  We have performed high resolution $3-$dimensional compressible hydrodynamics
simulations to investigate the effects of shocks and turbulence on energy
transport into hot Jupiter atmospheres, under a variety of shear gradients. We
focus on a local atmospheric region to accurately follow the small-scale
structures of turbulence and shocks. We find that the effects of turbulence
above and below a shear layer are different in scale and magnitude: below the
shear layer, the effects of turbulence on the vertical energy transfer are
local, generally $\lesssim2~\times(\text{scale height})$. However, turbulence
can have a spatially and thermally-large influence on almost the entire region
above the shear layer. We also find that shock formation is local and
transient. Once the atmosphere becomes steady, the time-averaged heat energy
flux at $P\sim 1$ bar is insignificant, on the order of 0.001\% of the incoming
stellar flux with a shear motion at $P\sim 1$ mbar, and 0.1\% with a deeper
shear layer at $P\sim 100$ mbar. Accordingly, the diffusion coefficient is
higher for the deeper shear layer. Therefore, our results suggest that
turbulence near less dense regions ($P\sim 1$ mbar) does not cause a
sufficiently deep and large penetration of thermal energy to account for radius
inflation in hot Jupiters, regardless of how violent the turbulence is.
However, as the shear layer gets deeper, heat energy transfer becomes more
effective throughout the atmosphere (upwards and downwards) due to a larger
kinetic energy budget. Therefore, it is more important how deep turbulence
occurs in the atmosphere, than how unstable the atmosphere is for effective
energy transfer.
