Finite-Larmor-Radius effect

The general FLR effect refers to the correction terms introduced when considering the field difference at the particle location and the guiding center location. More theoretical details can be found in Non-uniform E Field.

using TestParticle
using OrdinaryDiffEq
using StaticArrays
using LinearAlgebra: ×, ⋅, norm, normalize
using Tensors: laplace
import Tensors: Vec as Vec3
# using SpecialFunctions
using CairoMakie

function uniform_B(x)
    return SA[0, 0, 1e-8]
end

function nonuniform_E(x)
    return SA[1e-9*cos(0.3*x[1]), 0, 0]
end

# trace the orbit of the guiding center
function trace_gc!(dx, x, p, t)
    q2m, E, B, sol = p
    xu = sol(t)
    xp = @view xu[1:3]
    Bv = B(xp)
    b = normalize(Bv)
    v_par = (xu[4:6] ⋅ b) .* b # (v ⋅ b)b
    v_perp = xu[4:6] - v_par
    r4 = (norm(v_perp) / q2m / norm(Bv))^2 / 4
    EB(x) = (E(x) × B(x)) / norm(B(x))^2
    # dx[1:3] = EB(xp) + v_par
    dx[1:3] = EB(x) + r4*laplace.(EB, Vec3(x...)) + v_par

    # more accurate
    # dx[1:3] = besselj0(0.3*norm(v_perp)/q2m/norm(Bv))*EB(x) + v_par
end

# Initial condition
stateinit = let x0 = [1.0, 0, 0], v0 = [0.0, 1.0, 0.1]
    [x0..., v0...]
end
# Time span
tspan = (0, 20)
param = prepare(nonuniform_E, uniform_B, species=Proton)
prob = ODEProblem(trace!, stateinit, tspan, param)
sol = solve(prob, Vern9())

gc = get_gc(param)
gc_x0 = gc(stateinit)
prob_gc = ODEProblem(trace_gc!, gc_x0, tspan, (param..., sol))
sol_gc = solve(prob_gc, Tsit5(); save_idxs=[1,2,3])

# numeric result and analytic result
f = Figure(fontsize=18)
ax = Axis3(f[1, 1],
   title = "Finite Larmor Radius Effect",
   xlabel = "x [m]",
   ylabel = "y [m]",
   zlabel = "z [m]",
   aspect = :data,
   azimuth = 0.3π,
)

gc_plot(x, y, z, vx, vy, vz) = (gc(SA[x, y, z, vx, vy, vz])...,)

lines!(ax, sol, idxs=(1, 2, 3), color=Makie.wong_colors()[1])
lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6), color=Makie.wong_colors()[2])
lines!(ax, sol_gc, idxs=(1, 2, 3), color=Makie.wong_colors()[3])

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