Magnetic Field and Current Density Perturbations

Consider a tearing mode perturbation that has $m$ periods in the poloidal direction, and $n$ periods in the toroidal direction, where $m>0$, $n>0$, and $m\sim n\sim {\cal O}(1)$. We shall assume that all perturbed scalar and vector quantities vary as

$\displaystyle \delta A(r,\theta,z,t)$ $\displaystyle = \delta A(r,t)\,\exp[\,{\rm i}\,(m\,\theta-n\,\varphi)],$ (3.8)
$\displaystyle \delta {\bf A}(r,\theta,z,t)$ $\displaystyle = \delta {\bf A}(r,t)\,\exp[\,{\rm i}\,(m\,\theta-n\,\varphi)],$ (3.9)

respectively, where $\varphi=z/R_0$ is a simulated toroidal angle.

Given that tearing modes in tokamak plasmas are relatively low-amplitude (i.e., $\delta B/B\sim 10^{-4}$) [15], global (i.e., $\nabla_\perp \sim 1/a$) [see Equation (2.352)], relatively slowly-growing (i.e., $\partial/\partial t \sim \delta_i\,v_{t\,i}/a$) [see Equation (2.362)] instabilities, it follows from the analysis of Section 2.25 that they are governed by the linearized forms of the equations of marginally-stable ideal-MHD, (2.375)–(2.380). In particular, the linearized form of the curl of the force balance criterion, (2.377), combined with the linearized forms of Maxwell's equations, (2.349)–(2.351), give

$\displaystyle ({\bf B}\cdot\nabla)\,\delta {\bf j} +(\delta{\bf B}\cdot\nabla)\,{\bf j}- ({\bf j}\cdot\nabla)\,\delta{\bf B} -(\delta{\bf j}\cdot\nabla)\,{\bf B}$ $\displaystyle =0,$ (3.10)
$\displaystyle \nabla\cdot\delta {\bf B}$ $\displaystyle =0,$ (3.11)
$\displaystyle \mu_0\,\delta{\bf j}$ $\displaystyle =\nabla\times \delta {\bf B},$ (3.12)

where $\delta{\bf B}$ and $\delta {\bf j}$ are the perturbed magnetic field and current density, respectively.

Equations (3.1), (3.3), and (3.9)–(3.12) yield

$\displaystyle {\rm i}\,F\,r\,\delta\! j_r + {\rm i}\,(n\,\epsilon\,j_z-m\,j_\theta)\,\delta B_r$ $\displaystyle =0,$ (3.13)
$\displaystyle {\rm i}\,F\,r\,\delta \!j_z - B_z'\,r\,\delta\! j_r+ r\,j_z'\,\delta B_r + {\rm i}\,(n\,\epsilon\,j_z-m\,j_\theta)\,\delta B_z$ $\displaystyle =0,$ (3.14)


$\displaystyle (r\,\delta B_r)' +{\rm i}\,m\,\delta B_\theta - {\rm i}\,n\,\epsilon\,\delta B_z = 0,$ (3.15)


$\displaystyle \mu_0\,r\,\delta\!j_r$ $\displaystyle = {\rm i}\,m\,\delta B_z+{\rm i}\,n\,\epsilon\,\delta B_\theta,$ (3.16)
$\displaystyle \mu_0\,r\,\delta \!j_z$ $\displaystyle = (r\,\delta B_\theta)'-{\rm i}\,m\,\delta B_r,$ (3.17)


$\displaystyle \epsilon(r)$ $\displaystyle = \frac{r}{R_0},$ (3.18)
$\displaystyle F(r)$ $\displaystyle = \frac{m}{r}\,B_\theta - \frac{n}{R_0}\,B_z.$ (3.19)

If we write

$\displaystyle \delta B_r = {\rm i}\,\frac{m}{r}\,\delta\psi(r,t),$ (3.20)

then, after some algebra, Equations (3.13)–(3.17) reduce to

$\displaystyle \delta B_\theta$ $\displaystyle = -\frac{m^2}{m^2+(n\,\epsilon)^2}\,\delta\psi'
... (n\,\epsilon)^2}\,\frac{\mu_0\,(n\,\epsilon\,j_z-m\,j_\theta)}{F}\,\delta\psi,$ (3.21)
$\displaystyle \delta B_z$ $\displaystyle = \frac{m\,(n\,\epsilon)}{m^2+(n\,\epsilon)^2}\,\delta\psi'
... (n\,\epsilon)^2}\,\frac{\mu_0\,(n\,\epsilon\,j_z-m\,j_\theta)}{F}\,\delta\psi,$ (3.22)

and [6,10]

$\displaystyle \frac{1}{r}\,\frac{\partial}{\partial r}\!\left(f\,r\,\frac{\partial \delta\psi}{\partial r}\right) - g\,\delta\psi = 0,$ (3.23)


$\displaystyle f(r)$ $\displaystyle = \frac{m^2}{m^2+(n\,\epsilon)^2},$ (3.24)
$\displaystyle g(r)$ $\displaystyle =\frac{m}{r}\left\{\frac{m}{r} + \frac{\mu_0\,j_z'}{F} -\left[\fr...
...\,\epsilon)^2}\,\frac{\mu_0\,(n\,\epsilon\,j_z-m\,j_\theta)}{F}\right]' \right.$    
  $\displaystyle \phantom{=}\left.
- \frac{m}{r}\,\frac{n\,\epsilon}{m^2 + (n\,\ep...
..._z-m\,j_\theta)}{F}\,\frac{\mu_0\,(m\,j_z+ n\,\epsilon\,j_\theta)}{F}
\right\}.$ (3.25)

Now, a global tearing instability in a low-$\beta $, large aspect-ratio, tokamak plasma is characterized by [6]

$\displaystyle \frac{\epsilon}{q} = \frac{B_\theta}{B_z}$ $\displaystyle \ll 1,$ (3.26)
$\displaystyle \beta\equiv \frac{2\,\mu_0\,p}{B_z^{\,2}}$ $\displaystyle \sim{\cal O}\left(\frac{\epsilon}{q}\right)^2,$ (3.27)
$\displaystyle \frac{n\,\epsilon}{m}$ $\displaystyle \sim{\cal O}\left(\frac{\epsilon}{q}\right).$ (3.28)

It follows from Equations (3.4), (3.5), (3.7), and (3.19) that

$\displaystyle \frac{r\,B_z'}{B_z}$ $\displaystyle \sim{\cal O}\left(\frac{\epsilon}{q}\right)^2,$ (3.29)
$\displaystyle \frac{j_\theta}{j_z}$ $\displaystyle \sim {\cal O}\left(\frac{\epsilon}{q}\right),$ (3.30)
$\displaystyle \frac{\mu_0\,j_z}{F}$ $\displaystyle \sim {\cal O}(1).$ (3.31)

Thus, in the low-$\beta $, large aspect-ratio limit, Equations (3.20)–(3.25) simplify considerably to give

$\displaystyle \delta B_r$ $\displaystyle = {\rm i} \,\frac{m}{r}\,\delta\psi,$ (3.32)
$\displaystyle \delta B_\theta$ $\displaystyle \simeq - \delta\psi',$ (3.33)
$\displaystyle \delta B_z$ $\displaystyle \simeq \frac{n\,\epsilon}{m}\,\delta\psi' - \frac{\mu_0\,(n\,\epsilon\,j_z-m\,j_\theta)}{r\,F}\,\delta\psi,$ (3.34)


$\displaystyle \frac{1}{r}\,\frac{\partial}{\partial r}\!\left(r\,\frac{\partial...
... \frac{m^2}{r^2}\,\delta\psi- \frac{m\,\mu_0\,j_z'}{r\,F}\,\delta\psi\simeq 0 .$ (3.35)

It is also easily demonstrated that

$\displaystyle \mu_0\,\delta\! j_r$ $\displaystyle \simeq {\rm i}\,\frac{m}{r}\,\frac{\mu_0}{F}\left(\frac{m}{r}\,j_\theta-\frac{n}{R_0}\,j_z\right)\delta\psi,$ (3.36)
$\displaystyle \mu_0\,\delta\! j_\theta$ $\displaystyle \simeq \frac{m}{r}\,\frac{n}{R_0}\,\delta\psi -\frac{n}{R_0\,m}\,...
...u_0}{F}\left(\frac{m}{r}\,j_\theta-\frac{n}{R_0}\,j_z\right)\delta\psi\right]',$ (3.37)
$\displaystyle \mu_0\,\delta\! j_z$ $\displaystyle \simeq \frac{m^2}{r^2}\,\delta\psi - \frac{1}{r}\,(r\,\delta\psi')'.$ (3.38)

Hence, we conclude that the magnetic field and current density perturbations associated with a tearing mode in a low-$\beta $, large aspect-ratio, tokamak plasma are specified by Equations (3.32)–(3.38). From now on, we shall treat $B_z$ as approximately independent of $r$, in accordance with Equation (3.29).