Mass Balance Models to Derive Critical Loads of Nitrogen and Acidity for Terrestrial and Aquatic Ecosystems

Abstract

This chapter describes the standard approaches (mass balance models) to calculate critical loads of nutrient nitrogen (N) as well as for sulphur (S) and N acidity for both terrestrial and aquatic ecosystems. The description focuses on the so-called Simple Mass Balance (SMB) model for nutrient nitrogen and acidity for terrestrial ecosystems and on the First-order Acidity Balance (FAB) model for aquatic ecosystems. The model descriptions are in accordance with the methods for calculating critical loads under the LRTAP Convention. For both types of models, a discussion is presented on the required input data, data sources and standard model parameter values used in their application. For acidity, the chapter elaborates on the critical load function as there is no unique critical load of S and N acidity, and on the approach to assess critical load exceedances. The chapter ends with a discussion on the possible formulation of critical loads based on biodiversity criteria.

1 Introduction

The purpose of a model-based approach to calculating critical loads is to link, via mathematical equations, a chemical criterion (critical limit) with the maximum deposition(s) ‘below which significant harmful effects on specified sensitive elements of the environment do not occur’, i.e. for which the criterion is not violated. The quotations are from the definition of a critical load (Nilsson and Grennfelt 1988; see also Chap. 1). In most cases the ‘sensitive element of the environment’ will be of a biological nature (e.g., the vitality of a tree, a fish species in a lake), and thus the criterion should be a biological one. However, there is a dearth of simple yet reliable models that adequately describe the whole chain from deposition to biological impact. Therefore, chemical criteria are used instead, and physico-chemical relationships are used to derive critical loads. This simplifies the modelling process, but shifts the burden to find, or derive, appropriate chemical criteria (and critical limits) with established relationships to biological effects (see Chap. 2). The choice of the critical limit is an important step in deriving a critical load, and much of the uncertainty in critical load calculations stems from the uncertainty in the link between (soil or water) chemistry and biological impact.

The definition of a critical load does not include any time aspect. Obviously, the ‘harmful effects’ should not only be avoided in a particular year, but ‘forever’. Thus, a critical load is a time-independent quantity, and this simplifies its modelling, since only steady-state models need to be considered. On the other hand—in contrast to ‘classical’ cause-effect modelling, where ‘effect’ is the explained variable—critical load models require the ‘inverse’, i.e. ‘cause’ becomes the output. A critical load model computes the deposition(s), called critical load(s), which lead to a prescribed critical limit at a given site. This becomes more involved if the processes considered in the critical load model depend themselves on the deposition.

Here we concentrate on the so-called Simple Mass Balance (SMB) model for soils (Sverdrup et al. 1990; Sverdrup and De Vries 1994) and the First-order Acidity Balance (FAB) model for surface waters (Henriksen and Posch 2001; Posch et al. 2012). Both models are single-layer models, i.e. the soil (and the water body) is treated as a single homogeneous compartment . Furthermore, it is assumed that the soil depth is (at least) the depth of the rooting zone, which allows neglecting the nutrient cycle and to deal with net growth uptake (harvested biomass) only. Additional simplifying assumptions include:

• all evapotranspiration occurs on the top of the soil profile,

• percolation is constant through the soil profile and occurs only vertically,

• physico-chemical constants are assumed uniform throughout the soil profile.

Since the critical load models describe steady-state conditions, they require long-term averages for input fluxes. Short-term variations—e.g., episodic, seasonal, inter-annual, due to harvest and as a result of short-term natural perturbations—are not considered, but are assumed to be included in the calculation of long-term means. In this context ‘long-term’ is defined as about 100 years, i.e. at least one rotation period for forests. Ecosystem interactions and processes like competition , pests, herbivore influences etc. are not considered in the models. Although the models are formulated for undisturbed (semi-natural) ecosystems, the effects of extensive management, such as grazing and fires, could be included.

2 Critical Loads of Nutrient Nitrogen for Soils

2.1 The SMB Model for Critical Loads of Nutrient Nitrogen

Elevated inputs of N can lead to undesirable changes in vegetation and/or to increased amounts of N in the leachate that can cause problems to ground and surface waters (see Chap. 2). The starting point for calculating critical loads of nutrient N with the SMB model is the mass balance of total N for the soil compartment under consideration (inputs = sinks + outputs):

$$ {N_{dep}}+ {N_{fix}}= {N_{ad}}+ {N_i} + {N_u} + {N_{de}}+ {N_{eros}}+ {N_{fire}}+ {N_{vol}}+ {N_{le}}$$

(6.1)

where:

N _dep :

total N deposition

N _fix :

N ‘input’ by biological fixation

N _ad :

N adsorption

N _i :

long-term net immobilisation of N in soil organic matter

N _u :

net removal of N in harvested vegetation and animals

N _de :

flux of N to the atmosphere due to denitrification

N _eros :

N losses through erosion

N _fire :

N losses in smoke due to (wild or controlled) fires

N _vol :

N losses to the atmosphere via NH₃ volatilisation

N _le :

leaching of N below the root zone

The units used are eq ha⁻¹yr⁻¹. The following assumptions are frequently made:

• N adsorption, e.g., the adsorption of NH₄ by clay minerals, can temporarily lead to an accumulation of N in the soil; however it is stored/released only when the deposition changes, and can thus be neglected under steady-state considerations;

• N fixation is negligible in most (forest) ecosystems, except for N-fixing species;

• The loss of N due to fire, erosion and volatilisation is small for many ecosystems and therefore neglected in the following discussion;

• The leaching of ammonium (NH₄) can be neglected in forest ecosystems due to its (preferential) uptake and complete nitrification within the root zone (i.e. NH _4,le  = 0, N _le  = NO _3,le ).

Under these simplifying assumptions Eq. 6.1 becomes:

$$ {N_{dep}}= {N_i} + {N_u} + {N_{de}}+ {N_{le}}$$

(6.2)

From this equation a critical load is obtained by defining a critical or acceptable limit to the leaching of N, N _le,acc , the choice of this limit depending on the ‘sensitive element of the environment’ to be protected. Inserting an acceptable leaching into Eq. 6.2, the deposition of N becomes the critical load of nutrient nitrogen, CL _nut N:

$$ C{L_{nut}}N = {N_i} + {N_u} + {N_{de}}+ {N_{le,acc}}$$

(6.3)

In this derivation of a critical load it is assumed that the sources and sinks of N do not depend on the deposition of N. This is unlikely to be the case, and thus all quantities should be taken ‘at critical load’. However, to compute, e.g., ‘denitrification at critical load’ one needs to know the critical load, the very quantity one wants to compute. The only clean way to avoid this circular reasoning is to use a functional relationship between deposition and the sink of N, insert this function into Eq. 6.2 and solve for the deposition (to obtain the critical load). This has been done for denitrification: In the simplest case, denitrification is linearly related to the net input of N (De Vries et al. 1994):

$$ {N_{de}}= {f_{de}}\cdot {({{N_{dep}}-{N_i}-{N_u}})_ + } $$

(6.4)

where (x)₊ or x ₊ is a short-hand notation for max{x,0}, i.e. x ₊ = x for x > 0 and x ₊  = 0 for x ≤ 0; and the denitrification fraction f _de (0 ≤ f _de  < 1) is a site-specific quantity. Inserting Eq. 6.4 into Eq. 6.2 and solving for the deposition leads to the following expression for the critical load of nutrient N:

$$ C{L_{nut}}N = {N_i} + {N_u} + \frac{{{N_{le,acc}}}}{{1-{f_{de}}}}$$

(6.5)

Based on data in Solberg et al. (2009), Posch et al. (2011) derived critical load expressions with net growth uptake as the sum of a constant term and a term proportional to N deposition. Ideally, all N fluxes depending on N deposition should be modelled as such and used in deriving a critical load . However, these might be so involved that no (simple) explicit expression for CL _nut N can be obtained. Although this does not matter in principle, it would reduce the appeal and widespread use of critical loads. Therefore, when calculating critical loads from Eq. 6.3 or 6.5, the N fluxes are generally estimated as long-term averages derived from conditions not influenced by elevated anthropogenic N inputs.

2.2 Critical/Acceptable N Leaching

The value for the acceptable N leaching, N _le,acc , depends on the ‘harmful effect’ that should be avoided. In general, it is not the N leaching flux itself that is ‘harmful’, but the concentration of N in the leaching flux or a concentration in another compartment (e.g. tree foliage) that is related to the dissolved N concentration. This is related to the acceptable N leaching (in eq ha⁻¹yr⁻¹) by:

$$ {N_{le,acc}}= Q\cdot {[N]_{acc}}$$

(6.6)

where [N] _acc is the acceptable dissolved N concentration (eq m⁻³) and Q is the precipitation surplus (in m³ ha⁻¹yr⁻¹). Values for acceptable N concentrations are discussed in Chap. 2.

2.3 Sources and Derivation of Input Data

The obvious sources of input data for calculating critical loads are measurements at the site under consideration. However, in many cases these will not be available. A discussion on N sources and sinks can be found in De Vries (1993), Hornung et al. (1995) and UNECE (1995) . Some data sources, default values and procedures to derive critical loads of N are summarised below.

Nitrogen Immobilisation:

N _i refers to the long-term net immobilisation (accumula-­tion) of N in the root zone, i.e., the continuous build-up of stable C-N-compounds in (forest) soils. In other words, this immobilisation of N should not lead to significant changes in the prevailing C/N ratio. This has to be distinguished from the high amounts of N accumulated in the soils over many years (decades) due to the increased deposition of N, leading to a decrease in the C/N ratio in the (top)soil.

Using data from Swedish forest soil plots, Rosén et al. (1992) estimated the average annual N immobilisation since the last glaciation at 0.2–0.5 kgN ha⁻¹yr⁻¹. Considering that the immobilisation of N is probably higher in warmer climates, values of up to 1 kgN ha⁻¹yr⁻¹ could be used for N _i , without causing unsustainable accumulation of N in the soil. It should be pointed out, however, that even higher values (closer to present-day immobilisation rates) have been used in critical load calculations. Although studies on the capacity of forests to absorb N have been carried out (see, e.g. Sogn et al. 1999), there is no consensus on long-term sustainable immobilisation rates .

Nitrogen Uptake:

The uptake flux N _u equals the long-term average removal of N from the ecosystem. For unmanaged ecosystems (e.g., national parks) the long-term (steady-state) net uptake is basically zero, whereas for managed forests it is the long-term net growth uptake. The harvesting practice is of crucial importance, i.e., whether stems only, stems plus (parts of) branches or stems plus branches plus leaves/needles (whole-tree harvesting) are removed. The uptake of N is then calculated as:

$$ {N_u} = \frac{{{\text{N}}{\text{removed}}{\text{in}}{\text{harvested}}{\text{biomass}}{\text{(eq/ha)}}}}{{{\text{interval}}{\text{between}}{\text{harvests}}{\text{(rotation}}{\text{period)}}{\text{(a)}}}}$$

(6.7)

The amount of N in the harvested biomass (stems and branches) can be calculated as:

$$ {N_u} = {k_{gr}}\cdot {\rho_{st}}\cdot ({ct{N_{st}}+ {f_{br,st}}\cdot ct{N_{br}}+ {f_{lv,st}}\cdot ct{N_{lv}}}) $$

(6.8)

where k _gr is the average annual growth rate (m³ ha⁻¹yr⁻¹), ρ _st is the density of stem wood (kg m⁻³) and ctN is the N content in stems, branches and foliage (subscripts st, br and lv, respectively) (eq kg⁻¹). Furthermore, f _br,st and f _lv,st denote the (average) branch-to-stem and foliage-to-stem ratios (biomass expansion factors; kg kg⁻¹). Depending on the harvesting practice, the contribution of foliage and/or branches should be neglected.

Values for the quantities should be taken from local sources; alternatively they can be found in overview reports such as Kimmins et al. (1985) and Jacobsen et al. (2003). Growth rates used should be long-term average values, typical for the site. It has to be noted that recent growth rates are higher due to increased N input. Therefore it is recommended to use older investigations (yield tables), preferably from before the 1970s. Net uptake of N in non-forest (semi-)natural ecosystems is insignificant, unless they are used for extensive grazing.

Denitrification:

Dutch and Ineson (1990) reviewed data on denitrification rates, which can be used directly for the computation of critical loads (Eq. 6.3) or enable the verification of computations made with Eq. 6.4. Typical values of N _de for boreal and temperate ecosystems are in the range of 0.1–3.0 kgN ha⁻¹yr⁻¹ (=7.14–214.3 eq ha⁻¹yr⁻¹), where the higher values apply to wet(ter) soils; rates for well-drained soils are generally below 0.5 kgN ha⁻¹yr⁻¹. Concerning deposition-dependent denitrification, values for the denitrification fraction used by De Vries et al. (1994) were f _de  = 0.95 for peat soils, 0.8 for clay soils, 0.5 for sandy soils with gleyic features and 0.1 for sandy soils without gleyic features. Reinds et al. (2001) related the denitrification fraction to the drainage status of the soil, ranging from f _de  = 0.1 for well drained to 0.8 for (very) poorly drained soils.

Precipitation Surplus:

The precipitation surplus Q is the amount of water percolating from the root zone. It is conveniently calculated as the difference between precipitation and actual evapotranspiration and it should be the long-term annual mean value. In many cases evapotranspiration will have to be calculated by a model using basic meteorological input data (precipitation, temperature, radiation etc.) . For the basics of modelling evapotranspiration see Monteith and Unsworth (1990), and for an extensive collection of models see Burman and Pochop (1994).

3 Critical Loads of Acidity (N and S) for Soils

3.1 The SMB Model for Critical Loads of Acidity

Starting point for deriving critical loads of acidifying N and S for soils is the charge balance of the ions in the soil leaching flux:

$$ {H_{le}}+ A{l_{le}}+ B{C_{le}}+ N{H_{4,le}}= S{O_{4,le}}+ N{O_{3,le}}+ C{l_{le}}+ HC{O_{3,le}}+ Or{g_{le}}$$

(6.9)

where the subscript le stands for leaching, Al stands for the sum of all positively charged aluminium species, BC is the sum of base cations (BC = Ca+Mg+K+Na) and Org denotes the sum of organic anions. A leaching term is given by X _le  = Q·[X], where [X] is the soil solution concentration of ion X and Q is the precipitation surplus; again, all fluxes are expressed in eq ha⁻¹yr⁻¹. The concentrations of OH and CO ₃ can be neglected in the pH-ranges considered. Defining the (leaching of) Acid Neutralising Capacity (ANC) as:

$$ AN{C_{le}}= -{H_{le}}-A{l_{le}}+ HC{O_{3,le}}+ Or{g_{le}}$$

(6.10)

Equation 6.9 becomes:

$$ B{C_{le}}+ N{H_{4,le}}-S{O_{4,le}}-N{O_{3,le}}-C{l_{le}}= AN{C_{le}}$$

(6.11)

For more detailed discussions on the processes and concepts of (soil) chemistry encountered in the context of acidification see, e.g., the books by Reuss and Johnson (1986) or Ulrich and Sumner (1991).

Chloride is assumed to be a tracer, i.e. there are no sources or sinks of Cl within the soil compartment , and Cl leaching is therefore equal to Cl deposition (subscript dep):

$$ C{l_{le}}= C{l_{dep}}$$

(6.12)

At steady-state the leaching of base cations has to be balanced by their net inputs:

$$ B{C_{le}}= B{C_{dep}}+ B{C_w}-B{c_u} $$

(6.13)

where the subscripts w and u stand for weathering and net growth uptake, i.e. the net uptake by vegetation that is needed for long-term average growth. Bc stands for Ca+Mg+K, reflecting the fact that Na is not taken up by vegetation. The finite pool of base cations at exchange sites (cation exchange capacity) might buffer incoming acidity for decades, but will reach equilibrium with the soil solution under steady-state conditions, and thus need not be considered here.

The leaching of sulphate and nitrate can be linked to the deposition of these compounds by means of mass balances for S and N. For S this reads (De Vries 1991):

$$ {S_{le}}= {S_{dep}}-{S_{ad}}-{S_i}-{S_u}-{S_{re}}-{S_{pr}}$$

(6.14)

where the subscripts ad, i, re and pr refer to adsorption, immobilisation , reduction and precipitation, respectively. An overview of sulphur cycling in forests by Johnson (1984) suggests that uptake, immobilisation and reduction of S are generally insignificant. Adsorption (and in some cases precipitation with Al complexes) can temporarily lead to a strong accumulation of sulphate (e.g. Johnson et al. 1982); however, it is only stored at or released from the adsorption complex when the input (deposition) changes. Consequently, as with base cations at exchange sites, S adsorption will reach equilibrium with the soil solution under steady-state conditions, and thus need not be considered in critical load calculations. Since sulphur is completely oxidised in the soil profile, SO _4,le equals S _le , and thus:

$$ S{O_{4,le}}= {S_{dep}}$$

(6.15)

For nitrogen, the mass balance in the soil has been derived above (see Eq. 6.2) assuming complete nitrification (i.e. NH4,le = 0):

$$ N{O_{3,le}}= {N_{dep}}-{N_i}-{N_u}-{N_{de}}$$

(6.16)

Inserting all the mass balances into Eq. 6.11 leads to the following simplified charge balance for the soil compartment :

$$ {S_{dep}}+ {N_{dep}}= B{C_{dep}}-C{l_{dep}}+ B{C_w}-B{c_u} + {N_i} + {N_u} + {N_{de}}-AN{C_{le}}$$

(6.17)

Strictly speaking, we should replace NO _3,le in the charge balance not by the right-hand side of Eq. 6.16, but by max{N _dep –N _i –N _u –N _de ,0}, since leaching cannot be negative; and the same holds true for base cations. However, this would lead to unwieldy critical load expressions; therefore we go ahead with Eq. 6.17, keeping this constraint in mind.

For given values for the sources and sinks of S, N and BC, Eq. 6.17 allows the calculation of the leaching of ANC, and thus assessment of the acidification status of the soil. Conversely, critical loads of S, CLS, and N, CLN, can be computed by defining a critical ANC leaching, ANC _le,crit :

$$ CLS + CLN = B{C_{dep}}-C{l_{dep}}+ B{C_w}-B{c_u} + {N_i} + {N_u} + {N_{de}}-AN{C_{le,crit}}$$

(6.18)

Note that Eq. 6.18 does not give a unique critical load of S or N. However, nitrogen sinks cannot compensate incoming sulphur acidity , and therefore the maximum critical load of sulphur is given by:

$$ C{L_{max}}S = B{C_{dep}}-C{l_{dep}}+ B{C_w}-B{c_u}-AN{C_{le,crit}}$$

(6.19)

Equation 6.19 gives the critical load as long as the N deposition is lower than the sum of the N sinks, termed the minimum critical load of N:

$$ C{L_{min}}N = {N_i} + {N_u} + {N_{de}}$$

(6.20)

Finally, the maximum critical load of nitrogen (in the case of zero S deposition) is given by:

$$ C{L_{max}}N = C{L_{min}}N + C{L_{max}}S $$

(6.21)

The three quantities CL _max S, CL _min N and CL _max N define the so-called critical load function (CLF; Fig. 6.1) ; and every deposition pair (N _dep ,S _dep ) lying on the CLF are critical loads of acidifying N and S.

Fig. 6.1

Critical load function (CLF) of acidifying N and S, defined by the three quantities CL _max S, CL _min N and CL _max N in the (N _dep ,S _dep )-plane. Reducing emissions from point E one can reach the CLF, e.g., at points Z_N, Z, or Z_S, The grey area below the CLF denotes deposition pairs resulting in an ANC leaching greater than ANC _le,crit (non-exceedance of critical loads)

Deriving critical loads as above assumes that the sources and sinks of N do not depend on the N deposition. However, if we consider denitrification to be linearly related to the net input of N (see Eq. 6.4), we get the following expressions for CL _min N and CL _max N (CL _max S remains unchanged) :

$$ C{L_{min}}N = {N_i} + {N_u}\quad {\text{and}}\quad C{L_{max}}N = C{L_{min}}N + \frac{{C{L_{max}}S}}{{1-{f_{de}}}}$$

(6.22)

where f _de (0 ≤ f _de  < 1) is the denitrification fraction (see Eq. 6.4).

Since the main use of critical loads is to guide reductions in anthropogenic emissions (and thus depositions) of S and N, sea-salt derived sulphate should not be considered in the balance. To retain charge balance, this is achieved by applying a sea-salt correction to sulphate, chloride and base cations , which is often done by using either Cl or Na as a tracer, whichever can be (safer) assumed to derive from sea-salts only. Denoting sea-salt corrected depositions with an asterisk, one has either Cl ^* _dep  = 0 or Na ^* _dep  = 0 (and BC ^* _dep  = Bc ^* _dep ), respectively. For procedures to compute sea-salt corrections see the Annex to this chapter.

3.2 Chemical Criteria and the Critical Leaching of ANC

To compute a critical load, the critical ANC leaching , ANC _le,crit , has to be defined. Critical chemical limits for the derivation of acidity critical loads are discussed in Chap. 2. For surface waters , for example, a critical limit for ANC is most often used. For soils, however, critical limits are mostly expressed as critical concentrations of Al ([Al] _crit ) or H ([H] _crit ), sometimes in conjunction with base cations; and in the following we show how to derive ANC _le,crit from them.

The terms in the (leaching of) ANC, as given in Eq. 6.10, can all be expressed in terms of [H]. The relationship between [H] and [Al] is modelled as:

$$ [Al] = {K_{Alox}}\cdot {[H]^a} $$

(6.23)

with an equilibrium constant K _Alox and an exponent a. For a = 3 this describes the dissolution of gibbsite, but for a< 3 this is an empirical relationship accounting also for the complexation of Al with organic matter (Cronan et al. 1986; Mulder and Stein 1994), thus making it strongly dependent on soil type and soil depth. Values for K _Alox and a derived by Van der Salm and De Vries (2001) from several 100 Dutch forest soil solution samples show that a standard gibbsite constant (K _Alox  = K _gibb ) and a = 3 are reasonable for sandy soils. Very different values, however, are obtained for peat soils and, to a lesser extent, also for loess and clay soils (especially for shallow parts of the soil, where the organic matter content is highest). Data also show that there is a strong correlation between a and K _Alox , which emphasises that these two parameters cannot be chosen independently.

Bicarbonates (HCO₃) and organic anions have earlier been neglected in critical load calculations. However, it is not always a negligible quantity; and since they can both be expressed as functions of [H], their inclusion poses no real problem. The concentration of bicarbonates is computed as:

$$ [HC{O_3}] = \frac{{{K_1}\cdot {K_H}\cdot {p_{CO2}}}}{{[H]}}$$

(6.24)

where K ₁ is the first dissociation constant, K _H is Henry’s constant and p _CO2 is the partial pressure of CO₂ in the soil solution. The simplest model describing organic acids assumes that only mono-protic organic anions are produced by the dissociation of dissolved organic carbon :

$$ [Org] = \frac{{m\cdot DOC\cdot {K_{org}}}}{{{K_{org}}+ [H]}}$$

(6.25)

where DOC is the concentration of dissolved organic carbon (in molC m⁻³), m is the concentration of functional groups (the ‘charge density’, in mol molC⁻¹) and K _org the dissociation constant. Other models for the dissociation of organic acids have been suggested and are in use in dynamic models, such as di- and tri-protic models (see, e.g. Driscoll et al. 1994). In fact, any model for the dissociation of DOC can be used here as long as it depends only on [H] and/or [Al].

The combination of Eq. 6.10 with Eqs. 6.23–6.25 shows that ANC _le,crit can be calculated if [H] _crit is known; leaching fluxes are obtained by multiplying [H] as well as [Al], [HCO ₃ ] and [Org] (which all depend on [H]) with Q, the precipitation surplus in m³ ha⁻¹yr⁻¹ (see above). If [Al] _crit is known, [H] _crit can be obtained by inverting Eq. 6.23. In the following we discuss different criteria in turn.

Aluminium Criteria:

Aluminium criteria are considered most appropriate for mineral soils with low organic matter content. Three commonly used criteria are:

• a. Critical aluminium concentration: Critical limits for Al have been suggested for forest soils, not only for effects on roots, but especially for drinking water (ground water) protection (see Chap. 2).

• b. Critical base cation to aluminium ratio: Widely used for soils is the connection between soil chemical status and plant response (damage to fine roots) via a critical molar ratio of the concentrations of base cations (Bc = Ca+Mg+K) and Al in soil solution, denoted as (Bc/Al) _crit (see Chap. 2). The critical Al leaching is calculated from the leaching of Bc:

$$ A{l_{le,crit}}= 1.5\cdot \frac{{B{c_{le}}}}{{{{(Bc/Al)}_{crit}}}}= 1.5\cdot \frac{{B{c_{dep}}+ B{c_w}-B{c_u}}}{{{{(Bc/Al)}_{crit}}}}$$

(6.26)

The factor 1.5 arises from the conversion of moles to equivalents (assuming that K is divalent). Note that the expression Bc _le  = Bc _dep +Bc _w –Bc _u has to be non-negative. In fact, it has been suggested that it should be above a minimum leaching or, more precisely, there is a minimum concentration of base cations in the leachate, below which they cannot be taken up by vegetation, i.e., Bc _le is set equal to max{0,Bc _dep +Bc _w –Bc _u –Q⋅[Bc] _min }, with [Bc] _min in the order of 0.01 eq m⁻³. If considered more appropriate, a critical molar ratio of calcium to aluminium in soil solution can be used, by replacing all the Bc-terms in Eq. 6.26 with Ca-terms.

• c. Critical Al mobilisation rate: Critical ANC leaching can also be calculated using a criterion to prevent the depletion of secondary Al phases and complexes, which may cause structural changes in soils and a further pH decline. Aluminium depletion occurs when the acid deposition leads to an Al leaching in excess of the Al produced by the weathering of primary minerals . Thus the critical leaching of Al is given by:

$$ A{l_{le,crit}}= A{l_w} = q\cdot B{C_w} $$

(6.27)

where Al _w is the weathering of Al from primary minerals, and q is the stoichiometric ratio of Al to BC weathering in primary minerals (eq eq⁻¹), with a default value of q = 2 for typical mineralogy of northern European soils (range: 1.5–3.0).

Hydrogen Ion Criteria:

A hydrogen ion (proton) criterion is generally recommended for soils with high organic matter content. Two such criteria are:

• a. Critical pH: Critical pH limits have been suggested for forest soils, for example, pH _crit  = 4.0, corresponding to [H] _crit  = 0.1 eq m⁻³.

• b. Critical base cation to proton ratio: For organic soils that do not contain Al-(hydr)oxides (such as peat lands), it is suggested to use a critical molar base cation to proton ratio (Bc/H) _crit . In this case there is no Al leaching, and H _le,crit is given by:

$$ {H_{le,crit}}= 0.5\cdot \frac{{B{c_{dep}}+ B{c_w}-B{c_u}}}{{{{(Bc/H)}_{crit}}}}$$

(6.28)

where the factor 0.5 comes from converting moles to equivalents. For organic soils the weathering in Eq. 6.28 will probably be negligible (Bc _w  = 0). Values suggested for (Bc/H) _crit are expressed as multiples of (Bc/Al) _crit .

Critical Base Saturation:

Base saturation, i.e. the fraction of base cations at the cation exchange complex, is an indicator of the acidity status of a soil, and one may want to keep this pool above a certain level, e.g., to avoid nutrient deficiencies . Thus a base saturation could be chosen as a criterion for calculating critical loads of acidity (UNECE 2001). Base saturation is also used as criterion in the New England Governors/Eastern Canadian Premiers ‘Acid Rain Action Plan’ for calculating sustainable S and N depositions to upland forests (NEG/ECP 2001) .

To relate base saturation to [H] requires a model of the exchange of cations between the exchange complex and the soil solution. In general, this leads to a non-linear equation for [H] _crit that has to be solved numerically (see UBA 2004). However, using the Gapon model for the exchange between H, Al and Bc = Ca+Mg+K—as implemented in the SAFE (Warfvinge et al. 1993) and the VSD (Posch and Reinds 2009) dynamic soil models—and gibbsite dissolution (i.e. a = 3 in Eq. 6.23) the critical concentration [H] _crit can be found as:

$$ {[H]_{crit}}= {K_{Gap}}\cdot \sqrt {[Bc]} \cdot \left({\frac{1}{{{E_{Bc,crit}}}}-1}\right)\quad \quad {\text{with}}\quad {K_{Gap}}= \frac{1}{{{k_{HBc}}+ {k_{AlBc}}\cdot K_{gibb}^{1/3}}}$$

(6.29)

where E _Bc,crit is the selected critical base saturation, k _HBc and k _AlBc are the two (site-specific) selectivity coefficients describing the Gapon exchange, and [Bc] = Bc _le /Q. Assuming only Al and Bc exchange, Ouimet et al. (2006) used base saturation to select a chemical criterion for calculating acidity critical loads for eastern Canada. Values of selectivity coefficients for a range of (Dutch) soil types can be found in De Vries and Posch (2003) .

3.3 Sources and Derivation of Input Data

The obvious sources of input data for calculating acidity critical loads are measurements at the site under consideration. However, in many cases these will not be available. For data on the different N quantities see Sect. 6.2.3. Some data sources and default values for the other variables, and procedures to derive them, are summarised here.

Base Cation and Chloride Deposition:

The base cation and chloride depositions entering the critical load calculations should ideally be the (non-anthropogenic) deposition at critical load. Observations are available from the EMEP (www.emep.int) on a European scale, or from national sources. In a one-time study, EMEP has also modelled the deposition of base cations (for the year 2000) on a European scale (Van Loon et al. 2005).

Base Cation Weathering:

Weathering here refers to the release of base cations from minerals in the soil matrix due to chemical dissolution, and the neutralisation and production of alkalinity connected to this process. This has to be distinguished from the denudation of base cations from ion exchange complexes (cation exchange) and the degradation of soil organic matter. Many methods for determining weathering rates have been suggested, and here we list those with the highest potential for regional applications (in order of increasing complexity).

• a. The Skokloster assignment: This is a (semi-)empirical method devised in the early days of the critical loads development (UNECE Skokloster workshop, documented in Nilsson and Grennfelt 1988, p. 40, Table 1).

• b. The soil type—texture approximation: Since mineralogy controls weathering rates, weathering rate classes were assigned to European (forest) soils by De Vries et al. (1993), see also Posch et al. (2003), based on texture class and parent material class (Table 6.1). Texture class, as a function of clay and sand content, is defined according to Eurosoil (1999) and parent material class has been defined for each FAO soil type (for details see UBA 2004).

  Table 6.1 Weathering rate classes as function of texture and parent material classes

The weathering rate (in eq ha⁻¹yr⁻¹) for a non-calcareous soil of depth z (in m) is then computed as:

$$ B{C_w} = z\cdot 500\cdot (WRc-0.5)\cdot \exp \left({\frac{A}{{281}}-\frac{A}{{273 + T}}}\right) $$

(6.30)

where WRc is the weathering rate class (Table 6.1), T (℃) is the average annual (soil) temperature and A = 3600 K (Sverdrup 1990). For calcareous soil, for which critical loads are not really of interest, one could, e.g., set WRc = 20 in Eq. 6.30. The above equation gives weathering rates for BC = Ca+Mg+K+Na. However, in Eqs. 6.26 and 6.28 the weathering rate for Bc = Ca+Mg+K is needed; it can be approximated by multiplying BC _w with a factor between 0.70 for poor sandy soils and 0.85 for rich (sandy) soils.

Table 6.2 Ranges for K _gibb as a function of soil organic matter content

• c. The total base cation content correlation: Using the ‘zirconium method’, Olsson et al. (1993) derived from Swedish sites a correlation between historical average weathering rates of base cations and the total content of the respective element in the undisturbed bottom soil, with an additional temperature correction. For Ca, Mg and K the equations are (in eq ha⁻¹yr⁻¹):

$$ \begin{matrix} {C{a_w} = 0.13\cdot {{(Ca)}_{tot}}\cdot ETS-55.5} \\ {M{g_w} = 0.23\cdot {{(Mg)}_{tot}}\cdot ETS-24.1} \\ {{K_w} = 0.05\cdot {{(K)}_{tot}}\cdot ETS-79.8}\end{matrix} $$

(6.31)

where (X) _tot is the total content of element X (Ca, Mg or K in dry weight %) in the coarse fraction (< 2 mm) of the undisturbed C-horizon soil and ETS is the annual sum of daily temperatures above a threshold of + 5 ℃. Care has to be taken when applying these formulae, since they are based on Nordic geological history and they cover a limited number of soil types (mostly podzols). Using part of the Swedish data, this method was adapted in Finland for estimating weathering rates on a national scale (Johansson and Tarvainen 1997) .

• d. The calculation of weathering rates: Weathering rates can be computed with the kinetic Sverdrup-Warfvinge weathering model (Sverdrup and Warfvinge 1988) for (groups of) minerals in the soil. It is also integrated into the multi-layer steady-state model PROFILE (Warfvinge and Sverdrup 1992, 1995). Basic input data are the mineralogy of the site or a total element analysis, from which the mineralogy can be derived by a normative procedure or more general methods, such as the A2M model (Posch and Kurz 2007). The generic weathering rate of each mineral is modified by the pH, base cations, aluminium and organic anions as well as the partial pressure of CO₂ and temperature. The total weathering rate is proportional to soil depth and the wetted surface area of the minerals present. For the theoretical foundations of the weathering rate model, see Sverdrup (1990).

• e. Other methods: Weathering rates can also be estimated from budget studies of small catchments (see, e.g. Paces 1983). However, budget studies can easily overestimate soil weathering rates where there is significant cation release due to weathering of the bedrock. More methods are listed and described in Sverdrup et al. (1990).

Base Cation Uptake:

The uptake flux of base cations entering the critical load calculations, Bc _u , is the long-term average removal of base cations from the ecosystem. The data sources and calculations are the same as for the uptake of N (see Sect. 6.2.3). The (long-term) net uptake of base cations is limited by their availability through deposition and weathering (no depletion of exchangeable base cations at steady state). Furthermore, base cations might not be taken up below a certain concentration in soil solution, or due to other limiting factors, such a temperature. Thus the values entering critical load calculations should be constrained by:

$$ {Y_u}\le {Y_{dep}}+ {Y_w}-Q\cdot {[Y]_{min}}\quad {\text{for}}\quad Y = Ca,Mg,K $$

(6.32)

Suggested values are 5 meq m⁻³ for [Ca] _min and [Mg] _min , and zero for [K] _min (Warfvinge and Sverdrup 1992). It should also be taken into account that vegetation takes up nutrients in fairly constant (vegetation-specific) ratios. Thus, when adjusting the uptake value for one element, the values for the other elements (including N) should be adjusted proportionally .

Other Parameters:

The equilibrium constant relating the Al concentration to pH (Eq. 6.23) depends on the soil type, i.e. the type of Al (hydr)oxide. Table 6.2 presents ranges of K _gibb (and pK _gibb  = –log₁₀(K _gibb in (mol/l)⁻²) as a function of soil organic matter content. An often used default value is K _gibb  = 10⁸ (mol/l)⁻² = 300 m⁶ eq⁻².

If sufficient empirical data are available to derive the relationship between [H] and [Al], these should be used in preference to the gibbsite equilibrium.

The two constants in the bicarbonate equilibrium (Eq. 6.24) are weakly temperature-dependent, and the value for their product at 8 ℃ is K ₁⋅K _H  = 10^−1.7 = 0.02 eq² m⁻⁶atm⁻¹. For systems open to the atmosphere, p _CO2 is about 390 ppm or 3.9·10⁻⁴ atm (in the year 2010). However, in soils p _CO2 is generally higher (ranging from 10⁻² to 10⁻¹ atm; Bolt and Bruggenwert 1976), due to respiration and oxidation of below-ground organic matter. Respiratory production of CO₂ is temperature-dependent, and Gunn and Trudgill (1982) derived the following relationship:

$$ {\log_{10}}{p_{CO2}}= -2.38 + 0.031\cdot T $$

(6.33)

where T is the (soil) temperature (℃). Brook et al. (1983) present a similar regression equation based on data for 19 regions of the world.

Concerning organic acids, both DOC and m are site-specific quantities. While DOC estimates are often available, data for m are less easy to obtain. For example, Santore et al. (1995) report values of m between 0.014 for topsoil samples and 0.044 mol molC⁻¹ for a B-horizon in the Hubbard Brook experimental forest in New Hampshire. Since a single value of K _org does not always model the dissociation of organic acids satisfactorily, Oliver et al. (1983) derived an empirical relationship between K _org and pH:

$$ p{K_{org}}= -{\log_{10}}{K_{org}}= a + b\cdot pH-c\cdot {(pH)^2} $$

(6.34)

with a = 0.96, b = 0.90 and c = 0.039 (and m = 0.12 mol molC⁻¹). Note that Eq. 6.34 gives K _org in mol l⁻¹. In Fig. 6.2 the fraction of m⋅DOC dissociated is displayed as a function of pH for the Oliver model and a mono-protic acid with a ‘widely-used’ value of pK _org  = 4.5. It shows that the share of dissociated organic acids can be substantial, even at low pH.

Fig. 6.2

Fraction of organic acids dissociated as a function of pH for the Oliver model (solid line) and the mono-protic model (Eq. 6.25) with pK ₁ = 4.5 (dashed line)

In Fig. 6.3 the contribution of the individual ions to the ANC is illustrated. It shows the ANC concentration as a function of pH (thick solid line), with the ANC defined as in Eq. 6.10. Also shown is the ANC in the absence of (or neglecting) organic anions (thick dashed line), and the ANC solely defined as –[H]–[Al] (thick dotted line) as done in earlier works on critical loads (Sverdrup et al. 1990; Sverdrup and De Vries 1994) and which can never attain positive values.

Fig. 6.3

ANC concentration (in meq m⁻³) as a function of pH (thick solid line) with [ANC] = –[H]–[Al]+[HCO ₃]+[Org] (see Eq. 6.10). Also shown is the ANC in the absence of (or neglecting) organic anions (thick dashed line) and the ANC solely defined as –[H]–[Al] (thick dotted line). The thin dashed lines show [HCO ₃] and –[Al], and the thin solid line is [HCO ₃]+[Org], all as a function of pH (computed with a = 3, log₁₀ K _gibb  = 8 and m·DOC = 20 mol m⁻³).

4 Critical Loads of Acidity for Surface Waters

Critical loads for aquatic ecosystems estimate the maximum deposition(s) onto a catchment below which ‘significant harmful effects’ on biological species in the water body do not occur. Similar to terrestrial ecosystems, the links between water chemistry and biological impacts cannot be modelled adequately at present and thus (empirically derived) water quality criteria are used to determine critical loads for aquatic ecosystems. The models presented here are restricted to freshwater systems, since models for marine ecosystems do not seem to be documented in the open literature. Furthermore, only acidity critical loads are considered, since critical loads of nutrient N for surface waters have not been formulated/used to date; but see Smith et al. (1999), De Wit and Lindholm (2010) and Chap. 2 for a summary of effects and potential critical limits for N as a nutrient in oligotrophic surface waters.

In the following the most widely used models, the SSWC and the FAB model , are described. More details, especially also earlier work and references, can be found in the review by Henriksen and Posch (2001). As models of critical loads for surface waters also include processes taking place in their terrestrial catchment, it is advised to consult the sections above for some of the terminology and variables used in the context of critical loads for soils.

4.1 The Steady-State Water Chemistry (SSWC) Model

In the SSWC model (Henriksen et al. 1992; Henriksen and Posch 2001; Sverdrup et al. 1990) the critical load for a lake or stream is derived from annual mean values of present-day water chemistry. A critical load of acidity, CLA , is calculated from the same principles as in the SMB model for soils, i.e. it should not exceed the non-anthropogenic net base cation input minus a buffer to protect selected biota from being damaged:

$$ CLA = BC_0^*-AN{C_{limit}}= Q\cdot ({{{[B{C^*}]}_0}-{{[ANC]}_{limit}}}) $$

(6.35)

where the second expression gives the critical load in terms of the catchment runoff Q and concentrations ([X] = X/Q). BC ^* ₀ is the pre-acidification sea salt corrected catchment-average net base cation flux (equal to BC ^* _dep +BC _we –Bc _upt when everything was in equilibrium; compare Eq. 6.13) and ANC _limit is the lowest ANC-flux that does not damage the selected biota. For estimating the sea salt correction, see the Annex to this chapter.

Although one might be able to estimate a pre-industrial base cation deposition, and one can assume that the pre-industrial average net removal of base cations by harvest is negligible, it is the catchment-average weathering rate that poses the problem, especially on a regional scale. Thus Henriksen (1984) argued that the change over time t in base cation concentration is proportional to the change in sulphate (plus nitrate) concentration:

$$ {[B{C^*}]_t}-{[B{C^*}]_0} = F\cdot ({{{[SO_4^*]}_t}-{{[SO_4^*]}_0} + {{[N{O_3}]}_t}-{{[N{O_3}]}_0}}) $$

(6.36)

with F denoting the proportionality factor (the ‘F-factor’) and where the subscripts t and 0 refer to present and pre-acidification concentrations, respectively. If F = 1, all incoming protons are neutralised in the catchment (only soil acidification), at F = 0 none of the incoming protons are neutralised in the catchment (only water acidification). The F-factor was estimated empirically to be in the range 0.2–0.4, based on the analysis of historical data from Norway, Sweden, the U.S.A. and Canada (Henriksen 1984). Brakke et al. (1990) later suggested that the F-factor should be a function of the base cation concentration:

$$ F = \sin ({\tfrac{\pi }{2}{{{{[B{C^*}]}_t}}/{[S]}}}) $$

(6.37)

where [S] is the base cation concentration at which F = 1; and for [BC ^*] _t > [S] F is set to 1. For Norway [S] has been set to 0.4 eq m⁻³ (Brakke et al. 1990). In Eq. 6.37 the present base cation concentration is used for practical reasons. However, to make the F-factor independent of the present base cation concentration (and to simplify the functional form), Posch et al. (1993) suggested the following relationship between F and the pre-acidification base cation concentration [BC ^*]₀:

$$ F = 1-\exp ({-{{{{[B{C^*}]}_0}}/{[B]}}}) $$

(6.38)

where [B] is a scaling concentration estimated to be 0.131 eq m⁻³ from paleolimnological data from Finland (Posch et al. 1993). Inserting this expression into Eq. 6.36 gives a non-linear equation for [BC ^*]₀ which has to be solved by an iterative procedure (see the Supporting Information in Posch et al. 2012 for details). The two expressions for the F-factor give similar results when used to calculate critical loads for surface waters in Norway (Henriksen and Posch 2001). It has also been suggested to replace the concentrations in the F-factor expressions by the corresponding fluxes to better capture differences in weathering characteristics instead of differences in runoff (Henriksen and Posch 2001).

Using data from lakes in the United Kingdom, Battarbee et al. (1996) developed the ‘empirical diatom-based paleolimnological model’. The basic equation for the critical load of acidity in the empirical diatom model is:

$$ CLA = \frac{{{{[C{a^*}]}_0}}}{{CR}}$$

(6.39)

where CLA is in keq ha⁻¹yr⁻¹, [Ca ^*]₀ in meq m⁻³, and CR is the so-called critical ratio (e.g. 94:1). The pre-acidification Ca concentration is calculated with an F-factor approach according to Eqs. 6.36 and 6.37 (with BC replaced by Ca). For more details see Battarbee et al. (1996) and UBA (2004) .

The pre-acidification sulphate concentration in lakes, [SO ₄ ^*]₀, is assumed to consist of a constant atmospheric contribution and a geological contribution proportional to the concentration of base cations (Brakke et al. 1989):

$$ {[SO_4^*]_0} = a + b\cdot {[B{C^*}]_t} $$

(6.40)

The coefficients in this equation, estimated for different regions and by different authors, are summarised in Table 6.3. Larssen and Høgåsen (2003) suggested that the atmospheric contribution in Eq. 6.40 be derived from background S deposition (e.g. estimated by atmospheric transport models), i.e. a = S _dep,0/Q. Posch et al. (2012) also used background deposition to estimate a (with b set to zero).

Table 6.3 Constants to estimate the pre-acidification sulphate concentration (Eq. 6.40), derived from empirical data (N is the number of samples and r is the correlation coefficient)

The pre-acidification nitrate concentration [NO ₃]₀ is generally assumed zero, since the background deposition of N is low and will most likely be taken up in the soil and the water.

The SSWC model has been developed for, and is particularly applicable to, dilute oligotrophic waters located on granitic and gneissic bedrock with thin soils, such as in large parts of Fennoscandia, Scotland, Canada and Ireland. In such areas, surface waters are generally more sensitive to acid inputs than soils. In the model it is assumed that all sulphate in runoff originates from deposition alone, except for a small geologic contribution. In areas where geological conditions lead to more alkaline waters, the SSWC model has to be modified, since, e.g., significant amounts of sulphate from geological sources could be present in the runoff water.

The link between water chemistry and the impact on aquatic organisms is represented by the critical ANC-limit, [ANC] _limit (see Eq. 6.35). Mostly fish species have been considered as the aquatic organisms that should be protected, and it has been mostly ANC itself that has been correlated with fish status, leading to values of [ANC] _limit between zero and 50 meq m⁻³, depending on location and study. An overview of critical limits for surface waters can be found in Chap. 2.

4.2 The First-order Acidity Balance (FAB) Model

The FAB model for calculating critical loads of S and N for a lake/catchment is the equivalent of the SMB model for a soil site, as described above. The original version of the FAB model was developed and applied to Finland, Norway and Sweden by Henriksen et al. (1993) and was also described by Posch et al. (1997). A modified version was first reported by Hindar et al. (2000) and fully described by Henriksen and Posch (2001), as well as in the Mapping Manual (UBA 2004). The description here follows closely that given in the Supporting Information of Posch et al. (2012).

The system we consider consists of the catchment soils and the lake water body, but vegetation is not part of the system. Thus N and base cation uptake are treated as sinks, but at steady state only the net uptake, i.e. the removal by harvest (on an annual average basis) is considered. Other N processes considered are denitrification and net N immobilization (see the discussion under the SMB model above). The lake and its catchment are assumed to be in a (semi-)natural state, unaffected by direct pollution (e.g. from agriculture), with all precipitation falling on the catchment entering the lake, i.e. no groundwater recharge bypassing the lake . The total catchment area (lake + terrestrial catchment) A consists of the lake area A _l =A ₀ and m different sub-areas A _j (j=1,…,m), comprising the terrestrial catchment:

$$ A = {A_l} + \sum_{j = 1}^m {{A_j}}= \sum_{j = 0}^m {{A_j}}$$

(6.41)

where A ₁ could be forested area, A ₂ covered with grass, A ₃ the area of bare rocks, etc. The starting point for the derivation of the FAB model is the charge balance in the surface water running off the catchment:

$$ {S_{runoff}}+ {N_{runoff}}= \sum_Y {{Y_{runoff}}}-AN{C_{runoff}}$$

(6.42)

where Σ _Y stands for the sum of base cations minus chloride (Ca+Mg+K+Na–Cl). In the above equation we assume that the quantities are total amounts per time (e.g. eq yr⁻¹) . To derive critical loads we have to link the ions in the lake water to their depositions, taking into account their steady-state sources and sinks in the terrestrial catchment and the lake. Steady-state mass balances in the lake are given by:

$$ {X_{runoff}}= {X_{in}}-{X_{ret}},\quad \quad X = S,N,Ca,Mg,K,Na,Cl $$

(6.43)

where X _in is the total amount of ion X entering the lake and X _ret the amount of X retained in the lake. The in-lake retention is assumed to be proportional to the input of the respective ion into the lake:

$$ {X_{ret}}= {\rho_X}\cdot {X_{in}},\quad \quad X = S,N,Ca,Mg,K,Na,Cl $$

(6.44)

where 0 ≤ ρ _X  ≤ 1 is a dimensionless retention factor. The mass balances then become:

$$ {X_{runoff}}=(1-{\rho_X})\cdot {X_{in}},\quad \quad X = S,N,Ca,Mg,K,Na,Cl $$

(6.45)

The total amount of S entering the lake is given by:

$$ {S_{in}}= \sum_{j = 0}^m {{A_j}\cdot {S_{dep,j}}}$$

(6.46)

where S _dep,j is the total deposition of S per unit area onto sub-area j. It is assumed that all S deposited onto the catchment enters the lake, i.e. sources (e.g. weathering) or sinks (e.g. immobilization, reduction and uptake) are negligible in the terrestrial catchment. For N we assume that net uptake, net immobilization and denitrification can occur on all sub-areas, possibly at different rates . Thus the amount of N entering the lake is:

$$ {N_{in}}= \sum_{j = 0}^m {{A_j}\cdot {{({N_{dep,j}}-{N_{i,j}}-{N_{u,j}}-{N_{de,j}})}_ + }}$$

(6.47)

where N _dep,j is the total N deposition, N _i,j is the long-term net immobilization of N, N _u,j the net growth uptake of N, N _de,j is N lost by denitrification, all per unit area for sub-area j, and (x)₊ = max{x,0}. The effects of nutrient cycling are ignored in this model and the leaching of ammonium is considered negligible, i.e. it is assumed to be completely taken up and/or nitrified in the terrestrial catchment. Note that some of the terms in Eq. 6.47 can be zero for certain indices. Especially for j = 0, i.e. the lake itself, one mostly assumes N _i,0+N _u,0 = 0.

While we assume immobilization and uptake to be independent of N deposition, denitrification is modelled, as in the SMB model, as a fraction of the available N :

$$ {N_{de,j}}= {f_{de,j}}\cdot {({N_{dep,j}}-{N_{i,j}}-{N_{u,j}})_ + }\quad {\text{for}}\quad j = 0,\ldots{},m $$

(6.48)

where 0 ≤ f _de,j  < 1 is the denitrification fraction for sub-area j. Inserting Eq. 6.48 into Eq. 6.47 one obtains:

$$ {N_{in}}= \sum_{j = 0}^m {{A_j}\cdot (1-{f_{de,j}})\cdot {{({{N_{dep,j}}-{N_{i,j}}-{N_{u,j}}})}_ + }}$$

(6.49)

For base cations and chloride the amounts entering the lake are given by:

$$ {Y_{in}}= \sum_{j = 0}^m {{A_j}\cdot {{({Y_{dep,j}}+ {Y_{w,j}}-{Y_{u,j}})}_ + }},\quad Y = Ca,Mg,K,Na,Cl $$

(6.50)

where Y _w,j is the weathering rate and Y _u,j the net uptake of ion Y for area j.

To obtain critical loads, a link has to be established between a chemical variable and effects on aquatic biota. The most commonly used criterion is a critical ANC-limit, i.e. a minimum concentration of ANC derived to avoid ‘harmful effects’ on aquatic organisms: ANC _runoff,crit  = A·Q·[ANC] _limit , where Q is the catchment runoff. Other criteria, e.g., a critical pH or Al concentration can be employed, and the critical ANC concentration calculated from it.

Inserting Eqs. 6.46, 6.49 and 6.50 into Eqs. 6.45 and 6.42 and dividing by the total catchment area A yields the following equation to be fulfilled by critical depositions (loads) of S and N:

$$ (1-{\rho_S})\cdot \sum_{j = 0}^m {{c_j}\cdot {S_{dep,j}}}+ (1-{\rho_N})\cdot \sum_{j = 0}^m {{c_j}\cdot (1-{f_{de,j}})\cdot {{({{N_{dep,j}}-{N_{i,j}}-{N_{u,j}}})}_ + } = {L_{crit}}}$$

(6.51)

where we have defined the sub-area fractions c _j  = A _j /A and

$$ {L_{crit}}= \sum_Y {(1-{\rho_Y})\cdot \sum_{j = 0}^m {{c_j}\cdot {{({Y_{dep,j}}+ {Y_{w,j}}-{Y_{u,j}})}_ + }}}-Q\cdot {[ANC]_{limit}}$$

(6.52)

Denoting catchment average depositions with S _dep and N _dep , the depositions to the various sub-areas can be written as:

$$ {S_{dep,j}}= {s_j}\cdot {S_{dep}}\quad {\text{and}}\quad {N_{dep,j}}= {n_j}\cdot {N_{dep}},\quad j = 0,\ldots{},m $$

(6.53)

where s _j and n _j are dimensionless factors describing the enhanced (or reduced) deposition onto sub-area j. Inserting them into Eq. 6.51 yields:

$$ {a_S}\cdot {S_{dep}}+ (1-{\rho_N})\cdot \sum_{j = 0}^m {{c_j}\cdot (1-{f_{de,j}})\cdot {n_j}\cdot {{({N_{dep}}-{N_j})}_ + }}= {L_{crit}}$$

(6.54)

where we define the dimensionless parameter:

$$ {a_S} =(1-{\rho_S})\cdot \sum_{j = 0}^m {{c_j}\cdot {s_j}}$$

(6.55)

and the quantities:

$$ {N_j}: = \frac{{{N_{i,j}}+ {N_{u,j}}}}{{{n_j}}}{,}\quad j = 0,\ldots{},m $$

(6.56)

Equation 6.54 defines the critical load function (CLF, see Fig. 6.4) in the (N _dep , S _dep )-plane, and in the following we look at this function in more detail. We assume that the sub-areas of the catchment are indexed in such a way that

Fig. 6.4

Piece-wise linear critical load function (CLF) of S and acidifying N for a lake as defined by catchment properties, here shown for two land use classes characterized by N ₀ = 0, (N ₁,S ₁) and (N ₂,S ₂) (see Eqs. 6.56 and 6.64). The grey area below the CLF denotes deposition pairs resulting in an ANC leaching greater than Q·[ANC] _limitt (non-exceedance of critical loads; see Sect. 6.5 for the calculation of exceedances)

$$ {N_{j-1}}\le {N_j}\quad {\text{for}}\quad j = 1,\ldots{},m $$

(6.57)

Between two successive values of N _j the CLF is linear, but at N _j it changes the slope (another of the brackets (…)₊ in Eq. 6.54 becomes non-zero). The resulting piecewise linear function has (at most) m+2 segments, and every segment is of the form:

$$ {a_S}\cdot {S_{dep}}+ {a_{N,k}}\cdot {N_{dep}}= {L_{N,k}}+ {L_{crit}}\quad {\text{for}}\quad {N_{k-1}}\le {N_{dep}}\le {N_k},\quad k = 0,\ldots{},m + 1 $$

(6.58)

with the settings N _–1 = 0 and N _m+1 = ∞. In Eq. 6.58 we introduced the dimensionless parameters:

$$ {a_{N,0}}= 0,\quad {a_{N,k}}=(1-{\rho_N})\cdot \sum_{j = 0}^{k-1} {{c_j}\cdot (1-{f_{de,j}})\cdot {n_j}},\quad k = 1,\ldots{},m + 1 $$

(6.59)

and the terms:

$$ {L_{N,0}}= 0,\quad {L_{N,k}}=(1-{\rho_N})\cdot \sum_{j = 0}^{k-1} {{c_j}\cdot (1-{f_{de,j}})\cdot ({N_{i,j}}+ {N_{u,j}})},\quad k = 1,\ldots{},m + 1 $$

(6.60)

The maximum critical load of S is obtained by setting N _dep  = 0 in Eq. 6.54:

$$ C{L_{max}}S = {{{L_{crit}}}/{{a_S}}}$$

(6.61)

To compute the maximum critical load of N one has to find the segment of the CLF that intersects the horizontal axis. The first segment is horizontal (since a _N,0 = 0), and this segment extends till N _dep  = N ₀ (see Eqs. 6.56 and 6.57). Each of the following (at most) m+1 straight lines defined in Eq. 6.58 intersects the horizontal axis at (setting S _dep  = 0):

$$ {N_{0,k}}= {{({L_{N,k}}+ {L_{crit}})}/{{a_{N,k}}}},\quad k = 1,\ldots{},m + 1 $$

(6.62)

And the N _0,k that lies between the limits defined in Eq. 6.57 gives the maximum critical load of N. Denoting this specific index with K (1 ≤ K ≤ m+1), we have:

$$ C{L_{max}}N = {{{N_{0,K}}=({L_{N,K}}+ {L_{crit}})}/{{a_{N,K}}}}\quad {\text{where}}\quad {N_{K-1}}<{N_{0,K}}\le {N_K} $$

(6.63)

The first node of the CLF is (0,CL _max S), the second one (N ₀,CL _max S). Note that in most applications N uptake and immobilization in the lake is assumed zero, i.e. N ₀ = 0, and thus the second node coincides with the first. The next (maximum) K–1 nodes of this piecewise linear function are given by (N _k ,S _k ), where N _k is defined in Eq. 6.56 and the S _k are obtained as:

$$ {S_k} = {{{a_{N,k}}\cdot ({N_{0,k}}-{N_k})}/{{a_S}}},\quad k = 1,\ldots{},K-1 $$

(6.64)

And the last, at most (K+2)-nd, node is given by (CL _max N,0).

The base cation fluxes (Eq. 6.50) require, inter alia, estimates for the catchment weathering rates. Rapp and Bishop (2003) used the PROFILE model (Warfvinge and Sverdrup 1992, 1995) to estimate soil weathering rates in five Swedish catchments. In another study, Posch et al. (2007) used a geological weathering model (Lichtner 1992) to estimate base cation weathering in 100 high-altitude catchments in the southern Swiss Alps. Furthermore, the MAGIC model (e.g. Cosby et al. 2001) is routinely used to estimate average weathering rates for catchments, e.g. for the critical load calculations in Posch et al. (2012). The original—and by far most common—approach, however, is to estimate the net base cation flux with the SSWC model, using ‘only’ water chemistry data. This results in (compare Eq. 6.35):

$$ {L_{crit}}= Q\cdot ({ {{[B{C^*}]}_0}-{{[ANC]}_{lim}}}) $$

(6.65)

In addition to the input data required for the SSWC model, the FAB model needs also information on (a) the lake and catchment area, and land cover information, (b) terrestrial nitrogen sinks, and (c) parameters for in-lake retention of N and S .

While lake and catchment characteristics can be derived from (digital or paper) maps, data on the terrestrial N sinks are obtained in the same way as for the SMB model (see above), but expressed as catchment averages. In earlier FAB applications the denitrification fraction f _de has been estimated from the fraction of peat-lands, f _peat , in the catchment as f _de  = 0.1+0.7·f _peat (Posch et al. 1997).

Concerning in-lake processes, the retention factor for nitrogen ρ _N is modelled by a kinetic equation (Kelly et al. 1987):

$$ {\rho_N} = \frac{{{s_N}}}{{{s_N} + z/\tau }}= \frac{{{s_N}}}{{{s_N} + Q/r}}$$

(6.66)

where z is the mean lake depth, τ is the lake’s residence time, r = c ₀ is the lake:catchment ratio and s _N is the net mass transfer coefficient. Dillon and Molot (1990) give a range of 2–8 m yr⁻¹ for s _N . Values for Canadian and Norwegian catchments are given in Kaste and Dillon (2003). An equation analogous Eq. 6.66 for ρ _S —with a mass transfer coefficient s _S —is used to model the in-lake retention of S. Baker and Brezonik (1988) give a range of 0.2–0.8 m yr⁻¹ for s _S .

5 Calculation of Exceedances

Critical loads are derived to characterise the vulnerability (of parts/components) of ecosystems with respect to the deposition of one or more pollutants. In general, if the deposition of a pollutant at a given location is higher than the critical load of that pollutant at that location, it is said that the critical load is exceeded, and the difference is called exceedance (see Chap. 1 on the terminology). For a single pollutant, e.g., the critical load of nutrient N (see Eq. 6.5), the exceedance is thus defined as:

$$ E{x_{nut}}N = {N_{dep}}-C{L_{nut}}N $$

(6.67)

where N _dep is the (total) deposition of N. If the critical load is greater than or equal to the deposition, one says that it is not exceeded or there is non-exceedance of the critical load. An exceedance defined by Eq. 6.67 can obtain positive, negative or zero value. Since it is in most cases sufficient to know that there is non-exceedance (without being interested in the magnitude of non-exceedance), the exceedance of a single pollutant is also defined as:

$$ E{x_{nut}}N = {({{N_{dep}}-C{L_{nut}}N})_ + } = \left\{{\begin{matrix} {{N_{dep}}-C{L_{nut}}N}&{{N_{dep}}>C{L_{nut}}N} \\ 0&{{N_{dep}}\le C{L_{nut}}N}\end{matrix}}\right. $$

(6.68)

It should be noted that exceedances differ fundamentally from critical loads, as they are time-dependent. One can speak of the critical load (e.g. of nutrient N) for an ecosystem, but not of the exceedance of it. For exceedances the time, for which they have been calculated, has to be reported, since it is exceedances due to (past or future) anthropogenic depositions that are of interest .

Of course, the time-invariance of critical loads has its limitations, certainly when considering a geological time frame. But also during shorter time periods, such as decades or centuries, one can anticipate changes in the magnitude of critical loads due to global (climate) change, which influences the processes from which critical loads are derived. An example of a study of the (first-order) influence of temperature and precipitation changes on critical loads of acidity and nutrient N in Europe can be found in Posch (2002).

As shown (see Eqs. 6.19–6.21), there is no unique critical load of S and N acidity, and all deposition pairs (N _dep ,S _dep ) lying on the critical load function (Fig. 6.1) lead to the critical leaching of ANC . Similarly, there is no unique exceedance of acidity critical loads, as illustrated in Fig. 6.5: Let the point E denote the (current) deposition of N and S. By reducing N _dep substantially, one reaches the point Z_N and thus non-exceedance without reducing S _dep ; on the other hand one can reach non-exceedance by only reducing S _dep (by a smaller amount) until reaching Z_S; finally, with a reduction of both N _dep and S _dep , one can reach non-exceedance as well (e.g. point Z) .

Fig. 6.5

Critical load function (CLF) of acidifying N and S in the (N _dep ,S _dep )-plane (see also Fig. 6.1). The grey area below the CLF denotes deposition pairs for which there is non-exceedance. Reducing emissions from point E one can reach, e.g., points Z_N, Z, or Z_S, showing that there is no unique exceedance. The critical load exceedance is calculated by adding the N and S deposition reductions needed to reach the CLF via the shortest path (E→Z): Ex = ΔS+ΔN

Intuitively, the reduction required in N and S deposition to reach point Z (see Fig. 6.5), i.e., the shortest distance to the critical load function, seems a good measure for exceedance. Thus we define the exceedance for a given pair of depositions (N _dep ,S _dep ) as the sum of the N and S deposition reductions required to reach the critical load function by the ‘shortest’ path. Figure 6.6 depicts the five cases that can arise:

• a. The deposition falls on or below the critical load function (Region 0). In this case the exceedance is defined as zero (non-exceedance);

• b. The deposition falls into Region 1 (e.g. point E1). In this case the line perpendicular to the critical load function would yield a negative S _dep , and thus every exceedance in this region is defined as the sum of N and S deposition reduction needed to reach point Z1;

• c. The deposition falls into Region 2 (e.g. point E2): this is the ‘regular’ case, the exceedance is given by the sum of N and S deposition reduction, ExN+ExS, required to reach the point Z2, such that the line E2-Z2 is perpendicular to the critical load function;

• d. Region 3: every exceedance is defined as the sum of N and S deposition reduction needed to reach point Z3;

• e. Region 4: the exceedance is simply defined as S _dep –CL _max (S).

The exceedance function can be described by the following equation, in which the coordinates of point Z2 (see Fig. 6.6) are denoted by (N ₀,S ₀):

Fig. 6.6

The different cases for calculating the exceedance for a given critical load function

$$ Ex({N_{dep}},{S_{dep}}) = \left\{{\begin{matrix} 0&{{\text{if }}({N_{dep}},{S_{dep}})\in {\text{Region 0}}}\\ {{N_{dep}}-C{L_{max}}(N) +{S_{dep}}}&{{\text{if }}({N_{dep}},{S_{dep}})\in {\text{Region 1}}}\\ {{N_{dep}}-{N_0}+ {S_{dep}}- {S_0}}&{{\text{if }}({N_{dep}},{S_{dep}})\in {\text{Region 2}}}\\ {{N_{dep}}-C{L_{min}}(N) + {S_{dep}}-C{L_{max}}(S)}&{{\text{if }}({N_{dep}},{S_{dep}})\in {\text{Region 3}}}\\ {{S_{dep}}-C{L_{max}}(S)}&{{\text{if }}({N_{dep}},{S_{dep}})\in {\text{Region 4}}}\end{matrix}}\right. $$

(6.69)

The function thus defined fulfils the criteria of a meaningful exceedance function: it is zero, if there is no exceedance of critical loads, positive when there is exceedance, and increasing in value when the point (N _dep ,S _dep ) moves away from the critical load function.

Concerning critical loads for surface waters, exceedances of critical loads computed with the FAB model are defined in an analogous manner as exceedances for soils computed with the SMB model. Depending on the number of land cover classes within the catchment (and thus the number of nodes of the CLF ; see Fig. 6.4), there are more ‘cases’ (and more ‘regions’ as defined in Fig. 6.6) of exceedances than for SMB critical loads, but the principles of calculation are the same.

Matters are different for critical loads according to the SSWC model. In that model, sulphate is assumed to be a mobile anion (i.e. leaching equals deposition), whereas N is assumed to a large extent to be retained in the catchment by various processes. Therefore, only the so-called present-day exceedance can be calculated from the leaching of N, N _le , determined as the sum of the measured concentrations of nitrate and ammonia in runoff. This ‘present exceedance’ is defined as (Henriksen and Posch 2001):

$$ ExA = {S_{dep}}+ {N_{le}}-CLA $$

(6.70)

where CLA is the critical load of acidity as computed with Eq. 6.35. No N deposition data are required for this exceedance calculation; however, ExA quantifies only the exceedance at present rates of retention of N in the catchment. Nitrogen processes are modelled explicitly in the FAB model (see above), and thus only that model can be used for comparing the effects of different N deposition scenarios on surface waters.

To compute a single exceedance value for a grid cell or region—needed, e.g., for mapping and scenario assessments (see Chap. 25)—the so-called average accumulated exceedance (AAE) has been introduced (UBA 2004; Posch et al. 2001) . For n ecosystems within a grid cell or region it is defined as the area-weighted mean of the individual exceedances, i.e.:

$$ AAE ={{\sum_{i = 1}^n {{A_i}E{x_i}}}/{\sum_{i = 1}^n {{A_i}}}}$$

(6.71)

where Ex _i and A _i are the exceedance and ecosystem area, respectively, of ecosystem i.

6 Discussion and Outlook

The equations for computing critical loads of nutrient N as well as S and N acidity presented above are the standard forms widely applied in Europe and elsewhere (Forsius et al. 2010; Hettelingh et al. 1995, 2001, see also Chaps. 15–17) Needless to say, generalisations in many directions are possible. For example, the many N processes listed in Sect. 6.2 (e.g. N fixation) , neglected in the current formulation of critical loads, could play a role at certain locations and could thus be included in the critical load equations. Furthermore, not only denitrification can be (modelled as) a function of deposition (see Eq. 6.4), other N processes will, to some extent, also be functions of the N deposition, e.g. N growth uptake. Also, these dependences on the N deposition need not be linear, leading to non-linear equations for the critical load, which might not be explicitly solvable. The lack of explicit equations may reduce the ‘appeal’ of the concept, but does not otherwise impede the calculation of critical loads. Also base cation fluxes may depend on the inputs of S and N (e.g., weathering depends on the pH and thus the input of acidifying N and S), and this could be incorporated as well.

The critical loads derived in this chapter are for total nitrogen, owing to the assumption of complete nitrification of all deposited ammonium . This is largely justified, since significant quantities of ammonium are rarely measure below the rooting zone. However, if reduced N is a major contributor and/or the critical load refers to the top layer of the soil, separate critical loads for reduced and oxidised N could be defined; and an example of separate oxidized and reduced nutrient N critical loads can be found in De Vries (1988) and Bonten et al. (2011). In the case of acidity the distinction between oxidised and reduced nitrogen leads to a ‘critical load surface’ in the (NO _x,dep ,NH _y,dep ,S _dep )-space, with its shape depending on the nitrification model. In the acidity critical loads presented here base cation deposition has been taken as a constant at a given location (assuming that it is largely non-anthropogenic, as is the case in Europe). If, however, alkaline dust is part of the pollution problem, it makes sense to define critical load functions for which one of the axes is base cation deposition, thus providing a guide on how to ‘optimally’ combine sulphur and base cation deposition reductions. For an example of such an approach in China, see Zhao et al. (2007).

Traditionally, critical loads of acidity and nutrient N are treated separately. However, if both the nutrient and the acidity status of an ecosystem are an issue, the chemical criteria for both effects can be considered simultaneously, and a single critical load function can be derived (see Fig. 6.7a). By definition, a critical load represents an upper limit for deposition(s): every deposition below the critical load is acceptable: the chemical criterion (or criteria) is (are) not violated and thus no ‘harmful effects’ are to be expected. In contrast, when correlating the occurrence/abundance of plant species with abiotic factors this often results in finite ranges for the abiotic factors under consideration. In the simplest case these intervals of ‘good’ values are independent, e.g. [N] _lo < [N] < [N] _hi and pH _lo <pH<pH _hi . Using the critical load equations one can determine deposition pairs (N _dep ,S _dep ) for which both of these pH and [N] conditions are fulfilled (see Fig. 6.7b). Every such deposition pair—which is also limited from below—could be called an ‘optimal deposition’ or ‘optimal load’. Also in the case of correlated chemical limits, the critical load equations allow the determination of an ‘optimal load region’; and for examples and a further discussion see Posch et al. (2011).

Fig. 6.7

a Critical load function (CLF) defined by an acceptable [N] and a critical pH. b CLF defined by upper and lower bounds for the chemical criteria, i.e. [N] _lo < [N] < [N] _hi and pH _lo <pH<pH _hi . The grey areas denote those deposition pairs (N _dep ,S _dep )—the ‘optimal loads’—for which there is non-violation of the criteria

The appeal of critical loads lies in their simplicity, which makes them also applicable on a (large) regional scale and thus useful in the integrated assessment of emission reduction policies. As noted, critical loads are steady-state quantities, i.e. they do not have a time dimension. If depositions become equal to critical loads, one cannot say when the critical chemical value is attained, i.e. the danger of ‘harmful effects’ is averted. It may take a few years, or centuries, depending on the finite buffers in the soil, such as cation exchange capacity and N retention (these finite buffers do not influence steady state, i.e. critical loads). The time delays involved before a critical chemical value is attained under a given deposition (scenario) can only be estimated by dynamic models, i.e. models that take into account the finite buffers causing those delays; and some (simple) dynamic models and their applications are presented in Chap. 8.

7 Annex: Correcting for Sea Salts

Acidity critical loads are often compared with anthropogenic S deposition, i.e. the contribution due to sea spray is not included. If this is the case, the critical load of S has to be reduced by the S deposition originating from sea salts , i.e.

$$ C{L_{max}}({S^*}) = C{L_{max}}(S)-S{O_{4,dep,ss}}$$

(A.1)

where the asterisk indicates a sea salt corrected quantity and the subscript ‘ss’ stands for sea salt derived. Ignoring ions such as Br, F, Sr, boric acid and bicarbonate, which occur only in traces in seawater, the charge balance of sea salt derived deposition reads:

$$ S{O_{4,dep,ss}}= B{C_{dep,ss}}-C{l_{dep,ss}}$$

(A.2)

Subtracting this from the critical load equation (Eq. 6.19) yields for the sea salt corrected critical load:

$$ C{L_{max}}({S^*}) = BC_{dep}^*-Cl_{dep}^* + B{C_w}-B{c_u}-AN{C_{le,crit}}$$

(A.3)

The amounts of those ions in ocean water (unaffected by land drainage) are remarkably constant. This has been established by Dittmar (1884); and Dittmar’s results were so consistent that later investigations introduced only minor changes, mostly with respect to more accurate atomic weights. Here we report the values given in Sverdrup et al. (1946), which are in turn based on data by Lyman and Fleming (1940). Table A.1 lists the amounts of the six major ions in seawater, their atomic weights and the calculated equivalents.

Table A.1 Major ions in seawater and their abundance

Assuming that either the Na or the Cl deposition at a given location derives only from sea salts , and using their globally constant ratio in sea water, the depositions of base cations, sulphur and chloride (given in equivalents) are corrected for sea salts according to:

$$ X_{dep}^ *= {X_{dep}}-{r_{XY}}\cdot {Y_{dep}}$$

(A.4)

where X = Ca,Mg,K,Na,Cl or SO ₄, Y = Na or Cl and r _XY is the ratio of ions X to Y in seawater. Ratios r _XY can by computed from the last column of Table A.1 and are shown in Table A.2 with 3-decimal accuracy. Note that if Na (Cl) is chosen to correct for sea salts, Na ^* _dep  = 0 (Cl ^* _dep  = 0).

Table A.2 Ion ratios r _XY =[X]/[Y] (in eq/eq) in seawater. (Computed from Table A.1)

It should be noted that the above procedure assumes that all quantities involved disperse in the atmosphere in the same way, which is not entirely true, especially for chloride. Nevertheless, given the dearth of dispersion modelling results for sea salts, the above procedure is widely used for locations not too far from the sea.