A Model of Biogeochemical Cycling of Phosphorus

JOURNAL OF GEOPHYSICAL

RESEARCH, VOL. 94, NO. C2, PAGES 1979-2004, FEBRUARY 15, 1989

A Model of Biogeochemical Cycling of Phosphorus, Nitrogen, Oxygen, and Sulphur in the Ocean' One Step Toward a Global Climate Model GARY

SHAFFER

Oceano•IraphicInstitute, University of Gothenbur•t,Gothenbur•t,Sweden

A ocean model has been developed which, for prescribedphysics,deals with interrelationshipsbetween chemical distributions, biogeochemical sinks and sources,chemical reactions at redox fronts, and transports acrossthe air-sea and sediment-water interfaces. In its first application here, the model focuseson biogeochemicalcycling of phosphorus,nitrogen, oxygen, and sulphur in an ocean forced by river input of nutrients. This is a natural starting point for a global climate model sinceocean circulation and biology determine atmosphericCO 2 concentrationsfor a given inventory of inorganic C and oceanicproduction is controlled mainly by the availability of inorganic P and/or N. A general approach is taken to look at oxic versus anoxic conditions, P versus N limitation of primary production, with or without inorganic removal of phosphate to the sediments. As demanded by this approach, the model is nonlinear and continuous in a vertical coordinate. To focus on the biogeochemical aspects,ocean physics are kept as simple as possible. Cold, oxygen-rich water sinks at high latitudes and is upwelled with a constant velocity. Turbulent mixing is parameterized with a constant, vertical diffusion coefficient.The biogeochemical processesconsidered are new production, burial, nitrogen fixation, phosphorite formation, and

three types of organic decomposition: oxidation with 0 2, denitrification, and sulphate reduction. Organic matter is taken to consist of a high- and a low-reactive fraction. The chemical speciesconsidered

explicitlyare PO,,3- -P, NO 3--N, 0 2, NH 4+-N and H2S-S.Resultsindicatethat a changefromoxicto weakly anoxic conditions at middepths in a P-limited ocean would lead to strong local denitrification and low nitrate concentrations throughout the water column. New production would also become dominated by nitrogen fixers. Geological evidence implies that anoxic conditions in the water column have been rare in the Phanerozoic ocean. Both phosphorite formation (for P limitation) and denitrification (for N limitation) can strongly constrain primary production and the development of anoxia. N limitation, i.e., negligable nitrogen fixation, practically precludes anoxia but is unlikely for very long times scales.For P limitation and no phosphorite formation the model indicates that the redox state of the ocean may be most sensitiveto changesin ocean biology followed by changesin ocean circulation and mixing and finally by changesin oceantemperature.

1.

INTRODUCTION

A strong casehas been mounting for an intimate connection between the physics,chemistry,and biology in the ocean and climate on Earth. The links proposed are through the chemical composition of the atmosphere which influences the radiation balance of the Earth. For instance, a recent model

recent geochemical models suggestthat CO 2 may also be a viable controller of climate over much longer geological epochs [Berner et al., 1983' Lasa•la et al., 1985].

The pro 2 in surface seawater can be expressedas a function of concentrationsof HCO 3 - , CO 32- , temperature,and

salinity.The verticaldistributionsof HCO 3- and CO32- are

determined, as for any chemical speciesin the ocean, by ocean circulation and mixing, sourcesand sinks, and boundary conditions. Thus atmospheric pro 2 will depend upon distributions of the sourcesand sinks of these speciesin the interior of the ocean. A major complexity of the problem lies in the densation nuclei. Charlson et al. calculate that a 30% increase differences in these distributions from the primary in the rate of DMS production could lead to a global cooling of 1.3 øC. production-organic decompositioncycle versus the CaCO 3 The archetype for an ocean-climate connection pivots on cast production-cast dissolution cycle. Recently, the potential control of pCO 2 in the atmosphere the CO 2 "greenhouse"effect, however. For time scales > 1 by the physical and biological dynamics of high-latitude year, somemean pCO 2 (partial pressureof CO2) in the oceanoceans has been stressed [Sarmiento and To•t•tweiler, 1984' ic surfacelayer determinesatmosphericprO2, which strongly influencesthe infrared radiative propertiesof the atmosphere. Sie,qenthaler and Wenk, 1984' Knox and McElroy, 1984' Measurements of pCO 2 in polar ice cores indicate that an Broeckerand Takahashi, 1984]. The pro 2 of all these models increase from 200 to 280 ppm may be characteristic for the depends upon changesin circulation and exchange between a transitions from glacial to interglacial periods during the high-latitude, surface •'box" and the rest of the ocean. HowQuaternary [Delmas et al. 1980; Neftel et al., 1982]. A recent ever, little is known about such key aspects of this circulation general circulation model with a coupled ocean-atmosphere and exchange in the present ocean as the relative amount of predictshigh-latitudewarmingof • 10øCdue to sucha pCO2 deep convection recirculated in the deep ocean and upwelled increase[Manabe and Bryan, 1985]. Indeed, 5x80 data from at high latitudes. This helps to explain the current lack of polar ice cores imply that this degree of warming did occur consensusas to whether the net atmospherictransport of CO 2 is to or from high latitudes. Past pertubations about the presover these transitions [Lorius et al., 1985]. Furthermore, ent physical situation are unknown but probably follow

[Charlson et al., 1987] showshow an increasein ocean production can lead to an increasein cloud albedo and global cooling: marine phytoplankton produce dimethylsulphide (DMS) which appears to be the major source of cloud con-

Copyright 1989 by the American GeophysicalUnion.

changesin climate. However, for scales103 years a turnover time scale for the ocean, some type of resonant oscillations are conceivable [Broecker et al., 1985]. On the other hand, recent

Paper number 88JC03478. 0148-0227/89/88 JC-03478 $05.00 1979

1980

SHAFFER:A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

analysesof 813C and 8•80 in planktonic and benthic for-

possibleand adopt a continuous, one-dimensional advectiondiffusion model. This approach neglectsthe dynamicsof highnationslag decreases in surfaceto bottom 5•3C differences by latitude surface layers, and the model will be unable to fit > 103 years[Shackletonet al., 1983]. Given 5•3C fractionation observed chemical profiles in the ocean. In particular, deep during marine photosynthesisand the above arguments con- nutrient concentrations will be underestimated since precerning circulation and exchange, these results strongly sug- formednutrientsare set by low- and mid-latitudesurface gest a sequence like primary production change -• CO 2 values in any consistent, advection-diffusion model. In the change -• climate change. presentpaper, however, the emphasisis on developing a rigorOutside high latitudes, over about 90% of the surfaceof the ous and comprehensivetheoretical framework for dealing with ocean, primary production is almost certainly limited by the biogeochemical cycling in the ocean and on understanding supply of nutrients, in particular, inorganic P and N, to the interrelationships between chemical distributions and biogeolighted surface layer. Although present atmospheric pCO 2 is chemical processrates. For this goal it appears appropriate to •-1000 times less than in the atmosphere when life began, it start with the simplest possible model consistent with the may never have been so low as to impair oceanic primary above constraints. A more realistic ocean with high-latitude production [Lovelock and Whi(field, 1982]. Thus the oceanic dynamicswill be consideredin subsequentwork. carbon cycle is a strong function of primary production in the The paper is organized as follows: In section 2 the model is ocean but not vice versa. If climate on Earth is mainly conpresentedand functional forms for circulation and mixing and trolled by concentration of CO 2 and of other substancesaf- biogeochemicalprocessesare chosen and motivated. Overall fecting the radiation balance and, for a given river input of constraints are derived, the chemical regimes of the problem HCO 3-, theseconcentrationsare mainly controlledby ocean- are defined, and standard values are chosen for the physical, ic primary production and related interior degra- biologicaland geochemicalparameters.In Appendix A, generdation/dissolution processesand oceanic primary production al solutions are presented. Also boundary, matching and is mainly controlled by the availability of inorganic P and N, chemicalconsistencyconditionsare derivedand appliedto the then one logical starting point for a global climate model is a generalsolutions.In section3, specificsolutionsare presented model of biogeochemical cycling in the ocean as forced by for the different regimesand the sensitivityto variations of the external sources of P and N. The purpose of this paper is to parametersis studied.Also cycling of P, N, O, and sulphur is develop such a model which should satisfy the following con- discussedbasedon model output for rates of the biogeochemistraints: cal processesand of the chemical reactions at redox bound1. It should be able to deal quantitatively with an oxic as aries. Finally, in section4, the resultsare discussed,and direcwell as a partially anoxic ocean. At present,large portions of tions for future work are indicated. the ocean are nearly anoxic at middepths. If primary pro2. THE MODEL duction was 30-40% higher during Pleistocene glacials, as

aminifera in deep-seasediment cores show that ice age termi-

implied by the 5•3C analysescited above, partially anoxic 2.1. Problem Definition and Overall Constraints

conditions may have prevailed then. Also, enhanced anoxia may have existed in past epochs,i.e., during the middle Cretaceous [Holland, 1984]. 2. It should be able to deal with P, N, oxygen, and sulphur cycles simultaneously. In a partially anoxic ocean, anaerobic decompositionusesNO 3 as an oxidant where 0 2 is unavailable and SO4 as an oxidant where both 0 2 and NO 3 are unavailable (here both charges and element indications will be dropped for simplicity, i.e., NO 3 - -- N-• NO3, etc.). Neglect of the N cycle in a model of ocean anoxia leads to incomplete and misleadingresults.Furthermore, both P and N limitation should be consideredgiven the lack of knowledge as to where and when the one or the other might be applicable. 3. It should have high vertical resolution as follows from constraints 1 and 2: Whether oxidation with 0 2, denitrification, or sulphate reduction will act at a specific level will depend upon concentrationsof 02 and NO 3 at that level. But these concentrations depend upon the decomposition reactions; thus the systemis nonlinear. Also, exchangesacrossthe sediment-water interface will generally depend upon concentrationsjust above the interface. Local values may not be representableby vertical averagesand oceanic"box" models will not be suitable for the present problem. 4. It should include all important external sources and sinks of P and N for bona fide, steady state solutions. In addition to river input and burial the N cycle should include nitrogen fixation to counteract denitrification in the P-limited case.In addition to river input and burial the P cycle should allow for an inorganic P sink, i.e., through authigenic mineral

below. Also indicated in parenthesesare the abbreviations to be used and the nutrient element (P or N) affected by each process.Note that denitrification and nitrogen fixation have double roles denoted separately here by DN, NF and DN', NF'. The first role involves cycling of N through the large atmosphere-oceanreservoir of N2, and the second role involves cycling of P and N (and implicity organic carbon) through the primary production/organicdecompositioncycle. All processeslisted in Table 1 are mediated by the biota exceptphosphoriteformation which follows equilibrium thermodynamics.The chemical reactions at the redox boundaries

formation.

oxidation with 02 act in mutually exclusivelayers in the

Here I choose to keep the physics of the model as simple as

Here a continuously stratified, steady state ocean model is formulated which, for a given forcing by river input of nutrients,predictsprimary production, distributionsof PO,•, NO 3,

0 2, NH,,, and H2S as well as total ratesof the biogeochemical processes consideredand of the chemicalreactionsinvolvedin cycling P, N, O, and sulphur. Table 1 lists and classifiesthe role of the forcing, the biogeochemical processes,and the chemical

reactions.

Each of these will be discussed in detail

where 0 2• 0 will lead to line sourcesof NO 3 and line sinks of 02, NH 4, and H2S. Figure 1 shows a schematicview of the ocean with another classification

of the sinks and sources of P and N listed in

Table 1. River inflow, primary production, and nitrogen fixation act at the surface, while burial and some of the oxidation

of organic matter with 02 act on the ocean bottom. All of these will be included in the boundary conditions of the model. Denitrification, sulphate reduction, and the rest of the ocean interior. These decomposition processes,together with

SHAFFER; A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

TABLE

1.

1981

A Classification of the Role of the Forcing and the Biogeochemical Processes of the Model

as Well

as a List of the Reactions

at Redox

Net Gain

Boundaries Net

Loss

Inorganic P, N

Denitrification (DN; N) Phosphorite formation (PF; P)

River input (L; P, N) Organic P, N

Nitrogen fixation (NF; N)

Burial (BR; P, N)

Internal Cycling of Inorganic to Organic P, N

Internal Cycling of Organic to Inorganic P, N

Primary production (NP; P, N) (including nitrogen fixation (NF; P))

Oxidation with 02 (OX; P, N) Denitrification (DN; P, N) Sulphate reduction (SR; P, N)

Reaction

Number

Chemical

Reactions

at Redox

Boundaries

(R1)

NH4+ + 202--• NO3- + H20 + 2H +

(R2)

HeS + 202-• SO42-+ 2H+

the circulation and mixing and the chemical reactions of Table

1, determine the internal structure of the NO3, 02, NH,•, and H2S distributions. Phosphorite formation acts at the ocean sides, i.e., at intermediate depths where PO,• maxima occur. Thus PF and the decomposition reactions will determine the

internal structure of the PO,• distribution. Clearly, the nature and structure of the sources and sinks of P and N are quite different in general, and a linear relation between inorganic P and N in the ocean would be expected only in certain regimes. Here we deal with a one-dimensional model and neglect continental shelves.One may think of L as the net transport of P and N across the shelf edge into the deep ocean. The storage capacity of continental slope is limited [Hay and Southam,

1977], and for the scalesmost appropriate here, •-102-108 years, nutrient input from rivers may be a good approximation for total net transport of inorganic P to the deep

and for N,

L-- BR + NF-

DN = 0

(2)

For the internal cycling of P and N (and implicitly for C cycledthrough organiccompounds),we have NP -- OX -- DN'--

SR -- BR = 0

(3)

Note that (1)-(3) are all coupled through burial (BR). Here we are concerned with O and S cycles mainly to be able to treat N cycling properly. The model implies a net

productionof 0 2 in the ocean, proportional to BR, which entersthe large atmospherereservoir.Sulphur is only cycled internally from sulphateto H2S in anoxic intermediatelayers and back to sulphate upon reaction with 0 2 at the redox boundaries.

ocean. As we will see, the internal net sinks and sources of

inorganic N associated with DN and NF may dominate over river input. For a steady state, sources and sinks of P and N will balance and certain overall constraints can be formulated regardless of the physical model chosen. For the net flowthrough of these elements, we have for P, L -- BR -- PF = 0

PF

OX,

ON , SR

(1)

PF

BR,o'x Fig. 1. Schematic view of the model ocean indicating the forcing by river input of nutrients, L, and the sitesat the surface,bottom, and sides and in the interior of the biogeochemical processeslisted in Table

1.

2.2.

Physical Model and Governing Equations

The model has one-dimensional, advection-diffusion physics [Wyrtki, 1962; Munk, 1966]. Ocean circulation and mixing are specified by a constant upwelling velocity, w, and a constant coefficient of vertical turbulent

diffusion, k. The circu-

lation is considered closed by cooling and sinking to the bottom at high latitudes. Temperatures Ts and TB are prescribed at the surface water outside and within high latitudes, respectively.Temperature only enters the model for determining saturation concentrations for 02 and for choosing the molecular diffusion coefficientfor PO•. A constant salinity of 35%0is also assumedfor these purposes.Surface mixed layers and euphotic zones need not be treated explicitly but are included implicitly in subsequentboundary conditions. The level z = 0 here may be considered as the base of such layers. Onedimensional, advection-diffusion physics can only roughly approximate the physics of the real ocean with its significant interocean variability, its strong isopycnal exchange, and its variable bottom topography. Still, these physicsrepresent the simplest, consistent physical framework for present purposes and allow analytic solutions which are readily understandable and computationally efficient. For an advective-diffusive ocean the differential equation governing the vertical distribution of a chemical speciesGi in a vertical layer j is

kd2Gi.j/dz 2 -- wdGi.j/dz -- Si,j -- 0

(4)

1982

SHAFFER:A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

or SO4 as oxidants. 0 2 will be used when presentin sufficient layer j. Unless otherwise stated, the subscriptsi- 1, 2, 3, 4, quantity followed by NO 3 and finally by SO4 when 0 2 and and 5 will refer to PO4, NO3, 02, NH4, and H2S, respectively, NO 3 are depleted. Here we assumethat decomposition proand the vertical layers will be numbered from the surface ceeds with O2(OX) when 0 2 > 0, by denitrification (DN) downward.The problemwill be to solve(4) for each Gi,j(z) when 0 2 = 0 and NO 3 > 0 and by sulphate reduction (SR) given specifiedSio(z) and boundary,matchingor chemical when 02, NO3 --0. The validity of this simplification is disconsistencyconditions on Gi at the upper and lower limits of cussed below. Standard oxidation equations will be used to represent layerj. The Si.• will be associated with interior organicdecomposition as OX, DN, or SR and also with phosphorite these processes[Richards, 1965]: formation. Boundary conditions will include the forcing L and For OX

where Si.j(Z) is the distributionof sourcesor sinksof Gi in

the unknowns

NP

and NF

at the surface

and the unknowns

OX and BR at the bottom. Chemical consistencyconditions at internal boundaries between layers will derive from the chemical reactions

listed in Table

1. Functional

Here

nitrification

of ammonia

has been assumed

(7)

to occur es-

sentially simultaneously.

Primary Production and Burial

Here we consider the net role of primary production in cycling P and N (and, implicitly, organic C) between the surface and deeper layers. Therefore our primary production measure will be new production (NP [Epply and Petersen, 1979]) which is proportional to the net transport of limiting nutrient

-• 106CO2 q- 16HNO 3 q- H3PO 4 q- 122H20

forms for all these

processesare chosen in the next several sections. 2.3.

(CH20)•oo(NH3)•oH3PO 4 q- 13802

For

DN

(CH20)•o6(NH3)•6H3PO 4 q- 84.8HNO 3

•

106CO2 + 16NH 3 + 42.4N2 + H3PO,• + 148.4H20

For

(8)

SR

to the surface or

(CH20)106(NH3)16H3PO 4 q- 53H280 4

r/1.2NP-- L1.2 -- kdG1.2/dz

z= 0

(5)

Subscript 1 or 2 refers to P or N, whichever is limiting for

--• 106CO2 + 16NH 3 q- 53H2S + H3PO 4 + 106H20

(9)

production,q•.2 is the stoichiometricmole ratio of uptake of Alternative pathways to the Redfield-Richards equations may inorganic P or N to that of inorganic C, and NP is expressed be presented, i.e., for DN [Codispoti and Christensen, 1985]. in terms of C. Also, a general approach might have been taken for the deNew production is considered to consist of a high- and a composition of organic matter with C :N :P ratios of X: Y: 1, low-reactive fraction, where r/i(1- yi)NP and qiYiNP are where X and Y may have been functions of z, i.e., due to these fractions, respectively, for organically bound element i, fractionation. Nonstandard Redfield ratios have been implied by recent analysesof ocean chemistry [Takahashi et al., 1985], i.e., P or N. The low-reactive fraction is assumed to sink to the bottom intact, whereas the high-reactive fraction is assumed but such evidence remains somewhat ambiguous [Shaffer, to decomposemainly in the water column. This subdivision is 1987], and we retain the Redfield-Richards equations here. motivated in part by experimental work [e.g., Westrich and From (7)-(9) the Redfieldratios relative to C are r/1 -- 0.00943 for PO 4 -- P, t12= 0.151 for NO 3 -- N (appliesalso to release Berner, 1984] demonstrating such fractions in fresh phytofor 02 and plankton and in part by oceanic sediment trap data showing of NH 4--N during DN and SR), r/3=--1.30 rapidly decreasingorganic fluxes in the upper ocean and small •15- 0.5 for H2S -- S. As an exception to our numbering conbut vertically uniform organic fluxes in the deep ocean [Suess, vention,•/4 is the mole ratio of NO 3 -- N lossto the releaseof inorganicC during DN; thus r/4 - --0.800. The above r/1 and 1980; Noriki and Tsuno•Iai, 1986]. An alternative way of looking at this is to ascribe the "low-reactive fraction" to fast r/2 are also assumedto apply to primary production in (5). Given sourcesof 0 2 at the surfaceand to the bottom and sinking fecal pellets or marine snow. In general, Yi may vary from element to element due to the nature of the pelagic eco- the anticipation that • will be small, the above oxidation system.However, for simplicity, we take Yi- y = const, and equations and the criteria for the vertical zonation of OX, DN, and SR imply that the problem divides into three distinct both fractions have the same C:N:P ratios as total new production. The consideration of different C :N: P ratios of these chemical regimes for increasing nutrient input from land. For fractions in the model must await future work. small L, new production and organic decomposition within the water column are also small. The ocean remains oxiginaBurial of an organically bound element i is assumedto be ted throughout and OX will be the only oxidation reaction. B R = tliOtflNP (6) This one-layer situation will be called regime 1. If L increases where fl is the burial percentageof the low-reactive fraction on to L c, a critical nutrient input level, a correspondingNP•, a the seafloor. In general, fl may differ between elementsdue to critical new production level, is reached. These are the levels the nature of ecosystemstructure and geochemicalprocesses which lead to sufficientorganic decompositionto draw 0 2 to in the deep sediments,but for simplicity,we take fli = fl. Fur- zero at someintermediatedepth, z½.As L and NP exceedthese thermore, fl may be a function of variables such as temper- levels,a zone with 0 2 - 0, but NO 3 > 0 will expand about z½. ature or oxygen concentration of the bottom water [Emerson, This three-layer situation with the reaction sequenceOX-DN1985]. However, these variables do not vary extremely at the OX from ocean surface to bottom will be called regime 2. bottom of the deep ocean. Therefore we also take fl- const Finally as L and NP increase even more, another layer for and leave a more rigorous approach for modeling burial to which02, NO3 - 0 will developewithin the first intermediate future work. layer. This five-layer situation with the reaction sequenceOXDN-SR-DN-OX will be called regime 3. L½,NP½,z½,and anal2.4. Or,qanic Decomposition ogous values for the transition from regime 2 to regime 3 will Oxidation reactions involving decomposition of organic depend upon the physical and biological parameters of the

matter by microbesin the oceanmay proceedwith 0 2, NO3,

model.

SI-IAFFE•,: A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

Figure 2a illustrates these three regimes schematicallyand showswhich chemical speciesinhabit the various layers. PO 4 (not shown) is found throughout the water column as implied by (7)-(9). NH 4 producedin the DN and SR layers and H2S produced in the SR layer are diffused and advected to the upper and lower redox boundaries where they react with 0 2 to form NO 3 and SO4. The presenceof H2S in the DN layers does not imply sulphate reduction there but rather is due to physical transport through these layers. The depths of the redox boundaries between all the layers are unknowns to be solved for in the problem. The rate at which oxidation reactions proceed at any point in any layer may be a function of several variables including the amount of high-reactive organic material available, temperature, concentration of oxidant, and/or the reaction itself (OX, DN, or SR). One popular concept which may be in error is that anoxic decomposition proceeds much slower than oxic decomposition since SR is supposedto be energetically less efficient than OX [e.g., Claypool and Kaplan, 1974]. Recently, Westrich and Berner [1984] showed that decomposition rate is a stronger function of the reactivity of the organic material being oxidized than of the oxidant. They found SR rates to be only 2 and 3 times lessthan OX rates for fresh phytoplankton and argued, with reference to their experimental conditions,

(a.

INCREASING NUTRIENT INPUT FROM LAND

i

REfilME

1

I

I

INEREASINfi

NUTRIENT

INPUT

LAND

FROM

vation.

Organic decomposition in all layers for all chemical species considered

here is taken

to be

Si,j =/'/i(1 -- 0c)aNPexp (az)

(1O)

wherea- • is an e-foldinglengthfor the decreaseof the highreactive organic fraction due to bacterial oxidation. In (10) the assumption has been made that ai = a =const; modeling fractionation is left for future work; a may be derived as a first-order rate constant divided by a constant settling speed. Although approximate, this approach to modeling water column oxidation by 0 2, NO3, and SO4 has the advantageof introducing only one new parameter. This leads to an important simplification from (7)-(10): P cycling is decoupled from

the complex,internal N, O, and sulphur dynamicssincePO 4 regeneration at any depth is independent of the nature of the oxidation reaction there. At present, no firm guidance exists on temperature or 0 2 dependenceof organic decomposition in the ocean interior. Hopefully, resultsfrom recent and ongoing sediment trap studiesin the open ocean will provide guidance on this matter

".'.".':.:"."•':.".'::.':':::v.:':'.::"• I' •:.: ','.." '. 1.4, maximum NH 4 concentrationsalready exceedthose of NO 3. As L* increasesfurther, more and more NO 3 is formed by nitrification at z• and z4. For L* > 1.6, these line sourcesof NO3, marked by the growing discontinuitiesin the NO 3 gradients,have becomestrongenough to reversethe trend of decreasingNO 3 concentrationslocally. Despite increasing denitrification, even total water column NO 3 starts to increasedue mainly to NO 3 accumulation in the deep ocean. Ammonium accumulatesmuch faster, however, and for L* > 2.2 it forms the bulk of the inorganic N pool of the model ocean. While NH 4 gradients at the upper redox boundary continue to sharpen with still larger L*, those at the lower boundary remain quite constant. Figure 12a shows O2(z) and H2S(z) in regimes2 and 3 for the standard parameter set and NP*= 1.0, 1.2, ..-, 2.6; Figure 12b is a blowup of the upper left-hand corner. For increasingL*, 0 2 continuesto decreasein the upper and lower oxic layers. Gradients also sharpen at all levels in the layers to meet the increasingdemand for 0 2 in the interior by increased diffusionfrom the boundary sourcesof 0 2. As for NH 4 above, the gradients of 0 2 at the deep redox boundary tend to be constant for large L*, however. The anomalously small decrease in 0 2 between L* = 1 and L* = 1.2 reflects the small, overall 0 2 consumption associated with the DN layer in regime 2 as discussedabove. H2S accumulatesas L* increases,

exceeding100 x 10-3 mol m -3 for L* > 2.1. H2S profiles (Figure 11a) are similar but not identical in form to those of

due to the fact that NH 4 but not H2S is produced in DN layers. 3.3.

P-Limited Ocean With Phosphorite Formation

Here the simple phosphorite formation model is used to study the potential role of PF in controllingoceanproduction and chemistry. According to the model, PF commenses at

L• = L T, the level of river input which drives a new pro-

_ Regime: 1

NF, IDN) (10'%oI. Nr•Z• 1) _

I

_

--

I

1

2

LI

3

Fig. 9. Model solutionsfor nitrogen fixation (NF) as a function of scaled river input of phosphate for the case of P limitation with PF

("(PO0.)eq = 2.5X 10-3") andwithoutPF ("noPF"),bothfor standard parameters. As discussedin the text, these solutions also repre-

NH 4 (Figure 11); that is, they do not overlap exactlyif scaled sent total denitrification (DN). Also shown are boundarys between to the same maximum value for a given NP*. Differences are

regimes 1, 2, and 3 (see Figure 2a).

1992

SHAFFER:A MODEL OF BIOGEOCHEMICALCYCLING IN THE OCEAN

0 ,

0

1 I , • , I

(m) (a.

2

Lß

10•.•0•

I

--

SR

000_ -

- Z•

I Z3

2000Z•. Regime: 1

2

0

)

(b.

'

I

I

I

,

1000-

''

z; z•'

Region: 1

0

2

•

maxima or minima oFconcentrations coupled through organic decomposition, the results in both casesare similar. The solution For NP* as a Function oF L* in region 2 For standard parameters is shown in Figure 7 as the solid line

marked"(PO4)e,=2.5 x 10-3." Phosphorite formationsets an abrupt stop to the linear increase oF new production with the river input of phosphate. For L* > 1.22, NP* increases only slightly with L*. The solutionsForz•' and z2' as Functions oF L* are shown by the dotted lines in Figure 10b. The PF layer grows rapidly as region 2 is entered, but the growth rate

decreases ForlargerL*. For L* = 3 or L• = 8.25 x 10-•2 mol P m-2 s- •, the PF layeris about500m thick(z•' z2' = --418 m, -931

3

I •,

Forstandardparameters.The sensitivityoFNPT, LT, and zT to the parameters oFthe problem can be studied as For NPc, Lc, and • earlier. Since both casesdeal with criteria concerning

'

I ,

t' !'•-..C6 =1.5•10'q.... • -,•,,•:•:,,,;,•

Z•

m).

Figure 13a shows PO4(z) For the standard parameter set and For L* = 1.0, 1.1, 1.2, 1.5, 2.0, ..., 4.0; Figure 13b is a blowup oFthe PO 4 maximum. For L* < 1.22 (no PF), PO,• at every level increases linearly with L* as in Figure 3a. After

reachingthe barrieroFPO4 = (PO,•)e,ForL* = 1.22,the PO,• maximum grows only slightly with L*. By arresting the growth oF the maximum, phosphorite Formation in the relatively narrow PF layer limits the growth oFPO 4 at all depths since profile Form is fixed mainly by circulation, mixing, and organic decomposition.Simultaneously,the surfacePO,• gradient and thus the new production are also arrested. This explains why NP* only increasesslightly with L* in this case. From (12), each curve in the shaded part oF Figure 13b, when multiplied

by q = 7.5 x 10- • • s- •, givesthe verticaldistributionoFphosphorite Formation For given L*. For L* = 3, maximum PF per

unit depth is 1.36 x 10-13 mol P m -3 s-z. For L*- 3, NP* = 1.32 from Figure 7 and NP = 1.28 x 10- ? mol C m -2

I

Regime,Region:,

1

]

i

2

,

Fig. 10. Model solutionsfor depthsto redoxboundarieszl, z2, z•, and z• (solid lines) and for the depths of the boundaries of the phosphorire formation (PF) layer (dotted lines) as functions of river input of phosphate.Three different casesare shown' (a) P limitation without PF (b) P limitation with PF, and (c) N limitation. Note that

s-•. From (12) with z = --639 m, the localPO4 sourcefrom organic decompositionis 5.93 x 10-•3 mol P m -• s-• or nearly 5 times the local PF sink even For maximum local PF

and high levels oF phosphorite Formation.This explains why the Form oF the profiles in Figure 13 is barely influencedby

phosphorite Formation. Profiles oFNO•, 0 2, NH 4, and H2S and redox layer depths -' in Figure 10c. All solutions are for standard parameter values -•- z4 in a P-limited ocean with phosphorite Formation are except for the depths to PF layer boundariesin Figure 10c,where the obtained by first solving NP* For a given L* (Figure 7). Redox solutions are presented for •3= 1.5, 3.0, and 6.0 x 10-• m with (PO•)•q= 2.5x I0 -• toolm-•. Alsoshownareboundaries between layer depths and chemical profiles from regimes 1, 2, and 3 regm•es1, 2, and 3 and regions 1 and 2 (seeFigures2a and 2b). which correspond to this NP* can then be selected from the above solutions For these regimes. For the standard parameters, NP* = L* for L* _< 1.22, and the solutionsare as present-

L* represents scaledriver input of nitrate for the solutionof z•' and

duction, NP T, such that associatedorganic decomposition ed earlier. For L* > 1.22,the PF solution in Figure 7 discussed

leadsto a maximumoF PO4 = (PO4)½,at an intermediate above yields NP*. For instance, for L* = 3, NP* = 1.32 and depthzT. For L• > Lt, a layer developesaboutzt Fromwhich profiles For NO 3, NH 4, 0 2, and H2S lie betweenthose for PO 4 is extracted from the water column, across a diffusive sublayer, to Form phosphorite at the sediment surface on the

NP*- 1.2 and NP* = 1.4 in Figures 11 and 12. Phosphorite formation, by limiting new production, also limits the devel-

ocean sides.SolutionsFor NP, z•', and z2', the depths oF the PF layer boundaries, and the six constants for the general solutionsoFPO4 in the three layersdefinedby z•' and z2' For this case(region 2 in Figure 2b) are obtainedby solvingequations (BS)-(B16) oF Appendix B For given L• > L T. Top to bottom profiles oFPO 4 are obtained by substitutinginto (A1) and (A2) oF Appendix A and then by stacking the layers in order. Constant A•. 3 Followsdirectly from (B16). The remaining nonlinear set oF eight equations in eight unknowns was again solved with the simplex method. Here, however, we start at L• = L t and proceedinto region 2 in small stepsof L•. For the standardparametersof Table 2, LT = 3.36 x 10-•2 mol P

opment oF anoxia as illustrated by a comparisonof z•-

retained (lower and upper dotted lines, respectively,marked

m-2 s-• or L* = 1.22.From (18),zt = ZpM= constor --639 m

by "(PO4)eq = 2.5X 10-3").Clearly,the solutions are quite

z4

versus L* For the case of no PF (Figure 10a) and for the standard PF case (Figure 10b). When anoxia is arrested so are denitrification and nitrogen fixation as shown by the solution ForNF versusL* For the standard PF casein Figure 9 (curve

marked"(PO4)eq = 2.5 x 10-•"). Figure 7 shows tests oF the sensitivity oF the solutions in a P-limited ocean with phosphorite formation to variations in

theparameters/5 and(PO4)½q. First,equations (B8)-(B16)were solved For each oF two new diffusive sublayer thicknesses,1.5

and 6 x 10-4 m, while(PO•.)e q--2.5 X 10-3 mo1m-3 was

SHAFFER' A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

1993

N03, NHt.(1o'3m ot.i3)

NO3, NHdlo-•moJ ,/3)

0

16

32

48

0

2

4

6

8

[

I

I

i

I

i

I

I

i

(a.

(b.

o

0

(NP'-. 2.6)

-1000-

"

"

,..'...ß

-2000-

-200

(NP%1.2)

NP'=I.2)

NO3

-3000-

NH•.

.o.

.

-600

.' ß

ß

. ß ß

ß

-4000 -

-800

Fig. 11. Model solutionsfor NO3(z) and NH4(z) as functions of scaled new production levels NP*= 1.17, 1.2, 1.4, ---, 2.6 and for standardparameter values.Figure 1lb is a blowup of the upper left hand corner of Figure 1la. These solutionscorrespondto regime 3 in Figure 2a.

insensitiveto choicesof 6 or to choicesof the constants7 and k min q: If/5 is doubled, the transport capacity of the diffusive

20-25% decreasein this choice not only leads to phosphorite formation before anoxia but also prevents subsequentdevelsublayeris halvedas is q. Then(PO4 --(PO4)eq ) mustalmost opment of anoxia even for large river inputs of phosphateby double to remove the same amount of PO 4 from the water constraining new production below the critical level for the column. From Figure 13 for the range of L* considered, a onset of anoxia. On the other hand, the thickness of the doublingof (PO4 --(PO4)eq) only resultsin a small, L c implies constant burial from (6). Then, from overall equation (14), the excessof river input of POa over L c must be lost by inorganic processesin a steady state.

Figure 14 showsPOa(z) for the standard parametersand for

-/+000 Fig. 14. Model solutionsfor PO•(z) in an N-limited ocean as functions of scaled river input of phosphate, L* = 1.0, 1.05, 1.5, 2.0, ..., 4.0 and for standard parameter values. Also shown is the

levelPOa= (POa,)e q= 2.5x 10-3 mol m-3. Solutions for L* < 1, (L* < NP*), and L* > 1, (L* > NP*), correspondto regions 1 and 2 of Figure 2b.

SHAFFER'A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

tative model of a suggestion by Piper and Codispoti [1975] concerningphosphorite formation. SurfacePO 4 also increases above zero, consistent with the condition that N, not P, limits

lOO%

1995

Regime:

%OX+DN'

1

+SR

primary production in this case. Surface "mixed layers" have been drawn in Figure 14 as a visualization aid. Since NO3(z ) remains unchanged, N:P ratios fall below 16 throughout the water column for L* > 1. as in the P-limited, partially anoxic case. However, for N limitation this is due to increased PO 4 not decreasedNO 3. Solutions can be obtained for regime 3 but are not relevant since they would imply driving by an unrealistically large river input of inorganic N. Thus N limitation precludesocean-wideappearanceof H2S. The dotted curves in Figure 10c show the depths where

80-

60-

20-

PO4 = (PO4)eqasa functionof L* for diffusivesublayerthicknesses1.5, 3, and 6 x 10-4 m. Region 2 and regime 2 both start at L* = 1, as indicated above. The PF layer is narrower for small 5 (large q). For a given level of PF, increased transport capacity of the diffusive sublayer for small 5 would imply

decreased (PO4 - (PO4)eq), lesspenetrationof PO4 maximum, and a thinner P F layer. On the other hand, when g is held constant the depths z•' and z2' are essentiallyindependent of

variationsof (PO4)eq.The N-limited solutionsimply phosphorite formation in oxic layers (compare PF and DN layers in Figure 10c) and the diffusive sublayer approach used here may break down due to bioturbation. In any case, it is clear that phosphorite formation plays a passive role in an N-

limitedocean,for whichthe parameter(PO4)eq , a key onefor production and nutrient dynamics in a P-limited ocean, only determinesthe mean level of P04.

0

1

0

2

L*

3

Fig. 16. Model calculations for the portion of total organic decomposition due to oxidation with 0 2 (OX), denitrification (DN'), and sulphate reduction (SR) in a P-limited ocean, as functions of scaled river input of phosphate and for standard parameter values. The two cases shown are with PF (dashed lines) and without PF (solid lines).

and standard parameters, NP in terms of P or N is L•, L 2

(:zfi)-• or 333.3L•, L 2. After enteringwith river input, a P or N atom recirculates vertically over 300 times in the production-organic decomposition cycle before it is lost in burial.

3.5.

Phosphorus,Nitrogen, Oxygen, and Sulphur Cycling

For low to moderate nutrient inputs from land (regime 1 and region 1) and with L 2 = 16L•, the biogeochemicalprocesses acting in the model are just new production (NP), organic decomposition with oxygen (OX), and burial (BR), all of which increase linearly with L•. The release of PO 4 and NO 3 during OX is balancedby their transport into the surface layer. The consumption of 0 2 during OX is balanced by the deep, constant inflow of cold oxygenated water and by the surface source which increaseswith L, NP, and OX. From (15)

When the nutrient input from land increases sufficiently to lead to the onset of anoxia through increased NP and OX, this simple cycling picture becomes significantly altered. Figure 15 shows the relative proportion of total new production due to nitrogen-fixing plankton (NF') and nonnitrogen-fixing plankton (NP-NF') for the P-limited ocean, based on the results from Figures 7 and 9 with proper Redfield scaling. "Automatic" compensation by nitrogen fixation not only leads to complexities in the N cycling (a new DN-(N2)-NF-DN cycle competes with an OX-(NP-NF')-OX cycle), it also results in radical changes in the ecology of the model

lOO%

1

%NP

NP-NF'

80-

formation ( no PF)

I

ocean.

Just the increase

from

L* = 1 to 1.17 needed

to

traverse regime 2 leads to a state in which over half of the total new production is due to nitrogen fixers. This portion rises to • 70ø/,,and levels off for slightly larger L*. Phosphorite

Regime:

for

the

standard

PF

case has

not

hindered

this

ecological switchover(curve"(PO4)eq -- 2.5X 10-3").In a Nlimited ocean, NF0 by definition, and all primary production is due to nonnitrogen fixers. As anoxia developes in a P-limited ocean, the level of total organic decomposition, NP(1 -- eft), continues to be controled

60-

I

,,

NF'

20-

i

0

1

i

-

2

3

Lß

Fig. 15. Model calculations for the portion of total new production due to nitrogen-fixing (NF') and nonnitrogen-fixing (NP-NF') organismsin a P-limited ocean as functions of scaled river input of phosphate and for standard parameters. The two cases shown are

withPF ("(PO•.)e q= 2.5x i0 -3")andwithoutPF ("noPF").

by the transport of PO 4 to the surfacelayer. However, for a given level of NP the partition of total decomposition between oxidation with 0 2, denitrification, and sulphate reduction is a complex function of N-O-S dynamics. Figure 16 shows the relative proportions of each of these reactions in a P-limited ocean with standard parameters for varying river input of phosphate. These proportions were calculated using (11) and the integral of (10) given above solutionsfor z•-z 4 as functions of L*. Figure 16 confirms the impression of Figure 10a that a rapid transition from mainly oxic to mainly anoxic decomposition takes place at the beginning of regime 3 for the case of no phosphorite formation (solid lines). Between L* = 1.17 and L*-- 1.50, the proportion of anoxic decomposition increases

1996

SHAFFER' A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

lOO%

I

I

ø/oL1 80

balance. NO 3 sourcesare nitrification at redox boundaries z• and z4 and nitrification associatedwith oxic decomposition above z• and below z4 and on the bottom, and NO 3 sinksare denitrification(DN) betweenz• and z2 and betweenz3 and z4. An additional NO 3 sink in the upper layer is transport of NO 3 at z -- 0 into the surfacemixed layer to support a part of new production. All these processrates can be calculated from the solutionsfor NO3(z), NH4(z ) and z I - z4; for instance,the nitrification rate at the upper redox boundary is kdNH4/dz, Z•Z

Fig. 17. Model calculations for the portion of river input of phosphate lost to burial (BR) and phosphorite formation (PF) as functions of scaled river input of phosphate. The solid lines represent standard parameter solutions for oceans which are N-limited, P-limited with

PF ("(PO4)eq = 2.5X 10-3"),and P-limitedwithoutPF ("noPF"). Also presented are solutions for P-limited oceans with phosphorite

formationfor (PO4)eq = 1.5 and 3.5x 10-3 mol m-3 with g = 3 x 10-'• m -• (dashed lines) and for g = 1.5 and 6 x 10-'• m with

{PO4)eq = 2.5X 10-3 mol m-• (upperandlowerdottedlines).For the N-limited case, L* is defined in terms of river inflow of nitrate.

from 10% to 49%. At the transition to regime 3, DN already accounts for 10% of total decomposition and increases to 13% of the total (20% of the anoxic) decomposition at L* = 3. Thus denitrification plays a significant role in the cycling of nutrients and oxygen in a partially anoxic ocean despite low ambient NO 3 concentrations(Figure 11). Figure 16 also confirms the impression of Figure 10b that phosphorite formation for standard parameters (dashed lines) strongly constrains the development of anoxia. Anoxia is suppressedin the N-limited ocean and essentiallyall decomposition is due to OX. When phosphate input from land increasessufficiently to lead to the onset of phosphorite formation, the simple, overall cycling of inorganic P changes.Figure 17 shows the geologic fate of phosphorus entering with river inflow, expressedas the proportion going into phosphorite formation and into burial, for all the cases considered in Figure 7. From (14) and our scaling, the relative proportion of phosphorite formation is

(L*- NP*)(L*) -•

Alternatively, PF can be calculated from

1.

Figure 18 shows the size of these sources and sinks in each layer as a function of river input of phosphate for a P-limited ocean with standard parameters but no PF. A similar figure for the case with PF could be constructed, as was Figure 10b, by stretching the x axis according to the NP* versus L* solutions from Figure 7. Solid and dashed lines refer to the upper and deep layers, respectively.The dotted lines for L* = 0-1.17 give sourcesand sinks above and below -550 m, the depth at which splitting initiates for NP*= 1.17. Sources and sinks balance over the water column for all L* and in each layer for L* > 1.17 as required. Although the deep DN layer widens rapidly for large L* (Figure 10a), it also deepens and receives proportionally less decaying organic matter. The system is being controlled such that these two effects cancel to give nearly constant DN. Nitrification at z4 increasesrapidly and levels off to be about twice as large as total nitrification below Z4 ß

The N cycling in the upper layer is much more intense even though this layer becomesthin, O(100 m). Both DN and nitrication at z• increasestrongly with increasingL*, while nitrification above z• decreasesslightly. For L* = 3, 78% of NO 3 production in this layer stems from oxidation at z I of ammonia producedin anoxic decompositionbelow z•. The transport of NO 3 acrossz --0 is strongly reducedby the onset of

'

NO3 3-

(10'Smo•N.•2•

'

Regime:

1

2

I

3

2-

o--'"'"':::

above solutions' The area under each curve in the shadedpart

of Figure 13b,when multipliedby q = 7.5 x 10-1• s-•, gives PF for standard parameters. For the standard PF case (solid

linemarked"(PO4)½q = 2.5 X 10-3") andL* = 3, 55% of the total inflow of phosphate is precipitated as phosphorite. As for Figure 7, the results in Figure 17 are quite insensitiveto g but

I

I

i

very dependentupon (PO4)½qin a P-limited ocean.In the N-limited case, PF is essentially independent of both g and

(PO4)½qand the percent PF curve is almost exactly (L* -- 1)(L*)-•. If phosphoriteformation is the only other sink of P besides (constant) burial in a steady state, N-limited ocean, PF must take up excess P for L•- BR > 0 regardless of model

details.

For increasingnew production, the NO 3 profile splits into an upper layer above z2 and deep layer below z3 as happens for NP* > 1.17 for standard parameters (Figure 11). In a steady state, the sourcesand sinks in each of these layers must

0

1

2

L'

3

Fig. 18. Modelcalculations for the sources andsinksof NO: in a P-limited oceanwith no phosphoriteformation as functionsof scaled river input of phosphate and for standard parameter values. Results for the upper and deep NO: layersare indicatedby solid and dashed lines for L* > 1.17 (seeFigure 11). Dotted lines indicate resultsfrom

aboveand belowz• = -550 m for L* < 1.17.NO: sources are nitrification at redoxboundaries(curvea) and within the oxic part of each layer and on the bottomfor the deeplayer(curveb). NO3 sinksare denitrification(curvec) and the transportof •O• acrossz = 0 (curve d) to support part of the new production.

SHAFFER: A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

denitrification at L* = 1, goes through a minimum at L*-1.35, and increasesagain for larger L*. For L* = 3, total inter-

nal sourcesand sinksof NO 3 add up to _+4.39 x 10-8 mol N m- 2 s- x.This and total internalsourcesand sinksof PO 4 can also be calculated from new production and burial

or

_+r/x.2NP(1 --0•fi). For L* = 3 the total sourcesand sinksfor ammoniumproducedin the anoxiclayer are +3.45 x 10-8 mol N m-2 S-1.

For L* > 1 the 0 2 profile splitsinto an upper layer above z• and a deep layer below z4. Again sources and sinks must balance in each of these layers. Sinks of 02 are oxidation of NH 4 and H2S at z• and z4 and oxic decomposition (OX) above z• and below z4 and on the bottom for the upper and lower layers, respectively. The upper layer is supplied by transport across the base of the surface mixed layer and the lower layer is suppled by deep ventilation from high latitudes. These rates can be calculated from solutions for O2(z), NH(z), H2S(z), z1, and z4. For instance, the 0 2 transport from the

Regime:

3-

1

2

3

_

02

(10-7m0t 0•2g,) , e 2-

o

....... .

-1-

ß

oø1 I

-

d

t.'.' ...I-J. I I

ß--'

I/'

'>z2, the bona fide, steady state solutions presentedin ments after the latest opening of the Bosporus but are not this paper hold. Over these time scales, new production is forming at present despite strong anoxia there (S. Calvert, independent of circulation and mixing and, like the amount of personal communication, 1986). Indeed, results of the Deep PO 4 in the water column, is a function of river input of nutriSea Drilling Project indicate that holanoxic deposits (sulphate ents. It is important to remember, however, that the redox reduction in overlying water, no benthic macroorganisms) state of the ocean will depend on the physicsfor all r > r• have been rare in the Phanerozoic ocean [Ber•ter, 1981]. The (Figures 4 and 5). A scale z2 can be obtained as follows' present(precivilization) ocean is characterizedby lower L, Ts, Consider a steady state, P-limited ocean in region 1 (no phos-

and TB than earlier epochs,Still, large areas of the present phorite formation)with L 1 = L1ø. Then from (16), NPø= ocean approach anoxia at middepths. Thus a state of near or very weak anoxia at middepth may be a preferred one in the

L•ø01•y[S) - •. If at t = to, L 1 changes abruptlyto L• ø + ALl, conservationof phosphoruswill imply

ocean.

The development of a well-oxygenated atmosphere was probably associated with the rise of eukaryotes in the late Precambrian/early Phanerozoic ocean [Holland, 1984]. Above, it was shown that any significantanoxia at middepths in a P-limited ocean leads to a dominance of prokaryotic N fixers (Figure 15). Such a switchover might severly disrupt ocean ecosystemsbuilt around production by diatoms, dinoflagellates, etc. This is a point that has been overlooked by advocates of "automatic" compensation by nitrogen fixation. Above evidence suggeststhat such disruptions have been rare, however.

If it is true that anoxia is inhibited in the ocean, how might this work? One possibility is that the ocean is essentially N limited, which precludes extensive anoxia in a steady state. But N-fixing plankton are found in the present ocean, and extensive nitrogen fixation cannot be ruled out a priori. Given the apparent inactivity of marine cyanobacteria, the question of P versus N limitation may be one of time scales. A mismatch of "slow" NF and "fast" DN may have had consequencesfor climate over ice age time scales(G. Shaffer, Ocean primary production, mid-depth denitrification, continental shelves and ice age cycles, submitted to Nature, 1988). However, for longer time scales the cyanobacteria should be able to take advantage of increasing PO 4 in the surfacelayer and N limitation becomes unlikely. Other potential control mechanisms follow from Figures 4 and 5. An increase in burial percentage, •/•, and/or a decrease in the length scale for organic

decomposition,a-•, could maintain oxic conditionsthroughout the ocean despite increased L, Ts, and TB. Finally, the model shows how phosphorite formation can strongly limit new production and check the development of anoxia. At present, maximum PO 4 values at middepths in near-anoxic conditions lie near experimentally determined saturation concentrations in equilibrium with apatite [Atlas and Pytkowicz,

df•:,PO (23) dt 4dz=L•o+AL•--tl•fiNP(t) where --z, is the ocean depth. Integration of (17) yields a

relationship between1ø_:,PO4dzand NP for r >>r•, applicable over slow variations which characterize (23). Then d

-NP(t) = (L• o + AL•)a - • --ba - •NP(t) dt

(24)

where a and b follow from (23) and integration of (17). The solution to (24) with NP = NP o at t = to is

NP(t) = NP ø + ALb-•(1 -- e-ba-i(t-tø•)

(25)

from whichr2 may be identifiedas ab-1. Then for z• >>a-1, r 2 = {1 -- •[1 -- a(z, -- kw- •)(1 -- •)]}(aw•) -1 For our standardparameterchoices,r2 -- 1.173(Wa•)- • = 4.96 X 104 years. Both r• and z2 depend only upon the parametersof the problem. As long as oxic conditions prevail everywhere in the ocean, flow through cycles of P and N are similar' both enter in inorganic form with river input and leave in organic form in burial. Then, net cyclesand total ocean content of inorganic P

and N vary only over the "long" time scale z2. However, as anoxic conditions develop at middepth, the net cycle of N quickly becomes dominated by denitrification-nitrogen fixation in a P-limited ocean (see section 3.2 and Figures 9 and 15). Net cycling of N becomes coupled to the productionorganic decomposition-physical transport cycle and net

sourcesand sinks of N scaleup to be of order r/2NP not L 2.

Thus, even for weakly anoxic conditions, total ocean content of inorganic N can vary over the "short" time scale• or over only several hundred years. Indeed the denitrification rate for, say, L* = 1.1 (Figure 9) divided into the total NO 3 content in the ocean at the transition to anoxia (NP* -- 1 in Figure 3), 1977]. yields a time for sweepingthe ocean of NO 3 by denitrification Two intrinsic time scales follow from the structure of the of ,-,400 years. Even under present, "oxic" conditions in the model. One is a "short" time scale, r•, of the production- ocean, estimates of river input of inorganic P and N, and of organic decomposition-physical transport cycle and the other "background" denitrification and nitrogen fixation indicate a is a "long" time scale,'r2, of the net flow through of nutrients. shorter flow through time for N than for P [McElroy, 1983' The chemical profiles are linked to the first cycle and respond Smith, 1984]. In contrast, net cycles of P vary over the "long" on the scale r 1, to changesin w, k, or a. From dimensional time scale •2 even in the presenceof phosphorite formation analysis,r• is of order (aw)-•, (62k)- • or kw-2 or one to since this is an inorganic processcoupled to absolute PO4 several hundred years for standard parameter choices. All concentrationswhich vary with z2. The chemical profiles presented here help to illustrate an steady state solutions for chemical profiles presented above will hold for r >>r 1. Complete, quasi-steadysolutionsfor z2 >> important point' For given ocean physics, chemical con-

SHAFFER' A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

centrations in the ocean not only depend upon the size of sources and sinks but also upon their detailed distributions.

1999

with anoxic conditions. Finally, I was forced to make the questionable oversimplification that biological parameter values for the present-day pelagic ecosystem in the ocean dominated by eukaryotic primary producers also apply to an ecosystem based on cyanobacteria. Certainly, more information on the latter ecosystemis needed. What controls the growth of marine cyanobacteria and what happens to their remains? An extension of the present model is being pursued

Consider the NO 3 versusNH 4 distributions.The model prescribesthat all inorganic N releasedduring organic decomposition is transformed to NO 3 before it is lost to the surface layer or to denitrification. Then total NO 3 source/sinkrates are +__r/2NP(1- 0tfi).Since some nitrification occurs in oxic layers, it follows that total NH4 source/sink rates in the anoxic layer will be lessthan those of NO 3. For L* = 3, total whichwill allow the decomposition scale,a-•, and the portion rates for NH• are 79% of thosefor NO 3 in the model. On the of fast sinking particles, y, to assume different values for euother hand, NH• concentrationsgreatly exceedthose of NO 3 karyotic or prokaryotic new production. for strong anoxia (Figure 11). This is a reflection of the differAPPENDIX A; GENERAL SOLUTIONS AND BOUNDARY, ent distributionsof the sourcesand sinks of thesespecies.NH• MATCHING AND CHEMICAL CONSISTENCY CONDITIONS is produced within the anoxic layer but only consumed on boundaries,while NO 3 is producedin the oxic layers and on The advective-diffusive governing equation (4) with the spethe anoxic layer boundaries and is consumed in the denitrificific source/sink functions for organic decomposition and cation layers and at the surface (see Figure 18). Given the phosphorite formation (equations (10) and (12)) has the followlarge vertical separation of its sourcesand sinks, NH 4 ac- ing analytic solutions: cumulates to large concentrations to create vertical gradients large enough to support the diffusive transport from sources For phosphate to sinks required by steady state. For NO 3, sourcesand sinks G•,•(z)= A•,•+ B• exp(wk-•z)--•l•r•NPexp(az) (A1) are grouped more closely, and large concentrations are not for region 1 and layers 1 and 3 in region 2 (seeFigure 2b), and necessaryfor communication by way of the physics. It also follows

that concentrations

are not reliable

indica-

tors of rates. For instance, the shallow, upper layer with low

NO 3 concentrationsin Figure 11, quite like the NO 3 distri-

for r• = (1 - •)(kcr-- w)- •, and G1.2(z)= (PO4)eq-t-A1.2 exp(s•z)

+ B 1.2exp (s2z)-- •/•r2NP exp (az) (A2) bution in the anoxic Black Sea, is a site of intensive N cycling (Figure 18). The existenceof only low NO 3 concentrationsis for layer 2 of region 2 with sx, s2 =0.5[wk-•_+ (w2k-2 not a valid argument for discountingdenitrification as an im+4qk-•)•/2] and r2=(1 -00(ka-w--qa-•)• Roots s• portant decomposition reaction in oxic-anoxic systems.For and s2 are both real sincew, k, and q are positive. L* - 3, the typical residencetime of an NO 3 ion in the upper For nitrate layer of the model is only 6 months. In contrast, the residence times for an NH 4 ion in the anoxic layer and an NO 3 ion in G2.j(œ ) = A2.j -{-B2,j exp(wk-•) - r/2rlNPexp(az) (A3) the deep NO 3 layer are 41 years and 210 years, respectively, for L* - 3. for regime 1, layers 1 and 3 in regime 2, and layers 1 and 5 in What are the next steps toward the global climate model? regime 3 (seeFigure 2a)and Extensions to two and three dimensions to include highlatitude processesand perhapsother horizontal variations will be necessary. For instance, limited regions like the eastern tropical Pacific are anoxic at present, although the 0 2 minimum in global mean profiles substantially exceedszero. In the present model such an ocean is oxic (regime 1). However, current N budget estimates(seeabove) imply loss of inorganic N in the present ocean due mainly to denitrification in such limited regions. This might be interpreted as a symptom of nutrient overloadinglike in my regime 2. These estimatesof N sources and sinks are small, however (1 or 2 orders of magnitude lessthan thosedevelopingin the model's regime 2), and are certainly very rough. Observed, oceanic N' P ratios near Redfield values [Takahashi et al., 1985] speak against a major inbalance in N at present. Still, horizontal variations like in the real ocean would be expected to smear out transitions between model regimes and should be considered in future work.

The assumption of constantdecomposition lengthscalea- • should be reconsidered,since crmay not only depend on temperature and oxygen concentration and differ among elements due to fractionation, it may also depend upon ecosystem structure' Zooplankton which rework the detritus rain are absent in anoxic layers. It was shown that phosphorite formation may be an important mechanismfor limiting production and anoxia in the ocean, but a more complete PF model including pH dependency is needed to deal more rigorously

G2,j{z ) = A2,j d-B2,j exp(wk-lz)--•/•r•NPexp(az)

(A4)

for layer 2 in regime2 and layers2 and 4 in regime 3. For free oxygen

G3,j(z ) = A3,/ -I-B3,dexp(wk-•z)- r/3rlNPexp(az)

(A5)

for regime 1, layers 1 and 3 of regime 2, and layers 1 and 5 of regime 3. For ammonium

G4,j(z ) ---/t•4, j -{-B4,j exp(wk-•z)-- r/2rlNPexp(az)

(A6)

for layer 2 of regime 2 and layers 2, 3, and 4 of regime 3. For hydrogen sulphide

G5,•= A5,•+ B5,•exp(wk-•z)

(A7)

for layers 2 and 4 of regime 3 and

Gs,3(z) = As,3 + Bs,3 exp(wk-•z)- r/sr•NPexp(az)

(AS)

for layer 3 of regime 3.

For a given forcing by the nutrient input from land (L•), completesolutionsfor G•(z) are obtainedby applyingboundary, matching,and chemicalconsistencyconditionsand solving the resultingalgebraicequationsfor new production (NP),

the constantsA•.• and B•.•, and the depthsof the internal boundariesof the layers. Solutions will be sought for different

2000

SHAFFER' A MODELOF BIOGEOCHEMICAL CYCLINGIN THEOCEAN

casescorresponding to P or N limitation, regions 1 or 2 for

PO4 and regimes1, 2, and 3 for NO3, 02, NH4, and H2S. NP is controlled by the supply of inorganic phosphorus to the

62.2 -- 62.3 --- 0

Z = Z4

(A20)

Theseconditions prescribe that NO 3 is continuous acrosszx

ocean s•rface for"automatic" compensation bynitrogen fix- and z,. In the model,all chemicalprofilesare continuous.Any ation. Below the boundary, matching, and chemical consistency conditions are considered first for this P-limited case and

vertical discontinuity in concentrationwould be smoothedout by turbulent diffusion.

case. The boundaryandmatching conditions for NF andNO 3 in In region1, wherePO4 < (PO4)eq andphosphorate forma- regime2 alsohold in regime3 with appropriaterenumbering. The internalboundaryconditions at redoxboundaries z2 and tion is zero everywhere, then for the N-limited

23 are

•lxNP = L• -- kdGx/dz

z= 0

(A9)

The boundary conditionson phosphateare G1 =0

z=0

62.2 -- 0

z = z2

(A21)

62.•' = 0

z = z3

(A22)

(A10)

Thesecondition follow by definition. For free oxygen in regime 1 we have

w(G•- G•t•)- kdG•/dz --•/•NP[(1 -- fi)• + (1 - •) exp (-4000a)] = 0

(All)

z = - 4000 m

G3 = O2s

z= 0

(A23)

w(G3 -- O2•) -- kdG3/dz- t?3NP[(1-- fi)• + (1 - e)

Condition (A10) is appropriate for P limitation, and condition (All) prescribesthat the net flux of PO4 out of the bottom

ßexp (-4000a)]

= 0

z - -4000

(A24)

boundaryis equal to the total regenerationof PO4 due to

Condition(A23) setssurfaceoxygenvaluesto be are equal to

organic decomposition on the bottom. Substitution of (A1)

the saturationvaluesof 0 2 for surfacetemperatureTs and

into (A9)-(All) given Gx• = 0 yieldsthe algebraicequation 35%0salinity, and condition (A24) prescribesthat the differset(B1)-(B3)of AppendixB in the unknownsNP, A•, and B•. encein 0 2 flux into and out of the bottom is equal to the net In region 2, (A9)-(All) still hold at the surface and bottom 0 2 consumption theredue to organicdecomposition. O2• is of the ocean with appropriate renumbering.In addition, matchingconditionsfor PO4 at boundariesz•' and z2' are

Gx.x-- (PO4)eq -- 0

Z = Zx

the saturationvalueof 0 2 for high-latitudesurfacewater with temperatureT• and 35%,,salinity.Substitutionwith (A5) yields (B6)and(B7) of AppendixB in the unknowns,zl3 and B3. (A12) In regime 2, (A23) and (A24) also hold with appropriate

G•.2 -- (PO,,)eq-- 0

Z = Z•

(A13)

G1,2 - (PO4)eq -- 0

z = z2

(A14)

G1,3 -- (PO,,)eq -- 0

2 = z2

(A15)

Thesematchingconditions followby definition.In region2 we

nowhavesevenequations in thenineunknowns, NP, •}=•

renumbering. The internalboundaryconditionsat zx and z4 are

63. • = 0

z = zx

(A25)

63.3 = 0

z = z4

(A26)

These conditions follow by definition. Boundary conditions

A•.j, B•.j, z•' and z2' The two additionalconditionsneeded (A23)-(A26) alsohold in regime3 with appropriatenumbering changes.

are

Ammonium doesnot exist in regime 1; in regime2 its inter-

dGi.i/dz - dG•.2/dz= 0

z = zi'

(A16) nal boundary conditionsare

dGi.2/dz- dGi,3/dz= 0

z = z2'

(A17)

Conditions(A16) and (A17) prescribethat the transportof PO4 is continuousacrossz• and z2 sincethereare no sources or sinks of PO4 at theseboundaries.Theseboundaryand matchingconditionswith the use of (A1) and (A2) yield the algebraicequationset,(B8)-(B16)in AppendixB, requiredto solvefor NP and PO4(z) and to calculatetotal phosphorite formation.In the regime 1 where 0 2 > 0 everywhere,the boundaryconditionsfor nitrate are identicalto (A10) and (All) after appropriate subscriptchange.With the choice L2 --16L•, 62•-- 0. Substitutionwith (A3) yields(B4) and (B5)of AppendixB in theunknowns,zl2 andB2.(NP is known from the P cyclesolution.)In regime2 the sameboundary conditionshold with appropriaterenumbering. The equation for nitrogen fixation (NF) is

r/2NP-- L 2 -- NF + kdG2.•/dz = 0

z= 0

(A18)

Matchingconditions on NO3 at theredoxboundaries z• and z4 (where0 2 = 0) are

62.• -- 62.2 = 0

Z = Z•

(A19)

64.2 -- 0

Z = Z•

(A27)

64.2 -- 0

Z = Z4

(A28)

All NH4 reaching these redox boundariesis consumedin the nitrification

reaction.

In regime 2 after substitutionwith (A3)-(A6) we now have

11equations in the 15unknowns, NF, •j3.=x,A2,j' B2,j' Z:13,1 ' B3.1, /13.3, B3.3, /14.2, B4.2,Zl, and z4. Again NP entersas a parameter from the P cycle solutions.Four additional equations are neededwhich follow from chemicalconsistencyconditions,

dG2.•/dz- dG2.2/dz-- dG4.2/dz= 0

z = z• (A29)

dG2•2/dz- dG2.3/dz+ dG4,2/dz= 0

z = z4 (A30)

dG3,•/dz+ 2dG4.2/dz= 0

z = z•

(A31)

dG3.3/dz+ 2dG4,2/dz= 0

z- z4

(A32)

These conditions follow from the nitrification

reaction at the

redox boundariesz• and z4. Oxygen, transportedto these boundariesby advectionand mixingacrosslayers1 and 3, will react with ammonium,releasedby denitrificationin layer 2

SHAFFER' A MODEL OF BIDGEOCHEMICAL CYCLING IN THE OCEAN

and also transported to these boundaries, to produce nitrate. The mole ratio in nitrification is --2:--1:1 for O2:NH•: NO 3 (reaction (R1), Table 1). With 0 2 and NH• = 0 at z• and z• and the implied diffusive transport ratios of 2:1 for O2:NH •, (A31) and (A32) follow. Also total diffusive transport of NO 3 -- N + NH• -- N acrossz• and z• must be conserved implying (A29) and (A30). The boundary, matching, and chemical consistencyconditions considered yield the set of algebraic equations, (B17)-(B31) in Appendix B, required to solvefor NF, NO3(z), O2(z), and NH4(z) and to calculatetotal

2001

P cycle equations. Equation (A9) becomes the boundary con-

dition on PO•(z) at z-0'

that is, PO• is no longer con-

strained to be zero at the surface (equations (A10) and (B9) drop out of the problem). The boundary condition (All) at

the oceanbottomstill holdsbut now G•B= G• (z -- 0) > 0. In region 2, (A11) with the use of (A3) then yields (B16') in Appendix B, and the set of equations (B8), (B10)-(B15), and (B16') lead to the solution of PO4(z) and calculations of total phosphoritc formation in an N-limited, advective-diffusive ocean.

denitrification.

The internal boundary conditions(A27) and (A28) for NH4 and the chemical consistencyconditions (A29) and (A30) from regime 2 also apply to regime 3 with appropriate renumbering. Hydrogen sulphide is only found in regime 3 where its matching conditions at the internal redox boundaries z2 and 23 are

Gs,2 -- Gs,3 = 0

z: Z2

dGs,2/dz- dGs.3/dz= 0 Gs,• - Gs,3 = 0

(A33)

z = z2 z = z3

dGs.•/dz - dGs.3/dz= 0

z = z3

(A34)

APPENDIX

B

Sets of algebraic equations resulting from the application of boundary, matching, and chemical consistencyconditions to the general solutionsfor PO4(z), NO3(z), O2(z), NH4(z), and H2S(z) in regimes 1, 2, and 3 and in regions 1 and 2 (see

Figure2) are given.Ai,j, Bi,j are constants from generalsolutions,r/i are Redfieldratios with respectto carbon,O2s and O2Bare saturationconcentrations of oxygencorresponding to a salinity of 35%0and temperaturesTs and T•, r• = (1 -- •)(ka

(A35)

-- W)-1, r2 = (1 -- a)(ka-- w -- qa- •)- 1, and s1, S2 -0.5[wk-1-}- (w2k-2 q- 4qk-•)•/2]. All other symbolsare de-

(A36)

fined in Tables 1 and 2 except q, which is defined in section 2.6.

The conditionsfollow from continuity of H2S and its transport acrossz2 and z3. In regime 3 after substitution with (A3)-(A8) we now have

P, N, and O cycle, region 1, regime 1

t/•NP(1 -- karl) -- L• + wB• = 0

(B1)

A• + B• -- r/•r•NP = 0

(B2)

wA 1 -- t/l(1 -- fi)•NP = 0

(B3)

21 equations in the25 unknowns, NF, •j= •,2,•,s(A2,j,B2,j), A3.•, B3.•,A3.s,B3.s A4,2-4,B•.2-•, •}=2 (As,j,ns,j) and •: 1 zj- Againfouradditional equations areneeded. Thefirst two

are

A 2 -1-B2 -- t]2r •NP = 0

(B4)

dG2,2/dz-- 0

z = z2

(A37)

wA2 -- t/2(1 -- fi)•NP = 0

(B5)

dG2,•./dz= 0

z = z3

(A38)

,'t3 q- B3 -- t/3r•NP - O2s = 0

(B6)

At internal redox boundariesz2 and z 3 where NO 3 is zero by definition, its diffusive transport must also go to zero since no sourcesor sinks of NO 3 exist on these boundaries. The last two equations are chemical consistancyconditions

w(A3--0 2 --r/3(1--fi)czNP=0 P cycle, region 2

t/iNP(1 -- kari)-

z = zi

(A39)

dG3,3/dz+ 2dG4.2_4/dz + 2dGs,•/dz= 0

z = z•

(A40) All +Bll[exp(wk-iz

-- •llrl NP exp(azl') -- (PO•)½q =0

for NF, NO3(z), O2(z), NH•(z), and H2S(z) and to calculate total denitrification and sulphate reduction. For the N-limited case (NF --0), new production is an unknown in the equations for the coupled N, O, and sulphur cycles.NF is set to zero in (A18), but otherwise the boundary, matching, and chemical consistency conditions remain unchanged. Application of these conditions using the general solutions (A3)-(A8) yields the same set of equations in Appendix B for regime 2 and 3 of the P limitation case with the exception that NF = 0 in equations (B 17) and (B32). The solution for NP is then used as a parameter for the solution of the

(B8) (B9)

(B10)

/11.2 exp (SlZl') + B1.2 exp (s2z•')

boundariesz• and z•. The reasoningbehind (A39) and (A40) is analogous to that presented for (A31) and (A32) with respect to nitrification in regime 2. Then for regime 3 the boundary, matching, and chemical consistencyconditions considered above yield the set of algebraic equations, (B32)-(B56) in Appendix B, required to solve

Li + wBi.i = 0

Al. l + Bl. l -- •llr•NP = 0

dG3.•/dz + 2dG•.2_•/dz+ 2dGs,2/dz= 0

These conditions follow from nitrification and sulphide oxidation reactions (reactions (R1) and (R2), Table 1) at the redox

(B7)

--r/lr2NP exp (az•') = 0

(Bll)

A1.2 exp (siz2') + Bi. 2 exp (S222') --r/•r2NP exp (az2') = 0

(B12)

A1,3+ B1,3exp(wk-iz2') --r/•r•NP exp(az2')-- (PO,,)eq-- 0

(B13)

wk-•B•.• exp(wk-•z•') - s•/1•.2 exp(s•zi') --s2B•,2 exp (s2z•')-- r/1NPa(r• -- r2) exp (az•') = 0 (B14)

wk- •B1.3exp(wk- •22') -- SiAl, 2 exp(s1z2') --s2B•. 2 exp (s2z2')-- r/1NPa(rI -- r2) exp (az2') = 0 (•5)

2002

SHAFFER:A MODEL OF BIOGEOCHEMICAL CYCLING IN THE OCEAN

wA,. 3 -- •11(1-- fi)•NP = 0

(B16)

wk-l(B2.4 -- 82.5 -- 84,2 4) exp(wk-lz4) +crq4r•NP exp (crz4)= 0

w(A1.3-- A1,1 -- Bi,1)- /•1{(1 -- fi)(z-- wrl}NP = 0

'43.1+ 83.1-- q3rlNP -- O2s = 0

(B16')

N and O cycle, regime 2

•/2NP(1 -- kerr1)- L 2 -- NF + wB2.x = 0

(B17)

A2,1+ 82,1 -- t72r•NP= 0

(B•8)

A2,1-- A2,2 q-(82,1-- 82,2)exp(wk-lZl) --(r/2 -- r/4)rlNP exp (aZl) = 0

--(r/4 -- t72)r•NP exp (az4) = 0

wA2,3 -- r/2(1-- fi)aNP = 0

(B43)

'43.1+ 83.1exp(wk-lzl) - O3rlNPexp(az•) = 0

(B44)

A3.5 + 83.5 exp(wk-lz4) -- q3rlNP exp(az4)= 0

(B45)

w(A3.5-- O2B)-- •/3(1-- fi)czNP= 0

(B46)

A4.24 q-84.2 4 exp(wk-lzl) - t72rlNPexp(az•) = 0

(B47)

A4.2_ 4 q-84.2_4 exp(wk- lz4)- r/2r•NPexp(crz4) -- 0

(B48)

(B19)

'42.2-- '42,3+ (82,2-- 82,3)exp(wk-lz4)

As.2 q- 85.2 exp(wk-lzl)= 0

(B49)

As.2 -- As.3 + (85.2 -- 85,3)exp(wk-1Z2) (820) (821)

+•/sr•NP exp (crz2)-- 0

(BS•) (822)

As,3 -- As,4 + (85,3 -- 85,4)exp(wk-123)

Wk-1(82.2 -- 82.3 -- 84.2)exp(wk-lz4) + r/4rlaNP exp (az4) = 0

A3.1+ 83,1-- r/3rlNP-- O2S-- 0

+ •/sr xNP exp (crz3)-- 0 (823)

(825)

A3,3 q-83,3 exp(wk-1z4)- O3rlNPexp(crz4) =0

(826)

w(A3, 3 -- O2B)-- /73(1-- fi)yNP= 0

(827)

A4,2 q- 84,2 exp(wk-iz•)-r12r•NP exp(azi)= 0

(828)

(852)

Wk-1(85, 3 -- 85,4)exp(wk-•z3)+ a•/sr•NPexp(az3)= 0

(824)

A3,1+ 83,1exp(wk-lzl) -- q3rlNP exp(aZl)= 0

(B53)

As,4 + Bs,4 exp(wk-•z4)= 0

(B54)

wk-1(83,1+ 284.2_ 4 + 285.2)exp(wk-lzl) '"-- + Z'rl •" 2!• cxp (az)= 1

0

(855)

--a(r/3 + 2r/2) exp (az4)= 0

(856)

-- o [r 13

A4,2 + 84.2 exp(wk-124)- •12rlNPexp(az4)= 0

(B50)

wk-1(85, 2 -- 85,3)exp(wk-122)q-a•/srlNPexp(az2)= 0

wk-1(82.1-- 82,2 -- 84,2)exp(wk-lzl) + r/4rlaNP exp (aZl) = 0

(B42)

wk-•(B3.5+ 284,2_ 4 + 285,4)exp(wk-lz4)

(829)

wk- 1(83.• + 284.2)exp(wk- •z•) --rlCr(q3 + 2r/2)NP exp (aZl) = 0

(B30)

wk- •(83,3 + 284.2)exp(wk-lz4)

Acknowledgments.I wishto thank TeresaFierro S., AgnetaMalm, and Ola/•kerlund for extensivehelp in the preparationof the manuscript. This researchwas supportedby grants from the geoscience division of Swedish Natural

--rla(r/3 + 2r/2)NP exp (az4)= 0

Science Research Council.

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(Received March 28, 1988; acceptedJuly 14, 1988.)