Electromagnetic Torques

The flux-surface integrated poloidal and toroidal electromagnetic torque densities acting on the plasma can be written

$$T_\theta(r,t) = \left\{ r\,{\bf j}\times {\bf B}\cdot {\bf e}_\theta\right\}$$ (3.129)

$$T_z(r,t) = \left\{ R_0\,{\bf j}\times {\bf B}\cdot {\bf e}_z\right\},$$ (3.130)

respectively, where

$$\left\{\cdots\right\} \equiv \oint\oint r\,R_0\,\cdots\,d\theta\,d\varphi$$ (3.131)

is a flux-surface integration operator. However, according to the equations of marginally-stable ideal-MHD, (2.375)–(2.380), both the plasma equilibrium and the tearing perturbation satisfy the force balance criterion

$${\bf j}\times {\bf B} \simeq \nabla p.$$ (3.132)

Given that the scalar pressure is a single-valued function of $\theta$ and $\varphi$, it immediately follows that $T_\theta=T_z = 0$ throughout the plasma [4]. The only exception to this rule occurs in the immediate vicinity of the rational surface, where Equation (3.132) breaks down. It follows that we can write

$$T_\theta(r,t) = T_{\theta\,s}(t)\,\delta(r-r_s),$$ (3.133)

$$T_z(r,t) = T_{z\,s}(t)\,\delta(r-r_s),$$ (3.134)

where

$$T_{\theta\,s} = \frac{1}{4}\int_{r_{s-}}^{r_{s+}}\oint\oint R_0\,r^{\,2}\,(\del... ...!j_r\,\delta B_z^\ast-\delta\!j_r^{\,\ast}\,\delta B_z)\,dr\,d\theta\,d\varphi,$$ (3.135)

$$T_{z\,s} = \frac{1}{4}\int_{r_{s-}}^{r_{s+}}\oint\oint R_0^{\,2}\,r\,(\del... ...a B_r^{\,\ast} - \delta\! j_\theta^{\,\ast}\,\delta B_r)\,dr\,d\theta\,d\varphi$$ (3.136)

are the net poloidal and toroidal torques, respectively, acting at the rational surface. Note that the zeroth-order (in perturbed quantities) torques are zero because $B_r=j_r=0$. Furthermore, the linear (in perturbed quantities) torques average to zero over the flux-surface. Hence, the largest non-zero torques are quadratic in perturbed quantities.

Now, it follows from Equations (3.32)–(3.38) that

$$\delta\! j_z\,\delta B_r^{\,\ast} + \delta\! j_z^{\,\ast}\,\delta B_r -\delta\!j_r\,\delta B_z^\ast-\delta\!j_r^{\,\ast}\,\delta B_z \simeq \frac{{\rm i}\,m}{\mu_0\,r^2}\,\frac{\partial}{\partial r}... ...artial \delta\psi^\ast}{\partial r}\,\delta\psi\right)+{\cal O}(n\,\epsilon)^2,$$ (3.137)

$$\delta \!j_r\,\delta B_\theta^\ast+\delta \! j_r^{\,\ast}\,\delta... ...delta \! j_\theta\,\delta B_r^{\,\ast} - \delta\! j_\theta^{\,\ast}\,\delta B_r \simeq -\frac{{\rm i}\,n\,\epsilon}{\mu_0\,r^2}\frac{\partial}{\p... ...ta\psi^\ast- r\,\frac{\partial \delta\psi^\ast}{\partial r}\,\delta\psi\right).$$ (3.138)

The previous four equations yield [4]

$$T_{\theta\,s} = -\frac{2\pi^2\,R_0\,m}{\mu_0}\,{\rm Im}({\mit\Delta\Psi}_s\,{\mit\Psi}_s^{\,\ast}),$$ (3.139)

$$T_{z\,s} = \frac{2\pi^2\,R_0\,n}{\mu_0}\,{\rm Im}({\mit\Delta\Psi}_s\,{\mit\Psi}_s^{\,\ast}).$$ (3.140)

where use has been made of Equations (3.72) and (3.73).

The net poloidal and toroidal electromagnetic torques acting on the resistive wall can be written

$$T_{\theta\,w} = \frac{1}{4}\int_{r_{w-}}^{r_{w+}}\oint\oint R_0\,r^{\,2}\,(\del... ...!j_r\,\delta B_z^\ast-\delta\!j_r^{\,\ast}\,\delta B_z)\,dr\,d\theta\,d\varphi,$$ (3.141)

$$T_{z\,w} = \frac{1}{4}\int_{r_{w-}}^{r_{w+}}\oint\oint R_0^{\,2}\,r\,(\del... ... B_r^{\,\ast} - \delta\! j_\theta^{\,\ast}\,\delta B_r)\,dr\,d\theta\,d\varphi.$$ (3.142)

Making use of Equations (3.32), (3.82), and (3.105)–(3.107), we obtain [4]

$$T_{\theta\,w} = -\frac{2\pi^2\,R_0\,m}{\mu_0}\,{\rm Im}({\mit\Delta\Psi}_w\,{\mit\Psi}_w^{\,\ast}),$$ (3.143)

$$T_{z\,w} = \frac{2\pi^2\,R_0\,n}{\mu_0}\,{\rm Im}({\mit\Delta\Psi}_w\,{\mit\Psi}_w^{\,\ast}),$$ (3.144)

The net poloidal and toroidal electromagnetic torques acting on the magnetic field-coil can be written

$$T_{\theta\,c} = \frac{1}{4}\int_{r_{c-}}^{r_{c+}}\oint\oint R_0\,r^{\,2}\,(\del... ...!j_r\,\delta B_z^\ast-\delta\!j_r^{\,\ast}\,\delta B_z)\,dr\,d\theta\,d\varphi,$$ (3.145)

$$T_{z\,c} = \frac{1}{4}\int_{r_{c-}}^{r_{c+}}\oint\oint R_0^{\,2}\,r\,(\del... ... B_r^{\,\ast} - \delta\! j_\theta^{\,\ast}\,\delta B_r)\,dr\,d\theta\,d\varphi.$$ (3.146)

Making use of Equations (3.32), (3.109)–(3.111), (3.120), and (3.122), we obtain [4]

$$T_{\theta\,c} = -\frac{2\pi^2\,R_0\,m}{\mu_0}\,{\rm Im}({\mit\Delta\Psi}_c\,{\mit\Psi}_c^{\,\ast}),$$ (3.147)

$$T_{z\,c} = \frac{2\pi^2\,R_0\,n}{\mu_0}\,{\rm Im}({\mit\Delta\Psi}_c\,{\mit\Psi}_c^{\,\ast}).$$ (3.148)

It follows from Equations (3.123)–(3.125) that

$${\rm Im}({\mit\Delta\Psi}_s\,{\mit\Psi}_s^{\,\ast}) = E_{sw}\,{\rm Im}({\mit\Psi}_w\,{\mit\Psi}_s^{\,\ast}),$$ (3.149)

$${\rm Im}({\mit\Delta\Psi}_w\,{\mit\Psi}_w^{\,\ast}) = E_{ws}\,{\rm Im}({\mit\Psi}_s\,{\mit\Psi}_w^{\,\ast})+E_{wc}\,{\rm Im}({\mit\Psi}_c\,{\mit\Psi}_w^{\,\ast}),$$ (3.150)

$${\rm Im}({\mit\Delta\Psi}_c\,{\mit\Psi}_c^{\,\ast}) =E_{cw}\,{\rm Im}({\mit\Psi}_w\,{\mit\Psi}_c^{\,\ast}).$$ (3.151)

Thus,

$${\rm Im}({\mit\Delta\Psi}_s\,{\mit\Psi}_s^{\,\ast})+ {\rm Im}({\m... ...w\,{\mit\Psi}_w^{\,\ast}) + {\rm Im}({\mit\Delta\Psi}_c\,{\mit\Psi}_c^{\,\ast}) = (E_{sw}-E_{ws})\,{\rm Im}({\mit\Psi}_w\,{\mit\Psi}_s^{\,\ast})$$

$$\phantom{=}+ (E_{wc}-E_{cw})\,{\rm Im}({\mit\Psi}_c\,{\mit\Psi}_w^{\,\ast}).$$ (3.152)

However, according to Equations (3.90), (3.127), and (3.128), $E_{sw}=E_{ws}$ and $E_{wc}=E_{cw}$. We deduce that

$${\rm Im}({\mit\Delta\Psi}_s\,{\mit\Psi}_s^{\,\ast})+ {\rm Im}({\m... ...{\mit\Psi}_w^{\,\ast}) + {\rm Im}({\mit\Delta\Psi}_c\,{\mit\Psi}_c^{\,\ast})=0.$$ (3.153)

Hence, Equations (3.139), (3.140), (3.143), (3.144), (3.147), and (3.148) yield [4]

$$T_{\theta\,s} + T_{\theta\,w}+T_{\theta\,c} =0,$$ (3.154)

$$T_{z\,s} + T_{z\,w}+T_{z\,c} = 0.$$ (3.155)

In other words, the plasma/resistive wall/field-coil system exerts zero net poloidal electromagnetic torque, and zero net toroidal electromagnetic torque, on itself.