Internal
Public APIs
TestParticle.BiMaxwellian
— TypeType for BiMaxwellian velocity distributions with respect to the magnetic field.
TestParticle.BiMaxwellian
— Method BiMaxwellian(B::Vector{U}, u0::Vector{T}, ppar, pperp, 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.
TestParticle.Maxwellian
— TypeType for Maxwellian velocity distributions.
TestParticle.Maxwellian
— Method Maxwellian(u0::AbstractVector{T}, p, 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.
TestParticle.energy2velocity
— MethodReturn velocity magnitude from energy in [eV].
TestParticle.get_energy
— MethodCalculate the energy [eV] of a relativistic particle from γv.
TestParticle.get_energy
— MethodReturn the energy [eV] from relativistic sol
.
TestParticle.get_gc
— Method 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))
TestParticle.get_gyrofrequency
— FunctionReturn the gyrofrequency.
TestParticle.get_gyroperiod
— FunctionReturn the gyroperiod.
TestParticle.get_gyroradius
— MethodReturn the gyroradius.
TestParticle.get_velocity
— MethodReturn velocity from relativistic γv in sol
.
TestParticle.prepare
— Method 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 whenSpecies=User
.m::AbstractFloat=1.0
: particle mass. Only works whenSpecies=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
.
TestParticle.sample
— Method 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.
TestParticle.trace!
— 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.
TestParticle.trace
— Method 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.
TestParticle.trace_gc!
— Method 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).
TestParticle.trace_gc_1st!
— Method1st order approximation of guiding center equations.
TestParticle.trace_gc_drifts!
— Method 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
.
TestParticle.trace_normalized!
— Method 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.
TestParticle.trace_relativistic!
— Method trace_relativistic!(dy, y, p::TPTuple, t)
ODE equations for relativistic charged particle (x, γv) moving in static EM field with in-place form.
TestParticle.trace_relativistic
— Method 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.
TestParticle.trace_relativistic_normalized!
— Method 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.
TestParticle.trace_relativistic_normalized
— Method 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.
Private types and methods
TestParticle.AbstractTraceSolution
— TypeAbstract type for tracing solutions.
TestParticle.Field
— Type 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 offield_function
.
TestParticle.FullTPTuple
— TypeThe type of parameter tuple for full test particle problem.
TestParticle.GCTuple
— TypeThe type of parameter tuple for guiding center problem.
TestParticle.Species
— TypeType for the particles, Proton
, Electron
, Ion
, or User
.
TestParticle.TPNormalizedTuple
— TypeThe type of parameter tuple for normalized test particle problem.
TestParticle.TPTuple
— TypeThe type of parameter tuple for normal test particle problem.
TestParticle.TraceSolution
— MethodInterpolate solution at time x
. Forward tracing only.
TestParticle.VDF
— TypeAbstract type for velocity distribution functions.
TestParticle._boris!
— MethodApply Boris method for particles with index in irange
.
TestParticle._prepare
— MethodPrepare for advancing.
TestParticle.cross!
— MethodIn-place cross product.
TestParticle.dipole
— MethodCalculates the magnetic field from a dipole with magnetic moment M
at r
.
TestParticle.dipole_fieldline
— Function 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.
TestParticle.getB_CS_harris
— Function getB_CS_harris(B₀, L)
Return the magnetic field at location r
near a current sheet with magnetic strength B₀
and sheet length L
. The current sheet is assumed to lie in the z = 0 plane.
TestParticle.getB_bottle
— Method 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].
TestParticle.getB_current_loop
— Method 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.
TestParticle.getB_dipole
— MethodAnalytic magnetic field function for testing. Return in SI unit.
TestParticle.getB_mirror
— Method 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.
TestParticle.getB_tokamak_coil
— Method 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].
TestParticle.getB_tokamak_profile
— Method 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].
TestParticle.getE_dipole
— MethodAnalytic electric field function for testing.
TestParticle.get_rotation_matrix
— Method 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)
TestParticle.getchargemass
— Method getchargemass(species::Species, q::AbstractFloat, m::AbstractFloat)
Return charge and mass for species
. if species = User
, input q
and m
are returned.
TestParticle.getinterp
— Function 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 field 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.
TestParticle.getinterp_scalar
— Function getinterp_scalar(A, gridx, gridy, gridz, order::Int=1, bc::Int=1)
Return a function for interpolating scalar array A
on the grid given by gridx
, gridy
, and gridz
. Currently only 3D arrays are supported.
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.
TestParticle.guiding_center
— Method 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:
\[\mathbf{X}=\mathbf{x}-m\frac{\mathbf{v}\times\mathbf{B}}{qB}\]
TestParticle.is_time_dependent
— MethodJudge whether the field function is time dependent.
TestParticle.makegrid
— MethodReturn uniform range from 2D/3D CartesianGrid.
TestParticle.set_axes_equal
— Method 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.
TestParticle.solve
— Function 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 vectorxv
that determines whether to stop tracing.
TestParticle.sph2cart
— MethodConvert from spherical to Cartesian coordinates vector.
TestParticle.update_location!
— MethodUpdate location in one timestep dt
.
TestParticle.update_velocity!
— Method 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