  Geomagnetic secular acceleration is a unique window on the dynamics taking
place in Earth's core. In this study, the behaviours of the secular
acceleration and underlying core dynamics are examined in new numerical
simulations of the geodynamo that reside on a theoretical path in parameter
space connecting the region where most classical models are found to the
natural conditions. The typical time scale for geomagnetic acceleration is
found to be invariant along this path, at a value close to 10 years that
matches Earth's core estimates. Despite this invariance, the spatio-temporal
properties of secular acceleration show significant variability along the path,
with an asymptotic regime of rapid rotation reached after 30% of this path
(corresponding to a model Ekman number $E=3~10^{-7}$). In this regime, the
energy of secular acceleration is entirely found at periods longer than that of
planetary rotation, and the underlying flow acceleration patterns acquire a
two-dimensional columnar structure representative of the rapid rotation limit.
The spatial pattern of the secular acceleration at the core-mantle boundary
shows significant localisation of energy within an equatorial belt. Rapid
hydromagnetic wave dynamics is absent at the start of the path but can be
clearly exhibited in the asymptotic regime. This study reports on ubiquitous
axisymmetric geostrophic torsional waves of weak amplitude relatively to
convective transport, and also stronger, laterally limited, quasi-geostrophic
Alfv\'en waves propagating in the cylindrical radial direction from the tip of
convective plumes towards the core-mantle boundary. Quasi-geostrophic Alfv\'en
waves are shown to be an important carrier of flow acceleration to the core
surface that links with the generation of strong, short-lived and intermittent
equatorial pulses in the secular acceleration energy.
