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Woltjer's Theorem
In plasma physics, Woltjer's theorem states that force-free magnetic fields in a closed system with constant force-free parameter \alpha represent the state with lowest magnetic energy in the system and that the magnetic helicity is invariant under this condition. It is named after Lodewijk Woltjer who derived it in 1958.Kholodenko, A. L. (2013). Applications of contact geometry and topology in physics. World Scientific. The force-free field strength \mathbf equation is :\nabla \times \mathbf = \alpha \mathbf. The helicity \mathcal invariant is given by :\frac =0. where \mathcal is related to \mathbf=\nabla\times \mathbf through the vector potential \mathbf as below :\mathcal = \int_V \mathbf\cdot\mathbf\ dV = \int_V \mathbf \cdot (\nabla \times \mathbf) \ dV. See also *Chandrasekhar–Kendall function *Hydrodynamical helicity :''This page is about helicity in fluid dynamics. For helicity of magnetic fields, see magnetic helicity. For helicity in particle physics, see helicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plasma Physics
Plasma ()πλάσμα , Henry George Liddell, Robert Scott, ''A Greek English Lexicon'', on Perseus is one of the . It contains a significant portion of charged particles – s and/or s. The presence of these charged particles is what primarily sets plasma apart from the other fundamental states of matter. It is the most abundant form of |
Force-free Magnetic Field
A force-free magnetic field is a magnetic field in which the Lorentz force is equal to zero and the magnetic pressure greatly exceeds the plasma pressure such that non-magnetic forces can be neglected. For a force-free field, the electric current density is either zero or parallel to the magnetic field. Definition When a magnetic field is approximated as force-free, all non-magnetic forces are neglected and the Lorentz force vanishes. For non-magnetic forces to be neglected, it is assumed that the ratio of the plasma pressure to the magnetic pressure—the plasma ''β''—is much less than one, i.e., \beta \ll 1. With this assumption, magnetic pressure dominates over plasma pressure such that the latter can be ignored. It is also assumed that the magnetic pressure dominates over other non-magnetic forces, such as gravity, so that these forces can similarly be ignored. In SI units, the Lorentz force condition for a static magnetic field \mathbf can be expressed as :\mathbf \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Energy
Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet or magnetic moment \mathbf in a magnetic field \mathbf is defined as the mechanical work of the magnetic force (actually magnetic torque) on the re-alignment of the vector of the magnetic dipole moment and is equal to: E_\text = -\mathbf \cdot \mathbf while the energy stored in an inductor (of inductance L) when a current I flows through it is given by: E_\text = \frac LI^2. This second expression forms the basis for superconducting magnetic energy storage. Energy is also stored in a magnetic field. The energy per unit volume in a region of space of permeability \mu _0 containing magnetic field \mathbf is: u = \frac \frac More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates \mathbf and \mathbf, then it can be shown that the magnetic field stores an energy of E = \frac \int \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Helicity
In plasma physics, magnetic helicity is a measure of the linkage, twist, and writhe of a magnetic field. In ideal magnetohydrodynamics, magnetic helicity is conserved. When a magnetic field contains magnetic helicity, it tends to form large-scale structures from small-scale ones. This process can be referred as an inverse transfer in Fourier space. This second property makes magnetic helicity special: three-dimensional turbulent flows tend to "destroy" structure, in the sense that large-scale vortices break-up in smaller and smaller ones (a process called "direct energy cascade", described by Lewis Fry Richardson and Andrey Nikolaevich Kolmogorov). At the smallest scales, the vortices are dissipated in heat through viscous effects. Through a sort of "inverse cascade of magnetic helicity", the opposite happens: small helical structures (with a non-zero magnetic helicity) lead to the formation of large-scale magnetic fields. This is for example visible in the heliospheric curren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lodewijk Woltjer
Lodewijk Woltjer (26 April 1930 – 25 August 2019) was an astronomer, and the son of astronomer Jan Woltjer. He studied at the University of Leiden under Jan Oort earning a PhD in astronomy in 1957 with a thesis on the Crab Nebula. This was followed by post-doctoral research appointments to various American universities and the subsequent appointment of professor of theoretical astrophysics and plasma physics in the University of Leiden. From 1964 to 1974 he was Rutherford Professor of Astronomy and Chair of the Astronomy Department at Columbia University in New York. From 1975 to 1987 he was Director General of the European Southern Observatory (ESO), where he initiated the construction of the Very Large Telescope. In 1994–1997 he was President of the International Astronomical Union. Woltjer was honored in 1987 with the Karl Schwarzschild Medal. He was the first Editor-in-Chief of The Astronomy and Astrophysics Review, inaugurated in 1989; and also the Editor of the Astronomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vector potential'' is a C^2 vector field A such that \mathbf = \nabla \times \mathbf. Consequence If a vector field v admits a vector potential A, then from the equality \nabla \cdot (\nabla \times \mathbf) = 0 (divergence of the curl is zero) one obtains \nabla \cdot \mathbf = \nabla \cdot (\nabla \times \mathbf) = 0, which implies that v must be a solenoidal vector field. Theorem Let \mathbf : \R^3 \to \R^3 be a solenoidal vector field which is twice continuously differentiable. Assume that decreases at least as fast as 1/\, \mathbf\, for \, \mathbf\, \to \infty . Define \mathbf (\mathbf) = \frac \int_ \frac \, d^3\mathbf. Then, A is a vector potential for , that is, \nabla \times \mathbf =\mathbf. Here, \nabla_y \times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chandrasekhar–Kendall Function
Chandrasekhar–Kendall functions are the axisymmetric eigenfunctions of the curl operator derived by Subrahmanyan Chandrasekhar and P. C. Kendall in 1957 while attempting to solve the force-free magnetic fields. The functions were independently derived by both, and the two decided to publish their findings in the same paper. If the force-free magnetic field equation is written as \nabla\times\mathbf=\lambda\mathbf, where \mathbf is the magnetic field and \lambda is the force-free parameter, with the assumption of divergence free field, \nabla\cdot\mathbf=0, then the most general solution for the axisymmetric case is :\mathbf = \frac\nabla\times(\nabla\times\psi\mathbf) + \nabla \times \psi \mathbf where \mathbf is a unit vector and the scalar function \psi satisfies the Helmholtz equation, i.e., :\nabla^2\psi + \lambda^2\psi=0. The same equation also appears in Beltrami flows from fluid dynamics where, the vorticity vector is parallel to the velocity vector, i.e., \nabla\times\math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hydrodynamical Helicity
:''This page is about helicity in fluid dynamics. For helicity of magnetic fields, see magnetic helicity. For helicity in particle physics, see helicity (particle physics).'' In fluid dynamics, helicity is, under appropriate conditions, an invariant of the Euler equations of fluid flow, having a topological interpretation as a measure of linkage and/or knottedness of vortex lines in the flow. This was first proved by Jean-Jacques Moreau in 1961 and Moffatt derived it in 1969 without the knowledge of Moreau's paper. This helicity invariant is an extension of Woltjer's theorem for magnetic helicity. Let \mathbf(x,t) be the velocity field and \nabla\times\mathbf the corresponding vorticity field. Under the following three conditions, the vortex lines are transported with (or 'frozen in') the flow: (i) the fluid is inviscid; (ii) either the flow is incompressible (\nabla\cdot\mathbf = 0), or it is compressible with a barotropic relation p = p(\rho) between pressure p and density \rh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the heavenly bodies, rather than their positions or motions in space–''what'' they are, rather than ''where'' they are." Among the subjects studied are the Sun, other stars, galaxies, extrasolar planets, the interstellar medium and the cosmic microwave background. Emissions from these objects are examined across all parts of the electromagnetic spectrum, and the properties examined include luminosity, density, temperature, and chemical composition. Because astrophysics is a very broad subject, ''astrophysicists'' apply concepts and methods from many disciplines of physics, including classical mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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