Geometric calculus: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Chessfan
en>Quondum
→Differentiation: removing apparently nonsensical statement: "Unlike the exterior product, the exterior derivative is not even associative."
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
'''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]].
Video games are not just for kids! They offer a variety of games that can work for people of all ages. It is no wonder that modern pop culture is so fascinated with video gaming as a whole. This article will provide you with some great tips, tricks and even cheats to help enhance your gaming experience.<br><br>
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 ==
If you are having trouble seeing important details in your video game, adjust the brightness. Video game designers often want to set a mood in their games, but that mood comes with the price that important puzzle pieces may be missed in the dimness. Take back the control a bit by boosting that brightness in the setting's area.<br><br>Some video games can be educational. Consider these educational titles for children, and avoid those with questionable content. Look on the Internet for reviews from parents whose children have played the games that you are considering.<br><br>Modern video games are not without their dangers, and playing them obsessively might be the one to worry about most. If you know that a friend or family member is a gamer, be on the lookout for signs that they're spending an immoderate amount of time playing. If they're having trouble meeting their obligations at school, work, or in their social circle, it may be time to confront them about how they manage their gaming time.<br><br>Remember to take [http://Answers.Yahoo.com/search/search_result?p=regular+breaks&submit-go=Search+Y!+Answers regular breaks] when playing video games. When you are playing video games for extended periods of time without any breaks, your eyes can start to dry out and become sore. Therefore, it is important to take a 15 minute break from video games at least once every 2 hours.<br><br>Trade in your old games at a video game store. You may not know this, but you can trade your old games to the video game store, and you can get cash or credit toward new games. Check with a few different stores so you can get the best deal on your games, though.<br><br>Make sure that you do not let video games take over your life. Video games can be very additive, so you need to make sure that you moderate the time that you spending playing such games. If you spend too much time playing video game, your real life could start to fall apart.<br><br>Keep posture in mind while playing your games. While sitting at a video game, you might want to use a pillow or other device to support your spine. If you really get into your action games, you might find yourself sitting for hours on end. Try to schedule regular breaks to get up and move around.<br><br>Preorder specials offer a great way to get an effective video game buy. Sometimes, preorders are done at discounts to that video game publishers can have bigger opening day numbers and buzz. More often, premium additions or extra content is thrown in at no extra charge so you have more game to enjoy.<br><br>If you're a parent of a child who plays video games, you need to monitor them. Look at things like how long they are playing and what type of games they are playing. You need to know exactly what they are doing when they switch on that console or computer.<br><br>Consider games before you consider the system. Before you go out and spend hundreds of dollars on a gaming system, think about what kind of games are available on each system. While many games have a version for each system, there are still some games that are exclusive to an individual system. Choose wisely.<br><br>Online multiplayer games offer you the opportunity to interact, play, and team up with people all around the world. Keep this in mind when you're getting to know other players. Don't assume that they're in the same time zone, country, or age bracket as yourself. If they're not, don't snub them; you can learn some interesting things about other cultures by playing games with strangers.<br><br>There are many ways you can approach the world of video games and this article provided you with some of them. Use what you have learned today for a whole new approach to your favorite games or game store. There is no shame in looking to other place for some help time to time.<br><br>Here's more information on [http://whatisvideogameaddiction.com/video-games/how-to-find-the-best-video-game-prices/ video game addiction] check out our page.
 
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.&nbsp;111, 1983.
* J.R. Cary and R.G.Littlejohn, Annals of Physics Vol 151, 1983.
* T.S. Hahm, Physics of Fluids Vol 31 pp.&nbsp;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]]

Latest revision as of 17:06, 8 September 2014

Video games are not just for kids! They offer a variety of games that can work for people of all ages. It is no wonder that modern pop culture is so fascinated with video gaming as a whole. This article will provide you with some great tips, tricks and even cheats to help enhance your gaming experience.

If you are having trouble seeing important details in your video game, adjust the brightness. Video game designers often want to set a mood in their games, but that mood comes with the price that important puzzle pieces may be missed in the dimness. Take back the control a bit by boosting that brightness in the setting's area.

Some video games can be educational. Consider these educational titles for children, and avoid those with questionable content. Look on the Internet for reviews from parents whose children have played the games that you are considering.

Modern video games are not without their dangers, and playing them obsessively might be the one to worry about most. If you know that a friend or family member is a gamer, be on the lookout for signs that they're spending an immoderate amount of time playing. If they're having trouble meeting their obligations at school, work, or in their social circle, it may be time to confront them about how they manage their gaming time.

Remember to take regular breaks when playing video games. When you are playing video games for extended periods of time without any breaks, your eyes can start to dry out and become sore. Therefore, it is important to take a 15 minute break from video games at least once every 2 hours.

Trade in your old games at a video game store. You may not know this, but you can trade your old games to the video game store, and you can get cash or credit toward new games. Check with a few different stores so you can get the best deal on your games, though.

Make sure that you do not let video games take over your life. Video games can be very additive, so you need to make sure that you moderate the time that you spending playing such games. If you spend too much time playing video game, your real life could start to fall apart.

Keep posture in mind while playing your games. While sitting at a video game, you might want to use a pillow or other device to support your spine. If you really get into your action games, you might find yourself sitting for hours on end. Try to schedule regular breaks to get up and move around.

Preorder specials offer a great way to get an effective video game buy. Sometimes, preorders are done at discounts to that video game publishers can have bigger opening day numbers and buzz. More often, premium additions or extra content is thrown in at no extra charge so you have more game to enjoy.

If you're a parent of a child who plays video games, you need to monitor them. Look at things like how long they are playing and what type of games they are playing. You need to know exactly what they are doing when they switch on that console or computer.

Consider games before you consider the system. Before you go out and spend hundreds of dollars on a gaming system, think about what kind of games are available on each system. While many games have a version for each system, there are still some games that are exclusive to an individual system. Choose wisely.

Online multiplayer games offer you the opportunity to interact, play, and team up with people all around the world. Keep this in mind when you're getting to know other players. Don't assume that they're in the same time zone, country, or age bracket as yourself. If they're not, don't snub them; you can learn some interesting things about other cultures by playing games with strangers.

There are many ways you can approach the world of video games and this article provided you with some of them. Use what you have learned today for a whole new approach to your favorite games or game store. There is no shame in looking to other place for some help time to time.

Here's more information on video game addiction check out our page.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Geometric_calculus&oldid=262020"