Equilibrium scaling

The GS equation is given by Eq. (53), i.e.,

$$\frac{\partial^2 \Psi}{\partial Z^2} + R \frac{\partial}{\partial... ...i}{\partial R} \right) = - \mu_0 R^2 \frac{d P}{d \Psi} - \frac{d g}{d \Psi} g.$$ (84)

If a solution to the GS equation is obtained, the solution can be scaled to obtain a family of solutions. Given an equilibrium with $\Psi (R, Z)$, $P (\Psi)$, $g (\Psi)$, then it is ready to prove that $\Psi_2 = s \Psi$, $P_2 = s^2 P (\Psi)$, and $g_2 = \pm s g (\Psi)$ is also a solution to the GS equation, where $s$ is a constant. In this case, both the poloidal and toroidal magnetic fields are increased by a factor of $s$, and thus the safety factor remains unchanged. Also note that the pressure is increased by $s^2$ factor and thus the value of $\beta$ (the ratio of the therm pressure to magnetic pressure) remains unchanged. Note that $g_2 = \pm s g (\Psi)$, which indicates that the direction of the toroidal magnetic field can be reversed without breaking the force balance. Also note that $\Psi_2 = s \Psi$ and $s$ can be negative, which indicates that the direction of the toroidal current can also be reversed without breaking the force balance.

The second kind of scaling is to set $\Psi_2 = \Psi$, $P_2 = P (\Psi)$, and $g^2_2 = g^2 (\Psi) + c$. It is ready to prove that the scaled expression is still a solution to the GS equation because $g_2 g_2' = g g'$. This scaling keep the pressure and the poloidal field unchanged and thus the poloidal beta $\beta_p$ remains unchanged. This scaling scales the toroidal field and thus can be used to generate a series of equilibria with different profile of safety factor.

Another scaling, which is trivial, is to set $\Psi_2 = \Psi$, $P_2 = P (\Psi) + c$, and $g_2 = g (\Psi)$. This scaling can be used to test the effects of the pressure (not the pressure gradient) on various physical processes.

When a numerical equilibrium is obtained, one can use these scaling methods together to generate new equilibria that satisfy particular global conditions. Note that the shape of magnetic surfaces of the scaled equilibrium remains the same as the original one.