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Undersea warfare relies heavily on acoustic means to detect a submerged vessel. The frequency of the acoustic signal radiated by the vessel is typically very low, thus requires a large array aperture to achieve acceptable angular resolution. In this paper, we present a novel approach for low-frequency direction-of-arrival (DOA) estimation using miniature circular vector-sensor array mounted on the perimeter of a cylinder. Under this approach, we conduct beamforming using decomposition in the acoustic mode domain rather than frequency domain, to avoid the long wavelength constraints. We first introduce a multi-layer acoustic gradient scattering model to provide a guideline and performance predication tool for the mode beamformer design and algorithm. We optimize the array gain and frequency response with this model. We further develop the adaptive DOA estimation algorithm based on this model. We formulate the Capon spectra of the mode beamformer which is independent of the frequency band after the mode decomposition. Numerical simulations are conducted to quantify the performance and evaluate the theoretical results developed in this study.