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In this paper, we consider a family of piecewise constant solutions of the quasi-geostrophic shallow-water (QGSW) equation, also known as Hasegawa-Mima equation in plasma science. We derive the contour dynamics equation of the QGSW front, which is a nonlinear, nonlocal dispersive equation. Different from the SQG front (Hunter et al. (2021) [45]) or GSQG front (Córdoba et al. (2019) [18] and Hunter et al. (2024) [47]), the dispersion relation for QGSW front equation has a stationary phase at zero frequency, which may lead to slower decay. Fortunately, the nonlinearity has a null structure which can compensate the slower decay. Using this null structure, we prove the global existence of the solutions when the initial data is sufficiently close to a flat front. [ABSTRACT FROM AUTHOR]