  In fluid dynamics, the scaling behaviour of flow length scales is commonly
used to infer the governing force balance of a system. The key to a successful
approach is to measure length scales that are representative of the energy
contained in the solution (energetically relevant) and indicative of the
established force balance (dynamically relevant). In numerical simulations of
rotating convection and magneto-hydrodynamic dynamos in spherical shells, it
has remained difficult to measure length scales that are both energetically and
dynamically relevant, which has led to conflicting interpretations of the
underlying force balance. By analysing an extensive set of magnetic and
non-magnetic models, we focus on two length scales that achieve both energetic
and dynamical relevance. The first one is the peak of the poloidal kinetic
energy spectrum, which we successfully compare to crossover points on spectral
representations of the force balance. In most dynamo models, this result
confirms that the dominant length scale of the system is controlled by a
quasi-geostrophic (QG-) MAC (Magneto-Archimedean-Coriolis) balance. In
non-magnetic convection models, the analysis favours a QG-CIA
(Coriolis-Inertia-Archimedean) balance. In dynamo models, we introduce a second
energetically relevant length scale associated with the loss of axial
invariance in the flow. We again relate this length scale to a crossover point
in scale-dependent force balance diagrams, which marks the transition between
large-scale geostrophy (the equilibrium of Coriolis and pressure forces) and
small-scale magnetostrophy, where the Lorentz force overtakes the Coriolis
force. Scaling analysis of these two energetically and dynamically relevant
length scales suggests that the Earth's dynamo is controlled by a QG-MAC
balance at a dominant scale of about 200 km, while magnetostrophic effects are
deferred to scales smaller than 50 km.
