Derivation of electron-cyclotron current drive efficiency
(based on notes from Emmi Tholerus, UKAEA)
The local dimensionless electron-cyclotron (EC) current drive efficiency is
defined in , as
\[\zeta = \frac{e^3 \ln \Lambda}{16\pi\varepsilon_0^2} \frac{n_e}{T_e}
\frac{j_\mathrm{tor}}{Q_\mathrm{ec}},\]
where \(e\) is the electron charge in \(C\), \(\varepsilon_0\) the
vacuum permittivity, \(n_e\) the electron density in \(m^{-3}\),
\(T_e\) the electron temperature in \(J\),
\(j_\mathrm{tor} = \frac{dI_p}{dA}\) with A as the cross sectional area
inside the flux surface, and \(Q\) is the absorbed EC power density in
\(Wm^{-3}\).
Assume that the current drive and power absorption are localised in a region
\(\delta\rho\) around a flux surface at \(\rho\), meaning that the total
current driven is \(I = j_\mathrm{tor}\delta A\) and the total absorbed power
is \(P = Q \delta V\).
Then,
\[\frac{j_\mathrm{tor}}{Q_\mathrm{ec}} \approx
\frac{I_\mathrm{ec}}{P_\mathrm{ec}} \frac{\delta V}{\delta A} \approx
\frac{I_\mathrm{ec}}{P_\mathrm{ec}} \frac{V'}{A'} =
\frac{I_\mathrm{ec}}{P_\mathrm{ec}} \frac{2\pi}{\langle R^{-1} \rangle}\]
For conventional tokamaks, it is often acceptable to take
\(\langle R^{-1} \rangle \approx R^{-1}\); however, this is not valid for
high-beta strongly shaped plasmas like those found in STs.
Substituting in the above gives
\[\zeta = \frac{e^3 \ln \Lambda}{16\pi\varepsilon_0^2}
\frac{2\pi}{\langle R^{-1} \rangle} \frac{n_e}{T_e} \frac{I}{P} ,\]
with all variables in SI units.
As per , the EC-driven current is parallel to the magnetic field and
\(\langle J \cdot B \rangle\) is a flux function that can be written as
\[\langle J \cdot B \rangle = \frac{J}{B} \langle B^2 \rangle.\]
We can therefore write
\[ \begin{align}\begin{aligned}\begin{split}\langle J \cdot B \rangle &= \langle J_\phi B_\phi
\rangle + \langle \frac{J_\phi B_\mathrm{pol}^2}{B_\phi} \rangle =
F \langle \frac{J_\phi}{R}\rangle +
\frac{J}{B} \langle B_\mathrm{pol}^2 \rangle \\
&= F \langle \frac{J_\phi}{R}\rangle +
\frac{\langle J \cdot B \rangle}{\langle B^2 \rangle}
\langle B_\mathrm{pol}^2 \rangle \\
&= F \langle \frac{J_\phi}{R}\rangle
\left( 1 - \frac{ \langle B_\mathrm{pol}^2 \rangle}
{\langle B^2 \rangle} \right)^{-1} \\
&= F \langle \frac{J_\phi}{R}\rangle
\left( \frac{\langle B^2 \rangle- \langle B_\mathrm{pol}^2
\rangle}{\langle B^2 \rangle} \right)^{-1} \\
&= F \langle \frac{J_\phi}{R}\rangle
\left( \frac{\langle B^2 \rangle}{\langle B^2 \rangle-
\langle B_\mathrm{pol}^2 \rangle} \right) \\
&= F \langle \frac{J_\phi}{R}\rangle
\left( \frac{\langle B_\phi^2 \rangle +
\langle B_\mathrm{pol}^2 \rangle}{\langle B_\phi^2 \rangle} \right) \\\end{split}\\&= F \langle \frac{J_\phi}{R}\rangle\left( 1 +
\frac{\langle B_\mathrm{pol}^2\rangle}{\langle B_\phi^2 \rangle} \right)\end{aligned}\end{align} \]
We have \(\langle B_\phi^2 \rangle = F^2 \langle R^{-2} \rangle\), and
\[\begin{split}\langle B_\mathrm{pol}^2 \rangle = \frac{1}{4\pi^2} \left\langle
\frac{|\nabla \Psi_\mathrm{pol}|^2}{R^2} \right\rangle &= \frac{1}{4\pi^2}
\left(\frac{\Psi_\mathrm{pol}'}{V'}\right)^2\left\langle
\frac{|\nabla \Psi_\mathrm{pol}|^2}{R^2} \right\rangle\\
&= \frac{F^2 \langle R^{-2} \rangle^2
\langle \frac{|\nabla V|^2}{R^2} \rangle}{16\pi^4q^2}\end{split}\]
where we have used
\(q = \frac{F \langle R^{-2} \rangle V'}{2\pi \Psi'_\mathrm{pol}}\)
(derived from the definition of the safety factor
\(q = \frac{\partial \Psi_\mathrm{tor}}{\partial\Psi_\mathrm{pol}}\)).
Hence,
\[\langle J \cdot B \rangle = F \left\langle\frac{J_\phi}{R}\right\rangle
\left(1+ \frac{g_2 g_3}{16\pi^4 q^2}\right)\]
where \(g_2 = \left\langle \frac{|\nabla V|^2}{R^2} \right\rangle\) and \(g_3 = \langle R^{-2} \rangle\).