We analyze the power spectrum of the line-of-sight ionospheric velocity measured by the SuperDARN CUTLASS Finland radar and of contemporaneous magnetic field perturbations measured by nearby IMAGE magnetometers. The measurements come from 69--87¿ AACGM latitude and 0415--2230 MLT during 58 intervals of pulsed ionospheric flow between March 1995 and September 1996. The median power spectrum of both the velocity and magnetic field is of power law form with a best fit exponent of -0.5 and -1.3, respectively. By simulating the effect of the finite sample window on the measured spectral exponent and on the structure seen in the power spectral density of individual spectra, we show that the measured spectral exponent actually corresponds to a true exponent of -0.95 for the velocity and -1.8 for the magnetic field. Furthermore, it is shown that the difference in the exponents of the velocity and magnetic field spectra is at least in part due to a spatial smoothing effect that causes the spectral exponent of the ground magnetic spectrum to be less than that of the ionospheric current spectrum producing it. The power law power spectrum implies that there is no preferred timescale for ionospheric velocity and current fluctuations in the observed frequency range. It is proposed that this scale-free nature may arise from the intermittent, turbulent nature of the interplanetary magnetic field causing magnetic reconnection to occur at the magnetopause on a wide range of spatial and temporal scales.