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

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

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

$$({\bf B}\cdot\nabla)\,\delta {\bf j} +(\delta{\bf B}\cdot\nabla)\,{\bf j}- ({\bf j}\cdot\nabla)\,\delta{\bf B} -(\delta{\bf j}\cdot\nabla)\,{\bf B} =0,$$ (3.10)

$$\nabla\cdot\delta {\bf B} =0,$$ (3.11)

$$\mu_0\,\delta{\bf j} =\nabla\times \delta {\bf B},$$ (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

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

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

and

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

with

$$\mu_0\,r\,\delta\!j_r = {\rm i}\,m\,\delta B_z+{\rm i}\,n\,\epsilon\,\delta B_\theta,$$ (3.16)

$$\mu_0\,r\,\delta \!j_z = (r\,\delta B_\theta)'-{\rm i}\,m\,\delta B_r,$$ (3.17)

where

$$\epsilon(r) = \frac{r}{R_0},$$ (3.18)

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

If we write

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

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

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

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

and [6,10]

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

where

$$f(r) = \frac{m^2}{m^2+(n\,\epsilon)^2},$$ (3.24)

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

$$\phantom{=}\left. - \frac{m}{r}\,\frac{n\,\epsilon}{m^2 + (n\,\ep... ..._z-m\,j_\theta)}{F}\,\frac{\mu_0\,(m\,j_z+ n\,\epsilon\,j_\theta)}{F} \right\}.$$ (3.25)

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

$$\frac{\epsilon}{q} = \frac{B_\theta}{B_z} \ll 1,$$ (3.26)

$$\beta\equiv \frac{2\,\mu_0\,p}{B_z^{\,2}} \sim{\cal O}\left(\frac{\epsilon}{q}\right)^2,$$ (3.27)

$$\frac{n\,\epsilon}{m} \sim{\cal O}\left(\frac{\epsilon}{q}\right).$$ (3.28)

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

$$\frac{r\,B_z'}{B_z} \sim{\cal O}\left(\frac{\epsilon}{q}\right)^2,$$ (3.29)

$$\frac{j_\theta}{j_z} \sim {\cal O}\left(\frac{\epsilon}{q}\right),$$ (3.30)

$$\frac{\mu_0\,j_z}{F} \sim {\cal O}(1).$$ (3.31)

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

$$\delta B_r = {\rm i} \,\frac{m}{r}\,\delta\psi,$$ (3.32)

$$\delta B_\theta \simeq - \delta\psi',$$ (3.33)

$$\delta B_z \simeq \frac{n\,\epsilon}{m}\,\delta\psi' - \frac{\mu_0\,(n\,\epsilon\,j_z-m\,j_\theta)}{r\,F}\,\delta\psi,$$ (3.34)

and

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

It is also easily demonstrated that

$$\mu_0\,\delta\! j_r \simeq {\rm i}\,\frac{m}{r}\,\frac{\mu_0}{F}\left(\frac{m}{r}\,j_\theta-\frac{n}{R_0}\,j_z\right)\delta\psi,$$ (3.36)

$$\mu_0\,\delta\! j_\theta \simeq \frac{m}{r}\,\frac{n}{R_0}\,\delta\psi -\frac{n}{R_0\,m}\,... ...u_0}{F}\left(\frac{m}{r}\,j_\theta-\frac{n}{R_0}\,j_z\right)\delta\psi\right]',$$ (3.37)

$$\mu_0\,\delta\! j_z \simeq \frac{m^2}{r^2}\,\delta\psi - \frac{1}{r}\,(r\,\delta\psi')'.$$ (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).