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(1)• Review: where does D come from? ◦ one wave ◦ many waves • How to calculate D • How to approximate D • What to do with D.

(2) First: motion of a particle resonant with one wave. Start with the Hamiltonian of a particle in a B field: s H(x, P; t) = mc. 2. . 1+. P − qA(x)/c mc. where P = p + qA/c is the canonical momentum and A = Ao + Aw .. Recall:. dx ∂H = , dt ∂P. is equivalent to F = dp/dt.. dP ∂H =− dt ∂x. 2.

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(4) The wave with specified (ω, k) is described by Aw = A1 sin ψˆ e⊥ + A2 cos ψˆ e⊗ − A3 sin ψˆ ek , Z ψ=. k · dx − ωt,. where Ai = cEi /ω. The components of Ew and Bw are related by the usual Stix coefficients for an EM wave..

(5) Change variables from (x, Px , y, Py , z, Pz ) to guiding center variables (X, PX , φ, I, z˜, P˜z ), using a generating function.. I is essentially the first adiabatic invariant µ = p2⊥ /2mB, φ is the gyroangle, and z˜ = z.. Rewrite H in the new variables and • Taylor expand (to 1st order) in qAw /mc2 P∞ • use the expansion sin(a sin θ) = n=−∞ Jn (a) sin nθ • normalize the variables. After “a little” algebra . . ..

(6) ∞ X an H = γ + sin ξn , with 2 mc 2γ n=−∞. r γ=. 1+. P 2. 2Ωeq gI z + mc2 mc. Z ,. ξn = kx x +. r. kz dz − ωt + snφ,. i Ωeq gI h Pz an = − 2 ǫ3 Jn , (ǫ1 −ǫ2 )Jn−1 +(ǫ1 +ǫ2 )Jn+1 + 2s mc2 mc. dξn Ωc To lowest order, = ω − kk vk − sn . dt γ. s = q/|q|, qAi . ǫi = mc2.

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(8) Near the ℓth resonance, all sin ξn terms except n = ℓ can be dropped by gyroaveraging over φ. Then dH/dt = ∂H/∂t ⇒ ω dI/dt = sℓ dγ/dt.. If ω is constant, ωI = sℓγ(I, Pz , z) eliminates Pz and leads to K(I, ξ; z) = Ko (I, z) + ǫK1 (I, z) sin ξ with “time” z..

(9) For fixed z, the phase portrait of K(I, ξ; z) is like that of a plane pendulum . . .. but it varies with z. Let τ be the “time” for the island to move by W . 2 ∂ Ko /∂z∂I ∂Bo /∂z 1 ∼ The ratio of timescales is R ≡ . = 2 2 ω0 τ K1 (∂ Ko /∂I ) Bw.

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(13) This was for one wave, treating spatial dependence in detail. For broadband waves, get local D, then bounce average.. Heuristic treatment instead of the Vlasov equation: Dαα =. ∂α 2 dI ∂I. dt. ∆t. 2 . 2∆t. 1 an dI 2 = −sn cos ξ (from H equations) and cos ξ = . (RPA) with d(ωt) 2γ 2 d ω − k v + nΩe /γ , Instead of ∆t = π v dz π (coherence time with a wavepacket ∼ ∆k ) use ∆t = ∆k |v −∂ω/∂k |.

(14) Then. 2 Ω2c Φ2n π Bwave , Dn = 2 2 2 B γ ∆k |v −∂ω/∂k |. which can be written as 2 2 2 2 π Bwave Ωc k⊥ dk⊥ dk |Bk | Φn δ k v +snΩc /γ−ω Dn = . 2 2 2 2 B γ k⊥ dk⊥ dk |Bk | where Φ2n is related to an above. This D is the same as Kennel and Englemann [1966] before (k⊥ , k ) → (ω, θ) and bounce averaging. The two expressions for ∆t are physically different [Walker, 1993] but give the same answer for VLF with ω Ωe [Albert, 2001]..

(15) • Review: where does D come from? ◦ one wave ◦ many waves • How to calculate D • How to approximate D • What to do with D.

(16) 2 2 We usually take Bw (k⊥ , k ) → Bw (ω, θ), and model 2 Bw (ω, θ). −(ω−ωm )2 /(δω)2. ∼e. −(tan θ−tan θm )2 / tan2 (δθ). ×e. . N (ω). as long as ωLC ≤ ω ≤ ωU C and θmin ≤ θ ≤ θmax , and 0 otherwise. For a given particle, the resonance condition relates ω to θ. Since the diffusion coefficients are integrals over θ, it is helpful to know what subsets of θmin ≤ θ ≤ θmax give ωLC ≤ ω ≤ ωU C . These ranges of θ are the only ones that contribute to D, and are also useful for approximating D..

(17) Finding θ ranges 2. Write the resonance condition as either k (ω, θ) = (ω −. 2 2 sec ωn ) v2. θ. v2 ω2 ω2 2 or 2 2 (ω, θ) = 2 cos θ, and plot vs. ω at fixed θ. 2 k c c (ω − snΩc /γ). Need max[LHS] > min[RHS] and min[LHS] < max[RHS], e.g., k. 2. ⇒. 2 sec2 θ (ωU C , θ)>(ωU C −ωn ) v2 . and k. A cos4 θ + B cos2 θ + C > 0. ⇒. 2. 2 sec2 θ (ωLC , θ)<(ωLC −ωn ) v2 . θ ranges..

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(21) Diffusion Coefficients ∞ 2 X Dαα Ωc Bwave n , D = p2 γ 2 Bo2 n=−∞ αα. Z n Dαα =. θmax. sin θdθ∆n G1 G2 , θmin. and similarly for Dαp and Dpp .. π sec θ 2 (− sin2 α + snΩc /ωγ)2 Φn , ∆n (ω, θ) = 3 2 |vk /c| |1−(∂ω/∂kk )θ /vk | Φ2n ∼ Jn±1 (k⊥ ρ). ∆n is independent of the wave distribution in ω, θ..

(22) Z n Dαα. θmax. sin θdθ∆n G1 G2 ,. = θmin. Ωc B 2 (ω) G1 (ω) = R ωU C 2 ′ ′ B (ω )dω ωLC 2. −(ω−ωm )2 /δω 2. B (ω) = e. , ωLC ≤ ω ≤ ωU C. G1 depends only on the ω parameters..

(23) Z n Dαα. θmax. sin θdθ∆n G1 G2 ,. = θmin. G2 (ω, θ) = R θmax θmin. gω (θ) dθ′ sin θ′ Γ gω (θ′ ). −(tan θ−tan θm )2 / tan2 (δθ). gω (θ) = e. , θmin ≤ θ ≤ θmax. G2 depends on the θ parameters.. Γ ω(θ), θ. ′.

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(28) ∂(k⊥ , kk )

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(30) ∂µ

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(34) µ + ω .

(35) comes from

(36) ∂ω ∂(ω, θ)

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(38) Factors of D(θ).

(39) Factors of D(θ).

(40) Approximation (1) Z n Dαα. θmax. = θmin. With Γ ω(θ), θ. ′. . sin θdθ ∆n G1 R θmax θmin. ′. ′. . n Dαα. with. sin θ′. ≈. R θmax. R. Γ(ω(θ), θ′ ) gω (θ′ ). ′. dθ outside. dθ sin θ ∆n G1 gω (θ). θmin R θmax θmin. dθ sin θ Γ gω (θ). n This is a weighted average: Dαα. f (θ) ≡. dθ′. ≈ Γ ω(θ ), θ , can take R θmax. gω (θ). D∆ G E n 1 ≈ , Γ. dθ sin θ Γ ω(θ), θ) gω (θ)f (θ). θmin R θmax θmin. dθ sin θ Γ(ω(θ), θ) gω (θ). .. .. R. dθ:. ..

(41) Approximation (2) ∆ G ∆ G n 1 n 1 n = ≈ Mean value theorem: Dαα for some θ0 . Γ Γ θ0 Setting θ0 = 0 reproduces Summers [2005, 2007] (twice, in fact). Need θmin ≤ θ0 ≤ θmax , and also ωLC ≤ ω(θ0 ) ≤ ωU C . What values of θ0 correspond to ωLC ≤ ω ≤ ωU C ? The discussion above answers exactly this, giving “θ ranges” θa ≤ θ ≤ θb . Then take θ0 = (θa + θb )/2, or maybe tan θ0 = (tan θa + tan θb )/2. This includes oblique waves, and not just n = −1. It’s fast and fairly accurate..

(42) full θ integration, |n|<5. E (MeV). E (MeV). sec-1. Nightside chorus, L=4.5 (85 x 49 sets of <D>).

(43) full θ integration, |n|<5. θ=0, n=-1. Nightside chorus, L=4.5.

(44) full θ integration, |n|<5. θ=0, n=-1. Nightside chorus, L=4.5. θ0=(θa+θb)/2, |n|<5.

(45) full θ integration, |n|<5. CPU = 5 hours. θ=0, n=-1. CPU = 4 sec. Nightside chorus, L=4.5. θ0=(θa+θb)/2, |n|<5. CPU = 90 sec.

(46) Conclusion The “θ ranges” approach has most of the speed and simplicity of the parallel propagation approximation (θ = 0), and gives a better approximation to full integration over θ. The biggest payoff is for weakly trapped high energy electrons, where the “delicate balance between acceleration and loss” is accurately captured (while taking θ = 0 misses it entirely)..


(48) The radiation belts can be modeled by multidimensional diffusion. We would like to solve the full time-dependent diffusion equation: ∂f = ∂t. ". ∂/∂J1 ∂/∂J2 ∂/∂J3. #". D11 D12 D13. D12 D22 D23. D13 D23 D33. #". ∂f /∂J1 ∂f /∂J2 ∂f /∂J3. Dpp ↔ acceleration, Dα0 α0 ↔ loss. D ∆α ∆p E 0 relates ∆α0 to ∆p, and can be < 0. Dα0 p ∼ 2∆t. Simple finite differencing methods fail, mainly because the cross terms can be large and rapidly varying.. #.

(49) Do the cross terms really matter? Green's Function solutions with Dyy = 2 Dxx, Dxy = ε (Dxx Dyy)1/2. Yes..

(50) Do the cross terms really matter?. Yes..

(51) Do the cross terms really matter?. Yes..

(52) When are the cross terms large? Dαp Dpp − sin α cos α γ ω For a single resonance, = = , ∼ Dα0 α Dαp n Ωe sin2 α + nΩe /ωγ so typically Dα0 α0 > |Dα0 p | > Dpp .. For a single resonance, |Dαp | = R. dλ. P R n. dx enforces. p. |Dα0 p | <. Dαα Dpp .. p. Dα0 α0 Dpp ,. so |Dα0 p | is larger if the wave distribution is narrow in (θ, ω, λ)..

(53) One solution (in 2D): change variables Transform from (J1 , J2 ) to new variables (Q1 , Q2 ) with D˜12 = 0: ". #. ˜ 11 D 0. 0 ˜ 22 D. ". ∂Q1 ∂J1 ∂Q2 ∂J1. =.

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(56) dα0

(57). Choosing Q1 ≡ α0 leads to. D1 = Dα0 α0 ,. D2 =. ∂Q1 ∂J2 ∂Q2 ∂J2. ∂Q 2 2. ∂E. D11 D12. = Q2. DEE. #". #". Dα0 E Dα0 α0. D12 D22. ∂Q1 ∂J1 ∂Q1 ∂J2. ∂Q2 ∂J1 ∂Q2 ∂J2. # .. and. Dα2 0 E − . Dα0 α0. The 2D diffusion equation becomes . . ∂f ∂f ∂ 1 ∂ ∂f GD1 + GD2 , = ∂t G ∂Q1 ∂Q1 ∂Q2 ∂Q2.

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(60) ∂(J1 , J2 )

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(62) , G =

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(64) ∂(Q1 , Q2 )

(65). for which practically any finite difference scheme works..

(66) Starting with these values:.

(67) tracing the constant-Q2 curves gives:. Label them by something like Q2 ≡ log E(α0 = 90◦ ), and evaluate ∂Ji /∂Qj from finite differences of nearby curves..

(68) This results in.

(69) and. This model produces 1 MeV electrons in ∼ 1 day..

(70) 3D: Full Matrix " If. D11 D12 D13 ". D12 D22 D23. D13 D23 D33. A11 D12 0. D12 A22 0. # can be split into # " 0 0 0 + 0 0 0. 0 B22 D23. 0 D23 B33. #. ". C11 + 0 D13. 0 0 0. # D13 0 , C33. 2 2 2 with A11 A22 ≥ D12 , B22 B33 ≥ D23 , C11 C33 ≥ D13 ,. then each 2D operator can be treated sequentially as above. If not, there’s always the Monte Carlo approach..

(71) 2D + Radial Diffusion Including pure radial diffusion, ∂f = ∂t. ". ∂/∂J1 ∂/∂J2 ∂/∂J3. #". D11 D12 0. D12 D22 0. 0 0 D33. #". ∂f /∂J1 ∂f /∂J2 ∂f /∂J3. #. shouldn’t be hard. At each timestep, the previous approach will be used for the (J1 , J2 ) diffusion at each L, and then the D33 diffusion advanced (this is operator splitting). Coupling the grids at different L (at constant J1 , J2 ) requires only 1D interpolation, not 2D..

(72) Setting Up the 3D Grid. Well-known problem: Since radial diffusion occurs at constant (J1 , J2 ), a rectangular grid of (Q1 , Q2 ) points at L1 doesn’t map to a rectangular grid at L2 . Some interpolation is unavoidable.. The same thing happens in (α0 , E, L), or any variables except (J1 , J2 , J3 ), regardless of cross terms..

(73) But in a dipole p and y = sin α0 map according to Y (y1 ) Y (y2 ) = y2 y1. r. L2 , L1. y1 L1 3/2 . p2 = p1 y2 L2. y2 (y1 ) doesn’t depend on p1 , so points with different Q2 but the same Q1 at L1 map to points with the same Q1 at L2 . So Q2 diffusion at constant (Q1 , L) can be done without interpolation..

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(76) The grid is irregular: interpolate in Q2 for diffusion in Q1 . O(∆Q)2 methods become O(∆Q)..

(77) The 2nd approach seems easier to code. The low and high Q2 boundaries, mapped in L, will distort anyway. This way, the grid point (α0 , E) values don’t depend on D.. Either way, interpolation in Q2 is necessary.. The exact same considerations apply to working in (α0 , E, L) or any related variables, even neglecting the cross terms..

(78) 1: 2:. (Q1 , Q2 , L). (α0 , E, L). Model the waves, compute diffusion coefficients. Integrate 1D ODEs for Q2 , compute partials and D2 .. SAME Neglect cross terms.. 3:. Set up 3D grid (1D interpolation). 4:. Pick a numerical scheme (explicit, C-N, ADI, etc.) and time step.. SAME. Variable BCs and Bwave are OK. If ne or (ω, θ) change, go to 1. SAME. 5:. SAME.

(79) Prototype 3D Run ("equatorial" chorus + DLL).

(80) Summary. Can do diffusion simulations in 3D, including cross diffusion, using the variable transformation method from 2D. This approach really isn’t much more work than simply dropping the cross terms. Either way, 1D interpolation seems unavoidable so we should be careful about numerical diffusion..

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