Here, we investigate interannual to multidecadal response of oceanic circulation to volcanic eruptions and their possible impact on climate (e.g., droughts, glaciations). We also study other modes of coupled atmosphere-ocean climate variability (e.g., North Atlantic Oscillation) and establish linkages between these patterns and low-frequency perturbations of climate triggered by volcanic eruptions. For that, we study the Last Millennium Ensemble (https://www2.cesm.ucar.edu/models/experiments/LME). We analyze individually over 70 eruptions of both tropical and extratropical origin, with a particular focus on their regional impact (e.g., Baffin Island, Sahel, Australia) on long-term perturbation of precipitation or ice cover. We capture patterns of coupled atmosphere-ocean response to volcanic eruptions through multi-component composites, providing physical mechanisms behind regional climatic response to volcanic eruptions. Moreover, we apply a recently introduced technique called Nonlinear Laplacian Spectral Analysis. Through this technique, drawbacks associated with ad-hoc filtering are avoided as the extracted signals span many temporal scales without preprocessing the input data, enabling detection of low-frequency, low-amplitude and intermittent modes otherwise not accessible with classical approaches.