  This work concerns the zero Mach number limit of the compressible primitive
equations. The primitive equations with the incompressibility condition are
identified as the limiting equations. The convergence with well-prepared
initial data (i.e., initial data without acoustic oscillations) is rigorously
justified, and the convergence rate is shown to be of order $ \mathcal
O(\varepsilon) $, as $ \varepsilon \rightarrow 0^+ $, where $ \varepsilon $
represents the Mach number. As a byproduct, we construct a class of global
solutions to the compressible primitive equations, which are close to the
incompressible flows.
