PlasmaEquilibriumToolkit

AbstractMagneticCoordinates

Abstract supertype for different magnetic coordinates.

AbstractMagneticEquilibrium

Abstract supertype for different magnetic equilibrium representations (VMEC, SPEC,...).

BasisVectors{T} <: SArray{Tuple{3,3},T,2,9}

Convenience type for defining the Cartesian coordinate representation of 3D curvilinear vector fields. For a vector defined by inverse map $R(x,y,z) = (U(x,y,z),V(x,y,z),W(x,y,z))$, the components are represented by the 3×3 SArray:

R = [Ux Vx Wx;Uy Vy Wy;Uz Vz Wz]

where Ux is the x-component of U.

Examples

julia> using StaticArrays
julia> typeof(@SArray(ones(3,3))) <: BasisVectors{Float64}
true

BoozerCoordinates{T,A}(ψ::T,χ::A,ϕ::A) <: MagneticCoordinates

Coordinates on a magnetic flux surface, where ψ is the flux label and χ and ϕ are angle-like variables

ClebschCoordinates(α,β,η)

Coordintes (α,β,η) representing a divergence-free magnetic field $B = ∇α×∇β$ with a Jacobian defined by $√g = 1/(∇η ⋅ ∇α × ∇β)$.

CoordinateVector{T} <: SVector{3,T}

Convenience type for defining a 3-element StaticVector.

FluxCoordinates{T,A}(ψ::T,θ::A,ζ::A) <: MagneticCoordinates

Coordinates on a magnetic flux surface, where ψ is the physical toroidal flux divided by 2π and θ and ζ are angle-like variables

NullEquilibrium()

Empty subtype of MagneticEquilibrium to represent no equilibrium

PestCoordinates{T,A}(ψ::T,α::A,ζ::A) <: MagneticCoordinates

Coordinates on a magnetic flux surface, where ψ is the toroidal flux divided by 2π in the clockwise direction, α is the field line label, and ζ is the geometric toroidal angle advancing the clockwise direction.

abs(e::BasisVectors{T},component=0)

Compute the L2 norm of the basis vector e. If component > 0 and <= 3, the L2 norm of the component is returned.

Examples

julia> using StaticArrays

julia> e = @SArray([1 4 7;2 5 8;3 6 9]);

julia> abs(e)
16.881943016134134

julia> abs(e,1)
3.7416573867739413

MagneticCoordinateGrid(C::AbstractMagneticCoordinates,α::Union{AbstractArray,Real},β::Union{AbstractArray,Real},η::Union{AbstractArray,Real}

Generate a grid of magnetic coordiantes of type C, returning a StructArray{C} containing the coordinates

basis_vectors(b::BasisType,t::Transformation,x::AbstractArray{T},eq::AbstractMagneticEquilibrium) where T <: AbstractMagneticCoordinates

Generate basis vectors of type b for the magnetic coordinates defined by the transformation t at the points given by x using data from the magnetic equilibrium eq. The routine providing the point transformation designated by t is defined in the respective equilibrium module.

Examples

Generate covariant basis vectors for points along a field line defined by PestCoordinates using a VMEC equilibrium

julia> using PlasmaEquilibriumToolkit, VMEC, NetCDF
julia> wout = NetCDF.open("wout.nc"); vmec, vmec_data = readVmecWout(wout);
julia> s = 0.5; vmec_s = VmecSurface(s,vmec);
julia> x = PestCoordinates(s*vmec.phi(1.0)/2π*vmec.signgs,0.0,-π:2π/10:π);
julia> v = VmecFromPest()(x,vmec_s);
julia> e_v = basis_vectors(Covariant(),CartesianFromVmec(),v,vmec_s)
10-element Array{StaticArrays.SArray{Tuple{3,3},Float64,2,9},1}:
 ⋮

curvature_components(e::BasisVectors,gradB::CoordinateVector)

Computes the normal and geodesic curvature vectors for a stright field line coordinate system with basis vectors ∇X = e[:,1] and ∇Y = e[:,2].

See also: normalCurvature, geodesicCurvature

geodesic_curvature(b::CoordinateVector,gradB::CoordinateVector,gradX::CoordinateVector)

Computes the geodesic curvature component defined in straight fieldline coordinates (X,Y,Z) with basis vectors defined by (∇X,∇Y,∇Z) with the relation κ_g = -(B × ∇B) ⋅ ∇X/(B²|∇X|)

grad_B_projection(e::BasisVectors,∇B::CoordinateVector)

Computes the projection of B × ∇B/B² onto the perpendicular coordinate vectors given by ∇X = e[:,1] and ∇Y = e[:,2].

jacobian(t::BasisType,e::BasisVectors{T})

Compute the Jacobian of the covariant/contravariatn basis vectors e given by $J = e₁ ⋅ e₂ × e₃$ (covariant) or $J = 1/(e¹ ⋅ e² × e³)$ (contravariant).

metric2(e::BasisVectors)

Computes the metric tensor components gᵘᵛ or gᵤᵥ for a set of basis vectors e in a 6 element SVector: [g11 = e[:,1]*e[:,1], g12 = e[:,1]*e[:,2],…]

normalCurvature(B::CoordinateVector,∇B::CoordinateVector,∇X::CoordinateVector,∇Y::CoordinateVector)

Computes the normal curvature component defined in straight fieldline coordinates (X,Y,Z) with basis vectors defined by (∇X,∇Y,∇Z) with the relation κₙ = (B × ∇B) ⋅ ((∇X⋅∇X)∇Y - (∇X⋅∇Y)∇X)/(B³|∇X|)

transform_basis(t::Transformation,x::AbstracyArray{AbstractMagneticCoordinates},e::AbstractArray{BasisVectors{T}},eq::AbstractMagneticEquilibrium)

Perform a change of basis for magnetic coordinates denoted by the transformation t, using the provided coordinate charts and basis vectors.

transform_basis(t::BasisTransformation, e::BasisVectors)
transform_basis(t::BasisTransformation, e::BasisVectors, jacobian)
transform_basis(t::BasisTransformation, e::AbstractArray{BasisVectors})
transform_basis(t::BasisTransformation, e::AbstractArray{BasisVectors}, jacobian::AbstractArray)

Transform the covariant/contravariant basis given by e to a contravariant/covariant basis by computing the transformation Jacobian from the basis vectors; for ContravariantFromCovariant() the Jacobian is given by $J = e₁ ⋅ e₂ × e₃$ and for CovariantFromContravariant() the Jacobian is given by $J = 1.0/(e¹ ⋅ e² × e³)$.