Linear Growth-Rate Regimes

As before (see Section 5.7), we shall assume that $P_\varphi\sim P_\perp \sim P$, and $\tau\sim {\cal O}(1)$, for the sake of simplicity.

Suppose that $Q_\ast\gg P\,p^2$ and $P\gg Q_\ast\,D^2$. It follows that $k=0$, $\nu=1/2$, and

$$G= -\frac{Q_\ast^{\,2}}{(1+1/\tau)\,P_\perp}.$$ (6.11)

Hence,

$$\skew{6}\hat{\mit\Delta} = {\rm e}^{-{\rm i}\,\pi/2}\,\pi\,\frac{Q_\ast}{(1+1/\tau)^{1/2}\,P_\perp^{1/2}}\,\hat{\gamma},$$ (6.12)

and $p_\ast\sim P^{1/2}/Q_\ast$. This so-called resistive-inertial growth-rate regime is valid when $P\ll Q_\ast^{3/2}$ and $P\gg Q_\ast\,D^2$. Making use of Equations (5.48), (5.54), (5.65), (6.2), and (6.4), the corresponding tearing mode growth-rate is [3]

$$\gamma ={\rm e}^{\,{\rm i}\,\pi/2}\, \frac{E_{ss}}{\pi}\,\frac{(1+1/\tau)^{1/2}}{\omega_\ast\,\tau_H\,\tau_R^{1/2}\,\tau_\perp^{1/2}}.$$ (6.13)

Here, $\omega _\ast$ is the total diamagnetic frequency [see Equation (5.47)], $\tau _R$ is the resistive diffusion timescale [see Equation (5.49)], and $\tau _\perp$ the energy confinement timescale [see Equation (5.52)]. Note that $\tau$ and $\omega _\ast$ are both assumed to be positive quantities (which is always the case if the electron and ion equilibrium pressure profiles are monotonically decreasing functions of minor radius). According to the previous equation, the tearing mode is purely oscillatory in the resistive-inertial growth-rate regime.

Suppose that $Q_\ast \ll P\,p^2$ and $D^2\,p^2 \ll 1$. It follows that $k=3$, $\nu=1/6$, and

$$G = P_\varphi.$$ (6.14)

Hence,

$$\skew{6}\hat{\mit\Delta} = \frac{ 6^{2/3}\,\pi\,{\mit\Gamma}(5/6)}{{\mit\Gamma}(1/6)}\,P_\varphi^{1/6}\,\hat{\gamma},$$ (6.15)

and $p_\ast \sim P^{-1/6}$. This so-called viscous-resistive growth-rate regime is valid when $P\gg Q_\ast^{3/2}$ and $P\gg D^6$. Making use of Equation (5.53), the corresponding tearing mode growth-rate is [3]

$$\gamma = \frac{E_{ss}}{[6^{2/3}\,\pi\,{\mit\Gamma}(5/6)/{\mit\Gamma}(1/6)]}\,\frac{\tau_\varphi^{1/6}}{\tau_H^{1/3}\,\tau_R^{5/6}},$$ (6.16)

where $\tau _\varphi$ is the momentum confinement timescale [see Equation (5.50)]. We conclude that, in the viscous-resistive growth-rate regime, the tearing mode is a purely growing mode (in a frame of reference that is co-moving with the electron fluid at the rational surface) when the tearing stability index, $E_{ss}$, is positive, and a purely decaying mode otherwise [4].

Suppose that $Q_\ast\gg P\,p^2$ and $P\ll Q_\ast\,D^2$. It follows that $k=0$, $\nu=1/2$, and

$$G = -{\rm i}\,\frac{Q_\ast}{(1+1/\tau)\,D^2}.$$ (6.17)

Hence,

$$\skew{6}\hat{\mit\Delta} = {\rm e}^{-{\rm i}\,\pi/4}\,\pi\,\frac{Q_\ast^{1/2}}{(1+1/\tau)^{1/2}\,D}\,\hat{\gamma},$$ (6.18)

and $p_\ast\sim D/Q_\ast^{1/2}$. This so-called semi-collisional growth-rate regime is valid when $P\ll Q_\ast^{\,2}/D^2$ and $P\ll Q_\ast\,D^2$. Making use of Equation (5.66), the corresponding tearing mode growth-rate is [3]

$$\gamma = {\rm e}^{\,{\rm i}\,\pi/4}\,\frac{E_{ss}}{\pi}\,\frac{d_\beta/r_s}{\omega_\ast^{1/2}\,\tau_H\,\tau_R^{1/2}},$$ (6.19)

where $d_\beta$ is the ion sound radius [see Equation (4.75)], and $r_s$ the minor radius of the rational surface. We conclude that, in the semi-collisional growth-rate regime, the tearing mode is a growing oscillatory mode (in the electron fluid reference frame) when the tearing stability index is positive, and a decaying oscillatory mode otherwise.

Table: 6.1 Quantities controlling tearing mode growth-rates in a low-field and a high-field tokamak fusion reaction. Here, $\tau$ is the ratio of the electron to the ion pressure gradients at the rational surface, $\tau _H$ the hydromagnetic timescale, $\tau _R$ the resistive diffusion timescale, $\tau _\varphi$ the momentum confinement timescale, $\tau _\perp$ the energy confinement timescale, $\omega _\ast$ the diamagnetic frequency, $r_s$ the minor radius of the rational surface, and $d_\beta$ the ion sound radius.

	Low-Field	High-Field
$B({\rm T})$	5.0	12.0
$\tau$	1.0	1.0
$\tau_H({\rm s})$	$1.04\times 10^{-6}$	$4.31\times 10^{-7}$
$\tau_R({\rm s})$	$1.40\times 10^3$	$2.43\times 10^2$
$\tau_\varphi({\rm s})$	$1.60$	0.278
$\tau_\perp({\rm s})$	$4.89$	0.848
$\omega_{\ast}({\rm krad/s})$	$1.75$	4.19
$r_s({\rm m})$	$1.23$	$0.527$
$d_\beta({\rm m})$	$4.89\times 10^{-3}$	$2.04\times 10^{-3}$

Suppose, finally, that $Q_\ast \ll P\,p^2$ and $D^2\,p^2\gg 1$. It follows that $k=2$, $\nu=1/4$, and

$$G = \frac{P_\perp}{(1+1/\tau)\,D^2}.$$ (6.20)

Hence,

$$\skew{6}\hat{\mit\Delta} = \frac{2\pi\,{\mit\Gamma}(3/4)}{{\mit\Gamma}(1/4)}\frac{P_\perp^{1/4}}{(1+1/\tau)^{1/4}\,D^{1/2}}\,\hat{\gamma},$$ (6.21)

and $p_\ast \sim D^{1/2}/P^{1/4}$. This so-called diffusive-resistive growth-rate regime is valid when $P\gg Q_\ast^2/D^2$ and $P\ll D^6$. The corresponding tearing mode growth-rate is [3]

$$\gamma= \frac{E_{ss}}{[2\pi\,{\mit\Gamma}(3/4)/{\mit\Gamma}(1/4)]}\,\frac{(d_\beta/r_s)^{1/2}\,\tau_\perp^{1/4}}{\tau_H^{1/2}\,\tau_R^{3/4}}.$$ (6.22)

We conclude that, in the diffusive-resistive growth-rate regime, the tearing mode is a purely growing mode (in the electron fluid reference frame) when the tearing stability index is positive, and a purely decaying mode otherwise.

Table 5.1 gives estimates for all of the normalized quantities that appear in the layer equation, (6.5), for a low-field and a high-field fusion reactor. (See Chapter 1.) Likewise, Table 6.1 gives estimates for all of the unnormalized quantities that appear in the growth-rate formulae (6.13), (6.16), (6.19), and (6.22) [except for $E_{ss}$, which is ${\cal O}(1)$] for a low-field and a high-field fusion reactor. (See Chapter 1.)

According to Equations (6.1), (6.3), (6.13), (6.16), (6.19), and (6.22), a linear tearing mode is unstable (except in the resistive-inertial growth-rate regime, in which it is marginally stable) when the tearing stability index, $E_{ss}$, is positive, and is stable otherwise [4]. Moreover, the perturbed magnetic field associated with the mode co-rotates with the electron fluid at the resonant surface [1]. Finally, the mode grows on a hybrid timescale that is much greater than the hydrodmagnetic time, $\tau _H$, but much less than the resistive evolution time, $\tau _R$. Note that, in all cases, the growth-rate goes to zero as $\tau_R\rightarrow \infty$. This is not surprising because, as is clear from Equation (5.39), the perturbed helical magnetic flux at the resonant surface, $\tilde{\psi}(\hat{x}=0)$, is constrained to take the value zero in the limit that $S\rightarrow\infty$. In other words, magnetic reconnection at the resonant surface (which corresponds to a finite $\tilde{\psi}$ at $\hat{x}=0$) is impossible in the absence of plasma resistivity [4].

Figure: 6.1 Linear tearing mode growth-rate regimes in $Q_\ast$-$P$ space. The various regimes are the diffusive-resisitive (DR), the viscous-resistive (VR), the resistive-inertial (RI), and the semi-collisional (SC). The $\times$ and $+$ markers indicate the location of a low-field and a high-field tokamak fusion reactor, respectively, in $Q_\ast$-$P$ space.

There are three main factors, other than plasma inertia and resistivity, that affect the growth-rate of a tearing mode in a conventional tokamak plasma. First, the strength of diamagnetic flows in the plasma, which is parameterized by the diamagnetic frequency, $\omega _\ast$ (and by the normalized diamagnetic frequency, $Q_\ast$). Second, the anomalous perpendicular diffusion of momentum and particles, which is parameterized by the momentum and particle confinement timescales, $\tau _\varphi$ and $\tau _\perp$ (and by the magnetic Prandtl numbers, $P_\varphi$ and $P_\perp$). Third, finite ion sound radius effects, which are parameterized by the ion sound radius, $d_\beta$ (and by the normalized ion sound radius $D$).

There are four tearing-mode growth-rate regimes—the resistive-inertial, the viscous-resistive, the semi-collisional, and the diffusive-resistive—and their extents in $Q_\ast$-$P$ space are illustrated in Figure 6.1 [3]. Note that Figure 6.1 differs somewhat from Figures 5.1 and 5.2 because in the latter two figures it is assumed that $\vert Q-Q_E-Q_e\vert\sim \vert Q-Q_E\vert\sim \vert Q-Q_E-Q_i\vert$ whereas in the former figure it is assumed that $\vert Q-Q_E-Q_e\vert\ll \vert Q-Q_E\vert\sim \vert Q-Q_E-Q_i\vert$. This refined ordering eliminates the nonconstant-$\psi$ response regimes, and significantly modifies the extent of the resistive-inertial growth-rate regime. It is clear from Figure 6.1 that a low-field tokamak fusion reactor lies in the diffusive-resistive growth-rate regime, whereas a high-field tokamak fusion reactor lies in the viscous-resistive growth-rate regime. (See Section 5.13.)

The absence of nonconstant-$\psi$ response regimes in Figure 6.1 should come as no surprise. As we saw in Section 5.12, nonconstant-$\psi$ resonant layers are characterized by $\skew{6}\hat{\mit\Delta}\sim\hat{\delta}_\ast^{\,-1}= S^{-1/3}\,(r_s/\delta_s)$, where $\delta _s$ is the radial layer thickness. Hence, according to Equation (6.2), asymptotic matching of such a layer to the outer solution is only possible if $\vert E_{ss}\vert\sim r_s/\delta_s \gg 1$ (given that resonant layers in tokamak plasmas are invariably very thin compared to the minor radius of the plasma). However, low-$m$ tearing modes in conventional tokamak plasmas are characterized by $\vert E_{ss}\vert\sim {\cal O}(1)$ rather than $\vert E_{ss}\vert\gg 1$ [6]. (As before, we are neglecting $m = 1$ modes, which are characterized by $\vert E_{ss}\vert\gg 1$, because they are not really tearing modes.)