In the present paper, the time constant of the adjustment process following a changed P supply is derived. The steady-state (equilibrium) winter surface water P concentration in the Baltic proper that would be attained at the end of a restoration effort, is also derived. The salinity stratified Baltic proper has two layers (Fig. 4). Each autumn and winter the upper layer is vertically mixed down to the halocline whereby water from the lower layer is entrained into the surface layer. The lower layer is vertically stratified. For the physics of the Baltic Sea see e.g. Stigebrandt (2001) and Leppäranta and Myrberg (2009). The time-dependent phosphorus model is a mass balance model that considers sources and sinks, that all can be represented by fluxes through the external boundaries of the water mass, and storage changes in the water mass (Fig. 4). For an explanation of symbols used, see Table 1.
Following Stigebrandt et al. (2014), the time-dependent equation (time resolution 1 year) for the total content of phosphorus in the water column of the Baltic proper reads
$$ V\frac{{{\text{d}}\bar{c}}}{{{\text{d}}t}} = {\text{Extsource}} + {\text{Intsource}} + Q_{1} c_{0} - (Q_{f} + Q_{1} )\bar{c}_{1} - {\text{Intsink}} $$
(1)
here V is the volume of the Baltic proper and \( \bar{c} \) is the volume mean winter concentration of phosphorus so that V
\( \bar{c} \) is the total winter content of P in the water column. Extsource is the land-based external P source; Q
f
is the rate of freshwater supply; Q
1
is the rate of inflow of new deepwater from Kattegat c.f. Fig. 4. \( \bar{c}_{1} \) is the annual mean concentration in the surface layer and c
0 is the time mean concentration of phosphorus in the water flowing into the Baltic from the entrance area. Q
1
c
0 is the oceanic source of P, i.e. import from the Kattegat area, while (Q
f + Q
1
)
\( \bar{c}_{1} \) is the export of P. It is assumed that \( \bar{c}_{1} = \gamma c_{1}, \) where c
1 is the winter concentration of phosphorus in the surface layer (Wulff and Stigebrandt 1989). The internal source term, Intsource, constitutes the very important difference between the improved model described by Eq. (1) and the old model by Wulff and Stigebrandt (1989) where it is lacking. Using data for 1980 and 2005, Stigebrandt et al. (2014) found that Intsource = fs · A
anox where fs = 2.3 g P m−2 year−1 is the temporal and spatial mean-specific DIP flux from anoxic bottoms and A
anox is the area of anoxic bottoms. Please remember that the internal dynamics that regulate the exchange between the layers are not invoked in the P model.
The deepwater in the 100-m deep Bornholm Basin in the southern Baltic proper switches since the 1960s between oxic and anoxic conditions. Observations from this basin confirm that the internal P source is turned on only during anoxic conditions and turned off again when the bottom water is oxygenated (Stigebrandt et al. 2014). Please note that in that study it was possible to differentiate between the P flux from internal sources and the P flux from decomposition of fresh organic matter. Observations in the Baltic proper show that the top layer of the sediment was rapidly oxygenated by the recent major deepwater inflow (Rosenberg et al. 2016). Additional examples of oxygen control of the internal P source from anoxic bottoms were presented in the introductory section. Long-term effects of oxygenation on P fluxes were discussed by Stigebrandt et al. (2015a). P storage in sediments is further discussed in “Discussion” section.
The internal sink, Intsink = Intsink1 + Intsink2 (c.f. Fig 4), can be written as (Wulff and Stigebrandt 1989)
$$ {\text{Intsink}} = c_{1} vA. $$
(2)
here v (m year−1) is the so-called apparent settling velocity and A is the surface area of the Baltic proper. The annual removal rate of phosphorus from the surface water to internal sinks is related to the biological production and is thus assumed to be proportional to the upper layer winter concentration c
1 as explained by Wulff and Stigebrandt (1989). The external sink by export to Kattegat is also proportional to c
1 as discussed above. Thus, all sinks are proportional to the winter surface water concentration c
1
.
Using Eq. (2), and writing Totsource = Extsource + Intsource + Q
1
c
0, and introducing the quantity Total Removal Volume Flux to the sinks, defined by TRVF = νA + γ (Q
f + Q
1), the winter surface concentration c
1 (mmol P m−3) can be written as follows (from Eq. 1):
$$ c_{1} = \frac{{{\text{Totsource}} - V\frac{{{\text{d}}\bar{c}}}{{{\text{d}}t}}}}{\text{TRVF}} $$
(3)
The winter surface concentration c
1 in the Baltic proper is thus proportional to the total supply of phosphorus minus the rate of change of P stored in the water column. It can be seen that c
1 is inversely proportional to TRVF, the total removal volume flux to the sinks.
The numerical value of the denominator TRVF in Eq. (3) will be determined here. In 1980, the winter concentration c
1
of phosphorus (TP) in the surface layer was 0.8 mmol P m−3, Extsource = 60 000, Intsource = 46 000, \( V\frac{{{\text{d}}\bar{c}}}{{{\text{d}}t}} = 5000 \) and \( Q_{1} c_{0} = 1 1\,000 \, \left( {{\text{tonnes P year}}^{ - 1} } \right), \) see Table 2. The estimated oceanic import Q
1
c
0 is similar to other estimates as discussed in Stigebrandt et al. (2014). Inserting this in Eq. (3) one finds that TRVF = vA + γ (Qf + Q1) = 4520 km3 year−1. With Q
1 ≈ Q
f ≈ 450 km3 year−1 (e.g. Stigebrandt et al. 2014) and γ = 0.8 (Wulff and Stigebrandt 1989), the flushing term γ (Q
f + Q
1) equals 720 km3 year−1. The value of the internal sink term vA thus equals 3800 km3 year−1. According to the present model, the sink by export to Kattegat accounts for 16% (720/4520) while the internal sink accounts for 84% (3800/4520) of the total sink. The internal phosphorus sink, which in the present case also includes P exported to the Bothnian Sea (Stigebrandt et al. 2014), is thus 5.3 times greater than the export sink to Kattegat. Within the frame of the budget model, variations of the physical circulation system are not considered.
Table 2 Column 2 shows parameter values valid for 1980 for evaluation of TRVF
The error in c
1 as determined from the total P content in the upper layer is estimated to be ±0.05 mmol m−3, c.f. Fig. 2 in Stigebrandt et al. (2014). The error in TRVF is of the same magnitude as the error in Totsource—\( V\frac{{{\text{d}}\bar{c}}}{{{\text{d}}t}} \). The errors in both Extsource and \( V\frac{{{\text{d}}\bar{c}}}{{{\text{d}}t}} \) are discussed in Stigebrandt et al. (2014). Intsource was estimated in Stigebrandt et al. (2014) using data from 1980 and 2005 and the error should be of the same magnitude as the errors in the other source terms. Assume that the error in Totsource—\( V\frac{{{\text{d}}\bar{c}}}{{{\text{d}}t}} \) is ±20%. Then TRVF should be 4520 ± 950 km3 year−1.
With A = 250 000 km2, one finds that the apparent removal rate v ≈ 15 m year−1, c.f. Equation (2). This is comparable to 14–15 m usually found in lakes (Reckhov and Chapra 1983) but it is twice the value estimated by Wulff and Stigebrandt (1989). The lower value found by Wulff and Stigebrandt (1989) should be due to their neglection of the internal source, an explanation also suggested by them. With c
1 = 0.8 mmol P m−3, Intsink amounts to about 93 000 tonnes year−1 which is quite close to the value (93 750 tonnes year−1) estimated in Stigebrandt et al. (2014).
Below, two additional results of the phosphorus model are derived. The first result is a formula for the equilibrium concentration c
1e, which is the steady-state concentration that should occur after a sufficiently long time of constant P supply. The second result is the response time of c
1 to changes of the total P supply. The response time is system-dependent and determines the inherent restoration time of a system.
The equilibrium winter surface P concentration
For steady-state situations, the total sink = c
1e·TRVF equals the total source, Totsource, and the storage of P in the water column and the total supply do not change. For this situation, Eq. (3) shows that the equilibrium concentration in the upper layer c
1e equals
$$ c_{{1{\text{e}}}} = \frac{\text{Totsource}}{\text{TRVF}}. $$
(4)
here TRVF, the total removal volume flux, equals 4520 km3 year−1 for the Baltic proper as estimated above. This result for the equilibrium concentration, Eq. (4), is also displayed in Fig. 5. It should be extremely interesting from e.g. a management point of view. It is discussed in “The equilibrium solution” section.
The adjustment time to changed total P supply
The adjustment of the surface water winter concentration c1 towards the new equilibrium concentration c1e when the total P source Totsource is changed abruptly is estimated using Eq. (3). This equation contains the mean concentration in the water column in winter, \( \bar{c} \), which is defined by
$$ V\bar{c} = V_{1} c_{1} + V_{2} c_{2}. $$
(5)
here V = V
1
+ V
2
and (V
1
,V
2
) are the volumes and (c
1
, c
2
) the winter P concentrations of the two layers. For the understanding and description of the adjustment process, it is valuable to find analytical solutions if possible. In the following, two cases are studied. In Case 1, the lower layer has disappeared and there is only a surface layer. It is assumed that this case might occur during a period with low salinity mode of water renewal as discussed in “Introduction” section. It reminds of the case believed to occur when the Baltic Sea restores itself spontaneously as argued in “Introduction” section. In Case 2, the stratification is as usual, with a halocline at 60-m depth and the lower layer is kept oxygenated by some unspecified method.
Case 1
In this case, the lower layer has disappeared, i.e. V
2
= 0, due to only small inflows of new dense water. With V
2 = 0 it follows from Eq. (5) that V = V
1
and \( \bar{c} = c_{1} \). Equation (3) is then written.
$$ \frac{{{\text{d}}c_{1} }}{{{\text{d}}t}} = \frac{{{\text{Totsource}} - c_{1} \cdot {\text{TRVF}}}}{V}. $$
(6)
Using Eq. (4), Eq. (6) can be written.
$$ \frac{{{\text{d}}c_{1} }}{{{\text{d}}t}} = \frac{\text{TRVF}}{V}(c_{{1{\text{e}}}} - c_{1} ). $$
(7)
Make use of the following substitution of variables
$$ x = c_{1} - c_{{1{\text{e}}}}. $$
(8)
One then obtains
$$ \frac{{{\text{d}}x}}{{{\text{d}}t}} = - \frac{\text{TRVF}}{V}x. $$
(9)
Equation (9) has the following solution
$$ x = a \cdot e^{{ - \frac{t}{T}}} + b. $$
(10)
here T is the time constant for the adjustment process. T is defined by
$$ T = \frac{V}{\text{TRVF}}. $$
(11)
Change variables again using Eq. (8).
$$ c_{1} - c_{{1{\text{e}}}} = a \cdot e^{{ - \frac{t}{T}}} + b. $$
(12)
The temporal boundary conditions are applied to determine the two integration constants, a and b. For long times (t → ∞), we expect that c
1 attains the equilibrium concentration c
1e. This gives b = 0. At t = 0, i.e. at the start of the change of supply, c
1 equals the initial concentration c
1i. This gives a = c
1i − c
1e.
Finally, insertion of the integration constants a and b gives the solution of Eq. (6)
$$ c_{1} = c_{{1{\text{e}}}} + (c_{{1{\text{i}}}} - c_{{1{\text{e}}}} )e^{{ - \frac{t}{T}}}. $$
(13)
with V = 14 780 km3 (Stigebrandt 1987) and TRVF = 4520 km3 year−1 (given above in “Materials and methods” section), one obtains T = 3.27 year.
The inherent restoration time TR is here defined so that 5% of the difference between the initial and the equilibrium concentrations remain (Fig. 6). This gives TR = 3T. In Case 1, restoration may thus be performed in about 10 years.
Case 2
In this case, the 2-layer stratification is maintained and the lower layer is kept oxygenated. This should be possible to achieve by man-made and possibly also by natural oxygenation. During the period 1970–2010, there were about equal amounts of P in the upper and lower layers, i.e. V
1
c
1 = V
2
c
2, cf. Fig. 2 in Stigebrandt et al. (2014). Assuming that this is true for Case 2, one may replace V
\( \bar{c} \) by 2Vc
1
. Equation (3) is then rewritten.
$$ \frac{{{\text{d}}c_{1} }}{{{\text{d}}t}} = \frac{{{\text{Totsource}} - c_{1} \cdot {\text{TRVF}}}}{{2V_{1} }}. $$
(14)
The only difference between Eq. (6) and Eq. (14) is the value of the denominator on the right side; V in Eq. (6) has been replaced by 2V
1 in Eq. (14). The solution of Case 1 in Eq. (13) is valid also for Eq. (14) if the value of T in Eq. (11) is replaced by T = 2V
1
/TRVF. With V
1
= 10 790 km3 (Stigebrandt 1987), one obtains T = 4.77 year. The restoration time in Case 2 is thus about 14 years, c.f. Fig. 6. The reason why the response is somewhat slower in Case 2 than in Case 1 is because c
1
in Case 2 is lower than the vertical mean concentration, which makes the sink rates smaller in Case 2 than in Case 1.
When solving Eqs. (6) and (14), it was assumed that Totsource, which equals the land-based external supply plus the ocean supply, was constant. If Totsource changes with time during the restoration process, c
1e will change according to Eq. (4). However, solutions still follow Eq. (13) but with values of c
1i and c
1e that are changing step by step as Totsource changes.