Tau is a microtubule-associated protein that regulates axonal transport, stabilizes and spatially organizes microtubules in parallel networks. The Tau-microtubule pair is crucial for maintaining the architecture and integrity of axons. Therefore, it is essential to understand how these two entities interact to ensure and modulate the normal axonal functions. Based on evidence from several published experiments, we have developed a two-dimensional model that describes the interaction between a population of Tau proteins and a stabilized microtubule at the scale of the tubulin dimers (binding sites) as an adsorption-desorption dynamical process in which Tau can bind on the microtubule outer surface via two distinct modes: a longitudinal (along a protofilament) and lateral (across adjacent protofilaments) modes. Such a process yields a dynamical distribution of Tau molecules on the microtubule surface referred to as microtubule decoration that we have characterized at the equilibrium using two observables: the total microtubule surface coverage with Tau's and the distribution of nearest neighbors Tau's. Using both analytical and numerical approaches, we have derived expressions and computed these observables as a function of key parameters controlling the binding reaction: the stoichiometries of the Taus in the two binding modes, the associated dissociation constants and the ratio of the Tau concentration to that of microtubule tubulin dimers.