Topic
Induction equation
About: Induction equation is a research topic. Over the lifetime, 510 publications have been published within this topic receiving 12772 citations.
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1 Jan 1982
TL;DR: In this article, the basic equations of magnetohydrodynamics of the sun are presented, including the basic equation of magneto-hydrostatics, and the equations of the acceleration and deformation of the Sun's upper atmosphere.
Abstract: 1 A description of the sun- 11 Brief History- 12 Overall Properties- 121 Interior- 122 Outer Atmosphere- 13The Quiet Sun- 131 The Interior- A The Core- B A Model- C Convection Zone- 132 The Photosphere- A Motions- B Magnetic Field- C A Model- 133 The Chromosphere- 134 The Corona- A At Eclipses- B In X-rays- C Solar Wind- 14 Transient Features- 141 Active Regions- A Development- B Structure- C Loops- D Internal Motions- 142 Sunspots- A Development- B Umbra- C Penumbra- D Motion- E Solar Cycle- 143 Prominences- A Introduction- B Properties- C Development- D Structure- E Eruption- F Coronal Transients- 144 Solar Flares- A Basic Description- B Ground-Based Observations- C Space Observations- 2 The basic equations of magnetohydrodynamics- 21 Electromagnetic Equations- 211 Maxwell's Equations- 212 Ohm's Law- 213 Generalised Ohm's Law- 214 Induction Equation- 215 Electrical Conductivity- 22 Plasma Equations- 221 Mass Continuity- 222 Equation of Motion- 223 Perfect Gas Law- 23 Energy Equations- 231 Different Forms of Heat Equation- 232 Thermal Conduction- 233 Radiation- 234 Heating- 235 Energetics- 24 Summary of Equations- 241 Assumptions- 242 Reduced Forms of the Equations- 25 Dimensionless Parameters- 26 Consequences of the Induction Equation- 261 Diffusive Limit- 262 Perfectly Conducting Limit- 27 The Lorentz Force- 28 Some Theorems- 281 Cowling's Antidynamo Theorem- 282 Taylor-Proudman Theorem- 283 Ferraro's Law of Isorotation- 284 The Virial Theorem- 29 Summary of Magnetic Flux Tube Behaviour- 291 Definitions- 292 General Properties- 293 Flux Tubes in the Solar Atmosphere- 210 Summary of Current Sheet Behaviour- 2101 Processes of Formation- 2102 Properties- 3 Magnetohydrostatics- 31 Introduction- 32 Plasma Structure in a Prescribed Magnetic Field- 33 The Structure of Magnetic Flux Tubes (Cylindrically Symmetric)- 331 Purely Axial Field- 332 Purely Azimuthal Field- 333 Force-Free Fields- A Linear Field- B Nonlinear Fields- C Effect of Twisting a Tube- D Effect of Expanding a Tube- E A Tube of Non-Uniform Radius- 334 Magnetostatic Fields- 34 Current-Free Fields- 35 Force-Free Fields- 351 General Theorems- 352 Simple Constant-? Solutions- 353 General Constant- ? Solutions- 354 Non-Constant- ? Solutions- 355 Diffusion- 356 Coronal Evolution- 36 Magnetohydrostatic Fields- 4 Waves- 41 Introduction- 411 Fundamental Modes- 412 Basic Equations- 42 Sound Waves- 43 Magnetic Waves- 431 Shear Alfven Waves- 432 Compressional Alfven Waves- 44 Internal Gravity Waves- 45 Inertial Waves- 46 Magnetoacoustic Waves- 47 Acoustic-Gravity Waves- 48 Summary of Magnetoacoustic-Gravity Waves- 49 Five-Minute Oscillations- 491 Observations- 492 Models- A Photospheric Ringing- B Wave Trapping- 493 Wave Generation- 494 Strong Magnetic Field Regions- 495 The Future- 410 Waves in a Strongly Inhomogeneous Medium- 4101 Surface Waves on a Magnetic Interface- 4102 A Twisted Magnetic Flux Tube- 4103 A Stratified Atmosphere- 5 Shock waves- 51 Introduction- 511 Formation of a Hydrodynamic Shock- 512 Effects of a Magnetic Field- 52 Hydrodynamic Shocks- 53 Perpendicular Shocks- 54 Oblique Shocks- 541 Jump Relations- 542 Slow and Fast Shocks- 543 Switch-Off and Switch-On Shocks- 544 The Intermediate Wave- 6 Heating of the upper atmosphere- 61 Introduction- 62 Models for Atmospheric Structure- 621 Basic Model- 622 Magnetic Field Effects- 623 Additional Effects- 63 Acoustic Wave Heating- 631 Steepening- 632 Propagation and Dissipation- 64 Magnetic Heating- 641 Propagation and Dissipation of Magnetic Waves- 642 Nonlinear Coupling of Alfven Waves- 643 Resonant Absorption of Alfven Waves- 644 Magnetic Field Dissipation- A Order of Magnitude- B Current Sheets- C Current Filaments- 65 Coronal Loops- 651 Static Energy-Balance Models- A Uniform Pressure Loops- B Cool Cores- C Hydrostatic Equilibrium- 652 Flows in Coronal Loops- 7 Instability- 71 Introduction- 72 Linearised Equations- 73 Normal Mode Method- 731 Example: Rayleigh-Taylor Instability- A Plasma Supported by a Magnetic Field- B Uniform Magnetic Field B0(+) = B0(-)- 74 Variational (or Energy) Method- 741 Example: Kink Instability- 742 Use of the Energy Method- 75 Summary of Instabilities- 751 Interchange Instability- 752 Rayleigh-Taylor Instability- 753 Pinched Discharge- 754 Flow Instability- 755 Resistive Instability- 756 Convective Instability- 757 Radiatively-Driven Thermal Instability- 758 Other Instabilities- 8 Sunspots- 81 Magnetoconvection- 811 Physical Effects- 812 Linear Stability Analysis- 813 Magnetic Flux Expulsion and Concentration- 82 Magnetic Buoyancy- 821 Qualitative Effect- 822 Magnetic Buoyancy Instability- 823 The Rise of Flux Tubes in the Sun- 83 Cooling of Sunspots- 84 Equilibrium Structure of Sunspots- 841 Magnetohydrostatic Equilibrium- 842 Sunspot Stability- 85 The Sunspot Penumbra- 86 Evolution of a Sunspot- 861 Formation- 862 Decay- 87 Intense Flux Tubes- 871 Equilibrium of a Slender Flux Tube- 872 Intense Magnetic Field Instability- 873 Spicule Generation- 874 Tube Waves- 9 Dynamo theory- 91 Introduction- 92 Cowling's Theorem- 93 Qualitative Dynamo Action- 931 Generation of Toroidal and Poloidal Fields- 932 Phenomenological Model- 94 Kinematic Dynamos- 941 Nearly-Symmetric Dynamo- 942 Turbulent Dynamo: Mean-Field Electrodynamics- 943 Simple Solution: Dynamo Waves- 944 Solar Cycle Models: The ?-? Dynamo- 95 Magnetohydrodynamic Dynamos- 951 Modified Kinematic Dynamos- 952 Strange Attractors- 953 Convective Dynamos- 96 Difficulties with Dynamo Theory- 10 Solar flares- 101 Magnetic Reconnection- 1011 Unidirectional Field- 1012 Diffusion Region- 1013 The Petschek Mechanism- 1014 External Region- 102 Simple-Loop Flare- 1021 Emerging (or Evolving) Flux Model- 1022 Thermal Nonequilibrium- 1023 Kink Instability- 1024 Resistive Kink Instability- 103 Two-Ribbon Flare- 1031 Existence and Multiplicity of Force-Free Equilibria- 1032 Eruptive Instability- 1033 The Main Phase: 'Post'-Flare Loops- 11 Prominences- 111 Formation- 1111 Formation in a Loop (Active-Region Prominences)- 1112 Formation in a Coronal Arcade- 1113 Formation in a Current Sheet- A Thermal Nonequilibrium- B Line-Tying- 112 Magnetohydrostatics of Support in a Simple Arcade- 1121 Kippenhahn-Schluter Model- 1122 Generalised Kippenhahn-Schluter Model- 1123 The External Field- 1124 Magnetohydrodynamic Stability- 1125 Helical Structure- 113 Support in Configurations with Helical Fields- 1131 Support in a Current Sheet- 1132 Support in a Horizontal Field- 114 Coronal Transients- 1141 Twisted Loop Models- 1142 Untwisted Loop Models- 1143 Numerical Models- 1144 Conclusion- 12 The solar wind- 121 Introduction- 122 Parker's Solution- 123 Models for a Spherical Expansion- 1221 Energy Equation- 1222 Two-Fluid Model- 1223 Magnetic Field- 124 Streamers and Coronal Holes- 1241 Pneuman-Kopp Model- A Basic Model- B Angular Momentum Loss- C Current Sheet- 1242 Coronal Hole Models- 125 Extra Effects- Appendix I Units- Appendix II Useful Values and Expressions- Appendix III Notation- References
1,301 citations
TL;DR: An ecient and easy-to-implement three-dimensional shock- capturing scheme for ideal RHD that gives results comparable to those obtained with more sophisticated algorithms, even in ultrarelativistic multidimensional test problems.
Abstract: A third order shock-capturing numerical scheme for three-dimensional special relativistic magnetohydrodynamics (3-D RMHD) is presented and validated against several numerical tests. The simple and efficient central scheme described in Paper I (Del Zanna and Bucciantini, Astron. Astrophys., 390, 1177--1186, 2002) for relativistic hydrodynamics is here extended to the magnetic case by following the strategies prescribed for classical MHD by Londrillo and Del Zanna (Astrophys. J., 530, 508--524, 2000). The scheme avoids completely spectral decomposition into characteristic waves, computationally expensive and subject to many degenerate cases in the magnetic case, while it makes use of a two-speed Riemann solver that just require the knowledge of the two local fast magnetosonic velocities. Moreover, the onset of spurious magnetic monopoles, which is a typical problem for multi-dimensional MHD upwind codes, is prevented by properly taking into account the solenoidal constraint and the specific antisymmetric nature of the induction equation. Finally, the extension to generalized orthogonal curvilinear coordinate systems is included, thus the scheme is ready to incorporate general relativistic (GRMHD) effects.
371 citations
TL;DR: In this paper, a new numerical code, ECHO, based on an Eulerian Conservative High Order scheme for time dependent three-dimensional general relativistic magnetohydrodynamics (GRMHD) and magnetodynamics (GRMD), is presented.
Abstract: We present a new numerical code, ECHO, based on an Eulerian Conservative High Order scheme for time dependent three-dimensional general relativistic magnetohydrodynamics (GRMHD) and magnetodynamics (GRMD). ECHO is aimed at providing a shock-capturing conservative method able to work at an arbitrary level of formal accuracy (for smooth flows), where the other existing GRMHD and GRMD schemes yield an overall second order at most. Moreover, our goal is to present a general framework, based on the 3+1 Eulerian formalism, allowing for different sets of equations, different algorithms, and working in a generic space-time metric, so that ECHO may be easily coupled to any solver for Einstein's equations. Various high order reconstruction methods are implemented and a two-wave approximate Riemann solver is used. The induction equation is treated by adopting the Upwind Constrained Transport (UCT) procedures, appropriate to preserve the divergence-free condition of the magnetic field in shock-capturing methods. The limiting case of magnetodynamics (also known as force-free degenerate electrodynamics) is implemented by simply replacing the fluid velocity with the electromagnetic drift velocity and by neglecting the matter contribution to the stress tensor. ECHO is particularly accurate, efficient, versatile, and robust. It has been tested against several astrophysical applications, including a novel test on the propagation of large amplitude circularly polarized Alfven waves. In particular, we show that reconstruction based on a Monotonicity Preserving filter applied to a fixed 5-point stencil gives highly accurate results for smooth solutions, both in flat and curved metric (up to the nominal fifth order), while at the same time providing sharp profiles in tests involving discontinuities.
322 citations
TL;DR: In this article, the authors studied the mutual influence of thermal and magnetic evoluti on in a neutron star's crust in axial symmetry, and showed that the feedback between Joule heating and magnetic diffusion is strong, resulting in a faster dissipation of the stronger fields during the first 10 5 − 10 6 years of a star's life.
Abstract: Context. The presence of magnetic fields in the crust of neutron stars c auses a non-spherically symmetric temperature distribution. The strong temperature dependence of the magnetic diffusivity and thermal conductivity, together with the heat generated by magnetic dissipation, couple the magnetic and thermal evolution of NSs, that cannot be formulated as separated one‐dimensional problems. Aims. We study the mutual influence of thermal and magnetic evoluti on in a neutron star’s crust in axial symmetry. Taking into account realistic microphysical inputs, we find the heat rel eased by Joule effect consistent with the circulation of currents in the crust , and we incorporate its effects in 2‐dimensional cooling calculations. Methods. We solve the induction equation numerically using a hybrid method (spectral in angles, but a finite‐di fferences scheme in the radial direction), coupled to the thermal diffusion equation. To improve the boundary conditions, we also revisit the envelope stationary solutions updating the well known Tb− Ts‐relations to include the effect of 2‐D heat transfer calculations and new microphysical inputs. Results. We present the first long term 2‐dimensional simulations of t he coupled magneto-thermal evolution of neutron stars. This substantially improves previous works in which a very crude approximation in at least one of the parts (thermal or magnetic diffusion) has been adopted. Our results show that the feedback between Joule heating and magnetic diffusion is strong, resulting in a faster dissipation of the stronger fields during the first 10 5 − 10 6 years of a NS’s life. As a consequence, all neutron stars born with fields larger than a critical value (> 5×10 13 G) reach similar field strengths (≈ 2−3×10 13 G) at late times. Irrespectively of the initial magnetic field strength, after 10 6 years the temperature becomes so low that the magnetic diffusion timescale becomes longer than the typical ages of radio‐pulsars, thus resulting in apparently no diss ipation of the field in old NS. We also confirm the strong correl ation between the magnetic field and the surface temperature of relatively young NSs discussed in preliminary works. The effective temperature of models with strong internal toroidal components are systematically higher than those of models with purely poloidal fie lds, due to the additional energy reservoir stored in the toroidal field tha t is gradually released as the field dissipates.
296 citations
TL;DR: It is shown that the CT formalism, when fully exploited, can be used as a general guideline to design the reconstruction procedures of the B vector field, to adapt standard upwind procedures for the momentum and energy equations, avoiding the onset of numerical monopoles of O(1) size.
290 citations
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