Reduction Process

All variables in the resonant layer are assumed to be functions of

$\displaystyle \hat{x}$ $\displaystyle =\frac{r-r_s}{l},$ (4.35)
$\displaystyle \zeta$ $\displaystyle =m\,\theta-n\,\varphi,$ (4.36)

and $\hat{t}$, only. Here, $m$ and $n$ are the poloidal and toroidal mode numbers, respectively, of the tearing mode, $\varphi=z/R_0$ is a simulated toroidal angle, and $R_0$ is the simulated major radius of the plasma. (See Chapter 3.)


$\displaystyle {\bf n} = \frac{\epsilon^{(1)}}{q_s}\,{\bf e}_\theta + {\bf e}_z,$ (4.37)

where $\epsilon(r)=r/R_0$ and $q_s\equiv q(r_s)= m/n$. Here, $q(r)$ is the safety factor profile. [See Equation (3.2).] Moreover, the superscript $(1)$ indicates a quantity that is first order in our ordering scheme. [Zeroth order terms are left without superscripts, whereas second order terms are given the superscript (2).]

It follows that

$\displaystyle {\bf n}\cdot\hat{\nabla} A=0$ (4.38)

for any $A(\hat{x},\zeta,\hat{t})$. Furthermore,

$\displaystyle \hat{\nabla}\cdot{\bf n}$ $\displaystyle =0,$ (4.39)
$\displaystyle \hat{\nabla}\times {\bf n}$ $\displaystyle \simeq \frac{2\,\hat{\epsilon}^{(1)}}{q_s}\,{\bf n},$ (4.40)

where $\hat{\epsilon}^{(1)}=l/R_0$.

We can automatically satisfy Equation (4.32) by writing

$\displaystyle {\bf b} = \left(1+\delta b^{(1)}\right){\bf n} + \hat{\nabla}\tim...
...on^{(1)}}{q_s}\,\psi^{(1)}\right){\bf n} +\hat{\nabla}\psi^{(1)}\times {\bf n}.$ (4.41)

It follows that

$\displaystyle \psi^{(1)}(\hat{x},\zeta,\hat{t})= \frac{\psi_0(\hat{x}) + \delta\psi(\hat{x},\hat{t})\,{\rm e}^{\,{\rm i}\,\zeta}}{l\,B_z},$ (4.42)

where [see Equations (3.1) and (3.2)]

$\displaystyle \psi_0(\hat{x})=\frac{B_z}{R_0}\int_{r_s}^r r'\left[\frac{1}{q_s}-\frac{1}{q(r)}\right]dr',$ (4.43)

and $\delta\psi$ is defined in Equation (3.20). Note that

$\displaystyle {\bf b}\cdot\hat{\nabla}A^{(1)} = \left[A^{(1)},\psi^{(1)}\right],$ (4.44)


$\displaystyle [A,B] \equiv \hat{\nabla} A\times \hat{\nabla} B\cdot{\bf n}.$ (4.45)

Equations (4.33) and (4.41) give

$\displaystyle \hat{\bf E} =\hat{\nabla}\phi^{(1)}+\left(\hat{E}_\parallel^{(2)}...
\frac{\partial^{(1)}\chi^{(1)}}{\partial\hat{t}}\,{\bf n}\right),$ (4.46)


$\displaystyle \hat{\nabla}^2\chi^{(1)} =\delta b^{(1)}.$ (4.47)

Here, we have explicitly separated out the time dependent, spatially uniform part of $\psi^{(1)}$ which generates the (normalized) inductive electric field, $\hat{E}_\parallel^{(2)}$, that is responsible for maintaining the equilibrium parallel current density in the inner region against ohmic decay.

Equations (4.34) and (4.41) yield

$\displaystyle \hat{\bf j} = \left(-J^{(1)} + \frac{2\,\epsilon^{(1)}}{q_s}\,\de...
... n}
-\frac{2\,\hat{\epsilon}^{(1)}\,\epsilon^{(1)}}{q_s^{\,2}}\,{\bf e}_\theta,$ (4.48)


$\displaystyle J^{(1)} = -\frac{2\,\epsilon^{(1)}}{q_s} + \hat{\nabla}^2\psi^{(1)}.$ (4.49)

Let us write

$\displaystyle \hat{p} = \hat{p}_0 +\delta p^{(1)}.$ (4.50)

Note that the ordering of the plasma pressure adopted here is somewhat different to that adopted in Section 3.3. In fact, in Section 3.3, $\hat{p}$ was assumed to be second order. Here, we are assuming that $\hat{p}$ has a spatially constant component that is zeroth order, and a spatially varying component that is first order. This high pressure ordering is merely an artifice to aid the extraction of the compressible-Alfvén (i.e., fast magnetosonic) wave from the system of equations [3,4]. In fact, the ordering ensures that the compressible-Alfvén wave has a substantially different phase velocity than the shear-Alfvén and slow magnetosonic waves [2].

Equations (4.28)–(4.31), (4.41), (4.46), and (4.50) yield

$\displaystyle \hat{\bf V}_E$ $\displaystyle = \hat{\nabla}\phi^{(1)}\times {\bf b} -\hat{\nabla}\left(\frac{\partial^{(1)}\chi^{(1)}}{\partial\hat{t}}\right),$ (4.51)
$\displaystyle \hat{\bf V}_\ast$ $\displaystyle = \hat{d}_i\,{\bf b}\times \hat{\nabla}\delta p^{(1)},$ (4.52)
$\displaystyle \hat{\bf V}$ $\displaystyle = \hat{\bf V}_E + \hat{V}_\parallel^{(1)}\,{\bf b} +\hat{\nabla}{\mit\Upsilon}^{(2)},$ (4.53)
$\displaystyle \hat{\bf V}_i$ $\displaystyle = \hat{\bf V} + \frac{1}{1+\tau}\,\hat{\bf V}_\ast.$ (4.54)

Here, the additional factor involving ${\mit\Upsilon}^{(2)}$ in Equation (4.53) is needed because the plasma flow is slightly compressible. In fact, the velocity fields $\hat{\bf V}_E$, $\hat{\bf V}_\ast$, $\hat{\bf V}$, and $\hat{\bf V}_i$ all have normalized divergences that are second order.

Evaluating the normalized drift-MHD fluid equations, (4.25)–(4.27), up to second order, we obtain

$\displaystyle \hat{\nabla}\left(\delta p^{(1)}+ \delta b^{(1)}\right)+\left(\fr...
...l^{(1)},\phi^{(1)}\right] +\left[\psi^{(1)},\delta b^{(1)}\right]\right){\bf n}$    
$\displaystyle +\hat{\nabla}\left(\frac{\partial^{(1)}\phi^{(1)}}{\partial\hat{t...
... b^{(1)}\,\delta b^{(1)}+\frac{2\,\epsilon^{(1)}\,\epsilon^{(1)}}{q_s^2}\right)$    
$\displaystyle -\hat{\nabla}^2\phi^{(1)}\,\hat{\nabla}\phi^{(1)}+\frac{\hat{d}_i...
- {\mit\Xi}_{\perp}^{(1)}\,\hat{\nabla}^{2}{\bf V}_i^{(1)}=0,$ (4.55)
$\displaystyle \hat{d}_i\,\hat{\nabla}\left(\delta p^{(1)}+ \delta b^{(1)}\right)$    
$\displaystyle +\left(\hat{E}_\parallel^{(2)}-\frac{\partial^{(1)}\psi^{(1)}}{\p...
...i^{(1)},\delta b^{(1)}\right]+\hat{\eta}_\parallel^{(1)}\,J^{(1)}\right){\bf n}$    
$\displaystyle +\hat{\nabla}\left[{\mit\Upsilon}^{(2)}-\hat{\eta}_\perp^{(1)}\,(...
... b^{(1)}\,\delta b^{(1)}+\frac{2\,\epsilon^{(1)}\,\epsilon^{(1)}}{q_s^2}\right)$    
$\displaystyle +\hat{d}_i\,J^{(1)}\,\hat{\nabla}\psi^{(1)}-2\,\delta b^{(1)}\,\hat{\nabla}\phi^{(1)}=0,$ (4.56)
$\displaystyle \frac{3}{2}\,\frac{\partial^{(1)}\delta p^{(1)}}{\partial\hat{t}}...
$\displaystyle +\frac{5}{2}\,\lambda\,\hat{p}_0\,\hat{d}_i\,[\delta p^{(1)},\del...
...ight],\psi^{(1)}\right]-\hat{\chi}_\perp^{(1)}\,\hat{\nabla}^2\delta p^{(1)}=0.$ (4.57)

To first order, Equations (4.55) and (4.56) both yield

$\displaystyle \delta b^{(1)} = -\delta p ^{(1)},$ (4.58)

which is simply an expression of lowest-order equilibrium force balance [3,7].

Taking the scalar product of Equation (4.55) with ${\bf n}$ annihilates the first-order terms, leaving

$\displaystyle \frac{\partial^{(1)}\hat{V}_\parallel^{(1)}}{\partial\hat{t}}=\le...
...{(1)}\right] +\hat{\mit\Xi}_\perp^{(1)}\,\hat{\nabla}^2\hat{V}_\parallel^{(1)}.$ (4.59)

Moreover, taking the scalar product of Equation (4.56) with ${\bf n}$ annihilates the first-order terms, leaving

$\displaystyle \frac{\partial^{(1)}\psi^{(1)}}{\partial \hat{t}}= \left[\phi^{(1...
...i^{(1)}\right] + \hat{\eta}_\parallel^{(1)}\,J^{(1)} + \hat{E}_\parallel^{(2)}.$ (4.60)

Furthermore, taking the scalar product of the curl of Equation (4.55) with ${\bf n}$ annihilates the first-order terms, leaving

$\displaystyle \frac{\partial^{(1)}\,U^{(1)}}{\partial\hat{t}}$ $\displaystyle = \left[\phi^{(1)},U^{(1)}\right]+\frac{\hat{d}_i}{2\,(1+\tau)}
...ta p ^{(1)}\right]
+\left[\hat{\nabla}^2\delta p^{(1)},\phi^{(1)}\right]\right)$    
  $\displaystyle \phantom{=} +\left[J^{(1)},\psi^{(1)}\right] + \hat{\mit\Xi}_\per...
\phi^{(1)}-\frac{\hat{d}_i}{1+\tau}\,\delta p^{(1)}\right),$ (4.61)


$\displaystyle U^{(1)} = \hat{\nabla}^2\phi^{(1)}.$ (4.62)

Finally, taking the scalar product of the curl of Equation (4.56) with ${\bf n}$ annihilates the first-order terms, leaving

$\displaystyle \hat{\nabla}^2{\mit\Upsilon}^{(2)} = 2\left[\delta p^{(1)}, \phi^{(1)}\right] +\hat{d}_i\left[J^{(1)}, \psi^{(1)}\right].$ (4.63)

The previous equation can be combined with Equations (4.57) and (4.58) to give

$\displaystyle \frac{\partial^{(1)}\delta p^{(1)}}{\partial \hat{t}}$ $\displaystyle = \left[\phi^{(1)},\delta p^{(1)}\right]
...(1)},\psi^{(1)}\right] -c_\beta^{\,2}\,\hat{d}_i\left[J^{(1)},\psi^{(1)}\right]$    
  $\displaystyle \phantom{=}
...2}{3}\,(1-c_\beta^2)\,\hat{\chi}_\perp^{(1)}\,\hat{\nabla}^{\,2}\delta p^{(1)},$ (4.64)


$\displaystyle c_\beta$ $\displaystyle = \sqrt{\frac{\beta_\ast}{1+\beta_\ast}},$ (4.65)
$\displaystyle \beta_\ast$ $\displaystyle = \frac{5}{3}\,\frac{\mu_0\,p_0}{B_z^{\,2}}.$ (4.66)