Mesoscale eddies are ubiquitous in the global ocean. Most of them are
materially coherent: they advect a different water mass in their core
than in the surrounding water, according to studies based on in situ
observations and Lagrangian techniques. In parallel, laboratory
experiments have shown that eddies have the ability to locally modify
the stratification according to the thermal wind balance, without
necessarily contain heterogeneous water. These two types of density
anomalies associated with mesoscale eddies are often erroneously
confused in the literature. Here we propose a new theoretical
decomposition of the potential density field in the core of eddies to
assess their respective amplitude and dynamical effect. This allows the
modelling of their 3D shape and the estimation of the importance of each
term. This decomposition is applied to 6 anticyclonic eddies sampled
during the EUREC4A-OA, METEOR 124 and PHYSINDIEN 2011 in situ
experiments. We show that the anomaly corresponding to the slope of the
isopycnals is the largest contributor to the total density anomaly. Its
vertical shape is nearly Gaussian, but also depends on the local
background stratification. The horizontal density gradient associated
with the trapped water mass adds a second order term to the total
anomaly and can be neglected for the study of eddy dynamics. The
horizontal structures of the eddies studied are consistent with previous
studies and show an exponential-alpha shape.