Add approximate diamagnetic beta and stored-energy formulas

[HongSik-Yun-Fusion]

Summary

Add simple closed-form diamagnetic estimates to vaft.formula.equilibrium, without evaluating the full Shafranov/virial surface integrals.

Formulas

For a circular, high-aspect-ratio plasma,

$$\beta_{p,\mathrm{dia}} \simeq 1- \frac{8\pi B_{\phi0}\Phi_{\mathrm{dia}}} {\mu_0^2 I_p^2},$$ $$W_{\mathrm{dia}} \simeq \frac{3}{8}\mu_0 R_0 I_p^2\beta_{p,\mathrm{dia}}.$$

For an elongated elliptical approximation,

$$\beta_{p\perp,\mathrm{dia}} \simeq 1- \frac{1+\kappa^2}{2\kappa} \frac{8\pi B_{\phi0}\Phi_{\mathrm{dia}}} {\mu_0^2 I_p^2},$$ $$W_{\mathrm{dia}} \simeq \frac{3}{4}\mu_0 V \left(\frac{I_p}{L_C}\right)^2 \beta_{p\perp,\mathrm{dia}}.$$

Suggested pure helpers:

• beta_p_diamagnetic_from_flux_circular(...)
• stored_energy_diamagnetic_from_beta_p_circular(...)
• beta_p_diamagnetic_from_flux_elliptic(...)
• stored_energy_diamagnetic_from_beta_p_geometry(...)

Document the diamagnetic-flux sign convention explicitly; do not apply abs() implicitly.

Scope / limitations

These are closed-form approximations and should not be presented as valid spherical-tokamak formulas. Conventional high-aspect-ratio approximations can give large errors at low aspect ratio. No aspect-ratio argument or automatic validity correction is required in this issue; the limitation should simply be stated in the docstrings.

Tests should at least verify that the elliptical expressions reduce to the circular results for kappa = 1 and circular geometry.

References

• Lao et al., Nucl. Fusion 25, 1421 (1985).
• Bongard et al., Phys. Plasmas 23, 072508 (2016) — limitations at low aspect ratio.
• Yang et al., Rev. Sci. Instrum. 89, 103508 (2018) — VEST diamagnetic-flux measurement with low-aspect-ratio treatment.
• Giannone et al., ASDEX Upgrade diamagnetic-flux/stored-energy formulation.