ion density profile

[ddudt]

Resolves #2069 by adding an optional ion_density profile to the Equilibrium class. If the profile is not supplied, it defaults to the previous assumption of $n_i = n_e / Z_{eff}$ for backwards compatibility.

[ddudt]

I ran a test case comparing the old assumption to using the "correct" profiles. The difference is small, but I still think this PR is beneficial to give users more flexibility in prescribing accurate profiles.

Details:
I have profile data for each ion and impurity species (which obeys quasi-neutrality). In the old code, I had to assume $Z_{eff}\approx1$ to get the correct ion density. In the new code, I can set the ion density directly and use the correct atomic number profile of $Z_{eff}\approx1.2$. The only difference between the two cases is the Zeff profile. This plot shows the final bootstrap current profile, after optimizing for self-consistency. The differences in other equilibrium quantities are also negligible.

[ddudt]

I'm confused why there is no d_ni_d_s term related to $\partial_\rho n_i$ in our bootstrap computations. When comparing this dnds_term to equation A1 from Landreman et al., it looks like we are assuming that $\frac{d \ln n_e}{d \psi_t} = \frac{d \ln n_i}{d \psi_t}$?

[github-actions]

Memory benchmark result

|               Test Name                |      %Δ      |    Master (MB)     |      PR (MB)       |    Δ (MB)    |    Time PR (s)     |  Time Master (s)   |
| -------------------------------------- | ------------ | ------------------ | ------------------ | ------------ | ------------------ | ------------------ |
  test_objective_jac_w7x                 |    2.97 %    |     3.822e+03      |     3.935e+03      |    113.34    |       40.36        |       37.04        |
  test_proximal_jac_w7x_with_eq_update   |   -0.82 %    |     6.511e+03      |     6.458e+03      |    -53.13    |       164.94       |       164.64       |
  test_proximal_freeb_jac                |    0.25 %    |     1.321e+04      |     1.324e+04      |    33.61     |       88.85        |       85.72        |
  test_proximal_freeb_jac_blocked        |    0.66 %    |     7.492e+03      |     7.541e+03      |    49.25     |       75.39        |       74.79        |
  test_proximal_freeb_jac_batched        |   -0.38 %    |     7.531e+03      |     7.502e+03      |    -28.49    |       74.79        |       74.70        |
  test_proximal_jac_ripple               |   -3.73 %    |     3.592e+03      |     3.458e+03      |   -133.83    |       66.58        |       66.88        |
  test_proximal_jac_ripple_bounce1d      |   -0.06 %    |     3.566e+03      |     3.564e+03      |    -2.02     |       77.63        |       78.53        |
  test_eq_solve                          |   -5.52 %    |     2.059e+03      |     1.945e+03      |   -113.76    |       95.33        |       95.34        |

For the memory plots, go to the summary of Memory Benchmarks workflow and download the artifact.

[ddudt]

Remember to update this PR with the changes from #2046

[dpanici]

I ran a test case comparing the old assumption to using the "correct" profiles. The difference is small, but I still think this PR is beneficial to give users more flexibility in prescribing accurate profiles.

Details: I have profile data for each ion and impurity species (which obeys quasi-neutrality). In the old code, I had to assume Z e f f ≈ 1 to get the correct ion density. In the new code, I can set the ion density directly and use the correct atomic number profile of Z e f f ≈ 1.2 . The only difference between the two cases is the Zeff profile. This plot shows the final bootstrap current profile, after optimizing for self-consistency. The differences in other equilibrium quantities are also negligible.

How would these compare to say SFINCS for the "correct" profiles?