Tidy wade divertor model

documentation/source/eng-models/divertor.md

@@ -169,21 +169,11 @@ The interactive graph below can be used to investigate how changing the key para
 !!! Note ""
     `i_div_heat_load == 2`
 
-A divertor heat flux model is provided in Appendix A.II. of [^2].  This uses the Eich scaling 
-[^3] and S-factor [^4] to calculate the SOL width at the outboard divertor, mapped to the midplane:
+A divertor heat flux model is provided in Appendix A.II. of [^wade_21].  This model originally uses the [Eich scaling](../physics-models/plasma_scrape_off_layer.md#eich-2013-model--calculate_eich2013_sol_power_decay_length) 
+and the [Scrabosio S-factor](../physics-models/plasma_scrape_off_layer.md#scarabosio-2015--calculate_scarabosio2015_power_spreading_factor) to calculate the SOL width at the outboard divertor, mapped to the midplane:
 
 $$
-\lambda_{int} = \lambda_{q,Eich} + 1.64S
-$$
-
-where
-
-$$
-\lambda_{q,Eich} = 1.35 \, P_{\mathrm{SOL}}^{-0.02} \, R_{o}^{0.04} \, B_{p}^{-0.92} \, \epsilon^{0.42}
-$$
-
-$$
-S = 0.12(n_{e,mid}/10^{19})^{-0.02} \, P_{\mathrm{SOL}}^{-0.21} \, R_{o}^{0.71} \, B_{p}^{-0.82}.
+\lambda_{\text{int}} = \lambda_{\text{q,Eich}} + 1.64S
 $$
 
 This is then used to calculate the wetted area in the divertor
@@ -196,44 +186,40 @@ where $N_{div}$ is the number of divertors (1 or 2), $F_{exp}$ is the relevant f
 $\theta_{div}$ is the tilt of the separatrix relative to the target in the poloidal plane, and has the form
 
 $$
-\theta_{div} = \sin^{-1} [(1+1/\alpha_{div}^{2})\sin\beta_{div}],
+\theta_{\text{div}} = \sin^{-1} [(1+1/\alpha_{\text{div}}^{2})\sin(\beta_{\text{div}})],
 $$
 
 where
 
 $$
-\alpha_{div} = F_{exp}\alpha_{mid}
+\alpha_{\text{div}} = F_{\text{exp}}\alpha_{\text{mid}}
 $$
 
 $$
-\alpha_{mid} = \tan^{-1}\frac{B_{p,mid}}{B_{T,mid}}
+\alpha_{\text{mid}} = \tan^{-1}\frac{B_{\text{p,mid}}}{B_{\text{T,mid}}}
 $$
 
-where $B_{p,mid}$ and $B_{T,mid}$ are the poloidal and toroidal fields on the outer midplane. The 
+where $B_{\text{p,mid}}$ and $B_{\text{T,mid}}$ are the poloidal and toroidal fields on the outer midplane. The 
 parameter $\beta_{div}$ is the angle of incidence between the field line and the target.
 
-The divertor heat flux in $\mathrm{MW}/\mathrm{m^{2}}$ is then 
+The divertor heat flux in $[\mathrm{MW}/\mathrm{m^{2}}]$ is then 
 
 $$
-q_{div} = P_{\mathrm{SOL}}(1-f_{rad,div})/A_{wetted}
+q_{\text{div}} = P_{\mathrm{SOL}}(1-f_{\text{rad,div}})/A_{\text{wetted}}
 $$
 
-where $f_{rad,div}$ is the SOL radiative fraction.
+where $f_{\text{rad,div}}$ is the SOL radiative fraction.
 
 For the purposes of this model, the following are inputs:
 
-- Flux expansion $F_{exp}$  (`f_div_flux_expansion`, default = 2)  
-- Field line angle with respect to divertor target plate (degrees) $\beta_{div}$ (`deg_div_field_plate`), also 
+- Flux expansion $F_{\text{exp}}$  (`f_div_flux_expansion`, default = 2)  
+- Field line angle with respect to divertor target plate (degrees) $\beta_{\text{div}}$ (`deg_b_div_lower_outboard_grazing`), also 
   available as an iteration variable (170)  
-- SOL radiative fraction, $f_{rad,div}$ (`rad_fraction_sol`).
+- SOL radiative fraction, $f_{\text{rad,div}}$ (`rad_fraction_sol`).
 
 [^1]: N.A. Uckan and ITER Physics Group, 'ITER Physics Design Guidelines: 1989',
 ITER Documentation Series, No. 10, IAEA/ITER/DS/10 (1990)
 
-[^2]: M.R. Wade & J.A. Leuer, 'Cost Drivers for a Tokamak-Based Compact Pilot Plant, Fusion Science and Technology, 77:2, 119-143 (2021)
-
-[^3]: T. Eich et al, 'Scaling of the tokamak near the scrape-off layer H-mode power width and implications for ITER', Nucl. Fusion 53 093031 (2013)
-
-[^4]: A. Scarabosio et al, 'Scaling of the divertor power spreading (S-factor) in open and closed divertor operation in JET and ASDEX Upgrade, Journal of Nuclear Materials, Vol. 463, 49-54 (2015)
+[^wade_21]: M. R. Wade and J. A. Leuer, “Cost Drivers for a Tokamak-Based Compact Pilot Plant,” Fusion Science and Technology, vol. 77, no. 2, pp. 119–143, Feb. 2021, doi: 10.1080/15361055.2020.1858670.
 
 [^5]: Y.-K. M. Peng, J. B. Hicksand AEA Fusion, Culham (UK), "Engineering feasibility of tight aspect ratio Tokamak (spherical torus) reactors". 1990. https://inis.iaea.org/records/ey2rf-dah04

documentation/source/physics-models/plasma_scrape_off_layer.md

@@ -81,7 +81,34 @@ This can be found in Table 2 and Equation 4 from Thornton et.al [^mast_2014]
 
 The $R^2$ value for this fit is 0.55
 
---------
+-------------
+
+## Spreading Parameter
+
+The scrape-off layer (SOL) spreading parameter $S$ represents a Gaussian width that quantifies additional perpendicular heat spreading in the divertor leg. It works alongside the upstream heat flux decay length $\lambda_{q}$ to determine total target heat loads on the divertor.
+
+Unlike $\lambda_{q}$, which is governed by robust upstream parallel and perpendicular transport physics at the plasma midplane, $S$ is inherently a "local" divertor parameter. Deriving a single, absolute multi-machine formula for $S$ is incredibly difficult due to several overlapping regional variables:
+
+- Divertor Geometry: The path length from the X-point to the target tile heavily impacts how much the heat spreads radially.
+
+- Plasma Recycling Regimes: Low-recycling, high-recycling, and detached plasma conditions completely alter the cross-field diffusion rates.
+
+- Localized Radiation: Impurity seeding and neutral gas interactions dissipate power unevenly along the divertor leg, altering the effective Gaussian profile width.
+
+-----------
+
+### Scarabosio 2015 | `calculate_scarabosio2015_power_spreading_factor()`
+
+The H-mode SOL spreading factor, $S$ is given in $\text{m}$ by[^scarabosio_2015]:
+
+$$
+S = (0.12(\pm0.07)\times 10^{-3}) P_{\text{sep}}^{0.21(\pm0.11)}R_0^{0.71(\pm0.5)}B_{\text{p}}(a)^{-0.82(\pm0.27)}n_{\text{sep}}^{0.71(\pm0.5)}
+$$
+
+- This was fitted from ASDEX Upgrade and JET outer target data
+- The $R^2$ value of the regression fit was 0.65
+
+------------
 
 [^eich_2013]: T. Eich et al., “Scaling of the tokamak near the scrape-off layer H-mode power width and implications for ITER,” Nuclear Fusion, vol. 53, no. 9 p. 093031, Aug. 2013, doi: 10.1088/0029-5515/53/9/093031.
 
@@ -90,4 +117,6 @@ Plasma Physics and Controlled Fusion, vol. 56, no. 5, p. 055008, Apr. 2014, doi:
 
 [^stangeby_boundary]: P. C. Stangeby, “The Plasma Boundary of Magnetic Fusion Devices,” Jan. 2000, doi: 10.1201/9780367801489.
 
-[^henderson_step]: S. S. Henderson et al., “An overview of the STEP divertor design and the simple models driving the plasma exhaust scenario,” Nuclear Fusion, vol. 65, no. 1, pp. 016033–016033, Nov. 2024, doi: 10.1088/1741-4326/ad93e7.
+[^henderson_step]: S. S. Henderson et al., “An overview of the STEP divertor design and the simple models driving the plasma exhaust scenario,” Nuclear Fusion, vol. 65, no. 1, pp. 016033–016033, Nov. 2024, doi: 10.1088/1741-4326/ad93e7.
+
+[^scarabosio_2015]: A. Scarabosio et al., “Scaling of the divertor power spreading (S-factor) in open and closed divertor operation in JET and ASDEX Upgrade,” Journal of Nuclear Materials, vol. 463, pp. 49-54, Aug. 2015, doi: 10.1016/j.jnucmat.2014.11.076.

process/core/input.py

@@ -217,7 +217,9 @@ def bounds(self) -> tuple[NumberType | None, NumberType | None]:
     "bctmp": InputVariable("pulse", float, range=(1.0, 800.0)),
     "e_beam_kev": InputVariable("current_drive", float, range=(1.0, 1000000.0)),
     "dx_beam_duct": InputVariable("current_drive", float, range=(0.001, 5.0)),
-    "deg_div_field_plate": InputVariable("divertor", float, range=(0.0, 360.0)),
+    "deg_b_div_lower_outboard_grazing": InputVariable(
+        "divertor", float, range=(0.0, 360.0)
+    ),
     "beta_poloidal_max": InputVariable("constraints", float, range=(0.01, 2.0)),
     "betai": InputVariable("divertor", float, range=(0.0, 1.5707)),
     "betao": InputVariable("divertor", float, range=(0.0, 1.5707)),

process/core/io/obsolete_vars.py

@@ -302,7 +302,7 @@
     "wave_mode": "i_ecrh_wave_mode",
     "pheatfix": "p_hcd_secondary_extra_heat_mw",
     "ftar": "f_p_div_lower",
-    "beta_div": "deg_div_field_plate",
+    "beta_div": "deg_b_div_lower_outboard_grazing",
     "divclfr": "f_vol_div_coolant",
     "divdens": "den_div_structure",
     "divplt": "dx_div_plate",
@@ -473,6 +473,7 @@
     "i_cs_stress": None,
     "f_nd_alpha_electron": "f_nd_alpha_thermal_electron",
     "cost_model": "i_cost_model",
+    "deg_div_field_plate": "deg_b_div_lower_outboard_grazing",
 }
 
 OBS_VARS_HELP = {