Gyrokinetic Field Theory without Lie Transforms
|
|
- Sophia Stewart
- 5 years ago
- Views:
Transcription
1 ASDEX Upgrade Gyrokinetic Field Theory without Lie Transforms for realisable tokamak codes B. Scott Max Planck Institut für Plasmaphysik Euratom Association D Garching, Germany Lectures in Gyrokinetic Theory,
2 Outline of Gyrokinetics gyrokinetic ordering dynamics slower than gyrofrequency scale of background larger than gyroradius energy of field perturbation smaller than thermal plasma as a distribution of gyrocenter particles guiding center versus gyrocenter: representation, not approximation Principle of Least Action for both particles and fields particles have Lagrangian, fields have Lagrangian density approximations in Lagrangian preservation of exact energetic consistency energy theorem from time variations Noether s Theorem momentum theorem from angle variations Noether s Theorem symmetry principles preserve rigor consistent model guaranteed
3 Gyrokinetic Theory as a Gauge Transform not an orbit average over equations, but a set of operations on a Lagrangian L basics of a symplectic part and Hamiltonian Ldt = p dz Hdt for 6D phase-space coordinates z and time t this procedure closely follows Littlejohn s drift kinetic approach (JPP 1983) the mechanics involving flows is that introduced by Brizard (Phys Plasmas 1995) in our case, no separation between equil or dynamical ExB flow (hence u 0 v E ) maintain original gyrokinetic strategy: preserve canonical form all dependence on dynamical fields is moved to the time component results in all terms due to φ and A appearing only in H correspondence at large-scale small-flow to previous models capture of reduced MHD and tokamak equilibrium Lie-transform version is in Miyato et al, J Phys Soc Japan 78 (2009) this version recovers all those terms except 2nd order in ρ/r small by L R
4 Landau Lifshitz Lagrangian in the conventional form we assume particle positions x and t in 4-space, nonrelativistic conditions fields which depend on x and t, with evolution to be considered later construct the Lagrangian in the familiar way (linear interactions) L = m 2 ẋ ẋ+ e c A ẋ eφ change this to phase space using the Legendre transformation p L/ ẋ then H p ẋ L so that L = p ẋ H turn it into a fundamental one-form in canonical representation Ldt = p dx Hdt where H = m U2 2 +eφ mu = p e c A
5 you can get the Lorentz force equation of motion with this but it s easier to do it in a non-canonical representation by defining v Ldt = ( e c A+mv ) dx Hdt where H = m v2 2 +eφ vary x and v independently to find δx [ e c F ẋ ( m v+ e c ) A t H x ] = 0 δv ( mẋ H ) v = 0 these are solved to find in conventional form ẋ = v m v = E+ 1 c F v = E+ 1 c v B where E = 1 c A t φ with rank-3 Levi-Civita pseudotensor ǫ F = A ( A)T = ǫ B
6 low-frequency low-beta Kinetic Lagrangian we assume φ is a dynamical field but A evolves through small, shear-alfven disturbances A b we now assume A, b, and B are static functions of position they describe the background geometry the dynamical fields are now solely φ and A additionally, we assume gyromotion is fast, so that ExB vorticity is small this leads to c/e equivalently mc/e as a formal small parameter for expansion before any redefinition we will then start with ( e Ldt = c A+ e ) c A b+mv dx Hdt H = m v2 2 +eφ with v recast in terms of parallel streaming, gyromotion, and drift given by v = mv b+mw+mu 0
7 Canonical Form using A to anchor drifts, we won t be using canonical variables the gyrokinetic Lagrangian therefore represents a non-canonical transformation however, we do want canonical form, which means that the whole Lagrangian except for H is static depending on geometry, coordinates, and constants only the resulting phase-space Jacobian keeps time- and geometric symmetry there are no extra / t terms on fields in the kinetic equation we get canonical form using the gauge freedom of the transformation generating functions of the coordinate changes (ie, representation) additional freedom to add a gauge term (pure differential in the one-form) all time (and nonsymmetric geometry) dependence involving fields is moved into H and out of the symplectic part of L
8 parallel phase space variables the first step is the treatment of A no need to gyroaverage: A evolves through electrons and m e M i hence we re-define p z mv + e c A and then we have Ldt = ( e c A+p zb+mw+mu 0 ) dx Hdt H = 1 2m ( p z e ) 2 c A m + 2 w+u 0 2 +eφ note how A is moved out of the symplectic part and into H parallel phase space plane: p z and a spatial coordinate following b before treating gyro-drift motion, this establishes canonical form with respect to A
9 gyrokinetic representation flows (gradients of φ) will enter through u 0 drift kinetic: treat w but leave u 0 whose / t represents the polarisation drift gyro kinetic: treat w+u 0 together strictly moves φ into H maintaining canonical form polarisation enters as a density (field equation for φ) not a drift as we will show after doing the field theory, the representations are equivalent same expression for J = 0 gyrokinetic refers to the representation, not the FLR effects zero-flr with polarisation is still gyrokinetic specifically, the presence of a gyrocenter-charge Poisson equation for φ under quasineutrality = gyrokinetic representation it won t become obvious until we get the self-consistent field equations
10 expansion going from particles to gyrocenters choose gyrocenter positions R = x a assume a is smaller than the scale on which A,b,B varies maximal ordering on velocities: all components enter with A at O(a) hence to zeroth order we have the part with A and φ with dependence on R only L 0 dt = e c A(R) dr eφ(r)dt solve this to find lowest order drift recall F = A ( A) T = ǫ B Ṙ 0 = c B 2 φ F = u 0 = c B 2 φ F main step: setting this equal to u 0 in subsequent orders henceforth: gradients and spatial dependence in terms of R understood
11 next two orders we write L out order by order (0 then 1 then2) Ldt = e c A dr eφdt + e ( e ) c A da+ c a A+p zb+mw+mu 0 dr (ea φ+m U2 2 + m ) 2 w+u 0 2 ( e ) + c a A+p zb+mw+mu 0 da e 2 aa: φdt dt first step: add d(a A) and choose a to cancel all the da and dr terms except p z at first order second step: follow the consequences through second order define the meaning of fast gyromotion drop some terms using generally du 0 dw viz. / t Ω = eb/mc
12 first order subtracting d(a A) this cancels the da term leaving as our choice use d(a A) = A da+a (dr A), then dot with F and impose b a = 0 e c a F+m(w+u 0) = 0 = a = mc eb 2F (w+u 0) this is a directed gyro-drift radius which includes lowest order ExB motion also find a φ = mu 0 (w+u 0 ) this leaves the Lagrangian through first order as ( e ) L 0,1 dt = c A+p zb dr H 0,1 dt mu = p z e c A H 0,1 = m U2 2 +mw2 2 +eφ mu2 E 2 u 2 E = c2 B 2 φ 2 = u 2 0
13 why second order this is almost good enough however, we haven t specified w yet (still need w 2 in H) moreover, to get FLR and to have a well defined theory we specify gyromotion gyromotion we set up an auxiliary basis e 1,2 for the plane perp to b introduce the gyrophase angle ϑ and relate da to w due to large Ω the fast part of da is due solely to d(b w) contributions due to u 0 are down an order contributions due to e 1,2 give gyrophase invariance signs: sense of coordinate system is e 1 e 2 b = 1 with b out of paper sense of motion dϑ is clockwise for ions
14 coordinates for gyromotion these are often called guiding center coordinates in our case the gyromotion enters only at last order so we use simplest approximations express motion (Ω dt) as a geometric circle (dϑ) where we identify w as the directed gyration velocity and ϑ the gyrophase angle w = w(e 1 sinϑ+e 2 cosϑ) the e are two arbitrary vectors forming a right-hand basis e 1 e 2 = 0 (e 1,e 2 ) b = 0 e 1 e 2 b = 1 the sense of the fast gyromotion component is signed with e w da = w2 Ω (dϑ dr e 1 e 2 ) we will identify µ with the conserved quantity multiplying dϑ in the end
15 detail on gyromotion term we will have (w+u 0 ) da with (e/c)a F+m(w+u 0 ) = 0 large Ω: keep only the dw part of da so that da Ω 1 b dw the u 0 da piece averages to zero this leaves w b dw which is then worked through as dw w b w = w(e 1 sinϑ+e 2 cosϑ) dw = w(e 1 cosϑ e 2 sinϑ) dϑ wdr ( e 1 sinϑ+ e 2 cosϑ) w b = w(e 2 sinϑ e 1 cosϑ) finally we use e 1 e 2 + e 2 e 1 = 0 to express it as dw w b = w 2 (dϑ dr e 1 e 2 )
16 write second order line again second order ( e ) L 2 dt = c a A+p zb+mw+mu 0 da e 2 aa: φdt first step: subtract d(a A a) with 1/2 to symmetrise form use b da either zero or higher order L 2 dt+ds 2 = ( 1 2 ) e c a F+mw+mu 0 definition of a through a F above combines da e 2 aa: φdt L 2 dt+ds 2 = 1 2 m(w+u 0) da e 2 aa: φdt use gyro-drift motion approximation on the da in the first term L 2 dt+ds 2 = 1 2 mw 2 Ω (dϑ dr e 1 e 2 ) e 2 aa: φdt
17 second order (2) we have L 2 dt+ds 2 = 1 2 mw 2 Ω (dϑ dr e 1 e 2 ) e 2 aa: φdt average the gradient components on the φ term over direction of a L 2 dt+ds 2 = 1 2 mw 2 Ω (dϑ dr e 1 e 2 ) e a2 4 2 φdt now identify the magnetic moment (in the right units) µ = 1 2 mw 2 B L 2 dt+ds 2 = mc e µ(dϑ dr e 1 e 2 ) e a2 4 2 φdt and in the first order piece m w2 2 = µb
18 gyrophase invariance the piece due to e 1,2 is small but formally important the dϑ piece is not gyrophase invariant by itself L gy = mc e µ(dϑ W dr) where W = e 1 e 2 if ϑ ϑ+α(r) then the combination dϑ W dr is invariant follow it through with dα = dr α and the dependence of e 1,2 on ϑ in practice gyromotion drops out of the kinetic equation anyway the µw piece is a small (a/l B ) 2 correction to the (a/l B ) drifts no numerical simulation at present day keeps it
19 gyrokinetic Langrangian we have the definition of µ hence w 2 and also a 2 so put them in Ldt = ( e c A+p zb mc e µw ) dr+ mc e µdϑ Hdt with Hamiltonian H = m U2 2 +µb + ) (1+ a2 4 2 eφ m u2 E 2 where mu = p z e c A u 2 E = c2 B 2 φ 2 a 2 = 2µB +mu2 E mω 2 this can be shown to be a low-k and low-β version of the result of Hahm et al, Phys Fluids 31 (1988) 1940 for the field theory, the particle Lagrangian version of the gyrocenter one-form is L p = ( e c A+p zb mc e µw ) Ṙ+ mc e µ ϑ H
20 gyrokinetic field Langrangian the field theory embeds this into a phase space L = dλfl p + sp dvl f where for shear Alfven conditions the field Lagrangian density is L f = E2 B 2 8π L f = 1 8πR 2 ( ψ +A R ) 2 where R is the toroidal major radius and the equilibrium B is with I = constant, and we also use B = A = I ϕ+ ψ ϕ b R ϕ B I R
21 gyrokinetic field Langrangian (2) with these approximations and the neglect of W... the system Lagrangian is L = sp dλfl p + dvl f where keeping A but using b R ϕ viz. reduced conditions for the MHD part ( e ) L p = c A+p zr ϕ mc Ṙ+ e µ ϑ H L f = 1 ( 8πR 2 ψ +A R ) 2 H = m U2 2 +µb + ) (1+ a2 4 2 eφ m u2 E 2 mu = p z e c A u 2 E = c2 B 2 φ 2 a 2 = 2µB +mu2 E mω 2 this is the basis for my total-f tokamak models
22 Euler-Lagrange equations for gyrocenters these arise from varying R and z in L p, noting that is w.r.t. R holding z constant take the variation for general canonical-structure L p δr [ e cṙ B e c A ] z ż H δz [ e c A z Ṙ H z ] set these to zero and solve ( e A ) c z B ż = B H then def B e c A z B e c A z (Ṙ B ) = A z H then B Ṙ B ( e c A ) z Ṙ = A z H so that with the definition of B = A and above for B B Ṙ = A z H + H z B B ż = B H
23 Euler-Lagrange equations for gyrocenters (2) in the conventional (electrostatic) case we have z = mv with e c A = e c A+mv b e c A z = b B = b B H = m v2 2 +µb +ej 0φ A with which we rewrite the Euler-Lagrange equations as z H = c e b H B Ṙ = c e b H +v B B ( m v ) = B H geometric quantities and gradients of H and definitions B = B+ mc e v b B = B B = B + mc e v b b H z = v H = e (J 0 φ)+µ B
24 Euler-Lagrange equations for gyrocenters (3) in our case we have z = p z and b R ϕ and B I/R with e c A = e c A+p zr ϕ e c A z = R ϕ therefore B = B+ c e p z R ϕ B = B with which we rewrite the Euler-Lagrange equations as B Ṙ = H c e F B + H z B B ż = B H geometric quantities and gradients of H and definitions F = ǫ (I ϕ) eφ E = eφ m u2 E 2 H z = U ( 1 Ω ) E 2Ω H = e φ E +µ B E e c U A B E = B ( 1+ Ω ) E 2Ω Ω E = c B 2 φ
25 Gyrokinetic/Field Equation System embed this using Liouville theorem B f t + H c e F B f +B ( H z ) f f z H = 0 field equations, Euler-Lagrange equations for φ and A [ ne+ 2 P E + N E φ ] = 0 R 2 sp 1 ( R 2 ψ +A R ) = 4π c J R with N E = nmc2 B 2 ( 1 Ω ) E 2Ω P E = mc2 2eB 2 ( ) p +nm u2 E 2 where the moment quantities are n = dwf p = dw µbf J = sp dw eu f
26 Correspondence to Standard Forms our gyrocenter Lagrangian in z = p z representation L p = ( e c A+p zb mc e µw ) H = m U2 2 +µb + ) (1+ a2 4 2 mc Ṙ+ e µ ϑ H eφ m u2 E 2 mu = p z e c A u 2 E = c2 B 2 φ 2 a 2 = 2µB +mu2 E mω 2 neglect A so mu = p z, neglect W, neglect mu 2 E 2µB in FLR, and restore J 0 ( e ) L p = c A+p zb mc Ṙ+ e µ ϑ H H = m U2 2 +µb +ej 0(φ) m u2 E 2 this recovers the push potential (Eq. 2) of WW Lee J Comput Phys 72 (1987) 243
27 Correspondence to Standard Forms (2) start with Eq. (16) TS Hahm Phys Fluids 31 (1988) 2670 M = mc e µ L p = ( e c A+mUb ) Ψ = φ e 2Ω Ω = eb mc Ṙ+M ϑ ( M ) (m U2 2 +MΩ+eΨ φ2 + 1 ) S b φ Ω φ = φ φ switch to µ and p z = mu, go to low-k with J 0 neglect last term in Ψ (it contributes O δ to polarisation) ( e ) L p = c A+p zb mc Ṙ+ e µ ϑ [ p 2 z 2m +µb + S θ = φ = J 0 () ( 1+ µb ) ] 2mΩ 2 2 eφ m u2 E 2 this recovers our form with A, ρ 2 E, and W neglected
28 Correspondence to Reduced MHD in the Lagrangian neglect FLR, W, and use z = p z R L p = ( e c A+z ϕ ) mc Ṙ+ e µ ϑ H L f = 1 ( 8πR 2 ψ +A R ) 2 H = m U2 2 +µb +eφ mu2 E 2 u 2 E = c2 B 2 φ 2 mur = z e c A R B = B R = I R 2 = Bϕ B = B = A b = R ϕ important derivatives (of H, for the GK equation, of U, for the Ohm s law) H = e φ ( µb +mu 2) logr e c U R (A R) H z = U R U t = e mc A t time derivative of the polarisation (Poisson) equation gives the vorticity equation time derivative of the induction (Ampère) equation gives the Ohm s law
29 detail vorticity equation start with the polarisation equation, take / t t t ( ρm c 2 ) B 2 φ = sp dw e f t ρ M = sp dw mf use divergence form of the GK equation (note / z gets annihilated) t = sp dw 1 B [ ( H cfb f ) + ( euf ) R B z ] (efb H) in conventional notation (A terms and B give B, and φ terms give v E ) t + ( v E)+O(φ 2 J ) = B B c B b logr2 sp dw mu2 +µb 2 f
30 detail Ohm s law start with the Ampère s law (induction equation), take / t t ( ψ +A R ) = 4π c R sp dw ( eu f t +ef U t ) use ψ/ t = 0 and U/ t = (e/mc)( A / t) and bring to left side ( ω 2 p c 2 ) t ( ψ +A R ) = sp dw 4π c ReU f t ω 2 p = sp dw 4πe2 f m neglect all mass ratio corrections, finite c 2 /ω 2 p, find two-fluid Ohm s law 1 c A t = φ+ 1 n e e P e P e = dw (mu 2 f) e neglect two-fluid effect (p e ), add collisions, to find resistive Ohm s law
31 detail MHD equilibrium equilibrium state of the vorticity equation t + ( v E)+O(φ 2 J ) = B B c B b logr2 sp dw mu2 +µb 2 f no flow, no A, so B B and P p and 0 B J B = cr B ϕ logr2 p put in Ampère s law (induction equation), use p = p(ψ) and B = I/R B ( ψ) = 4π IR B ϕ logr2 p = 4π p ψ ψ ϕ R2 this is the Grad-Shafranov equilibrium along field lines B ( flux function+ ψ +4πR 2 p ) ψ = 0
32 Gyrokinetic Theory as a Gauge Transform not an orbit average over equations, but a set of operations on a Lagrangian L basics of a symplectic part and Hamiltonian Ldt = p dz Hdt for 6D phase-space coordinates z and time t this procedure closely follows Littlejohn s drift kinetic approach (JPP 1983) the mechanics involving flows is that introduced by Brizard (Phys Plasmas 1995) in our case, no separation between equil or dynamical ExB flow (hence u 0 v E ) maintain original gyrokinetic strategy: preserve canonical form all dependence on dynamical fields is moved to the time component results in all terms due to φ and A appearing only in H correspondence at large-scale small-flow to previous models capture of reduced MHD and tokamak equilibrium Lie-transform version is in Miyato et al, J Phys Soc Japan 78 (2009) energy/momentum theorems/consistency is in Phys Plasmas 17 (2010)
Total-f Gyrokinetic Energetics and the Status of the FEFI Code
ASDEX Upgrade Total-f Gyrokinetic Energetics and the Status of the FEFI Code B. Scott Max Planck Institut für Plasmaphysik Euratom Association D-85748 Garching, Germany JAEA Rokkasho, Feb 2015 Outline
More informationTurbulence in Tokamak Plasmas
ASDEX Upgrade Turbulence in Tokamak Plasmas basic properties and typical results B. Scott Max Planck Institut für Plasmaphysik Euratom Association D-85748 Garching, Germany Uni Innsbruck, Nov 2011 Basics
More informationGyrokinetic simulations of magnetic fusion plasmas
Gyrokinetic simulations of magnetic fusion plasmas Tutorial 2 Virginie Grandgirard CEA/DSM/IRFM, Association Euratom-CEA, Cadarache, 13108 St Paul-lez-Durance, France. email: virginie.grandgirard@cea.fr
More informationMonte-Carlo finite elements gyrokinetic simulations of Alfven modes in tokamaks.
Monte-Carlo finite elements gyrokinetic simulations of Alfven modes in tokamaks. A. Bottino 1 and the ORB5 team 1,2,3 1 Max Planck Institute for Plasma Physics, Germany, 2 Swiss Plasma Center, Switzerland,
More informationParticle flux representations with FLR effects in the gyrokinetic model
Particle flux representations with FLR effects in the gyrokinetic model ジャイロ運動論モデルにおける有限ラーマ 半径効果を含む粒子フラックスの表現 N. Miyato 1), M. Yagi 1), B. Scott 2) 1) Japan Atomic Energy Agency 2) Max Planck Institut
More informationGeometric Gyrokinetic Theory and its Applications to Large-Scale Simulations of Magnetized Plasmas
Geometric Gyrokinetic Theory and its Applications to Large-Scale Simulations of Magnetized Plasmas Hong Qin Princeton Plasma Physics Laboratory, Princeton University CEA-EDF-INRIA School -- Numerical models
More informationGyrokinetic Theory and Dynamics of the Tokamak Edge
ASDEX Upgrade Gyrokinetic Theory and Dynamics of the Tokamak Edge B. Scott Max Planck Institut für Plasmaphysik D-85748 Garching, Germany PET-15, Sep 2015 these slides: basic processes in the dynamics
More informationOverview of Gyrokinetic Theory & Properties of ITG/TEM Instabilities
Overview of Gyrokinetic Theory & Properties of ITG/TEM Instabilities G. W. Hammett Princeton Plasma Physics Lab (PPPL) http://w3.pppl.gov/~hammett AST559: Plasma & Fluid Turbulence Dec. 5, 2011 (based
More informationLectures on basic plasma physics: Hamiltonian mechanics of charged particle motion
Lectures on basic plasma physics: Hamiltonian mechanics of charged particle motion Department of applied physics, Aalto University March 8, 2016 Hamiltonian versus Newtonian mechanics Newtonian mechanics:
More informationarxiv: v1 [physics.plasm-ph] 3 Apr 2011
A comparison of Vlasov with drift kinetic and gyrokinetic theories arxiv:1104.0427v1 [physics.plasm-ph] 3 Apr 2011 H. Tasso 1, G. N. Throumoulopoulos 2 1 Max-Planck-Institut für Plasmaphysik, Euratom Association,
More informationSummer College on Plasma Physics August Introduction to Nonlinear Gyrokinetic Theory
2052-24 Summer College on Plasma Physics 10-28 August 2009 Introduction to Nonlinear Gyrokinetic Theory T.S. Hahm Princeton Plasma Physics Laboratory Princeton University USA Introduction to Nonlinear
More informationMotion of Charged Particles in Fields
Chapter Motion of Charged Particles in Fields Plasmas are complicated because motions of electrons and ions are determined by the electric and magnetic fields but also change the fields by the currents
More informationTokamak Edge Turbulence background theory and computation
ASDEX Upgrade Tokamak Edge Turbulence background theory and computation B. Scott Max Planck Institut für Plasmaphysik Euratom Association D-85748 Garching, Germany Krakow, Sep 2006 Outline Basic Concepts
More informationGlobal particle-in-cell simulations of Alfvénic modes
Global particle-in-cell simulations of Alfvénic modes A. Mishchenko, R. Hatzky and A. Könies Max-Planck-Institut für Plasmaphysik, EURATOM-Association, D-749 Greifswald, Germany Rechenzentrum der Max-Planck-Gesellschaft
More informationElectric and Magnetic Forces in Lagrangian and Hamiltonian Formalism
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger
More informationTURBULENT TRANSPORT THEORY
ASDEX Upgrade Max-Planck-Institut für Plasmaphysik TURBULENT TRANSPORT THEORY C. Angioni GYRO, J. Candy and R.E. Waltz, GA The problem of Transport Transport is the physics subject which studies the physical
More informationPenning Traps. Contents. Plasma Physics Penning Traps AJW August 16, Introduction. Clasical picture. Radiation Damping.
Penning Traps Contents Introduction Clasical picture Radiation Damping Number density B and E fields used to increase time that an electron remains within a discharge: Penning, 936. Can now trap a particle
More informationKinetic theory of ions in the magnetic presheath
Kinetic theory of ions in the magnetic presheath Alessandro Geraldini 1,2, Felix I. Parra 1,2, Fulvio Militello 2 1. Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, Oxford
More informationAMSC 663 Project Proposal: Upgrade to the GSP Gyrokinetic Code
AMSC 663 Project Proposal: Upgrade to the GSP Gyrokinetic Code George Wilkie (gwilkie@umd.edu) Supervisor: William Dorland (bdorland@umd.edu) October 11, 2011 Abstract Simulations of turbulent plasma in
More informationFluid equations, magnetohydrodynamics
Fluid equations, magnetohydrodynamics Multi-fluid theory Equation of state Single-fluid theory Generalised Ohm s law Magnetic tension and plasma beta Stationarity and equilibria Validity of magnetohydrodynamics
More informationLecture 16 March 29, 2010
Lecture 16 March 29, 2010 We know Maxwell s equations the Lorentz force. Why more theory? Newton = = Hamiltonian = Quantum Mechanics Elegance! Beauty! Gauge Fields = Non-Abelian Gauge Theory = Stard Model
More informationSingle Particle Motion
Single Particle Motion C ontents Uniform E and B E = - guiding centers Definition of guiding center E gravitation Non Uniform B 'grad B' drift, B B Curvature drift Grad -B drift, B B invariance of µ. Magnetic
More informationImplementation of Guiding Center Rotation Drifts in Simulations of Tokamak Plasmas with Large Flows
University of Colorado, Boulder CU Scholar Undergraduate Honors Theses Honors Program Spring 2016 Implementation of Guiding Center Rotation Drifts in Simulations of Tokamak Plasmas with Large Flows Nikola
More informationWhat place for mathematicians in plasma physics
What place for mathematicians in plasma physics Eric Sonnendrücker IRMA Université Louis Pasteur, Strasbourg projet CALVI INRIA Nancy Grand Est 15-19 September 2008 Eric Sonnendrücker (U. Strasbourg) Math
More informationSingle particle motion
Single particle motion Plasma is a collection of a very large number of charged particles moving in, and giving rise to, electromagnetic fields. Before going to the statistical descriptions, let us learn
More information0 Magnetically Confined Plasma
0 Magnetically Confined Plasma 0.1 Particle Motion in Prescribed Fields The equation of motion for species s (= e, i) is written as d v ( s m s dt = q s E + vs B). The motion in a constant magnetic field
More informationEntropy evolution and dissipation in collisionless particle-in-cell gyrokinetic simulations
Max-Planck-Insititut für Plasmaphysik Entropy evolution and dissipation in collisionless particle-in-cell gyrokinetic simulations A. Bottino Objectives Develop a numerical tool able to reproduce and predict
More informationThe Virial Theorem, MHD Equilibria, and Force-Free Fields
The Virial Theorem, MHD Equilibria, and Force-Free Fields Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 10 12, 2014 These lecture notes are largely
More informationGyrokinetics an efficient framework for studying turbulence and reconnection in magnetized plasmas
Frank Jenko Gyrokinetics an efficient framework for studying turbulence and reconnection in magnetized plasmas Max-Planck-Institut für Plasmaphysik, Garching Workshop on Vlasov-Maxwell Kinetics WPI, Vienna,
More informationarxiv: v1 [physics.plasm-ph] 23 Jan 2019
Gauge-free electromagnetic rokinetic theory J. W. Burby,3 and A. J. Brizard 2 Courant Institute of Mathematical Sciences, New York, NY 2, USA 2 Department of Physics, Saint Michael s College, Colchester,
More information2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1. Waves in plasmas. T. Johnson
2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasma physics Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas Transverse waves
More informationM2A2 Problem Sheet 3 - Hamiltonian Mechanics
MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,
More informationPHY 5246: Theoretical Dynamics, Fall September 28 th, 2015 Midterm Exam # 1
Name: SOLUTIONS PHY 5246: Theoretical Dynamics, Fall 2015 September 28 th, 2015 Mierm Exam # 1 Always remember to write full work for what you do. This will help your grade in case of incomplete or wrong
More informationDispersive Media, Lecture 7 - Thomas Johnson 1. Waves in plasmas. T. Johnson
2017-02-14 Dispersive Media, Lecture 7 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasmas as a coupled system Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas
More informationModels for Global Plasma Dynamics
Models for Global Plasma Dynamics F.L. Waelbroeck Institute for Fusion Studies, The University of Texas at Austin International ITER Summer School June 2010 Outline 1 Models for Long-Wavelength Plasma
More informationProblems in Magnetostatics
Problems in Magnetostatics 8th February 27 Some of the later problems are quite challenging. This is characteristic of problems in magnetism. There are trivial problems and there are tough problems. Very
More informationWeek 1, solution to exercise 2
Week 1, solution to exercise 2 I. THE ACTION FOR CLASSICAL ELECTRODYNAMICS A. Maxwell s equations in relativistic form Maxwell s equations in vacuum and in natural units (c = 1) are, E=ρ, B t E=j (inhomogeneous),
More informationEnergy and Equations of Motion
Energy and Equations of Motion V. Tanrıverdi tanriverdivedat@googlemail.com Physics Department, Middle East Technical University, Ankara / TURKEY Abstract. From the total time derivative of energy an equation
More informationMHD Linear Stability Analysis Using a Full Wave Code
US-Japan JIFT Workshop on Progress of Extended MHD Models NIFS, Toki,Japan 2007/03/27 MHD Linear Stability Analysis Using a Full Wave Code T. Akutsu and A. Fukuyama Department of Nuclear Engineering, Kyoto
More informationCharged particle motion in external fields
Chapter 2 Charged particle motion in external fields A (fully ionized) plasma contains a very large number of particles. In general, their motion can only be studied statistically, taking appropriate averages.
More informationSolutions: Homework 5
Ex. 5.1: Capacitor Solutions: Homework 5 (a) Consider a parallel plate capacitor with large circular plates, radius a, a distance d apart, with a d. Choose cylindrical coordinates (r,φ,z) and let the z
More informationTheory and Simulation of Neoclassical Transport Processes, with Local Trapping
Theory and Simulation of Neoclassical Transport Processes, with Local Trapping Daniel H. E. Dubin Department of Physics, University of California at San Diego, La Jolla, CA USA 92093-0319 Abstract. Neoclassical
More informationPlasmas as fluids. S.M.Lea. January 2007
Plasmas as fluids S.M.Lea January 2007 So far we have considered a plasma as a set of non intereacting particles, each following its own path in the electric and magnetic fields. Now we want to consider
More informationSimple examples of MHD equilibria
Department of Physics Seminar. grade: Nuclear engineering Simple examples of MHD equilibria Author: Ingrid Vavtar Mentor: prof. ddr. Tomaž Gyergyek Ljubljana, 017 Summary: In this seminar paper I will
More informationSMR/ Summer College on Plasma Physics. 30 July - 24 August, Introduction to Magnetic Island Theory.
SMR/1856-1 2007 Summer College on Plasma Physics 30 July - 24 August, 2007 Introduction to Magnetic Island Theory. R. Fitzpatrick Inst. for Fusion Studies University of Texas at Austin USA Introduction
More informationThe Principle of Least Action
The Principle of Least Action In their never-ending search for general principles, from which various laws of Physics could be derived, physicists, and most notably theoretical physicists, have often made
More informationRelativistic Dynamics
Chapter 4 Relativistic Dynamics The most important example of a relativistic particle moving in a potential is a charged particle, say an electron, moving in an electromagnetic field, which might be that
More informationPlasma waves in the fluid picture I
Plasma waves in the fluid picture I Langmuir oscillations and waves Ion-acoustic waves Debye length Ordinary electromagnetic waves General wave equation General dispersion equation Dielectric response
More informationMacroscopic plasma description
Macroscopic plasma description Macroscopic plasma theories are fluid theories at different levels single fluid (magnetohydrodynamics MHD) two-fluid (multifluid, separate equations for electron and ion
More informationGyrokinetic simulations including the centrifugal force in a strongly rotating tokamak plasma
Gyrokinetic simulations including the centrifugal force in a strongly rotating tokamak plasma F.J. Casson, A.G. Peeters, Y. Camenen, W.A. Hornsby, A.P. Snodin, D. Strintzi, G.Szepesi CCFE Turbsim, July
More informationIdeal MHD Equilibria
CapSel Equil - 01 Ideal MHD Equilibria keppens@rijnh.nl steady state ( t = 0) smoothly varying solutions to MHD equations solutions without discontinuities conservative or non-conservative formulation
More informationTurbulent Transport of Toroidal Angular Momentum in Low Flow Gyrokinetics
PSFC/JA-09-27 Turbulent Transport of Toroidal Angular Momentum in Low Flow Gyrokinetics Felix I. Parra and P.J. Catto September 2009 Plasma Science and Fusion Center Massachusetts Institute of Technology
More informationGlobal gyrokinetic particle simulations with kinetic electrons
IOP PUBLISHING Plasma Phys. Control. Fusion 49 (2007) B163 B172 PLASMA PHYSICS AND CONTROLLED FUSION doi:10.1088/0741-3335/49/12b/s15 Global gyrokinetic particle simulations with kinetic electrons Z Lin,
More informationLimitations of gyrokinetics on transport time scales
PSFC/JA-08-16 Limitations of gyrokinetics on transport time scales Parra, F.I. and Catto, P.J February 2008 Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge MA 02139 USA
More informationThe Particle-Field Hamiltonian
The Particle-Field Hamiltonian For a fundamental understanding of the interaction of a particle with the electromagnetic field we need to know the total energy of the system consisting of particle and
More informationGyrokinetic theory for particle transport in fusion plasmas
Gyrokinetic theory for particle transport in fusion plasmas Matteo Valerio Falessi 1,2, Fulvio Zonca 3 1 INFN - Sezione di Roma Tre, Via della Vasca Navale, 84 (00146) Roma (Roma), Italy 2 Dipartimento
More informationGTC Simulation of Turbulence and Transport in Tokamak Plasmas
GTC Simulation of Turbulence and Transport in Tokamak Plasmas Z. Lin University it of California, i Irvine, CA 92697, USA and GPS-TTBP Team Supported by SciDAC GPS-TTBP, GSEP & CPES Motivation First-principles
More informationA.G. PEETERS UNIVERSITY OF BAYREUTH
IN MEMORIAM GRIGORY PEREVERZEV A.G. PEETERS UNIVERSITY OF BAYREUTH ESF Workshop (Garching 2013) Research areas Grigory Pereverzev. Current drive in magnetized plasmas Transport (ASTRA transport code) Wave
More informationCHARGED PARTICLE MOTION IN CONSTANT AND UNIFORM ELECTROMAGNETIC FIELDS
CHARGED PARTICLE MOTION IN CONSTANT AND UNIFORM ELECTROMAGNETIC FIELDS In this and in the following two chapters we investigate the motion of charged particles in the presence of electric and magnetic
More informationAMSC 664 Final Report: Upgrade to the GSP Gyrokinetic Code
AMSC 664 Final Report: Upgrade to the GSP Gyrokinetic Code George Wilkie (gwilkie@umd.edu) Supervisor: William Dorland (bdorland@umd.edu) May 14, 2012 Abstract Simulations of turbulent plasma in a strong
More informationPhysics 5153 Classical Mechanics. Canonical Transformations-1
1 Introduction Physics 5153 Classical Mechanics Canonical Transformations The choice of generalized coordinates used to describe a physical system is completely arbitrary, but the Lagrangian is invariant
More informationFinite-Orbit-Width Effect and the Radial Electric Field in Neoclassical Transport Phenomena
1 TH/P2-18 Finite-Orbit-Width Effect and the Radial Electric Field in Neoclassical Transport Phenomena S. Satake 1), M. Okamoto 1), N. Nakajima 1), H. Sugama 1), M. Yokoyama 1), and C. D. Beidler 2) 1)
More informationLecture II: Vector and Multivariate Calculus
Lecture II: Vector and Multivariate Calculus Dot Product a, b R ' ', a ( b = +,- a + ( b + R. a ( b = a b cos θ. θ convex angle between the vectors. Squared norm of vector: a 3 = a ( a. Alternative notation:
More informationMassachusetts Institute of Technology Department of Physics. Final Examination December 17, 2004
Massachusetts Institute of Technology Department of Physics Course: 8.09 Classical Mechanics Term: Fall 004 Final Examination December 17, 004 Instructions Do not start until you are told to do so. Solve
More informationMHD RELATED TO 2-FLUID THEORY, KINETIC THEORY AND MAGANETIC RECONNECTION
MHD RELATED TO 2-FLUID THEORY, KINETIC THEORY AND MAGANETIC RECONNECTION Marty Goldman University of Colorado Spring 2017 Physics 5150 Issues 2 How is MHD related to 2-fluid theory Level of MHD depends
More informationHamiltonian and Non-Hamiltonian Reductions of Charged Particle Dynamics: Diffusion and Self-Organization
NNP2017 11 th July 2017 Lawrence University Hamiltonian and Non-Hamiltonian Reductions of Charged Particle Dynamics: Diffusion and Self-Organization N. Sato and Z. Yoshida Graduate School of Frontier Sciences
More informationCOMPLETE ALL ROUGH WORKINGS IN THE ANSWER BOOK AND CROSS THROUGH ANY WORK WHICH IS NOT TO BE ASSESSED.
BSc/MSci EXAMINATION PHY-304 Time Allowed: Physical Dynamics 2 hours 30 minutes Date: 28 th May 2009 Time: 10:00 Instructions: Answer ALL questions in section A. Answer ONLY TWO questions from section
More informationThe Accelerator Hamiltonian in a Straight Coordinate System
Hamiltoninan Dynamics for Particle Accelerators, Lecture 2 The Accelerator Hamiltonian in a Straight Coordinate System Andy Wolski University of Liverpool, and the Cockcroft Institute, Daresbury, UK. Given
More informationMicrotearing Simulations in the Madison Symmetric Torus
Microtearing Simulations in the Madison Symmetric Torus D. Carmody, P.W. Terry, M.J. Pueschel - University of Wisconsin - Madison dcarmody@wisc.edu APS DPP 22 Overview PPCD discharges in MST have lower
More informationBRIEF COMMUNICATION. Near-magnetic-axis Geometry of a Closely Quasi-Isodynamic Stellarator. Greifswald, Wendelsteinstr. 1, Greifswald, Germany
BRIEF COMMUNICATION Near-magnetic-axis Geometry of a Closely Quasi-Isodynamic Stellarator M.I. Mikhailov a, J. Nührenberg b, R. Zille b a Russian Research Centre Kurchatov Institute, Moscow,Russia b Max-Planck-Institut
More informationfor changing independent variables. Most simply for a function f(x) the Legendre transformation f(x) B(s) takes the form B(s) = xs f(x) with s = df
Physics 106a, Caltech 1 November, 2018 Lecture 10: Hamiltonian Mechanics I The Hamiltonian In the Hamiltonian formulation of dynamics each second order ODE given by the Euler- Lagrange equation in terms
More informationFundamentals of Magnetic Island Theory in Tokamaks
Fundamentals of Magnetic Island Theory in Tokamaks Richard Fitzpatrick Institute for Fusion Studies University of Texas at Austin Austin, TX, USA Talk available at http://farside.ph.utexas.edu/talks/talks.html
More informationThe gyrokinetic turbulence code GENE - Numerics and applications
Contributors: T. Dannert (1), F. Jenko (1),F. Merz (1), D. Told (1), X. Lapillonne (2), S. Brunner (2), and others T. Görler (1) The gyrokinetic turbulence code GENE - Numerics and applications (1) Max-Planck-Institut
More informationCaltech Ph106 Fall 2001
Caltech h106 Fall 2001 ath for physicists: differential forms Disclaimer: this is a first draft, so a few signs might be off. 1 Basic properties Differential forms come up in various parts of theoretical
More informationIntroduction to Plasma Physics
Introduction to Plasma Physics Hartmut Zohm Max-Planck-Institut für Plasmaphysik 85748 Garching DPG Advanced Physics School The Physics of ITER Bad Honnef, 22.09.2014 A simplistic view on a Fusion Power
More informationPhysical Processes in the Tokamak Edge/Pedestal
ASDEX Upgrade Physical Processes in the Tokamak Edge/Pedestal B. Scott Max Planck Institut für Plasmaphysik Boltzmannstr 2 D-85748 Garching, Germany PSI Summer School, MEPhI, Moscow, July 2016 Magnetic
More informationHybrid Kinetic-MHD simulations with NIMROD
simulations with NIMROD 1 Yasushi Todo 2, Dylan P. Brennan 3, Kwang-Il You 4, Jae-Chun Seol 4 and the NIMROD Team 1 University of Washington, Seattle 2 NIFS, Toki-Japan 3 University of Tulsa 4 NFRI, Daejeon-Korea
More informationOn the Physics of the L/H Transition
ASDEX Upgrade On the Physics of the L/H Transition B. Scott Max Planck Institut für Plasmaphysik Euratom Association D-85748 Garching, Germany EFDA-TTG Workshop, Sep 2010, updated Apr 2012 Outline Physical
More information[variable] = units (or dimension) of variable.
Dimensional Analysis Zoe Wyatt wyatt.zoe@gmail.com with help from Emanuel Malek Understanding units usually makes physics much easier to understand. It also gives a good method of checking if an answer
More informationMHD equilibrium calculations for stellarators. Antoine Cerfon, MIT PSFC with F. Parra, J. Freidberg (MIT NSE)
MHD equilibrium calculations for stellarators Antoine Cerfon, MIT PSFC with F. Parra, J. Freidberg (MIT NSE) March 20, 2012 MAGNETIC FIELD LINE HAMILTONIAN Consider a general toroidal coordinate system
More informationPlasma physics in noninertial frames
Plasma physics in noninertial frames PHYSICS OF PLASMAS 16, 092506 2009 A. Thyagaraja and K. G. McClements EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon OX14 3DB, United Kingdom Received
More information[#1] R 3 bracket for the spherical pendulum
.. Holm Tuesday 11 January 2011 Solutions to MSc Enhanced Coursework for MA16 1 M3/4A16 MSc Enhanced Coursework arryl Holm Solutions Tuesday 11 January 2011 [#1] R 3 bracket for the spherical pendulum
More informationHamiltonian Dynamics In The Theory of Abstraction
Hamiltonian Dynamics In The Theory of Abstraction Subhajit Ganguly. email: gangulysubhajit@indiatimes.com Abstract: This paper deals with fluid flow dynamics which may be Hamiltonian in nature and yet
More informationThe Linear Theory of Tearing Modes in periodic, cyindrical plasmas. Cary Forest University of Wisconsin
The Linear Theory of Tearing Modes in periodic, cyindrical plasmas Cary Forest University of Wisconsin 1 Resistive MHD E + v B = ηj (no energy principle) Role of resistivity No frozen flux, B can tear
More informationThe Hamiltonian formulation of gauge theories
The Hamiltonian formulation of gauge theories I [p, q] = dt p i q i H(p, q) " # q i = @H @p i =[q i, H] ṗ i = @H =[p @q i i, H] 1. Symplectic geometry, Hamilton-Jacobi theory,... 2. The first (general)
More informationDirect drive by cyclotron heating can explain spontaneous rotation in tokamaks
Direct drive by cyclotron heating can explain spontaneous rotation in tokamaks J. W. Van Dam and L.-J. Zheng Institute for Fusion Studies University of Texas at Austin 12th US-EU Transport Task Force Annual
More informationRelativistic magnetohydrodynamics. Abstract
Relativistic magnetohydrodynamics R. D. Hazeltine and S. M. Mahajan Institute for Fusion Studies, The University of Texas, Austin, Texas 78712 (October 19, 2000) Abstract The lowest-order description of
More information20. Alfven waves. ([3], p ; [1], p ; Chen, Sec.4.18, p ) We have considered two types of waves in plasma:
Phys780: Plasma Physics Lecture 20. Alfven Waves. 1 20. Alfven waves ([3], p.233-239; [1], p.202-237; Chen, Sec.4.18, p.136-144) We have considered two types of waves in plasma: 1. electrostatic Langmuir
More informationPhysics 452 Lecture 33: A Particle in an E&M Field
Physics 452 Lecture 33: A Particle in an E&M Field J. Peatross In lectures 31 and 32, we considered the Klein-Gordon equation for a free particle. We would like to add a potential to the equation (since
More informationMagnetic Deflection of Ionized Target Ions
Magnetic Deflection of Ionized Target Ions D. V. Rose, A. E. Robson, J. D. Sethian, D. R. Welch, and R. E. Clark March 3, 005 HAPL Meeting, NRL Solid wall, magnetic deflection 1. Cusp magnetic field imposed
More informationNonlinear hydrid simulations of precessional Fishbone instability
Nonlinear hydrid simulations of precessional Fishbone instability M. Faganello 1, M. Idouakass 1, H. L. Berk 2, X. Garbet 3, S. Benkadda 1 1 Aix-Marseille University, CNRS, PIIM UMR 7345, Marseille, France
More informationProblem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension
105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1
More informationGuiding Center Orbit Studies in a Tokamak Edge Geometry Employing Boozer and Cartesian Coordinates
Contrib. Plasma Phys. 48, No. -3, 4 8 (8) / DOI./ctpp.839 Guiding Center Orbit Studies in a Tokamak Edge Geometry Employing Boozer and Cartesian Coordinates Y. Nishimura,Y.Xiao,and Z. Lin Department of
More informationFINAL EXAM GROUND RULES
PHYSICS 507 Fall 2011 FINAL EXAM Room: ARC-108 Time: Wednesday, December 21, 10am-1pm GROUND RULES There are four problems based on the above-listed material. Closed book Closed notes Partial credit will
More informationMHD turbulence in the solar corona and solar wind
MHD turbulence in the solar corona and solar wind Pablo Dmitruk Departamento de Física, FCEN, Universidad de Buenos Aires Turbulence, magnetic reconnection, particle acceleration Understand the mechanisms
More informationElectromagnetic theory of turbulent acceleration of parallel flow and momentum conservation. Abstract
Electromagnetic theory of turbulent acceleration of parallel flow and momentum conservation Shuitao Peng and Lu Wang * State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School
More informationGinzburg-Landau length scales
597 Lecture 6. Ginzburg-Landau length scales This lecture begins to apply the G-L free energy when the fields are varying in space, but static in time hence a mechanical equilibrium). Thus, we will be
More informationxkcd.com It IS about physics. It ALL is.
xkcd.com It IS about physics. It ALL is. Introduction to Space Plasmas The Plasma State What is a plasma? Basic plasma properties: Qualitative & Quantitative Examples of plasmas Single particle motion
More informationZ. Lin University of California, Irvine, CA 92697, USA. Supported by SciDAC GPS-TTBP, GSEP & CPES
GTC Framework Development and Application Z. Lin University of California, Irvine, CA 92697, USA and dgpsttbp GPS-TTBP Team Supported by SciDAC GPS-TTBP, GSEP & CPES GPS-TTBP Workshop on GTC Framework
More informationMagnetostatics III Magnetic Vector Potential (Griffiths Chapter 5: Section 4)
Dr. Alain Brizard Electromagnetic Theory I PY ) Magnetostatics III Magnetic Vector Potential Griffiths Chapter 5: Section ) Vector Potential The magnetic field B was written previously as Br) = Ar), 1)
More information