Postprocessing

In this notebook we take a deeper look at postprocessing, using the python interface of GVEC.

GVEC is parallelised with OpenMP and we can set the number of OpenMP threads before importing the gvec package.

import os

os.environ["OMP_NUM_THREADS"] = "2"

This tutorial requires matplotlib to be installed in addition to gvec. numpy and xarray are dependencies of gvec and should therefore be already installed.

import matplotlib.pyplot as plt
import numpy as np
import xarray as xr

import gvec

In this tutorial we will use the equilibrium of the elliptic stellarator, computed in the elliptic stellarator tutorial. To obtain a gvec.State object for this equilibrium, gvec.load_state can be used with specific paths for the parameter and statefile. Alternatively, when each equilibrium is saved in a separate directory, gvec.find_state can load the state with only the directory provided.

state = gvec.find_state("./run_ellipstell/")

Derived Quantities

One of the central features of the GVEC python bindings is the ability to compute a variety of derived quantities. A list of computable quantities is returned by gvec.table_of_quantities(markdown=True) and also shown at the end of this notebook.

The requested quantities will be stored in an xarray.Dataset, which groups several varaibles, their coordinates and metadata, similar to a pandas Dataframe. Such a Dataset can also be stored directly as netCDF. GVEC will also automatically determine which quantities are necessary to compute the desired quantities and add them to the Dataset, thereby also caching them for future computations.

The grid on which the quantities are to be evaluated, can be specified explicitly (with an array or float) or automatically with either an integer for linear spacing (within one field period) or "int" for the integration points required by GVEC.

The state.evaluate method can be used to create a new Dataset and compute desired quantities in a single step:

ev = state.evaluate("pos", "B", rho=[0.1, 0.5, 0.9], theta=20, zeta=0.0)
ev

<xarray.Dataset> Size: 27kB
Dimensions:     (rad: 3, pol: 20, tor: 1, xyz: 3)
Coordinates:
  * rho         (rad) float64 24B 0.1 0.5 0.9
  * theta       (pol) float64 160B 0.0 0.3142 0.6283 ... 5.341 5.655 5.969
  * zeta        (tor) float64 8B 0.0
  * xyz         (xyz) <U1 12B 'x' 'y' 'z'
Dimensions without coordinates: rad, pol, tor
Data variables: (12/44)
    X1          (rad, pol, tor) float64 480B 3.092 3.086 3.068 ... 3.946 4.128
    dX1_dr      (rad, pol, tor) float64 480B 1.217 1.153 0.9674 ... 1.652 1.838
    dX1_dt      (rad, pol, tor) float64 480B 0.0 -0.03914 ... 0.754 0.3979
    dX1_dz      (rad, pol, tor) float64 480B 0.0 0.02456 ... -0.5363 -0.2756
    dX1_drr     (rad, pol, tor) float64 480B -0.02021 -0.06073 ... 3.7 3.349
    dX1_drt     (rad, pol, tor) float64 480B 0.0 -0.4048 ... 0.777 0.4011
    ...          ...
    Jac         (rad, pol, tor) float64 480B 0.2556 0.2549 0.2527 ... 3.559 4.34
    e_theta     (xyz, rad, pol, tor) float64 1kB 0.0 -0.03914 ... 0.4543 0.5512
    e_zeta      (xyz, rad, pol, tor) float64 1kB 0.0 0.02456 ... 1.495 1.643
    B_contra_t  (rad, pol, tor) float64 480B 0.08812 0.08847 ... 0.09179 0.08512
    B_contra_z  (rad, pol, tor) float64 480B 0.1215 0.122 ... 0.06608 0.05889
    B           (rad, pol, tor, xyz) float64 1kB 0.0 -0.3757 ... -0.2431 0.1437

xarray.Dataset

• Dimensions:
  • rad: 3
  • pol: 20
  • tor: 1
  • xyz: 3
• Coordinates: (4)
  • rho
    (rad)
    float64
    0.1 0.5 0.9

    long_name :

    Logical radial coordinate

    symbol :

    \rho

    integration_points :

    False

    array([0.1, 0.5, 0.9])

  • theta
    (pol)
    float64
    0.0 0.3142 0.6283 ... 5.655 5.969

    long_name :

    Logical poloidal angle

    symbol :

    \theta

    integration_points :

    False

    array([0.      , 0.314159, 0.628319, 0.942478, 1.256637, 1.570796, 1.884956,
           2.199115, 2.513274, 2.827433, 3.141593, 3.455752, 3.769911, 4.08407 ,
           4.39823 , 4.712389, 5.026548, 5.340708, 5.654867, 5.969026])

  • zeta
    (tor)
    float64
    0.0

    long_name :

    Logical toroidal angle

    symbol :

    \zeta

    integration_points :

    False

    array([0.])

  • xyz
    (xyz)
    <U1
    'x' 'y' 'z'

    long_name :

    cartesian vector components

    symbol :

    (x,y,z)

    array(['x', 'y', 'z'], dtype='<U1')

• Data variables: (44)
  • X1
    (rad, pol, tor)
    float64
    3.092 3.086 3.068 ... 3.946 4.128

    long_name :

    first reference coordinate

    symbol :

    X^1

    array([[[3.09221255],
            [3.0860084 ],
            [3.06806121],
            [3.04028358],
            [3.00559738],
            [2.96757662],
            [2.93002788],
            [2.89657746],
            [2.87032732],
            [2.85361591],
            [2.84788378],
            [2.85361591],
            [2.87032732],
            [2.89657746],
            [2.93002788],
            [2.96757662],
            [3.00559738],
            [3.04028358],
            [3.06806121],
            [3.0860084 ]],
    ...
           [[4.19133503],
            [4.1282799 ],
            [3.94570571],
            [3.66306661],
            [3.31122793],
            [2.92897973],
            [2.55741097],
            [2.23353144],
            [1.98545175],
            [1.8308367 ],
            [1.77846102],
            [1.8308367 ],
            [1.98545175],
            [2.23353144],
            [2.55741097],
            [2.92897973],
            [3.31122793],
            [3.66306661],
            [3.94570571],
            [4.1282799 ]]])

  • dX1_dr
    (rad, pol, tor)
    float64
    1.217 1.153 0.9674 ... 1.652 1.838

    long_name :

    radial derivative of the first reference coordinate

    symbol :

    \frac{\partial X^1}{\partial \rho}

    array([[[ 1.21690145],
            [ 1.15270203],
            [ 0.96744803],
            [ 0.6821681 ],
            [ 0.32867134],
            [-0.05486358],
            [-0.42903273],
            [-0.75800961],
            [-1.01298131],
            [-1.17371542],
            [-1.22854909],
            [-1.17371542],
            [-1.01298131],
            [-0.75800961],
            [-0.42903273],
            [-0.05486358],
            [ 0.32867134],
            [ 0.6821681 ],
            [ 0.96744803],
            [ 1.15270203]],
    ...
           [[ 1.9016538 ],
            [ 1.83827454],
            [ 1.65235265],
            [ 1.35366712],
            [ 0.95282396],
            [ 0.4623265 ],
            [-0.09332905],
            [-0.66475833],
            [-1.17429988],
            [-1.53080788],
            [-1.65934524],
            [-1.53080788],
            [-1.17429988],
            [-0.66475833],
            [-0.09332905],
            [ 0.4623265 ],
            [ 0.95282396],
            [ 1.35366712],
            [ 1.65235265],
            [ 1.83827454]]])

  • dX1_dt
    (rad, pol, tor)
    float64
    0.0 -0.03914 ... 0.754 0.3979

    long_name :

    poloidal derivative of the first reference coordinate

    symbol :

    \frac{\partial X^1}{\partial \theta}

    array([[[ 0.00000000e+00],
            [-3.91394205e-02],
            [-7.40883493e-02],
            [-1.01180066e-01],
            [-1.17720235e-01],
            [-1.22285608e-01],
            [-1.14814758e-01],
            [-9.64789057e-02],
            [-6.93871889e-02],
            [-3.62339436e-02],
            [-1.43257768e-17],
            [ 3.62339436e-02],
            [ 6.93871889e-02],
            [ 9.64789057e-02],
            [ 1.14814758e-01],
            [ 1.22285608e-01],
            [ 1.17720235e-01],
            [ 1.01180066e-01],
            [ 7.40883493e-02],
            [ 3.91394205e-02]],
    ...
           [[ 0.00000000e+00],
            [-3.97912344e-01],
            [-7.54001981e-01],
            [-1.02877677e+00],
            [-1.19020932e+00],
            [-1.22110182e+00],
            [-1.12447341e+00],
            [-9.22413839e-01],
            [-6.47639054e-01],
            [-3.32176440e-01],
            [-1.30458044e-16],
            [ 3.32176440e-01],
            [ 6.47639054e-01],
            [ 9.22413839e-01],
            [ 1.12447341e+00],
            [ 1.22110182e+00],
            [ 1.19020932e+00],
            [ 1.02877677e+00],
            [ 7.54001981e-01],
            [ 3.97912344e-01]]])

  • dX1_dz
    (rad, pol, tor)
    float64
    0.0 0.02456 ... -0.5363 -0.2756

    long_name :

    toroidal derivative of the first reference coordinate

    symbol :

    \frac{\partial X^1}{\partial \zeta}

    array([[[ 0.00000000e+00],
            [ 2.45567019e-02],
            [ 4.69180219e-02],
            [ 6.50012169e-02],
            [ 7.69955363e-02],
            [ 8.15729771e-02],
            [ 7.81021860e-02],
            [ 6.67918138e-02],
            [ 4.87086188e-02],
            [ 2.56633517e-02],
            [ 1.01776990e-17],
            [-2.56633517e-02],
            [-4.87086188e-02],
            [-6.67918138e-02],
            [-7.81021860e-02],
            [-8.15729771e-02],
            [-7.69955363e-02],
            [-6.50012169e-02],
            [-4.69180219e-02],
            [-2.45567019e-02]],
    ...
           [[ 0.00000000e+00],
            [ 2.75595712e-01],
            [ 5.36280003e-01],
            [ 7.60174181e-01],
            [ 9.18513626e-01],
            [ 9.84109411e-01],
            [ 9.42707235e-01],
            [ 7.99320262e-01],
            [ 5.75426084e-01],
            [ 2.99789321e-01],
            [ 1.18367681e-16],
            [-2.99789321e-01],
            [-5.75426084e-01],
            [-7.99320262e-01],
            [-9.42707235e-01],
            [-9.84109411e-01],
            [-9.18513626e-01],
            [-7.60174181e-01],
            [-5.36280003e-01],
            [-2.75595712e-01]]])

  • dX1_drr
    (rad, pol, tor)
    float64
    -0.02021 -0.06073 ... 3.7 3.349

    long_name :

    second radial derivative of the first reference coordinate

    symbol :

    \frac{\partial^2 X^1}{\partial \rho^2}

    array([[[-0.02020568],
            [-0.06073134],
            [-0.1707709 ],
            [-0.31738448],
            [-0.45281201],
            [-0.52825112],
            [-0.51302082],
            [-0.41107276],
            [-0.26427398],
            [-0.13693795],
            [-0.08679421],
            [-0.13693795],
            [-0.26427398],
            [-0.41107276],
            [-0.51302082],
            [-0.52825112],
            [-0.45281201],
            [-0.31738448],
            [-0.1707709 ],
            [-0.06073134]],
    ...
           [[ 3.21742397],
            [ 3.34883452],
            [ 3.6995497 ],
            [ 4.12539641],
            [ 4.37323455],
            [ 4.14041444],
            [ 3.22093432],
            [ 1.67107321],
            [-0.12515854],
            [-1.5721426 ],
            [-2.12739206],
            [-1.5721426 ],
            [-0.12515854],
            [ 1.67107321],
            [ 3.22093432],
            [ 4.14041444],
            [ 4.37323455],
            [ 4.12539641],
            [ 3.6995497 ],
            [ 3.34883452]]])

  • dX1_drt
    (rad, pol, tor)
    float64
    0.0 -0.4048 ... 0.777 0.4011

    long_name :

    radial-poloidal derivative of the first reference coordinate

    symbol :

    \frac{\partial^2 X^1}{\partial \rho\partial \theta}

    array([[[ 0.00000000e+00],
            [-4.04755981e-01],
            [-7.63294711e-01],
            [-1.03573857e+00],
            [-1.19415574e+00],
            [-1.22633571e+00],
            [-1.13650604e+00],
            [-9.42459397e-01],
            [-6.70015538e-01],
            [-3.47106281e-01],
            [-1.36845079e-16],
            [ 3.47106281e-01],
            [ 6.70015538e-01],
            [ 9.42459397e-01],
            [ 1.13650604e+00],
            [ 1.22633571e+00],
            [ 1.19415574e+00],
            [ 1.03573857e+00],
            [ 7.63294711e-01],
            [ 4.04755981e-01]],
    ...
           [[ 0.00000000e+00],
            [-4.01060834e-01],
            [-7.77044845e-01],
            [-1.11884430e+00],
            [-1.42695686e+00],
            [-1.68344306e+00],
            [-1.82802885e+00],
            [-1.76779242e+00],
            [-1.42599297e+00],
            [-8.02132831e-01],
            [-3.25383374e-16],
            [ 8.02132831e-01],
            [ 1.42599297e+00],
            [ 1.76779242e+00],
            [ 1.82802885e+00],
            [ 1.68344306e+00],
            [ 1.42695686e+00],
            [ 1.11884430e+00],
            [ 7.77044845e-01],
            [ 4.01060834e-01]]])

  • dX1_drz
    (rad, pol, tor)
    float64
    0.0 0.2407 ... -1.329 -0.7352

    long_name :

    radial-toroidal derivative of the first reference coordinate

    symbol :

    \frac{\partial^2 X^1}{\partial \rho\partial \zeta}

    array([[[ 0.00000000e+00],
            [ 2.40704078e-01],
            [ 4.62491625e-01],
            [ 6.45847767e-01],
            [ 7.71612396e-01],
            [ 8.23854851e-01],
            [ 7.93565028e-01],
            [ 6.81367872e-01],
            [ 4.98011730e-01],
            [ 2.62656710e-01],
            [ 1.04194239e-16],
            [-2.62656710e-01],
            [-4.98011730e-01],
            [-6.81367872e-01],
            [-7.93565028e-01],
            [-8.23854851e-01],
            [-7.71612396e-01],
            [-6.45847767e-01],
            [-4.62491625e-01],
            [-2.40704078e-01]],
    ...
           [[ 0.00000000e+00],
            [ 7.35151132e-01],
            [ 1.32855561e+00],
            [ 1.70046370e+00],
            [ 1.85297252e+00],
            [ 1.83807849e+00],
            [ 1.70322890e+00],
            [ 1.45817344e+00],
            [ 1.08626534e+00],
            [ 5.85407515e-01],
            [ 2.34333343e-16],
            [-5.85407515e-01],
            [-1.08626534e+00],
            [-1.45817344e+00],
            [-1.70322890e+00],
            [-1.83807849e+00],
            [-1.85297252e+00],
            [-1.70046370e+00],
            [-1.32855561e+00],
            [-7.35151132e-01]]])

  • dX1_dtt
    (rad, pol, tor)
    float64
    -0.1269 -0.1201 ... -1.024 -1.222

    long_name :

    second poloidal derivative of the first reference coordinate

    symbol :

    \frac{\partial^2 X^1}{\partial \theta^2}

    array([[[-0.12686503],
            [-0.12005278],
            [-0.10046937],
            [-0.07055614],
            [-0.03398185],
            [ 0.00494309],
            [ 0.04197994],
            [ 0.07361114],
            [ 0.09741437],
            [ 0.11205469],
            [ 0.11697885],
            [ 0.11205469],
            [ 0.09741437],
            [ 0.07361114],
            [ 0.04197994],
            [ 0.00494309],
            [-0.03398185],
            [-0.07055614],
            [-0.10046937],
            [-0.12005278]],
    ...
           [[-1.28894397],
            [-1.22195978],
            [-1.02374975],
            [-0.70810232],
            [-0.31015894],
            [ 0.1118366 ],
            [ 0.49111436],
            [ 0.77722114],
            [ 0.95463093],
            [ 1.04100436],
            [ 1.06527077],
            [ 1.04100436],
            [ 0.95463093],
            [ 0.77722114],
            [ 0.49111436],
            [ 0.1118366 ],
            [-0.31015894],
            [-0.70810232],
            [-1.02374975],
            [-1.22195978]]])

  • dX1_dtz
    (rad, pol, tor)
    float64
    0.07934 0.07582 ... 0.7852 0.8628

    long_name :

    poloidal-toroidal derivative of the first reference coordinate

    symbol :

    \frac{\partial^2 X^1}{\partial \theta\partial \zeta}

    array([[[ 0.07934173],
            [ 0.07582084],
            [ 0.06542241],
            [ 0.04872657],
            [ 0.02691514],
            [ 0.00188274],
            [-0.0238688 ],
            [-0.04756297],
            [-0.06658601],
            [-0.07886719],
            [-0.08310722],
            [-0.07886719],
            [-0.06658601],
            [-0.04756297],
            [-0.0238688 ],
            [ 0.00188274],
            [ 0.02691514],
            [ 0.04872657],
            [ 0.06542241],
            [ 0.07582084]],
    ...
           [[ 0.88422422],
            [ 0.86279335],
            [ 0.78517478],
            [ 0.62442276],
            [ 0.36850134],
            [ 0.04116063],
            [-0.30190204],
            [-0.59898409],
            [-0.81061344],
            [-0.92939264],
            [-0.96654547],
            [-0.92939264],
            [-0.81061344],
            [-0.59898409],
            [-0.30190204],
            [ 0.04116063],
            [ 0.36850134],
            [ 0.62442276],
            [ 0.78517478],
            [ 0.86279335]]])

  • dX1_dzz
    (rad, pol, tor)
    float64
    0.09173 0.107 ... -1.923 -2.448

    long_name :

    second toroidal derivative of the first reference coordinate

    symbol :

    \frac{\partial^2 X^1}{\partial \zeta^2}

    array([[[ 0.09172721],
            [ 0.10696369],
            [ 0.15041165],
            [ 0.21574033],
            [ 0.29382861],
            [ 0.37460297],
            [ 0.44897293],
            [ 0.51029424],
            [ 0.5548978 ],
            [ 0.58157879],
            [ 0.59041087],
            [ 0.58157879],
            [ 0.5548978 ],
            [ 0.51029424],
            [ 0.44897293],
            [ 0.37460297],
            [ 0.29382861],
            [ 0.21574033],
            [ 0.15041165],
            [ 0.10696369]],
    ...
           [[-2.62962447],
            [-2.44815761],
            [-1.92314414],
            [-1.11716588],
            [-0.14069451],
            [ 0.85997647],
            [ 1.7388122 ],
            [ 2.39631475],
            [ 2.80802587],
            [ 3.01407052],
            [ 3.07370214],
            [ 3.01407052],
            [ 2.80802587],
            [ 2.39631475],
            [ 1.7388122 ],
            [ 0.85997647],
            [-0.14069451],
            [-1.11716588],
            [-1.92314414],
            [-2.44815761]]])

  • X2
    (rad, pol, tor)
    float64
    0.0 0.02097 ... -0.3398 -0.1804

    long_name :

    second reference coordinate

    symbol :

    X^2

    array([[[ 0.00000000e+00],
            [ 2.09705629e-02],
            [ 3.97618776e-02],
            [ 5.44569839e-02],
            [ 6.36185853e-02],
            [ 6.64287822e-02],
            [ 6.27382778e-02],
            [ 5.30326166e-02],
            [ 3.83375102e-02],
            [ 2.00902555e-02],
            [ 7.95298899e-18],
            [-2.00902555e-02],
            [-3.83375102e-02],
            [-5.30326166e-02],
            [-6.27382778e-02],
            [-6.64287822e-02],
            [-6.36185853e-02],
            [-5.44569839e-02],
            [-3.97618776e-02],
            [-2.09705629e-02]],
    ...
           [[ 0.00000000e+00],
            [ 1.80373819e-01],
            [ 3.39763791e-01],
            [ 4.61010601e-01],
            [ 5.33141594e-01],
            [ 5.51706309e-01],
            [ 5.17661452e-01],
            [ 4.35963206e-01],
            [ 3.14716396e-01],
            [ 1.64893678e-01],
            [ 6.52800060e-17],
            [-1.64893678e-01],
            [-3.14716396e-01],
            [-4.35963206e-01],
            [-5.17661452e-01],
            [-5.51706309e-01],
            [-5.33141594e-01],
            [-4.61010601e-01],
            [-3.39763791e-01],
            [-1.80373819e-01]]])

  • dX2_dr
    (rad, pol, tor)
    float64
    0.0 0.2137 ... -0.2008 -0.09554

    long_name :

    radial derivative of the second reference coordinate

    symbol :

    \frac{\partial X^2}{\partial \rho}

    array([[[ 0.00000000e+00],
            [ 2.13677202e-01],
            [ 4.03916737e-01],
            [ 5.50557924e-01],
            [ 6.39281710e-01],
            [ 6.62983299e-01],
            [ 6.21842305e-01],
            [ 5.22340374e-01],
            [ 3.75699188e-01],
            [ 1.96237797e-01],
            [ 7.75955249e-17],
            [-1.96237797e-01],
            [-3.75699188e-01],
            [-5.22340374e-01],
            [-6.21842305e-01],
            [-6.62983299e-01],
            [-6.39281710e-01],
            [-5.50557924e-01],
            [-4.03916737e-01],
            [-2.13677202e-01]],
    ...
           [[ 0.00000000e+00],
            [ 9.55359319e-02],
            [ 2.00837433e-01],
            [ 3.15593366e-01],
            [ 4.25308718e-01],
            [ 5.05412682e-01],
            [ 5.30962799e-01],
            [ 4.86545261e-01],
            [ 3.71789328e-01],
            [ 2.01190013e-01],
            [ 8.04827014e-17],
            [-2.01190013e-01],
            [-3.71789328e-01],
            [-4.86545261e-01],
            [-5.30962799e-01],
            [-5.05412682e-01],
            [-4.25308718e-01],
            [-3.15593366e-01],
            [-2.00837433e-01],
            [-9.55359319e-02]]])

  • dX2_dt
    (rad, pol, tor)
    float64
    0.06794 0.0644 ... 0.4543 0.5512

    long_name :

    poloidal derivative of the second reference coordinate

    symbol :

    \frac{\partial X^2}{\partial \theta}

    array([[[ 0.06793642],
            [ 0.06439593],
            [ 0.05420452],
            [ 0.03857741],
            [ 0.0193107 ],
            [-0.00149767],
            [-0.02173398],
            [-0.03950302],
            [-0.05327891],
            [-0.06197265],
            [-0.06494108],
            [-0.06197265],
            [-0.05327891],
            [-0.03950302],
            [-0.02173398],
            [-0.00149767],
            [ 0.0193107 ],
            [ 0.03857741],
            [ 0.05420452],
            [ 0.06439593]],
    ...
           [[ 0.58572448],
            [ 0.55122335],
            [ 0.45425149],
            [ 0.31179589],
            [ 0.14511247],
            [-0.02633639],
            [-0.18772564],
            [-0.32807268],
            [-0.43797471],
            [-0.50861017],
            [-0.5330517 ],
            [-0.50861017],
            [-0.43797471],
            [-0.32807268],
            [-0.18772564],
            [-0.02633639],
            [ 0.14511247],
            [ 0.31179589],
            [ 0.45425149],
            [ 0.55122335]]])

  • dX2_dz
    (rad, pol, tor)
    float64
    0.5625 0.5579 ... 1.495 1.643

    long_name :

    toroidal derivative of the second reference coordinate

    symbol :

    \frac{\partial X^2}{\partial \zeta}

    array([[[ 0.562512  ],
            [ 0.55791791],
            [ 0.54462713],
            [ 0.5240596 ],
            [ 0.49839983],
            [ 0.47033249],
            [ 0.44271025],
            [ 0.41821579],
            [ 0.39908864],
            [ 0.38696317],
            [ 0.38281418],
            [ 0.38696317],
            [ 0.39908864],
            [ 0.41821579],
            [ 0.44271025],
            [ 0.47033249],
            [ 0.49839983],
            [ 0.5240596 ],
            [ 0.54462713],
            [ 0.55791791]],
    ...
           [[ 1.69315477],
            [ 1.64287257],
            [ 1.49530833],
            [ 1.26210546],
            [ 0.96644304],
            [ 0.6429554 ],
            [ 0.33165349],
            [ 0.06789375],
            [-0.12587525],
            [-0.24153682],
            [-0.27963329],
            [-0.24153682],
            [-0.12587525],
            [ 0.06789375],
            [ 0.33165349],
            [ 0.6429554 ],
            [ 0.96644304],
            [ 1.26210546],
            [ 1.49530833],
            [ 1.64287257]]])

  • dX2_drr
    (rad, pol, tor)
    float64
    0.0 0.0735 0.1151 ... 1.215 0.7589

    long_name :

    second radial derivative of the second reference coordinate

    symbol :

    \frac{\partial^2 X^2}{\partial \rho^2}

    array([[[ 0.00000000e+00],
            [ 7.35031240e-02],
            [ 1.15079509e-01],
            [ 1.05776186e-01],
            [ 4.72246269e-02],
            [-3.91131974e-02],
            [-1.20499677e-01],
            [-1.65607439e-01],
            [-1.56304116e-01],
            [-9.42211803e-02],
            [-3.89782942e-17],
            [ 9.42211803e-02],
            [ 1.56304116e-01],
            [ 1.65607439e-01],
            [ 1.20499677e-01],
            [ 3.91131974e-02],
            [-4.72246269e-02],
            [-1.05776186e-01],
            [-1.15079509e-01],
            [-7.35031240e-02]],
    ...
           [[ 0.00000000e+00],
            [-7.58881762e-01],
            [-1.21517690e+00],
            [-1.23023757e+00],
            [-8.87096861e-01],
            [-4.12670548e-01],
            [-2.94723913e-02],
            [ 1.57427969e-01],
            [ 1.72488646e-01],
            [ 9.87427074e-02],
            [ 3.94032612e-17],
            [-9.87427074e-02],
            [-1.72488646e-01],
            [-1.57427969e-01],
            [ 2.94723913e-02],
            [ 4.12670548e-01],
            [ 8.87096861e-01],
            [ 1.23023757e+00],
            [ 1.21517690e+00],
            [ 7.58881762e-01]]])

  • dX2_drt
    (rad, pol, tor)
    float64
    0.693 0.6547 ... 0.3542 0.3163

    long_name :

    radial-poloidal derivative of the second reference coordinate

    symbol :

    \frac{\partial^2 X^2}{\partial \rho\partial \theta}

    array([[[ 0.69295492],
            [ 0.65474276],
            [ 0.54552428],
            [ 0.38035237],
            [ 0.18070995],
            [-0.02966969],
            [-0.22871651],
            [-0.39868924],
            [-0.52718741],
            [-0.6067362 ],
            [-0.63361554],
            [-0.6067362 ],
            [-0.52718741],
            [-0.39868924],
            [-0.22871651],
            [-0.02966969],
            [ 0.18070995],
            [ 0.38035237],
            [ 0.54552428],
            [ 0.65474276]],
    ...
           [[ 0.29769221],
            [ 0.31627444],
            [ 0.35416719],
            [ 0.36846103],
            [ 0.31641236],
            [ 0.17974946],
            [-0.02557163],
            [-0.25736976],
            [-0.46525847],
            [-0.60711518],
            [-0.65719113],
            [-0.60711518],
            [-0.46525847],
            [-0.25736976],
            [-0.02557163],
            [ 0.17974946],
            [ 0.31641236],
            [ 0.36846103],
            [ 0.35416719],
            [ 0.31627444]]])

  • dX2_drz
    (rad, pol, tor)
    float64
    0.9567 0.9091 ... 2.063 2.294

    long_name :

    radial-toroidal derivative of the second reference coordinate

    symbol :

    \frac{\partial^2 X^2}{\partial \rho\partial \zeta}

    array([[[ 0.95666155],
            [ 0.90910192],
            [ 0.77169634],
            [ 0.55979509],
            [ 0.29721162],
            [ 0.01315845],
            [-0.26205   ],
            [-0.50147767],
            [-0.68475678],
            [-0.79900656],
            [-0.83772147],
            [-0.79900656],
            [-0.68475678],
            [-0.50147767],
            [-0.26205   ],
            [ 0.01315845],
            [ 0.29721162],
            [ 0.55979509],
            [ 0.77169634],
            [ 0.90910192]],
    ...
           [[ 2.37734629],
            [ 2.2942426 ],
            [ 2.06281801],
            [ 1.72260468],
            [ 1.30564152],
            [ 0.82034544],
            [ 0.26604363],
            [-0.33157884],
            [-0.89509937],
            [-1.30718329],
            [-1.4592927 ],
            [-1.30718329],
            [-0.89509937],
            [-0.33157884],
            [ 0.26604363],
            [ 0.82034544],
            [ 1.30564152],
            [ 1.72260468],
            [ 2.06281801],
            [ 2.2942426 ]]])

  • dX2_dtt
    (rad, pol, tor)
    float64
    0.0 -0.02231 ... 0.3919 0.216

    long_name :

    second poloidal derivative of the second reference coordinate

    symbol :

    \frac{\partial^2 X^2}{\partial \theta^2}

    array([[[ 0.00000000e+00],
            [-2.23071553e-02],
            [-4.19173919e-02],
            [-5.65996965e-02],
            [-6.49273264e-02],
            [-6.64088429e-02],
            [-6.14060967e-02],
            [-5.09022272e-02],
            [-3.62199226e-02],
            [-1.87859256e-02],
            [-7.41008010e-18],
            [ 1.87859256e-02],
            [ 3.62199226e-02],
            [ 5.09022272e-02],
            [ 6.14060967e-02],
            [ 6.64088429e-02],
            [ 6.49273264e-02],
            [ 5.65996965e-02],
            [ 4.19173919e-02],
            [ 2.23071553e-02]],
    ...
           [[ 0.00000000e+00],
            [-2.16023416e-01],
            [-3.91946501e-01],
            [-5.03329295e-01],
            [-5.47331329e-01],
            [-5.36342745e-01],
            [-4.85410764e-01],
            [-4.03139716e-01],
            [-2.91756922e-01],
            [-1.54102851e-01],
            [-6.12486555e-17],
            [ 1.54102851e-01],
            [ 2.91756922e-01],
            [ 4.03139716e-01],
            [ 4.85410764e-01],
            [ 5.36342745e-01],
            [ 5.47331329e-01],
            [ 5.03329295e-01],
            [ 3.91946501e-01],
            [ 2.16023416e-01]]])

  • dX2_dtz
    (rad, pol, tor)
    float64
    0.0 -0.02898 ... 0.615 0.3185

    long_name :

    poloidal-toroidal derivative of the second reference coordinate

    symbol :

    \frac{\partial^2 X^2}{\partial \theta\partial \zeta}

    array([[[ 0.00000000e+00],
            [-2.89832756e-02],
            [-5.48664028e-02],
            [-7.48973864e-02],
            [-8.70153854e-02],
            [-9.01349417e-02],
            [-8.42756010e-02],
            [-7.04643221e-02],
            [-5.04333385e-02],
            [-2.62434912e-02],
            [-1.03624279e-17],
            [ 2.62434912e-02],
            [ 5.04333385e-02],
            [ 7.04643221e-02],
            [ 8.42756010e-02],
            [ 9.01349417e-02],
            [ 8.70153854e-02],
            [ 7.48973864e-02],
            [ 5.48664028e-02],
            [ 2.89832756e-02]],
    ...
           [[ 0.00000000e+00],
            [-3.18480186e-01],
            [-6.14975911e-01],
            [-8.57363676e-01],
            [-1.00616807e+00],
            [-1.03140968e+00],
            [-9.31160390e-01],
            [-7.35998700e-01],
            [-4.93610935e-01],
            [-2.43472506e-01],
            [-9.41449714e-17],
            [ 2.43472506e-01],
            [ 4.93610935e-01],
            [ 7.35998700e-01],
            [ 9.31160390e-01],
            [ 1.03140968e+00],
            [ 1.00616807e+00],
            [ 8.57363676e-01],
            [ 6.14975911e-01],
            [ 3.18480186e-01]]])

  • dX2_dzz
    (rad, pol, tor)
    float64
    0.0 0.07067 ... -1.459 -0.7254

    long_name :

    second toroidal derivative of the second reference coordinate

    symbol :

    \frac{\partial^2 X^2}{\partial \zeta^2}

    array([[[ 0.00000000e+00],
            [ 7.06738282e-02],
            [ 1.36619444e-01],
            [ 1.92647417e-01],
            [ 2.33116206e-01],
            [ 2.52613940e-01],
            [ 2.47139322e-01],
            [ 2.15337295e-01],
            [ 1.59309322e-01],
            [ 8.46969438e-02],
            [ 3.36929814e-17],
            [-8.46969438e-02],
            [-1.59309322e-01],
            [-2.15337295e-01],
            [-2.47139322e-01],
            [-2.52613940e-01],
            [-2.33116206e-01],
            [-1.92647417e-01],
            [-1.36619444e-01],
            [-7.06738282e-02]],
    ...
           [[ 0.00000000e+00],
            [ 7.25372560e-01],
            [ 1.45929439e+00],
            [ 2.15847426e+00],
            [ 2.71749315e+00],
            [ 3.00960955e+00],
            [ 2.94952919e+00],
            [ 2.53391645e+00],
            [ 1.83473659e+00],
            [ 9.57408599e-01],
            [ 3.78082856e-16],
            [-9.57408599e-01],
            [-1.83473659e+00],
            [-2.53391645e+00],
            [-2.94952919e+00],
            [-3.00960955e+00],
            [-2.71749315e+00],
            [-2.15847426e+00],
            [-1.45929439e+00],
            [-7.25372560e-01]]])

  • pos
    (xyz, rad, pol, tor)
    float64
    3.092 3.086 ... -0.3398 -0.1804

    long_name :

    position vector

    symbol :

    \mathbf{x}

    array([[[[ 3.09221255e+00],
             [ 3.08600840e+00],
             [ 3.06806121e+00],
             [ 3.04028358e+00],
             [ 3.00559738e+00],
             [ 2.96757662e+00],
             [ 2.93002788e+00],
             [ 2.89657746e+00],
             [ 2.87032732e+00],
             [ 2.85361591e+00],
             [ 2.84788378e+00],
             [ 2.85361591e+00],
             [ 2.87032732e+00],
             [ 2.89657746e+00],
             [ 2.93002788e+00],
             [ 2.96757662e+00],
             [ 3.00559738e+00],
             [ 3.04028358e+00],
             [ 3.06806121e+00],
             [ 3.08600840e+00]],
    ...
            [[ 0.00000000e+00],
             [ 1.80373819e-01],
             [ 3.39763791e-01],
             [ 4.61010601e-01],
             [ 5.33141594e-01],
             [ 5.51706309e-01],
             [ 5.17661452e-01],
             [ 4.35963206e-01],
             [ 3.14716396e-01],
             [ 1.64893678e-01],
             [ 6.52800060e-17],
             [-1.64893678e-01],
             [-3.14716396e-01],
             [-4.35963206e-01],
             [-5.17661452e-01],
             [-5.51706309e-01],
             [-5.33141594e-01],
             [-4.61010601e-01],
             [-3.39763791e-01],
             [-1.80373819e-01]]]])

  • e_q1
    (xyz, rad, pol, tor)
    float64
    1.0 1.0 1.0 1.0 ... 0.0 0.0 0.0 0.0

    long_name :

    first reference tangent basis vector

    symbol :

    \mathbf{e}_{q^1}

    array([[[[ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.],
             [ 1.]],
    ...
            [[ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.],
             [ 0.]]]])

  • e_q2
    (xyz, rad, pol, tor)
    float64
    0.0 0.0 0.0 0.0 ... 1.0 1.0 1.0 1.0

    long_name :

    second reference tangent basis vector

    symbol :

    \mathbf{e}_{q^2}

    array([[[[0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.],
             [0.]],
    ...
            [[1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.],
             [1.]]]])

  • e_q3
    (xyz, rad, pol, tor)
    float64
    -0.0 -0.0 -0.0 -0.0 ... 0.0 0.0 0.0

    long_name :

    toroidal reference tangent basis vector

    symbol :

    \mathbf{e}_{q^3}

    array([[[[-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ],
             [-0.        ]],
    ...
            [[ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ],
             [ 0.        ]]]])

  • iota
    (rad)
    float64
    0.6285 0.7125 0.9085

    long_name :

    rotational transform

    symbol :

    \iota

    array([0.6285, 0.7125, 0.9085])

  • LA
    (rad, pol, tor)
    float64
    0.0 -0.007488 ... 0.0721 0.02747

    long_name :

    straight field line potential

    symbol :

    \lambda

    array([[[ 0.00000000e+00],
            [-7.48794599e-03],
            [-1.42219376e-02],
            [-1.95201547e-02],
            [-2.28466041e-02],
            [-2.38779041e-02],
            [-2.25448193e-02],
            [-1.90318566e-02],
            [-1.37336396e-02],
            [-7.18616121e-03],
            [-2.84308571e-18],
            [ 7.18616121e-03],
            [ 1.37336396e-02],
            [ 1.90318566e-02],
            [ 2.25448193e-02],
            [ 2.38779041e-02],
            [ 2.28466041e-02],
            [ 1.95201547e-02],
            [ 1.42219376e-02],
            [ 7.48794599e-03]],
    ...
           [[ 0.00000000e+00],
            [-2.74728879e-02],
            [-7.20998300e-02],
            [-1.38707265e-01],
            [-2.15270632e-01],
            [-2.78629540e-01],
            [-3.06140693e-01],
            [-2.85738112e-01],
            [-2.19130677e-01],
            [-1.18342949e-01],
            [-4.72744252e-17],
            [ 1.18342949e-01],
            [ 2.19130677e-01],
            [ 2.85738112e-01],
            [ 3.06140693e-01],
            [ 2.78629540e-01],
            [ 2.15270632e-01],
            [ 1.38707265e-01],
            [ 7.20998300e-02],
            [ 2.74728879e-02]]])

  • dLA_dr
    (rad, pol, tor)
    float64
    0.0 -0.07578 ... -0.04945 -0.05337

    long_name :

    radial derivative of the straight field line potential

    symbol :

    \frac{\partial \lambda}{\partial \rho}

    array([[[ 0.00000000e+00],
            [-7.57821674e-02],
            [-1.43997152e-01],
            [-1.97586209e-01],
            [-2.30784884e-01],
            [-2.40114979e-01],
            [-2.25132922e-01],
            [-1.88441142e-01],
            [-1.34852085e-01],
            [-7.01302050e-02],
            [-2.76832243e-17],
            [ 7.01302050e-02],
            [ 1.34852085e-01],
            [ 1.88441142e-01],
            [ 2.25132922e-01],
            [ 2.40114979e-01],
            [ 2.30784884e-01],
            [ 1.97586209e-01],
            [ 1.43997152e-01],
            [ 7.57821674e-02]],
    ...
           [[ 0.00000000e+00],
            [ 5.33712071e-02],
            [ 4.94488629e-02],
            [-4.49738420e-02],
            [-2.23183527e-01],
            [-4.37118485e-01],
            [-6.12204829e-01],
            [-6.74423532e-01],
            [-5.80000827e-01],
            [-3.35650095e-01],
            [-1.37240113e-16],
            [ 3.35650095e-01],
            [ 5.80000827e-01],
            [ 6.74423532e-01],
            [ 6.12204829e-01],
            [ 4.37118485e-01],
            [ 2.23183527e-01],
            [ 4.49738420e-02],
            [-4.94488629e-02],
            [-5.33712071e-02]]])

  • dLA_dt
    (rad, pol, tor)
    float64
    -0.02424 -0.02302 ... -0.1079

    long_name :

    poloidal derivative of the straight field line potential

    symbol :

    \frac{\partial \lambda}{\partial \theta}

    array([[[-0.02424241],
            [-0.02302355],
            [-0.01948068],
            [-0.01396076],
            [-0.00704216],
            [ 0.00051343],
            [ 0.0078729 ],
            [ 0.01427808],
            [ 0.01916337],
            [ 0.02219281],
            [ 0.02321556],
            [ 0.02219281],
            [ 0.01916337],
            [ 0.01427808],
            [ 0.0078729 ],
            [ 0.00051343],
            [-0.00704216],
            [-0.01396076],
            [-0.01948068],
            [-0.02302355]],
    ...
           [[-0.07683021],
            [-0.10788915],
            [-0.17903567],
            [-0.23828019],
            [-0.236167  ],
            [-0.15459738],
            [-0.01397681],
            [ 0.14273375],
            [ 0.27458211],
            [ 0.35803296],
            [ 0.38602498],
            [ 0.35803296],
            [ 0.27458211],
            [ 0.14273375],
            [-0.01397681],
            [-0.15459738],
            [-0.236167  ],
            [-0.23828019],
            [-0.17903567],
            [-0.10788915]]])

  • dLA_dz
    (rad, pol, tor)
    float64
    -0.07923 -0.07993 ... -0.381

    long_name :

    toroidal derivative of the straight field line potential

    symbol :

    \frac{\partial \lambda}{\partial \zeta}

    array([[[-0.07922588],
            [-0.07992827],
            [-0.08205826],
            [-0.08563845],
            [-0.09058212],
            [-0.09658762],
            [-0.10310124],
            [-0.10937515],
            [-0.11459962],
            [-0.11805985],
            [-0.11927036],
            [-0.11805985],
            [-0.11459962],
            [-0.10937515],
            [-0.10310124],
            [-0.09658762],
            [-0.09058212],
            [-0.08563845],
            [-0.08205826],
            [-0.07992827]],
    ...
           [[-0.44044753],
            [-0.38102792],
            [-0.23191698],
            [-0.06333068],
            [ 0.05344944],
            [ 0.08248584],
            [ 0.03418106],
            [-0.05152066],
            [-0.13321568],
            [-0.18658659],
            [-0.20450816],
            [-0.18658659],
            [-0.13321568],
            [-0.05152066],
            [ 0.03418106],
            [ 0.08248584],
            [ 0.05344944],
            [-0.06333068],
            [-0.23191698],
            [-0.38102792]]])

  • dLA_drr
    (rad, pol, tor)
    float64
    0.0 -0.008466 ... -0.03052 0.03538

    long_name :

    second radial derivative of the straight field line potential

    symbol :

    \frac{\partial^2 \lambda}{\partial \rho^2}

    array([[[ 0.00000000e+00],
            [-8.46610828e-03],
            [-2.31901236e-02],
            [-4.13875179e-02],
            [-5.10121760e-02],
            [-4.03981796e-02],
            [-9.84188413e-03],
            [ 2.52274136e-02],
            [ 4.34248080e-02],
            [ 3.27041836e-02],
            [ 1.44099805e-17],
            [-3.27041836e-02],
            [-4.34248080e-02],
            [-2.52274136e-02],
            [ 9.84188413e-03],
            [ 4.03981796e-02],
            [ 5.10121760e-02],
            [ 4.13875179e-02],
            [ 2.31901236e-02],
            [ 8.46610828e-03]],
    ...
           [[ 0.00000000e+00],
            [-3.53776492e-02],
            [ 3.05206306e-02],
            [ 1.66065796e-01],
            [ 2.04403086e-01],
            [-1.35937028e-02],
            [-4.69476669e-01],
            [-9.24294552e-01],
            [-1.05983972e+00],
            [-7.09257404e-01],
            [-3.03354965e-16],
            [ 7.09257404e-01],
            [ 1.05983972e+00],
            [ 9.24294552e-01],
            [ 4.69476669e-01],
            [ 1.35937028e-02],
            [-2.04403086e-01],
            [-1.66065796e-01],
            [-3.05206306e-02],
            [ 3.53776492e-02]]])

  • dLA_drt
    (rad, pol, tor)
    float64
    -0.2453 -0.2331 ... -0.1485 0.105

    long_name :

    radial-poloidal derivative of the straight field line potential

    symbol :

    \frac{\partial^2 \lambda}{\partial \rho\partial \theta}

    array([[[-0.24528207],
            [-0.23312334],
            [-0.19735978],
            [-0.14068408],
            [-0.06877592],
            [ 0.00961569],
            [ 0.08433444],
            [ 0.14662691],
            [ 0.19141696],
            [ 0.21756482],
            [ 0.22605068],
            [ 0.21756482],
            [ 0.19141696],
            [ 0.14662691],
            [ 0.08433444],
            [ 0.00961569],
            [-0.06877592],
            [-0.14068408],
            [-0.19735978],
            [-0.23312334]],
    ...
           [[ 0.20303457],
            [ 0.10499893],
            [-0.14847566],
            [-0.44916863],
            [-0.65903415],
            [-0.66184257],
            [-0.41184963],
            [ 0.04012743],
            [ 0.55751686],
            [ 0.96588484],
            [ 1.12065057],
            [ 0.96588484],
            [ 0.55751686],
            [ 0.04012743],
            [-0.41184963],
            [-0.66184257],
            [-0.65903415],
            [-0.44916863],
            [-0.14847566],
            [ 0.10499893]]])

  • dLA_drz
    (rad, pol, tor)
    float64
    0.1264 0.1227 ... -0.8597 -1.353

    long_name :

    radial-toroidal derivative of the straight field line potential

    symbol :

    \frac{\partial^2 \lambda}{\partial \rho\partial \zeta}

    array([[[ 0.12637681],
            [ 0.12270325],
            [ 0.10999992],
            [ 0.08442102],
            [ 0.04283779],
            [-0.01430298],
            [-0.08172721],
            [-0.15023288],
            [-0.20908974],
            [-0.2487155 ],
            [-0.26267252],
            [-0.2487155 ],
            [-0.20908974],
            [-0.15023288],
            [-0.08172721],
            [-0.01430298],
            [ 0.04283779],
            [ 0.08442102],
            [ 0.10999992],
            [ 0.12270325]],
    ...
           [[-1.54133897],
            [-1.35305949],
            [-0.85971725],
            [-0.23940295],
            [ 0.31007234],
            [ 0.65712434],
            [ 0.77584608],
            [ 0.72754498],
            [ 0.60896701],
            [ 0.50453286],
            [ 0.46448207],
            [ 0.50453286],
            [ 0.60896701],
            [ 0.72754498],
            [ 0.77584608],
            [ 0.65712434],
            [ 0.31007234],
            [-0.23940295],
            [-0.85971725],
            [-1.35305949]]])

  • dLA_dtt
    (rad, pol, tor)
    float64
    0.0 0.0077 ... 0.2382 0.1852

    long_name :

    second poloidal derivative of the straight field line potential

    symbol :

    \frac{\partial^2 \lambda}{\partial \theta^2}

    array([[[ 0.00000000e+00],
            [ 7.69966463e-03],
            [ 1.46711209e-02],
            [ 2.01605638e-02],
            [ 2.34743479e-02],
            [ 2.41757453e-02],
            [ 2.22672087e-02],
            [ 1.82073716e-02],
            [ 1.27179288e-02],
            [ 6.49252550e-03],
            [ 2.54503066e-18],
            [-6.49252550e-03],
            [-1.27179288e-02],
            [-1.82073716e-02],
            [-2.22672087e-02],
            [-2.41757453e-02],
            [-2.34743479e-02],
            [-2.01605638e-02],
            [-1.46711209e-02],
            [-7.69966463e-03]],
    ...
           [[ 0.00000000e+00],
            [-1.85206555e-01],
            [-2.38229875e-01],
            [-1.11011412e-01],
            [ 1.34454755e-01],
            [ 3.73033431e-01],
            [ 4.97934999e-01],
            [ 4.77111976e-01],
            [ 3.49893512e-01],
            [ 1.78273688e-01],
            [ 6.93891551e-17],
            [-1.78273688e-01],
            [-3.49893512e-01],
            [-4.77111976e-01],
            [-4.97934999e-01],
            [-3.73033431e-01],
            [-1.34454755e-01],
            [ 1.11011412e-01],
            [ 2.38229875e-01],
            [ 1.85206555e-01]]])

  • dLA_dtz
    (rad, pol, tor)
    float64
    0.0 -0.004486 ... -0.5479 -0.3618

    long_name :

    poloidal-toroidal derivative of the straight field line potential

    symbol :

    \frac{\partial^2 \lambda}{\partial \theta\partial \zeta}

    array([[[ 0.00000000e+00],
            [-4.48636529e-03],
            [-9.09191545e-03],
            [-1.36604165e-02],
            [-1.76541223e-02],
            [-2.02810643e-02],
            [-2.07817203e-02],
            [-1.87209763e-02],
            [-1.41524752e-02],
            [-7.61396323e-03],
            [-3.04025722e-18],
            [ 7.61396323e-03],
            [ 1.41524752e-02],
            [ 1.87209763e-02],
            [ 2.07817203e-02],
            [ 2.02810643e-02],
            [ 1.76541223e-02],
            [ 1.36604165e-02],
            [ 9.09191545e-03],
            [ 4.48636529e-03]],
    ...
           [[ 0.00000000e+00],
            [ 3.61798250e-01],
            [ 5.47870360e-01],
            [ 4.85448557e-01],
            [ 2.38331024e-01],
            [-4.69227256e-02],
            [-2.37732342e-01],
            [-2.84838149e-01],
            [-2.22416347e-01],
            [-1.14265116e-01],
            [-4.43533856e-17],
            [ 1.14265116e-01],
            [ 2.22416347e-01],
            [ 2.84838149e-01],
            [ 2.37732342e-01],
            [ 4.69227256e-02],
            [-2.38331024e-01],
            [-4.85448557e-01],
            [-5.47870360e-01],
            [-3.61798250e-01]]])

  • dLA_dzz
    (rad, pol, tor)
    float64
    0.0 -0.1016 ... 1.484 0.9025

    long_name :

    second toroidal derivative of the straight field line potential

    symbol :

    \frac{\partial^2 \lambda}{\partial \zeta^2}

    array([[[ 0.00000000e+00],
            [-1.01621481e-01],
            [-1.94504636e-01],
            [-2.70422696e-01],
            [-3.22155948e-01],
            [-3.44023947e-01],
            [-3.32546183e-01],
            [-2.87234448e-01],
            [-2.11316389e-01],
            [-1.12011715e-01],
            [-4.45178740e-17],
            [ 1.12011715e-01],
            [ 2.11316389e-01],
            [ 2.87234448e-01],
            [ 3.32546183e-01],
            [ 3.44023947e-01],
            [ 3.22155948e-01],
            [ 2.70422696e-01],
            [ 1.94504636e-01],
            [ 1.01621481e-01]],
    ...
           [[ 0.00000000e+00],
            [-9.02508130e-01],
            [-1.48410613e+00],
            [-1.61118647e+00],
            [-1.39158380e+00],
            [-1.06486177e+00],
            [-8.21235037e-01],
            [-6.88342790e-01],
            [-5.61262449e-01],
            [-3.32159366e-01],
            [-1.37881102e-16],
            [ 3.32159366e-01],
            [ 5.61262449e-01],
            [ 6.88342790e-01],
            [ 8.21235037e-01],
            [ 1.06486177e+00],
            [ 1.39158380e+00],
            [ 1.61118647e+00],
            [ 1.48410613e+00],
            [ 9.02508130e-01]]])

  • dPhi_dr
    (rad)
    float64
    0.03183 0.1592 0.2865

    long_name :

    toroidal magnetic flux gradient

    symbol :

    \frac{d\Phi}{d\rho}

    array([0.03183099, 0.15915494, 0.2864789 ])

  • Jac_h
    (rad, pol, tor)
    float64
    3.092 3.086 3.068 ... 3.946 4.128

    long_name :

    reference Jacobian determinant

    symbol :

    \mathcal{J}_h

    array([[[3.09221255],
            [3.0860084 ],
            [3.06806121],
            [3.04028358],
            [3.00559738],
            [2.96757662],
            [2.93002788],
            [2.89657746],
            [2.87032732],
            [2.85361591],
            [2.84788378],
            [2.85361591],
            [2.87032732],
            [2.89657746],
            [2.93002788],
            [2.96757662],
            [3.00559738],
            [3.04028358],
            [3.06806121],
            [3.0860084 ]],
    ...
           [[4.19133503],
            [4.1282799 ],
            [3.94570571],
            [3.66306661],
            [3.31122793],
            [2.92897973],
            [2.55741097],
            [2.23353144],
            [1.98545175],
            [1.8308367 ],
            [1.77846102],
            [1.8308367 ],
            [1.98545175],
            [2.23353144],
            [2.55741097],
            [2.92897973],
            [3.31122793],
            [3.66306661],
            [3.94570571],
            [4.1282799 ]]])

  • Jac_l
    (rad, pol, tor)
    float64
    0.08267 0.08259 ... 0.902 1.051

    long_name :

    logical Jacobian determinant

    symbol :

    \mathcal{J}_l

    array([[[0.08267193],
            [0.08259252],
            [0.08236558],
            [0.08202176],
            [0.08160327],
            [0.08115548],
            [0.08072126],
            [0.08033849],
            [0.08003925],
            [0.07984873],
            [0.07978331],
            [0.07984873],
            [0.08003925],
            [0.08033849],
            [0.08072126],
            [0.08115548],
            [0.08160327],
            [0.08202176],
            [0.08236558],
            [0.08259252]],
    ...
           [[1.11384518],
            [1.05131477],
            [0.90201548],
            [0.74674297],
            [0.64447304],
            [0.60498434],
            [0.61457381],
            [0.66688513],
            [0.75509894],
            [0.84541505],
            [0.8845168 ],
            [0.84541505],
            [0.75509894],
            [0.66688513],
            [0.61457381],
            [0.60498434],
            [0.64447304],
            [0.74674297],
            [0.90201548],
            [1.05131477]]])

  • Jac
    (rad, pol, tor)
    float64
    0.2556 0.2549 0.2527 ... 3.559 4.34

    long_name :

    Jacobian determinant

    symbol :

    \mathcal{J}

    array([[[0.25563918],
            [0.25488122],
            [0.25270266],
            [0.24936942],
            [0.24526657],
            [0.24083511],
            [0.23651555],
            [0.23270667],
            [0.22973886],
            [0.2278576 ],
            [0.22721359],
            [0.2278576 ],
            [0.22973886],
            [0.23270667],
            [0.23651555],
            [0.24083511],
            [0.24526657],
            [0.24936942],
            [0.25270266],
            [0.25488122]],
    ...
           [[4.66849834],
            [4.34012166],
            [3.55908763],
            [2.73536925],
            [2.13399712],
            [1.77198686],
            [1.5717178 ],
            [1.48950889],
            [1.49921251],
            [1.5478169 ],
            [1.57307866],
            [1.5478169 ],
            [1.49921251],
            [1.48950889],
            [1.5717178 ],
            [1.77198686],
            [2.13399712],
            [2.73536925],
            [3.55908763],
            [4.34012166]]])

  • e_theta
    (xyz, rad, pol, tor)
    float64
    0.0 -0.03914 ... 0.4543 0.5512

    long_name :

    poloidal tangent basis vector

    symbol :

    \mathbf{e}_\theta

    array([[[[ 0.00000000e+00],
             [-3.91394205e-02],
             [-7.40883493e-02],
             [-1.01180066e-01],
             [-1.17720235e-01],
             [-1.22285608e-01],
             [-1.14814758e-01],
             [-9.64789057e-02],
             [-6.93871889e-02],
             [-3.62339436e-02],
             [-1.43257768e-17],
             [ 3.62339436e-02],
             [ 6.93871889e-02],
             [ 9.64789057e-02],
             [ 1.14814758e-01],
             [ 1.22285608e-01],
             [ 1.17720235e-01],
             [ 1.01180066e-01],
             [ 7.40883493e-02],
             [ 3.91394205e-02]],
    ...
            [[ 5.85724480e-01],
             [ 5.51223348e-01],
             [ 4.54251494e-01],
             [ 3.11795893e-01],
             [ 1.45112470e-01],
             [-2.63363893e-02],
             [-1.87725643e-01],
             [-3.28072677e-01],
             [-4.37974710e-01],
             [-5.08610175e-01],
             [-5.33051702e-01],
             [-5.08610175e-01],
             [-4.37974710e-01],
             [-3.28072677e-01],
             [-1.87725643e-01],
             [-2.63363893e-02],
             [ 1.45112470e-01],
             [ 3.11795893e-01],
             [ 4.54251494e-01],
             [ 5.51223348e-01]]]])

  • e_zeta
    (xyz, rad, pol, tor)
    float64
    0.0 0.02456 0.04692 ... 1.495 1.643

    long_name :

    toroidal tangent basis vector

    symbol :

    \mathbf{e}_\zeta

    array([[[[ 0.00000000e+00],
             [ 2.45567019e-02],
             [ 4.69180219e-02],
             [ 6.50012169e-02],
             [ 7.69955363e-02],
             [ 8.15729771e-02],
             [ 7.81021860e-02],
             [ 6.67918138e-02],
             [ 4.87086188e-02],
             [ 2.56633517e-02],
             [ 1.01776990e-17],
             [-2.56633517e-02],
             [-4.87086188e-02],
             [-6.67918138e-02],
             [-7.81021860e-02],
             [-8.15729771e-02],
             [-7.69955363e-02],
             [-6.50012169e-02],
             [-4.69180219e-02],
             [-2.45567019e-02]],
    ...
            [[ 1.69315477e+00],
             [ 1.64287257e+00],
             [ 1.49530833e+00],
             [ 1.26210546e+00],
             [ 9.66443042e-01],
             [ 6.42955398e-01],
             [ 3.31653490e-01],
             [ 6.78937468e-02],
             [-1.25875252e-01],
             [-2.41536818e-01],
             [-2.79633287e-01],
             [-2.41536818e-01],
             [-1.25875252e-01],
             [ 6.78937468e-02],
             [ 3.31653490e-01],
             [ 6.42955398e-01],
             [ 9.66443042e-01],
             [ 1.26210546e+00],
             [ 1.49530833e+00],
             [ 1.64287257e+00]]]])

  • B_contra_t
    (rad, pol, tor)
    float64
    0.08812 0.08847 ... 0.09179 0.08512

    long_name :

    poloidal magnetic field

    symbol :

    B^\theta

    array([[[0.0881227 ],
            [0.08847248],
            [0.0895035 ],
            [0.09115686],
            [0.09332334],
            [0.09583426],
            [0.09846114],
            [0.10093091],
            [0.10295862],
            [0.10429206],
            [0.10475725],
            [0.10429206],
            [0.10295862],
            [0.10093091],
            [0.09846114],
            [0.09583426],
            [0.09332334],
            [0.09115686],
            [0.0895035 ],
            [0.08847248]],
    ...
           [[0.08277715],
            [0.08511801],
            [0.0917947 ],
            [0.10178113],
            [0.11478644],
            [0.13354254],
            [0.15936317],
            [0.18464184],
            [0.19905754],
            [0.20268495],
            [0.20269384],
            [0.20268495],
            [0.19905754],
            [0.18464184],
            [0.15936317],
            [0.13354254],
            [0.11478644],
            [0.10178113],
            [0.0917947 ],
            [0.08511801]]])

  • B_contra_z
    (rad, pol, tor)
    float64
    0.1215 0.122 ... 0.06608 0.05889

    long_name :

    toroidal magnetic field

    symbol :

    B^\zeta

    array([[[0.12149675],
            [0.12201027],
            [0.12350839],
            [0.12586388],
            [0.12886726],
            [0.13223708],
            [0.13564263],
            [0.13873893],
            [0.14120805],
            [0.14279711],
            [0.14334513],
            [0.14279711],
            [0.14120805],
            [0.13873893],
            [0.13564263],
            [0.13223708],
            [0.12886726],
            [0.12586388],
            [0.12350839],
            [0.12201027]],
    ...
           [[0.05664962],
            [0.05888566],
            [0.06608125],
            [0.07977594],
            [0.10254092],
            [0.13667709],
            [0.17972363],
            [0.21978325],
            [0.24355512],
            [0.25135259],
            [0.25241389],
            [0.25135259],
            [0.24355512],
            [0.21978325],
            [0.17972363],
            [0.13667709],
            [0.10254092],
            [0.07977594],
            [0.06608125],
            [0.05888566]]])

  • B
    (rad, pol, tor, xyz)
    float64
    0.0 -0.3757 ... -0.2431 0.1437

    long_name :

    magnetic field

    symbol :

    \mathbf{B}

    array([[[[ 0.00000000e+00, -3.75693772e-01,  7.43301201e-02]],
    
            [[-4.66591590e-04, -3.76524720e-01,  7.37689822e-02]],
    
            [[-8.36396967e-04, -3.78931313e-01,  7.21175169e-02]],
    
            [[-1.04195135e-03, -3.82661898e-01,  6.94767713e-02]],
    
            [[-1.06384177e-03, -3.87323087e-01,  6.60295579e-02]],
    
            [[-9.32179172e-04, -3.92423658e-01,  6.20518658e-02]],
    
            [[-7.10806068e-04, -3.97436678e-01,  5.79104293e-02]],
    
            [[-4.71078937e-04, -4.01868049e-01,  5.40357345e-02]],
    
            [[-2.65959740e-04, -4.05313337e-01,  5.08690078e-02]],
    
            [[-1.14260028e-04, -4.07488119e-01,  4.87939687e-02]],
    
    ...
    
            [[ 3.43460554e-18, -4.48908269e-01, -1.78629622e-01]],
    
            [[-8.02565559e-03, -4.60185547e-01, -1.63798535e-01]],
    
            [[-1.12305275e-02, -4.83566933e-01, -1.17839732e-01]],
    
            [[-5.36101790e-03, -4.90892799e-01, -4.56540339e-02]],
    
            [[ 9.77287393e-03, -4.59627195e-01,  2.96894177e-02]],
    
            [[ 2.85638263e-02, -4.00324431e-01,  8.43602456e-02]],
    
            [[ 4.24346597e-02, -3.39536369e-01,  1.15756906e-01]],
    
            [[ 4.40664554e-02, -2.92224580e-01,  1.32420588e-01]],
    
            [[ 3.37753370e-02, -2.60737160e-01,  1.40509723e-01]],
    
            [[ 1.76408727e-02, -2.43096492e-01,  1.43660674e-01]]]])

• Indexes: (4)
  • rho
    PandasIndex

    PandasIndex(Index([0.1, 0.5, 0.9], dtype='float64', name='rho'))

  • theta
    PandasIndex

    PandasIndex(Index([               0.0, 0.3141592653589793, 0.6283185307179586,
           0.9424777960769379, 1.2566370614359172, 1.5707963267948966,
           1.8849555921538759,  2.199114857512855, 2.5132741228718345,
            2.827433388230814,  3.141592653589793, 3.4557519189487724,
           3.7699111843077517,  4.084070449666731,   4.39822971502571,
             4.71238898038469,  5.026548245743669,  5.340707511102648,
            5.654866776461628,  5.969026041820607],
          dtype='float64', name='theta'))

  • zeta
    PandasIndex

    PandasIndex(Index([0.0], dtype='float64', name='zeta'))

  • xyz
    PandasIndex

    PandasIndex(Index(['x', 'y', 'z'], dtype='object', name='xyz'))

• Attributes: (0)

An existing dataset can be extended using state.compute:

state.compute(ev, "J", "V")

The individual quantities, within an xarray.Dataset can be accessed using ev.B or ev["B"] and are enriched with coordinate information which can be used to a variety of operations, e.g.:

mean_B = ev.B.mean(dim=("rad", "pol", "tor"))
B2 = xr.dot(ev.B, ev.B, dim="xyz")
JxB = xr.cross(ev.J, ev.B, dim="xyz")

Indexing is best done using ev.B.sel(rho=0.5) (by value) or ev.B.isel(rad=0) (by position) and the raw numpy array can be extracted with ev.B.values. To ensure a particular order, use ev.B.transpose("rad", "pol", "tor", "xyz").values. To convert a scalar (e.g. the volume V) into a python scalar use the ev.V.item() method. The .squeeze() method can be used to remove any dimensions of length 1.

For more information, see the xarray documentation.

Computing integrals

Specifying "int" for rho, theta and zeta in the Evaluations function, chooses the grid to be the integration points used by GVEC internally. The functions radial_integral, fluxsurface_integral and volume_integral can then be used to perform integration with the apropriate weights. E.g. to compute the volume averaged plasma beta, one could use:

ev = state.evaluate("mod_B", "mu0", "p", "Jac", "V", rho="int", theta="int", zeta="int")
beta = ev.p / (ev.mod_B**2 / (2 * ev.mu0))
beta_avg = gvec.volume_integral(beta * ev.Jac) / ev.V
beta_avg.item()

0.008802419491901461

For supported quantities which require integration (e.g. the volume V computed above), an auxiliary dataset is created to compute them if the grid does not conform to the integration points.

Boozer transform

To evaluate the equilibrium in Boozer angles, you can use state.evaluate_sfl(..., sfl="boozer"), which performs a Boozer transform to obtain a set of \(\vartheta,\zeta\) points which correspond to your desired grid in \(\vartheta_B,\zeta_B\). The evaluations with this new dataset work the same as above, note however that the suffixes t and z still refer to components/derivatives with respect to \(\vartheta,\zeta\). Some additional quantities, like B_contra_t_B or e_zeta_B are now also available.

The argument MNfactor specifies how many fourier modes should be used for the Boozer potential \(\nu_B\) (and recomputed straight fieldline potential \(\lambda\)) relative to maximum fourier modes used for computing the equilibrium solution. This parameter has a strong influence on the accuracy and required computational effort!

ev = state.evaluate_sfl(
    "mod_B",
    "B_contra_t_B",
    "B_contra_z_B",
    rho=1.0,
    theta=40,
    zeta=50,
    sfl="boozer",
).squeeze()

fig, axs = plt.subplots(1, 2, figsize=(15, 6), layout="constrained")
fig.suptitle(r"Streamlines of $\mathbf{B}$ in boozer and logical coordinates at $\rho=1.0$.")

streamplot_kwargs = dict(
    color="black",
    broken_streamlines=False,
    density=1,
    integration_direction="both",
    start_points=np.vstack(
        [
            np.linspace(0, 2 * np.pi / state.nfp, 22)[1:-1],
            np.linspace(2 * np.pi, 0, 22)[1:-1],
        ]
    ).T,
)

ax = axs[0]
c = ax.contour(ev.zeta_B, ev.theta_B, ev.mod_B, levels=21, cmap="plasma")
fig.colorbar(c, ax=[axs[0], axs[1]], label=f"${ev.mod_B.attrs['symbol']}$")
ax.streamplot(
    ev.zeta_B.values,
    ev.theta_B.values,
    ev.B_contra_z_B,
    ev.B_contra_t_B,
    **streamplot_kwargs,
)
ax.set_xlabel(r"$\zeta_B$")
ax.set_ylabel(r"$\theta_B$")
ax.set_title("Grid in boozer coordinates")

ev = state.evaluate("mod_B", "B_contra_t", "B_contra_z", rho=1.0, theta=40, zeta=50).squeeze()

ax = axs[1]
ax.contour(ev.zeta, ev.theta, ev.mod_B, levels=21, cmap=c.cmap, norm=c.norm)
ax.streamplot(
    ev.zeta.values,
    ev.theta.values,
    ev.B_contra_z.values,
    ev.B_contra_t.values,
    **streamplot_kwargs,
)
ax.set_xlabel(r"$\zeta$")
ax.set_ylabel(r"$\theta$")
ax.set_title("Grid in logical coordinates");

Note

The Boozer transform recomputes \(\lambda\) with a higher resolution (to satisfy the integrability condition for \(\nu_B\))! Therefore some quantities will differ between the equilibrium evaluation and Boozer evaluation.

In particular \(\langle B_\vartheta \rangle, \langle B_\zeta \rangle\) will differ from \(B_{\vartheta_B},B_{\zeta_B}\) by an offset.

Note

Currently the Boozer transform is performed for each surface individually and radial derivatives are therfore not available. This means that that \(\frac{\partial \mathbf{B}}{\partial \rho}\) and \(\mathbf{J}\) are not available!

Field-aligned grid

The state.evaluate_sfl method and EvaluationsBoozer factory function can also be used to generate a non-tensorproduct grid by providing (up to 3D) arrays for the values of \(\theta_B,\zeta_B\). This can be used for example to create a field-aligned grid:

rho = [0.5, 1.0]  # radial positions
alpha = np.linspace(0, 2 * np.pi, 100, endpoint=False)  # fieldline label
phi = np.linspace(0, 2 * np.pi / state.nfp, 101)  # angle along the fieldline

# evaluate the rotational transform (fieldline angle) on the desired surfaces
iota = state.evaluate("iota", rho=rho, theta=None, zeta=None).iota

# 3D toroidal and poloidal arrays that correspond to fieldline coordinates for each surface
theta_B = alpha[None, :, None] + iota.data[:, None, None] * phi[None, None, :]

# create the grid
ev = gvec.EvaluationsBoozer(rho=rho, theta_B=theta_B, zeta_B=phi, state=state, MNfactor=5)

# set the fiedline label as poloidal coordinate & index (not necessary, but good practice)
ev["alpha"] = ("pol", alpha)
ev["alpha"].attrs = dict(symbol=r"\alpha", long_name="fieldline label")
ev = ev.set_coords("alpha").set_xindex("alpha")

state.compute(ev, "B", "B_contra_t_B", "B_contra_z_B", "mod_B")

ev = ev.sel(rho=0.5)

fig, ax = plt.subplots(figsize=(8, 6), layout="constrained")

c = ax.contour(ev.zeta_B, ev.alpha, ev.mod_B, levels=21, cmap="plasma")
fig.colorbar(c, ax=ax, label=f"${ev.mod_B.attrs['symbol']}$")
ax.set_xlabel(f"${ev.zeta_B.attrs['symbol']}$")
ax.set_ylabel(f"${ev.alpha.attrs['symbol']}$")
ax.set_title(f"${ev.mod_B.attrs['symbol']}$ on a field-aligned grid at $\\rho=0.5$");

---

Available quantities for evaluation

The following table contains the quantities that can be evaluated with the python bindings, it can be generated with

gvec.table_of_quantities(markdown=True)

label	long name	symbol
B	magnetic field	\(\mathbf{B}\)
B_contra_t	poloidal magnetic field	\(B^\theta\)
B_contra_t_B	contravariant poloidal magnetic field in Boozer coordinates	\(B^{\theta_B}\)
B_contra_z	toroidal magnetic field	\(B^\zeta\)
B_contra_z_B	contravariant toroidal magnetic field in Boozer coordinates	\(B^{\zeta_B}\)
B_theta_avg	average poloidal magnetic field	\(\overline{B_\theta}\)
B_zeta_avg	average toroidal magnetic field	\(\overline{B_\zeta}\)
F	MHD force	\(F\)
F_r_avg	radial force balance	\(\overline{F_\rho}\)
I_pol	poloidal current	\(I_{pol}\)
I_tor	toroidal current	\(I_{tor}\)
J	current density	\(\mathbf{J}\)
J_contra_r	contravariant radial current density	\(J^{\rho}\)
J_contra_t	contravariant poloidal current density	\(J^{\theta}\)
J_contra_z	contravariant toroidal current density	\(J^{\zeta}\)
Jac	Jacobian determinant	\(\mathcal{J}\)
Jac_B	Jacobian determinant in Boozer coordinates	\(\mathcal{J}_B\)
Jac_P	Jacobian determinant in PEST coordinates	\(\mathcal{J}_P\)
Jac_h	reference Jacobian determinant	\(\mathcal{J}_h\)
Jac_l	logical Jacobian determinant	\(\mathcal{J}_l\)
LA	straight field line potential	\(\lambda\)
N_FP	number of field periods	\(N_{FP}\)
Phi	toroidal magnetic flux	\(\Phi\)
Phi_edge	toroidal magnetic flux at the edge	\(\Phi_0\)
V	plasma volume	\(V\)
W_MHD	total MHD energy	\(W_{MHD}\)
X1	first reference coordinate	\(X^1\)
X2	second reference coordinate	\(X^2\)
chi	poloidal magnetic flux	\(\chi\)
dB_contra_t_dr	radial derivative of the poloidal magnetic field	\(\frac{\partial B^\theta}{\partial \rho}\)
dB_contra_t_dt	poloidal derivative of the poloidal magnetic field	\(\frac{\partial B^\theta}{\partial \theta}\)
dB_contra_t_dz	toroidal derivative of the poloidal magnetic field	\(\frac{\partial B^\theta}{\partial \zeta}\)
dB_contra_z_dr	radial derivative of the toroidal magnetic field	\(\frac{\partial B^\zeta}{\partial \rho}\)
dB_contra_z_dt	poloidal derivative of the toroidal magnetic field	\(\frac{\partial B^\zeta}{\partial \theta}\)
dB_contra_z_dz	toroidal derivative of the toroidal magnetic field	\(\frac{\partial B^\zeta}{\partial \zeta}\)
dJac_dr	radial derivative of the Jacobian determinant	\(\frac{\partial \mathcal{J}}{\partial \rho}\)
dJac_dt	poloidal derivative of the Jacobian determinant	\(\frac{\partial \mathcal{J}}{\partial \theta}\)
dJac_dz	toroidal derivative of the Jacobian determinant	\(\frac{\partial \mathcal{J}}{\partial \zeta}\)
dJac_h_dr	radial derivative of the reference Jacobian determinant	\(\frac{\partial \mathcal{J}_h}{\partial \rho}\)
dJac_h_dt	poloidal derivative of the reference Jacobian determinant	\(\frac{\partial \mathcal{J}_h}{\partial \theta}\)
dJac_h_dz	toroidal derivative of the reference Jacobian determinant	\(\frac{\partial \mathcal{J}_h}{\partial \zeta}\)
dJac_l_dr	radial derivative of the logical Jacobian determinant	\(\frac{\partial \mathcal{J}_l}{\partial \rho}\)
dJac_l_dt	poloidal derivative of the logical Jacobian determinant	\(\frac{\partial \mathcal{J}_l}{\partial \theta}\)
dJac_l_dz	toroidal derivative of the logical Jacobian determinant	\(\frac{\partial \mathcal{J}_l}{\partial \zeta}\)
dLA_dr	radial derivative of the straight field line potential	\(\frac{\partial \lambda}{\partial \rho}\)
dLA_drr	second radial derivative of the straight field line potential	\(\frac{\partial^2 \lambda}{\partial \rho^2}\)
dLA_drt	radial-poloidal derivative of the straight field line potential	\(\frac{\partial^2 \lambda}{\partial \rho\partial \theta}\)
dLA_drz	radial-toroidal derivative of the straight field line potential	\(\frac{\partial^2 \lambda}{\partial \rho\partial \zeta}\)
dLA_dt	poloidal derivative of the straight field line potential	\(\frac{\partial \lambda}{\partial \theta}\)
dLA_dtt	second poloidal derivative of the straight field line potential	\(\frac{\partial^2 \lambda}{\partial \theta^2}\)
dLA_dtz	poloidal-toroidal derivative of the straight field line potential	\(\frac{\partial^2 \lambda}{\partial \theta\partial \zeta}\)
dLA_dz	toroidal derivative of the straight field line potential	\(\frac{\partial \lambda}{\partial \zeta}\)
dLA_dzz	second toroidal derivative of the straight field line potential	\(\frac{\partial^2 \lambda}{\partial \zeta^2}\)
dNU_B_dt	poloidal derivative of the Boozer potential computed from the magnetic field	\(\left.\frac{\partial \nu_B}{\partial \theta}\right|\)
dNU_B_dz	toroidal derivative of the Boozer potential computed from the magnetic field	\(\left.\frac{\partial \nu_B}{\partial \zeta}\right|\)
dPhi_dr	toroidal magnetic flux gradient	\(\frac{d\Phi}{d\rho}\)
dPhi_drr	toroidal magnetic flux curvature	\(\frac{d^2\Phi}{d\rho^2}\)
dV_dPhi_n	derivative of the plasma volume w.r.t. normalized toroidal magnetic flux	\(\frac{dV}{d\Phi_n}\)
dV_dPhi_n2	second derivative of the plasma volume w.r.t. normalized toroidal magnetic flux	\(\frac{d^2V}{d\Phi_n^2}\)
dX1_dr	radial derivative of the first reference coordinate	\(\frac{\partial X^1}{\partial \rho}\)
dX1_drr	second radial derivative of the first reference coordinate	\(\frac{\partial^2 X^1}{\partial \rho^2}\)
dX1_drt	radial-poloidal derivative of the first reference coordinate	\(\frac{\partial^2 X^1}{\partial \rho\partial \theta}\)
dX1_drz	radial-toroidal derivative of the first reference coordinate	\(\frac{\partial^2 X^1}{\partial \rho\partial \zeta}\)
dX1_dt	poloidal derivative of the first reference coordinate	\(\frac{\partial X^1}{\partial \theta}\)
dX1_dtt	second poloidal derivative of the first reference coordinate	\(\frac{\partial^2 X^1}{\partial \theta^2}\)
dX1_dtz	poloidal-toroidal derivative of the first reference coordinate	\(\frac{\partial^2 X^1}{\partial \theta\partial \zeta}\)
dX1_dz	toroidal derivative of the first reference coordinate	\(\frac{\partial X^1}{\partial \zeta}\)
dX1_dzz	second toroidal derivative of the first reference coordinate	\(\frac{\partial^2 X^1}{\partial \zeta^2}\)
dX2_dr	radial derivative of the second reference coordinate	\(\frac{\partial X^2}{\partial \rho}\)
dX2_drr	second radial derivative of the second reference coordinate	\(\frac{\partial^2 X^2}{\partial \rho^2}\)
dX2_drt	radial-poloidal derivative of the second reference coordinate	\(\frac{\partial^2 X^2}{\partial \rho\partial \theta}\)
dX2_drz	radial-toroidal derivative of the second reference coordinate	\(\frac{\partial^2 X^2}{\partial \rho\partial \zeta}\)
dX2_dt	poloidal derivative of the second reference coordinate	\(\frac{\partial X^2}{\partial \theta}\)
dX2_dtt	second poloidal derivative of the second reference coordinate	\(\frac{\partial^2 X^2}{\partial \theta^2}\)
dX2_dtz	poloidal-toroidal derivative of the second reference coordinate	\(\frac{\partial^2 X^2}{\partial \theta\partial \zeta}\)
dX2_dz	toroidal derivative of the second reference coordinate	\(\frac{\partial X^2}{\partial \zeta}\)
dX2_dzz	second toroidal derivative of the second reference coordinate	\(\frac{\partial^2 X^2}{\partial \zeta^2}\)
dchi_dr	poloidal magnetic flux gradient	\(\frac{d\chi}{d\rho}\)
dchi_drr	poloidal magnetic flux curvature	\(\frac{d^2\chi}{d\rho^2}\)
dg_rr_dr	radial derivative of the rr component of the metric tensor	\(\frac{\partial g_{\rho\rho}}{\partial \rho}\)
dg_rr_dt	poloidal derivative of the rr component of the metric tensor	\(\frac{\partial g_{\rho\rho}}{\partial \theta}\)
dg_rr_dz	toroidal derivative of the rr component of the metric tensor	\(\frac{\partial g_{\rho\rho}}{\partial \zeta}\)
dg_rt_dr	radial derivative of the rt component of the metric tensor	\(\frac{\partial g_{\rho\theta}}{\partial \rho}\)
dg_rt_dt	poloidal derivative of the rt component of the metric tensor	\(\frac{\partial g_{\rho\theta}}{\partial \theta}\)
dg_rt_dz	toroidal derivative of the rt component of the metric tensor	\(\frac{\partial g_{\rho\theta}}{\partial \zeta}\)
dg_rz_dr	radial derivative of the rz component of the metric tensor	\(\frac{\partial g_{\rho\zeta}}{\partial \rho}\)
dg_rz_dt	poloidal derivative of the rz component of the metric tensor	\(\frac{\partial g_{\rho\zeta}}{\partial \theta}\)
dg_rz_dz	toroidal derivative of the rz component of the metric tensor	\(\frac{\partial g_{\rho\zeta}}{\partial \zeta}\)
dg_tt_dr	radial derivative of the tt component of the metric tensor	\(\frac{\partial g_{\theta\theta}}{\partial \rho}\)
dg_tt_dt	poloidal derivative of the tt component of the metric tensor	\(\frac{\partial g_{\theta\theta}}{\partial \theta}\)
dg_tt_dz	toroidal derivative of the tt component of the metric tensor	\(\frac{\partial g_{\theta\theta}}{\partial \zeta}\)
dg_tz_dr	radial derivative of the tz component of the metric tensor	\(\frac{\partial g_{\theta\zeta}}{\partial \rho}\)
dg_tz_dt	poloidal derivative of the tz component of the metric tensor	\(\frac{\partial g_{\theta\zeta}}{\partial \theta}\)
dg_tz_dz	toroidal derivative of the tz component of the metric tensor	\(\frac{\partial g_{\theta\zeta}}{\partial \zeta}\)
dg_zz_dr	radial derivative of the zz component of the metric tensor	\(\frac{\partial g_{\zeta\zeta}}{\partial \rho}\)
dg_zz_dt	poloidal derivative of the zz component of the metric tensor	\(\frac{\partial g_{\zeta\zeta}}{\partial \theta}\)
dg_zz_dz	toroidal derivative of the zz component of the metric tensor	\(\frac{\partial g_{\zeta\zeta}}{\partial \zeta}\)
diota_dr	rotational transform gradient	\(\frac{d\iota}{d\rho}\)
diota_drr	rotational transform curvature	\(\frac{d^2\iota}{d\rho^2}\)
dp_dr	pressure gradient	\(\frac{dp}{d\rho}\)
dp_drr	pressure curvature	\(\frac{d^2p}{d\rho^2}\)
e_q1	first reference tangent basis vector	\(\mathbf{e}_{q^1}\)
e_q2	second reference tangent basis vector	\(\mathbf{e}_{q^2}\)
e_q3	toroidal reference tangent basis vector	\(\mathbf{e}_{q^3}\)
e_rho	radial tangent basis vector	\(\mathbf{e}_\rho\)
e_theta	poloidal tangent basis vector	\(\mathbf{e}_\theta\)
e_theta_B	poloidal tangent basis vector in Boozer coordinates	\(\mathbf{e}_{\theta_B}\)
e_zeta	toroidal tangent basis vector	\(\mathbf{e}_\zeta\)
e_zeta_B	toroidal tangent basis vector in Boozer coordinates	\(\mathbf{e}_{\zeta_B}\)
g_rr	rr component of the metric tensor	\(g_{\rho\rho}\)
g_rt	rt component of the metric tensor	\(g_{\rho\theta}\)
g_rz	rz component of the metric tensor	\(g_{\rho\zeta}\)
g_tt	tt component of the metric tensor	\(g_{\theta\theta}\)
g_tz	tz component of the metric tensor	\(g_{\theta\zeta}\)
g_zz	zz component of the metric tensor	\(g_{\zeta\zeta}\)
gamma	adiabatic index	\(\gamma\)
grad_rho	radial reciprocal basis vector	\(\nabla\rho\)
grad_theta	poloidal reciprocal basis vector	\(\nabla\theta\)
grad_theta_P	poloidal reciprocal basis vector in PEST coordinates	\(\nabla \theta_P\)
grad_zeta	toroidal reciprocal basis vector	\(\nabla\zeta\)
iota	rotational transform	\(\iota\)
iota_0	geometric contribution to the rotational transform	\(\iota_0\)
iota_avg	average rotational transform	\(\bar{\iota}\)
iota_curr	toroidal current contribution to the rotational transform	\(\iota_{curr}\)
iota_curr_0	factor to the toroidal current contribution to the rotational transform	\(\iota_{curr,0}\)
major_radius	major radius	\(r_{maj}\)
minor_radius	minor radius	\(r_{min}\)
mod_B	modulus of the magnetic field	\(\left|\mathbf{B}\right|\)
mod_F	modulus of the MHD force	\(\left|F\right|\)
mod_J	modulus of the current density	\(\left|\mathbf{J}\right|\)
mod_e_rho	modulus of the radial tangent basis vector	\(\left|\mathbf{e}_\rho\right|\)
mod_e_theta	modulus of the poloidal tangent basis vector	\(\left|\mathbf{e}_\theta\right|\)
mod_e_zeta	modulus of the toroidal tangent basis vector	\(\left|\mathbf{e}_\zeta\right|\)
mod_grad_rho	modulus of the radial reciprocal basis vector	\(\left|\nabla\rho\right|\)
mod_grad_theta	modulus of the poloidal reciprocal basis vector	\(\left|\nabla\theta\right|\)
mod_grad_zeta	modulus of the toroidal reciprocal basis vector	\(\left|\nabla\zeta\right|\)
mu0	magnetic constant	\(\mu_0\)
p	pressure	\(p\)
pos	position vector	\(\mathbf{x}\)
shear	global magnetic shear	\(s_g\)
theta_P	poloidal angle in PEST coordinates	\(\theta_P\)
xyz	cartesian vector components	\((x,y,z)\)