|
Grad–Shafranov Equation
The Grad–Shafranov equation ( H. Grad and H. Rubin (1958); Vitalii Dmitrievich Shafranov (1966)) is the equilibrium equation in ideal magnetohydrodynamics (MHD) for a two dimensional plasma, for example the axisymmetric toroidal plasma in a tokamak. This equation takes the same form as the Hicks equation from fluid dynamics.Smith, S. G. L., & Hattori, Y. (2012)Axisymmetric magnetic vortices with swirl. Communications in Nonlinear Science and Numerical Simulation 17(5), 2101-2107. This equation is a two-dimensional, nonlinear, elliptic partial differential equation obtained from the reduction of the ideal MHD equations to two dimensions, often for the case of toroidal axisymmetry (the case relevant in a tokamak). Taking (r,\theta,z) as the cylindrical coordinates, the flux function \psi is governed by the equation,where \mu_0 is the magnetic permeability, p(\psi) is the pressure, F(\psi)=rB_ and the magnetic field and current are, respectively, given by\begin \mathbf &= \fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Harold Grad
Harold Grad (January 23, 1923 in New York City – November 17, 1986) was an American applied mathematician. His work specialized in the application of statistical mechanics to plasma physics and magnetohydrodynamics. Work In statistical mechanics he had developed in his thesis new methods for the solution of the Boltzmann equation. He derived the Boltzmann equation from Liouville equation using BBGKY hierarchy under certain limits, known as Boltzmann–Grad limit. Harold Grad was the founder of the Magneto-fluid Dynamics Division of the Courant Institute and served as its head until shortly before his death From 1964 to 1967 and 1974 to 1977 he was a member of the Advisory Committee for Fusion Energy at Oak Ridge National Laboratory. Grad was a critic and supporter of many early fusion schemes including picket fences, magnetic mirrors and Biconic cusps. Recognition In 1970, Grad became a member of the National Academy of Sciences. He was an invited speaker at the Internat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Magnetic Permeability
In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. It is the ratio of the magnetic induction B to the magnetizing field H in a material. The term was coined by William Thomson, 1st Baron Kelvin in 1872, and used alongside permittivity by Oliver Heaviside in 1885. The reciprocal of permeability is magnetic reluctivity. In SI units, permeability is measured in henries per meter (H/m), or equivalently in newtons per ampere squared (N/A2). The permeability constant ''μ''0, also known as the magnetic constant or the permeability of free space, is the proportionality between magnetic induction and magnetizing force when forming a magnetic field in a classical vacuum. A closely related property of materials is magnetic susceptibility, which is a dimensionless proportionality factor that indicates the degree of magnet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Magnetohydrodynamics
In physics and engineering, magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single Continuum mechanics, continuous medium. It is primarily concerned with the low-frequency, large-scale, magnetic behavior in Plasma (physics), plasmas and liquid metals and has applications in multiple fields including space physics, geophysics, astrophysics, and engineering. The word ''magnetohydrodynamics'' is derived from ' meaning magnetic field, ' meaning water, and ' meaning movement. The field of MHD was initiated by Hannes Alfvén, for which he received the Nobel Prize in Physics in 1970. History The MHD description of electrically conducting fluids was first developed by Hannes Alfvén in a 1942 paper published in Nature (journal), ''Nature'' titled "Existence of Electromagnetic–Hydrodynamic Waves" which outlined his discovery ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Balance Equation
In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ... in and out of states or set of states. Global balance The global balance equations (also known as full balance equations) are a set of equations that characterize the equilibrium distribution (or any stationary distribution) of a Markov chain, when such a distribution exists. For a continuous time Markov chain with state space \mathcal, transition rate from state i to j given by q_ and equilibrium distribution given by \pi, the global balance equations are given by ::\pi_i = \sum_ \pi_j q_, or equivalently :: \pi_i \sum_ q_ = \sum_ \pi_j q_. for all i \in S. Here \pi_i q_ represents the probability f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Field Line
A field line is a graphical Scientific visualization, visual aid for visualizing vector fields. It consists of an imaginary integral curve which is tangent to the field Euclidean vector, vector at each point along its length. A diagram showing a representative set of neighboring field lines is a common way of depicting a vector field in scientific and mathematical literature; this is called a field line diagram. They are used to show electric fields, magnetic fields, and gravitational fields among many other types. In fluid mechanics, field lines showing the velocity field of a fluid flow are called Streamlines, streaklines, and pathlines, streamlines. Definition and description A vector field defines a direction and magnitude at each point in space. A field line is an integral curve for that vector field and may be constructed by starting at a point and tracing a line through space that follows the direction of the vector field, by making the field line tangent line, tang ... [...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 \mathbf, a ''vector potential'' is a C^2 vector field \mathbf such that \mathbf = \nabla \times \mathbf. Consequence If a vector field \mathbf admits a vector potential \mathbf, 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 \mathbf 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 \mathbf(\mathbf) decreases at least as fast as 1/\, \mathbf\, for \, \mathbf\, \to \infty . Define \mathbf (\mathbf) = \frac \int_ \frac \, d^3\mathbf where \nabla_y \times denotes curl with respect to variab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Reversed Field Pinch
A reversed-field pinch (RFP) is a device used to produce and contain near-thermonuclear plasmas. It is a toroidal pinch that uses a unique magnetic field configuration as a scheme to magnetically confine a plasma, primarily to study magnetic confinement fusion. Its magnetic geometry is somewhat different from that of a tokamak. As one moves out radially, the portion of the magnetic field pointing toroidally reverses its direction, giving rise to the term ''reversed field''. This configuration can be sustained with comparatively lower fields than that of a tokamak of similar power density. One of the disadvantages of this configuration is that it tends to be more susceptible to non-linear effects and turbulence. This makes it a useful system for studying non-ideal (resistive) magnetohydrodynamics. RFPs are also used in studying astrophysical plasmas, which share many common features. The largest Reversed Field Pinch device presently in operation is the RFX (R/a = 2/0.46) in P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and even by industry. Further, both spellings are often used ''within'' a particular industry or country. Industries in British English-speaking countries typically use the "gauge" spelling. is the pressure relative to the ambient pressure. Various #Units, units are used to express pressure. Some of these derive from a unit of force divided by a unit of area; the International System of Units, SI unit of pressure, the Pascal (unit), pascal (Pa), for example, is one newton (unit), newton per square metre (N/m2); similarly, the Pound (force), pound-force per square inch (Pound per square inch, psi, symbol lbf/in2) is the traditional unit of pressure in the imperial units, imperial and United States customary units, US customary systems. Pressure ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Toroid
In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus. The term ''toroid'' is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A ''g''-holed ''toroid'' can be seen as approximating the surface of a torus having a topological genus, ''g'', of 1 or greater. The Euler characteristic χ of a ''g'' holed toroid is 2(1−''g''). The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and are not toroids. Toroidal structures occur in both natural and synthetic materials. Equations A toroid is specified by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Vitaly Shafranov
Vitaly Dmitrievich Shafranov (; December 1, 1929 – June 9, 2014) was a Russian theoretical physicist and Academician who worked with plasma physics and thermonuclear fusion research. Life Vitaly Dmitrievich Shafranov was born in the village of Mordvinovo in Ryazan region in 1929. During World War II, Schafranov attended the school and worked together with his father building roads. In 1943 he got his first national award at the age 14. From 1946, Schafranov studied at the Physics Department of the Moscow State University. After graduating in 1951, he started to work with nuclear fusion in the Theory Department headed by Mikhail Aleksandrovich Leontovich at LIPAN (Laboratory of Measuring Instruments of the USSR Academy of Sciences) as today's Russian Research Centre "Kurchatov Institute" was known at the time. He examined tokamaks stability and gave some parameter estimation for Soviet tokamak experiments. He also dealt with shock waves in plasmas and interaction of electromagne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Elliptic Partial Differential Equation
In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which generally model phenomena that change in time. The canonical examples of elliptic PDEs are Laplace's Equation and Poisson's Equation. Elliptic PDEs are also important in pure mathematics, where they are fundamental to various fields of research such as differential geometry and optimal transport. Definition Elliptic differential equations appear in many different contexts and levels of generality. First consider a second-order linear PDE for an unknown function of two variables u = u(x,y), written in the form Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0, where , , , , , , and are functions of (x,y), using subscript notation for the partial derivatives. The PDE is called elliptic if B^2-AC 0 are hyperbolic. For a general linear second-order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Nonlinear
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |