Eulerian and Lagrangian diagnostics of the dynamical properties of the water masses sampled during the Tara Pacific Expedition 2016-2018

In order to provide a description of the dynamical properties of the water masses sampled, different Eulerian and Lagrangian diagnostics were calculated. 

For each of the 246 stations sampled, we proceeded as follows.

We identified the water mass sampled at the given station. This was considered as a stadium shape with the two semi-circles centered on the starting and ending points of the transect, respectively. The radius of the stadium semi-circles was considered 0.1°, which is in accordance with previous studies25,29,30. The stadium was filled with virtual particles separated by 0.01°.

For each virtual particle inside the stadium shape, we calculated an Eulerian or Lagrangian diagnostic (described above). The Eulerian diagnostics were extracted directly from the velocity field of the day of sampling. Concerning the Lagrangian diagnostics, these were obtained by advecting the virtual particle backward in time for an amount of time 𝞽 from the day of sampling day_S. For the Lagrangian betweenness, the advection was performed between day_S+𝞽/2 and day_S-𝞽/2, so that the advective time window was centered on the sampling day (details in25).

For the Lagrangian diagnostics, we used the following advective times 𝞽: 5, 10, 15, 20, 30, and 60 days. The only exception is the retention time, which, by construction, was calculated only with the largest advective time, namely 𝞽=60 days.

Once that, a given diagnostic (Eulerian or Lagrangian) was calculated for all the virtual particles filling the stadium shape, we calculated the mean value, and the 25, 50, and 75 percentiles. The percentiles were calculated in order to quantify the spatial variation of the diagnostic inside the stadium shape. Therefore, we associated each station with four values (mean, 25, 50, and 75 percentiles) of a given diagnostic.

 Furthermore, two different velocity fields were used, which are described as follows. 

Both the velocity fields were downloaded from E.U. Copernicus Marine Environment Monitoring Service (CMEMS, http://marine.copernicus.eu/). The first velocity field used was MULTIOBS_GLO_PHY_REP_015_004 [GlobEkmanDt]. This was produced by combining the altimetry derived geostrophic velocities and modeled Ekman surface currents. It had a spatial resolution of 0.25° and a temporal resolution of one day. The second velocity field was GLOBAL_REANALYSIS_PHY_001_030 [GloryS12]. It was obtained by a NEMO model assimilating altimetry and other observations. It had a spatial resolution of 1/12° and a temporal resolution of 1 day.

The following Eulerian diagnostics were calculated:

Absolute velocity ([Uabs], m s-1): sqrt(u2+v2), where u and v are the zonal and meridional components of the horizontal velocity field used (described below)

Kinetic energy ([Ekin], m2 .s-2): 0.5*(u2+v2)

Divergence ([EulerDiverg], d-1): du/dx + dv/dy

Vorticity ([Vorticity], d-1): dv/dx - du/dy

Okubo-Weiss ([OW], d-2): s2-vorticity2, where s2 is (du/dx-dv/dy)2 + (dv/dx+du/dy)2. If negative, it indicates that the station sampled was inside an eddy.

The following Lagrangian diagnostics were calculated:

Finite-Time Lyapunov Exponents ([Ftle], d-1): it indicates the rate of horizontal stirring, and it is a means to quantify the intensity of turbulence in a given region. FTLE are commonly used to identify Lagrangian Coherent Structures, i.e. barriers to transport. In this case, a strong FTLE value indicates a region separating water masses which were far away backward in time.

Lagrangian betweenness ([betw], adimensional): this diagnostic draws inspiration from Lagrangian Flow Network Theory26. It can identify regions which act as bottlenecks for the circulation, in that they receive waters coming from different origins, and that are then spread over several different destinations. These can represent possible hotspots driving biodiversity25.

Lagrangian Divergence ([LagrDiverg], d-1). This diagnostic was calculated by integrating the Eulerian divergence along the backward trajectories. If positive, it indicates a water mass that, during the previous days, was subjected to a strong divergence, thus to a possible upwelling. If negative, it indicates a strong convergence, thus possible downwelling.

Retention Time ([RetentionTime], d). This diagnostic indicates how many days a water mass has spent inside an eddy in the previous period. If the water mass is outside an eddy, then its retention time is set to zero.