Derivation of electron-cyclotron current drive efficiency
The local dimensionless electron-cyclotron (EC) current drive efficiency is given as ,
\[\zeta = \frac{e^3}{\varepsilon_0^2} \frac{n_e}{T_e} R_0 \frac{dI^\mathrm{ec}_\mathrm{tor}}{dP^\mathrm{ec}_\mathrm{absorbed}},\]
where \(e\) is the electron charge, \(\varepsilon_0\) the vacuum permittivity, \(n_e\) the electron density in \(m^{-3}\), \(T_e\) the electron temperature in \(J\), and \(R_0\) the device major radius in \(m\).
\(dI^\mathrm{ec}_\mathrm{tor}\) is defined as the toroidal EC current driven in the elemental area between two flux surfaces, \(dA\), and \(dP^\mathrm{ec}_\mathrm{absorbed}\) is the EC power absorbed in the elemental volume between two flux surfaces, \(dV\).
Defining the flux-surface averaged toroidal current density as:
\[j^\mathrm{ec}_\mathrm{tor} = \frac{\partial I^\mathrm{ec}_\mathrm{tor}}{\partial A},\]
and setting \(dP = Q^\mathrm{ec} dV = 2\pi R_0 Q^\mathrm{ec} dA\) gives:
\[\zeta = \frac{e^3}{\varepsilon_0^2} \frac{n_e}{T_e} \frac{j^\mathrm{ec}_\mathrm{tor}}{2\pi Q^\mathrm{ec}}.\]
From the ASTRA manual ,
\[\begin{split}\langle \boldsymbol{j}^\mathrm{ec} \cdot \boldsymbol{B} \rangle &= 2\pi R_0 B_0 J^2 \frac{\partial}{\partial V} \left[\frac{I^\mathrm{ec}_\mathrm{tor}}{J}\right], \\
&= 2\pi F^2 \frac{\partial}{\partial V} \left[\frac{I_p}{F}\right], \\
&= 2\pi \left( F \frac{\partial I^\mathrm{ec}_\mathrm{tor}}{\partial V} - I^\mathrm{ec}_\mathrm{tor} \frac{\partial F}{\partial V} \right), \\
\langle \boldsymbol{j}^\mathrm{ec} \cdot \boldsymbol{B} \rangle &= 2\pi \left( F \frac{j^\mathrm{ec}_\mathrm{tor}}{2\pi R_0} - I^\mathrm{ec}_\mathrm{tor} \frac{\frac{\partial F}{\partial \rho}}{\frac{\partial V}{\partial \rho}} \right),\end{split}\]
where \(J = \frac{F}{R_0 B_0} = \frac{RB_\phi}{R_0 B_0}\).
Assume the second term is small, i.e.
\[\frac{F j^\mathrm{ec}_\mathrm{tor}}{2\pi R_0} \gg \frac{I^\mathrm{ec}_\mathrm{tor} \frac{\partial F}{\partial \rho}}{\frac{\partial V}{\partial \rho}}.\]
In practice, testing for various devices, we found that typically
\(\frac{I^\mathrm{ec}_\mathrm{tor} \frac{\partial F}{\partial \rho}}{\frac{\partial V}{\partial \rho}} \propto 1e^{-3}-1e^{-5} \times \frac{F j^\mathrm{ec}_\mathrm{tor}}{2\pi R_0}\).
Then:
\[\langle \boldsymbol{j}^\mathrm{ec} \cdot \boldsymbol{B} \rangle = \frac{F}{R_0} j^\mathrm{ec}_\mathrm{tor},\]
and so:
\[\zeta = \frac{e^3}{\varepsilon_0^2} \frac{R_0}{2\pi F} \frac{n_e}{T_e} \frac{\langle \boldsymbol{j}^\mathrm{ec} \cdot \boldsymbol{B} \rangle}{Q^\mathrm{ec}}.\]
References