Geometric calculus - Revision history

en>Quondum: /* Differentiation */ removing apparently nonsensical statement: "Unlike the exterior product, the exterior derivative is not even associative."

2014-09-08T16:06:56Z

Differentiation: removing apparently nonsensical statement: "Unlike the exterior product, the exterior derivative is not even associative."

← Older revision		Revision as of 18:06, 8 September 2014
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en>Slawekb: Align is fixed now.

2014-02-09T11:35:50Z

Align is fixed now.

Show changes

en>Chessfan: /* Relation to differential forms */ Disambiguation

2014-01-29T13:04:55Z

Relation to differential forms: Disambiguation

← Older revision		Revision as of 15:04, 29 January 2014
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	~~There~~ is ~~nothing~~ to ~~say about myself I think~~.<br>~~Nice to~~ be ~~here~~ and a ~~part~~ of this ~~community~~.<br>I ~~really hope I~~'~~m useful~~ in ~~one way~~ .<br><br>~~Feel free~~ to ~~surf~~ to ~~my web site~~ ... [http://~~pbhm~~.com/~~cheapuggs~~/ ~~cheap uggs~~]		'''Gyrokinetics''' is a theoretical framework to study plasma behavior on perpendicular spatial scales comparable to the [[gyroradius]] and timescales much slower than the particle [[Cyclotron resonance\|cyclotron frequencies]].
			These particular scales have been experimentally shown to be appropriate for modeling plasma turbulence.<ref>G.R. McKee, C.C. Petty, et al. Non-dimensional scaling of turbulence characteristics and turbulent diffusivity. Nuclear Fusion, 41(9):1235, 2001.</ref> The trajectory of charged particles in a magnetic field is a helix that winds around the field line. This trajectory can be decomposed into a relatively slow motion of the [[guiding center]] along the field line and a fast circular motion, called gyromotion. For most plasma behavior, this gyromotion is irrelevant. Averaging over this gyromotion reduces the equations to six dimensions (3 spatial, 2 velocity, and time) rather than the seven (3 spatial, 3 velocity, and time). Because of this simplification, gyrokinetics governs the evolution of charged rings with a [[guiding center]] position, instead of gyrating charged particles.

			== Derivation of the gyrokinetic equation ==

			Fundamentally, the gyrokinetic model assumes the plasma is strongly magnetized ( <math>\rho_{i} \ll L_{plasma}</math> ), the perpendicular spatial scales are comparable to the gyroradius ( <math>k_{\perp} \rho_{i} \sim 1</math> ), and the behavior of interest has low frequencies ( <math>\omega \ll \Omega_{i} \ll \Omega_{e}</math> ). We must also expand the [[distribution function]], <math>f_{s} = f_{s 0} + f_{s 1} + \ldots</math>, and assume the perturbation is small compared to the background (<math>f_{s 1} \ll f_{s 0}</math>).<ref>G.G. Howes, S.C. Cowley, W. Dorland, G.W. Hammett, E. Quataert, and A.A. Schekochihin. Astrophysical gyrokinetics: Basic equations and linear theory. ApJ, 651(1):590, 2006.</ref> The starting point is the [[Fokker–Planck equation]] and [[Maxwell's equations]]. The first step is to change spatial variables from the particle position <math>\vec{r}</math> to the guiding center position <math>\vec{R}</math>. Then, we change velocity coordinates from <math>(v_{x},v_{y},v_{z})</math> to the velocity parallel <math>v_{\|\|} \equiv \vec{v} \cdot \hat{b}</math>, the [[magnetic moment]] <math>\mu \equiv \frac{m_{s} v_{\perp}^{2}}{2 B}</math>, and the gyrophase angle <math>\varphi</math>. Here parallel and perpendicular are relative to <math>\vec{b} \equiv \vec{B}/B</math>, the direction of the magnetic field, and <math>m_{s}</math> is the mass of the particle. Now, we can average over the gyrophase angle at constant guiding center position, denoted by <math>\left\langle \ldots \right\rangle_{\varphi}</math>, yielding the gyrokinetic equation.

			The electrostatic gyrokinetic equation, in the absence of large plasma flow, is given by<ref>I. G. Abel, G. G. Plunk, E. Wang, M. Barnes,
			S. C. Cowley, W. Dorland, and A. A. Schekochihin. Multiscale Gyrokinetics for Rotating Tokamak Plasmas: Fluctuations, Transport and Energy Flows. http://arxiv.org/pdf/1209.4782.pdf</ref>

			<math>\frac{\partial h_{s}}{\partial t} + \left( v_{\|\|} \hat{b} + \vec{V}_{d s} + \left\langle \vec{V}_{\phi} \right\rangle_{\varphi} \right) \cdot \vec{\nabla}_{\vec{R}} h_{s} - \sum_{s'} \left\langle C \left[ h_{s}, h_{s'} \right] \right\rangle_{\varphi} = \frac{Z_{s} e f_{s 0}}{T_{s}} \frac{\partial \left\langle \phi \right\rangle_{\varphi}}{\partial t} - \frac{\partial f_{s 0}}{\partial \psi} \left\langle \vec{V}_{\phi} \right\rangle_{\varphi} \cdot \vec{\nabla} \psi </math>.

			Here the first term represents the change in the perturbed distribution function, <math>h_{s} \equiv f_{s 1} + \frac{Z_{s} e \phi}{T_{s}} f_{s 0}</math>, with time. The second term represents particle streaming along the magnetic field line. The third term contains the effects of cross-field particle drifts, including the [[curvature drift]], the [[grad-B drift]], and the lowest order [[Guiding center#Electric field\|E-cross-B drift]]. The fourth term represents the nonlinear effect of the perturbed <math>\vec{E} \times \vec{B}</math> drift interacting with the distribution function perturbation. The fifth term uses a collision operator to include the effects of collisions between particles. The sixth term represents the Maxwell–Boltzmann response to the perturbed electric potential. The last term includes temperature and density gradients of the background distribution function, which drive the perturbation. These gradients are only significant in the direction across flux surfaces, parameterized by <math>\psi</math>, the [[magnetic flux]].

			The gyrokinetic equation, together with gyro-averaged Maxwell's equations, give the distribution function and the perturbed electric and magnetic fields. In the electrostatic case we only require [[Gauss's law]] (which takes the form of the quasineutrality condition), given by<ref>F.I. Parra, M. Barnes, and A.G. Peeters. Up-down symmetry of the turbulent transport of toroidal angular momentum in tokamaks. Phys. Plasmas, 18(6):062501, 2011.</ref>

			<math>\sum_{s} Z_{s} e B \int d v_{\|\|} d\mu d\varphi h_{s} \left(\vec{R}\right) = \sum_{s} \frac{Z_{s}^{2} e^{2} n_{s} \phi}{T_{s}}</math>.

			Usually solutions are found numerically with the help of [[supercomputers]], but in simplified situations analytic solutions are possible.

			== Notes ==
			{{reflist}}

			== References ==

			* J.B. Taylor and R.J. Hastie, Stability of general plasma equilibria - I formal theory. Plasma Phys. 10:479, 1968.
			* P.J. Catto, Linearized gyro-kinetics. Plasma Physics, 20(7):719, 1978.
			* R.G. LittleJohn, Journal of Plasma Physics Vol 29 pp. 111, 1983.
			* J.R. Cary and R.G.Littlejohn, Annals of Physics Vol 151, 1983.
			* T.S. Hahm, Physics of Fluids Vol 31 pp. 2670, 1988.
			* A.J. Brizard and T.S. Hahm, Foundations of Nonlinear Gyrokinetic Theory, Rev. Modern Physics 79, PPPL-4153, 2006.

			==External links==
			* [http://gyrokinetics.sourceforge.net/gs2_documentation/ GS2:] A numerical continuum code for the study of [[turbulence]] in [[nuclear fusion\|fusion]] plasmas.
			* [http://www.physics.uiowa.edu/~ghowes/astrogk/ AstroGK:] A code based on GS2 (above) for studying turbulence in [[astrophysics\|astrophysical]] plasmas.
			* [http://www.ipp.mpg.de/~fsj/gene/ GENE:] A semi-global continuum turbulence simulation code, for fusion plasmas.
			* [http://cips.colorado.edu/simulation/index.htm GEM:] A particle in cell turbulence code, for fusion plasmas.
			* [http://gkw.googlecode.com GKW:] A semi-global continuum gyrokinetic code, for turbulence in fusion plasmas.
			* [https://fusion.gat.com/theory/Gyrooverview GYRO:] A semi-global continuum turbulence code, for fusion plasmas.
			* [http://www-math.u-strasbg.fr/ae_fusion/spip.php?article116 GYSELA:] A semi-lagrangian code, for turbulence in fusion plasmas.
			* [http://physics.aalto.fi/groups/fusion/research/elmfire/ ELMFIRE:] Particle in cell monte-carlo code, for fusion plasmas.
			* [http://epsppd.epfl.ch/Warsaw/pdf/P4_040.pdf GT5D]: A global continuum code, for turbulence in fusion plasmas.
			* [http://epsppd.epfl.ch/Sofia/pdf/O2_005.pdf ORB5] (Or NEMORB): Global particle in cell code, for turbulence in fusion plasmas.
			* [http://www.ipp.mpg.de/~bds/presentations/ (d)FEFI]: Homepage for the author of continuum gyrokinetic codes, for turbulence in fusion plasmas.
			* [http://airex.tksc.jaxa.jp/pl/dr/AA0064635000 GKV]: A local continuum gyrokinetic code, for turbulence in fusion plasmas.
			* [http://phoenix.ps.uci.edu/gtc/ GTC]: A global gyrokinetic particle in cell simulation for fusion plasmas in toroidal and cylindrical geometries.

			===See also ===
			* [[GYRO]]
			* [[Gyrokinetic ElectroMagnetic]]
			* [[List of plasma (physics) articles]]

			[[Category:Kinetics]]
			[[Category:Plasma physics]]
			[[Category:Theoretical physics]]

en>Quondum: Stokes theorem → Stokes' theorem

2012-09-01T19:43:23Z

Stokes theorem → Stokes' theorem

New page

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