  We revisit the effect of non-linear Landau (NL) damping on the electrostatic
instability of blazar-induced pair beams, using a realistic pair-beam
distribution. We employ a simplified 2D model in ${\bf k}$-space to study the
evolution of the electric-field spectrum and to calculate the relaxation time
of the beam. We demonstrate that the 2D model is an adequate representation of
the 3D physics. We find that non-linear Landau damping, once it operates
efficiently, transports essentially the entire wave energy to small wavenumbers
where wave driving is weak or absent. The relaxation time also strongly depends
on the IGM temperature, $T_\mathrm{IGM}$, and for $T_\mathrm{IGM}\ll10$ eV, and
in the absence of any other damping mechanism, the relaxation time of the pair
beam is longer than the inverse Compton (IC) scattering time. The weak
late-time beam energy losses arise from the accumulation of wave energy at
small $k$, that non-linearly drains the wave energy at the resonant
$\mathbf{k}$ of the pair-beam instability. Any other dissipation process
operating at small $k$ would reduce that wave-energy drain and hence lead to
stronger pair-beam energy losses. As an example, collisions reduce the
relaxation time by an order of magnitude, although their rate is very small.
Other non-linear processes, such as the modulation instability, could provide
additional damping of the non-resonant waves and dramatically reduce the
relaxation time of the pair beam. An accurate description of the spectral
evolution of the electrostatic waves is crucial for calculating the relaxation
time of the pair beam.
