  Context: The anomalously large radii of hot Jupiters has long been a mystery.
However, by combining both theoretical arguments and 2D models, a recent study
has suggested that the vertical advection of potential temperature leads to an
adiabatic temperature profile in the deep atmosphere hotter than the profile
obtained with standard 1D models. Aims: In order to confirm the viability of
that scenario, we extend this investigation to three dimensional,
time-dependent, models. Methods: We use a 3D GCM, DYNAMICO to perform a series
of calculations designed to explore the formation and structure of the driving
atmospheric circulations, and detail how it responds to changes in both upper
and deep atmospheric forcing. Results: In agreement with the previous, 2D,
study, we find that a hot adiabat is the natural outcome of the long-term
evolution of the deep atmosphere. Integration times of order $1500$ years are
needed for that adiabat to emerge from an isothermal atmosphere, explaining why
it has not been found in previous hot Jupiter studies. Models initialised from
a hotter deep atmosphere tend to evolve faster toward the same final state. We
also find that the deep adiabat is stable against low-levels of deep heating
and cooling, as long as the Newtonian cooling time-scale is longer than $\sim
3000$ years at $200$ bar. Conclusions: We conclude that the steady-state
vertical advection of potential temperature by deep atmospheric circulations
constitutes a robust mechanism to explain hot Jupiter inflated radii. We
suggest that future studies of hot Jupiters are evolved for a longer time than
currently done, and, when possible, include models initialised with a hot deep
adiabat. We stress that this mechanism stems from the advection of entropy by
irradiation induced mass flows and does not require (finely tuned) dissipative
process, in contrast with most previously suggested scenarios.
