Ohmic heating of a giant planet ($q=J^2/\sigma$)

Atmospheric winds across a magnetic field can induce electric currents. Their Ohmic heating converts mechanical energy into heat, at local rate $q=\mathbf{J}\cdot \mathbf{E}=J^2/\sigma$ in the simplest resistive description. To maintain hot-Jupiter radius inflation, current paths and deposited heat must influence the deep interior; dissipation restricted to optically thin upper layers need not maintain the planet's entropy.