COLLISIONLESS PLASMA PHYSICS TAKE-HOME EXAM

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1 Honou School of Mathematical and Theoetical Physics Pat C Maste of Science in Mathematical and Theoetical Physics COLLISIONLESS PLASMA PHYSICS TAKE-HOME EXAM HILARY TERM 18 TUESDAY, 13TH MARCH 18, 1noon to THURSDAY, 15 MARCH 18, 1noon You should submit answes to all questions. Answe booklets ae povided fo you to use but you may type you answes if you wish. Typed answes should be pinted single-sided and the pages secuely fastened togethe. You may efe to books and othe souces when completing the exam but should not discuss the exam with anyone else. Do not tun this page until you ae told that you may do so Page 1 of 5

2 z ˆ ˆ R B ˆ y (a x (b Figue 1: (a Dipola magnetic field aound a planet. (b Spheical coodinates. 1. Conside the dipola magnetic field aound a spheical planet of adius R shown in figue 1(a. The magnetic field of a dipole is whee the flux function ψ(, θ is 1 ψ B(, θ = B ˆ B θˆθ = sin θ θ ˆ 1 ψ sin θ ˆθ, (1 ψ(, θ = A sin θ. ( Hee {, θ, φ} ae the usual spheical coodinates, shown in figue 1(b, and {ˆ, ˆθ, ˆφ} ae the usual unit vectos associated to the spheical coodinates, also shown in figue 1(b. Paticles of chage Ze and mass m move in the dipola magnetic field (1. The electic field E is negligibly small. Assume that the paticles ae magnetized, that is, thei chaacteistic gyoadius ρ is small compaed to the chaacteistic size of the system, ρ/r 1. Using the lowest ode guiding cente equations, answe the following questions. (a [5 maks] Paticles move along the magnetic field line on which they stated. Show that this is equivalent to moving along lines of constant ψ(, θ and constant φ. Show that the kinetic enegy K = mv / of each paticle is conseved. (b [1 maks] To detemine the position of each paticle, we use the coodinates {ψ, θ, φ}. Plot B as a function of θ fo fixed ψ and φ, and using this plot, descibe the motion of a paticle with kinetic enegy K and magnetic moment µ. Give a fomula fo the points θ b in which the paallel velocity vanishes. (c [5 maks] Show that paticles with ψ A /L and L R collide with the planet when mµ K R3 A. (3 Page of 5

3 (d [1 maks] Assume that any paticle that collides with the planet is lost, and hence thee ae no paticles that satisfy condition (3. Using the lowest ode dift kinetic equation, calculate the distibution function of the paticles f(, θ, v eveywhee given the distibution function at θ = π/, which is f(, θ = π/, v = N fo mµ/k > R 3 / A and L < < L, and ( m πt 3/ exp ( mv. (4 T f(, θ = π/, v = (5 othewise. Hee the density N, the tempeatue T and the length L ae constants. Using the distibution function that you have obtained, calculate the density n = f d 3 v eveywhee, and in paticula nea = R.. By answeing the following questions, you ae going to detemine the behavio of whistle waves in the plasma confined by the Eath s dipola magnetic field. Conside a plasma composed of one ion species with chage Ze and mass m i, and electons with chage e and mass m e. (a [1 maks] Whistle waves ae chaacteized by fequencies Ω i Ω e ω Ω e, whee Ω i = ZeB/m i and Ω e = eb/m e ae the ion and electon gyofequencies. Assume ω pe Ω e and m e /m i 1, whee ω pe = e n e /ɛ m e is the electon plasma fequency, n e is the electon numbe density and ɛ is the vacuum pemittivity. Show that fo whistle waves, the cold plasma dielectic tenso becomes ɛ = ωpe/ω e iωpe/ω e ω iωpe/ω e ω ωpe/ω e ωpe/ω. (6 When you neglect tems (fo example, the ion contibutions to the cold plasma dielectic tenso, justify you decision with ode of magnitude estimates. (b [8 maks] Assuming that kd e 1, whee d e = c/ω pe is the electon skin depth, show that the whistle wave satisfies the dispesion elation ω = k kd eω e. (7 (c [ maks] Whistles get thei name fom thei dispesive natue: waves ae geneated in the ionosphee and each fequency popagates towads obseves in Eath at a diffeent velocity, leading to a signal with time-dependent fequency o whistle. Ague that the goup velocity of the whistles appoximately scales as ω and hence obseves on Eath eceive a signal with a fequency that deceases with time. (d [1 maks] Whistle popagates towads Eath along ovedense plasma egions that ae aligned with the dipola magnetic field. To study this popagation, conside a system with a unifom magnetic field B = Bẑ and electon density n e (x = n e (1 x /L, whee {x, y, z} ae the usual Catesian coodinates, and {ˆx, ŷ, ẑ} ae the unit vectos in the diections of the Catesian axes. The whistle wave is launched fom the oigin with wavevecto k = k ˆx k ẑ. Using ay tacing equations, show that k is constant along ays and find k as a function of x. Descibe the popagation of the whistle wave. Page 3 of 5 Tun Ove

4 z ẑ ˆ ˆ B = B(ˆ Figue : z-pinch and cylindical coodinates. 3. The z-pinch is a magnetic confinement cylindical configuation in which the magnetic field closes in azimuthal loops (see Figue. The magnitude of the magnetic field depends on adius, that is, B = B(ˆθ. Hee we use the usual cylindical coodinates {, θ, z} and thei associated unit vectos {ˆ, ˆθ, ẑ}, shown in figue. The plasma contained in the z-pinch is composed of electons with chage e and mass m e, and one ion species with chage Ze and mass m i. The distibution functions fo both species (s = i, e ae Maxwellians that only depend on the adial position, ( ( 3/ ms f s ϕ = f Ms (, v, µ = n s ( exp m s(v / µb(. (8 πt s ( T s ( The equilibium electic field is zeo, E =. We conside the stability of this configuation to axisymmetic kinetic MHD modes, that is, to infinitely small petubations of the fom δ f s ϕ = g s (, v, µ exp(ikz iωt, δb = B( exp(ikz iωt, δv E = ṽ E ( exp(ikz iωt, δe = Ẽ ( exp(ikz iωt. (a [3 maks] Befoe consideing the petubations, show that the equilibium must satisfy the equation d (P B B =, (9 d µ µ whee P = n i T i n e T e is the total plasma pessue. (b [7 maks] Using the pependicula plasma displacement = ˆ z ẑ = ṽ E /( iω, show that the kinetic MHD induction equation gives [ 1 d B = ( d ( ik z µ ] dp B Bˆθ. (1 d [Hint: use the esult deived in pat (a to eliminate db/d fom the equation.] Show that the petubation δb to the magnitude of the magnetic field is δb θ, and that the petubation δκ to the magnetic field line cuvatue κ = ˆb ˆb vanishes. Page 4 of 5

5 (c [1 maks] Using the kinetic MHD dift kinetic equation, obtain the distibution function g s. Integate it to obtain the petubed paallel pessues, p s = = m s v g s d 3 v B θ B n st s [ ( 1 d n s T s d ( ik z and the petubed pependicula pessues, p s = = m s µb g s d 3 v B θ B n st s [ ( 1 d n s T s d ( ik z ( ns T s d(n ] st s, (11 d ( n st s d(n ] st s. (1 d (d [5 maks] Show that the kinetic MHD momentum consevation equation gives [ (B ω n i m i = d ( 1 d P ( d µ d ( ] ik B z P µ 1 [( ( B 1 d P ( µ d ( ik 4B z µ 3P dp d ω n i m i z = ik [ (B µ P ( 1 d d ( ( ik B z P µ ], (13 ]. (14 (e [1 maks] To solve the equations in pat (d, we use a vaiational pinciple. Show that the eigenvalues ω ae extema of the functional Λ[ η, η z ] = N[ η, η z ] D[ η, η z ]. (15 whee N[ η, η z ] = D[ η, η z ] = ( B 1 d P µ d ( η ik η z B µ P η B µ P d ( 7B 5µ P P B µ P dp η d, (16 d n i m i ( η η z d. (17 In othe wods, show that fo the solutions η = and η z = z, Λ[, z ] = ω and that Λ[ δ η, z δ η z ] Λ[, z ] = to fist ode in δ η and δ η z z. (f [5 maks] The smallest eigenvalue is the minimum of the functional Λ[ η, η z ]. Hence, one possible stategy to pove that the plasma is unstable, that is, that at least one eigenvalue satisfies ω <, is to find functions η and η z that make Λ[ η, η z ] negative. Ague that D[ η, η z ] is always positive. Then minimize N[ η, η z ] by choosing an appopiate function η z. The final esult should be that N[ η, η z ] and hence Λ[ η, η z ] become negative fo sufficiently lage dp/d, and as a esult, the plasma is unstable fo sufficiently lage dp/d. Find the citical value of dp/d fo which the plasma becomes unstable. Page 5 of 5 End of Last Page