The equivalence of Joule dissipation and frictional heating in the collisional ionosphere

We address what is meant by Joule dissipation in the context of the coupled magnetosphere-ionosphere-thermosphere system. We first show that the ionospheric Ohm's law that includes the classical Pedersen and Hall conductivities can be extended to include other terms such as plasma inertia and pressure. Within the ionosphere the Pedersen and Hall conductivities usually dominate. We note, however, that this does not mean that the electric field causes the currents. We next investigate the role of friction in heating the different species within the plasma. The frictional electron-ion collisional heating rate is proportional to j(2), where j is the current density, and as such could be considered to be Ohmic in nature. The heating from electron-neutral collisions can be recast to include a term that includes j(2), but only part of the electron collisional heating is Ohmic in the sense that it depends on j(2). We also show that the Joule dissipation in the neutral frame equals the total amount of frictional heating for all the fluids (electrons, ions, and neutrals) with additional terms to account for the rate at which work is done by pressure gradient forces, and the power used to accelerate the plasma. In general, these additional terms are small within the ionosphere, and most of the Joule dissipation in the neutral frame results in heating, mainly by initially increasing the ion fluid temperature relative to the neutrals, while the neutral atmosphere temperature increases with a much slower rate.