Global models of solar turbulent convection have recently shown some success to explain the origin of the observed differential rotation (DR) and surface magnetic flux, at the origin of the 11-years activity cycle. However, the amplitude of giant convective cells needed to reproduce these solar patterns is currently too high when compared to observational constraints coming from helioseismology. This over-estimation of convective velocities can even result in a global reversal of the DR profile when the turbulence of models is increased to reproduce the solar small convective scales, which is known as the Convective Conundrum. It then important to better understand how energy is transported by the solar convection and distributed among the different scales of the turbulence.

In this talk, we will present our current understanding of this paradox and its possible origin, resulting from a parametric study using high-performance numerical simulations. In particular, we propose a method to control the amplitude of convective transport, while maintaining the overall transport of solar luminosity at the solar rotation rate and being closer to what is expected from current solar surface observations. We further study scale by scale force and energy balances in different models, showing how the different balances vary with the Nusselt number.