Improved Torque Balance Model

It is clear from the calculation presented in the previous section that plasma inertia and the transient wall eddy current are only important during the rotation collapse. At all other times, it should be possible to accurately model the wall response using Equation (10.65), which specifies the response of a wall of arbitrary thickness to a steadily rotating island chain, with $\gamma=-{\rm i}\,\omega$. This observation leads us to formulate the following generalization of the torque balance model presented in the Section 10.2:

$\displaystyle \frac{dx}{dT}$ $\displaystyle =1-x+\beta_l\left(\frac{{\cal R}}{{\cal R}^2+{\cal I}^2}\right)
\frac{\cal I}{{\cal R}^{\,2}+{\cal I}^{\,2}}\right)^2,$ (10.113)
$\displaystyle \zeta\,\frac{dy}{dT}$ $\displaystyle = 1-y -\xi_l\,x^4\left(\frac{\cal I}{{\cal R}^{\,2}+{\cal I}^{\,2}}\right),$ (10.114)


$\displaystyle {\cal G} = 1 + \sqrt{\frac{-{\rm i}\,y\,\hat{\tau}_w}{\hat{\zeta}_w}}\,\tanh\!\sqrt{-{\rm i}\,y\,\hat{\tau}_w\,\hat{\zeta}_w}.$ (10.115)

Here, $x$ is the width of the magnetic island chain relative to its saturated width when the wall is perfectly conducting, $y$ is the island rotation frequency relative to its value when there is no interaction with the wall, and $T$ is time normalized to the typical time required for the island chain to attain is final saturated width. Moreover, $\beta_l$, $\gamma_l'$, $\delta_l'$, $\xi_l$, ${\cal R}$, ${\cal I}$, $\hat{\tau}_w$, and $\hat{\zeta}_w$ are defined in Equations (10.20), (10.102), (10.103), (10.110), (10.91), (10.92), (10.98), and (10.101), respectively. The improved torque balance model should be capable of accurately predicting the critical island width that must be exceeded in order to trigger a collapse in the island rotation frequency, even when the thin-wall approximation breaks down, despite not being able to accurately predict the timescale on which the rotation collapse occurs.

Figure 10.9: Critical island width required to trigger rotation collapse calculated as a function of the wall thickness, for various unperturbed island rotation frequencies, using the improved torque balance model. The left-hand/right-hand panels correspond to a low-field/high-field tokamak fusion reactor

Figure 10.9 shows the critical island width required to trigger a rotation collapse calculated as a function of the wall thickness, for various different unperturbed island rotation frequencies, using the improved torque balance model. Separate calculations are made for a low-field and a high-field tokamak fusion reactor. (See Chapter 1.) The calculation parameters are determined 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), ${\mit\Xi}_{\perp\,i}= \chi_{\perp\,e}= \chi_{\perp\,i} = 1\,{\rm m^2/s}$, $\tau =1$, $\eta_e=\eta_i$, and $dp/dr=-p/a$. The plasma equilibrium is assumed to be of the Wesson type (see Section 9.4), with $q(0)=0.8$ and $q(a)=6.0$. The poloidal and toroidal mode numbers of the tearing mode are $m=2$ and $n=1$, respectively. It follows that $r_s=0.560\,a$. The perfect-wall saturated island width is $W_{pw}/a= 0.272$. The wall radius and resistivity are assumed to be $r_w=1.2\,a$ and $\eta_w= 6.9\times 10^{-7}\,{\rm\Omega\,m}$ (which is the electrical resistivity of stainless steel), respectively.

As is clear from Figure 10.9, the critical island width required to trigger rotation collapse increases with increasing wall thickness (because the wall becomes less electrically resistive) until a critical thickness is reached above which the critical island width becomes independent of the wall thickness. Of course, the critical wall thickness is that above which the thin-wall approximation breaks down. The eddy current induced by the rotating island chain in a wall whose thickness is greater than the critical thickness (which corresponds to a skin-depth in the wall material) only penetrates a skin-depth into the wall from its inner boundary, which implies that the effective thickness of the wall becomes the skin-depth, rather than its actual thickness. According to the figure, for a plasma with diamagnetic levels of ion fluid rotation (i.e., $\omega_{\perp\,i}\sim\omega_{\ast\,e}$), the critical island width is below 10% of the plasma minor radius for thin (i.e., $\delta_w/a\sim 10^{-3}$) resistive walls. On the other hand, the critical island width is about twice this value for thick (i.e., $\delta_w/a\sim 10^{-1}$) conducting walls. As before, it is apparent that a low-field tokamak fusion reactor is more susceptible to rotation braking than a high-field fusion reactor because of its lower diamagnetic frequency, and consequent lower ion fluid rotation.