  We show that, in the context of patchy reionisation, an accurate description
of the angular power spectrum of the kinetic Sunyaev-Zel'dovich (kSZ) effect is
not possible with simple scaling relations between the amplitude of the
spectrum and global parameters, such as the reionisation midpoint and its
duration. We introduce a new parameterisation of this spectrum, based on a
novel description of the power spectrum of the free electrons density contrast
Pee (k,z) in terms of the reionisation global history and morphology. We
directly relate features of the spectrum to the typical ionised bubble size at
different stages in the process and, subsequently, to the angular scale at
which the patchy kSZ power spectrum reaches its maximum. We successfully
calibrated our results on a custom set of advanced radiative hydrodynamical
simulations and later found our parameterisation to be a valid description of a
wide range of other simulations and, therefore, reionisation physics. In the
end, and as long as the global reionisation history is known, two parameters
are sufficient to derive the angular power spectrum. Such an innovative
framework applied to cosmic microwave background data and combined with 21cm
intensity mapping will allow a first consistent detection of the amplitude and
shape of the patchy kSZ signal, giving in turn access to the physics of early
light sources.
