Ordering Scheme

Let us assume that

$$\epsilon_\beta\sim \epsilon_p\sim \epsilon_c\sim \epsilon_R\sim \epsilon_\varphi \sim \epsilon_\perp \sim\epsilon_\parallel \sim \epsilon,$$ (8.47)

where $\epsilon \ll 1$.

Table 5.1 gives estimates for the values of the various parameters that characterize the rescaled reduced drift-MHD model, (8.16)–(8.20), in a low-field and a high-field tokamak fusion reactor. (See Chapter 1.) These estimates are made using the following assumptions: $B=5\,{\rm T}$ (low-field) or $B=12\,{\rm T}$ (high-field), $\beta=0.02$, $T_e=T_i=7\,{\rm keV}$, $m_i=(m_D+m_T)/2$ (where $m_D$ and $m_T$ are the deuteron and triton masses, respectively), ${\mit\Xi}_{\perp\,i}= \chi_{\perp\,e}= \chi_{\perp\,i} = 1\,{\rm m^2/s}$, $m=2$, $n=1$, $r_s=a/2$ (where $a$ is the minor radius of the plasma), $s_s=1$, $\tau =1$, $\eta_e=\eta_i=1$, and $dp/dr=-p/a$. It is clear from Table 5.1 and Equations (8.25)–(8.34) that all of the $\epsilon$ parameters appearing in the rescaled, reduced, drift-MHD equations are less than unity, in accordance with Equation (8.47), provided that the radial width of the magnetic island chain that develops in the inner region exceeds a few centimeters.

Table: 8.1 Parameters characterizing the rescaled reduced drift-MHD model for a low-field and a high-field tokamak reactor. [See Equations (8.23)–(8.34) and (8.84).]

	Low-Field	High-Field
$B({\rm T})$	5.0	12.0
$\epsilon_\beta$	$1.64\times 10^{-2}$	$1.64\times 10^{-2}$
$\epsilon_p$	$7.47\times 10^{-2}$	$7.47\times 10^{-2}$
$w_c({\rm m})$	$1.79\times 10^{-2}$	$7.46\times 10^{-3}$
$w_R({\rm m})$	$8.09\times 10^{-4}$	$5.22\times 10^{-4}$
$w_\varphi({\rm m})$	$2.39\times 10^{-2}$	$1.54\times 10^{-2}$
$w_\perp({\rm m})$	$1.37\times 10^{-2}$	$8.84\times 10^{-3}$
$w_\parallel({\rm m})$	$7.21\times 10^{-3}$	$3.01\times 10^{-3}$
$w_d({\rm m})$	$6.97\times 10^{-3}$	$3.89\times 10^{-3}$

The orderings $\epsilon_\beta\ll 1$ and $\epsilon_p\ll 1$ are a direct consequence of the fact that conventional tokamak fusion reactors have large aspect-ratios (i.e., $R_0\gg r_s$), and confine low-$\beta$ plasmas (i.e., $\beta\sim 0.02$). As will become apparent, the ordering $\epsilon_c\ll 1$ ensures that the island chain is sufficiently wide that ion acoustic waves propagating parallel to the magnetic field are able to smooth out any variations in the normalized plasma pressure, ${\cal N}$, around magnetic flux-surfaces [9]. The orderings $\epsilon_R\ll 1$, $\epsilon_\varphi\ll 1$, and $\epsilon_\perp\ll 1$ ensure that the island chain is sufficiently wide that the perpendicular diffusion of magnetic flux, momentum, and energy are not dominant effects in the rescaled, reduced, drift-MHD equations. Finally, the ordering $\epsilon_\parallel\,\epsilon_\perp\ll 1$ ensures that the island chain is sufficiently wide that parallel energy transport smooths out any variations in the normalized plasma pressure, ${\cal N}$, around magnetic flux-surfaces [3].

Finally, suppose that

$$\frac{\partial}{\partial T} \sim {\cal O}(\epsilon^3),$$ (8.48)

and let us expand the various fields in our rescaled model as follows:

$${\mit\Psi} = {\mit\Psi}_0+\epsilon^2\,{\mit\Psi}_2+ \epsilon^3\,{\mit\Psi}_3 + \cdots,$$ (8.49)

$${\mit\Phi} = {\mit\Phi}_0+\epsilon^2\,{\mit\Phi}_2+ \epsilon^3\,{\mit\Phi}_3 + \cdots,$$ (8.50)

$${\cal N} = {\cal N}_0+\epsilon^2\,{\cal N}_2+ \epsilon^3\,{\cal N}_3 + \cdots,$$ (8.51)

$${\cal V} ={\cal V}_0 +\epsilon\,{\cal V}_1 +\cdots,$$ (8.52)

$${\cal J} = {\cal J}_0 + \epsilon\,{\cal J}_1 +\cdots.$$ (8.53)

Here, ${\mit\Psi}_0$, ${\mit\Psi}_1$, et cetera are assumed to be ${\cal O}(1)$ in the inner region.