Reduction Process

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

$$\hat{x} =\frac{r-r_s}{l},$$ (4.35)

$$\zeta =m\,\theta-n\,\varphi,$$ (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.)

Let

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

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

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

$$\hat{\nabla}\cdot{\bf n} =0,$$ (4.39)

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

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

We can automatically satisfy Equation (4.32) by writing

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

It follows that

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

where [see Equations (3.1) and (3.2)]

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

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

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

where

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

Equations (4.33) and (4.41) give

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

where

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

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

where

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

Let us write

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

$$\hat{\bf V}_E = \hat{\nabla}\phi^{(1)}\times {\bf b} -\hat{\nabla}\left(\frac{\partial^{(1)}\chi^{(1)}}{\partial\hat{t}}\right),$$ (4.51)

$$\hat{\bf V}_\ast = \hat{d}_i\,{\bf b}\times \hat{\nabla}\delta p^{(1)},$$ (4.52)

$$\hat{\bf V} = \hat{\bf V}_E + \hat{V}_\parallel^{(1)}\,{\bf b} +\hat{\nabla}{\mit\Upsilon}^{(2)},$$ (4.53)

$$\hat{\bf V}_i = \hat{\bf V} + \frac{1}{1+\tau}\,\hat{\bf V}_\ast.$$ (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

$$\hat{\nabla}\left(\delta p^{(1)}+ \delta b^{(1)}\right)+\left(\fr... ...l^{(1)},\phi^{(1)}\right] +\left[\psi^{(1)},\delta b^{(1)}\right]\right){\bf n}$$

$$+\hat{\nabla}\left(\frac{\partial^{(1)}\phi^{(1)}}{\partial\hat{t... ... b^{(1)}\,\delta b^{(1)}+\frac{2\,\epsilon^{(1)}\,\epsilon^{(1)}}{q_s^2}\right)$$

$$-\hat{\nabla}^2\phi^{(1)}\,\hat{\nabla}\phi^{(1)}+\frac{\hat{d}_i... ...\nabla}\psi^{(1)} - {\mit\Xi}_{\perp}^{(1)}\,\hat{\nabla}^{2}{\bf V}_i^{(1)}=0,$$ (4.55)

$$\hat{d}_i\,\hat{\nabla}\left(\delta p^{(1)}+ \delta b^{(1)}\right)$$

$$+\left(\hat{E}_\parallel^{(2)}-\frac{\partial^{(1)}\psi^{(1)}}{\p... ...i^{(1)},\delta b^{(1)}\right]+\hat{\eta}_\parallel^{(1)}\,J^{(1)}\right){\bf n}$$

$$+\hat{\nabla}\left[{\mit\Upsilon}^{(2)}-\hat{\eta}_\perp^{(1)}\,(... ... b^{(1)}\,\delta b^{(1)}+\frac{2\,\epsilon^{(1)}\,\epsilon^{(1)}}{q_s^2}\right)$$

$$+\hat{d}_i\,J^{(1)}\,\hat{\nabla}\psi^{(1)}-2\,\delta b^{(1)}\,\hat{\nabla}\phi^{(1)}=0,$$ (4.56)

$$\frac{3}{2}\,\frac{\partial^{(1)}\delta p^{(1)}}{\partial\hat{t}}... ...{V}_\parallel^{(1)},\psi^{(1)}\right]+\hat{\nabla}^2{\mit\Upsilon}^{(2)}\right)$$

$$+\frac{5}{2}\,\lambda\,\hat{p}_0\,\hat{d}_i\,[\delta p^{(1)},\del... ...ight],\psi^{(1)}\right]-\hat{\chi}_\perp^{(1)}\,\hat{\nabla}^2\delta p^{(1)}=0.$$ (4.57)

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

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

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

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

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

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

$$\frac{\partial^{(1)}\,U^{(1)}}{\partial\hat{t}} = \left[\phi^{(1)},U^{(1)}\right]+\frac{\hat{d}_i}{2\,(1+\tau)} \... ...ta p ^{(1)}\right] +\left[\hat{\nabla}^2\delta p^{(1)},\phi^{(1)}\right]\right)$$

$$\phantom{=} +\left[J^{(1)},\psi^{(1)}\right] + \hat{\mit\Xi}_\per... ...hat{\nabla}^4\left( \phi^{(1)}-\frac{\hat{d}_i}{1+\tau}\,\delta p^{(1)}\right),$$ (4.61)

where

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

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

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

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

$$\frac{\partial^{(1)}\delta p^{(1)}}{\partial \hat{t}} = \left[\phi^{(1)},\delta p^{(1)}\right] -c_\beta^{\,2}\left[\hat... ...(1)},\psi^{(1)}\right] -c_\beta^{\,2}\,\hat{d}_i\left[J^{(1)},\psi^{(1)}\right]$$

$$\phantom{=} +\frac{2}{3}\,(1-c_\beta^2)\,\hat{\chi}_\parallel^{(-... ...2}{3}\,(1-c_\beta^2)\,\hat{\chi}_\perp^{(1)}\,\hat{\nabla}^{\,2}\delta p^{(1)},$$ (4.64)

where

$$c_\beta = \sqrt{\frac{\beta_\ast}{1+\beta_\ast}},$$ (4.65)

$$\beta_\ast = \frac{5}{3}\,\frac{\mu_0\,p_0}{B_z^{\,2}}.$$ (4.66)