Context: Observations of asymptotic giant branch (AGB) stars with increasing spatial resolution reveal new layers of complexity of atmospheric processes on a variety of scales. Aim: To analyze the physical mechanisms that cause asymmetries and surface structures in observed images, we use detailed 3D dynamical simulations of AGB stars; these simulations self-consistently describe convection and pulsations. Methods: We used the CO5BOLD radiation-hydrodynamics code to produce an exploratory grid of global star-in-a-box models of the outer convective envelope and the inner atmosphere of AGB stars to study convection, pulsations, and shock waves and their dependence on stellar and numerical parameters. Results: The model dynamics are governed by the interaction of long-lasting giant convection cells, short-lived surface granules, and strong, radial, fundamental-mode pulsations. Radial pulsations and shorter wavelength, traveling, acoustic waves induce shocks on various scales in the atmosphere. Convection, waves, and shocks all contribute to the dynamical pressure and, thus, to an increase of the stellar radius and to a levitation of material into layers where dust can form. Consequently, the resulting relation of pulsation period and stellar radius is shifted toward larger radii compared to that of non-linear 1D models. The dependence of pulsation period on luminosity agrees well with observed relations. The interaction of the pulsation mode with the non-stationary convective flow causes occasional amplitude changes and phase shifts. The regularity of the pulsations decreases with decreasing gravity as the relative size of convection cells increases. The model stars do not have a well-defined surface. Instead, the light is emitted from a very extended inhomogeneous atmosphere with a complex dynamic pattern of high-contrast features.
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