PEST coordinate system basis vectors

[unalmis]

Generalizes theta_PEST basis vectors for toroidal stream function and changes e_phi docstring since it is somewhat misleading.

While debugging a test that only assumes nested flux surfaces, I checked the computation of the pest sfl Jacobian which looked wrong. We were computing

# e_rho fixed theta and zeta, e_vartheta fixed rho and zeta, e_phi at fixed rho and theta
dot(data["e_rho|t,z"], cross(data["e_vartheta|r,z"], data["e_phi|r,t"]))

when we should have been computing

# e_rho fixed vartheta and phi, e_vartheta at fixed rho and phi, e_phi at fixed rho and vartheta
dot(data["e_rho|v,p"], cross(data["e_vartheta|r,p"], data["e_phi|r,v"]))

Assuming $\zeta = \phi$, only the second of these vectors are the same between the two compuations; the first and third are of course not. However, the triple scalar product suppresses the differences so that the final result is equal when $\zeta = \phi$.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 95.23%. Comparing base (8085dc7) to head (2dad8f8).
Report is 1829 commits behind head on master.

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1090   +/-   ##
=======================================
  Coverage   95.23%   95.23%           
=======================================
  Files          87       87           
  Lines       21908    21918   +10     
=======================================
+ Hits        20863    20873   +10     
  Misses       1045     1045           

Files with missing lines	Coverage Δ
desc/compute/_basis_vectors.py	100.00% <100.00%> (ø)
desc/compute/_metric.py	100.00% <100.00%> (ø)
desc/compute/_surface.py	100.00% <ø> (ø)

[github-actions]

|             benchmark_name             |         dt(%)          |         dt(s)          |        t_new(s)        |        t_old(s)        | 
| -------------------------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- |
 test_build_transform_fft_lowres         |     +0.89 +/- 6.57     | +4.49e-03 +/- 3.31e-02 |  5.08e-01 +/- 3.2e-02  |  5.04e-01 +/- 6.3e-03  |
 test_build_transform_fft_midres         |     +0.84 +/- 1.74     | +4.92e-03 +/- 1.02e-02 |  5.91e-01 +/- 7.5e-03  |  5.87e-01 +/- 6.9e-03  |
 test_build_transform_fft_highres        |     +1.24 +/- 3.87     | +1.20e-02 +/- 3.77e-02 |  9.86e-01 +/- 3.6e-02  |  9.74e-01 +/- 1.2e-02  |
 test_equilibrium_init_lowres            |     +0.93 +/- 0.64     | +3.37e-02 +/- 2.33e-02 |  3.65e+00 +/- 1.5e-02  |  3.62e+00 +/- 1.8e-02  |
 test_equilibrium_init_medres            |     +0.14 +/- 0.87     | +5.87e-03 +/- 3.55e-02 |  4.08e+00 +/- 2.0e-02  |  4.08e+00 +/- 3.0e-02  |
 test_equilibrium_init_highres           |     +0.16 +/- 0.59     | +8.91e-03 +/- 3.22e-02 |  5.47e+00 +/- 2.3e-02  |  5.46e+00 +/- 2.3e-02  |
 test_objective_compile_dshape_current   |     +0.40 +/- 0.92     | +1.52e-02 +/- 3.48e-02 |  3.78e+00 +/- 3.4e-02  |  3.76e+00 +/- 4.5e-03  |
 test_objective_compile_atf              |     +0.02 +/- 3.74     | +1.89e-03 +/- 3.04e-01 |  8.11e+00 +/- 2.1e-01  |  8.11e+00 +/- 2.2e-01  |
 test_objective_compute_dshape_current   |     +0.29 +/- 4.92     | +3.63e-06 +/- 6.23e-05 |  1.27e-03 +/- 2.4e-05  |  1.27e-03 +/- 5.7e-05  |
 test_objective_compute_atf              |     -0.97 +/- 5.77     | -4.07e-05 +/- 2.42e-04 |  4.15e-03 +/- 1.8e-04  |  4.19e-03 +/- 1.7e-04  |
 test_objective_jac_dshape_current       |     +1.18 +/- 10.20    | +4.22e-04 +/- 3.65e-03 |  3.62e-02 +/- 2.2e-03  |  3.58e-02 +/- 2.9e-03  |
 test_objective_jac_atf                  |     -1.75 +/- 3.22     | -3.25e-02 +/- 5.97e-02 |  1.83e+00 +/- 4.2e-02  |  1.86e+00 +/- 4.2e-02  |
 test_perturb_1                          |     +0.00 +/- 1.37     | +5.67e-04 +/- 1.78e-01 |  1.30e+01 +/- 1.7e-01  |  1.30e+01 +/- 3.6e-02  |
 test_perturb_2                          |     +0.04 +/- 0.89     | +7.20e-03 +/- 1.59e-01 |  1.78e+01 +/- 5.6e-02  |  1.78e+01 +/- 1.5e-01  |
 test_proximal_jac_atf                   |     +0.84 +/- 1.03     | +6.13e-02 +/- 7.48e-02 |  7.32e+00 +/- 4.6e-02  |  7.26e+00 +/- 5.9e-02  |
 test_proximal_freeb_compute             |     +1.15 +/- 0.79     | +2.00e-03 +/- 1.37e-03 |  1.77e-01 +/- 1.1e-03  |  1.75e-01 +/- 8.1e-04  |
 test_proximal_freeb_jac                 |     +1.17 +/- 1.54     | +8.40e-02 +/- 1.11e-01 |  7.28e+00 +/- 7.2e-02  |  7.19e+00 +/- 8.4e-02  |
 test_solve_fixed_iter                   |     +0.66 +/- 4.38     | +1.18e-01 +/- 7.81e-01 |  1.80e+01 +/- 5.9e-01  |  1.78e+01 +/- 5.1e-01  |

[dpanici]

remove e_phi|R,Z

The base branch was changed.

[unalmis]

Just fixed merge conflicts since last review

[dpanici]

Add notation on basis to list of variables

[ddudt]

Do we really care about backwards compatibility here? I doubt a lot of users have code that depends on computing "e_phi"

[unalmis]

We could deprecate it later in that case

done in clebsch pr