Internal

Type for BiMaxwellian velocity distributions with respect to the magnetic field.

BiMaxwellian(B::Vector{U}, u0::Vector{T}, ppar::T, pperp::T, n; m=mᵢ)

Construct a BiMaxwellian distribution with magnetic field B, bulk velocity u0, parallel thermal pressure ppar, perpendicular thermal pressure pperp, and number density n in SI units. The default particle is proton.

Type for Maxwellian velocity distributions.

Maxwellian(u0::Vector{T}, p::T, n; m=mᵢ)

Construct a Maxwellian distribution with bulk velocity u0, thermal pressure p, and number density n in SI units. The default particle is proton.

Return velocity magnitude from energy in [eV].

Calculate the energy [eV] of a relativistic particle from γv.

Return the energy [eV] from relativistic sol.

get_gc(param::Union{TPTuple, FullTPTuple})

Get three functions for plotting the orbit of guiding center.

For example:

param = prepare(E, B; species=Proton)
gc = get_gc(param)
# The definitions of stateinit, tspan, E and B are ignored.
prob = ODEProblem(trace!, stateinit, tspan, param)
sol = solve(prob, Vern7(); dt=2e-11)

f = Figure(fontsize=18)
ax = Axis3(f[1, 1], aspect = :data)
gc_plot(x,y,z,vx,vy,vz) = (gc([x,y,z,vx,vy,vz])...,)
lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6))

Return the gyrofrequency.

Return the gyroperiod.

Return the gyroradius.

Return velocity from relativistic γv in sol.

prepare(grid::CartesianGrid, E, B; kwargs...) -> (q2m, E, B)

Return a tuple consists of particle charge-mass ratio for a prescribed species and interpolated EM field functions.

keywords

• order::Int=1: order of interpolation in [1,2,3].

• bc::Int=1: type of boundary conditions, 1 -> NaN, 2 -> periodic.

• species::Species=Proton: particle species.

• q::AbstractFloat=1.0: particle charge. Only works when Species=User.

• m::AbstractFloat=1.0: particle mass. Only works when Species=User.

  prepare(grid::CartesianGrid, E, B, F; species=Proton, q=1.0, m=1.0) -> (q, m, E, B, F)

Return a tuple consists of particle charge, mass for a prescribed species of charge q and mass m, interpolated EM field functions, and external force F.

prepare(x::AbstractRange, y::AbstractRange, z::AbstractRange, E, B; kwargs...) -> (q2m, E, B)
prepare(x, y, E, B; kwargs...) -> (q2m, E, B)

prepare(x::AbstractRange, E, B; kwargs...) -> (q2m, E, B)

1D grid. An additional keyword dir is used for specifying the spatial direction, 1 -> x, 2 -> y, 3 -> z.

Direct range input for uniform grid in 2/3D is also accepted.

prepare(E, B; kwargs...) -> (q2m, E, B)

Return a tuple consists of particle charge-mass ratio for a prescribed species of charge q and mass m and analytic EM field functions. Prescribed species are Electron and Proton; other species can be manually specified with species=Ion/User, q and m.

prepare(E, B, F; kwargs...) -> (q, m, E, B, F)

Return a tuple consists of particle charge, mass for a prescribed species of charge q and mass m, analytic EM field functions, and external force F.

sample(vdf::Maxwellian)

Sample a 3D velocity from a Maxwellian distribution vdf using the Box-Muller method.

sample(vdf::BiMaxwellian)

Sample a 3D velocity from a BiMaxwellian distribution vdf using the Box-Muller method.

trace!(dy, y, p::TPTuple, t)
trace!(dy, y, p::FullTPTuple, t)

ODE equations for charged particle moving in static EM field with in-place form.

ODE equations for charged particle moving in static EM field and external force field with in-place form.

trace(y, p::TPTuple, t) -> SVector{6, Float64}
trace(y, p::FullTPTuple, t) -> SVector{6, Float64}

ODE equations for charged particle moving in static EM field with out-of-place form.

ODE equations for charged particle moving in static EM field and external force field with out-of-place form.

trace_gc!(dy, y, p::TPTuple, t)

Guiding center equations for nonrelativistic charged particle moving in static EM field with in-place form. Variable y = (x, y, z, u), where u is the velocity along the magnetic field at (x,y,z).

1st order approximation of guiding center equations.

trace_gc_drifts!(dx, x, p, t)

Equations for tracing the guiding center using analytical drifts, including the grad-B drift, curvature drift, and ExB drift. Parallel velocity is also added. This expression requires the full particle trajectory p.sol.

trace_normalized!(dy, y, p::TPNormalizedTuple, t)

Normalized ODE equations for charged particle moving in static EM field with in-place form. If the field is in 2D X-Y plane, periodic boundary should be applied for the field in z via the extrapolation function provided by Interpolations.jl.

trace_relativistic!(dy, y, p::TPTuple, t)

ODE equations for relativistic charged particle (x, γv) moving in static EM field with in-place form.

trace_relativistic(y, p::TPTuple, t) -> SVector{6}

ODE equations for relativistic charged particle (x, γv) moving in static EM field with out-of-place form.

trace_relativistic_normalized!(dy, y, p::TPNormalizedTuple, t)

Normalized ODE equations for relativistic charged particle (x, γv) moving in static EM field with in-place form.

trace_relativistic_normalized(y, p::TPNormalizedTuple, t)

Normalized ODE equations for relativistic charged particle (x, γv) moving in static EM field with out-of-place form.

Abstract type for tracing solutions.

Field{itd, F} <: AbstractField{itd}

A representation of a field function f, defined by:

time-independent field

\[\mathbf{F} = F(\mathbf{x}),\]

time-dependent field

\[\mathbf{F} = F(\mathbf{x}, t).\]

Arguments

• field_function::Function: the function of field.
• itd::Bool: whether the field function is time dependent.
• F: the type of field_function.

The type of parameter tuple for full test particle problem.

The type of parameter tuple for guiding center problem.

Type for the particles, Proton, Electron, Ion, or User.

The type of parameter tuple for normalized test particle problem.

The type of parameter tuple for normal test particle problem.

Interpolate solution at time x. Forward tracing only.

Abstract type for velocity distribution functions.

Apply Boris method for particles with index in irange.

Prepare for advancing.

In-place cross product.

Calculates the magnetic field from a dipole with magnetic moment M at r.

dipole_fieldline(ϕ, L=2.5, nP=100)

Creates nP points on one field line of the magnetic field from a dipole. In a centered dipole magnetic field model, the path along a given L shell can be described as r = L*cos²λ, where r is the radial distance (in planetary radii) to a point on the line, λ is its co-latitude, and L is the L-shell of interest.

getB_CS_harris(B₀, L)

Return the magnetic field at location r near a current sheet with magnetic strength B₀ and sheet length L.

getB_bottle(x, y, z, distance, a, b, I1, I2) -> StaticVector{Float64, 3}

Get magnetic field from a magnetic bottle. Reference: https://en.wikipedia.org/wiki/Magneticmirror#Magneticbottles

Arguments

• x,y,z::Float: particle coordinates in [m].
• distance::Float: distance between solenoids in [m].
• a::Float: radius of each side coil in [m].
• b::Float: radius of central coil in [m].
• I1::Float: current in the solenoid times number of windings in side coils in [A].
• I2::Float: current in the central solenoid times number of windings in the

central loop in [A].

getB_current_loop(x, y, z, cl::Currentloop) -> StaticVector{Float64, 3}

Get magnetic field at [x, y, z] from a magnetic mirror generated from two coils.

Arguments

• x,y,z::Float: particle coordinates in [m].
• distance::Float: distance between solenoids in [m].
• a::Float: radius of each side coil in [m].
• I1::Float: current in the solenoid times number of windings in side coils.

Analytic magnetic field function for testing. Return in SI unit.

getB_mirror(x, y, z, distance, a, I1) -> StaticVector{Float64, 3}

Get magnetic field at [x, y, z] from a magnetic mirror generated from two coils.

Arguments

• x,y,z::Float: particle coordinates in [m].
• distance::Float: distance between solenoids in [m].
• a::Float: radius of each side coil in [m].
• I1::Float: current in the solenoid times number of windings in side coils.

getB_tokamak_coil(x, y, z, a, b, ICoils, IPlasma) -> StaticVector{Float64, 3}

Get the magnetic field from a Tokamak topology consists of 16 coils. Original: https://github.com/BoschSamuel/Simulation-of-a-Tokamak-Fusion-Reactor/blob/master/Simulation2.m

Arguments

• x,y,z::Float: location in [m].
• a::Float: radius of each coil in [m].
• b::Float: radius of central region in [m].
• ICoil::Float: current in the coil times number of windings in [A].
• IPlasma::Float: current of the plasma in [A].

getB_tokamak_profile(x, y, z, q_profile, a, R₀, Bζ0) -> StaticVector{Float64, 3}

Reconstruct the magnetic field distribution from a safe factor(q) profile. Reference: Tokamak, 4th Edition, John Wesson.

Arguments

• x,y,z::Float: location in [m].
• q_profile::Function: profile of q. The variable of this function must be the normalized radius.
• a::Float: minor radius [m].
• R₀::Float: major radius [m].
• Bζ0::Float: toroidal magnetic field on axis [T].

Analytic electric field function for testing.

get_rotation_matrix(axis::AbstractVector, angle::Real) --> SMatrix{3,3}

Create a rotation matrix for rotating a 3D vector around a unit axis by an angle in radians. Reference: Rotation matrix from axis and angle

Example

using LinearAlgebra
v = [-0.5, 1.0, 1.0]
v̂ = normalize(v)
θ = deg2rad(-74)
R = get_rotation_matrix(v̂, θ)

getchargemass(species::Species, q::AbstractFloat, m::AbstractFloat)

Return charge and mass for species. if species = User, input q and m are returned.

getinterp(A, gridx, gridy, gridz, order::Int=1, bc::Int=1)
getinterp(A, gridx, gridy, order::Int=1, bc::Int=1)
getinterp(A, gridx, order::Int=1, bc::Int=1, dir::Int=1)

Return a function for interpolating array A on the grid given by gridx, gridy, and gridz.

Arguments

• order::Int=1: order of interpolation in [1,2,3].
• bc::Int=1: type of boundary conditions, 1 -> NaN, 2 -> periodic, 3 -> Flat.
• dir::Int: 1/2/3, representing x/y/z direction.

guiding_center(xu, param::Union{TPTuple, FullTPTuple})

Calculate the coordinates of the guiding center according to the phase space coordinates of a particle. Reference: https://en.wikipedia.org/wiki/Guiding_center

A simple definition:

\[\mathbf{X}=\mathbf{x}-m\frac{\mathbf{v}\times\mathbf{B}}{qB}\]

Judge whether the field function is time dependent.

Return uniform range from 2D/3D CartesianGrid.

set_axes_equal(ax)

Set 3D plot axes to equal scale for Matplotlib. Make axes of 3D plot have equal scale so that spheres appear as spheres and cubes as cubes. Required since ax.axis('equal') and ax.set_aspect('equal') don't work on 3D.

solve(prob::TraceProblem; trajectories::Int=1,
   savestepinterval::Int=1, isoutofdomain::Function=ODE_DEFAULT_ISOUTOFDOMAIN)

Trace particles using the Boris method with specified prob.

keywords

• trajectories::Int: number of trajectories to trace.
• savestepinterval::Int: saving output interval.
• isoutofdomain::Function: a function with input of position and velocity vector xv that determines whether to stop tracing.

Convert from spherical to Cartesian coordinates vector.

Update location in one timestep dt.

update_velocity!(xv, paramBoris, param, dt, t)

Update velocity using the Boris method, Birdsall, Plasma Physics via Computer Simulation. Reference: https://apps.dtic.mil/sti/citations/ADA023511