<p>Mesoscale eddies are persistent and dynamic structures that are ubiquitous across the global ocean. Most of these eddies are observed to be in geostrophic equilibrium, where the Coriolis force is balanced by the horizontal pressure gradient. However, a subset exhibits more nonlinear behaviors, requiring a more complex framework for describing their azimuthal flows. To account for the curvature of the streamlines, several studies have applied the cyclo-geostrophic velocity, incorporating not only the Coriolis and pressure gradient forces but also the centrifugal acceleration. Despite this correction, the equilibrium solution often diverges near the eddy core, failing to accurately represent the velocity field in these regions. This study evaluates the geostrophic and cyclo-geostrophic characteristics of nine mesoscale eddies sampled in various regions. A theoretical analysis is provided to explain why the cyclo-geostrophic adjustment can lead to divergences in eddy cores. Our results show that while geostrophy remains a robust approximation for the overall dynamics of mesoscale eddies, it does not fully capture the three-dimensional structure. Additionally, the ageostrophic components of eddy flows are not solely attributable to cyclostrophic effects, suggesting the need to consider other factors influencing eddy behavior.</p>