For our analysis of the Arctic overturning circulation, we use output from a global NEMO ocean hindcast simulation ORCA0083-N06;. The simulation is based on NEMO-LIM2 with a horizontal resolution of 1/12°, which is approximately 3–5 km in the Arctic, and 75 unevenly spaced depth layers. The model is forced with atmospheric and river runoff data from the DRAKKAR forcing set 5.2 and was run from 1958 to 2015. To prevent excessive drifts in salinity, sea-surface salinity is relaxed towards climatology . We consider output from 1979 to 2015 in this study. We make use of the monthly mean velocity and tracer output.

ORCA0083-N06 has previously been used to study the Arctic Ocean, and the upper ocean circulation, sea surface height and mixed-layer depth has been found to match well with observations . Furthermore, NEMO has been extensively validated in the Arctic Ocean, and previous and similar versions have found to generally perform well . Nevertheless, we evaluate the model against observational estimates of net volume transports at the gateways in Table . Acknowledging that the observations cover different periods than the model, the net volume transports at the Arctic gateways are generally consistent with observational estimates, although ORCA0083-N06 overestimates the net outflow through Fram Strait and underestimates the outflow through Davis Strait compared to the most recent estimates. However, the inflow of Atlantic water through Fram Strait (T>2 °C, S>34.7) for the period 1997–2015 is 3.0 Sv, close to the estimate of 3.0±0.2 Sv by . The inflow of Atlantic Water (T>3 °C) through the Barents Sea Opening between 75–73.5°N is 2.4 Sv, consistent with estimates from mooring data of 2.1±1.0 Sv . The temperature and depth of the Atlantic Water core in the Arctic Ocean is similar to estimates from the PHC3.0 climatology , but with a cold bias in the Makarov Basin (Fig. ). We conduct a more thorough evaluation of the Arctic overturning streamfunction and Atlantic Water pathways below, and show that the hindcast is able to realistically simulate both.

To analyze the pathways, timescales, and locations of water mass transformation of the Arctic overturning circulation in the ORCA0083-N06 hindcast, we trace waters entering the Arctic Ocean through the four gateways using TRACMASS version 7.0 . TRACMASS solves the trajectories of parcels through each model grid cell analytically by assuming that the velocity field varies linearly in time (between output time steps) and space (between opposite grid cell walls). An important property of the algorithm is that it conserves mass, or in the case of NEMO, volume, such that each trajectory retains a constant volume transport throughout the tracking. This makes it possible to calculate Lagrangian streamfunctions, and thereby decompose the total Eulerian Arctic overturning streamfunction into contributions from different pathways . We do not add any stochastic diffusion to the trajectories as this would break volume conservation, and instead follow the resolved advective pathways in the model.

For the Lagrangian tracing experiment, we calculate the average annual cycle over 1979–2015. We start trajectories at all grid cells across the four Arctic gateways every month. To obtain a sufficient resolution, we assign each trajectory a maximum volume transport of 2500 m3s-1. If the volume transport in a model grid cell exceeds this value, multiple trajectories are started in that grid cell. In total, around 150 000 trajectories are started. We track the trajectories forward in time using the climatological monthly velocity and tracer fields. We chose to use climatology as input to represent an average picture of the Arctic overturning circulation. We have tested using monthly and 5-daily time-evolving input and looping the period 1979–2015 instead, and the main results did not qualitatively change (not shown). Trajectories are terminated either if they exit the Arctic by returning to the inflow gateways, or after 500 years. We calculate Lagrangian streamfunctions following . To quantify water mass transformation along the pathways, we calculate the Lagrangian mass, heat and salt divergence for each grid box, following and .

We will first quantify the time-mean water mass transformation for the Arctic Ocean, with a particular focus on the overturning circulation. We start by considering the overturning streamfunction in density (σ) space (Fig. a), obtained by integrating M(σ*) over σ. The overturning streamfunction has a local minimum and a local maximum, indicating the presence of both a negative and a positive overturning cell, which are referred to as the estuarine and thermal overturning cells, respectively . The estuarine cell has a minimum of -0.96Sv at σ0=27.0kgm-3, while the thermal cell reaches a maximum of 3.09Sv at σ0=27.95kgm-3. The Arctic Ocean thus densifies inflowing waters at a rate approximately three times greater than it lightens them. Compared to the observation-based estimate of , the thermal cell is slightly stronger but agrees well in density structure, whereas the estuarine cell is weaker and peaks at a lower density, as well as extending through a much lower density level of approximately σ0=25.0kgm-3. Finally, find a third cell, which represents the transformation of Pacific Waters to a denser outflow. This third cell is likely obscured in the hindcast by the large amounts of export of waters between σ0=25.0 kgm-3 and σ0=26.5 kgm-3. Overall, however, there is good agreement between ORCA0083-N06 and that inferred from available observations.

The volume tendency dVσdt in the Arctic Ocean is generally small, except for σ0>27.75–28.0 kgm-3, where there is net water mass formation (Fig. b), and σ0>28.0–28.1 kgm-3, where there is net water mass destruction. This hints at a long term trend of buoyancy gain in the Arctic Ocean deep waters. The source of this buoyancy gain is not investigated here, but we note that it is consistent with an observed warming of deep waters in Fram Strait . The diapycnal transformation due to surface buoyancy fluxes, Fσsfc, is directed toward higher densities across nearly all density classes and reaches a maximum of -2.32 Sv at σ0=27.75kgm-3 (Fig. a), peaking at a lower density than the overturning. Surface forcing contributes to the destruction of water masses lighter than σ0<27.75kgm-3, and to the production of water masses in the range σ0=27.75–28.1kgm-3. Notably, it contributes substantially to the formation of dense waters around σ0=28.0kgm-3, which are exported from the Arctic as part of the dense outflow, suggesting that surface buoyancy forcing is most important for the thermal cell. We find that cooling as a result of both air–sea heat loss and ice melt dominates the surface-forced densification signal (Fig. ). The heat-flux component drives densification at a rate of approximately -3.7 Sv at the peak of Fσsfc. This is nearly three times larger in magnitude than the freshwater contribution, which acts to lighten dense waters at a rate of approximately 1.4 Sv.

The residual transformation Fσres (interpreted as mixing) shows a more complex structure (Fig. b). At lower densities, specifically below σ0=27.7kgm-3, mixing drives water mass formation with the characteristics of lighter surface polar waters, supplying the majority of the waters exported as part of the estuarine circulation cell. Mixing then acts destructively over a broad intermediate density range, with pronounced water mass destruction between σ0=27.7–27.95kgm-3 (Fig. b), corresponding to a part of the Atlantic Water inflow. The lighter fraction of this destruction contributes to the estuarine cell by mixing Atlantic Water upward into fresher surface layers, while the denser fraction, along with the destruction observed at the upper end of the density range for σ0>28.05kgm-3, leads to a net convergence of transformation around σ0=28.0–28.05kgm-3. This convergence corresponds to the majority of dense water mass exports, suggesting that while surface fluxes can provide much of the required densification to sustain the Arctic overturning circulation, preconditioning the inflowing waters, mixing sets the final properties of the dense waters that are exported from the Arctic Ocean through its deeper branches, analogous to the findings of for the formation of North Atlantic Deep Water.

While the water mass transformation in density space offers general insights into the transformations required to sustain the Arctic overturning circulation, a diagnosis of the water mass transformation in temperature-salinity (T–S) space permits a more detailed, process based analysis . The time-mean volume distribution of the Arctic Ocean T–S as well as some of the key water masses are visualized in Fig. . In this section, as before, we place a special focus on the thermal branch of the Arctic overturning circulation and the water mass transformation processes that support it. We will start by considering the time-mean overturning divergence MSΘ (Fig. , colors) and the water mass transformation GSΘ (Fig. , vectors) required to support it. The use of thermohaline coordinates reveals the presence of the Pacific cell , where relatively warm and fresh Bering Strait inflow (Θ=-2–7 °C and S=32–33 gkg-1), undergoes a predominantly cooling transformation shown by the vectors in Fig. . This cell's presence is obscured in σ-space (Fig. ).

We first turn our focus to the estuarine circulation cell, which governs the net freshwater export from the Arctic Ocean . In contrast to the comparatively small Pacific cell, the estuarine cell exhibits a more persistent structure in both σ space and T–S space. In T–S space, however, it is characterized by two distinct branches (visible as two blue curves converging on the freezing line in Fig. ), corresponding to different Arctic outflow pathways. Both branches export Polar Waters, defined here by near-freezing temperatures and relatively low salinities (S=31–34 gkg-1). While their exported properties are broadly similar, the branches differ in their source waters and transformation pathways.

The warmer branch originates primarily from Atlantic Water inflows (S>34.5 gkg-1, Θ>5°C) entering through the Barents Sea Opening and Davis Strait (not shown). In T–S space, the pathway of this branch reflects progressive modification of warm, saline Atlantic inflows through cooling and dilution by fresher surface waters, sea-ice melt, and runoff. This results in outflowing Polar Waters of approximately S≈31–33 gkg-1. The colder branch is fed predominantly by denser and cooler Atlantic Waters (S>35 gkg-1, Θ>0°C). These waters undergo substantial modification through mixing with fresher local waters as well as freshwater inputs from ice melt, producing an outflow that likewise emerges as Polar Water but with slightly higher maximum salinities (S≈34 gkg-1).

Next, we consider the thermal overturning cell, which occupies a relatively narrow density range (σ=27.5–28.2), yet dominates the Arctic's volume transport. This cell is primarily driven by the inflow of warm, saline Atlantic waters (Θ>2 °C and S>35 gkg-1), which separate into two distinct branches in T–S space: the denser, cooler, and slightly fresher Fram Strait Branch (Θ=2–5 °C) and the lighter, warmer (Θ>4 °C), and saltier Barents Sea Branch. The water mass transformation, GSΘ, highlights the transformation from inflow to outflow properties (Fig. , vectors). While both branches undergo net cooling with slight freshening, the warmer waters entering as part of the Barents Sea Branch experience the most substantial transformation, ultimately converging toward the cold, dense Fram Strait outflow (-1<Θ<1°C, 35.0<S<35.1g kg-1). This outflow forms the primary deep limb of the Arctic overturning circulation.

As the thermal cell dominates volume transports and concerns mainly inflowing Atlantic Water and densified outflows of similar salinity, it is instructive for a process-based analysis to limit our focus to the specific transformation processes in the range of T and S values corresponding to the thermal overturning cell. To that end, we consider the components of the decomposed WMT: the surface forced WMT FSΘsfc and the residual WMT FSΘres (Eq. , Fig. ). The decomposition reveals that in the time-mean, the Atlantic Water inflowing as part of the Barents Sea Branch (S>35 gkg-1, Θ>4 °C), undergoes substantial cooling due to air-sea heat fluxes. Surface cooling is the dominant process until they reach approximately 2.5 °C, below which ice melt becomes increasingly important, indicated by the transformation vectors bending towards lower salinities due to the addition of freshwater (Fig. a). This transformation results in the highly effective destruction of large volumes of warm inflow waters, consistent with a large surface forced diapycnal transformation in Fig. . As such, the Barents Sea expectedly dominates the surface forced component. Conversely, the Fram Strait Branch is cooled to a lesser extent, though both branches ultimately contribute to the formation of substantial volumes of dense waters (σ>27.75kgm-3) spanning a broad range of T–S characteristics. We further see both strong formation and destruction of waters near the freezing point, which is related to the seasonal melting and freezing cycle. Finally, surface forcing is responsible for the production of the densest waters found in the Arctic: the highly saline, near freezing point Barents Sea dense waters (S>35.1 gkg-1, σ>28.1kgm-3) .

The residual component, on the other hand, is much more clearly associated with a net freshening of the inflowing salty Atlantic Water (Fig. b). Here we interpret the residual component as the contribution from ocean mixing, but make no explicit distinction between vertical mixing or horizontal mixing. However, the orientation of the thermohaline transformation vectors with respect to the isopycnals drawn in Fig. b leaves clues as to the nature of the relevant mixing processes. Transformation vectors aligned parallel to isopycnals imply purely isopycnal mixing, while those oriented perpendicular to isopycnals are indicative of diapycnal mixing.

Firstly, mixing leads to a seasonal transformation near the freezing point that opposes the effect of surface forcing. These opposing effects largely cancel, resulting in little net impact of much of the melting and freezing cycle on the Arctic WMT. Meanwhile, both inflow branches experience freshening, but there is more destruction of waters that have properties associated with the Fram Strait Branch. Notably, there is substantial net destruction of waters with temperature characteristics (Θ=1.5–3 °C) corresponding to Fram Strait inflow waters. While these waters undergo both isopycnal and diapycnal modification, the distinct isopycnal component suggests that isopycnal mixing plays an important role in modulating the Fram Strait Branch properties. Parts of the inflowing Fram Strait Branch waters are only slightly modified (cooled and freshened) to produce the mixing convergence at S>35 gkg-1, Θ=1.5–2 °C, which corresponds to the outflow on the same thermohaline properties in Fig. . We identify this as recirculating Atlantic Water, flowing westwards relatively quickly after entering the Arctic Ocean. These Atlantic Waters are likely mixed with colder and fresher halocline waters or shelf waters near the shelf break . In agreement with observations , the curvature of the transformation arrows in T–S space towards higher densities additionally suggests the possibility of cabbeling as a relevant transformation and densification process for the Fram Strait Atlantic Water.

Figure  furthermore shows the destruction through mixing of a cold and salty water mass (S=34.85–35.1 gkg-1, Θ<-0.5 °C). A regional decomposition (not shown) shows this water mass is seasonally produced during winter in the Barents Sea (see also Fig. ). Vectors indicate that these cold and salty Barents Sea waters mix with Atlantic Water originating from Fram Strait. Being denser than the Fram Strait inflow, these waters likely contribute to the additional densification required for the transformation of Atlantic Water to the outflow density. Finally, mixing leads to a general convergence around the dense outflow waters (Θ=-1–0 °C, S≈35.1 gkg-1) that ultimately exit through Fram Strait. This convergence requires contributions from both diapycnal and isopycnal mixing processes. Thus, surface forcing preconditions the inflowing Atlantic Water, generating large volumes of water near the outflow density, particularly from Barents Sea inflows. However, mixing ultimately sets the final T–S properties of the dense outflow by homogenizing waters from the two inflow branches. Importantly, the relevant mixing processes are largely obscured by considering only the WMT in density space.

After analyzing the Arctic overturning in density and T–S space, we next analyze the contributions of the different inflows at the gateways and the pathways of waters between the gateways. First, we decompose the total Eulerian overturning streamfunction into the net contribution from each of the four gateways (Fig. a). Most of the lighter waters (σ0<27 kgm-3) enter the Arctic through Bering Strait and the Barents Sea, and exit the Arctic through Davis Strait and Fram Strait. More lighter waters exit Fram Strait in the hindcast than in the estimates from . Most of the denser, Atlantic Waters (27.4kgm-3<σ0<27.8kgm-3) enter the Arctic through the Barents Sea and Fram Strait, and the Dense Waters (σ0>27.95kgm-3) leave through Fram Strait. The net dense overturning across the Barents Sea Opening and Fram Strait is in good agreement with the estimates from , showing that the dense overturning and outflow of densified waters is dominated by Fram Strait.

The analysis in T–S space indicated that both the Fram Strait and the Barents Sea branch contribute to the dense overturning across Fram Strait. To more accurately quantify the relative importance of the two branches, and along which pathways the waters are transformed, we use Lagrangian trajectories (Sect. 2.3), and explicitly track waters entering the four gateways. We calculate a Lagrangrian streamfunction in a similar way to the Eulerian streamfunction by binning the volume transport as trajectories enter the Arctic (inflow) and as they exit the Arctic (outflow) into density bins (Δσ0=0.01 kgm-3), calculating the net transport for each density bin, and integrating over density. The total Lagrangian streamfunction of all trajectories entering the Arctic (Fig. b) is very similar to the Eulerian streamfunction (Fig. a). Note that the Lagrangian streamfunctions are closed, since the trajectories conserve their volume transport along the way. We then decompose the streamfunction into the different inflow pathways by selecting only the trajectories that start at the different gateways. Dense overturning is dominated by the Barents Sea Branch (e.g. water entering the Barents Sea and exiting Fram Strait), while the Fram Strait Branch has a major contribution at very high densities (σ0>27.8 kgm-3; Fig. b). Comparing the values of the streamfunctions at the density of maximum overturning (27.95 kgm-3), approximately 60 % of Dense Water produced in the Arctic Ocean originates from the Barents Sea, and approximately 40 % originates from Fram Strait itself. Furthermore, most of the water in the estuarine cell (σ0<27.5 kgm-3) originates from the Barents Sea and Davis Strait, which has mixed with fresher Arctic origin waters. Fram Strait waters also contribute to the estuarine cell, but at higher densities (σ0>27 kgm-3) and therefore mostly to the colder estuarine branch found in Fig. . Finally, the Pacific overturning cell identified by and in Fig.  emerges in the Lagrangian decomposition, but is hidden in the total overturning streamfunction.

The pathways of trajectories entering the Arctic at the two Atlantic Water gateways are shown in Fig. c and d. Approximately 60 % of the water entering Fram Strait recirculates and quickly flows back south. The rest largely follows the Atlantic Water Boundary Current (AWBC) eastwards along the rim of the Eurasian Basin (as indicated in Fig. ). Waters originating in the Barents Sea mostly enter the Arctic Ocean at St. Anna trough, where they merge with the AWBC. Further into the Arctic, north of the Laptev Sea, part of the Atlantic Water turns north and follows either the Lomonosov Ridge or the Gakkel Ridge, thereby recirculating in the Eurasian basin, whereas another part continues along the AWBC into the Amerasian Basin. These pathways are consistent with existing studies of the pathways of Atlantic Water in the Arctic e.g..

Next, we briefly analyze the time scales of the Arctic overturning circulation. We calculate the Lagrangian streamfuncion as before, but now decompose it into different transit times, based on how long the trajectories take to exit the Arctic (Fig. ). Waters densifying and recirculating north of Fram Strait within less than 10 years contribute approximately 10 % to the total maximum overturning. Waters recirculating in the Eurasian basin and following either the Lomonosov or Gakkel ridge back to Fram Strait take 10–20 years and contribute another 25 % to the total overturning. This time scale is consistent with tracer studies of intermediate Atlantic Water circulation in the Eurasian basin . Approximately 65 % of the total Arctic overturning is made up of water that takes longer than 20 years to exit the Arctic Ocean. In addition to trajectories circulating in the Eurasian basin for several decades, this also includes trajectories entering the Amerasian Basin and circumnavigating the entire deep Arctic basin. This indicates that most Atlantic Water will take multiple decades to exit Fram Strait as Dense Water and contribute to the lower limb of the AMOC.

The Arctic estuarine cell (σ0=27.0kgm-3) has much shorter time scales, and most of the lighter waters exit the Arctic after 10–20 years. Here, the shortest pathway is the transpolar drift, a surface circulation feature that connects the Siberian Arctic to Fram Strait (not shown).

The results in Sect. 3.1 show that the transformation of Atlantic Waters into Dense Waters is dominated by surface cooling (Fig. ). Mixing is driving the transformation into lighter polar waters and also plays a role in producing very dense overflow waters. Here, we remap the diapycnal water mass transformation from density space into geographical space to understand the importance of the two main pathways of Atlantic Water in the Arctic. First, we remap the surface-forced water mass transformation into geographical space. Second, we calculate where transformation occurs along the pathways of the water entering the Arctic Ocean via Fram Strait and the Barents Sea, using the Lagrangian trajectories. Third, to distinguish between the surface forced and the internal mixing component, we compare this total (Lagrangian) transformation with the (Eulerian) surface transformation, thereby combining the Lagrangian and Eulerian approaches.

We start with the surface forced water mass transformation (Fig. ). The strongest transformation occurs in the Barents Sea and northwest of Svalbard, along the main inflow pathways of the Atlantic Water through Fram Strait and the Barents Sea Opening . Most of the transformation in these areas occurs through surface cooling and through interaction with sea-ice melt and freeze. Negative surface transformation through sea-ice melt is largest north of Fram Strait. Further into the Arctic, the surface forcing is small, as those areas are permanently sea-ice covered and the warm Atlantic Waters are sheltered from the surface by the cold halocline .

Next, we look at the total water mass transformation in the Lagrangian experiment. For a given grid cell, we sum up the density change for all trajectories that pass through that grid cell, obtaining spatial maps of density divergence, or mass flux (Fig. ). For water entering via Fram Strait, a large part of the water mass transformation occurs close to Fram Strait (Fig. a). Waters flowing into the Arctic immediately northwest of Svalbard gain density due to surface cooling (Fig. ), as these pathways are partly ice-free in winter . Waters leaving the Arctic on the western side of Fram Strait also gain density on their way, while waters flowing north along the shelf break around the Yermak Plateau lose density. Because northwestern Fram Strait is usually ice-covered, this density change is likely through sea-ice melt (Fig. ) and interaction between the inflowing Atlantic Waters and the outflowing Polar Waters or modified and recirculating Atlantic Waters. Mixing of these water masses cools and freshens the inflowing Atlantic Waters, and warms and salinifies the outflowing waters (Figs.  and ). Overall, the salinity change dominates the density changes, such that Atlantic Water is becoming lighter and the outflowing waters become denser.

Further downstream, the Fram Strait branch waters slowly become less dense as they transit through the Arctic. Notable exceptions are found along the AWBC in the Nansen Basin, where waters gain density north of Franz Joseph Land and east of St. Anna trough. Those locations coincide with troughs in the shelf break in the northern Barents and Kara seas, where the Fram Strait branch interacts with waters originating in the Barents Sea , and where the mass divergence shows a density loss for Barents branch waters (Fig. b). This indicates that the interaction between the two Atlantic Water branches transforms some of the Fram Strait branch waters into denser waters, and contributes to the dense water formation along the Fram Strait pathway, consistent with results in Sect. 3.2. Analysis of the heat and salinity change shows that the heat exchange between the two branches dominates the density signal in these regions (Figs. b, , and ).

Waters entering the Barents Sea undergo a strong density increase in the Barents Sea as they are cooled (Fig. b), consistent with Fig. . Beyond the Barents Sea, waters decrease their density along the Siberian shelves, most strongly close to the outflows of Siberian rivers, where they mix with the fresh riverine water, decreasing their salinity (Fig. ). Water that becomes lighter will mostly follow the transpolar drift and eventually exit the Arctic with a lower density, contributing to the estuarine cell. Before exiting the Arctic, just north of Fram Strait, those waters increase their density again, most likely by mixing with the inflowing and recirculating Atlantic Waters in the central Fram Strait, which, as shown above, experience a decrease in density (Fig. a). Lastly, waters from both branches decrease their density in the boundary current north of Alaska (Fig. a and b). Here, they mostly interact with fresher Pacific Waters originating from Bering Strait that enter the Amerasian Basin from the Chukchi Shelf, or that circulate in the Beaufort Gyre, and ventilate the Arctic halocline .

Next, we focus on the water mass transformation that contributes to the dense overturning circulation by only selecting trajectories that contribute to net overturning at σ0=27.95 kgm-3 (the density of maximum overturning; Fig. c and d). Apart from the Barents Sea and the areas north of Svalbard, most locations of transformation for both branches are now over the Atlantic Water Boundary Current (AWBC). Waters originating from Fram Strait increase their density along most of their path along the AWBC, and notably north of Franz Joseph Land and St. Anna trough, where they, as discussed, likely interact with Barents Sea waters. Waters originating from the Barents Sea increase in density within the Barents Sea, and decrease in density downstream where they interact with the Fram Strait branch.

Further downstream in the Laptev Sea, Barents Sea origin waters in the boundary current increase their density (Fig. d). The Laptev shelf is a known area of high sea-ice production and formation of dense shelf waters , which could mix with the Atlantic Waters in the AWBC and increase their density. An analysis of dense water production over the Laptev shelf in ORCA0083-N06 confirms that there are dense waters produced, although slightly lighter than the density of maximum Arctic overturning (not shown). Another source for the density increase in this region is Barents and Kara Sea origin waters that flow through Vilkitsky Strait, gain density on the shelf, and then mix with the waters in the AWBC (Fig. d; ).

As a last step, we combine the surface forced transformation in Fig.  and the total Lagrangian transformation in Fig. c and d to produce a rough estimate of the relative contributions of surface forcing and internal mixing in driving water mass transformation at the density of maximum overturning. We do so by determining the regions with substantial surface forcing (more than +1×10-6 kgs-1m-2; Fig. ). The areas meeting this criterion, e.g. the Barents Sea and the region northwest of Svalbard, are indicated in Fig. c and d (hatched areas). Assuming that water in those regions will predominantly transform through surface forcing, we sum the transformation from Fig. c and d inside the regions and compare them to the transformation outside the regions, which we then assume to be due to internal mixing. Based on this calculation, we estimate that for the dense overturning cell, approximately 15 % of total transformation for the Fram Strait branch is surface-forced, and approximately 85 % driven by internal mixing. For the Barents Sea branch, approximately 85 % of the transformation is surface-driven, and only approximately 15 % is through internal mixing. These rough estimates are robust to the exact choice of threshold used to define regions of surface forcing. Taking the two branches together gives an estimated total contribution of approximately 25 % from internal mixing to the dense overturning. We emphasize that this is only a rough estimate of the relative role of mixing, and that the exact value will most likely vary over time. However, our rough estimate is consistent with the earlier result that surface forcing dominates the transformation, but internal mixing plays a role at high densities (Figs.  and ).