2.3.1. Heat Budget
 From the internal energy and salt equations for a fixed volume [Gill, 1982] the equations expressing the heat and freshwater budgets of an ocean volume can be derived [Serreze et al., 2006, 2007]. The volumes considered in this study extend from the seafloor to the sea surface, and are horizontally limited by the boundaries of the subregions defined in section 2.1. It is assumed that the net flow with the major branches toward and away from the volume is captured by calculating the flow through a finite number (N) of cross-sections at the boundary of the volume (cross-sections are shown in Figure 2). Most of the cross-sections have a vertical extent from the sea surface to the seafloor, but for Sections 1 and 2, which are located in the interior of the Nordic seas and not on the open boundary, only the Atlantic Water has been considered. The flow through the Faroe Shetland Channel has been split into flow from the North Atlantic to the Nordic seas and return flow. Hence the flow of Atlantic Water (Section 3) and overflow water (Section 14) will be calculated separately. The heat budget for each individual subregion may then be written
where the bar denotes the time mean ( ≡ a dt/Δt). ρw and ρi are respectively the density of seawater and sea ice, cp the specific heat capacity of seawater, Lf the latent heat of fusion, all assumed constant, and Tf is the freezing temperature of seawater (−1.9°C). Vj and Ij are respectively the oceanic volume and sea ice transports through each cross-section (j = 1,…, N), positive toward the volume. Tj is the transport weighted average temperature value of the oceanic part of the flow through the cross-section under consideration (Section j), and is calculated from the relation
where x1 and x2 are the boundary points that define the cross-section, T and v are respectively the oceanic temperature and velocity normal to the section, and h is the thickness of the layer. The reference temperature, Tref, is arbitrary for a closed volume/mass budget and is here set equal to 0°C. N is the number of cross-sections at the boundary of the ocean volume required to cover all branches of the mean current field. is the mean surface heat flux integrated over the sea surface area, given as the sum of longwave and shortwave radiation and turbulent sensible and latent heat fluxes, defined to be positive upward.
 The smallest terms of the heat budget have been included in the mean heat budget residual, . Since an approximately steady state is assumed (Δt ∼ 10 years), the mean rate of change in heat content has been included in , together with mean ocean heat fluxes caused by molecular diffusion, mean heat fluxes due to precipitation, evaporation and runoff from land, mean heat flux due to temperature variations of sea ice, and the mean heat flux through the seafloor (e.g. thermal heating). In addition to the smallest terms of the heat budget, also the mean ocean heat flux through the part of the open boundary where the flow has not been measured has been included in , since this term cannot be calculated from the available data sets. When the flow with the major branches is captured by the N cross-sections this contribution will be due to the eddy heat flux, ρwcp, where the prime denotes time fluctuations.
2.3.2. Freshwater Budget
 The freshwater budget of the ocean volume, averaged over Δt, can be expressed
where and are respectively the mean precipitation and evaporation rates integrated over the sea surface, is the mean total runoff from land toward the ocean volume, and Sj is the transport weighted average salinity value of the mean flow through the cross-section under consideration. Sj is given as
where S is the ocean salinity. Sice is the salinity of sea ice, assumed to be constant, and Sref is the reference salinity, here set equal to 34.9. Since we in this study have simplified the mass budget to a balance between ocean and sea ice transports through the open boundaries, the value chosen for Sref in equation (3) is in principle not arbitrary. However, freshwater fluxes are not very sensitive to the choice of reference salinity. For example, increasing Sref from 34.9 to 35 will modify the freshwater flux by less than 0.3%. The residual term, Rf, includes the ocean fluxes due to molecular freshwater diffusion, the freshwater flux through the seafloor, the rate of change in depth-integrated freshwater content and the eddy freshwater flux through the part of the open boundary where the flow has not been measured.
2.3.3. Estimating the Terms of the Heat and Freshwater Budgets
 The surface heat, Fsfc, and freshwater, P − E, fluxes are calculated from the two re-analysis products presented in section 2.2, and estimates of the runoff term, R, are available from the literature (see Table 5). Transport of sea ice takes place through Sections 7, 10 and 11. Estimates of sea ice fluxes from the literature applied in this study are shown in Table 3. The estimate across Section 7 also includes the ice flux through the opening between Svalbard and Franz Josef Land. For Section 11, no direct observations of sea ice fluxes are available. The sea ice flux through the section is obtained by assuming that the melt rate of sea ice flowing through the Fram Strait is proportional to the ice flux, (d I/d y = c I) and that the sea ice flux is reduced by approximately 50% from 79°N to 73°N, as suggested by Aagaard and Carmack . This gives a sea ice transport of approximately 505 km3 yr−1 from the Nordic seas to the North Atlantic Ocean. Heat and freshwater fluxes due to sea ice are presented in Table 4.
Table 4. Mean Ocean Volume Fluxes, V (Sv; 1 Sv = 106 m3 s−1), Heat Fluxes, H (TW), and Freshwater Fluxes, FW (mSv), Through Sections 1–14 (See Figure 2), Split Into the Contribution From Mean Ocean Currents (Vo, Ho, Fo) and From Sea Ice (Vi, Hi, Fi)a
 For most of the cross-sections considered in this study oceanic heat and freshwater fluxes have been presented in the literature. Fluxes through Sections 2, 3, 5, 7 and 14 are calculated from the values and studies presented in Table 2. Also the oceanic heat fluxes through Sections 8 and 10 are calculated from the values listed in the table. Although some water flows through the eastern part of the Fram Strait in the lower layer [Schauer et al., 2004], all transport of deep water through the strait (northward and southward) has in this study been included in the volume flux through Section 10. This is because we assume that the deep circulation does not to play an active role in the heat and freshwater budgets: deep circulation through Section 8 will be due to a volume flux from the Greenland/Iceland Sea, caused by a leakage from the internal circulation in the Greenland Basin [Voet et al., 2010], and we will assume that this water will enter and leave the Norwegian Sea at the same temperature and salinity. The freshwater fluxes through the Fram Strait in liquid form follows Rabe et al. . Converted to Sref = 34.9 the liquid freshwater fluxes through Sections 8 and 10 become 2 mSv and 78 mSv, respectively (for information about the direction of the fluxes see Table 4).
 The freshwater flux through Section 6 is calculated from the values listed in Table 2, but in order to obtain a closed volume budget for the Barents Sea we have modified the volume transport through Section 6 by 0.1 Sv, from −0.7 Sv to −0.6 Sv. This value is within the range indicated by Loeng et al. . The average temperature of the flow through the Kara Strait is set equal to 0.0°C [Karcher et al., 2004]. For the Atlantic Water flowing through Section 1 we apply mean weighted average temperature and salinity values of 6.0°C and 35, as indicated by the results of Østerhus et al. .
 In the circulation scheme assumed in the present study there is no net volume transport through Section 4, but a part of the water that crosses the Greenland-Scotland ridge takes a detour into the North Sea where it mixes with fresher and colder water before it returns to the Nordic seas as the Norwegian Coastal Current. This results in non-zero heat and freshwater transports through the section. The freshwater flux of 22 mSv is taken from Aure and Østensen . To estimate the heat flux it is assumed that the temperature is reduced by approximately 1.5°C in the North Sea, as suggested by Mauritzen [1996a], whereas a flow of 1 Sv is applied [Furnes, 1980]. This gives a heat flux equal to −6 TW. For the flow of Atlantic Water through Section 9 we apply a volume flux equal to −0.5 Sv [Cisewski et al., 2003] and assume that this water has the same temperature and salinity as the Atlantic Water flowing through Section 8 (∼2.3°C, 34.9 [Schauer et al., 2004; Rabe et al., 2009]). In order to obtain mass balance for the Norwegian and Greenland/Iceland seas the same amount of water must be transported in the opposite direction. The model results of Spall  indicate that such a flow will take place in the deepest openings. As it is assumed that there is no flow of deep water through Section 8, this flow must then contribute to the transport through Sections 13 or 14. For the deep flow through Section 9 we apply the same temperature as for the deep water in the Fram Strait (∼−0.4°C [Schauer et al., 2004]), whereas a salinity of 34.9 is used [Farrelly et al., 1985]. Due to the compensating flows there will be no net flow through the cross-section.
 The flow through Section 12 consists of a relatively fresh surface flow, and transport of more saline and colder water in the deep and intermediate layers. The volume flux in the surface layer is about 0.8 Sv and the water has a salinity close to 34.7 [Jónsson, 2007]. Based on the hydrography and velocity maps shown in Jónsson  a mean temperature of 2°C for the surface flow is assumed. For the deep and intermediate part of the flow through Section 12 (∼1.7 Sv) we again apply the same temperature and salinity values as for the deep water in the Fram Strait of −0.4°C and 34.9, which appears to be in agreement with the temperature and salinity maps presented by Jónsson . For the temperature and salinity of water flowing through Section 13 we apply the same values as used for the Faroe Bank Channel Overflow (Section 14).
 In order to give an estimate of the heat and freshwater fluxes in liquid form through Section 11, where only the deep part of the flow has been extensively measured, we calculate the ocean flow field using the thermal wind equation following the method of Mork and Skagseth . In combination with temperature and salinity observations the heat and freshwater fluxes may be estimated, since we are then able to calculate the mean weighted average temperature and salinity values from equations (2) and (4). Using the thermal wind equation the geostrophic velocity component normal to the cross-section is given as
where vs is the geostrophic velocity at the sea surface (z = zs), f is the Coriolis parameter, g is the acceleration due to gravity and ∂ρ/∂x is the density gradient along the section. Estimates of the geostrophic component of the surface velocity is obtained from drifter observations, whereas the density gradient in the thermal wind balance is calculated from hydrographic observations. Due to sparse data coverage, only data from summer and autumn has been applied. In particular, observations on the shelf (shallower than 400 m depth) are from September 1991 only.
 In order to obtain mass balance for the Nordic seas, we assume an oceanic volume flux equal to −5.2 Sv through Section 11. To achieve this value for the derived volume flux vs is multiplied by a constant factor of 0.85, thus decreasing the surface velocities obtained from drifters with 15%, without changing the baroclinic part of the flow. The mean weighted average temperature and salinity of the water flowing through the Denmark Strait are then calculated from equations (2) and (4). Using this method we find that the transport weighted temperature and salinity of the flow through the Denmark Strait are equal to approximately 0.25°C and 34.12 PSU, respectively.
 The constants ρw, ρi, cp, Lf and Sice in equations (1) and (3) are set equal to respectively 1027 kg m−3, 917 kg m−3, 4000 J kg−1 ° C−1, 3.35 × 105 J kg−1 and 3 PSU. Obtained values for ocean heat (sensible and latent) and freshwater (solid and liquid) fluxes through Sections 1–14 are shown in Table 4.