The structure of dust spirals in the nuclei of the SAab-type Liner galaxies NGC 4450 and NGC 4736 is studied using archival HST PC images. The spirals are typically only several hundredths of a magnitude fainter than the neighboring disks, so unsharp mask techniques are used to highlight them. The ambient extinction is estimated to be less than 0.1 mag from the intensity decrements of the dust features and from the spiral surface filling factor, which is about constant for all radii and sizes. The nuclear dust spirals differ from main-disk spirals in several respects: the nuclear spirals have no associated star formation, they are very irregular with both trailing and leading components that often cross, they become darker as they approach the center, they completely fill the inner disks with a constant areal density, making the number of distinct spirals (the azimuthal wavenumber m) increase linearly with radius, and their number decreases with increasing arm width as a power law. Fourier transform power spectra of the spirals, taken in the azimuthal direction, show a power law behavior with a slope of -5/3 over the range of frequencies where the power stands above the pixel noise. This is the same slope as that found for the one-dimensional power spectra of HI emission in the Large Magellanic Cloud, and also the slope expected for a thin turbulent disk. All of these properties suggest that the dust spirals are a manifestation of acoustic turbulence in the inner gas disks of these galaxies. Such turbulence should dissipate orbital energy and transfer angular momentum outward, leading to a steady accretion of gas toward the nucleus.
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