APPENDIX Z. USEFUL FORMULAS 1 Appendix Z Useful Formulas
APPENDIX Z. USEFUL FORMULAS 2 Key Vector Relations A B = B A, A B = B A, A A = 0, A B C) = A B) C A B C) = B A C) C A B), bac-cab rule A B) C D) = A C)B D) A D)B C) A B) C D) = C A B D) D A B C) A = A ˆb + A with ˆb B/B A B A/B = ˆb A A B B A)/B 2 = ˆb ˆb A) A =B )A /B)+A /B) B)+ A A = A [ ln B +ˆb )ˆb ]+1/B) ˆb B A ) For A = B f/b 2, ˆb B A )= ˆb f)ˆb ˆb) fg) = g f + f g f 2 f fa) = f A + f A f = 0 fa) = f A + f A A 2 A ft) = f T + f T = A) A) ft) = f T + f T A = 0 B )A C) = C B ) A + A B ) C A B) = A B)+B A)+A ) B +B ) A AB) = B A)+A ) B A B) = B A A B A B) = A B) B A)+B ) A A ) B For the general coordinate x xê x + yê y + zê z and x x 2 + y 2 + z 2, x =3, x = 0, x = I, I = 0, I = 0, A I = A, x = x/ x, 1/ x ) = x/ x 3, 2 1/ x ) = 4πδx), I A = A. For a volume V enclosed by a closed, continuous surface S, V d3 x A = ds A, divergence, Gauss theorem. S For an open surface S bounded by a closed, continuous contour C, S ds A = dl A, Stokes theorem. C
APPENDIX Z. USEFUL FORMULAS 3 Explicit Forms Of Vector Differentiation Operators for orthogonal curvilinear coordinates u i, ê i u i / u i, A i ê i A) Cartesian coordinates: u i =x, y, z), d 3 x = dx dy dz, f f = ê x x + ê f y y + ê f z A = A x x + A y y + A z [ Az A = ê x y A y 2 f = 2 f x 2 + 2 f y 2 + 2 f 2 ] [ Ax + ê y A z x ] [ Ay + ê z x A ] x y Cylindrical coordinates: u i =r, θ, z), d 3 x = rdr 2π dθ 0 0 dz, with r x 2 + y 2, θ arctan y/x), z z, and inverse relations x = r sin θ, y = r cos θ, z = z, f f = ê r r + ê 1 f θ r θ + ê f z A = 1 r r ra r)+ 1 A θ r θ + A z [ 1 A z A = ê r r θ A ] [ θ Ar + ê θ A ] z r [ 1 +ê z r r ra θ) A ] r θ 2 f = 1 r f ) + 1 2 f r r r r 2 θ 2 + 2 f 2 Spherical coordinates: u i =r, ϑ, ϕ), d 3 x = r 2 dr π 0 0 dϑ sin ϑ 2π dϕ, 0 with r x 2 + y 2 + z 2, ϑ arctan x 2 + y 2 /r), ϕ arctan y/x), and inverse relations x = r sin ϑ cos ϕ, y = r sin ϑ sin ϕ, z = r cos ϑ, f f = ê r r + ê 1 f ϑ r ϑ + ê 1 f ϕ r sin ϑ ϕ A = 1 r 2 r r2 A r )+ 1 1 A = ê r r sin ϑ [ +ê ϑ 2 f = 1 r 2 r r sin ϑ ϑ sin ϑa ϑ)+ 1 r sin ϑ ] [ ϑ sin ϑa ϕ) A ϑ ϕ 1 A r r sin ϑ ϕ 1 ] [ r r ra 1 ϕ) + ê ϕ r r 2 f ) 1 + r r 2 sin ϑ f ) + sin ϑ ϑ ϑ A ϕ ϕ r ra ϑ) A r ϑ 1 2 f r 2 sin 2 ϑ ϕ 2 ]
APPENDIX Z. USEFUL FORMULAS 4 Physical Constants m e electron mass 9.11 10 31 kg, 511 kev m p proton mass 1.67 10 27 kg, 938 MeV m p /m e mass ratio 1836 = 42.85) 2 e elementary charge 1.602 10 19 C = J/eV) c speed of light in vacuum 3 10 8 m/s = 1/ µ 0 ɛ 0 µ 0 permeability of vacuum 4π 10 7 N/A 2 ɛ 0 permittivity of vacuum 8.85 10 12 F/m, 4πɛ 0 10 10 h Planck constant 6.626 10 34 J s N A Avogadro constant 6.022 10 23 #/mol e/k B Boltzmann constant 11 604 K/eV Key Plasma Formulas Quantities are in SI mks) units except temperature and energy which are expressed in ev; Z i is the ion charge state; A i m i /m p is the atomic mass number. Frequencies electron plasma ω pe ion gyrofrequency ω ci q i B m i electron collision ν e Lengths electron Debye λ De ion gyroradius ϱ i v Ti ω ci n e e 2 m e ɛ 0 56 n e rad/s, f pe 9 n e Hz 0.96 10 8 Z i B A i rad/s 4 3 π νv Te) 5 10 11 n e Z i [T e ev)] 3/2 ɛ0 T e Te ev) 7.4 103 n e e2 n e Ti ev) A i 1.4 10 4 Z i B electron collision λ e = v Te 1.2 10 16 [T eev)] 2 ν e n e Z i m 17 ln Λ ) ln Λ 17 m ) m Speeds, Velocities electron thermal v Te 2 T e /m e 5.9 10 5 T e ev) m/s ion thermal v Ti 2 T i /m i 1.4 10 4 T i ev)/a i m/s ion acoustic T e >> T i ) c S Z i T e /m i 10 4 Z i T e ev)/a i m/s Alfvén c A B/ µ 0 ρ m 2.2 10 16 B/ n i A i m/s electron diamagnetic ) T e 1 dn e flow dt e /dx =0) V e ê y = T eev) ê y q e B n e dx BL n m/s electron drift in Bx) average, low β) v de = 2T ) e 1 db ê y = 2 T eev) ê y q e B B dx BL B m/s s 1
APPENDIX Z. USEFUL FORMULAS 5 Drift, flow velocities for ϱ << 1, ω<<ω c ) perpendicular to B : particle drift velocities plasma species flow velocities v F = F B/qB 2 general force V F = F B/qB 2 v E = E B/B 2 E B V E = E B/B 2 v µ = B µ B/qB 2 µ grad-b µ mv 2/2B v κ = B mv 2 κ/qb2 curvature κ ˆb )ˆb = R C /RC 2 diamagnetic V = B p/nqb 2 v p = B mdv d /dt)/qb 2 polarization V p = B mdv/dt)/qb 2 friction V η = R B/nqB 2 viscosity V π = B π)/nqb 2 v d = v E + v µ + v κ + v p total V = V E + V + V p + V η + V π Diffusivities no magnetic field ν e λ 2 e vte/ν 2 e 7 10 21 [T 5/2 ) eev)] 17 m 2 /s n e Z i ln Λ magnetic field η/µ 0 m e ν e /n e e 2 )/µ 0 = ν e c/ω pe ) ) 2 ) 1.4 10 3 Z i ln Λ m 2 /s [T e ev)] 3/2 17 classical ν e ϱ 2 e = β e η/µ 0 ) 5.6 10 22 n e Z i B 2 [T e ev)] 1/2 Dimensionless ) ln Λ m 2 /s 17 number of electrons in Debye cube n e λ 3 De 4.1 10 11 [T e ev)] 3/2 / n e ) λ D Coulomb logarithm ln Λ ln max { b cl min,bqm min } b cl min = Z i/12πn e λ 2 De ) 5 10 10 Z i /T e ev) m b qm min = h/4πm ev) 1.1 10 10 /[T e ev)] 1/2 m plasma to magnetic pressure β Lundquist number S a2 /η/µ 0 ) L /c A P B 2 /2µ 0 = n e T e + n i T i B 2 /2µ 0 4.0 10 25 ne B 2 )[ T e ev) + n i n e T i ev) 1.6 10 13 a2 B [T e ev)] 3/2 L Z i ni A i ) 17 ln Λ ]
APPENDIX Z. USEFUL FORMULAS 6 Fundamental Equations of Physics Mechanics m a mdv/dt = F, v dx/dt Newton s second law F = q E + v B) Lorentz force H = p qa 2 /2m + qφ, p = mv + qa Hamiltonian, energy dp/dt = H/ q, dq/dt = H/ p Hamilton s equations Electrodynamics E = ρ q /ɛ 0 Gauss s law E = B/ t Faraday s law B = 0 no magnetic monopoles B = µ 0 J + ɛ 0 E/ t) Ampere s law, µ 0 ɛ 0 =1/c 2 0 = ρ q / t + J charge continuity equation E = φ A/ t, B = A potential representations Plasma Physics Plasma kinetic equation PKE) for distribution function f f s x, v,t): f/ t + v f/ x +q/m)e + v B) f/ v = C{f}. Density, flow moments and charge, current densities: n s d 3 vf s, V s d 3 v v f s /n s, ρ q s n sq s, J s n sq s V s. Gibb s A: adiabatic) distribution of plasma species with temperature T: ) 3/2 f A = n m 0 2πT e H/T ; n A x,t)=n 0 e qφ/t, Boltzmann relation. Maxwellian M: collisional equilibrium) distribution v T 2T/m): ) 3/2 f M = n m 2πT exp m v 2 ) /v 2 2T = ne v 2 T π 3/2 vt 3, v v V. Species fluid moment equations density, momentum, energy): n/ t + nv =0, nt d 3 v mv 2 /3) f, mn dv/dt) =nq E + V B) p π + R, d/dt / t + V, 3/2)n dt/dt)+p V = q π : V + Q, p nt. Magnetohydrodynamics plasma fluid description, isotropic pressure and isentropic responses for plasma species, ρ m s n sm s, V s n sm s V s /ρ m ): ρ m / t + ρ m V =0, E + V B = ηj, P s p s, ρ m dv/dt) =J B P, d ln P/ρm) Γ /dt =Γ 1) ηj 2 /P 0.