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We analyze the 1D spatial power spectra of dust surface density and mid tofar-infrared emission at $24-500\,\mu$m in the LMC, SMC, M31, and M33. Byforward-modelling the point-spread-function (PSF) on the power spectrum, wefind that nearly all power spectra have a single power-law and point sourcecomponent. A broken power-law model is only favoured for the LMC 24 $\mu$m MIPSpower spectrum and is due to intense dust heating in 30 Doradus. We also testfor local power spectrum variations by splitting the LMC and SMC maps into$820$ pc boxes. We find significant variations in the power-law index with nostrong evidence for breaks. The lack of a ubiquitous break suggests that thespatial power spectrum does not constrain the disc scale height. Thiscontradicts claims of a break where the turbulent motion changes from 3D to 2D.The power spectrum indices in the LMC, SMC, and M31 are similar ($2.0-2.5$).M33 has a flatter power spectrum ($1.3$), similar to more distant spiralgalaxies with a centrally-concentrated H$_{2}$ distribution. We compare thepower spectra of HI, CO, and dust in M31 and M33 and find that HI power spectraare consistently flatter than CO power spectra. These results cast doubt on theidea that the spatial power spectrum traces large scale turbulent motion innearby galaxies. Instead, we find that the spatial power spectrum is influencedby (1) the PSF on scales below $\sim3$ times the FWHM, (2) bright compactregions (30 Doradus), and (3) the global morphology of the tracer (anexponential CO disc).

Spatial Power Spectra of Dust across the Local Group: No Constraint on Disc Scale Height