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\title{\begin{center}
Auxiliary Material for:
\end{center}The contribution of the Weddell Gyre to the lower limb of the Global Overturning Circulation
\begin{center}
Journal of Geophysical Research
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%  AUTHORS AND AFFILIATIONS
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\author[1,2,3]{Lo\"{\i}c Jullion\thanks{ljullion@fsu.edu}}
\author[1,3]{Alberto C. Naveira Garabato}
\author[3]{Sheldon Bacon}
\author[4,5]{Michael P. Meredith}
\author[4,6]{Peter J. Brown}
\author[3]{Sine Torres-Vald\'{e}s}
\author[2]{Kevin G. Speer}
\author[4]{Paul R. Holland}
\author[2]{Jun Dong}
\author[6]{Doroth\'ee Bakker}
\author[7]{Mario Hoppema}
\author[4]{Hugh J. Venables}
\author[8]{William J. Jenkins}
\author[10]{Marie-Jos\'e Messias}
\author[7]{Eberhard Fahrbach}
%\authors{L. Jullion\altaffilmark{1,2,3}, A. C. Naveira Garabato\altaffilmark{1,2}, S. Bacon\altaffilmark{2}, M. P. Meredith\altaffilmark{4,5}, P.J. Brown \altaffilmark{4,5}, S. Torres-Valdes\altaffilmark{2}, K. Speer\altaffilmark{3}, P.H. Holland\altaffilmark{4},  J. Dong\altaffilmark{3}, D. Bakker \altaffilmark{5}, M. Hoppema\altaffilmark{7}, W.J. Jenkins\altaffilmark{8}  and E. Fahrbach\altaffilmark{7}}

\affil[1]{University of Southampton, National Oceanography Centre, Southampton, U.K.}
\affil[2]{Geophysical Fluid Dynamics Institute, Tallahassee, U.S.A.}

\affil[3]{National Oceanography Centre, Southampton, U.K.}
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\affil[4]{British Antarctic Survey, Cambridge, U.K.}
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\affil[5]{Scottish Association for Marine Science, Oban, U.K.}
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\affil[6]{University of East Anglia, Norwich, U.K.}
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\affil[7]{Alfred Weneger Institute, Bremerhaven, Germany}
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\affil[8]{University of Rhode Island, Narragansett, U.S.A.}

\affil[9]{Woods Hole Oceanographic Institution, Woods Hole, U.S.A.}

\affil[10]{University of Exeter, Exeter, U.K.}

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\section{Introduction}

Here we outline the basic formulation and implementation of our box inverse model of the Weddell Gyre.
 
\section{Box inverse modelling}
	\subsection{Model set up}

	We enforce conservation of mass, heat and salt (approximated as volume, potential temperature and salinity in the model, where the anomalies are defined as deviation from section-mean potential temperature, $\overline{\Theta}$, and salinity, $\overline{S}$) within the region enclosed by the four sections described in the main text and the Antarctic coast. The sections are divided vertically into 10 layers separated by neutral density interfaces \citep{Jackett:1997p370}. The layer interfaces are selected to correspond with the boundaries of the major water masses in the region. The ocean surface is considered to be permeable to exchanges of heat and freshwater between the ocean and the overlying atmosphere or ice. The conservation statement for any given tracer \emph{C}  can be written in discretized form as
	\begin{equation}\label{conservation}
\sum_{\jmath=1}^{m}[\sum_{\imath=1}^{n}[\delta_{\imath}L_{\imath}D_{\imath \jmath}(V_{\imath\jmath}+b_{\imath})\rho_{\imath\jmath}C_{\imath\jmath}] + \nu_{\jmath}(C) - [A\overline{\rho C\omega_{c}^{\ast}}^{\gamma}]_{\gamma_{\jmath}}^{\gamma_{\jmath+1}} + F_{\jmath}^{A-S}(C) + F_{\jmath}^{SI}(C)]  = 0 ,
\end{equation}
\noindent where $n$ is the number of station pairs; $m$ is the number of layers; $\delta_{i}$ adopts the value +1 or -1 depending on whether flow is directed into or out of the box; $L_{\imath}$ and $D_{\imath\jmath}$ are the distance between successive stations and the layer thickness at each station pair, respectively; V$_{\imath\jmath}$ is the baroclinic velocity at the station pair $i$ and layer $j$; b$_{\imath}$ is the barotropic velocity at station pair $i$; $\rho_{\imath\jmath}$ is \textit{in situ} density; $\emph{A}$ is the area of the layer interface within the box; $\omega_{C}^{\ast}$ is the diapycnal velocity for tracer \emph{C} \citep{mcintoshandrintoul97,sloyan00}; $F_{\jmath}^{A-S}(C)$ and $F_{\jmath}^{SI}(C)$ are the fluxes of tracer \emph{C} associated with air-sea interactions and sea ice, respectively;  $\nu_{\jmath}(C) = \rho_{\jmath}[\overline{v'C'}^{t}\overline{h}^{t} + \overline{v'h'}^{t}\overline{C}^{t}]_{\jmath}$ is the eddy-induced flux of tracer \emph{C} for the layer $\jmath$, which consists of advective (the first) and diffusive (the second) components; $v'$ and $h'$ are the deviation from the time-averaged mean velocity and isopycnal layer thickness calculated from SOSE (see below); $\overline{(\cdot)}^{\gamma}$ and $\overline{(\cdot)}^{t}$ denote the area-mean operator over a layer interface and the time-mean operator, respectively. We note that the model incorporates two sets of terms (sea ice-mediated and eddy-induced transports) that are not normally represented in box inverse models, but that are important in the context of the Weddell gyre. 

	In previous inverse models, the eddy contribution was usually neglected or treated as a source of uncertainty \citep{Ganachaud:2003p28}. Anticipating eddy-induced transports to contribute significantly to exchanges across the rim of the Weddell Gyre, particularly at the gyre's eastern edge \citep{Schroder:tt}, we represent explicitly the transports of volume, potential temperature and salinity across the ANDREX / I6S section by adding an eddy transport variable to each of the layer equations. Across the SR4 section, the eddy term is not calculated due to the poor performance of SOSE in reproducing the structure of the Antarctic Slope Front.

	Advection of mobile sea ice conveys heat and freshwater, and thus requires consideration in the potential temperature and salinity budgets of the gyre. Air-sea interactions lead to diapycnal mixing and water mass transformations, particularly notable in AABW formation regions. An advantage of our particular choice of model domain is that the areas where AABW is produced and all model layers outcrop are excluded from the domain (Figure 1 of the paper). Within the box, only the uppermost two layers (encompassing SW and WW) are exposed to air-sea heat and freshwater fluxes, with the extensive sea ice in the region limiting oceanic heat loss. Therefore, we do not represent air-sea-induced diapycnal fluxes explicitly in the model.

	Further to the conservation of volume, potential temperature and salinity within the model box, we additionally prescribe that the net full-depth transports of volume and salinity across each coast-to-coast section (ANDREX / I6S and SR4) be approximately zero. We also require that the net full-depth volume and salinity transports associated with the ACC in the northeastern corner of the model box across the ANDREX section be $0\pm 5$ Sv. 

	\subsection{Model initialization}
	
	The barotropic component of the geostrophic velocity  for each station pair is estimated by fitting the geostrophic shear profiles referenced to the deepest common level to the averaged velocity profiles for the same station pair obtained from a combination of shipboard ADCP (SADCP) and lowered ADCP (LADCP) data. Specifically, the quasi-zonal section along the northern rim of the gyre between the Antarctic Peninsula and $30^{\circ}$E was referenced using SADCP, because of issues with the LADCP during most of the JR239 and JC30 cruises. A comparison between the LADCP- and SADCP-referenced geostrophic velocities was only possible west of the South Orkney Islands, where excellent agreement between the two sets of velocities was found. SADCP data were also used to reference the geostrophic shear on the SR4 section, because LADCP data were not available. On the I6S section, good-quality data were obtained with both SADCP and LADCP, and we used the LADCP data to reference the geostrophic shear. All direct velocity measurements are de-tided using the AntPen tidal model near the Antarctic Peninsula, and the Circum-Antarctic tidal model elsewhere \citep{Padman:2002p1403}. 
	
	Bottom triangle velocities are routinely determined by extrapolating the referenced velocity at the deepest common level. It is only in a few areas of high near-bottom geostrophic shear, such as the deep gaps in the South Scotia Ridge \citep{NaveiraGarabato:2002p219} and the western edge of the SR4 section \citep{Thompson:2008p1207}, that the vertical gradient of geopotential anomaly at the shallowest station of a pair is extrapolated down to the depth of the deepest station of that pair, and geostrophic shear is subsequently calculated within the bottom triangle as per \cite{Thompson:2008p1207}.
	
	All diapycnal velocities are initialized to zero. The area, mean potential temperature and mean salinity of each layer interface in the interior of the Weddell gyre are extracted from the WOCE Global Hydrographic Climatology \citep{gouretskiandkoltermann04} in order to calculate the posterior diapycnal fluxes.
	
	The sea ice volume transport term is initially calculated from the 6-year mean of the daily products of sea ice concentration and velocity across each section are multiplied by the climatological sea ice thickness distribution and the array of distances between data points (see section 2.2 of the main text). We estimate  sea ice volume transport to be $18\pm9$ mSv into the box (northward) across the SR4 section, and to $16\pm8$ mSv out of the box (northward) across the ANDREX / I6S section. The corresponding salinity anomaly flux  is calculated by multiplying the volume transport by a characteristic salinity anomaly of sea ice (5 - $\overline{S}$ \citep{Eicken92}). We calculate a $0.5\pm0.2$ Sv salinity anomaly flux (or $15\pm7$ mSv freshwater equivalent) across the SR4 transect, and a $0.4\pm0.2$ Sv salinity anomaly flux (or $14\pm7$ mSv freshwater equivalent) across the ANDREX / I6S section. The potential temperature transport associated with the sea ice motion is the sum of the latent and sensible potential temperature fluxes due to sea ice production and advection, and is initially set to $5\pm3$ TW and $5\pm3$ TW across the ANDREX / I6S and SR4 sections, respectively.
	
	The addition of meteoric water due to precipitation and glacial melt is initialised by reference to the regional atmospheric reanalysis of \cite{Lenaerts:2012vd}, who estimate the net annual precipitation to be approximately $400\pm200$ mm yr$^{-1}$ in our model domain and $200\pm100$ mm yr$^{-1}$ south of the SR4 section.  We expect the bulk of the glacial runoff entering the Weddell gyre to do so to the south of the SR4 section, where the largest ice shelves and glaciers are located. Accordingly, we estimate the glacial runoff to be $10\pm5$ mSv \citep{Rignot:2008p994}.

	The set of eddy transports is initialized by applying a Reynolds decomposition to the time-dependent model layer thickness (h), potential temperature, salinity and cross-section velocity (v) at the ANDREX / I6S section location in SOSE. The resulting eddy fluxes conform to our general expectations. For example, eddy-induced transports of volume are modest ($<1$ Sv), both in individual model layers and full-depth. The full-depth eddy potential temperature flux ($4\pm2$ TW) is directed into the gyre, and occurs primarily in association with the inflow of relatively warm CDW across the gyre's eastern rim. Since these initial eddy transport values may be affected by the thermohaline biases in SOSE, we ascribe sizeable prior uncertainties of $50\%$ of the initial values to all eddy transport variables.
	
		\subsection{Weighting and solution method}
	
	We express the set of equations (\ref{conservation}) as	
\begin{equation}\label{matrix_form}
 E\mathbf{x} + \mathbf{n} = \mathbf{y},
 \end{equation}
where $E$ is the matrix of conservation statements, $\mathbf{x} = [b_{\imath}, \omega_{C}^{\ast}, F_{\jmath}^{A-S}(C),  F_{\jmath}^{SI}(C), \nu_{\jmath}(C)]$ groups the unknowns, $\mathbf{y}$ contains the observation-based prior imbalances in the conservation equations, and $\mathbf{n}$ is the noise term, which amalgamates the prior uncertainties in each of the unknowns and conservation statements. The model is underdetermined, having a total of 238 unknowns (175 barotropic velocities; 27 diapycnal velocities; 30 eddy flux terms; 2 sea ice transport terms, across ANDREX/I6S and SR4; 2 air-sea heat flux terms, one in the box and one south of SR4; and 2 air-sea freshwater terms, one in the box and one south of SR4) and only 40 equations (conservation of mass, heat and salt in 10 isopycnal layers and full depth, plus full-depth conservation of mass and salt across ANDREX/I6S, SR4 and within the ACC). There are a variety of methods at hand to solve underdetermined problems. We chose to solve  Eq. \ref{matrix_form} using singular value decomposition \citep{wunsch96}.
	
	Row and column weighting are applied to the model (Eqn. \ref{matrix_form}) in order to weight constraints and unknowns, respectively. The weighting procedure reflects our understanding of the uncertainties in the conservation statements and in our prior estimates of the unknowns (for example, our expectation about the barotropic velocity in the middle of the abyssal plain compared to
velocity in the western boundary current). Whereas the choice of weights in general least square minimization problems is left up to the investigator�s judgment, we chose to formulate the column weighting of the system in a form that makes the methods essentially equivalent to the Gauss-Markov statistical estimation \citep{wunsch96}. The inverse of the square root of the prior solution covariance matrix ($\mathbf{W^c} = 1/\sqrt{R_{xx}}$), is commonly taken as diagonal in the absence of prior knowledge of its structure \citep{wunsch96}. Following the same approach, the row weighting is chosen to be the inverse of the square root of the model covariance matrix ($\mathbf{W^{r}}= 1/\sqrt{R_{nn}}$) nn and is taken as diagonal.
	
	The prior covariance matrices are defined as follow. For the column weighting, the barotropic velocities ($b_{\imath}$) prior uncertainties reflect the temporal variability of the flow across the model box's boundaries. We compute a time series of barotropic velocity at each station pair by conducting a least squares fit of the geostrophic shear profile to the velocity profile in SOSE (essentially mimicking the referencing procedure followed with the observations). The rms values of the time series of barotropic velocity were adopted as the prior uncertainties in those variables. The diapycnal velocities ($\omega_{C}^{\ast}$) are weighted by the square root of their prior uncertainty normalised by the area of the pertinent layer interface \citep{mcintoshandrintoul97}, where in the absence of independent estimates of the $\omega_{C}^{\ast}$ terms their prior uncertainties  are set to a relatively large value ($1\times10^{-6}$ m s$^{-1}$). The same {\it ad hoc} strategy of assigning ample prior uncertainties (of 50\% to 100\% of their initial estimates) was followed for the exchange of volume and heat between the ocean and the overlying atmosphere and glacial ice, the sea ice volume and heat transports, and the eddy transport terms.
	
	For the row weighting, we follow the recommendations of \cite{Ganachaud:2003p28} and ascribe the largest (smallest) uncertainties  to the shallowest (deepest) model layers. For volume conservation equations, the resulting weights are expressed as  $\mathbf{W^{r}_{mm}} = 1/\epsilon_{m}$, where $m$ is the row number and $\epsilon_{m}$ is the prior uncertainty in the relevant volume conservation equation (Table 2 in the main article). For conservation of heat and salt, prior uncertainties are estimated as double the product of the prior uncertainties in volume conservation and the standard deviation of each variable (potential temperature or salinity) within the pertinent layer, yielding $\mathbf{W^{r}_{mm}} = 1/2\eta^{c}_{m}\epsilon_{m}$ with $\eta^{c}$ as the standard deviation of property $c$. A final element of row weighting is the normalisation of each weight by the cross-sectional area of the relevant layer. This ensures that differences in thickness between the various model layers do not influence the relative extent to which conservation is enforced.

		\subsection{Closure of the model box}

	A difficulty in defining the model domain is the absence of land boundaries between the segments of the ANDREX section conducted during cruises JC30 and JR239 (between stations 66 and 67), and between the ANDREX and I6S sections (station 90 and 91). The former case does not pose a major problem, as the JC30 / JR239 junction is located within the Weddell gyre, where mesoscale activity is weak. This is confirmed by comparison of co-located hydrographic profiles from the two cruises (not shown), which reveals quasi-identical hydrographic properties and geostrophic shear. Thus, we conclude that simply joining the JC30 and JR239 lines at their geographical intersection does not result in significant transport bias.

	In contrast, the junction between the ANDREX and I6S sections calls for a careful examination, as it is located within an ACC sector of elevated mesoscale activity. We calculate the dynamic height at 500 dbar referenced to 3000 dbar ($\Phi_{3000}$), which has been shown to accurately track the ACC frontal positions \citep{NaveiraGarabato:2009p301}, for the I6S section and for three stations along that line that were re-occupied during JC30 (Figure \ref{junction_station}). Despite being at the exact same geographical location, the I6S $\Phi_{3000}$ values (8 - 9 m$^{2}$ s$^{-2}$) are significantly higher than the JC30 values (7.5 m$^{2}$ s$^{-2}$), suggesting the presence in I6S of waters originating from farther north. Maps of altimetric SSH (not shown) confirm the occurrence of a southward meander of the Polar Front during I6S, which is absent from JC30. We thus surmise that choosing one of the three repeat stations as the junction between the ANDREX and I6S sections would incur dynamical inconsistency. In order to close the model domain in a consistent manner, we construct an I6S junction station (corresponding to station 91 on Figure \ref{junction_station}) with $\Phi_{3000}$ matching the value of the station 90 in the ANDREX section (Figure \ref{junction_station}), by linearly interpolating between the two I6S stations bounding that $\Phi_{3000}$ value. While there is a sizeable geographical gap between this synthetic station and the first station of the ANDREX section (Figure 1 of the paper), these two stations sample the same water mass (as evidenced in vertical profiles of various water mass properties, see Figure \ref{junction_station}), so there is no transport between them.
			  	

	\subsection{Posterior uncertainties}
	
	This specific choice of weights described in section 2c allows us to use a statistical interpretation of the inverse problem, which has the advantage of providing an error estimate for the solution based on the Gauss-Markov theorem. The Gauss-Markov posterior coveriance matrix is given by $\mathbf{P} = \mathbf{R_{xx}}-\mathbf{R_{xx}}E^{T}(E\mathbf{R_{xx}}E^{T}-\mathbf{R_{nn}}$ (see \cite{wunsch96} for a full derivation). The posterior covariance matrix are an updated version of the prior covariance matrices $R_{xx}$ and $R_{nn}$, which takes into account the introduction of new information represented by the vector $\mathbf{b}$ and include the null space (the part of the solution unresolved by the equations) \citep{mcintoshandrintoul97}.
	
	The Gauss-Markov framework provides an estimate of the error for each unknown or equation independently from the other unknowns or equations due to the diagonal structure of the \textit{a priori} covariance matrices. However, the assumed statistical independence of these errors leads to gross overestimation of the uncertainties in the transports across specific regions or water masses because the integral of these errors is itself a random process. In order to address this issue, we estimate posterior uncertainties in the unknowns by using an empirical approach based on their formal uncertainties similar to a Monte Carlo approach. For each unknown ($\mathbf{x}$ in Eq. \ref{matrix_form}), we generate 1000 random \textit{a priori} estimates, assuming a Gaussian distribution centered around the inverse model solution and a standard deviation equal to the posterior formal error (Figure \ref{montecarlo} shows the distribution of the 1000 batrotropic velocities as well as the inverted solution). We generate another 1000 \textit{a priori} solutions in which only initial barotropic velocities are being modified (keeping all the other unknowns constant). These 2000 initial solutions represent the impact of local uncorrelated noise (apart from the spurious correlations generated by the noise) and we run them through the inverse model in order to close the mass balance. This provides 2000 inverse solutions from which we derive our uncertainties in the transport estimates. For example, we calculate the cumulative transport across the ANDREX/I6S  and SR4 sections in the 1000 solutions and use the standard deviations of these runs as the uncertainty, represented by the grey shading in the figures of the paper. The uncertainties obtained most likely represent an upperbound limit in particular because of the lack of correlation in the noise introduced simulatenously in the barotropic velocities and diapycnal fluxes. A more detailed analysis of the error budget requires a better understanding of the covariance matrices.

%\bibliographystyle{../../../../templates_journals/AGU/agu08}
%\bibliography{andrex_bib}
\begin{thebibliography}{14}
\providecommand{\natexlab}[1]{#1}
\expandafter\ifx\csname urlstyle\endcsname\relax
  \providecommand{\doi}[1]{doi:\discretionary{}{}{}#1}\else
  \providecommand{\doi}{doi:\discretionary{}{}{}\begingroup
  \urlstyle{rm}\Url}\fi

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\end{thebibliography}

\pagebreak

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%  END ARTICLE
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% 
\begin{figure}[t]
 %\noindent\includegraphics[width=35pc]{fs01.eps}
 \caption{Closing the model box between the ANDREX and I6s sections: a) dynamic height referenced to 3000 m along the I6S line (black line) and the ANDREX JC30 section (grey line). The lower values of dynamic height during cruise JC30 suggest that the ACC fronts had migrated northward relative to their positions during the I6S cruise. Vertical profiles of: b) potential temperature, c) salinity, d) oxygen and e) neutral density for the junction station labeled "stat 91" on Figure \ref{junction_station}a.}\label{junction_station}
 \end{figure}
 
\begin{figure}[t]
% \noindent\includegraphics[width=35pc]{fs02.eps}
 \caption{Initial barotropic velocities for the 1000 Montecarlo runs (black lines) and the final barotropic velocities diagnosed by the inverse model (red curve) for the ANDREX/I6S (left) and SR4 (right) sections. The bathymetry is shown as geographic reference.}\label{montecarlo}
 \end{figure}
%  
%\begin{table}[t]
%\caption{Model layer definitions and characteristics. The mean and standard deviation of the $\theta$ and S of each layer are quoted, and water mass equivalences indicated (SW: Surface Water; WW: Winter Water; CDW: Circumpolar Deep Water; WSDW: Weddell Sea Deep Water; WSBW: Weddell Sea Bottom Water).}\label{table3}
%\begin{center}
%\begin{tabular}{|c|c|c|c|c|c|}
%\hline
%$\gamma_{n}$ limits & Layer no. & Uncertainty &$\overline{\theta}\pm std(\theta)$  & $\overline{S}\pm std(S)$   & Water mass \\ 
% (kg m$^{-3}$) & &  (Sv) & $(^{\circ}$C) &    &\\ \hline
%Sea surface					       &  	       &                  & &  &	  \\ 
%	 					       &  1	       & 4               & -0.3995 $\pm$ 0.7190 &  33.9468 $\pm$0.1670& SW	  \\ 
%27.70					       &  	       &                    & &  	  & \\ \hline
%27.70					       &  	       &    & & & 	  \\
%	 					       & 2 	       & 4 & -0.6888  $\pm$ 0.8942& 34.3846$\pm$0.0678& WW          \\ 
%28.00					       &  	       & & & & 	  \\ \hline
%28.00					       &  	       & & & & 	  \\ 
% 	 					       & 3  	       & 3 &-0.1810 $\pm$ 0.8928& 34.5684$\pm$0.0693&     	 \\ 
% 28.10					       &              & & &  &	  \\
% 	        					&  4 	       & 3 &  0.3815 $\pm$ 0.3098&34.6663 $\pm$0.0301& CDW 	 \\
% 28.20  	        				       &  	       & & & &  	  \\
% 	        					       &  5 	       & 2 &0.1718 $\pm$ 0.2128 & 34.6732$\pm$0.0203& 	 \\
% 28.26   	        				       & 	       & & & & 	  \\ \hline
% 28.26   	        				       & 	       & & & &  	  \\
%	        					       & 6 	       & 2 &  -0.0919 $\pm$ 0.0672& 34.6679$\pm$0.0059& 	 \\
%28.31   	        				       &  	       & & & & 	  \\
%	       					       & 7 	       & 2 & -0.3310 $\pm$0.0785 & 34.6606$\pm$0.0056&	 \\
% 28.35  	        				       &  	       & & &  &  	  \\
% 	        					       &  8 	       & 2 & -0.4793  $\pm$0.0824 & 34.6564$\pm$0.0044& WSDW 	 \\
%28.37   	        				       &  	       & & & & 	  \\
%	        					       &  9 	       & 2& -0.5917 $\pm$ 0.0830& 34.6529$\pm$0.0044&	 \\
%28.40   	        				       &  	       & & &  &	  \\ \hline
%28.40   	        				       &  	       & & &  &	  \\
%       						       & 10  	       & 1& -0.7309  $\pm$ 0.0594& 34.6478$\pm$0.0051& WSBW	 \\
%Sea floor	        				       &               & & & &  	 \\ \hline
%Full-depth                                                 &     11      & 1 & -0.1377 $\pm$ 0.3301 & 34.6027$\pm$0.0903& \\\hline
%\end{tabular}
%\end{center}
%\end{table}   
%   

 \end{document}
