Rescaled Reduced Neoclassical Drift-MHD Model

Following Sections 5.2 and 8.2, it is convenient to set the normalization scale-length, $l$, in our reduced neoclassical drift-MHD model equal to the minor radius of the rational surface, $r_s$. It is also convenient to work in a frame of reference that co-rotates with the magnetic island chain that develops in the inner region. This goal can be achieved by making the transformation $\phi\rightarrow\phi + (\hat{\omega}/m)\,\hat{x}$ and $\hat{V}_\parallel\rightarrow \hat{V}_\parallel + (q_s/\epsilon_s)\,(\hat{\omega}/m)$, where $\omega = \hat{\omega}\,V_A/r_s$ is the rotation frequency of the tearing mode in the laboratory frame. In the co-rotating reference frame, the normalized reconnected flux at the rational surface, $\hat{\mit\Psi}_s(\hat{t})$ [see Equations (3.72) and (3.184)], is assumed to be a positive real quantity. It is helpful to define the reduced (by a factor four) radial width of the magnetic island chain: $w = (L_s\,R_0\,\hat{\mit\Psi}_s)^{1/2}$ [see Equation (5.129)]. Here, $R_0$ is the plasma major radius (see Section 3.2), and $L_s$ is the magnetic shear-length at the rational surface [see Equation (5.27)].

As before (see Section 8.2), it is assumed that $\delta_s\ll w\ll r_s$, where $\delta _s$ is the linear layer width. (See Chapter 6.) In other words, the width of the island chain is assumed to be much greater than the linear layer width, but much less than the minor radius of the rational magnetic flux-surface. Let $\hat{w}= w/r_s$. Reusing the analysis of Sections 5.3 and 8.2, we find that

$$\psi(\hat{x},\zeta,\hat{t}) \rightarrow \frac{\hat{x}^{\,2}}{2\,\hat{L}_s} + \hat{R}_0\,\hat{\mit\Psi}_s(\hat{t})\,\cos\zeta,$$ (11.58)

$$\delta p(\hat{x},\zeta,\hat{t}) \rightarrow \frac{\hat{V}_\ast}{\hat{d}_i}\,\hat{x},$$ (11.59)

$$\phi(\hat{x},\zeta,\hat{t}) \rightarrow \left[\frac{\hat{\omega}}{m}- \hat{V}_E(\hat{t})\right]\hat{x} - \frac{\varsigma}{4}\,\hat{V}_E'(\hat{t})\,\hat{x}^{\,2},$$ (11.60)

$$\hat{V}_\parallel(\hat{x},\zeta,\hat{t}) \rightarrow \frac{q_s}{\epsilon_s}\left[\frac{\hat{\omega}}{m}-\h... ...a_i}{1+\eta_i}\right)-\frac{\varsigma}{2}\,\hat{V}_E'(\hat{t})\,\hat{x}\right],$$ (11.61)

$$J(\hat{x},\zeta,\hat{t}) \rightarrow -\left(\frac{2}{s_s}-1\right)\frac{1}{\hat{L}_s},$$ (11.62)

in the limit $\vert\hat{x}\vert/\hat{w}\gg 1$ (i.e., many island widths from the rational surface). Here, $\hat{L}_s=L_s/r_s$, $\hat{R}_0=R_0/r_s$, $\hat{V}_E= V_E(r_s)/V_A$, $\hat{V}_\ast= V_\ast(r_s)/V_A$, and $s_s=s(r_s)$, where $V_E(r)$ is the E-cross-B velocity profile in the outer region [see Equation (5.21)], $V_\ast(r)$ the diamagnetic velocity profile [see Equation (5.29)], and $s(r)$ the magnetic shear profile [see Equation (5.28)]. Moreover, $\varsigma = {\rm sgn}(\hat{x})$, and $\hat{V}_E' = [r\,dV_E/dr]_{r_{s-}}^{r_{s+}}/V_A$. The parameter $\hat{V}_E'$ is introduced into the analysis in order to take into account the fact that the E-cross-B velocity profile in the outer region (i.e., everywhere in the plasma apart from the immediate vicinity of the magnetic island chain) develops a gradient discontinuity at the rational surface in response to the localized electromagnetic torque that emerges at the surface. (See Section 8.2.) Note, finally, that in neglecting any dependance of $\hat{\mit\Psi}_s$ on $\hat{x}$ in Equation (11.58) we are making use of the so-called constant-$\psi$ approximation [9,19], which is valid as long as $\vert{\mit\Delta\hat{\Psi}}_s\vert\,\hat{w}/\hat{\mit\Psi}_s\ll 1$ [4,6]. Here, ${\mit\Delta\hat{\Psi}}_s$ is defined in Equations (3.73) and (3.183).

Equations (11.58)–(11.62) are analogous to the boundary conditions in our previous reduced non-neoclassical drift-MHD model, (8.2)–(8.6), apart from Equation (11.61). The latter equation is derived on the assumption that ion neoclassical poloidal flow damping relaxes the ion poloidal velocity profile to its neoclassical value (see Section 2.18) many island widths away from the rational surface.

Let $X = \hat{x}/\hat{w}$ and $T = \omega_\ast\,t= \hat{\omega}_\ast\,\hat{t}$, where $\omega _\ast$ is the diamagnetic frequency at the rational surface [see Equation (5.47)], and $\hat{\omega}_\ast = \omega_\ast/(V_A/r_s)$. It follows that $\vert X\vert\sim {\cal O}(1)$ in the immediate vicinity of the island chain. It is helpful to define the rescaled fields ${\mit\Psi}(X,\zeta,T)$, ${\cal N}(X,\zeta,T)$, ${\mit\Phi}(X,\zeta,T)$, ${\cal V}(X,\zeta,T)$, and ${\cal J}(X,\zeta,T)$, where

$$\psi = \left(\frac{\hat{w}^{\,2}}{\hat{L}_s}\right){\mit\Psi},$$ (11.63)

$$\delta p = -\left(\frac{\hat{\omega}_\ast\,\hat{w}}{m\,\hat{d}_i}\right){\cal N},$$ (11.64)

$$\phi =\left(\frac{\hat{\omega}_\ast\,\hat{w}}{m}\right){\mit\Phi},$$ (11.65)

$$\hat{V}_\parallel = \left(\frac{\hat{L}_s\,\hat{\omega}_\ast^{\,2}}{m^2\,c_\beta^{\,2}\,\hat{d}_i}\right){\cal V},$$ (11.66)

$$J =-\left(\frac{2}{s_s}-1\right)\frac{1}{\hat{L}_s} + \left(\frac{\hat{L}_s\,\hat{\omega}_\ast^{\,2}}{m^2\,\hat{w}^{\,2}}\right){\cal J}.$$ (11.67)

These fields are (essentially) the same as those used in our previous rescaled reduced non-neoclassical drift-MHD model. (See Section 8.2.)

Equations (11.50)–(11.56) rescale to give

$$\frac{d(\ln \hat{w}^2)}{dT}\,\cos\zeta = \left\{{\mit\Phi}-\frac{\tau}{1+\tau}\,{\cal N}, {\mit\Psi}\rig... ... \zeta_{bs}\,\epsilon_\beta\,\epsilon_c\,\epsilon_R\,(\partial_X {\cal N} - 1),$$ (11.68)

$$\frac{\partial {\cal N}}{\partial T} = \{{\mit\Phi}, {\cal N}\} + \{{\cal V},{\mit\Psi}\} + \epsilon_p... ...\epsilon_p\,\epsilon_c\left\{{\mit\Phi}-\frac{\tau}{1+\tau}\,{\cal N},X\right\}$$

$$\phantom{=} -\zeta_{nc}\,\epsilon_p\,\epsilon_c\,\{\partial_X {\c... ...\{{\cal N},{\mit\Psi}\}, {\mit\Psi}\}+\epsilon_\perp\,\partial_X^{\,2}{\cal N},$$ (11.69)

$$\frac{\partial(\partial_X^{\,2}\,{\mit\Phi})}{\partial T} = \partial_X\!\left\{{\mit\Phi} + \frac{{\cal N}}{1+\tau},\partial_X{\mit\Phi}\right\} + \{{\cal J},{\mit\Psi}\} -\zeta_g\,\{{\cal N},X\}$$ (11.70)

$$\phantom{=} +\frac{\epsilon_\theta}{\epsilon_q}\,\partial_X\!\lef... ...on_\varphi \,\partial_X^{\,4}\left({\mit\Phi} + \frac{{\cal N}}{1+\tau}\right),$$

$$\epsilon_c\,\frac{\partial {\cal V}}{\partial T} =\epsilon_c\, \{{\mit\Phi},{\cal V}\}+\{{\cal N}, {\mit\Psi}\}+\f... ...\zeta_g\,\epsilon_p\,\epsilon_c^{\,2}\,X+\epsilon_c\,\epsilon_\beta\,{\cal N}\}$$

$$\phantom{=}-\epsilon_c\,\epsilon_\theta\left({\cal V} -\xi\,\part... ...{\cal N}\right]\right)+\epsilon_c\,\epsilon_\varphi\,\partial_X^{\,2} {\cal V},$$ (11.71)

$$\partial_X^{\,2}{\mit\Psi} = 1+ \epsilon_\beta\,\epsilon_c\,{\cal J}.$$ (11.72)

Here, $\partial_X\equiv \partial/\partial X$, $\{A,B\} \equiv (\partial A/\partial X)\,(\partial B/\partial\zeta) - (\partial A/\partial \zeta)\,(\partial B/\partial X)$, and we have set

$$\hat{E}_\parallel =\left[ \left(\frac{2}{s_s}-1\right)\frac{1}{\hat{L}_s}+\alpha_{bs}\,\frac{\hat{V}_\ast}{\hat{d}_i}\right]\hat{\eta}_\parallel.$$ (11.73)

We have also made use of the identity (8.39). Furthermore,

$$\epsilon_q = \left(\frac{\epsilon_s}{q_s}\right)^2,$$ (11.74)

$$\epsilon_\theta = \left(\frac{\epsilon_s}{q_s}\right)^2\,\frac{1}{\omega_\ast\,\tau_\theta},$$ (11.75)

and

$$\xi =\sqrt{\frac{\epsilon_p}{\epsilon_q}}=\frac{L_p}{L_s}\,\frac{q_s}{\epsilon_s},$$ (11.76)

$$\zeta_{bs} =f_{t\,s}\,\xi\left[\beta_{11}\left(1-\alpha_1\,\frac{1}{1+\tau}\... ...ta_e}\right]\left(\frac{w}{d_\beta}\right)^2 = \left(\frac{w}{w_{bs}}\right)^2,$$ (11.77)

$$\zeta_{nc} =f_{t\,s}\,\xi\left[\left(\beta_{11}+\frac{2}{5}\,\epsilon_1\righ... ...\eta_e}{1+\eta_e}-\frac{\alpha_2}{\tau'}\right]\left(\frac{w}{d_\beta}\right)^2$$

$$=\left(\frac{w}{w_{nc}}\right)^2,$$ (11.78)

$$\zeta_g = 2\,\frac{L_p}{L_c}\left(\frac{w}{d_\beta}\right)^2 = \left(\frac{w}{w_g}\right)^2,$$ (11.79)

where

$$w_{bs} = d_\beta\left\{f_{t\,s}\,\xi\left[\beta_{11}\left(1-\alpha_1\,\f... ... -\beta_{12}\,\frac{\tau}{1+\tau}\frac{\eta_e}{1+\eta_e}\right]\right\}^{-1/2},$$ (11.80)

$$w_{nc} = d_\beta\left\{f_{t\,s}\,\xi\left[\left(\beta_{11}+\frac{2}{5}\,... ...u}{1+\tau}\frac{\eta_e}{1+\eta_e}-\frac{\alpha_2}{\tau'}\right]\right\}^{-1/2},$$ (11.81)

$$w_g = d_\beta\left(\frac{L_c}{2\,L_p}\right)^{1/2}.$$ (11.82)

Here, $L_p$ is the effective pressure gradient scale-length at the rational surface [see Equation (8.35)], $d_\beta$ is the ion sound radius [see Equation 4.75)], and the dimensionless quantities $\epsilon_\beta$, $\epsilon_p$, $\epsilon_c$, $\epsilon_R$, $\epsilon_\varphi$, $\epsilon_\perp$, and $\epsilon_\parallel$ are defined in Section 8.3.

Equations (11.68)–(11.72) must be solved subject to the boundary conditions [see Equations (11.58)–(11.62) and (11.63)–(11.67)]

$${\mit\Psi}(X,\zeta,T) \rightarrow \frac{X^{2}}{2} + \cos\zeta,$$ (11.83)

$${\cal N}(X,\zeta,T) \rightarrow X,$$ (11.84)

$${\mit\Phi}(X,\zeta,T) \rightarrow v(T)\,X +\frac{\varsigma\,v'(T)\,X^{2}}{2},$$ (11.85)

$${\cal V}(X,\zeta,T) \rightarrow \xi\left[v(T) + \frac{1}{1+\tau}\left(1-\alpha_\theta\,\frac{\eta_i}{1+\eta_i}\right)+\varsigma\,v'(T)\,X\right],$$ (11.86)

$${\cal J}(X,\zeta,T) \rightarrow 0,$$ (11.87)

as $\vert X\vert\rightarrow\infty$. Here,

$$v(T) = \frac{\omega-\omega_E}{\omega_\ast},$$ (11.88)

$$v'(T) = -\frac{\hat{w}}{2\,\omega_\ast}\left[r\,\frac{d\omega_E}{dr}\right]_{r_{s-}}^{r_{s+}},$$ (11.89)

where $\omega_E= (m/r_s)\,V_E(r_s)$ is the E-cross-B frequency at the rational surface. Note that ${\mit\Psi}$, ${\cal N}$, ${\mit\Phi}$, ${\cal V}$, and ${\cal J}$ are all ${\cal O}(1)$ quantities in the inner region. Note, further, that the boundary conditions (11.83)–(11.87), as well as the symmetry of the rescaled reduced neoclassical drift-MHD equations, (11.68)–(11.72), ensure that ${\mit\Psi}$, ${\cal V}$, and ${\cal J}$ are even functions of $X$, whereas ${\cal N}$ and ${\mit\Phi}$ are odd functions.

Finally, asymptotic matching between the inner region and the surrounding plasma yields (see Section 8.10)

$${\rm Re}\!\left(\frac{{\mit\Delta}\hat{\mit\Psi}_s}{\hat{\mit\Psi}_s}\right) =\frac{2\,\epsilon_\beta\,\epsilon_c}{\hat{w}}\int_{-\infty}^{\infty}\oint {\cal J}\,\cos\zeta\,\frac{d\zeta}{2\pi}\,dX,$$ (11.90)

$${\rm Im}\!\left(\frac{{\mit\Delta}\hat{\mit\Psi}_s}{\hat{\mit\Psi}_s}\right) =-\frac{2\,\epsilon_\beta\,\epsilon_c}{\hat{w}}\int_{-\infty}^{\infty}\oint {\cal J}\,\sin\zeta\,\frac{d\zeta}{2\pi}\,dX.$$ (11.91)