An asymptotic analysis of a quasi-geostrophic model is presented to investigate the vertically propagating internal Rossby wave structure in the presence of a sharp density interface. For the case of a nonconstant density gradient, the pattern of the vertical structure is nontrivial and depends on the parameter of varying stratification, denoted by $\delta$. As this parameter approaches a critical value from above (i.e., $\delta \approx\delta_{cri}$ ), the internal wave amplitudes increase continually until wave breaking occurs. When $\delta < \delta_{cri}$, two cases are considered. For the case without a turning-point, the requirement of appropriate interfacial conditions provides a general matching solution, available only for a certain range of $\delta$. In the presence of a turning-point, which describes the solution transition from monotonic to oscillatory behavior or vice versa, three asymptotic forms of the solutions are derived, even for small values of $\delta$.
