) Anisotropy in
MHD turbulence due to a mean magnetic ... of shear - Alfvén wave packets :
Implications for weak magnetohydrodynamic turbulence in astrophysical plasmas
.

Advances in Turbulence 7

Author: Uriel Frisch

Publisher: Springer Science & Business Media

ISBN:

Page: 613

View: 875

This volume contains an overview of the state of turbulence research with some bias towards work done in Europe. It represents an almost complete collection of the invited and contributed papers delivered at the Seventh European Turbulence Conference, sponsored by EUROMECH and ERCOFTAC and organized by the Observatoire de la Cote d'Azur. High-Reynolds number experiments combined with techniques of imaging, non-intrusive probing, processing and simulation provide high-quality data which put significant constraints on possible theories. For the first time, it has been shown, for a class of passive scalar problems, why dimensional analysis sometimes gives the wrong answers and how anomalous intermittency corrections can be calculated from first principles. The volume is thus geared towards specialists in the area of flow turbulence who could not attend the conference as well as anybody interested in this rapidly-moving field.

Numerical Simulation of Magnetohydrodynamic Turbulence in Confined Domains

In this thesis we developed a Fourier pseudo-spectral code coupled with the volume penalization for the simulation of turbulent MHD flows with non-periodic boundary conditions.

Numerical Simulation of Magnetohydrodynamic Turbulence in Confined Domains

Author: Salah Neffaa

Publisher:

ISBN:

Page: 139

View: 249

In this thesis we developed a Fourier pseudo-spectral code coupled with the volume penalization for the simulation of turbulent MHD flows with non-periodic boundary conditions. This method was validated with classical and academic test-cases. Then we studied the influence of the confinement with walls in a 2D configuration for decaying MHD and we found four decaying regimes which depend on the initial conditions. We also discussed the phenomenon of spontaneous rotation, or spin-up, in 2D non-axisymmetric geometries, originally discovered in hydrodynamic flows. We showed the influence of the Reynolds number and the magnetic pressure on this phenomenon. Finally, simulations of MHD flows in wall bounded three-dimensional domains were addressed. The first results concerning the simulation of decaying MHD turbulence in a cylinder imposing Dirichlet boundary conditions for both the velocity and the magnetic field showed the validity of the code and suggest good prospects for developing more physically justified boundary conditions for the magnetic field.

Magnetic Helicity in Space and Laboratory Plasmas

Montgomery, D., L. Turner, and G. Vahala, Three dimensional
magnetohydrodynamic turbulence in cylindrical geometry, Phys. Fluids 21, 757-
764, 1978. Montgomery, D., and L. Phillips, Minimum dissipation rates in
magnetohydrodynamics, ...

Magnetic Helicity in Space and Laboratory Plasmas

Author: Michael R. Brown

Publisher: American Geophysical Union

ISBN:

Page: 304

View: 430

Published by the American Geophysical Union as part of the Geophysical Monograph Series, Volume 111. Using the concept of magnetic helicity, physicists and mathematicians describe the topology of magnetic fields: twisting, writhing, and linkage. Mathematically, helicity is related to linking integrals, which Gauss introduced in the 19th century to describe the paths of asteroids in the sky. In the late 1970s the concept proved to be critical to understand laboratory plasma experiments on magnetic reconnection, dynamos, and magnetic field relaxation. In the late 1980s it proved equally important in understanding turbulence in the solar wind and the interplanetary magnetic field. During the last five years interest in magnetic helicity has grown dramatically in solar physics, and it will continue to grow as observations of vector magnetic fields become increasingly sophisticated.