The Ostrovsky equation is used to describe the oceanic waves affected by Earth rotation. The auxiliary elliptic equation was employed as a direct method to construct the evolution solutions for the Ostrovsky equation in this manuscript. Detailed analysis of evolution solutions of the Ostrovsky equation is presented. The periodic solutions, which read Jacobi elliptic function solutions, hyperbolic solutions and trigonal solutions, are obtained. More parameters are included in the evolution solutions, the parameter space is enlarged. This method adds a new route to explore evolution solutions of nonlinear differential equation in a perspective from auxiliary elliptic equation.