Convectively coupled waves are prevalent in the tropical atmosphere. To understand how clouds respond to large-scale vertical motion across varying periods, we develop a microscopic cloud-ensemble model comprising multiple independent cloud members and force it with background vertical motion of the second-baroclinic-mode vertical structure. The lifecycle of each cloud member is simulated as a dual-threshold process of the boundary-layer equivalent potential temperature, which alternates between the shallow- and deep-convection stages. The stage alternation is characterized by the elimination of CIN and the exhaustion of CAPE. The cloud response exhibits two regimes. When the wave period is close to the convective lifecycle timescale, the cloud ensemble exhibits a resonant response, with individual clouds being synchronized to the wave. The rainfall rate lags the low-level large-scale vertical velocity. When the wave period is much longer than the convective lifecycle timescale, the cloud ensemble exhibits a quasi-equilibrium response. The rainfall rate does not significantly lag the low-level large-scale vertical velocity, and synchronization is weak. Cloud synchronization results from phase and frequency adjustments of individual cloud members, driven by temperature modulation and moisture vertical advection in the boundary layer. Based on the microscopic cloud ensemble model, an analytical theory of the rainfall response amplitude and response phase is derived. The amplitude agrees well with cloud-permitting simulations, but the phase lag is overestimated. 

convection,wave,equilibrium,synchronization