🎨 Update plasma_beta.md for clarity and consistency in beta limit des…

…criptions

documentation/proc-pages/physics-models/plasma_beta/plasma_beta.md

@@ -46,6 +46,8 @@ $$
 
 Models for the fast alpha particle pressure contribution can be found [here](plasma_alpha_beta_contribution.md).
 
+The calculation of the beta component given by neutral beams is calculated in the neutral beam fusion calculations in [`beam_fusion()`](../fusion_reactions/beam_reactions.md#beam-slowing-down-properties--beam_fusion). A description can be found [here](../fusion_reactions/beam_reactions.md#derivation-of-beam-slowing-down-rate-and-critical-energy).
+
 ----------------
 
 ## Derivation of plasma beta parameter
@@ -90,20 +92,18 @@ $$
 
 ------------------------
 
-## Beta Limit
+## Troyon Beta Limit
 
-The plasma beta limit[^1] is given by 
+The Troyon plasma beta limit is given by[^0][^1]:
 
 $$\begin{aligned}
-\beta < 0.01\, g \, \frac{I(\mbox{MA})}{a(\mbox{m}) \, B_0(\mbox{T})}
+\beta < 0.01\, g \, \frac{I \ [\mbox{MA}]}{a \ [\mbox{m}] \, B_0 \ [\mbox{T}]}
 \end{aligned}$$
 
 where $B_0$ is the axial vacuum toroidal field. The beta
-coefficient $g$ is set using input parameter `beta_norm_max`. To apply the beta limit, 
-constraint equation 24 should be turned on with iteration variable 36
-(`fbeta_max`). 
+coefficient $g$ is set using input parameter `beta_norm_max`. To apply the Troyon limit please see [Beta upper limit](#beta-upper-limit).
 
-By default, $\beta$ is defined with respect to the total equilibrium B-field [^2]. 
+By default, $\beta$ is defined with respect to the total equilibrium B-field. This can be changed depending on the setting of `i_beta_component` seen below:
 
 | `i_beta_component` | Description |
 | :-: | - |
@@ -112,20 +112,70 @@ By default, $\beta$ is defined with respect to the total equilibrium B-field [^2
 | 2 | Apply the $\beta$ limit to only the thermal plus neutral beam contributions to beta |
 | 3 | Apply the $\beta$ limit to the total toroidal beta (including the contribution from fast alphas and neutral beams) |
 
+------------
+
 ### Setting the Beta $g$ Coefficient
 
-Switch `iprofile` determines how the beta $g$ coefficient `beta_norm_max` should 
-be calculated.
+Switch `iprofile` determines how the beta $g$ coefficient `beta_norm_max` should
+be calculated. The following switch options are available below:
 
-| `iprofile` | Description |
-| :-: | - |
-| 0 | `alphaj`, `rli` and `beta_norm_max` are inputs. |
-| 1 (default) | `alphaj`, `rli` and `beta_norm_max` are calulcated consistently. `beta_norm_max` calculated using $g=4l_i$ [^3].  This is only recommended for high aspect ratio tokamaks.|
-| 2 | `alphaj` and `rli` are inputs. `beta_norm_max` calculated using $g=2.7(1+5\epsilon^{3.5})$ (which gives g = 3.0 for aspect ratio = 3) |
-| 3 | `alphaj` and `rli` are inputs. `beta_norm_max` calculated using $g=3.12+3.5\epsilon^{1.7}$ [^4]|
-| 4 | `alphaj` and `beta_norm_max` are inputs. `rli` calculated from elongation [^4]. This is only recommended for spherical tokamaks.|
-| 5 | `alphaj` is an input.  `rli` calculated from elongation and `beta_norm_max` calculated using $g=3.12+3.5\epsilon^{1.7}$ [^4]. This is only recommended for spherical tokamaks.|
-| 6 | `alphaj` and `c_beta` are inputs.  `rli` calculated from elongation and `beta_norm_max` calculated using $C_{\beta}=(g-3.7)F_p / 12.5-3.5F_p$, where $F_p$ is the pressure peaking and $C_{\beta}$ is the destabilisation papermeter (default 0.5)[^5]. See Section 2.4 of Tholerus et al. (2024) for a more detailed description.  <u> This is only recommended for spherical tokamaks <u>.|
+---------
+
+#### User Input
+
+This can be activated by stating `iprofile = 0` in the input file.
+
+`alphaj`, `rli` and `beta_norm_max` are inputs.
+
+---------
+
+#### Wesson Relation
+
+This can be activated by stating `iprofile = 1` in the input file.
+
+`alphaj`, `rli` and `beta_norm_max` are calculated consistently. `beta_norm_max` calculated using $g=4l_i$ [^3].  This is only recommended for high aspect ratio tokamaks.
+
+---------
+
+#### Original Scaling Law
+
+This can be activated by stating `iprofile = 2` in the input file.
+
+`alphaj` and `rli` are inputs. `beta_norm_max` calculated using $g=2.7(1+5\epsilon^{3.5})$ (which gives g = 3.0 for aspect ratio = 3)
+
+---------
+
+#### Menard Beta Relation
+
+This can be activated by stating `iprofile = 3` in the input file.
+
+ `alphaj` and `rli` are inputs. `beta_norm_max` calculated using $g=3.12+3.5\epsilon^{1.7}$ [^4]
+
+---------
+
+#### Menard Inductance Relation
+
+This can be activated by stating `iprofile = 4` in the input file.
+
+`alphaj` and `beta_norm_max` are inputs. `rli` calculated from elongation [^4]. This is only recommended for spherical tokamaks.
+
+---------
+
+#### Menard Beta & Inductance Relation
+
+This can be activated by stating `iprofile = 5` in the input file.
+
+`alphaj` is an input.  `rli` calculated from elongation and `beta_norm_max` calculated using $g=3.12+3.5\epsilon^{1.7}$ [^4]. This is only recommended for spherical tokamaks.
+
+---------
+
+#### Tholerus Relation
+
+This can be activated by stating `iprofile = 6` in the input file.
+
+`alphaj` and `c_beta` are inputs.  `rli` calculated from elongation and `beta_norm_max` calculated using $C_{\beta}=(g-3.7)F_p / 12.5-3.5F_p$, where $F_p$ is the pressure peaking and $C_{\beta}$ is the destabilisation papermeter (default 0.5)[^5]. See Section 2.4 of Tholerus et al. (2024) for a more detailed description.  <u> This is only recommended for spherical tokamaks </u>
+
+---------
 
 Further details on the calculation of `alphaj` and `rli` is given in [Plasma Current](./plasma_current.md).
 
@@ -172,7 +222,10 @@ This constraint can be activated by stating `icc = 24` in the input file.
 
 It is the general setting of the $\beta$ limit depending on the $\beta_{\text{N}}$ value calculated in the [beta limit](#beta-limit) calculations.
 
-The upper limit value of beta is calculated by `calculate_beta_limit()`. The scaling value `fbeta_max` can be varied also.
+The upper limit value of beta is calculated by `calculate_beta_limit()`. The beta
+coefficient $g$ can be set using `beta_norm_max`, depending on the setting of [`iprofile`](#setting-the-beta--coefficient). It can be set directly or follow some relation.
+
+The scaling value `fbeta_max` can be varied also.
 
 **It is recommended to have this constraint on as it is a plasma stability model**
 
@@ -192,6 +245,8 @@ This constraint can be activated by stating `icc = 84` in the input file.
 
 The value of `beta_max` can be set to the desired minimum total beta. The scaling value `fbeta_min` can be varied also.
 
+[^0]: F. Troyon et.al,  “Beta limit in tokamaks. Experimental and computational status,” Plasma Physics and Controlled Fusion, vol. 30, no. 11, pp. 1597–1609, Oct. 1988, doi: https://doi.org/10.1088/0741-3335/30/11/019.
+
 [^1]: N.A. Uckan and ITER Physics Group, 'ITER Physics Design Guidelines: 1989',
 
 [^2]: D.J. Ward, 'PROCESS Fast Alpha Pressure', Work File Note F/PL/PJK/PROCESS/CODE/050
@@ -202,4 +257,4 @@ The value of `beta_max` can be set to the desired minimum total beta. The scalin
 
 [^5]: Tholerus et al. (2024), arXiv:2403.09460
 
-[^6]: M. E. Mauel et al., “Operation at the tokamak equilibrium poloidal beta-limit in TFTR,” Nuclear Fusion, vol. 32, no. 8, pp. 1468–1473, Aug. 1992. doi:10.1088/0029-5515/32/8/i14 
+[^6]: M. E. Mauel et al., “Operation at the tokamak equilibrium poloidal beta-limit in TFTR,” Nuclear Fusion, vol. 32, no. 8, pp. 1468–1473, Aug. 1992. doi:https://dx.doi.org/10.1088/0029-5515/32/8/I14