F. Schütte et al.: Hidden vortices: near-equatorial low-oxygen extremes driven by high-baroclinic-mode vortices 125 
E(z) = ) an On). 
n=1 
(5) 
Here, K — co expresses the exact solution with an infinite 
number of vertical modes for a continuously stratified ocean. 
The expansion coefficients x, are the modal amplitudes. The 
modal amplitudes are obtained by projecting the observed 
fields onto the structure functions computed from the World 
Ocean Atlas. The projection is preferred over resolving a 
jeast-square problem, which sometimes leads to unrealis- 
ic modal amplitudes into the high modes (Vic and Ferron, 
2023). The modal amplitudes x, are calculated via a scalar 
product: 
30 
/ nz) SetD(z)dz. 
80 
CXn = 
(6) 
These amplitudes are then normalized by dividing with 
-30 
S Von (z)”*dz. This analysis is restricted to the depth range 
-980 
from 30 to 980m in order to exclude the surface mixed 
layer while retaining the majority of available profiles along 
23° W. The barotropic mode assumed to be zero. A vertical 
resolution of 10m is used, with both the CTD profiles and 
World Ocean Atlas data interpolated accordingly. After com- 
puting the contribution of one mode, it is subtracted from the 
displacement profile: E’(z) = E(z) — xnY.(z) and the proce- 
dure is repeated for the next mode. This recursive removal re- 
duces cross-talk between modes caused by the limited verti- 
cal resolution and incomplete depth coverage. Since the order 
of mode extraction may influence the result, the decomposi- 
tion is repeated M = 100 times with random permutations of 
modes n = 1 ton =20, and the final modal amplitudes are 
calculated as the mean over all realizations, with associated 
standard errors. 
3.2 Potential vorticity and Rossby number 
Subsurface eddies exhibit signatures of high or low poten- 
tial vorticity (PV), depending on their stratification anomaly 
and rotation direction (D’Asaro, 1988; McWilliams, 1985; 
Molemaker et al., 2015). In the absence of mixing, PV is a 
conserved quantity and serves as an effective tracer to differ- 
entiate water masses and track eddy pathways. 
We refer to Ertels PV (Gill, 1982), being one of the most 
complete formulations for PV conservation, and take its ver- 
cal approximation (see e.g. Thomsen et al., 2016), which is 
given by 
Q0=(&+f)-N* 
where &, = Zu -_ Ze is the vertical component of the rela- 
tive vorticity with u and v being the zonal and meridional 
https:/doi.org/10.5194/o0s-22-119-2026 
velocity, respectively, and f is the Coriolis parameter. The 
term &, + f represents the absolute vorticity. The approxi- 
mation given by Eg. (7) is valid in case of nearly horizon- 
tally orientated isopycnal surfaces (Thomsen et al., 2016). 
Counter-clockwise and clockwise rotating eddies correspond 
to positive and negative relative vorticity, respectively. In the 
northern hemisphere, anticyclonic eddies rotate clockwise 
and have negative relative vorticity (vice versa for the south- 
ern hemisphere, which is not further considered throughout 
this study). 
In the case of geostrophic balance, the Rossby number 
ne 
Lf F 
where U is characteristic velocity and L is characteristic 
length scale, is smaller than one and PV is always positive. 
PV can be reduced by either a reduction of N? (weakened 
stratification) or by a gain of anticyclonic relative vorticity 
(D’Asaro, 1988). The explanation also applies vice versa, 
i.e. PV can be increased by a strengthening in stratification 
or a gain of cyclonic relative vorticity. The Rossby number 
becomes larger than one for submesoscale dynamics in the 
ageostrophic range. 
For the propagation speeds we followed an approach by 
Nof (1981) and Rubino et al. (2009), who formulated the 
westward translation of isolated high baroclinic eddies on 
a plane, which is given as a function of the th baroclinic 
Rossby radius of deformation and the Rossby number: 
lem A 1 
Cn= 36Ran(l Ro) 
with ß being the meridional derivative of the Coriolis param- 
eter. 
3.3 Eddy identification algorithms 
3.3.1 Eddy identification from shipboard observations 
Horizontal velocity data from the vmADCP system (see 
Sect. 2.1) is used to detect eddies along the 23° W merid- 
ian between 6 and 12°N, following the methodology from 
Bendinger et al. (2025). This methodology is based on an ide- 
alized eddy solution, known as Rankine vortex characterized 
by solid-body rotation in its inner core, 1.e., a linear increase 
of velocity with increasing distance from the eddy center. We 
do so through the conversion from Cartesian into cylindri- 
cal coordinates in areas that are suspected to cross eddies. 
Every point in the horizontal plane is defined by the radial 
distance, r, to the origin (eddy center) and the azimuthal an- 
gle, 0, Le., 
vr = u cosO + vsinG 
vn = —usinG + vcos5 
(10) 
an 
where vr and vo are the radial and azimuthal veloci- 
ties, respectively. Following Casteläo and Johns (2011) and 
Ocean Seci.. 22. 119-143. 2026