\[e=mc^{2}\] \[\text{this is a tes} \tag{1}\]

^3cef19

\(\frac{ \partial^{2} \psi }{ \partial r^{2} } -\frac{1}{r} \frac{ \partial \psi }{ \partial r } + \frac{ \partial^{2} \psi }{ \partial z^{2} } = -\mu_{0}r^{2} \frac{\text{d} p }{\text{d}\psi} - \frac{1}{2} \frac{\text{d} F^{2} }{\text{d}\psi} \tag{*}\) ^3de88a

[[ #^3cef19 ]]

[[ #^3de88a ]] this is a reference i hope works!

where: $\psi$ is the poloidal flux surface “label” $p(\psi)$ is the pressure $F(\psi)=rB_\theta$ $\frac{\text{d} F^{2} }{\text{d}\psi}$ is ff-prime or ff’

or from [[ Plasma II - APPH6102 ]]:

where

• the Stokes’ operator: $\equiv \frac{1}{R}\Delta^*\hat\phi=\frac{1}{R}\partial_z^2\Psi - \partial_R\left(\frac{1}{R}\partial_R\Psi \right)$
• $F=RB_\phi=\frac{\mu_0I_{pol}}{2\pi}$
• $F\frac{\partial F}{\partial \Psi}$ = FF’ (eff eff prime)
• $\mathcal{L}[\Psi]=P(\Psi)+F(\Psi)$
• in words, G-S is:
  • some operator (Stoke’s) operating on 2-D $\Psi$ (flux) function
  • =
  • 1-D poloidal function
  • +
  • 1-D flux function
• what is $\Psi_{toroidal}$ (Zohm uses $\Phi$)? $\Psi_{poloidal}$?
  • $\Psi_T = \int B_\phi dS\approx B_\phi r^2$
    • $\phi$ points along axis of donut (toroidally)
    • often used as a “pseudo radius”, because $B_\phi$ is constant
  • $\Psi_P=\int B_\theta dS$
    • normalized poloidal flux; $\Psi_N$ = 0 at center, 1 at the edge
    • add constant so center is at 0, then scale to edge is =1
    • $\theta$ points around donut the short way

Shafronov shift