# Reconstructing the Three-Dimensional Structure of Loop Current Rings from Satellite Altimetry and In Situ Data Using the Gravest Empirical Modes Method

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## Abstract

**:**

## 1. Introduction

## 2. Data

#### 2.1. In Situ Data

#### 2.2. Altimetry Data

#### 2.3. Glider Data

## 3. Methods

#### 3.1. The Gravest Empirical Modes (GEM) Method

**A.**The steric height relative to 2000-dbar is computed for each in situ profile of temperature and salinity;**B**. All profiles are sorted according to their steric height. The sorted ARGO temperature and salinity profiles are shown in Figure 2, where a pattern already emerges, showing the clear relationship between steric height, pressure, and both temperature and salinity;**C**. A regular pressure grid is defined ((0–2000 dbar) with a vertical grid-step of 2 dbar) and for each reference pressure value, a spline interpolant is fitted to the functions $T\left({\eta}_{2000}\right){|}_{p}$ and $S\left({\eta}_{2000}\right){|}_{p}$, where T and S are temperature and salinity, ${\eta}_{2000}$ is the 2000 dbar-referenced steric height, and p is the pressure at which the variables are evaluated;**D**. The relationship between 2000-dbar referenced steric height and dynamic height is assessed by comparing local ADT and ${\eta}_{2000}$ to ensure that the empirical relationship obtained from in situ steric height holds when using ADT (Figure 1d).

#### 3.2. Eddy Detection and Edge Definition

## 4. Validation Using Independent Observations

## 5. The 3D Structure of an Average LCR

**∇**is the horizontal gradient operator, $\mathbf{k}$ is the vertical unit vector, and $\rho $ is in situ density.

## 6. Heat, Salt, and Energy Statistical Properties

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**): Location of all available ARGO profiles in the Gulf of Mexico. The local 2000 dbar-referenced steric height is color coded. (

**b**): T-S diagram of all ARGO profiles. The color code is the same as in panel (

**a**). The black contours are isopleths of spice and potential density. (

**c**): Distributions (normalized PDF) of the 2000 dbar-referenced steric height computed from in situ data (blue bars) and the gridded absolute dynamic topography (ADT) data (orange line). (

**d**): 2000 dbar-referenced steric height against ADT.

**Figure 2.**Steric height-sorted raw temperature (

**a**) and salinity (

**b**) profiles for the whole ARGO dataset. The x-axis represents 2000 dbar-referenced steric heigh (${\eta}_{2000}$) and the y-axis is pressure. The 2000 dbar-referenced steric height closely matches absolute dynamic topography (ADT), as shown in Figure 1c,d.

**Figure 3.**Yearly averaged gravest empirical mode (GEM) fields for temperature (

**a**), salinity (

**b**), and potential density (

**c**). The x-axis represents dynamic height, and the y-axis represents pressure. Dynamic height was shown to be equal to the 2000 dbar-referenced steric height ${\eta}_{2000}$ and absolute dynamic topography (ADT) in Figure 1d. The root mean squared (RMS) errors are shown in panels (

**d**–

**f**) for temperature, salinity, and potential density, respectively. The black dashed lines represent the limits of the SUW potential density range (1024–1026.5 kg m${}^{-3}$), while the red or green dashed lines represent the depth range of the mixed layer.

**Figure 4.**Example of seasonal variation of GEM fields for a given value of sea surface height (SSH; 0.8 m here). The x-axis is the time of year while the y axis is pressure. Temperature is shown in panel (

**a**), salinity in panel (

**b**), and potential density in panel (

**c**).

**Figure 5.**(

**a**): Map of the edge contours of the 40 detected Loop Current rings (LCRs) one day after detachment. The maximum absolute dynamic topography (ADT) value within each eddy is color coded. (

**b**): Same as (

**a**) in an eddy-centric frame: the x and y axis are the distance (in km) from the eddy’s rotation axis. The color code is the same as in (

**a**). (

**c**): Maximum ADT value (at the eddy’s center) against eddy’s radius. (

**d**): Volume anomaly (surface-integral of the ADT anomaly) against distance from the Yucatan channel.

**Figure 6.**(

**a**,

**b**): Temperature sections across a Loop Current ring measured by the glider and reconstructed using the GEM. in each panel, the GEM-reconstructed sections are flipped laterally in order to appear as a mirror image of the glider section. (

**c**,

**d**): Same as (

**a**,

**b**) for salinity. (

**e**,

**f**): Same as (

**a**,

**b**) for geostrophic velocity.

**Figure 7.**(

**a**,

**b**): Horizontalprofiles of depth-averaged temperature (red) and salinity (blue) anomalies. The glider observations are plotted as dotted lines, while the GEM-reconstructed profile is plotted as plain lines. The difference between the glider and GEM sections are plotted as dotted lines. (

**c**,

**d**): Same as (

**a**,

**b**) for the geostrophic velocity.

**Figure 8.**Selected examples of LCRs three-dimensional structure reconstructed from altimetry and the GEM method. (

**First row**): maps of absolute dynamic topography (ADT). The dotted red line represents the trajectory of the virtual vertical sections. (

**Second row**): Vertical sections of salinity. (

**Third row**): Vertical sections of temperature. (

**Fourth row**): Vertical sections of cyclogeostrophic velocity.

**Figure 9.**(

**a**): Radial profiles of sea surface height (SSH) for the 40 detached Loop Current rings (LCRs) (gray lines) and mean universal SSH profile (Equation (3)) computed using the mean parameters of the 40 least-square fits (black line). (

**b**): Coefficient of determination (${R}^{2}$) of the observed SSH profiles and the fitted universal profiles.

**Figure 10.**Parameters of the least-square fit to the so-called universal stream function (Equation (3)) for each detected Loop Current ring. The x-axis is the radial length scale and the y-axis is the maximum ADT anomaly ${\eta}_{m}$.

**Figure 11.**Vertical profiles of temperature (

**a**) and salinity (

**b**) for the reconstructed average Loop Current ring (LCR). The average LCR is computed using the universal sea surface height (SSH) profile and the Gravest Empirical Mode (GEM) fields. The parameters used in the universal profile are the mean from the 40 least-square fits.

**Figure 12.**Same as Figure 11 for geostrophic velocity (

**a**), cyclogeostrophic velocity (

**b**), geostrophic relative vorticity (

**c**), and cyclogeostrophic relative vorticity (

**d**). The contour interval is 0.1 m s${}^{-1}$. Note that the color bars have a different range in each panel.

**Figure 13.**(

**a**): Same as Figure 11 for potential vorticity. (

**b**): same as (

**a**) for the squared buoyancy frequency.

**Figure 14.**Same as Figure 11 for available potential energy density (

**a**), and kinetic energy density (

**b**).

**Figure 15.**(

**a**): Total heat content of each detached Loop Current ring (bar plot). The light blue bars represent values computed using the maximum velocity contour as the LCR’s edge, while dark blue bars were obtained using the last closed absolute dynamic topography (ADT) contour. The orange curves, referenced on the right-hand side y-axis, represent the cumulative heat input of the LCRs over time. The plain and dotted lines represent the maximum velocity and the last closed contour criteria, respectively. (

**b**): same as panel (

**a**) for the salt input. (

**c**): Total heat content of each detached LCR against total salt content. (

**d**): Total heat content of each LCR against its total area.

**Figure 16.**(

**a**): Bar graph of the total mechanical energy (sum of the kinetic and available potential energy) of each detached Loop Current rings (LCRs). The light blue bars represent values computed using the maximum velocity contour as the LCR’s edge, while dark blue bars were obtained using the last closed absolute dynamic topography (ADT) contour. The orange curves, referenced on the right-hand side y-axis, represent the cumulative energy input of the LCRs over time. The plain, dotted, and dashed lines represent total mechanical energy (TE), available potential energy (APE), and kinetic energy (KE), respectively. (

**b**): KE against APE for each of the 40 detached LCRs. The black line represents equal partition of energy (Burger number equals to one).

**Figure 17.**Hypothetical evolution of temperature (

**a**), salinity (

**b**), sea surface height (

**c**), and energy density (

**d**) if the heat, salt, and energy inputs of Loop Current rings into the Gulf of Mexico (GoM) were not balanced at all. The bar plots represent the effect of individual eddies while the orange lines represent their cumulative effects over time. The plain line represents the scenario where the excess heat, salt and energy are mixed homogeneously within the entire GoM’s volume. The dashed and dotted lines represent scenarios where the excess of heat, salt and energy are mixed within the top 2000 and 1000 m, respectively.

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**MDPI and ACS Style**

Meunier, T.; Pérez-Brunius, P.; Bower, A. Reconstructing the Three-Dimensional Structure of Loop Current Rings from Satellite Altimetry and In Situ Data Using the Gravest Empirical Modes Method. *Remote Sens.* **2022**, *14*, 4174.
https://doi.org/10.3390/rs14174174

**AMA Style**

Meunier T, Pérez-Brunius P, Bower A. Reconstructing the Three-Dimensional Structure of Loop Current Rings from Satellite Altimetry and In Situ Data Using the Gravest Empirical Modes Method. *Remote Sensing*. 2022; 14(17):4174.
https://doi.org/10.3390/rs14174174

**Chicago/Turabian Style**

Meunier, Thomas, Paula Pérez-Brunius, and Amy Bower. 2022. "Reconstructing the Three-Dimensional Structure of Loop Current Rings from Satellite Altimetry and In Situ Data Using the Gravest Empirical Modes Method" *Remote Sensing* 14, no. 17: 4174.
https://doi.org/10.3390/rs14174174