Contents. Part 1. Fluid Mechanics 1 1. Vorticity 1

Transcription

1 PLASMA ERNEST YEUNG Abstract. Everything about plasmas. The end goal is to understand Field Reverse Configuration (FRC) for fusion. I wanted to begin at an elementary level. 0

2 PLASMA 1 Contents Part 1. Fluid Mechanics 1 1. Vorticity 1 Part 2. Electromagnetism (em) Ideal MHD (Magnetohydrodynamics) equations approximation or MHD approximation Flux 3 References 4 Part 1. Fluid Mechanics 1. Vorticity The beginning of Ch. 14 Vorticity of Thorne and Blandford (2016) [1] says that vorticity must be mastered to understand magneto hydrodynamics. For spacetime manifold M, magnetic field B is a 2-form, i.e. B Ω 2 (M) such that locally B 1 2 B ijdx i dx j 1 2 ɛ ijkb k dx i dx j. B is the formal analogue to the 2-form vorticity ω, defined as ω : du Ω 2 (M), where u is the fluid velocity vector field u X(M) T M, u Ω 1 (M) T M is the 1-form, through the cotangent-tangent isomorphism (or musical isomorphism), and d is the exterior derivative with respect to only the spatial coordinates of M. Part 2. Electromagnetism (em) Ch. 19 Magnetohydrodynamics of Thorne and Blandford (2016) [1] says to start with Maxwell s equations (in SI): E ρ e ɛ 0 B 0 E B µ 0 j + µ 0 ɛ 0 In cgs, and accounting for induced polarization P and possible inherent magnetization M, if B 0 then E 1 ( ) c if E 4πρ total then B 1 ( ) P + 4π c + 4πJ free + 4πc M 1.1. Ideal MHD (Magnetohydrodynamics) equations approximation or MHD approximation. The following follows Thorne and Blandford (2016) [1], Sec Maxwell s Equations in the MHD Approximation and Goedbloed and Poedts (2004) [4], Ch. 4 The MHD model. Recall the form of a Lorentz transformation, in particular, the Lorentz boost in the x-axis of positive β velocity, i.e. observer in frame L of (t, x, y, z) and (inertial) frame L of (t, x, y, z ) has observer moving with velocity β +β: t γ(t βx) x γ(x βt) y y z z

3 2 ERNEST YEUNG In general, for a Lorentz boost in an arbitrary direction, t γ(t β x) x x (β x)β β 2 ( ) (x β)β + γ β 2 tβ Recall how the electric E and magnetic B fields transform, with denoting the component of the field parallel to the boost and denoting the component of the field perpendicular to the boost: E E B B E γ(e + β B ) B γ(b β E ) In differential forms, E is a 1-form and B is a 2-form on spacetime manifold M, and the electromagnetic field tensor F is a 2-form: F, B Ω 2 (M) E Ω 1 (M) F B + E dt E E i dx i (locally) B 1 2 B ijdx i dx j 1 2 ɛ ijkb k dx i dx j The Lorentz transformation in the language of differential forms becomes thus: E γ(e + (β B)) B γ(b (β E)) Indeed, this general differential form reproduces the Lorentz transformation in flat Minkowski space R R 3 g β B β k dx k B ij (3 2)! 2 gil g jm ɛ lmn dx n B ij β k g 2 gil g jm ɛ lmn dx k dx n (β B) g g ij δ ij B ij gβk (3 2)! 2 gil g jm ɛ lmn g kp g nq ɛ pqr dx r g1 B ij β k 2 ɛ ijnɛ knr dx r B ij β k 2 (δ irδ jk δ ik δ jr )dx r β j B ij 2 dxi β i B ij 2 dxj β j B ij dx i β B e i ɛ ijk β j B k e i ɛ ijk β j 1 2 ɛklm B lm e i β j (B ij B ji ) 1 2 e iβ j B ij Nevertheless, the first major assumption is this: Thus Suppose β 1 (fluid moves with nonrelativistic speed). Then γ 1. E (E + β B) B (B β E) or E E + (β B) B B (β E) The second major assumption is this: Consider E E. Why? Consider this: boost into rest frame of fluid moving at β. To the fluid, due to the fluid s high conductivity, the current will flow quickly to annihilate E in fluid s rest frame. Thus, E β B B B β E or E (β B) B (B (β E)) Now E βb and so B β E B + β 2 B i.e. B + O(β 2 ) B. So Also, B B E 4πρ total 4πρ total βb L β4πj free where L is the length scale on which E, B vary. Also, of consequence, E E, so ρ total ρ total, and so ρ total ρ e βj.

4 PLASMA 3 Recall how current J Ω 1 (M) transforms under Lorentz transformation; assume for simplicity that j j 1 dx 1 j x dx and we are boosting in this x direction: J ρdt + j i dx i ρ (x ) µ d(x ) µ x i + j i (x ) µ d(x ) µ ρ(γdt + γβd(x ) 1 ) + j 1 (βγdt + γd(x ) 1 ) and so in general In the approximation(s) above, then and so nonrelativistically J γ( ρ + j β)dt + γ( ρβ i + j i )d(x ) i j γj γβρ j βρ j O(β 2 ) j j Now recall that Ohm s law, relating current density to electric field E, is formulated only in the rest frame of the conductor. j κ e E The conductor in consideration is a conducting fluid. So in fluid rest frame, (1) j j κ e (E + β B) Now Start from Ampere s law, So B 4πj B β L E β2 B L + 4πj or B 4πj B 4πj + O ( β 2 B L (2) j 1 4π B (sincee βb) ) B 4πj or From Ohm s law, Eq. 1, and the approximation on Ampere s law, Eq. 2, E j κ e β B 1 B β B Then using E, we d want to apply curl to the above equation. Note that (β B) e i ɛ ijk j ɛ klm β l B m e i [ j (β i B j ) j (β j B i )] e i [( j β i )B j + 0 ( j β j )B i β j j B i ] (B j j β i ( j β j )B i β j j B i )e i where B 0 was used in the last equality. Thus (3) + βj j B B j j β ( β)b + 1 B Compressibility or incompressibility was not assumed: ( β) factor is included above. B formally obeys the same vorticity equation as vorticity (cf. Ch. 14, Thorne and Blandford (2016) [1]) Flux. : flux : magnetic flux : tubes

5 4 ERNEST YEUNG Boundary conditions at the wall. Boundary conditions at the wall: (1) Wall is perfect i.e. absorbs neither plasma nor emits gas (impurities) Then on surface W, for normal n W on W, n W β 0 (2) require wall acts as perfect conductor; i.e. short circuits tangential E t (at wall, on perfect conductor surface!) This condition isn t a restriction on kinds of materials a wall could be made of, but on resistivity of plasma itself: if wall isolator, thin perfectly conducting plasma layer in front of wall serves same purpose of short circuiting E n W E n W (E+β B) n W E t +(n W B)β (n W β)b 0 n W B 0 (on W ) B don t intersect wall, so plasma prevented from flowing along field lines hitting wall. (4) Ψ : For notation, Ψ tor δ d tor toroidal pol poloidal ( d (β B) + 1 ) B (β B) + 1 Ψ pol S tor S B (β B) (β B)+ 1 (β B) dl pol κ e j dl pol δb ( B) dl pol (β B)+ 1 ( B) (β B) dl tor + 1 ( B) dl tor S tor S tor S tor S tor References [1] Kip S. Thorne, Roger D. Blandford, Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics, Princeton University Press (July 5, 2016). ISBN-13: Applications of Classical Physics notes.html [2] John Baez (Author), Javier P Muniain (Author), Gauge Fields, Knots, and Gravity (Series on Knots and Everything), World Scientific Publishing Company (October 24, 1994), ISBN-13: [3] Bernard F. Schutz, Geometrical Methods of Mathematical Physics, Cambridge University Press, ISBN-13: [4] J. P. Hans Goedbloed, Stefaan Poedts. Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas. Cambridge University Press (August 30, 2004). ISBN-13: QC718.5.M36G [5] P. Goedbloed, Rony Keppens, Stefaan Poedts. Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas. Cambridge University Press; 1 edition (June 7, 2010). ISBN-13: QC718.5.M36G address: ernestyalumni@gmail.com