  Structure imprinted in foreground extragalactic point sources by ionospheric
refraction has the potential to contaminate Epoch of Reionisation (EoR) power
spectra of the 21~cm emission line of neutral hydrogen. The alteration of the
spatial and spectral structure of foreground measurements due to total electron
content (TEC) gradients in the ionosphere create a departure from the expected
sky signal. We present a general framework for understanding the signatures of
ionospheric behaviour in the two-dimensional (2D) neutral hydrogen power
spectrum measured by a low-frequency radio interferometer. Two primary classes
of ionospheric behaviour are considered, corresponding to dominant modes
observed in Murchison Widefield Array (MWA) EoR data; namely, anisotropic
structured wave behaviour, and isotropic turbulence. Analytic predictions for
power spectrum bias due to this contamination are computed, and compared with
simulations. We then apply the ionospheric metric described in Jordan et al.
(2017) to study the impact of ionospheric structure on MWA data, by dividing
MWA EoR datasets into classes with good and poor ionospheric conditions, using
sets of matched 30-minute observations from 2014 September. The results are
compared with the analytic and simulated predictions, demonstrating the
observed bias in the power spectrum when the ionosphere is active (displays
coherent structures or isotropic turbulence). The analysis demonstrates that
unless ionospheric activity can be quantified and corrected, active data should
not be included in EoR analysis in order to avoid systematic biases in
cosmological power spectra. When data are corrected with a model formed from
the calibration information, bias reduces below the expected 21~cm signal
level. Data are considered `quiet' when the median measured source position
offsets are less than 10-15~arcseconds.
