Supporting Information for ‘The thermodynamic balance of the Weddell Gyre’, by A.
C. Naveira Garabato, J. D. Zika, L. Jullion, P. J. Brown, P. R. Holland, M. P.
Meredith and S. Bacon
A box inverse model of the Weddell Gyre
A box inverse model of the Weddell Gyre was constructed by Jullion et al. [2014] using the
four hydrographic sections shown in Figure S1 and summarized in Table S1 (note that the
18O data analysed in this study were sampled solely at the sections on the outer rim of the
gyre, and were not considered by the inverse model). Box inverse modelling [Wunsch,
1996] provides a simple yet effective technique to estimate the general circulation of the
ocean by combining observations in a theoretical framework enforcing conservation of
mass, heat and salt.
A difficulty in defining the model domain was the absence of land boundaries between the
segments of the ANDREX section conducted during cruises JC30 and JR239, and between
the ANDREX and I6S sections. The former case did not pose a significant problem, as the
JC30 / JR239 junction is located within the Weddell Gyre, where mesoscale activity is
weak. In contrast, the junction between the ANDREX and I6S sections called for a careful
examination, as it is located within an ACC sector of elevated mesoscale activity. To avoid
the introduction of significant horizontal transport biases linked to the asynopticity of the
two transects, we closed the northeastern corner of the model domain in a dynamically
consistent manner. Rather than matching the sections geographically, we selected end
stations for the sections with an identical dynamic height and water mass structure, such
that no horizontal transport occurs between these end stations. See Jullion et al. (2014) for
further details.
The sections were divided into 10 layers separated by neutral density surfaces, selected to
correspond to the boundaries of the major water masses in the gyre. Within the box
enclosed by the four sections, conservation of volume, potential temperature anomaly and
salinity anomaly (approximately equivalent to mass, heat and salt) was enforced, in the
form given by
𝑛
̅̅̅̅̅̅̅∗ 𝛾 𝛾𝑗+1 + 𝐹𝑗𝐴−𝑆 (𝐶) + 𝐹𝑗𝑆𝐼 (𝐶) = 0 ,
∑𝑚
𝑗=1 ∑𝑖=1[𝛿𝑖 𝐿𝑖 𝐷𝑖𝑗 (𝑉𝑖𝑗 + 𝑏𝑖 )𝜌𝑖𝑗 𝐶𝑖𝑗 ] + 𝑣𝑗 (𝐶) − [𝐴𝜌𝐶𝜔𝐶 ]𝛾𝑗
(1)
where m is the number of layers; n is the number of station pairs; i adopts the value 1 or -1
depending on whether flow is directed into or out of the model box, respectively; Li is the
horizontal distance between successive stations; Dij is the layer thickness at station pair i
and layer j; Vij is the baroclinic geostrophic velocity at station pair i and layer j; bi is the
barotropic velocity at station pair i; ij is the in situ density at station pair i and layer j; A is
the area of a layer interface within the model box; 𝜔𝐶∗ is the diapycnal velocity for tracer C
[McIntosh and Rintoul, 1997]; 𝐹𝑗𝐴−𝑆 (𝐶) and 𝐹𝑗𝑆𝐼 (𝐶) are the fluxes of tracer C into layer j
′𝐶′𝑡ℎ
̅𝑡 +
̅̅̅̅̅̅
associated with air-ocean interactions and sea ice, respectively; 𝑣𝑗 (𝐶) = 𝜌𝑗 [𝑣
𝑡
̅̅̅̅̅
𝑣 ′ ℎ′ 𝐶̅ 𝑡 ]𝑗 is the eddy-induced flux of tracer C into layer j, and consists of diffusive (the
first) and advective (the second) components; v’ and h’ are deviations from the time-mean
cross-section velocity and isopycnal layer thickness calculated from the eddy-permitting
Southern Ocean State Estimate (SOSE, Mazloff et al. [2010]; see Jullion et al. [2014] for
details); ̅ 𝛾 and ̅ 𝑡 denote the averaging operators over a layer interface and time,
respectively. The model incorporates two sets of terms (sea ice-mediated and eddy-induced
transports) that are not usually represented in box inverse models, but that are potentially
significant in the context of the Weddell Gyre.
The model is underdetermined: it has 238 unknowns (175 barotropic velocities, 27
diapycnal velocities, 30 eddy flux terms, 2 sea ice transport terms, 2 air-ocean heat flux
terms, and 2 air-ocean freshwater flux terms) and only 40 equations (conservations of
volume, potential temperature anomaly and salinity anomaly in 10 isopycnal layers and full
depth, supplemented by additional constraints on the volume and salinity anomaly
transports across each of the two coast-to-coast sections). Row and column weighting are
applied to the model to account for prior uncertainties in the model’s equations and
unknowns. Row weights are selected by reference to SOSE, and reflect the uncertainty in
the model’s annual-mean conservation statements introduced by sampling oceanic
variability with asynoptic sections with a summer focus. Column weights reflect the
expected variability of each unknown (assessed with SOSE) and our level of confidence in
the prior estimate of that unknown. The model is solved using singular value
decomposition. A solution rank of 28 is selected, as the lowest solution rank providing a
dynamically acceptable solution within uncertainties. Because heat and freshwater transport
calculations are only meaningful with a closed mass budget, a second inversion is
performed to eliminate the small full-depth volume and salinity anomaly transport residuals
present in the solution to the first inversion. A complete description of the model procedure
is given by Jullion et al. [2014].
Estimation of uncertainties in transport terms and derived variables
The uncertainty in each transport term (such as the residual transports calculated using
expression (1)) and derived variables (such as the residual-transport-weighted-mean
variables computed using (2)) was estimated using a Monte Carlo approach. This entailed
creating 2000 alternative solutions to the box inverse model, each of which was obtained
by subjecting initial best estimates of all unknowns to random perturbations with a standard
deviation equal to the prior uncertainties of those unknowns (determined from SOSE,
regional atmospheric reanalyses, and ice data sets, as discussed by Brown et al. [2014]),
and running these perturbed fields through the inverse model. Uncertainties in transports
and derived variables were estimated as the standard deviation of the pertinent variable in
all 2000 solutions. Overall, the general robustness of our results stems from the inverse
machinery’s dampening of the effect of prior uncertainties in the unknowns. Although
these uncertainties are substantial, the conservation statements in the model diminish their
impact on the solution. For this reason, the variables and layers that are directly constrained
by the model’s conservation equations are associated with smaller uncertainties than
variables and layers for which conservation is not explicitly enforced by the model.
References
Bacon, S., and L. Jullion (2011), RRS James Cook: Antarctic Deep Water Rates of EXport
(ANDREX), Tech. Rep. 8, Natl. Oceanogr. Cent., Southampton, U. K.
Fahrbach, E. (2005), Expedition FS Polarstern ANT-XXII/3, Tech. Rep. 533, AlfredWegener-Inst. für Polar- und Meeresforschung in der Helmholtz-Gemeinschaft,
Bremerhaven, Germany.
McIntosh, P. C., and S. R. Rintoul (1997), Do box inverse models work?, J. Phys.
Oceanogr., 27, 291–308.
Meredith, M. (2010), Cruise report, RRS James Clark Ross, Tech. Rep. jr235/236/239, Br.
Antarct. Surv., Cambridge, U. K.
Speer, K. G., and T. Dittmar (2008), Cruise report, RV Revelle, Tech. Rep. 33rr20080204,
Fla. State Univ., Tallahassee, Fla.
Wunsch, C. (1996), The Ocean Circulation Inverse Problem, 442 pp., Cambridge Univ.
Press, Cambridge, U. K.
Table S1. Summary of the cruise data sets entering the box inverse model of Jullion et al.
[2014]. Station numbers for each section in the model box are indicated.
Year
Month
Vessel
2005
Jan – Feb
2008
Feb
Mar
2009
Jan
2010
Mar
Apr
Cruise no.
Section
id.
PFS
ANTXXII/3
SR4
Polarstern
– RV Roger 33RR20080204 I6S
Revelle
RRS
James
Cook
– RRS
James
Clark
Ross
JC30
ANDREX
JR239
ANDREX
Box
Reference
station
no.
126 – 178 Fahrbach
[2005]
91 – 125 Speer and
Dittmar
[2008]
67 – 90
Bacon and
Jullion
[2011]
1 – 66
Meredith
[2010]
Figure S1. Map of the bathymetry (from GEBCO, depth in m) of the Weddell Gyre region.
The three sections entering the box inverse model of Jullion et al. (2014) are indicated by
blue diamonds and labelled.