Rotation Braking by a Thick Conducting Wall

The calculation presented in the previous section has a number of deficiencies. The first, and most important, deficiency is that the criterion (10.3) for the validity of the thin-wall approximation (i.e., that the radial variation of the perturbed magnetic flux across the wall is relatively weak) is not always satisfied in practice. For instance, in a low-field tokamak fusion reactor (see Section 1.5), with diamagnetic levels of ion fluid rotation (i.e., $\omega_{\perp\,i}\simeq \omega_{\ast\,e}$), the critical wall thickness above which the thin-wall approximation fails is $\delta_{w\,{\rm max}}\simeq 9.7\times 10^{-4}/\tau_w({\rm s})$ m. For the case of a high-field reactor, the critical wall thickness is $\delta_{w\,{\rm max}}\simeq 1.7\times 10^{-4}/\tau_w({\rm s})$ m. It follows that the thin-wall approximation holds for resistive walls (i.e., $\tau_w< 10^{-1}\,{\rm s}$), but not for highly conducting walls (i.e., $\tau_w> 10^{-1}\,{\rm s}$), because the latter type of wall would have to be impossibly thin.

A second deficiency in our previous calculation is that it neglects plasma inertia. This neglect is reasonable when the magnetic island chain is rotating steadily, but not when the chain's rotation frequency collapses, because such a collapse is associated with a rapid deceleration of the chain, and a consequent rapid deceleration of the plasma in the vicinity of the resonant surface.

A third deficiency in our previous calculation is that it assumes that the eddy current excited in the wall has a simple ${\rm e}^{-{\rm i}\,\omega\,t}$ time dependence, where $\omega$ is the instantaneous rotation frequency of the island chain. This assumption is reasonable when the island chain is rotating steadily, but not when its rotation frequency collapses, in which case we expect the rapid deceleration of the chain to excite a transient eddy current in the wall [4].

The aim of this section is to generalize the analysis of the previous section in order to take into account thick walls, plasma inertia, and a transient component of the wall eddy current. Note that, by a “thick” wall, we mean one in which the skin-depth in the wall material is less than the wall's radial thickness. It is still reasonable to assume that the wall's thickness, $\delta_w$, is much less than its minor radius, $r_w$.

Suppose that the wall extends from $r=r_w$ to $r=r_w+\delta_w$, where $\delta_w\ll r_w$. Here, $r$ is a conventional cylindrical coordinate. Let $\delta\psi(r,t)= R_0\,B_z\,\delta\hat{\psi}(r,t)$ be the perturbed magnetic flux within the wall [see Equation (3.20)]. Here, $B_z$ is the equilibrium toroidal magnetic field-strength. Ohm's law inside the wall yields [see Equation (3.101)]

$$\frac{\partial^2\delta\hat{\psi}}{\partial r^2} \simeq \frac{\mu_0}{\eta_w}\,\frac{\partial\delta\hat{\psi}}{\partial t},$$ (10.31)

where $\eta_w$ is the electrical resistivity of the wall material. The previous equation must be solved subject to the boundary conditions

$$\delta\hat{\psi}(r_w,t) = \hat{\mit\Psi}_w(t),$$ (10.32)

$$\frac{\partial\ln\delta\hat{\psi}(r_w+\delta_w,t)}{\partial\ln r} = - m.$$ (10.33)

The first boundary condition follows from Equations (3.82) and (3.192). Note that we have effectively redefined $\hat{\mit\Psi}_w$ to be the normalized helical magnetic flux that penetrates the inner (in $r$) boundary of the wall. The second boundary condition follows because, in the vacuum region outside the wall, a well-behaved solution of the cylindrical tearing mode equation, (3.60), varies as $r^{-m}$.

Let

$$\rho =\frac{r-r_w}{\delta_w},$$ (10.34)

$$\delta\hat{\psi}(r,t) = \hat{\mit\Psi}_w(t)\,F(\rho,t).$$ (10.35)

Equations (10.31)–(10.33) yield

$$\frac{\partial^2 F}{\partial \rho^2} =\tau_w\,\zeta_w\left(\gamma\,F + \frac{\partial F}{\partial t}\right),$$ (10.36)

$$F(0,t) =1,$$ (10.37)

$$\frac{\partial \ln F(1,t)}{\partial \rho} \simeq -m\,\zeta_w,$$ (10.38)

where

$$\gamma = \frac{d\ln\hat{\mit\Psi}_w}{dt},$$ (10.39)

$$\tau_w = \frac{\mu_0\,r_w\,\delta_w}{\eta_w},$$ (10.40)

$$\zeta_w = \frac{\delta_w}{r_w}\ll 1.$$ (10.41)

Let us, first, search for a solution of

$$\frac{d^2F_0}{d\rho^2} =\gamma\,\tau_w\,\zeta_w\,F_0,$$ (10.42)

subject to the boundary conditions

$$F_0(0) =1,$$ (10.43)

$$\frac{d\ln F_0(1)}{d\rho} = -m\,\zeta_w.$$ (10.44)

We obtain

$$F_0(\rho) = \frac{\alpha_w\,\cosh[\alpha_w\,(\rho-1)]-m\,\zeta_w\,\sinh[\alpha_w\,(\rho-1)]}{\alpha_w\,\cosh\alpha_w + m\,\zeta_w\,\sinh\alpha_w},$$ (10.45)

where

$$\alpha_w = \!\sqrt{\gamma\,\tau_w\,\zeta_w}.$$ (10.46)

However, in the physically relevant limit,

$$\vert\gamma\vert\,\tau_w\gg \zeta_w,$$ (10.47)

expression (10.45) simplifies to give

$$F_0(\rho) \simeq \frac{\cosh[\alpha_w\,(\rho-1)]}{\cosh \alpha_w}.$$ (10.48)

Incidentally, it is clear from Equation (10.48) that if the inequality (10.47) is satisfied then the boundary condition (10.44) effectively reduces to

$$\frac{d F_0(1)}{d \rho}\simeq 0.$$ (10.49)

Let us write

$$F(\rho,t) = F_0(\rho)+F_1(\rho,t),$$ (10.50)

where

$$F_1(0,t) =0,$$ (10.51)

$$\frac{\partial \ln F_1(1,t)}{\partial \rho} = -m\,\zeta_w.$$ (10.52)

Expression (10.50) automatically satisfies the boundary conditions (10.37) and (10.38). It is clear, by analogy with Equation (10.49), that if the inequality (10.47) is satisfied then the boundary condition (10.52) effectively reduces to

$$\frac{\partial F_1(1,t)}{\partial \rho}\simeq 0.$$ (10.53)

Let us write

$$F_1(\rho,t) = \sum_{j=1,\infty} f_j(t)\,\frac{\sin(\beta_{w\,j}\,\rho)}{\beta_{w\,j}},$$ (10.54)

where

$$\beta_{w\,j} = \left(j-\frac{1}{2}\right)\pi.$$ (10.55)

Expression (10.54) automatically satisfies the boundary conditions (10.51) and (10.53). Equations (10.36) and (10.50) can be combined to give

$$\sum_{k=1,\infty}\frac{df_{k}}{dt}\,\frac{\sin(\beta_{w\,k}\,\rho... ...\infty}(\gamma+\lambda_k)\,f_{k}\,\frac{\sin(\beta_{w\,k}\,\rho)}{\beta_{w\,k}}$$ (10.56)

$$\phantom{=======}-\frac{\alpha_w}{2\,\gamma}\,\frac{d\gamma}{dt}\... ...w\,(\rho-1)]\,\sinh\alpha_w}{\cosh^2\alpha_w}\right\} \sin(\beta_{w\,k}\,\rho),$$

where

$$\lambda_j = \frac{(j-1/2)^2\,\pi^2}{\tau_w\,\zeta_w}.$$ (10.57)

Finally, multiplying Equation (10.56) by $2\,\sin(\beta_{w\,j}\,\rho)$, and integrating from $\rho=0$ to $\rho=1$, we obtain [4]

$$\frac{df_j}{dt} +(\gamma+\lambda_j)\,f_j =\sigma_j\,\frac{d\gamma}{dt},$$ (10.58)

where

$$\sigma_j = \frac{2\,\lambda_j}{(\gamma+\lambda_j)^2}.$$ (10.59)

Here, use has been made of the easily demonstrated results:

$$\int_0^12\,\sin(\beta_{w\,j}\,\rho)\,\sin(\beta_{w\,k}\,\rho)\,d\rho = \delta_{jk},$$ (10.60)

$$\int_0^1\left\{ \frac{(\rho-1)\,\sinh[\alpha_w\,(\rho-1)]}{\cosh\... ...ho-1)]\,\sinh\alpha_w}{\cosh^2\alpha_w}\right\} \sin(\beta_{w\,j}\,\rho)\,d\rho =$$

$$- \frac{2\,\alpha_w\,\beta_{w\,j}}{(\alpha_w^2+\beta_{w\,j}^{\,2})^2} .$$ (10.61)

Equation (3.83) generalizes to give

$${\mit\Delta\hat{\Psi}}_w = \left[r\,\frac{\partial \delta\hat{\ps... ...si}_w}{\zeta_w}\left[\frac{\partial F}{\partial \rho}\right]_{\rho=0}^{\rho=1},$$ (10.62)

where use has been made of Equations (3.192), (3.193), (10.34), (10.35), and (10.41). Note that ${\mit\Delta\hat{\Psi}}_w$ is a measure of the normalized net eddy current induced in the wall. It follows from Equations (10.48), (10.46), (10.50), and (10.54) that [4]

$$\frac{{\mit\Delta\hat{\Psi}}_w}{\hat{\mit\Psi}_w} = G(t),$$ (10.63)

where

$$G(t) = \sqrt{\frac{\gamma\,\tau_w}{\zeta_w}}\,\tanh\left(\!\sqrt{\gamma\,\tau_w\,\zeta_w}\right) - \frac{1}{\zeta_w}\sum_{j=1,\infty} f_j(t).$$ (10.64)

Clearly, Equation (10.63) specifies the relation between the net eddy current induced in the wall and the helical magnetic flux that penetrates the inner boundary of the wall.

The first term on the right-hand side of the Equation (10.64),

$$G_0(\gamma)= \sqrt{\frac{\gamma\,\tau_w}{\zeta_w}}\,\tanh\left(\!\sqrt{\gamma\,\tau_w\,\zeta_w}\right),$$ (10.65)

specifies the net eddy current induced in the wall by a steadily rotating island chain. In the thin-wall limit [see Equations (3.104), (10.39), (10.41), and (10.47)] [4],

$$\zeta_w\ll\vert\gamma\vert\,\tau_w \ll \frac{1}{\zeta_w},$$ (10.66)

the steady-state wall response reduces to [see Equation (3.102)]

$$G_0(\gamma)\simeq \gamma\,\tau_w,$$ (10.67)

which is consistent with the analysis employed in the previous section. On the other hand, in the thick-wall limit [4],

$$\vert\gamma\vert\,\tau_w \gg \frac{1}{\zeta_w},$$ (10.68)

we obtain

$$G_0(\gamma)\simeq \sqrt{\frac{\gamma\,\tau_w}{\zeta_w}}.$$ (10.69)

The previous expression represents a steady-state wall response in which the eddy current only penetrates a distance of order the skin-depth into the wall from its inner boundary.

The second term on the right-hand side of Equation (10.64),

$$G_1(t) =- \frac{1}{\zeta_w}\sum_{j=1,\infty} f_j(t),$$ (10.70)

specifies the transient eddy current excited in the wall. As is clear from Equation (10.58), a transient eddy current is excited when the island rotation frequency (which is proportional to $\gamma$) changes in time.

Equation (3.188) (with $\hat{I}_c=0$, because there is no error-field) and Equation (10.63) yield

$$\hat{\mit\Psi}_w = \frac{E_{sw}\,\hat{\mit\Psi}_s}{G+(-\tilde{E}_{ww})}.$$ (10.71)

Writing

$$\hat{\mit\Psi}_s(t)= \left\vert\hat{\mit\Psi}_s\right\vert\!(t)\,\exp\left(-{\rm i}\int_0^t\omega(t')\,dt'\right),$$ (10.72)

where $\omega(t)$ is the instantaneous island rotation frequency, Equations (10.39) and (10.71) imply that

$$\gamma(t) = -{\rm i}\,\omega + 2\,\frac{d\ln w}{dt} - \frac{dG/dt}{G+(-\tilde{E}_{ww})}.$$ (10.73)

Here, we have made use of the fact that $w\propto \vert\hat{\mit\Psi}_s\vert^{1/2}$ [see Equation (8.1)]. Equations (3.187) and (10.71) give

$$\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}= {\mit\Delta}_{... ...lta}_{pw}(0)\right]\left[\frac{(-\tilde{E}_{ww})}{G+ (-\tilde{E}_{ww})}\right],$$ (10.74)

where use has been made of Equations (7.4) and (7.5), as well as the island saturation theory presented in Section 9.4. Hence,

$${\rm Re}\left(\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}\right) = {\mit\Delta}_{pw}(0)\left(1-\frac{w}{w_{pw}}\right) + \left[{\m... ...elta}_{pw}(0)\right]\left(\frac{\cal R}{{\cal R}^{\,2}+ {\cal I}^{\,2}}\right),$$ (10.75)

$${\rm Im}\left(\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}\right) =\left[{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)\right]\left(\frac{\cal I}{{\cal R}^{\,2}+ {\cal I}^{\,2}}\right),$$ (10.76)

where

$${\cal R} = 1 + \frac{{\rm Re}(G)}{(-\tilde{E}_{ww})},$$ (10.77)

$${\cal I} = - \frac{{\rm Im}(G)}{(-\tilde{E}_{ww})}.$$ (10.78)

Equations (8.108), (10.75), and (10.76) yield the following modified Rutherford island width evolution equation, which is a generalization of Equation (10.9):

$$I_1\,\tau_R\,\frac{d}{dt}\!\left(\frac{4\,w}{r_s}\right) = {\mit\Delta}_{pw}(0)\left(1-\frac{w}{w_{pw}}\right) + \left[{\m... ...Delta}_{pw}(0)\right]\left(\frac{\cal R}{{\cal R}^{\,2}+ {\cal I}^{\,2}}\right)$$

$$\phantom{=} +I_2\left[{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)\r... ...frac{w}{r_s}\right)^2\left(\frac{\cal I}{{\cal R}^{\,2}+ {\cal I}^{\,2}}\right)$$

$$\phantom{=}+I_3\left[{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)\ri... ...c{w}{r_s}\right)^7\left(\frac{\cal I}{{\cal R}^{\,2}+ {\cal I}^{\,2}}\right)^2.$$ (10.79)

All of the parameters appearing in this equation are defined in the previous section.

As in the previous section, the instantaneous island rotation frequency can be written [see Equation (10.10)]

$$\omega = \omega_{\perp\,i} -\sum_{p=1,\infty}(\alpha_p+\beta_p).$$ (10.80)

However, according to Equations (3.190) and (3.191), the time evolution of the quantities $\alpha_p$ and $\beta_p$ is specified by

$$\tau_M\,\frac{d\alpha_p}{dt} + \left(\zeta_\theta+ j_{1p}^{\,2}\right)\alpha_p = g_{\theta\,p}\left(\frac{\tau_\varphi}{\tau_H^{\,2}}\right) \le... ...right)^4{\rm Im}\left(\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}\right),$$ (10.81)

$$\tau_M \,\frac{d\beta_p}{dt} + j_{0p}^{\,2}\,\beta_p = g_{\varphi\,p}\left(\frac{\epsilon_s}{q_s}\right)^2\left(\frac{... ...right)^4{\rm Im}\left(\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}\right),$$ (10.82)

where all of the parameters appearing in the previous two equations are defined in the previous section, except for

$$\tau_M = \left(\frac{a}{r_s}\right)^2\tau_\varphi,$$ (10.83)

$$\zeta_\theta =\frac{\tau_M}{\tau_\theta},$$ (10.84)

$$g_{\theta\,p} = \left[\frac{J_1(j_{1p}\,r_s/a)}{J_2(j_{1p})}\right]^2,$$ (10.85)

$$g_{\varphi\,p} = \left[\frac{J_0(j_{0p}\,r_s/a)}{J_1(j_{0p})}\right]^2.$$ (10.86)

Here, $J_0(z)$ and $J_1(z)$ are standard Bessel functions, and $j_{np}$ denotes the $p$th zero of the $J_n(z)$ Bessel function [1]. Note that the terms involving $d/dt$ in Equations (10.81) and (10.82) represent plasma inertia.

Let $x=w/w_{pw}$, $y=\omega/\vert\omega_{\perp\,i}\vert$, and $T=t\,\vert\omega_{\perp\,i}\vert$. Thus, $x$ 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 the magnitude of its value when there is no interaction with the wall, and $T$ is time normalized to the typical time required for the island chain complete a full rotation. Equations (10.73) and (10.75)–(10.82) can be converted into the following closed set of normalized equations that govern the time evolution of the island chain's rotation frequency:

$$y = {\rm sgn}(\omega_{\perp\,i})-\sum_{p=1,\infty} (\hat{\alpha}_p+\hat{\beta}_p),$$ (10.87)

where

$$\hat{\tau}_{pw}\,\frac{dx}{dT} = 1- x+\beta_l\left(\frac{\cal R}{{\cal R}^{\,2}+ {\cal I}^{\,2}}... ...ht)+\delta_l'\,x^7\left(\frac{\cal I}{{\cal R}^{\,2}+ {\cal I}^{\,2}}\right)^2,$$ (10.88)

$$\hat{\tau}_M\,\frac{d\hat{\alpha}_p}{dT} = -\left(\zeta_\theta + j_{1p}^{\,2}\right)\hat{\alpha}_p + g_{\t... ...p}\,\epsilon_l'\,x^4\left(\frac{\cal I}{{\cal R}^{\,2}+ {\cal I}^{\,2}}\right),$$ (10.89)

$$\hat{\tau}_M\,\frac{d\hat{\beta}_p}{dT} = -j_{0p}^{\,2}\,\hat{\beta}_p + g_{\varphi\,p}\,\theta_l\,x^4\left(\frac{\cal I}{{\cal R}^{\,2}+ {\cal I}^{\,2}}\right).$$ (10.90)

Here,

$${\cal R} = {\rm Re}({\cal G}),$$ (10.91)

$${\cal I} = -{\rm Im}({\cal G}),$$ (10.92)

where

$${\cal G} = 1+\sqrt{\frac{\hat{\gamma}\,\hat{\tau}_w}{\hat{\zeta}_w}}\tanh\... ...{\hat{\gamma}\,\hat{\tau}_w\,\hat{\zeta}_w}\right) -\sum_{j=1,\infty}\hat{f}_j.$$ (10.93)

Moreover,

$$\hat{\gamma} = -{\rm i}\,y + 2\,\frac{d\ln x}{dT} - \frac{d\ln {\cal G}}{dT},$$ (10.94)

$$\frac{d\hat{f}_j}{dT} = -\left(\hat{\gamma}+\skew{3}\hat{\lambda}_j\right)\hat{f}_j + \hat{\sigma}_j\,\frac{d\hat{\gamma}}{dT},$$ (10.95)

$$\skew{3}\hat{\lambda}_j = \frac{(j-1/2)^2\,\pi^2}{\hat{\tau}_w\,\hat{\zeta}_w},$$ (10.96)

$$\hat{\sigma}_j = \frac{2\,\skew{3}\hat{\lambda}_j}{\hat{\zeta}_w\,(\hat{\gamma}+\skew{3}\hat{\lambda}_j)^2}.$$ (10.97)

Finally, $\beta_l$ is specified in Equation (10.20), $\zeta_\theta$ is specified in Equation (10.84), and

$$\hat{\tau}_w = \frac{\tau_w\,\vert\omega_{\perp\,i}\vert}{(-\tilde{E}_{ww})},$$ (10.98)

$$\hat{\tau}_M = \tau_M\,\vert\omega_{\perp\,i}\vert,$$ (10.99)

$$\hat{\tau}_{pw} = \tau_{pw}\,\vert\omega_{\perp\,i}\vert,$$ (10.100)

$$\hat{\zeta}_w = \frac{\delta_w\,(-\tilde{E}_{ww})}{r_w},$$ (10.101)

$$\gamma_l' = I_2\,\beta_l\left(\frac{c_\beta}{1+\tau}\right)\left(\frac{L_s}... ...ight)\left(\frac{\tau_\varphi}{\tau_H}\right)\left(\frac{w_{pw}}{r_s}\right)^2,$$ (10.102)

$$\delta_l' = I_3\,{\mit\Delta}_{pw}(0)\,\beta_l^{\,2}\left(\frac{\tau_\varphi}{\tau_H}\right)^2\left(\frac{w_{pw}}{r_s}\right)^7,$$ (10.103)

$$\epsilon_l' = \left[{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)\right]\left(\fr... ..._H^{\,2}\,\vert\omega_{\perp\,i}\vert}\right)\left(\frac{w_{pw}}{r_s}\right)^4,$$ (10.104)

$$\theta_l = \left(\frac{\epsilon_s}{q_s}\right)^2\epsilon_l'.$$ (10.105)

Note that $x$, $y$, $\hat{\alpha}_p$ and $\hat{\beta}_p$ are real quantities, whereas $\hat{\gamma}$ and $f_j$ are complex.

The type of rotation braking calculation discussed in this section is far more computationally intensive than the type discussed in the previous section, because the former type involves the solution of a great many more differential equations than the latter. However, the new calculation is an improvement on the previous one because it allows us to determine the time scale on which rotating braking occurs. Our previous calculation is unable to achieve this goal because it neglects plasma inertia.

Let us investigate a specific example. Consider a high-field tokamak fusion reactor (see Chapter 1) characterized by $B=12\,{\rm T}$, $\beta=0.02$, $T_e=T_i=7\,{\rm keV}$, $m_i=(m_D+m_T)/2$ (where $m_D$ and $m_T$ are the deuteron and triton masses, respectively), ${\mit\Xi}_{\perp\,i}= \chi_{\perp\,e}= \chi_{\perp\,i} = 1\,{\rm m^2/s}$, $\tau =1$, $\eta_e=\eta_i$, $dp/dr=-p/a$, $\omega_{\perp\,i}=4\,\omega_{\ast\,e}$. The wall parameters are $\eta_w= 6.9\times 10^{-7}\,{\rm\Omega\,m}$ (which is the electrical resistivity of stainless steel), $r_w=1.2\,a$, and $\delta_w=0.1\,a$. 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 poloidal flow-damping time is $\tau_\theta=4.59\times 10^{-5}$ s, the wall time-constant is $\tau_w=0.24\,{\rm s}$, the momentum confinement time is $\tau_M = 1.1\,{\rm s}$ [see Equation (3.180)], and the typical type required for the magnetic island to attain its final saturated width is $\tau_{pw} = 6.6\times 10^1\,{\rm s}$. The normalized parameters that characterize our model take the values $\hat{\tau}_w= 431$, $\hat{\tau}_M= 8.33\times 10^3$, $\hat{\tau}_{pw} = 4.95\times 10^5$, $\zeta_\theta=2.42\times 10^4$, $\hat{\zeta}_w=0.352$, $\beta_l=6.95\times 10^{-2}$, $\gamma_l'=3.28\times 10^{-2}$, $\delta_l'=1.66\times 10^2$, $\epsilon_l'=1.72\times 10^4$, and $\theta_l = 1.50\times 10^2$. We conclude that the effective L/R time of the wall is about $500$ times larger than the typical time required for the unperturbed magnetic island chain to complete a full rotation (i.e., $\hat{\tau}_w\sim 500$), the momentum confinement time is about $10^4$ times larger than the island rotation time (i.e., $\hat{\tau}_M\sim 10^4$), and the island saturation time is about $5\times 10^5$ times larger than the island rotation time (i.e., $\hat{\tau}_{pw}\sim 5\times 10^5$).

Figure: 10.5 Simulation of island rotation braking in a high-field tokamak fusion reactor with a thick wall. Here, $x$ and $y$ are normalized island width and rotation frequency, respectively.

It turns out that 100 poloidal and toroidal velocity harmonics are sufficient to describe the time evolution of the plasma poloidal and toroidal rotation profiles in a reasonably accurate manner. Consequently, we shall neglect all $\alpha_p$ and $\beta_p$ variables with $p>p_{\rm max}=100$ in our calculation. In order to compensate for the truncation of the sum in Equation (10.87), we shall replace this equation by

$$y = {\rm sgn}(\omega_{\perp\,i}) - \sum_{p=1,p_{\rm max}}\left(\frac{\hat{\alpha}_p}{S_\theta} + \frac{\beta_p}{S_\varphi}\right),$$ (10.106)

where

$$S_\theta = 4\left(\frac{r_s}{a}\right)\sqrt{\zeta_\theta}\sum_{p=1,p_{\rm max}}\frac{g_{\theta\,p}}{\zeta_\theta+j_{1p}^{\,2}},$$ (10.107)

$$S_\varphi =\frac{2}{\ln(a/r_s)}\sum_{p=1,p_{\rm max}}\frac{g_{\varphi\,p}}{j_{0p}^{\,2}}.$$ (10.108)

[See Equations (7.34) and (7.35).] For the calculation in hand, $S_\theta = 0.708$ and $S_\varphi = 0.997$.

For the transient wall harmonics, an examination of Equations (10.95) and (10.97) implies that, roughly speaking, that all harmonics in the range $1\leq j< 2\, j_{\rm crit}$, where $\hat{\lambda}_{j_{\rm crit}} \sim \vert\hat{\gamma}\vert$, are important in the calculation. Hence, given that $\vert\hat{\gamma}\vert\leq 1$, we deduce from Equation (10.96) that $j_{\rm crit}\sim (\hat{\tau}_w\,\hat{\zeta}_w)^{1/2}/\pi\sim 7$. This result merely implies that when the island chain is rotating at its unperturbed rotation frequency the transient eddy current induced in the wall only penetrates radially into about the inner 7th part of the wall. Hence, we need to retain all transient wall harmonics up to of order $j=14$ in order to resolve this relatively thin current distribution. In the following, we shall keep all transient wall harmonics up to $j=20$ (i.e., we shall neglect $\hat{f}_j$ variables with $j>j_{\rm max}=20$ in our calculation), so as to ensure that all important transient wall harmonics are retained in the calculation. It follows that the final set of coupled, first-order, ordinary differential equations that makes up our model consists of 243 real equations.

Figure 10.5 shows the numerical solution of our set of differential equations. The solution is qualitatively similar to that obtained for a thin wall. (See Figure 10.2.) As before, it can be seen that as the normalized width, $x$, of the island chain grows in time, the chain's normalized rotation frequency, $y$, is gradually reduced, until it has been reduced to about half of its original value, at which point there is a sudden collapse in the rotation frequency to a very low value. The rotation collapse occurs when $x=0.69$, which corresponds to $W/a=0.19$. It is clear from the right-hand panel of Figure 10.5 that the rotation collapse takes place over a time interval of about 100 normalized time units, which corresponds to about 15 ms. This timescale is similar to the hybrid timescale $(\tau_\theta\,\tau_M)^{1/2} = 7$ ms. Hence, it is plausible that the timescale for the rotation collapse is determined by a combination of poloidal flow damping and perpendicular viscosity.

Figure: 10.6 Simulation of island rotation braking in a high-field tokamak fusion reactor with a thick wall. Here, $y$ and ${\cal T}$ are the normalized rotation frequency and a torque balance diagnostic, respectively.

We can construct a torque balance diagnostic:

$${\cal T}(T) = y -1 + \xi_l\,x^4\left(\frac{\cal I}{{\cal R}^{\,2}+ {\cal I}^{\,2}}\right),$$ (10.109)

where

$$\xi_l = \left[{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)\right]\left(\fr... ..._H^{\,2}\,\vert\omega_{\perp\,i}\vert}\right)\left(\frac{w_{pw}}{r_s}\right)^4,$$ (10.110)

and $\tau_V$ is specified in Equation (10.12). The quantity ${\cal T}$ takes the value zero when the plasma is in torque balance. In other words, when the electromagnetic braking torque exerted at the rational surface is exactly balanced by the viscous restoring torque. Obviously, plasma inertia plays no role in the rotation braking process when the plasma is in torque balance. On the other hand, if ${\cal T}$ is non-zero then the electromagnetic braking torque is not balanced by the viscous restoring torque, which indicates that plasma inertia is playing a role in the braking process. Figure 10.6 displays the time evolution of the torque balance diagnostic in the rotation braking simulation shown in Figure 10.5. It can be seen that the plasma is in torque balance to a very good approximation both before and after the rotation collapse. However, during the rotation collapse, the plasma is clearly not in torque balance, indicating that plasma inertia plays an important role in the rotation collapse. According to the right-hand panel of Figure 10.6, after torque balance breaks down during the rotation collapse, it takes a time interval of order 4000 normalized time units, which corresponds to about $0.53$ s, for torque balance to be reestablished. This timescale is similar to the momentum confinement timescale, $\tau_M=1.1$ s. Hence, it is plausible that torque balance is reestablished by plasma viscosity.

Figure: 10.7 Simulation of island rotation braking in a high-field tokamak fusion reactor with a thick wall. Here, $\hat{V}_\theta$ and $\hat{V}_\varphi$ are measures of the poloidal and toroidal ion fluid velocities, respectively, at the rational surface.

Figure 10.7 shows the time evolution of the quantities

$$\hat{V}_\theta(T) = \sum_{p=1,p_{\rm max}}\frac{\hat{\alpha}_p}{S_\theta},$$ (10.111)

$$\hat{V}_\varphi(T) =\sum_{p=1,p_{\rm max}}\frac{\hat{\beta}_p}{S_\varphi},$$ (10.112)

in the rotation braking simulation shown in Figure 10.5. Now, the rotation braking process causes the normalized rotation frequency of the island chain, $y$, to decrease from unity (assuming that $\omega_{\perp\,i}>0$) to a value that is very much smaller than unity. In other words, ${\mit\Delta}y \simeq -1$. Hence, it is clear from Equations (10.106), (10.111), and (10.112) that ${\mit\Delta} \hat{V}_\theta+{\mit\Delta}\hat{V}_\varphi = 1$. Here, ${\mit\Delta}\hat{V}_\theta$ is the fraction of the decrease in the island rotation frequency that is due to a shift in the poloidal ion fluid angular velocity at the rational surface, whereas ${\mit\Delta}\hat{V}_\varphi$ is the fraction of the decrease that is due to a shift in the toroidal ion fluid angular velocity at the rational surface. In fact, it is apparent from Figure 10.7 that about 53% of the decrease in the rotation frequency is due to a poloidal velocity shift, the remaining 47% being due to a toroidal velocity shift. It is also apparent from the figure's right-hand panel that the rotation collapse, which takes place on a timescale of about 100 normalized time units [i.e., $(\tau_\theta\,\tau_M)^{1/2}$], is due to a sudden shift in the poloidal angular velocity at the rational surface. In fact, this sudden shift is responsible for the loss of torque balance during the rotation collapse. The corresponding shift in the toroidal angular velocity at the rational surface takes place on a timescale of 4000 normalized time units (i.e., $\tau_M$). Note that, after the sudden shift that is associated with rotation collapse, the poloidal velocity subsequently readjusts to its final value on the $\tau_M$ timescale.

Figure: 10.8 Simulation of island rotation braking in a high-field tokamak fusion reactor with a thick wall. Here, $y$ and the $f_j$ are the normalized rotation frequency and wall transient harmonics, respectively.

Figure 10.8 shows the time evolution of the transient wall harmonics in the rotation braking simulation shown in Figure 10.5. It can be seen that the transient wall harmonics are only important [i.e., $\vert f_j\vert\sim {\cal O}(1)$] during the rotation collapse. Note that $\vert f_{j_{\rm max}}\vert\ll \vert f_{1}\vert$, $\vert f_{j_{\rm crit}}\vert$ at all times, indicating that our calculation has included all of the important transient wall harmonics. Prior to the rotation collapse, $\vert f_{j_{\rm crit}}\vert \gg \vert f_{1}\vert$, indicating that the (very small) transient eddy current induced in the wall is localized to within a skin-depth of the inner boundary of the wall. However, during the rotation collapse, the low-$j$ transient wall harmonics become dominant, indicating that the transient eddy current has penetrated to the outer boundary of the wall. It can been seen from the right-hand panel of Figure 10.8 that the longest-wavelength $f_1$ transient wall harmonic excited by the rotation collapse decays away after a time interval of about 1000 normalized time units, which corresponds to 0.13 s. This timescale is similar to the time-constant of the wall, $\tau_w= 0.24$ s. Hence, it is plausible that the transient eddy current induced by the rotation collapse decays away after a time interval of order the wall time-constant.