Add ITER normalised Inductance l_i(3)

documentation/physics-models/plasma_current/plasma_inductance.md

@@ -42,7 +42,7 @@ This is only recommended for high aspect ratio tokamaks[^2].
 ---------
 
 
-#### Menard Inductance Relation
+### Menard Inductance Relation
 
 This can be activated by stating `ind_plasma_internal_norm = 2` in the input file.
 
@@ -60,9 +60,43 @@ This relation is based off of data from NSTX for $l_i$ in the range of 0.4-0.85.
 
 **It is recommended to use this switch with [`i_beta_norm_max = 3`](../plasma_beta/plasma_beta.md#menard-beta-relation) as they are self-consistent with each other.**
 
+--------------
+
+## ITER Definitions
+
+The generally agreed definition of $l_i$ is:
+
+$$
+l_i = \frac{\langle B_{\text{p}}^2 \rangle}{B_{\text{p}}(a)^2}
+$$
+
+where $B_{\text{p}}(a)^2$ is the square of the average poloidal field at the plasma edge. In the second term, the average on the
+boundary is made explicit, with Lp the line integral of the poloidal circumference of the last closed flux surface.
+
+In the unpublished ITER Physics Guidelines document[^4], different approximations for the denominator are used.
+
+Although the motivation for these approximations is not explicitly stated, it is likely due to the challenges involved in generating free-boundary equilibria that include an X-point at that time[^5][^6].
+
+### ITER Version 3 | `calculate_normalised_internal_inductance_iter_3()`
+
+$$
+\overbrace{l_i(3)}^{\texttt{ind_plasma_internal_norm_iter_3}} = \frac{2V\langle B_{\text{p}}^2 \rangle}{\mu_0^2I_{\text{p}}^2R_0}
+$$
+
+
+
+
+
 [^1]: T. T. S et al., “Profile Optimization and High Beta Discharges and Stability of High Elongation Plasmas in the DIII-D Tokamak,” Osti.gov, Oct. 1990. https://www.osti.gov/biblio/6194284 (accessed Dec. 19, 2024).
 
 [^2]: Tokamaks 4th Edition, Wesson, page 116
 
 [^3]: J. E. Menard et al., “Fusion nuclear science facilities and pilot plants based on the spherical tokamak,” Nuclear Fusion, vol. 56, no. 10, p. 106023, Aug. 2016, doi: https://doi.org/10.1088/0029-5515/56/10/106023.
 
+[^4]: N. A. Uckan, International Atomic Energy Agency, Vienna (Austria) and ITER Physics Group, "ITER physics design guidelines: 1989", no. No. 10. Feb. 1990
+
+[^5]: T. C. Luce, D. A. Humphreys, G. L. Jackson, and W. M. Solomon, “Inductive flux usage and its optimization in tokamak operation,”
+Nuclear Fusion, vol. 54, no. 9, p. 093005, Jul. 2014, doi: https://doi.org/10.1088/0029-5515/54/9/093005.
+
+[^6]: G. L. Jackson et al., “ITER startup studies in the DIII-D tokamak,” Nuclear Fusion, vol. 48, no. 12, p. 125002, Nov. 2008, doi: https://doi.org/10.1088/0029-5515/48/12/125002.
+

process/data_structure/physics_variables.py

@@ -1132,6 +1132,9 @@
 ind_plasma_internal_norm: float = None
 """Plasma normalised internal inductance"""
 
+ind_plasma_internal_norm_iter_3: float = None
+"""Plasma normalised internal inductance (ITER type 3)"""
+
 
 ind_plasma_internal_norm_wesson: float = None
 """Wesson-like plasma normalised internal inductance"""
@@ -1583,6 +1586,7 @@ def init_physics_variables():
     global ind_plasma_internal_norm
     global ind_plasma_internal_norm_wesson
     global ind_plasma_internal_norm_menard
+    global ind_plasma_internal_norm_iter_3
     global i_ind_plasma_internal_norm
     global ind_plasma
     global rmajor
@@ -1840,6 +1844,7 @@ def init_physics_variables():
     ind_plasma_internal_norm = 0.9
     ind_plasma_internal_norm_wesson = 0.0
     ind_plasma_internal_norm_menard = 0.0
+    ind_plasma_internal_norm_iter_3 = 0.0
     i_ind_plasma_internal_norm = 0
     ind_plasma = 0.0
     rmajor = 8.14

process/io/plot_proc.py

@@ -2674,7 +2674,7 @@ def plot_main_plasma_information(
         f"$V_{{\\text{{loop}}}}$: {mfile_data.data['v_plasma_loop_burn'].get_scan(scan):.4f} V\n"
         f"$\\Omega_{{\\text{{p}}}}$: {mfile_data.data['res_plasma'].get_scan(scan):.4e} $\\Omega$\n"
         f"Plasma resistive diffusion time: {mfile_data.data['t_plasma_res_diffusion'].get_scan(scan):,.4f} s\n"
-        f"Plasma inductance: {mfile_data.data['ind_plasma'].get_scan(scan):.4e} H\n"
+        f"Plasma inductance: {mfile_data.data['ind_plasma'].get_scan(scan):.4e} H | ITER $l_i(3)$: {mfile_data.data['ind_plasma_internal_norm_iter_3'].get_scan(scan):.4f}\n"
         f"Plasma stored magnetic energy: {mfile_data.data['e_plasma_magnetic_stored'].get_scan(scan) / 1e9:.4f} GJ\n"
         f"Plasma normalised internal inductance: {mfile_data.data['ind_plasma_internal_norm'].get_scan(scan):.4f}"
     )

process/physics.py

@@ -1680,6 +1680,15 @@ def physics(self):
             self.calculate_internal_inductance_menard(kappa=physics_variables.kappa)
         )
 
+        physics_variables.ind_plasma_internal_norm_iter_3 = (
+            self.calculate_normalised_internal_inductance_iter_3(
+                b_plasma_poloidal_vol_avg=physics_variables.b_plasma_poloidal_average,
+                c_plasma=physics_variables.plasma_current,
+                vol_plasma=physics_variables.vol_plasma,
+                rmajor=physics_variables.rmajor,
+            )
+        )
+
         # Map calculation methods to a dictionary
         ind_plasma_internal_norm_calculations = {
             0: physics_variables.ind_plasma_internal_norm,
@@ -3657,6 +3666,47 @@ def phyaux(
             f_alpha_energy_confinement,
         )
 
+    @staticmethod
+    def calculate_normalised_internal_inductance_iter_3(
+        b_plasma_poloidal_vol_avg: float,
+        c_plasma: float,
+        vol_plasma: float,
+        rmajor: float,
+    ) -> float:
+        """
+        Calculate the normalised internal inductance using ITER-3 scaling li(3).
+
+        :param b_plasma_poloidal_vol_avg: Volume-averaged poloidal magnetic field (T).
+        :type b_plasma_poloidal_vol_avg: float
+        :param c_plasma: Plasma current (A).
+        :type c_plasma: float
+        :param vol_plasma: Plasma volume (m^3).
+        :type vol_plasma: float
+        :param rmajor: Plasma major radius (m).
+        :type rmajor: float
+
+        :returns: The li(3) normalised internal inductance.
+        :rtype: float
+
+        :references:
+            - T. C. Luce, D. A. Humphreys, G. L. Jackson, and W. M. Solomon,
+            “Inductive flux usage and its optimization in tokamak operation,”
+            Nuclear Fusion, vol. 54, no. 9, p. 093005, Jul. 2014,
+            doi: https://doi.org/10.1088/0029-5515/54/9/093005.
+
+            - G. L. Jackson et al., “ITER startup studies in the DIII-D tokamak,”
+            Nuclear Fusion, vol. 48, no. 12, p. 125002, Nov. 2008,
+            doi: https://doi.org/10.1088/0029-5515/48/12/125002.
+        ‌
+        """
+
+        return (
+            2
+            * vol_plasma
+            * b_plasma_poloidal_vol_avg**2
+            / (constants.RMU0**2 * c_plasma**2 * rmajor)
+        )
+
     @staticmethod
     def plasma_ohmic_heating(
         f_c_plasma_inductive: float,
@@ -4411,6 +4461,14 @@ def outplas(self):
                 physics_variables.ind_plasma_internal_norm_menard,
                 "OP ",
             )
+            po.ovarrf(
+                self.outfile,
+                "ITER li(3) plasma normalised internal inductance",
+                "(ind_plasma_internal_norm_iter_3)",
+                physics_variables.ind_plasma_internal_norm_iter_3,
+                "OP ",
+            )
+
         else:
             po.ovarrf(
                 self.outfile,

tests/unit/test_physics.py

@@ -3459,3 +3459,11 @@ def test_calculate_beta_norm_max_stambaugh():
         f_c_plasma_bootstrap, kappa, aspect
     )
     assert result == pytest.approx(3.840954484207041, abs=0.00001)
+
+
+def test_calculate_internal_inductance_iter_3():
+    """Test calculate_normalised_internal_inductance_iter_3."""
+    result = Physics.calculate_normalised_internal_inductance_iter_3(
+        b_plasma_poloidal_vol_avg=1.0, c_plasma=1.5e7, vol_plasma=1000.0, rmajor=6.2
+    )
+    assert result == pytest.approx(0.9078959099585583, abs=0.00001)