.. _ec-derivation:

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 [1]_, [2]_ as

.. math::

   \zeta = \frac{e^3 \ln \Lambda}{16\pi\varepsilon_0^2} \frac{n_e}{T_e}
   \frac{j_\mathrm{tor}}{Q_\mathrm{ec}},

where :math:`e` is the electron charge in :math:`C`, :math:`\varepsilon_0` the
vacuum permittivity, :math:`n_e` the electron density in :math:`m^{-3}`,
:math:`T_e` the electron temperature in :math:`J`,
:math:`j_\mathrm{tor} = \frac{dI_p}{dA}` with `A` as the cross sectional area
inside the flux surface, and :math:`Q` is the absorbed EC power density in
:math:`Wm^{-3}`.

Assume that the current drive and power absorption are localised in a region
:math:`\delta\rho` around a flux surface at :math:`\rho`, meaning that the total
current driven is :math:`I = j_\mathrm{tor}\delta A` and the total absorbed power
is :math:`P = Q \delta V`.
Then,

.. math::

     \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
:math:`\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

.. math::

   \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 [1]_, the EC-driven current is parallel to the magnetic field and
:math:`\langle J \cdot B \rangle` is a flux function that can be written as

.. math::

    \langle J \cdot B \rangle = \frac{J}{B} \langle B^2 \rangle.

We can therefore write

.. math::

    \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) \\

    &= F \langle \frac{J_\phi}{R}\rangle\left( 1 +
    \frac{\langle B_\mathrm{pol}^2\rangle}{\langle B_\phi^2 \rangle} \right)

We have :math:`\langle B_\phi^2 \rangle = F^2 \langle R^{-2} \rangle`, and

.. math::

    \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}


where we have used
:math:`q = \frac{F \langle R^{-2} \rangle V'}{2\pi \Psi'_\mathrm{pol}}`
(derived from the definition of the safety factor
:math:`q = \frac{\partial \Psi_\mathrm{tor}}{\partial\Psi_\mathrm{pol}}`).

Hence,

.. math::

    \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 :math:`g_2 = \left\langle \frac{|\nabla V|^2}{R^2} \right\rangle` and :math:`g_3 =  \langle R^{-2} \rangle`.


.. [1] Lin-Liu, Y. R., Chan, V. S., & Prater, R. (2003). `Electron cyclotron current drive efficiency in general tokamak geometry <https://doi.org/10.1063/1.1610472>`_. Physics of Plasmas, 10(10), 4064-4071.

.. [2] Luce, T. C., Lin-Liu, Y. R., Harvey, R. W., Giruzzi, G., Politzer, P. A., Rice, B. W., Lohr, J. M., Petty, C. C., and Prater, R. (1999). `Generation of Localized Noninductive Current by Electron Cyclotron Waves on the DIII-D Tokamak <https://doi.org/10.1103/PhysRevLett.83.4550>`_. Phys. Rev. Lett. (83), 4550.