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Hasegawa–Mima Equation
In plasma physics, the Hasegawa–Mima equation, named after Akira Hasegawa and Kunioki Mima, is an equation that describes a certain regime of plasma, where the time scales are very fast, and the distance scale in the direction of the magnetic field is long. In particular the equation is useful for describing turbulence in some tokamaks. The equation was introduced in Hasegawa and Mima's paper submitted in 1977 to ''Physics of Fluids'', where they compared it to the results of the ATC tokamak. Assumptions * The magnetic field is large enough that: :: \frac\frac \ll 1 :for all quantities of interest. When the particles in the plasma are moving through a magnetic field, they spin in a circle around the magnetic field. The frequency of oscillation, \omega_ known as the cyclotron frequency or gyrofrequency, is directly proportional to the magnetic field. * The particle density follows the quasineutrality condition: :: n_e \approx Z n_i \, :where Z is the number of protons in the ... [...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 |
Navier–Stokes Equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes). The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ''viscous flow''. The difference between them and the closely related Euler equations is that Navier–Stokes equations take ... [...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 |
Turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe. The onset of turbulence can be predicted by the dimensionless Rey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plasma (physics)
Plasma () 1, where \nu_ is the electron gyrofrequency and \nu_ is the electron collision rate. It is often the case that the electrons are magnetized while the ions are not. Magnetized plasmas are ''anisotropic'', meaning that their properties in the direction parallel to the magnetic field are different from those perpendicular to it. While electric fields in plasmas are usually small due to the plasma high conductivity, the electric field associated with a plasma moving with velocity \mathbf in the magnetic field \mathbf is given by the usual Lorentz force, Lorentz formula \mathbf = -\mathbf\times\mathbf, and is not affected by Debye shielding. Mathematical descriptions To completely describe the state of a plasma, all of the particle locations and velocities that describe the electromagnetic field in the plasma region would need to be written down. However, it is generally not practical or necessary to keep track of all the particles in a plasma. Therefore, plasma physicist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Navier–Stokes Equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes). The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ''viscous flow''. The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enstrophy
In fluid dynamics, the enstrophy can be interpreted as another type of potential density; or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It is particularly useful in the study of turbulent flows, and is often identified in the study of thrusters as well as the field of combustion theory. Given a domain \Omega \subseteq \R^n and a once-weakly differentiable vector field u \in H^1(\R^n)^n which represents a fluid flow, such as a solution to the Navier-Stokes equations, its enstrophy is given by: \mathcal(u) := \int_\Omega , \nabla \mathbf, ^2 \, dx, where , \nabla \mathbf, ^2 = \sum_^n \left, \partial_i u^j \^2 . This quantity is the same as the squared seminorm , \mathbf, _^2of the solution in the Sobolev space H^1(\Omega)^n. In the case that the flow is incompressible, or equivalently that \nabla \cdot \mathbf = 0 , the enstrophy can be described as the integral of the square ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest. Formally, a kinetic energy is any term in a system's Lagrangian which includes a derivative with respect to time. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2. In relativistic mechanics, this is a good approximation only when ''v'' is much less than the speed of light. The standard unit of kinetic energy is the joule, while the English unit of kinetic energy is the foot-pound. History and etymology The adjective ''kinetic'' has its roots in the Greek word κίνησις ''kinesis'', m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called ''canonical transformations'', which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by q_i and p_i, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself H =H(q, p, t) as one of the new canonical momentum coordinates. In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxwell–Boltzmann Distribution
In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used for describing particle speeds in idealized gases, where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions in which they exchange energy and momentum with each other or with their thermal environment. The term "particle" in this context refers to gaseous particles only (atoms or molecules), and the system of particles is assumed to have reached thermodynamic equilibrium.''Statistical Physics'' (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, The energies of such particles follow what is known as Maxwell–Boltzmann statistics, and the statistical distribution of speeds is derived by equating particle energies with kinetic energy. Mathematica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, salt water, and electrolytes. 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. The fundamental concept behind MHD is that magnetic fields can induce currents in a moving conductive fluid, which in turn polarizes the fluid and reciprocally changes the magnetic field itself. The set of equations that describe MHD are a combination of the Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. These differential equations must be solved simultaneously, either analytically or numerically. History The first record ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroradius
The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field. In SI units, the non-relativistic gyroradius is given by :r_ = \frac where m is the mass of the particle, v_ is the component of the velocity perpendicular to the direction of the magnetic field, q is the electric charge of the particle, and B is the strength of the magnetic field. The angular frequency of this circular motion is known as the gyrofrequency, or cyclotron frequency, and can be expressed as :\omega_ = \frac in units of radians/second. Variants It is often useful to give the gyrofrequency a sign with the definition :\omega_ = \frac or express it in units of hertz with :f_ = \frac. For electrons, this frequency can be reduced to :f_ = (2.8\times10^\,\mathrm/\mathrm)\times B. In cgs-units the gyroradius :r_ = \frac and the corresponding gyrofrequency :\omega_ = \frac include a factor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
