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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] |
Eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as Df = \lambda f for some scalar eigenvalue \lambda. The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvector. Eigenfunctions In general, an eigenvector of a linear operator ''D'' defined on some vector space is a nonzero vector in the domain of ''D'' that, when ''D'' acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where ''D'' is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function ''f'' is an eigenfunction of ''D'' if it satisfies the equation where λ is a scalar. The solutions to Equation may also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curl (mathematics)
In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. is a notation common today to the United States and Americas. In many European countries, particularly in classic scientific literature, the alternative notation is traditionally used, which is spelled as "rotor", and comes from the "rate of rotation", which it rep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar (; ) (19 October 1910 – 21 August 1995) was an Indian-American theoretical physicist who spent his professional life in the United States. He shared the 1983 Nobel Prize for Physics with William A. Fowler for "...theoretical studies of the physical processes of importance to the structure and evolution of the stars". His mathematical treatment of stellar evolution yielded many of the current theoretical models of the later evolutionary stages of massive stars and black holes. Many concepts, institutions, and inventions, including the Chandrasekhar limit and the Chandra X-Ray Observatory, are named after him. Chandrasekhar worked on a wide variety of problems in physics during his lifetime, contributing to the contemporary understanding of stellar structure, white dwarfs, stellar dynamics, stochastic process, radiative transfer, the quantum theory of the hydrogen anion, hydrodynamic and hydromagnetic stability, turbulence, equilibrium and the stabi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Helmholtz Equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenvalue, and is the (eigen)function. When the equation is applied to waves, is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences. Motivation and uses The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. For example, consider the wave equation \left(\nabla^2-\frac\frac\right) u(\mathbf,t)=0. Separation of variables begins by assumi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beltrami Flow
In fluid dynamics, Beltrami flows are flows in which the vorticity vector \mathbf and the velocity vector \mathbf are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881. Description Since the vorticity vector \boldsymbol and the velocity vector \mathbf are parallel to each other, we can write :\boldsymbol\times\mathbf=0, \quad \boldsymbol = \alpha(\mathbf,t) \mathbf, where \alpha(\mathbf,t) is some scalar function. One immediate consequence of Beltrami flow is that it can never be a planar or axisymmetric flow because in those flows, vorticity is always perpendicular to the velocity field. The other important consequence will be realized by looking at the incompressible vorticity equation :\frac + (\mathbf\cdot\nabla)\bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. It is an important quantity in the dynamical theory of fluids and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of vortex rings. Mathematically, the vorticity \vec is the curl of the flow velocity \vec: :\vec \equiv \nabla \times \vec\,, where \nabla is the nabla operator. Conceptually, \vec could be determined by marking parts of a continuum in a small neighborhood of the point in question, and watching their ''relative'' displacements as they move along the flow. The vorticity \vec would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule. In a two-dimensional fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poloidal–toroidal Decomposition
In vector calculus, a topic in pure and applied mathematics, a poloidal–toroidal decomposition is a restricted form of the Helmholtz decomposition. It is often used in the spherical coordinates analysis of solenoidal vector fields, for example, magnetic fields and incompressible fluids. Definition For a three-dimensional vector field F with zero divergence : \nabla \cdot \mathbf = 0, this F can be expressed as the sum of a toroidal field T and poloidal vector field P :\mathbf = \mathbf + \mathbf where r is a radial vector in spherical coordinates (''r'', ''θ'', ''φ''). The toroidal field is obtained from a scalar field, ''Ψ''(''r'', ''θ'', ''φ''), as the following curl, : \mathbf = \nabla \times (\mathbf \Psi(\mathbf)) and the poloidal field is derived from another scalar field Φ(''r'', ''θ'', ''φ''), as a twice-iterated curl, : \mathbf = \nabla \times (\nabla \times (\mathbf \Phi (\mathbf)))\,. This decomposition is symmetric in that the curl of a toroid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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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