Mesoscale eddies are fundamental components of global ocean circulation. In situ observations and Lagrangian analyses have shown that most eddies are materially coherent, transporting a water mass within their core that differs from the surrounding environment. Additionally, laboratory experiments indicate that eddies locally modify stratification in accordance with thermal wind balance, regardless of whether they trap a water mass. These two mechanisms drive, respectively, spiciness mode anomalies and heaving mode anomalies associated with mesoscale eddies. In this study, aiming to quantitatively assess the physical processes governing mesoscale eddy dynamics, we introduce a novel theoretical decomposition of the potential density field within eddy cores to account for both effects. This framework is applied to six anticyclonic eddies sampled during the EUREC4A-OA, METEOR 124, and Physindien 2011 oceanographic campaigns. Unlike previous studies, we evaluate not only the amplitude of these anomalies but also their vertical structure. Our results confirm that \textit{heaving mode anomalies} dominate the total density anomaly. However, in contrast to previous assumptions, we demonstrate that their vertical structure is dictated by the local background stratification and often follows a nearly Gaussian profile. Meanwhile, spiciness anomalies provide only a second-order contribution to the total potential density anomaly, rendering them negligible in most dynamical processes governing mesoscale eddies. By bridging experimental results with observational eddy datasets, this study refines our understanding of mesoscale eddy vertical structure, offering a more accurate predictive framework for their shape and role in oceanic property transport.