$beta$ profiles and new average values

[chris-ashe]

Description of issue / requirement to address

With the addition of pressure profiles coming in #3870 it would be good to visualise the toroidal and poloidal $\beta$ profiles across the plasma. This is simply done as:

$$ \beta(r) = \frac{2\mu_0 p(r)}{B^2(r)} $$

With the soon to be added plasma pressures the only component needed is the toroidal and poloidal field profile. We normally assume that the toroidal field from the TF coils i slike that from an infintely straight current wire so that the toroidal field varies across the plasma with only $\propto 1/R$. The poloidal field component is not as simple though still easily achievable.

Using Amperes law and assuming the plasma is a perfect circle (which its not, though we dont model 2D flux) the poloidal field function is given by:

$$ B_p(r) = \mu_0 \int_0^r J_p(r) dr $$

We already assume a parabolic form for the current profile and self-consistently match it with its profile index and the total plasma current value:

At current we seem to have a very rough definition of our avaerge poloidal beta given in calculate_poloidal_field() in physics.py. Pretty much in all cases where i_plasma_current !=2 the following expression is used for the average poloidal beta:

$$ \langle B_p \rangle \approx B_p(a) = \frac{\mu_0 I_p}{2\pi a} $$

Though in the current implementation the poloidal perimeter is used instead of assuming a circle given by the $2\pi a$ term. This approximation only holds true if we assume that the current density is relatively flat (which we dont, its parabolic mainly)

What we should be doing is just integrating the current density profile we assume and just take an average:

$$ \langle B_p \rangle = \frac{\mu_0}{\pi a^2} \int_0^a \left(\int_0^r J(r) 2 \pi r dr \right) dr $$

Proposed solution

Once #3870 is merged, output the toridal field variation across the plasma, do the current desnity integral for the poloidal field and plot the toroidal and poloidal $\beta$ distributions across the plasma. Implement the new avergae poloidal beta for the models.

Again all of this only holds true in our 1D case for our profiles and that we have no $Z$ dependance and that our plasma is a circle, which isnt true in relatity. We also already use the large tokamak circular(ish) model for the required vertical equilibrium field calculation in the PF coils. If we try to extend this to 2D we will be in GS territory. This should now atleast allow us to make self consistent interprotations of our profiles.