🔄 Rename classic physics variables 2

documentation/eng-models/heating_and_current_drive/NBI/culham_nb.md

@@ -71,7 +71,7 @@ plus correction terms outlined in Culham Report AEA FUS 172.
 | $\mathtt{alphat}$, $\alpha_T$       | temperature profile factor  |
 |  $\mathtt{aspect}$, $A$      |   aspect ratio                            |
 |  $\mathtt{nd_plasma_electrons_vol_avg}$, $n_{\text{e}}$     |    volume averaged electron density $(\text{m}^{-3})$                           |
-|  $\mathtt{nd_electron_line}$, $n_{\text{e,0}}$      |    line averaged electron density $(\text{m}^{-3})$                           |
+|  $\mathtt{nd_plasma_electron_line}$, $n_{\text{e,0}}$      |    line averaged electron density $(\text{m}^{-3})$                           |
 |  $\mathtt{e_beam_kev}$      |  neutral beam energy $(\text{keV})$                             |
 |  $\mathtt{f_radius_beam_tangency_rmajor}$      |   R_tangent / R_major for neutral beam injection                            |
 |  $\mathtt{fshine}$      |  shine-through fraction of beam                             |

documentation/physics-models/fusion_reactions/beam_reactions.md

@@ -103,8 +103,8 @@ Please see the [H&CD section](../../eng-models/heating_and_current_drive/heating
 3. **Set the plasma tritium and ion densities**
 
     $$
-    \mathtt{deuterium\_density = nd_fuel_ions * f\_deuterium\_plasma} \\
-    \mathtt{tritium\_density = nd_fuel_ions * f\_tritium\_plasma}
+    \mathtt{deuterium\_density = nd_plasma_fuel_ions_vol_avg * f\_deuterium\_plasma} \\
+    \mathtt{tritium\_density = nd_plasma_fuel_ions_vol_avg * f\_tritium\_plasma}
     $$
 
 4. **Calculate the beam alpha powers, density and deposited energy**

documentation/physics-models/fusion_reactions/plasma_reactions.md

@@ -55,10 +55,10 @@ profiles
 ## Setting of plasma fuel composition
 
 The fractional composition of the 'fuel' ions ($\text{D}$, $\text{T}$ and $^3\text{He}$) is
-controlled using the three variables `f_deuterium`, `f_tritium` and `f_helium3`, respectively.
+controlled using the three variables `f_plasma_fuel_deuterium`, `f_plasma_fuel_tritium` and `f_plasma_fuel_helium3`, respectively.
 More information about setting seeded impurities and simulating first wall sputtering can be found in the [composition and impurities section](../plasma_composition.md).
 
-It is also possible to optimise on the deuterium-tritium fuel mixture ratio. For this, `f_tritium` **must** be set as an iteration variable with `ixc = 173`. More info can be found [here](#fuel-ions-mixture-self-consistency).
+It is also possible to optimise on the deuterium-tritium fuel mixture ratio. For this, `f_plasma_fuel_tritium` **must** be set as an iteration variable with `ixc = 173`. More info can be found [here](#fuel-ions-mixture-self-consistency).
 
 !!! note "Reactions not calculated"
 
@@ -238,7 +238,7 @@ The value of `big_q_plasma_min` can be set to the minimum desired $Q_{\text{plas
 
 This constraint can be activated by stating `icc = 92` in the input file.
 
-This constraint should be activated if the user wishes to allow the ratio of the fuel ions to be iterated upon. For this, `f_tritium` has to be set as an iteration variable with `ixc = 173`. This constraint ensure that the summation of the fuel fractions always sums to 1.0 and will modify the other compositions depending on the value of `f_tritium`.
+This constraint should be activated if the user wishes to allow the ratio of the fuel ions to be iterated upon. For this, `f_plasma_fuel_tritium` has to be set as an iteration variable with `ixc = 173`. This constraint ensure that the summation of the fuel fractions always sums to 1.0 and will modify the other compositions depending on the value of `f_plasma_fuel_tritium`.
 
 [^1]: H.-S. Bosch and G. M. Hale, “Improved formulas for fusion cross-sections and thermal reactivities,” Nuclear Fusion, vol. 32, no. 4, pp. 611–631, Apr. 1992, doi: https://doi.org/10.1088/0029-5515/32/4/i07.
 [^2]: I. P. E. G. on E. Drive and I. P. B. Editors, “Chapter 5: Physics of energetic ions,” Nuclear Fusion, vol. 39, no. 12, pp. 2471–2495, Dec. 1999, doi: https://doi.org/10.1088/0029-5515/39/12/305.

documentation/physics-models/plasma_beta/plasma_beta.md

@@ -5,9 +5,19 @@
 The efficiency of confinement of plasma pressure by the magnetic field is represented by the ratio:
 
 $$
-\beta = \frac{2\mu_0p}{B^2}
+\beta(\rho) = \frac{2\mu_0p(\rho)}{\left(B(\rho)\right)^2}
 $$
 
+Where $\beta$ is a function of normalised minor radius across the plasma ($\rho$), due to the change in pressure and magnetic field strength. 
+
+The standard $\beta$ term used for comparison and to represent the plasma as a whole in many calculations is the volume averaged value given by:
+
+$$
+\langle \beta \rangle = \frac{2\mu_0 \langle p \rangle}{\langle B \rangle^2}
+$$
+
+Where $\langle p \rangle$ is the volume averaged plasma pressure and $\langle B \rangle$ is the average field.
+
 There are several different measures of this type, arising from different choices of definition and from the need to quantify different equilibrium properties.
 
 In its expanded form of magnetic field components:
@@ -92,6 +102,32 @@ $$
 
 ------------------------
 
+## Definitions
+
+### Volume averaged thermal toroidal beta
+
+We define $B_{\text{T,on-axis}}$ as the toroidal field at the plasma major radius, $R_0$
+
+$$
+\overbrace{\langle \beta_t \rangle_{\text{V}}}^{\texttt{beta_toroidal_thermal_vol_avg}} = \frac{2\mu_0 \overbrace{\langle p_{\text{thermal}} \rangle}^{\texttt{pres_plasma_thermal_vol_avg}}}{\underbrace{B_{\text{T,on-axis}}^2}_{\texttt{b_plasma_toroidal_on_axis}}}
+$$
+
+### Volume averaged thermal poloidal beta
+
+
+
+$$
+\overbrace{\langle \beta_p \rangle_{\text{V}}}^{\texttt{beta_poloidal_thermal_vol_avg}} = \frac{2\mu_0 \overbrace{\langle p_{\text{thermal}} \rangle}^{\texttt{pres_plasma_thermal_vol_avg}}}{\underbrace{\langle B_{\text{P,average}} \rangle^2}_{\texttt{b_plasma_poloidal_average}}}
+$$
+
+### Volume averaged thermal beta
+
+$$
+\overbrace{\langle \beta \rangle_{\text{V}}}^{\texttt{beta_thermal_vol_avg}} = \frac{2\mu_0 \overbrace{\langle p_{\text{thermal}} \rangle}^{\texttt{pres_plasma_thermal_vol_avg}}}{\sqrt{\langle B_{\text{P,average}} \rangle^2+B_{\text{T,on-axis}}^2}}
+$$
+
+------------------
+
 ## Troyon Beta Limit
 
 The Troyon plasma beta limit is given by[^0][^1]:
@@ -350,7 +386,13 @@ This constraint can be activated by stating `icc = 1` in the input file.
 Ensures the relationship between $\beta$, density, temperature and total magnetic field is withheld by checking the fixed input or iteration variable $\mathtt{beta}$ is consistent in value with the rest of the physics parameters
 
 $$
-\mathtt{beta} \equiv \frac{2\mu_0 \langle n_{\text{e}}T_{\text{e}}+n_{\text{i}}T_{\text{i}}\rangle}{B^2} + \beta_{\alpha} + \beta_{\text{beam}}
+\texttt{beta_total_vol_avg} \equiv \frac{2\mu_0 \langle n_{\text{e}}T_{\text{e}}+n_{\text{i}}T_{\text{i}}\rangle}{B^2} + \beta_{\alpha} + \beta_{\text{beam}}
+$$
+
+Here the calculation of the volume averaged pressure of the ions and electrons has to use the density weighted temperature for each. This is because $\langle nT \rangle_{\text{V}} \neq \langle n \rangle_{\text{V}} \langle T \rangle_{\text{V}}$, where $\text{V}$ denotes the volume averaged value. The true value is, $\langle nT \rangle_{\text{V}} = \langle n \rangle_{\text{V}} \langle T \rangle_{\text{n}}$, where $\text{n}$ is the density weighted averaged. For example:
+
+$$
+\langle n_{\text{e}}T_{\text{e}} \rangle_{\text{V}} = \overbrace{\langle n_{\text{e}} \rangle_{\text{V}}}^{\texttt{nd_plasma_electrons_vol_avg}} \times  \overbrace{\langle T_{\text{e}} \rangle_{\text{n}}}^{\texttt{temp_plasma_electron_density_weighted_kev}}
 $$
 
 **It is highly recommended to always have this constraint on as it is a global consistency checker**
@@ -403,7 +445,7 @@ The value of `beta_poloidal_max` can be set to the desired maximum poloidal beta
 
 This constraint can be activated by stating `icc = 84` in the input file.
 
-The value of `beta_min` can be set to the desired minimum total beta. The scaling value `fbeta_min` can be varied also.
+The value of `beta_vol_avg_min` 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.
 

documentation/physics-models/plasma_composition.md

@@ -36,7 +36,7 @@ using input array `f_nd_impurity_electrons(1,...,14)`. The available species alo
 As stated above, the number density fractions for hydrogen (all isotopes) and
 helium should not be set, as they are calculated by the code. This is to ensure 
 plasma quasi-neutrality taking into account the fuel ratios
-`f_deuterium`, `f_tritium` and `f_helium3`, and the alpha particle fraction `f_nd_alpha_electron` which may 
+`f_plasma_fuel_deuterium`, `f_plasma_fuel_tritium` and `f_plasma_fuel_helium3`, and the alpha particle fraction `f_nd_alpha_electron` which may 
 be input by the user or selected as an iteration variable.
 
 !!! note "Location of impurities"
@@ -59,7 +59,7 @@ All of the plasma composites are normally given as a fraction of the volume aver
 
 1. **Alpha Ash Portion Calculation**
 
-    - Calculate the number density of alpha particles (`nd_alphas`) using the electron density (`nd_plasma_electrons_vol_avg`) and the alpha particle to electron ratio (`f_nd_alpha_electron`).
+    - Calculate the number density of alpha particles (`nd_plasma_alphas_vol_avg`) using the electron density (`nd_plasma_electrons_vol_avg`) and the alpha particle to electron ratio (`f_nd_alpha_electron`).
         - `f_nd_alpha_electron` can be set as an iteration variable (`ixc = 109`) or set directly.
 
     $$
@@ -68,19 +68,19 @@ All of the plasma composites are normally given as a fraction of the volume aver
 
 
 2. **Protons Calculation**
-    - The calculation of proton density (`nd_protons`) depends on whether the alpha rate density has been calculated. This should only happen in the first function call as the rates are calculated later on in the code.
+    - The calculation of proton density (`nd_plasma_protons_vol_avg`) depends on whether the alpha rate density has been calculated. This should only happen in the first function call as the rates are calculated later on in the code.
 
     - If the alpha rate density is not yet calculated, use a rough estimate.
 
     $$
-    \texttt{nd_protons} | n_{\text{p}} = \\
+    \texttt{nd_plasma_protons_vol_avg} | n_{\text{p}} = \\
     \text{max}\left[\texttt{f_nd_protium_electrons} \times n_{\text{e}}, n_{\alpha}\times \left(f_{\text{3He}} + 0.001\right)\right]
     $$
 
     - Otherwise, use the calculated proton rate density.
 
     $$
-    \texttt{nd_protons} | n_{\text{p}} = \\
+    \texttt{nd_plasma_protons_vol_avg} | n_{\text{p}} = \\
     \text{max}\left[\texttt{f_nd_protium_electrons} \times n_{\text{e}}, n_{\alpha}\times \frac{r_{\text{p}}}{r_{\alpha,\text{total}}}\right]
     $$
 
@@ -116,7 +116,7 @@ All of the plasma composites are normally given as a fraction of the volume aver
     \mathtt{nd\_fuel\_ions} | n_{\text{i}} = \frac{\mathtt{znfuel}}{1+f_{\text{3He}}}
     $$
 
-    - Calculate the fuel ion density (`nd_fuel_ions`).
+    - Calculate the fuel ion density (`nd_plasma_fuel_ions_vol_avg`).
 
 7. **Set Hydrogen and Helium Impurity Fractions**
 
@@ -132,15 +132,15 @@ All of the plasma composites are normally given as a fraction of the volume aver
 
 8. **Total Impurity Density Calculation**
 
-    - Calculate the total impurity density (`nd_impurities`).
+    - Calculate the total impurity density (`nd_plasma_impurities_vol_avg`).
 
     $$
     \mathtt{nd\_impurities} | n_{\text{impurities}} = \sum_j n_{\text{e}} f_j
     $$
 
 9. **Total Ion Density Calculation**
 
-    - Calculate the total ion density (`nd_ions_total`).
+    - Calculate the total ion density (`nd_plasma_ions_total_vol_avg`).
 
     $$
     \mathtt{nd\_ions\_total} | n_{\text{i,total}} = n_{\text{i}} + n_{\alpha}+n_{\text{protons}}+ n_{\text{beam}}+n_{\text{impurities}}

documentation/physics-models/profiles/plasma_density_profile.md

@@ -13,7 +13,7 @@ The density profile class is organised around a central runner function that is
 ### Calculate core values | `set_physics_variables()`
 
 The core electron density is calculated using the [`ncore`](plasma_density_profile.md#electron-core-density-of-a-pedestalised-profile--ncore) method.
-The core ion density is then set from $n_{\text{i}}$ (`nd_ions_total`) which is the total ion density such as:
+The core ion density is then set from $n_{\text{i}}$ (`nd_plasma_ions_total_vol_avg`) which is the total ion density such as:
 
 $$
 n_{\text{i0}} = \left(\frac{n_\text{i}}{n_\text{e}}\right)n_{\text{e0}}

documentation/physics-models/profiles/plasma_profiles.md

@@ -282,23 +282,23 @@ The line averaged density is then calculated for the profile paramaters
 Line averaged electron density is calculated by integrating the profile across the normalised width of the profile and then dividing by the width of the integration bounds
 
 $$
-\overbrace{\bar{n_{\text{e}}}}^{\texttt{nd_electron_line}} = \frac{\int^1_0 n_0(1-\rho^2)^{\alpha_n} \ d\rho}{\rho}
+\overbrace{\bar{n_{\text{e}}}}^{\texttt{nd_plasma_electron_line}} = \frac{\int^1_0 n_0(1-\rho^2)^{\alpha_n} \ d\rho}{\rho}
 $$
 
 This can be more easily represented in radial coordinates and returning $\rho = \frac{r}{a}$
 
 $$
-\overbrace{\bar{n_{\text{e}}}}^{\texttt{nd_electron_line}} = \frac{\int^a_0 n_0(1-r^2/a^2)^{\alpha_n} \ dr}{a}
+\overbrace{\bar{n_{\text{e}}}}^{\texttt{nd_plasma_electron_line}} = \frac{\int^a_0 n_0(1-r^2/a^2)^{\alpha_n} \ dr}{a}
 $$
 
 $$
-\overbrace{\bar{n_{\text{e}}}}^{\texttt{nd_electron_line}} = \frac{\left[\frac{an_0}{2}\frac{\Gamma(1/2)\Gamma(\alpha_n+1)}{\Gamma(\alpha_n+3/2)}\right]}{a}
+\overbrace{\bar{n_{\text{e}}}}^{\texttt{nd_plasma_electron_line}} = \frac{\left[\frac{an_0}{2}\frac{\Gamma(1/2)\Gamma(\alpha_n+1)}{\Gamma(\alpha_n+3/2)}\right]}{a}
 $$
 
 $\Gamma$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function) and is calculated in the code with [scipy.special.gamma()](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gamma.html)
 
 $$
-\overbrace{\bar{n_{\text{e}}}}^{\texttt{nd_electron_line}} = \frac{n_0}{2}\frac{\Gamma(1/2)\Gamma(\alpha_n+1)}{\Gamma(\alpha_n+3/2)}
+\overbrace{\bar{n_{\text{e}}}}^{\texttt{nd_plasma_electron_line}} = \frac{n_0}{2}\frac{\Gamma(1/2)\Gamma(\alpha_n+1)}{\Gamma(\alpha_n+3/2)}
 $$
 
 This is in agreement with the derivation from the ITER Physics Design 1989 [^2]
@@ -385,7 +385,7 @@ $$\begin{aligned}
 
 $$\begin{aligned}
 \texttt{nd_plasma_electron_on_axis} = n_{\text{e}} \times (\alpha_n+1) \\
-\texttt{nd_plasma_ions_on_axis} = \texttt{nd_ions_total} \times (\alpha_n+1)
+\texttt{nd_plasma_ions_on_axis} = \texttt{nd_plasma_ions_total_vol_avg} \times (\alpha_n+1)
 \end{aligned}$$
 
 -----
@@ -605,7 +605,7 @@ $$
 Calculate the line averaged electron density by integrating the normalised profile using the class [`integrate_profile_y()`](./plasma_profiles_abstract_class.md#calculate-the-profile-integral-value-integrate_profile_y) function
 
 $$
-\texttt{nd_electron_line} = \int_0^1{n(\rho) \ d\rho}
+\texttt{nd_plasma_electron_line} = \int_0^1{n(\rho) \ d\rho}
 $$
 
 A divertor variable `prn1` is set to be equal to the separatrix density over the mean density: