# Equilibrium.jl

Equilibrium.jl provides tools for working with solutions of the Grad-Shafranov Equation.

## AbstractEquilibrium API

```
using Equilibrium
typeof(S) <: AbstractEquilibrium
psi = S(r, z) # Poloidal flux at r,z
gradpsi = psi_gradient(S, r, z)
B = Bfield(S, r, z)
Bp = poloidal_Bfield(S, r, z)
J = Jfield(S, r, z)
Jp = poloidal_Jfield(S, r, z)
F = poloidal_current(S, psi)
Fprime = poloidal_current_gradient(S, psi)
p = pressure(S, psi)
pprime = pressure_gradient(S, psi)
V = electric_potential(S, psi)
gradV = electric_potential_gradient(S, psi)
q = safety_factor(S, psi)
maxis = magnetic_axis(S)
btip = B0Ip_sign(S)
rlims, zlims = limits(S)
psi_lims = psi_limits(S)
cc = cocos(S) # Return COCOS structure
fs = flux_surface(S, psi) # returns a boundary object
```

## Solov'ev Equilibrium

Solov'ev Equilibrium are analytic solutions to the Grad-Shafranov equation where the p' and FF' are constant.
The resulting Grad-Shafranov equation takes the form `Δ⋆ψ = α + (1-α)x²`

where `α`

is some constant.
The boundary conditions are found using a plasma shape parameterization.

```
# ITER parameters
δ = 0.33 # Triangularity
ϵ = 0.32 # Inverse aspect ratio a/R0
κ = 1.7 # Elongation
B0 = 5.3 # Magnitude of Toroidal field at R0 [T]
R0 = 6.2 # Major Radius [m]
qstar = 1.57 # Kink safety factor
alpha = -0.155 # constant
S = solovev(B0, MillerShape(R0, ϵ, κ, δ), alpha, qstar, B0_dir=1, Ip_dir=1)
SolovevEquilibrium
B0 = 2.0 [T]
S = MillerShape{Float64}(6.2, 0.32, 1.7, 0.33)
α = -0.155
q⋆ = 1.57
βp = 1.1837605469381924
βt = 0.049177281028224634
σ = 1
diverted = false
symmetric = true
```

## EFIT Equilibrium

EFIT geqdsk files are a commonly used file format. Here we provide routines for converting the GEQDSK files into an Equilibrium object.

```
using EFIT
g = readg("g000001.01000")
M = efit(g, clockwise_phi=false) # direction of phi needed to determine COCOS ID
wall = Wall(g)
in_vessel(wall, r, z)
# or
# M, wall = read_geqdsk("g000001.01000",clockwise_phi=false)
```

## COCOS: Tokamak Coordinate Conventions

We provide routines for working determining, transforming, and checking COCOS's.

```
julia> cocos(3)
COCOS = 3
e_Bp = 0
σ_Bp = -1
σ_RΦZ = (R,Φ,Z): 1
σ_ρθΦ = (ρ,Φ,θ): -1
Φ from top: CCW
θ from front: CCW
ψ_ref: Decreasing assuming +Ip, +B0
sign(q) = -1 assuming +Ip, +B0
sign(p') = 1 assuming +Ip, +B0
julia> transform_cocos(3,1)
Dict{Any, Any} with 14 entries:
"Z" => 1.0
"Q" => -1
"P" => 1.0
"B" => 1.0
"F_FPRIME" => -1.0
"ψ" => -1.0
"TOR" => 1.0
"Φ" => 1.0
"PSI" => -1.0
"I" => 1.0
"J" => 1.0
"R" => 1.0
"F" => 1.0
"PPRIME" => -1.0
```

## Boundaries

Equilibrium.jl also provides routines for working with boundries such as walls or flux surfaces. Internally boundaries are stored as a list of points forming a polygon.

```
fs = flux_surface(S, psi) # alternatively fs = boundary(S, psi)
in_plasma(fs, r, z) # or in_vessel(fs, r, z), in_boundary(fs, r, z)
cicumference(fs)
area(fs) # Area enclosed by the boundary
volume(fs) # assuming toroidal symmetry. F can be a vector with the same length as fs or a function of (r,z)
average(fs, F) # Average F over the boundary
area_average(fs, F) # average F over the area
volume_average(fs, F) # average F over the volume
```