Equilibrium Magnetic Field

The equilibrium magnetic field is written [5,14] [see Equation (2.130)]

$${\bf B} =B_0\,R_0\left[ f(r)\,\nabla\varphi\times \nabla r + g(r)\,\nabla\varphi\right],$$ (14.17)

where $B_0$ is a convenient scale magnetic field-strength, and

$$q(r)= \frac{r\,g}{R_0\,f}$$ (14.18)

is the safety-factor profile. [See Equation (2.128).] Here, $g(r)$ and $f(r)$ are dimensionless functions.

It follows from Equations (14.1)–(14.5), (14.17), and (14.18) that

$$B^{\,r} =0,$$ (14.19)

$$B^{\,\theta} =B_0\,R_0^{\,2}\, \frac{f}{r\,R^{2}},$$ (14.20)

$$B^{\,\varphi} = B_0\,R_0\,\frac{g}{R^{2}} =B_0\,R_0^{\,2}\, \frac{f\,q}{r\,R^{2}} ,$$ (14.21)

$$B_r = - B_0\,r\,f\,\nabla r\cdot\nabla\theta,$$ (14.22)

$$B_\theta = B_0\,r\,f\,\vert\nabla r\vert^{2},$$ (14.23)

$$B_\varphi = B_0\,R_0\,g.$$ (14.24)

The equilibrium poloidal magnetic flux (divided by $2\pi$), ${\mit\Psi}_p(r)$, satisfies,

$$\frac{d{\mit\Psi}_p}{dr} = B_0\,R_0\,f(r),$$ (14.25)

where, by convention, ${\mit\Psi}_p(r_{100})=0$. [Note that ${\mit\Psi}_p$ is equivalent to the quantity $\psi$ defined in Equation (2.130).] The normalized poloidal magnetic flux, ${\mit\Psi}_N(r)$, is defined such that

$${\mit\Psi}_N(r)=1-\frac{{\mit\Psi}_p(r)}{{\mit\Psi}_p(0)}.$$ (14.26)

Hence, ${\mit\Psi}_N(0)=0$ and ${\mit\Psi}_N(r_{100})=1$. In general, $q(r_{100})=\infty$. Finally, if ${\mit\Psi}_N(r_{95})=0.95$ then $q_{95} = q(r_{95})$. Thus, $q_{95}$ is the safety-factor of the magnetic flux-surface that encloses 95% of the poloidal magnetic flux enclosed by the last closed magnetic flux-surface.