It has been proposed that the geomagnetic field-aligned component of the perturbation Poynting vector above the ionosphere, as obtained from the cross product of the electric and magnetic perturbation fields observed on a spacecraft, may be used to estimate the field line-integrated electromagnetic energy dissipation in the ionosphere below. This paper clarifies conditions under which this approximation may be either valid or invalid. It is shown that the downward field-aligned component of the perturbation Poynting vector can underestimate the electromagnetic energy dissipation in regions of high ionospheric Pedersen conductance, and it can significantly overestimate the dissipation in regions of low conductance. Local values of upward perturbation Poynting vector do not necessarily correspond to net ionospheric generation of electromagnetic energy along that geomagnetic field line. An Equipotential Boundary Poynting Flux (EBPF) theorem is presented for quasi-static electromagnetic fields as follows: when a volume of the ionosphere is bounded on the sides by an equipotential surface and on the bottom by the base of the conducting ionosphere, then the area integral of the downward normal component of the perturbation Poynting vector over the top of that volume equals the energy dissipation within the volume. This equality does not apply to volumes with arbitrary side boundaries. However, the EBPF theorem can be applied separately to different components of the electric potential, such as the large- and small-scale components. Since contours of the small-scale component of potential tend to close over relatively localized regions, the associated small-scale structures of downward perturbation Poynting vector tend to be dissipated locally.
