A Model of Multicomponent Di usion in a Reactive Acid Soil Michael Hauhs (BITOK) Ulrich Hornung (UniBwM) Holger Lange (BITOK)
Abstract
In structured soils, a considerable amount of soil water can be immobile. As this immobile water is con ned to the smallest pore classes, it may be nevertheless in contact with most of the reactive surface areas of the solid phase. A model is proposed for the following general case: i) The mobile soil water is characterized by a relatively low pH and high Al3+ concentrations due to internal or external inputs of acids; ii) the solid phase consists of silicate minerals (e.g. feldspars) that weather irreversibly and relatively independent of pH by releasing base cations, and iii) all soil water is exposed to a high partial pressure of CO2 . In the simplest case we model this situation as a one-dimensional multicomponent di usion problem in which the connections to the mobile phase and the solid phase are described by Dirichlet and Neumann boundary conditions, respectively. Precipitation of secondary minerals (Al(OH)3) in the water column is linked to decreased di usion coecients of dissolved ions. The resulting set of coupled non-linear partial di erential equations has a nonstandard form, since a reaction-di usion system is coupled to a condition of electric neutrality everywhere in space. Therefore, special care has to be taken when applying numerical techniques for parabolic systems. Test calculations show that the model exhibits a new mechanism for SO24? accumulation in acid soils that interferes in a novel way with variations of NO?3 in the mobile water.
1
Introduction
The process of soil acidi cation in natural systems has been a focus of attention in ecology since several years. For a range in spatial and temporal scales a number of modeling attempts have been made, especially for simulating streamwater chemistry as a result of soil chemical processes. A crucial distinction between the models is whether or not chemical equilibrium conditions are assumed. When whole catchments are considered, it is customary to use yearly or monthly timesteps only. Here, the equilibrium assumption may be justi ed. This is the case, e.g., in common models of soil acidi cation like MAGIC [2], BEM [11], ALCHEMI [4], or PROFILE [18]. However, there are indications that short-term or even seasonal variations in concentrations of acidifying anions in streamwater may be due to non-equilibrium situations in soil solution [8]. Hence, a description of soil chemical processes based on homogeneity and/or 1
Advective Zone
Solid Phase Immobile Fraction Silicate Weathering
Diffusion
D
N
Figure 1: Di usion in the immobile zone equilibrium assumptions may not be sucient. If the intention is identi cation of soil key processes, e.g. the sulfate accumulation mechanism, abandoning the equilibrium assumption may be even crucial. Especially for structured soils with a wide range of pore sizes, it is known that there are chemical gradients between pore size classes [7], which can be conceptualized as two portions of soil water: a mobile and an immobile fraction. This is further corroborated by the hydrologic observation that water predominantly ows through the largest pores. The relative importance of the contact times of the mobile pore water with the reactive mineral surfaces along advective owpaths versus the accessibility of the reactive surfaces in contact with immobile water remains unclear. In our model, we therefore consider the description of chemical processes in soil solution as an example of a reaction/di usion problem. The soil is viewed as consisting of three di erent regions: the advective ow zone (mobile water fraction), the solid phase and the immobile water fraction. The communication between advective zone and solid phase is mediated by di usion processes within the immobile zone (see Fig. 1). We concentrate on major ions typical of soil acidi cation processes. The model includes H , Ca and Al as cations and OH ? , HCO? , NO? and SO ? as anions in the liquid (immobile) phase. Here, Ca is considered as representing all base cations ; thus, the behavior of K , Na or Mg , which also could have been included, is considered similar. In B- and C-horizons of acid soils, transport of Al is the dominant mechanism regulating the acidity of soil water [13]. In acid soils several solid phases are involved in the control of dissolved Al-species. However, the e ective solubility of Al can often be described by a reaction with gibbsite [3]. Here, Al(OH ) constitutes the only solid phase which is in contact with the immobile pore water. For the model this reaction represents the secondary mineral and thus the potential source or sink for Al in solution. +
2+
3+
3
2+
1
+
+
3
2 4
2+
3+
3+
3
3+
1
In soil chemistry, this term refers to cations which are associated with weak acid anions.
2
The model is investigated to analyze the concurrent di usion processes under typical chemical gradients expected for immobile soil water. It is hypothesized that this multicomponent di usion problem implies novel mechanisms for transient sulfate accumulation in soils. A crucial test for this hypothesis is the correlation between SO ? and NO? signals in mobile water. In acid forest soils a seasonal variation of NO? is often induced by biological cycles of nitrogen uptake and mineralization. In B- or C-horizons of such soils this seasonal variation is paralleled by Al-species [8]. The observed synchronicity of seasonal variations of SO ? and NO? could not be well reproduced with existing (equilibrium) models [8]. The relevant processes are controlled by very di erent time scales: Processes such as cation exchange controlling Al and H concentration in the mobile phase are fast compared with di usion [17]; silicate weathering on the other hand is much slower compared to di usion. For the range of pH values considered here it may be viewed as an externally given constant input ux to the solution [6, 14]. The two boundary conditions are prescribed accordingly. 2 4
3
3
2 4
3
3+
2 2.1
+
Description of the Model Processes
According to the picture of the system developed in the introduction, ions in the immobile water regime are subject to di usion along chemical gradients and reactions. Advective transport through this part of the water- lled pore space is considered irrelevant and hence neglected. The di usion of the di erent species is coupled by the electro-neutrality and zero charge ux conditions. A second type of coupling may be introduced by concentration-dependent di usion coecients by which an increased tortuosity or partial blocking of the di usive pathways by precipitation of secondary minerals (here, Al(OH ) ) can be simulated. These dependencies result in an inherent nonlinearity. Another source of nonlinearity as well as coupling are the reaction terms. These are based on the fact that in soil solution with pH < 5, Al is the dominating cation and thus transports acidity, whereas HCO? concentrations are negligible; on the other hand, in solutions with pH > 5, HCO? prevails [13]. To simulate this observation, the following reactions are included in the model: * Al + 3H O (1) 3H + Al(OH ) ) and 3
3+
3
3
+
3
3+
2
CO + H O * (2) ) HCO? + H Thus, we provide unlimited sinks and sources of ions transporting alkalinity or acidity by assuming pertinent contact of the di usion region with the corresponding gas or solid phases. Via this contact the required amount of CO , which is not explicitly contained in the model, is provided. The value of the CO concentration uCO aq = 0:3% (i.e., ten times the present natural atmospheric concentration) is assumed to be constant in this work. Over typical time scales of di usive processes the e ect of cation exchange is indistinguishable from a reversible reaction of the type of (1). In order to keep the model in its rst stage as simple as possible cation exchange is not included. 2
2
3
+
2
2
3
2(
)
In contrast to most other models of soil chemistry, we allow the reactions to deviate from equilibrium conditions. The e ective reaction rates are explicitly quanti ed; they are not assumed to be equal in both directions. As the focus of investigation is process identi cation, a one-dimensional approach as a rst approximation may be sucient to re ect the observed micro-scalic spatial heterogeneity in soils (see Fig. 1). On the left side, rapid water ow and intensive mixing takes place. This region is therefore best characterized by xed concentrations on time scales relevant for di usive processes (Dirichlet boundary). The right side of Fig. 1 represents the mineral surface, on which weathering processes take place. On the time scales relevant for this investigation, these are given by xed ion uxes (Neumann boundary). To stay as simple as possible, we restrict ourselves to one single weathering ux: Ca is delivered from the surface with a rate ?f=2 (f is constant) and the corresponding eux of H is consumed at this interface with a rate f . The Ca pool and H sink at this boundary are thus arbitrarily large. 2+
+
2+
2.2
+
Realization
2.2.1 System of Equations
The model is realized as a system of partial di erential equations for the concentrations in the immobile phase. As discussed above, the equations include time derivatives for the concentrations, uxes for the dissolved species and also reaction terms for some of the ions: @tuAl OH = ?RAl (uH ; uAl ) = ?@xFH ? RH O (uH ; uOH ? ) @tuH +RCO (uH ; uHCO? ) ? 3RAl (uH ; uAl ) @tuCa = ?@xFCa = ?@xFAl + RAl(uH ; uAl ) @tuAl (3) @tuHCO? = ?@xFHCO? + RCO (uH ; uHCO? ) @tuSO ? = ?@xFSO ? @tuOH ? = ?@xFOH ? ? RH O (uH ; uOH ? ) @tuNO? = ?@xFNO? (
3+
+
)3
+
+
+
2
2+
+
2
+
3
3+
2+
+
3+
3+
2 4
+
2
3
3
3+
3
2 4
+
2
3
3
2.2.2 Reaction rates
The nonlinearity of these equations stems from two sources. Firstly, the reaction terms include second and third powers of the concentrations uj [mol=cm ]: 3
(
= 0?;rAl(uAl ? kAl (uH ) ); ifelseuAl > 0 RAl(uH ; uAl ) (4) RCO (uH ; uHCO? ) = kB uCO aq ? kF uH uHCO? RH O (uH ; uOH ? ) = rH O (uH uOH ? ? cH O ) where rAl, kAl , rH O , cH O , and kB ; kF are constants that are input to the model. Estimates for equilibrium constants and the kinetics of these reactions are derived from [10, 15]. We xed the values kF = 64 cm =mol=s, kB = 0:0123 mol=cm =s, kAl = 389 cm =mol , rH O = 1:35 10 cm =mol=s and cH O = 10? mol =cm . 2
2
+
2(
3
+
3
3+
3
2
2
2
6
+
)
+
2
2
+
3+
3+
+
2
2
3
5
3
3
2
8
2
6
The dissociation/recombination reaction of water is very fast in relation to the other reactions in this model. It is included here because the self-di usion constants for these two ions are di erent. 2
4
The reaction rate for aluminum sensibly controls the dynamics of the system. Estimates of this rate constant were taken from [3]. Here, we used rAl = 10? =s. To study the sensitivity of this model on this parameter, values ranging from 10? =s to 10? =s were investigated. The pH value is 3 ? log (uH =(mol=cm )). All concentrations and reaction rates are given for the temperature 25 C . 3
3
3
+
10
1
0
2.2.3 Fluxes
The ux rates appearing in equation (3) are given by
FH FCa FAl FHCO? FSO ? FOH ? FNO? +
2+
3+
3
2 4
3
= ?DH (uAl OH )(@xuH + 1uH @x) = ?DCa (uAl OH )(@xuCa + 2uCa @x) = ?DAl (uAl OH )(@xuAl + 3uAl @x) = ?DHCO? (uAl OH )(@xuHCO? ? 1uHCO? @x) = ?DSO ? (uAl OH )(@xuSO ? ? 2uSO ? @x) = ?DOH ? (uAl OH )(@xuOH ? ? 1uOH ? @x) = ?DNO? (uAl OH )(@xuNO? ? 1uNO? @x) +
(
2+
(
3+
3
2+
)3
(
2+
3+
)3
(
3
2 4
+
+
)3
)3
(
)3
(
)3
(
)3
3+
3
(5)
3
2 4
2 4
3
3
or in more general form
Fj = ?Dj (uAl OH )(@xuj + j uj @x); j 2 J = fH ; : : :; NO? g (
+
)3
3
(Al(OH ) not included in J ) where the factors j are the respective ion charges. Electro-neutrality is guaranteed by the electric potential . Its gradient is used to describe the average force on any ion due to Coulomb interactions with all other ions undergoing di usion [9]. is characterized by the condition 3
X
j 2J
j Fj = 0:
Writing formula (3) as
@tuj = ?@xFj + R^j ; j 2 J where the R^ j can be read o from (3), we get X ^ j Rj = 0
(6)
j 2J
and thus
X j 2J
j @tuj = 0:
(7)
This equation will be useful for an alternate formulation of the di erential equations, see section 2.2.7.
2.2.4 Di usion Rates
The second source of nonlinearity of the system (3) is the proposed dependence of the di usion coecients on the concentration of Al(OH ) already mentioned above. This dependence allows a feedback between precipitation of the solid phase that reacts with 3
5
D 10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0
u
0.0 0.0
1.0
2.0
Figure 2: Di usion rate for H
+
dissolved Al and di usive transport in the immobile water. Here, we assume the following behavior 3+
Dj (uAl OH (
(
)3
d; ) = djj b
(
a
uAl(OH )3
+
if uAl OH < uAl OH ; else )3
(
(8)
)3
with positive constants dj , a and b and a given threshold uAl OH . Choosing the parameter values such that a b + uAl OH < 1 leads to discontinous di usion coecients, see Fig. 2. The threshold uAl OH is assumed to be 0:5 mol=cm . For a and b values of 1.0 and 2.0 are used. The self-di usion constants dj [10? cm =s] for dissolved ions are taken from [9] and shown in the following table. j dj H 9:34 Ca 0:793 Al 0:559 (9) ? HCO 0:9 SO ? 1:07 OH ? 5:5 NO? 1:92 Di usion constants in solid phases can be several orders of magnitude below those of self-di usion in water[12] . Here, the coecients a and b yield only a slight reduction in di usivities (factor 0.4) to study the potential feedback e ect qualitatively. They may be considered as free parameters of the model. (
(
)3
(
3
5
)3
2
+
2+
3+
2 4
3
3
6
)3
2.2.5 Column Length and Boundary Conditions The boundary point D (contact to the mobile phase) is assumed to be at x = 0 and the point N (contact to the weathering silicates) at x = L. Water held in pores by
capillary forces of less than -200 hPa is regarded as immobile. The equivalent radius characterizing the mobile/immobile interface is in this case 7.5 m. Estimates of the total inner surface between the solid phase and the soil water in a given soil vary by orders of magnitude and depend on the techniques used [1]. Here, the relation between this inner surface A and the volume V of pores lled with immobile water is assumed to vary as V =A: : The total surface is set to 10 cm =cm and the fraction of immobile water to 40%. Under these conditions a di usion length scale of L = 1 : : : 5 m is used as typical for the model. We use the lower limit of this interval. Dirichlet boundary D: At x = 0 we use the following time-independent values uj [mol=cm ]. These concentrations, which are also used as initial conditions within the whole column, match the equilibrium for all speci ed reactions at pH 4.39. 1 25
4
0
2
3
3
uj [mol=cm ] j H 41:0 10? Ca 144:3 10? Al 26:3 10? ? HCO 1:41 10? SO ? 183:1 10? OH ? 2:46 10? ? NO 41:0 10? At this Dirichlet boundary the e ect of an (observed) external variation of Al and NO? in the mobile water on the evolution of concentration gradients within the column is investigated. In the model this slow variation (weeks to months) is superimposed to the values given by the table above : 0
3
+
3
2+
3
3+
3
2 4
3
3
3
8
3
3
3+
3
uAl (t) = uAl + uAl sin(!t) (10) In order to maintain electro-neutrality and equilibrium with respect to Al(OH ) dissolution at the Dirichlet boundary, this aluminum signal must be accompanied by appropriate H and NO? variations. From the electro-neutrality condition we deduce NO? = 3Al + H : (11) We therefore set uNO? (t) = uNO? + 3uAl sin(!t) + uH (t): (12) The equilibrium condition for reaction (1) yields a condition for H (t), namely uH (t) = (kAl )? = (uAl ) = (1 + uuAl sin(!t)) = (13) Al which provides an approximate expression for uH (t), namely uAl sin(!t) ? (uAl ) sin (!t)) (14) uH (t) = (kAl)? = ( 3( uAl ) = 9(uAl ) = 0
3+
3+
3+
3
+
3
3+
3
0
3
3+
3
1 3
+
+
0
3+
+
3+
1 3
0
1 3
3+
+
+
3+
3+
1 3
0
3+
2 3
7
0
3+
2
5 3
2
with a maximum deviation of the concentrations from their equilibrium values of less than 0.2% . For uAl we use 8 10? mol=cm , guided by the observed variability in nature. Neumann boundary N: For x = L we prescribe 3
3+
3
FH = f; FCa = ?f=2 +
2+
with a xed positive rate f :
f = 4:8 10? mol=cm =s 10
2
and
FAl = FHCO? = FSO ? = FOH ? = FNO? = 0: This estimate for the weathering ux into the column is derived from laboratory experiments on the dissolution of feldspars. Its order of magnitude corresponds to the sum of base cations released in such studies [6]. Values derived under these conditions are usually much larger than the net release of base cations in eld soils [14, 16, 18]. 3+
2 4
3
3
2.2.6 Initial Conditions
For t = 0 and 0 < x < L we use the values that are given at the Dirichlet boundary. (Of course, this means that the initial and the boundary conditions are not compatible at t = 0, x = 0.) For the solid phase we use uAl OH (t = 0; 0 < x < L) = u^Al OH with prescribed u^Al OH satisfying 0 < uAl OH ? u^Al OH uAl OH , e.g., u^Al OH = 0:95uAl OH . (
(
(
)3
(
(
)3
(
)3
)3
(
)3
(
)3
)3
)3
2.2.7 Alternate Formulation
The di erential equations can be transformed in the following way. Writing the uxes Fj as Fj = ?Dj (uAl OH )(@xuj + j uj @x); one gets after summation (
X
k2J
)3
Dk (uAl OH ) k uk @x = ? (
and after division
)3
2
X
k2J
Dk (uAl OH ) k @xuk (
)3
P D (u @x = ? Pk2J Dk (uAl(OH ) ) ) k @2 uxuk : 3
k2J
This expression is plugged into
k Al(OH )3
k k
?@xFj = @x(Dj (uAl OH )(@xuj + j uj @x)); (
and one obtains a term of the form
)3
X
?@xFj = @x( ajk (uAl OH )@xuk ) (
k2J
with
)3
) j k uj ) ajk (uAl OH ) = Dj (uAl OH )(jk ? PDk (uDAl(OH i2J i uAl OH ) i ui (
)3
(
(
)3
)3 (
8
)3
2
(15)
(jk is Kronecker's delta). In this way, from equation (6) one gets for j 2 J di erential equations of the form X @tuj = @x( ajk (uAl OH )@xuk ) + R^ j (uH ; : : :; uSO ? ); (16) (
k2J
with
Fj = ?
X k2J
+
)3
2 4
ajk (uAl OH )@xuk : (
)3
The matrix formed by the coecients ajk from equation (15) has the property (see equation (7)) X j ajk = 0 8k 2 J: j 2J
2.2.8 Dimensionless Formulation
System (6) can be made dimensionless in the standard way. One uses t = T t~ with a characteristic time T [s], x = Lx~, uj = C u~j with a characteristic concentration ~ ^ C~ C [mol=cm ], Fj = LC T Fj , and Rj = T Rj . In this way one obtains 3
@tu~j = ?@xF~j + R~j ; j 2 J: ~
~
In the same way, system (16) can be made dimensionless X @tu~j = @x( a~jk @xu~k ) + R~j ; j 2 J ~
~
~
k2J
by using ajk = LT a~jk . 2
2.2.9 Discretization
The nonlinear system (16) of parabolic equations is solved using a fully nonlinear implicit scheme in time and nite di erences in space. The nonlinear system of equations to be solved in each time step is treated by Newton's method. For simplicity, the coecients ajk are held xed in each Newton iteration step, whereas the reaction terms Rj are linearized in the standard way. The linear systems to be solved within this iteration have block tridiagonal structure; they are solved using a direct tridiagonal block solver. Due to the incompatibility of the initial and the boundary conditions, at the beginning very small time steps have to used. Later, 20 time steps per period length of the boundary values (see eqs. (10) - (14)) are sucient.
3
Results
Typical pro les across the immobile zone were rst calculated without the hypothesized feedback mechanism by which precipitated Al lowers the di usion constants (Case A). As a result an almost uniform gradient for Ca develops (Fig. 3). After 100 days, the
ux of H establishes a gradient in pH; at the Neumann boundary pH rises above the value 6.0. As a consequence of this gradient HCO? has disappeared from the solution (it is converted into CO ) in the region of the column where pH < 5:0. Thus the strong 2+
+
3
2
9
u(t,x) 0.8
Ca
0.7
0.1pH
0.6
SO4 0.5
0.4
0.3
0.2
0.1
HCO3NO3 Al(OH)3 x
0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 3: Spatial distribution of j j juj ( Al OH = 0:05) after 100 days. Case A (
)3
u(t,x) 3.0
Ca
HCO3 2.0
1.0
0.1pH SO4
NO3 Al(OH)3 x
0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 4: Spatial distribution of j j juj ( Al OH = 0:1) after 100 days. Case B (
10
)3
u(t,x) Ca
0.5
0.1pH
SO4
0.4
0.3
Al(OH)3 0.2
0.1
NO3 HCO3 t [d]
0.0 0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0 100.0
Figure 5: Time evolution of j j juj ( Al OH = 0:05) at the point x = 0:5 during 100 days. Case A (
)3
acid anions SO ? and NO? are needed to compensate the cation charge in this part of the column (Fig. 3). The coupled problem of counterdi using H and Ca results in a net in ux of SO ? into the column during evolution (Fig. 5). This in ux is driven by the gradient in charge potential and runs upstream the concentration gradient of this ion. It has ceased however after a few days when quasi-stationary gradients are achieved within the column. In the acid region of the column (pH < 5:0) the ux of H is accompanied by a net
ux of Al . At the interface with the HCO? regime Al is precipitated into the solid phase. This ux does not cease and thus no stationary solution was found. However, all dissolved species attain quasi-stationary pro les after a couple of days. Thereafter only Al(OH ) continues to precipitate (in nitely) (Fig. 5) and the peak moves slowly but permanently towards the Dirichlet boundary. The variation in NO ; Al , and H concentrations superimposed at the Dirichlet boundary leads to a concurrent displacement cycle of SO ? within the column in the acid region and of HCO? in the region with pH > 5:0 (Fig. 5) . Thus the evolution of these concentrations within the mobile phase may become coupled without the direct involvement of an adsorbed phase of SO ? . This is a novel mechanism by which these ions may become correlated at the macroscopic scale. No quasi-stationary concentration pro les were achieved when the feedback mechanism controlling di usivities within the column was acitivated (Case B). Even a moderate jump in the di usion constant at the threshold value of Al(OH ) caused large changes in the dynamics of the problem (Figs. 4 and 6). After 100 days the concentration of SO ? in the immobile water has nearly doubled from its initial value. In deeply-weathered soils this transient pool of dissolved SO ? may thus exceed the yearly net transport rate through the mobile water. The chemical composition of mobile and immobile soil water deviates signi cantly (Fig. 3) and (Fig. 4). and is separated by the zone of lowered di usivities. In the 2 4
3
+
2+
2 4
+
3+
3
3+
3
3
3+
+
2 4
3
2 4
3
2 4
11
2 4
u(t,x) Ca
2.0
HCO3 Al(OH)3 1.0
0.1pH SO4
NO3 t [d]
0.0 0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0 100.0
Figure 6: Time evolution of j j juj ( Al OH = 2:0) at the point x = 0:5 during 100 days. Case B (
)3
0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02
20.0
0.01 0.0
10.0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
Figure 7: Space-time pro le of uNO? 3
12
1.0 90.0 80.0 70.0 60.0
0.0
10.0 0.0
0.1
0.2
0.4
30.0 20.0
0.3
0.5
0.6
0.7
0.9
0.8
1.0
50.0 40.0
0.0 10.0 20.0
Figure 8: Space-time pro le of 3uAl shown backwards in time 3+
inner section of the column high levels of Ca and HCO? prevail with a relatively sharp transition zone towards the Dirichlet boundary. The superimposed weekly variations in concentrations at one end are still a ecting the pro les throughout the length of the column (Figs. 6...9). The e ect at the other end, however, becomes gradually damped as the amount of precipitated Al(OH ) increases along the di usion paths (Fig. 6). In this case SO ? varies in the same direction as NO? (Fig. 9) whereas antiparallel variations in HCO? provide electroneutrality. Figure 9 shows the phase shift in the SO ? response near the right boundary. Depending on the chosen parameter set various types of interaction among the strong acid anions can be generated. 2+
3
3
2 4
3
4
3
2 4
Conlcusions and Perspectives
This study shows that by introducing chemical heterogeneity within the soils correlations among the time evolution of the strong acid anions can be achieved. The speculative addition of a feedback mechanism between the di usion constants and the precipitation of secondary minerals within the immobile zone leads to a wide range of such interactions. Depending on the weathering input ux into pores with immobile water the zone of Al precipitation is moved towards the mobile phase. As acids enter most soils through the mobile water in larger pores the resulting gradients may be quite general when local mixing of soil water is not achieved by hydrological processes. This study was initiated by the failure of existing equilibrium models to explain streamwater chemistry at Lange Bramke (Harz) [8]. It suggests novel mechanisms for the close correlation in the variation of SO ? and NO? at this site. Recently, new techniques have been developed that allow to study these processes in the eld at suciently small scales [5]. Now additional eld work at such scales is needed to test the predictions provided by the model. 2 4
13
3
0.8 0.7 0.6 0.5 0.4 0.3 0.2 20.0
0.1 0.0
10.0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
Figure 9: Space-time pro le of 2uSO ? 2 4
Acknowledgment. We thank H. Nietfeld (Univ. of Gottingen) for valuable com-
ments on multicomponent di usion in soils. This work was funded by the German Ministry of Education and Research (BMBF) under grant No. PT BEO 51-0339476B.
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Addresses of the Authors:
Uni Bayreuth, O kologische ModellbilMichael HAUHS, Holger LANGE, BITOK, dung, Dr.-Hans-Frisch-Str. 1-3, D-95448 Bayreuth, Germany, e-mail:
[email protected],
[email protected] Ulrich HORNUNG, Department of Computer Science, UniBwM, D-85577 Neubiberg, Germany, e-mail:
[email protected], URL: http://www.informatik.unibw-muenchen.de/informatik/institute/inst1/uhg. html 15
