As mentioned in the introduction we are interested in the response of the Baltic Sea salinity to perturbations in precipitation/runoff and boundary salinity. Moreover, in this section we restrict our attention to study this response in a statistical steady state.

Let S‾ denote a long-time averaged salinity property, for example the basin mean salinity, defined through 1S‾=1V(t2-t1)∫t1t2dt∫VsdV, where V is the volume of the basin, t1 and t2 the start and end times of the averaging period and s the salinity. To study the response of S‾ to the aforementioned perturbations we approximate S‾ using a second order Taylor polynomial 2S‾≈S‾(SB0,RP0)+∂S‾∂SB(SB-SB0)+∂S‾∂RP(RP-RP0)+12∂2S‾∂SB2(SB-SB0)2+12∂2S‾∂RP2(RP-RP0)2+∂2S‾∂SB∂RP(SB-SB0)(RP-RP0), where SB and RP denote the boundary salinity and runoff+precipitation respectively, and the 0 subscript denotes the unperturbed state (i.e. long-time averages from a historical simulation). RP is normalized with the runoff + precipitation of the unperturbed state.

The choice of a second order Taylor polynomial as opposed to a simpler first order one is motivated by the fact that it gives us a chance to quantify the interaction between the two forcing terms through the cross-derivative term. To compute the coefficients for the Taylor polynomial, six runs are needed in addition to the unperturbed hindcast giving the S‾(SB0,RP0) term. The scheme is as follows: 3∂S‾∂SB=(2h)-1(S‾(SB0+h,RP0)-S‾(SB0-h,RP0)),4∂S‾∂RP=(2k)-1(S‾(SB0,RP0+k)-S‾(SB0,RP0-k)),5∂2S‾∂SB2=h-2(S‾(SB0+h,RP0)-2S‾(SB0,RP0)+S‾(SB0-h,RP0)),6∂2S‾∂RP2=k-2(S‾(SB0,RP0+k)-2S‾(SB0,RP0)+S‾(SB0,RP0-k)),7∂2S‾∂SBdRP=(2hk)-1(S‾(SB0+h,RP0+k)-S‾(SB0+h,RP0)-S‾(SB0,RP0+k)+2S‾(SB0,RP0)-S‾(SB0-h,RP0)-S‾(SB0,RP0-k)+S‾(SB0-h,RP0-k)), where h is a salinity perturbation on the outer boundary, and k is a fractional perturbation of the precipitation and runoff. This means that in addition to the reference case S‾(SB0,RP0), the following integrations are needed: S‾(SB0+h,RP0), S‾(SB0-h,RP0), S‾(SB0,RP0+k), S‾(SB0,RP0-k), S‾(SB0+h,RP0+k) and S‾(SB0-h,RP0-k).

Note that a relative change in precipitation and runoff is not the same as a relative change in net freshwater input to a system, Qf, since a large fraction of the precipitation is lost to evaporation, making Qf/Qf0 smaller than RP. Evaporation is relatively constant between the different runs in this experiment, so the change in net freshwater input is about equal to the change in precipitation and runoff, but since Qf is smaller than precipitation + runoff, the relative change in Qf is about 28 % when the change in RP is 20 %.

Analysing transports through a transect in salinity coordinates rather than Eulerian vertical coordinates (e.g. Walin, 1977; MacCready, 2011; Burchard et al., 2018) enables quantification of water mass transformation and estuarine water exchange in a way very similar to the ideas behind the Knudsen theorem (Knudsen, 1900). The time averaged advective inflows of volume and salt mass in water with salinity larger than s can be written as 8Q(s)=∫A(s)udA‾, and 9F(s)=∫A(s)sudA‾, where A(s) at any time is the transect area with salinity larger than s, u is the horizontal velocity component normal to the transect pointing into the estuary, and overbar denotes time averaging. These can be integrated into in- and outflows through 10Qin=∫0∞max⁡0,-∂Q∂sds,11Qout=∫0∞max⁡0,∂Q∂sds,12Fin=∫0∞max⁡0,-∂F∂sds,13Fout=∫0∞max⁡0,∂F∂sds, where 14max⁡(x,y)=x,x≥yy,x<y. Now, mean inflow and outflow salinities can be calculated as 15Sin=FinQin,Sout=FoutQout. When averaged over long time relative to the residence times of salt and freshwater in the system, and changes to inflow and freshwater input, the accumulation of volume and salt are small relative to the in- and outflow transports, and a steady state volume and salt budget analysis results in the Knudsen relations (see Burchard et al., 2018, for a discussion) 16Qin=SoutSin-SoutQf,17Qout=SinSin-SoutQf, where Qf is the mean integrated freshwater input (runoff + precipitation - evaporation) to the estuary inside the investigated transect.

For two transects, A and B, where B is located closest to the open ocean, part of the outflow through section A, QRA, is recirculated and contribute to the inflow through section A, while the rest, QoutA-QRA, flows out through section B. Similarly, part of the inflow through section B, QRB, is recirculated and contribute to the outflow through section B, while the rest, QinB-QRB, contributes to the inflow though section A. The corresponding salt and volume inflows through section A can be written as 18QinASinA=(QinB-QRB)SinB+QRASoutA,19QinA=QinB-QRB+QRA+γΔQf, where ΔQf is the freshwater input between section A and B and γ is the fraction of this that enters the Baltic Sea. The recirculation fluxes can now be found as 20QRA=QinA(SinB-SinA)-γΔQfSinBSinB-SoutA,21QRB=QoutB(SoutB-SoutA)+(1-γ)ΔQfSoutASinB-SoutA. These expressions are similar to the efflux/reflux expressions first presented by Cokelet and Stewart (1985), except for the addition of local runoff. They basically express that the water masses flowing out from the region bounded by the A and B sections are a result of turbulent mixing between the water masses flowing into the region. In- and outflows, and recirculation fluxes, were calculated as described above for two transects, Fig. 1, one in northern Kattegat (section B), and one that includes the Darss- and Drogden sills at the rim of the Baltic Sea (section A). The recirculation fluxes are therefore caused by water mass modifications in Kattegat or in the Belts and the Sound. Temporal averages were based on the period 1990–2017.

Our experiments use a NEMO 4.2 configuration of the North Sea and Baltic Sea region (see Fig. 1). The configuration builds on a previous NEMO 3.6 configuration with evaluations of ocean and sea-ice parameters presented in Hordoir et al. (2019) and Pemberton et al. (2017), respectively. The horizontal resolution is 0.055° in the zonal and 0.033° in the meridional direction, which amounts to a nominal resolution of 3.7 km (2 nmi). To adapt the configuration to NEMO 4.2 we have changed the tracer advection scheme to 4th order Flux Corrected Transport (FCT) scheme, and the lateral viscosity formulation from a laplacian to a bi-laplacian operator. To improve the salinity dynamics of the model configuration we have changed the vertical coordinate system from the geopotential Z-partial steps formulation (used in Hordoir et al., 2019) to the terrain-following Multi-Envelope s-coordinate (MEs) system developed by Bruciaferri et al. (2018). The MEs coordinates are configured to use 2 envelopes. The upper envelope ranges between the surface and 250 m and uses 43 levels, and the lower envelope covers all depths below 250 m using 13 levels. We thus retain the 56 vertical levels used in Hordoir et al. (2019) but gain a higher vertical resolution in the Baltic Sea. In a separate study (Pemberton et al., 2026) we make a thorough evaluation of the impact of the vertical coordinate system on the Baltic Sea hydrography and circulation. Here we instead restrict our evaluation to compare the mean vertical salinity distribution and surface/bottom salinity evolution for the hindcast run to observations at three stations.

The hindcast run is the reference simulation for the perturbation runs described in Sect. 2.4. It is a run from 1961 to 2017 based on the best available forcing. Before performing the hindcast and perturbation runs, we spun up the model circulation by using initial conditions for salinity and temperature from the Janssen et al. (1999) climatology, with ocean currents and sea surface height set to zero. Then we ran the model three cycles repeating the same 10 years period using atmospheric, runoff and open boundary forcing from 1961–1970 so that model dynamics reached near-equlibrium level. The meteorological forcing was derived from the UERRA regional reanalysis (Dahlgren et al., 2016), which offers a spatial resolution of 11 km and a temporal resolution of 1 h for parameters such as wind, air pressure, air temperature, humidity, and both solar and long-wave downward radiation. Precipitation (rain and snow) is provided at a temporal resolution of 12 h. River runoff forcing was provided as daily values from a dedicated simulation with the Hydrological Predictions for the Environment model with the European application v.3.1.8 (E-HYPE; Donnelly et al., 2016). The open boundary forcing includes barotropic currents, sea level, nine tidal constituents, and monthly salinity and temperature data. The barotropic currents and sea level were calculated using the 2D North Atlantic Model (NOAMOD; She et al., 2007) storm-surge model. These values were adjusted for baroclinic influences by incorporating monthly sea level data from the European Centre for Medium-Range Weather Forecasts' Ocean Reanalysis System 4 (ORAS4), which enhances the description of the North Sea ocean circulation. The salinity and temperature profiles are monthly mean values interpolated from the ORAS4 configuration (Balmaseda et al., 2013).

To illustrate the model performance, observations were taken from the Swedish archive for oceanographic data (SHARK, https://www.smhi.se, last access: 12 January 2026), at three selected stations in the Baltic Sea, located at different distances from the entrance (see Fig. 1). Vertical salinity profiles from the three stations (Fig. 2) show that the salinity stratification and the range of variability are well described by the model at the three stations. The temporal variability in the surface water and in the bottom water (Fig. 3) show that surface water salinity is well represented by the model both with respect to short-term variability and long-term trends. Note that since the observational stations are often located near the deepest point in a basin, the bottom water comparison was done somewhat above the bottom to make the time series representative for the water volume below the halocline rather than for just a small local bottom water volume. Both model and observations show a decrease in surface water salinity between 1980 and 2000. Before 1980 the model surface salinities are slightly larger than the observed salinities, which may be due to an adjustment from too high initial salinities. Near-bottom water model salinities also show similar temporal variability as in observations with a long period between 1980 and 1993 without strong inflows of saline water. Generally, the model salinities compare as well with observations as those of the better models in the Baltic Sea Model Intercomparison Project (Gröger et al., 2022) and the model is therefore well suited for the sensitivity study which is the focus of this work.

To investigate the salinity dynamics of the Baltic Sea in response to variations in freshwater input and boundary salinity using a Taylor expansion approach (Sect. 2.1), we performed a series of model simulations in which these two factors were systematically perturbed over the period from 1961 to 2017, see Table 1. In these runs, either boundary salinities were maintained the same as in the reference run while freshwater input was varied by ±20 % relative to the reference scenario (runs RP+ and RP-), or boundary salinities (meaning the salinities at the open boundaries in the Channel and between the Orkney Islands and Norway, Fig. 1) were altered by ±0.5 psu compared to the reference experiment while runoff and precipitation was maintained equal to those of the reference run (runs SB+ and SB-). These choices for size of perturbations were somewhat arbitrarily chosen to yield a span containing the long-term variability and expected effects of climate change. For river runoff, 20 % is in the upper end of the expected increase in runoff by 2100 (e.g. Saraiva et al., 2019). For boundary salinity 0.5 psu represents rather well the expected variability in changes at the outer boundaries of the North Sea (e.g. Quante and Colijn, 2016). However, the variability in projected salinity changes between different GCMs is rather in the range of 1–2 psu, so in hindsight we could have chosen a larger perturbation value.

The fractional changes in freshwater input were applied using NEMO namelist options for runoff and precipitation multipliers. To capture interaction terms, two experiments were performed (RP+SB+ and RP-SB-). Finally, three additional runs were carried out to evaluate the polynomial's validity outside the bounds of the runs used to determine the polynomial. These involved; a 50 % increase in freshwater input and a 1.0 psu rise in boundary salinity from the reference run (RP++SB++), a 10 % increase in freshwater input and 0.6 psu rise (RP+10SB+06) and 0.6 psu decrease (RP+10SB-06) in boundary salinity.

The time series of yearly mean salinities averaged over the whole Baltic Sea volume inside the sill transect (Fig. 1) are shown in Fig. 4. It is clearly seen that the salinities show a large sensitivity towards freshwater forcing, with a difference from the hindcast run of about 2 psu. When freshwater forcing increases (red lines), the salinities decrease relative to the hindcast (black solid line), and when the freshwater forcing decreases (blue lines), the salinities increase. The green lines are the additional runs (runs 7–9 in Table 1) and the solid green line corresponds to the most extreme case with 50 % increase in runoff and precipitation. The changes caused by changing boundary salinity (dashed and dotted lines) are much smaller, maybe due to the choice of perturbation values. The adjustment to new forcing is seen to be largest during the first 20–30 years and then level out relative to the hindcast run. In the following, the period 1990–2017 is therefore chosen to represent the “steady state” conditions where the runs have adjusted to the changed forcing.

Figure 5 shows maps of depth averaged salinity changes relative to the hindcast (CTL) run, averaged over the period 1990–2017. For the precipitation and evaporation change (RP+ and RP-) runs, the salinity changes are larger inside the Baltic Sea than outside and rather homogeneous, however gradually somewhat decreasing as we move north. The changes for these runs are almost equal but of opposite sign, except for the Gulf of Bothnia where the salinity decrease is smaller for increasing runoff than the salinity increase for decreasing runoff. In Kattegat, changes are mainly seen in the shallow areas towards the Danish coast and close to the Swedish coast, and in Skagerrak and the North Sea, changes are mainly concentrated in the coastal areas.

The boundary salinity changes (SB+ and SB-) manifest themselves rather uniformly in the North Sea, Skagerrak and Kattegat. Inside the Baltic Sea, the changes are much smaller but still distributed rather uniformly.

For combinations of the perturbations, the changes caused by runoff and precipitation dominate in the Baltic Sea, Kattegat and along the coasts of Skagerrak and the North Sea, while the boundary salinity changes dominate in the open parts of the North Sea and Skagerrak.

Figure 6, shows the second order Taylor polynomial for S‾. The coefficients of the polynomial are given in Table 2. It is quite clear from the figure that the response to perturbations is close to linear in h (boundary salinity change), and nonlinear in k (runoff and freshwater input change). The latter is seen through the increasing distance between level curves as k increases. The interaction term, the cross derivative, is not very important. It is negative (Table 2), meaning that for positive changes in RP and SB, the salinity decrease will be slightly larger with than without the interaction term. The change is, however, small, which can easiest be seen by comparing polynomial plots with and without (not shown) the interaction term. These two plots are very similar. It is also clear from Fig. 4 that boundary salinity changes are damped in the basin. The first derivative of S‾ with respect to SB is 0.36, indicating that a salinity change of 1 psu at the boundary gives a change in S‾ of 0.36 psu in the Baltic Sea.

The interpolation and extrapolation qualities of the polynomial are shown in Fig. 7. The scatter plot shows modeled S‾ against S‾ derived from the Taylor polynomial. The point with the largest deviation is from to the S‾(SB0+2h,RP0+2.5k) run (RP++SB++) which is a large extrapolation. Even so the quality is rather good. Note also that only the point S‾(SB0,RP0) is exactly equal for the polynomial and the model, but almost all the other points also fall on a near perfect line, indicating very good interpolation properties. Looking beyond S‾ one can also estimate Taylor polynomials for other quantities based on the present experiment, for example surface salinities, bottom salinities, or higher percentiles of the Baltic Sea salinity distribution. As an example, the Taylor polynomial for the 95th percentile of the Baltic Sea spatial salinity distribution gives a larger response to boundary salinity change than the mean salinity, with first derivative of the 95 % percentile with respect to SB being 0.57 where it is 0.36 for S‾. However, qualitatively, polynomials for different percentiles are relatively similar.

The inflow functions, Q and F, for volume and salt, defined in Eqs. (8) and (9), and averaged over the period 1990 to 2017, are shown in Figs. 8 and 9.

For the hindcast run, the net outflow, seen as Q at s=0, Fig. 8, is 15 700 m3s-1 at the sill transect and 16 400 m3s-1 at the northern Kattegat transect. The main outflows at the sill transect, seen as the part of Q with positive gradient with respect to salinity, have salinities between 7 and 11 psu, and the main inflows have salinities between 11 and 28 psu. At the northern Kattegat transect, the main outflows have salinities between 15 and 31 psu while the inflows happen at 31–36 psu. The strength of the inflow, seen as the maximum value of Q, is about 13 500 m3s-1 at the sill transect and 45 000 m3s-1 at northern Kattegat.

For increased runoff and precipitation, RP+, RP+SB+, and RP++SB++, the inflow strengths through the sill transect are reduced and the salinities of inflows and outflows are also decreased. For the northern Kattegat transect, the inflow strength increases for these cases, the outflow salinities decrease, whereas the inflow salinities are less affected.

The salinity and inflow strength changes for cases with decreasing runoff and precipitation are opposite to those for the cases with increasing runoff and precipitation.

Changing boundary salinities are mainly affecting the salinities of inflows and outflows at the northern Kattegat boundary, but do cause less changes to the inflows and outflows at the sill transect, and only small changes to the inflow and outflow strengths.

Figure 9 clearly illustrates that the overturning of salt in Kattegat and the Belts and Sound is much larger than within the Baltic Sea. Otherwise, it shows many of the same features as seen in Fig. 8. In a steady state condition, the net cumulated inflow of salt at small salinities should be zero. It is, however, seen that many of the curves have a small negative value there, i.e. a net outflow of salt, which corresponds to the negative mean salinity trend in the period 1990–2017 seen in Fig. 2.

The in- and out fluxes through the transects and the recirculation calculated from Eqs. (20) and (21) are shown for the hindcast run in Fig. 10. The recirculation fluxes are calculated using γ=0, i.e. assuming the local freshwater input between transects A and B is being added to the outflow through transect B. It is seen that the inflowing water to the Baltic Sea through transect A consists of about 64 % outflowing Baltic Sea water and 36 % inflowing Skagerrak water, and that the outflow through transect B consists of about 64 % inflowing Skagerrak water, 1 % local freshwater and 35 % outflowing Baltic Sea water.

When modifying the runoff, precipitation and boundary salinity, these fluxes change. The changes are small for the SB+ and SB- experiments (not shown), but for the RP+ and RP- experiments the changes are larger. Figure 11 shows the changes to these fluxes for an increase in Qf of 100 m3s-1, calculated from the central differences derivative of these fluxes with respect to Qf based on the RP+ and RP- experiments. It is seen that 54 % of the increased net inflow of freshwater to the Baltic Sea cause increased outflows from the Baltic, while the remaining 46 % cause decreased inflows to the Baltic Sea. The decreased inflow is caused by about 2/3 decreased recirculation and 1/3 decreased Skagerrak water. This means that the inflowing waters continue to consist of about 2/3 Baltic Sea water and 1/3 Skagerrak water also when the freshwater forcing changes. While the overturning of salt inside the Baltic Sea decreases due to decreasing inflows, the overturning within Kattegat increases, since both the inflows and outflows through the northern Kattegat transect increase. One way to explain this increased overturning may be the increased baroclinic pressure gradients between Skagerrak and Kattegat when the salinities in Kattegat decrease with increasing freshwater forcing.

Figure 12 shows the probability density functions (pdf) for inflows and outflows through the sill transect for the hindcast and changed runoff and precipitation cases. It is seen that the changed freshwater basically shifts the pdf towards larger or smaller inflows while maintaining the same shape.

Since the curve is almost symmetric around zero, though shifted somewhat towards the negative side, shifting the curve laterally causes almost the same decrease or increase on the positive side as the increase or decrease on the negative side. This explains why the changes in inflows and outflows (Fig. 11) are of about the same magnitude.

The dashed lines in Fig. 12 are pdfs obtained from the hindcast run time series with an addition or subtraction of a constant volume flux corresponding to the mean change in Qf for the RP+ and RP- cases. These are quite similar to the corresponding RP+ and RP- curves. This indicates that the changed in- and outflows are basically caused by a shift up or down of the time series with the mean change in Qf. This also means that there is a larger relative change to intermediate strength in- and outflows than to low and high strength in- and outflows where the change in frequency is small when changing the volume flux with the change in Q.

One way to explain that an increased input of freshwater causes a more or less constant change to the fluctuating inflows is that they cause a raised sea level relative to the reference situation, Fig. 13, which causes an extra outward barotropic flow. Similarly, a decreased input of freshwater will cause a lowered sea level relative to the reference situation and therefore smaller outward barotropic flows and increase inward barotropic flows.