We use the “box model” formalism, where the box is a region of ocean completely enclosed on its sides by coastline and/or hydrographic sections. The box includes the seabed as its impermeable base and the sea surface as its permeable lid. The sides of our Arctic Ocean box are comprised of the four ocean gateways–Bering, Davis, and Fram Straits, and the BSO, and the intervening coasts of the islands of Greenland and Svalbard, and of the continents of Eurasia and North America. There is one extremely small gap in this continuous boundary, Fury and Hecla Strait, between Baffin Island and the Canadian mainland, which will be described further in section 3.5. The permeable lid accommodates the air-sea heat flux, and the freshwater fluxes resulting from evaporation, precipitation and river/meltwater runoff. The ocean is assumed to be in quasi-steady state, and in hydrostatic and geostrophic balance; it is assumed to be mass- and salinity-conserving. By use of a closed box and the application of mass and salinity conservation constraints, our inverse model generates perturbations–within a priori uncertainties–to the initial horizontal and vertical velocities that are consistent with the conservation conditions.
2.1. Conservation Conditions
 We first examine mass balance. A mass imbalance in an enclosed region might be generated (for example) by a transient wind event near the region's boundary. In order to estimate the adjustment timescales over which stationarity (in this case, mass balance) may be assumed, we need to calculate relevant barotropic wave propagation speeds. Kelvin and gravity waves are fast, with phase speeds , where g is acceleration due to gravity and H is water depth. For g = 10 m s−2 and H ∼ 1 km, consequent speeds of 100 m s−1 result in waves which travel 1,000 km (scale distance for the Arctic) in ∼3 h. Barotropic Rossby waves are slower, with an upper bound on phase speed c given by
where β is the meridional gradient of the Coriolis parameter f such that β = 2Ωcosθ/RE, Ω is the rotation rate of the Earth, RE is the radius of the Earth and θ latitude, and f0 = 2Ωsinθ. For 80°N, β ∼ 5 × 10−12 m−1 s−1, f0 ∼ 1.5 × 10−4 s−1, and as above gH ∼ 104 m2 s−2, yielding a phase speed c ∼ 2 m s−1, and a timescale to travel 1,000 km of ∼6 days [see, e.g., Gill, 1983]. Mass conservation is allowed as long as we do not use an unfeasibly short timescale.
 We express the Arctic Ocean mass balance as follows. For an enclosed volume of ocean, the net rate of addition (or removal) of FW at the sea surface by all processes is denoted by F. Assuming conservation of mass (volume), and denoting net ocean flux through the sides of the volume by V0
where v = v(x, z) is the distribution of ocean velocity normal to the sides of the volume, xis the along-side distance coordinate,z is the vertical (depth) coordinate, area A represents the side area and dA = dxdz is an area element.
 We now derive the calculation of surface FW flux F from volume and salinity fluxes, assuming volume and salinity conservations. The net import (or export) of salinity (S) through the ocean boundaries of the enclosed region is equal to the rate of change of salinity storage inside the region, assuming no significant surface or seabed pathways for salinity addition or removal
where d(vol) is a volume element. Now we decompose v and S around the boundary into means (overbar) and deviations from means (prime)
Using the above to expand (3), we have
since the cross-terms (the means multiplied by the integrals of the anomalies) are identically zero. Applying(2) to (6) and rearranging, we have
The first term on the right-hand side is similar to the conventional expression for the estimation of FW flux, but the term implicitly performing the role of “reference salinity” is a clearly defined quantity: the boundary-mean salinity. The second term is “storage,” the rate of change of internal salinity (scaled by boundary-mean salinity), which we callFstor. For the application of the inverse model, we assume Fstor = 0. This assumption is examined in section 3.5.
 Finally, we consider the possibility of some limited application of heat conservation, given that monthly/annual mean air-sea heat fluxes are not well known. Not wishing to constrain by heat conservation the circulation of any part of the ocean that may be in contact with the atmosphere in any part of the year, we inspect the (near-) surface distribution of density in winter (Figure 1), when surface densities are at a maximum and contact with the atmosphere has (ultimately) its deepest influence on the ocean. Figure 1 shows the winter potential density distribution at 10 m depth based on the Polar Science Centre Hydrographic T/S Climatology (PHC); [Steele et al., 2001]. The densest surface outcrop is in the Barents Sea, where maximum densities reach σ0 ∼ 27.97 kg m−3. Denser waters have been observed in the Barents Sea: Schauer et al. [2002a] reported bottom water in the vicinity of St. Anna Trough as dense as 28.05 kg m−3. Although of high density, these waters mainly ventilate layers of lower density as a consequence of turbulent mixing on exiting the Barents Sea. They contribute to the sub-surface Atlantic Water (AW) layer in the central Arctic Ocean; the core of this layer resides at depths around 500–700 m [e.g.,Carmack, 2000], and the 0°C isotherm is found at depths ∼800 m [Carmack et al., 1997]. Therefore we will assume that potential temperature (heat flux) constraints can be applied to below the density σ1.0 > 32.750 kg m−3 (almost equivalent to 28.035 σ0). This isopycnal surface corresponds to depths greater than ∼1,000 m and relevant only, therefore, to Fram Strait and a tiny part of the Storfjordrenna between Bear Island and Svalbard in the BSO.
2.2. Data and Model Output
 The inverse model domain is a single box bounded by CTD observations in four major gateways enclosing the Arctic Ocean: Davis, Fram, and Bering Straits and the BSO (see Figure 2). The data used in this study comprise 131 finely spaced hydrographic stations, and 16 GCM grid cells in the BSO which function as CTD stations in regions of absent data. The CTD data were obtained as follows: 16 stations during 5–10 September 2005 in Davis Strait [Lee et al., 2004]; 74 stations during 16 August to 9 September 2005 in Fram Strait [Fahrbach and Lemke, 2005]; 29 stations during 9–14 August 2005 in the BSO [Skagseth et al., 2008]; and 12 stations on 21 August 2005 in Bering Strait [Woodgate et al., 2008]. A total of 131 stations were collected within 32 days, between 9 August and 10 September 2005. They span an oceanic distance of 1803 km, comprising 1464 km of measurements supplemented by 340 km of GCM grid points. The total (vertical) section area is 1,050 km2, of which 1,024 km2 is covered by measurements and 26 km2 by the GCM. The (horizontal) surface area of ocean enclosed by the sections is 11.3 × 1012 m2 [see Jakobsson, 2002], with allowance for different definitions of Baffin Bay.
Figure 2. Bathymetric configuration in Davis, Fram, and Bering Straits and the Barents Sea Opening (BSO), showing CTD stations (red cross), OGCM model grid points (green cross), mooring locations (blue diamond), and station numbers (including model grid points). Bathymetric contour intervals (CI) are shown for each strait; the CI for the Arctic figure is 1000 m.
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 Velocity data from 31 moorings deployed in these straits are used to initialize the reference velocities. The distribution of these moorings is: 4 in western Fram Strait (Norwegian Polar Institute, Tromsø) [de Steur et al., 2009]; 12 in central and eastern Fram Strait (Alfred Wegener Institute, Bremerhaven) [Schauer et al., 2008]; 5 in the BSO (Institute for Marine Research, Bergen) [Skagseth et al., 2008]; 2 in Bering Strait (University of Washington) [Woodgate et al., 2006]. In Davis Strait, weekly averaged optimal interpolation (OI) velocity fields based on temperature, salinity and velocity observations from 8 moorings are used [see Curry et al., 2011]. Over Belgica Bank in the western Fram Strait where moored observations are lacking, vessel-mounted ADCP (VMADCP) data are used [Fahrbach and Lemke, 2005]. VMADCP data are collected from the same cruise as the CTD observations in Fram Strait. The timing of the CTD observations and the locations of the moorings around the Arctic boundary are shown in Figure 3.
Figure 3. Observational periods of CTD stations and moored current meters in each strait (Davis, Fram, Bering Straits and the BSO). The height of each figure is scaled to the width of each strait. Crosses show the timing of CTD observations in 2005; dotted lines show the observational periods of moored current meters; solid lines show the 3-week averaging period used to obtain initial estimate of reference velocities.
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 Model output is used to fill small gaps in the BSO where observations are lacking. The model is an implementation of the Nucleus for European Modeling of the Ocean (NEMO) coupled ice-ocean GCM at NOC, Southampton [Barnier et al., 2006]. The model's global mean spatial resolution is 0.25° but the tripolar grid increases the local (Arctic) resolution to 20 km. Model output is available between January 1958 and December 2007 every 5 days. In this study, we analyze temperature, salinity and velocity output in summer 2005. The hydrographic data gaps on the BSO section lie south of Svalbard, north of Bear Island (74.3°N), and near the Norwegian coast (Figure 2), where 16 NEMO grid points act as substitute CTD stations. We also employ NEMO velocity output as initial velocity estimates where hydrographic data are present but in situ velocity data are absent: for north of Bear Island and south of 71.5°N in the BSO. NEMO has been used successfully in several high-latitude northern-hemisphere analyses, for example: in the Arctic Ocean and Nordic Seas, concerning FW fluxes east and west of Greenland [Lique et al., 2009, 2010; Marsh et al., 2010]; concerning Arctic primary production [Popova et al., 2010]; and concerning North Atlantic ocean heat fluxes [Grist et al., 2010].
 The NEMO output of temperature and salinity in the relevant areas of the BSO are compared with available historical CTD data from the World Ocean Database 2009 [Boyer et al., 2009] and Hydrobase 2 [Curry, 2001], which includes the Barents and Kara Seas Oceanographic Database [Golubev et al., 2000]. All CTD data since 1995 during July–September near the NEMO grid are used (Figure 4). In the Norwegian coastal region, 5 historical CTD profiles are available within 19.0–21.0°E, 70.0–70.4°N. NEMO salinity (34.5–34.7) is systematically higher than CTD data by ∼0.5, while NEMO temperatures lie within the range (6–10°C) of the CTD data [see also Skagseth et al., 2011]. This positive salinity bias in the NEMO representation in the Norwegian Coastal Current (NCC) results from an over-saline simulated outflow from the Skagerrak compared with the hydrography [e.g.,Røed and Albretsen, 2007]. Therefore, the 3 NEMO salinity profiles in this region are corrected by subtracting 0.5. To the north of Bear Island, 50 historical CTD profiles are available within 18.5–19.5°E, 74.5–75.6°N. NEMO salinities are slightly higher than the historical observations but still fall broadly within the envelope of observed values, while NEMO temperatures show no systematic difference from the CTD data; so no corrections are introduced. To the south of Svalbard, 8 historical CTD profiles are available within 18.0–19.0°E, 76.7–77.6°N. NEMO salinities and temperatures all fall within the range of observations, so no corrections are introduced.
Figure 4. (a) T-S plots of available historical CTD data (green) and NEMO output of temperature and salinity (red) in the Norwegian coastal region along the BSO. Profile locations are shown in the map, which is inserted in the T-S diagram. Bathymetry is also shown with contour interval of 200 m. (b) Same as Figure 4a but for north of Bear Island. (c) Same as Figure 4a but for south of Svalbard.
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 NEMO velocity output in the BSO is compared with the small number of available velocity observations. In the Norwegian coastal region, Skagseth et al. shows the seasonal cycle of velocity during July 2007 to July 2008 observed by bottom-mounted upward-looking ADCPs at 71.1°N, 24.0°E. The NEMO bottom velocity (6 cm s−1) agrees well with their summer mean velocity at 172–188 m (6.6 cm s−1). Between Bear Island and Svalbard, 13 hydrographic and VMADCP sections were occupied during July 1997 to November 1999 [O'Dwyer et al., 2001]. NEMO velocities in this region in the upper 200 m are compared with their results [O'Dwyer et al., 2001, Figure 4]. NEMO velocities associated with the topographic recirculation in Storfjordrenna (3–5 cm s−1) are generally weaker than measured (8–12 cm s−1), but as O'Dwyer et al.  observe, the region is shallow and transports are small, so no adjustment is introduced to bottom velocities between Bear Island and Svalbard. The impact of introducing NEMO model output to the inverse model is examined in section 3.5.
 The circum-Arctic distributions of potential temperature and salinity are shown inFigure 5. Their (area-weighted) mean values are 1.159°C and 34.662, respectively, where the latter includes the mobile sea ice area with salinity of 6 (seesection 2.5). The sub-division of the water column employs density criteria based onRudels et al. ; six layers are selected, which we name Surface, Subsurface, Upper AW, AW, Intermediate Water and Deep Water (Figure 5 and Table 1). Table 1also includes the further sub-division of these layers for use in the inverse model, which is described insection 2.3; also included in Table 1 are conventional central Arctic Ocean water masses, for reference.
Figure 5. (top) Potential temperature section and (middle) salinity section along Davis and Fram Straits the BSO and Bering Strait; bold black lines show defined water mass boundaries, and the color bar scale is nonlinear. (bottom) The distribution of defined water masses and layer boundaries along the section. These corresponding densities are labeled: 26.0 σ0, 27.1 σ0, 27.5 σ0, 30.28 σ0.5, and 32.75 σ1.0. The pressure axis is expanded between 0 and 50 dbar and 50 and 500 dbar and station numbers are shown along the base of each plot.
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Table 1. Definitions of Model Layers, Observed Water Masses, and Common Water Masses in the Central Arctic Ocean
|Layer||Upper Interface||Lower Interface||Layer Group||Central Arctica||Davis||Fram||BSO||Bering|
|1||Surface||24.700 σ0||Surface Water||MLW, UHW||WGSW, SBICW||PSW, PSWw||—||ACCW, sPacW|
|2||24.700 σ0||25.500 σ0|| || || || || || |
|3||25.500 σ0||26.000 σ0|| || || || || || |
|4||26.000 σ0||27.000 σ0||Subsurface Water||UHW, LHW||WGSW, ArcW||PSW, PSWw||NCCW, ESCW||sPacW|
|5||27.000 σ0||27.100 σ0|| || || || || || |
|6||27.100 σ0||27.300 σ0||Upper AW||LHW||WGIW||PSW, PSWs, AW||AW||—|
|7||27.300 σ0||27.500 σ0|| || || || || || |
|8||27.500 σ0||27.700 σ0||AW||AW, ASW, PIW||TrW||AW||AW||—|
|9||27.700 σ0||30.280 σ0.5|| || || || || || |
|10||30.280 σ0.5||30.320 σ0.5||Intermediate Water||UIW, LIW||—||AIW||BSW, BrSW||—|
|11||30.320 σ0.5||32.750 σ1.0|| || || || || || |
|12||32.750 σ1.0||35.126 σ1.5||Deep Water||ADW||—||DW||BSW, BrSW||—|
|13||35.126 σ1.5||35.142 σ1.5|| || || || || || |
|14||35.142 σ1.5||37.457 σ2.0|| || || || || || |
|15||37.457 σ2.0||Bottom|| || || || || || |
 Since water masses are generally defined using potential temperature and salinity classes, there is generally no unique relationship between water masses and the layers defined by density. Also, our naming of layers is admittedly imperfect: for example, the Subsurface and Upper AW layers can be found at the surface, particularly in Fram Strait and the BSO; and the AW layer can include water other than AW. Nevertheless, these are simple and useful categories that capture dominant features. The water masses occupying the circum-Arctic section are next briefly described, for each part of the section, and are illustrated on theθ-S plot (Figure 6).
Figure 6. The θ-S plot from all sections; the color code is as follows: Davis Strait is red, Fram Strait is green, BSO is blue, and Bering Strait is cyan. Defined major water mass divisions (density contours) are in black. These corresponding densities are 26.0σ0, 27.1 σ0, 27.5 σ0, 30.28 σ0.5, and 32.75 σ1.0. Some conventional water masses (MLW, UHW, LHW, and AW) and water mass layers (Mixed Layer, Halocline Water, and Arctic Intermediate Water) in the central Arctic are shown based on Aksenov et al. [2010, Table 2].
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 Bering Strait is shallow (∼50 m depth); it is occupied by fresh (<33), warm (2–8°C) summer Pacific Water (sPacW), and the Alaskan Coastal Current Water (ACCW) is seen on the east side of the strait as warmer (>6°C), fresher (<32) water [Steele et al., 2004; Woodgate and Aagaard, 2005]. sPacW occurs in both the Surface and Subsurface layers, while the ACCW is found only in the Surface layer.
 In the BSO, warm (3–7°C) and saline (35.0–35.2) AW appears in the middle of the section as a dominant feature, and is found in both Upper AW and AW layers. Warmer (<8°C) and fresher (34–35) Norwegian Coastal Current Water (NCCW) is present in the southern part of the section in both NEMO grid points and CTD stations, and it appears in the Subsurface layer. Barents Sea Water (BSW) occupies the deepest parts to the south of Bear Island, and is in the Intermediate Water layer. Brine-enriched shelf water (BrSW) at the freezing point temperature appears in the northern part of Storfjordrenna between Bear Island and Svalbard in the Intermediate Water and Deep Water Layers. Finally, the cold, fresh and well-stratified waters between Svalbard and Bear Island appear to lack a conventional name, so we refer to this as East Spitzbergen Current Water (ESCW), and it occupies three layers (Subsurface, Upper AW, AW) [seeLoeng, 1991; Sætre, 1999; Furevik, 2001; Schauer et al., 2002a; Fer et al., 2003; Ingvaldsen et al., 2004].
 In Fram Strait, very fresh Polar Surface Waters (PSW; 28–34.5) near the freezing point of seawater appear over Belgica Bank and in the East Greenland Current (EGC); the more saline fraction (∼30–34.5) of these waters resemble Arctic Halocline water, and occupy the Surface and Subsurface layers. The Upper AW layer is very thin; beneath it over Belgica Bank is a water mass in the AW layer that resembles Lower Halocline water. In the EGC, the AW layer contains modes of recirculated and/or returned AW which have arrived there either via the “short circuit,” recirculating close north of Fram Strait, or via the longer circuit around the Eurasian Basin. Slightly colder (−0.5 to 2°C) and fresher (∼34.85) water is found in the Intermediate layer, which is likely highly modified AW that has taken the long circuit around the whole Arctic Ocean. The dominant water in the upper layers of eastern Fram Strait is warm (3–6°C) and saline (35.0–35.2) AW in the West Spitzbergen Current (WSC), which is partly overlain by a mixture of Arctic- and Atlantic-sourced waters in the Subsurface layer. The Intermediate layer here contains intermediate waters from the Nordic Sea. The central region of Fram Strait exists between the north-going warm and saline regime to the east and the south-going cold and fresh regime to the west, and as such, it represents a transition region between these two extremes, displaying evidence of transient eddies, quasi-stationary meanders, and substantial local recirculation. Water in Fram Strait deeper than ∼1,000 dbar (in the Deep Water layer) has small ranges of temperature and salinity: −0.7 ± 0.1°C and 34.90 ± 0.01, and contains a mixture of various deep waters sourced from the Nordic Seas and Arctic Ocean [seeRudels et al., 2002, 2005].
 Much of Davis Strait is occupied by relatively fresh (<34.5) water. A temperature minimum (close to the freezing temperature) appears in the western part of the strait; it corresponds to Arctic Halocline water, and is found in the Subsurface layer. Warmer water (4–5°C) exists in the eastern part of Davis Strait, possibly reflecting the Atlantic Ocean origin of the West Greenland Current; it is found in the Surface and Subsurface layers. In the Surface layer on the west side is an otherwise unnamed feature of moderate temperature and very low salinity which may be influenced by surface meltwater runoff; we refer to it as Surface Baffin Island Current Water (SBICW). The AW layer contains a relatively warm and saline water mass that has been called West Greenland Intermediate Water (WGIW), although this regional name may not accurately reflect its origins [see Tang et al., 2004; Cuny et al., 2005; Curry et al., 2011].
2.3. Inverse Model Setup
 The inverse model used in this study is formulated with 15 layers defined using isopycnal surfaces. Different reference depths are used to calculate potential density depending on the average depth of the surface. Model layers are listed in Table 2. The following constraints are applied to the inverse model: full-depth conservation of volume and salinity anomaly transport (1 constraint each); conservation of volume transport and of salinity anomaly transport for each layer; and conservation of potential temperature anomaly transport in the four deepest layers (σ1.0 > 32.750 kg m−3; see section 2.1) that do not outcrop in winter. Therefore 36 constraints in total are prescribed. Salinity anomaly and potential temperature anomaly are obtained by subtracting the mean property value around the boundary of the model domain (cf. section 2.1), which improves the conditioning of the inversion [McIntosh and Rintoul, 1997; Ganachaud, 1999]. The resulting conservation equations for transport T of volume or of some property C are of the general form
where j and m refer to station pair and layer interface indices respectively, N is the total number of station pairs, Δx is the station spacing, hm and hm+1 are the depths of the upper and lower interfaces of model layer m, C is property concentration either around the boundary (Cj) or over an interface (Cm), v is geostrophic velocity calculated from hydrography, b is the barotropic velocity, wmC is the effective interfacial velocity for each property C, and m is the layer interface area within the domain. For each layer therefore, the transport Tm of volume or of property C is the sum of the transports through the sides of the layer, and into and out of the upper and lower interfaces. McIntosh and Rintoul  and Sloyan and Rintoul showed that property-specific diapycnal velocities across each layer interface are effective parameterizations of net diapycnal fluxes in inverse models.
Table 2. A Priori Uncertainties in Volume Conservation and the Mean and Standard Deviation of Potential Temperature and of Salinity Anomaly in Each Layer
|Model Layer||Uncertainty (Sv)||Potential Temperature (θ, °C)||Salinity Anomaly|
|1||4.0||0.716 ± 3.052||−4.097 ± 0.311|
|2||4.0||0.637 ± 2.859||−3.155 ± 0.403|
|3||4.0||1.650 ± 2.546||−2.356 ± 0.250|
|4||4.0||0.822 ± 3.887||−1.331 ± 0.605|
|5||4.0||2.176 ± 3.633||−0.688 ± 0.421|
|6||3.0||3.050 ± 3.453||−0.443 ± 0.426|
|7||3.0||2.426 ± 3.070||−0.265 ± 0.379|
|8||2.0||3.486 ± 2.612||0.081 ± 0.331|
|9||2.0||3.396 ± 1.381||0.368 ± 0.208|
|10||1.0||1.976 ± 0.622||0.325 ± 0.093|
|11||1.0||0.497 ± 0.617||0.259 ± 0.052|
|12||0.5||−0.511 ± 0.196||0.245 ± 0.020|
|13||0.5||−0.783 ± 0.120||0.254 ± 0.015|
|14||0.5||−0.896 ± 0.105||0.257 ± 0.016|
|15||0.5||−0.922 ± 0.131||0.263 ± 0.012|
|Full depth||1.0||1.160 ± 2.673||0.017 ± 0.975|
 The first layer is a special case because it receives in the model the net FW input and accommodates the Fram Strait sea ice flux. The model layer 1 transports, T′1, are
where the ice is treated as a rectangular plate, Cice is the relevant property concentration, δh is sea ice thickness, δx is the sea ice width, and u is the sea ice advection speed. The surface input of FW is represented as a transport by the product of a scale area Asurf to represent the Arctic Ocean surface area and is set to 107 km2, and a velocity parameter q, which is to be determined.
 The depth-independent adjustment to the relative velocitybj provides 143 unknowns, one for each station pair j. The model includes diapycnal velocities in the ocean interior for each of the 14 layer interfaces for volume and salinity, which provides 28 unknowns, and 4 unknowns are set for potential temperature between the four deepest (non-outcropping) layers. Since potential temperature anomaly conservation is not required for the remaining layers, no diapycnal potential temperature anomaly velocities are derived between them. Sea ice advection velocityu in Fram Strait (1 unknown) and surface FW input q (1 unknown) are included in layer 1 and in the full depth volume and salt anomaly equations. Therefore the model comprises a total of 177 unknowns.
 As is conventional, these equations are represented in matrix form
where A is M × N and contains information about properties and geometry, x is N × 1 and contains unknown barotropic, diapycnal, sea ice and FW velocities, and d is M × 1 and contains initial estimates of transports; M is the number of conservation equations (36), and N is the number of unknowns (177).
2.4. Weighting and Uncertainties
 Row and column weighting are applied to the model before inversion to weight constraints and unknowns (respectively), using the row weighting matrix W (M × M) and column weighting matrix E (N × N)
The weighted system of equations A′x′ = d′ is then solved using singular value decomposition [Wunsch, 1996] with A′ = WAE, x′ = E−1x and d′ = Wd. W and E contain only diagonal components. For volume conservation,
where εm is the a priori volume transport uncertainty for each layer, and for property transports,
where ηmC is the standard deviation of property variations within the relevant layer. All uncertainties and standard deviations are listed in Table 2. The factor 2 in (13) is set according to Ganachaud and Wunsch ; it is an ad hoc best guess to account for possible correlations between the section averaged and mesoscale components of the noise in property conservation equations. The weighting term for full-depth salinity anomaly transport conservation is set 4 times larger than(13)to account for the higher standard deviation of full-depth salinity anomaly (0.98) compared with either the AW layer (0.21–0.33) or the IW layer (0.05–0.09).
 Column weighting employs a priori uncertainties for all unknowns: for barotropic, diapycnal, sea ice and surface FW velocities (δb, δw, δu and δq),
Aj means station pair area for reference velocity, j layer interface area for diapycnal velocity, Aicemobile sea ice cross-sectional area in Fram Strait. is section mean salinity as before. Use of station pair area and layer interface area is normal for the column norms for the reference and diapycnal velocities (respectively). However, for the sea ice advection term, the salinity anomaly of sea ice is ∼30, so we employ as a representative column norm for sea ice velocity. Similarly, the surface FW input term is normalized by .
 The a priori uncertainty in the reference velocity is estimated as the standard deviation of moored velocity data over 3 months (0.02–0.05 m s−1). The uncertainties are linearly interpolated onto each station pair as appropriate. In the case of Belgica Bank and in the BSO, larger a priori uncertainties are provided (0.06 m s−1) where direct measurements are lacking. Smaller a priori uncertainties (0.02 m s−1) are provided for Bering Strait to take account of the observation that the flux “first guesses” are similar to the estimation of Woodgate et al. , which is based on long-term sustained observations. The a priori uncertainty in the diapycnal velocities is set as 1 × 10−5 m s−1, near the upper end of the range of vertical velocities inferred from observed ocean mixing rates. The a priori uncertainty in the sea ice advection velocity (u) is set to 50% magnitude of its initial estimate. The a priori uncertainty of the total surface FW flux velocity parameter (q) is set to 50% magnitude of its initial estimate.
 The a posteriori uncertainties are calculated as the square root of the diagonal component of the error covariance matrix P, which is estimated using the Gauss-Markov formalism [Wunsch, 1996]
2.5. Inverse Model Velocity Initialization
 The initial state of the model must be specified. At the position of each station pair, the reference velocity is initialized from the deepest available moored velocity measurement, from the VMADCP data over Belgica Bank, or from NEMO model grid cells in parts of the BSO. The cross-sectional moored velocity components are averaged over 3 weeks in order to eliminate higher frequency variability. The moorings are spaced more widely than the stations, so average velocities are then linearly interpolated onto station pair locations.
 Ideally, the 3 week averaging period would center on the hydrographic observations, but in practice the period depends on data availability because the time of the hydrographic observation is also the time when moorings were recovered and replaced. Therefore the averaging period is selected to be as close as possible to the date of the hydrographic observations near each mooring (Figure 3). In Davis Strait, the averaging period is 8–29 August 2005, just before the hydrographic observations (5–10 September). In western Fram Strait, the 3 week averaging period spans 4–27 September depending on the data availability of each mooring, just after the hydrographic observation. In eastern Fram Strait, the 3 week averaging period is just before the hydrographic observations, spanning 21 July to 24 August. In the BSO, the averaging periods are 5–26 August 2005, during the hydrographic observations (9–14 August). In Bering Strait, the averaging periods are 21 August–11 September (just after hydrographic observations) and 10–30 August (during hydrographic observations). NEMO velocities are averaged over 20 days because the output is recorded as 5-day means. The averaging period in the BSO is 3–23 August, which sits in the middle of the hydrographic observations.
 All diapycnal velocities are initialized to zero. The area, mean potential temperature and mean salinity of each layer interface in the interior of the Arctic are extracted from the PHC summer data set.
 In Fram Strait, the initial sea ice volume flux is set at 50.2 mSv, with salinity anomaly flux of 1.44 Sv, equivalent to 41.5 mSv FW flux. These are calculated as follows. The zonal extent of mobile sea ice is taken to lie between 12 and 3°W. There is a stationary region of fast ice between the Greenland coast and 12°W. The eastern edge of 3°W is selected as a simple version of the sea ice thickness parameterization of Kwok . The mean sea ice thickness of 1.8 m is as observed by upward-looking sonar (ULS) in August 2005 at 5°W (E. Hansen, Thinning of Arctic multiyear and ridged sea ice 1990–2010, manuscript in preparation, 2012). Sea ice salinity is set to 6. Summer 2005 is a difficult time to estimate sea ice volume flux. This year featured widespread change in thickness composition [Kwok, 2007; Nghiem et al., 2007] and a peak in the Fram Strait sea ice export [Kwok, 2007]. We focus on August and September (AS) 2005 and estimate its sea ice volume flux during the period when the hydrographic observations were conducted. Kwok  estimated sea ice volume flux of four summer months (JJAS) during 1991–1999 as 30.6 ± 7.8 mSv with sea ice area flux of 115 ± 20 × 103 km2 (over JJAS) based on the sea level pressure gradient across Fram Strait, and a thickness parameterization based on ULS sea ice thickness observations [Vinje et al., 1998]. Based on the pressure gradient, Kwok  also estimated sea ice area fluxes of 140 × 103 km2 over 4 months (JJAS) during 2000–2006 and 250 × 103 km2 over 4 months (JJAS) in 2005, with a large area flux estimation of 180 × 103 km2 over 2 months, AS 2005. Hansen (manuscript in preparation, 2012) has shown significant sea ice thickness reduction from the 1990s of 3.3 ± 0.5 m to the 2000s of 2.2 ± 0.6 m, including 1.8 m in August 2005 based on ULS observations. Combining the Kwok  sea ice area flux estimate and the Hansen (manuscript in preparation, 2012) sea ice thickness measurement, the sea ice volume flux estimate for the four summer (JJAS) months in 2000–2006 is 23.9 mSv, for the four summer (JJAS) months 2005 is 34.9 mSv, and for AS 2005 is 50.2 mSv. We employ 50.2 mSv as the initial sea ice volume flux estimate for our inversion, which requires therefore a mean advection velocity of 0.15 m s−1.
 Figure 7 shows initial volume, salinity anomaly and potential temperature anomaly transport imbalances for each layer. The net initial imbalances are: volume, 5.22 Sv deficit; salinity anomaly, 0.51 Sv excess (equivalent to 15 mSv FW deficit); and potential temperature anomaly, 44.3°C × Sv.
Figure 7. Initial imbalances for (left) volume transport, (middle) salinity anomaly transport, and (right) potential temperature (θ) anomaly transport for each model layer. The total initial imbalances for these parameters are shown beneath each figure.
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