We consider an isolated, materially coherent vortex in a hydrostatically balanced ocean with no fronts, incropping, or outcropping. A cylindrical coordinate system (r,θ,z) is used, where r is the radial coordinate, θ the azimuthal angle, and z the vertical coordinate. The conservative temperature and absolute salinity fields are denoted as T̃(r,θ,z) and S̃(r,θ,z), respectively, with the associated potential density σ̃(r,θ,z) at atmospheric pressure. The reference profiles, representing the climatological state of the ocean, are given by T‾̃(z), S‾̃(z), and σ‾̃(z). The stratification is assumed to be stable for invertibility. These reference quantities vary only with depth. For clarity, we distinguish function definitions (e.g., σ̃) from their evaluated values (e.g., σ=σ̃(r,θ,z)). The potential density deviation is referenced against pure water density (1000 kgm-3). Under these assumptions, σ̃ and σ‾̃ are bijective in z and invertible, allowing us to define the reciprocal functions Z̃ and Z‾̃ that provide the depth z corresponding to a given potential density σ such that 1z=Z̃(r,θ,σ̃(r,θ,z)),z=Z‾̃(σ‾̃(z)).

Since two water masses of equal density can have different temperature and salinity, we define isopycnal thermohaline anomalies to distinguish trapped water from surrounding water: 2∀σ,ΔσT̃(r,θ,σ)=T̃(r,θ,σ)-T‾̃(σ),3∀σ,ΔσS̃(r,θ,σ)=S̃(r,θ,σ)-S‾̃(σ). These anomalies quantify temperature and salinity variations along isopycnals, representing the so-called spiciness mode anomalies , commonly used to estimate eddy-induced heat and salt transport .

Next, we define modified temperature and salinity fields: 4∀z,T^̃(r,θ,z)=T̃(r,θ,z)-ΔσT̃r,θ,Z̃(r,θ,σ̃(z)),5∀z,S^̃(r,θ,z)=S̃(r,θ,z)-ΔσS̃r,θ,Z̃(r,θ,σ̃(z)). These fields share the same isopleths as the isopycnal temperature and salinity fields. Since each isopycnal now corresponds to a unique isotherm and isohaline, these fields quantify the so-called heaving mode anomalies .

Given a temperature–salinity field, the potential density σ at atmospheric pressure is determined by the nonlinear function F̃: 6σ̃(r,θ,z)=F̃T̃(r,θ,z),S̃(r,θ,z). This function is computed using the Thermodynamic Equation of Seawater – 2010 . Introducing T^̃ and S^̃ in a Taylor expansion, we write 7σ̃=σ^̃+δsσ̃, where 8σ^̃=F̃T^̃,S^̃,δsσ̃=∂F̃∂TS,zΔσT̃+∂F̃∂ST,zΔσS̃9+oΔσT̃+oΔσS̃. Physically, δsσ̃ represents the density anomaly due to spiciness mode anomalies. This term is computed as the difference between σ̃ and σ^̃ at geopotential levels.

By further subtracting the background stratification σ‾̃, we define the first-order density anomaly: 10∀z,δσ̃(r,θ,z)=σ^̃(r,θ,z)-σ‾̃(z). This term represents the isopycnal displacement due to heaving mode anomalies. In mesoscale eddies, these anomalies manifest as shallowing or deepening of isopycnals within the core. Thus, the total potential density σ̃ is decomposed into three components (see Fig. ): 11σ̃(r,θ,z)=σ‾̃(z)+δσ̃(r,θ,z)+δsσ̃(r,θ,z).

Beyond analyzing the potential density anomalies, this study seeks to characterize the spatial structure of these anomalies within the eddy core. To formulate expressions for δσ̃ and δsσ̃, we separate variables such that 23δσ̃(r,θ,z)=δhσ̃(r,θ)δzσ̃(z),24δsσ̃(r,θ,z)=δhsσ̃(r,θ)δzsσ̃(z), where subscripts h and z denote the horizontal and vertical components, respectively. In the following, we omit the notation ̃ for simplicity.

The data analyzed in this study were collected during three oceanographic cruises conducted in distinct regions of the world ocean: the EUREC4A-OA campaign along the northern coast of South America in the Atlantic Ocean, the FS METEOR M124 expedition in the South Atlantic, and the Physindien 2011 experiment in the Arabian Sea. The objective was to obtain a representative dataset of mesoscale eddies at different life cycle stages and in varying oceanic environments to investigate their vertical extent. Each campaign provided hydrological and velocity measurements with sufficient spatial coverage (both horizontally and vertically) to enable a detailed characterization of eddy structure.

In this study, structures with a velocity maximum located above (below) the pycnocline are classified as surface (subsurface) eddies. Three of the six analyzed eddies were sampled using two vertical sections rather than a single section. Figures  and and Sect.  focus on hydrological and velocity sections that cross the eddy as close to the center as possible. Additional sections are discussed in Sect.  to examine axisymmetry and horizontal extent.

To minimize spurious effects, only linear interpolations were applied in the radial x and vertical z directions. The data were then smoothed using a fourth-order low-pass filter (scipy.signal.filt in Python) to ensure a consistent representation of mesoscale features. The selection of the filter threshold is subjective and depends on the characteristic scales of interest. Since this study focuses on mesoscale dynamics, submesoscale processes are neglected, and the filtering thresholds are set to approximately 10 km for the horizontal scale (Lx) and 10 m for the vertical scale (Lz). The cutoff scale is chosen to be larger than the spatial sampling interval to preserve relevant mesoscale structures.

The grid resolution for interpolated data (Δx,Δz) and the corresponding cutoff scales (Lx and Lz) for each cruise are presented in Table . Figure  shows vertical cross-sections of the eddy core potential density, corresponding to the velocity sections shown in Fig. , following the application of the low-pass filter. The second sections of eddies in panels (a), (e), and (f) will be used in the Discussion section.

The selection of reference profiles for potential density σ‾(z), temperature T‾(z), and salinity S‾(z) is a critical step in our decomposition method. This study follows the approach developed by and for anticyclonic eddies.

To construct the climatological mean profiles, temperature, salinity, and potential density are averaged over geopotential levels within a domain surrounding the sampled eddy. The eddy center is first identified using the methodology of . A 0.5°×0.5° square is then defined around the estimated center, ensuring that the center lies at the intersection of the diagonals.

Within this region, temperature, salinity, and potential density profiles from Argo profiling floats are extracted from the Coriolis.eu.org database. These profiles are averaged over a 20-year period for the corresponding month to account for seasonal variability.

For a two-dimensional vertical section passing through the eddy center, with the horizontal axis denoted as x and the vertical axis as z (as for in situ data collected from ships), the potential density decomposition follows 51σ(x,z)=σ‾(z)+δσ(x,z)+δsσ(x,z).

To compute δσ and δsσ, a series of steps is performed in a specific order. First, the reference profiles of potential density σ‾(z), temperature T‾(z), and salinity S‾(z) are determined using the methodology described in the previous section, incorporating Argo float data. The thermohaline anomalies ΔσT and ΔσS are then computed on isopycnal surfaces using Eqs. () and (). The values of T^(x,z) and S^(x,z) are subsequently obtained via interpolation at geopotential levels, as defined in Eqs. () and ().

The next step involves computing σ^(x,z) using the TEOS-10 equation of state for seawater. The in situ potential density σ(x,z) is then subtracted from σ^(x,z) at geopotential levels, yielding δsσ(x,z) in accordance with Eq. (). The value of δσ(x,z) is then obtained by subtracting σ‾(z) from σ^(x,z) at geopotential levels, as described in Eq. (). Once δσ(x,z) is determined, it can either be divided by dσ‾/dz to obtain the isopycnal displacement ηz or integrated to obtain the streamfunction Ψv, which is analyzed as a function of the stretched coordinate ζ(z).

Since the available data consist of two-dimensional vertical sections, direct analysis of azimuthal modes is not possible. However, the dominant circular mode can be extracted, and residual modes can be estimated. For eddies in panels (a), (e), and (f) in Fig. , a second section is used to assess axisymmetry and validate the shape of the circular mode. The next section focuses on the horizontal circular mode, while axisymmetry is discussed in the later Discussion section.

Following the separation of variables, the density decomposition can be rewritten as 52σ(x,z)=σ‾(z)-ϕ(x)δzσ(z)+χ(x)δzsσ(z), where the horizontal functions ϕ(x) and χ(x) are given by 53ϕ(x)=exp⁡-x-x1R1α1,54χ(x)=exp⁡-x-x2R2α2, with x1 and x2 representing the locations of the maxima ϕ and χ, respectively, along the ship's course. In this article, α1 and α2 are enforced to be integers. For brevity, this function is referred to as the “alpha exponential”. The vertical components δzσ and δzsσ are given by Eqs. (45)–(50).

During the analysis, both ηz and δzsσ exhibited an offset, indicating a shift by a constant. To account for this, the following adjustments are introduced: 55ηz(z)=ηz0z-z1H1exp⁡-(z-z1)2H12+B,56δzsσ(z)=ξ0exp⁡-(z-z2)2H22+D.

Here, B and D represent small correction terms that do not alter the shape of the anomalies.

For each eddy, these generic expressions are fitted to the data using nonlinear least squares optimization (scipy.optimize.curve_fit in Python), minimizing the root mean square (RMS) error. The parameters estimated include

z1, H1, ηz0 for ηz(z),

The optimization is performed at the vortex center for both ηz and δzsσ, as isopycnal displacements are maximal at this location (where δσ is also maximal). Here, the vortex center is the appearing eddy center on a two-dimensional cross-section. This is the vertically averaged location where the orthogonal velocity to the ship transect is zero, not the eddy center determined using the routine. In the case that the location of the eddy center varied with depth, we took the average location. For ϕ and χ, the optimization is performed at the depths where the amplitudes of ηz and δzsσ are maximal. Consequently, vertical optimizations are conducted prior to horizontal ones.

Finally, to assess the accuracy of the proposed expressions, the RMS (root mean square) difference between the data and the theoretical predictions is computed.

In this section, we apply the potential density decomposition methodology to the subsurface AC sampled during the EUREC4A-OA experiment. While this approach has been applied to all analyzed eddies, we focus on two representative cases for clarity: the subsurface-intensified AC from EUREC4A-OA (this section) and the surface-intensified AC from Physindien 2011 (next section).

The smoothed in situ temperature and salinity fields are shown in Fig. a and d, along with the computed potential density σ and isopycnals (dark lines). The separation of isopycnals, particularly those at 26.5 and 27 kgm-3, indicates the anticyclonic nature of the eddy.

The thermohaline anomalies on isopycnals are calculated and interpolated to geopotential levels, as shown in Fig. c and f. The eddy core exhibits a distinct negative anomaly in temperature and salinity, indicative of a trapped water mass that is colder and less saline than its surroundings. The temperature and salinity anomalies reach approximately -1.6 °C and -0.3 gkg-1, respectively. These anomalies are most pronounced at the eddy center and decrease rapidly at its periphery.

The removal of these anomalies from the total in situ thermohaline fields yields the (T^/S^) fields associated with heaving mode anomalies as shown in Fig. b and e. The associated potential density σ^ is also computed, with its isopycnals resembling those in the σ field. However, small deviations beyond the 26.5 and 27 kgm-3 isopycnals highlight the impact of spiciness mode anomalies on density. Once these anomalies are removed, the eddy core appears warmer and more saline, consistent with their influence.

Figure  depicts the potential density decomposition for the EUREC4A-OA eddy. Panel (a) shows the total potential density field computed from the in situ T/S fields, while panel (b) shows the modified potential density field derived from the T^/S^ fields. The heaving mode anomaly, δσ, representing isopycnal displacements in the absence of trapped water, is shown in panel (c). The spiciness mode anomaly, δsσ, illustrating the thermohaline anomaly effect, is displayed in panel (d).

The σ and σ^ fields exhibit strong similarities, with comparable amplitudes and isopycnal displacements. The spiciness mode anomaly δsσ, shown in panel (d), has small values, reaching only -0.015 kgm-3 in the eddy core; this is 4 times smaller than the heaving mode anomaly δσ, which reaches approximately 0.5 kgm-3 in panel (c). This result aligns with theoretical expectations.

The δσ field exhibits a sign change at approximately 250 m depth, consistent with the theoretical expectation that isopycnals in an anticyclonic eddy are shallower in the upper core and deeper in the lower core relative to the surrounding water. The modest amplitude of δsσ is attributed to the incomplete compensation of temperature and salinity anomalies on isopycnals. At depths below 300 m, the modulus of δsσ is greater within the eddy core than in the surrounding water, whereas above this depth, the modulus is smaller, as evident from the color contrast in panel (d).

The results for the surface-intensified AC sampled during the Physindien 2011 experiment are presented in Fig. , with panel configurations similar to those in Fig. .

The eddy exhibits a positive spiciness mode anomaly in both temperature and salinity, indicating a trapped water mass that is warmer and saltier than its surroundings. The anomalies reach values of approximately 0.6 °C and 0.1 gkg-1, respectively. The vertical section reveals a secondary positive maximum for both thermal and haline spiciness mode anomalies at -250 m, while between -70 and -250 m, both anomalies decrease to a minimum at approximately -150 m. The eddy is surrounded by a ring of colder and fresher water, as evidenced by the negative anomaly at approximately 400 and 550 km in panels (c) and (f).

Figure  presents the potential density decomposition for this eddy. The δsσ anomaly is larger than that of the EUREC4A-OA eddy but exhibits weaker fluctuations

Figure a and b illustrate the similarity between σ and σ^0, although minor discrepancies are evident. Notably, differences appear between depths of -50 and -100 m, coinciding with the location of the warm water. Panel (c) shows that the density anomaly associated with the isopycnal displacement, δσ, reaches approximately 0.4 kgm-3, while panel (d) depicts a relatively constant density anomaly of approximately -0.15 kgm-3. Compared to the EUREC4A-OA eddy, δsσ is larger for this anticyclonic eddy. However, its variations are less pronounced, indicating a smoother spatial distribution of the spiciness mode anomaly within the eddy core.

In contrast with previous studies , the values of α1 and α2 in our analysis frequently exceed 2 (Gaussian eddies) and, in some cases, even 3 (cubic-exponential profiles). This parameter is critical as it governs the horizontal gradient of δσ, which in turn determines the velocity field. A higher α1 results in a steeper horizontal gradient and increased velocity maximum. Among the six eddies analyzed, only two exhibit Gaussian profiles (Fig. a and d). In the remaining cases, the density anomaly remains nearly constant in the core and decreases rapidly at the periphery. Such profiles may induce barotropic instabilities as previously observed by , though stratification stabilized the flow.

Gaussian shapes correspond to self-similar solutions associated with diffusive processes, as exemplified by the well-known Lamb–Oseen vortex in unsteady incompressible flow . However, mesoscale eddies in the global ocean are rarely fully isolated. Their boundaries are subject to not only diffusion but also advection and shear from the background flow. This external forcing can erode the rotating flow, steepening the velocity profile and increasing α1. As a result, the vortex diffuses less momentum into the background flow . Vertically, Figs.  and show that the observed anomalies lack symmetry about the median plane, in contrast with vortices produced in laboratory experiments under constant stratification .

Figures  and demonstrate that both models Flierl1987 and Bara2024 adequately capture the vertical structure of the potential density anomaly in the eddy cores. While model Bara2024 yields more accurate results, it introduces an additional parameter compared to model Flierl1987. The former is derived from self-similar diffusivity considerations, whereas the latter is based on quasi-geostrophic theory. Model Bara2024, supported by both experimental and observational studies, predicts that density anomalies vanish at depth, unlike model Flierl1987. However, model Flierl1987 is built on stronger theoretical foundations. Further investigation is needed to determine whether inertia or diffusion is the primary driver of coherent eddy shape.

Our study raises several important questions, many of which merit dedicated investigation in future work, including our own ongoing research. In particular, elucidating the physical interpretation of the parameters B and D, as well as the variability of R and H, is essential for a better understanding of eddy dynamics.

This section presents additional vertical sections of eddies in panels (a), (e), and (f) from Fig. , analyzed using the same methodology. The fitted models and reconstructed density fields using the Bara2024 model are shown in Fig. . The low RMS errors indicate that the models successfully predict density variations in these additional sections.

While the horizontal profile maintains the same α1 as in Fig. , other parameter values (see Table ) vary between sections. This suggests that the eddy shapes are not perfectly axisymmetric, despite appearing so in single ship transects. This discrepancy in parameter values may arise from differences in the estimated eddy center across sections. In a single transect, the center is identified as the point where the orthogonal velocity is zero, but this estimate does not necessarily coincide with those obtained using or from another section. The influence of center misalignment is quantified in Appendix  for an idealized circular vortex.

Higher-order azimuthal modes appear weak in coherent eddies, as indicated by the low RMS errors in the horizontal reconstruction of δσ in both sections of eddies in panels (a), (e), and (f) in Fig. . Although true circular vortices are rare, the radial profile remains largely unchanged when modeling a slightly non-circular vortex. The key advantage of our model is its variable separation approach, commonly used for advection–diffusion equations, governing density anomalies. Introducing angular variations θ does not fundamentally alter the reasoning. Figure  confirms that these eddies are not perfectly circular, yet our methodology and formulas accurately describe their structure (Fig. ). The variations of R1 across sections suggest that the horizontal shape of the density anomaly can be approximated by 58ϕ(r,θ)=exp⁡-rR1(θ)α1, where the eddy radius R1 varies with the orientation θ of the section.

Vertically, parameter values differ between sections (see Table ). While the magnitudes remain consistent, notable discrepancies persist. These variations may result from differences in the estimated eddy center or the temporal gap between section acquisitions. Since eddies can move or evolve during this period, vertical structure sampling may vary, as observed in previous studies .

As illustrated in Figs.  and and previously noted by , heaving mode anomalies are the dominant contributors to the total potential density anomaly in eddy cores. Consequently, the radial gradient of δσ is significantly larger than that of δsσ, leading to stronger azimuthal geostrophic flow via thermal wind balance. Since mesoscale eddies are generally in geostrophic equilibrium , thermal wind is often the primary driver of horizontal rotational flow. The flow generated by heaving mode anomalies therefore determines the intensity of the geostrophic velocity, the shape of the trapped water, and, consequently, the horizontal structure of spiciness mode anomalies. This interaction explains why the characteristic parameters of both density anomaly types are nearly identical. While thermohaline anomalies of trapped water remain largely conserved during advection, their influence on the eddy's dynamics is minimal.

Previous studies have identified similar horizontal or vertical structures for mesoscale eddies of both polarities . In the cruise analyzed, the lack of data does not permit us to study in detail the structure of cyclonic eddies. In the quasi-geostrophic framework, there should be no difference in the horizontal and vertical structures between cyclonic and anticyclonic eddies, except for the sign of their anomalies. The methodology and formulas proposed in this article can thus be applied to cyclonic eddies.