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Ground observatory and satellite-based determinations of temporal variationsin the geomagnetic field probe a decadal to annual time scale range whereEarth's core slow, inertialess convective motions and rapidly propagating,inertia-bearing hydromagnetic waves are in interplay. Here we numerically modeland jointly investigate these two important features with the help of ageodynamo simulation that (to date) is the closest to the dynamical regime ofEarth's core. This model also considerably enlarges the scope of a previousasymptotic scaling analysis. Three classes of hydrodynamic and hydromagneticwaves are identified in the model output, all with propagation velocity largelyexceeding that of convective advection: axisymmetric, geostrophic Alfv\'entorsional waves, and non-axisymmetric, quasi-geostrophic Alfv\'en and Rossbywaves. The contribution of these waves to the geomagnetic acceleration amountsto an enrichment and flattening of its energy density spectral profile atdecadal time scales, thereby providing a constraint on the extent of the$f^{-4}$ range observed in the geomagnetic frequency power spectrum. The flowand magnetic acceleration energies carried by waves both linearly increase withthe ratio of the magnetic diffusion time scale to the Alfv\'en time scale,highlighting the dominance of Alfv\'en waves in the signal and the stabilisingcontrol of magnetic dissipation at non-axisymmetric scales. Extrapolation ofthe results to Earth's core conditions supports the detectability of Alfv\'enwaves in geomagnetic observations, either as axisymmetric torsionaloscillations or through the geomagnetic jerks caused by non-axisymmetric waves.In contrast, Rossby waves appear to be too fast and carry too little magneticenergy to be detectable in geomagnetic acceleration signals of limitedspatio-temporal resolution.

The interplay of fast waves and slow convection in geodynamo simulations nearing Earth's core conditions