Conclusion:

The thermal-wind balance, which relies on the assumptions of **geostrophy** (Coriolis balance) and **hydrostasy**, **breaks down near the equator** ($f \to 0$).

When $f$ is very small, the dominant terms in the horizontal momentum equations are no longer the Coriolis force, but rather the **inertial terms** ( $\frac{D\mathbf{u}}{Dt}$) and potentially **friction**. Thus, the flow is no longer geostrophic. In equatorial dynamics, the flow is instead dominated by a balance between the pressure gradient force and the inertial acceleration, or the flow may be directly forced by wind stress.