  The effectiveness of multiple electron cyclotron resonance (ECR) harmonics
has been thoroughly investigated in context of high current drive efficiency,
generally observed in fully non-inductive operation of the low aspect ratio
EXL-50 spherical tokamak (ST) powered by electron cyclotron (EC) waves. The
Fokker-Plank equation is numerically solved to obtain electron distribution
function, under steady state of the relativistic nonlinear Coulomb collision
and quasi-linear diffusion operators, for calculating plasma current driven by
the injected EC wave. For the extra-ordinary EC wave, simulation results unfold
a mechanism by which electrons moving around the cold second harmonic ECR layer
strongly resonate with higher harmonics via the relativistic Doppler shifted
resonance condition. This feature is in fact evident above a certain value of
input EC wave power in simulation, indicating it to be a non-linear phenomenon.
Similar to the experimental observation, high efficiency in current drive (over
1 A/W) has indeed been found in simulation for a typical low density ($\sim
1\times10^{18}~m^{-3}$), low temperature ($\lesssim 100$ eV) plasma of EXL-50
by taking into account multi-pass absorptions in our simulation model. However,
such characteristic is not found in the ordinary EC-wave study for both
single-pass and multi-pass simulations, suggesting it as inefficient in driving
current on our ST device.
