  The inflated radii of giant short-period extrasolar planets collectively
indicate that the interiors of hot Jupiters are heated by some anomalous energy
dissipation mechanism. Although a variety of physical processes have been
proposed to explain this heating, recent statistical evidence points to the
confirmation of explicit predictions of the Ohmic dissipation theory, elevating
this mechanism as the most promising candidate for resolving the radius
inflation problem. In this work, we present an analytic model for the
dissipation rate and derive a simple scaling law that links the magnitude of
energy dissipation to the thickness of the atmospheric weather layer. From this
relation, we find that the penetration depth influences the Ohmic dissipation
rate by an order of magnitude. We further investigate the weather layer depth
of hot Jupiters from the extent of their inflation and show that, depending on
the magnetic field strength, hot Jupiter radii can be maintained even if the
circulation layer is relatively shallow. Additionally, we explore the evolution
of zonal wind velocities with equilibrium temperature by matching our analytic
model to statistically expected dissipation rates. From this analysis, we
deduce that the wind speed scales approximately as
$1/\sqrt{T_\mathrm{eq}-T_0}$, where $T_0$ is a constant that equals $T_0 \sim
1000~\mathrm{K}-1800~\mathrm{K}$ depending on planet-specific parameters
(radius, mass, etc.). This work outlines inter-related constraints on the
atmospheric flow and the magnetic field of hot Jupiters and provides a
foundation for future work on the Ohmic heating mechanism.
