Rotation Braking by a Thin Conducting Wall

Let us, first of all, investigate the rotation braking of a nonlinear magnetic island chain due to electromagnetic interaction with a thin conducting wall (in the absence of an error-field). Note that, by a “thin” wall, we mean one in which the skin-depth in the wall material is greater than the wall's radial thickness.

Assuming that the chain possesses the rotation frequency $\omega$ in the laboratory frame, so that $d/dt\equiv - {\rm i}\,\omega$, and that $\hat{I}_c=0$ (i.e., there is no error-field), Equation (3.188) yields

$$\hat{\mit\Psi}_w = \frac{E_{sw}/(-\tilde{E}_{ww})\,\hat{\mit\Psi}_s}{1-{\rm i}\,\tau_{LR}\,\omega}.$$ (10.1)

Here, $\hat{\mit\Psi}_s$ is the normalized reconnected magnetic flux at the rational surface [see Equations (3.72) and (3.184)], $\hat{\mit\Psi}_w$ is the normalized helical magnetic flux that penetrates the wall [see Equations (3.82) and (3.192)], the real dimensionless parameters $E_{sw}$ and $\tilde{E}_{ww}$ are defined in Equations (3.87) and (3.195), respectively,

$$\tau_{LR} = \frac{\tau_w}{(-\tilde{E}_{ww})}$$ (10.2)

is the effective $L/R$ time of the wall, and $\tau_w$ the time-constant of the wall [see Equation (3.103)]. Note that Equation (10.1) is only valid in the so-called thin-wall limit,

$$\frac{\delta_w}{r_w}\ll \frac{1}{\vert\omega\vert\,\tau_w}$$ (10.3)

[see Equation (3.104)], where $\delta_w$ is the radial thickness of the wall, and $r_w$ its minor radius. The previous inequality ensures that the perturbed radial magnetic field only exhibits weak radial variation across the wall.

Equations (3.187) and (10.1) yield

$$\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s} = {\mit\Delta}_... ...)\left(\frac{1+{\rm i}\,\tau_{LR}\,\omega}{1+\tau_{LR}^{\,2}\,\omega^2}\right),$$ (10.4)

where use has been made of Equations (7.4) and (7.5). Here, ${\mit\Delta}_{pw}$ is the (real, dimensionless) tearing stability index when the wall is perfectly conducting (i.e., $\tau_w\rightarrow\infty$), whereas ${\mit\Delta}_{nw}$ is the (real, dimensionless) tearing stability index when there is no wall (i.e., $\tau_w\rightarrow 0$). We expect ${\mit\Delta}_{nw}-{\mit\Delta}_{pw}>0$. In other words, we expect the magnetic island chain to be stabilized by the presence of a perfectly conducting wall [3,10].

It follows from the previous equation that

$${\rm Re}\left(\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}\right) = {\mit\Delta}_{pw} + ({\mit\Delta}_{nw}-{\mit\Delta}_{pw})\left(\frac{1}{1+\tau_{LR}^{\,2}\,\omega^2}\right),$$ (10.5)

$${\rm Im}\left(\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}\right) =({\mit\Delta}_{nw}-{\mit\Delta}_{pw})\left(\frac{\tau_{LR}\,\omega}{1+\tau_{LR}^{\,2}\,\omega^2}\right).$$ (10.6)

Finally, making use of the island saturation theory presented in Section 9.4, we obtain

$${\rm Re}\left(\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}\right) = {\mit\Delta}_{pw}(0)\left(1-\frac{w}{w_{pw}}\right) + \left[{\m... ...-{\mit\Delta}_{pw}(0)\right]\left(\frac{1}{1+\tau_{LR}^{\,2}\,\omega^2}\right),$$ (10.7)

$${\rm Im}\left(\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}\right) =\left[{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)\right]\left(\frac{\tau_{LR}\,\omega}{1+\tau_{LR}^{\,2}\,\omega^2}\right).$$ (10.8)

Here, ${\mit\Delta}_{pw}(0)$ is the perfect-wall tearing stability index when the width of the magnetic island chain at the rational surface is zero. Likewise, ${\mit\Delta}_{nw}(0)$ is the no-wall tearing stability index when the width of the magnetic island chain at the rational surface is zero. Finally, $w_{pw}=W_{pw}/4$, where $W_{pw}$ is the saturated radial magnetic island width when the wall is perfectly conducting [see Equation (9.21)].

Equations (8.108), (10.7), and (10.8) yield the following modified Rutherford island width evolution equation [13]:

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

$$\phantom{=} +I_2\left[{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)\r... ...r_s}\right)^2\left(\frac{\tau_{LR}\,\omega}{1+\tau_{LR}^{\,2}\,\omega^2}\right)$$

$$\phantom{=}+I_3\left[{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)\ri... ...}\right)^7\left(\frac{\tau_{LR}\,\omega}{1+\tau_{LR}^{\,2}\,\omega^2}\right)^2.$$ (10.9)

Here, $r_s$ is the minor radius of the rational surface, $\tau _R$ the resistive evolution time [see Equation (5.49)], $\tau _H$ the hydromagnetic time [see Equation (5.43)], $\tau _\varphi$ the toroidal momentum confinement time [see Equation (5.50)], $c_\beta$ a dimensionless measure of the plasma pressure at the rational surface [see Equation (4.65)], $\tau$ the ratio of the electron and ion pressure gradients at the rational surface [see Equation (4.5)], $4\,w$ the radial width of the magnetic island chain, $d_\beta= c_\beta\,d_i$ the ion sound radius, $d_i$ the collisionless ion skin-depth at the rational surface [see Equation (4.24)], $L_s$ the magnetic shear-length at the rational surface [see Equation (5.27)], and $L_p$ the effective pressure gradient scale-length at the rational surface [see Equation (8.35)]. Furthermore, $I_1=0.8227$, $I_2= 0.04889$, and $I_3=0.02944$. (See Section 8.11.)

The first term on the right-hand side of the previous equation governs the growth and saturation of the magnetic island chain when its rotation frequency is sufficiently large that the wall acts as a perfect conductor. The second term describes the loss of wall stabilization when the chain's rotation frequency is sufficiently small that perturbed magnetic field can penetrate through the wall [3,10]. The third and fourth terms represent the destabilizing effect of the ion polarization current induced in the vicinity of the rational surface when the ion fluid is diverted around the island chain's magnetic separatrix [7,13]. Unlike the case of an isolated island chain (see Chapter 9), the polarization effect is non-zero because, according to Equation (8.74), (8.87), (8.101), and (10.8), the electromagnetic braking torque exerted on the plasma in the immediate vicinity of the rational surface, due to interaction with the conducting wall, generates finite ion flow in the island rest frame.

Assuming that the island chain co-rotates with the ion fluid at the rational surface (see Section 9.5), Equations (3.189) and (9.31) imply that

$$\omega = \omega_{\perp\,i} -\sum_{p=1,\infty}(\alpha_p+\beta_p),$$ (10.10)

where $\omega_{\perp\,i}$, which is defined in Equation (9.32), is the unperturbed (by any electromagnetic torques that develop at the rational surface) rotation frequency. Here, we are treating $\alpha_p$ and $\beta_p$ as constants because the electromagnetic torque that develops at the rational surface has no explicit time dependance (assuming that the width of the island chain grows on a timescale that is much greater than $\tau_{LR}$ and $\tau _\varphi$). Equations (3.180), (3.190), (3.191), (4.23), (5.27), (5.43), (5.50), (7.28), and (7.34)–(7.35) can be combined to give

$$\sum_{p=1,\infty}(\alpha_p+\beta_p)= \left(\frac{\epsilon_s}{q_s}... ...right)^4{\rm Im}\left(\frac{{\mit\Delta\hat{\Psi}}_s}{\hat{\mit\Psi}_s}\right),$$ (10.11)

where

$$\tau_V = \frac{1}{4}\left[\left(\frac{q_s}{\epsilon_s}\right)^2\l... ...tau_\varphi}\right)^{1/2} +2\,\ln\left(\frac{a}{r_s}\right)\right]\tau_\varphi.$$ (10.12)

Here, $\tau_\theta$ is the poloidal flow-damping time [see Equation (7.28)], $\epsilon_s=r_s/R_0$, $R_0$ the simulated major radius of the plasma, $a$ the minor radius of the plasma, $q_s=m/n$, $m$ the poloidal mode number of the island chain, and $n$ the toroidal mode number of the island chain. Equations (10.8), (10.10), and (10.11) can be combined to give the torque balance equation [6]

$$\omega_{\perp\,i}-\omega = \left[{\mit\Delta}_{nw}(0)-{\mit\Delta... ..._s}\right)^4\left(\frac{\tau_{LR}\,\omega}{1+\tau_{LR}^{\,2}\,\omega^2}\right).$$ (10.13)

The left-hand side of the previous equation represents the viscous restoring torque that acts to prevent changes in the plasma rotation at the rational surface, whereas the right-hand side represents the electromagnetic braking torque acting on the plasma in the vicinity of the rational surface due to the eddy current induced in the conducting wall.

Figure: 10.1 Steady-state solutions of the torque balance equation, (10.18). The solid lines show solution curves for various different values of $\alpha _l$. The dashed line shows the critical solution curve, $\alpha _l=\sqrt {27}$. The dash-dotted line shows the boundary of the forbidden band of island rotation frequencies. The high-rotation solution branch lies above the critical solution curve, to the right of the forbidden band. The low-rotation solution branch lies above the critical solution curve, to the left of the forbidden band. Within the forbidden band, all of the solutions of the torque balance equation are dynamically unstable. When a high-rotation/low-rotation solution reaches the boundary of the forbidden band there is a bifurcation to a low-rotation/high-rotation solution with the same value of $\epsilon_l\,x^4/8$.

Let

$$x = \frac{w}{w_{pw}},$$ (10.14)

$$y = \frac{\omega}{\omega_{\perp\,i} },$$ (10.15)

$$T = \frac{t}{\tau_{pw}},$$ (10.16)

where $\tau_{pw}$ is defined in Equation (9.22). 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 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. The modified Rutherford equation, (10.9), and the torque balance equation, (10.13), can be written in the non-dimensional forms [13]

$$\frac{dx}{dT} = 1 - x +\beta_l\left(\frac{1}{1+\alpha_l^2\,y^2}\right) + \gamma... ...,y^2}\right) +\delta_l \left[\frac{x^7\,y^2}{(1+\alpha_l^2\,y^2)^{\,2}}\right],$$ (10.17)

$$\zeta\,\frac{dy}{dT} = 1- y- \epsilon_l\left(\frac{x^4\,y}{1+\alpha_l^2\,y^2}\right),$$ (10.18)

respectively, where

$$\alpha_l =\tau_{LR} \,\vert\omega_{\perp\,i}\vert,$$ (10.19)

$$\beta_l = \frac{{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)}{{\mit\Delta}_{pw}(0)},$$ (10.20)

$$\gamma_l = I_2\,\alpha_l\,\beta_l\left(\frac{c_\beta}{1+\tau}\right)\left(... ...ight)\left(\frac{\tau_\varphi}{\tau_H}\right)\left(\frac{w_{pw}}{r_s}\right)^2,$$ (10.21)

$$\delta_l = I_3\,{\mit\Delta}_{pw}(0)\,(\alpha_l\,\beta_l)^{2}\left(\frac{\tau_\varphi}{\tau_H}\right)^2\left(\frac{w_{pw}}{r_s}\right)^7,$$ (10.22)

$$\epsilon_l = \left[{\mit\Delta}_{nw}(0)-{\mit\Delta}_{pw}(0)\right]\left(\fr... ...\frac{\tau_{LR}\,\tau_V}{\tau_H^{\,2}}\right)\left(\frac{w_{pw}}{r_s}\right)^4,$$ (10.23)

and $0<\zeta\ll 1$. Here, we have added artificial plasma inertia (i.e., the $\zeta\,dy/dT$ term) into the torque balance equation, (10.18), in order to distinguish between dynamically-stable and dynamically-unstable solutions [6].

The steady-state solutions of the torque balance equation, (10.18), correspond to the roots of the cubic polynomial

$$\alpha_l^{\,2}\,y^3 - \alpha_l^{\,2}\,y^2 +\left(1+\epsilon_l\,x^4\right)y - 1 = 0.$$ (10.24)

If $\alpha_l<\sqrt{27}$ then the previous equation only possesses one real root. In this situation, the normalized island rotation frequency, $y$, slows down in a smooth reversible fashion as the normalized island width, $x$, increases [6]. On the other hand, if $\alpha_l>\sqrt{27}$ then the previous equation possesses three real roots. However, the intermediate root is dynamically unstable. In this situation, the torque balance equation possesses two branches of dynamically-stable solutions [6]. The high-rotation solution branch is characterized by $1/3\leq y\leq 1$. On the other hand, the low-rotation solution branch is characterized by $0\leq y\leq 1/3$. The two solution branches are separated by a forbidden band of island rotation frequencies. The existence of this forbidden band has been verified experimentally [8]. The high-rotation solution branch ceases to exist when the normalized island width exceeds a critical value, and there is a bifurcation to the low-rotation solution branch. Likewise, the low-rotation solution branch ceases to exist when the normalized island width falls below a different critical value, and there is a bifurcation to the high-rotation solution branch. Thus, if $\alpha_l>\sqrt{27}$ then the slowing down of the island rotation frequency with increasing island width is neither smooth nor reversible [6]. The steady-state solutions of the torque balance equation are illustrated in Figure 10.1.

In the limit $\alpha_l\gg \sqrt{27}$, we can find approximate solutions of the torque balance equation, (10.18). The high-rotation solution branch is characterized by $1/2\leq y\leq 1$, and is such that

$$y \simeq \frac{1}{2}\left(1+\left[1-\left(\frac{x}{x_h}\right)^4\right]^{1/2}\right),$$ (10.25)

where

$$x_h =\left(\frac{\alpha_l^2}{4\,\epsilon_l}\right)^{1/4}.$$ (10.26)

The high-rotation solution branch ceases to exist when the normalized island width, $x$, exceeds the critical value, $x_h$—at which point the island rotation frequency has been reduced to half of its original value—and there is a bifurcation to the low-rotation solution branch. The low-rotation solution branch is characterized by $0\leq y\leq 1/\alpha_l$, and is such that

$$y \simeq \frac{1}{\alpha_l}\left\{\left(\frac{x}{x_l}\right)^4-\left[\left(\frac{x}{x_l}\right)^8-1\right]^{1/2}\right\},$$ (10.27)

where

$$x_l=\left(\frac{2\,\alpha_l}{\epsilon_l}\right)^{1/4}.$$ (10.28)

The low-rotation solution branch ceases to exist when the normalized island width falls below the critical value $x_l$, and there is a bifurcation to the high-rotation solution branch. Note that $x_h\gg x_l$, which indicates that, once the width of the island chain has grown sufficiently wide to trigger a high-rotation to low-rotation bifurcation, the chain is unlikely to ever attain a high-rotation state again (because its width would have to shrink by a considerable factor to trigger the reverse bifurcation). In the limit $\alpha_l\gg \sqrt{27}$, the forbidden band of normalized island rotation frequencies corresponds to $1/\alpha_l < y < 1/2$. (See Figure 10.1.)

Figure: 10.2 Simulation of island rotation braking in a low-field (left panel) and a high-field (right panel) tokamak fusion reactor with a thin wall. Here, $x$ and $y$ are normalized island width and rotation frequency, respectively.

It is clear from Equation (10.17) that the slowing down of the island chain's rotation due to the eddy current induced in the conducting wall has a destabilizing effect on the chain. In fact, if there is no substantial slowing down then the normalized modified Rutherford equation (10.17) reduces to

$$\frac{dx}{dT} \simeq 1 - x,$$ (10.29)

assuming that $\alpha_l\gg \beta_l^{1/2}$, $\gamma_l^{1/2}$, $\delta_l^{1/4}$, indicating that the normalized island width, $x$, saturates at its perfect-wall value, unity. On the other hand, on the low-rotation solution branch, assuming that $\alpha_l\gg \sqrt{27}$ and $x\gg x_l$, Equation (10.17) reduces to

$$\frac{dx}{dT} \simeq 1 - x + \beta_l +\frac{\gamma_l}{\epsilon_l\,x^2}+\frac{\delta_l}{\epsilon_l^{\,2}\,x},$$ (10.30)

indicating that the normalized island width saturates at a value that is greater than unity. (Because $\beta_l$, $\gamma_l$, $\delta_l$, and $\epsilon_l$ are all positive quantities.) The island chain is destabilized by the loss of wall stabilization, and also by the ion polarization effect associated with rotation braking.

Figure 10.2 shows a numerical solution of Equations (10.17) and (10.18) made for a low-field and a high-field tokamak fusion reactor. (See Chapter 1.) The simulation 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), ${\mit\Xi}_{\perp\,i}= \chi_{\perp\,e}= \chi_{\perp\,i} = 1\,{\rm m^2/s}$, $\tau =1$, $\eta_e=\eta_i=1$, $dp/dr=-p/a$, $\omega_{\perp\,i}=4\,\omega_{\ast\,e}$. Here, $\omega_{\ast\,e}$ is the electron diamagnetic frequency [see Equation (5.45)]. The poloidal and toroidal mode numbers of the magnetic island chain are $m=2$ are $n=1$, respectively. The wall parameters are $\tau_w= 10^{-1}\,{\rm s}$ and $r_w/a=1.1$, which correspond to a moderately conducting, close-fitting wall. The plasma equilibrium is assumed to be of the Wesson type (see Section 9.4), with $q(0)=0.8$ and $q(a)=6.0$. It follows that $r_s=0.560\,a$. The perfect-wall saturated island width is $W_{pw}/a= 0.264$. Finally, the various dimensionless parameters appearing in Equations (10.17) and (10.18) take the values $\alpha_l=7.21\times 10^1$ (low-field) or $\alpha_l=1.73\times 10^2$ (high-field), $\beta_l=0.104$, $\gamma_l=7.98$, $\delta_l=8.61\times 10^6$, $\epsilon_l=2.05\times 10^4$, and $\zeta=0.1$.

It can be seen from Figure 10.2 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 collapse in the rotation frequency causes the chain to be further destabilized due to the loss of wall stabilization, and ion polarization effects. Consequently, the final saturated width of the island chain is greater than the perfect-wall saturated island width (i.e., $x>1$). Note 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 (see Table 6.1), and consequent lower ion fluid rotation. The slowing-down curves shown in the figure are similar in form to those observed experimentally when a wide magnetic island chain interacts electromagnetically with a resistive wall in a toroidal confinement device [4].

Figure: 10.3 Results of a series of simulations of the type shown in Figure 10.2 in which the radius of the rational surface is scanned over a range of values (by changing the edge safety-factor). $W_{crit}$ is the critical island chain width above which the chain's rotation frequency collapses. $W_{pw}$ is the perfect-wall saturated island width. $W_\infty$ is the final saturated island width. $y_\infty$ is the final normalized island rotation frequency. The left-hand/right-hand panels correspond to a low-field/high-field tokamak fusion reactor.

Figure 10.3 displays the results of a series of simulations of the type shown in Figure 10.2 in which the radius of the rational surface is scanned over a range of values [by changing $q(a)$ while keeping $q(0)=0.8$ fixed]. The wall parameters are $r_w/a=1.2$ and $\tau_w= 5\times 10^{-2}\,{\rm s}$. It can be seen that if the rational surface lies well inside the plasma boundary then the island chain attains its final, perfect-wall saturated width without a collapse in its rotation frequency. On the other hand, if the rational surface lies closer to the plasma boundary then a collapse in the rotation frequency is triggered before the chain attains its final saturated width. The critical island width at which the rotation collapse is triggered lies between 20% and 10% of the plasma minor radius. Moreover, the final saturated width of the island chain exceeds the perfect-wall value due to the loss of wall stabilization, and ion polarization effects. Finally, it is again clear that a low-field tokamak fusion reactor is more susceptible to rotation braking than a high-field fusion reactor.

Figure: 10.4 Results of a series of simulations of the type shown in Figure 10.2 in which the unperturbed island rotation frequency is scanned over a range of values for various different wall time-constants. $W_{crit}$ is the critical island chain width above which the chain's rotation frequency collapses. The left-hand/right-hand panels correspond to a low-field/high-field tokamak fusion reactor.

Figure 10.4 shows the results of a series of simulations of the type shown in Figure 10.2 in which the unperturbed island chain rotation frequency is scanned over a range of values for various different wall time-constants. The plasma parameters are $q(0)=0.8$ and $q(a)=7.0$, which corresponds to $r_s=0.517\,a$, whereas the wall radius is $r_w=1.2\,a$. It is clear that if the wall is very resistive (i.e., $\tau_w\lesssim10^{-3}$ s) then the critical island width above which the chain's rotation frequency collapses is only weakly dependent on the unperturbed rotation frequency, and is about 10% of the plasma minor radius. On the other hand, for the case of a highly conducting wall (i.e., $\tau_w\lesssim 1$ s), diamagnetic levels of plasma rotation are sufficient to completely suppress the collapse in the island rotation frequency, unless the unperturbed rotation frequency lies close to zero. As before, it is clear that a low-field tokamak fusion reactor is more susceptible to rotation braking than a high-field fusion reactor.