Effect of Electromagnetic Torques

We have seen that a nonlinear $n=1$ tearing mode that reconnects magnetic flux principally at a given rational surface in our example tokamak discharge generates comparatively small reconnected fluxes at the other rational surfaces, as a consequence of sheared plasma rotation. However, the small, but nonzero, reconnected fluxes driven at the other surfaces give rise to localized electromagnetic torques. [See Equations (14.69) and (14.70).] In principle, such torques can modify the plasma rotation, and may even lead to the collapse of the rotation shear that is responsible for the strong shielding of different rational surfaces from one another [14]. Let us investigate this effect.

We shall adopt model plasma equations of poloidal and toroidal angular motion that are analogous to those introduced in Section 3.14. Let us write

$${\mit\Omega}_\theta(r,t)= {\mit\Omega}_{\theta\,0}(r) + {\mit\Delta\Omega}_\theta(r,t),$$ (14.153)

$${\mit\Omega}_\varphi(r,t) = {\mit\Omega}_{\varphi\,0}(r) +{\mit\Delta\Omega}_\varphi(r,t).$$ (14.154)

Here, $r$ is the flux-surface label introduced in Section 14.2. Moreover, ${\mit\Omega}_\theta(r,t)$ and ${\mit\Omega}_\varphi(r,t)$ are the majority ion poloidal and toroidal angular velocity profiles, respectively. Furthermore, ${\mit\Omega}_{\theta\,0}(r)$ and ${\mit\Omega}_{\varphi\,0}(r)$ are the majority ion poloidal and toroidal angular velocity profiles, respectively, in the absence of electromagnetic torques at the rational surfaces, whereas, ${\mit\Delta\Omega}_\theta(r,t)$ and ${\mit\Delta\Omega}_\varphi(r,t)$ are the respective changes in these profiles induced by the electromagnetic torques. The modifications to the angular velocity profiles are governed by poloidal and toroidal angular equations of motion that take the respective forms [8,16]:

$$4\pi^2\,R_0\left[\rho\,r^3\,\frac{\partial{\mit\Delta\Omega}_\the... ...rp\,i}\,r^3\,\frac{\partial{\mit\Delta\Omega}_\theta}{\partial r}\right)\right] = \sum_{k=1,K} T_{\theta\,k}\,\delta(r-r_k),$$ (14.155)

$$4\pi^2\,R_0^{\,3}\left[\rho\,r\,\frac{\partial{\mit\Delta\Omega}_... ...erp\,i}\,r\,\frac{\partial{\mit\Delta\Omega}_\varphi}{\partial r}\right)\right] = \sum_{k=1,K} T_{\varphi\,k}\,\delta(r-r_k),$$ (14.156)

and are subject to the spatial boundary conditions

$$\frac{\partial{\mit\Delta\Omega}_\theta(0,t)}{\partial r} = \frac{\partial{\mit\Delta\Omega}_\varphi(0,t)}{\partial r} =0,$$ (14.157)

$${\mit\Delta\Omega}_\theta(r_{100},t) ={\mit\Delta\Omega}_\varphi(r_{100},t)=0.$$ (14.158)

Here,

$$\tau_\theta(r) = \frac{r^2}{q^2\,R_0^{\,2}}\,\frac{\tau_{ii}}{\mu_{i\,11}},$$ (14.159)

where $\tau_{ii}$ and $\mu_{i\,11}$ are defined in Equations (A.23) and (A.53), respectively, is the poloidal flow-damping time profile. Equation (14.159) is a generalization of Equation (2.332) that does not assume that the faction of trapped particles is small, or that the plasma is in the banana collisionality regime. Finally, the electromagnetic torques, $T_{\theta\,k}(t)$ and $T_{\varphi\,k}(t)$, are specified in Equations (14.69) and (14.70), respectively.

Following the analysis of Section 3.15, it is convenient to write

$${\mit\Delta\Omega}_\theta(r,t) = \sum_{k=1,K} {\mit\Delta\Omega}_{\theta\,k}(r,t),$$ (14.160)

$${\mit\Delta\Omega}_\varphi(r,t) = \sum_{k=1,K} {\mit\Delta\Omega}_{\varphi\,k}(r,t),$$ (14.161)

where

$$4\pi^{2}\,R_0\left[\rho\,r^{\,3}\,\frac{\partial{\mit\Delta\Omega... ...r^{3}\,\frac{\partial{\mit\Delta\Omega}_{\theta\,k}}{\partial r}\right) \right] =T_{\theta\,k}\,\delta (r-r_k),$$ (14.162)

$$4\pi^{2}\,R_0^{\,3}\left[\rho\,r\,\frac{\partial{\mit\Delta\Omega... ...}\,r\,\frac{\partial{\mit\Delta\Omega}_{\varphi\,k}}{\partial r}\right) \right] = T_{\varphi\,k}\,\delta(r-r_k),$$ (14.163)

and

$$\frac{\partial{\mit\Delta\Omega}_{\theta\,k}(0,t)}{\partial r} = \frac{\partial{\mit\Delta\Omega}_{\varphi\,k}(0,t)}{\partial r} =0,$$ (14.164)

$${\mit\Delta\Omega}_{\theta\,k}(r_{100},t) ={\mit\Delta\Omega}_{\varphi\,k}(r_{100},t)=0.$$ (14.165)

Now, the modified angular velocity profiles, ${\mit\Delta\Omega}_{\theta\,k}$ and ${\mit\Delta\Omega}_{\varphi\,k}$, are mostly localized in the vicinity of the $k$th rational surface. Hence, it is a reasonable approximation to express Equations (14.162) and (14.163) in the simplified forms [16]

$$4\pi^{2}\,R_0\left[\rho_k\,r^{3}\,\frac{\partial{\mit\Delta\Omega... ...r^{3}\,\frac{\partial{\mit\Delta\Omega}_{\theta\,k}}{\partial r}\right) \right] =T_{\theta\,k}\,\delta (r-r_k),$$ (14.166)

$$4\pi^{2}\,R_0^{\,3}\left[\rho_k\,r\,\frac{\partial{\mit\Delta\Ome... ...t( r\,\frac{\partial{\mit\Delta\Omega}_{\varphi\,k}}{\partial r}\right) \right] = T_{\varphi\,k}\,\delta(r-r_k),$$ (14.167)

where $\rho_k=\rho(r_k)$, $\tau_{\theta\,k}=\tau_\theta(r_k)$, and ${\mit\Xi}_k={\mit\Xi}_{\perp\,i}(r_k)$.

Let us write [4,17,16]

$${\mit\Delta\Omega}_{\theta\,k}(r,t) = - \frac{1}{m_k}\sum_{p=1,\infty} \alpha_{k,p}(t)\,\frac{y_p(r)}{y_p(r_k)},$$ (14.168)

$${\mit\Delta\Omega}_{\varphi\,k}(r,t) = \frac{1}{n}\sum_{p=1,\infty} \beta_{k,p}(t)\,\frac{z_p(r)}{z_p(r_k)},$$ (14.169)

where

$$y_p(r) = \frac{J_1(j_{1p}\,r/r_{100})}{r/r_{100}},$$ (14.170)

$$z_p(r) = J_0(j_{0p}\,r/r_{100}).$$ (14.171)

Here, $J_m(z)$ is a Bessel function, and $j_{mp}$ denotes its $p$th zero [1]. Note that Equations (14.168)–(14.171) automatically satisfy the boundary conditions (14.164) and (14.165).

It is easily demonstrated that [33]

$$\frac{d}{dr}\!\left(r^{3}\,\frac{dy_p}{dr}\right) = -\frac{j_{1p}^{\,2}\,r^{3}\,y_p}{r_{100}^{2}},$$ (14.172)

$$\frac{d}{dr}\!\left(r\,\frac{dz_p}{dr}\right) = -\frac{j_{0p}^{\,2}\,r\,z_p}{r_{100}^{2}},$$ (14.173)

and

$$\int_0^{r_{100}} r^{3}\,y_p(r)\,y_q(r)\,dr = \frac{r_{100}^{4}}{2}\,[J_2(j_{1p})]^{\,2}\,\delta_{pq},$$ (14.174)

$$\int_0^{r_{100}} r\,z_p(r)\,z_q(r)\,dr = \frac{r_{100}^{2}}{2}\,[J_1(j_{0p})]^{\,2}\,\delta_{pq}.$$ (14.175)

Equations (14.69), (14.70), and (14.166)–(14.175) yield

$$\frac{d\alpha_{k,p}}{dt} + \left(\frac{1}{\tau_{\theta\,k}}+\frac{j_{1p}^{\,2}}{\tau_{M\,k}}\right)\alpha_{k,p} = \frac{m_k^{2}\,[J_1(j_{1p}\,r_k/r_{100})]^{\,2}}{\tau_{A\,k}^{\... ..._{1p})]^{\,2}}\, {\rm Im}({\mit\Delta\hat{\Psi}_k}\,\hat{\mit\Psi}_k^{\,\ast}),$$ (14.176)

$$\frac{d\beta_{k,p}}{dt} + \frac{j_{0p}^{\,2}}{\tau_{M\,k}}\,\beta_{k,p} = \frac{n^{2}\,[J_0(j_{0p}\,r_k/r_{100})]^{\,2}}{\tau_{A\,k}^{\,2... ..._{0p})]^{\,2}}\, {\rm Im}({\mit\Delta\hat{\Psi}_k}\,\hat{\mit\Psi}_k^{\,\ast}).$$ (14.177)

Here,

$$\tau_{M\,k} = \frac{r_{100}^{\,2}}{{\mit\Xi}_k},$$ (14.178)

$$\tau_{A\,k} = \left(\frac{\mu_0\,\rho_k\,r_{100}^{\,2}}{B_0^{\,2}}\right)^{1/2},$$ (14.179)

$$\epsilon_k =\frac{r_k}{R_0},$$ (14.180)

$${\mit\Delta\hat{\Psi}}_k =\frac{{\mit\Delta\Psi}_k}{R_0\,B_0},$$ (14.181)

$$\hat{\mit\Psi}_k =\frac{{\mit\Psi}_k}{R_0\,B_0}.$$ (14.182)

The values of $\tau_{A\,k}$, $\tau_{M\,k}$, $\tau_{\theta\,k}$, and $\epsilon_k$ at the $n=1$ rational surfaces in our example tokamak discharge are specified in Table 14.8. Incidentally, Equations (14.176) and (14.177) are the toroidal generalizations of the cylindrical equations (3.190) and (3.191), respectively.

Table: 14.8 Alfvén times, momentum confinement times, poloidal flow-damping times, and inverse aspect-ratios at the $n=1$ rational surfaces in KSTAR discharge #18594 at time $t=6450$ ms.

$k$	$\tau_{A\,k}({\rm s})$	$\tau_{M\,k}({\rm s})$	$\tau_{\theta\,k}({\rm s})$	$\epsilon_k$
1	$1.21\times 10^{-7}$	$3.55\times 10^{-1}$	$2.30\times 10^{-5}$	$0.229$
2	$1.11\times 10^{-7}$	$3.55\times 10^{-1}$	$1.81\times 10^{-5}$	$0.282$
3	$1.07\times 10^{-7}$	$3.55\times 10^{-1}$	$1.74\times 10^{-5}$	$0.311$
4	$5.47\times 10^{-8}$	$3.55\times 10^{-1}$	$5.22\times 10^{-5}$	$0.329$

Let us define the frequency shifts that develop at the various rational surfaces in the plasma in response to the electromagnetic torques:

$${\mit\Delta\Omega}_k(t) = m_k\,{\mit\Delta\Omega}_{\theta}(r_k,t)-n\,{\mit\Delta\Omega}_\varphi(r_k,t).$$ (14.183)

It follows from Equations (14.160), (14.161), and (14.168)–(14.171) that

$${\mit\Delta\Omega}_k = -\sum_{k'=1,K}\sum_{p=1,\infty} \alpha_{k'... ...fty}\beta_{k',p}\,\frac{J_0(j_{0p}\,r_k/r_{100})}{J_0(j_{0p}\,r_{k'}/r_{100})}.$$ (14.184)

The frequency shifts are ultimately due to changes in the E-cross-B rotation frequency at the various rational surfaces (because the diamagnetic component of the majority ion poloidal and toroidal rotation frequencies are not directly affected by the electromagnetic torques). (See Section A.7.) It follows that

$$-n\,{\mit\Delta\Omega}_E(r_k,t)= {\mit\Delta\Omega}_k(t)$$ (14.185)

at each rational surface in the plasma. This is the essence of the no-slip constraint introduced in Section 3.16 [8].

Consider an $n=1$ tearing mode that reconnects magnetic flux principally at the $k$th rational surface in our example tokamak discharge. Let ${\mit\hat{\Psi}}_k$ be the normalized magnetic flux reconnected at the $k$th surface, and let us suppose that $\vert{\mit\hat{\Psi}}_k\vert$ is sufficiently large that the plasma response at the $k$th surface lies in the nonlinear regime. Because of the assumed strong shielding at the other rational surfaces in the plasma, we expect the plasma responses at these surfaces to lie in the linear regime.

We can determine the normalized reconnected magnetic fluxes, ${\mit\hat{\Psi}}_{k'}$, driven at the other rational surfaces from Equations (14.119), (14.123), (14.146), and (14.182). We find that

$$\sum_{k''=1,K}^{k''\neq k}({\mit\Delta}_{k'}\,\delta_{k' k''}-E_{k'k''})\,{\mit\hat{\Psi}}_{k''} = E_{k'k}\,{\mit\hat{\Psi}}_k$$ (14.186)

for $k'\neq k$. Hence,

$${\mit\hat{\Psi}}_{k'}(t) = p_{k'}\,{\mit\hat{\Psi}}_k$$ (14.187)

for $k'\neq k$, where

$$p_{k'}(t) = \sum_{k''=1,K}^{k''\neq k} ({\mit\Delta}_{k'}\,\delta_{k'k''}-E_{k'k''})^{-1}\,E_{k''k}.$$ (14.188)

The normalized electromagnetic torque acting at the $k'$th (where $k'\neq k$) rational surface is

$${\rm Im}({\mit\Delta\hat{\Psi}_{k'}}\,\hat{\mit\Psi}_{k'}^{\,\ast... ...rm Im}({\mit\Delta}_{k'})\,\vert p_{k'}\vert^2\,\vert{\mit\hat{\Psi}}_k\vert^2,$$ (14.189)

where use has been made of Equations (14.123), (14.146), (14.181), (14.182), and (14.187). The normalized electromagnetic torque acting at the $k$th rational surface is obtained from angular momentum conservation (see Section 14.11):

$${\rm Im}({\mit\Delta\hat{\Psi}_k}\,\hat{\mit\Psi}_k^{\,\ast})= -\... ...rm Im}({\mit\Delta}_{k'})\,\vert p_{k'}\vert^2\,\vert{\mit\hat{\Psi}}_k\vert^2.$$ (14.190)

According to Equation (14.151), the frequency shift at the $k$th rational surface modifies the real frequency of the tearing mode. In fact,

$$\omega = \omega_{{\rm nonlinear}\,k} + {\mit\Delta\Omega}_k.$$ (14.191)

Figure: 14.4 $n=1$ natural frequencies in KSTAR discharge #18594 at time $t=6450$ ms calculated as functions of the normalized $m=2/n=1$ magnetic island width.

The normalized linear layer response indicies, ${\mit\Delta}_{k'}$ (where $k'\neq k$), that appear in Equation (14.188), are functions of nine normalized layer parameters. (See Table 14.3.) Two of these parameters, $Q_{k'}$ and $Q_{E\,k'}$, are modified by the frequency shifts induced by the electromagnetic torques. In fact, it is clear from Equations (14.130), (14.131), (14.185), and (14.191) that

$$Q_{k'} = Q_{k'}^{\,(0)} + S_{k'}^{\,1/3}\,{\mit\Delta\Omega}_k\,\tau_{H\,k'},$$ (14.192)

$$Q_{E\,k'} =Q_{E\,k'}^{\,(0)} + S_{k'}^{\,1/3}\,{\mit\Delta\Omega}_{k'}\,\tau_{H\,k'},$$ (14.193)

where the superscript $(0)$ indicates a quantity that is unaffected by the electromagnetic torques.

Equations (14.176), (14.177), (14.184), (14.188)–(14.190), (14.192), (14.193)—together with the numerical solution of the Riccati differential equation, (5.121), subject to the boundary conditions (5.122) and (5.123), which determines the ${\mit\Delta}_{k'}(Q_{k'},Q_{E\,k'})$— form a closed set of equations that allow us to determine the reconnected magnetic fluxes driven at the various rational surfaces in our example tokamak discharge by a tearing mode that reconnects magnetic flux primarily at the $k$th rational surface. The only free parameter in the model is the normalized reconnected magnetic flux at the $k$th rational surface, $\vert{\mit\hat{\Psi}_k}\vert(t)$.

Consider an $n=1$ tearing mode in our example tokamak discharge that reconnects magnetic flux principally at the $m=2$ (i.e., $k=1$) rational surface. Such a mode could represent an $m=2/n=1$ neoclassical tearing mode. It is helpful to define the natural frequencies at the various $n=1$ rational surfaces in the plasma:

$$\omega_{0\,1} =\omega_{{\rm nonlinear}\,1} + {\mit\Delta\Omega}_1,$$ (14.194)

$$\omega_{0\,k'} =\omega_{{\rm linear}\,k'} + {\mit\Delta\Omega}_{k'},$$ (14.195)

for $k'=2$, 3, 4, where $\omega_{{\rm nonlinear}\,k}$ and $\omega_{{\rm linear}\,k}$ are specified in Table 14.5. Here, we are taking into account the fact that the plasma response at the $m=2$ rational surface lies in the nonlinear regime (because the $m=2/n=1$ island width is assumed to be greater than the corresponding linear layer width), whereas the plasma responses at the other rational surfaces lie in the linear regime (because the driven island widths are assumed to be less than the corresponding linear layer widths). Moreover, we are also taking into account the modifications to the natural frequencies generated by the electromagnetic torques that develop at the rational surfaces. Recall that the natural frequency at a given rational surface is the preferred rotation frequency of reconnected magnetic flux at that surface.

Figure: 14.5 Normalized $n=1$ magnetic island widths in KSTAR discharge #18594 at time $t=6450$ ms calculated as functions of the normalized $m=2/n=1$ magnetic island width.

Figure 14.4 shows the $n=1$ natural frequencies in our example tokamak discharge calculated as functions of the normalized $m=2/n=1$ magnetic island width, $W_1/r_{100}$, in a simulation in which the island width is slowly ramped up. This calculation is made using the previously mentioned closed set of equations, as well as the data given in Tables 14.1, 14.2, 14.3, 14.5, and 14.8. It can be seen that if $W_1/r_{100}$ lies close to zero then the natural frequencies all take the unperturbed values specified in Table 14.5. On the other hand, as $W_1/r_{100}$ increases, the electromagnetic torques that develop at the rational surfaces modify the natural frequencies. In particular, the torques cause the $m=2$ natural frequency, $\omega_{0\,1}$, and the $m=3$ natural frequency, $\omega_{0\,2}$, to approach one another. At a critical value of $W_1/r_{100}$, which is approximately 0.41, the two natural frequencies suddenly snap together, indicating a sudden loss of shielding at the $m=3$ rational surface [14].

The aforementioned sudden loss of shielding is illustrated in Figure 14.5, which shows the driven normalized magnetic island widths at the $m=3$, 4, and 5 rational surfaces in our example discharge as functions of the normalized $m=2$ island width. It can be seen that, prior to the loss of shielding, comparatively narrow magnetic island chains are driven at the $m=3$, 4, and 5 rational surfaces. However, as soon as the shielding at the $m=3$ rational surface is lost, there is a very significant increase in the width of the magnetic island chain driven at the $m=3$ surface.

Figure: 14.6 Normalized electromagnetic torques, $T_k\equiv {\rm Im}({\mit\Delta\hat{\Psi}_{k}}\,\hat{\mit\Psi}_{k}^{\,\ast})$, at the $n=1$ rational surfaces in KSTAR discharge #18594 at time $t=6450$ ms calculated as functions of the normalized $m=2/n=1$ magnetic island width.

Finally, Figure 14.6 shows the normalized electromagnetic torques that develop at the $n=1$ rational surfaces in our example discharge close to the time at which shielding is lost at the $m=3$ rational surface. It can be seen that the sudden loss of shielding is due to comparatively large transient electromagnetic torques that develop at the $m=2$ and $m=3$ rational surfaces, and drive the corresponding natural frequencies together.

The calculation that we have just performed indicates that the shielding of $n=1$ rational surfaces from one another, as a consequence of sheared rotation, in our example tokamak discharge is a very robust effect. In fact, the shielding only breaks down when a tearing mode grows to a sufficient amplitude that the width of its magnetic island chain becomes a substantial fraction of the plasma minor radius. However, when shielding breaks down, it does so in a sudden and catastrophic manner [14]. In fact, in our example calculation, the loss of shielding at the $m=3$ rational surface drives a magnetic island chain at that surface whose width is sufficient that the chain would overlap with the chain at the $m=2$ rational surface, leading to the sudden destruction of magnetic flux-surfaces [26], which could quite conceivably trigger a disruption [7].